Boundary value problems for semilinear differential inclusions of fractional order in a Banach space

On the Existence of a Unique Solution for a Class of Fractional Differential Inclusions in a Hilbert Space

“…where λ ∈ R, f : [0, T] → R is a continuous function. By a solution of this problem, we mean a continuous function x : [0, T] → R satisfying condition (11) whose fractional derivative C D q x is also continuous and satisfies Equation (10). It is known (see [1], Example 4.9) that the unique solution of this equation has the form…”

“…(see, for example, [32,33]). Let us recall (see, for example, [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]) that a mild solution to problems (15) and 16is a function x ∈ C([0, T], H) of the form…”

“…Moreover, it is known (see [5,8,[10][11][12][13][14]) that the family (34) has compact convex values and is condensing with respect to the MNC ν in C([0, T]; H) (see Section 2). Since the multioperators λF satisfy conditions (F1)-(F5) independently on λ, by applying Theorem 1, we conclude that there exists a constant C(T) such that all solutions to problems (35) and (36) satisfy the a priori estimate…”

“…The increasing interest in this subject is explained by its numerous applications in various branches of applied mathematics, physics, engineering, biology, economics, and other sciences (see, e.g., the monographs of [1][2][3]). A number of various approaches to the solving of boundary value problems for differential equations and inclusions in the case of fractional order q ∈ (0, 1) have been widely described in the literature (see [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and references therein).…”

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

“…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. However, this method can not be extended directly to the case of functional differential inclusions.…”