This paper investigates the existence of at least two positive solutions for the following high-order fractional semipositone boundary value problem (SBVP, for short) with coupled integral boundary value conditions:where n − 1 < α n, n 3, 0 < η 1 , η 2 1, λ, λ 1 , λ 2 are parameters and satisfy λ 1 λ 2 (η 1 η 2 ) α < Γ 2 (α + 1), D α 0 + is the standard Riemann-Liouville derivative, and f, g are continuous and semipositone. By using the nonlinear alternative of Leray-Schauder type, Krasnoselskii's fixed point theorems, and the theory of fixed point index on cone, we establish some existence results of multiple positive solutions to the considered fractional SBVP. As applications, two examples are presented to illustrate our main results. c 2017 All rights reserved.Keywords: Fractional differential equations, semipositone boundary value problem, coupled integral boundary value conditions, fixed point index. 2010 MSC: 34A08, 34B15, 34B18.

This paper deals with the exact controllability for a class of fractional evolution systems in a Banach space. First, we introduce a new concept of exact controllability and give notion of the mild solutions of the considered evolutional systems via resolvent operators. Second, by utilizing the semigroup theory, the fixed point strategy and Kuratowski's measure of noncompactness, the exact controllability of the evolutional systems is investigated without Lipschitz continuity and growth conditions imposed on nonlinear functions. The results are established under the hypothesis that the resolvent operator is differentiable and analytic, respectively, instead of supposing that the semigroup is compact. An example is provided to illustrate the proposed results.2000 Mathematics Subject Classification. Primary: 47D06, 93B05; Secondary: 34K30, 35R11.

This paper investigates the existence of positive solutions for the following high-order nonlinear fractional differential boundary value problem (BVP, for short)where n − 1 < α ≤ n, n ≥ 3, 0 ≤ λ < 2, D α 0 + is the Caputo fractional derivative. By using the monotone method, the theory of fixed point index on cone for differentiable operators and the properties of Green's function, some new uniqueness and existence criteria for the considered fractional BVP are established. As applications, some examples are worked out to demonstrate the main results. c 2016 All rights reserved.Keywords: Fractional differential equations, differentiable operators, fixed point index theorems on cone, integral boundary value conditions. 2010 MSC: 34A08, 34B15, 34B18.

By using the theory of fixed-point index on cone for differentiable operators, spectral radii of some related linear integral operators, and properties of Green’s function, the existence of multiple positive solutions to a nonlinear fractional differential system with integral boundary value conditions and a parameter is established. At last, some examples are also provided to illustrate the validity of our main results.

In the present paper, sufficient conditions ensuring the complete controllability for a class of semilinear fractional nonlocal evolution systems with finite delay in Banach spaces are derived. The new results are obtained under a weaker definition of complete controllability we introduced, and then the Lipschitz continuity and other growth conditions for the nonlinearity and nonlocal item are not required in comparison with the existing literatures. In addition, an appropriate complete space and a corresponding time delay item are introduced to conquer the difficulties caused by time delay. Our main tools are properties of resolvent operators, theory of measure of noncompactness, and Mönch fixed point theorem.