On a Nonlinear Volterra-Fredholm Integrodifferential Equation on Time Scales

The main aim in this work is to obtain an integral inequality with a clear estimate on time scales. The obtained inequality is used as a tool to investigate some basic qualitative properties of solutions to certain nonlinear Volterra-Fredholm integrodifferential equations on time scales.

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Noori, M. and Mahmood, A. (2020) On a Nonlinear Volterra-Fredholm Integrodifferential Equation on Time Scales. Open Access Library Journal, 7, 1-10. doi: 10.4236/oalib.1106103. 1. Introduction

The theory of time scales had been begun in 1988 by Stefan Hilger , in order to develop a theory that can standardize a continuous and discrete analysis. Recently several authors in this field have investigated various forms of integral and integrodifferential equations under different hypotheses by using different ways, see     . In this article we consider the nonlinear integrodifferential equation of the following form

$\begin{array}{l}{y}^{\Delta }\left(t\right)=h\left(t,y\left(t\right),{y}^{\Delta }\left(t\right),{\int }_{\alpha }^{t}{h}_{1}\left(t,z,y\left(z\right),{y}^{\Delta }\left(z\right)\right)\Delta z,{\int }_{\alpha }^{\beta }{h}_{2}\left(t,z,y\left(z\right),{y}^{\Delta }\left(z\right)\right)\Delta z\right),\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in {J}_{\mathbb{T}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{with the initial condition}\text{\hspace{0.17em}}y\left(\alpha \right)={y}_{0},\end{array}$ (1.1)

where y is unknown function and $h:{J}_{\mathbb{T}}×{ℝ}^{n}×{ℝ}^{n}×{ℝ}^{n}×{ℝ}^{n}\to {ℝ}^{n}$, ${h}_{1},{h}_{2}:{J}_{\mathbb{T}}^{2}×{ℝ}^{n}×{ℝ}^{n}\to {ℝ}^{n}$ and $h,{h}_{1},{h}_{2}$ are given functions, assuming them to be rd-continuous functions, $\alpha <\beta ,z\le t$ and ${J}_{\mathbb{T}}=J\cap \mathbb{T},J=\left[\alpha ,\infty \right)$. We denote a time scale by $\mathbb{T}$ which is nonempty closed subset of $ℝ$. ${ℝ}^{n}$ denotes Euclidean space with a suitable norm defined by $|\text{ }.\text{ }|$.

We can investigate the existence and uniqueness results for (1.1) by using the technique present in .

2. Preliminaries

The operators $\sigma \left(t\right)$ and $\rho \left(t\right)$ denote the forward and backward operators respectively which are defined by $\sigma \left(t\right)=\mathrm{inf}\left\{s\in \mathbb{T}:s>t\right\}\in \mathbb{T}$ and $\rho \left(t\right)=\mathrm{sup}\left\{s\in \mathbb{T}:s, for all $t\in \mathbb{T}$.

For $t\in \mathbb{T}$, If $t<\mathrm{sup}\mathbb{T}$ and $\sigma \left(t\right)=t$, then t is said to be right-dense; while If $t>\mathrm{inf}\mathbb{T}$ and $\rho \left(t\right)=t$, then t is said to be left-dense. The graininess $\mu :\mathbb{T}\to \left[0,\infty \right)$ is defined by $\mu \left(t\right)=\sigma \left(t\right)-t$. The set ${\mathbb{T}}^{k}$ is denoted by

${\mathbb{T}}^{k}=\left\{\begin{array}{l}\mathbb{T}\\left(\rho \left(\mathrm{sup}\mathbb{T}\right),\mathrm{sup}\mathbb{T}\right]\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}\mathrm{sup}\mathbb{T}<\infty \\ \mathbb{T}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{otherwise}\end{array}$

Let $z:\mathbb{T}\to ℝ$, $t\in {\mathbb{T}}^{k}$, then ${z}^{\Delta }\left(t\right)$ denotes the delta derivative of z at t which is exist with the property that given $\epsilon >0$ there is a neighbourhood U of t such that $|z\left(\sigma \left(t\right),\tau \right)-z\left(s,\tau \right)-{z}^{\Delta }\left(t,\tau \right)\left(\sigma \left(t\right)-s\right)|\le \epsilon |\sigma \left(t\right)-s|$ for all $s\in U$. Then $g\left(t\right)={\int }_{\alpha }^{t}z\left(t,\tau \right)\Delta \tau$ implies ${g}^{\Delta }\left(t\right)={\int }_{\alpha }^{t}{z}^{\Delta }\left(t,\tau \right)\Delta \tau +z\left(\sigma \left(t\right),t\right)$. If a function $g:\mathbb{T}\to ℝ$ is continuous at any right-dense point $t\in \mathbb{T}$ and the left-hand limits exists (finite) at any left-dense point $t\in \mathbb{T}$, then g is said to be rd-continuous. ${C}_{rd}$ denotes the class of all rd-continuous functions. We denote the class of all regressive functions by $\mathcal{R}$ which is defined by

$\mathcal{R}=\left\{p\in {C}_{rd}\left(\mathbb{T},ℝ\right)\text{\hspace{0.17em}}\text{ }\text{and}\text{\hspace{0.17em}}\text{ }1+p\left(t\right)\mu \left(t\right)\ne 0,\forall t\in \mathbb{T}\right\}$

For $p\in \mathcal{R}$, we define ${e}_{p}\left(t,s\right)=\mathrm{exp}\left({\int }_{s}^{t}{\xi }_{\mu }\left(\tau \right)\left(p\left(\tau \right)\right)\Delta \tau \right)$ for $t,s\in \mathbb{T}$, with the cylinder transformation ${\xi }_{h}\left(\tau \right)=\left\{\begin{array}{l}\frac{\mathrm{log}\left(1+hz\right)}{h}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}h\ne 0\\ z\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}h=0\end{array}$.

For more basic information about time scales calculus, see  .

We need the following result given in .

Lemma 2.1. suppose $\nu ,b\in {C}_{rd}$ and $a\in {\mathcal{R}}^{+}$. Then

${\nu }^{\Delta }\left(t\right)\le a\left(t\right)\nu \left(t\right)+b\left(t\right)$, for all $t\in \mathbb{T}$

Implies $\nu \left(t\right)\le \nu \left(\alpha \right){e}_{a}\left(t,\alpha \right)+{\int }_{\alpha }^{t}{e}_{a}\left(t,\sigma \left(\tau \right)\right)b\left(\tau \right)\Delta \tau$, for all $t\in \mathbb{T}$.

3. Main Results

In the following result we establish an integral inequality on time scales.

