Existence of solutions for hybrid fractional pantograph equations

Abstract

In this paper, we study the existence of the hybrid fractional pantograph equation {Dα0+[x(t)/f(t,x(t),(μt))]=g(t,x(t),x(σt)), 0 < t < 1, x+(0)= 0, where α, μ, σ ε (0, 1) and Dα0+ denotes the Riemann-Liouville fractional derivative. The results are obtained using the technique of measures of noncompactness in the Banach algebras and a fixed point theorem for the product of two operators verifying a Darbo type condition. Some examples are provided to illustrate our results.