A Study on Solutions for a Class of Higher-Order System of Singular Boundary Value Problem

Abstract

In this article, we propose a fourth-order non-self-adjoint system of singular boundary value problems (SBVPs), which arise in the theory of epitaxial growth by considering hte equation $\frac{1}{r^{\beta }}{\left\{r^{\beta }{\left[\frac{1}{r^{\beta }}{\left(r^{\beta }{\Theta }^{\prime }\right)}^{{}^{\prime }}\right]}^{\prime }\right\}}^{{}^{\prime }}=\frac{1}{2r^{\beta }}\left(K_{11}\left({\mu }^{\prime }{\Theta }^{\prime 2}+2\mu {\Theta }^{{}^{\prime }}{\Theta }^{{}^{″}}\right)+K_{12}\left({\mu }^{\prime }{\phi }^{\prime 2}+2\mu {\phi }^{{}^{\prime }}{\phi }^{{}^{″}}\right)\right)+{\lambda }_1{G}_1\left(r\right),$$\frac{1}{r^{\beta }}{\left\{r^{\beta }{\left[\frac{1}{r^{\beta }}{\left(r^{\beta }{\phi }^{\prime }\right)}^{{}^{\prime }}\right]}^{\prime }\right\}}^{{}^{\prime }}=\frac{1}{2r^{\beta }}\left(K_{21}\left({\mu }^{\prime }{\Theta }^{\prime 2}+2\mu {\Theta }^{{}^{\prime }}{\Theta }^{{}^{″}}\right)+K_{22}\left({\mu }^{\prime }{\phi }^{\prime 2}+2\mu {\phi }^{{}^{\prime }}{\phi }^{{}^{″}}\right)\right)+{\lambda }_2{G}_2\left(r\right),$ where ${\lambda }_1\ge 0$ and ${\lambda }_2\ge 0$ are two parameters, $\mu =p{r}^{2\beta -2},p\in {\mathbb{R}}^{+}$, $G_1,G_2\in L^1[0,1]$ such that $M_1^{*}\ge G_1\left(r\right)\ge M_1>0,M_2^{*}\ge G_2\left(r\right)\ge M_2>0$ and $K_{12}>0$, $K_{11}\ge 0$, and $K_{21}>0$, $K_{22}\ge 0$ are constants that are connected by the relation $(K_{12}+K_{22})\ge (K_{11}+K_{21})$ and $\beta >1$. To study the governing equation, we consider three different types of homogeneous boundary conditions. We use the transformation $t=\frac{r^{1+\beta }}{1+\beta }$ to deduce the second-order singular boundary value problem. Also, for $\beta =p=G_1\left(r\right)=G_2\left(r\right)=1$, it admits dual solutions. We show the existence of at least one solution in continuous space. We derive a sign of solutions. Furthermore, we compute the approximate bound of the parameters to point out the region of nonexistence. We also conclude bounds are symmetric with respect to two different transformations.

1. Introduction

In epitaxial growth, various methods are used to produce thin films [1], such as solar cells and microwaves from crystalline substrates under high vacuum. This process is very useful in the semiconductor industry. There are many types of epitaxial growth techniques, such as chemical beam epitaxy [2], hybrid vapor phase epitaxy [3], and molecular beam epitaxy [1]. Here, we focus on the model resulting from molecular beam epitaxy. We [4,5,6,7] consider the following equation

$$\frac{\partial F}{\partial t}={\left(1+{(\nabla F)}^2\right)}^{\frac{1}{2}}\left[-\frac{\delta I(F)}{\delta F}+\eta (x,t)\right],$$

(1)

where [4,5,6,7] $I(F)$ is given as:

$$I(F)={\int }_{\mathsf{\Omega }}R(z){\left(1+{(\nabla F)}^2\right)}^{\frac{1}{2}}dx,$$

(2)

and F is the epitaxial growth function such that $F:\mathsf{\Omega }\subset {\mathbb{R}}^N\times {\mathbb{R}}^{+}\to \mathbb{R}$. We choose $N=2$. Therefore, after similar calculation as in [4,5,6,7], we have

$${\Delta }^{2}F=\det (D^{2}F)+\lambda H(x),x\in \mathsf{\Omega }\subset {\mathbb{R}}^2,$$

(3)

where $H(x)\mathrm{equivalent}\mathrm{to}\eta (x,t)$ is well known [8]. Equations (1) and (3) have a clear geometrical meaning [4,5,6,7]). We set the following two types of limits and analyze Equation (3):

The Dirichlet boundary condition:

$${\displaystyle F=0,\frac{\partial F}{\partial n}=0\mathrm{on}\partial \mathsf{\Omega },}$$

(4)

and Navier boundary condition:

$${\displaystyle F=0,\Delta F=0\mathrm{on}\partial \mathsf{\Omega }.}$$

(5)

Let $r=|x|$ and $F(x)=\Theta (|x|)$. Equation (3) is then simplified to form

$$\begin{array}{cc}\hfill & \frac{1}{r}{\left\{r{\left[\frac{1}{r}{(r{\Theta }^{\prime })}^{{}^{\prime }}\right]}^{\prime }\right\}}^{{}^{\prime }}=\frac{1}{r}{\Theta }^{{}^{\prime }}{\Theta }^{{}^{″}}+\lambda H(r),\hfill \end{array}$$

(6)

where ${\displaystyle {}^{\prime }=\frac{d}{dr}}$. The differential Equation (6) is non-self-adjoint, nonlinear, and singular, and its solution is not known. So, it is difficult to study the solution. Verma et al. [9] analyzed the solutions of Equation (6) with a monotone iterative technique and computed the bounds of the growth parameter $\lambda$. Recently, they [10] developed the Adomian decomposition method and simulated the solution of (6). All these results and properties make Equation (6) very interesting and attractive for researchers. In [11], Verma et al. proposed a new system of SBVPs.

$$\left\{\begin{array}{c}\frac{1}{r}{\left\{r{\left[\frac{1}{r}{\left(r{\Theta }^{\prime }\right)}^{{}^{\prime }}\right]}^{\prime }\right\}}^{{}^{\prime }}=\frac{1}{r}\left(K_{11}{\Theta }^{\prime }{\Theta }^{″}+K_{12}{\phi }^{{}^{\prime }}{\phi }^{{}^{″}}\right)+{\lambda }_1,\hfill \\ \frac{1}{r}{\left\{r{\left[\frac{1}{r}{\left(r{\phi }^{\prime }\right)}^{{}^{\prime }}\right]}^{\prime }\right\}}^{{}^{\prime }}=\frac{1}{r}\left(K_{21}{\Theta }^{\prime }{\Theta }^{″}+K_{22}{\phi }^{{}^{\prime }}{\phi }^{{}^{″}}\right)+{\lambda }_2,\hfill \end{array}\right.$$

(7)

They [11] studied the existence of solutions as well as derived the bounds ${\lambda }_1$ and ${\lambda }_2$. Here, we try to generalize Equation (7). In view of [12] and (7), we propose the system of fourth-order singular differential equation given by

$$\left\{\begin{array}{c}\frac{1}{r^{\beta }}{\left\{r^{\beta }{\left[\frac{1}{r^{\beta }}{\left(r^{\beta }{\Theta }^{\prime }\right)}^{{}^{\prime }}\right]}^{\prime }\right\}}^{{}^{\prime }}=\frac{1}{2r^{\beta }}\left(K_{11}\left({\mu }^{\prime }{\Theta }^{\prime 2}+2\mu {\Theta }^{{}^{\prime }}{\Theta }^{{}^{″}}\right)+K_{12}\left({\mu }^{\prime }{\phi }^{\prime 2}+2\mu {\phi }^{{}^{\prime }}{\phi }^{{}^{″}}\right)\right)+{\lambda }_1{G}_1,\hfill \\ \frac{1}{r^{\beta }}{\left\{r^{\beta }{\left[\frac{1}{r^{\beta }}{\left(r^{\beta }{\phi }^{\prime }\right)}^{{}^{\prime }}\right]}^{\prime }\right\}}^{{}^{\prime }}=\frac{1}{2r^{\beta }}\left(K_{21}\left({\mu }^{\prime }{\Theta }^{\prime 2}+2\mu {\Theta }^{{}^{\prime }}{\Theta }^{{}^{″}}\right)+K_{22}\left({\mu }^{\prime }{\phi }^{\prime 2}+2\mu {\phi }^{{}^{\prime }}{\phi }^{{}^{″}}\right)\right)+{\lambda }_2{G}_2,\hfill \end{array}\right.$$

(8)

where ${\lambda }_1\ge 0$ and ${\lambda }_2\ge 0$ are two parameters, $\mu \left(r\right)=p{r}^{2\beta -2},p\in {\mathbb{R}}^{+}$, $G_1\left(r\right),G_2\left(r\right)\in L^1[0,1]$ such that $M_1^{*}\ge G_1\left(r\right)\ge M_1>0,M_2^{*}\ge G_2\left(r\right)\ge M_2>0$ and $K_{12}>0$, $K_{11}\ge 0$, and $K_{21}>0$, $K_{22}\ge 0$ are constants that are connected by the relation $(K_{12}+K_{22})\ge (K_{11}+K_{21})$ and $\beta >1$. It is trivial to show that Equations (7) and (8) are the same for $\beta =1$ and $G_1=G_2=1$.

To analyze (8), we consider the following three types of homogeneous boundary conditions

$$\left\{\begin{array}{c}{\displaystyle \underset{r\to 0}{\lim }{r}^{\beta }{\Theta }^{‴}\left(r\right)=0,{\Theta }^{\prime }\left(0\right)=0,\Theta \left(1\right)=0,{\Theta }^{\prime }\left(1\right)=0,}\hfill \\ {\displaystyle \underset{r\to 0}{\lim }{r}^{\beta }{\phi }^{‴}\left(r\right)=0,{\phi }^{\prime }\left(0\right)=0,\phi \left(1\right)=0,{\phi }^{\prime }\left(1\right)=0,}\hfill \end{array}\right.$$

(9)

and

$$\left\{\begin{array}{c}{\displaystyle \underset{r\to 0}{\lim }{r}^{\beta }{\Theta }^{‴}\left(r\right)=0,{\Theta }^{\prime }\left(0\right)=0,\Theta \left(1\right)=0,{\Theta }^{″}\left(1\right)=0,}\hfill \\ {\displaystyle \underset{r\to 0}{\lim }{r}^{\beta }{\phi }^{‴}\left(r\right)=0,{\phi }^{\prime }\left(0\right)=0,\phi \left(1\right)=0,{\phi }^{″}\left(1\right)=0,}\hfill \end{array}\right.$$

(10)

and

$$\left\{\begin{array}{c}{\displaystyle \underset{r\to 0}{\lim }{r}^{\beta }{\Theta }^{‴}\left(r\right)=0,{\Theta }^{\prime }\left(0\right)=0,\Theta \left(1\right)=0,{\Theta }^{\prime }\left(1\right)+{\Theta }^{″}\left(1\right)=0,}\hfill \\ {\displaystyle \underset{r\to 0}{\lim }{r}^{\beta }{\phi }^{‴}\left(r\right)=0,{\phi }^{\prime }\left(0\right)=0,\phi \left(1\right)=0,{\phi }^{\prime }\left(1\right)+{\phi }^{″}\left(1\right)=0.}\hfill \end{array}\right.$$

(11)

The vanishing derivatives ${\Theta }^{\prime }\left(0\right)=0$ and ${\phi }^{\prime }\left(0\right)=0$ show that the origin is a point of maxima or minima. The fourth boundary condition, i.e., ${\displaystyle \underset{r\to 0}{\lim }{r}^{\beta }{\Theta }^{‴}\left(r\right)=0}$ and ${\displaystyle \underset{r\to 0}{\lim }{r}^{\beta }{\phi }^{‴}\left(r\right)=0}$, assures a higher regularity at the origin. These conditions restrict $\beta$ to be greater than 1. The conditions ${\displaystyle \underset{r\to 0}{\lim }{r}^{\beta }{\Theta }^{‴}\left(r\right)=0}$, ${\displaystyle \underset{r\to 0}{\lim }{r}^{\beta }{\phi }^{‴}\left(r\right)=0}$ give rise to the following system.

$$\left\{\begin{array}{c}{r}^{\beta }{\left[\frac{1}{r^{\beta }}{\left(r^{\beta }{\Theta }^{\prime }\right)}^{{}^{\prime }}\right]}^{\prime }=\frac{\mu }{2}\left(K_{11}{\left({\Theta }^{{}^{\prime }}\right)}^2+K_{12}{\left({\phi }^{{}^{\prime }}\right)}^2\right)+{\int }_0^r{\lambda }_1{t}^{\beta }{G}_1\left(t\right)dt,\hfill \\ r^{\beta }{\left[\frac{1}{r^{\beta }}{\left(r^{\beta }{\phi }^{\prime }\right)}^{{}^{\prime }}\right]}^{\prime }=\frac{\mu }{2}\left(K_{21}{\left({\Theta }^{{}^{\prime }}\right)}^2+K_{22}{\left({\phi }^{{}^{\prime }}\right)}^2\right)+{\int }_0^r{\lambda }_2{t}^{\beta }{G}_2\left(t\right)dt.\hfill \end{array}\right.$$

(12)

We put $x=r^{\beta }{\Theta }^{\prime }$ and $y=r^{\beta }{\phi }^{\prime }$. Therefore, from Equation (12), we have

$$\left\{\begin{array}{c}{r}^2{x}^{″}-\beta r{x}^{\prime }=\frac{p\left(K_{11}{x}^2+K_{12}{y}^2\right)}{2}+r^2{\int }_0^r{\lambda }_1{t}^{\beta }{G}_1\left(t\right)dt,\hfill \\ r^2{y}^{″}-\beta r{y}^{\prime }=\frac{p\left(K_{21}{x}^2+K_{22}{y}^2\right)}{2}+r^2{\int }_0^r{\lambda }_2{t}^{\beta }{G}_2\left(t\right)dt.\hfill \end{array}\right.$$

(13)

Therefore, the homogeneous boundary condition is (9)

$$\left\{\begin{array}{c}{x}^{\prime }\left(0\right)=0=y^{\prime }\left(0\right),\hfill \\ x\left(1\right)=0=y\left(1\right),\hfill \end{array}\right.$$

(14)

Equation (11) becomes

$$\left\{\begin{array}{c}{x}^{\prime }\left(0\right)=0=y^{\prime }\left(0\right)\hfill \\ \left(\beta -1\right)x\left(1\right)=x^{\prime }\left(1\right),\left(\beta -1\right)y\left(1\right)=y^{\prime }\left(1\right),\hfill \end{array}\right.$$

(15)

and Equation (10) leads to the following

$$\left\{\begin{array}{c}{x}^{\prime }\left(0\right)=0=y^{\prime }\left(0\right)\hfill \\ \beta x\left(1\right)=x^{\prime }\left(1\right),\beta y\left(1\right)=y^{\prime }\left(1\right).\hfill \end{array}\right.$$

(16)

Now, we set $\tilde{\delta }=\frac{1}{1+\beta }$, $t=r^{1+\beta }\tilde{\delta }$, $x\left(r\right)=\mathsf{\Psi }\left(t\right)$, $y\left(r\right)=\mathsf{\Phi }\left(t\right)$, and then (13) gives rise to

$$\left\{\begin{array}{c}{\mathsf{\Psi }}^{″}=K_1(t,\mathsf{\Psi },\mathsf{\Phi })+{(1+\beta )}^{-\tilde{\beta }}{t}^{-\tilde{\beta }}{\int }_0^{L\left(t\right)}{\lambda }_1{s}^{\beta }{G}_1\left(s\right)ds,\hfill \\ {\mathsf{\Phi }}^{″}=K_2(t,\mathsf{\Psi },\mathsf{\Phi })+{(1+\beta )}^{-\tilde{\beta }}{t}^{-\tilde{\beta }}{\int }_0^{L\left(t\right)}{\lambda }_2{s}^{\beta }{G}_2\left(s\right)ds,\hfill \end{array}\right.$$

(17)

where $\tilde{\beta }=\frac{2\beta }{1+\beta }$, $L\left(t\right)={(1+\beta )}^{\frac{1}{1+\beta }}{t}^{\frac{1}{1+\beta }}$, and $K_i(t,\mathsf{\Psi },\mathsf{\Phi })=\frac{p\left(K_{i1}{\mathsf{\Psi }}^2+K_{i2}{\mathsf{\Phi }}^2\right)}{2{(1+\beta )}^2{t}^2},i=1,2$ for $t\in \left(0,\tilde{\delta }\right].$ Thus, (17) gives rise to the following boundary value problems (BVPs):

$$\mathrm{System}1:\left\{\begin{array}{c}{\mathsf{\Psi }}^{″}=K_1(t,\mathsf{\Psi },\mathsf{\Phi })+{(1+\beta )}^{-\tilde{\beta }}{t}^{-\tilde{\beta }}{\int }_0^{L\left(t\right)}{\lambda }_1{s}^{\beta }{G}_1\left(s\right)ds,\hfill \\ {\mathsf{\Phi }}^{″}=K_2(t,\mathsf{\Psi },\mathsf{\Phi })+{(1+\beta )}^{-\tilde{\beta }}{t}^{-\tilde{\beta }}{\int }_0^{L\left(t\right)}{\lambda }_2{s}^{\beta }{G}_2\left(s\right)ds,\hfill \\ \underset{t\to 0^{+}}{\lim }{t}^{\beta \tilde{\delta }}{\mathsf{\Psi }}^{\prime }\left(t\right)=0,\mathsf{\Psi }\left(\tilde{\delta }\right)=0,\hfill \\ \underset{t\to 0^{+}}{\lim }{t}^{\beta \tilde{\delta }}{\mathsf{\Phi }}^{\prime }\left(t\right)=0,\mathsf{\Phi }\left(\tilde{\delta }\right)=0,\hfill \end{array}\right.$$

