On Noncompact Fractional Order Differential Inclusions with Generalized Boundary Condition and Impulses in a Banach Space

Abstract

We provide existence results for a fractional differential inclusion with nonlocal conditions and impulses in a reflexive Banach space. We apply a technique based on weak topology to avoid any kind of compactness assumption on the nonlinear term. As an example we consider a problem in population dynamic described by an integro-partial-differential inclusion.

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

“…In this paper we prove the existence of mild solutions to problem (1.3), obtaining the corresponding existence of solutions to (1.1) and (1.2) with z ∈ C([0, b], L 2 (Ω, R)). We extend to semilinear differential inclusions a recent result obtained in [7] given for fully nonlinear inclusions. It is worth noting that considering a semilinear inclusion instead of a fully nonlinear one it is not a trivial generalization.…”

“…We will study problem (1. Notice that under conditions (F 1) -(F 3), by Proposition 3.1 in [7] the superposition multioperator…”

Time fractional gradient flows: Theory and numerics

“…From a modeling point of view, their advantages have been observed in [15,12]. Some other types of nonlinear problems have been studied in [8,40,2,24,23,39,45] and [31,38] where, for a particular type of nonlinear problem other "energy dissipation inequalities" than those we obtain are derived. Regularity properties for nonlinear problems with fractional time derivatives have been obtained in [22,14,21,1,44,43,42,41].…”

Evolution fractional differential problems with impulses and 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.…”