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

S. Kumar, R. K. Vats, H. K. Nashine

We present here a new type of three-point nonlinear fractional boundary value problem of arbitrary order of the form \begin{align*} &^{c}D^{q}u(t) = f(t,u(t)), t ı [0,1], &u(\eta) = u^{rime}(0)= u^{rimerime}(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 a continuous function and $\eta^{n-1} \neq \frac{\Gamma(n)}{(p+n-1)(p+n-2)\dots(p+1)}$. We give new existence and uniqueness results using Banach contraction principle, Krasnoselskii, Scheafer's fixed point theorem and Leray-Schauder degree theory. To justify the results, we give some illustrative examples.