On a Nonlinear Volterra-Fredholm Integrodifferential Equation on Time Scales

Mohammed I. Noori, Akram H. Mahmood

Department of Mathematic, College of Education for Pure Science, University of Mosul, Mosul, Iraq.

DOI: 10.4236/oalib.1106103                

Abstract

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.

Keywords

Integrodifferential Equations, Time Scales, Integral Inequality, Estimate on the Solutions

1. Introduction

The theory of time scales had been begun in 1988 by Stefan Hilger [1], 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 [4] [5] [6] [7] [8]. In this article we consider the nonlinear integrodifferential equation of the following form

$\begin{array}{l}{y}^{\Delta }\left(t\right)=h\left(t,y\left(t\right),y^{\Delta }\left(t\right),{\displaystyle {\int }_{\alpha }^t{h}_1\left(t,z,y\left(z\right),y^{\Delta }\left(z\right)\right)\Delta z},{\displaystyle {\int }_{\alpha }^{\beta }{h}_2\left(t,z,y\left(z\right),y^{\Delta }\left(z\right)\right)\Delta z}\right),\\ t\in J_{\mathbb{T}}\text{with the initial condition}y\left(\alpha \right)=y_0,\end{array}$ (1.1)

where y is unknown function and $h:J_{\mathbb{T}}\times {ℝ}^n\times {ℝ}^n\times {ℝ}^n\times {ℝ}^n\to {ℝ}^n$, $h_1,h_2:J_{\mathbb{T}}^2\times {ℝ}^n\times {ℝ}^n\to {ℝ}^n$ 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 $\left|.\right|$.

We can investigate the existence and uniqueness results for (1.1) by using the technique present in [6].

2. Preliminaries

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

For $t\in \mathbb{T}$, If $t<\sup \mathbb{T}$ and $\sigma \left(t\right)=t$, then t is said to be right-dense; while If $t>\inf \mathbb{T}$ 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

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

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 $\left|z\left(\sigma \left(t\right),\tau \right)-z\left(s,\tau \right)-z^{\Delta }\left(t,\tau \right)\left(\sigma \left(t\right)-s\right)\right|\le \epsilon \left|\sigma \left(t\right)-s\right|$ for all $s\in U$. Then $g\left(t\right)={\displaystyle {\int }_{\alpha }^{t}z\left(t,\tau \right)\Delta \tau }$ implies $g^{\Delta }\left(t\right)={\displaystyle {\int }_{\alpha }^t{z}^{\Delta }\left(t,\tau \right)\Delta \tau }+z\left(\sigma \left(t\right),t\right)$. 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

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

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

For more basic information about time scales calculus, see [1] [3].

We need the following result given in [2].

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 $\nu \left(t\right)\le \nu \left(\alpha \right)e_a\left(t,\alpha \right)+{\displaystyle {\int }_{\alpha }^t{e}_a\left(t,\sigma \left(\tau \right)\right)b\left(\tau \right)\Delta \tau }$, 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 $\nu ,r,b_1,b_2,p,q,g,d\in C_{rd}\left(J_{\mathbb{T}},ℝ+\right)$ and assume that

$\begin{array}{l}\nu \left(t\right)\le r\left(t\right)+b_1\left(t\right){\displaystyle {\int }_{\alpha }^t\{\left[b_2\left(t\right)p\left(\tau \right)+1\right]\nu \left(\tau \right)+b_2\left(\tau \right){\displaystyle {\int }_{\alpha }^{\tau }p\left(z\right)\nu \left(z\right)\Delta z}}\\ +q\left(\tau \right){\displaystyle {\int }_{\alpha }^{\beta }g\left(z\right)\nu \left(z\right)\Delta z}\}\Delta \tau +d\left(t\right){\displaystyle {\int }_{\alpha }^{\beta }g\left(z\right)\nu \left(z\right)\Delta z},t\in J_{\mathbb{T}},\end{array}$ (3.1)

If

$N={\displaystyle {\int }_{\alpha }^{\beta }g\left(\gamma \right)N_2\left(\gamma \right)\Delta \gamma }<1,$ (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

$\begin{array}{c}{N}_1\left(t\right)=r\left(t\right)+b_1\left(t\right){\displaystyle {\int }_{\alpha }^t\left[b_2^{\sigma }\left(\tau \right)p\left(\tau \right)+1\right][r\left(\tau \right)+b\left(\tau \right){\displaystyle {\int }_{\alpha }^{\tau }{e}_{\left(b_2^{\sigma }p+p+1\right)b}\left(\tau ,\sigma \left(z\right)\right)}}\\ \underset{}{\overset{}{}}\times \left(b_2^{\sigma }\left(z\right)p\left(z\right)+p\left(z\right)+1\right)r\left(z\right)\Delta z]\Delta \tau ,\end{array}$ (3.4)

$\begin{array}{c}{N}_2\left(t\right)=d\left(t\right)+b_1\left(t\right){\displaystyle {\int }_{\alpha }^t\{\left[b_2^{\sigma }\left(\tau \right)p\left(\tau \right)+1\right][d\left(\tau \right)+b\left(\tau \right){\displaystyle {\int }_{\alpha }^{\tau }{e}_{\left(b_2^{\sigma }p+p+1\right)b}\left(\tau ,\sigma \left(z\right)\right)}}\\ \underset{}{\overset{}{}}\times \left(\left[b_2^{\sigma }\left(z\right)p\left(z\right)+p\left(z\right)+1\right]d\left(z\right)+q\left(z\right)\right)\Delta z]+q\left(\tau \right)\}\Delta \tau ,\end{array}$ (3.5)

for $t\in J_{\mathbb{T}}$,

$b_2^{\sigma }\left(t\right)=b_2\left(\sigma \left(t\right)\right)=b_2\circ \sigma ,b\left(t\right)={\max }_{t\in J_{\mathbb{T}}}\left\{b_1\left(t\right),b_2\left(t\right)+b_2^{\Delta }\left(t\right)\right\},$ (3.6)

