New Results on the Quasilinearization Method for Time Scales

Abstract

We have developed the generalized quasilinearization method (QM) for an initial value problem (IVP) of dynamic equations on time scales by using comparison theorems with a coupled lower solution (LS) and upper solution (US) of the natural type. Under some conditions, we observed that the solutions converged to the unique solution of the problem uniformly and monotonically, and the rate of convergence was investigated.

1. Introduction

The theory of calculus on time scales was studied by Stefan Hilger [1] in 1990 to connect continuous and discrete analyses. Examining dynamic equations on time scales is a research area of great interest. Many researchers are devoted to the qualitative study of ordinary differential equations on time scale.

The QM, combined with the LS and US methods, provides a clear analytical representation for the solution of the nonlinear differential equation, providing pointwise upper and lower estimates for the solution of the problem. Using these generalized concepts given by Lakshmikantham and Köksal, [2] it was developed on time scales by Kaymakçalan and Lawrence in [3]. Bohner and Peterson [4] discussed basic information, definitions and theorems about time scale. They also dealt first and second order linear equation on time scale. In recent years, studies and articles have been argued, and extraordinary results have been obtained. Some articles and books that can be found are referenced in this article.

Akın et al. [5] applied QM to the unique solution of BVP on time scales with monotone sequences of LS and US, and the authors obtained the convergence rate. Atici and Topal [6] improved the convergence of monotone sequences on time scales for nonlinear dynamical equations. They generated two sequences that, under some favorable conditions, converged to the solution of BVP.

Daneev and Sizykh [7] investigated the convergence issues of the developed algorithm method for minimizing the generalized work functional. The authors demonstrated the algorithm using numerical stabilization of a two-dimensional oscillatory linear control system.

Yang and Vatsala [8] developed a generalized QM for reaction diffusion systems where the forcing functions are the sum of convex and concave functions. They showed the solutions of the corresponding linear systems converge monotonically, uniformly and quadratically to the unique solution of the nonlinear problem.

Idiz et al. [9] examined the numerical solution of a fractional Lane–Emdan-type equation that occurs mainly in astrophysics applications. They improved a numerical approach using QM and solved some example problems to compare with other numerical techniques in the literature.

Jyoti and Singh [10] obtained a coupled iterative approach based on a QM approximation for evaluating the solutions of nonlinear two-point Dirichlet BVP.

Lakshmikantham and Vatsala [11] applied the QM to an IVP and obtained LS and US on ordinary differential equations.

Mohammadi et al. [12] proposed a method to solve some classes of singular fractional nonlinear Lane–Emdan-type equations using the QM. They compared the results produced using the method with some well-known results to demonstrate its accuracy.

Mwakilama et al. [13] offered and applied a multivariate spectral local QM to model the transport and interaction of soil. They conducted systematic analyses on the influence of model parameters on the distribution profiles to check the accuracy of the solution. The findings correspond to the theory and literature and thus demonstrate the feasibility of solving coupled nonlinear PDEs with environmental fluid dynamics applications.

In 2023, Radhika improved the generalized QM for an IVP using the LS and US of type I and obtained the solution of a periodic boundary value problem. He also obtained the generalized quasilinearization technique BVP using the method developed for IVP for Caputo fractional differential equations.

Safari et al. [14] attempted to solve variable-order fractional diffusion problems with a fourth-order derivative term. They obtained the solution of the problem in terms of the primary approximation, as well as the corresponding correction functions at each time step by using a radial basis in functions.

Verma and Urus [15] developed a monotone iterative technique with LS and US for a class of four-point nonlinear BVP. They offered Green’s function and iterative sequences for the corresponding linear problem by using the QM.

Wang and Tian [16] studied nonlinear BVP for difference equations with causal operators. By using the methosd of LS and US together with the monotone iterative technique, they showed the existence of extremal solutions.

Wang and Wu [17] dealt with convergence for a class with a given IVP involving the sum of two terms as a right-hand function on time scales. They obtained second-order convergence of approximate solutions for approximately iterative generated series by using the generalized QM.

Torkaman et al. [18] studied the numerical solution of nonlinear multi-dimensional Volterra integral equations. The authors reduced the problem to a sequence of linear Volterra integral equations by using the QM.

Tunitsky [19] obtained necessary and sufficient conditions for local quasilinearizability of elliptic Monge-Ampère systems.

Yakar [20] investigated the qualitative behavior of perturbed dynamical systems on time scales that differed in initial time compared to unperturbed dynamical systems. The author proposed the classical concept of stability with the concept of initial time difference stability on time scales. Yakar and Arslan [21] extended the generalized QM for nonlinear terminal value problems. They obtained several conditions to apply the QM to the nonlinear problem involving Riemann–Liouville fractional derivatives.

Yüzbaşı and Izadi [22] improved two numerical methods based on Bessel polynomials. In the first method, they transformed the HIV problem into a system of nonlinear equations by using the Bessel polynomial. The other method was used to transform the HIV problem into a sequence of linear equations using the QM.

Zare et al. [23] introduced a combination of QMs to approximate the solution of nonlinear Volterra integral equations. Moreover, the authors solved the obtained linear equation in each iteration using these method.

We have given two results, with the solutions showing the practical applicability of the conditions. We believe that our study will add a new perspective to the literature.

In this paper, our aim is to elucidate the LS and US of dynamic non-linear IVP,

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

(1)

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

2. Preliminaries

This section reviews some basic concepts and fundamental theorems on LS and US that are crucial to obtain our results in the other sections below.

Definition 1 

([24]). Any nonempty closed subset T of the real numbers is called a time scale. $R, Z, N, N_0,[2,3]\cup [4,5]$ are examples of time scales. Furthermore, $u^{\Delta }=\frac{du}{dz}$ is the usual derivative if $T=R$ is taken.

Definition 2 

([11]). A function $\varpi \in C_{rd}\left[J,R\right]$ is said to be a LS of

$$u^{\Delta }=f\left(z,u\right)$$

(2)

if

$${\varpi }^{\Delta }\le f\left(z,\varpi \right),\varpi \left(z_0\right)\le u_0$$

and similarly, $\epsilon \in C_{rd}\left[J,R\right]$ is said to be an US if

$${\epsilon }^{\Delta }\ge f\left(z,\epsilon \right),\epsilon \left(z_0\right)\ge u_0.$$

Definition 3 

([11]). Let ϖ and ε be rd-continuously differentiable functions such that $\varpi \le \epsilon$ on $T^k.$ Then, ϖ and ε are called natural LS and US of (1) if

$$\begin{array}{ccc}\hfill {\varpi }^{\Delta }& \le & f\left(z,\varpi \right)+g\left(z,\varpi \right)+h\left(z,\varpi \right)\hfill \\ \hfill \varpi \left(z_0\right)& \le & u_0\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \epsilon {\left(z\right)}^{\Delta }& \ge & f\left(z,\epsilon \right)+g\left(z,\epsilon \right)+h\left(z,\epsilon \right)\hfill \\ \hfill \epsilon \left(z_0\right)& \ge & u_0.\hfill \end{array}$$

Theorem 1 

([11]). Let $\varpi ,\epsilon \in C_{rd}\left[J,R\right]$ be the lower and upper solutions of (2), respectively, and suppose that

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

whenever $u_1\ge u_2$ for some $L>0.$ Then, $\varpi \left(z_0\right)\le \epsilon \left(z_0\right)$ implies $\varpi \left(z\right)\le \epsilon \left(z\right).$

Theorem 2 

([11]). Let $\varpi ,\epsilon \in C_{rd}\left[J,R\right]$ be the LS and US of (2), respectively, such that $\varpi \left(z\right)\le \epsilon \left(z\right)$ on $T^k$ and $f\in C_{rd}\left[\Omega ,R\right]$ will be bounded on Ω, where $\Omega =\left\{\left(z,u\right):\varpi \left(z\right)\le u\le \epsilon \left(z\right),z\in J\right\}$. Then, there exists a solution $u\left(z\right)$ of (2) satisfying $\varpi \left(z\right)\le u\left(z\right)\le \epsilon \left(z\right)$ on $z\in J$ whenever $\varpi \left(z_0\right)\le u\left(z_0\right)\le \epsilon \left(z_0\right)$.

Theorem 3 

(Arzela Ascoli Theorem [25]). Let $f_n\left(z\right)$ be a sequence of functions defined on a compact set J that is equicontinuous and uniformly bounded on J. Then, there exists a subsequence $\left\{f_n\right\},n=1,2,...$, which is uniformly convergent on J.

