Asymptotic Stability and Dependency of a Class of Hybrid Functional Integral Equations

Abstract

Here, we discuss the solvability of a class of hybrid functional integral equations by applying Darbo’s fixed point theorem and the technique of the measure of noncompactness (MNC). This study has been located in space BC $\left(R_{+}\right)$. Furthermore, we prove the asymptotic stability of the solution of our problem on $R_{+}.$ We introduce the idea of asymptotic dependency of the solutions on some parameters for that class. Moreover, general discussion, examples, and remarks are demonstrated.

1. Introduction

The study of delay functional integral equations has received much consideration over the last few decades. Further studies and results for such kinds of problems may be found in [1,2,3] and the references therein.

The technique of MNC [4] in the Banach space $BC\left(R_{+}\right)$ had been effectively utilized by J. Banaś (see [5,6]) for demonstrating that asymptotic stable solutions for various functional equations have been established (see [7,8]).

Numerous practical real-world applications of quadratic integral equations are established. For various studies on the solvability of different classes of nonlinear equations, see [1,9,10,11,12,13].

Quadratic integral equations continuously emerge in numerous problems, such as the theory of radiative transfer, the kinetic theory of gases, the theory of neutron transport, the queuing theory, and the traffic theory.

Although the existence results in each of these monographs are included [1,11,12,13], their primary goal was to show a unique method or strategy as well as results pertaining to different existence for particular quadratic integral equations.

The significance of the investigations of hybrid functional integral and quadratic functional integral problems locates within the reality that this type involves different dynamic systems in particular cases. This class of hybrid differential equations involves the perturbations of original differential equations in several ways. A sharp classification of distinctive sorts of perturbations of differential equations shows up in Dhage [14], which can be treated with hybrid fixed point theory.

The authors in [13,15] discussed the two equations

$$\begin{array}{c}\hfill \frac{d}{d\mathfrak{r}}\left(\frac{\vartheta \left(\mathfrak{r}\right)-h(\mathfrak{r},\vartheta \left({\varphi }_1\left(\mathfrak{r}\right)\right))}{g(\mathfrak{r},\vartheta \left({\varphi }_2\left(\mathfrak{r}\right)\right))}\right)=f\left(\mathfrak{r},D^{\alpha }\left(\frac{\vartheta \left(\mathfrak{r}\right)-h(\mathfrak{r},\vartheta \left({\varphi }_1\left(\mathfrak{r}\right)\right))}{g(\mathfrak{r},\vartheta \left({\varphi }_2\left(\mathfrak{r}\right)\right))}\right)\right)a.e.\end{array}$$

$$\begin{array}{c}\hfill \frac{\vartheta \left(\mathfrak{r}\right)-\mathfrak{h}(\mathfrak{r},\vartheta (\mathfrak{r}\left)\right)}{g(\mathfrak{r},\vartheta (\mathfrak{r}\left)\right)}=f_1\left(\mathfrak{r},\frac{\vartheta \left(\mathfrak{r}\right)-\mathfrak{h}(\mathfrak{r},\vartheta (\mathfrak{r}\left)\right)}{g(\mathfrak{r},\vartheta (\mathfrak{r}\left)\right)},{\int }_0^{\varphi \left(\mathfrak{r}\right)}{f}_2\left(\mathfrak{r},s,\frac{\vartheta \left(s\right)-\mathfrak{h}(s,\vartheta (s\left)\right)}{g(s,\vartheta (s\left)\right)}\right)ds\right),\end{array}$$

in a bounded interval. Also, they studied the solvability of these problems using the technique of MNC in a finite interval and also discussed the continuous dependency.

Existence and stability results for Atangana–Baleanu fractional problems are established in [16], BVP for a nonlinear Hadamard fractional differential equation is discussed in [17,18], and, for fractional systems, see [19,20,21].

Here, consider the class of hybrid functional integral equation

$$\begin{array}{ccc}\hfill & & \frac{\vartheta \left(t\right)-h(t,\vartheta (t\left)\right)}{g(t,\vartheta (t\left)\right)}=f(t,g_1\left(t,\frac{\vartheta \left({\beta }_1\left(t\right)\right)-h(t,\vartheta \left({\beta }_1\left(t\right)\right))}{g(t,\vartheta \left({\beta }_1\left(t\right)\right))}\right),\lambda {\int }_0^{{\beta }_2\left(t\right)}{f}_1\left(t,\varsigma ,\frac{\vartheta \left(\varsigma \right)-h(\varsigma ,\vartheta (\varsigma \left)\right)}{g(\varsigma ,\vartheta (\varsigma \left)\right)}\right)d\varsigma ,\hfill \\ \hfill \\ & & g_2\left(t,\frac{\vartheta \left({\beta }_3\left(t\right)\right)-h(t,\vartheta \left({\beta }_3\left(t\right)\right))}{g(t,\vartheta \left({\beta }_3\left(t\right)\right))}\right)·{\int }_0^{{\beta }_4\left(t\right)}{f}_2\left(t,\varsigma ,\frac{\vartheta \left(\varsigma \right)-h(\varsigma ,\vartheta (\varsigma \left)\right)}{g(\varsigma ,\vartheta (\varsigma \left)\right)}\right)d\varsigma ),t\ge 0.\hfill \end{array}$$

(1)

Our aim here is to establish the solvability and discuss some asymptotic stability facts of the solution $\vartheta \in BC\left(R_{+}\right)$ of (1). The main tool in our study is applying Darbo’s fixed point [4] and MNC technique.

Furthermore, the asymptotic dependency of $\vartheta \in BC\left(R_{+}\right)$ on the parameter $\lambda \ge 0$ and on the functions $g_1,g_2,{\beta }_2$, and ${\beta }_4$ has been studied. Some special cases and examples have been discussed.

Let

$$\begin{array}{c}\hfill \frac{\vartheta \left(t\right)-h(t,\vartheta (t\left)\right)}{g(t,\vartheta (t\left)\right)}=\nu \left(t\right),\end{array}$$

and, easily, we obtain

$$\begin{array}{ccc}\hfill & & \vartheta \left(t\right)=h(t,\vartheta (t\left)\right)+\nu \left(t\right)g(t,\vartheta (t\left)\right),t\ge 0\hfill \end{array}$$

(2)

as a solution of (1), which implies that $\nu$ is a solution of

$$\nu \left(t\right)=f\left(t,g_1(t,\nu \left({\beta }_1\left(t\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t\right)}{f}_1(t,\varsigma ,\nu \left(\varsigma \right))d\varsigma ,g_2(t,\nu \left({\beta }_3\left(t\right)\right))·{\int }_0^{{\beta }_4\left(t\right)}{f}_2(t,\varsigma ,\nu \left(\varsigma \right))d\varsigma \right),t\ge 0.$$

(3)

We arrange our article just like that: we conclude the solvability of (3) in $BC\left(R_{+}\right)$, and then the asymptotic stability of the solution $\nu \in BC\left(R_{+}\right)$ of (3) is discussed in Section 2. The main theorems for the existence of the solutions $\vartheta \in BC\left(R_{+}\right)$ and the asymptotic stability and dependency of the solution $\vartheta \in BC\left(R_{+}\right)$ on the parameter $\lambda \ge 0$, and on the functions $g_1,g_2,{\beta }_2$ and ${\beta }_4$ have been established. Finally, some general remarks and comments will be provided.

The class $BC\left(R_{+}\right)$ of all bounded and continuous functions in $R_{+}$, with an internal composition law is noted by

$$\begin{array}{c}\hfill (.):X\times X⟶X,(x,y)⟶x.y,\end{array}$$

which is associative and bilinear.

A normed algebra is an algebra endowed with a norm satisfying the following property:

For all $x,y\in X$, we have

$$\begin{array}{c}\hfill \parallel x.y\parallel \le \parallel x\parallel .\parallel y\parallel .\end{array}$$

A complete normed algebra is called a Banach algebra.

Now, let $\vartheta \in Y\subseteq BC\left(R_{+}\right)$ and $\epsilon \ge 0$ be provided, defined as ${\omega }^{\mathfrak{T}}(\vartheta ,\epsilon )$, $\mathfrak{T}\ge 0$, the modulus of continuity of the function $\vartheta$ on $[0,\mathfrak{T}]$

$$\begin{array}{c}\hfill {\omega }^{\mathfrak{T}}(\vartheta ,\epsilon )=sup\left[\right|\vartheta \left(t\right)-\vartheta \left(\varsigma \right)|:t,\varsigma \in [0,\mathfrak{T}],|t-\varsigma |\le \epsilon ]\end{array}$$

and

$$\begin{array}{c}\hfill {\omega }^{\mathfrak{T}}(Y,\epsilon )=sup[{\omega }^{\mathfrak{T}}(\vartheta ,\epsilon ):\vartheta \in Y].\end{array}$$

Also,

$$\begin{array}{c}\hfill {\omega }_0^{\mathfrak{T}}\left(Y\right)=\underset{\epsilon \to 0}{lim}{\omega }^{\mathfrak{T}}(Y,\epsilon ),{\omega }_0\left(Y\right)=\underset{\mathfrak{T}\to \infty }{lim}{\omega }_0^{\mathfrak{T}}\left(Y\right)\end{array}$$

and

$$\begin{array}{c}\hfill diamY\left(t\right)=sup\left\{\right|\vartheta \left(t\right)-\nu \left(t\right)|,\vartheta ,\nu \in Y\}.\end{array}$$

The MNC on $BC\left(R_{+}\right)$ has the form [22,23]

$$\begin{array}{c}\hfill \mu \left(Y\right)={\omega }_0\left(Y\right)+\underset{t\to \infty }{lim}supdiamY\left(t\right).\end{array}$$

Theorem 1 

([4]). Let C be a nonempty, bounded, closed, and convex subset of a Banach space ε and let $Y:C⟶C$ be a continuous mapping. Assume that there exists a constant $K\in [0,1)$ such that $\mu (Y\wedge )\le K\mu (\wedge )$ for any nonempty subset ∧ of C, where μ is an MNC defined in ε. Then, Y has at least one fixed point in C.

2. Existence of Solutions

To achieve our goals, assume that

(i)

${\beta }_{\iota }:R_{+}\to R_{+}$, ${\beta }_{\iota }\left(t\right)\le t$, $\iota =1,2,3,4$ are continuous.

(ii)

$f:R_{+}\times R\times R\times R\to R$ is continuous in $t\in R_{+}$, $\forall \xi ,\upsilon ,w\in R$ and satisfies Lipschitz condition,

$$\begin{array}{ccc}\hfill & & |f(t,\xi ,\upsilon ,w)-f(t,{\xi }_1,{\upsilon }_1,w_1)|\le b(|\xi -{\xi }_1|+|\upsilon -{\upsilon }_1|+|w-w_1\left|\right),\hfill \\ \\ & & \forall (t,\xi ,\upsilon ,w),(t,{\xi }_1,{\upsilon }_1,w_1)\in R_{+}\times R\times R\times R,b>0.\hfill \end{array}$$

(4)

(iii)

$g_i:R_{+}\times R\to R$, $i=1,2$ are continuous in $t\in R_{+}$, $\forall \xi \in R$ and satisfy Lipschitz condition,

$$\begin{array}{ccc}\hfill & & |g_i(t,\xi )-g_i(t,{\xi }_1)|\le b_i|\xi -{\xi }_1|,\hfill \\ \\ & & \forall (t,\xi ),(t,{\xi }_1)\in R_{+}\times R,b_i>0.\hfill \end{array}$$

(5)

(iv)