$A=\frac{1}{1-N}{\displaystyle {\int }_{\alpha }^{\beta }g\left(\gamma \right)N_1\left(\gamma \right)\Delta \gamma },$ (3.7)

Proof. Let

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

we shall define functions $B_1\left(t\right)$ and $B_2\left(t\right)$ by

$\begin{array}{c}{B}_1\left(t\right)={\displaystyle {\int }_{\alpha }^t\{\left[b_2\left(t\right)p\left(\tau \right)+1\right]\nu \left(\tau \right)+b_2\left(\tau \right){\displaystyle {\int }_{\alpha }^{\tau }p\left(z\right)\nu \left(z\right)\Delta z}}\\ +q\left(\tau \right){\displaystyle {\int }_{\alpha }^{\beta }g\left(z\right)\nu \left(z\right)\Delta z}\}\Delta \tau ,\end{array}$ (3.9)

$B_2\left(t\right)=B_1\left(t\right)+{\displaystyle {\int }_{\alpha }^{t}p\left(z\right)\left[r\left(z\right)+b_1\left(z\right)B_1\left(z\right)+d\left(z\right)\lambda \right]\Delta z},$ (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

$\begin{array}{l}{B}_1^{\Delta }\left(t\right)={\displaystyle {\int }_{\alpha }^t{b}_2^{\Delta }\left(t\right)p\left(\tau \right)\nu \left(\tau \right)\Delta \tau }+\left[b_2^{\sigma }\left(t\right)p\left(t\right)+1\right]\nu \left(t\right)\\ +b_2\left(t\right){\displaystyle {\int }_{\alpha }^{t}p\left(z\right)\nu \left(z\right)\Delta z}+q\left(t\right){\displaystyle {\int }_{\alpha }^{\beta }g\left(z\right)\nu \left(z\right)\Delta z}\\ \le \left[b_2^{\sigma }\left(t\right)p\left(t\right)+1\right]r\left(t\right)+\left[b_2^{\sigma }\left(t\right)p\left(t\right)+1\right]b_1\left(t\right)B_1\left(t\right)\\ +\left[b_2^{\sigma }\left(t\right)p\left(t\right)+1\right]d\left(t\right)\lambda +\left[b_2\left(t\right)+b_2^{\Delta }\left(t\right)\right]{\displaystyle {\int }_{\alpha }^{t}p\left(z\right)\nu \left(z\right)\Delta z}+q\left(t\right)\lambda \end{array}$

$\begin{array}{l}\le \left[b_2^{\sigma }\left(t\right)p\left(t\right)+1\right]r\left(t\right)+b_2^{\sigma }\left(t\right)p\left(t\right)b\left(t\right)B_1\left(t\right)\\ +b\left(t\right)\left[B_1\left(t\right)+{\displaystyle {\int }_{\alpha }^{t}p\left(z\right)\nu \left(z\right)\Delta z}\right]+\left[b_2^{\sigma }\left(t\right)p\left(t\right)+1\right]d\left(t\right)\lambda +q\left(t\right)\lambda \\ \le \left[b_2^{\sigma }\left(t\right)p\left(t\right)+1\right]\left[r\left(t\right)+b\left(t\right)B_2\left(t\right)+d\left(t\right)\lambda \right]+q\left(t\right)\lambda ,\end{array}$ (3.12)

integrating the inequality (3.12) and using $B_1\left(\alpha \right)=0$, we have

$B_1\left(t\right)\le {\displaystyle {\int }_{\alpha }^t\left\{\left[b_2^{\sigma }\left(\tau \right)p\left(\tau \right)+1\right]\left[r\left(\tau \right)+b\left(\tau \right)B_2\left(\tau \right)+d\left(\tau \right)\lambda \right]+q\left(\tau \right)\lambda \right\}\Delta \tau },$ (3.13)

therefore

$\begin{array}{c}{B}_2^{\Delta }\left(t\right)=B_1^{\Delta }\left(t\right)+p\left(t\right)\left[r\left(t\right)+b_1\left(t\right)B_1\left(t\right)+d\left(t\right)\lambda \right]\\ \le \left[b_2^{\sigma }\left(t\right)p\left(t\right)+1\right]\left[r\left(t\right)+b\left(t\right)B_2\left(t\right)+d\left(t\right)\lambda \right]+q\left(t\right)\lambda \\ +p\left(t\right)\left[r\left(t\right)+b\left(t\right)B_2\left(t\right)+d\left(t\right)\lambda \right]\\ =\left[b_2^{\sigma }\left(t\right)p\left(t\right)+p(t)+1\right]b\left(t\right)B_2\left(t\right)\\ +\left[b_2^{\sigma }\left(t\right)p\left(t\right)+p(t)+1\right]\left[r\left(t\right)+d\left(t\right)\lambda \right]+q\left(t\right)\lambda ,\end{array}$ (3.14)

now applying lemma 2.1, we get

$\begin{array}{l}{B}_2\left(t\right)\le {\displaystyle {\int }_{\alpha }^t{e}_{\left(b_2^{\sigma }p+p+1\right)b}\left(t,\sigma \left(z\right)\right)}\\ \times \left(\left[b_2^{\sigma }\left(z\right)p\left(z\right)+p\left(z\right)+1\right]\left[a\left(z\right)+d\left(z\right)\lambda \right]+q\left(z\right)\lambda \right)\Delta z,\end{array}$ (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