Theorem 3.1. Let $\nu ,r,{b}_{1},{b}_{2},p,q,g,d\in {C}_{rd}\left({J}_{\mathbb{T}},ℝ\text{ }+\right)$ and assume that

$\begin{array}{l}\nu \left(t\right)\le r\left(t\right)+{b}_{1}\left(t\right){\int }_{\alpha }^{t}\left\{\left[{b}_{2}\left(t\right)p\left(\tau \right)+1\right]\nu \left(\tau \right)+{b}_{2}\left(\tau \right){\int }_{\alpha }^{\tau }p\left(z\right)\nu \left(z\right)\Delta z\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+q\left(\tau \right){\int }_{\alpha }^{\beta }g\left(z\right)\nu \left(z\right)\Delta z\right\}\Delta \tau +d\left(t\right){\int }_{\alpha }^{\beta }g\left(z\right)\nu \left(z\right)\Delta z,\text{\hspace{0.17em}}t\in {J}_{\mathbb{T}},\end{array}$ (3.1)

If

$N={\int }_{\alpha }^{\beta }g\left(\gamma \right){N}_{2}\left(\gamma \right)\Delta \gamma <1,$ (3.2)

Implies

$\nu \left(t\right)\le {N}_{1}\left(t\right)+A{N}_{2}\left(t\right),t\in {J}_{\mathbb{T}},$ (3.3)

where

$\begin{array}{c}{N}_{1}\left(t\right)=r\left(t\right)+{b}_{1}\left(t\right){\int }_{\alpha }^{t}\left[{b}_{2}^{\sigma }\left(\tau \right)p\left(\tau \right)+1\right]\left[r\left(\tau \right)+b\left(\tau \right){\int }_{\alpha }^{\tau }{e}_{\left({b}_{2}^{\sigma }p+p+1\right)b}\left(\tau ,\sigma \left(z\right)\right)\\ \underset{}{\overset{}{}}×\left({b}_{2}^{\sigma }\left(z\right)p\left(z\right)+p\left(z\right)+1\right)r\left(z\right)\Delta z\right]\Delta \tau ,\end{array}$ (3.4)

$\begin{array}{c}{N}_{2}\left(t\right)=d\left(t\right)+{b}_{1}\left(t\right){\int }_{\alpha }^{t}\left\{\left[{b}_{2}^{\sigma }\left(\tau \right)p\left(\tau \right)+1\right]\left[d\left(\tau \right)+b\left(\tau \right){\int }_{\alpha }^{\tau }{e}_{\left({b}_{2}^{\sigma }p+p+1\right)b}\left(\tau ,\sigma \left(z\right)\right)\\ \underset{}{\overset{}{}}×\left(\left[{b}_{2}^{\sigma }\left(z\right)p\left(z\right)+p\left(z\right)+1\right]d\left(z\right)+q\left(z\right)\right)\Delta z\right]+q\left(\tau \right)\right\}\Delta \tau ,\end{array}$ (3.5)

for $t\in {J}_{\mathbb{T}}$,

${b}_{2}^{\sigma }\left(t\right)={b}_{2}\left(\sigma \left(t\right)\right)={b}_{2}\circ \sigma ,b\left(t\right)={\mathrm{max}}_{t\in {J}_{\mathbb{T}}}\left\{{b}_{1}\left(t\right),{b}_{2}\left(t\right)+{b}_{2}^{\Delta }\left(t\right)\right\},$ (3.6)

$A=\frac{1}{1-N}{\int }_{\alpha }^{\beta }g\left(\gamma \right){N}_{1}\left(\gamma \right)\Delta \gamma ,$ (3.7)

Proof. Let

$\lambda ={\int }_{\alpha }^{\beta }g\left(z\right)\nu \left(z\right)\Delta z,$ (3.8)

we shall define functions ${B}_{1}\left(t\right)$ and ${B}_{2}\left(t\right)$ by

$\begin{array}{c}{B}_{1}\left(t\right)={\int }_{\alpha }^{t}\left\{\left[{b}_{2}\left(t\right)p\left(\tau \right)+1\right]\nu \left(\tau \right)+{b}_{2}\left(\tau \right){\int }_{\alpha }^{\tau }p\left(z\right)\nu \left(z\right)\Delta z\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+q\left(\tau \right){\int }_{\alpha }^{\beta }g\left(z\right)\nu \left(z\right)\Delta z\right\}\Delta \tau ,\end{array}$ (3.9)

${B}_{2}\left(t\right)={B}_{1}\left(t\right)+{\int }_{\alpha }^{t}p\left(z\right)\left[r\left(z\right)+{b}_{1}\left(z\right){B}_{1}\left(z\right)+d\left(z\right)\lambda \right]\Delta z,$ (3.10)

then ${B}_{1}\left(\alpha \right)=0$, ${B}_{2}\left(\alpha \right)=0$, ${B}_{1}\left(t\right)\le {B}_{2}\left(t\right)$ and we have

$\nu \left(t\right)\le r\left(t\right)+{b}_{1}\left(t\right){B}_{1}\left(t\right)+d\left(t\right)\lambda ,$ (3.11)

from (3.9), we get

$\begin{array}{l}{B}_{1}^{\Delta }\left(t\right)={\int }_{\alpha }^{t}{b}_{2}^{\Delta }\left(t\right)p\left(\tau \right)\nu \left(\tau \right)\Delta \tau +\left[{b}_{2}^{\sigma }\left(t\right)p\left(t\right)+1\right]\nu \left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{b}_{2}\left(t\right){\int }_{\alpha }^{t}p\left(z\right)\nu \left(z\right)\Delta z+q\left(t\right){\int }_{\alpha }^{\beta }g\left(z\right)\nu \left(z\right)\Delta z\\ \le \left[{b}_{2}^{\sigma }\left(t\right)p\left(t\right)+1\right]r\left(t\right)+\left[{b}_{2}^{\sigma }\left(t\right)p\left(t\right)+1\right]{b}_{1}\left(t\right){B}_{1}\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left[{b}_{2}^{\sigma }\left(t\right)p\left(t\right)+1\right]d\left(t\right)\lambda +\left[{b}_{2}\left(t\right)+{b}_{2}^{\Delta }\left(t\right)\right]{\int }_{\alpha }^{t}p\left(z\right)\nu \left(z\right)\Delta z+q\left(t\right)\lambda \end{array}$

