# Article ID: MTJPAM-D-21-00021

## Title: On Some Classes of Fredholm-Volterra Integral Equations in Two Variables

Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-21-00021; Volume 4 / Issue 3 / Year 2022 (Special Issue), Pages 25-32

Document Type: Research Paper

Author(s): Adrian Petruşel a , Ioan A. Rus b

aDepartment of Mathematics, Babeş-Bolyai University Cluj-Napoca and Academy of Romanian Scientists Bucharest, Romania

bDepartment of Mathematics, Babeş-Bolyai University Cluj-Napoca, Romania

Received: 26 February 2021, Accepted: 16 June 2021, Published: 9 July 2021.

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Abstract

Let a, b, c ∈ ℝ2, ai < ci < bi, i ∈ {1, 2}, [a, b]:=[a1, b1]×[a2, b2], let $(\mathbb{B},|\cdot |)$ be a (real or complex) Banach space, $K\in C([a,b]\times [a,c]\times \mathbb{B},\mathbb{B})$, $H\in C([a,b]\times [a,b]\times \mathbb{B},\mathbb{B})$ and $g\in C([a,b],\mathbb{B})$. In this paper we study the following integral equation

$u(x)=\int\limits_{[a,c]}K(x,s,u(s))ds +\int\limits_{[a,x]}H(x,s,u(s))ds +g(x),\ x=(x_1,x_2)\in [a,b].$

Using the Fibre Contraction Principle we give existence and uniqueness results, and we prove the convergence of the successive approximations. By the weakly Picard operator theory (in the framework of the ordered Banach space $\mathbb{B}$) we give Gronwall lemma type results and comparison theorems. Some other similar type of Fredholm-Volterra integral equations are also studied.

Keywords: Fredholm-Volterra integral equation, Existence and uniqueness, Successive approximations, Integral inequality, Gronwall lemma, Comparison lemma, Fibre contraction principle

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