(18)

$$\mathrm{System}2:\left\{\begin{array}{c}{\mathsf{\Psi }}^{″}=K_1(t,\mathsf{\Psi },\mathsf{\Phi })+{(1+\beta )}^{-\tilde{\beta }}{t}^{-\tilde{\beta }}{\int }_0^{L\left(t\right)}{\lambda }_1{s}^{\beta }{G}_1\left(s\right)ds,\hfill \\ {\mathsf{\Phi }}^{″}=K_2(t,\mathsf{\Psi },\mathsf{\Phi })+{(1+\beta )}^{-\tilde{\beta }}{t}^{-\tilde{\beta }}{\int }_0^{L\left(t\right)}{\lambda }_2{s}^{\beta }{G}_2\left(s\right)ds,\hfill \\ \underset{t\to 0^{+}}{\lim }{t}^{\beta \tilde{\delta }}{\mathsf{\Psi }}^{\prime }\left(t\right)=0,(\beta -1)\mathsf{\Psi }\left(\tilde{\delta }\right)={\mathsf{\Psi }}^{\prime }\left(\tilde{\delta }\right),\hfill \\ \underset{t\to 0^{+}}{\lim }{t}^{\beta \tilde{\delta }}{\mathsf{\Phi }}^{\prime }\left(t\right)=0,(\beta -1)\mathsf{\Phi }\left(\tilde{\delta }\right)={\mathsf{\Phi }}^{\prime }\left(\tilde{\delta }\right),\hfill \end{array}\right.$$

(19)

and

$$\mathrm{System}3:\left\{\begin{array}{c}{\mathsf{\Psi }}^{″}=K_1(t,\mathsf{\Psi },\mathsf{\Phi })+{(1+\beta )}^{-\tilde{\beta }}{t}^{-\tilde{\beta }}{\int }_0^{L\left(t\right)}{\lambda }_1{s}^{\beta }{G}_1\left(s\right)ds,\hfill \\ {\mathsf{\Phi }}^{″}=K_2(t,\mathsf{\Psi },\mathsf{\Phi })+{(1+\beta )}^{-\tilde{\beta }}{t}^{-\tilde{\beta }}{\int }_0^{L\left(t\right)}{\lambda }_2{s}^{\beta }{G}_2\left(s\right)ds,\hfill \\ \underset{t\to 0^{+}}{\lim }{t}^{\beta \tilde{\delta }}{\mathsf{\Psi }}^{\prime }\left(t\right)=0,\beta \mathsf{\Psi }\left(\tilde{\delta }\right)={\mathsf{\Psi }}^{\prime }\left(\tilde{\delta }\right),\hfill \\ \underset{t\to 0^{+}}{\lim }{t}^{\beta \tilde{\delta }}{\mathsf{\Phi }}^{\prime }\left(t\right)=0,\beta \mathsf{\Phi }\left(\tilde{\delta }\right)={\mathsf{\Phi }}^{\prime }\left(\tilde{\delta }\right),\hfill \end{array}\right.$$

(20)

$\mathrm{for}t\in \left(0,\tilde{\delta }\right].$ Therefore, we can obtain the following integral equation (IE):

• Corresponding to (18), the IE is

  $$\left\{\begin{array}{c}\mathsf{\Psi }\left(t\right)=-\left(\tilde{\delta }-t\right){\int }_0^{t}s(1+\beta )K_1(s,\mathsf{\Psi },\mathsf{\Phi })ds-t{\int }_t^{\tilde{\delta }}(1+\beta )K_1(s,\mathsf{\Psi },\mathsf{\Phi })\left(\tilde{\delta }-s\right)ds\hfill \\ -\left(\tilde{\delta }-t\right){\int }_0^t{(1+\beta )}^{\tilde{\gamma }}{s}^{\tilde{\gamma }}{\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\beta }{G}_1\left(y\right)dyds\hfill \\ -t{\int }_t^{\tilde{\delta }}{(1+\beta )}^{\tilde{\gamma }}{s}^{-\tilde{\beta }}\left(\tilde{\delta }-s\right){\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\beta }{G}_1\left(y\right)dyds,0\le t\le \tilde{\delta },\hfill \\ \mathsf{\Phi }\left(t\right)=-\left(\tilde{\delta }-t\right){\int }_0^{t}s(1+\beta )K_2(s,\mathsf{\Psi },\mathsf{\Phi })ds-t{\int }_t^{\tilde{\delta }}(1+\beta )K_2(s,\mathsf{\Psi },\mathsf{\Phi })\left(\tilde{\delta }-s\right)ds\hfill \\ -\left(\tilde{\delta }-t\right){\int }_0^t{(1+\beta )}^{\tilde{\gamma }}{s}^{\tilde{\gamma }}{\int }_0^{L\left(s\right)}{\lambda }_2{y}^{\beta }{G}_2\left(y\right)dyds\hfill \\ -t{\int }_t^{\tilde{\delta }}{(1+\beta )}^{\tilde{\gamma }}{s}^{-\tilde{\beta }}\left(\tilde{\delta }-s\right){\int }_0^{L\left(s\right)}{\lambda }_2{y}^{\beta }{G}_2\left(y\right)dyds,0\le t\le \tilde{\delta },\hfill \end{array}\right.$$
• Corresponding to (19), the IE is

  $$\left\{\begin{array}{c}\mathsf{\Psi }\left(t\right)=-\left(1+\frac{(\beta -1)(\beta +1)}{2}t\right){\int }_0^{t}s{K}_1(s,\mathsf{\Psi },\mathsf{\Phi })ds-t{\int }_t^{\tilde{\delta }}{K}_1(s,\mathsf{\Psi },\mathsf{\Phi })\left(1+\frac{(\beta -1)(\beta +1)}{2}s\right)ds\hfill \\ -\left(1+\frac{(\beta -1)(\beta +1)}{2}t\right){\int }_0^t{(1+\beta )}^{-\tilde{\beta }}{s}^{\tilde{\gamma }}{\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\beta }{G}_1\left(y\right)dyds\hfill \\ -t{\int }_t^{\tilde{\delta }}{(1+\beta )}^{-\tilde{\beta }}{s}^{-\tilde{\beta }}\left(1+\frac{(\beta -1)(\beta +1)}{2}s\right){\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\beta }{G}_1\left(y\right)dyds,0\le t\le \tilde{\delta },\hfill \\ \mathsf{\Phi }\left(t\right)=-\left(1+\frac{(\beta -1)(\beta +1)}{2}t\right){\int }_0^{t}s{K}_2(s,\mathsf{\Psi },\mathsf{\Phi })ds-t{\int }_t^{\tilde{\delta }}{K}_2(s,\mathsf{\Psi },\mathsf{\Phi })\left(1+\frac{(\beta -1)(\beta +1)}{2}s\right)ds\hfill \\ -\left(1+\frac{(\beta -1)(\beta +1)}{2}t\right){\int }_0^t{(1+\beta )}^{-\tilde{\beta }}{s}^{\tilde{\gamma }}{\int }_0^{L\left(s\right)}{\lambda }_2{y}^{\beta }{G}_2\left(y\right)dyds\hfill \\ -t{\int }_t^{\tilde{\delta }}{(1+\beta )}^{-\tilde{\beta }}{s}^{-\tilde{\beta }}\left(1+\frac{(\beta -1)(\beta +1)}{2}s\right){\int }_0^{L\left(s\right)}{\lambda }_2{y}^{\beta }{G}_2\left(y\right)dyds,0\le t\le \tilde{\delta },\hfill \end{array}\right.$$
• Corresponding to (20), the IE is

  $$\left\{\begin{array}{c}\mathsf{\Psi }\left(t\right)=-\left(t+\frac{1}{(\beta +1)\beta }\right){\int }_0^{t}s\beta (1+\beta )K_1(s,\mathsf{\Psi },\mathsf{\Phi })ds-t{\int }_t^{\tilde{\delta }}\beta (1+\beta )K_1(s,\mathsf{\Psi },\mathsf{\Phi })\times \hfill \\ \left(s+\frac{1}{(\beta +1)\beta }\right)ds-\left(t+\frac{1}{(1+\beta )\beta }\right){\int }_0^t\beta {(1+\beta )}^{\tilde{\gamma }}{s}^{\tilde{\gamma }}{\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\beta }{G}_1\left(y\right)dyds\hfill \\ -t{\int }_t^{\tilde{\delta }}\beta {(1+\beta )}^{\tilde{\gamma }}{s}^{-\tilde{\beta }}\left(s+\frac{1}{(1+\beta )\beta }\right){\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\alpha }{G}_1\left(y\right)dyds,0\le t\le \tilde{\delta },\hfill \\ \mathsf{\Phi }\left(t\right)=-\left(t+\frac{1}{(\beta +1)\beta }\right){\int }_0^{t}s\beta (1+\beta )K_2(s,\mathsf{\Psi },\mathsf{\Phi })ds-t{\int }_t^{\tilde{\delta }}\beta (1+\beta )K_2(s,\mathsf{\Psi },\mathsf{\Phi })\times \hfill \\ \left(s+\frac{1}{(\beta +1)\beta }\right)ds-\left(t+\frac{1}{(1+\beta )\beta }\right){\int }_0^t\beta {(1+\beta )}^{\tilde{\gamma }}{s}^{\tilde{\gamma }}{\int }_0^{L\left(s\right)}{\lambda }_2{y}^{\beta }{G}_2\left(y\right)dyds\hfill \\ -t{\int }_t^{\tilde{\delta }}\beta {(1+\beta )}^{\tilde{\gamma }}{s}^{-\tilde{\beta }}\left(s+\frac{1}{(1+\beta )\beta }\right){\int }_0^{L\left(s\right)}{\lambda }_2{y}^{\beta }{G}_2\left(y\right)dyds,0\le t\le \tilde{\delta },\hfill \end{array}\right.$$

  where $\tilde{\gamma }=\frac{1-\beta }{1+\beta }.$ Now, we define

$$C_{loc}^2\left(\left(0,\tilde{\delta }\right];\mathbb{R}\right)=\left\{\mathsf{\Psi }:\left(0,\tilde{\delta }\right]\to \mathbb{R}|\mathsf{\Psi }\in C^2\left([a,b],\mathbb{R}\right)\mathrm{for} \mathrm{each} \mathrm{interval} [a,b]\subset \left(0,\tilde{\delta }\right]\right\}.$$

We are looking for a system of solutions $(\mathsf{\Psi },\mathsf{\Phi })\in E\times E$ that satisfy (17), where $E\times E=C_{loc}^2\left(\left(0,\frac{1}{2}\right],\mathbb{R}\right)\times C_{loc}^2\left(\left(0,\frac{1}{2}\right],\mathbb{R}\right)$. Again, we denote $\tilde{X}=C\left(\left[0,\frac{1}{2}\right],\mathbb{R}\right)$.

These results [11] inspired us to consider (8) for $\beta >1$ subject to boundary conditions (9)–(11) and analyze them. The differential Equation (8) is non-self-adjoint, nonlinear, and singular, and its solution is not known. So, it is difficult to study the solution. In this paper, we prove there is at least one solution in $E\times E$. We derive some results to establish the sign of solutions. We also estimate of the parameters ${\lambda }_1$ and ${\lambda }_2$ to show the nonexistence of solutions.

Many researchers such as Omari et al. [13,14,15,16,17,18], Pandey et al. [19,20], Xinguange et al. [21,22], Shah et al. [23], Li [24], Dunninger et al. [25,26], Cabada et al. [27,28,29,30,31], Zhang [32], Barilla et al. [33], etc., have studied the existence and nonexistence of solutions for a class of singular boundary value problems with the help of upper and lower solutions. Readers can refer to Reference [34] to learn more about the literature regarding the origin of singular BVPs and the various methodologies.

We have organized the rest of the paper as follows. We derive a sign of solutions in Section 2. We show the existence of solutions in Section 3. Section 4 is placed by the derivation of the region of the nonexistence of solutions. Finally, we conclude the work in Section 5.

2. Sign of Solutions

For non-negative values of the parameters ${\lambda }_1$ and ${\lambda }_2$, here, we derive some properties of the system of solution $(\mathsf{\Psi },\mathsf{\Phi })\in E\times E$, which satisfy the inequality

$$\left\{\begin{array}{c}{\mathsf{\Psi }}^{″}\ge K_1(t,\mathsf{\Psi },\mathsf{\Phi })+{(1+\beta )}^{-\tilde{\beta }}{t}^{-\tilde{\beta }}{\int }_0^{L\left(t\right)}{\lambda }_1{s}^{\beta }{G}_1\left(s\right)ds,\hfill \\ {\mathsf{\Phi }}^{″}\ge K_2(t,\mathsf{\Psi },\mathsf{\Phi })+{(1+\beta )}^{-\tilde{\beta }}{t}^{-\tilde{\beta }}{\int }_0^{L\left(t\right)}{\lambda }_2{s}^{\beta }{G}_2\left(s\right)ds,\hfill \end{array}\right.$$

(21)

for $t\in \left(0,\tilde{\delta }\right]$ and ${}^{\prime }$ denotes ${\displaystyle \frac{d}{dt}}$.

Lemma 1.

Assume the system of solutions $(\mathsf{\Psi },\mathsf{\Phi })\in E\times E$ of (17) satisfies the conditions ${\displaystyle \underset{t\to 0^{+}}{\lim }{t}^{\beta \tilde{\delta }}{\mathsf{\Phi }}^{\prime }\left(t\right)=0}$ and ${\displaystyle \underset{t\to 0^{+}}{\lim }{t}^{\beta \tilde{\delta }}{\mathsf{\Psi }}^{\prime }\left(t\right)=0}$ along with (21), then

$$\underset{t\to 0^{+}}{\lim }\mathsf{\Psi }\left(t\right)=0\&\underset{t\to 0^{+}}{\lim }\mathsf{\Phi }\left(t\right)=0.$$

(22)

Proof. 

The proof is similar as in Lemma 2.0.1 in References [11,35]. □

Lemma 2.

Assume the system of solutions $(\mathsf{\Psi },\mathsf{\Phi })\in E\times E$ of (17) satisfies the conditions $\mathsf{\Psi }\left(\tilde{\delta }\right)=0$, $\mathsf{\Phi }\left(\tilde{\delta }\right)=0$ and Inequality (21) along with (22), then

$$\mathsf{\Psi }\left(t\right)\le 0\&\mathsf{\Phi }\left(t\right)\le 0,\forall t\in \left(0,\tilde{\delta }\right].$$

(23)

Proof. 

It follows from Lemma 2.0.2 in References [11,35]. □

Lemma 3.

Assume the system of solutions $(\mathsf{\Psi },\mathsf{\Phi })\in E\times E$ of (17) satisfies the conditions $\beta \mathsf{\Phi }\left(\tilde{\delta }\right)={\mathsf{\Phi }}^{\prime }\left(\tilde{\delta }\right)$, $\beta \mathsf{\Psi }\left(\tilde{\delta }\right)={\mathsf{\Psi }}^{\prime }\left(\tilde{\delta }\right)$ and Inequality (21) along with (22), then $\mathsf{\Psi }\left(t\right)\le 0\&\mathsf{\Phi }\left(t\right)\le 0$ $\forall t\in \left(0,\tilde{\delta }\right]$.

Proof. 

It follows from Lemma 2.0.3 in References [11,35]. □

Lemma 4.

Assume the system of solutions $(\mathsf{\Psi },\mathsf{\Phi })\in E\times E$ of (17) satisfies the conditions $(\beta -1)\mathsf{\Phi }\left(\tilde{\delta }\right)={\mathsf{\Phi }}^{\prime }\left(\tilde{\delta }\right)$, $(\beta -1)\mathsf{\Psi }\left(\tilde{\delta }\right)={\mathsf{\Psi }}^{\prime }\left(\tilde{\delta }\right)$, Inequality (21) and Equation (22), then $\mathsf{\Psi }\left(t\right)\le 0$ and $\mathsf{\Phi }\left(t\right)\le 0$ $\forall t\in \left(0,\tilde{\delta }\right]$.

Proof. 

It follows from Lemma 3. □

Corollary 1.

Assume the system of solutions $(\mathsf{\Psi },\mathsf{\Phi })\in E\times E$ of (17) satisfies the conditions $\mathsf{\Psi }\left(t\right)\le 0$, $\mathsf{\Phi }\left(t\right)\le 0$ and Inequality (21) along with (22), then ${\displaystyle \underset{t\to 0^{+}}{\lim }{t}^{\beta \tilde{\delta }}{\mathsf{\Psi }}^{\prime }\left(t\right)=0}$ and ${\displaystyle \underset{t\to 0^{+}}{\lim }{t}^{\beta \tilde{\delta }}{\mathsf{\Phi }}^{\prime }\left(t\right)=0}$ if and only if ${\displaystyle \underset{t\to 0^{+}}{\lim }\frac{\mathsf{\Psi }\left(t\right)}{t^{\tilde{\delta }}}=0}$ and ${\displaystyle \underset{t\to 0^{+}}{\lim }\frac{\mathsf{\Phi }\left(t\right)}{t^{\tilde{\delta }}}=0}$.

Proof. 

It follows from Corollary 2.0.3 in References [11,35]. □

Lemma 5.

Assume the system of solutions $(\mathsf{\Psi },\mathsf{\Phi })\in E\times E$ of (17) satisfies the conditions ${\displaystyle \underset{t\to 0^{+}}{\lim }\frac{\mathsf{\Psi }\left(t\right)}{t^{\tilde{\delta }}}=0}$ and ${\displaystyle \underset{t\to 0^{+}}{\lim }\frac{\mathsf{\Phi }\left(t\right)}{t^{\tilde{\delta }}}=0}$. Then, for every $\mu \in [0,1)$, we have

$$\underset{t\to 0^{+}}{\lim }{t}^{1-\mu }{\int }_t^{\tilde{\delta }}\frac{{\mathsf{\Psi }}^2}{s^2}ds=0\&\underset{t\to 0^{+}}{\lim }{t}^{1-\mu }{\int }_t^{\tilde{\delta }}\frac{{\mathsf{\Phi }}^2}{s^2}ds=0.$$

(24)

Proof. 

