POSITIVE SOLUTION FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION WITH NONLOCAL MULTI-POINT CONDITION

In this paper, we study and consider the positive solution of fractional differential equation with nonlocal multi-point conditions of the from: D-RL(0+)q u(t) + g(t)f(t, u(t)) = 0, t is an element of (0, 1) u((k))(0) = 0, u(1) = Sigma(m)(i=1) beta(i) (RL)D(0+)(pi)u(eta(i)) where n - 1 < q < n, n >= 2, n - 1 < p(i) < n, q > p(i) m,n is an element of N, k = 0,1, ..., n - 2, 0 < eta(1)< eta(2) < ... < kappa beta(i) <= 0, kappa is an element of (0, 1], D-RL(0+)q, D-RL(0+)pi are the Riemann-Liouville fractional derivative of order q, pi, f: [0, 1] x C([0, 1], E) -> E, E be Banach space and g: (0, 1) -> R+ are continuous functions. The main tools for finding positive solutions of the above problem are the fixed point theorems of Guo-Krasnoselskii and of Boyd and Wong. An example is included to illustrate the applicability of our results.