3. Results

3.1. The Semi Quadraticity of the Convergence

We achieved monotone sequences on the time scale that uniformly converges to the solution of (1). Auxiliary linear IVPs were constructed for this purpose. By using QM with LS and US, under some appropriate conditions, we indicated that this convergence is semi quadratic. In addition to this, we will prefer $\Delta t$ instead of $dt$ in the proof, because we are studying the time scale instead of an ordinary differential equation. Let

$${\Omega }_0=\left\{\left(z,u\right):{\varpi }_0\left(z\right)\le u\le {\epsilon }_0\left(z\right)z\in T\right\},$$

where ${\varpi }_0\left(z\right),{\epsilon }_0\left(z\right)\in C_{rd}\left[T,R\right].$

Theorem 4. 

Assume that the following hypotheses hold:

(A₁) 

Let ${\varpi }_0\left(z\right),{\epsilon }_0\left(z\right)\in C_{rd}\left[J,R\right]$ be the natural LS and US of (1) such that ${\varpi }_0\left(z\right)\le {\epsilon }_0\left(z\right)$ on J.

(A₂) 

$f_u\left(z,u\right),$$g_u\left(z,u\right),$$h_u\left(z,u\right),$$f_{uu}\left(z,u\right),$$g_{uu}\left(z,u\right),$$h_{uu}\left(z,u\right)$ exist as continuous functions on $J\times R$, where $f\left(z,u\right),g\left(z,u\right),h\left(z,u\right)\in C_{rd}^2\left[J\times R,R\right]$, and they satisfy $f_{uu}\left(z,u\right)+{\varphi }_{uu}\left(z,u\right)\ge 0,$ $g_{uu}\left(z,u\right)+{\psi }_{uu}\left(z,u\right)\le 0$ whenever ${\varphi }_{uu}$ and ${\psi }_{uu}$ exist and ${\varphi }_{uu}\left(z,u\right)>0,{\psi }_{uu}\left(z,u\right)<0$ on $\Omega .$ And, $h\left(z,u\right)$ satisfies the following inequality. For $u_1\ge u_2\in {\Omega }_0,$

$$h\left(z,u_1\right)-h\left(z,u_2\right)\ge -k\left(u_1-u_2\right),$$

where $k\ge 0$ is a Lipschitz constant. Then, there exist monotone sequences $\left\{{\varpi }_n\left(z\right)\right\}$ and $\left\{{\epsilon }_n\left(z\right)\right\}$, and these sequences converge uniformly to the unique solution of (1). The rate of convergence is semi-quadratic.

Proof of Theorem 4. 

Let us take

$$\begin{array}{ccc}\hfill F\left(z,u\right)& =& f\left(z,u\right)+\varphi \left(z,u\right),\hfill \\ \hfill G\left(z,u\right)& =& g\left(z,u\right)+\psi \left(z,u\right).\hfill \end{array}$$

In view of (A₂), it is clear that for ${\varpi }_0\left(z\right)\le u_2\le u_1\le {\epsilon }_0\left(z\right),$

$$\begin{array}{ccc}\hfill -L_1\left(u_1-u_2\right)& \le & f\left(z,u_1\right)-f\left(z,u_2\right)\le L_1\left(u_1-u_2\right),L_1>0,\hfill \\ \hfill -L_2\left(u_1-u_2\right)& \le & g\left(z,u_1\right)-g\left(z,u_2\right)\le L_2\left(u_1-u_2\right),L_2>0,\hfill \\ \hfill -L_3\left(u_1-u_2\right)& \le & h\left(z,u_1\right)-h\left(z,u_2\right),L_3>0.\hfill \end{array}$$

(3)

Since $f_{uu}\left(z,u\right)+{\varphi }_{uu}\left(z,u\right)\ge 0,g_{uu}\left(z,u\right)+{\psi }_{uu}\left(z,u\right)\le 0$, and because of the properties of $h\left(z,u\right)$, we note the following inequalities. For $u\ge v$ and $u,v\in \Omega$,

$$\begin{array}{ccc}\hfill f\left(z,u\right)& \ge & f\left(z,v\right)+\left[f_u\left(z,v\right)+{\varphi }_u\left(z,v\right)\right]\left(u-v\right)+\varphi \left(z,v\right)-\varphi \left(z,u\right)\hfill \\ & =& F\left(z,v\right)+F_u\left(z,v\right)\left(u-v\right)-\varphi \left(z,u\right),\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill g\left(z,u\right)& \le & g\left(z,v\right)+\left[g_u\left(z,v\right)+{\psi }_u\left(z,v\right)\right]\left(u-v\right)+\psi \left(z,v\right)-\psi \left(z,u\right)\hfill \\ & =& G\left(z,v\right)+G_u\left(z,v\right)\left(u-v\right)-\psi \left(z,u\right),\hfill \end{array}$$

(5)

and

$$h\left(z,u\right)-h\left(z,v\right)\ge -k\left(u-v\right).$$

(6)

Now, we consider the following auxiliary IVPs

$$\begin{array}{ccc}\hfill {\varpi }_1^{\Delta }& =& f\left(z,{\varpi }_0\right)+g\left(z,{\varpi }_0\right)+h\left(z,{\varpi }_0\right)\hfill \\ & & +\left[f_u\left(z,{\varpi }_0\right)+{\varphi }_u\left(z,{\varpi }_0\right)+g_u\left(z,{\epsilon }_0\right)+{\psi }_u\left(z,{\epsilon }_0\right)-{\varphi }_u\left(z,{\epsilon }_0\right)-{\psi }_u\left(z,{\varpi }_0\right)-k\right]\left({\varpi }_1-{\varpi }_0\right)\hfill \\ & =& N\left(z,{\varpi }_0\right)+\left[F_u\left(z,{\varpi }_0\right)+G_u\left(z,{\epsilon }_0\right)-{\varphi }_u\left(z,{\epsilon }_0\right)-{\psi }_u\left(z,{\varpi }_0\right)-k\right]\left({\varpi }_1-{\varpi }_0\right)\hfill \\ \hfill {\varpi }_1\left(0\right)& =& u_0\hfill \end{array}$$

(7)

and

$$\begin{array}{ccc}\hfill {\epsilon }_1^{\Delta }& =& f\left(z,{\epsilon }_0\right)+g\left(z,{\epsilon }_0\right)+h\left(z,{\epsilon }_0\right)\hfill \\ & & +\left[f_u\left(z,{\varpi }_0\right)+{\varphi }_u\left(z,{\varpi }_0\right)+g_u\left(z,{\epsilon }_0\right)+{\psi }_u\left(z,{\epsilon }_0\right)-{\varphi }_u\left(z,{\epsilon }_0\right)-{\psi }_u\left(z,{\varpi }_0\right)-k\right]\left({\epsilon }_1-{\epsilon }_0\right)\hfill \\ & =& N\left(z,{\epsilon }_0\right)+\left[F_u\left(z,{\varpi }_0\right)+G_u\left(z,{\epsilon }_0\right)-{\varphi }_u\left(z,{\epsilon }_0\right)-{\psi }_u\left(z,{\varpi }_0\right)-k\right]\left({\epsilon }_1-{\epsilon }_0\right)\hfill \\ \hfill {\epsilon }_1\left(0\right)& =& u_0.\hfill \end{array}$$

(8)

Now, we will show that ${\varpi }_0\left(z\right)\le {\varpi }_1\left(z\right)$ on $J.$ Let $p={\varpi }_0-{\varpi }_1,$ then $p\left(0\right)={\varpi }_0\left(0\right)-{\varpi }_1\left(0\right)\le 0.$ Taking the delta derivative of both sides, we can obtain the following:

$$\begin{array}{ccc}\hfill p^{\Delta }& =& {\varpi }_0^{\Delta }-{\varpi }_1^{\Delta }\le N\left(z,{\varpi }_0\right)\hfill \\ & & -\left\{N\left(z,{\varpi }_0\right)+\left[F_u\left(z,{\varpi }_0\right)+G_u\left(z,{\epsilon }_0\right)-{\varphi }_u\left(z,{\epsilon }_0\right)-{\psi }_u\left(z,{\varpi }_0\right)-k\right]\left({\varpi }_1-{\varpi }_0\right)\right\}\hfill \\ & =& \left[F_u\left(z,{\varpi }_0\right)+G_u\left(z,{\epsilon }_0\right)-{\varphi }_u\left(z,{\epsilon }_0\right)-{\psi }_u\left(z,{\varpi }_0\right)-k\right]\left({\varpi }_0-{\varpi }_1\right)\hfill \\ & =& \left[F_u\left(z,{\varpi }_0\right)+G_u\left(z,{\epsilon }_0\right)-{\varphi }_u\left(z,{\epsilon }_0\right)-{\psi }_u\left(z,{\varpi }_0\right)-k\right]p.\hfill \end{array}$$