$f_i:R_{+}\times R_{+}\times R\to R$, $i=1,2$ are Carathéodory functions, which are measurable in $t,\varsigma \in R_{+}\times R_{+}$, $\forall \nu \in R$ and continuous in $\nu \in R$, $\forall t,\varsigma \in R_{+}\times R_{+}$, and there exist measurable and bounded functions $k_i,c_i:R_{+}\times R_{+}\to R$, where

$$\begin{array}{c}\hfill |f_i(t,\varsigma ,\nu )|\le k_i(t,\varsigma )+c_i(t,\varsigma )\left|\nu \right|,\forall (t,\varsigma )\in R_{+}\times R_{+}\end{array}$$

and

$$\begin{array}{c}\hfill \underset{\mathfrak{r}\to \infty }{lim}{\int }_0^{{\beta }_j\left(\mathfrak{r}\right)}{k}_i(\mathfrak{r},\varsigma )d\varsigma =0,\underset{\mathfrak{r}\to \infty }{lim}{\int }_0^{{\beta }_j\left(\mathfrak{r}\right)}{c}_i(\mathfrak{r},\varsigma )d\varsigma =0,j=2,4,\\ \hfill \underset{\mathfrak{r}\in R_{+}}{sup}{\int }_0^{{\beta }_j\left(\mathfrak{r}\right)}{k}_i(\mathfrak{r},\varsigma )d\varsigma =k_i,\underset{\mathfrak{r}\in R_{+}}{sup}{\int }_0^{{\beta }_j\left(t\right)}{c}_i(t,\varsigma )d\varsigma =c_i,j=2,4.\end{array}$$

$\left(v\right)$

For a positive constant r satisfying the equation

$$\begin{array}{c}\hfill b{b}_2{c}_2{r}^2+(b{b}_1+b\lambda c_1+b{c}_2{m}_2^{*}+b{b}_2{k}_2-1)r+m^{*}+b{m}_1^{*}+b\lambda k_1+b{k}_2{m}_2^{*}=0.\end{array}$$

From Equation (4), we receive

$$\begin{array}{ccc}\hfill \left|f\right(\mathfrak{r},\xi ,u,w\left)\right|-\left|f\right(\mathfrak{r},0,0,0\left)\right|& & \le \left|f\right(\mathfrak{r},\xi ,u,w)-f(\mathfrak{r},0,0,0\left)\right|\hfill \\ & & \le b\left(\right|\xi |+|u|+|w\left|\right),\hfill \\ \hfill \left|f\right(\mathfrak{r},\xi ,u,w\left)\right|& & \le \left|f\right(\mathfrak{r},0,0,0\left)\right|+b\left(\right|\xi |+|u|+|w\left|\right)\hfill \\ \hfill \left|f\right(\mathfrak{r},\xi ,u,w\left)\right|& & \le \left|m\right(\mathfrak{r}\left)\right|+b\left(\right|\xi |+|u|+|w\left|\right),\hfill \end{array}$$

where

$m\left(t\right)=\left|f(t,0,0,0)\right|,m\in BC\left(R_{+}\right)$ and $m^{*}=\underset{t\in R_{+}}{sup}\left|m\left(t\right)\right|<\infty .$

In the same manner, from Equation (5), we receive

$$\begin{array}{ccc}& & |g_i(t,\xi )|\le |m_i\left(t\right)|+b_i\left|\xi \right|,\hfill \end{array}$$

where

$m_i\left(t\right)=\left|g_i(t,0)\right|,m_i\in BC\left(R_{+}\right)$ and $m_i^{*}=\underset{t\in R_{+}}{sup}\left|m_i\left(t\right)\right|<\infty .$

Theorem 2. 

Suppose that $\left(i\right)-\left(iv\right)$ hold. Then, we have a solution $\nu \in BC\left(R_{+}\right)$ for (3).

Proof. 

Let

$$\begin{array}{ccc}& & Q_r=\{\nu \in BC\left(R_{+}\right):\parallel \nu \parallel \le r\},\hfill \\ & & r=m^{*}+b{m}_1^{*}+b{b}_{1}r+b\lambda k_1+b\lambda c_{1}r+b{m}_2^{*}{k}_2+b{m}_2^{*}{c}_{2}r+b{b}_2{k}_{2}r+b{b}_2{c}_2{r}^2.\hfill \end{array}$$

Associate the operator

$F\nu \left(t\right)=$

$$\begin{array}{c}\hfill f\left(t,g_1(t,\nu \left({\beta }_1\left(t\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t\right)}{f}_1(t,\varsigma ,\nu \left(\varsigma \right))d\varsigma ,g_2(t,\nu \left({\beta }_3\left(t\right)\right))·{\int }_0^{{\beta }_4\left(t\right)}{f}_2(t,\varsigma ,\nu \left(\varsigma \right))d\varsigma \right),t\ge 0.\end{array}$$

For $\nu \in Q_r$, then

$$\begin{array}{ccc}\hfill \left|F\nu \right(t\left)\right|& =& \\ & & \left|f\left(t,g_1(t,\nu \left({\beta }_1\left(t\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t\right)}{f}_1(t,\varsigma ,\nu \left(\varsigma \right))d\varsigma ,g_2(t,\nu \left({\beta }_3\left(t\right)\right))·{\int }_0^{{\beta }_4\left(t\right)}{f}_2(t,\varsigma ,\nu \left(\varsigma \right))d\varsigma \right)\right|\hfill \\ & \le & \left|m\left(t\right)\right|+b\left(|m_1\left(t\right)|+b_1\left|\nu \left({\beta }_1\left(t\right)\right)\right|\right)+b\lambda \left({\int }_0^{{\beta }_2\left(t\right)}(k_1(t,\varsigma )+c_1(t,\varsigma )\left|\nu \left(\varsigma \right)\right|)d\varsigma \right)\hfill \\ & +& b\left(|m_2\left(t\right)|+b_2\left|\nu \left({\beta }_3\left(t\right)\right)\right|\right)·{\int }_0^{{\beta }_4\left(t\right)}(k_2(t,\varsigma )+c_2(t,\varsigma )\left|\nu \left(\varsigma \right)\right|)d\varsigma ,\hfill \end{array}$$

then

$$\begin{array}{ccc}\hfill \parallel F\nu \parallel & \le & m^{*}+b{m}_1^{*}+b{b}_1\parallel \nu \parallel +b\lambda \left({\int }_0^{{\beta }_2\left(t\right)}{k}_1(t,\varsigma )d\varsigma +\parallel \nu \parallel {\int }_0^{{\beta }_2\left(t\right)}{c}_1(t,\varsigma )d\varsigma \right)\hfill \\ & +& b\left((m_2^{*}+b_2\parallel \nu \parallel )·({\int }_0^{{\beta }_4\left(t\right)}{k}_2(t,\varsigma )d\varsigma +\parallel \nu \parallel {\int }_0^{{\beta }_4\left(t\right)}{c}_2(t,\varsigma )d\varsigma )\right)\hfill \\ & \le & m^{*}+b{m}_1^{*}+b{b}_{1}r+b\lambda (k_1+c_{1}r)+(b{m}_2^{*}+b{b}_{2}r)·(k_2+c_{2}r)\hfill \\ & \le & m^{*}+b{m}_1^{*}+b{b}_{1}r+b\lambda k_1+b\lambda c_{1}r+b{m}_2^{*}{k}_2+b{m}_2^{*}{c}_{2}r+b{b}_2{k}_{2}r+b{b}_2{c}_2{r}^2.\hfill \end{array}$$

Thus, the mapping F draws the set $Q_r$ into $Q_r$.

Then, take $\delta >0$ and take ${\nu }_1,{\nu }_2\in Q_r$, such that $\parallel {\nu }_2-{\nu }_1\parallel \le \delta$, and then

$|F{\nu }_2\left(t\right)-F{\nu }_1\left(t\right)|=$

$$\begin{array}{ccc}& & |f\left(t,g_1(t,{\nu }_2\left({\beta }_1\left(t\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t\right)}{f}_1(t,\varsigma ,{\nu }_2\left(\varsigma \right))d\varsigma ,g_2(t,{\nu }_2\left({\beta }_3\left(t\right)\right))·{\int }_0^{{\beta }_4\left(t\right)}{f}_2(t,\varsigma ,{\nu }_2\left(\varsigma \right))d\varsigma \right)\hfill \\ & -& f\left(t,g_1(t,{\nu }_1\left({\beta }_1\left(t\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t\right)}{f}_1(t,\varsigma ,{\nu }_1\left(\varsigma \right))d\varsigma ,g_2(t,{\nu }_1\left({\beta }_3\left(t\right)\right))·{\int }_0^{{\beta }_4\left(t\right)}{f}_2(t,\varsigma ,{\nu }_1\left(\varsigma \right))d\varsigma \right)|\hfill \end{array}$$

$$\begin{array}{ccc}\hfill & \le & b|g_1(t,{\nu }_2\left({\beta }_1\left(t\right)\right))-g_1\left({\nu }_1\left({\beta }_1\left(t\right)\right)\right)|+b\lambda {\int }_0^{{\beta }_2\left(t\right)}|f_1(t,\varsigma ,{\nu }_2\left(\varsigma \right))-f_1(t,\varsigma ,{\nu }_1\left(\varsigma \right))|d\varsigma \hfill \\ & +& b|g_2(t,{\nu }_2\left({\beta }_3\left(t\right)\right))-g_2\left({\nu }_1\left({\beta }_3\left(t\right)\right)\right)|·{\int }_0^{{\beta }_4\left(t\right)}\left|f_2(t,\varsigma ,{\nu }_2\left(\varsigma \right))\right|d\varsigma \hfill \\ & +& b|g_2(t,{\nu }_1\left({\beta }_3\left(t\right)\right))|·{\int }_0^{{\beta }_4\left(t\right)}|f_2(t,\varsigma ,{\nu }_2\left(\varsigma \right))-f_2(t,\varsigma ,{\nu }_1\left(\varsigma \right))|d\varsigma .\hfill \end{array}$$

(6)

$(i*)$ Choose $\mathfrak{T}>0$, satisfying $t\ge \mathfrak{T}$, and

$$\begin{array}{ccc}\hfill \parallel F{\nu }_2-F{\nu }_1\parallel & \le & b{b}_1\parallel {\nu }_2-{\nu }_1\parallel +2b\lambda \left({\int }_0^{{\beta }_2\left(t\right)}{k}_1(t,\varsigma )d\varsigma +r{\int }_0^{{\beta }_2\left(t\right)}{c}_1(t,\varsigma )d\varsigma \right)\hfill \\ & +& b{b}_2\parallel {\nu }_2-{\nu }_1\parallel ·\left({\int }_0^{{\beta }_4\left(t\right)}{k}_2(t,\varsigma )d\varsigma +r{\int }_0^{{\beta }_4\left(t\right)}{c}_2(t,\varsigma )d\varsigma \right)\hfill \\ & +& b(m_2^{*}+b_{2}r)(2{\int }_0^{{\beta }_4\left(t\right)}{k}_2(t,\varsigma )d\varsigma +2r{\int }_0^{{\beta }_4\left(t\right)}{c}_2(t,\varsigma )d\varsigma )\hfill \\ & \le & b{b}_1\delta +2b\lambda ({ϵ}_1+r{ϵ}_2)+b{b}_2\delta ({ϵ}_3+{ϵ}_{4}r)+2b(m_2^{*}+b_{2}r)({ϵ}_3+{ϵ}_{4}r)=ϵ.\hfill \end{array}$$

$(ii*)$ Also, for $\mathfrak{T}>0$, $t\in [0,\mathfrak{T}]$; thus, from (6), we find

$$\begin{array}{ccc}\hfill \parallel F{\nu }_2-F{\nu }_1\parallel & \le & b{b}_1\parallel {\nu }_2-{\nu }_1\parallel +b\lambda {\int }_0^{{\beta }_2\left(t\right)}\omega \left(\mathcal{E}\right)ds+b{b}_2\parallel {\nu }_2-{\nu }_1\parallel ({\int }_0^{{\beta }_4\left(t\right)}{k}_2(t,\varsigma )d\varsigma \hfill \\ & +& r{\int }_0^{{\beta }_4\left(t\right)}{c}_2(t,\varsigma )d\varsigma )+b(m_2^{*}+b_{2}r){\int }_0^{{\beta }_4\left(t\right)}\omega \left(\mathcal{E}\right)d\varsigma \hfill \\ \hfill & \le & b{b}_1\delta +b\lambda \omega \left(\mathcal{E}\right)\mathfrak{T}+b{b}_2({ϵ}_3+{ϵ}_{4}r)\delta +b(m_2^{*}+b_{2}r)\omega \left(\mathcal{E}\right)\mathfrak{T}\hfill \end{array}$$

where $\omega \left(\mathcal{E}\right)=sup\left\{\right|f_i(\mathfrak{r},s,{\nu }_2\left(\varsigma \right))-f_i(\mathfrak{r},s,{\nu }_1\left(\varsigma \right))|:t,s\in [0,\mathfrak{T}],{\nu }_1,{\nu }_2\in [-r,r],|{\nu }_2-{\nu }_1|<\delta \}$, $i=1,2$. Then, from uniform continuity of the functions $f_i$ on $[0,\mathfrak{T}]\times [0,\mathfrak{T}]\times [-r,r]$; therefore, $\omega \left(\mathcal{E}\right)\to 0$ as $\delta \to \infty$. In consequence, F maps $Q_r$ continuously.