It follows from Lemma 2.0.5 in References [11,35]. □

Lemma 6.

If the system of solutions of (18) is $(\mathsf{\Psi },\mathsf{\Phi })\in E\times E$ if and only if it lies in $\tilde{X}\times \tilde{X}$ and satisfies IE (21) along with the conditions

$$\underset{t\to 0^{+}}{\lim }\frac{\left|\mathsf{\Psi }\right(t\left)\right|}{t}<+\infty \&\underset{t\to 0^{+}}{\lim }\frac{\left|\mathsf{\Phi }\right(t\left)\right|}{t}<+\infty .$$

(25)

Proof. 

Assume, $(\mathsf{\Psi },\mathsf{\Phi })\in E\times E$ is the solutions of (18). Thus, Green’s function of (18) is $-(1+\beta )s\left(\tilde{\delta }-t\right)$ when $0\le s\le t$ and $-(1+\beta )t\left(\tilde{\delta }-s\right)$ for all $t\le s\le \tilde{\delta }$. So, from (18), we can have (21). Conversely, let $(\mathsf{\Psi },\mathsf{\Phi })\in \tilde{X}\times \tilde{X}$ satisfy (21). Therefore, we have (18) and $(\mathsf{\Psi },\mathsf{\Phi })\in E\times E$. Now, we take $F_1\left(t\right)=\frac{{\mathsf{\Psi }}^2}{t^{\tilde{\delta }}} \mathrm{and}{F}_2\left(t\right)=\frac{1}{t^{\beta \tilde{\delta }}},\forall t\in \left(0,\tilde{\delta }\right].$ Since ${\displaystyle \underset{t\to 0^{+}}{\lim }\frac{\mathsf{\Psi }\left(t\right)}{t^{\tilde{\delta }}}=0}$ and ${\displaystyle \underset{t\to 0^{+}}{\lim }\mathsf{\Psi }\left(t\right)=0}$, we have $F_1\in L\left(0,\tilde{\delta }\right]$. Hence, $F_1{F}_2=\frac{{\mathsf{\Psi }}^2}{t^{(1+\beta )\tilde{\delta }}}=\frac{{\mathsf{\Psi }}^2}{t}\in L\left(0,\tilde{\delta }\right]$ i.e.,

$${\int }_0^{\tilde{\delta }}\frac{{\mathsf{\Psi }}^2}{s}ds<\infty .$$

(26)

Again, we take $F_3\left(t\right)=F_1{F}_2\times \frac{1}{t^{\mu }}=\frac{{\mathsf{\Psi }}^2}{t^{1+\mu }} \mathrm{and}{F}_4\left(t\right)=\frac{1}{t^{1-\mu }},\forall t\in \left(0,\tilde{\delta }\right],\mu \in (0,1).$ So, we have $F_3\in L\left(0,\tilde{\delta }\right]$ and $F_4\in L\left(0,\tilde{\delta }\right]$. Consequently, $F_3{F}_4\in L\left(0,\tilde{\delta }\right]$ subject to the condition $\mu \in (0,1)$, i.e.,

$${\int }_0^{\tilde{\delta }}\frac{{\mathsf{\Psi }}^2}{s^2}ds<\infty .$$

(27)

By a similar analysis, we have

$${\int }_0^{\tilde{\delta }}\frac{{\mathsf{\Phi }}^2}{s}ds<\infty \mathrm{and}{\int }_0^{\tilde{\delta }}\frac{{\mathsf{\Phi }}^2}{s^2}ds<\infty .$$

(28)

Now,

$$\begin{array}{ccc}\hfill & & \underset{t\to 0^{+}}{\lim }\frac{\left|\mathsf{\Psi }\right(t\left)\right|}{t}=\left|{\int }_0^{\tilde{\delta }}(1+\beta )K_1(s,\mathsf{\Psi },\mathsf{\Phi })\left(\tilde{\delta }-s\right)ds+{\int }_0^{\tilde{\delta }}{(1+\beta )}^{\tilde{\gamma }}{s}^{-\tilde{\beta }}\left(\tilde{\delta }-s\right){\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\beta }{G}_{1}dyds\right|,\hfill \\ & & \underset{t\to 0^{+}}{\lim }\frac{\left|\mathsf{\Phi }\right(t\left)\right|}{t}=\left|{\int }_0^{\tilde{\delta }}(1+\beta )K_2(s,\mathsf{\Psi },\mathsf{\Phi })\left(\tilde{\delta }-s\right)ds+{\int }_0^{\tilde{\delta }}{(1+\beta )}^{\tilde{\gamma }}{s}^{-\tilde{\beta }}\left(\tilde{\delta }-s\right){\int }_0^{\tilde{\delta }}{\lambda }_2{y}^{\beta }{G}_{2}dyds\right|.\hfill \end{array}$$

After substituting the values of $K_1(s,\mathsf{\Psi },\mathsf{\Phi })$ and $K_2(s,\mathsf{\Psi },\mathsf{\Phi })$ and using Inequalities (26)–(28), we have (25). □

The following proof of two lemmas follows from Lemma 6.

Lemma 7.

The system of solutions of (19) is $(\mathsf{\Psi },\mathsf{\Phi })\in E\times E$ if and only if it lies in $\tilde{X}\times \tilde{X}$ and satisfies IE (21) along with Condition (25).

Lemma 8.

The system of solutions of (20) is $(\mathsf{\Psi },\mathsf{\Phi })\in E\times E$ if and only if it lies in $\tilde{X}\times \tilde{X}$ and satisfies IE (21) along with Condition (25).

3. Results on Existence

Here, we go through the following lemmas to prove the main results. Now, for $k,m\in \mathbb{R}$, Equations (18)–(20) lead to

$$\mathrm{System}1a:\left\{\begin{array}{c}{\displaystyle {\mathsf{\Psi }}^{″}+k\mathsf{\Psi }=H_1(t,\mathsf{\Psi },\mathsf{\Phi }),}\hfill \\ {\displaystyle {\mathsf{\Phi }}^{″}+m\mathsf{\Phi }=H_2(t,\mathsf{\Psi },\mathsf{\Phi }),}\hfill \\ {\displaystyle \underset{t\to 0^{+}}{\lim }{t}^{\beta \tilde{\delta }}{\mathsf{\Psi }}^{\prime }\left(t\right)=0,\mathsf{\Psi }\left(\tilde{\delta }\right)=0,}\hfill \\ {\displaystyle \underset{t\to 0^{+}}{\lim }{t}^{\beta \tilde{\delta }}{\mathsf{\Phi }}^{\prime }\left(t\right)=0,\mathsf{\Phi }\left(\tilde{\delta }\right)=0,}\hfill \end{array}\right.$$

(29)

$$\mathrm{System}2a:\left\{\begin{array}{c}{\displaystyle {\mathsf{\Psi }}^{″}+k\mathsf{\Psi }=H_1(t,\mathsf{\Psi },\mathsf{\Phi }),}\hfill \\ {\displaystyle {\mathsf{\Phi }}^{″}+m\mathsf{\Phi }=H_2(t,\mathsf{\Psi },\mathsf{\Phi }),}\hfill \\ {\displaystyle \underset{t\to 0^{+}}{\lim }{t}^{\beta \tilde{\delta }}{\mathsf{\Psi }}^{\prime }\left(t\right)=0,(\beta -1)\mathsf{\Psi }\left(\tilde{\delta }\right)={\mathsf{\Psi }}^{\prime }\left(\tilde{\delta }\right),}\hfill \\ {\displaystyle \underset{t\to 0^{+}}{\lim }{t}^{\beta \tilde{\delta }}{\mathsf{\Phi }}^{\prime }\left(t\right)=0,(\beta -1)\mathsf{\Phi }\left(\tilde{\delta }\right)={\mathsf{\Phi }}^{\prime }\left(\tilde{\delta }\right),}\hfill \end{array}\right.$$

(30)

and

$$\mathrm{System}3a:\left\{\begin{array}{c}{\displaystyle {\mathsf{\Psi }}^{″}+k\mathsf{\Psi }=H_1(t,\mathsf{\Psi },\mathsf{\Phi }),}\hfill \\ {\displaystyle {\mathsf{\Phi }}^{″}+m\mathsf{\Phi }=H_2(t,\mathsf{\Psi },\mathsf{\Phi }),}\hfill \\ {\displaystyle \underset{t\to 0^{+}}{\lim }{t}^{\beta \tilde{\delta }}{\mathsf{\Psi }}^{\prime }\left(t\right)=0,\beta \mathsf{\Psi }\left(\tilde{\delta }\right)={\mathsf{\Psi }}^{\prime }\left(\tilde{\delta }\right),}\hfill \\ {\displaystyle \underset{t\to 0^{+}}{\lim }{t}^{\beta \tilde{\delta }}{\mathsf{\Phi }}^{\prime }\left(t\right)=0,\beta \mathsf{\Phi }\left(\tilde{\delta }\right)={\mathsf{\Phi }}^{\prime }\left(\tilde{\delta }\right),}\hfill \end{array}\right.$$

(31)

for $0<t\le \tilde{\delta }$ such that

$$\begin{array}{ccc}& & \hfill {\displaystyle H_1(t,\mathsf{\Psi },\mathsf{\Phi })=K_1(t,\mathsf{\Psi },\mathsf{\Phi })+{(1+\beta )}^{-\tilde{\beta }}{t}^{-\tilde{\beta }}{\int }_0^{L\left(t\right)}{\lambda }_1{s}^{\beta }{G}_1\left(s\right)ds+k\mathsf{\Psi },}\hfill \\ & & \hfill {\displaystyle H_2(t,\mathsf{\Psi },\mathsf{\Phi })=K_2(t,\mathsf{\Psi },\mathsf{\Phi })+{(1+\beta )}^{-\tilde{\beta }}{t}^{-\tilde{\beta }}{\int }_0^{L\left(t\right)}{\lambda }_2{s}^{\beta }{G}_2\left(s\right)ds+m\mathsf{\Phi },}\hfill \end{array}$$

Now, we define

• $H_0=\{k<0,m<0\}$;
• $H_1^{*}=\left\{{\displaystyle L<0,\left(\sqrt{\left|L\right|}cosh\left(\sqrt{\left|L\right|}\tilde{\delta }\right)-(\beta -1)sinh\left(\sqrt{\left|L\right|}\tilde{\delta }\right)\right)>0\mathrm{for} L=k,m}\right\}$;
• $H_2^{*}=\left\{{\displaystyle L<0,\sqrt{\left|L\right|}cosh\left(\sqrt{\left|L\right|}\tilde{\delta }\right)-\beta sinh\left(\sqrt{\left|L\right|}\tilde{\delta }\right)>0\mathrm{for} L=k,m}\right\}$.

By using Lemma $3.0.1$ in [11], we can prove the following lemmas.

Lemma 9.

Let $H_0$ be true. The solution of System $1a$ is $(\mathsf{\Psi },\mathsf{\Phi })\in E\times E$ if and only if it lies in $\tilde{X}\times \tilde{X}$ and satisfies the following IE

$$\mathsf{\Psi }\left(t\right)={\int }_0^{\tilde{\delta }}{H}_1(s,\mathsf{\Psi },\mathsf{\Phi })G_D\left[k\right](s,t)ds\&\mathsf{\Phi }\left(t\right)={\int }_0^{\tilde{\delta }}{H}_2(s,\mathsf{\Psi },\mathsf{\Phi })G_D\left[m\right](s,t)ds$$

(32)

for ${\displaystyle 0\le t\le \tilde{\delta }}$. Also, Green’s functions $G_D\left[L\right](s,t)$ are nonpositive for  $L=m,k$ over the domain $0\le s\le \tilde{\delta }$ and $0\le t\le \tilde{\delta }$.

Lemma 10.

Let $H_1^{*}$ be true. The solution of System $2a$ is $(\mathsf{\Psi },\mathsf{\Phi })\in E\times E$ if and only if it lies in $\tilde{X}\times \tilde{X}$ and satisfies the following IE

$$\mathsf{\Psi }\left(t\right)={\int }_0^{\tilde{\delta }}{H}_1(s,\mathsf{\Psi },\mathsf{\Phi })G_{N_1}\left[k\right](s,t)ds\&\mathsf{\Phi }\left(t\right)={\int }_0^{\tilde{\delta }}{H}_2(s,\mathsf{\Psi },\mathsf{\Phi })G_{N_1}\left[m\right](s,t)ds$$

(33)

for ${\displaystyle 0\le t\le \tilde{\delta }}$. Also, Green’s functions $G_{N_1}\left[L\right](s,t)$ are nonpositive for $L=m,k$ over the domain $0\le s\le \tilde{\delta }$ and $0\le t\le \tilde{\delta }$.

Lemma 11.

Let $H_2^{*}$ be true. The solution of System $3a$ is $(\mathsf{\Psi },\mathsf{\Phi })\in E\times E$ if and only if it lies in $\tilde{X}\times \tilde{X}$ and satisfies the following IE

$$\mathsf{\Psi }\left(t\right)={\int }_0^{\tilde{\delta }}{H}_1(s,\mathsf{\Psi },\mathsf{\Phi })G_{N_2}\left[k\right](s,t)ds\&\mathsf{\Phi }\left(t\right)={\int }_0^{\tilde{\delta }}{H}_2(s,\mathsf{\Psi },\mathsf{\Phi })G_{N_2}\left[m\right](s,t)ds$$

(34)

for ${\displaystyle 0\le t\le \tilde{\delta }.}$ Also, Green’s functions $G_{N_2}\left[L\right](s,t)$ are nonpositive for $L=m,k$ over the domain $0\le s\le \tilde{\delta }$ and $0\le t\le \tilde{\delta }$.

Now, we define $(u_0,v_0)$ and $({\alpha }_0,{\beta }_0)$ such that $u_0\le {\alpha }_0$ and $v_0\le {\beta }_0$ for Equations (18)–(20).

Definition 1

([36]). Pair functions $({\alpha }_0,{\beta }_0)\in E\times E$ are called the upper solutions of Equation (18)–(20)) if

$$\left\{\begin{array}{c}{\displaystyle {\alpha }_0^{″}\le K_1\left(t,{\alpha }_0,{\beta }_0\right)+{(1+\beta )}^{-\tilde{\beta }}{t}^{-\tilde{\beta }}{\int }_0^{L\left(t\right)}{\lambda }_1{s}^{\beta }{G}_1\left(s\right)ds,}\hfill \\ {\displaystyle {\beta }_0^{″}\le K_2\left(t,{\alpha }_0,{\beta }_0\right)+{(1+\beta )}^{-\tilde{\beta }}{t}^{-\tilde{\beta }}{\int }_0^{L\left(t\right)}{\lambda }_2{s}^{\beta }{G}_2\left(s\right)ds,}\hfill \end{array}\right.$$

(35)

for all ${\displaystyle t\in \left(0,\tilde{\delta }\right]}$ with ${\displaystyle \underset{t\to 0^{+}}{\lim }\frac{{\beta }_0\left(t\right)}{t^{\tilde{\delta }}}=0}$, ${\displaystyle \underset{t\to 0^{+}}{\lim }\frac{{\alpha }_0\left(t\right)}{t^{\tilde{\delta }}}=0}$, ${\alpha }_0\left(\tilde{\delta }\right)\ge 0$ and ${\beta }_0\left(\tilde{\delta }\right)\ge 0$ (respectively, $(\beta -1){\alpha }_0\left(\tilde{\delta }\right)\le {\alpha }_0^{\prime }\left(\tilde{\delta }\right)$, $(\beta -1){\beta }_0\left(\tilde{\delta }\right)\le {\beta }_0^{\prime }\left(\tilde{\delta }\right)$ and $\beta {\alpha }_0\left(\tilde{\delta }\right)\le {\alpha }_0^{\prime }\left(\tilde{\delta }\right)$), $\beta {\beta }_0\left(\tilde{\delta }\right)\le {\beta }_0^{\prime }\left(\tilde{\delta }\right)$).

Definition 2

([36]). Pair functions $(u_0,v_0)\in E\times E$ are called the lower solutions of Equation (18) (respectively, (19) and (20)) if

$$\left\{\begin{array}{c}{\displaystyle u_0^{″}\ge K_1\left(t,u_0,v_0\right)+{(1+\beta )}^{-\tilde{\beta }}{t}^{-\tilde{\beta }}{\int }_0^{L\left(t\right)}{\lambda }_1{s}^{\beta }{G}_1\left(s\right)ds,}\hfill \\ {\displaystyle v_0^{″}\ge K_2\left(t,u_0,v_0\right)+{(1+\beta )}^{-\tilde{\beta }}{t}^{-\tilde{\beta }}{\int }_0^{L\left(t\right)}{\lambda }_2{s}^{\beta }{G}_2\left(s\right)ds,}\hfill \end{array}\right.$$

(36)

for all ${\displaystyle t\in \left(0,\tilde{\delta }\right]}$ such that ${\displaystyle \underset{t\to 0^{+}}{\lim }\frac{v_0\left(t\right)}{t^{\tilde{\delta }}}=0}$, ${\displaystyle \underset{t\to 0^{+}}{\lim }\frac{u_0\left(t\right)}{t^{\tilde{\delta }}}=0}$, $u_0\left(\tilde{\delta }\right)\le 0$ and $v_0\left(\tilde{\delta }\right)\le 0$ (respectively, $(\beta -1)u_0\left(\tilde{\delta }\right)\ge (\beta +1)u_0^{\prime }\left(\tilde{\delta }\right)$, $(\beta -1)v_0\left(\tilde{\delta }\right)\ge v_0^{\prime }\left(\tilde{\delta }\right)$ and $\beta u_0\left(\tilde{\delta }\right)\ge (\beta +1)u_0^{\prime }\left(\tilde{\delta }\right)$), $\beta v_0\left(\tilde{\delta }\right)\ge v_0^{\prime }\left(\tilde{\delta }\right)$).