Since $p^{\Delta }\le Mp$ $p\left(0\right)\le 0$ then yields $p\left(z\right)={\varpi }_0\left(z\right)-{\varpi }_1\left(z\right)\le 0$. Similarly, we will show that ${\varpi }_1\left(z\right)\le {\epsilon }_0\left(z\right).$ Let $p={\varpi }_1-{\epsilon }_0$, then $p\left(0\right)={\varpi }_1\left(0\right)-{\epsilon }_0\left(0\right)\le 0.$ Taking the delta derivative of both sides and using (4)–(6), we can write the following:

$$\begin{array}{ccc}\hfill p^{\Delta }& \le & \left[F_u\left(z,{\varpi }_0\right)+G_u\left(z,{\epsilon }_0\right)-{\varphi }_u\left(z,{\epsilon }_0\right)-{\psi }_u\left(z,{\varpi }_0\right)-k\right]\left({\varpi }_1-{\epsilon }_0\right)\hfill \\ & \le & \left[F_u\left(z,{\varpi }_0\right)+G_u\left(z,{\epsilon }_0\right)-{\varphi }_u\left(z,{\epsilon }_0\right)-{\psi }_u\left(z,{\varpi }_0\right)-k\right]p.\hfill \end{array}$$

Therefore, $p\left(z\right)\le 0.$ Hence, ${\varpi }_1\left(z\right)\le {\epsilon }_0\left(z\right)$ on J. Now, it can be shown that ${\varpi }_1\left(z\right)\le {\epsilon }_1\left(z\right).$ By using the inequalities given in (7) and (4)–(6), and using the mean value theorem, the following can be obtained:

$$\begin{array}{ccc}\hfill {\varpi }_1^{\Delta }& =& N\left(z,{\varpi }_0\right)+\left[F_u\left(z,{\varpi }_0\right)+G_u\left(z,{\epsilon }_0\right)-{\varphi }_u\left(z,{\epsilon }_0\right)-{\psi }_u\left(z,{\varpi }_0\right)-k\right]\left({\varpi }_1-{\varpi }_0\right)\hfill \\ & =& \left[f\left(z,{\varpi }_0\right)+F_u\left(z,{\varpi }_0\right)\left({\varpi }_1-{\varpi }_0\right)\right]+\left[g\left(z,{\varpi }_0\right)+G_u\left(z,{\epsilon }_0\right)\left({\varpi }_1-{\varpi }_0\right)\right]\hfill \\ & & -\left[{\varphi }_u\left(z,{\epsilon }_0\right)+{\psi }_u\left(z,{\varpi }_0\right)+k\right]\left({\varpi }_1-{\varpi }_0\right)+h\left(z,{\varpi }_0\right)\hfill \\ & \le & N\left(z,{\varpi }_1\right).\hfill \end{array}$$

Therefore, ${\varpi }_1$ is an LS. Similarly, using (8), we obtain the following:

$$\begin{array}{ccc}\hfill {\epsilon }_1^{\Delta }& =& N\left(z,{\epsilon }_0\right)+\left[F_u\left(z,{\varpi }_0\right)+G_u\left(z,{\epsilon }_0\right)-{\varphi }_u\left(z,{\epsilon }_0\right)-{\psi }_u\left(z,{\varpi }_0\right)-k\right]\left({\epsilon }_1-{\epsilon }_0\right)\hfill \\ & =& \left[f\left(z,{\epsilon }_0\right)+F_u\left(z,{\varpi }_0\right)\left({\epsilon }_1-{\epsilon }_0\right)\right]+\left[g\left(z,{\epsilon }_0\right)+G_u\left(z,{\epsilon }_0\right)\left({\epsilon }_1-{\epsilon }_0\right)\right]\hfill \\ & & -\left[{\varphi }_u\left(z,{\epsilon }_0\right)+{\psi }_u\left(z,{\varpi }_0\right)+k\right]\left({\epsilon }_1-{\epsilon }_0\right)+h\left(z,{\epsilon }_0\right)\hfill \\ & \ge & N\left(z,{\epsilon }_1\right).\hfill \end{array}$$

Hence, ${\epsilon }_1$ is a US. Since ${\varpi }_1,{\epsilon }_1$ are the LS and US of (1) with ${\varpi }_1\left(0\right)\le {\epsilon }_1\left(0\right)$, Theorem 1 yields ${\varpi }_1\left(z\right)\le {\epsilon }_1\left(z\right)$. Therefore, we have established the following inequalities:

$${\varpi }_0\left(z\right)\le {\varpi }_1\left(z\right)\le {\epsilon }_1\left(z\right)\le {\epsilon }_0\left(z\right).$$

To generalize this situation, let us define $p_n\left(z\right)={\varpi }_n\left(z\right)-{\varpi }_{n+1}\left(z\right)$. It is clear that $p_n\left(0\right)={\varpi }_n\left(0\right)-{\varpi }_{n+1}\left(0\right)=0$. Taking the delta derivative of both sides and using inequalities (4)–(6) and the mean value theorem, we can obtain the following:

$$\begin{array}{ccc}\hfill p_n^{\Delta }& =& {\varpi }_n^{\Delta }-{\varpi }_{n+1}^{\Delta }\hfill \\ & =& f\left(z,{\varpi }_{n-1}\right)+g\left(z,{\varpi }_{n-1}\right)+h\left(z,{\varpi }_{n-1}\right)\hfill \\ & & +\left[F_u\left(z,{\varpi }_{n-1}\right)+G_u\left(z,{\epsilon }_{n-1}\right)-{\varphi }_u\left(z,{\epsilon }_{n-1}\right)-{\psi }_u\left(z,{\varpi }_{n-1}\right)-k\right]\left({\varpi }_n-{\varpi }_{n-1}\right)\hfill \\ & & -f\left(z,{\varpi }_n\right)-g\left(z,{\varpi }_n\right)-h\left(z,{\varpi }_n\right)\hfill \\ & & -\left[F_u\left(z,{\varpi }_n\right)+G_u\left(z,{\epsilon }_n\right)-{\varphi }_u\left(z,{\epsilon }_n\right)-{\psi }_u\left(z,{\varpi }_n\right)-k\right]\left({\varpi }_{n+1}-{\varpi }_n\right)\hfill \\ & \le & \left[F_u\left(z,{\varpi }_{n-1}\right)+G_u\left(z,{\varpi }_n\right)-{\varphi }_u\left(z,{\varpi }_n\right)-{\psi }_u\left(z,{\varpi }_{n-1}\right)-k\right]\left({\varpi }_n-{\varpi }_{n-1}\right)\hfill \\ & & -\left[F_u\left(z,{\varpi }_{n-1}\right)-{\varphi }_u\left(z,{\varpi }_n\right)+G_u\left(z,{\varpi }_n\right)-{\psi }_u\left(z,{\varpi }_{n-1}\right)-k\right]\left({\varpi }_n-{\varpi }_{n-1}\right)\hfill \\ & & -\left[F_u\left(z,{\varpi }_n\right)+G_u\left(z,{\epsilon }_n\right)-{\varphi }_u\left(z,{\epsilon }_n\right)-{\psi }_u\left(z,{\varpi }_n\right)-k\right]\left({\varpi }_{n+1}-{\varpi }_n\right)\hfill \\ & =& \left[F_u\left(z,{\varpi }_n\right)+G_u\left(z,{\epsilon }_n\right)-{\varphi }_u\left(z,{\epsilon }_n\right)-{\psi }_u\left(z,{\varpi }_n\right)-k\right]\left({\varpi }_n-{\varpi }_{n+1}\right).\hfill \end{array}$$