$\begin{array}{l}\le \left[{b}_{2}^{\sigma }\left(t\right)p\left(t\right)+1\right]r\left(t\right)+{b}_{2}^{\sigma }\left(t\right)p\left(t\right)b\left(t\right){B}_{1}\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+b\left(t\right)\left[{B}_{1}\left(t\right)+{\int }_{\alpha }^{t}p\left(z\right)\nu \left(z\right)\Delta z\right]+\left[{b}_{2}^{\sigma }\left(t\right)p\left(t\right)+1\right]d\left(t\right)\lambda +q\left(t\right)\lambda \\ \le \left[{b}_{2}^{\sigma }\left(t\right)p\left(t\right)+1\right]\left[r\left(t\right)+b\left(t\right){B}_{2}\left(t\right)+d\left(t\right)\lambda \right]+q\left(t\right)\lambda ,\end{array}$ (3.12)

integrating the inequality (3.12) and using ${B}_{1}\left(\alpha \right)=0$, we have

${B}_{1}\left(t\right)\le {\int }_{\alpha }^{t}\left\{\left[{b}_{2}^{\sigma }\left(\tau \right)p\left(\tau \right)+1\right]\left[r\left(\tau \right)+b\left(\tau \right){B}_{2}\left(\tau \right)+d\left(\tau \right)\lambda \right]+q\left(\tau \right)\lambda \right\}\Delta \tau ,$ (3.13)

therefore

$\begin{array}{c}{B}_{2}^{\Delta }\left(t\right)={B}_{1}^{\Delta }\left(t\right)+p\left(t\right)\left[r\left(t\right)+{b}_{1}\left(t\right){B}_{1}\left(t\right)+d\left(t\right)\lambda \right]\\ \le \left[{b}_{2}^{\sigma }\left(t\right)p\left(t\right)+1\right]\left[r\left(t\right)+b\left(t\right){B}_{2}\left(t\right)+d\left(t\right)\lambda \right]+q\left(t\right)\lambda \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+p\left(t\right)\left[r\left(t\right)+b\left(t\right){B}_{2}\left(t\right)+d\left(t\right)\lambda \right]\\ =\left[{b}_{2}^{\sigma }\left(t\right)p\left(t\right)+p\left(t\right)+1\right]b\left(t\right){B}_{2}\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left[{b}_{2}^{\sigma }\left(t\right)p\left(t\right)+p\left(t\right)+1\right]\left[r\left(t\right)+d\left(t\right)\lambda \right]+q\left(t\right)\lambda ,\end{array}$ (3.14)

now applying lemma 2.1, we get

$\begin{array}{l}{B}_{2}\left(t\right)\le {\int }_{\alpha }^{t}{e}_{\left({b}_{2}^{\sigma }p+p+1\right)b}\left(t,\sigma \left(z\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}×\left(\left[{b}_{2}^{\sigma }\left(z\right)p\left(z\right)+p\left(z\right)+1\right]\left[a\left(z\right)+d\left(z\right)\lambda \right]+q\left(z\right)\lambda \right)\Delta z,\end{array}$ (3.15)

from (3.11), (3.13) and (3.15), we obtain that

$\nu \left(t\right)\le {N}_{1}\left(t\right)+\lambda {N}_{2}\left(t\right),$ (3.16)

and from (3.8) and (3.16) we observe that

$\lambda \le A,$ (3.17)

using (3.17) in (3.16) we obtain (3.3). □

We provide the result that includes the estimate on the solutions of (1.1) as follows.

Theorem 3.2. Assume that the following conditions satisfied

$|h\left(t,{\nu }_{1},{\nu }_{2},{\nu }_{3},{\nu }_{4}\right)|\le L\left[|{\nu }_{1}|+|{\nu }_{2}|+|{\nu }_{3}|+|{\nu }_{4}|\right],$ (3.18)

$|{h}_{1}\left(t,z,u,\nu \right)|\le {c}_{1}\left(t\right){s}_{1}\left(z\right)\left[|u|+|\nu |\right],$ (3.19)

$|{h}_{2}\left(t,z,u,\nu \right)|\le {c}_{2}\left(t\right){s}_{2}\left(z\right)\left[|u|+|\nu |\right],$ (3.20)

for the functions $h,{h}_{1},{h}_{2}$ in (1.1), where $0\le L<1$ is a constant and

${c}_{1},{s}_{1},{c}_{2},{s}_{2}\in {C}_{rd}\left({J}_{\mathbb{T}},ℝ+\right)$

If $y\left(t\right)$ is a solution of (1.1) on ${J}_{\mathbb{T}}$, then

$|y\left(t\right)|+|{y}^{\Delta }\left(t\right)|\le {M}_{1}\left(t\right)+{D}_{1}{M}_{2}\left(t\right),t\in {J}_{\mathbb{T}},$ (3.21)

where

$\begin{array}{l}{M}_{1}\left(t\right)=\frac{|{y}_{0}|}{1-L}+\frac{L}{1-L}{\int }_{\alpha }^{t}\left[{c}_{1}^{\sigma }\left(\tau \right){s}_{1}\left(\tau \right)+1\right]\left[\frac{|{y}_{0}|}{1-L}+b\left(\tau \right){\int }_{\alpha }^{\tau }{e}_{\left({c}_{1}^{\sigma }{s}_{1}+{s}_{1}+1\right)b}\left(\tau ,\sigma \left(z\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}×\left({c}_{1}^{\sigma }\left(z\right){s}_{1}\left(z\right)+{s}_{1}\left(z\right)+1\right)\frac{|{y}_{0}|}{1-L}\Delta z\right]\Delta \tau ,t\in {J}_{\mathbb{T}}\end{array}$ (3.22)

$\begin{array}{l}{M}_{2}\left(t\right)=\frac{L}{1-L}{c}_{2}\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{L}{1-L}{\int }_{\alpha }^{t}\left\{\left[{c}_{1}^{\sigma }\left(\tau \right){s}_{1}\left(\tau \right)+1\right]\left[\frac{L}{1-L}{c}_{2}\left(\tau \right)+b\left(\tau \right){\int }_{\alpha }^{\tau }{e}_{\left({c}_{1}^{\sigma }{s}_{1}+{s}_{1}+1\right)b}\left(\tau ,\sigma \left(z\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}×\left(\left[{c}_{1}^{\sigma }\left(z\right){s}_{1}\left(z\right)+{s}_{1}\left(z\right)+1\right]\frac{L}{1-L}{c}_{2}\left(z\right)+{c}_{2}\left(z\right)\right)\Delta z\right]+{c}_{2}\left(\tau \right)\right\}\Delta \tau ,t\in {J}_{\mathbb{T}}\end{array}$

(3.23)

Assume that

$b\left(t\right)={\mathrm{max}}_{t\in {J}_{\mathbb{T}}}\left\{\frac{L}{1-L},{c}_{1}\left(t\right)+{c}_{1}^{\Delta }\left(t\right)\right\},$ (3.24)

