Qualitative Outcomes on Monotone Iterative Technique and Quasilinearization Method on Time Scale

Abstract

In this paper, a nonlinear dynamic equation with an initial value problem (IVP) on a time scale is considered. First, applying comparison results with a coupled lower solution (LS) and an upper solution (US), we improved the quasilinearization method (QLM) for the IVP. Unlike other studies, we consider the LS and US pair of the seventh type instead of the natural type. It was determined that the solutions of the dynamic equation converge uniformly and monotonically to the unique solution of the IVP, and the convergence is quadratic. Moreover, we will use the delta derivative $\left(\mathrm{\Delta }\gamma \right)$ instead of the classical derivative $\left(d\gamma \right)$ in the proof because it studies a time scale. In the second part of the paper, we applied the monotone iterative technique (MIT) coupled with the LS and US, which is an effective method, proving a clear analytical representation for the solution of the equation when the relevant functions are monotonically non-decreasing and non-increasing. Then an example is given to illustrate the results obtained.

1. Introduction

A time scale is a system that combines discrete and continuous analysis. Stefan Hilger was the first to study this subject in 1990 [1]. Although it is a relatively new subject, it has become a field that has attracted more and more attention due to dynamic equations that allow differential equations and difference equations to be considered simultaneously. Some books, articles, and other studies on this subject are given with their references.

Kaymakçalan and Lawrence in [2] discussed the monotone iterative technique with the classical method for the IVP in a unified setting.

Akın et al. [3] used the QLM approximately on a time scale for the unique solution of the BVP that divides from above and below with the monotone convergent sequences of the LS and US.

Bhaskar and McRae [4] proved a fundamental theorem regarding the existence of coupled maximal and minimal solutions of dynamical systems.

West and Vatsala [5] examined some theorems for the type I pair of the LS and US. They obtained natural monotone sequences starting from the coupled LS and US of type I of the equation and observed that the results varied based on the iterative steps used to develop the arrays.

In their book, Lakshmikantham and Vatsala [6] presented a systematic development of a generalized QLM, illustrating the concepts and technical challenges encountered in the combined approach. They significantly increased the usefulness of the QLM, which has proven to be very effective in various research fields and applications.

Denton and Vatsala [7] presented comparison results of a nonlinear Riemann–Liouville fractional differential equation. The authors improved a monotone method for finite systems of q-order fractional equations using the coupled LS and US.

Ramírez and Vatsala [8] investigated a generalized MIT using the pair of the LS and US of Caputo fractional differential equations. They developed results that yield natural monotone sequences or convolute monotone sequences that converge uniformly and monotonically to minimal and maximal solutions.

Khavanin [9] extended the mixed monotony method to generate monotone sequences that converge to the unique solution of the IVP with a delay difference equation.

Daneev and Sizykh [10] proposed a new approach to real-time multifunctional automatic control systems depending on the combined use of dynamic technologies.

Jyoti and Singh [11] demonstrated a combined iterative approach based on the QLM and the Krasnoselskii–Mann approximation to evaluate solutions of nonlinear Dirichlet BVPs. In addition to this, the authors reduced the nonlinear problems to a set of linear equations using the QLM with Green’s function.

Wu and Shu [12] introduced an improved framework based on some important insights from geometry that is called geometric QLM. They proposed the fundamental theory of the QLM through the geometric features of convex areas to construct sequences.

Heydari et al. [13] presented a combination of quasilinearization to approximate the solution of nonlinear functional Volterra integral equations. They solved the linear integral equation, which is obtained from each iteration using the Legendre ordering method.

Izadi and Roul [14] offered a Vieta–Fibonacci matrix technique to find the solution of nonlinear and multiple singularity third-order Emden–Fowler equations. The authors applied the QLM to the underlying model problem to obtain an efficacious approach.

Verma and Urus [15] improved an MIT with lower and upper solutions for four-point nonlinear BVPs. They established an iterative design using quasilinearization and under convenient conditions, which proved that the resulting sequences converge uniformly to a solution in a given region.

Idiz et al. [16] investigated numerical solutions of fractional Lane–Emden-type equations that arise in astrophysics applications. They offered a numerical approach with the QLM using Legendre wavelets.

Yakar and Arslan [17] created new definitions for a causal extreme value problem with Riemann–Liouville fractional derivatives and examined the unique solution by combining the QLM.

Consider the following dynamic non-linear IVP:

$$\begin{array}{ccc}\hfill u^{\mathrm{\Delta }}& =& f\left(\gamma ,u\right)+g\left(\gamma ,u\right)+h\left(\gamma ,u\right)\hfill \\ \hfill u\left(0\right)& =& u_0,\hfill \end{array}$$

(1)

in the particular case of $f\left(\gamma ,u\right),g\left(\gamma ,u\right),h\left(\gamma ,u\right)\in C_{rd}\left[J\times R,R\right]$, where $J=[0,T].$

2. Preliminaries

This part provides some basic concepts regarding the calculus on time scales and fundamental theorems needed in the next sections. We will introduce some useful tools for proofs.

Definition 1

([18]). Any nonempty closed subset of the real numbers is called a time scale. A set of real numbers, integers, and natural numbers, $[4,5]\cup [6,7]$, are examples of a time scale. In contrast, sets of rational and irrational numbers are not a time scale. In addition to this, $u^{\mathrm{\Delta }}={\displaystyle \frac{du}{d\gamma }}$ is the usual derivative if the time scale is taken as R.

Definition 2

([6]). A function ϰ is called LS of

$$\begin{array}{ccc}\hfill u^{\mathrm{\Delta }}& =& f\left(\gamma ,u\right)\hfill \\ \hfill u\left(0\right)& =& u_0\hfill \end{array}$$

(2)

if

$${\varkappa }^{\mathrm{\Delta }}\le f\left(\gamma ,\varkappa \right),\varkappa \left({\gamma }_0\right)\le u_0,$$

and similarly, ϱ is called US if

$${\varrho }^{\mathrm{\Delta }}\ge f\left(\gamma ,\varrho \right),\varrho \left({\gamma }_0\right)\ge u_0.$$

Definition 3

([6]). Let ϰ and ϱ be rd-continuously differentiable functions such that $\varkappa \le \varrho$ on T. Then ϰ and ϱ are called a couple of seventh-type LS and US of (1) if

$$\begin{array}{ccc}\hfill {\varkappa }^{\mathrm{\Delta }}& \le & f\left(\gamma ,\varrho \right)+g\left(\gamma ,\varrho \right)+h\left(\gamma ,\varrho \right)\hfill \\ \hfill \varkappa \left({\gamma }_0\right)& \le & u_0\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\varrho }^{\mathrm{\Delta }}& \ge & f\left(\gamma ,\varkappa \right)+g\left(\gamma ,\varkappa \right)+h\left(\gamma ,\varkappa \right)\hfill \\ \hfill \varrho \left({\gamma }_0\right)& \ge & u_0\hfill \end{array}$$

and called a couple of third-type LS and US if

$$\begin{array}{ccc}\hfill {\varkappa }^{\mathrm{\Delta }}& \le & f\left(\gamma ,\varkappa \right)+g\left(\gamma ,\varkappa \right)+h\left(\gamma ,\varrho \right)\hfill \\ \hfill \varkappa \left({\gamma }_0\right)& \le & u_0\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\varrho }^{\mathrm{\Delta }}& \ge & f\left(\gamma ,\varrho \right)+g\left(\gamma ,\varrho \right)+h\left(\gamma ,\varkappa \right)\hfill \\ \hfill \varrho \left({\gamma }_0\right)& \ge & u_0.\hfill \end{array}$$

Theorem 1

([6]). Let $\varkappa ,\varrho$ be the third-type LS and US of (1), respectively, and assume that

$$f\left(\gamma ,u_1\right)-f\left(\gamma ,u_2\right)\le L\left(u_1-u_2\right),$$

$u_1\ge u_2$ for some $L>0$ whenever $\varkappa \left({\gamma }_0\right)\le \varrho \left({\gamma }_0\right).$ Then this implies $\varkappa \left(\gamma \right)\le \varrho \left(\gamma \right).$

Theorem 2

([6]). Let $\varkappa ,\varrho$ be the LS and US of (1), respectively, such that $\varkappa \left(\gamma \right)\le \varrho \left(\gamma \right)$. Then there exists a solution $u\left(\gamma \right)$ of (1), satisfying $\varkappa \left(\gamma \right)\le u\left(\gamma \right)\le \varrho \left(\gamma \right)$ and $\varkappa \left({\gamma }_0\right)\le u\left({\gamma }_0\right)\le \varrho \left({\gamma }_0\right)$.

Theorem 3

([19] Arzela–Ascoli theorem). Let $f_n\left(\gamma \right)$ be a sequence of functions defined on a compact set J, which is equicontinuous and uniformly bounded on J. Then there exists a subsequence $\left\{f_n\right\},n=1,2,\dots$, which is uniformly convergent on J.

