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$$\begin{array}{rlr}& {}^c{D}^{q}u(t)=f(t,u(t)),t\in [0,1],& u(\eta )=u^{\prime }(0)=u^{\prime \prime }(0)=\cdots =u^{n-2}(0)=0,I^{p}u(1)=0,0<\eta <1,\end{array}$$where $n-1<q\le n$, $n\in \mathbb{N}$, $n\ge 3$ and${}^c{D}^q$ denotes the Caputo fractional derivative of order $q$,$I^p$ is the Riemann-Liouville fractional integral of order $p$,$f:[0,1]\times \mathbb{R}\to \mathbb{R}$ is acontinuous function and ${\eta }^{n-1}\ne \frac{\mathrm{\Gamma }(n)}{(p+n-1)(p+n-2)\dots (p+1)}$. 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.