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

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 \(00,\\ 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.