3. Results

3.1. The Quasilinearization Method for the Problem

Let

$${\Omega }_0=\left\{\left(\gamma ,u\right):{\varkappa }_0\left(\gamma \right)\le u\le {\varrho }_0\left(\gamma \right)\gamma \in T\right\}$$

where ${\varkappa }_0\left(\gamma \right),{\varrho }_0\left(\gamma \right)\in C_{rd}\left[J,R\right]$. Under some convenient conditions by using comparison theorems with the QLM, the monotone sequences that converge to the solution of (1) were obtained and showed the convergence rate.

Theorem 4.

Suppose that the following hypotheses hold:

(A₁) Let ${\varkappa }_0\left(\gamma \right),{\varrho }_0\left(\gamma \right)\in C_{rd}\left[J,R\right]$ be coupled with the seventh-type LS and US of (1) such that ${\varkappa }_0\left(\gamma \right)\le {\varrho }_0\left(\gamma \right)$ on J.

(A₂) Assume that $f_{uu}\left(\gamma ,u\right)\ge 0,g_{uu}\left(\gamma ,u\right)+{\varphi }_{uu}\left(\gamma ,u\right)\ge 0$ whenever ${\varphi }_{uu}$ exists and ${\varphi }_{uu}\left(\gamma ,u\right)>0$ and $h_{uu}\left(\gamma ,u\right)\le 0$ on ${\Omega }_0,$ where $f\left(\gamma ,u\right),g\left(\gamma ,u\right),h\left(\gamma ,u\right)\in C_{rd}^2\left[J\times R,R\right]$ and

(A₃) $f_u(\gamma ,u)\le 0,g_u(\gamma ,u)\le 0$ and $h_u(\gamma ,u)\le 0.$

Then there exist monotone sequences $\left\{{\varkappa }_n\left(\gamma \right)\right\}$ and $\left\{{\varrho }_n\left(\gamma \right)\right\},$ which converge to the unique solution of (1) uniformly and monotonically, and the convergence is quadratic.

Proof of Theorem 4.

In view of the condition (A₂), the Lipschitz conditions below are satisfied. For ${\varkappa }_0\left(\gamma \right)\le u_2\le u_1\le {\varrho }_0\left(\gamma \right),$

$$\begin{array}{ccc}\hfill -L\left(u_1-u_2\right)& \le & f\left(\gamma ,u_1\right)-f\left(\gamma ,u_2\right)\le L\left(u_1-u_2\right)\hfill \\ \hfill -L\left(u_1-u_2\right)& \le & g\left(\gamma ,u_1\right)-g\left(\gamma ,u_2\right)\le L\left(u_1-u_2\right)\hfill \\ \hfill -L\left(u_1-u_2\right)& \le & h\left(\gamma ,u_1\right)-h\left(\gamma ,u_2\right)\le L\left(u_1-u_2\right),L>0\hfill \end{array}$$

(3)

Since $f_{uu}\left(\gamma ,u\right)\ge 0,g_{uu}\left(\gamma ,u\right)+{\varphi }_{uu}\left(\gamma ,u\right)\ge 0$ and $h_{uu}\left(\gamma ,u\right)\le 0$, the following inequalities, which are very important for the proof, can be written, respectively. For $u\ge v$,

$$f\left(\gamma ,u\right)\ge f\left(\gamma ,v\right)+f_u\left(\gamma ,v\right)\left(u-v\right),$$

(4)

$$g\left(\gamma ,u\right)\ge g\left(\gamma ,v\right)+\left[g_u\left(\gamma ,v\right)+{\varphi }_u\left(\gamma ,v\right)\right]\left(u-v\right)-\left[\varphi \left(\gamma ,u\right)-\varphi \left(\gamma ,v\right)\right]$$

(5)

and

$$h\left(\gamma ,u\right)\ge h\left(\gamma ,v\right)+h_u\left(\gamma ,u\right)\left(u-v\right).$$

(6)

Consider auxiliary linear IVP as follows:

$$\begin{array}{ccc}\hfill u^{\mathrm{\Delta }}& =& w\left(\gamma ,{\varkappa }_0,{\varrho }_0;v\right)=f\left(\gamma ,{\varrho }_0\right)+f_u\left(\gamma ,{\varrho }_0\right)\left(v-{\varrho }_0\right)\hfill \\ & & +g\left(\gamma ,{\varrho }_0\right)+\left[g_u\left(\gamma ,{\varrho }_0\right)+{\varphi }_u\left(\gamma ,{\varrho }_0\right)\right]\left(v-{\varrho }_0\right)-\left[\varphi \left(\gamma ,v\right)-\varphi \left(\gamma ,{\varrho }_0\right)\right]\hfill \\ & & +h\left(\gamma ,{\varrho }_0\right)+h_u\left(\gamma ,{\varrho }_0\right)\left(v-{\varrho }_0\right),\hfill \\ \hfill u\left(0\right)& =& u_0,\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill v^{\mathrm{\Delta }}& =& W\left(\gamma ,{\varkappa }_0,{\varrho }_0;u\right)=f\left(\gamma ,{\varkappa }_0\right)+f_u\left(\gamma ,{\varkappa }_0\right)\left(u-{\varkappa }_0\right)\hfill \\ & & +g\left(\gamma ,{\varkappa }_0\right)+\left[g_u\left(\gamma ,{\varkappa }_0\right)+{\varphi }_u\left(\gamma ,{\varkappa }_0\right)\right]\left(u-{\varkappa }_0\right)-\left[\varphi \left(\gamma ,u\right)-\varphi \left(\gamma ,{\varkappa }_0\right)\right]\hfill \\ & & +h\left(\gamma ,{\varkappa }_0\right)+h_u\left(\gamma ,{\varkappa }_0\right)\left(u-{\varkappa }_0\right),\hfill \\ \hfill v\left(0\right)& =& u_0\hfill \end{array}$$

(8)

Due to the condition (A₁) in the hypothesis, it is clear that ${\varkappa }_0^{\mathrm{\Delta }}\le f\left(\gamma ,{\varrho }_0\right)+g\left(\gamma ,{\varrho }_0\right)+h\left(\gamma ,{\varrho }_0\right)=w\left(\gamma ,{\varkappa }_0,{\varrho }_0;{\varrho }_0\right).$ Namely, ${\varkappa }_0$ is an LS for (7). Similarly, it is known to be ${\varrho }_0^{\mathrm{\Delta }}\ge f\left(\gamma ,{\varkappa }_0\right)+g\left(\gamma ,{\varkappa }_0\right)+h\left(\gamma ,{\varkappa }_0\right)$, and using the inequalities given in (4)–(6), we can write

$$\begin{array}{ccc}\hfill {\varrho }_0^{\mathrm{\Delta }}& \ge & f\left(\gamma ,{\varrho }_0\right)+f_u\left(\gamma ,{\varrho }_0\right)\left({\varkappa }_0-{\varrho }_0\right)\hfill \\ & & +g\left(\gamma ,{\varrho }_0\right)+\left[g_u\left(\gamma ,{\varrho }_0\right)+{\varphi }_u\left(\gamma ,{\varrho }_0\right)\right]\left({\varkappa }_0-{\varrho }_0\right)-\left[\varphi \left(\gamma ,{\varkappa }_0\right)-\varphi \left(\gamma ,{\varrho }_0\right)\right]\hfill \\ & & +h\left(\gamma ,{\varrho }_0\right)+h_u\left(\gamma ,{\varkappa }_0\right)\left({\varkappa }_0-{\varrho }_0\right)\hfill \\ & \ge & f\left(\gamma ,{\varrho }_0\right)+f_u\left(\gamma ,{\varrho }_0\right)\left({\varkappa }_0-{\varrho }_0\right)\hfill \\ & & +g\left(\gamma ,{\varrho }_0\right)+\left[g_u\left(\gamma ,{\varrho }_0\right)+{\varphi }_u\left(\gamma ,{\varrho }_0\right)\right]\left({\varkappa }_0-{\varrho }_0\right)-\left[\varphi \left(\gamma ,{\varkappa }_0\right)-\varphi \left(\gamma ,{\varrho }_0\right)\right]\hfill \\ & & +h\left(\gamma ,{\varrho }_0\right)+h_u\left(\gamma ,{\varrho }_0\right)\left({\varkappa }_0-{\varrho }_0\right)\hfill \\ & =& w\left(\gamma ,{\varkappa }_0,{\varrho }_0;{\varkappa }_0\right)\hfill \end{array}$$