Since $p_n^{\Delta }\le Mp$ $p\left(0\right)\le 0$ then yields $p\left(z\right)={\varpi }_0\left(z\right)-{\varpi }_1\left(z\right)\le 0$. Now, we will show ${\varpi }_{n+1}\le {\epsilon }_{n+1}.$ Based on (7), the following can be written:

$$\begin{array}{ccc}\hfill {\varpi }_{n+1}^{\Delta }& =& N\left(z,{\varpi }_n\right)+\left[F_u\left(z,{\varpi }_n\right)+G_u\left(z,{\epsilon }_n\right)-{\varphi }_u\left(z,{\epsilon }_n\right)-{\psi }_u\left(z,{\varpi }_n\right)-k\right]\left({\varpi }_{n+1}-{\varpi }_n\right)\hfill \\ & =& \left[f\left(z,{\varpi }_n\right)+F_u\left(z,{\varpi }_n\right)\left({\varpi }_{n+1}-{\varpi }_n\right)\right]+\left[g\left(z,{\varpi }_n\right)+G_u\left(z,{\epsilon }_n\right)\left({\varpi }_{n+1}-{\varpi }_n\right)\right]\hfill \\ & & -\left[{\varphi }_u\left(z,{\epsilon }_n\right)+{\psi }_u\left(z,{\varpi }_n\right)+k\right]\left({\varpi }_{n+1}-{\varpi }_n\right)+h\left(z,{\varpi }_n\right).\hfill \end{array}$$

By using (4)–(6), we obtain the following:

$$\begin{array}{ccc}\hfill {\varpi }_{n+1}^{\Delta }& \le & f\left(z,{\varpi }_{n+1}\right)+\varphi \left(z,{\varpi }_{n+1}\right)-\varphi \left(z,{\varpi }_n\right)+g\left(z,{\varpi }_{n+1}\right)+\psi \left(z,{\varpi }_{n+1}\right)-\psi \left(z,{\varpi }_n\right)\hfill \\ & & -{\varphi }_u\left(z,{\epsilon }_n\right)\left({\varpi }_{n+1}-{\varpi }_n\right)-{\psi }_u\left(z,{\varpi }_n\right)\left({\varpi }_{n+1}-{\varpi }_n\right)\hfill \\ & & -k\left({\varpi }_{n+1}-{\varpi }_n\right)+h\left(z,{\varpi }_{n+1}\right)+k\left({\varpi }_{n+1}-{\varpi }_n\right)\hfill \\ & =& \left[f\left(z,{\varpi }_{n+1}\right)+\varphi \left(z,{\varpi }_{n+1}\right)-\varphi \left(z,{\varpi }_n\right)\right]+\left[g\left(z,{\varpi }_{n+1}\right)+\psi \left(z,{\varpi }_{n+1}\right)-\psi \left(z,{\varpi }_n\right)\right]\hfill \\ & & -{\varphi }_u\left(z,{\epsilon }_n\right)\left({\varpi }_{n+1}-{\varpi }_n\right)-{\psi }_u\left(z,{\varpi }_n\right)\left({\varpi }_{n+1}-{\varpi }_n\right)+h\left(z,{\varpi }_{n+1}\right).\hfill \end{array}$$

When the mean value theorem is applied, and because of the properties of ${\varphi }_u\left(z,u\right)$ and ${\psi }_u\left(z,u\right)$, the following inequality can be obtained:

$$\begin{array}{ccc}\hfill {\varpi }_{n+1}^{\Delta }& \le & f\left(z,{\varpi }_{n+1}\right)+{\varphi }_u\left(z,{\epsilon }_n\right)\left({\varpi }_{n+1}-{\varpi }_n\right)\hfill \\ & & +g\left(z,{\varpi }_{n+1}\right)+{\psi }_u\left(z,{\varpi }_n\right)\left({\varpi }_{n+1}-{\varpi }_n\right)\hfill \\ & & -{\varphi }_u\left(z,{\epsilon }_n\right)\left({\varpi }_{n+1}-{\varpi }_n\right)-{\psi }_u\left(z,{\varpi }_n\right)\left({\varpi }_{n+1}-{\varpi }_n\right)+h\left(z,{\varpi }_{n+1}\right)\hfill \\ & =& f\left(z,{\varpi }_{n+1}\right)+g\left(z,{\varpi }_{n+1}\right)+h\left(z,{\varpi }_{n+1}\right)=N\left(z,{\varpi }_{n+1}\right).\hfill \end{array}$$

Hence, ${\varpi }_{n+1}$ is an LS. Similarly, we can show that ${\epsilon }_{n+1}$ is a US. Using (8),

$$\begin{array}{ccc}\hfill {\epsilon }_{n+1}^{\Delta }& =& N\left(z,{\epsilon }_n\right)+\left[F_u\left(z,{\varpi }_n\right)+G_u\left(z,{\epsilon }_n\right)-{\varphi }_u\left(z,{\epsilon }_n\right)-{\psi }_u\left(z,{\varpi }_n\right)-k\right]\left({\epsilon }_{n+1}-{\epsilon }_n\right)\hfill \\ & =& \left[f\left(z,{\epsilon }_n\right)+F_u\left(z,{\varpi }_n\right)\left({\epsilon }_{n+1}-{\epsilon }_n\right)\right]+\left[g\left(z,{\epsilon }_n\right)+G_u\left(z,{\epsilon }_n\right)\left({\epsilon }_{n+1}-{\epsilon }_n\right)\right]\hfill \\ & & -{\varphi }_u\left(z,{\epsilon }_n\right)\left({\epsilon }_{n+1}-{\epsilon }_n\right)-{\psi }_u\left(z,{\varpi }_n\right)\left({\epsilon }_{n+1}-{\epsilon }_n\right)-k\left({\epsilon }_{n+1}-{\epsilon }_n\right)+h\left(z,{\epsilon }_n\right)\hfill \end{array}$$

can be written. In a similar fashion, the following can be obtained:

$$\begin{array}{ccc}\hfill {\epsilon }_{n+1}^{\Delta }& \ge & \left[f\left(z,{\epsilon }_{n+1}\right)+\varphi \left(z,{\epsilon }_{n+1}\right)-\varphi \left(z,{\epsilon }_n\right)\right]+\left[g\left(z,{\epsilon }_{n+1}\right)+\psi \left(z,{\epsilon }_{n+1}\right)-\psi \left(z,{\epsilon }_n\right)\right]\hfill \\ & & -{\varphi }_u\left(z,{\epsilon }_n\right)\left({\epsilon }_{n+1}-{\epsilon }_n\right)-{\psi }_u\left(z,{\varpi }_n\right)\left({\epsilon }_{n+1}-{\epsilon }_n\right)\hfill \\ & & -k\left({\epsilon }_{n+1}-{\epsilon }_n\right)+h\left(z,{\epsilon }_{n+1}\right)+k\left({\epsilon }_{n+1}-{\epsilon }_n\right)\hfill \\ & \ge & f\left(z,{\epsilon }_{n+1}\right)+{\varphi }_u\left(z,{\epsilon }_n\right)\left({\epsilon }_{n+1}-{\epsilon }_n\right)+g\left(z,{\epsilon }_{n+1}\right)+{\psi }_u\left(z,{\varpi }_n\right)\left({\epsilon }_{n+1}-{\epsilon }_n\right)\hfill \\ & & -{\varphi }_u\left(z,{\epsilon }_n\right)\left({\epsilon }_{n+1}-{\epsilon }_n\right)-{\psi }_u\left(z,{\varpi }_n\right)\left({\epsilon }_{n+1}-{\epsilon }_n\right)+h\left(z,{\epsilon }_{n+1}\right)\hfill \\ & =& f\left(z,{\epsilon }_{n+1}\right)+g\left(z,{\epsilon }_{n+1}\right)+h\left(z,{\epsilon }_{n+1}\right)=N\left(z,{\epsilon }_{n+1}\right).\hfill \end{array}$$

Consequently, ${\epsilon }_{n+1}$ is a US. Since $N\left(z,u\right)$ satisfies the Lipschitz condition, from Theorem 1, it follows that ${\varpi }_{n+1}\le {\epsilon }_{n+1}$. Through mathematical induction, we have

$${\varpi }_0\le {\varpi }_1\le ···\le {\varpi }_n\le {\varpi }_{n+1}\le {\epsilon }_{n+1}\le {\epsilon }_n\le ···\le {\epsilon }_1\le {\epsilon }_0$$

for all $n.$ Since the sequences $\left\{{\varpi }_n\left(z\right)\right\}$ and $\left\{{\epsilon }_n\left(z\right)\right\}$ are equicontinuous and uniformly bounded by Theorem 3, it is clear that the sequences uniformly converge to the unique solution of (1) on J. Now, we will prove this convergence is semi-quadratic on J. To show this, we define the following:

$$\begin{array}{c}\hfill p_{n+1}\left(z\right)=u\left(z\right)-{\varpi }_{n+1}\left(z\right)\ge 0,\\ \hfill q_{n+1}\left(z\right)={\epsilon }_{n+1}\left(z\right)-u\left(z\right)\ge 0.\end{array}$$

