# Applications of fibre contraction principle to some classes of functional integral equations

## Abstract

Let $$a<c<b)$$ real numbers, $$(\mathbb{B},|\cdot|)$$ a (real or complex) Banach space, $$H\in C([a,b]\times [a,c]\times\mathbb{B},\mathbb{B})$$, $$K\in C([a,b]^{2}\times\mathbb{B},\mathbb{B})$$,  $$g\in C([a,b]),\mathbb{B} ,A:C([a,c],\mathbb{B})\rightarrow C([a,c],\mathbb{B})$$ and $$B:C([a,b],\mathbb{B})\rightarrow C([a,b],\mathbb{B})$$.

In this paper we study the following functional integral equation,
$x (t) =\int_a^c H( t,s,A) ( x) ( s)) ds \int_a^t K (t,s,B) (x) (s))ds g(t), \quad t\in [a,b]$

By a new variant of fibre contraction principle (A. Petrusel, I.A. Rus, M.A. Serban, Some variants of fibre contraction principle and applications: from existence to the convergence of successive approximations, Fixed Point Theory, 22 (2021), no. 2, 795-808) we give existence, uniqueness and convergence of successive approximations results for this equation.

In the case of ordered Banach space $$\mathbb{B}$$, Gronwall-type and comparison-type results are also given.

## Authors

Veronica Ilea
Babes-Bolyai University, Faculty of Mathematics and Computer Science, Cluj-Napoca, Romania

Diana Otrocol
Technical University of Cluj-Napoca, Cluj-Napoca, Romania,
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania

Ioan A. Rus
Babes-Bolyai University, Faculty of Mathematics and Computer Science,Cluj-Napoca, Romania

Babes-Bolyai University, Faculty of Mathematics and Computer Science, Cluj-Napoca, Romania

## Keywords

Functional integral equation, Volterra operator, Picard operator, fibre contraction principle, Gronwall lemma, comparison lemma.

## Paper coordinates

V. Ilea, D. Otrocol, I.A. Rus, M.-A. Serban, Applications of fibre contraction principle to some classes of functional integral equations, Fixed Point Theory, 23 (2022) no. 1, 279-292, http://doi.org/10.24193/fpt-ro.2022.1.18

## PDF

##### Journal

Fixed Point Theory

##### Publisher Name

Casa Cărţii de Ştiinţă Cluj-Napoca
(House of the Book of Science Cluj-Napoca)

1583-5022

2066-9208