$\lambda ={\int }_{\alpha }^{\beta }{s}_{2}\left(\gamma \right){M}_{2}\left(\gamma \right)\Delta \gamma <1,$ (3.25)

${D}_{1}=\frac{1}{1-\lambda }{\int }_{\alpha }^{\beta }{s}_{2}\left(\gamma \right){M}_{1}\left(\gamma \right)\Delta \gamma ,$ (3.26)

Proof. Let $a\left(t\right)=|y\left(t\right)|+|{y}^{\Delta }\left(t\right)|,t\in {J}_{\mathbb{T}}$, since $y\left(t\right)$ is a solution of (1.1), then by using this and the hypotheses, we get

$\begin{array}{l}a\left(t\right)=|{y}_{0}+{\int }_{\alpha }^{t}h\left(\tau ,y\left(\tau \right),{y}^{\Delta }\left(\tau \right),{\int }_{\alpha }^{\tau }{h}_{1}\left(\tau ,z,y\left(z\right),{y}^{\Delta }\left(z\right)\right)\Delta z,\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}{\int }_{\alpha }^{\beta }{h}_{2}\left(\tau ,z,y\left(z\right),{y}^{\Delta }\left(z\right)\right)\Delta z\right)\Delta \tau |\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+|h\left(t,y\left(t\right),{y}^{\Delta }\left(t\right),{\int }_{\alpha }^{t}{h}_{1}\left(t,z,y\left(z\right),{y}^{\Delta }\left(z\right)\right)\Delta z,{\int }_{\alpha }^{\beta }{h}_{2}\left(t,z,y\left(z\right),{y}^{\Delta }\left(z\right)\right)\Delta z\right)|\\ \le |{y}_{0}|+{\int }_{\alpha }^{t}L\left[a\left(\tau \right)+{\int }_{\alpha }^{\tau }{c}_{1}\left(\tau \right){s}_{1}\left(z\right)a\left(z\right)\Delta z+{\int }_{\alpha }^{\beta }{c}_{2}\left(\tau \right){s}_{2}\left(z\right)a\left(z\right)\Delta z\right]\Delta \tau \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+L\left[a\left(t\right)+{\int }_{\alpha }^{t}{c}_{1}\left(t\right){s}_{1}\left(z\right)a\left(z\right)\Delta z+{\int }_{\alpha }^{\beta }{c}_{2}\left(t\right){s}_{2}\left(z\right)a\left(z\right)\Delta z\right]\end{array}$

from the above inequality, we have

$\begin{array}{c}a\left(t\right)\le \frac{|{y}_{0}|}{1-L}+\frac{L}{1-L}{\int }_{\alpha }^{t}\left\{\left[{c}_{1}\left(t\right){s}_{1}\left(\tau \right)+1\right]a\left(\tau \right)+{c}_{1}\left(\tau \right){\int }_{\alpha }^{\tau }{s}_{1}\left(z\right)a\left(z\right)\Delta z\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{c}_{2}\left(\tau \right){\int }_{\alpha }^{\beta }{s}_{2}\left(z\right)a\left(z\right)\Delta z\right\}\Delta \tau +\frac{L}{1-L}{c}_{2}\left(t\right){\int }_{\alpha }^{\beta }{s}_{2}\left(z\right)a\left(z\right)\Delta z,\end{array}$ (3.27)

Now applying theorem 3.1 in (3.27) we obtain (3.21). □

Remark 3.3. Since $y\left(t\right)$ is a solution of (1.1). Then (3.21) yields the bounds on $y\left(t\right)$ and ${y}^{\Delta }\left(t\right)$. If the estimate in (3.21) is bounded, implies the solution $y\left(t\right)$ and ${y}^{\Delta }\left(t\right)$ are also bounded on ${J}_{\mathbb{T}}$.

Consider (1.1) with the following corresponding equation

$\begin{array}{l}{Y}^{\Delta }\left(t\right)=H\left(t,Y\left(t\right),{Y}^{\Delta }\left(t\right),{\int }_{\alpha }^{t}{h}_{1}\left(t,z,Y\left(z\right),{Y}^{\Delta }\left(z\right)\right)\Delta z,\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\int }_{\alpha }^{\beta }{h}_{2}\left(t,z,Y\left(z\right),{Y}^{\Delta }\left(z\right)\right)\Delta z\right),t\in {J}_{\mathbb{T}},\end{array}$

with the initial condition

$Y\left(\alpha \right)={Y}_{0}$ (3.28)

where $H\in {C}_{rd}\left({J}_{\mathbb{T}}×{ℝ}^{n}×{ℝ}^{n}×{ℝ}^{n}×{ℝ}^{n},{ℝ}^{n}\right),{h}_{1},{h}_{2}$ as in (1.1).

The next result concerning the closeness of solution of (1.1) and (3.28).

Theorem 3.4. Suppose that the following conditions satisfied

$\begin{array}{l}|h\left(t,{\nu }_{1},{\nu }_{2},{\nu }_{3},{\nu }_{4}\right)-h\left(t,{\stackrel{¯}{\nu }}_{1},{\stackrel{¯}{\nu }}_{2},{\stackrel{¯}{\nu }}_{3},{\stackrel{¯}{\nu }}_{4}\right)|\\ \le L\left[|{\nu }_{1}-{\stackrel{¯}{\nu }}_{1}|+|{\nu }_{2}-{\stackrel{¯}{\nu }}_{2}|+|{\nu }_{3}-{\stackrel{¯}{\nu }}_{3}|+|{\nu }_{4}-{\stackrel{¯}{\nu }}_{4}|\right],\end{array}$ (3.29)

$|{h}_{1}\left(t,z,u,\nu \right)-{h}_{1}\left(t,z,\stackrel{¯}{u},\stackrel{¯}{\nu }\right)|\le {c}_{1}\left(t\right){s}_{1}\left(z\right)\left[|u-\stackrel{¯}{u}|+|\nu -\stackrel{¯}{\nu }|\right],$ (3.30)

$|{h}_{2}\left(t,z,u,\nu \right)-{h}_{2}\left(t,z,\stackrel{¯}{u},\stackrel{¯}{\nu }\right)|\le {c}_{2}\left(t\right){s}_{2}\left(z\right)\left[|u-\stackrel{¯}{u}|+|\nu -\stackrel{¯}{\nu }|\right],$ (3.31)

where the functions $h,{h}_{1},{h}_{2}$ in (1.1), and $0\le L<1$ is a constant.