It is obvious that $p_{n+1}\left(0\right)=u\left(0\right)-{\varpi }_{n+1}\left(0\right)=0$ and $q_{n+1}\left(0\right)=0$. If we take the delta derivative of both sides, then we have the following:

$$\begin{array}{ccc}\hfill p_{n+1}^{\Delta }\left(z\right)& =& u^{\Delta }\left(z\right)-{\varpi }_{n+1}^{\Delta }\left(z\right)\hfill \\ & =& f\left(z,u\right)+g\left(z,u\right)+h\left(z,u\right)-f\left(z,{\varpi }_n\right)-g\left(z,{\varpi }_n\right)-h\left(z,{\varpi }_n\right)\hfill \\ & & -\left[F_u\left(z,{\varpi }_n\right)+G_u\left(z,{\epsilon }_n\right)-{\varphi }_u\left(z,{\epsilon }_n\right)-{\psi }_u\left(z,{\varpi }_n\right)-k\right]\left({\varpi }_{n+1}-{\varpi }_n\right)\hfill \\ & & +\varphi \left(z,u\right)+\psi \left(z,u\right)+\varphi \left(z,{\varpi }_n\right)+\psi \left(z,{\varpi }_n\right)\hfill \\ & & -\varphi \left(z,u\right)-\psi \left(z,u\right)-\varphi \left(z,{\varpi }_n\right)-\psi \left(z,{\varpi }_n\right)\hfill \\ & =& F\left(z,u\right)-F\left(z,{\varpi }_n\right)+G\left(z,u\right)-G\left(z,{\varpi }_n\right)\hfill \\ & & -\left[F_u\left(z,{\varpi }_n\right)+G_u\left(z,{\epsilon }_n\right)-{\varphi }_u\left(z,{\epsilon }_n\right)-{\psi }_u\left(z,{\varpi }_n\right)-k\right]\left({\varpi }_{n+1}-{\varpi }_n\right)\hfill \\ & & -\left[\varphi \left(z,u\right)-\varphi \left(z,{\varpi }_n\right)\right]-\left[\psi \left(z,u\right)-\psi \left(z,{\varpi }_n\right)\right]+h\left(z,u\right)-h\left(z,{\varpi }_n\right).\hfill \end{array}$$

When the mean value theorem is applied several times and when using the properties of the functions,

$$\begin{array}{ccc}\hfill p_{n+1}^{\Delta }\left(z\right)& \le & F_{uu}\left(z,\gamma \right)\left(u-{\varpi }_n\right)\left(u-{\varpi }_n\right)-G_{uu}\left(z,\sigma \right)\left({\epsilon }_n-{\varpi }_n\right)\left(u-{\varpi }_n\right)\hfill \\ & & +{\varphi }_{uu}\left(z,\delta \right)\left({\epsilon }_n-{\varpi }_n\right)\left(u-{\varpi }_n\right)-{\psi }_{uu}\left(z,\eta \right)\left(u-{\varpi }_n\right)\left(u-{\varpi }_n\right)+2k\left(u-{\varpi }_n\right)\hfill \\ & & -\left[F_u\left(z,{\varpi }_n\right)+G_u\left(z,{\epsilon }_n\right)-{\varphi }_u\left(z,{\epsilon }_n\right)-{\psi }_u\left(z,{\varpi }_n\right)-k\right]\left({\varpi }_{n+1}-u\right)\hfill \\ & =& \left[F_{uu}\left(z,\gamma \right)-{\psi }_{uu}\left(z,\eta \right)\right]{\left(u-{\varpi }_n\right)}^2+\left[{\varphi }_{uu}\left(z,\delta \right)-G_{uu}\left(z,\sigma \right)\right]\left({\epsilon }_n-{\varpi }_n\right)\left(u-{\varpi }_n\right)\hfill \\ & & +\left[F_u\left(z,{\varpi }_n\right)+G_u\left(z,{\epsilon }_n\right)-{\varphi }_u\left(z,{\epsilon }_n\right)-{\psi }_u\left(z,{\varpi }_n\right)-k\right]\left(u-{\varpi }_{n+1}\right)+2k\left(u-{\varpi }_n\right)\hfill \\ & =& \left[f_{uu}\left(z,\gamma \right)+{\varphi }_{uu}\left(z,\gamma \right)-{\psi }_{uu}\left(z,\eta \right)\right]{\left(u-{\varpi }_n\right)}^2\hfill \\ & & +\left[{\varphi }_{uu}\left(z,\delta \right)-g_{uu}\left(z,\sigma \right)-{\psi }_{uu}\left(z,\sigma \right)\right]\left({\epsilon }_n-{\varpi }_n\right)\left(u-{\varpi }_n\right)\hfill \\ & & +\left[\begin{array}{c}{f}_u\left(z,{\varpi }_n\right)+{\varphi }_u\left(z,{\varpi }_n\right)+g_u\left(z,{\epsilon }_n\right)+{\psi }_u\left(z,{\epsilon }_n\right)\\ -{\varphi }_u\left(z,{\epsilon }_n\right)-{\psi }_u\left(z,{\varpi }_n\right)-k\end{array}\right]\left(u-{\varpi }_{n+1}\right)\hfill \\ & & +2k\left(u-{\varpi }_n\right),\hfill \end{array}$$

where ${\varpi }_n<\gamma <u,$${\varpi }_n<\sigma <{\epsilon }_n,$${\varpi }_n<\delta <{\epsilon }_n,$${\varpi }_n<\eta <u,$ can be obtained. Thus,

$$\begin{array}{ccc}\hfill p_{n+1}^{\Delta }\left(z\right)& \le & \left(A+B+D\right)p_n^2+\left(B+C+D\right)p_n^2+\left(B+C+D\right)\left(p_n{q}_n\right)\hfill \\ & & +\left(E+2R+z^{*}+2S\right)p_{n+1}+2k{p}_n,\hfill \end{array}$$

where $\left|f_{uu}\right|<A,\left|{\varphi }_{uu}\right|<B,\left|g_{uu}\right|<C,\left|{\psi }_{uu}\right|<D,\left|f_u\right|<E,\left|g_u\right|<T^{*},\left|{\varphi }_u\right|<R,\left|{\psi }_u\right|<S.$ When the Cauchy inequality is applied to term $\left(B+C+D\right)\left(p_n{q}_n\right),$ then we have the following:

$$\begin{array}{ccc}\hfill p_{n+1}^{\Delta }\left(z\right)& \le & \left(A+\frac{5B}{2}+\frac{3C}{2}+\frac{5D}{2}\right)p_n^2+\left(\frac{B}{2}+\frac{C}{2}+\frac{D}{2}\right)q_n^2\hfill \\ & & +\left(E+2R+T^{*}+2S\right)p_{n+1}+2k{p}_n\hfill \\ & =& M{p}_n^2+N{q}_n^2+W{p}_{n+1}+2k{p}_n,\hfill \end{array}$$

where $A+\frac{5B}{2}+\frac{3C}{2}+\frac{5D}{2}=M,\frac{B}{2}+\frac{C}{2}+\frac{D}{2}=N,E+2R+T^{*}+2S=W$. Since

$$p_{n+1}^{\Delta }\left(z\right)-W{p}_{n+1}\le M{p}_n^2+N{q}_n^2+2k{p}_n$$

is linear in $p_{n+1}$, we have to use Gronwall’s inequality. Hence, it can be written as follows:

$$\begin{array}{ccc}\hfill 0& \le & p_{n+1}\le e^{Wz}\underset{0}{\overset{z}{\int }}\left[M{p}_n^2+N{q}_n^2+2k{p}_n\right]e^{-Ws}\Delta s\hfill \\ & =& \underset{0}{\overset{z}{\int }}\left[M{p}_n^2+N{q}_n^2+2k{p}_n\right]e^{W\left(z-s\right)}\Delta s\hfill \\ & =& \left[M{p}_n^2+N{q}_n^2+2k{p}_n\right]\frac{e^{Wz}-1}{W}\hfill \\ & \le & \left[M{p}_n^2+N{q}_n^2+2k{p}_n\right]\frac{e^{Wz}}{W}.\hfill \end{array}$$

Therefore, one can see the following:

$$\begin{array}{ccc}\hfill \underset{J}{max}\left|p_{n+1}\right|& \le & \left\{\underset{J}{Mmax}\left|p_n^2\right|+\underset{J}{Nmaks}\left|q_n^2\right|+\underset{J}{2kmax}\left|p_n\right|\right\}\frac{e^{Wz}}{W}\hfill \\ \hfill \underset{J}{max}\left|u-{\varpi }_{n+1}\right|& \le & \left\{\underset{J}{Mmax}{\left|u-{\varpi }_n\right|}^2+\underset{J}{Nmax}{\left|{\epsilon }_n-u\right|}^2+\underset{J}{2kmax}\left|u-{\varpi }_n\right|\right\}\frac{e^{Wz}}{W}.\hfill \end{array}$$

This shows us that $\left\{{\varpi }_n\left(z\right)\right\}$ converges to the solution of (1) semi-quadratically. Similarly, it can be easily shown that the convergence of $\left\{{\epsilon }_n\left(z\right)\right\}$ is also semi-quadratical. This completes the proof. □

3.2. The Weak Quadraticity of the Convergence

Under different conditions, we examined whether the rate of quadraticity changes. By placing another condition on the function $h\left(z,u\right)$, the sequences converged as weakly quadratic.

