Abstract

Keywords

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

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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)