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



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


	Fixed Point Theory, Volume 23, No. 1, 2022, 279-292, February 1st, 2022 DOI: 10.24193/fpt-ro.2022.1.18 Authors: V. Ilea, D. Otrocol, I.A. Rus, M.A. Șerban Abstract: Let a < c < b real numbers, (𝔹,\| ⋅ \|) a (real or complex) Banach space, H ∈ C([a,b] × [a,c] × 𝔹,𝔹), K ∈ C([a,b]² × 𝔹,𝔹), g ∈ C([a,b],𝔹), A:C([a,c],𝔹) → C([a,c],𝔹) and B:C([a,b],𝔹) → C([a,b],𝔹). In this paper we study the following functional integral equation, $x(t)=\int _{a}^{c}H(t,s,A(x)(s))ds \int _{a}^{t}K(t,s,B(x)(s))ds g(t),\ t\in\lbrack a,b].$ By a new variant of fibre contraction principle (A. Petrușel, I.A. Rus, M.A. Șerban, 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 𝔹, Gronwall-type and comparison-type results are also given. Key Words and Phrases: Functional integral equation, Volterra operator, Picard operator, fibre contraction principle, Gronwall lemma, comparison lemma. 2010 Mathematics Subject Classification: 47N05, 47H10, 45D05, 47H09, 54H25. Published on-line: February 1st, 2022. Abstract pdf Fulltext pdf Back to volume's table of contents