On generalized boundary value problems for a class of fractional differential inclusions

Abstract

We prove existence of mild solutions to a class of semilinear fractional differential inclusions with non local conditions in a reflexive Banach space. We are able to avoid any kind of compactness assumptions both on the nonlinear term and on the semigroup generated by the linear part. We apply the obtained theoretical results to two diffusion models described by parabolic partial integro-differential inclusions.

On antiperiodic boundary value problem for semilinear fractional differential inclusion with deviating argument in Banach space

“…Many physical, economical, biological and engineering problems, first of all related with processes in dynamical systems, lead one to boundary value problems for differential equations and inclusions of fractional order, see monographs [8], [16], [19], [20], [23] and paper [17]. During recent years, a large set of problems related with equations and inclusions of fractional order is very intensively studied in Russia and abroad, see paper [1]- [5], [10]- [15], [21], [22]. In the present work we study the solvability of a boundary value problem for a semi-linear differential inclusion of a fractional order in a separable Banach space :…”

Evolution fractional differential problems with impulses and nonlocal conditions

“…In [28], see Remark 3.2, the authors declare that the mild solutions for impulsive Caputo differential equations can be expressed only by using piecewise functions, however with this approach we are able to give a unique formula for the solution. This definition takes into account the fact that the families of operators {S α (t)} t∈[0,b] and {T α (t)} t∈[0,b] defined respectively in (5) and (6) do not satisfy the semigroup properties, that the solutions of an impulsive equation are no longer continuous and that the Caputo derivative strongly depends on the initial time. In our opinion it is particularly suitable to prove existence results in the presence of nonlocal conditions.…”

“…With this approach, we avoid the compactness of the semigroup generated by the linear part and we do not need to assume any hypotheses of monotonicity, Lipschizianity, or compactness neither on the nonlinear term F , nor on the impulse functions, nor on the nonlocal condition. We apply a similar approach in the framework of fractional differential inclusion in [4,5] and in comparison with the literature on the subject, this is the main novelty of the paper. For instance, in [2,8,12,29] the existence, uniqueness and controllability ( [29]) of the solution of a problem similar to (2) via fixed point theorems is proved under Lipschitz regularity assumptions on the nonlinear part, the nonlocal condition and the impulse functions; applying the monotone iterative technique in the presence of upper and lower solutions, in [24] the existence of extremal solutions is obtained under monotonicity and compactness like assumptions on the nonlinear term and on the nonlocal condition and under monotonicity assumptions on the impulse functions; in [1] the compactness of the α-resolvent family generated by the linear part is assumed; in [7] and in [28] the Lipschitz regularity of the nonlinear term, the nonlocal condition and the impulse functions, or alternatively the compactness of the α−resolvent family generated by the linear part, of the nonlinear term, of the nonlocal condition and of the impulse function are taken as main hypotheses.…”

On a Periodic Boundary Value Problem for a Fractional–Order Semilinear Functional Differential Inclusions in a Banach Space

“…To verify condition (13), let us estimate the norm of the operator G(T). By using estimate (29), we have…”

“…Among a large amount of papers dedicated to fractional-order equations and inclusions in Banach spaces, let us mention works [5][6][7][8][9][10][11][12][13][14][15] where existence results of various types were obtained. In particular, in the authors' paper [6], the periodic boundary value problem for fractional-order semilinear differential inclusions in Banach spaces was studied by the method of translation multioperator along the trajectories of the inclusion.…”