Solutions For Two Classes BVPs Of Higher Order Fractional Differential Equations

In this paper, by using the Schauder fixed point theorem, Krasnosel’skiT fixed point theorem, upper-lower solutions method and topological degree theory, we discuss the existence of solutions for two classes boundary value problems of higher order fractional differential equations andFirstly, we present the progress of boundary value problem of fractional differen-tial equation, then introduce the main works of this paper and prove some preliminary knowledge.Secondly, we study the existence of solutions to the problem (P). Firstly, we obtain positive solutions of (P) when A=0, by analyzing we derive the corresponding Green’s function and give its some properties, and then by using Krasnosel’skii fixed point the-orem, existences criteria of one and two positive solutions are established. Secondly, we discuss the dependence of solutions on the parameter A of the problem (P), by em-ploying the Schauder fixed point theorem, the problem has at least one solution for sufficiently small|λ|. Secondly, under the assumption f∈C([0,1]×[0,∞),[0,∞)) and by using Krasnosel’skii fixed point theorem, we obtain the problem has at least one positive solution for sufficiently small λ when the nonlinear term f is superlinear, no positive solution for sufficiently larger λ; the problem has at least one positive solution for any λ∈[0,∞) when the nonlinear term f is sublinear. The second part, under suit-able conditions, by using upper-lower solutions method and topological degree theory, there exists a positive number λ*such that the above problem has at least two positive solutions for O<λ<λ*, at least one positive solution for λ=λ*and no solution for λ> λ*.Finally, we establish the existence of positive solution to the problem (P*), where λ>0, the nonlinearity may change sign, that is f(t, u)∈C((0,1) x [0,∞),(-∞,+∞)), f(t,u)≥-e(t), and e∈C(0,1)∩L(0,1). By means of the Krasnosel’skii fixed point theorem, some existence results of positive solutions are obtained.