Hence, ${\varrho }_0$ is a US for (7). Then there exists a unique solution of (7) called ${\varkappa }_1\left(\gamma \right)$ such that ${\varkappa }_0\left(\gamma \right)\le {\varkappa }_1\left(\gamma \right)\le {\varrho }_0\left(\gamma \right)$ with the help of Theorem 2. Similarly, to guarantee the solution ${\varrho }_1\left(\gamma \right),$ we will show that ${\varkappa }_0$ and ${\varrho }_0$ are the LS and US of (8), respectively. Since ${\varkappa }_0$ and ${\varrho }_0$ are the seventh-type LS and US, we have ${\varkappa }_0^{\mathrm{\Delta }}\le f\left(\gamma ,{\varrho }_0\right)+g\left(\gamma ,{\varrho }_0\right)+h\left(\gamma ,{\varrho }_0\right).$ When the inequalities obtained from (4)–(6) is used, the following expression is obtained:

$$\begin{array}{ccc}\hfill {\varkappa }_0^{\mathrm{\Delta }}& \le & f\left(\gamma ,{\varkappa }_0\right)+f_u\left(\gamma ,{\varkappa }_0\right)\left({\varrho }_0-{\varkappa }_0\right)\hfill \\ & & +g\left(\gamma ,{\varkappa }_0\right)+\left[g_u\left(\gamma ,{\varrho }_0\right)+{\varphi }_u\left(\gamma ,{\varrho }_0\right)\right]\left({\varrho }_0-{\varkappa }_0\right)-\left[\varphi \left(\gamma ,{\varrho }_0\right)-\varphi \left(\gamma ,{\varkappa }_0\right)\right]\hfill \\ & & +h\left(\gamma ,{\varkappa }_0\right)+h_u\left(\gamma ,{\varkappa }_0\right)\left({\varrho }_0-{\varkappa }_0\right)\hfill \\ & =& f\left(\gamma ,{\varkappa }_0\right)+f_u\left(\gamma ,{\varrho }_0\right)\left({\varrho }_0-{\varkappa }_0\right)\hfill \\ & & +g\left(\gamma ,{\varkappa }_0\right)+\left[g_u\left(\gamma ,{\varrho }_0\right)+{\varphi }_u\left(\gamma ,{\varrho }_0\right)\right]\left({\varrho }_0-{\varkappa }_0\right)-\left[\varphi \left(\gamma ,{\varrho }_0\right)-\varphi \left(\gamma ,{\varkappa }_0\right)\right]\hfill \\ & & +h\left(\gamma ,{\varkappa }_0\right)+h_u\left(\gamma ,{\varkappa }_0\right)\left({\varrho }_0-{\varkappa }_0\right)\hfill \\ & =& W\left(\gamma ,{\varkappa }_0,{\varrho }_0;{\varrho }_0\right)\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\varrho }_0^{\mathrm{\Delta }}& \ge & f\left(\gamma ,{\varkappa }_0\right)+g\left(\gamma ,{\varkappa }_0\right)+h\left(\gamma ,{\varkappa }_0\right)\hfill \\ & =& W\left(\gamma ,{\varkappa }_0,{\varrho }_0;{\varkappa }_0\right).\hfill \end{array}$$

Thus, ${\varkappa }_0$ and ${\varrho }_0$ are the LS and US of (8); then there exists a unique solution of (8) called ${\varrho }_1$ such that ${\varkappa }_0\left(\gamma \right)\le {\varrho }_1\left(\gamma \right)\le {\varrho }_0\left(\gamma \right).$ Now we have to show that ${\varkappa }_1\le {\varrho }_1.$ By using (7),

$$\begin{array}{ccc}\hfill {\varkappa }_1^{\mathrm{\Delta }}& =& w\left(\gamma ,{\varkappa }_0,{\varrho }_0;{\varrho }_1\right)=f\left(\gamma ,{\varrho }_0\right)+f_u\left(\gamma ,{\varrho }_0\right)\left({\varrho }_1-{\varrho }_0\right)\hfill \\ & & +g\left(\gamma ,{\varrho }_0\right)+\left[g_u\left(\gamma ,{\varrho }_0\right)+{\varphi }_u\left(\gamma ,{\varrho }_0\right)\right]\left({\varrho }_1-{\varrho }_0\right)-\left[\varphi \left(\gamma ,{\varrho }_1\right)-\varphi \left(\gamma ,{\varrho }_0\right)\right]\hfill \\ & & +h\left(\gamma ,{\varrho }_0\right)+h_u\left(\gamma ,{\varrho }_0\right)\left({\varrho }_1-{\varrho }_0\right)\hfill \end{array}$$

can be written. If the inequalities obtained from (4)–(6) are substituted into the equation above, we obtain

$$\begin{array}{ccc}\hfill {\varkappa }_1^{\mathrm{\Delta }}& \le & f\left(\gamma ,{\varrho }_1\right)+f_u\left(\gamma ,{\varrho }_0\right)\left({\varrho }_0-{\varrho }_1\right)+f_u\left(\gamma ,{\varrho }_0\right)\left({\varrho }_1-{\varrho }_0\right)\hfill \\ & & +g\left(\gamma ,{\varrho }_1\right)+\left[g_u\left(\gamma ,{\varrho }_0\right)+{\varphi }_u\left(\gamma ,{\varrho }_0\right)\right]\left({\varrho }_0-{\varrho }_1\right)-\left[\varphi \left(\gamma ,{\varrho }_0\right)-\varphi \left(\gamma ,{\varrho }_1\right)\right]\hfill \\ & & +\left[g_u\left(\gamma ,{\varrho }_0\right)+{\varphi }_u\left(\gamma ,{\varrho }_0\right)\right]\left({\varrho }_1-{\varrho }_0\right)-\left[\varphi \left(\gamma ,{\varrho }_1\right)-\varphi \left(\gamma ,{\varrho }_0\right)\right]\hfill \\ & & +h\left(\gamma ,{\varrho }_1\right)+h_u\left(\gamma ,{\varrho }_1\right)\left({\varrho }_0-{\varrho }_1\right)+h_u\left(\gamma ,{\varrho }_0\right)\left({\varrho }_1-{\varrho }_0\right)\hfill \\ & \le & f\left(\gamma ,{\varrho }_1\right)+g\left(\gamma ,{\varrho }_1\right)+h\left(\gamma ,{\varrho }_1\right).\hfill \end{array}$$

Similarly, from (8),

$$\begin{array}{ccc}\hfill {\varrho }_1^{\mathrm{\Delta }}& =& W\left(\gamma ,{\varkappa }_0,{\varrho }_0;{\varkappa }_1\right)=f\left(\gamma ,{\varkappa }_0\right)+f_u\left(\gamma ,{\varkappa }_0\right)\left({\varkappa }_1-{\varkappa }_0\right)\hfill \\ & & +g\left(\gamma ,{\varkappa }_0\right)+\left[g_u\left(\gamma ,{\varkappa }_0\right)+{\varphi }_u\left(\gamma ,{\varkappa }_0\right)\right]\left({\varkappa }_1-{\varkappa }_0\right)-\left[\varphi \left(\gamma ,{\varkappa }_1\right)-\varphi \left(\gamma ,{\varkappa }_0\right)\right]\hfill \\ & & +h\left(\gamma ,{\varkappa }_0\right)+h_u\left(\gamma ,{\varkappa }_0\right)\left({\varkappa }_1-{\varkappa }_0\right).\hfill \end{array}$$

By using (4)–(6),

$$\begin{array}{ccc}\hfill {\varrho }_1^{\mathrm{\Delta }}& \ge & f\left(\gamma ,{\varkappa }_1\right)+f_u\left(\gamma ,{\varkappa }_1\right)\left({\varkappa }_0-{\varkappa }_1\right)+f_u\left(\gamma ,{\varkappa }_0\right)\left({\varkappa }_1-{\varkappa }_0\right)\hfill \\ & & +g\left(\gamma ,{\varkappa }_1\right)+\left[g_u\left(\gamma ,{\varkappa }_1\right)+{\varphi }_u\left(\gamma ,{\varkappa }_1\right)\right]\left({\varkappa }_0-{\varkappa }_1\right)-\left[\varphi \left(\gamma ,{\varkappa }_0\right)-\varphi \left(\gamma ,{\varkappa }_1\right)\right]\hfill \\ & & +\left[g_u\left(\gamma ,{\varkappa }_0\right)+{\varphi }_u\left(\gamma ,{\varkappa }_0\right)\right]\left({\varkappa }_1-{\varkappa }_0\right)-\left[\varphi \left(\gamma ,{\varkappa }_1\right)-\varphi \left(\gamma ,{\varkappa }_0\right)\right]\hfill \\ & & +h\left(\gamma ,{\varkappa }_1\right)+h_u\left(\gamma ,{\varkappa }_0\right)\left({\varkappa }_0-{\varkappa }_1\right)+h_u\left(\gamma ,{\varkappa }_0\right)\left({\varkappa }_1-{\varkappa }_0\right)\hfill \\ & =& f\left(\gamma ,{\varkappa }_1\right)+g\left(\gamma ,{\varkappa }_1\right)+h\left(\gamma ,{\varkappa }_1\right).\hfill \end{array}$$

