EXISTENCE AND UNIQUENESS RESULTS FOR THREE-POINT NONLINEAR FRACTIONAL (ARBITRARY ORDER) BOUNDARY VALUE PROBLEM

We present here a new type of three-point nonlinear fractional boundary value problem of arbitrary order of the form (c)D(q)u(t) = f(t, u(t)), t is an element of[0,1], u(eta) = u'(0) = u ''(0) = . . . = u(n-2)(0) = 0, I(p)u(1) = 0, 0 < eta < 1, where n - 1 < q <= n, n is an element of N, n >= 3 and D-c(q) denotes the Caputo fractional derivative of order q, I-p is the Riemann-Liouville fractional integral of order p, f : [0, 1] x R -> R is a continuous function and eta(n-1) not equal Gamma(n)/(p+n-1)(p+n-2)...(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.