We consider the Banach space $\tilde{X}$ such that

$$\underset{t\in \left[0,\tilde{\delta }\right]}{max}\left|\mathsf{\Psi }\left(t\right)\right|=∥\mathsf{\Psi }∥.$$

(37)

In view of (37), we consider $\tilde{X}\times \tilde{X}$ such that

$$∥\mathsf{\Psi }∥+∥\mathsf{\Phi }∥={\Vert (\mathsf{\Psi },\mathsf{\Phi })\Vert }_1.$$

(38)

Now, we define $T_D:\tilde{X}\times \tilde{X}\to \tilde{X}\times \tilde{X}$ corresponding to Equation (29), which is given by

$$T_D(\mathsf{\Psi },\mathsf{\Phi })\left(t\right)=\left(X_D(\mathsf{\Psi },\mathsf{\Phi })\left(t\right),Y_D(\mathsf{\Psi },\mathsf{\Phi })\left(t\right)\right),$$

(39)

where $X_D(\mathsf{\Psi },\mathsf{\Phi })\left(t\right),Y_D(\mathsf{\Psi },\mathsf{\Phi })\left(t\right):\tilde{X}\times \tilde{X}\to \tilde{X}$ are the operators defined by

$$X_D(\mathsf{\Psi },\mathsf{\Phi })\left(t\right)={\int }_0^{\tilde{\delta }}{H}_1(s,\mathsf{\Psi },\mathsf{\Phi })G_D\left[k\right](s,t)ds\&Y_D(\mathsf{\Psi },\mathsf{\Phi })\left(t\right)={\int }_0^{\tilde{\delta }}{H}_2(s,\mathsf{\Psi },\mathsf{\Phi })G_D\left[m\right](s,t)ds$$

(40)

for all ${\displaystyle 0\le t\le \tilde{\delta }}$. By a similar manner, we can define $T_{N_1},T_{N_2},X_{N_1},X_{N_2},Y_{N_1}$, and $Y_{N_2}$ corresponding to Equations (30) and (31), respectively. If we show $T_D$ has a fixed point $(\mathsf{\Psi },\mathsf{\Phi })\in \tilde{X}\times \tilde{X}$, then by using Lemma 9, we obtain the system of solutions $(\mathsf{\Psi },\mathsf{\Phi })$ for some values of k and m. Hence, we have $(\mathsf{\Psi },\mathsf{\Phi })$ is the system of solutions of Equation (18)

Lemma 12.

Let $H_0$ be true, then the operator $T_D:\tilde{X}\times \tilde{X}\to \tilde{X}\times \tilde{X}$ is completely continuous.

Proof. 

Let us consider $\left\{({\zeta }_n,{\Im }_n)\right\}$ as any sequence in $\tilde{X}\times \tilde{X}$ that converge to the pair of functions $(\zeta ,\Im )$ in $\tilde{X}\times \tilde{X}$. Therefore, we have ${\displaystyle \underset{n\to \infty }{\lim }({\zeta }_n,{\Im }_n)=(\zeta ,\Im )}$. So, there exists $m_1\in \mathbb{N}$ such that

$${∥({\zeta }_n,{\Im }_n)-(\zeta ,\Im )∥}_1\to 0 \mathrm{for} \mathrm{all} n\ge m_1.$$

(41)

Now, in view of the definition of the norm, it is easy to show that ${\zeta }_n\to \zeta \& {\Im }_n\to \Im \mathrm{as} n\to \infty .$ Now,

$$\begin{array}{ccc}& & \left|X_D({\zeta }_n,{\Im }_n\left(t\right)-X_D(\zeta ,\Im )\left(t\right)\right|\hfill \\ & & \hfill \end{array}$$

(42)

$$\begin{array}{ccc}& & =\left|{\int }_0^{\tilde{\delta }}{H}_1(s,{\zeta }_n,{\Im }_n)G_D\left[k\right](s,t)ds-{\int }_0^{\tilde{\delta }}{H}_1(s,\zeta ,\Im )G_D\left[k\right](s,t)ds\right|\hfill \\ & & \hfill \end{array}$$

(43)

$$\begin{array}{ccc}& & =\left|{\int }_0^{\tilde{\delta }}{G}_D\left[k\right](s,t)\left(\frac{p{K}_{11}({\zeta }_n^2-{\zeta }^2)}{2{(1+\beta )}^2{s}^2}+\frac{p{K}_{12}({\Im }_n^2-{\Im }^2)}{2{(1+\beta )}^2{s}^2}+k({\zeta }_n-\zeta )\right)ds\right|.\hfill \\ & & \hfill \end{array}$$

(44)

By using Lemmas 1 and 5, we have

$$∥X_D({\zeta }_n,{\Im }_n)\left(t\right)-X_D(\zeta ,\Im )\left(t\right)∥\to 0 \mathrm{as} n\to \infty .$$

(45)

Similarly,

$$∥Y_D({\zeta }_n,{\Im }_n)\left(t\right)-Y_D(\zeta ,\Im )\left(t\right)∥\to 0 \mathrm{as} n\to \infty .$$

(46)

Therefore,

$${∥T_D({\zeta }_n,{\Im }_n)\left(t\right)-T_D(\zeta ,\Im )\left(t\right)∥}_1\to 0 \mathrm{as} n\to \infty .$$

(47)

It follows that the operator $T_D$ is continuous. Now, we consider any bounded subset B of $\tilde{X}\times \tilde{X}$ such that $\{(\zeta ,\Im )\in \tilde{X}\times \tilde{X}:{\parallel (\zeta ,\Im )\parallel }_1\le m_2\}$. Now,

$$\begin{array}{ccc}& & ∥X_D(\zeta ,\Im )\left(t\right)∥\hfill \\ & & =\underset{t\in \left[0,\tilde{\delta }\right]}{max}\left|{\int }_0^{\tilde{\delta }}{H}_1(s,\zeta ,\Im )G_D\left[k\right](s,t)ds\right|\hfill \\ & & \le \underset{(t,s,\zeta ,\Im )\in \left[0,\tilde{\delta }\right]\times \left[0,\tilde{\delta }\right]\times [0,m_2]\times [0,m_2]}{max}\left|G_D\left[k\right](s,t)\tilde{\delta }\left(K_1\left(s,\zeta ,\Im \right)+{(1+\beta )}^{-\tilde{\beta }}{s}^{-\tilde{\beta }}\times \right.\right.\hfill \\ & & \left.\left.{\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\beta }{G}_1\left(y\right)dy+k\zeta \right)\right|,\hfill \\ & & <\infty .\hfill \end{array}$$

Hence, we have $X_D(\mathsf{\Psi },\mathsf{\Phi })\left(t\right)$ as uniformly bounded. Similarly, we can show $Y_D(\mathsf{\Psi },\mathsf{\Phi })\left(t\right)$ and $T_D$ is uniformly bounded. Now,

$$\begin{array}{ccc}& & \left|X_D(\mathsf{\Psi },\mathsf{\Phi })\left(t_1\right)-X_D(\mathsf{\Psi },\mathsf{\Phi })\left(t_2\right)\right|\hfill \end{array}$$

(48)

$$\begin{array}{ccc}& & =\left|{\int }_0^{\tilde{\delta }}{H}_1(s,\mathsf{\Psi },\mathsf{\Phi })G_D\left[k\right](s,t_1)ds-{\int }_0^{\tilde{\delta }}{H}_1(s,\mathsf{\Psi },\mathsf{\Phi })G_D\left[k\right](s,t_2)ds\right|,\hfill \end{array}$$

(49)

Since Green’s function is continuous, we have

$$∥X_D(\mathsf{\Psi },\mathsf{\Phi })\left(t_1\right)-X_D(\mathsf{\Psi },\mathsf{\Phi })\left(t_2\right)∥\to 0 \mathrm{as} t_1\to t_2.$$

(50)

Similarly,

$$∥Y_D(\mathsf{\Psi },\mathsf{\Phi })\left(t_1\right)-Y_D(\mathsf{\Psi },\mathsf{\Phi })\left(t_2\right)∥\to 0 \mathrm{as} t_1\to t_2.$$

(51)

So, $T_D$ is equicontinuous. It follows that $T_D:\tilde{X}\times \tilde{X}\to \tilde{X}\times \tilde{X}$ is completely continuous. Similarly, we can show $X_D(\mathsf{\Psi },\mathsf{\Phi })\left(t\right),Y_D(\mathsf{\Psi },\mathsf{\Phi })\left(t\right):\tilde{X}\times \tilde{X}\to \tilde{X}$ are completely continuous. Hence, the fixed point of $T_D$ lies in $\tilde{X}\times \tilde{X}$ such that

$$(\mathsf{\Psi },\mathsf{\Phi })\left(t\right)=T_D(\mathsf{\Psi },\mathsf{\Phi })\left(t\right)=\left(X_D(\mathsf{\Psi },\mathsf{\Phi })\left(t\right),Y_D(\mathsf{\Psi },\mathsf{\Phi })\left(t\right)\right),\forall (\mathsf{\Psi },\mathsf{\Phi })\left(t\right)\in \tilde{X}\times \tilde{X}.$$

(52)

Hence, the proof is complete. □

Similarly, we prove that $T_{N_1},T_{N_2}:\tilde{X}\times \tilde{X}\to \tilde{X}\times \tilde{X}$ and $X_{N_1},X_{N_2},Y_{N_1},Y_{N_2}:\tilde{X}\times \tilde{X}\to \tilde{X}$ are also completely continuous. We consider

$${\mathsf{\Omega }}_1=\left\{\mathsf{\Psi }\in E:u_0\left(t\right)\le \mathsf{\Psi }\left(t\right)\le {\alpha }_0\left(t\right),\forall t\in \left[0,\tilde{\delta }\right]\right\},$$

$${\mathsf{\Omega }}_2=\left\{\mathsf{\Phi }\in E:v_0\left(t\right)\le \mathsf{\Phi }\left(t\right)\le {\beta }_0\left(t\right),\forall t\in \left[0,\tilde{\delta }\right]\right\},$$

and

$$\mathsf{\Omega }=\left\{(\mathsf{\Psi },\mathsf{\Phi })\in E\times E:(u_0,v_0)\left(t\right)\le (\mathsf{\Psi },\mathsf{\Phi })\left(t\right)\le ({\alpha }_0,{\beta }_0)\left(t\right),\forall t\in \left[0,\tilde{\delta }\right]\right\},$$

such that $u_0\le \mathsf{\Psi }\le {\alpha }_0$ and $v_0\le \mathsf{\Phi }\le {\beta }_0$. Again, we assume the following conditions:

$$\left\{\begin{array}{c}{\displaystyle \underset{t\to 0^{+}}{\lim }{\alpha }_0\left(t\right)=0,\underset{t\to 0^{+}}{\lim }{\displaystyle \frac{|{\alpha }_0\left(t\right)|}{t}}<\infty ,{\alpha }_0\left(t\right)\le 0, \mathrm{for} t\in \left(0,\tilde{\delta }\right],}\hfill \\ {\displaystyle \underset{t\to 0^{+}}{\lim }{\beta }_0\left(t\right)=0,\underset{t\to 0^{+}}{\lim }{\displaystyle \frac{|{\beta }_0\left(t\right)|}{t}}<\infty ,{\beta }_0\left(t\right)\le 0, \mathrm{for} t\in \left(0,\tilde{\delta }\right],}\hfill \\ {\displaystyle \underset{t\to 0^{+}}{\lim }{u}_0\left(t\right)=0,\underset{t\to 0^{+}}{\lim }{\displaystyle \frac{|u_0\left(t\right)|}{t}}<\infty ,}\hfill \\ {\displaystyle \underset{t\to 0^{+}}{\lim }{v}_0\left(t\right)=0,\underset{t\to 0^{+}}{\lim }{\displaystyle \frac{|v_0\left(t\right)|}{t}}<\infty .}\hfill \end{array}\right.$$

(53)

Lemma 13.

Consider an operator $T_D^{\mathsf{\Omega }}:\mathsf{\Omega }\subset \tilde{X}\times \tilde{X}\to \tilde{X}\times \tilde{X}$ by

$$T_D^{\mathsf{\Omega }}(\mathsf{\Psi },\mathsf{\Phi })\left(t\right)=\left(X_D^{\mathsf{\Omega }}(\mathsf{\Psi },\mathsf{\Phi })\left(t\right),Y_D^{\mathsf{\Omega }}(\mathsf{\Psi },\mathsf{\Phi })\left(t\right)\right),$$

(54)

where $X_D^{\mathsf{\Omega }}(\mathsf{\Psi },\mathsf{\Phi })\left(t\right)$ and $Y_D^{\mathsf{\Omega }}(\mathsf{\Psi },\mathsf{\Phi })\left(t\right)$ are defined by (40) such that m and k are negative real numbers. Then,

i. 

$$\left(X_D^{\mathsf{\Omega }}(u_0,v_0)\left(t\right),Y_D^{\mathsf{\Omega }}(u_0,v_0)\left(t\right)\right)=T_D^{\mathsf{\Omega }}(u_0,v_0)\left(t\right)\ge (u_0,v_0)\left(t\right),$$

$$\left(X_D^{\mathsf{\Omega }}({\alpha }_0,{\beta }_0)\left(t\right),Y_D^{\mathsf{\Omega }}({\alpha }_0,{\beta }_0)\left(t\right)\right)=T_D^{\mathsf{\Omega }}({\alpha }_0,{\beta }_0)\left(t\right)\le ({\alpha }_0,{\beta }_0)\left(t\right),$$

where $(u_0,v_0)$ and $({\alpha }_0,{\beta }_0)$ are given by (35) and (36).

ii. 

and $T_D^{\mathsf{\Omega }}\left(\mathsf{\Omega }\right)\subseteq \mathsf{\Omega }$.

Proof. 

From (36), we have

$${\displaystyle u_0^{\prime \prime }\ge K_1(t,u_0,v_0)+{(1+\beta )}^{-\tilde{\beta }}{t}^{-\tilde{\beta }}{\int }_0^{L\left(t\right)}{\lambda }_1{s}^{\beta }{G}_1\left(s\right)ds,}$$

(55)

for ${\displaystyle t\in \left(0,\tilde{\delta }\right].}$ Equivalently, there exists a non-negative function $f\left(t\right)\in L^1\left[0,\tilde{\delta }\right]$ such that

$$\begin{array}{c}\hfill {\displaystyle u_0^{″}+k{u}_0=K_1\left(t,u_0,v_0\right)+{(1+\beta )}^{-\tilde{\beta }}{t}^{-\tilde{\beta }}{\int }_0^{L\left(t\right)}{\lambda }_1{s}^{\beta }{G}_1\left(s\right)ds+k{u}_0+f\left(t\right),}\end{array}$$

for $t\in \left(0,\tilde{\delta }\right].$ By using Lemma 9, we obtain

$$\begin{array}{ccc}& & u_0={\int }_0^{\tilde{\delta }}{G}_D\left[k\right](s,t)\left(K_1\left(s,u_0,v_0\right)+{(1+\beta )}^{-\tilde{\beta }}{s}^{-\tilde{\beta }}{\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\beta }{G}_1\left(y\right)dy+k{u}_0+f\left(s\right)\right)ds,\hfill \end{array}$$

for $t\in \left(0,\tilde{\delta }\right].$ Since $G_D\left[k\right](s,t)$ is nonpositive for all $0\le t\le \tilde{\delta }$ and $0\le s\le \tilde{\delta }$, we have

$$\begin{array}{ccc}\hfill {\displaystyle u_0}& & \le {\int }_0^{\tilde{\delta }}{G}_D\left[k\right](s,t)\left(K_1\left(s,u_0,v_0\right)+{(1+\beta )}^{-\tilde{\beta }}{s}^{-\tilde{\beta }}{\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\beta }{G}_1\left(y\right)dy+k{u}_0\right)ds,\hfill \\ & & =X_D^{\mathsf{\Omega }}(u_0,v_0)\left(t\right), \mathrm{for} t\in \left(0,\tilde{\delta }\right].\hfill \end{array}$$

(56)

By a similar manner, we can obtain $v_0\left(t\right)\le Y_D^{\mathsf{\Omega }}(u_0,v_0)\left(t\right) \mathrm{for} \mathrm{all} t\in \left(0,\tilde{\delta }\right].$ Again, from (35), there exists a non-negative function $f\left(t\right)\in L^1\left[0,\tilde{\delta }\right]$ such that

$$\begin{array}{ccc}\hfill {\displaystyle {\alpha }_0^{″}+k{\alpha }_0}& & =K_1\left(t,{\alpha }_0,{\beta }_0\right)+{(1+\beta )}^{-\tilde{\beta }}{t}^{-\tilde{\beta }}{\int }_0^{L\left(t\right)}{\lambda }_1{s}^{\beta }{G}_1\left(s\right)ds+k{\alpha }_0-f\left(t\right),\hfill \end{array}$$

for all $t\in \left(0,\tilde{\delta }\right].$ Therefore, by a similar analysis, we have

$$\begin{array}{ccc}\hfill {\displaystyle {\alpha }_0}& & \ge {\int }_0^{\tilde{\delta }}{G}_D\left[k\right](s,t)\left(K_1\left(s,{\alpha }_0,{\beta }_0\right)+{(1+\beta )}^{-\tilde{\beta }}{s}^{-\tilde{\beta }}{\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\beta }{G}_1\left(y\right)dy+k{\alpha }_0\right)ds,\hfill \\ & & =X_D^{\mathsf{\Omega }}({\alpha }_0,{\beta }_0)\left(t\right), \mathrm{for} \mathrm{all} t\in \left(0,\tilde{\delta }\right].\hfill \end{array}$$

(57)

Similarly, we have ${\beta }_0\left(t\right)\ge Y_D^{\mathsf{\Omega }}({\alpha }_0,{\beta }_0)\left(t\right), \mathrm{for} t\in \left(0,\tilde{\delta }\right].$ Hence, we have

$$\begin{array}{ccc}\hfill & & \left(X_D^{\mathsf{\Omega }}(u_0,v_0)\left(t\right),Y_D^{\mathsf{\Omega }}(u_0,v_0)\left(t\right)\right)=T_D^{\mathsf{\Omega }}(u_0,v_0)\left(t\right)\ge (u_0,v_0)\left(t\right),\hfill \end{array}$$