Next, let $Y\in Q_r$ be nonempty. Then, for any ${\nu }_2,{\nu }_1\in Y$ and $\mathfrak{T}>0$ satisfying $t\ge \mathfrak{T}$, from (6), we find

$|F{\nu }_2\left(t\right)-F{\nu }_1\left(t\right)|=$

$$\begin{array}{ccc}& \le & b{b}_1|{\nu }_2\left({\beta }_1\left(t\right)\right)-{\nu }_1\left({\beta }_1\left(t\right)\right)|+2b\lambda \left({\int }_0^{{\beta }_2\left(t\right)}{k}_1(t,\varsigma )d\varsigma +r{\int }_0^{{\beta }_2\left(t\right)}{c}_1(t,\varsigma )d\varsigma \right)\hfill \\ & +& b{b}_2|{\nu }_2\left({\beta }_3\left(t\right)\right)-{\nu }_1\left({\beta }_3\left(t\right)\right)|·{\int }_0^{{\beta }_4\left(t\right)}{k}_2(t,\varsigma )d\varsigma +r{\int }_0^{{\beta }_4\left(t\right)}{c}_2(t,\varsigma )d\varsigma \hfill \\ & +& 2b(m_2^{*}+b_{2}r)({\int }_0^{{\beta }_4\left(t\right)}{k}_2(t,\varsigma )d\varsigma +r{\int }_0^{{\beta }_4\left(t\right)}{c}_2(t,\varsigma )d\varsigma )\hfill \\ \hfill & \le & b{b}_{1}diamY\left(t\right)+2b\lambda \left({\int }_0^{{\beta }_2\left(t\right)}{k}_1(t,\varsigma )d\varsigma +r{\int }_0^{{\beta }_2\left(t\right)}{c}_1(t,\varsigma )d\varsigma \right)\hfill \\ & +& b{b}_{2}diamY\left(t\right)({\int }_0^{{\beta }_4\left(t\right)}{k}_2(t,\varsigma )d\varsigma +r{\int }_0^{{\beta }_4\left(t\right)}{c}_2(t,\varsigma )d\varsigma )\hfill \\ & +& 2b(m_2^{*}+b_{2}r)·({\int }_0^{{\beta }_4\left(t\right)}{k}_2(t,\varsigma )d\varsigma +r{\int }_0^{{\beta }_4\left(t\right)}{c}_2(t,\varsigma )d\varsigma ).\hfill \end{array}$$

Hence, we obtain

$$\begin{array}{ccc}\hfill diamFY\left(t\right)& \le & b{b}_{1}diamY\left(t\right)+2b\lambda (k_1+c_{1}r)+b{b}_2(k_2+c_{2}r)diamY\left(t\right)\hfill \\ & +& 2b(m_2^{*}+b_{2}r)(k_2+c_{2}r)\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \underset{n\to \infty }{lim}supdiamFY\left(t\right)& \le & \left(b{b}_1+b{b}_2(k_2+c_{2}r)\right)\underset{n\to \infty }{lim}supdiamY\left(t\right).\hfill \end{array}$$

(7)

Let $\mathfrak{T}>0$ and $\delta >0$ be taken and select a mapping $\nu \in Y$ and $t\in [0,\mathfrak{T}]$, where $|t_2-t_1|<\delta$, $t_1\le t_2.$ Take ${\theta }_f\left(\delta \right),{\theta }_{g_i}\left(\delta \right),{\theta }_{f_i}\left(\delta \right)$ as defined in [9,13]. Therefore,

$$\begin{gathered} |F\nu \left(t_2\right)-F\nu \left(t_1\right)|= \\ \begin{array}{ccc}& & |f\left(t_2,g_1(t_2,\nu \left({\beta }_1\left(t_2\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t_2\right)}{f}_1(t_2,\varsigma ,\nu \left(\varsigma \right))d\varsigma ,g_2(t_2,\nu \left({\beta }_3\left(t_2\right)\right))·{\int }_0^{{\beta }_4\left(t_2\right)}{f}_2(t_2,\varsigma ,\nu \left(\varsigma \right))d\varsigma \right)\hfill \\ & -& f\left(t_1,g_1(t_1,\nu \left({\beta }_1\left(t_1\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t_1\right)}{f}_1(t_1,\varsigma ,\nu \left(\varsigma \right))d\varsigma ,g_2(t_1,\nu \left({\beta }_3\left(t_1\right)\right))·{\int }_0^{{\beta }_4\left(t_1\right)}{f}_2(t_1,\varsigma ,\nu \left(\varsigma \right))d\varsigma \right)|\hfill \\ \hfill & \le & |f\left(t_2,g_1(t_2,\nu \left({\beta }_1\left(t_2\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t_2\right)}{f}_1(t_2,\varsigma ,\nu \left(\varsigma \right))d\varsigma ,g_2(t_2,\nu \left({\beta }_3\left(t_2\right)\right)).{\int }_0^{{\beta }_4\left(t_2\right)}{f}_2(t_2,\varsigma ,\nu \left(\varsigma \right))d\varsigma \right)\hfill \\ & -& f\left(t_1,g_1(t_2,\nu \left({\beta }_1\left(t_2\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t_2\right)}{f}_1(t_2,\varsigma ,\nu \left(\varsigma \right))d\varsigma ,g_2(t_2,\nu \left({\beta }_3\left(t_2\right)\right))·{\int }_0^{{\beta }_4\left(t_2\right)}{f}_2(t_2,\varsigma ,\nu \left(\varsigma \right))d\varsigma \right)\hfill \\ & +& f\left(t_1,g_1(t_2,\nu \left({\beta }_1\left(t_2\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t_2\right)}{f}_1(t_2,\varsigma ,\nu \left(\varsigma \right))d\varsigma ,g_2(t_2,\nu \left({\beta }_3\left(t_2\right)\right))·{\int }_0^{{\beta }_4\left(t_2\right)}{f}_2(t_2,\varsigma ,\nu \left(\varsigma \right))d\varsigma \right)\hfill \\ & -& f\left(t_1,g_1(t_1,\nu \left({\beta }_1\left(t_2\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t_2\right)}{f}_1(t_2,\varsigma ,\nu \left(\varsigma \right))d\varsigma ,g_2(t_2,\nu \left({\beta }_3\left(t_2\right)\right))·{\int }_0^{{\beta }_4\left(t_2\right)}{f}_2(t_2,\varsigma ,\nu \left(\varsigma \right))d\varsigma \right)\hfill \\ & +& f\left(t_1,g_1(t_1,\nu \left({\beta }_1\left(t_2\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t_2\right)}{f}_1(t_2,\varsigma ,\nu \left(\varsigma \right))d\varsigma ,g_2(t_2,\nu \left({\beta }_3\left(t_2\right)\right))·{\int }_0^{{\beta }_4\left(t_2\right)}{f}_2(t_2,\varsigma ,\nu \left(\varsigma \right))d\varsigma \right)\hfill \\ & -& f\left(t_1,g_1(t_1,\nu \left({\beta }_1\left(t_1\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t_2\right)}{f}_1(t_2,\varsigma ,\nu \left(\varsigma \right))d\varsigma ,g_2(t_2,\nu \left({\beta }_3\left(t_2\right)\right))·{\int }_0^{{\beta }_4\left(t_2\right)}{f}_2(t_2,\varsigma ,\nu \left(\varsigma \right))d\varsigma \right)\hfill \\ & +& f\left(t_1,g_1(t_1,\nu \left({\beta }_1\left(t_1\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t_2\right)}{f}_1(t_2,\varsigma ,\nu \left(\varsigma \right))d\varsigma ,g_2(t_2,\nu \left({\beta }_3\left(t_2\right)\right))·{\int }_0^{{\beta }_4\left(t_2\right)}{f}_2(t_2,\varsigma ,\nu \left(\varsigma \right))d\varsigma \right)\hfill \\ & -& f\left(t_1,g_1(t_1,\nu \left({\beta }_1\left(t_1\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t_1\right)}{f}_1(t_2,\varsigma ,\nu \left(\varsigma \right))d\varsigma ,g_2(t_2,\nu \left({\beta }_3\left(t_2\right)\right))·{\int }_0^{{\beta }_4\left(t_2\right)}{f}_2(t_2,\varsigma ,\nu \left(\varsigma \right))d\varsigma \right)\hfill \\ & +& f\left(t_1,g_1(t_1,\nu \left({\beta }_1\left(t_1\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t_1\right)}{f}_1(t_2,\varsigma ,\nu \left(\varsigma \right))d\varsigma ,g_2(t_2,\nu \left({\beta }_3\left(t_2\right)\right))·{\int }_0^{{\beta }_4\left(t_2\right)}{f}_2(t_2,\varsigma ,\nu \left(\varsigma \right))d\varsigma \right)\hfill \\ & -& f\left(t_1,g_1(t_1,\nu \left({\beta }_1\left(t_1\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t_1\right)}{f}_1(t_1,\varsigma ,\nu \left(\varsigma \right))d\varsigma ,g_2(t_2,\nu \left({\beta }_3\left(t_2\right)\right))·{\int }_0^{{\beta }_4\left(t_2\right)}{f}_2(t_2,\varsigma ,\nu \left(\varsigma \right))d\varsigma \right)\hfill \\ & +& f\left(t_1,g_1(t_1,\nu \left({\beta }_1\left(t_1\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t_1\right)}{f}_1(t_1,\varsigma ,\nu \left(\varsigma \right))d\varsigma ,g_2(t_2,\nu \left({\beta }_3\left(t_2\right)\right))·{\int }_0^{{\beta }_4\left(t_2\right)}{f}_2(t_2,\varsigma ,\nu \left(\varsigma \right))d\varsigma \right)\hfill \\ & -& f\left(t_1,g_1(t_1,\nu \left({\beta }_1\left(t_1\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t_1\right)}{f}_1(t_1,\varsigma ,\nu \left(\varsigma \right))d\varsigma ,g_2(t_1,\nu \left({\beta }_3\left(t_2\right)\right))·{\int }_0^{{\beta }_4\left(t_2\right)}{f}_2(t_2,\varsigma ,\nu \left(\varsigma \right))d\varsigma \right)\hfill \\ & +& f\left(t_1,g_1(t_1,\nu \left({\beta }_1\left(t_1\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t_1\right)}{f}_1(t_1,\varsigma ,\nu \left(\varsigma \right))d\varsigma ,g_2(t_1,\nu \left({\beta }_3\left(t_2\right)\right))·{\int }_0^{{\beta }_4\left(t_2\right)}{f}_2(t_2,\varsigma ,\nu \left(\varsigma \right))d\varsigma \right)\hfill \\ & -& f\left(t_1,g_1(t_1,\nu \left({\beta }_1\left(t_1\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t_1\right)}{f}_1(t_1,\varsigma ,\nu \left(\varsigma \right))d\varsigma ,g_2(t_1,\nu \left({\beta }_3\left(t_1\right)\right))·{\int }_0^{{\beta }_4\left(t_2\right)}{f}_2(t_2,\varsigma ,\nu \left(\varsigma \right))d\varsigma \right)\hfill \\ & +& f\left(t_1,g_1(t_1,\nu \left({\beta }_1\left(t_1\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t_1\right)}{f}_1(t_1,\varsigma ,\nu \left(\varsigma \right))d\varsigma ,g_2(t_1,\nu \left({\beta }_3\left(t_1\right)\right))·{\int }_0^{{\beta }_4\left(t_2\right)}{f}_2(t_2,\varsigma ,\nu \left(\varsigma \right))d\varsigma \right)\hfill \\ & -& f\left(t_1,g_1(t_1,\nu \left({\beta }_1\left(t_1\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t_1\right)}{f}_1(t_1,\varsigma ,\nu \left(\varsigma \right))d\varsigma ,g_2(t_1,\nu \left({\beta }_3\left(t_1\right)\right))·{\int }_0^{{\beta }_4\left(t_1\right)}{f}_2(t_2,\varsigma ,\nu \left(\varsigma \right))d\varsigma \right)\hfill \\ & +& f\left(t_1,g_1(t_1,\nu \left({\beta }_1\left(t_1\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t_1\right)}{f}_1(t_1,\varsigma ,\nu \left(\varsigma \right))d\varsigma ,g_2(t_1,\nu \left({\beta }_3\left(t_1\right)\right))·{\int }_0^{{\beta }_4\left(t_1\right)}{f}_2(t_2,\varsigma ,\nu \left(\varsigma \right))d\varsigma \right)\hfill \\ & -& f\left(t_1,g_1(t_1,\nu \left({\beta }_1\left(t_1\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t_1\right)}{f}_1(t_1,\varsigma ,\nu \left(\varsigma \right))d\varsigma ,g_2(t_1,\nu \left({\beta }_3\left(t_1\right)\right))·{\int }_0^{{\beta }_4\left(t_1\right)}{f}_2(t_1,\varsigma ,\nu \left(\varsigma \right))d\varsigma \right)|\hfill \\ & \le & {\theta }_f\left(\delta \right)+b{\theta }_{g_1}\left(\delta \right)+b{b}_1|\nu \left({\beta }_1\left(t_2\right)\right)-\nu \left({\beta }_1\left(t_1\right)\right)|+b\lambda {\int }_{{\beta }_2\left(t_1\right)}^{{\beta }_2\left(t_2\right)}\left|f_1(t_2,\varsigma ,\nu \left(\varsigma \right))\right|d\varsigma \hfill \\ & +& b\lambda {\int }_0^{{\beta }_2\left(t_2\right)}|f_1(t_2,\varsigma ,\nu \left(\varsigma \right))-f_1(t_1,\varsigma ,\nu \left(\varsigma \right))|d\varsigma \hfill \\ & +& b{\theta }_{g_2}\left(\delta \right)·{\int }_0^{{\beta }_4\left(t_2\right)}|f_2(t_2,\varsigma ,\nu \left(\varsigma \right)|d\varsigma +b{b}_2|\nu \left({\beta }_3\left(t_2\right)\right)-\nu \left({\beta }_3\left(t_1\right)\right)|·{\int }_0^{{\beta }_4\left(t_2\right)}\left|f_2(t_2,\varsigma ,\nu \left(\varsigma \right))\right|d\varsigma \hfill \\ & +& b(m_2^{*}+b_{2}r){\int }_{{\beta }_4\left(t_1\right)}^{{\beta }_4\left(t_2\right)}|f_2(t_2,\varsigma ,\nu \left(\varsigma \right))|d\varsigma +b(m_2^{*}+b_{2}r){\int }_0^{{\beta }_4\left(t_1\right)}|f_2(t_2,\varsigma ,\nu \left(\varsigma \right))-f_2(t_1,\varsigma ,\nu \left(\varsigma \right))|d\varsigma \hfill \\ \hfill & \le & {\theta }_f\left(\delta \right)+b{\theta }_{g_1}\left(\delta \right)+b{b}_{1}sup\left\{\right|\nu \left({\beta }_1\left(t_2\right)\right)-\nu \left({\beta }_1\left(t_1\right)\right)\left|\right\}+b\lambda {\int }_{{\beta }_2\left(t_1\right)}^{{\beta }_2\left(t_2\right)}\left|f_1(t_2,\varsigma ,\nu \left(\varsigma \right))\right|d\varsigma \hfill \\ & +& b\lambda {\theta }_{f_1}\left(\delta \right){\beta }_2\left(\mathfrak{T}\right)+b{\theta }_{g_2}\left(\delta \right)(k_2+c_{2}r)+b{b}_2(k_2+c_{2}r)sup\left\{\right|\nu \left({\beta }_3\left(t_2\right)\right)-\nu \left({\beta }_3\left(t_1\right)\right)\left|\right\}\hfill \\ & +& b(m_2^{*}+b_{2}r){\int }_{{\beta }_4\left(t_1\right)}^{{\beta }_4\left(t_2\right)}\left|f_2(t_2,\varsigma ,\nu \left(\varsigma \right))\right|d\varsigma +b(m_2^{*}+b_{2}r){\theta }_{f_2}\left(\delta \right){\beta }_4\left(\mathfrak{T}\right)\hfill \\ \hfill & \le & {\theta }_f\left(\delta \right)+b{\theta }_{g_1}\left(\delta \right)+b{b}_1{\omega }^{\mathfrak{T}}(\nu ,{\omega }^{\mathfrak{T}}({\beta }_1,ϵ))+b\lambda {\int }_{{\beta }_2\left(t_1\right)}^{{\beta }_2\left(t_2\right)}\left|f_1(t_2,\varsigma ,\nu \left(\varsigma \right))\right|d\varsigma \hfill \\ & +& b\lambda {\theta }_{f_1}\left(\delta \right){\beta }_2\left(\mathfrak{T}\right)+b{\theta }_{g_2}\left(\delta \right)(k_2+c_{2}r)+b{b}_2(k_2+c_{2}r){\omega }^{\mathfrak{T}}(\nu ,{\omega }^{\mathfrak{T}}({\beta }_3,ϵ))\hfill \\ & +& b(m_2^{*}+b_{2}r)·{\int }_{{\beta }_4\left(t_1\right)}^{{\beta }_4\left(t_2\right)}\left|f_2(t_2,\varsigma ,\nu \left(\varsigma \right))\right|d\varsigma +b(m_2^{*}+b_{2}r){\theta }_{f_2}\left(\delta \right){\beta }_4\left(\mathfrak{T}\right).\hfill \end{array} \end{gathered}$$