$\left|h\left(t,{\nu }_1,{\nu }_2,{\nu }_3,{\nu }_4\right)\right|\le L\left[\left|{\nu }_1\right|+\left|{\nu }_2\right|+\left|{\nu }_3\right|+\left|{\nu }_4\right|\right],$ (3.18)

$\left|h_1\left(t,z,u,\nu \right)\right|\le c_1\left(t\right)s_1\left(z\right)\left[\left|u\right|+\left|\nu \right|\right],$ (3.19)

$\left|h_2\left(t,z,u,\nu \right)\right|\le c_2\left(t\right)s_2\left(z\right)\left[\left|u\right|+\left|\nu \right|\right],$ (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

$\left|y\left(t\right)\right|+\left|y^{\Delta }\left(t\right)\right|\le M_1\left(t\right)+D_1{M}_2\left(t\right),t\in J_{\mathbb{T}},$ (3.21)

where

$\begin{array}{l}{M}_1\left(t\right)=\frac{\left|y_0\right|}{1-L}+\frac{L}{1-L}{\displaystyle {\int }_{\alpha }^t\left[c_1^{\sigma }\left(\tau \right)s_1\left(\tau \right)+1\right]}[\frac{\left|y_0\right|}{1-L}+b\left(\tau \right){\displaystyle {\int }_{\alpha }^{\tau }{e}_{\left(c_1^{\sigma }{s}_1+s_1+1\right)b}\left(\tau ,\sigma \left(z\right)\right)}\\ \times \left(c_1^{\sigma }\left(z\right)s_1\left(z\right)+s_1\left(z\right)+1\right)\frac{\left|y_0\right|}{1-L}\Delta z]\Delta \tau ,t\in J_{\mathbb{T}}\end{array}$ (3.22)

$\begin{array}{l}{M}_2\left(t\right)=\frac{L}{1-L}{c}_2\left(t\right)\\ +\frac{L}{1-L}{\displaystyle {\int }_{\alpha }^t\{\left[c_1^{\sigma }\left(\tau \right)s_1\left(\tau \right)+1\right][\frac{L}{1-L}{c}_2\left(\tau \right)+b\left(\tau \right){\displaystyle {\int }_{\alpha }^{\tau }{e}_{\left(c_1^{\sigma }{s}_1+s_1+1\right)b}\left(\tau ,\sigma \left(z\right)\right)}}\\ \times \left(\left[c_1^{\sigma }\left(z\right)s_1\left(z\right)+s_1\left(z\right)+1\right]\frac{L}{1-L}{c}_2\left(z\right)+c_2\left(z\right)\right)\Delta z]+c_2\left(\tau \right)\}\Delta \tau ,t\in J_{\mathbb{T}}\end{array}$

(3.23)

Assume that

$b\left(t\right)={\max }_{t\in J_{\mathbb{T}}}\left\{\frac{L}{1-L},c_1\left(t\right)+c_1^{\Delta }\left(t\right)\right\},$ (3.24)

$\lambda ={\displaystyle {\int }_{\alpha }^{\beta }{s}_2\left(\gamma \right)M_2\left(\gamma \right)\Delta \gamma }<1,$ (3.25)

$D_1=\frac{1}{1-\lambda }{\displaystyle {\int }_{\alpha }^{\beta }{s}_2\left(\gamma \right)M_1\left(\gamma \right)\Delta \gamma },$ (3.26)

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

$\begin{array}{l}a\left(t\right)=|y_0+{\displaystyle {\int }_{\alpha }^{t}h(\tau ,y\left(\tau \right),y^{\Delta }\left(\tau \right),{\displaystyle {\int }_{\alpha }^{\tau }{h}_1\left(\tau ,z,y\left(z\right),y^{\Delta }\left(z\right)\right)\Delta z},}\\ {\displaystyle {\int }_{\alpha }^{\beta }{h}_2\left(\tau ,z,y\left(z\right),y^{\Delta }\left(z\right)\right)\Delta z})\Delta \tau |\\ +\left|h\left(t,y\left(t\right),y^{\Delta }\left(t\right),{\displaystyle {\int }_{\alpha }^t{h}_1\left(t,z,y\left(z\right),y^{\Delta }\left(z\right)\right)\Delta z},{\displaystyle {\int }_{\alpha }^{\beta }{h}_2\left(t,z,y\left(z\right),y^{\Delta }\left(z\right)\right)\Delta z}\right)\right|\\ \le \left|y_0\right|+{\displaystyle {\int }_{\alpha }^{t}L\left[a\left(\tau \right)+{\displaystyle {\int }_{\alpha }^{\tau }{c}_1\left(\tau \right)s_1\left(z\right)a\left(z\right)\Delta z}+{\displaystyle {\int }_{\alpha }^{\beta }{c}_2\left(\tau \right)s_2\left(z\right)a\left(z\right)\Delta z}\right]\Delta \tau }\\ +L\left[a\left(t\right)+{\displaystyle {\int }_{\alpha }^t{c}_1\left(t\right)s_1\left(z\right)a\left(z\right)\Delta z}+{\displaystyle {\int }_{\alpha }^{\beta }{c}_2\left(t\right)s_2\left(z\right)a\left(z\right)\Delta z}\right]\end{array}$

from the above inequality, we have

$\begin{array}{c}a\left(t\right)\le \frac{\left|y_0\right|}{1-L}+\frac{L}{1-L}{\displaystyle {\int }_{\alpha }^t\{\left[c_1\left(t\right)s_1\left(\tau \right)+1\right]a\left(\tau \right)+c_1\left(\tau \right){\displaystyle {\int }_{\alpha }^{\tau }{s}_1\left(z\right)a\left(z\right)\Delta z}}\\ +c_2\left(\tau \right){\displaystyle {\int }_{\alpha }^{\beta }{s}_2\left(z\right)a\left(z\right)\Delta z}\}\Delta \tau +\frac{L}{1-L}{c}_2\left(t\right){\displaystyle {\int }_{\alpha }^{\beta }{s}_2\left(z\right)a\left(z\right)\Delta z},\end{array}$ (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