Also ${c}_{1},{s}_{1},{c}_{2},{s}_{2}\in {C}_{rd}\left({J}_{\mathbb{T}},ℝ\text{ }+\right)$, and

$|h\left(t,{\nu }_{1},{\nu }_{2},{\nu }_{3},{\nu }_{4}\right)-H\left(t,{\nu }_{1},{\nu }_{2},{\nu }_{3},{\nu }_{4}\right)|\le \epsilon ,$ (3.32)

$|{y}_{0}-{Y}_{0}|\le \delta ,$ (3.33)

where ${y}_{0}$ and $H,{Y}_{0}$ as in (1.1) and (3.28) respectively.

If $y\left(t\right)$ and $Y\left(t\right)$ be solutions of (1.1) and (3.28) on ${J}_{\mathbb{T}}$, then

$|y\left(t\right)-Y\left(t\right)|+|{y}^{\Delta }\left(t\right)-{Y}^{\Delta }\left(t\right)|\le {M}_{3}\left(t\right)+{D}_{2}{M}_{2}\left(t\right),t\in {J}_{\mathbb{T}},$ (3.34)

where ${M}_{3}\left(t\right)$ is described by the right side of (3.22) by substituting $m\left(t\right)=\delta +\epsilon \left(1+t-\alpha \right)$ instead of $|{y}_{0}|$, ${M}_{2}\left(t\right),b\left(t\right)$ and $\lambda$ be as in (3.23), (3.24) and (3.25) respectively and

${D}_{2}=\frac{1}{1-\lambda }{\int }_{\alpha }^{\beta }{r}_{2}\left(\gamma \right){M}_{3}\left(\gamma \right)\Delta \gamma ,$ (3.35)

Proof. Let $w\left(t\right)=|y\left(t\right)-Y\left(t\right)|+|{y}^{\Delta }\left(t\right)-{Y}^{\Delta }\left(t\right)|,t\in {J}_{\mathbb{T}}$, we have

$\begin{array}{l}w\left(t\right)\le |{y}_{0}-{Y}_{0}|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{\alpha }^{t}|h\left(\tau ,y\left(\tau \right),{y}^{\Delta }\left(\tau \right),{\int }_{\alpha }^{\tau }{h}_{1}\left(\tau ,z,y\left(z\right),{y}^{\Delta }\left(z\right)\right)\Delta z,{\int }_{\alpha }^{\beta }{h}_{2}\left(\tau ,z,y\left(z\right),{y}^{\Delta }\left(z\right)\right)\Delta z\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-h\left(\tau ,Y\left(\tau \right),{Y}^{\Delta }\left(\tau \right),{\int }_{\alpha }^{\tau }{h}_{1}\left(\tau ,z,Y\left(z\right),{Y}^{\Delta }\left(z\right)\right)\Delta z,{\int }_{\alpha }^{\beta }{h}_{2}\left(\tau ,z,Y\left(z\right),{Y}^{\Delta }\left(z\right)\right)\Delta z\right)|\Delta \tau \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{\alpha }^{t}|h\left(\tau ,Y\left(\tau \right),{Y}^{\Delta }\left(\tau \right),{\int }_{\alpha }^{\tau }{h}_{1}\left(\tau ,z,Y\left(z\right),{Y}^{\Delta }\left(z\right)\right)\Delta z,{\int }_{\alpha }^{\beta }{h}_{2}\left(\tau ,z,Y\left(z\right),{Y}^{\Delta }\left(z\right)\right)\Delta z\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-H\left(\tau ,Y\left(\tau \right),{Y}^{\Delta }\left(\tau \right),{\int }_{\alpha }^{r}{h}_{1}\left(\tau ,z,Y\left(z\right),{Y}^{\Delta }\left(z\right)\right)\Delta z,{\int }_{\alpha }^{\beta }{h}_{2}\left(\tau ,z,Y\left(z\right),{Y}^{\Delta }\left(z\right)\right)\Delta z\right)|\Delta \tau \end{array}$

$\begin{array}{l}+|h\left(t,y\left(t\right),{y}^{\Delta }\left(t\right),{\int }_{\alpha }^{t}{h}_{1}\left(t,z,y\left(z\right),{y}^{\Delta }\left(z\right)\right)\Delta z,{\int }_{\alpha }^{\beta }{h}_{2}\left(t,z,y\left(z\right),{y}^{\Delta }\left(z\right)\right)\Delta z\right)\\ -h\left(t,Y\left(t\right),{Y}^{\Delta }\left(t\right),{\int }_{\alpha }^{t}{h}_{1}\left(t,z,Y\left(z\right),{Y}^{\Delta }\left(z\right)\right)\Delta z,{\int }_{\alpha }^{\beta }{h}_{2}\left(t,z,Y\left(z\right),{Y}^{\Delta }\left(z\right)\right)\Delta z\right)|\\ +|h\left(t,Y\left(t\right),{Y}^{\Delta }\left(t\right),{\int }_{\alpha }^{t}{h}_{1}\left(t,z,Y\left(z\right),{Y}^{\Delta }\left(z\right)\right)\Delta z,{\int }_{\alpha }^{\beta }{h}_{2}\left(t,z,Y\left(z\right),{Y}^{\Delta }\left(z\right)\right)\Delta z\right)\\ -H\left(t,Y\left(t\right),{Y}^{\Delta }\left(t\right),{\int }_{\alpha }^{t}{h}_{1}\left(t,z,Y\left(z\right),{Y}^{\Delta }\left(z\right)\right)\Delta z,{\int }_{\alpha }^{\beta }{h}_{2}\left(t,z,Y\left(z\right),{Y}^{\Delta }\left(z\right)\right)\Delta z\right)|\end{array}$

$\begin{array}{l}\le \delta +{\int }_{\alpha }^{t}L\left[w\left(\tau \right)+{\int }_{\alpha }^{\tau }{c}_{1}\left(\tau \right){s}_{1}\left(z\right)w\left(z\right)\Delta z+{\int }_{\alpha }^{\beta }{c}_{2}\left(\tau \right){s}_{2}\left(z\right)w\left(z\right)\Delta z\right]\Delta \tau \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{\alpha }^{t}\epsilon \Delta \tau +L\left[w\left(t\right)+{\int }_{\alpha }^{t}{c}_{1}\left(t\right){s}_{1}\left(z\right)w\left(z\right)\Delta z+{\int }_{\alpha }^{\beta }{c}_{2}\left(t\right){s}_{2}\left(z\right)w\left(z\right)\Delta z\right]+\epsilon \\ =m\left(t\right)+L{\int }_{\alpha }^{t}\left[w\left(\tau \right)+{c}_{1}\left(\tau \right){\int }_{\alpha }^{\tau }{s}_{1}\left(z\right)w\left(z\right)\Delta z+{c}_{2}\left(\tau \right){\int }_{\alpha }^{\beta }{s}_{2}\left(z\right)w\left(z\right)\Delta z\right]\Delta \tau \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+L\left[w\left(t\right)+{\int }_{\alpha }^{t}{c}_{1}\left(t\right){s}_{1}\left(z\right)w\left(z\right)\Delta z+{c}_{2}\left(t\right){\int }_{\alpha }^{\beta }{s}_{2}\left(z\right)w\left(z\right)\Delta z\right]\end{array}$

