On a quadratic integral equation with supremum involving Erdélyi-Kober fractional order

Abstract

We introduce Erdélyi-Kober fractional quadratic integral equation with supremum, namely x(t)=f(t)+α(Tx)(t)Γ(β)∫0tsα-1u(t,s,x(s),max[0,σ(s)]|x(τ)|)tα-sα1-βds,0≤t≤1,α>0,β∈(0,1). This equation contains as special cases numerous integral equations studied by other authors. We show that there exists at least one monotonic solution belonging to C[0, 1] of our equation. The main tools in our analysis are Darbo fixed point theorem and the measure of noncompactness related to monotonicity which was introduced by Banaś and Olszowy. Finally, we present an example for illustrating the natural realizations of our abstract results