POSITIVE SOLUTIONS OF NONLOCAL BOUNDARY VALUE PROBLEM FOR HIGHER ORDER FRACTIONAL DIFFERENTIAL SYSTEM

In this paper, we study existence and multiplicity results for a coupled system of nonlinear nonlocal boundary value problems for higher order fractional differential equations of the type    cDα 0+u(t) = λa(t)f(u(t), v(t)), cD β 0+v(t) = µb(t)g(u(t), v(t)), u ′ (0) = u ′′(0) = u ′′′(0) = · · · = u (n−1)(0) = 0, u(1) = ξ1u(η1), v ′ (0) = v ′′(0) = v ′′′(0) = · · · = v (n−1)(0) = 0, v(1) = ξ2v(η2), where λ, µ > 0, n − 1 < α, β ≤ n for n ∈ N; ξi , ηi ∈ (0, 1) for i = 1, 2 and cDα 0+ is Caputo fractional derivative. We employ the Guo-Krasnosel’skii fixed point theorem to establish existence and multiplicity results for positive solutions. We derive explicit intervals for the parameters λ and µ for which the system possess the positive solutions or multiple positive solutions. Examples are included to show the applicability of the main results.