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 ($\psi, \theta, \varphi$)-weak contraction condition. The theoretical result is applied to the existence solution following fractional hybrid differential equation involving Riemann-Liouville and integral operators orders $0<\alpha<1$ $\beta>0:$ \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 $D^{\alpha}$ derivative order $\alpha,$ $I^{\beta}$ operator $\beta>0,$ $J=[t_{0}, t_{0}+a],$ for some fixed $t_{0}\in \mathbb{R},$ $a>0$ functions $f:J\times \mathbb{R}\rightarrow \mathbb{R}$ $g:J\times \mathbb{R}\times satisfy certain conditions. An example also furnished illustrate hypotheses abstract paper.