$\begin{array}{l}{Y}^{\Delta }\left(t\right)=H(t,Y\left(t\right),Y^{\Delta }\left(t\right),{\displaystyle {\int }_{\alpha }^t{h}_1\left(t,z,Y\left(z\right),Y^{\Delta }\left(z\right)\right)\Delta z},\\ {\displaystyle {\int }_{\alpha }^{\beta }{h}_2\left(t,z,Y\left(z\right),Y^{\Delta }\left(z\right)\right)\Delta z}),t\in J_{\mathbb{T}},\end{array}$

with the initial condition

$Y\left(\alpha \right)=Y_0$ (3.28)

where $H\in C_{rd}\left(J_{\mathbb{T}}\times {ℝ}^n\times {ℝ}^n\times {ℝ}^n\times {ℝ}^n,{ℝ}^n\right),h_1,h_2$ 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

$\begin{array}{l}\left|h\left(t,{\nu }_1,{\nu }_2,{\nu }_3,{\nu }_4\right)-h\left(t,{\bar{\nu }}_1,{\bar{\nu }}_2,{\bar{\nu }}_3,{\bar{\nu }}_4\right)\right|\\ \le L\left[\left|{\nu }_1-{\bar{\nu }}_1\right|+\left|{\nu }_2-{\bar{\nu }}_2\right|+\left|{\nu }_3-{\bar{\nu }}_3\right|+\left|{\nu }_4-{\bar{\nu }}_4\right|\right],\end{array}$ (3.29)

$\left|h_1\left(t,z,u,\nu \right)-h_1\left(t,z,\bar{u},\bar{\nu }\right)\right|\le c_1\left(t\right)s_1\left(z\right)\left[\left|u-\bar{u}\right|+\left|\nu -\bar{\nu }\right|\right],$ (3.30)

$\left|h_2\left(t,z,u,\nu \right)-h_2\left(t,z,\bar{u},\bar{\nu }\right)\right|\le c_2\left(t\right)s_2\left(z\right)\left[\left|u-\bar{u}\right|+\left|\nu -\bar{\nu }\right|\right],$ (3.31)

where the functions $h,h_1,h_2$ in (1.1), and $0\le L<1$ is a constant.

Also $c_1,s_1,c_2,s_2\in C_{rd}\left(J_{\mathbb{T}},ℝ+\right)$, and

$\left|h\left(t,{\nu }_1,{\nu }_2,{\nu }_3,{\nu }_4\right)-H\left(t,{\nu }_1,{\nu }_2,{\nu }_3,{\nu }_4\right)\right|\le \epsilon ,$ (3.32)

$\left|y_0-Y_0\right|\le \delta ,$ (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

$\left|y\left(t\right)-Y\left(t\right)\right|+\left|y^{\Delta }\left(t\right)-Y^{\Delta }\left(t\right)\right|\le M_3\left(t\right)+D_2{M}_2\left(t\right),t\in J_{\mathbb{T}},$ (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 $\left|y_0\right|$, $M_2\left(t\right),b\left(t\right)$ and $\lambda$ be as in (3.23), (3.24) and (3.25) respectively and

$D_2=\frac{1}{1-\lambda }{\displaystyle {\int }_{\alpha }^{\beta }{r}_2\left(\gamma \right)M_3\left(\gamma \right)\Delta \gamma },$ (3.35)

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

$\begin{array}{l}w\left(t\right)\le \left|y_0-Y_0\right|\\ +{\displaystyle {\int }_{\alpha }^t|h\left(\tau ,y\left(\tau \right),y^{\Delta }\left(\tau \right),{\displaystyle {\int }_{\alpha }^{\tau }{h}_1\left(\tau ,z,y\left(z\right),y^{\Delta }\left(z\right)\right)\Delta z},{\displaystyle {\int }_{\alpha }^{\beta }{h}_2\left(\tau ,z,y\left(z\right),y^{\Delta }\left(z\right)\right)\Delta z}\right)}\\ -h\left(\tau ,Y\left(\tau \right),Y^{\Delta }\left(\tau \right),{\displaystyle {\int }_{\alpha }^{\tau }{h}_1\left(\tau ,z,Y\left(z\right),Y^{\Delta }\left(z\right)\right)\Delta z},{\displaystyle {\int }_{\alpha }^{\beta }{h}_2\left(\tau ,z,Y\left(z\right),Y^{\Delta }\left(z\right)\right)\Delta z}\right)|\Delta \tau \\ +{\displaystyle {\int }_{\alpha }^t|h\left(\tau ,Y\left(\tau \right),Y^{\Delta }\left(\tau \right),{\displaystyle {\int }_{\alpha }^{\tau }{h}_1\left(\tau ,z,Y\left(z\right),Y^{\Delta }\left(z\right)\right)\Delta z},{\displaystyle {\int }_{\alpha }^{\beta }{h}_2\left(\tau ,z,Y\left(z\right),Y^{\Delta }\left(z\right)\right)\Delta z}\right)}\\ -H\left(\tau ,Y\left(\tau \right),Y^{\Delta }\left(\tau \right),{\displaystyle {\int }_{\alpha }^r{h}_1\left(\tau ,z,Y\left(z\right),Y^{\Delta }\left(z\right)\right)\Delta z},{\displaystyle {\int }_{\alpha }^{\beta }{h}_2\left(\tau ,z,Y\left(z\right),Y^{\Delta }\left(z\right)\right)\Delta z}\right)|\Delta \tau \end{array}$