Since ${\varkappa }_1,{\varrho }_1$ are the seventh-type LS and US with ${\varkappa }_1\left(0\right)\le$${\varrho }_1\left(0\right)$, then by Theorem 1, we obtain ${\varkappa }_1\left(\gamma \right)\le$${\varrho }_1\left(\gamma \right).$ Consequently, the inequality below can be written as

$${\varkappa }_0\left(\gamma \right)\le {\varkappa }_1\left(\gamma \right)\le {\varrho }_1\left(\gamma \right)\le {\varrho }_0\left(\gamma \right).$$

Contemplate the following initial value problems to obtain the next level:

$$\begin{array}{ccc}\hfill u^{\mathrm{\Delta }}& =& w\left(\gamma ,{\varkappa }_1,{\varrho }_1;v\right)=f\left(\gamma ,{\varrho }_1\right)+f_u\left(\gamma ,{\varrho }_1\right)\left(v-{\varrho }_1\right)\hfill \\ & & +g\left(\gamma ,{\varrho }_1\right)+\left[g_u\left(\gamma ,{\varrho }_1\right)+{\varphi }_u\left(\gamma ,{\varrho }_1\right)\right]\left(v-{\varrho }_1\right)-\left[\varphi \left(\gamma ,v\right)-\varphi \left(\gamma ,{\varrho }_1\right)\right]\hfill \\ & & +h\left(\gamma ,{\varrho }_1\right)+h_u\left(\gamma ,{\varrho }_1\right)\left(v-{\varrho }_1\right),\hfill \\ \hfill u\left(0\right)& =& u_0,\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill v^{\mathrm{\Delta }}& =& W\left(\gamma ,{\varkappa }_1,{\varrho }_1;u\right)=f\left(\gamma ,{\varkappa }_1\right)+f_u\left(\gamma ,{\varkappa }_1\right)\left(u-{\varkappa }_1\right)\hfill \\ & & +g\left(\gamma ,{\varkappa }_1\right)+\left[g_u\left(\gamma ,{\varkappa }_1\right)+{\varphi }_u\left(\gamma ,{\varkappa }_1\right)\right]\left(u-{\varkappa }_1\right)-\left[\varphi \left(\gamma ,u\right)-\varphi \left(\gamma ,{\varkappa }_1\right)\right]\hfill \\ & & +h\left(\gamma ,{\varkappa }_1\right)+h_u\left(\gamma ,{\varkappa }_1\right)\left(u-{\varkappa }_1\right),\hfill \\ \hfill v\left(0\right)& =& u_0.\hfill \end{array}$$

(10)

By (7), it is clear that

$${\varkappa }_1^{\mathrm{\Delta }}\le f\left(\gamma ,{\varrho }_1\right)+g\left(\gamma ,{\varrho }_1\right)+h\left(\gamma ,{\varrho }_1\right)=w\left(\gamma ,{\varkappa }_1,{\varrho }_1;{\varrho }_1\right),$$

and with (4)–(6), we have

$$\begin{array}{ccc}\hfill {\varrho }_1^{\mathrm{\Delta }}& \ge & f\left(\gamma ,{\varkappa }_1\right)+g\left(\gamma ,{\varkappa }_1\right)+h\left(\gamma ,{\varkappa }_1\right)\hfill \\ & \ge & f\left(\gamma ,{\varrho }_1\right)+g_u\left(\gamma ,{\varrho }_1\right)\left({\varkappa }_1-{\varrho }_1\right)\hfill \\ & & +g\left(\gamma ,{\varrho }_1\right)+\left[g_u\left(\gamma ,{\varrho }_1\right)+{\varphi }_u\left(\gamma ,{\varrho }_1\right)\right]\left({\varkappa }_1-{\varrho }_1\right)-\left[\varphi \left(\gamma ,{\varkappa }_1\right)-\varphi \left(\gamma ,{\varrho }_1\right)\right]\hfill \\ & & +h\left(\gamma ,{\varrho }_1\right)+h_u\left(\gamma ,{\varrho }_1\right)\left({\varkappa }_1-{\varrho }_1\right)\hfill \\ & =& w\left(\gamma ,{\varkappa }_1,{\varrho }_1;{\varkappa }_1\right).\hfill \end{array}$$

Therefore, ${\varkappa }_1$ and ${\varrho }_1$ are the LS and US that are the seventh type of (9). In a similar way, it can be shown that ${\varkappa }_1$ and ${\varrho }_1$ are the coupled seventh-type lower and upper solutions of (10). Based on Theorem 2, there exist the unique solutions ${\varkappa }_2$ and ${\varrho }_2$ of (9) and (10), respectively, such that ${\varkappa }_1\left(\gamma \right)\le {\varkappa }_2\left(\gamma \right)\le {\varrho }_1\left(\gamma \right)$ and ${\varkappa }_1\left(\gamma \right)\le {\varrho }_2\left(\gamma \right)\le {\varrho }_1\left(\gamma \right)$. Therefore, it can be obtained as follows:

$${\varkappa }_0\left(\gamma \right)\le {\varkappa }_1\left(\gamma \right)\le {\varkappa }_2\left(\gamma \right)\le {\varrho }_2\left(\gamma \right)\le {\varrho }_1\left(\gamma \right)\le {\varrho }_0\left(\gamma \right)$$

If this process continues like this, we can write

$${\varkappa }_0\left(\gamma \right)\le {\varkappa }_1\left(\gamma \right)\le {\varkappa }_2\left(\gamma \right)\le ···\le {\varkappa }_n\left(\gamma \right)\le {\varrho }_n\left(\gamma \right)\le ···\le {\varrho }_2\left(\gamma \right)\le {\varrho }_1\left(\gamma \right)\le {\varrho }_0\left(\gamma \right).$$

Each element of $\left\{{\varkappa }_n\left(\gamma \right)\right\}$ and $\left\{{\varrho }_n\left(\gamma \right)\right\}$ sequences is respectively a solution of the following system:

$$\begin{array}{ccc}\hfill {\varkappa }_{n+1}^{\mathrm{\Delta }}& =& w\left(\gamma ,{\varkappa }_n,{\varrho }_n;{\varrho }_{n+1}\right),\hfill \\ \hfill {\varkappa }_{n+1}\left(0\right)& =& u_0\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\varrho }_{n+1}^{\mathrm{\Delta }}& =& W\left(\gamma ,{\varkappa }_n,{\varrho }_n;{\varkappa }_{n+1}\right),\hfill \\ \hfill {\varrho }_{n+1}\left(0\right)& =& u_0.\hfill \end{array}$$

Because the solutions ${\varkappa }_0\left(\gamma \right)$ and ${\varrho }_0\left(\gamma \right)$ are bounded, the sequences $\left\{{\varkappa }_n\left(\gamma \right)\right\}$ and $\left\{{\varrho }_n\left(\gamma \right)\right\}$ are also uniformly bounded on T. Moreover, it can be easily shown that ${\varkappa }_0\left(\gamma \right)$ and ${\varrho }_0\left(\gamma \right)$ are equicontinuous. According to Theorem 3, the sequences converge to the unique solution of (1) uniformly. Now, it will be shown that this convergence is quadratic. For this purpose, define

$$\begin{array}{ccc}\hfill p_n\left(\gamma \right)& =& u\left(\gamma \right)-{\varkappa }_n\left(\gamma \right)\ge 0\hfill \\ \hfill p_n\left(0\right)& =& 0,\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill q_n\left(\gamma \right)& =& {\varrho }_n\left(\gamma \right)-u\left(\gamma \right)\ge 0\hfill \\ \hfill q_n\left(0\right)& =& 0.\hfill \end{array}$$

(12)

If we take the delta derivative of both sides of (11) defined above and take $g\left(\gamma ,u\right)+\varphi \left(\gamma ,u\right)=G\left(\gamma ,u\right)$ then we have

$$\begin{array}{ccc}\hfill p_n^{\mathrm{\Delta }}& =& [f\left(\gamma ,u\right)-f\left(\gamma ,{\varrho }_{n-1}\right)]-\left[f_u\left(\gamma ,{\varrho }_{n-1}\right)\right]\left({\varrho }_n-{\varrho }_{n-1}\right)\hfill \\ & & +[G\left(\gamma ,u\right)-G\left(\gamma ,{\varrho }_{n-1}\right)]-G_u\left(\gamma ,{\varrho }_{n-1}\right)\left({\varrho }_n-{\varrho }_{n-1}\right)\hfill \\ & & +\left[\varphi \left(\gamma ,{\varrho }_n\right)-\varphi \left(\gamma ,u\right)\right]-\left[g_u\left(\gamma ,{\varrho }_{n-1}\right)+{\varphi }_u\left(\gamma ,{\varrho }_{n-1}\right)\right]\left({\varrho }_n-{\varrho }_{n-1}\right)\hfill \\ & & +[h\left(\gamma ,u\right)-h\left(\gamma ,{\varrho }_{n-1}\right)]-h_u\left(\gamma ,{\varkappa }_{n-1}\right)\left({\varrho }_n-{\varrho }_{n-1}\right).\hfill \end{array}$$