Theorem 5. 

Suppose that (A₁) and (A₂) of Theorem 4 hold. Assume further that $h\left(z,u\right)$ is a continuous function. Then, there exist monotone sequences $\left\{{\varpi }_n\left(z\right)\right\}$ and $\left\{{\epsilon }_n\left(z\right)\right\}$, which converge uniformly to the unique solution of (1), and the convergence is weakly quadratic.

Proof of Theorem 5. 

In view of (4) and (5), we consider the following IVPs:

$$\begin{array}{ccc}\hfill u^{\Delta }& =& m\left(z,u;{\varpi }_0,{\epsilon }_0\right)=f\left(z,{\varpi }_0\right)+g\left(z,{\varpi }_0\right)+h\left(z,u\right)\hfill \\ & & +\left[f_u\left(z,{\varpi }_0\right)+{\varphi }_u\left(z,{\varpi }_0\right)-{\varphi }_u\left(z,{\epsilon }_0\right)\right]\left(u-{\varpi }_0\right)\hfill \\ & & +\left[g_u\left(z,{\epsilon }_0\right)+{\psi }_u\left(z,{\epsilon }_0\right)-{\psi }_u\left(z,{\varpi }_0\right)\right]\left(u-{\varpi }_0\right)\hfill \\ \hfill u\left(0\right)& =& u_0,\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill v^{\Delta }& =& M\left(z,v;{\varpi }_0,{\epsilon }_0\right)=f\left(z,{\epsilon }_0\right)+g\left(z,{\epsilon }_0\right)+h\left(z,v\right)\hfill \\ & & +\left[f_u\left(z,{\varpi }_0\right)+{\varphi }_u\left(z,{\varpi }_0\right)-{\varphi }_u\left(z,{\epsilon }_0\right)\right]\left(v-{\varpi }_0\right)\hfill \\ & & +\left[g_u\left(z,{\epsilon }_0\right)+{\psi }_u\left(z,{\epsilon }_0\right)-{\psi }_u\left(z,{\varpi }_0\right)\right]\left(v-{\varpi }_0\right)\hfill \\ \hfill v\left(0\right)& =& u_0.\hfill \end{array}$$

(10)

Since ${\varpi }_0$ is the LS of (1),

$${\varpi }_0^{\Delta }\le f\left(z,{\varpi }_0\right)+g\left(z,{\varpi }_0\right)+h\left(z,{\varpi }_0\right).$$

${\epsilon }_0$ is the US of (1). Based on the inequalities of (4) and (5), it can be written as follows

$$\begin{array}{ccc}\hfill {\epsilon }_0^{\Delta }& \ge & f\left(z,{\varpi }_0\right)+\left[f_u\left(z,{\varpi }_0\right)+{\varphi }_u\left(z,{\varpi }_0\right)\right]\left({\epsilon }_0-{\varpi }_0\right)-{\varphi }_u\left(z,{\epsilon }_0\right)\left({\epsilon }_0-{\varpi }_0\right)\hfill \\ & & +g\left(z,{\varpi }_0\right)+\left[g_u\left(z,{\epsilon }_0\right)+{\psi }_u\left(z,{\epsilon }_0\right)\right]\left({\epsilon }_0-{\varpi }_0\right)-{\psi }_u\left(z,{\varpi }_0\right)\left({\epsilon }_0-{\varpi }_0\right)\hfill \\ \hfill +h\left(z,{\epsilon }_0\right)& =& m\left(z,{\epsilon }_0;{\varpi }_0,{\epsilon }_0\right).\hfill \end{array}$$

Therefore, ${\varpi }_0$ and ${\epsilon }_0$ are the LS and US of (9). Based on Theorem 2, there exists a unique solution of (9) called ${\varpi }_1$ such that ${\varpi }_0\left(z\right)\le {\varpi }_1\left(z\right)\le {\epsilon }_0\left(z\right),z\in T^k.$ Similarly, we can write

$$\begin{array}{ccc}\hfill {\varpi }_0^{\Delta }& \le & f\left(z,{\epsilon }_0\right)+\left[f_u\left(z,{\varpi }_0\right)+{\varphi }_u\left(z,{\varpi }_0\right)\right]\left({\varpi }_0-{\epsilon }_0\right)-{\varphi }_u\left(z,{\epsilon }_0\right)\left({\varpi }_0-{\epsilon }_0\right)\hfill \\ & & +g\left(z,{\epsilon }_0\right)+\left[g_u\left(z,{\epsilon }_0\right)+{\psi }_u\left(z,{\epsilon }_0\right)\right]\left({\varpi }_0-{\epsilon }_0\right)-{\psi }_u\left(z,{\varpi }_0\right)\left({\varpi }_0-{\epsilon }_0\right)\hfill \\ & & +h\left(z,{\epsilon }_0\right)\hfill \\ & =& M\left(z,{\varpi }_0;{\varpi }_0,{\epsilon }_0\right)\hfill \end{array}$$

and

$${\epsilon }_0^{\Delta }\ge f\left(z,{\epsilon }_0\right)+g\left(z,{\epsilon }_0\right)+h\left(z,{\epsilon }_0\right)=M\left(z,{\epsilon }_0;{\varpi }_0,{\epsilon }_0\right).$$

Hence, ${\varpi }_0$ and ${\epsilon }_0$ are the natural LS and US of (10). Thus, there exists a unique solution of (10) called ${\epsilon }_1$ such that ${\varpi }_0\left(z\right)\le {\epsilon }_1\left(z\right)\le {\epsilon }_0\left(z\right),$$z\in T^k.$ Now, it will be shown that ${\varpi }_1\left(z\right)\le {\epsilon }_1\left(z\right).$ By using (9), we obtain the following:

$$\begin{array}{ccc}\hfill {\varpi }_1^{\Delta }& \le & f\left(z,{\varpi }_1\right)+\left[f_u\left(z,{\varpi }_0\right)+{\varphi }_u\left(z,{\varpi }_0\right)\right]\left({\varpi }_0-{\varpi }_1\right)+{\varphi }_u\left(z,{\epsilon }_0\right)\left({\varpi }_1-{\varpi }_0\right)\hfill \\ & & +\left[f_u\left(z,{\varpi }_0\right)+{\varphi }_u\left(z,{\varpi }_0\right)\right]\left({\varpi }_1-{\varpi }_0\right)-{\varphi }_u\left(z,{\epsilon }_0\right)\left({\varpi }_1-{\varpi }_0\right)\hfill \\ & & +g\left(z,{\varpi }_1\right)+\left[g_u\left(z,{\varpi }_1\right)+{\psi }_u\left(z,{\varpi }_1\right)\right]\left({\varpi }_0-{\varpi }_1\right)+{\psi }_u\left(z,{\varpi }_0\right)\left({\varpi }_1-{\varpi }_0\right)\hfill \\ & & +\left[g_u\left(z,{\varpi }_1\right)+{\psi }_u\left(z,{\varpi }_1\right)\right]\left({\varpi }_1-{\varpi }_0\right)-{\psi }_u\left(z,{\varpi }_0\right)\left({\varpi }_1-{\varpi }_0\right)\hfill \\ & & +h\left(z,{\varpi }_1\right)\hfill \\ & =& f\left(z,{\varpi }_1\right)+g\left(z,{\varpi }_1\right)+h\left(z,{\varpi }_1\right).\hfill \end{array}$$