$\begin{array}{l}+|h\left(t,y\left(t\right),y^{\Delta }\left(t\right),{\displaystyle {\int }_{\alpha }^t{h}_1\left(t,z,y\left(z\right),y^{\Delta }\left(z\right)\right)\Delta z},{\displaystyle {\int }_{\alpha }^{\beta }{h}_2\left(t,z,y\left(z\right),y^{\Delta }\left(z\right)\right)\Delta z}\right)\\ -h\left(t,Y\left(t\right),Y^{\Delta }\left(t\right),{\displaystyle {\int }_{\alpha }^t{h}_1\left(t,z,Y\left(z\right),Y^{\Delta }\left(z\right)\right)\Delta z},{\displaystyle {\int }_{\alpha }^{\beta }{h}_2\left(t,z,Y\left(z\right),Y^{\Delta }\left(z\right)\right)\Delta z}\right)|\\ +|h\left(t,Y\left(t\right),Y^{\Delta }\left(t\right),{\displaystyle {\int }_{\alpha }^t{h}_1\left(t,z,Y\left(z\right),Y^{\Delta }\left(z\right)\right)\Delta z},{\displaystyle {\int }_{\alpha }^{\beta }{h}_2\left(t,z,Y\left(z\right),Y^{\Delta }\left(z\right)\right)\Delta z}\right)\\ -H\left(t,Y\left(t\right),Y^{\Delta }\left(t\right),{\displaystyle {\int }_{\alpha }^t{h}_1\left(t,z,Y\left(z\right),Y^{\Delta }\left(z\right)\right)\Delta z},{\displaystyle {\int }_{\alpha }^{\beta }{h}_2\left(t,z,Y\left(z\right),Y^{\Delta }\left(z\right)\right)\Delta z}\right)|\end{array}$

$\begin{array}{l}\le \delta +{\displaystyle {\int }_{\alpha }^{t}L\left[w\left(\tau \right)+{\displaystyle {\int }_{\alpha }^{\tau }{c}_1\left(\tau \right)s_1\left(z\right)w\left(z\right)\Delta z}+{\displaystyle {\int }_{\alpha }^{\beta }{c}_2\left(\tau \right)s_2\left(z\right)w\left(z\right)\Delta z}\right]\Delta \tau }\\ +{\displaystyle {\int }_{\alpha }^t\epsilon \Delta \tau }+L\left[w\left(t\right)+{\displaystyle {\int }_{\alpha }^t{c}_1\left(t\right)s_1\left(z\right)w\left(z\right)\Delta z}+{\displaystyle {\int }_{\alpha }^{\beta }{c}_2\left(t\right)s_2\left(z\right)w\left(z\right)\Delta z}\right]+\epsilon \\ =m\left(t\right)+L{\displaystyle {\int }_{\alpha }^t\left[w\left(\tau \right)+c_1\left(\tau \right){\displaystyle {\int }_{\alpha }^{\tau }{s}_1\left(z\right)w\left(z\right)\Delta z}+c_2\left(\tau \right){\displaystyle {\int }_{\alpha }^{\beta }{s}_2\left(z\right)w\left(z\right)\Delta z}\right]\Delta \tau }\\ +L\left[w\left(t\right)+{\displaystyle {\int }_{\alpha }^t{c}_1\left(t\right)s_1\left(z\right)w\left(z\right)\Delta z}+c_2\left(t\right){\displaystyle {\int }_{\alpha }^{\beta }{s}_2\left(z\right)w\left(z\right)\Delta z}\right]\end{array}$

then we get

$\begin{array}{c}w\left(t\right)\le \frac{m\left(t\right)}{1-L}+\frac{L}{1-L}{\displaystyle {\int }_{\alpha }^t\{\left[c_1\left(t\right)s_1\left(\tau \right)+1\right]w\left(\tau \right)+c_1\left(\tau \right){\displaystyle {\int }_{\alpha }^{\tau }{s}_1\left(z\right)w\left(z\right)\Delta z}}\\ +c_2\left(\tau \right){\displaystyle {\int }_{\alpha }^{\beta }{s}_2\left(z\right)w\left(z\right)\Delta z}\}\Delta \tau +\frac{L}{1-L}{c}_2\left(t\right){\displaystyle {\int }_{\alpha }^{\beta }{s}_2\left(z\right)w\left(z\right)\Delta z},\end{array}$ (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

$\begin{array}{l}{y}^{\Delta }\left(t\right)=h(t,y\left(t\right),y^{\Delta }\left(t\right),{\displaystyle {\int }_{\alpha }^t{h}_1\left(t,z,y\left(z\right),y^{\Delta }\left(z\right)\right)\Delta z},\\ {\displaystyle {\int }_{\alpha }^{\beta }{h}_2\left(t,z,y\left(z\right),y^{\Delta }\left(z\right)\right)\Delta z}),t\in J_{\mathbb{T}},\end{array}$

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

$\left|y_1\left(t\right)-y_2\left(t\right)\right|+\left|y_1^{\Delta }\left(t\right)-y_2^{\Delta }\left(t\right)\right|\le M_4\left(t\right)+D_3{M}_2\left(t\right),t\in J_{\mathbb{T}},$ (3.38)

where $M_4\left(t\right)$ is described by the right side of (3.22) by substituting $\left|c_1-c_2\right|$ instead of $\left|y_0\right|$, $M_2\left(t\right),b\left(t\right)$ and $\lambda$ be as in (3.23), (3.24) and (3.25) respectively and

