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

Marcel-Adrian Serban
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

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About this paper

Journal

Fixed Point Theory

Publisher Name

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

Print ISSN

1583-5022

Online ISSN

2066-9208

google scholar link

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