On Fredholm type integral equation in two variables

Abstract

where f ,g are given functions and u is the unknown function to be found. Let R+ = [0,∞) , Ia = [0,a] , Ib = [0,b] (a > 0,b > 0) be the given subsets of R , the set of real numbers, Δ = Ia × Ib and C(A,B) denote the class of continuous functions from the set A to the set B . The partial derivatives of a function z(x,y) (x,y ∈ R) with respect to x and y are denoted by D1z(x,y) = ∂ ∂x z(x,y),D2z(x,y) = ∂ ∂y z(x,y). Throughout, we assume that f ∈C(Δ,R) , g∈C(Δ2 ×R3,R) and Di f ∈C(Δ,R) , Dig∈ C(Δ2×R3,R) for i = 1,2. In fact, the study of qualitative properties of solutions of equation (1.2) is challenging and requires new ideas in handling the equations of the form (1.2). The main objective of the present paper is to study some fundamental qualitative properties of solutions of equation (1.2) under some suitable conditions on the