Now, let $t_1,t_2\in [0,\mathfrak{T}]$, $|t_2-t_1|<\delta$. Then, we deduce that

$$\begin{array}{ccc}\hfill {\omega }^{\mathfrak{T}}(FY,ϵ)& \le & {\theta }_f\left(\delta \right)+b{\theta }_{g_1}\left(\delta \right)+b{b}_1{\omega }^{\mathfrak{T}}(Y,{\omega }^{\mathfrak{T}}({\beta }_1,ϵ))+b\lambda {\int }_{{\beta }_2\left(t_1\right)}^{{\beta }_2\left(t_2\right)}\left|f_1(t_2,\varsigma ,\nu \left(\varsigma \right))\right|d\varsigma \hfill \\ & +& b\lambda {\theta }_{f_1}\left(\delta \right){\beta }_2\left(\mathfrak{T}\right)+b{\theta }_{g_2}\left(\delta \right)(k_2+c_{2}r)+b{b}_2(k_2+c_{2}r){\omega }^{\mathfrak{T}}(Y,{\omega }^{\mathfrak{T}}({\beta }_3,ϵ))\hfill \\ & +& b(m_2^{*}+b_{2}r)·{\int }_{{\beta }_4\left(t_1\right)}^{{\beta }_4\left(t_2\right)}\left|f_2(t_2,\varsigma ,\nu \left(\varsigma \right))\right|d\varsigma +b(m_2^{*}+b_{2}r){\theta }_{f_2}\left(\delta \right){\beta }_4\left(\mathfrak{T}\right).\hfill \end{array}$$

From our assumptions, $f,g_i$, $i=1,2$ are uniform continuity, and ${\omega }^{\mathfrak{T}}({\beta }_1,ϵ)\to 0$, ${\omega }^{\mathfrak{T}}({\beta }_3,ϵ)\to 0$ as $ϵ\to 0$, we have achieved that ${\theta }_f\left(\delta \right)\to 0$, ${\theta }_{g_i}\left(\delta \right)\to 0$, ${\theta }_{f_i}\left(\delta \right)\to 0$, $i=1,2$ as $\delta \to 0$ independent of $\nu \in Q_r$. Consequently, we obtain

$$\begin{array}{ccc}\hfill {\omega }_0^{\mathfrak{T}}\left(FY\right)& \le & \left(b{b}_1+b{b}_2(k_2+c_{2}r)\right){\omega }_0^{\mathfrak{T}}\left(Y\right)\hfill \end{array}$$

and as $\mathfrak{T}\to \infty$

$$\begin{array}{ccc}\hfill {\omega }_0\left(FY\right)& \le & \left(b{b}_1+b{b}_2(k_2+c_{2}r)\right){\omega }_0\left(Y\right).\hfill \end{array}$$

(8)

Now, from (7) and (8),

$$\begin{array}{ccc}\hfill \mu \left(FY\right)& =& \left(b{b}_1+b{b}_2(k_2+c_{2}r)\right)\mu \left(Y\right).\hfill \end{array}$$

Since $(b{b}_1+b{b}_2(k_2+c_{2}r))<1$, F is a contraction regarding MNC $\left(\mu \right)$, which implies that $\nu \in Q_r$ is a solution of (3). □

Now, we discuss the asymptotic stability of the solution of (3).

Theorem 3. 

The solution $\nu \in BC\left(R_{+}\right)$ of Equation (3) is asymptotically stable; that is, for any ${ϵ}^{*}>0$, there exist $T\left({ϵ}^{*}\right)>0$ and $r>0$, satisfying for any two solutions $\nu ,\overline{\nu }\in Q_r$, and then $|\nu \left(t\right)-\overline{\nu }\left(t\right)|\le {ϵ}^{*}$ for $t\ge T\left({ϵ}^{*}\right)$.

Proof. 

For two solutions $\nu ,\overline{\nu }\in Q_r$ of Equation (3), therefore,

$$\begin{array}{l}|\nu \left(t\right)-\overline{\nu }\left(t\right)|=\\ |f\left(t,g_1(t,\nu \left({\beta }_1\left(t\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t\right)}{f}_1(t,\varsigma ,\nu \left(\varsigma \right))d\varsigma ,g_2(t,\nu \left({\beta }_3\left(t\right)\right))·{\int }_0^{{\beta }_4\left(t\right)}{f}_2(t,\varsigma ,\nu \left(\varsigma \right))d\varsigma \right)\\ -f\left(t,g_1(t,\overline{\nu }\left({\beta }_1\left(t\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t\right)}{f}_1(t,\varsigma ,\overline{\nu }\left(\varsigma \right))d\varsigma ,g_2(t,\overline{\nu }\left({\beta }_3\left(t\right)\right))·{\int }_0^{{\beta }_4\left(t\right)}{f}_2(t,\varsigma ,\overline{\nu }\left(\varsigma \right))d\varsigma \right)|\\ \le b{b}_1|\nu \left({\beta }_1\left(t\right)\right)-\overline{\nu }\left({\beta }_1\left(t\right)\right)|+2b\lambda \left({\int }_0^{{\beta }_2\left(t\right)}{k}_1(t,\varsigma )d\varsigma +r{\int }_0^{{\beta }_2\left(t\right)}{c}_1(t,\varsigma )d\varsigma \right)\\ +b{b}_2|\nu \left({\beta }_3\left(t\right)\right)-\overline{\nu }\left({\beta }_3\left(t\right)\right)|·(k_2+c_{2}r)+2b(m_2^{*}+b_{2}r)·\left({\int }_0^{{\beta }_4\left(t\right)}{k}_2(t,\varsigma )d\varsigma +r{\int }_0^{{\beta }_4\left(t\right)}{c}_2(t,\varsigma )d\varsigma \right),\end{array}$$

and then

$$\begin{array}{c}\parallel \nu -\overline{\nu }\parallel \\ \le & b{b}_1\parallel \nu -\overline{\nu }\parallel +2b\lambda ({ϵ}_1+r{ϵ}_2)+b{b}_2\parallel \nu -\overline{\nu }\parallel ·(k_2+c_{2}r)+2b(m_2^{*}+b_{2}r)·({ϵ}_3+r{ϵ}_4)\hfill \\ \le & \frac{2b\lambda ({ϵ}_1+r{ϵ}_2)+2b(m_2^{*}+b_{2}r).({ϵ}_3+r{ϵ}_4)}{(1-(b{b}_1+b{b}_2(k_2+c_{2}r)))}\le {ϵ}^{*},\hfill \end{array}$$

that is,

$$\begin{array}{ccc}\hfill |\nu \left(t\right)-\overline{\nu }\left(t\right)|& \le & \parallel \nu -\overline{\nu }\parallel \le {ϵ}^{*}.\hfill \end{array}$$

Therefore $\nu \in BC\left(R_{+}\right)$ is asymptotically stable. □

3. Main Theorems

Here, we examine the existence of $\vartheta \in BC\left(R_{+}\right)$ as a solution for the functional Equation (2). Take into account the following assumptions:

(vi)

$h:R_{+}\times R\to R$ and $g:R_{+}\times R\to R\setminus \left\{0\right\}$ satisfy the following;

they are continuous in $t\in R_{+}$, $\forall \vartheta \in R$ satisfy Lipschitz condition,

$$\begin{array}{ccc}\hfill & & |h(t,\vartheta )-h(t,{\vartheta }_1)|\le l_1|\vartheta -{\vartheta }_1\left|and\right|g(t,\vartheta )-g(t,{\vartheta }_1)|\le l_2|\vartheta -{\vartheta }_1|,\hfill \\ \\ \hfill & & \forall (t,\vartheta ),(t,{\vartheta }_1)\in R_{+}\times R,l_1,l_2>0.\hfill \end{array}$$

(9)

(vii)

$l_1+M{l}_2<1$.

