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 ($\psi ,\theta ,\phi$)-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 $0<\alpha <1$ and $\beta >0:$ \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 $D^{\alpha }$ is the Riemann-Liouville fractional derivative order of $\alpha ,$ $I^{\beta }$ is Riemann-Liouville fractional integral operator order of $\beta >0,$ $J=[t_0,t_0+a],$ for some fixed $t_0\in \mathbb{R},$ $a>0$ and the functions $f:J\times \mathbb{R}\to \mathbb{R}$ and $g:J\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ satisfy certain conditions. An example is also furnished to illustrate the hypotheses and the abstract result of this paper.