Semilinear Differential Inclusions Research Articles

Relaxation and weak solutions of nonlocal semilinear evolution systems

We give a relatively short proof of the fact that the solution set of a nonlocal semilinear differential inclusion is dense in the weak solution set of the corresponding convexified system. Moreover, we prove a similar result for the solutions with continuous pseudoderivatives when the right-hand side is continuous with nonempty convex closed and bounded values.

Existence results for second-order stochastic differential inclusions driven by Lévy noise

In this paper we prove the existence of mild solutions for a second-order impulsive semilinear stochastic differential inclusion with a standard cylindrical Wiener process and Poisson jumps. We consider the case in which the right hand side can be either convex-valued.

Topological method for coupled systems of impulsive stochastic semilinear differential inclusions with fractional Brownian motion

Topological method for coupled systems of impulsive stochastic semilinear differential inclusions with fractional Brownian motion

Solvability of inclusions of Hammerstein type

A fairly general continuation theorem of Leray-Schauder type for the class of so-called admissible multimaps is set forth. This result is then used to establish a universal rule for solving operator inclusions of Hammerstein type in Lebesgue-Bochner spaces. Examples illustrating the legitimacy of this approach include the initial value problem for perturbation of $m$-accretive mutivalued differential equations, the anti-periodic problem for semilinear differential inclusions, abstract integral inclusions of Fredholm and Volterra type and the two-point boundary value problem for nonlinear evolution inclusions.

О разрешимости каузальных функциональных включений с бесконечным запаздыванием

In the present article we develop the results of works devoted to the study of problems for functional differential equations and inclusions with causal operators, in case of infinite delay. In the introduction of the article we substantiates the relevance of the research topic and provides links to relevant works A. N. Tikhonov, C. Corduneanu, A. I. Bulgakov, E. S. Zhukovskii, V. V. Obukhovskii and P. Zecca. In section two we present the necessary information from the theory of condensing multivalued maps and measures of noncompactness, also introduced the concept of a multivalued causal operator with infinite delay and illustrated it by examples. In the next section we formulate the Cauchy problem for functional inclusion, containing the composition of multivalued and single-valued causal operators; we study the properties of the multiopera-tor whose fixed points are solutions of the problem. In particular, sufficient conditions under which this multioperator is condensing on the respective measures of noncompacness. On this basis, in section four we prove local and global results and continuous dependence of the solution set on initial data. Next the case of inclusions with lower semicontinuous causal multioperators is considered. In the last section we generalize some results for semilinear differential inclusions and Volterra integrodifferential inclusions with infinite delay.

Approximate controllability for abstract semilinear impulsive functional differential inclusions based on Hausdorff product measures

A second order semilinear impulsive functional differential inclusion in a separable Hilbert space is considered. Without imposing hypotheses of the compactness on the cosine families of operators, some sufficient conditions of approximate controllability are formulated in the case where the multivalued nonlinearity of the inclusion is a completely continuous map dominated by a function. By the use of resolvents of controllability Gramian operators and developing appropriate computing techniques for the Hausdorff product measures of noncompactness, the results of approximate controllability for position and velocity are derived. An example is also given to illustrate the application of the obtained results.

Existence results for impulsive semilinear differential inclusions with nonlinear boundary conditions

In this paper, we discuss the nonlinear boundary problem for first-order impulsive semilinear differential inclusions. We establish existence results by using Martelli’s fixed point theorem with upper and lower solutions method. We find that by giving different definitions of lower and upper solutions we can get all existence results. We also present an example.

On Some Boundary Value Problems for Fractional Feedback Control Systems

We consider a nonlocal boundary value problem for a feedback control system governed by a fractional degenerate (Sobolev type) semilinear differential inclusion in a Banach space. To solve this problem, we introduce a multivalued integral operator whose fixed points determine its solutions and study the properties of this operator. It is demonstrated, in particular, that the operator is condensing with respect to an appropriate measure of noncompactness in a functional space. This makes it possible to formulate a general existence principle (Theorem 33) in terms of the topological degree theory. Theorem 34 gives an example of a concrete realization of this principle. As its corollary we get the existence of an optimal solution to our problem (Theorem 35). Some important particular cases including a nonlocal Cauchy problem, periodic and anti-periodic boundary value problems are presented. As example, we consider the existence of an optimal periodic solution for a fractional diffusion type degenerate control system.

Controllability for noninstantaneous impulsive semilinear functional differential inclusions without compactness

The first part of this paper considers the controllability for a functional semilinear differential inclusion governed by a family of operators {A(t):t∈[0,b]} generating an evolution operator in a Banach space in the presence of noninstantaneous impulse effects. In the second part of this paper we study the controllability for a fractional noninstantaneous impulsive semilinear differential inclusion with delay, where the linear part is an infinitesimal generator of a C0−semigroup. Using a weakly convergent criterion in the space of piecewise continuous functions and weak topology theory (for weak sequentially closed graph operators) we establish sufficient conditions to guarantee controllability results. Examples are given to illustrate the abstract results.

Lyapunov Pairs in Semilinear Differential Problems with State-Dependent Impulses

The paper deals with semilinear differential inclusions with state-dependent impulses in Banach spaces. Defining a suitable Banach space in which all the solutions can be embedded we prove the first existence result for at least one global mild solution of the problem considered. Then we characterize this result by means of a new definition of Lyapunov pairs. To this aim we give sufficient conditions for the existence of Lyapunov pairs in terms of a new concept of contingent derivative.

