Inclusions In Banach Spaces Research Articles

Controllability for systems governed by semilinear evolution inclusions without compactness

In this paper we study the controllability for a class of semilinear differential inclusions in Banach spaces. Since we assume the regularity of the nonlinear part with respect to the weak topology, we do not require the compactness of the evolution operator generated by the linear part. As well we are not posing any conditions on the multivalued nonlinearity expressed in terms of measures of noncompactness. We are considering the usual assumption on the controllability of the associated linear problem. Notice that, in infinite dimensional spaces, the above mentioned compactness of the evolution operator and linear controllability condition are in contradiction with each other. We suppose that the nonlinear term has convex, closed, bounded values and a weakly sequentially closed graph when restricted to its second argument. This regularity setting allows us to solve controllability problem under various growth conditions. As application, a controllability result for hyperbolic integro-differential equations and inclusions is obtained. In particular, we consider controllability of a system arising in a model of nonlocal spatial population dispersal and a system governed by the second order one-dimensional telegraph equation.

Stable Perturbed Iterative Algorithms for Solving New General Systems of Nonlinear Generalized Variational Inclusion in Banach Spaces

We introduce and study a new general system of nonlinear variational inclusions involving generalizedm-accretive mappings in Banach space. By using the resolvent operator technique associated with generalizedm-accretive mappings due to Huang and Fang, we prove the existence theorem of the solution for this variational inclusion system in uniformly smooth Banach space, and discuss convergence and stability of a class of new perturbed iterative algorithms for solving the inclusion system in Banach spaces. Our results presented in this paper may be viewed as an refinement and improvement of the previously known results.

Abstract inclusions in Banach spaces with boundary conditions of periodic type

Abstract inclusions in Banach spaces with boundary conditions of periodic type

An iterative algorithm for random quasi-variational-like inclusions in Banach spaces

An iterative algorithm for random quasi-variational-like inclusions in Banach spaces

Direct and inverse problems for linear relations

Direct and inverse problem related to linear differential inclusions in Banach spaces are studied.The abstract results are applied to degenerate partial dierential equations of parabolic type.

Differential inclusions with measurable-pseudo-Lipschitz right-hand side

We obtain existence theorems and Filippov-Wazewski type relaxation theorems for differential inclusions in Banach spaces with measurable-pseudo-Lipschitz right-hand side. For the solution sets of these differential inclusions, we also describe some properties that extend classical theorems on continuous dependence and on differentiation of solutions with respect to initial data.

Approximate controllability of fractional nonlinear differential inclusions

Fractional differential equations have wide applications in both physical and social sciences. This paper addresses the issue of approximate controllability for a class of fractional nonlinear differential inclusions in Banach spaces. A new set of sufficient conditions are formulated and proved for the approximate controllability of fractional nonlinear differential inclusions. In particular, the results are established with the assumption that the associated linear part of system is approximately controllable. Further, the result is extended to obtain the conditions for the solvability of controllability results for fractional inclusions with nonlocal conditions. Finally, an example is presented to illustrate the theory of the obtained result.

Structure of the solution set to Volterra integral inclusions and applications

The topological and geometric structure of the solution set to Volterra integral inclusions in Banach spaces is investigated. It is shown that the set of solutions in the sense of the Aumann integral is nonempty compact acyclic in the space of continuous functions or is even an Rδ-set provided some appropriate conditions on the Banach space are imposed. Applications to the periodic problem for this type of inclusions are given.

Controllability of Impulsive Neutral Functional Differential Inclusions in Banach Spaces

We investigate the controllability of impulsive neutral functional differential inclusions in Banach spaces. Our main aim is to find an effective method to solve the controllability problem of impulsive neutral functional differential inclusions with multivalued jump sizes in Banach spaces. Based on a fixed point theorem with regard to condensing map, sufficient conditions for the controllability of the impulsive neutral functional differential inclusions in Banach spaces are derived. Moreover, a remark is given to explain less conservative criteria for special cases, and work is improved in the previous literature.

On the Controllability of Impulsive Fractional Evolution Inclusions in Banach Spaces

In this paper, we deal with the controllability of a class of impulsive fractional evolution inclusions in Banach spaces. We establish some sufficient conditions of controllability by use of the well-known fixed point theorem for multivalued maps due to Dhage associated with an evolution system. At the end of the paper, a concrete application is given to illustrate our main results.