(58)

$$\begin{array}{ccc}& & \left(X_D^{\mathsf{\Omega }}({\alpha }_0,{\beta }_0)\left(t\right),Y_D^{\mathsf{\Omega }}({\alpha }_0,{\beta }_0)\left(t\right)\right)=T_D^{\mathsf{\Omega }}({\alpha }_0,{\beta }_0)\left(t\right)\le ({\alpha }_0,{\beta }_0)\left(t\right),\hfill \end{array}$$

(59)

Here, we choose an element $(\mathsf{\Psi },\mathsf{\Phi })$ in $\mathsf{\Omega }$. Since $u_0\le \mathsf{\Psi }\le 0$ and $v_0\le \mathsf{\Phi }\le 0$, we have

$$u_0^2\ge {\mathsf{\Psi }}^2\&v_0^2\ge {\mathsf{\Phi }}^2.$$

(60)

So, we have

$$\begin{array}{ccc}\hfill {\displaystyle }& & X_D^{\mathsf{\Omega }}(\mathsf{\Psi },\mathsf{\Phi })\left(t\right)\hfill \\ & & ={\int }_0^{\tilde{\delta }}{G}_D\left[k\right](s,t)\left(K_1\left(s,\mathsf{\Psi },\mathsf{\Phi }\right)+{(1+\beta )}^{-\tilde{\beta }}{s}^{-\tilde{\beta }}{\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\beta }{G}_1\left(y\right)dy+k\mathsf{\Psi }\right)ds,\hfill \\ & & \ge {\int }_0^{\tilde{\delta }}{G}_D\left[k\right](s,t)\left(K_1\left(s,u_0,v_0\right)+{(1+\beta )}^{-\tilde{\beta }}{s}^{-\tilde{\beta }}{\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\beta }{G}_1\left(y\right)dy+k{u}_0\right)ds\hfill \\ & & \ge u_0\left(t\right), \mathrm{for} \mathrm{all} 0<t\le \tilde{\delta }.\hfill \end{array}$$

Similarly, we can have $Y_D^{\mathsf{\Omega }}(\mathsf{\Psi },\mathsf{\Phi })\left(t\right)\ge v_0\left(t\right)$. Again, since $\mathsf{\Psi }\le {\alpha }_0\le 0$ and $\mathsf{\Phi }\le {\beta }_0\le 0$, we have

$${\mathsf{\Psi }}^2\ge {\alpha }_0^2\&{\mathsf{\Phi }}^2\ge {\beta }_0^2.$$

(61)

By using (61), we have $X_D^{\mathsf{\Omega }}(\mathsf{\Psi },\mathsf{\Phi })\left(t\right)\le {\alpha }_0\left(t\right)$ and $Y_D^{\mathsf{\Omega }}(\mathsf{\Psi },\mathsf{\Phi })\left(t\right)\le {\beta }_0\left(t\right)$. Hence, we have

$$(u_0,v_0)\left(t\right)\le T_D^{\mathsf{\Omega }}(\mathsf{\Psi },\mathsf{\Phi })\left(t\right)\le ({\alpha }_0,{\beta }_0)\left(t\right),\forall (\mathsf{\Psi },\mathsf{\Phi })\in \mathsf{\Omega }.$$

(62)

Hence, $T_D^{\mathsf{\Omega }}\left(\mathsf{\Omega }\right)\subseteq \mathsf{\Omega }$. The proof is complete. □

Again, $\mathsf{\Omega }$ is a subset of $\tilde{X}\times \tilde{X}$. So, by using Lemmas 12 and 13 and Assumption (53), we can define $T_D^{\mathsf{\Omega }}$ as

$$(\mathsf{\Psi },\mathsf{\Phi })\left(t\right)=T_D^{\mathsf{\Omega }}(\mathsf{\Psi },\mathsf{\Phi })\left(t\right)=(X_D^{\mathsf{\Omega }}(\mathsf{\Psi },\mathsf{\Phi })\left(t\right),Y_D^{\mathsf{\Omega }}(\mathsf{\Psi },\mathsf{\Phi })\left(t\right)),\forall (\mathsf{\Psi },\mathsf{\Phi })\in \mathsf{\Omega }.$$

(63)

In similar manner, we conclude that

$$\begin{array}{ccc}& & T_{N_1}^{\mathsf{\Omega }},T_{N_2}^{\mathsf{\Omega }}:\mathsf{\Omega }\to \mathsf{\Omega },\hfill \\ & & X_D^{\mathsf{\Omega }},X_{N_1}^{\mathsf{\Omega }},X_{N_2}^{\mathsf{\Omega }}:\mathsf{\Omega }\to {\mathsf{\Omega }}_1,\hfill \\ & & Y_D^{\mathsf{\Omega }},Y_{N_1}^{\mathsf{\Omega }},Y_{N_2}^{\mathsf{\Omega }}:\mathsf{\Omega }\to {\mathsf{\Omega }}_2,\hfill \end{array}$$

are also completely continuous. Now, we define two sequences $\left\{(u_n,v_n)\right\}$ and $\left\{({\alpha }_n,{\beta }_n)\right\}$ in the region $\mathsf{\Omega }$ as follows:

$$\begin{array}{ccc}\hfill & & T_D^{\mathsf{\Omega }}(u_n,v_n)\left(t\right)=(u_{n+1},v_{n+1})\left(t\right)=(X_D^{\mathsf{\Omega }}(u_n,v_n)\left(t\right),Y_D^{\mathsf{\Omega }}(u_n,v_n)\left(t\right)),\hfill \end{array}$$

(64)

$$\begin{array}{ccc}& & T_D^{\mathsf{\Omega }}({\alpha }_n,{\beta }_n)\left(t\right)=({\alpha }_{n+1},{\beta }_{n+1})\left(t\right)=(X_D^{\mathsf{\Omega }}({\alpha }_n,{\beta }_n)\left(t\right),Y_D^{\mathsf{\Omega }}({\alpha }_n,{\beta }_n)\left(t\right)),\hfill \end{array}$$

(65)

where $n=0,1,\cdots ,$$(u_0,v_0)$ and $({\alpha }_0,{\beta }_0)$ are defined by (58) and (59), respectively, i.e.,

$$\left\{\begin{array}{c}{\displaystyle u_0\le {\int }_0^{\tilde{\delta }}{G}_D\left[k\right](s,t)\left(K_1\left(s,u_0,v_0\right)+{(1+\beta )}^{-\tilde{\beta }}{s}^{-\tilde{\beta }}{\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\beta }{G}_1\left(y\right)dy+k{u}_0\right)ds,}\hfill \\ {\displaystyle v_0\le {\int }_0^{\tilde{\delta }}{G}_D\left[m\right](s,t)\left(K_2\left(s,u_0,v_0\right)+{(1+\beta )}^{-\tilde{\beta }}{s}^{-\tilde{\beta }}{\int }_0^{L\left(s\right)}{\lambda }_2{y}^{\beta }{G}_2\left(y\right)dy+m{v}_0\right)ds,}\hfill \end{array}\right.$$

(66)

and

$$\left\{\begin{array}{c}{\displaystyle {\alpha }_0\ge {\int }_0^{\tilde{\delta }}{G}_D\left[k\right](s,t)\left(K_1\left(s,{\alpha }_0,{\beta }_0\right)+{(1+\beta )}^{-\tilde{\beta }}{s}^{-\tilde{\beta }}{\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\beta }{G}_1\left(y\right)dy+k{\alpha }_0\right)ds,}\hfill \\ {\displaystyle {\beta }_0\ge {\int }_0^{\tilde{\delta }}{G}_D\left[m\right](s,t)\left(K_2\left(s,{\alpha }_0,{\beta }_0\right)+{(1+\beta )}^{-\tilde{\beta }}{s}^{-\tilde{\beta }}{\int }_0^{L\left(s\right)}{\lambda }_2{y}^{\beta }{G}_2\left(y\right)dy+m{\beta }_0\right)ds,}\hfill \end{array}\right.$$

(67)

for all ${\displaystyle t\in \left(0,\tilde{\delta }\right].}$

Theorem 1.

Assume $H_0$ holds (respectively, $H_1$ and $H_2$) and ${\lambda }_1,{\lambda }_2\in \mathbb{R}$. Let $(u_0,v_0)$ and $({\alpha }_0,{\beta }_0)\in E\times E$ be the lower and upper system of solutions of System 1 (respectively, System 2 and System 3) that satisfy the properties (53) such that $(u_0,v_0)\le ({\alpha }_0,{\beta }_0)$. Then, the sequences $\left\{(u_n,v_n)\right\}$ and $\left\{({\alpha }_n,{\beta }_n)\right\}$ defined by (64) and (65), in a given order, converge uniformly to $({\mathsf{\Psi }}_{*},{\mathsf{\Phi }}_{*})$ and $({\mathsf{\Psi }}^{*},{\mathsf{\Phi }}^{*})$, which are the solutions of (63) in Ω such that

$$({\mathsf{\Psi }}_{*},{\mathsf{\Phi }}_{*})\left(t\right)\le (\mathsf{\Psi },\mathsf{\Phi })\left(t\right)\le ({\mathsf{\Psi }}^{*},{\mathsf{\Phi }}^{*})\left(t\right),\forall t\in \left[0,\tilde{\delta }\right].$$

(68)

Consequently, $({\mathsf{\Psi }}_{*},{\mathsf{\Phi }}_{*})$, $({\mathsf{\Psi }}^{*},{\mathsf{\Phi }}^{*})\in E\times E$ are also the system of solutions of System 1 (respectively, System 2 and System 3).

Proof. 

We claim that

$$\begin{array}{c}\hfill (u_n,v_n) \mathrm{satisfy} \left(66\right),(u_{n+1},v_{n+1})\ge (u_n,v_n) \mathrm{and} ({\alpha }_0,{\beta }_0)\ge (u_{n+1},v_{n+1})\forall n\in \mathbb{N}.\end{array}$$

(69)

In view of Lemma 13, we obtain $(u_0,v_0)$ satisfy (66). From (64),

$$\begin{array}{ccc}& & (u_1,v_1)\hfill \\ & & =T_D^{\mathsf{\Omega }}(u_0,v_0)\left(t\right)=(X_D^{\mathsf{\Omega }}(u_0,v_0)\left(t\right),Y_D^{\mathsf{\Omega }}(u_0,v_0)\left(t\right))\hfill \\ & & =\left({\displaystyle {\int }_0^{\tilde{\delta }}{G}_D\left[k\right](s,t)\left(K_1\left(s,u_0,v_0\right)+{(1+\beta )}^{-\tilde{\beta }}{s}^{-\tilde{\beta }}{\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\beta }{G}_1\left(y\right)dy+k{u}_0\right)ds,}\right.\hfill \\ & & \left.{\displaystyle {\int }_0^{\tilde{\delta }}{G}_D\left[m\right](s,t)\left(K_2\left(s,u_0,v_0\right)+{(1+\beta )}^{-\tilde{\beta }}{s}^{-\tilde{\beta }}{\int }_0^{L\left(s\right)}{\lambda }_2{y}^{\beta }{G}_2\left(y\right)dy+m{v}_0\right)ds}\right).\hfill \end{array}$$

Now, in view of Account (66), we have

$$\begin{array}{ccc}& & u_0-u_1\hfill \\ & & {\displaystyle \le {\int }_0^{\tilde{\delta }}{G}_D\left[k\right](s,t)\left(K_1\left(s,u_0,v_0\right)+{(1+\beta )}^{-\tilde{\beta }}{s}^{-\tilde{\beta }}{\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\beta }{G}_1\left(y\right)dy+k{u}_0\right)ds}\hfill \\ & & {\displaystyle -{\int }_0^{\tilde{\delta }}{G}_D\left[k\right](s,t)\left(K_1\left(s,u_0,v_0\right)+{(1+\beta )}^{-\tilde{\beta }}{s}^{-\tilde{\beta }}{\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\beta }{G}_1\left(y\right)dy+k{u}_0\right)ds,}\hfill \\ & & =0.\hfill \end{array}$$

Hence, we have $u_0\le u_1$. Similarly, we can prove $v_0\le v_1$. Now,

$$\begin{array}{ccc}& & u_1-{\alpha }_0\hfill \\ & & {\displaystyle \le {\int }_0^{\tilde{\delta }}{G}_D\left[k\right](s,t)\left(K_1\left(s,u_0,v_0\right)+{(1+\beta )}^{-\tilde{\beta }}{s}^{-\tilde{\beta }}{\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\beta }{G}_1\left(y\right)dy+k{u}_0\right)ds}\hfill \\ & & {\displaystyle -{\int }_0^{\tilde{\delta }}{G}_D\left[k\right](s,t)\left(K_1\left(s,{\alpha }_0,{\beta }_0\right)+{(1+\beta )}^{-\tilde{\beta }}{s}^{-\tilde{\beta }}{\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\beta }{G}_1\left(y\right)dy+k{\alpha }_0\right)ds.}\hfill \end{array}$$

Now,

$$u_0\le {\alpha }_0 \& v_0\le {\beta }_0 \mathrm{implies} u_0^2\ge {\alpha }_0^2 \& v_0^2\ge {\beta }_0^2.$$

(70)

Since $G_D\left[k\right](s,t)$ is nonpositive, by using (70), we can conclude $u_1-{\alpha }_0\le 0.$ Similarly, we have $v_1-{\beta }_0\le 0$ and $\left(u_1,v_1\right)\left(t\right)\le \left({\alpha }_0,{\beta }_0\right)\left(t\right).$ Hence, Equation (69) is held for $n=0$. Let Equation (69) be held up to $n=\tilde{m}$. Again, from (64),

$$\begin{array}{ccc}& & (u_{\tilde{m}+1},v_{\tilde{m}+1})\hfill \\ & & =T_D^{\mathsf{\Omega }}(u_{\tilde{m}},v_{\tilde{m}})\left(t\right)=(X_D^{\mathsf{\Omega }}(u_{\tilde{m}},v_{\tilde{m}})\left(t\right),Y_D^{\mathsf{\Omega }}(u_{\tilde{m}},v_{\tilde{m}})\left(t\right))\hfill \\ & & =\left({\displaystyle {\int }_0^{\tilde{\delta }}{G}_D\left[k\right](s,t)\left(K_1\left(s,u_{\tilde{m}},v_{\tilde{m}}\right)+{(1+\beta )}^{-\tilde{\beta }}{s}^{-\tilde{\beta }}{\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\beta }{G}_1\left(y\right)dy+k{u}_{\tilde{m}}\right)ds,}\right.\hfill \\ & & \left.{\displaystyle {\int }_0^{\tilde{\delta }}{G}_D\left[m\right](s,t)\left(K_2\left(s,u_{\tilde{m}},v_{\tilde{m}}\right)+{(1+\beta )}^{-\tilde{\beta }}{s}^{-\tilde{\beta }}{\int }_0^{L\left(s\right)}{\lambda }_2{y}^{\beta }{G}_2\left(y\right)dy+m{v}_{\tilde{m}}\right)ds}\right).\hfill \end{array}$$

So, we have,

$$\begin{array}{c}\hfill u_{\tilde{m}+1}\le {\int }_0^{\tilde{\delta }}{G}_D\left[k\right](s,t)\left(K_1\left(s,u_{\tilde{m}+1},v_{\tilde{m}+1}\right)+{(1+\beta )}^{-\tilde{\beta }}{s}^{-\tilde{\beta }}{\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\beta }{G}_1\left(y\right)dy+k{u}_{\tilde{m}+1}\right)ds,\end{array}$$

and

$$\begin{array}{c}{v}_{\tilde{m}+1}\le {\int }_0^{\tilde{\delta }}{G}_D\left[m\right](s,t)\left(K_2\left(s,u_{\tilde{m}+1},v_{\tilde{m}+1}\right)+{(1+\beta )}^{-\tilde{\beta }}{s}^{-\tilde{\beta }}{\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\beta }{G}_1\left(y\right)dy+m{v}_{\tilde{m}+1}\right)ds.\hfill \end{array}$$

Now, by using (71), we have

$$\begin{array}{ccc}& & u_{\tilde{m}+1}-u_{\tilde{m}+2}\hfill \\ & & {\displaystyle \le {\int }_0^{\tilde{\delta }}{G}_D\left[k\right](s,t)\left(K_1\left(s,u_{\tilde{m}+1},v_{\tilde{m}+1}\right)+{(1+\beta )}^{-\tilde{\beta }}{s}^{-\tilde{\beta }}{\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\beta }{G}_1\left(y\right)dy+k{u}_{\tilde{m}+1}\right)ds}\hfill \\ & & {\displaystyle -{\int }_0^{\tilde{\delta }}{G}_D\left[k\right](s,t)\left(K_1\left(s,u_{\tilde{m}+1},v_{\tilde{m}+1}\right)+{(1+\beta )}^{-\tilde{\beta }}{s}^{-\tilde{\beta }}{\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\beta }{G}_1\left(y\right)dy+k{u}_{\tilde{m}+1}\right)ds,}\hfill \\ & & =0.\hfill \end{array}$$

Hence, $u_{\tilde{m}+1}$ is less than equal to $u_{\tilde{m}+2}$. Similaly, we can prove $v_{\tilde{m}+1}$ is less than equal to $v_{\tilde{m}+2}$. Now,

$$\begin{array}{ccc}& & u_{\tilde{m}+2}-{\alpha }_0\hfill \\ & & {\displaystyle \le {\int }_0^{\tilde{\delta }}{G}_D\left[k\right](s,t)\left(K_1\left(s,u_{\tilde{m}+1},v_{\tilde{m}+1}\right)+{(1+\beta )}^{-\tilde{\beta }}{s}^{-\tilde{\beta }}{\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\beta }{G}_1\left(y\right)dy+k{u}_{\tilde{m}+1}\right)ds}\hfill \\ & & {\displaystyle -{\int }_0^{\tilde{\delta }}{G}_D\left[k\right](s,t)\left(K_1\left(s,{\alpha }_0,{\beta }_0\right)+{(1+\beta )}^{-\tilde{\beta }}{s}^{-\tilde{\beta }}{\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\beta }{G}_1\left(y\right)dy+k{\alpha }_0\right)ds.}\hfill \end{array}$$