If the definitions of ${\varkappa }_n$ and ${\varrho }_n$ are taken into account and applying the mean value theorem then the above expression can be written as

$$\begin{array}{ccc}\hfill p_n^{\mathrm{\Delta }}& =& -f_u\left(\gamma ,a\right)\left({\varrho }_{n-1}-u\right)-\left[f_u\left(\gamma ,{\varrho }_{n-1}\right)\right]\left({\varrho }_n-{\varrho }_{n-1}\right)\hfill \\ & & -G_u\left(\gamma ,b\right)\left({\varrho }_{n-1}-u\right)+{\varphi }_u\left(\gamma ,c\right)\left({\varrho }_n-u\right)\hfill \\ & & -G_u\left(\gamma ,{\varrho }_{n-1}\right)\left({\varrho }_n-{\varrho }_{n-1}\right)-h_u\left(\gamma ,d\right)\left({\varrho }_{n-1}-u\right)-h_u\left(\gamma ,{\varkappa }_{n-1}\right)\left({\varrho }_n-{\varrho }_{n-1}\right)\hfill \\ & \le & -f_{1u}(\gamma ,u)q_{n-1}-\left[f_{1u}\left(\gamma ,{\varrho }_{n-1}\right)\right]\left(q_n-q_{n-1}\right)\hfill \\ & & -G_u\left(\gamma ,u\right)q_{n-1}+{\varphi }_u\left(\gamma ,u\right)q_n\hfill \\ & & -G_u\left(\gamma ,{\varrho }_{n-1}\right)\left(q_n-q_{n-1}\right)-f_{3u}\left(\gamma ,{\varrho }_{n-1}\right)q_{n-1}-f_{3u}\left(\gamma ,{\varkappa }_{n-1}\right)\left(q_n-q_{n-1}\right)\hfill \end{array}$$

where $u<a<{\varrho }_{n-1},u<b<{\varrho }_{n-1},u<c<{\varrho }_n,$ $u<d<{\varrho }_{n-1}.$ Therefore,

$$\begin{array}{ccc}\hfill p_n^{\mathrm{\Delta }}& \le & q_{n-1}[f_u\left(\gamma ,{\varrho }_{n-1}\right)-f_u(\gamma ,u)]+q_{n-1}[G_u\left(\gamma ,{\varrho }_{n-1}\right)-G_u\left(\gamma ,u\right)]\hfill \\ & & +q_{n-1}[h_u\left(\gamma ,{\varkappa }_{n-1}\right)-h_u\left(\gamma ,{\varrho }_{n-1}\right)]\hfill \\ & & +q_n[-f_u\left(\gamma ,{\varrho }_{n-1}\right)+{\varphi }_u\left(\gamma ,u\right)-g_u\left(\gamma ,u\right)-{\varphi }_u\left(\gamma ,u\right)-h_u\left(\gamma ,{\varkappa }_{n-1}\right)\hfill \\ & \le & q_{n-1}^2{f}_{uu}\left(\gamma ,\delta \right)+q_{n-1}^2{G}_{uu}\left(\gamma ,\zeta \right)-\left(q_{n-1}\right)h_{uu}(\gamma ,\eta )\left({\varrho }_{n-1}-{\varkappa }_{n-1}\right)\hfill \\ & & +q_n[-f_u\left(\gamma ,{\varrho }_{n-1}\right)-g_u\left(\gamma ,u\right)-h_u\left(\gamma ,{\varkappa }_{n-1}\right)]\hfill \\ & =& q_{n-1}^2{f}_{uu}\left(\gamma ,\delta \right)+q_{n-1}^2{G}_{uu}\left(\gamma ,\zeta \right)-\left(q_{n-1}\right)h_{uu}(\gamma ,\eta )\left(q_{n-1}+p_{n-1}\right)\hfill \\ & & +q_n[-f_u\left(\gamma ,{\varrho }_{n-1}\right)-g_u\left(\gamma ,u\right)-h_u\left(\gamma ,{\varkappa }_{n-1}\right)]\hfill \\ & =& q_{n-1}^2[f_{uu}\left(\gamma ,\delta \right)+G_{uu}\left(\gamma ,\zeta \right)-h_{uu}(\gamma ,\eta )]\hfill \\ & & +q_n[-f_u\left(\gamma ,{\varrho }_{n-1}\right)-g_u\left(\gamma ,u\right)-h_u\left(\gamma ,{\varkappa }_{n-1}\right)]\hfill \\ & & -\left(p_{n-1}\right)\left(q_{n-1}\right)h_{uu}(\gamma ,\eta ),\hfill \end{array}$$

where $u<\delta ,\zeta <{\varrho }_{n-1},{\varkappa }_{n-1}<\eta <{\varrho }_n.$ Since continuous functions are bounded in a closed set, we can write

$$p_n^{\mathrm{\Delta }}\le q_{n-1}^{2}A+q_{n}B-\left(p_{n-1}\right)\left(q_{n-1}\right)C,$$

where A, B, and C are positive constants and

$$\begin{array}{ccc}\hfill \left|f_{uu}\left(\gamma ,\delta \right)+G_{uu}\left(\gamma ,\zeta \right)-h_{uu}(\gamma ,\eta )\right|& \le & A,\hfill \\ \hfill \left|f_u\left(\gamma ,{\varrho }_{n-1}\right)+g_u\left(\gamma ,u\right)+h_u\left(\gamma ,{\varkappa }_{n-1}\right)\right|& \le & B,\hfill \\ \hfill \left|h_{uu}(\gamma ,\eta )\right|& \le & C.\hfill \end{array}$$

When Cauchy inequality is applied to the term $\left(p_{n-1}\right)\left(q_{n-1}\right)C$, then

$$\begin{array}{ccc}\hfill p_n^{\mathrm{\Delta }}& \le & q_{n-1}^{2}A+q_{n}B-{\displaystyle \frac{p_{n-1}^2}{2}}C-{\displaystyle \frac{q_{n-1}^2}{2}}C\hfill \\ & =& -{\displaystyle \frac{p_{n-1}^2}{2}}C+q_{n-1}^2[A-{\displaystyle \frac{C}{2}}]+q_{n}B\hfill \end{array}$$

(13)

is obtained. Similarly, taking the delta derivative of both sides of (12),

$$\begin{array}{ccc}\hfill q_n^{\mathrm{\Delta }}& =& {\varrho }_n^{\mathrm{\Delta }}-u^{\mathrm{\Delta }}\hfill \\ & =& [f\left(\gamma ,{\varkappa }_{n-1}\right)-f\left(\gamma ,u\right)]+f_u\left(\gamma ,{\varrho }_{n-1}\right)\left({\varkappa }_n-{\varkappa }_{n-1}\right)\hfill \\ & & +[g\left(\gamma ,{\varkappa }_{n-1}\right)+\varphi \left(\gamma ,{\varkappa }_{n-1}\right)]-[g\left(\gamma ,u\right)+\varphi \left(\gamma ,u\right)]+[\varphi \left(\gamma ,u\right)-\varphi \left(\gamma ,{\varkappa }_n\right)]\hfill \\ & & +\left[g_u\left(\gamma ,{\varkappa }_{n-1}\right)+{\varphi }_u\left(\gamma ,{\varkappa }_{n-1}\right)\right]\left({\varkappa }_n-{\varkappa }_{n-1}\right)\hfill \\ & & +[h\left(\gamma ,{\varkappa }_{n-1}\right)-h\left(\gamma ,u\right)]+h_u\left(\gamma ,{\varkappa }_{n-1}\right)\left({\varkappa }_n-{\varkappa }_{n-1}\right)\hfill \end{array}$$

can be written. Similarly, by applying the above procedures, we obtain

$$\begin{array}{ccc}\hfill q_n^{\mathrm{\Delta }}& \le & p_{n-1}{f}_{uu}\left(\gamma ,\delta \right)\left({\varrho }_{n-1}-{\varkappa }_{n-1}\right)\hfill \\ & & +p_n[-f_u\left(\gamma ,{\varrho }_{n-1}\right)-G_u\left(\gamma ,{\varkappa }_{n-1}\right)+{\varphi }_u\left(\gamma ,u\right)-h_u\left(\gamma ,{\varkappa }_{n-1}\right)]\hfill \\ & & -p_{n-1}{h}_{uu}(\gamma ,\eta )\left(u-{\varkappa }_{n-1}\right)\hfill \\ & =& p_{n-1}^2[f_{uu}\left(\gamma ,\delta \right)-h_{uu}(\gamma ,\eta )]+\left(p_{n-1}\right)\left(q_{n-1}\right)f_{uu}\left(\gamma ,\delta \right)\hfill \\ & & +p_n[-f_u\left(\gamma ,{\varrho }_{n-1}\right)-G_u\left(\gamma ,{\varkappa }_{n-1}\right)+{\varphi }_u\left(\gamma ,u\right)-h_u\left(\gamma ,{\varkappa }_{n-1}\right)].\hfill \end{array}$$