Equation (9) implies

$$\left|h(t,\vartheta )\right|-\left|h(t,0)\right|\le |h(t,\vartheta )-h(t,0)|\le l_1\left|\vartheta \right|,$$

$$\left|h(t,\vartheta )\right|\le \left|h(t,0)\right|+l_1\left|\vartheta \right|$$

and

$$\left|h(t,\vartheta )\right|\le |a_1\left(t\right)|+l_1\left|\vartheta \right|,$$

where

$$a_1\left(t\right)=\left|h(t,0)\right|,a_1\in BC\left(R_{+}\right)and{a}_1^{*}=\underset{t\in R_{+}}{sup}\left|a_1\left(t\right)\right|<\infty .$$

Analogously, we can obtain

$$\left|g(t,\vartheta )\right|\le |a_2\left(t\right)|+l_2\left|\vartheta \right|,$$

where

$$a_2\left(t\right)=\left|g(t,0)\right|,a_2\in BC\left(R_{+}\right)and{a}_2^{*}=\underset{t\in R_{+}}{sup}\left|a_2\left(t\right)\right|<\infty .$$

Theorem 4. 

Suppose that Theorem 2 is verified. Let $\left(vi\right)$ and $\left(vii\right)$ occur and $\nu \in BC\left(R_{+}\right)$ is a solution of (3), and then (2) has a solution $\vartheta \in BC\left(R_{+}\right).$

Proof. 

Assume that $\nu \in BC\left(R_{+}\right)$, the solution of (3), exists and the set $B_{\rho }$ in form

$$\begin{array}{c}\hfill B_{\rho }=\{\vartheta \in BC\left(R_{+}\right):\parallel \vartheta \parallel \le \rho \},\rho =\frac{a_1^{*}+M{a}_2^{*}}{1-(l_1+M{l}_2)}.\end{array}$$

Define A by

$$\begin{array}{c}\hfill A\vartheta \left(t\right)=h(t,\vartheta (t\left)\right)+\nu \left(t\right)g(t,\vartheta (t\left)\right),t\ge 0.\end{array}$$

Let $\vartheta \in BC\left(R_{+}\right)$ and $M=\underset{t\in R_{+}}{sup}\left|\nu \left(t\right)\right|=r$, and then

$$\begin{array}{ccc}\hfill \left|A\vartheta \right(t\left)\right|& =& \left|h\right(t,\vartheta \left(t\right))+\nu (t\left)g\right(t,\vartheta \left(t\right)\left)\right|\hfill \\ & \le & |a_1\left(t\right)|+l_1\left|\vartheta \left(t\right)\right|+\left|\nu \left(t\right)\right|\left(\right|a_2\left(t\right)|+l_2\left|\vartheta \left(t\right)\right|)\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \parallel A\vartheta \parallel & \le & a_1^{*}+l_1\parallel \vartheta \parallel +M(a_2^{*}+l_2\parallel \vartheta \parallel )\hfill \\ & \le & a_1^{*}+l_1\rho +M(a_2^{*}+l_2\rho )=\rho .\hfill \end{array}$$

Then, the operator A maps $B_{\rho }$ into itself.

Now, let $\rho >0$ be given and take ${\vartheta }_1,{\vartheta }_2\in B_{\rho }$, such that $\parallel {\vartheta }_2-{\vartheta }_1\parallel \le \delta$, and then, for every $\nu \in BC\left(R_{+}\right)$, we have

$$\begin{array}{ccc}\hfill |A{\vartheta }_2\left(t\right)-A{\vartheta }_1\left(t\right)|& =& |h(t,{\vartheta }_2\left(t\right))+\nu \left(t\right)g(t,{\vartheta }_2\left(t\right))-h(t,{\vartheta }_1\left(t\right))-\nu \left(t\right)g(t,{\vartheta }_1\left(t\right))|\hfill \\ & \le & |h(t,{\vartheta }_2\left(t\right))-h(t,{\vartheta }_1\left(t\right))|+|\nu \left(t\right)\left|\right|g(t,{\vartheta }_2\left(t\right))-g(t,{\vartheta }_1\left(t\right))|.\hfill \end{array}$$

(10)

$\left(i^{★}\right)$ Choose $\mathfrak{T}>0$, where $t\ge \mathfrak{T}$, and then

$$\begin{array}{ccc}\hfill |A{\vartheta }_2\left(t\right)-A{\vartheta }_1\left(t\right)|& \le & l_1\parallel {\vartheta }_2-{\vartheta }_1\parallel +M{l}_2\parallel {\vartheta }_2-{\vartheta }_1\parallel \hfill \\ & \le & (l_1+M{l}_2)\parallel {\vartheta }_2-{\vartheta }_1\parallel \hfill \\ & \le & (l_1+M{l}_2)\delta =ϵ.\hfill \end{array}$$

$\left(i{i}^{★}\right)$ Also, for $\mathfrak{T}>0$, $t\in [0,\mathfrak{T}]$, from (10), we receive

$$\begin{array}{ccc}\hfill |A{\vartheta }_2\left(t\right)-A{\vartheta }_1\left(t\right)|& \le & l_1\parallel {\vartheta }_2-{\vartheta }_1\parallel +M{l}_2\parallel {\vartheta }_2-{\vartheta }_1\parallel \hfill \\ & \le & (l_1+M{l}_2)\delta =ϵ.\hfill \end{array}$$

Then, the mapping A is continuous.

Now, take nonempty $X\subset B_{\rho }$. Then, for any ${\vartheta }_1,{\vartheta }_2\in X$ and $\mathfrak{T}>0$, $t\ge \mathfrak{T}$, (10) yields

$$\begin{array}{ccc}\hfill |A{\vartheta }_2\left(t\right)-A{\vartheta }_1\left(t\right)|& \le & l_1|{\vartheta }_2\left(t\right)-{\vartheta }_1\left(t\right)|+M{l}_2|{\vartheta }_2\left(t\right)-{\vartheta }_1\left(t\right)|\hfill \\ \hfill & \le & l_{1}sup\left\{\right|{\vartheta }_2\left(t\right)-{\vartheta }_1\left(t\right)|,{\vartheta }_1,{\vartheta }_2\in X\}+M{l}_{2}sup\left\{\right|{\vartheta }_2\left(t\right)-{\vartheta }_1\left(t\right)|,{\vartheta }_1,{\vartheta }_2\in X\}\hfill \\ \hfill & \le & l_{1}diamX\left(t\right)+M{l}_{2}diamX\left(t\right),\hfill \end{array}$$

and then

$$\begin{array}{ccc}\hfill diamAX\left(t\right)& \le & (l_1+M{l}_2)diamX\left(t\right)\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \underset{n\to \infty }{lim}supdiamAX\left(t\right)& \le & (l_1+M{l}_2)\underset{n\to \infty }{lim}supdiamX\left(t\right).\hfill \end{array}$$

(11)

Arbitrarily take $\vartheta \in X$, $t\in [0,\mathfrak{T}]$, where $|t_2-t_1|<\delta$, $t_1\le t_2$, and define ${\theta }_h\left(\delta \right),{\theta }_g\left(\delta \right)$ as in [9,13].

Then, we have

$$\begin{array}{ccc}\hfill |A\vartheta \left(t_2\right)-A\vartheta \left(t_1\right)|& =& |h(t_2,\vartheta \left(t_2\right))+\nu \left(t_2\right)g(t_2,\vartheta \left(t_2\right))-h(t_1,\vartheta \left(t_1\right))-\nu \left(t_1\right)g(t_1,\vartheta \left(t_1\right))|\hfill \\ \hfill & \le & {\theta }_h\left(\delta \right)+l_1|\vartheta \left(t_2\right)-\vartheta \left(t_1\right)|+|\nu \left(t_2\right)-\nu \left(t_1\right)|(a_2^{*}+l_2\rho )+M{\theta }_g\left(\delta \right)\hfill \\ & +& M{l}_2|\vartheta \left(t_2\right)-\vartheta \left(t_1\right)|,\hfill \end{array}$$

and then

$$\begin{array}{ccc}\hfill |A\vartheta \left(t_2\right)-A\vartheta \left(t_1\right)|& \le & {\theta }_h\left(\delta \right)+l_{1}sup\left\{\right|\vartheta \left(t_2\right)-\vartheta \left(t_1\right)\left|\right\}+(a_2^{*}+l_2\rho )ϵ\hfill \\ & +& M{\theta }_g\left(\delta \right)+M{l}_{2}sup\left\{\right|\vartheta \left(t_2\right)-\vartheta \left(t_1\right)\left|\right\}.\hfill \end{array}$$

Now, we deduce that

$$\begin{array}{ccc}\hfill {\omega }^{\mathfrak{T}}(AX,ϵ)& \le & (l_1+M{l}_2){\omega }^{\mathfrak{T}}(X,ϵ)+{\theta }_h\left(\delta \right)+M{\theta }_g\left(\delta \right)+(a_2^{*}+l_2\rho )ϵ\hfill \\ \hfill {\omega }_0^{\mathfrak{T}}\left(AX\right)& \le & (l_1+M{l}_2){\omega }_0^{\mathfrak{T}}\left(X\right)\hfill \end{array}$$

and as $\mathfrak{T}\to \infty$

$$\begin{array}{ccc}\hfill {\omega }_0\left(AX\right)& \le & (l_1+M{l}_2){\omega }_0\left(X\right).\hfill \end{array}$$

(12)

Now, from (11) and (12), we obtain

$$\begin{array}{ccc}\hfill \mu \left(AX\right)& =& (l_1+M{l}_2)\mu \left(X\right).\hfill \end{array}$$

Since $l_1+M{l}_2<1$, A is a contraction operator regarding MNC; thus, Equation (2) has a solution $\vartheta \in B_{\rho }$. □

Consequently, we deduce the next result.

Corollary 1. 

For each solution $\nu \in BC\left(R_{+}\right)$ of (3), consequently, the uniqueness of solution of (2) holds.

Proof. 

