Semilinear Differential Inclusions Research Articles

Mild Solutions for w-Weighted, Φ-Hilfer, Non-Instantaneous, Impulsive, w-Weighted, Fractional, Semilinear Differential Inclusions of Order μ ∈ (1,2) in Banach Spaces

The aim of this work is to obtain novel and interesting results for mild solutions to a semilinear differential inclusion involving a w-weighted, Φ-Hilfer, fractional derivative of order μ∈(1,2) with non-instantaneous impulses in Banach spaces with infinite dimensions when the linear term is the infinitesimal generator of a strongly continuous cosine family and the nonlinear term is a multi-valued function. First, we determine the formula of the mild solution function for the considered semilinear differential inclusion. Then, we give sufficient conditions to ensure that the mild solution set is not empty or compact. The desired results are achieved by using the properties of both the w-weighted Φ-Laplace transform, w-weighted ψ-convolution and the measure of non-compactness. Since the operator, the w-weighted Φ-Hilfer, includes well-known types of fractional differential operators, our results generalize several recent results in the literature. Moreover, our results are novel because no one has previously studied these types of semilinear differential inclusions. Finally, we give an illustrative example that supports our theoretical results.

Strong Solutions and Mild Solutions for Sturm-Liouville Differential Inclusions

Existence results for a Cauchy problem driven by a semilinear differential Sturm-Liouville inclusion are achived by proving, in a preliminary way, an existence theorem for a suitable integral inclusion. In order to obtain this proposition we use a recent fixed point theorem that allows us to work with the weak topology and the De Blasi measure of weak noncompactness. So we avoid requests of compactness on the multivalued terms. Then, by requiring different properties on the map p involved in the Sturm-Liouville inclusion, we are able to establish the existence of both mild solutions and strong ones for the problem examinated. Moreover we focus our attention on the study of controllability for a Cauchy problem governed by a suitable Sturm-Liouville equation. Finally we precise that our results are able to study problems involving a more general version of a semilinear differential Sturm-Liouville inclusion.

Topological Properties of Solution Sets for τ-Fractional Non-Instantaneous Impulsive Semi-Linear Differential Inclusions with Infinite Delay

The knowledge of fractional calculus can be useful in developing models that allow us to better understand and manage some phenomena. In the present paper, we study the topological structure of the mild solution set for a semi-linear differential inclusion containing the τ-Caputo fractional derivative in the presence of non-instantaneous impulses and an infinite delay. We demonstrate that this set is non-empty and an Rδ-set. We use a recent result regarding the existence of solutions for τ-Caputo fractional semi-linear differential inclusions. Unlike many results, we do not suppose that the generating semigroup is compact. An illustrative example is given as an application of our results.

Об антипериодической краевой задаче для полулинейного дифференциального включения дробного порядка 2<q<3

On antiperiodic boundary value problem for a semilinear differential inclusion of a fractional order q. The investigation of control systems with nonlinear units forms a complicated and very important part of contemporary mathematical control theory and harmonic analysis, which has numerous applications and attracts the attention of a number of researchers around the world. In turn, the development of the theory of differential inclusions is associated with the fact that they provide a convenient and natural tool for describing control systems of various classes, systems with discontinuous characteristics, which are studied in various branches of the optimal control theory, mathematical physics, radio physics, acoustics etc. One of the best approaches to the study of this kind of problems is provided by the methods of multivalued and nonlinear analysis, which are distinguished as very powerful, effective and useful. However, the solving of these problems within the frameworks of existing theories is often a very difficult problem, since many of them find sufficiently adequate description in terms of differential equations and inclusions with fractional derivatives. The theory of differential equations of fractional order originates from the ideas of Leibniz and Euler, but only by the end of the XX century, interest in this topic increased significantly. In the 70s - 80s, this direction was greatly developed by the works of A.A. Kilbas, S.G. Samko, O.I. Marichev, I. Podlubny, K.S. Miller, B. Ross, R. Hilfer, F. Mainardi, H. M. Srivastava. Notice that the research in this direction will open up prospects and new opportunities for the studying of non-standard systems that specialists encounter while describing the development of physical and chemical processes in porous, rarefied and fractal media. It is known that a periodic boundary value problem is one of the classical boundary value problems of differential equations and inclusions. At the same time, in recent years, along with periodic boundary value problems, antiperiodic boundary value problems are of great interest due to their applications in physics and interpolation problems.
 In this paper, we study an antiperiodic boundary value problem for semilinear differential inclusions with Caputo fractional derivative of order q in Banach spaces. We assume that the nonlinear part is a multivalued map obeying the conditions of the Caratheodory type, boundedness on bounded sets, and the regularity condition expressed in terms of measures of noncompactness. In the first section, we present a necessary information from fractional analysis, Mittag -- Leffler function, theory of measures of noncompactness, and multivalued condensing maps. In the second section, we construct the Green's function for the given problem, then, we introduce into consideration a resolving multivalued integral operator in the space of continuous functions. The solutions to the boundary value problem are fixed points of the resolving multioperator. Therefore, we use a generalization of Sadovskii type theorem to prove their existence. Then, we first prove that the resolving multioperator is upper semicontinuous and condensing with respect to the two-component measure of noncompactness in the space of continuous functions. In a proof of a main theorem of the paper, we show that a resolving multioperator transforms a closed ball into itself. Thus, we obtain that the resolving multioperator obeys all the conditions of the fixed point theorem and we prove the existence of solutions to the antiperiodic boundary value problem.

