# Existence of Solution for Non-Linear Functional Integral Equations of Two Variables in Banach Algebra

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

- (N1)
- $\left|\right|x\left|\right|\ge 0$ and $\left|\right|x\left|\right|=0\iff x=0;$
- (N2)
- $\left|\right|\alpha x\left|\right|=\left|\alpha \right|\left|\right|x\left|\right|;$
- (N3)
- $\left|\right|x+y\left|\right|\le \left|\right|x\left|\right|+\left|\right|y\left|\right|$ for all $x,y\in X$ and $\alpha \in \mathbb{R}.$

**Definition**

**2.**

- (A1)
- $\left(xy\right)z=x\left(yz\right),$
- (A2)
- $x(y+z)=xy+xz,$
- (A3)
- $(x+y)z=xz+yz,$
- (A4)
- $\alpha \left(xy\right)=\left(\alpha x\right)y=x\left(\alpha y\right)$ for all $x,y,z\in A$ and scalars $\alpha .$

**Definition**

**3.**

- (i)
- for all $X\in {\mathcal{M}}_{\overline{E}},$ we have that $\mu \left(X\right)=0$ implies that X is precompact.
- (ii)
- the family ker $\mu =\left\{X\in {\mathcal{M}}_{\overline{E}}:\mu \left(X\right)=0\right\}$ is non-empty, and ker $\mu \subset {\mathcal{N}}_{\overline{E}}.$
- (iii)
- $X\subseteq Z\Rightarrow \mu \left(X\right)\le \mu \left(Z\right).$
- (iv)
- $\mu \left(\overline{X}\right)=\mu \left(X\right).$
- (v)
- $\mu \left(convX\right)=\mu \left(X\right)$ where $convX$ is the convex closure of set $X.$
- (vi)
- $\mu \left(\lambda X+\left(1-\lambda \right)Z\right)\le \lambda \mu \left(X\right)+\left(1-\lambda \right)\mu \left(Z\right)$ for $\lambda \in \left[0,1\right].$
- (vii)
- if ${X}_{n}\in {\mathcal{M}}_{\overline{E}},{X}_{n}={\overline{X}}_{n},\phantom{\rule{0.222222em}{0ex}}{X}_{n+1}\subset {X}_{n}$ for $n=1,2,3,\dots $ and $\underset{n\to \infty}{lim}\mu \left({X}_{n}\right)=0$, then ${\bigcap}_{n=1}^{\infty}{X}_{n}\ne \varphi .$

**Definition**

**4**

**.**Let E be a Banach space. Consider a non-empty subset X of E and a continuous operator $T:X\to E$ transforming the bounded subset of X to the bounded ones. We say that T satisfies the Darbo condition with a constant k with respect to measure μ provided $\mu \left(TY\right)\le k\mu \left(Y\right)$ for each $Y\in {\mathcal{M}}_{E}$ such that $Y\subset X.$ If $k<1,$ then T is called a contraction with respect to $\mu .$

**Remark**

**1.**

**Theorem**

**1**

**.**Assume that Z is a non-empty, closed, bounded, and convex subset of a Banach space $\overline{E}.$ Let $S:Z\to Z$ be a continuous mapping. Suppose that there is a constant $k\in \left[0,1\right)$ such that:

**Theorem**

**2**

**.**Suppose that X is a non-empty, bounded, convex, and closed subset of a Banach algebra $E,$ and the operators P and T transform continuously the set X into E such that $P\left(X\right)$ and $T\left(X\right)$ are bounded. Furthermore, suppose that the operator $S=P.T$ transforms X into itself. If P and T satisfy on the set X the Darbo condition with respect to the measure of noncompactness μ with the constants ${k}_{1}$ and ${k}_{2},$ respectively, then S satisfies on X the Darbo condition with constant $\Vert P\left(X\right)\Vert {k}_{2}+\Vert T\left(X\right)\Vert {k}_{1}.$ Particularly, if:

