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In this paper, we develop the theory of fractional hybrid differential equations with linear perturbations of second type involving Riemann-Liouville differential operators of order 0 < q < 1. An existence theorem for fractional hybrid differential equations is proved under the ϕ-Lipschitz condition. Some fundamental fractional differential inequalities which are utilized to prove the existence of extremal solutions are also established. Necessary tools are considered and the comparison principle which will be useful for further study of qualitative behavior of solutions is proved.

In this paper, we study the existence of positive solutions for the nonlinear fractional boundary value problem with a p-Laplacian operator. By the properties of Green's function, the Guo-Krasnosel'skii fixed-point theorem, the Leggett-Williams fixed-point theorem, and the upper and lower solutions method, some new results on the existence of positive solutions are obtained. As applications, examples are presented to illustrate the main results.

In this paper, we study boundary-value problems for the following nonlinear fractional differential equations involving the Caputo fractional derivative:continuous function and m ∈ R, n -1 < α < n (n ≥ 2), 0 < β < 1 is a real number. By means of the Banach fixed-point theorem and the Schauder fixed-point theorem, some solutions are obtained, respectively. As applications, some examples are presented to illustrate our main results. MSC: 34A08; 34B10

In this article, we investigate the Sturm-Liouville boundary value problems of fractional differential equations with p-LaplacianBy means of the properties of the Green's function, Leggett-Williams fixed-point theorems, and fixed-point index theory, several new sufficient conditions for the existence of at least two or at least three positive solutions are obtained. As an application, an example is given to demonstrate the main result.