Now,

$$\begin{array}{c}\hfill u_{\tilde{m}+1}\le {\alpha }_0\&v_{\tilde{m}+1}\le {\beta }_0\mathrm{implies}{u}_{\tilde{m}+1}^2\ge {\alpha }_0^2\&v_{\tilde{m}+1}^2\ge {\beta }_0^2.\end{array}$$

(71)

Since $G_D\left[k\right](s,t)$ is nonpositive, by using (71), we can conclude $u_{\tilde{m}+2}-{\alpha }_0\le 0.$ Similarly, $v_{\tilde{m}+2}-{\beta }_0\le 0$. Thus, $\left(u_{\tilde{m}+2},v_{\tilde{m}+2}\right)\left(t\right)\le \left({\alpha }_0,{\beta }_0\right)\left(t\right).$ Hence, Equation (69) is true for $n=\tilde{m}+1$. Finally, we conclude Equation (69) holds for all $n\in \mathbb{N}$. By a similar analysis, we claim that

$$\begin{array}{c}\hfill ({\alpha }_n,{\beta }_n) \mathrm{satisfy} \left(67\right),({\alpha }_{n+1},{\beta }_{n+1})\le ({\alpha }_n,{\beta }_n) \mathrm{and} ({\alpha }_{n+1},{\beta }_{n+1})\ge (u_0,v_0)\forall n\in \mathbb{N}.\end{array}$$

(72)

To show that

$$(u_n,v_n)\le ({\alpha }_n,{\beta }_n),\forall n\in \mathbb{N}.$$

(73)

We assume Equation (73) is true for $n=\tilde{m}$. Now,

$$\begin{array}{ccc}& & u_{\tilde{m}+1}-{\alpha }_{\tilde{m}+1}\hfill \\ & & {\displaystyle \le {\int }_0^{\tilde{\delta }}{G}_D\left[k\right](s,t)\left(K_1\left(s,u_{\tilde{m}},v_{\tilde{m}}\right)+{(1+\beta )}^{-\tilde{\beta }}{s}^{-\tilde{\beta }}{\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\beta }{G}_1\left(y\right)dy+k{u}_{\tilde{m}}\right)ds}\hfill \\ & & {\displaystyle -{\int }_0^{\tilde{\delta }}{G}_D\left[k\right](s,t)\left(K_1\left(s,{\alpha }_m,{\beta }_m\right)+{(1+\beta )}^{-\tilde{\beta }}{s}^{-\tilde{\beta }}{\int }_0^{L\left(s\right)}{\lambda }_1{y}^{\beta }{G}_1\left(y\right)dy+k{\alpha }_{\tilde{m}}\right)ds.}\hfill \end{array}$$

Now,

$$u_{\tilde{m}}\le {\alpha }_{\tilde{m}} \& v_{\tilde{m}}\le {\beta }_{\tilde{m}} \mathrm{implies} u_{\tilde{m}}^2\ge {\alpha }_{\tilde{m}}^2 \& v_{\tilde{m}}^2\ge {\beta }_{\tilde{m}}^2.$$

(74)

So, $u_{m+1}\le {\alpha }_{m+1}$ and $v_{m+1}\le {\beta }_{m+1}$ implies that $(u_{m+1},v_{m+1})\le ({\alpha }_{m+1},{\beta }_{m+1})$. Hence,

$$({\alpha }_n,{\beta }_n)\ge (u_n,v_n),\forall n\in \mathbb{N}.$$

Thus, we obtain the following inequalities

$$(u_0,v_0)\le \cdots \le (u_n,v_n)\le \cdots \le ({\alpha }_n,{\beta }_n)\le \cdots \le ({\alpha }_0,{\beta }_0).$$

(75)

Therefore, the sequences $\left\{(u_n,v_n)\right\}$ and $\left\{({\alpha }_n,{\beta }_n)\right\}$ are monotone and bounded uniformly for all $t\in \left[0,\tilde{\delta }\right]$. So, the sequences $\left\{(u_n,v_n)\right\}$ and $\left\{({\alpha }_n,{\beta }_n)\right\}$ converge uniformly to $\left\{({\mathsf{\Psi }}_{*},{\mathsf{\Phi }}_{*})\right\}$ and $\left\{({\mathsf{\Psi }}^{*},{\mathsf{\Phi }}^{*})\right\}$ in $\mathsf{\Omega }$, i.e.,

$$\underset{n\to \infty }{\lim }(u_n,v_n)=({\mathsf{\Psi }}_{*},{\mathsf{\Phi }}_{*}) \& \underset{n\to \infty }{\lim }({\alpha }_n,{\beta }_n)=({\mathsf{\Psi }}^{*},{\mathsf{\Phi }}^{*}).$$

(76)

Hence, from (64) and (65), we obtain

$$\left\{\begin{array}{c}{\displaystyle ({\mathsf{\Psi }}_{*},{\mathsf{\Phi }}_{*})\left(t\right)=T_D^{\mathsf{\Omega }}({\mathsf{\Psi }}_{*},{\mathsf{\Phi }}_{*})\left(t\right)=(X_D^{\mathsf{\Omega }}({\mathsf{\Psi }}_{*},{\mathsf{\Phi }}_{*})\left(t\right),Y_D^{\mathsf{\Omega }}({\mathsf{\Psi }}_{*},{\mathsf{\Phi }}_{*}))\left(t\right),\forall ({\mathsf{\Psi }}_{*},{\mathsf{\Phi }}_{*})\in \mathsf{\Omega },}\hfill \\ {\displaystyle ({\mathsf{\Psi }}^{*},{\mathsf{\Phi }}^{*})\left(t\right)=T_D^{\mathsf{\Omega }}({\mathsf{\Psi }}^{*},{\mathsf{\Phi }}^{*})\left(t\right)=(X_D^{\mathsf{\Omega }}({\mathsf{\Psi }}^{*},{\mathsf{\Phi }}^{*})\left(t\right),Y_D^{\mathsf{\Omega }}({\mathsf{\Psi }}^{*},{\mathsf{\Phi }}^{*})\left(t\right)),\forall ({\mathsf{\Psi }}^{*},{\mathsf{\Phi }}^{*})\in \mathsf{\Omega }.}\hfill \end{array}\right.$$

(77)

Since $\mathsf{\Omega }$ is a subset of $E\times E$, so from Lemma 9, we show that $({\mathsf{\Psi }}_{*},{\mathsf{\Phi }}_{*})$, $({\mathsf{\Psi }}^{*},{\mathsf{\Phi }}^{*})\in E\times E$ are the solutions of System 1. This completes the proof. □

4. Bounds of ${\mathit{\lambda }}_{\mathbf{1}} \& {\mathit{\lambda }}_{\mathbf{2}}$

In this section, we derive the bounds of ${\lambda }_1$ and ${\lambda }_2$. From (17), we have

$$\left\{\begin{array}{c}{\left(t{\mathsf{\Psi }}^{\prime }-\mathsf{\Psi }\right)}^{\prime }=t{K}_1\left(t,\mathsf{\Psi },\mathsf{\Phi }\right)+{(1+\beta )}^{-\tilde{\beta }}{t}^{\tilde{\gamma }}{\int }_0^{L\left(t\right)}{\lambda }_1{s}^{\beta }{G}_1\left(s\right)ds,\hfill \\ \\ {\left(t{\mathsf{\Phi }}^{\prime }-\mathsf{\Phi }\right)}^{\prime }=t{K}_2\left(t,\mathsf{\Psi },\mathsf{\Phi }\right)+{(1+\beta )}^{-\tilde{\beta }}{t}^{\tilde{\gamma }}{\int }_0^{L\left(t\right)}{\lambda }_2{s}^{\beta }{G}_2\left(s\right)ds,\hfill \end{array}\right.$$

(78)

for ${\displaystyle t\in \left(0,\tilde{\delta }\right].}$ By using ${\displaystyle z\left(t\right)=-\frac{\mathsf{\Phi }\left(t\right)}{t}}$ and ${\displaystyle w\left(t\right)=-\frac{\mathsf{\Psi }\left(t\right)}{t}}$, we have

$$z\left(t\right)\ge 0 \& w\left(t\right)\ge 0,\forall t\in \left(0,\tilde{\delta }\right].$$

(79)

Integrating (78) from 0 to t, we have

$$\begin{array}{ccc}\hfill & & -t^2{w}^{\prime }={\int }_0^t{s}^3{K}_1\left(s,w,z\right)ds+{\int }_0^t{(1+\beta )}^{-\tilde{\beta }}{y}^{\tilde{\gamma }}{\int }_0^{L\left(y\right)}{\lambda }_1{s}^{\beta }{G}_1\left(s\right)dsdy,\hfill \\ & & \hfill \end{array}$$

(80)

$$\begin{array}{ccc}& & -t^2{z}^{\prime }={\int }_0^t{s}^3{K}_2\left(s,w,z\right)ds+{\int }_0^t{(1+\beta )}^{-\tilde{\beta }}{y}^{\tilde{\gamma }}{\int }_0^{L\left(y\right)}{\lambda }_2{s}^{\beta }{G}_2\left(s\right)dsdy.\hfill \end{array}$$

(81)

So, from (18), we deduce

$$w\left(\tilde{\delta }\right)=0 \& z\left(\tilde{\delta }\right)=0,$$

(82)

from (19), we derive

$$w\left(\tilde{\delta }\right)=-\frac{1}{2}{w}^{\prime }\left(\tilde{\delta }\right) \& z\left(\tilde{\delta }\right)=-\frac{1}{2}{z}^{\prime }\left(\tilde{\delta }\right),$$

(83)

and from (20), we obtain

$$w\left(\tilde{\delta }\right)=-w^{\prime }\left(\tilde{\delta }\right) \& z\left(\tilde{\delta }\right)=-z^{\prime }\left(\tilde{\delta }\right).$$

(84)

Lemma 14.

Assume there exists a solution $\left(u,v\right)$ of (17) satisfying ${\displaystyle \underset{t\to 0^{+}}{\lim }\frac{u\left(t\right)}{t^{\tilde{\delta }}}=0}$, ${\displaystyle \underset{t\to 0^{+}}{\lim }\frac{v\left(t\right)}{t^{\tilde{\delta }}}=0}$, $v\left(t\right)\le 0$ and $u\left(t\right)\le 0$. Then, ${\lambda }_1\ge 0$ and ${\lambda }_2\ge 0$ satisfies the inequality

$${\lambda }_1{M}_1+{\lambda }_2{M}_2\le \frac{2{(1+\beta )}^4(3+\beta ){\pi }^2}{p(K_{11}+K_{21})}.$$

(85)

Proof. 

From Equations (80) and (81), we have

$$\begin{array}{ccc}\hfill w^{\prime }+z^{\prime }& & =-\frac{1}{t^2}{\int }_0^t\frac{p\left(\left(K_{11}+K_{21}\right)w^2+\left(K_{12}+K_{22}\right)z^2\right)s}{2{(1+\beta )}^2}ds\hfill \\ & & -\frac{1}{t^2}{\int }_0^t{(1+\beta )}^{-\tilde{\beta }}{y}^{\tilde{\gamma }}{\int }_0^{L\left(y\right)}\left({\lambda }_1{G}_1+{\lambda }_2{G}_2\right)s^{\beta }dsdy.\hfill \end{array}$$

(86)

Now, on account of Equation (79), $G_1\left(y\right)\ge M_1$ and $G_2\left(y\right)\ge M_2$, from Equation (86), we have

$$\begin{array}{ccc}\hfill (w^{\prime }+z^{\prime })& & \le -\frac{p\left(\left(K_{11}+K_{21}\right)w^2+\left(K_{12}+K_{22}\right)z^2\right)}{4{(1+\beta )}^2}\hfill \end{array}$$

(87)

$$\begin{array}{ccc}& & -\frac{1}{t^2}{\int }_0^t{(1+\beta )}^{-\tilde{\beta }}{s}^{\tilde{\gamma }}{\int }_0^{L\left(s\right)}\left({\lambda }_1{M}_1+{\lambda }_2{M}_2\right)y^{\beta }dyds,\hfill \end{array}$$

(88)

$$\begin{array}{ccc}& & =-\frac{p\left(\left(K_{11}+K_{21}\right)w^2+\left(K_{12}+K_{22}\right)z^2\right)}{4{(1+\beta )}^2}-\frac{\left({\lambda }_1{M}_1+{\lambda }_2{M}_2\right){(1+\beta )}^{\tilde{\gamma }}{t}^{\tilde{\gamma }}}{(3+\beta )}.\hfill \end{array}$$

(89)

Therefore,

$$\begin{array}{ccc}\hfill w^{\prime }+z^{\prime }& & \le -\frac{p\left(\left(K_{11}+K_{21}\right)w^2+\left(K_{12}+K_{22}\right)z^2\right)}{4{(1+\beta )}^2}-\frac{\left({\lambda }_1{M}_1+{\lambda }_2{M}_2\right)}{(3+\beta )},\hfill \\ & & \le -\frac{p\left(K_{11}+K_{21}\right){(w+z)}^2}{8{(1+\beta )}^2}-\frac{\left({\lambda }_1{M}_1+{\lambda }_2{M}_2\right)}{(3+\beta )},\forall t\in \left(0,\tilde{\delta }\right].\hfill \end{array}$$

(90)

Now, we can have

$$-\frac{w^{\prime }+z^{\prime }}{{(w+z)}^2+\frac{8{(1+\beta )}^2\left({\lambda }_1{M}_1+{\lambda }_2{M}_2\right)}{(K_{11}+K_{21})(3+\beta )p}}\ge \frac{p(K_{11}+K_{21})}{8{(1+\beta )}^2},\forall t\in \left(0,\tilde{\delta }\right].$$

(91)

From Inequality (91), we obtain

$$-{\int }_t^{\tilde{\delta }}\frac{w^{\prime }+z^{\prime }}{{(w+z)}^2+\frac{8{(1+\beta )}^2\left({\lambda }_1{M}_1+{\lambda }_2{M}_2\right)}{(K_{11}+K_{21})(3+\beta )p}}ds\ge \frac{p(K_{11}+K_{21})}{8{(1+\beta )}^2}\left(\tilde{\delta }-t\right)$$

(92)

for all ${\displaystyle t\in \left(0,\tilde{\delta }\right].}$ Equivalently, we have

$${\int }_{w\left(\tilde{\delta }\right)+z\left(\tilde{\delta }\right)}^{w\left(t\right)+z\left(t\right)}\frac{dx}{x^2+\frac{8{(1+\beta )}^2\left({\lambda }_1{M}_1+{\lambda }_2{M}_2\right)}{(K_{11}+K_{21})(3+\beta )p}}\ge \frac{p(K_{11}+K_{21})}{8{(1+\beta )}^2}\left(\tilde{\delta }-t\right)$$

(93)

for all ${\displaystyle t\in \left(0,\tilde{\delta }\right].}$ Thus

$$\begin{array}{ccc}\hfill {\int }_0^{\infty }\frac{dx}{x^2+\frac{8{(1+\beta )}^2\left({\lambda }_1{M}_1+{\lambda }_2{M}_2\right)}{(K_{11}+K_{21})(3+\beta )p}}& & \ge \underset{t\to 0^{+}}{\lim }{\int }_{v\left(\tilde{\delta }\right)}^{v\left(t\right)}\frac{dx}{x^2+\frac{8{(1+\beta )}^2\left({\lambda }_1{M}_1+{\lambda }_2{M}_2\right)}{(K_{11}+K_{21})(3+\beta )p}}\ge \frac{p(K_{11}+K_{21})}{8{(1+\beta )}^3}.\hfill \end{array}$$

(94)

Hence, from (94), we have

$${\lambda }_1{M}_1+{\lambda }_2{M}_2\le \frac{2{(1+\beta )}^4(3+\beta ){\pi }^2}{p(K_{11}+K_{21})}.$$

(95)

Hence, we obtain the results. □

Lemma 15.

If BVPs (18)–(20) are solvable for some pair $({\lambda }_1^c,{\lambda }_2^c)\ge (0,0)$, where the constants ${\lambda }_2^c$ and ${\lambda }_1^c$ are the critical values of the parameters ${\lambda }_2$ and ${\lambda }_1$, in a given order, then the solution of these BVPs also exists for every order pair $({\lambda }_1,{\lambda }_2)$ such that $(0,0)\le ({\lambda }_1,{\lambda }_2)\le ({\lambda }_1^c,{\lambda }_2^c)$.

Proof. 

It follows from Lemma 4.0.2 in [11]. □

We now upgrade bounds of the parameters ${\lambda }_1$ and ${\lambda }_2$ corresponding to the problems (18)–(20).

Lemma 16.

Assume there exists $(w,z)\in E\times E$ satisfying (80) and (81) such that $w\left(t\right)\ge 0$, $z\left(t\right)\ge 0$, $w\left(\tilde{\delta }\right)=0$ and $z\left(\tilde{\delta }\right)=0$. Then

$${\lambda }_1{M}_1+{\lambda }_2{M}_2\le \frac{12{(1+\beta )}^4(\beta +3)}{p(K_{11}+K_{21})}$$

(96)

and

$$w\left(t\right)+z\left(t\right)\ge c\left(\tilde{\delta }-t\right),0<t\le \tilde{\delta }.$$

(97)

Proof. 