$D_3=\frac{1}{1-\lambda }{\displaystyle {\int }_{\alpha }^{\beta }{r}_2\left(\gamma \right)M_4\left(\gamma \right)\Delta \gamma },$ (3.39)

Proof. Let $n\left(t\right)=\left|y_1\left(t\right)-y_2\left(t\right)\right|+\left|y_1^{\Delta }\left(t\right)-y_2^{\Delta }\left(t\right)\right|,t\in J_{\mathbb{T}}$, we get

$\begin{array}{l}n\left(t\right)\le \left|c_1-c_2\right|\\ +{\displaystyle {\int }_{\alpha }^t|h\left(\tau ,y_1\left(\tau \right),y_1^{\Delta }\left(\tau \right),{\displaystyle {\int }_{\alpha }^{\tau }{h}_1\left(\tau ,z,y_1\left(z\right),y_1^{\Delta }\left(z\right)\right)\Delta z},{\displaystyle {\int }_{\alpha }^{\beta }{h}_2\left(\tau ,z,y_1\left(z\right),y_1^{\Delta }\left(z\right)\right)\Delta z}\right)}\\ -h\left(\tau ,y_2\left(\tau \right),y_2^{\Delta }\left(\tau \right),{\displaystyle {\int }_{\alpha }^{\tau }{h}_1\left(\tau ,z,y_2\left(z\right),y_2^{\Delta }\left(z\right)\right)\Delta z},{\displaystyle {\int }_{\alpha }^{\beta }{h}_2\left(\tau ,z,y_2\left(z\right),y_2^{\Delta }\left(z\right)\right)\Delta z}\right)|\Delta \tau \\ +|h\left(t,y_1\left(t\right),y_1^{\Delta }\left(t\right),{\displaystyle {\int }_{\alpha }^t{h}_1\left(t,z,y_1\left(z\right),y_1^{\Delta }\left(z\right)\right)\Delta z},{\displaystyle {\int }_{\alpha }^{\beta }{h}_2\left(t,z,y_1\left(z\right),y_1^{\Delta }\left(z\right)\right)\Delta z}\right)\\ -h\left(t,y_2\left(t\right),y_2^{\Delta }\left(t\right),{\displaystyle {\int }_{\alpha }^t{h}_1\left(t,z,y_2\left(z\right),y_2^{\Delta }\left(z\right)\right)\Delta z},{\displaystyle {\int }_{\alpha }^{\beta }{h}_2\left(t,z,y_2\left(z\right),y_2^{\Delta }\left(z\right)\right)\Delta z}\right)|\end{array}$

$\begin{array}{l}\le \left|c_1-c_2\right|+L{\displaystyle {\int }_{\alpha }^t\left[n\left(\tau \right)+c_1\left(\tau \right){\displaystyle {\int }_{\alpha }^{\tau }{s}_1\left(z\right)n\left(z\right)\Delta z}+c_2\left(\tau \right){\displaystyle {\int }_{\alpha }^{\beta }{s}_2\left(z\right)n\left(z\right)\Delta z}\right]\Delta \tau }\\ +L\left[n\left(t\right)+{\displaystyle {\int }_{\alpha }^t{c}_1\left(t\right)s_1\left(z\right)n\left(z\right)\Delta z}+c_2\left(t\right){\displaystyle {\int }_{\alpha }^{\beta }{s}_2\left(z\right)n\left(z\right)\Delta z}\right]\end{array}$

then

$\begin{array}{l}n\left(t\right)\le \frac{\left|c_1-c_2\right|}{1-L}+\frac{L}{1-L}{\displaystyle {\int }_{\alpha }^t\{\left[c_1\left(t\right)s_1\left(\tau \right)+1\right]n\left(\tau \right)+c_1\left(\tau \right){\displaystyle {\int }_{\alpha }^{\tau }{s}_1\left(z\right)n\left(z\right)\Delta z}}\\ +c_2\left(\tau \right){\displaystyle {\int }_{\alpha }^{\beta }{s}_2\left(z\right)n\left(z\right)\Delta z}\}\Delta \tau +\frac{L}{1-L}{c}_2\left(t\right){\displaystyle {\int }_{\alpha }^{\beta }{s}_2\left(z\right)n\left(z\right)\Delta z},\end{array}$ (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

$\begin{array}{l}{Y}^{\Delta }\left(t\right)=h(t,Y\left(t\right),Y^{\Delta }\left(t\right),{\displaystyle {\int }_{\alpha }^t{h}_1\left(t,z,Y\left(z\right),Y^{\Delta }\left(z\right)\right)\Delta z},\\ {\displaystyle {\int }_{\alpha }^{\beta }{h}_2\left(t,z,Y\left(z\right),Y^{\Delta }\left(z\right)\right)\Delta z},\mu ),t\in J_{\mathbb{T}},Y\left(\alpha \right)=Y_0,\end{array}$ (3.41)

$\begin{array}{l}{Y}^{\Delta }\left(t\right)=h(t,Y\left(t\right),Y^{\Delta }\left(t\right),{\displaystyle {\int }_{\alpha }^t{h}_1\left(t,z,Y\left(z\right),Y^{\Delta }\left(z\right)\right)\Delta z},\\ {\displaystyle {\int }_{\alpha }^{\beta }{h}_2\left(t,z,Y\left(z\right),Y^{\Delta }\left(z\right)\right)\Delta z},{\mu }_0),t\in J_{\mathbb{T}},Y\left(\alpha \right)=Y_0,\end{array}$ (3.42)

where $h\in C_{rd}\left(J_{\mathbb{T}}\times {ℝ}^n\times {ℝ}^n\times {ℝ}^n\times {ℝ}^n,{ℝ}^n\right)$ 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