Topological Structure of the Solution Sets for Impulsive Fractional Neutral Differential Inclusions with Delay and Generated by a Non-Compact Demi Group

In this paper, we give an affirmative answer to a question about the sufficient conditions which ensure that the set of mild solutions for a fractional impulsive neutral differential inclusion with state-dependent delay, generated by a non-compact semi-group, are not empty compact and an Rδ-set. This means that the solution set may not be a singleton, but it has the same homology group as a one-point space from the point of view of algebraic topology. In fact, we demonstrate that the solution set is an intersection of a decreasing sequence of non-empty compact and contractible sets. Up to now, proving that the solution set for fractional impulsive neutral semilinear differential inclusions in the presence of impulses and delay and generated by a non-compact semigroup is an Rδ-set has not been considered in the literature. Since fractional differential equations have many applications in various fields such as physics and engineering, the aim of our work is important. Two illustrative examples are given to clarify the wide applicability of our results.

On semilinear fractional differential inclusions with a nonconvex-valued right-hand side in Banach spaces

On semilinear fractional differential inclusions with a nonconvex-valued right-hand side in Banach spaces

Trajectory approximately controllability and optimal control for noninstantaneous impulsive inclusions without compactness

In this paper, a noninstantaneous impulsive differential inclusion model is established based on the heating phenomenon of the rod. The controllability problem for this system governed by a semilinear differential inclusion with noninstantaneous impulses is studied in a Banach space and in this differential inclusion system we assume that the semigroup generated by the linear part of the inclusion is not compact. We suppose that the set-valued nonlinearity satisfies a regularity condition expressed in terms of the Hausdorff measure of noncompactness and some sufficient conditions for approximately controllability for both upper and almost lower semicontinuous types of nonlinearity are presented. Also we discuss existence and the stability of optimal control. As an application, the controllability for a differential inclusion system governed by a heat equation is considered.

Anti-periodic problem for semilinear differential inclusions involving Hille-Yosida operators

In this paper we are interested in the anti-periodic problem governed by a class of semilinear differential inclusions with linear parts generating integrated semigroups. By adopting the Lyapunov-Perron method and the fixed point argument for multivalued maps, we prove the existence of anti-periodic solutions. Furthermore, we study the long-time behavior of mild solutions in connection with anti-periodic solutions. Consequently, as the nonlinearity is of single-valued, we obtain the exponential stability of anti-periodic solutions. An application of theoretical results to a class of partial differential equations will be given.

Stepanov pseudo‐almost periodic functions and applications

In this work, we present basic results and applications of Stepanov pseudo‐almost periodic functions with measure. Using only the continuity assumption, we prove a new composition result of μ‐pseudo‐almost periodic functions in Stepanov sense. Moreover, we present different applications to semilinear differential equations and inclusions with weak regular forcing terms in Banach spaces. We prove the existence and uniqueness of μ‐pseudo‐almost periodic solutions (in the strong sense) to a class of semilinear fractional inclusions and semilinear evolution equations respectively, provided that the nonlinear forcing terms are only Stepanov μ‐pseudo‐almost periodic in the first variable and not a uniformly strict contraction with respect to the second argument. Our results are obtained using the Meir–Keeler principle and the Banach fixed point principle respectively. Some examples of fractional and nonautonomous partial differential equations illustrating our theoretical results are also presented.

Asymptotically periodic behavior of solutions to fractional non-instantaneous impulsive semilinear differential inclusions with sectorial operators

In this paper, we prove two results concerning the existence of S-asymptotically ω-periodic solutions for non-instantaneous impulsive semilinear differential inclusions of order 1<alpha <2 and generated by sectorial operators. In the first result, we apply a fixed point theorem for contraction multivalued functions. In the second result, we use a compactness criterion in the space of bounded piecewise continuous functions defined on the unbounded interval J=[0,infty ). We adopt the fractional derivative in the sense of the Caputo derivative. We provide three examples illustrating how the results can be applied.