Hence, one can see

$$q_n^{\mathrm{\Delta }}\le p_{n-1}^{2}K+\left(p_{n-1}\right)\left(q_{n-1}\right)L+p_{n}M,$$

where K, L,and M are positive constants and

$$\begin{array}{ccc}\hfill \left|f_{uu}\left(\gamma ,\delta \right)-h_{uu}(\gamma ,\eta )\right|& \le & K,\hfill \\ \hfill \left|f_{uu}\left(\gamma ,\delta \right)\right|& \le & L,\hfill \\ \hfill \left|f_u\left(\gamma ,{\varrho }_{n-1}\right)+G_u\left(\gamma ,{\varkappa }_{n-1}\right)+{\varphi }_u\left(\gamma ,u\right)+h_u\left(\gamma ,{\varkappa }_{n-1}\right)\right|& \le & M.\hfill \end{array}$$

If Cauchy inequality is implemented to the term $\left(p_{n-1}\right)\left(q_{n-1}\right)L$, then

$$\begin{array}{ccc}\hfill q_n^{\mathrm{\Delta }}& \le & p_{n-1}^{2}K+{\displaystyle \frac{p_{n-1}^2}{2}}L+{\displaystyle \frac{q_{n-1}^2}{2}}L+p_{n}M\hfill \\ & =& p_{n-1}^2[K+{\displaystyle \frac{L}{2}}]+{\displaystyle \frac{q_{n-1}^2}{2}}L+p_{n}M.\hfill \end{array}$$

(14)

When (13) and (14) are considered together, it can be written as

$$r_n^{\mathrm{\Delta }}\le P{r}_n+R{r}_{n-1}^2,$$

which is linear in $r_n$, where $r_n=\left[\begin{array}{c}{p}_n\\ q_n\end{array}\right],P=\left[\begin{array}{cc}0& B\\ M& 0\end{array}\right],R=\left[\begin{array}{cc}{\displaystyle \frac{C}{2}}& A+{\displaystyle \frac{C}{2}}\\ K+{\displaystyle \frac{L}{2}}& {\displaystyle \frac{L}{2}}\end{array}\right].$ Now, if Gronwall’s inequality is applied and integrated from 0 to $\gamma$, then it can be obtained that

$$\begin{array}{ccc}\hfill r_n{e}^{-P\gamma }& \le & r_n\left(0\right)+\underset{0}{\overset{\gamma }{\int }}R{r}_{n-1}^2{e}^{-Ps}\mathrm{\Delta }s\hfill \\ & =& \underset{0}{\overset{\gamma }{\int }}R{r}_{n-1}^2{e}^{P\left(\gamma -s\right)}\mathrm{\Delta }s.\hfill \end{array}$$

Consequently, we have

$$\begin{array}{ccc}\hfill \underset{J}{max}\left|r_n\right|& \le & \underset{J}{max}\left|R{r}_{n-1}^2\right|\underset{0}{\overset{\gamma }{\int }}{e}^{P\left(\gamma -s\right)}\mathrm{\Delta }s\hfill \\ & \le & \underset{J}{max}\left|r_{n-1}^{2}R\right|{\displaystyle \frac{e^{P\gamma }-1}{P}}\hfill \\ & \le & \underset{J}{max}\left|r_{n-1}^2\right|R{\displaystyle \frac{e^{P\gamma }}{P}}.\hfill \end{array}$$

That is,

$$\begin{array}{ccc}\hfill \underset{J}{max}\left|u-{\varkappa }_n\right|& \le & R{\displaystyle \frac{e^{P\gamma }}{P}}\underset{J}{max}{\left|u-{\varkappa }_n\right|}^2,\hfill \\ \hfill \underset{J}{max}\left|u-{\varrho }_n\right|& \le & R{\displaystyle \frac{e^{P\gamma }}{P}}\underset{J}{max}{\left|{\varrho }_n-u\right|}^2.\hfill \end{array}$$

This result shows that the sequences $\left\{{\varkappa }_n\left(\gamma \right)\right\}$ and $\left\{{\varrho }_n\left(\gamma \right)\right\}$ converge to the unique solution $u\left(\gamma \right)$ of (1) quadratically, and this completes the proof. □

3.2. The Monotone Iterative Technique for the Problem

Let us consider the following IVP:

$$\begin{array}{ccc}\hfill u^{\mathrm{\Delta }}& =& f\left(\gamma ,u\right)+g\left(\gamma ,u\right)-h\left(\gamma ,u\right)\hfill \\ \hfill u\left(0\right)& =& u_0.\hfill \end{array}$$

(15)

It has been determined that the sequences obtained with the monotone iteration technique converge to the extremal solutions of the problem (15) uniformly and monotonically.

Theorem 5.

(A₁) Let ${\varkappa }_0\left(\gamma \right),{\varrho }_0\left(\gamma \right)\in C_{rd}\left[T,R\right]$ be coupled with the third-type LS and US of (15) such that ${\varkappa }_0\left(\gamma \right)\le {\varrho }_0\left(\gamma \right)$ on J.

(A₂) Suppose that the functions $f\left(\gamma ,u\right),g\left(\gamma ,u\right),h\left(\gamma ,u\right)$ are non-decreasing with respect to u. Then there exist the monotone sequences $\left\{{\varkappa }_n\left(\gamma \right)\right\}$ and $\left\{{\varrho }_n\left(\gamma \right)\right\}$ converging uniformly and monotonically to the extremal solutions ρ (minimal solution) and r (maximal solution), respectively, which are solutions of (15).

Proof of Theorem 5.

We note that since ${\varkappa }_0\left(\gamma \right),{\varrho }_0\left(\gamma \right)$ are the couple of the third-type LS and US, then

$$\begin{array}{ccc}\hfill {\varkappa }_0^{\mathrm{\Delta }}& \le & f\left(\gamma ,{\varkappa }_0\right)+g\left(\gamma ,{\varkappa }_0\right)-h\left(\gamma ,{\varrho }_0\right),{\varkappa }_0\left(0\right)\le u_0,\hfill \\ \hfill {\varrho }_0^{\mathrm{\Delta }}& \ge & f\left(\gamma ,{\varrho }_0\right)+g\left(\gamma ,{\varrho }_0\right)-h\left(\gamma ,{\varkappa }_0\right),{\varrho }_0\left(0\right)\ge u_0.\hfill \end{array}$$

(16)

Let us consider the following systems:

$$\begin{array}{ccc}\hfill {\varkappa }_{m+1}^{\mathrm{\Delta }}& =& f\left(\gamma ,{\varkappa }_m\right)+g\left(\gamma ,{\varkappa }_m\right)-h\left(\gamma ,{\varrho }_m\right),{\varkappa }_{m+1}\left(0\right)=u_0,\hfill \\ \hfill {\varrho }_{m+1}^{\mathrm{\Delta }}& =& f\left(\gamma ,{\varrho }_m\right)+g\left(\gamma ,{\varrho }_m\right)-h\left(\gamma ,{\varkappa }_m\right),{\varrho }_{m+1}\left(0\right)=u_0.\hfill \end{array}$$

Now, it will be shown that ${\varkappa }_0\le {\varkappa }_1.$ For this purpose, define $p={\varkappa }_0-{\varkappa }_1.$ It is trivial that $p\left(0\right)={\varkappa }_0\left(0\right)-{\varkappa }_1\left(0\right)\le 0.$ If we take the delta derivative of both sides, then we have

$$\begin{array}{ccc}\hfill p^{\mathrm{\Delta }}& =& {\varkappa }_0^{\mathrm{\Delta }}-{\varkappa }_1^{\mathrm{\Delta }}\le f\left(\gamma ,{\varkappa }_0\right)+g\left(\gamma ,{\varkappa }_0\right)-h\left(\gamma ,{\varrho }_0\right)-f\left(\gamma ,{\varkappa }_0\right)-g\left(\gamma ,{\varkappa }_0\right)+h\left(\gamma ,{\varrho }_0\right)\hfill \\ & =& 0.\hfill \end{array}$$

Due to $p^{\mathrm{\Delta }}\le 0$ with $p\left(0\right)\le 0,$ then based on the comparison theorem, $p\left(\gamma \right)\le 0.$ Hence, ${\varkappa }_0\le {\varkappa }_1.$ Similarly, we show ${\varrho }_1\le {\varrho }_0.$ Let $q={\varrho }_1-{\varrho }_0,$$q\left(0\right)={\varrho }_1\left(0\right)-{\varrho }_0\left(0\right)\le 0.$ Taking the delta derivative, we obtain

$$\begin{array}{ccc}\hfill q^{\mathrm{\Delta }}& =& {\varrho }_1^{\mathrm{\Delta }}-{\varrho }_0^{\mathrm{\Delta }}\le \left[f\left(\gamma ,{\varrho }_0\right)+g\left(\gamma ,{\varrho }_0\right)-h\left(\gamma ,{\varkappa }_0\right)\right]-\left[f\left(\gamma ,{\varrho }_0\right)+g\left(\gamma ,{\varrho }_0\right)-h\left(\gamma ,{\varkappa }_0\right)\right]\hfill \\ & =& 0.\hfill \end{array}$$