$\begin{array}{l}\left|h\left(t,{\nu }_1,{\nu }_2,{\nu }_3,{\nu }_4,\mu \right)-h\left(t,{\bar{\nu }}_1,{\bar{\nu }}_2,{\bar{\nu }}_3,{\bar{\nu }}_4,\mu \right)\right|\\ \le L\left[\left|{\nu }_1-{\bar{\nu }}_1\right|+\left|{\nu }_2-{\bar{\nu }}_2\right|+\left|{\nu }_3-{\bar{\nu }}_3\right|+\left|{\nu }_4-{\bar{\nu }}_4\right|\right],\end{array}$ (3.43)

$\left|h\left(t,{\nu }_1,{\nu }_2,{\nu }_3,{\nu }_4,\mu \right)-h\left(t,{\nu }_1,{\nu }_2,{\nu }_3,{\nu }_4,{\mu }_0\right)\right|\le k\left(t\right)\left|\mu -{\mu }_0\right|,$ (3.44)

where $0\le L<1$ is a constant and $k\in C_{rd}\left(J_{\mathbb{T}},ℝ+\right)$. 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

$\left|Y_1\left(t\right)-Y_2\left(t\right)\right|+\left|Y_1^{\Delta }\left(t\right)-Y_2^{\Delta }\left(t\right)\right|\le M_5\left(t\right)+D_4{M}_2\left(t\right),t\in J_{\mathbb{T}},$ (3.45)

where $M_5\left(t\right)$ is described by the right side of (3.22) by substituting $\left|\mu -{\mu }_0\right|\bar{k}\left(t\right)$ instead of $\left|y_0\right|$, $M_2\left(t\right),b\left(t\right)$ and $\lambda$ be as in (3.23), (3.24) and (3.25) respectively.

Let

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

$D_4=\frac{1}{1-\lambda }{\displaystyle {\int }_{\alpha }^{\beta }{r}_2\left(\gamma \right)M_5\left(\gamma \right)\Delta \gamma },$ (3.47)

Proof. Let $P\left(t\right)=\left|Y_1\left(t\right)-Y_2\left(t\right)\right|+\left|Y_1^{\Delta }\left(t\right)-Y_2^{\Delta }\left(t\right)\right|,t\in J_{\mathbb{T}}$, we have

$\begin{array}{l}P\left(t\right)\\ \le {\displaystyle {\int }_{\alpha }^t|h\left(\tau ,Y_1\left(\tau \right),Y_1^{\Delta }\left(\tau \right),{\displaystyle {\int }_{\alpha }^{\tau }{h}_1\left(\tau ,z,Y_1\left(z\right),Y_1^{\Delta }\left(z\right)\right)\Delta z},{\displaystyle {\int }_{\alpha }^{\beta }{h}_2\left(\tau ,z,Y_1\left(z\right),Y_1^{\Delta }\left(z\right)\right)\Delta z},\mu \right)}\\ -h\left(\tau ,Y_2\left(\tau \right),Y_2^{\Delta }\left(\tau \right),{\displaystyle {\int }_{\alpha }^{\tau }{h}_1\left(\tau ,z,Y_2\left(z\right),Y_2^{\Delta }\left(z\right)\right)\Delta z},{\displaystyle {\int }_{\alpha }^{\beta }{h}_2\left(\tau ,z,Y_2\left(z\right),Y_2^{\Delta }\left(z\right)\right)\Delta z},\mu \right)|\Delta \tau \\ +{\displaystyle {\int }_{\alpha }^t|h\left(\tau ,Y_2\left(\tau \right),Y_2^{\Delta }\left(\tau \right),{\displaystyle {\int }_{\alpha }^{\tau }{h}_1\left(\tau ,z,Y_2\left(z\right),Y_2^{\Delta }\left(z\right)\right)\Delta z},{\displaystyle {\int }_{\alpha }^{\beta }{h}_2\left(\tau ,z,Y_2\left(z\right),Y_2^{\Delta }\left(z\right)\right)\Delta z},\mu \right)}\\ -h\left(\tau ,Y_2\left(\tau \right),Y_2^{\Delta }\left(\tau \right),{\displaystyle {\int }_{\alpha }^{\tau }{h}_1\left(\tau ,z,Y_2\left(z\right),Y_2^{\Delta }\left(z\right)\right)\Delta z},{\displaystyle {\int }_{\alpha }^{\beta }{h}_2\left(\tau ,z,Y_2\left(z\right),Y_2^{\Delta }\left(z\right)\right)\Delta z},{\mu }_0\right)|\Delta \tau \end{array}$

$\begin{array}{l}+|h\left(t,Y_1\left(t\right),Y_1^{\Delta }\left(t\right),{\displaystyle {\int }_{\alpha }^t{h}_1\left(t,z,Y_1\left(z\right),Y_1^{\Delta }\left(z\right)\right)\Delta z},{\displaystyle {\int }_{\alpha }^{\beta }{h}_2\left(t,z,Y_1\left(z\right),Y_1^{\Delta }\left(z\right)\right)\Delta z},\mu \right)\\ -h\left(t,Y_2\left(t\right),Y_2^{\Delta }\left(t\right),{\displaystyle {\int }_{\alpha }^t{h}_1\left(t,z,Y_2\left(z\right),Y_2^{\Delta }\left(z\right)\right)\Delta z},{\displaystyle {\int }_{\alpha }^{\beta }{h}_2\left(t,z,Y_2\left(z\right),Y_2^{\Delta }\left(z\right)\right)\Delta z},\mu \right)|\\ +|h\left(t,Y_2\left(t\right),Y_2^{\Delta }\left(t\right),{\displaystyle {\int }_{\alpha }^t{h}_1\left(t,z,Y_2\left(z\right),Y_2^{\Delta }\left(z\right)\right)\Delta z},{\displaystyle {\int }_{\alpha }^{\beta }{h}_2\left(t,z,Y_2\left(z\right),Y_2^{\Delta }\left(z\right)\right)\Delta z},\mu \right)\\ -h\left(t,Y_2\left(t\right),Y_2^{\Delta }\left(t\right),{\displaystyle {\int }_{\alpha }^t{h}_1\left(t,z,Y_2\left(z\right),Y_2^{\Delta }\left(z\right)\right)\Delta z},{\displaystyle {\int }_{\alpha }^{\beta }{h}_2\left(t,z,Y_2\left(z\right),Y_2^{\Delta }\left(z\right)\right)\Delta z},{\mu }_0\right)|\end{array}$