Take $\nu \in BC\left(R_{+}\right)$ as a solution of (3) and ${\vartheta }_1,{\vartheta }_2$ are two solutions of (2), and then

$$\begin{array}{ccc}\hfill |{\vartheta }_2\left(t\right)-{\vartheta }_1\left(t\right)|& \le & |h(t,{\vartheta }_2\left(t\right))+\nu \left(t\right)g(t,{\vartheta }_2\left(t\right))-h(t,{\vartheta }_1\left(t\right))-\nu \left(t\right)g(t,{\vartheta }_1\left(t\right))|\hfill \\ & \le & |h(t,{\vartheta }_2\left(t\right))-h(t,{\vartheta }_1\left(t\right))|+|\nu \left(t\right)\left|\right|g(t,{\vartheta }_2\left(t\right))-g(t,{\vartheta }_1\left(t\right))|,\hfill \end{array}$$

and then

$$\begin{array}{ccc}\hfill \parallel {\vartheta }_2-{\vartheta }_1\parallel & \le & l_1\parallel {\vartheta }_2-{\vartheta }_1\parallel +M{l}_2\parallel {\vartheta }_2-{\vartheta }_1\parallel \hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \parallel {\vartheta }_2-{\vartheta }_1\parallel (1-(l_1+M{l}_2))& \le & 0.\hfill \end{array}$$

Then, ${\vartheta }_2={\vartheta }_1$ and the solution of (2) are unique. □

Theorem 5. 

The solution $\vartheta \in BC\left(R_{+}\right)$ of (2) is asymptotically stable; that is, for any $ϵ>0$, there exist $T\left(ϵ\right)>0$ and $\rho >0.$ Moreover, for $\vartheta ,\overline{\vartheta }\in B_{\rho }$, there are any two solutions, and then $|\vartheta \left(t\right)-\overline{\vartheta }\left(t\right)|\le ϵ$ for $t\ge T\left(ϵ\right)$.

Proof. 

Take $\vartheta ,\overline{\vartheta }\in B_{\rho }$, any two solutions of (2), and thus

$$\begin{array}{ccc}\hfill |\vartheta \left(t\right)-\overline{\vartheta }\left(t\right)|& =& |h(t,\vartheta \left(t\right))+\nu \left(t\right)g(t,\vartheta \left(t\right))-h(t,\overline{\vartheta }\left(t\right))-\overline{\nu }\left(t\right)g(t,\overline{\vartheta }\left(t\right))|\hfill \\ & \le & |h(t,\vartheta \left(t\right))-h(t,\overline{\vartheta }\left(t\right))|+|\nu \left(t\right)-\overline{\nu }\left(t\right)\left|\right|g(t,\vartheta \left(t\right))|\hfill \\ & +& |\overline{\nu }\left(t\right)\left|\right|g(t,\vartheta \left(t\right))-g(t,\overline{\vartheta }\left(t\right))|\hfill \\ & \le & l_1|\vartheta \left(t\right)-\overline{\vartheta }\left(t\right)|+|\nu \left(t\right)-\overline{\nu }\left(t\right)|(a_2^{*}+l_2\rho )+M{l}_2|\vartheta \left(t\right)-\overline{\vartheta }\left(t\right)|.\hfill \end{array}$$

Theorem 3 implies that

$$\begin{array}{c}\hfill |\nu \left(t\right)-\overline{\nu }\left(t\right)|\le {ϵ}^{*},t\ge T\left({ϵ}^{*}\right),\end{array}$$

and then

$$\begin{array}{ccc}\hfill \parallel \vartheta -\overline{\vartheta }\parallel & \le & l_1\parallel \vartheta -\overline{\vartheta }\parallel +(a_2^{*}+l_2\rho ){ϵ}^{*}+M{l}_2\parallel \vartheta -\overline{\vartheta }\parallel .\hfill \end{array}$$

Hence,

$$\begin{array}{ccc}\hfill \parallel \vartheta -\overline{\vartheta }\parallel & \le & \frac{(a_2^{*}+l_2\rho ){ϵ}^{*}}{1-(l_1+M{l}_2)}=ϵ;\hfill \end{array}$$

that is,

$$\begin{array}{ccc}\hfill |\vartheta \left(t\right)-\overline{\vartheta }\left(t\right)|& \le & \parallel \vartheta -\overline{\vartheta }\parallel \le ϵ.\hfill \end{array}$$

Consequently, $\vartheta \in BC\left(R_{+}\right)$ is asymptotically stable of Equation (2). □

Asymptotic Dependency

Now, replace the assumption $\left(iv\right)$ by ${\left(iv\right)}^{*}$ as follows:

${\left(iv\right)}^{*}$

$f_i:R_{+}\times R_{+}\times R\to R$, $i=1,2$ are Carathéodory functions and satisfy Lipschitz condition,

$$\begin{array}{ccc}& & |f_i(t,\varsigma ,\nu )-f_i(t,\varsigma ,{\nu }_1)|\le {\zeta }_i\left(t\right)|\nu -{\nu }_1|,i=1,2,\hfill \\ & & \forall (t,\varsigma ,\nu ),(t,\varsigma ,{\nu }_1)\in R_{+}\times R\times R.\hfill \end{array}$$

where

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}{\int }_0^{{\beta }_j\left(t\right)}{\zeta }_i\left(t\right)dt=0,\underset{t\in R_{+}}{sup}{\int }_0^{{\beta }_j\left(t\right)}{\zeta }_i\left(t\right)dt={\zeta }_i^{*},i=1,2,j=2,4.\end{array}$$

Theorem 6. 

Suppose that Theorem 4 is verified and then the asymptotic dependency of the solution of (2) on the function $g_1$ occurred; that is,

$$\forall ϵ>0,\exists \delta \left(ϵ\right),where|g_1(t,\nu \left({\beta }_1\left(t\right)\right)-g_1^{*}(t,\nu \left({\beta }_1\left(t\right)\right)|<\delta ,t>T\left(ϵ\right)$$

$$andthen\parallel \vartheta -{\vartheta }^{*}\parallel <ϵ,$$

where ${\vartheta }^{*}$ is a solution of

$${\vartheta }^{*}\left(t\right)=h(t,{\vartheta }^{*}\left(t\right))+{\nu }^{*}\left(t\right)g(t,{\vartheta }^{*}\left(t\right)),t\ge 0$$

and ${\nu }^{*}$ is the solution of

${\nu }^{*}\left(t\right)=$

$$\begin{array}{c}\hfill f\left(t,g_1^{*}(t,{\nu }^{*}\left({\beta }_1\left(t\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t\right)}{f}_1(t,\varsigma ,{\nu }^{*}\left(\varsigma \right))d\varsigma ,g_2(t,{\nu }^{*}\left({\beta }_3\left(t\right)\right)).{\int }_0^{{\beta }_4\left(t\right)}{f}_2(t,\varsigma ,{\nu }^{*}\left(\varsigma \right))d\varsigma \right),t\ge 0.\end{array}$$

Proof. 

Assume that ${\vartheta }^{*}$ satisfies (2) blending with the function $g_1^{*}$, and then

$$\begin{array}{ccc}\hfill |\vartheta \left(t\right)-{\vartheta }^{*}\left(t\right)|& =& |h(t,\vartheta \left(t\right))+\nu \left(t\right)g(t,\vartheta \left(t\right))-h(t,{\vartheta }^{*}\left(t\right))-{\nu }^{*}\left(t\right)g(t,{\vartheta }^{*}\left(t\right))|\hfill \\ & \le & |h(t,\vartheta \left(t\right))-h(t,{\vartheta }^{*}\left(t\right))|+|\nu \left(t\right)-{\nu }^{*}\left(t\right)\left|\right|g(t,\vartheta \left(t\right))|\hfill \\ & +& |{\nu }^{*}\left(t\right)\left|\right|g(t,\vartheta \left(t\right))-g(t,{\vartheta }^{*}\left(t\right))|\hfill \\ & \le & l_1|\vartheta \left(t\right)-{\vartheta }^{*}\left(t\right)|+|\nu \left(t\right)-{\nu }^{*}\left(t\right)\left|\right|g(t,\vartheta \left(t\right))|+M{l}_2|\vartheta \left(t\right)-{\vartheta }^{*}\left(t\right)|,\hfill \end{array}$$

and then

$$\begin{array}{ccc}\hfill \parallel \vartheta -{\vartheta }^{*}\parallel & \le & l_1\parallel \vartheta -{\vartheta }^{*}\parallel +\parallel \nu -{\nu }^{*}\parallel (\parallel a_2\parallel +l_2\rho )+M{l}_2\parallel \vartheta -{\vartheta }^{*}\parallel \hfill \\ & \le & \frac{\parallel \nu -{\nu }^{*}\parallel (a_2^{*}+l_2\rho )}{1-(l_1+M{l}_2)}.\hfill \end{array}$$

But,

$$\begin{gathered} |\nu \left(t\right)-{\nu }^{*}\left(t\right)|= \\ \begin{array}{cc}& |f\left(t,g_1(t,\nu \left({\beta }_1\left(t\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t\right)}{f}_1(t,\varsigma ,\nu \left(\varsigma \right))d\varsigma ,g_2(t,\nu \left({\beta }_3\left(t\right)\right))·{\int }_0^{{\beta }_4\left(t\right)}{f}_2(t,\varsigma ,\nu \left(\varsigma \right))d\varsigma \right)\hfill \\ -& f\left(t,g_1^{*}(t,{\nu }^{*}\left({\beta }_1\left(t\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t\right)}{f}_1(t,\varsigma ,{\nu }^{*}\left(\varsigma \right))d\varsigma ,g_2(t,{\nu }^{*}\left({\beta }_3\left(t\right)\right))·{\int }_0^{{\beta }_4\left(t\right)}{f}_2(t,\varsigma ,{\nu }^{*}\left(\varsigma \right))d\varsigma \right)|\hfill \\ \le & b|g_1(t,\nu \left({\beta }_1\left(t\right)\right))-g_1^{*}(t,\nu \left({\beta }_1\left(t\right)\right))|+b{b}_1|\nu \left({\beta }_1\left(t\right)\right)-{\nu }^{*}\left({\beta }_1\left(t\right)\right)|\hfill \\ +& b\lambda {\int }_0^{{\beta }_2\left(t\right)}|f_1(t,\varsigma ,\nu \left(\varsigma \right))-f_1(t,\varsigma ,{\nu }^{*}\left(\varsigma \right))|d\varsigma +b{b}_2|\nu \left({\beta }_3\left(t\right)\right)-{\nu }^{*}\left({\beta }_3\left(t\right)\right)|·{\int }_0^{{\beta }_4\left(t\right)}\left|f_2(t,\varsigma ,\nu \left(\varsigma \right))\right|d\varsigma \hfill \\ +& b|g_2(t,{\nu }^{*}\left({\beta }_3\left(t\right)\right))|·{\int }_0^{{\beta }_4\left(t\right)}|f_2(t,\varsigma ,\nu \left(\varsigma \right))-f_2(t,\varsigma ,{\nu }^{*}\left(\varsigma \right))|d\varsigma ,\hfill \end{array} \end{gathered}$$

and then

$$\begin{array}{c}\parallel \nu -{\nu }^{*}\parallel \\ \le & b\delta +b{b}_1\parallel \nu -{\nu }^{*}\parallel +b{\zeta }_1^{*}\lambda \parallel \nu -{\nu }^{*}\parallel +b{b}_2\parallel \nu -{\nu }^{*}\parallel (k_2+c_{2}r)+b(m_2^{*}+b_{2}r)({\zeta }_2^{*}\parallel \nu -{\nu }^{*}\parallel )\hfill \\ \le & \frac{b\delta }{1-(b{b}_1+b{\zeta }_1^{*}\lambda +b{b}_2(k_2+c_{2}r)+b{\zeta }_2^{*}(m_2^{*}+b_{2}r))}={ϵ}_5.\hfill \end{array}$$

Then,

$$\begin{array}{ccc}\hfill \parallel \vartheta -{\vartheta }^{*}\parallel & \le & \frac{{ϵ}_5(a_2^{*}+l_2\rho )}{1-(l_1+M{l}_2)}=ϵ.\hfill \end{array}$$

By the same manner, we can prove the asymptotic dependency on the function $g_2$. □

Example 1. 

