Solvability for two dimensional functional integral equations via Petryshyn’s fixed point theorem

Abstract

This paper aims to use the Petryshyn's fixed point theorem associated with the measure of non-compactness to prove the existence of solutions of two-dimensional functional integral equations in the Banach algebra of continuous functions on the interval C([0, a] × [0, â], R), a, â > 0. Our existence results contains many functional integral equations as special case that arise in nonlinear analysis. Finally, we present some examples which show that our result is useful for various class of equations.

An existence result for implicit functional equations

“…Also, one can combine functional lists (as in Corollary 2.4) to form an integral equation under appropriate conditions that satisfy conditions (II)-(III) and obtain a fixed-point existence result about (integral) functional equations in C(I). Since Theorem 2.1 works for every bounded cube I ⊂ R r , one can obtain a multidimensional version of the above examples, for instance, Example 2.7 is a two-dimensional case of [21] with a few changes (see also [14]). Many other results, such as Hadamard-type fractional integral equations, fractional stochastic integral equations (even in product type) and so on, can be obtained in this way, for instance, some of them are [11-15, 17-21, 23, 25, 31].…”

On the solvability of a class of nonlinear functional integral equations involving Erdélyi–Kober fractional operator

“…It is worthwhile mentioning that Erdélyi-Kober integrals are applied to describe processes with continuously distributed scaling, medium with noninteger mass dimension, modeling of viscoelastic materials, and advection and dispersion of solutes in porous media. Quadratic integral equations have many potential applications in describing a great number of events, and problems of the real world, that can benefit both the understanding of profound complexities in the application and the field of fractional calculus itself [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40].…”