Fractional differential inclusions with fractional separated boundary conditions

Abstract

This paper studies a new class of boundary value problems of nonlinear fractional differential inclusions of order q ∈ (1, 2] with fractional separated boundary conditions. New existence results are obtained for this class of problems by using some standard fixed point theorems. A possible generalization for the inclusion problem with fractional separated integral boundary conditions is also discussed. Some illustrative examples are presented.MSC 2010 : Primary 34A60: Secondary 26A33, 34A08

A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations

“…With this advantage, fractional-order models have become more realistic and practical than the corresponding classical integer-order models. For some recent development on the topic, see [1], [2], [4], [5], [6], [7], [8], [9], [10], [11], [12], [14], [17], and the references therein. The study of coupled systems of fractional order differential equations is also very significant as such systems appear in a variety of problems of applied nature, especially in biosciences.…”

Filippov Lemma for a Class of Hadamard-Type Fractional Differential Inclusions

“…( [1, e]), x(1) = 0, x(e) = I β x(η). (1.2) In the last years we may see a strong development of the study of boundary value problems associated to fractional differential equations and inclusions ( [1,2,3,4] derivatives. Another type of fractional derivative is the one introduced by Hadamard ( [7]) which differs from the others in the sense that the kernel of the integral contains a logarithmic function of arbitrary exponent.…”

“…( [1, e]), x(1) = 0, x(e) = I β x(η), (1.1) where D α is the Hadamard fractional derivative of order α, α ∈ (1,2], I β is the Hadamard integral of order β, β, γ > 0, F : [1, e] × R × R → P(R) is a set-valued map and η ∈ (1, e).…”

On a fractional differential inclusion with integral boundary conditions in Banach space

“…Recently several qualitative results for fractional differential inclusion have been obtained in [1,3,9,10,19]. It should be noted that most of the papers on fractional differential equations or inclusions are devoted to the solvability in the cases wherein the nonlinear terms not depend on derivatives of unknown function.…”

“…Further, there are few works considering such problems in the general context of Banach spaces. In the present paper, with E is a separable Banach space, we consider the following prob- where α ∈ (1,2], β ∈ [0, 2−α] are given constants, Γ is the gamma function, D α is the standard Riemann-Liouville fractional derivative and F : [0, 1] × E ×E → 2 E is a closed valued multifunction. In the case of α = 2, (1.1) is a second order differential inclusion which has been studied by many authors.…”