then we get

$\begin{array}{c}w\left(t\right)\le \frac{m\left(t\right)}{1-L}+\frac{L}{1-L}{\int }_{\alpha }^{t}\left\{\left[{c}_{1}\left(t\right){s}_{1}\left(\tau \right)+1\right]w\left(\tau \right)+{c}_{1}\left(\tau \right){\int }_{\alpha }^{\tau }{s}_{1}\left(z\right)w\left(z\right)\Delta z\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{c}_{2}\left(\tau \right){\int }_{\alpha }^{\beta }{s}_{2}\left(z\right)w\left(z\right)\Delta z\right\}\Delta \tau +\frac{L}{1-L}{c}_{2}\left(t\right){\int }_{\alpha }^{\beta }{s}_{2}\left(z\right)w\left(z\right)\Delta z,\end{array}$ (3.36)

Now applying theorem 3.1, yields (3.34). □

The following theorem provide the continuous depends of solutions of (1.1) on given initial values.

Theorem 3.5. Assume that the conditions (3.29), (3.30) and (3.31) are satisfied for the functions $h,{h}_{1},{h}_{2}$ in (1.1). Let ${y}_{1}\left(t\right)$ and ${y}_{2}\left(t\right)$ be the solutions of equation

$\begin{array}{l}{y}^{\Delta }\left(t\right)=h\left(t,y\left(t\right),{y}^{\Delta }\left(t\right),{\int }_{\alpha }^{t}{h}_{1}\left(t,z,y\left(z\right),{y}^{\Delta }\left(z\right)\right)\Delta z,\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\int }_{\alpha }^{\beta }{h}_{2}\left(t,z,y\left(z\right),{y}^{\Delta }\left(z\right)\right)\Delta z\right),t\in {J}_{\mathbb{T}},\end{array}$

with the given initial values

${y}_{1}\left(\alpha \right)={c}_{1}$ and ${y}_{2}\left(\alpha \right)={c}_{2}$, (3.37)

where $h,{h}_{1},{h}_{2}$ as in (1.1), ${c}_{1}$ and ${c}_{2}$ are constants. Then

$|{y}_{1}\left(t\right)-{y}_{2}\left(t\right)|+|{y}_{1}^{\Delta }\left(t\right)-{y}_{2}^{\Delta }\left(t\right)|\le {M}_{4}\left(t\right)+{D}_{3}{M}_{2}\left(t\right),t\in {J}_{\mathbb{T}},$ (3.38)

where ${M}_{4}\left(t\right)$ is described by the right side of (3.22) by substituting $|{c}_{1}-{c}_{2}|$ instead of $|{y}_{0}|$, ${M}_{2}\left(t\right),b\left(t\right)$ and $\lambda$ be as in (3.23), (3.24) and (3.25) respectively and

${D}_{3}=\frac{1}{1-\lambda }{\int }_{\alpha }^{\beta }{r}_{2}\left(\gamma \right){M}_{4}\left(\gamma \right)\Delta \gamma ,$ (3.39)

Proof. Let $n\left(t\right)=|{y}_{1}\left(t\right)-{y}_{2}\left(t\right)|+|{y}_{1}^{\Delta }\left(t\right)-{y}_{2}^{\Delta }\left(t\right)|,t\in {J}_{\mathbb{T}}$, we get

$\begin{array}{l}n\left(t\right)\le |{c}_{1}-{c}_{2}|\\ \text{ }+{\int }_{\alpha }^{t}|h\left(\tau ,{y}_{1}\left(\tau \right),{y}_{1}^{\Delta }\left(\tau \right),{\int }_{\alpha }^{\tau }{h}_{1}\left(\tau ,z,{y}_{1}\left(z\right),{y}_{1}^{\Delta }\left(z\right)\right)\Delta z,{\int }_{\alpha }^{\beta }{h}_{2}\left(\tau ,z,{y}_{1}\left(z\right),{y}_{1}^{\Delta }\left(z\right)\right)\Delta z\right)\\ \text{ }-h\left(\tau ,{y}_{2}\left(\tau \right),{y}_{2}^{\Delta }\left(\tau \right),{\int }_{\alpha }^{\tau }{h}_{1}\left(\tau ,z,{y}_{2}\left(z\right),{y}_{2}^{\Delta }\left(z\right)\right)\Delta z,{\int }_{\alpha }^{\beta }{h}_{2}\left(\tau ,z,{y}_{2}\left(z\right),{y}_{2}^{\Delta }\left(z\right)\right)\Delta z\right)|\Delta \tau \\ \text{ }+|h\left(t,{y}_{1}\left(t\right),{y}_{1}^{\Delta }\left(t\right),{\int }_{\alpha }^{t}{h}_{1}\left(t,z,{y}_{1}\left(z\right),{y}_{1}^{\Delta }\left(z\right)\right)\Delta z,{\int }_{\alpha }^{\beta }{h}_{2}\left(t,z,{y}_{1}\left(z\right),{y}_{1}^{\Delta }\left(z\right)\right)\Delta z\right)\\ \text{ }-h\left(t,{y}_{2}\left(t\right),{y}_{2}^{\Delta }\left(t\right),{\int }_{\alpha }^{t}{h}_{1}\left(t,z,{y}_{2}\left(z\right),{y}_{2}^{\Delta }\left(z\right)\right)\Delta z,{\int }_{\alpha }^{\beta }{h}_{2}\left(t,z,{y}_{2}\left(z\right),{y}_{2}^{\Delta }\left(z\right)\right)\Delta z\right)|\end{array}$

