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

TL;DR

This paper extends Krasnoselskii's fixed-point theorem to a new weak contraction form and applies it to prove the existence of solutions for a class of fractional hybrid differential equations involving Riemann-Liouville operators.

Contribution

It introduces a novel version of Krasnoselskii's fixed-point theorem under a ($\\psi, \\theta, \\varphi$)-weak contraction condition and applies it to fractional hybrid differential equations.

Findings

Established a new fixed-point theorem under weak contraction conditions.

Proved the existence of solutions for fractional hybrid differential equations.

Provided an illustrative example validating the theoretical results.

Abstract

In this paper we prove a new version of Kransoselskii's fixed-point theorem under a ($\psi, \theta, \varphi$)-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^{\alpha}[x(t)-f(t, x(t))]=g(t, x(t), I^{\beta}(x(t))), \,\,\, \text{a.e.} \,\,\, t\in J,\,\, \beta>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 $f:J\times \mathbb{R}\rightarrow \mathbb{R}$ 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.