Solvability of a fractional hybrid initial value problem with supremum by using measures of noncompactness in Banach algebras

Abstract

In this paper, we study the existence of solutions for the following fractional hybrid initial value problem with supremumD0 + αx(t) f(t,x(t),max0 ≤τ≤t |x(τ)|)=g(t,x(t)),0<t<1,x(0)=0, where 0<α≤1 and D0 + α denotes the Riemann-Liouville fractional derivative. The main tool in our study is the technique of measures of noncompactness in the Banach algebras. Also, we present some examples to illustrate our results.