Therefore, we obtain $q^{\mathrm{\Delta }}\le 0$, $p\left(0\right)\le 0$, and by the comparison theorem, $q\left(0\right)={\varrho }_1\left(0\right)-{\varrho }_0\left(0\right)\le 0\Rightarrow {\varrho }_1\le {\varrho }_0.$ Now, it will be shown that ${\varkappa }_1\le {\varrho }_1.$ For this, let $p={\varkappa }_1-{\varrho }_1$; then $p\left(0\right)={\varkappa }_1\left(0\right)-{\varrho }_1\left(0\right)=0.$ Taking the delta derivative of both sides and using the properties of the functions $f,g,h$, we can write the following:

$$\begin{array}{ccc}\hfill p^{\mathrm{\Delta }}& =& {\varkappa }_1^{\mathrm{\Delta }}-{\varrho }_1^{\mathrm{\Delta }}=f\left(\gamma ,{\varkappa }_0\right)+g\left(\gamma ,{\varkappa }_0\right)-h\left(\gamma ,{\varrho }_0\right)-f\left(\gamma ,{\varrho }_0\right)-g\left(\gamma ,{\varrho }_0\right)+h\left(\gamma ,{\varkappa }_0\right)\hfill \\ & \le & 0.\hfill \end{array}$$

Since $p^{\mathrm{\Delta }}\le 0$, $p\left(0\right)\le 0$, again by the comparison theorem, $p\left(\gamma \right)\le 0.$ Therefore, ${\varkappa }_1\le {\varrho }_1$. Now, let us show ${\varkappa }_1\le u\le {\varrho }_1.$ Let $q=u-{\varrho }_1$ and $p={\varkappa }_1-u$ with $q\left(0\right)=u\left(0\right)-{\varrho }_1\left(0\right)=0$ and $p\left(0\right)={\varkappa }_1\left(0\right)-u\left(0\right)=0.$ Since the functions are non-decreasing, we obtain

$$\begin{array}{ccc}\hfill q^{\mathrm{\Delta }}& =& u^{\mathrm{\Delta }}-{\varrho }_1^{\mathrm{\Delta }}=f\left(\gamma ,u\right)+g\left(\gamma ,u\right)-h\left(\gamma ,u\right)-f\left(\gamma ,{\varrho }_0\right)-g\left(\gamma ,{\varrho }_0\right)+h\left(\gamma ,{\varkappa }_0\right)\hfill \\ & \le & 0.\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill p^{\mathrm{\Delta }}& =& {\varkappa }_1^{\mathrm{\Delta }}-u^{\mathrm{\Delta }}=f\left(\gamma ,{\varkappa }_0\right)+g\left(\gamma ,{\varkappa }_0\right)-h\left(\gamma ,{\varrho }_0\right)-f\left(\gamma ,u\right)-g\left(\gamma ,u\right)+h\left(\gamma ,u\right)\hfill \\ & \le & 0.\hfill \end{array}$$

Hence,

$${\varkappa }_0\le {\varkappa }_1\le u\le {\varrho }_1\le {\varrho }_0.$$

We will use mathematical induction to generalize this. For $k=1$, it is clear that ${\varkappa }_{k-1}\le {\varkappa }_k\le u\le {\varrho }_k\le {\varrho }_{k-1}$ is true. Assume that, for $k>1,$${\varkappa }_{k-1}\le {\varkappa }_k\le u\le {\varrho }_k\le {\varrho }_{k-1}$ is satisfied. We will show that ${\varkappa }_k\le {\varkappa }_{k+1}\le u\le {\varrho }_{k+1}\le {\varrho }_k$ is also satisfied. For this, we define $q={\varrho }_{k+1}-{\varrho }_k$ and $p={\varkappa }_k-{\varkappa }_{k+1}.$ We take the delta derivative on both sides, respectively. Then we have

$$\begin{array}{ccc}\hfill q^{\mathrm{\Delta }}& =& {\varrho }_{k+1}^{\mathrm{\Delta }}-{\varrho }_k^{\mathrm{\Delta }}\le \left[f\left(\gamma ,{\varrho }_k\right)+g\left(\gamma ,{\varrho }_k\right)-h\left(\gamma ,{\varkappa }_k\right)\right]-\left[f\left(\gamma ,{\varrho }_{k-1}\right)+g\left(\gamma ,{\varrho }_{k-1}\right)-h\left(\gamma ,{\varkappa }_{k-1}\right)\right]\hfill \\ & \le & 0\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill p^{\mathrm{\Delta }}& =& {\varkappa }_k^{\mathrm{\Delta }}-{\varkappa }_{k+1}^{\mathrm{\Delta }}\le f\left(\gamma ,{\varkappa }_{k-1}\right)+g\left(\gamma ,{\varkappa }_{k-1}\right)-h\left(\gamma ,{\varrho }_{k-1}\right)-f\left(\gamma ,{\varkappa }_k\right)-g\left(\gamma ,{\varkappa }_k\right)+h\left(\gamma ,{\varrho }_k\right)\hfill \\ & \le & 0.\hfill \end{array}$$

Therefore, we can say that ${\varrho }_{k+1}\le {\varrho }_k$ and ${\varkappa }_k\le {\varkappa }_{k+1}.$ Following similar steps, it can be shown that $u\le {\varrho }_{k+1}$ and ${\varkappa }_{k+1}\le u.$ Therefore, the following conclusion is reached:

$${\varkappa }_0\le {\varkappa }_1\le ···\le {\varkappa }_m\le {\varkappa }_{m+1}\le u\le {\varrho }_{m+1}\le {\varrho }_m\le ···\le {\varrho }_1\le {\varrho }_0.$$

To show that these sequences converge uniformly, we must show that they are uniformly bounded and equicontinuous. Since ${\varkappa }_0$ and ${\varrho }_0$ are bounded, then there exists $L_1,L_2>0$ such that, for every $\gamma \in N,$$\left|{\varkappa }_0\right|\le L_1$ and $\left|{\varrho }_0\right|\le L_2.$ Since, for every $m>0,$${\varkappa }_0\le {\varrho }_m\le {\varrho }_0$, then it can be written that $0\le \left|{\varrho }_m\left(\gamma \right)-{\varkappa }_0\left(\gamma \right)\right|\le \left|{\varrho }_0\left(\gamma \right)-{\varkappa }_0\left(\gamma \right)\right|.$ Therefore, $\left|{\varrho }_m\left(\gamma \right)\right|\le M,M>0$ and $\left\{{\varrho }_m\left(\gamma \right)\right\}$ are uniformly bounded. Similarly, it can be shown that $\left\{{\varkappa }_m\left(\gamma \right)\right\}$ is uniformly bounded. Now, we will show that $\left\{{\varrho }_m\left(\gamma \right)\right\}$ is equicontinuous. For this, we will find any $\delta >0$ when $\left|{\gamma }_1-{\gamma }_2\right|<\delta ,$$\left|{\varkappa }_m\left({\gamma }_1\right)-{\varkappa }_m\left({\gamma }_2\right)\right|<\epsilon$ is satisfied. For $0\le {\gamma }_1\le {\gamma }_2,$$m>0$

$$\begin{array}{ccc}\hfill \left|{\varrho }_m\left({\gamma }_1\right)-{\varrho }_m\left({\gamma }_2\right)\right|& =& \left|u_0+\underset{0}{\overset{{\gamma }_1}{\int }}\left[f(s,{\varrho }_{m-1}\left(s\right))+g(s,{\varrho }_{m-1}\left(s\right))-h(s,{\varkappa }_{m-1}\left(s\right))\right]\mathrm{\Delta }s\right.\hfill \\ & & \left.-u_0-\underset{0}{\overset{{\gamma }_2}{\int }}\left[f(s,{\varrho }_{m-1}\left(s\right))+g(s,{\varrho }_{m-1}\left(s\right))-h(s,{\varkappa }_{m-1}\left(s\right))\right]\mathrm{\Delta }s\right|\hfill \\ & =& \left|\underset{{\gamma }_2}{\overset{{\gamma }_1}{\int }}\left[f(s,{\varrho }_{m-1}\left(s\right))+g(s,{\varrho }_{m-1}\left(s\right))-h(s,{\varkappa }_{m-1}\left(s\right))\right]\mathrm{\Delta }s\right|\hfill \\ & \le & \underset{{\gamma }_2}{\overset{{\gamma }_1}{\int }}\left|\left[f(s,{\varrho }_{m-1}\left(s\right))+g(s,{\varrho }_{m-1}\left(s\right))-h(s,{\varkappa }_{m-1}\left(s\right))\right]\right|\mathrm{\Delta }s.\hfill \end{array}$$