$\begin{array}{l}\le |{c}_{1}-{c}_{2}|+L{\int }_{\alpha }^{t}\left[n\left(\tau \right)+{c}_{1}\left(\tau \right){\int }_{\alpha }^{\tau }{s}_{1}\left(z\right)n\left(z\right)\Delta z+{c}_{2}\left(\tau \right){\int }_{\alpha }^{\beta }{s}_{2}\left(z\right)n\left(z\right)\Delta z\right]\Delta \tau \\ \text{ }+L\left[n\left(t\right)+{\int }_{\alpha }^{t}{c}_{1}\left(t\right){s}_{1}\left(z\right)n\left(z\right)\Delta z+{c}_{2}\left(t\right){\int }_{\alpha }^{\beta }{s}_{2}\left(z\right)n\left(z\right)\Delta z\right]\end{array}$

then

$\begin{array}{l}n\left(t\right)\le \frac{|{c}_{1}-{c}_{2}|}{1-L}+\frac{L}{1-L}{\int }_{\alpha }^{t}\left\{\left[{c}_{1}\left(t\right){s}_{1}\left(\tau \right)+1\right]n\left(\tau \right)+{c}_{1}\left(\tau \right){\int }_{\alpha }^{\tau }{s}_{1}\left(z\right)n\left(z\right)\Delta z\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{c}_{2}\left(\tau \right){\int }_{\alpha }^{\beta }{s}_{2}\left(z\right)n\left(z\right)\Delta z\right\}\Delta \tau +\frac{L}{1-L}{c}_{2}\left(t\right){\int }_{\alpha }^{\beta }{s}_{2}\left(z\right)n\left(z\right)\Delta z,\end{array}$ (3.40)

Now applying theorem 3.1 in (3.40) we obtain (3.38). □

Remark 3.6. The inequality (3.38) gives the uniqueness of solutions of (3.37). If we have ${c}_{1}={c}_{2}=0$, then we get ${M}_{5}\left(t\right)=0$ and ${D}_{3}=0$, implies the right hand side of (3.37) is equal to zero.

Now consider the initial value problems

$\begin{array}{l}{Y}^{\Delta }\left(t\right)=h\left(t,Y\left(t\right),{Y}^{\Delta }\left(t\right),{\int }_{\alpha }^{t}{h}_{1}\left(t,z,Y\left(z\right),{Y}^{\Delta }\left(z\right)\right)\Delta z,\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\int }_{\alpha }^{\beta }{h}_{2}\left(t,z,Y\left(z\right),{Y}^{\Delta }\left(z\right)\right)\Delta z,\mu \right),t\in {J}_{\mathbb{T}},Y\left(\alpha \right)={Y}_{0},\end{array}$ (3.41)

$\begin{array}{l}{Y}^{\Delta }\left(t\right)=h\left(t,Y\left(t\right),{Y}^{\Delta }\left(t\right),{\int }_{\alpha }^{t}{h}_{1}\left(t,z,Y\left(z\right),{Y}^{\Delta }\left(z\right)\right)\Delta z,\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\int }_{\alpha }^{\beta }{h}_{2}\left(t,z,Y\left(z\right),{Y}^{\Delta }\left(z\right)\right)\Delta z,{\mu }_{0}\right),t\in {J}_{\mathbb{T}},Y\left(\alpha \right)={Y}_{0},\end{array}$ (3.42)

where $h\in {C}_{rd}\left({J}_{\mathbb{T}}×{ℝ}^{n}×{ℝ}^{n}×{ℝ}^{n}×{ℝ}^{n},{ℝ}^{n}\right)$ and $\mu ,{\mu }_{0}$ are parameters.

The dependency of solutions of (3.41) and (3.42) on parameters follows in the next theorem.

Theorem 3.7. Suppose that the conditions (3.30) and (3.31) are satisfied and

$\begin{array}{l}|h\left(t,{\nu }_{1},{\nu }_{2},{\nu }_{3},{\nu }_{4},\mu \right)-h\left(t,{\stackrel{¯}{\nu }}_{1},{\stackrel{¯}{\nu }}_{2},{\stackrel{¯}{\nu }}_{3},{\stackrel{¯}{\nu }}_{4},\mu \right)|\\ \le L\left[|{\nu }_{1}-{\stackrel{¯}{\nu }}_{1}|+|{\nu }_{2}-{\stackrel{¯}{\nu }}_{2}|+|{\nu }_{3}-{\stackrel{¯}{\nu }}_{3}|+|{\nu }_{4}-{\stackrel{¯}{\nu }}_{4}|\right],\end{array}$ (3.43)

$|h\left(t,{\nu }_{1},{\nu }_{2},{\nu }_{3},{\nu }_{4},\mu \right)-h\left(t,{\nu }_{1},{\nu }_{2},{\nu }_{3},{\nu }_{4},{\mu }_{0}\right)|\le k\left(t\right)|\mu -{\mu }_{0}|,$ (3.44)

where $0\le L<1$ is a constant and $k\in {C}_{rd}\left({J}_{\mathbb{T}},ℝ\text{ }+\right)$. Let ${Y}_{1}\left(t\right)$ and ${Y}_{2}\left(t\right)$ be respectively, the solutions of (3.41) and (3.42) on ${J}_{\mathbb{T}}$, then

$|{Y}_{1}\left(t\right)-{Y}_{2}\left(t\right)|+|{Y}_{1}^{\Delta }\left(t\right)-{Y}_{2}^{\Delta }\left(t\right)|\le {M}_{5}\left(t\right)+{D}_{4}{M}_{2}\left(t\right),t\in {J}_{\mathbb{T}},$ (3.45)

where ${M}_{5}\left(t\right)$ is described by the right side of (3.22) by substituting $|\mu -{\mu }_{0}|\stackrel{¯}{k}\left(t\right)$ instead of $|{y}_{0}|$, ${M}_{2}\left(t\right),b\left(t\right)$ and $\lambda$ be as in (3.23), (3.24) and (3.25) respectively.

Let

$\stackrel{¯}{k}\left(t\right)=k\left(t\right)+{\int }_{\alpha }^{t}k\left(r\right)\Delta \tau ,$ (3.46)

${D}_{4}=\frac{1}{1-\lambda }{\int }_{\alpha }^{\beta }{r}_{2}\left(\gamma \right){M}_{5}\left(\gamma \right)\Delta \gamma ,$ (3.47)

Proof. Let $P\left(t\right)=|{Y}_{1}\left(t\right)-{Y}_{2}\left(t\right)|+|{Y}_{1}^{\Delta }\left(t\right)-{Y}_{2}^{\Delta }\left(t\right)|,t\in {J}_{\mathbb{T}}$, we have

