Notes on Krasnoselskii-type fixed-point theorems and their application to fractional hybrid differential problems

Abstract

In this paper we prove a new version of Kransoselskii's fixed-point theorem under a ($ψ, θ, φ$)-weak contraction condition. The theoretical result is applied to prove the existence of a solution of the following fractional hybrid differential equation involving the Riemann-Liouville differential and integral operators orders of $0<α<1$ and $β>0:$ \begin{equation}\nonumber \left\{\begin{array}{ll} D^α[x(t)-f(t, x(t))]=g(t, x(t), I^β(x(t))), \,\,\, \text{a.e.} \,\,\, t\in J,\,\, β>0,\\ x(t_{0})=x_{0}, \end{array} \right. \end{equation} where $D^α$ is the Riemann-Liouville fractional derivative order of $α,$ $I^β$ is Riemann-Liouville fractional integral operator order of $β>0,$ $J=[t_{0}, t_{0}+a],$ for some fixed $t_{0}\in \mathbb{R},$ $a>0$ and the functions $f:J\times \mathbb{R}\rightarrow \mathbb{R}$ and $g:J\times \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}$ satisfy certain conditions. An example is also furnished to illustrate the hypotheses and the abstract result of this paper.