Nonconvex Values Research Articles

On the solutions of a Sturm-Liouville type system of differential inclusions with nonlocal integral boundary conditions

We consider a Sturm-Liouville type system of differential inclusions with nonlocal integral boundary conditions and we obtain an existence result for this problem in the case when the set-valued maps have nonconvex values.

A study of nonlocal fractional neutral stochastic integrodifferential inclusions of order 1

This paper considers a class of nonlocal fractional neutral stochastic integrodifferential inclusions of order 1 < α < 2 with impulses in a Hilbert space. We study the existence of the mild solution for the cases when the multi-valued map has convex and non-convex values. The results are obtained by combining fixed-point theorems with the fractional order cosine family, semigroup theory, and stochastic techniques. A new set of sufficient conditions is developed to demonstrate the approximate controllability of the system. Finally, an example is given to illustrate the obtained results.

On a Hilfer generalized proportional fractional integro-differential inclusion

Existence of solutions for a class of boundary value problems associated to a Hilfer generalized proportional fractional integro-differential inclusion is studied. A new result is obtained when the right hand side has nonconvex values.

On a Hilfer type fractional integro-differential inclusion

A boundary value problem associated to a Hilfer generalized proportional fractional integro-differential inclusion is studied. The existence of solutions is established in the case when the set-valued map has nonconvex values.

EXISTENCE AND RELAXATION OF SOLUTIONS FOR A DIFFERENTIAL INCLUSION WITH MAXIMAL MONOTONE OPERATORS AND PERTURBATIONS

A differential inclusion with a time-dependent maximal monotone operator and a perturbation is studied in a separable Hilbert space. The perturbation is the sum of a time-dependent single-valued operator and a multivalued mapping with closed nonconvex values. A particular feature of the single-valued operator is that its sum its with the identity operator multiplied by a positive square-integrable function is a monotone operator. The multivalued mapping is Lipschitz continuous with respect to the phase variable. We prove the existence of a solution and the density in the corresponding topology of the solution set of the initial inclusion in the solution set of the inclusion with the convexified multivalued mapping. For these purposes, new distances between maximal monotone operators are introduced.

Affine-Periodic Boundary Value Problem for a Fractional Differential Inclusion

In the article, affine-periodic boundary value problem involving fractional derivative is considered. Existence of solutions to a Caputo-type fractional differential inclusion is researched by some fixed-point theorems and set-valued analysis theory. Specifically, we consider two cases in which the multifunction has convex values and nonconvex values, respectively.

Existence of Solutions of Impulsive Partial Hyperbolic Differential Inclusion of Fractional Order

This paper is concerned with the existence of solutions of a class of Cauchy problems for hyperbolic partial fractional differential inclusions (HPFD) involving the Caputo fractional derivative with an impulse whose right hand side is convex and non-convex valued. Our results are achieved within the framework of the nonlinear alternative of Leray-Schauder type and contraction multivalued maps. A detailed example was provided to support the theorem.

Upper semicontinuous convex-valued selectors of a Nemytskii operator with nonconvex values and evolution inclusions with maximal monotone operators

On the space of continuous functions, we consider a multivalued superposition operator, a Nemytskii operator valued in the space of p-integrable functions with 1≤p<∞. The Nemytskii operator is generated by a multivalued mapping defined on the direct product of a segment of the real line and a separable reflexive Banach space and having closed values in this space. At each point of the phase variable, the mapping either has a closed graph and is convex-valued or it is lower semicontinuous in some neighborhood of this point. In recent years, such mappings are called mappings with mixed semicontinuity properties. We prove that the restriction of the Nemytskii operator onto any compact subset of the space of continuous functions with the sup-norm has a multivalued selector with convex closed values that is upper semicontinuous in the weak topology of the space of p-integrable functions. This result generalizes some well-known results of the author valid in the case when the multivalued mapping generating the Nemytskii operator has compact values. A multivalued selector theorem is used to study an evolution inclusion in a separable Hilbert space with a time-dependent maximal monotone operator and a composite perturbation that is the sum of two multivalued mappings. The values of the first multivalued mapping are closed, bounded, not necessarily convex sets. The second multivalued mapping has closed values and possesses mixed semicontinuity properties. The existence of an absolutely continuous solution is proved. The proof is based on the theorem on an upper semicontinuous convex-valued selector and the author's theorem on selectors continuous in parameter and passing through fixed points of a parameter-dependent multivalued mapping with closed, nonconvex, decomposable values in the space of p-integrable functions as well as on the classical Ky Fan's fixed point theorem. Our approach is fairly simple, concise and clear as compared to other known approaches based on discretization.

Existence study of semilinear fractional differential equations and inclusions for multi–term problem under Riemann–Liouville operators

In the present paper, we study the existence and uniqueness of the solution for multi–term fractional boundary value problem under Riemann–Liouville fractional operators by using Banach's fixed point theorem. Afterward, we investigate some existence results for a semilinear fractional differential inclusions multi–term problem by using some notions and properties on set‐valued maps together with classical fixed point theorems due to Leray–Schauder. The cases when the set‐valued function has convex as well as nonconvex values are considered. Finally, some examples are given to illustrate our main results.