Since $\left\{{\varrho }_m\left(\gamma \right)\right\}$ and $\left\{{\varkappa }_m\left(\gamma \right)\right\}$ are uniformly bounded and the functions $f(\gamma ,u),$ $g(\gamma ,u),$$h(\gamma ,u)$ are bounded, then there exists $\tilde{\Gamma }$ such that

$$\left|{\varkappa }_m\left({\gamma }_1\right)-{\varkappa }_m\left({\gamma }_2\right)\right|\le \underset{{\gamma }_2}{\overset{{\gamma }_1}{\int }}\left|\left[f(s,{\varrho }_{m-1}\left(s\right))+g(s,{\varrho }_{m-1}\left(s\right))-h(s,{\varkappa }_{m-1}\left(s\right))\right]\right|\mathrm{\Delta }s\le \tilde{\Gamma }\left|{\gamma }_1-{\gamma }_2\right|.$$

If we choose $\delta ={\displaystyle \frac{\epsilon }{\tilde{\Gamma }}},$ then $\left\{{\varrho }_m\left(\gamma \right)\right\}$ is equicontinuous. Similarly, it can be shown that $\left\{{\varkappa }_m\left(\gamma \right)\right\}$ is equicontinuous. By Theorem 3, the sequences $\left\{{\varkappa }_{m_k}\left(\gamma \right)\right\}$ and $\left\{{\varrho }_{mk}\left(\gamma \right)\right\}$ converge to the extremal solutions $\rho \left(\gamma \right)$ and $r\left(\gamma \right)$ of (15) uniformly and monotonically. Finally, it will be shown that $\rho \left(\gamma \right)$ and $r\left(\gamma \right)$ are coupled with third-type minimal and maximal solutions of (15). From previous findings, we have

$$\begin{array}{ccc}\hfill \underset{m\to \infty }{lim}{\varkappa }_m\left(\gamma \right)& =& \rho \left(\gamma \right),\hfill \\ \hfill \underset{m\to \infty }{lim}{\varrho }_m\left(\gamma \right)& =& r\left(\gamma \right).\hfill \end{array}$$

If we integrate the system (16) from 0 to $\gamma$, then it can be obtained that

$$\begin{array}{ccc}\hfill {\varkappa }_{m+1}\left(\gamma \right)& =& u_0+\underset{0}{\overset{\gamma }{\int }}\left[f(s,{\varkappa }_m\left(s\right))+g(s,{\varkappa }_m\left(s\right))-h(s,{\varrho }_m\left(s\right))\right]\mathrm{\Delta }s,\hfill \\ \hfill {\varrho }_{m+1}\left(\gamma \right)& =& u_0+\underset{0}{\overset{\gamma }{\int }}\left[f(s,{\varrho }_{m-1}\left(s\right))+g(s,{\varrho }_{m-1}\left(s\right))-h(s,{\varkappa }_{m-1}\left(s\right))\right]\mathrm{\Delta }s\hfill \end{array}$$

and take the limit while $m\to \infty ,$

$$\begin{array}{ccc}\hfill \underset{m\to \infty }{lim}{\varkappa }_{m+1}\left(\gamma \right)& =& \underset{m\to \infty }{lim}\left\{u_0+\underset{0}{\overset{\gamma }{\int }}\left[f(s,{\varkappa }_m\left(s\right))+g(s,{\varkappa }_m\left(s\right))-h(s,{\varrho }_m\left(s\right))\right]\mathrm{\Delta }s\right\}\hfill \\ \hfill \rho & =& u_0+\underset{0}{\overset{\gamma }{\int }}\left[f(s,\rho (s\left)\right)+g(s,\rho (s\left)\right)-h(s,r(s\left)\right)\right]\mathrm{\Delta }s.\hfill \end{array}$$

After taking the delta derivative of both sides, then we obtain

$${\rho }^{\mathrm{\Delta }}=f\left(\gamma ,\rho \right)+g\left(\gamma ,\rho \right)-h\left(\gamma ,r\right).$$

Similarly, it can be written that

$$\begin{array}{ccc}\hfill \underset{m\to \infty }{lim}{\varrho }_{m+1}\left(\gamma \right)& =& \underset{m\to \infty }{lim}\left\{u_0+\underset{0}{\overset{\gamma }{\int }}\left[f(s,{\varrho }_{m-1}\left(s\right))+g(s,{\varrho }_{m-1}\left(s\right))-h(s,{\varkappa }_{m-1}\left(s\right))\right]\mathrm{\Delta }s\right\}\hfill \\ \hfill r& =& u_0+\underset{0}{\overset{{\gamma }_1}{\int }}\left[f(s,r(s\left)\right)+g(s,r(s\left)\right)-h(s,\rho (s\left)\right)\right]\mathrm{\Delta }s\hfill \end{array}$$

and

$$r^{\mathrm{\Delta }}=f\left(\gamma ,r\right)+g\left(\gamma ,r\right)-h\left(\gamma ,\rho \right).$$

Consequently, it can be obtained that

$$\begin{array}{ccc}\hfill {\rho }^{\mathrm{\Delta }}& =& f\left(\gamma ,\rho \right)+g\left(\gamma ,\rho \right)-h\left(\gamma ,r\right),\rho \left(0\right)=u_0,\hfill \\ \hfill r^{\mathrm{\Delta }}& =& f\left(\gamma ,r\right)+g\left(\gamma ,r\right)-h\left(\gamma ,\rho \right),r\left(0\right)=u_0\hfill \end{array}$$

and

$${\varkappa }_0\le {\varkappa }_1\le ···\le \rho \le u\le r\le ···\le {\varrho }_1\le {\varrho }_0.$$

Therefore, $\rho \left(\gamma \right)$ and $r\left(\gamma \right)$ are coupled with third-type minimal and maximal solutions of (15). This completes the proof.

□

3.3. An Example

For the particular case of the system (15), consider the following dynamic initial value problem:

$$u^{\mathrm{\Delta }}={\displaystyle \frac{u^2}{4}}+{\displaystyle \frac{u^2}{2}}-{\displaystyle \frac{u^2}{3}},$$

where $f\left(\gamma ,u\right)={\displaystyle \frac{u^2}{4}},$$g\left(\gamma ,u\right)={\displaystyle \frac{u^2}{2}}$$,h\left(\gamma ,u\right)={\displaystyle \frac{u^2}{3}},\gamma \in \left[0,1\right].$ Let ${\varkappa }_0\left(\gamma \right)=-1,{\varrho }_0\left(\gamma \right)={\displaystyle \frac{1}{2}}.$ Therefore, the following inequalities are satisfied:

$${\varkappa }_0^{\mathrm{\Delta }}=0\le f\left(\gamma ,{\varkappa }_0\right)+g\left(\gamma ,{\varkappa }_0\right)-h\left(\gamma ,{\varrho }_0\right)={\displaystyle \frac{11}{12}},$$

and

$${\varrho }_0^{\mathrm{\Delta }}=0\ge f\left(\gamma ,{\varrho }_0\right)+g\left(\gamma ,{\varrho }_0\right)-h\left(\gamma ,{\varkappa }_0\right)=-{\displaystyle \frac{17}{96}}.$$

Hence, ${\varkappa }_0\left(\gamma \right),{\varrho }_0\left(\gamma \right)$ are coupled with the third-type LS and US of (15). If the functions ${\varkappa }_1\left(\gamma \right),{\varrho }_1\left(\gamma \right)$ are selected as ${\varkappa }_1\left(\gamma \right)={\displaystyle \frac{11}{12}}\gamma -1$ and ${\varrho }_1\left(\gamma \right)=-{\displaystyle \frac{17}{96}}\gamma +{\displaystyle \frac{1}{2}}$, then

$${\varkappa }_0\left(\gamma \right)\le {\varkappa }_1\left(\gamma \right)\le {\varrho }_1\left(\gamma \right)\le {\varrho }_0\left(\gamma \right),\gamma \in \left[0,1\right]$$

is satisfied. If we continue in this way, ${\varkappa }_2\left(\gamma \right),{\varkappa }_3\left(\gamma \right),···,{\varkappa }_n\left(\gamma \right)$ and ${\varrho }_2\left(\gamma \right),{\varrho }_3\left(\gamma \right),···,{\varrho }_n\left(\gamma \right)$ can be obtained. Therefore, all the conditions of Theorem 5 are satisfied.

4. Conclusions

In this work, we studied a unique solution by combining techniques from the generalized QLM, using comparison results with coupled seventh-type lower and upper solutions, and improved the QLM to given nonlinear differential equations on a time scale. Under convenient conditions, it was observed that the monotone sequences converge to the unique solution of the original problem uniformly and monotonically. Furthermore, we observed that this convergence is quadratic. Additionally, we applied a monotone iterative technique with coupled third-type lower and upper solutions when the functions involved were monotonically non-decreasing. The sequences obtained converge to the extremal solutions of the problem uniformly and monotonically. Finally, we calculated a numerical solution for a dynamic IVP based on the MIT.