## 3. Main Result

- (1)
- The functions $U:[0,a]\times [0,a]\times \mathbb{R}\to \mathbb{R},$$F:[0,a]\times [0,a]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$, and $G:[0,a]\times [0,a]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are continuous, and there exist nonnegative constants $L,M$ such that:$$\left|U(l,h,0)\right|\le L,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\left|F(l,h,{M}_{1},0)\right|\le M\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}and\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\left|F(l,h,{M}_{2},0)\right|\le M,$$
- (2)
- Let ${A}_{i}:[0,a]\times [0,a]\to {\mathbb{R}}_{+}$$(i=1,2,3,4,5)$ be continuous functions such that:$$\left|U(l,h,{x}_{1})-U(l,h,{x}_{2})\right|\le {A}_{1}(l,h)\left|{x}_{1}-{x}_{2}\right|,$$$$\left|F(l,h,y,{x}_{1})-F(l,h,y,{x}_{2})\right|\le {A}_{2}(l,h)\left|{x}_{1}-{x}_{2}\right|,$$$$\left|G(l,h,y,{x}_{1})-G(l,h,y,{x}_{2})\right|\le {A}_{3}(l,h)\left|{x}_{1}-{x}_{2}\right|,$$$$\left|F(l,h,{y}_{1},x)-F(l,h,{y}_{2},x)\right|\le {A}_{4}(l,h)\left|{y}_{1}-{y}_{2}\right|$$$$\left|G(l,h,{y}_{1},x)-G(l,h,{y}_{2},x)\right|\le {A}_{5}(l,h)\left|{y}_{1}-{y}_{2}\right|,$$$$K=max\{{A}_{i}(l,h):i=1,2,3,4,5;l,h\in [0,a]\},$$
- (3)
- The functions $P,\phantom{\rule{0.277778em}{0ex}}Q$ are continuous functions from $[0,a]\times [0,a]\times [0,a]\times [0,a]\times \mathbb{R}$ to $\mathbb{R}.$
- (4)
- Furthermore, $4\alpha \beta <1$ for $\alpha =2k,\phantom{\rule{0.277778em}{0ex}}\beta =L+M.$

**Theorem**

**3.**

**Proof.**

## 4. An Illustrative Example

**Example**

**1.**

## 5. An Iterative Algorithm Created by a Coupled Semi-Analytic Method to Find the Solution of the Integral Equation

Algorithm 1. Algorithm of calculating ${\mathit{u}}_{\mathit{k}}(\mathit{l},\mathit{h})$ |

${u}_{0}(l,h)=0,$ |

${u}_{1}(l,h)={A}_{0}(l,h){\widehat{A}}_{0}(l,h),$ |

${u}_{k}(l,h)={\displaystyle \sum _{i=0}^{k-1}}{A}_{i}(l,h){\widehat{A}}_{k-1-i}(l,h),\phantom{\rule{0.277778em}{0ex}}k=2,3,\cdots .$ |

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Srivastava, H.M.; Das, A.; Hazarika, B.; Mohiuddine, S.A.
Existence of Solution for Non-Linear Functional Integral Equations of Two Variables in Banach Algebra. *Symmetry* **2019**, *11*, 674.
https://doi.org/10.3390/sym11050674

**AMA Style**

Srivastava HM, Das A, Hazarika B, Mohiuddine SA.
Existence of Solution for Non-Linear Functional Integral Equations of Two Variables in Banach Algebra. *Symmetry*. 2019; 11(5):674.
https://doi.org/10.3390/sym11050674

**Chicago/Turabian Style**

Srivastava, Hari M., Anupam Das, Bipan Hazarika, and S. A. Mohiuddine.
2019. "Existence of Solution for Non-Linear Functional Integral Equations of Two Variables in Banach Algebra" *Symmetry* 11, no. 5: 674.
https://doi.org/10.3390/sym11050674