We can easily see that

$$w^{\prime }\left(t\right)\le 0\&z^{\prime }\left(t\right)\le 0,0<t\le \tilde{\delta }.$$

(98)

Now,

$$\begin{array}{ccc}\hfill w^{\prime }+z^{\prime }& & \le -\frac{1}{t^2}{\int }_0^t{(1+\beta )}^{-\tilde{\beta }}{s}^{\tilde{\gamma }}{\int }_0^{L\left(s\right)}\left({\lambda }_1{M}_1+{\lambda }_2{M}_2\right)y^{\beta }dyds,\hfill \end{array}$$

(99)

$$\begin{array}{ccc}& & \le -\frac{({\lambda }_1{M}_1+{\lambda }_2{M}_2)}{(\beta +3)},\mathrm{since} t\le \tilde{\delta }.\hfill \end{array}$$

(100)

So, we have

$$w^{\prime }+z^{\prime }\le -\frac{{\lambda }_1{M}_1+{\lambda }_2{M}_2}{(\beta +3)},\forall t\in \left(0,\tilde{\delta }\right].$$

(101)

Integrate (101) from t to $\tilde{\delta }$, we have

$$w\left(\tilde{\delta }\right)+z\left(\tilde{\delta }\right)-w\left(t\right)-z\left(t\right)\le -{\int }_t^{\tilde{\delta }}\frac{({\lambda }_1{M}_1+{\lambda }_2{M}_2)}{(\beta +3)}dt.$$

(102)

Therefore, we have

$$w\left(t\right)+z\left(t\right)\ge \frac{({\lambda }_1{M}_1+{\lambda }_2{M}_2)}{(\beta +3)}\left(\tilde{\delta }-t\right),\forall t\in \left(0,\tilde{\delta }\right].$$

(103)

Let

$$c_1=\frac{({\lambda }_1{M}_1+{\lambda }_2{M}_2)}{(\beta +3)} \mathrm{and} w\left(t\right)+z\left(t\right)\ge c_n\left(\tilde{\delta }-t\right),\forall t\in \left(0,\tilde{\delta }\right].$$

(104)

So, by using (86) and (104), we have

$$\begin{array}{ccc}& & w^{\prime }+z^{\prime }\hfill \\ & & \le -\frac{p(K_{11}+K_{21})}{4{(1+\beta )}^2{t}^2}{\int }_0^t{c}_n^2{\left(\tilde{\delta }-s\right)}^{2}sds-\frac{\left({\lambda }_1{M}_1+{\lambda }_2{M}_2\right)}{(\beta +3)},\hfill \\ & & =-\frac{p{c}_n^2(K_{11}+K_{21})}{16{(1+\beta )}^2}\left({\left(t-\frac{4}{3(1+\beta )}\right)}^2+\frac{2}{9{(1+\beta )}^2}\right)-\frac{\left({\lambda }_1{M}_1+{\lambda }_2{M}_2\right)}{(\beta +3)},\hfill \\ \hfill & & \le -\frac{p{c}_n^2(K_{11}+K_{21})}{48{(1+\beta )}^4}-\frac{\left({\lambda }_1{M}_1+{\lambda }_2{M}_2\right)}{(\beta +3)},\hfill \end{array}$$

(105)

since ${\displaystyle -{\left(t-\frac{4}{3(1+\beta )}\right)}^2\le -\frac{1}{9{(1+\beta )}^2}}$ for all $0<t\le \tilde{\delta }$. Thus

$$w\left(t\right)+z\left(t\right)\ge c_{n+1}\left(\tilde{\delta }-t\right),\forall t\in \left(0,\tilde{\delta }\right],$$

(106)

where

$$c_{n+1}=\frac{p{c}_n^2(K_{11}+K_{21})}{48{(1+\beta )}^4}+\frac{\left({\lambda }_1{M}_1+{\lambda }_2{M}_2\right)}{(\beta +3)}.$$

(107)

We claim that the sequence $\left\{c_n\right\}$ is convergent. Now, from (107), we obtain

$$c_2=\frac{p{c}_1^2(K_{11}+K_{21})}{48{(1+\beta )}^4}+\frac{\left({\lambda }_1{M}_1+{\lambda }_2{M}_2\right)}{(\beta +3)}\ge c_1.$$

(108)

We assume there exists $n\in \mathbb{N}$ such that $c_n\ge c_{n-1}\ge 0$. From (107), we have

$$\begin{array}{ccc}\hfill c_{n+1}& & =\frac{p{c}_n^2(K_{11}+K_{21})}{48{(1+\beta )}^4}+\frac{\left({\lambda }_1{M}_1+{\lambda }_2{M}_2\right)}{(\beta +3)}\hfill \end{array}$$

(109)

$$\begin{array}{ccc}& & \ge \frac{p{c}_{n-1}^2(K_{11}+K_{21})}{48{(1+\beta )}^4}+\frac{\left({\lambda }_1{M}_1+{\lambda }_2{M}_2\right)}{(\beta +3)}=c_n.\hfill \end{array}$$

(110)

Therefore, by using (104), we can have the sequence $\left\{c_n\right\}$ converge to c. Thus, from (106), we have (97) such that

$$c=\frac{p{c}^2(K_{11}+K_{21})}{48{(1+\beta )}^4}+\frac{\left({\lambda }_1{M}_1+{\lambda }_2{M}_2\right)}{(\beta +3)}.$$

(111)

Hence, we obtain Result (96). □

Lemma 17.

Assume there exists $(w,z)\in E\times E$ satisfying (79), (80), and (84). Then

$${\lambda }_1{M}_1+{\lambda }_2{M}_2\le \frac{12{(1+\beta )}^4(\beta +3)}{p(K_{11}+K_{21})(6{\beta }^2+16\beta +11)},$$

(112)

and

$$w\left(t\right)+z\left(t\right)\ge c\left((\beta +2)\tilde{\delta }-t\right),0<t\le \tilde{\delta }.$$

(113)

Proof. 

From (84) and (102), we have

$$-w^{\prime }\left(\tilde{\delta }\right)-z^{\prime }\left(\tilde{\delta }\right)-w\left(t\right)-z\left(t\right)\le -{\int }_t^{\tilde{\delta }}\frac{{\lambda }_1{M}_1+{\lambda }_2{M}_2}{(\beta +3)}dt$$

(114)

for all ${\displaystyle t\in \left(0,\tilde{\delta }\right].}$ So, we have

$$-w\left(t\right)-z\left(t\right)\le -\frac{{\lambda }_1{M}_1+{\lambda }_2{M}_2}{(\beta +3)}\left(\tilde{\delta }-t\right)+w^{\prime }\left(\tilde{\delta }\right)+z^{\prime }\left(\tilde{\delta }\right)$$

(115)

for all ${\displaystyle 0<t\le \tilde{\delta }.}$ By using (101), from (115), we obtain

$$c_1\left((\beta +2)\tilde{\delta }-t\right)\le w\left(t\right)+z\left(t\right), \mathrm{for} \mathrm{all} 0<t\le \tilde{\delta } \mathrm{where} c_1=\frac{{\lambda }_1{M}_1+{\lambda }_2{M}_2}{(\beta +3)}.$$

(116)

Let

$$c_n\left((\beta +2)\tilde{\delta }-t\right)\le w\left(t\right)+z\left(t\right), \mathrm{for} \mathrm{all} 0<t\le \tilde{\delta }.$$

(117)

From (80) and (117), we have

$$\begin{array}{ccc}\hfill & & w^{\prime }+z^{\prime }\hfill \end{array}$$

(118)

$$\begin{array}{ccc}& & \le -\frac{p(K_{11}+K_{21})}{4{(1+\beta )}^2{t}^2}{\int }_0^t{c}_n^2{\left((\beta +2)\tilde{\delta }-s\right)}^{2}sds-\frac{{\lambda }_1{M}_1+{\lambda }_2{M}_2}{(\beta +3)},\hfill \end{array}$$

(119)

$$\begin{array}{ccc}& & =-\frac{p{c}_n^2(K_{11}+K_{21})}{16{(1+\beta )}^2}\left({\left(t-\frac{4(\beta +2)}{3(\beta +1)}\right)}^2+\frac{2{(\beta +2)}^2}{9{(1+\beta )}^2}\right)-\frac{{\lambda }_1{M}_1+{\lambda }_2{M}_2}{(\beta +3)}.\hfill \end{array}$$

(120)

Since ${\displaystyle \left(t-\frac{4(\beta +2)}{3(\beta +1)}\right)\le \frac{-4\beta -5}{3(\beta +1)}}$, we have

$$\begin{array}{c}\hfill w^{\prime }+z^{\prime }\le -\frac{p(6{\beta }^2+16\beta +11)c_n^2(K_{11}+K_{21})}{48{(1+\beta )}^4}-\frac{{\lambda }_1{M}_1+{\lambda }_2{M}_2}{(\beta +3)}.\end{array}$$

(121)

So, we have

$$w^{\prime }+z^{\prime }\le -c_{n+1},\forall t\in \left(0,\tilde{\delta }\right],$$

(122)

where $\left\{c_n\right\}$ is given by

$$c_{n+1}=\frac{p(6{\beta }^2+16\beta +11)c_n^2(K_{11}+K_{21})}{48{(1+\beta )}^4}+\frac{{\lambda }_1{M}_1+{\lambda }_2{M}_2}{(\beta +3)}.$$

(123)

After simplifying (84) and (122),

$$w\left(t\right)+z\left(t\right)\ge c_{n+1}\left((\beta +2)\tilde{\delta }-t\right), \mathrm{for} \mathrm{all} 0<t\le \tilde{\delta },$$

(124)

Similarly, we obtain the sequence $\left\{c_n\right\}$ converge to c in $\mathbb{R}$. Hence, from (123), we have

$$c=\frac{p(6{\beta }^2+16\beta +11)c^2(K_{11}+K_{21})}{48{(1+\beta )}^4}+\frac{{\lambda }_1{M}_1+{\lambda }_2{M}_2}{(\beta +3)}.$$

(125)

Finally, from (125), we conclude

$${\lambda }_1{M}_1+{\lambda }_2{M}_2\le \frac{12{(1+\beta )}^4(\beta +3)}{p(6{\beta }^2+16\beta +11)(K_{11}+K_{21})}.$$

(126)

Hence, the proof is complete. □

Lemma 18.

Assume there exists $(w,z)\in E\times E$ satisfying (79), (80), and (83). Then,

$${\lambda }_1{M}_1+{\lambda }_2{M}_2\le \frac{24{(1+\beta )}^4(\beta +3)}{p(3{\beta }^2+10\beta +9)(K_{11}+K_{21})}$$

(127)

and

$$w\left(t\right)+z\left(t\right)\ge c\left(\frac{(\beta +3)\tilde{\delta }}{2}-t\right), for all 0<t\le \tilde{\delta }.$$

(128)

Proof. 

From (83) and (102), we obtain

$$-\frac{1}{2}{w}^{\prime }\left(\tilde{\delta }\right)-\frac{1}{2}{z}^{\prime }\left(\tilde{\delta }\right)-w\left(t\right)-z\left(t\right)\le -{\int }_t^{\tilde{\delta }}\frac{{\lambda }_1{M}_1+{\lambda }_2{M}_2}{(\beta +3)}ds$$

(129)

for all ${\displaystyle t\in \left(0,\tilde{\delta }\right].}$ After calculating as in Lemma 17, we have

$$w\left(t\right)+z\left(t\right)\ge c_1\left(\frac{(\beta +3)\tilde{\delta }}{2}-t\right), \mathrm{for} \mathrm{all} 0<t\le \tilde{\delta } \mathrm{subject} \mathrm{to} c_1=\frac{{\lambda }_1{M}_1+{\lambda }_2{M}_2}{(\beta +3)}.$$

(130)

Again, we assume

$$w\left(t\right)+z\left(t\right)\ge c_n\left(\frac{(\beta +3)\tilde{\delta }}{2}-t\right), \mathrm{for} \mathrm{all} 0<t\le \tilde{\delta }.$$

(131)

So, from (80) and (131), we have

$$\begin{array}{ccc}\hfill & & w^{\prime }\left(t\right)+z^{\prime }\left(t\right)\hfill \end{array}$$

(132)

$$\begin{array}{ccc}& & \le -\frac{p(K_{11}+K_{21})}{4{(1+\beta )}^2{t}^2}{\int }_0^t{c}_n^2{\left(\frac{(\beta +3)\tilde{\delta }}{2}-s\right)}^{2}sds-\frac{{\lambda }_1{M}_1+{\lambda }_2{M}_2}{(\beta +3)},\hfill \end{array}$$

(133)

$$\begin{array}{ccc}& & =-\frac{p{c}_n^2(K_{11}+K_{21})}{16{(1+\beta )}^2}\left({\left(t-\frac{2(\beta +3)}{3(\beta +1)}\right)}^2+\frac{{(\beta +3)}^2}{18{(\beta +1)}^2}\right)-\frac{{\lambda }_1{M}_1+{\lambda }_2{M}_2}{(\beta +3)}.\hfill \end{array}$$

(134)

Since $\left(t-\frac{2(\beta +3)}{3(\beta +1)}\right)\le \frac{-2\beta -3}{3(\beta +1)}$, we obtain

$$\begin{array}{c}\hfill w^{\prime }+z^{\prime }\le -\frac{p(3{\beta }^2+10\beta +9)c_n^2(K_{11}+K_{21})}{96{(1+\beta )}^4}-\frac{{\lambda }_1{M}_1+{\lambda }_2{M}_2}{(\beta +3)}.\end{array}$$

(135)

So, we have

$$w^{\prime }+z^{\prime }\le -c_{n+1},\forall t\in \left(0,\tilde{\delta }\right],$$

(136)

where

$$c_{n+1}=\frac{p(3{\beta }^2+10\beta +9)c_n^2(K_{11}+K_{21})}{96{(1+\beta )}^4}+\frac{{\lambda }_1{M}_1+{\lambda }_2{M}_2}{(\beta +3)}.$$

(137)

Therefore, from (83) and (136),

$$w\left(t\right)+z\left(t\right)\ge c_{n+1}\left(\frac{(\beta +3)\tilde{\delta }}{2}-t\right), \mathrm{for} \mathrm{all} 0<t\le \tilde{\delta }.$$

(138)

Thus, $\left\{c_n\right\}$ converges to c. Hence, from (137), we obtain

$$c=\frac{p(3{\beta }^2+10\beta +9)c^2(K_{11}+K_{21})}{96{(1+\beta )}^4}+\frac{{\lambda }_1{M}_1+{\lambda }_2{M}_2}{(\beta +3)}.$$

(139)

Finally, from Equation (139), we have

$${\lambda }_1{M}_1+{\lambda }_2{M}_2\le \frac{24{(1+\beta )}^4(\beta +3)}{p(3{\beta }^2+10\beta +9)(K_{11}+K_{21})}.$$

(140)

This completes the proof. □

Lemma 19.

Assume ${\lambda }_1$ and ${\lambda }_2$ satisfy conditions

$$0\le {\lambda }_1\le \frac{C{2}^{\tilde{\delta }}(2+\beta )}{M_1^{*}{(1+\beta )}^{\tilde{\delta }}},0\le {\lambda }_2\le \frac{C{2}^{\tilde{\delta }}(2+\beta )}{M_2^{*}{(1+\beta )}^{\tilde{\delta }}}$$

(141)

$$and0\le C\le \frac{8(2+\beta ){(1+\beta )}^2}{p{2}^{\tilde{\delta }}{(1+\beta )}^{\beta \tilde{\delta }}}min\left\{{\displaystyle \frac{1}{K_{11}+K_{12}}},{\displaystyle \frac{1}{K_{21}+K_{22}}}\right\}.$$

(142)

Then, there is at least one function $\left(\mathsf{\Psi },\mathsf{\Phi }\right)\in E\times E$ satisfying (36), (53), $\mathsf{\Psi }\left(\tilde{\delta }\right)=0$, $\mathsf{\Phi }\left(\tilde{\delta }\right)=0$, $\mathsf{\Psi }\left(t\right)\le 0$, and $\mathsf{\Phi }\left(t\right)\le 0$.

Proof. 