Then, we obtain the following:

$${\varpi }_1^{\Delta }\le f_1\left(z,{\varpi }_1\right)+f_2\left(z,{\varpi }_1\right)+f_3\left(z,{\varpi }_1\right).$$

Similarly, by using (8) with inequalities (2) and (3), we obtain the following:

$$\begin{array}{ccc}\hfill {\epsilon }_1^{\Delta }& \ge & f\left(z,{\epsilon }_1\right)+F_u\left(z,{\epsilon }_1\right)\left({\epsilon }_0-{\epsilon }_1\right)+\varphi \left(z,{\epsilon }_1\right)-\varphi \left(z,{\epsilon }_0\right)\hfill \\ & & +F_u\left(z,{\varpi }_0\right)\left({\epsilon }_1-{\epsilon }_0\right)-{\varphi }_u\left(z,{\epsilon }_0\right)\left({\epsilon }_1-{\epsilon }_0\right)\hfill \\ & & +g\left(z,{\epsilon }_1\right)+G_u\left(z,{\epsilon }_0\right)\left({\epsilon }_0-{\epsilon }_1\right)+\psi \left(z,{\epsilon }_1\right)-\psi \left(z,{\epsilon }_0\right)\hfill \\ & & +G_u\left(z,{\epsilon }_0\right)\left({\epsilon }_1-{\epsilon }_0\right)-{\psi }_u\left(z,{\varpi }_0\right)\left({\epsilon }_1-{\epsilon }_0\right)+h\left(z,{\epsilon }_1\right).\hfill \end{array}$$

With the condition (A₂), using the mean value theorem, we have the following:

$$\begin{array}{ccc}\hfill {\epsilon }_1^{\Delta }& \ge & f\left(z,{\epsilon }_1\right)+F_u\left(z,{\varpi }_0\right)\left({\epsilon }_0-{\epsilon }_1\right)+{\varphi }_u\left(z,{\epsilon }_0\right)\left({\epsilon }_1-{\epsilon }_0\right)\hfill \\ & & +F_u\left(z,{\varpi }_0\right)\left({\epsilon }_1-{\epsilon }_0\right)-{\varphi }_u\left(z,{\epsilon }_0\right)\left({\epsilon }_1-{\epsilon }_0\right)\hfill \\ & & +g\left(z,{\epsilon }_1\right)+G_u\left(z,{\epsilon }_0\right)\left({\epsilon }_0-{\epsilon }_1\right)+{\psi }_u\left(z,{\varpi }_0\right)\left({\epsilon }_1-{\epsilon }_0\right)\hfill \\ & & +G_u\left(z,{\epsilon }_0\right)\left({\epsilon }_1-{\epsilon }_0\right)-{\psi }_u\left(z,{\varpi }_0\right)\left({\epsilon }_1-{\epsilon }_0\right)+h\left(z,{\epsilon }_1\right)\hfill \\ & =& f\left(z,{\epsilon }_1\right)+g\left(z,{\epsilon }_1\right)+h\left(z,{\epsilon }_1\right).\hfill \end{array}$$

Based on Theorem 1, since the condition (3) holds and ${\varpi }_1,$${\epsilon }_1$ are the LS and US of (1), respectively, ${\varpi }_1\left(0\right)\le {\epsilon }_1\left(0\right)$ yields ${\varpi }_1\left(z\right)\le {\epsilon }_1\left(z\right).$ Therefore, we have established the following inequalities:

$${\varpi }_0\left(z\right)\le {\varpi }_1\left(z\right)\le {\epsilon }_1\left(z\right)\le {\epsilon }_0\left(z\right).$$

If this process continues, we have the following:

$${\varpi }_0\left(z\right)\le {\varpi }_1\left(z\right)\le {\varpi }_2\left(z\right)\le ···\le {\varpi }_n\left(z\right)\le {\epsilon }_n\left(z\right)\le ···\le {\epsilon }_2\left(z\right)\le {\epsilon }_1\left(z\right)\le {\epsilon }_0\left(z\right).$$

Here, the elements of the monotone sequences $\left\{{\varpi }_n\left(z\right)\right\}$ and $\left\{{\epsilon }_n\left(z\right)\right\}$ are the unique solutions of the following linear IVPs:

$$\begin{array}{ccc}\hfill {\varpi }_{n+1}^{\Delta }& =& g\left(z,{\varpi }_n,{\epsilon }_n;{\varpi }_{n+1}\right),\hfill \\ \hfill {\varpi }_{n+1}\left(0\right)& =& u_0,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\epsilon }_{n+1}^{\Delta }& =& G\left(z,{\varpi }_n,{\epsilon }_n;{\epsilon }_{n+1}\right),\hfill \\ \hfill {\epsilon }_{n+1}\left(0\right)& =& u_0.\hfill \end{array}$$

Since the sequences $\left\{{\varpi }_n\left(z\right)\right\}$ and $\left\{{\epsilon }_n\left(z\right)\right\}$ are uniformly bounded and equicontinuous, based on Theorem 3 (the Arzela Ascoli theorem), these sequences converge to the unique solution (1) uniformly. Now, we shall show that the convergence of the sequences $\left\{{\varpi }_n\left(z\right)\right\}$ and $\left\{{\epsilon }_n\left(z\right)\right\}$ to the unique solution $u\left(z\right)$ of (1) is weakly quadratic on $T^k.$ To show this, let us define the following:

$$\begin{array}{ccc}\hfill p_{n+1}\left(z\right)& =& u\left(z\right)-{\varpi }_{n+1}\left(z\right)\ge 0\hfill \\ \hfill p_{n+1}\left(0\right)& =& 0,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill q_{n+1}\left(z\right)& =& {\epsilon }_{n+1}\left(z\right)-u\left(z\right)\ge 0\hfill \\ \hfill q_{n+1}\left(0\right)& =& 0.\hfill \end{array}$$

When we take the delta derivative of both sides, and using the definition ${\varpi }_n,$${\epsilon }_n$ together with applying the mean value theorem, the following can be obtained:

$$\begin{array}{ccc}\hfill p_{n+1}^{\Delta }& =& u^{\Delta }-{\varpi }_{n+1}^{\Delta }\hfill \\ & =& F\left(z,u\right)-F\left(z,{\varpi }_n\right)+G\left(z,u\right)-G\left(z,{\varpi }_n\right)-F_u\left(z,{\varpi }_n\right)\left({\varpi }_{n+1}-u\right)-F_u\left(z,{\varpi }_n\right)\left(u-{\varpi }_n\right)\hfill \\ & & -G_u\left(z,{\epsilon }_n\right)\left({\varpi }_{n+1}-u\right)-G_u\left(z,{\epsilon }_n\right)\left(u-{\varpi }_n\right)+{\varphi }_u\left(z,{\epsilon }_n\right)\left({\varpi }_{n+1}-u\right)+{\varphi }_u\left(z,{\epsilon }_n\right)\left(u-{\varpi }_n\right)\hfill \\ & & +{\psi }_u\left(z,{\varpi }_n\right)\left({\varpi }_{n+1}-u\right)+{\psi }_u\left(z,{\varpi }_n\right)\left(u-{\varpi }_n\right)+\varphi \left(z,{\varpi }_n\right)-\varphi \left(z,u\right)+\psi \left(z,{\varpi }_n\right)-\psi \left(z,u\right)\hfill \\ & & +h\left(z,u\right)-h\left(z,{\varpi }_{n+1}\right)\hfill \\ & \le & \left[F_u\left(z,u\right)-F_u\left(z,{\varpi }_n\right)\right]\left(u-{\varpi }_n\right)+\left[G_u\left(z,{\varpi }_n\right)-G_u\left(z,{\epsilon }_n\right)\right]\left(u-{\varpi }_n\right)\hfill \\ & & +\left[F_u\left(z,{\varpi }_n\right)-{\varphi }_u\left(z,{\epsilon }_n\right)\right]\left(u-{\varpi }_{n+1}\right)+\left[G_u\left(z,{\epsilon }_n\right)-{\psi }_u\left(z,{\varpi }_n\right)\right]\left(u-{\varpi }_{n+1}\right)\hfill \\ & & +\left[{\varphi }_u\left(z,{\epsilon }_n\right)-{\varphi }_u\left(z,u\right)\right]\left(u-{\varpi }_n\right)+h\left(z,u\right)-h\left(z,{\varpi }_{n+1}\right)\hfill \\ & \le & F_{uu}\left(z,a\right)\left(u-{\varpi }_n\right)\left(u-{\varpi }_n\right)-G_{uu}\left(z,b\right)\left({\epsilon }_n-{\varpi }_n\right)\left(u-{\varpi }_n\right)\hfill \\ & & +\left[F_u\left(z,{\varpi }_n\right)-{\varphi }_u\left(z,{\epsilon }_n\right)\right]\left(u-{\varpi }_{n+1}\right)+\left[G_u\left(z,{\epsilon }_n\right)-{\psi }_u\left(z,{\varpi }_n\right)\right]\left(u-{\varpi }_{n+1}\right)\hfill \\ & & +{\varphi }_{uu}\left(z,c\right)\left({\epsilon }_n-u\right)\left(u-{\varpi }_n\right)+h\left(z,u\right)-h\left(z,{\varpi }_{n+1}\right),\hfill \end{array}$$