On (k,ψ)-Hilfer Fractional Differential Equations and Inclusions with Mixed (k,ψ)-Derivative and Integral Boundary Conditions

In this paper we study single-valued and multi-valued (k,ψ)-Hilfer-type boundary value problems of fractional order in (1,2], subject to nonlocal boundary conditions involving (k,ψ)-Hilfer-type derivative and integral operators. The results for single-valued case are established by using Banach and Krasnosel’skiĭ fixed point theorems as well as Leray–Schauder nonlinear alternative. In the multi-valued case, we establish an existence result for the convex valued right-hand side of the inclusion via Leray–Schauder nonlinear alternative for multi-valued maps, while the second one when the right-hand side has non-convex values is obtained by applying Covitz–Nadler fixed point theorem for multi-valued contractions. Numerical examples illustrating the obtained theoretical results are also presented.

"On certain boundary value problems associated to some fractional integro-differential inclusions"

"Two classes of fractional integro-differential inclusions with certain boundary conditions are studied. The existence of solutions is established in the case when the set-valued map has nonconvex values."

On a self-adjoint coupled system of second-order differential inclusions

A self-adjoint coupled system of second-order differential inclusions with nonlocal multi-point boundary conditions is considered. An existence result is established when the set-valued maps have nonconvex values.

A Note on a Coupled System of Hilfer Fractional Differential Inclusions

A coupled system of Hilfer fractional differential inclusions with nonlocal integral boundary conditions is considered. An existence result is established when the set-valued maps have non-convex values. We treat the case when the set-valued maps are Lipschitz in the state variables and we avoid the applications of fixed point theorems as usual. An illustration of the results is given by a suitable example.

Multi-Point Boundary Value Problems for (k, ϕ)-Hilfer Fractional Differential Equations and Inclusions

In this paper we initiate the study of boundary value problems for fractional differential equations and inclusions involving (k,ϕ)-Hilfer fractional derivative of order in (1,2]. In the single-valued case the existence and uniqueness results are established by using classical fixed-point theorems, such as Banach, Krasnoselskiĭ and Leray-Schauder. In the multivalued case we consider both cases, when the right-hand side has convex or non-convex values. In the first case, we apply the Leray–Schauder nonlinear alternative for multivalued maps, and in the second, the Covit–Nadler fixed-point theorem for multivalued contractions. All results are well illustrated by numerical examples.

On a second-order differential inclusion with certain integral and multi-strip boundary conditions

We study a second-order differential inclusion with integral and multi-strip boundary conditions defined by a set-valued map with nonconvex values. We obtain an existence result and we prove the arcwise connectedness of the solution set of the considered problem.

Existence and Relaxation Results for Second Order Multivalued Systems

We consider nonlinear systems driven by a general nonhomogeneous differential operator with various types of boundary conditions and with a reaction in which we have the combined effects of a maximal monotone term A(x) and of a multivalued perturbation F(t,x,y) which can be convex or nonconvex valued. We consider the cases where D(A)neq mathbb{R}^{N} and D(A)= mathbb{R}^{N} and prove existence and relaxation theorems. Applications to differential variational inequalities and control systems are discussed.

Reduction method for functional nonconvex differential inclusions

Our aim in this paper is to present a reduction method that solves first order functional differential inclusion in the nonconvex case. This approach is based on a discretization of the time interval, a construction of approximate solutions by reducing the problem to a problem without delay and an application of known results in this case. We generalises earlier results, the right hand side of the inclusion has nonconvex values and satisfies a linear growth condition instead to be integrably bounded. The lack of convexity is replaced by the topological properties of decomposable sets, that represents a good alternative in the absence of convexity.

A note on a coupled system of Caputo-Fabrizio fractional differential inclusions

A coupled system of Caputo-Fabrizio fractional differential inclusions is studied and the existence of solutions is obtained when the set-valued maps that define the problem have nonconvex values, but there are Lipschitz in the state variables.

Bogolyubov’s theorem for a controlled system related to a variational inequality

We consider the problem of minimizing an integral functional on the solutions of a controlled system described by a non-linear differential equation in a separable Banach space and a variational inequality. The variational inequality determines a hysteresis operator whose input is a trajectory of the controlled system and whose output occurs in the right-hand side of the differential equation, in the constraint on the control, and in the functional to be minimized. The constraint on the control is a multivalued map with closed non-convex values and the integrand is a non-convex function of the control. Along with the original problem, we consider the problem of minimizing the integral functional with integrand convexified with respect to the control, on the solutions of the controlled system with convexified constraints on the control (the relaxed problem). By a solution of the controlled system we mean a triple: the output of the hysteresis operator, the trajectory, and the control. We establish a relation between the minimization problem and the relaxed problem. This relation is an analogue of Bogolyubov’s classical theorem in the calculus of variations. We also study the relation between the solutions of the original controlled system and those of the system with convexified constraints on the control. This relation is usually referred to as relaxation. For a finite-dimensional space we prove the existence of an optimal solution in the relaxed optimization problem.

Approximate controllability results for Sobolev‐type delay differential system of fractional order without uniqueness

AbstractThe main aim of this article is to focus on approximate controllability for Sobolev‐type fractional control problems in Hilbert spaces without uniqueness. By using the fixed point theorem for multivalued maps with nonconvex values, the main results of our article are proved. Moreover, we obtain some results on the continuity of the solution map and the topological structure of the solution set of the considered Sobolev‐type fractional differential system. Finally, we provide a theoretical application to assist in the effectiveness of our discussion.