Existence and uniqueness results for three-point nonlinear fractional (arbitrary order)

Abstract

We present here a new type of three-point nonlinear fractionalboundary value problem of arbitrary order of the form\begin{align*}&^{c}D^{q}u(t) = f(t,u(t)),\ \ t \in [0,1], &u(\eta) = u^{\prime}(0)= u^{\prime\prime}(0) = \dots = u^{n-2}(0)= 0,\ I^{p}u(1) = 0,\ \ \ \ 0 < \eta < 1,\end{align*}where $n-1 < q \leq n$, $n \in \mathbb{N}$, $n \geq 3$ and$^{c}D^{q}$ denotes the Caputo fractional derivative of order $q$,$I^{p}$ is the Riemann-Liouville fractional integral of order $p$,$f : [0,1] \times \mathbb{R} \rightarrow \mathbb{R}$ is acontinuous function and $\eta^{n-1} \neq\frac{\Gamma(n)}{(p+n-1)(p+n-2)\dots(p+1)}$. We give new existenceand uniqueness results using Banach contraction principle,Krasnoselskii, Scheafer's fixed point theorem and Leray-Schauderdegree theory. To justify the results, we give some illustrativeexamples.