Finite-time attractivity of solutions for a class of fractional differential inclusions with finite delay

Our aim in this paper is to give a sufficient condition ensuring the finite-time attractivity for the zero solution to semilinear functional differential inclusions in Banach spaces, in the case where the nonlinearity function possibly has superlinear growth. Our analysis is based on the semigroup theory, the fixed point principle for condensing multi-valued maps, and local estimates of solutions. The abstract results will be applied to a class of polytope inclusions in $$C_0$$ setting.

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

We obtained results on the existence and uniqueness of a mild solution for a fractional-order semi-linear differential inclusion in a Hilbert space whose right-hand side contains an unbounded linear monotone operator and a Carathéodory-type multivalued nonlinearity satisfying some monotonicity condition in the phase variables. We used the Yosida approximations of the linear part of the inclusion, the method of a priori estimates of solutions, and the topological degree method for condensing vector fields. As an example, we considered the existence and uniqueness of a solution to the Cauchy problem for a system governed by a perturbed subdifferential inclusion of a fractional diffusion type.

Existence results on impulsive stochastic semilinear differential inclusions

In this paper, we present some existence results of mild solutions and study the topological structure of solution sets for the following first-order impulsive stochastic semilinear differential inclusions driven by Poisson jumps with periodic boundary conditions.We consider the cases in which the right hand side can be either convex . The results are obtained by using fixed point theorems for multivalued mappings, more precisely, the technique is based on fixed point theorem a nonlinear alternative of Leray-Schauder's fixed point theorem in generalised metric and Banach spaces.

Existence results on impulsive stochastic semilinear differential inclusions

Existence results on impulsive stochastic semilinear differential inclusions

О существовании решения периодической краевой задачи для полулинейных дифференциальных включений дробного порядка в банаховых пространствах

In this paper, we study a periodic boundary value problem for a class of semilinear differential inclusions of fractional order in a Banach space for which the multivalued nonlinearity satisfies the regularity condition expressed in terms of measures of noncompactness. To prove the existence of solutions to the problem, we first construct the corresponding Green function. Then we introduce into consideration a multivalued resolving operator in the space of continuous functions and reduce the posed problem to the existence of fixed points of the resolving multioperator. To prove the existence of a fixed point, a generalized theorem of B.N. Sadovskii type for a condensing multivalued map is used.

Solvability of Fractional Differential Inclusion with a Generalized Caputo Derivative

This paper is devoted to the investigation of a kind of generalized Caputo semilinear fractional differential inclusions with deviated-advanced nonlocal conditions. Solvability of the problem is established by means of the Leray-Schauder’s alternative approach with the help of the Lagrange mean-value classical theorem. Finally, some examples are given to delineate the efficient of theoretical results.

Nonlocal fractional semilinear differential inclusions with noninstantaneous impulses and of order α ∈ (1, 2)

Abstract In this paper, we establish the existence of mild solutions for nonlocal fractional semilinear differential inclusions with noninstantaneous impulses of order α ∈ (1,2) and generated by a cosine family of bounded linear operators. Moreover, we show the compactness of the solution set. We consider both the case when the values of the multivalued function are convex and nonconvex. Examples are given to illustrate the theory.

Existence of mild solutions to semilinear fractional differential inclusion with deviated advanced nonlocal conditions

This paper is devoted to study the existence of at least one continuous mild solution to semilinear fractional differential inclusion with deviated-advanced nonlocal conditions. We develop the results obtained, by exchanging the deviated-advanced nonlocal condition to an integral form. Our results will be accomplished by using the nonlinear Leray-Schauder’s alternative fixed point theorem. The main results are well illustrated with the aid of an example.

Controllability of Hilfer fractional noninstantaneous impulsive semilinear differential inclusions with nonlocal conditions

In this paper, we investigate the controllability of nonlocal Hilfer-type fractional differential inclusions with noninstantaneous impulsive conditions in Banach spaces.

Global attracting solutions to Hilfer fractional differential inclusions of Sobolev type with noninstantaneous impulses and nonlocal conditions

In this paper, we establish the existence of decay mild solutions on an unbounded interval of nonlocal fractional semilinear differential inclusions with noninstantaneous impulses and involving the Hilfer derivative. Our argument uses fixed point theorems, semigroup theory, multi-functions and a measure of noncompactness on the space of piecewise weighted continuous functions defined on an unbounded interval. An example is provided to illustrate our results.