## Turkish Journal of Mathematics

#### Article Title

### Solvability, stability, smoothness and compactness of the set of solutions for a nonlinear functional integral equation

#### DOI

10.3906/mat-2101-102

#### Abstract

This paper is devoted to the study of the following nonlinear functional integral equation \begin{equation} f(x)=\sum\limits_{i=1}^{q}\alpha _{i}(x)f(\tau_{i}(x))+\int_{0}^{\sigma_{1}(x)}\Psi \left( x,t,f(\sigma _{2}(t)),\int_{0}^{\sigma_{3}(t)}f(s)ds\right) dt+g(x),\text{ }\forall x\in \lbrack 0,1], \tag{E} \label{E} \end{equation} where $\tau _{i},$ $\sigma _{1},$ $\sigma _{2},$ $\sigma _{3}:[0,1]\rightarrow \lbrack 0,1];$ $\alpha _{i},$ $g:[0,1]\rightarrow \mathbb{R};$ $\Psi :[0,1]\times \lbrack 0,1]\times \mathbb{R}^{2}\rightarrow \mathbb{R}$ are the given continuous functions and $f:[0,1]\,\rightarrow \mathbb{R}$ is an unknown function. First, two sufficient conditions for the existence and some properties of solutions of Eq. (E) are proved. By using Banach's fixed point theorem, we have the first sufficient condition yielding existence, uniqueness and stability of the solution. By applying Schauder's fixed point theorem, we have the second sufficient condition for the existence and compactness of the solution set. An example is also given in order to illustrate the results obtained here. Next, in the case of $\Psi \in C^{2}([0,1]\times \lbrack 0,1]\times \mathbb{R}^{2}; \mathbb{R}),$ we investigate the quadratic convergence for the solution of Eq. (E). Finally, the smoothness of the solution depending on data is established.

#### Keywords

Nonlinear functional integral equation, quadratic convergence, stability and smoothness of solutions, compactness of the solution set

#### First Page

1386

#### Last Page

1406

#### Recommended Citation

THUC, NGUYEN DAT; NGOC, LE THI PHUONG; and LONG, NGUYEN THANH
(2021)
"Solvability, stability, smoothness and compactness of the set of solutions for a nonlinear functional integral equation,"
*Turkish Journal of Mathematics*: Vol. 45:
No.
3, Article 19.
https://doi.org/10.3906/mat-2101-102

Available at:
https://journals.tubitak.gov.tr/math/vol45/iss3/19