An example of $g_1$ can be

$$\begin{array}{ccc}\hfill g_1(\mathfrak{r},\mathfrak{u})& =& \gamma \mathfrak{r}{e}^{-\mathfrak{r}}+b_2\mathfrak{u},\hfill \end{array}$$

and then

$$\begin{array}{ccc}\hfill g_1^{*}(\mathfrak{r},\mathfrak{u})& =& {\gamma }^{*}\mathfrak{r}{e}^{-\mathfrak{r}}+b_2\mathfrak{u};\hfill \end{array}$$

the function $g_1$ satisfies Lipschitz condition

$$\begin{array}{c}\hfill |g_1(\mathfrak{r},\mathfrak{u})-g_1(\mathfrak{r},v)|\le b_2|\mathfrak{u}-v|\end{array}$$

and

$$\begin{array}{ccc}\hfill |g_1(\mathfrak{r},\mathfrak{u})-g_1^{*}(\mathfrak{r},\mathfrak{u})|& \le & |\gamma -{\gamma }^{*}|\mathfrak{r}{e}^{-\mathfrak{r}}\hfill \\ & \le & |\gamma -{\gamma }^{*}|\le \delta .\hfill \end{array}$$

Theorem 7. 

Presume that Theorem 4 is established; therefore, the asymptotic dependency of the solution of (2) on ${\beta }_2$ yields

$$\forall ϵ>0,\exists \delta \left(ϵ\right),where|{\beta }_2-{\beta }_2^{*}|<\delta ,t>T\left(ϵ\right)$$

$$,andthen\parallel \vartheta -{\vartheta }^{*}\parallel <ϵ$$

where ${\vartheta }^{*}$ satisfies

$${\vartheta }^{*}\left(t\right)=h(t,{\vartheta }^{*}\left(t\right))+{\nu }^{*}\left(t\right)g(t,{\vartheta }^{*}\left(t\right)),t\ge 0$$

and ${\nu }^{*}$ is the solution of

$$\begin{array}{ccc}& & {\nu }^{*}\left(t\right)=f(t,g_1(t,{\nu }^{*}\left({\beta }_1\left(t\right)\right)),\lambda {\int }_0^{{\beta }_2^{*}\left(t\right)}{f}_1(t,\varsigma ,{\nu }^{*}\left(\varsigma \right))d\varsigma ,\hfill \\ \\ & & g_2(t,{\nu }^{*}\left({\beta }_3\left(t\right)\right))·{\int }_0^{{\beta }_4\left(t\right)}{f}_2(t,\varsigma ,{\nu }^{*}\left(\varsigma \right))d\varsigma ),t\ge 0.\hfill \end{array}$$

Proof. 

Given $\delta >0$, where $|{\beta }_2-{\beta }_2^{*}|<\delta .$, take ${\vartheta }^{*}$ as satisfying (2) regarding the function ${\beta }_2^{*}$, so

$$\begin{array}{ccc}\hfill |\vartheta \left(t\right)-{\vartheta }^{*}\left(t\right)|& =& |h(t,\vartheta \left(t\right))+\nu \left(t\right)g(t,\vartheta \left(t\right))-h(t,{\vartheta }^{*}\left(t\right))-{\nu }^{*}\left(t\right)g(t,{\vartheta }^{*}\left(t\right))|\hfill \\ & \le & |h(t,\vartheta \left(t\right))-h(t,{\vartheta }^{*}\left(t\right))|+|\nu \left(t\right)-{\nu }^{*}\left(t\right)\left|\right|g(t,\vartheta \left(t\right))|\hfill \\ & +& |{\nu }^{*}\left(t\right)\left|\right|g(t,\vartheta \left(t\right))-g(t,{\vartheta }^{*}\left(t\right))|\hfill \\ & \le & l_1|\vartheta \left(t\right)-{\vartheta }^{*}\left(t\right)|+|\nu \left(t\right)-{\nu }^{*}\left(t\right)\left|\right|g(t,\vartheta \left(t\right))|+M{l}_2|\vartheta \left(t\right)-{\vartheta }^{*}\left(t\right)|,\hfill \end{array}$$

and then

$$\begin{array}{ccc}\hfill \parallel \vartheta -{\vartheta }^{*}\parallel & \le & l_1\parallel \vartheta -{\vartheta }^{*}\parallel +\parallel \nu -{\nu }^{*}\parallel (a_2^{*}+l_2\rho )+M{l}_2\parallel \vartheta -{\vartheta }^{*}\parallel \hfill \\ & \le & \frac{\parallel \nu -{\nu }^{*}\parallel (a_2^{*}+l_2\rho )}{1-(l_1+M{l}_2)}.\hfill \end{array}$$

But,

$$\begin{array}{ccc}\hfill |\nu \left(t\right)-{\nu }^{*}\left(t\right)|=& & \\ & & |f\left(t,g_1(t,\nu \left({\beta }_1\left(t\right)\right)),\lambda {\int }_0^{{\beta }_2\left(t\right)}{f}_1(t,\varsigma ,\nu \left(\varsigma \right))d\varsigma ,g_2(t,\nu \left({\beta }_3\left(t\right)\right)).{\int }_0^{{\beta }_4\left(t\right)}{f}_2(t,\varsigma ,\nu \left(\varsigma \right))d\varsigma \right)\hfill \end{array}$$

$$\begin{array}{ccc}& -& f\left(t,g_1(t,{\nu }^{*}\left({\beta }_1\left(t\right)\right)),\lambda {\int }_0^{{\beta }_2^{*}\left(t\right)}{f}_1(t,\varsigma ,{\nu }^{*}\left(\varsigma \right))d\varsigma ,g_2(t,{\nu }^{*}\left({\beta }_3\left(t\right)\right)).{\int }_0^{{\beta }_4\left(t\right)}{f}_2(t,\varsigma ,{\nu }^{*}\left(\varsigma \right))d\varsigma \right)|\hfill \\ & \le & b{b}_1|\nu \left({\beta }_1\left(t\right)\right)-{\nu }^{*}\left({\beta }_1\left(t\right)\right)|+b\lambda {\int }_{{\beta }_2^{*}\left(t\right)}^{{\beta }_2\left(t\right)}\left|f_1(t,\varsigma ,\nu \left(\varsigma \right))\right|d\varsigma \hfill \\ & & +b\lambda {\int }_0^{{\beta }_2^{*}\left(t\right)}|f_1(t,\varsigma ,\nu \left(\varsigma \right)-f_1(t,\varsigma ,{\nu }^{*}\left(\varsigma \right))|d\varsigma \hfill \\ & +& b{b}_2|\nu \left({\beta }_3\left(t\right)\right)-{\nu }^{*}\left({\beta }_3\left(t\right)\right)|·{\int }_0^{{\beta }_4\left(t\right)}\left|f_2(t,\varsigma ,\nu \left(\varsigma \right))\right|d\varsigma \hfill \\ & +& b(m_2^{*}+b_{2}r)·{\int }_0^{{\beta }_4\left(t\right)}|f_2(t,\varsigma ,\nu \left(\varsigma \right)-f_2(t,\varsigma ,{\nu }^{*}\left(\varsigma \right))|d\varsigma ,\hfill \end{array}$$

and then

$$\begin{array}{ccc}\hfill \parallel \nu -{\nu }^{*}\parallel & \le & b{b}_1\parallel \nu -{\nu }^{*}\parallel +b\lambda (k_1+c_{1}r)|{\beta }_2-{\beta }_2^{*}|+b\lambda {\zeta }_1^{*}\parallel \nu -{\nu }^{*}\parallel \hfill \\ & +& b{b}_2\parallel \nu -{\nu }^{*}\parallel (k_2+c_{2}r)+b(m_2^{*}+b_{2}r)({\zeta }_2^{*}\parallel \nu -{\nu }^{*}\parallel )\hfill \\ & \le & b{b}_1\parallel \nu -{\nu }^{*}\parallel +b\lambda (k_1+c_{1}r)\delta +b\lambda {\zeta }_1^{*}\parallel \nu -{\nu }^{*}\parallel +b{b}_2\parallel \nu -{\nu }^{*}\parallel (k_2+c_{2}r)\hfill \\ & +& b(m_2^{*}+b_{2}r)({\zeta }_2^{*}\parallel \nu -{\nu }^{*}\parallel )\hfill \\ & \le & \frac{b\lambda (k_1+c_{1}r)\delta }{1-(b{b}_1+b\lambda {\zeta }_1^{*}+b{b}_2(k_2+c_{2}r)+b{\zeta }_2^{*}(m_2^{*}+b_{2}r)}={ϵ}_6.\hfill \end{array}$$

Then,

$$\begin{array}{ccc}\hfill \parallel \vartheta -{\vartheta }^{*}\parallel & \le & \frac{{ϵ}_6(a_2^{*}+l_2\rho )}{1-(l_1+M{l}_2)}=ϵ.\hfill \end{array}$$

Analogously, the asymptotic dependency on the function ${\beta }_4$ can be proved. □

Example 2. 

An example of ${\beta }_2$ in the form

$$\begin{array}{ccc}\hfill {\beta }_2\left(\mathfrak{r}\right)& =& \mathfrak{r}-\gamma \mathfrak{r}{e}^{-\mathfrak{r}}\hfill \end{array}$$

and then

$$\begin{array}{ccc}\hfill {\beta }_2^{*}\left(\mathfrak{r}\right)& =& \mathfrak{r}-{\gamma }^{*}\mathfrak{r}{e}^{-\mathfrak{r}}\hfill \end{array}$$

It is clear that our assumption of ${\beta }_2$ satisfies Lipschitz condition and

$$\begin{array}{ccc}\hfill |{\beta }_2\left(\mathfrak{r}\right)-{\beta }_2^{*}\left(\mathfrak{r}\right)|& \le & |\gamma -{\gamma }^{*}|\mathfrak{r}{e}^{-\mathfrak{r}}\hfill \\ & \le & |\gamma -{\gamma }^{*}|\le \delta .\hfill \end{array}$$

Theorem 8. 

Let Theorem 4 be verified, and then the asymptotic dependency of the solution of (2) on the parameter λ occurred; that is,

$$\forall ϵ>0,\exists \delta \left(ϵ\right),where|\lambda -{\lambda }^{*}|<\delta \Rightarrow \parallel \vartheta -{\vartheta }^{*}\parallel <ϵ,t>T\left(ϵ\right)$$

and where ${\vartheta }^{*}$ is the solution of

$${\vartheta }^{*}\left(t\right)=h(t,{\vartheta }^{*}\left(t\right))+{\nu }^{*}\left(t\right)g(t,{\vartheta }^{*}\left(t\right)),t\ge 0$$

and ${\nu }^{*}$ is the solution of

$$\begin{array}{ccc}\hfill & & {\nu }^{*}\left(t\right)=\hfill \\ \hfill & & f\left(t,g_1(t,{\nu }^{*}\left({\beta }_1\left(t\right)\right)),{\lambda }^{*}{\int }_0^{{\beta }_2\left(t\right)}{f}_1(t,\varsigma ,{\nu }^{*}\left(\varsigma \right))d\varsigma ,g_2(t,{\nu }^{*}\left({\beta }_3\left(t\right)\right)).{\int }_0^{{\beta }_4\left(t\right)}{f}_2(t,\varsigma ,{\nu }^{*}\left(\varsigma \right))d\varsigma \right),t\ge 0.\hfill \end{array}$$

Proof. 

Let $\delta >0$ be given such that $|\lambda -{\lambda }^{*}|\le \delta$ and take ${\vartheta }^{*}$ as a solution of (2) regarding parameter ${\lambda }^{*}$; thus,

$$\begin{array}{ccc}\hfill \parallel \vartheta -{\vartheta }^{*}\parallel & \le & \frac{\parallel \nu -{\nu }^{*}\parallel (a_2^{*}+l_2\rho )}{1-(l_1+M{l}_2)},\hfill \end{array}$$

and, as in Theorem 6, we obtain

$$\begin{array}{ccc}\hfill \parallel \nu -{\nu }^{*}\parallel & \le & \frac{b\delta (k_1+c_{1}r)}{1-(b{b}_1+b{\lambda }^{*}{\zeta }_1^{*}+b{b}_2(k_2+c_{2}r)+b{\zeta }_2^{*}(m_2^{*}+b_{2}r))}={ϵ}_7\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \parallel \vartheta -{\vartheta }^{*}\parallel & \le & \frac{{ϵ}_7(a_2^{*}+l_2\rho )}{1-(l_1+M{l}_2)}=ϵ.\hfill \end{array}$$

□

4. Comments and Remarks

The problem (1) contains several key problems that appear in classical analysis.