Nonlocal Integro-Differential Equations Without the Assumption of Equicontinuity on the Resolvent Operator in Banach Spaces

In this work, we study the existence of mild solutions for the nonlocal integro-differential equation $$\begin{aligned} \left\{ \begin{array}{l} x'(t)=Ax(t)+\displaystyle \int _{0}^{t}B(t-s)x(s)\text {d}s+f(t,x_{t})\quad \text {for}\;\; t\in [0,b]\\ x_{0}=\phi +g(x)\in C([-r,0];X), \end{array} \right. \end{aligned}$$ without the assumption of equicontinuity on the resolvent operator and without the assumption of separability on the Banach space X. The nonlocal initial condition is assumed to be compact. Our main result is new and its proof is based on a measure of noncompactness developed in Kamenskii et al. (Condensing multivalued maps and semilinear differential inclusions in Banach spaces. Walter De Gruyter, Berlin, 2001) together with the well-known Monch fixed point Theorem. To illustrate our result, we provide an example in which the resolvent operator is not equicontinuous.

Almost periodic solutions at infinity of differential equations and inclusions

We study the notion of a function almost periodic at infinity and present some spectral conditions of the almost periodicity at infinity for bounded solutions of a differential inclusions governed by closed multivalued linear operators in Banach spaces. Applications to degenerate differential equations and semilinear differential inclusions are given. As example, we consider the problem of stabilization of solutions to a heat equation.

Existence of mild solution for nonlocal impulsive fractional semilinear differential inclusion in Banach space

In this paper, we deal with the existence of PC-mild solutions of nonlocal impulsive differential inclusions in Banach space when the values of the orient field is convex (P). By using methods and results of semilinear differential inclusions, and techniques of fixed point theorems, we establish sufficient conditions that guarantee the existence of PC-mild solutions of (P). Our results develop and extend various results proved recently.

On Almost Periodic Solutions of Abstract Semilinear Fractional Inclusions with Weyl-Liouville Derivatives of Order γ ∈ (0, 1

The main aim of this paper is to examine the existence and uniqueness of almost periodic solutions for a class of (semilinear) fractional relaxation inclusions with Stepanov almost periodic coefficients. We deal with the Weyl-Liouville fractional derivatives of order γ ∈ (0, 1], paying special attention to the analysis of semilinear differential inclusions of first order. We use the results from the theory of fractional powers of sectorial multivalued linear operators to achieve our goals, providing an interesting application to semilinear fractional Poisson heat equation in Lp-spaces.

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

In the present paper, we show that the solution set of a fractional order semilinear differential inclusion in a Banach space has the topological structure of an -set. This result allows to apply a fixed point result for condensing multimaps to the translation multioperator along the trajectories of such inclusion and to prove the existence of solutions satisfying periodic and anti-periodic boundary value conditions. An example concerning with a fractional order feedback control system is presented.

Hausdorff product measures and $C^1$-solution sets of abstract semilinear functional differential inclusions

A second order semilinear neutral functional differential inclusion with nonlocal conditions and multivalued impulse characteristics in a separable Banach space is considered. By developing appropriate computing techniques for the Hausdorff product measures of noncompactness, the topological structure of $C^1$-solution sets is established; and some interesting discussion is offered when the multivalued nonlinearity of the inclusion is a weakly upper semicontinuous map satisfying a condition expressed in terms of the Hausdorff measure.

Existence theory for semilinear evolution inclusions involving measures

We provide existence results for semilinear differential inclusions involving measures: urn:x-wiley:0025584X:media:mana201600162:mana201600162-math-0001where A is the infinitesimal generator of a C0‐semigroup of contractions on a separable Banach space X and is a right‐continuous non‐decreasing function. The existence of mild solutions, as well as the compactness of the solution set are obtained via Kakutani–Ky Fan's fixed point theorem in the space of regulated functions endowed with weak, respectively strong topologies. Some examples of special cases covered by our existence results have been included.

Existence of Semi Linear Impulsive Neutral Evolution InclusionsWith Infinite Delay in Frechet Spaces

In this paper, sufficient conditions are given to investigate the existence of mild solutions on a semi-infinite interval for first order semi linear impulsive neutral functional differential evolution inclusions with infinite delay using a recently developed nonlinear alternative for contractive multivalued maps in Frechet spaces due to Frigon combined with semigroup theory. The existence result has been proved without assumption of compactness of the semigroup. We introduced a new phase space for impulsive system with infinite delay and claim that the phase space considered by different authors are not correct.

Viability for quasi-autonomous semilinear evolution inclusions

We provide some invariance properties for a closed graph with respect to quasi-autonomous semilinear differential inclusion. As applications, we study a Lipshitz regularity of the solution set and we give a relaxation result. Furthermore, we get null controllability results for semilinear control systems.

Existence of Semi Linear Impulsive Neutral Evolution Inclusions with Infinite Delay in Frechet Spaces

In this paper, sufficient conditions are given to investigate the existence of mild solutions on a semi-infinite interval for first order semi linear impulsive neutral functional differential evolution inclusions with infinite delay using a recently developed nonlinear alternative for contractive multivalued maps in Frechet spaces due to Frigon combined with semigroup theory. The existence result has been proved without assumption of compactness of the semigroup. We introduced a new phase space for impulsive system with infinite delay and claim that the phase space considered by different authors are not correct.