Set

$$\mathsf{\Psi }\left(t\right)=-Ct\left(A-{\left(2t\right)}^{\tilde{\delta }}\right),\mathsf{\Phi }\left(t\right)=-Ct\left(B-{\left(2t\right)}^{\tilde{\delta }}\right),\mathrm{and}C\ge 0$$

(143)

for all ${\displaystyle t\in \left(0,\tilde{\delta }\right].}$ Clearly, $\mathsf{\Psi }\left(t\right)$ and $\mathsf{\Phi }\left(t\right)$ satisfies (53). Now, $\mathsf{\Psi }\left(\tilde{\delta }\right)=0$ and $\mathsf{\Phi }\left(\tilde{\delta }\right)=0$ implies ${\displaystyle A=2^{\tilde{\delta }}{\left(\tilde{\delta }\right)}^{\tilde{\delta }}}$ and ${\displaystyle B=2^{\tilde{\delta }}{\left(\tilde{\delta }\right)}^{\tilde{\delta }}}$. Therefore, $\mathsf{\Psi }\left(t\right)\le 0$ and $\mathsf{\Phi }\left(t\right)\le 0$ are also fulfilled. Now, we have

$$\begin{array}{ccc}& & {\mathsf{\Psi }}^{″}-K_1\left(t,\mathsf{\Psi },\mathsf{\Phi }\right)-{(1+\beta )}^{-\tilde{\beta }}{t}^{-\tilde{\beta }}{\int }_0^{L\left(t\right)}{\lambda }_1{s}^{\beta }{G}_1\left(s\right)ds,\hfill \\ & & \ge \frac{2^{\tilde{\delta }}C{t}^{-\beta \tilde{\delta }}(2+\beta )}{{(1+\beta )}^2}-\frac{p(K_{11}+K_{12})C^2{\left(2^{\tilde{\delta }}{\left(\tilde{\delta }\right)}^{\tilde{\delta }}-{\left(2t\right)}^{\tilde{\delta }}\right)}^2}{2{(1+\beta )}^2}-\lambda M_1^{*}{(1+\beta )}^{-\tilde{\beta }}{t}^{\tilde{\gamma }}\hfill \\ & & \hfill \\ & & \ge \frac{C{t}^{-\beta \tilde{\delta }}{2}^{\tilde{\delta }}(2+\beta )}{{(1+\beta )}^{(1+2\beta )\tilde{\delta }}}\left(\frac{1}{{(1+\beta )}^{\tilde{\delta }}}-\frac{\lambda M_1^{*}{(1+\beta )}^{\tilde{\delta }}{t}^{\tilde{\delta }}}{C{2}^{\tilde{\delta }}(2+\beta )}\right)-\frac{p(K_{11}+K_{12})C^2{2}^{2\tilde{\delta }}}{2{(1+\beta )}^2}{\left({\left(\tilde{\delta }\right)}^{\tilde{\delta }}-{\left(t\right)}^{\tilde{\delta }}\right)}^2\hfill \\ & & \ge \frac{C{t}^{-\beta \tilde{\delta }}{2}^{\tilde{\delta }}(2+\beta )}{{(1+\beta )}^{(1+2\beta )\tilde{\delta }}}\left(\frac{1}{{(1+\beta )}^{\tilde{\delta }}}-t^{\tilde{\delta }}\right)-\frac{p(K_{11}+K_{12})C^2{2}^{2\tilde{\delta }}}{2{(1+\beta )}^2}{\left({\left(\tilde{\delta }\right)}^{\tilde{\delta }}-{\left(t\right)}^{\tilde{\delta }}\right)}^2,\hfill \\ & & \left(\mathrm{Since} \lambda \le \frac{C{2}^{\tilde{\delta }}(2+\beta )}{M_1^{*}{(1+\beta )}^{\tilde{\delta }}}\right)\hfill \\ & & =A_1\left(\frac{2(2+\beta ){(1+\beta )}^2}{(K_{11}+K_{12})PC{2}^{\tilde{\delta }}{(1+\beta )}^{(2+\beta )\tilde{\delta }}}-\frac{1}{{(1+\beta )}^{\tilde{\delta }}}{t}^{\tilde{\delta }}+{\left(t^{\tilde{\delta }}\right)}^2\right),\hfill \end{array}$$

(144)

where

$$A_1=\frac{(K_{11}+K_{12})\left({\tilde{\delta }}^{\tilde{\delta }}-t^{\tilde{\delta }}\right)p{C}^2{2}^{\tilde{\gamma }}}{{(1+\beta )}^{(1+3\beta )\tilde{\delta }}{t}^{\tilde{\delta }}}.$$

(145)

So, from (144), we obtain

$$\begin{array}{ccc}& & {\mathsf{\Psi }}^{″}-K_1\left(t,\mathsf{\Psi },\mathsf{\Phi }\right)-{(1+\beta )}^{-\tilde{\beta }}{t}^{-\tilde{\beta }}{\int }_0^{L\left(t\right)}{\lambda }_1{s}^{\beta }{G}_1\left(s\right)ds,\hfill \\ & & \ge A_1\left({\left(t^{\tilde{\delta }}-\frac{1}{2}{\left(\tilde{\delta }\right)}^{\tilde{\delta }}\right)}^2+\frac{8(2+\beta ){(1+\beta )}^2-(K_{11}+K_{12})pC{2}^{\tilde{\delta }}{(1+\beta )}^{\beta \tilde{\delta }}}{(K_{11}+K_{12})4pC{2}^{\tilde{\delta }}{(1+\beta )}^{(\beta +2)\tilde{\delta }}}\right)\hfill \\ & & \ge 0, \mathrm{since} C\le \frac{8(2+\beta ){(1+\beta )}^2}{p{2}^{\tilde{\delta }}{(\beta +1)}^{\beta \tilde{\delta }}}min\left\{\frac{1}{K_{11}+K_{12}},\frac{1}{K_{21}+K_{22}}\right\}.\hfill \end{array}$$

Again, we have

$$\begin{array}{ccc}& & {\mathsf{\Phi }}^{″}-K_2\left(t,\mathsf{\Psi },\mathsf{\Phi }\right)-{(1+\beta )}^{-\tilde{\beta }}{t}^{-\tilde{\beta }}{\int }_0^{L\left(t\right)}{\lambda }_2{s}^{\beta }{G}_2\left(s\right)ds,\hfill \\ & & \ge \frac{C{t}^{-\beta \tilde{\delta }}{2}^{\tilde{\delta }}(2+\beta )}{{(1+\beta )}^{(1+2\beta )\tilde{\delta }}}\left(\frac{1}{{(1+\beta )}^{\tilde{\delta }}}-\frac{\lambda M_2^{*}{(1+\beta )}^{\tilde{\delta }}{t}^{\tilde{\delta }}}{C{2}^{\tilde{\delta }}(2+\beta )}\right)-\frac{p(K_{21}+K_{22})C^2{2}^{2\tilde{\delta }}}{2{(1+\beta )}^2}{\left({\left(\tilde{\delta }\right)}^{\tilde{\delta }}-{\left(t\right)}^{\tilde{\delta }}\right)}^2\hfill \\ & & \ge \frac{C{t}^{-\beta \tilde{\delta }}{2}^{\tilde{\delta }}(2+\beta )}{{(1+\beta )}^{(1+2\beta )\tilde{\delta }}}\left(\frac{1}{{(1+\beta )}^{\tilde{\delta }}}-t^{\tilde{\delta }}\right)-\frac{p(K_{21}+K_{22})C^2{2}^{2\tilde{\delta }}}{2{(1+\beta )}^2}{\left({\left(\tilde{\delta }\right)}^{\tilde{\delta }}-{\left(t\right)}^{\tilde{\delta }}\right)}^2,\hfill \\ & & \left(\mathrm{Since} \lambda \le \frac{C{2}^{\tilde{\delta }}(2+\beta )}{M_2^{*}{(1+\beta )}^{\tilde{\delta }}}\right)\hfill \end{array}$$

(146)

$$\begin{array}{ccc}& & =A_1^{*}\left(\frac{2(2+\beta ){(1+\beta )}^2}{(K_{21}+K_{22})PC{2}^{\tilde{\delta }}{(1+\beta )}^{(2+\beta )\tilde{\delta }}}-\frac{1}{{(1+\beta )}^{\tilde{\delta }}}{t}^{\tilde{\delta }}+{\left(t^{\tilde{\delta }}\right)}^2\right),\hfill \end{array}$$

(147)

where

$$A_1^{*}=\frac{(K_{21}+K_{22})\left(\frac{1}{{(1+\beta )}^{\tilde{\delta }}}-t^{\tilde{\delta }}\right)p{C}^2{2}^{\tilde{\gamma }}}{{(1+\beta )}^{(1+3\beta )\tilde{\delta }}{t}^{\beta \tilde{\delta }}}.$$

(148)

Similarly, from (147), we derive

$$\begin{array}{ccc}& & {\mathsf{\Phi }}^{″}-K_2\left(t,\mathsf{\Psi },\mathsf{\Phi }\right)-{(1+\beta )}^{-\tilde{\beta }}{t}^{-\tilde{\beta }}{\int }_0^{L\left(t\right)}{\lambda }_2{s}^{\beta }{G}_2\left(s\right)ds,\hfill \\ & & \ge A_1^{*}\left({\left(t^{\tilde{\delta }}-\frac{1}{2}{\left(\tilde{\delta }\right)}^{\tilde{\delta }}\right)}^2+\frac{8(2+\beta ){(1+\beta )}^2-(K_{21}+K_{22})pC{2}^{\tilde{\delta }}{(1+\beta )}^{\beta \tilde{\delta }}}{(K_{21}+K_{22})4pC{2}^{\tilde{\delta }}{(1+\beta )}^{(\beta +2)\tilde{\delta }}}\right)\hfill \\ & & \ge 0, \mathrm{since} C\le \frac{8(2+\beta ){(1+\beta )}^2}{p{2}^{\tilde{\delta }}{(\beta +1)}^{\beta \tilde{\delta }}}min\left\{\frac{1}{K_{11}+K_{12}},\frac{1}{K_{21}+K_{22}}\right\}.\hfill \end{array}$$

(149)

Hence, the proof is complete. □

Lemma 20.

Assume ${\lambda }_1$ and ${\lambda }_2$ satisfy conditions

$$0\le {\lambda }_1\le \frac{(2+\beta )C}{M_1^{*}{(\beta +1)}^{\tilde{\delta }}{2}^{\beta \tilde{\delta }}},0\le {\lambda }_2\le \frac{(2+\beta )C}{M_2^{*}{(\beta +1)}^{\tilde{\delta }}{2}^{\beta \tilde{\delta }}}$$

(150)

and

$$C\le \frac{(\beta +2){(1+\beta )}^{(\beta +2)\tilde{\delta }}}{p{2}^{\tilde{\delta }}}min\left\{\frac{1}{K_{11}+K_{12}},\frac{1}{K_{21}+K_{22}}\right\}.$$

(151)

Then, there is at least one function $(\mathsf{\Psi },\mathsf{\Phi })\in E\times E$ satisfying (36), (53), $\beta \mathsf{\Psi }\left(\tilde{\delta }\right)={\mathsf{\Psi }}^{\prime }\left(\tilde{\delta }\right)$, $\beta \mathsf{\Phi }\left(\tilde{\delta }\right)={\mathsf{\Phi }}^{\prime }\left(\tilde{\delta }\right)$, $\mathsf{\Psi }\left(t\right)\le 0$, and $\mathsf{\Phi }\left(t\right)\le 0$.

Proof. 

We put

$$\mathsf{\Psi }\left(t\right)=-Ct\left(A-{\left(2t\right)}^{\tilde{\delta }}\right),\mathsf{\Phi }\left(t\right)=-Ct\left(B-{\left(2t\right)}^{\tilde{\delta }}\right), \mathrm{and} C\ge 0.$$

(152)

for all $t\in \left(0,\tilde{\delta }\right].$ Thus, Condition (53) is held. Now, $\beta \mathsf{\Psi }\left(\tilde{\delta }\right)={\mathsf{\Phi }}^{\prime }\left(\tilde{\delta }\right)$ and $\beta \mathsf{\Phi }\left(\tilde{\delta }\right)={\mathsf{\Phi }}^{\prime }\left(\tilde{\delta }\right)$ implies

$${\displaystyle A=2^{(\beta +2)\tilde{\delta }}{\left(\tilde{\delta }\right)}^{\tilde{\delta }} \& B=2^{(2+\beta )\tilde{\delta }}{\left(\tilde{\delta }\right)}^{\tilde{\delta }}.}$$

Therefore, $\mathsf{\Psi }\left(t\right)\le 0$ and $\mathsf{\Phi }\left(t\right)\le 0$ are also fulfilled. Hence, we obtain the results. □

Lemma 21.

Assume ${\lambda }_1$ and ${\lambda }_2$ satisfy conditions

$$0\le {\lambda }_1\le \frac{C{2}^{(\beta +2)\tilde{\delta }}(2+\beta )}{3M_1^{*}{(\beta +1)}^{\tilde{\delta }}},0\le {\lambda }_2\le \frac{C{2}^{(\beta +2)\tilde{\delta }}(2+\beta )}{3M_2^{*}{(\beta +1)}^{\tilde{\delta }}},$$

(153)

and

$$C\le \frac{2^{(6\beta +5)\tilde{\delta }}(\beta +2){(1+\beta )}^{(\beta +2)\tilde{\delta }}}{27p}min\left\{{\displaystyle \frac{1}{K_{11}+K_{12}}},{\displaystyle \frac{1}{K_{21}+K_{22}}}\right\}.$$

(154)

Then, there is a solution $(\mathsf{\Psi },\mathsf{\Phi })\in E\times E$ satisfying (36), (53), $(\beta -1)\mathsf{\Psi }\left(\tilde{\delta }\right)={\varphi }^{\prime }\left(\tilde{\delta }\right)$, $(\beta -1)\mathsf{\Phi }\left(\tilde{\delta }\right)={\mathsf{\Phi }}^{\prime }\left(\tilde{\delta }\right)$, $\mathsf{\Psi }\left(t\right)\le 0$, and $\mathsf{\Phi }\left(t\right)\le 0$.

Proof. 

We put

$$\mathsf{\Psi }\left(t\right)=-Ct\left(A-{\left(2t\right)}^{\tilde{\delta }}\right),\mathsf{\Phi }\left(t\right)=-Ct\left(B-{\left(2t\right)}^{\tilde{\delta }}\right), \mathrm{and} C\ge 0.$$

(155)

for $t\in \left(0,\tilde{\delta }\right].$ Now, the boundary conditions $(\beta -1)\mathsf{\Phi }\left(\tilde{\delta }\right)={\mathsf{\Phi }}^{\prime }\left(\tilde{\delta }\right)$ and $(\beta -1)\mathsf{\Psi }\left(\tilde{\delta }\right)={\mathsf{\Psi }}^{\prime }\left(\tilde{\delta }\right)$ provide the following values as

$${\displaystyle A=3\times 2^{-\beta \tilde{\delta }}{\left(\tilde{\delta }\right)}^{\tilde{\delta }} \& B=3\times 2^{-\beta \tilde{\delta }}{\left(\tilde{\delta }\right)}^{\tilde{\delta }}.}$$

Therefore, $\mathsf{\Psi }\left(t\right)\le 0$ and $\mathsf{\Phi }\left(t\right)\le 0$ are also fulfilled. Hence, after calculating as in Lemma 19, we obtain the results. □

Theorem 2.

Assume the conditions ${\lambda }_1^c>{\lambda }_1\ge 0$ and ${\lambda }_2^c>{\lambda }_1\ge 0$ hold. Then, the differential Equation (13) with respect to (14)–(16) is solvable. Also, the solutions of these problems do not exist if ${\lambda }_1>{\lambda }_1^c$ and ${\lambda }_2>{\lambda }_2^c$. Furthermore, every solution of (13) corresponding to (14)–(16) is nonpositive and gives the value zero at the origin.

Proof. 

The proof can be achived by using Lemmas 1–4, 14 and 15. □

Proposition 1.

Bounds of ${\lambda }_1^c$ and ${\lambda }_2^c$ for the system (14) are defined by the following estimation

$$\begin{array}{ccc}\hfill & & {\lambda }_1^c\ge \frac{8{(2+\beta )}^2(1+\beta ){\tilde{L}}_1}{p{M}_1^{*}},{\lambda }_2^c\ge \frac{8{(2+\beta )}^2(1+\beta ){\tilde{L}}_1}{p{M}_2^{*}}\hfill \end{array}$$

(156)

$$\begin{array}{ccc}& & \& {\lambda }_1^c{M}_1+{\lambda }_2^c{M}_2\le \frac{12{(1+\beta )}^4(\beta +3)}{p(K_{11}+K_{21})},\hfill \end{array}$$

(157)

where

$${\tilde{L}}_1=min\left\{{\displaystyle \frac{1}{K_{11}+K_{12}}},{\displaystyle \frac{1}{K_{21}+K_{22}}}\right\}.$$

(158)

Proof. 

The results can be deduced from Lemmas 16 and 19. □

Proposition 2.

Bounds of ${\lambda }_1^c$ and ${\lambda }_2^c$ for the system (15) are defined by the following estimation

$$\begin{array}{ccc}\hfill & & {\lambda }_1^c\ge \frac{128{(2+\beta )}^2(\beta +1){\tilde{L}}_1}{81p{M}_1^{*}},{\lambda }_2^c\ge \frac{128{(2+\beta )}^2(\beta +1){\tilde{L}}_1}{81p{M}_2^{*}}\hfill \end{array}$$

(159)

$$\begin{array}{ccc}& & and {\lambda }_1^c{M}_1+{\lambda }_2^c{M}_2\le \frac{24{(1+\beta )}^4(\beta +3)}{p(3{\beta }^2+10\beta +9)(K_{11}+K_{21})}.\hfill \end{array}$$

(160)

Proof. 

The results can be deduced from Lemmas 18 and 21. □

Proposition 3.

Bounds of ${\lambda }_1^c$ and ${\lambda }_2^c$ for the system (16) are defined by the following estimation

$$\begin{array}{ccc}\hfill & & {\lambda }_1^c\ge \frac{{(2+\beta )}^2(\beta +1){\tilde{L}}_1}{2p{M}_1^{*}},{\lambda }_2^c\ge \frac{{(2+\beta )}^2(\beta +1){\tilde{L}}_1}{2p{M}_2^{*}}\hfill \end{array}$$

(161)

$$\begin{array}{ccc}& & and {\lambda }_1^c{M}_1+{\lambda }_2^c{M}_2\le \frac{12{(1+\beta )}^4(\beta +3)}{p(K_{11}+K_{21})(6{\beta }^2+16\beta +11)}.\hfill \end{array}$$

(162)

Proof. 

The results can be deduce from Lemmas 17 and 20. □

Remark 1.

We can also reduce Equation (13) into a second-order differential equation similar to (17) by using the transformation ${\displaystyle t=\frac{r^{1+\beta }}{2}}$. Hence, for every $\beta >1$ the domain is transformed to $\left[0,{\displaystyle \frac{1}{2}}\right]$ instead of $\left[0,\tilde{\delta }\right]$, and similar results can be obtained. An important point to note is that the bounds of the parameters remain the same under this transformation.

Remark 2.

For $\beta =1$, it is easy to see that the bounds of the parameters ${\lambda }_1$ and ${\lambda }_2$ are well matched with respective bounds of the same parameters in the reference [11]. In [11], the authors placed the numerical examples to verify the theoretical results.

5. Conclusions

We proposed a system of fourth-order ordinary differential equations that arise in real life. We successfully converted the fourth-order SBVP into a system of integral equations. We classified the nature of solutions with respect to the space $C_{loc}^2\left[0,\frac{1}{2}\right]$. We showed the existence of at least one solution. We observed that the parameters ${\lambda }_1$ and ${\lambda }_2$ are dependent with each other. We calculated the relationship between these two parameters, which helps us find areas where there is no solution. We noticed that the range of the parameters is symmetric with respect to the transformations $t=\frac{r^{1+\beta }}{1+\beta }$ and $t=\frac{r^{1+\beta }}{2}$.