Here, we provide a few special problems.

• If $f(\mathfrak{r},\xi ,u,w)=f(\mathfrak{r},\xi ,u)$ taking $g_2=0$, then we have

  $$\begin{array}{ccc}& & \frac{\vartheta \left(\mathfrak{r}\right)-h(\mathfrak{r},\vartheta (\mathfrak{r}\left)\right)}{g(\mathfrak{r},\vartheta (\mathfrak{r}\left)\right)}=\hfill \\ & & f\left(\mathfrak{r},g_1\left(\mathfrak{r},\frac{\vartheta \left({\beta }_1\left(\mathfrak{r}\right)\right)-h(\mathfrak{r},\vartheta \left({\beta }_1\left(\mathfrak{r}\right)\right))}{g(\mathfrak{r},\vartheta \left({\varphi }_1\left(\mathfrak{r}\right)\right))}\right),\lambda {\int }_0^{{\beta }_4\left(\mathfrak{r}\right)}{f}_2\left(\mathfrak{r},\varsigma ,\frac{\vartheta \left(\varsigma \right)-h(\varsigma ,\vartheta (\varsigma \left)\right)}{g(\varsigma ,\vartheta (\varsigma \left)\right)}\right)d\varsigma \right),\mathfrak{r}\ge 0.\hfill \end{array}$$
• If $f(\mathfrak{r},\xi ,u,w)=f(\mathfrak{r},\xi )$ taking $\lambda ,g_1(\mathfrak{r},\vartheta )=0$, then we have

  $$\begin{array}{ccc}& & \frac{\vartheta \left(\mathfrak{r}\right)-h(\mathfrak{r},\vartheta (\mathfrak{r}\left)\right)}{g(\mathfrak{r},\vartheta (\mathfrak{r}\left)\right)}=\hfill \\ & & f\left(\mathfrak{r},g_2\left(\mathfrak{r},\frac{\vartheta \left({\beta }_3\left(\mathfrak{r}\right)\right)-h(\mathfrak{r},\vartheta \left({\beta }_3\left(\mathfrak{r}\right)\right))}{g(\mathfrak{r},\vartheta \left({\beta }_3\left(\mathfrak{r}\right)\right))}\right)·{\int }_0^{{\beta }_4\left(\mathfrak{r}\right)}{f}_2\left(\mathfrak{r},\varsigma ,\frac{\vartheta \left(\varsigma \right)-h(\varsigma ,\vartheta (\varsigma \left)\right)}{g(\varsigma ,\vartheta (\varsigma \left)\right)}\right)d\varsigma \right),\mathfrak{r}\ge 0.\hfill \end{array}$$
• For $\lambda$ and $g_2(\mathfrak{r},\vartheta )=0$, then we have

  $$\frac{\vartheta \left(\mathfrak{r}\right)-h(\mathfrak{r},\vartheta (\mathfrak{r}\left)\right)}{g(\mathfrak{r},\vartheta (\mathfrak{r}\left)\right)}=f\left(\mathfrak{r},g_1\left(\mathfrak{r},\frac{\vartheta \left({\beta }_1\left(\mathfrak{r}\right)\right)-h(\mathfrak{r},\vartheta \left({\beta }_1\left(\mathfrak{r}\right)\right))}{g(\mathfrak{r},\vartheta \left({\beta }_1\left(\mathfrak{r}\right)\right))}\right)\right),\mathfrak{r}\ge 0.$$
• For $g_1(\mathfrak{r},\vartheta )=g_2(\mathfrak{r},\vartheta )=0$, then we have

  $$\frac{\vartheta \left(\mathfrak{r}\right)-h(\mathfrak{r},\vartheta (\mathfrak{r}\left)\right)}{g(\mathfrak{r},\vartheta (\mathfrak{r}\left)\right)}=f\left(\mathfrak{r},\lambda {\int }_0^{{\beta }_2\left(\mathfrak{r}\right)}{f}_1\left(\mathfrak{r},\varsigma ,\frac{\vartheta \left(\varsigma \right)-h(\varsigma ,\vartheta (\varsigma \left)\right)}{g(\varsigma ,\vartheta (\varsigma \left)\right)}\right)d\varsigma \right),\mathfrak{r}\ge 0.$$
• For $f(\mathfrak{r},\xi ,u,w)=1$, therefore

  $$\vartheta \left(\mathfrak{r}\right)=h(\mathfrak{r},\vartheta (\mathfrak{r}\left)\right)+g(\mathfrak{r},\vartheta (\mathfrak{r}\left)\right),\mathfrak{r}\ge 0$$

Moreover, let $h=0$, and then we have

$$\vartheta \left(\mathfrak{r}\right)=g(\mathfrak{r},\vartheta (\mathfrak{r}\left)\right),\mathfrak{r}\ge 0.$$

Example 3. 

Taking into account the next problem

$$\begin{array}{ccc}& & \frac{\vartheta \left(\mathfrak{r}\right)-[\frac{e^{-\mathfrak{r}}sin\mathfrak{r}}{20}+\frac{\sqrt{\vartheta \left(\mathfrak{r}\right)}}{2}]}{\frac{e^{-\mathfrak{r}}}{3}+\frac{\vartheta \left(\mathfrak{r}\right)}{2}}=\frac{cos\mathfrak{r}}{e^{3\mathfrak{r}}}+\frac{1}{3}(e^{-2\mathfrak{r}}+\frac{1}{2}\left(\frac{\vartheta \left(\gamma \mathfrak{r}\right)-[\frac{e^{-\mathfrak{r}}sin\mathfrak{r}}{20}+\frac{\sqrt{\vartheta \left(\gamma \mathfrak{r}\right)}}{2}]}{\frac{e^{-\mathfrak{r}}}{3}+\frac{\vartheta \left(\gamma \mathfrak{r}\right)}{2}}\right)\hfill \\ & +& \frac{1}{4}{\int }_0^{\alpha \mathfrak{r}}{e}^{-(\mathfrak{r}+s)}(1+\frac{1}{2}\left(\frac{\vartheta \left(s\right)-[\frac{e^{-s}sins}{20}+\frac{\sqrt{\vartheta \left(s\right)}}{2}]}{\frac{e^{-s}}{3}+\frac{\vartheta \left(s\right)}{2}}\right))ds\hfill \\ & +& (\frac{e^{-\mathfrak{r}}}{5}+\frac{1}{2}\left(\frac{\vartheta \left(\varpi \mathfrak{r}\right)-[\frac{e^{-\mathfrak{r}}sin\mathfrak{r}}{20}+\frac{\sqrt{\vartheta \left(\varpi \mathfrak{r}\right)}}{2}]}{\frac{e^{-\mathfrak{r}}}{3}+\frac{\vartheta \left(\varpi \mathfrak{r}\right)}{2}}\right))·{\int }_0^{\psi \mathfrak{r}}sins{e}^{-(\mathfrak{r}+s)}(1+\frac{1}{2}\left(\frac{\vartheta \left(s\right)-[\frac{e^{-s}sins}{20}+\frac{\sqrt{\vartheta \left(s\right)}}{2}]}{\frac{e^{-s}}{3}+\frac{\vartheta \left(s\right)}{2}}\right))ds,\mathfrak{r}\ge 0.\hfill \end{array}$$

Set

$$\begin{array}{ccc}\hfill h(\mathfrak{r},\vartheta )& =& \frac{e^{-\mathfrak{r}}sin\mathfrak{r}}{20}+\frac{\sqrt{\vartheta \left(\mathfrak{r}\right)}}{2},\hfill \\ \hfill g(\mathfrak{r},\vartheta )& =& \frac{e^{-\mathfrak{r}}}{3}+\frac{\vartheta \left(\mathfrak{r}\right)}{2},\hfill \\ \hfill f(\mathfrak{r},\mathfrak{u}(\mathfrak{r}),z(\mathfrak{r}),w(\mathfrak{t}\left)\right)& =& \frac{cos\mathfrak{r}}{e^{3\mathfrak{r}}}+\frac{1}{3}(\mathfrak{u}\left(\mathfrak{r}\right),z\left(\mathfrak{r}\right),w\left(\mathfrak{r}\right)),\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill \mathfrak{u}\left(\mathfrak{r}\right)& =& e^{-2\mathfrak{r}}+\frac{1}{2}(\frac{\vartheta \left(\gamma \mathfrak{r}\right)-[\frac{e^{-\mathfrak{r}}sin\mathfrak{r}}{20}+\frac{\sqrt{\vartheta \left(\gamma \mathfrak{r}\right)}}{2}]}{\frac{e^{-\mathfrak{r}}}{3}+\frac{\vartheta \left(\gamma \mathfrak{r}\right)}{2}},\gamma \left(\mathfrak{r}\right)<1,\hfill \\ \hfill z\left(\mathfrak{r}\right)& =& \frac{1}{4}{\int }_0^{\alpha \mathfrak{r}}{e}^{-(\mathfrak{r}+s)}(1+\frac{1}{2}\left(\frac{\vartheta \left(s\right)-[e^{-s}sins+\frac{\sqrt{\vartheta \left(s\right)}}{2}]}{e^{-s}+\frac{\vartheta \left(s\right)}{200}}\right))ds,\alpha \left(\mathfrak{r}\right)<1,\hfill \\ \hfill w\left(t\right)& =& (\frac{e^{-\mathfrak{r}}}{5}+\frac{1}{2}\left(\frac{\vartheta \left(\varpi t\right)-[\frac{e^{-\mathfrak{r}}sin\mathfrak{r}}{20}+\frac{\sqrt{\vartheta \left(\varpi \mathfrak{r}\right)}}{2}]}{\frac{e^{-\mathfrak{r}}}{3}+\frac{\vartheta \left(\varpi \right(\mathfrak{r})}{2}}\right))\hfill \\ & ·& {\int }_0^{\psi \mathfrak{r}}sins{e}^{-(\mathfrak{r}+s)}(1+\frac{1}{2}\left(\frac{\vartheta \left(s\right)-[\frac{e^{-s}sins}{20}+\frac{\sqrt{\vartheta \left(s\right)}}{2}]}{\frac{e^{-s}}{3}+\frac{\vartheta \left(s\right)}{2}}\right))ds,\varpi \left(\mathfrak{r}\right),\psi \left(\mathfrak{r}\right)<1.\hfill \end{array}$$

Putting

$$m^{*}=\frac{1}{e^3},m_1^{*}=\frac{1}{e^2},m_2^{*}=\frac{1}{5e},b=\frac{1}{3}{b}_1=b_2=\frac{1}{2},k_1=k_2=\frac{1}{e},\lambda =\frac{1}{4},c_1=c_2=\frac{1}{2e},$$

$$a_1^{*}=\frac{1}{20},a_2^{*}=\frac{1}{3},l_1=\frac{1}{2},l_2=\frac{1}{2},andM=\frac{1}{e}.$$

We can find that

$$l_1+M{l}_2=0.6839397206<1andb{b}_1+b{b}_2(k_2+c_{2}r)=0.9739931049<1.$$

5. Conclusions

The solvability of various problems in some spaces of bounded continuous functions defined on the half-axis has been discussed by many scholars by applying MNC [24,25], for example, [3,26,27], and for global asymptotic stability (see [24]).

In this investigation, the asymptotic stability and dependency of the solutions for an implicit delays hybrid quadratic functional integral equation have been established on $R_{+}$. Firstly, we proved the existence of the solutions $\nu \in BC\left(R_{+}\right)$ of (3), and then we studied the asymptotic stability of the solutions $\nu \in BC\left(R_{+}\right)$ of (3). Next, we investigated the existence and the stability of $\vartheta \in BC\left(R_{+}\right)$ on $R_{+}$, by applying MNC in $BC(R+)$, and then we studied some asymptotic dependency of $\vartheta \in BC\left(R_{+}\right)$ on the parameter $\lambda \ge 0$, and on the functions $g_1,g_2,{\beta }_2$ and ${\beta }_4$. Furthermore, we can discuss other asymptotic dependency results on the other parameters of (3).

Finally, we discussed the exceptional cases, and examples are provided to illustrate our results.