where ${\varpi }_n<a<u,{\varpi }_n<b<{\epsilon }_n,u<c<{\epsilon }_n.$ Therefore, the following can be written:

$$\begin{array}{ccc}\hfill p_{n+1}^{\Delta }& \le & A{p}_n^2+B{p}_n{q}_n+B{p}_n^2+C{p}_{n+1}+D{p}_{n+1}+E{p}_n{q}_n+M\hfill \\ & =& (A+B)p_n^2+(B+E)p_n{q}_n+(C+D)p_{n+1}+M\hfill \end{array}$$

where $\left|F_{uu}\left(z,u\right)\right|\le A,\left|G_{uu}\left(z,u\right)\right|\le B,\left|F_u\left(z,u\right)-{\varphi }_u\left(z,u\right)\right|\le C,\left|G_u\left(z,u\right)-{\psi }_u\left(z,u\right)\right|\le D,$$\left|{\varphi }_{uu}\left(z,u\right)\right|\le E,\left|h\left(z,u\right)-h\left(z,{\varpi }_{n+1}\right)\right|\le M,$ and $A,B,C,D,E,M$ are positive constants. If the Cauchy inequality is applied to the term $B{p}_n{q}_n$, then we have

$$\begin{array}{ccc}\hfill p_{n+1}^{\Delta }& \le & (A+B)p_n^2+\frac{B+E}{2}{p}_n^2+\frac{B+E}{2}{q}_n^2+(C+D)p_{n+1}+M\hfill \\ & =& \frac{2A+3B}{2}{p}_n^2+\frac{B+E}{2}{q}_n^2+(C+D)p_{n+1}+M\hfill \end{array}$$

which is linear in $p_{n+1}.$ Now, if Gronwall’s inequality is used, then we have the following:

$$\begin{array}{ccc}\hfill p_{n+1}{e}^{-\left(C+D\right)z}& \le & p_{n+1}\left(0\right)+\underset{0}{\overset{z}{\int }}\left[\left(\frac{2A+3B}{2}\right)p_n^2+\left(\frac{B+E}{2}\right)q_n^2+M\right]e^{\left(C+D\right)\left(z-s\right)}\Delta s\hfill \\ & =& \left[\left(\frac{2A+3B}{2}\right)p_n^2+\left(\frac{B+E}{2}\right)q_n^2+M\right]\underset{0}{\overset{z}{\int }}{e}^{\left(C+D\right)\left(z-s\right)}\Delta s.\hfill \end{array}$$

Therefore, the following can be written:

$$\begin{array}{ccc}\hfill \underset{J}{max}\left|p_{n+1}\right|& \le & \underset{J}{max}\left|\left[\left(\frac{2A+3B}{2}\right)p_n^2+\left(\frac{B+E}{2}\right)q_n^2+M\right]\frac{e^{\left(C+D\right)z}}{C+D}\right|\hfill \\ & =& \left(\frac{2A+3B}{2}\right)\frac{e^{\left(C+D\right)z}}{C+D}\underset{J}{max}\left|p_n^2\right|+\left(\frac{B+E}{2}\right)\frac{e^{\left(C+D\right)z}}{C+D}\underset{J}{max}\left|q_n^2\right|\hfill \\ & & M\frac{e^{\left(C+D\right)z}}{C+D}.\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \underset{J}{max}\left|u-{\varpi }_{n+1}\right|& \le & \left(\frac{2A+3B}{2}\right)\frac{e^{\left(C+D\right)z}}{C+D}\underset{J}{max}{\left|u-{\varpi }_n\right|}^2\hfill \\ & & +\left(\frac{B+E}{2}\right)\frac{e^{\left(C+D\right)z}}{C+D}\underset{J}{max}{\left|{\epsilon }_n-u\right|}^2+M\frac{e^{\left(C+D\right)z}}{C+D}.\hfill \end{array}$$

This shows that $\left\{{\varpi }_n\left(z\right)\right\}$ converges (1) weakly quadratically. Similarly, it can be easily shown that the convergence of $\left\{{\epsilon }_n\left(z\right)\right\}$ is also weakly quadratical. This completes the proof. □

3.3. An Example

For the particular case of system (1), let us consider the dynamic initial value problem:

$$\begin{array}{ccc}\hfill u^{\Delta }& =& z^2-\frac{z}{3}-z^3,\hfill \\ \hfill u\left(0\right)& =& 1,z\in \left[0,1\right],\hfill \end{array}$$

(11)

where $f\left(z,u\right)=z^2,$$g\left(z,u\right)=-\frac{z}{3}$$,h\left(z,u\right)=-z^3$ and $\varphi \left(z,u\right)=z,$$\phi \left(z,u\right)=z.$ Let

$${\varpi }_0\left(z\right)=-1,{\epsilon }_0\left(z\right)=1,$$

for all $z\in \left[0,1\right].$ It can be determined that

$$\begin{array}{ccc}\hfill {\varpi }_0^{\Delta }& =& 0\le f\left(z,{\varpi }_0\right)+g\left(z,{\varpi }_0\right)+h\left(z,{\varpi }_0\right)=\frac{7}{3},\hfill \\ \hfill {\varpi }_0\left(0\right)& =& -1\le u\left(0\right)=1\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\epsilon }_0^{\Delta }& =& 0\ge f\left(z,{\epsilon }_0\right)+g\left(z,{\epsilon }_0\right)+h\left(z,{\epsilon }_0\right)=-\frac{1}{3},\hfill \\ \hfill {\epsilon }_0\left(0\right)& =& 1\ge u\left(0\right)=1.\hfill \end{array}$$

This implies that ${\varpi }_0$ (LS) and ${\epsilon }_0$ (US) are of a natural type for (1). Consider the functions

$$\begin{array}{ccc}\hfill {\varpi }_1\left(z\right)& =& \frac{1}{16}\left(27e^{\frac{-16z}{3}}-9\right),\hfill \\ \hfill {\varpi }_1\left(0\right)& =& 1\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\epsilon }_1\left(z\right)& =& \frac{1}{16}\left(e^{\frac{-16z}{3}}+15\right),\hfill \\ \hfill {\epsilon }_1\left(0\right)& =& 1.\hfill \end{array}$$

These satisfy the following inequality:

$${\varpi }_0\left(z\right)\le {\varpi }_1\left(z\right)\le {\epsilon }_1\left(z\right)\le {\epsilon }_0\left(z\right),z\in \left[0,1\right].$$

Similarly, ${\varpi }_2\left(z\right),{\varpi }_3\left(z\right),···,{\varpi }_n\left(z\right)$ and ${\epsilon }_2\left(z\right),{\epsilon }_3\left(z\right),···,{\epsilon }_n\left(z\right)$ can be obtained if it can be continued like this. Then, all the conditions of Theorem 5 are satisfied. Thus, we obtain the existence of monotone sequences that converge uniformly to the unique solution of (1).

4. Discussion

We have applied the QM to the given nonlinear differential equations on time scales. Under certain conditions, we have constructed monotone sequences that converge uniformly and monotonically to the unique solution of the problem. The most important advantage of this method is that each element of the monotone function sequence is the solution of linear differential equations. Also, it has been shown that the convergence is semi-quadratic in the first result and weakly quadratic in the other.

5. Conclusions

In this paper, by using the technique of upper and lower solutions, under convenient conditions, we showed that the convergence is semi-quadratic and weakly quadratic. We observed that similar results were obtained in parallel with the results given by the classical derivative. The novelty of the applied quasilinearization method is the change in the convergence speed under different conditions. While the rate of convergence is semi-quadratic in Section 3, it is weakly quadratic in Section 4. Therefore, it was seen that the rate of convergence may also change if the conditions of the functions are changed.