$\begin{array}{l}P\left(t\right)\\ \le {\int }_{\alpha }^{t}|h\left(\tau ,{Y}_{1}\left(\tau \right),{Y}_{1}^{\Delta }\left(\tau \right),{\int }_{\alpha }^{\tau }{h}_{1}\left(\tau ,z,{Y}_{1}\left(z\right),{Y}_{1}^{\Delta }\left(z\right)\right)\Delta z,{\int }_{\alpha }^{\beta }{h}_{2}\left(\tau ,z,{Y}_{1}\left(z\right),{Y}_{1}^{\Delta }\left(z\right)\right)\Delta z,\mu \right)\\ \text{ }-h\left(\tau ,{Y}_{2}\left(\tau \right),{Y}_{2}^{\Delta }\left(\tau \right),{\int }_{\alpha }^{\tau }{h}_{1}\left(\tau ,z,{Y}_{2}\left(z\right),{Y}_{2}^{\Delta }\left(z\right)\right)\Delta z,{\int }_{\alpha }^{\beta }{h}_{2}\left(\tau ,z,{Y}_{2}\left(z\right),{Y}_{2}^{\Delta }\left(z\right)\right)\Delta z,\mu \right)|\Delta \tau \\ \text{ }+{\int }_{\alpha }^{t}|h\left(\tau ,{Y}_{2}\left(\tau \right),{Y}_{2}^{\Delta }\left(\tau \right),{\int }_{\alpha }^{\tau }{h}_{1}\left(\tau ,z,{Y}_{2}\left(z\right),{Y}_{2}^{\Delta }\left(z\right)\right)\Delta z,{\int }_{\alpha }^{\beta }{h}_{2}\left(\tau ,z,{Y}_{2}\left(z\right),{Y}_{2}^{\Delta }\left(z\right)\right)\Delta z,\mu \right)\\ \text{ }-h\left(\tau ,{Y}_{2}\left(\tau \right),{Y}_{2}^{\Delta }\left(\tau \right),{\int }_{\alpha }^{\tau }{h}_{1}\left(\tau ,z,{Y}_{2}\left(z\right),{Y}_{2}^{\Delta }\left(z\right)\right)\Delta z,{\int }_{\alpha }^{\beta }{h}_{2}\left(\tau ,z,{Y}_{2}\left(z\right),{Y}_{2}^{\Delta }\left(z\right)\right)\Delta z,{\mu }_{0}\right)|\Delta \tau \end{array}$

$\begin{array}{l}\text{ }+|h\left(t,{Y}_{1}\left(t\right),{Y}_{1}^{\Delta }\left(t\right),{\int }_{\alpha }^{t}{h}_{1}\left(t,z,{Y}_{1}\left(z\right),{Y}_{1}^{\Delta }\left(z\right)\right)\Delta z,{\int }_{\alpha }^{\beta }{h}_{2}\left(t,z,{Y}_{1}\left(z\right),{Y}_{1}^{\Delta }\left(z\right)\right)\Delta z,\mu \right)\\ \text{ }-h\left(t,{Y}_{2}\left(t\right),{Y}_{2}^{\Delta }\left(t\right),{\int }_{\alpha }^{t}{h}_{1}\left(t,z,{Y}_{2}\left(z\right),{Y}_{2}^{\Delta }\left(z\right)\right)\Delta z,{\int }_{\alpha }^{\beta }{h}_{2}\left(t,z,{Y}_{2}\left(z\right),{Y}_{2}^{\Delta }\left(z\right)\right)\Delta z,\mu \right)|\\ \text{ }+|h\left(t,{Y}_{2}\left(t\right),{Y}_{2}^{\Delta }\left(t\right),{\int }_{\alpha }^{t}{h}_{1}\left(t,z,{Y}_{2}\left(z\right),{Y}_{2}^{\Delta }\left(z\right)\right)\Delta z,{\int }_{\alpha }^{\beta }{h}_{2}\left(t,z,{Y}_{2}\left(z\right),{Y}_{2}^{\Delta }\left(z\right)\right)\Delta z,\mu \right)\\ \text{ }-h\left(t,{Y}_{2}\left(t\right),{Y}_{2}^{\Delta }\left(t\right),{\int }_{\alpha }^{t}{h}_{1}\left(t,z,{Y}_{2}\left(z\right),{Y}_{2}^{\Delta }\left(z\right)\right)\Delta z,{\int }_{\alpha }^{\beta }{h}_{2}\left(t,z,{Y}_{2}\left(z\right),{Y}_{2}^{\Delta }\left(z\right)\right)\Delta z,{\mu }_{0}\right)|\end{array}$

$\begin{array}{l}\le {\int }_{\alpha }^{t}L\left[P\left(\tau \right)+{\int }_{\alpha }^{\tau }{c}_{1}\left(\tau \right){s}_{1}\left(z\right)P\left(z\right)\Delta z+{\int }_{\alpha }^{\beta }{c}_{2}\left(\tau \right){s}_{2}\left(z\right)P\left(z\right)\Delta z\right]\Delta \tau \\ \text{ }+{\int }_{\alpha }^{t}k\left(\tau \right)|\mu -{\mu }_{0}|\Delta \tau \\ \text{ }+L\left[P\left(t\right)+{\int }_{\alpha }^{t}{c}_{1}\left(t\right){s}_{1}\left(z\right)P\left(z\right)\Delta z+{\int }_{\alpha }^{\beta }{c}_{2}\left(t\right){s}_{2}\left(z\right)P\left(z\right)\Delta z\right]+k\left(t\right)|\mu -{\mu }_{0}|\\ =|\mu -{\mu }_{0}|\stackrel{¯}{k}\left(t\right)+L{\int }_{\alpha }^{t}\left[P\left(\tau \right)+{c}_{1}\left(\tau \right){\int }_{\alpha }^{\tau }{s}_{1}\left(z\right)P\left(z\right)\Delta z+{c}_{2}\left(\tau \right){\int }_{\alpha }^{\beta }{s}_{2}\left(z\right)P\left(z\right)\Delta z\right]\Delta \tau \\ \text{ }+L\left[P\left(t\right)+{\int }_{\alpha }^{t}{c}_{1}\left(t\right){s}_{1}\left(z\right)P\left(z\right)\Delta z+{\int }_{\alpha }^{\beta }{c}_{2}\left(t\right){s}_{2}\left(z\right)P\left(z\right)\Delta z\right]\end{array}$

then we have

$\begin{array}{l}P\left(t\right)\le \frac{|\mu -{\mu }_{0}|\stackrel{¯}{k}\left(t\right)}{1-L}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{L}{1-L}{\int }_{\alpha }^{t}\left\{\left[{c}_{1}\left(t\right){s}_{1}\left(\tau \right)+1\right]P\left(\tau \right)+{c}_{1}\left(\tau \right){\int }_{\alpha }^{\tau }{s}_{1}\left(z\right)P\left(z\right)\Delta z\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{c}_{2}\left(\tau \right){\int }_{\alpha }^{\beta }{s}_{2}\left(z\right)P\left(z\right)\Delta z\right\}\Delta \tau +\frac{L}{1-L}{c}_{2}\left(t\right){\int }_{\alpha }^{\beta }{s}_{2}\left(z\right)P\left(z\right)\Delta z,\end{array}$ (3.48)

Now applying theorem 3.1, we have (3.45). □

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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