$\begin{array}{l}\le {\displaystyle {\int }_{\alpha }^{t}L\left[P\left(\tau \right)+{\displaystyle {\int }_{\alpha }^{\tau }{c}_1\left(\tau \right)s_1\left(z\right)P\left(z\right)\Delta z}+{\displaystyle {\int }_{\alpha }^{\beta }{c}_2\left(\tau \right)s_2\left(z\right)P\left(z\right)\Delta z}\right]\Delta \tau }\\ +{\displaystyle {\int }_{\alpha }^{t}k\left(\tau \right)\left|\mu -{\mu }_0\right|\Delta \tau }\\ +L\left[P\left(t\right)+{\displaystyle {\int }_{\alpha }^t{c}_1\left(t\right)s_1\left(z\right)P\left(z\right)\Delta z}+{\displaystyle {\int }_{\alpha }^{\beta }{c}_2\left(t\right)s_2\left(z\right)P\left(z\right)\Delta z}\right]+k\left(t\right)\left|\mu -{\mu }_0\right|\\ =\left|\mu -{\mu }_0\right|\bar{k}\left(t\right)+L{\displaystyle {\int }_{\alpha }^t\left[P\left(\tau \right)+c_1\left(\tau \right){\displaystyle {\int }_{\alpha }^{\tau }{s}_1\left(z\right)P\left(z\right)\Delta z}+c_2\left(\tau \right){\displaystyle {\int }_{\alpha }^{\beta }{s}_2\left(z\right)P\left(z\right)\Delta z}\right]\Delta \tau }\\ +L\left[P\left(t\right)+{\displaystyle {\int }_{\alpha }^t{c}_1\left(t\right)s_1\left(z\right)P\left(z\right)\Delta z}+{\displaystyle {\int }_{\alpha }^{\beta }{c}_2\left(t\right)s_2\left(z\right)P\left(z\right)\Delta z}\right]\end{array}$

then we have

$\begin{array}{l}P\left(t\right)\le \frac{\left|\mu -{\mu }_0\right|\bar{k}\left(t\right)}{1-L}\\ +\frac{L}{1-L}{\displaystyle {\int }_{\alpha }^t\{\left[c_1\left(t\right)s_1\left(\tau \right)+1\right]P\left(\tau \right)+c_1\left(\tau \right){\displaystyle {\int }_{\alpha }^{\tau }{s}_1\left(z\right)P\left(z\right)\Delta z}}\\ +c_2\left(\tau \right){\displaystyle {\int }_{\alpha }^{\beta }{s}_2\left(z\right)P\left(z\right)\Delta z}\}\Delta \tau +\frac{L}{1-L}{c}_2\left(t\right){\displaystyle {\int }_{\alpha }^{\beta }{s}_2\left(z\right)P\left(z\right)\Delta z},\end{array}$ (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.

References

[1]	Bohner, M. and Georgiev, S.G. (2016) Multivariable Dynamic Calculus on Time Scales. Springer, Berlin. https://doi.org/10.1007/978-3-319-47620-9
[2]	Bohner, M. and Peterson, A. (2001) Dynamic Equations on Time Scales: An Introduction with Applications. Springer Science & Business Media. https://doi.org/10.1007/978-1-4612-0201-1
[3]	Georgiev, S.G. (2016) Integral Equations on Time Scales, Integral Equations on Time Scales. https://doi.org/10.2991/978-94-6239-228-1
[4]	Pachpatte, D. (2009) On a Nonstandard Volterra Type Dynamic Integral Equation on Time Scales. Electronic Journal of Qualitative Theory of Differential Equations, 72, 1-14. http://www.kurims.kyoto-u.ac.jp/EMIS/journals/EJQTDE/p459.pdf https://doi.org/10.14232/ejqtde.2009.1.72
[5]	Pachpatte, D.B. (2010) Fredholm Type Integrodifferential Equation on Time Scales. Electronic Journal of Differential Equations (EJDE), 2010, 1-10. http://www.kurims.kyoto-u.ac.jp/EMIS/journals/EJDE/Volumes/2010/140/pachpatte.pdf
[6]	Pachpatte, D.B. (2013) Properties of Some Dynamic Integral Equation on Time Scales. Annals of Functional Analysis, 4, 12–26. https://doi.org/10.15352/afa/1399899522 http://emis.ams.org/journals/AFA/AFA-tex_v4_n2_a2.pdf
[7]	Reinfelds, A. and Christian, S. (2019) Volterra Integral Equations on Unbounded Time Scales. International Journal of Difference Equations, 14, 169-177. https://doi.org/10.37622/IJDE/14.2.2019.169-177 http://campus.mst.edu/ijde/contents/v14n2p5.pdf
[8]	Salih, O.M. and Mahmood, A.H. (2018) Approximate Solutions for Certain Integrodifferential Equations of Volterra_Fredholm Type. International Journal of Enhanced Research in Science, Technology & Engineering, 7, 37-42.