Existence and Controllability Results for Nonlocal Fractional Impulsive Differential Inclusions in Banach Spaces

We firstly deal with the existence of mild solutions for nonlocal fractional impulsive semilinear differential inclusions involving Caputo derivative in Banach spaces in the case when the linear part is the infinitesimal generator of a semigroup not necessarily compact. Meanwhile, we prove the compactness property of the set of solutions. Secondly, we establish two cases of sufficient conditions for the controllability of the considered control problems.

Differential inclusions and multivalued integrals

In this paper we consider the nonlocal (nonstandard) Cauchy problem for dierential inclusions in Banach spaces x 0 (t)2 F (t;x(t)); x(0) = g(x); t2 [0;T ] = I: Investigation over some multivalued integrals allow us to prove the existence of solutions for considered problem. We concentrate on the problems for which the assumptions are expressed in terms of the weak topology in a Banach space. We recall and improve earlier papers of this type. The paper is complemented by a short survey about multivalued integration including Pettis and Henstock-Kurzweil-Pettis multivalued integrals.

Strong Convergence of Iterative Algorithm for a New System of GeneralizedH·,·-η-Cocoercive Operator Inclusions in Banach Spaces

We introduce and study a new system of generalizedH·,·-η-cocoercive operator inclusions in Banach spaces. Using the resolvent operator technique associated withH·,·-η-cocoercive operators, we suggest and analyze a new generalized algorithm of nonlinear set-valued variational inclusions and establish strong convergence of iterative sequences produced by the method. We highlight the applicability of our results by examples in function spaces.

Approximate Controllability for Impulsive Riemann-Liouville Fractional Differential Inclusions

We study the control systems governed by impulsive Riemann-Liouville fractional differential inclusions and their approximate controllability in Banach space. Firstly, we introduce thePC1-α-mild solutions for the impulsive Riemann-Liouville fractional differential inclusions in Banach spaces. Secondly, by using the fractional power of operators and a fixed point theorem for multivalued maps, we establish sufficient conditions for the approximate controllability for a class of Riemann-Liouville fractional impulsive differential inclusions, which is a generalization and continuation of the recent results on this issue. At the end, we give an example to illustrate the application of the abstract results.

Existence results for fractional semilinear differential inclusions in Banach spaces

We consider the existence of mild solutions for fractional semilinear differential inclusions involving a nonconvex set-valued map in Banach spaces. First, we study the continuous property of the solution map for an auxiliary fractional differential equation. Then the main result is obtained by using this solution map, selection theorems from multivalued analysis and Schauder’s fixed point theorem. Finally an example to illustrate the applications of the main result is also given.

H(.,.)−η -cocoercive operators and variational-like inclusions in Banach spaces

H(.,.)−η -cocoercive operators and variational-like inclusions in Banach spaces

Existence Results for a Second Order Impulsive Neutral Functional Integrodierential Inclusions in Banach Spaces with Innite Delay

A fixed point theorem for condensing maps due to Martelli combined with theories of a strongly continuous cosine family of bounded linear operators is used to investigate the existence of solutions to second order impulsive neutral functional integrodifferential inclusions with infinite delay in Banach spaces.

Generic Properties of Continuous Differential Inclusions and the Tonelli Method of Approximate Solutions

We study Cauchy problems for differential inclusions in Banach spaces and show that most such problems (in the sense of Baire’s categories) have solutions. We consider separately the cases where the point images of the right-hand side are compact and convex, and where they are merely bounded, closed and convex.

A New Class of Monotone Mappings and a New Class of Variational Inclusions in Banach Spaces

In this paper, we introduce and study a new class of variational inclusions in Banach spaces. As it concerns the methods of solution, we introduce a new class of monotone mappings. We define a proximal mapping associated with this mappings and show its Lipschitz continuity. By using the technique of proximal mapping, we construct a new iterative algorithm. Under some suitable conditions, we prove the convergence of iterative sequences generated by the algorithm. Our results improve and generalize many known results.

On Stepanov almost-periodic oscillations and their discretizations

The relationship between Carathéodory almost-periodic (a.p.) solutions and their discretizations is clarified for differential equations and inclusions in Banach spaces. Our investigation was stimulated by an old result of Meisters [Proc. Am. Math. Soc. 10 (1959), pp. 113–119] about Bohr a.p. solutions which we generalize in several directions. Unlike for functions, Stepanov and Bohr a.p. sequences are shown to coincide. A particular attention is paid to purely (i.e. non-uniformly continuous) Stepanov a.p. solutions. Many ideas are explained in detail by means of examples illustrated.