Approximation of Attainable Sets of an Evolution Inclusion of Subdifferential Type

In a separable Hilbert space we consider an evolution inclusion with a multivalued perturbation and evolution operators that are subdifferentials of a proper convex lower semicontinuous function depending on time. Along with the original inclusion, we consider a sequence of approximating evolution inclusions with the same perturbation and the evolution operators that are subdifferentials of the Moreau–Yosida regularizations of the original function. We show that the attainable set of the original inclusion, regarded as a multivalued function of time, is the uniform (in time) limit in the Hausdorff metric of the sequence of attainable sets of the approximating inclusions. As an application we consider an example of a control system with discontinuous nonlinearity.     

Solvability and continuous dependence results for second order nonlinear evolution inclusions with a Volterra-type operator

The paper deals with second order nonlinear evolution inclusions and their applications. We study evolution inclusions involving a Volterra-type integral operator, which are considered within the framework of an evolution triple of spaces. First, we deliver a result on the unique solvability of the Cauchy problem for the inclusion by combining a surjectivity result for multivalued pseudomonotone operators and the Banach contraction principle. Next, we provide a theorem on the continuous dependence of the solution to the inclusion with respect to the operators involved in the problem. Finally, we consider a dynamic frictional contact problem of viscoelasticity for materials with long memory and indicate how the result on evolution inclusion is applicable to the model of the contact problem.     

Solutions of evolution inclusions generated by a difference of subdifferentials

An evolution inclusion with the right-hand side containing the difference of subdifferentials of proper convex lower semicontinuous functions and a multivalued perturbation whose values are nonconvex closed sets is considered in a separable Hilbert space. In addition to the original inclusion, we consider an inclusion with convexified perturbation and a perturbation whose values are extremal points of the convexified perturbation that also belong to the values of the original perturbation. Questions of the existence of solutions under various perturbations are studied and relations between solutions are established. The primary focus is on the weakening of assumptions on the perturbation as compared to the known assumptions under which existence and relaxation theorems are valid. All our assumptions, in contrast to the known assumptions, concern the convexified rather than original perturbation.     

Extremal solutions of quasilinear parabolic inclusions with generalized Clarke's gradient

In this paper we consider an initial boundary value problem for a parabolic inclusion whose multivalued nonlinearity is characterized by Clarke's generalized gradient of some locally Lipschitz function, and whose elliptic operator may be a general quasilinear operator of Leray-Lions type. Recently, extremality results have been obtained in case that the governing multivalued term is of special structure such as, multifunctions given by the usual subdifferential of convex functions or subgradients of so-called dc-functions. The main goal of this paper is to prove the existence of extremal solutions within a sector of appropriately defined upper and lower solutions for quasilinear parabolic inclusions with general Clarke's gradient. The main tools used in the proof are abstract results on nonlinear evolution equations, regularization, comparison, truncation, and special test function techniques as well as tools from nonsmooth analysis.     

Existence and comparison results for quasilinear parrabolic inclusions woth state-dependent subdifferentials ∗

In this paper we consider initial boundary value problems for quasilinear parabolic differential inclusions governed by nonmonotone operators of Leray-Lions type and state-dependent subdifferentials. Our main goal is to prove the existence of solutions lying in a sector formed by appropriately defined upper and lower solutions. The main tools used in the proof are abstract results on nonlinear evolution equations, regularization, comparison and truncation techniques as well as special test function techniques     

On certain classes of functional inclusions with causal operators in Banach spaces

We introduce the notion of a multivalued causal operator and consider an abstract Cauchy problem in a Banach space for various classes of functional inclusions with causal operators. The methods of the topological degree theory for condensing maps are applied to obtain local and global existence results for this problem and to study the continuous dependence of a solution set on initial data. As application we generalize some existence results for semilinear functional differential inclusions and Volterra integro-differential inclusions with delay.     

Existence results for evolutionary inclusions and variational–hemivariational inequalities

We consider an abstract first-order evolutionary inclusion in a reflexive Banach space. The inclusion contains the sum of L-pseudomonotone operator and a maximal monotone operator. We provide an existence theorem which is a generalization of former results known in the literature. Next, we apply our result to the case of nonlinear variational–hemivariational inequalities considered in the setting of an evolution triple of spaces. We specify the multivalued operators in the problem and obtain existence results for several classes of variational–hemivariational inequality problems. Finally, we illustrate our existence result and treat a class of quasilinear parabolic problems under nonmonotone and multivalued flux boundary conditions.     

On the essential spectra of some matrix of linear relations

In this paper, we investigate a detailed treatment of some subsets of essential spectra of a closed multivalued linear operator. On the following, we will establish some results on perturbation theory of 2 × 2 matrix of multivalued linear operators. Copyright © 2013 John Wiley & Sons, Ltd.     

Estimates for restrictions of monotone operators on the cone of decreasing functions in Orlicz space

We introduce a number of notions related to the Lyapunov transformation of linear differential operators with unbounded operator coefficients generated by a family of evolution operators. We prove statements about similar operators related to the Lyapunov transformation and describe their spectral properties. One of the main results of the paper is a similarity theorem for a perturbed differential operator with constant operator coefficient, an operator which is the generator of a bounded group of operators. For the perturbation, we consider the operator ofmultiplication by a summable operator function. The almost periodicity (at infinity) of the solutions of the corresponding homogeneous differential equation is established.     

Banach空间一类H-增生算子的混合拟变分包含的邻近算子方程

本文在Banach空间中引入一类H-增生算子的混合拟变分包含,并提出求该变分包含问题解的邻近点法.通过H-增生算子的预解算子技术,建立了混合拟变分包含问题与邻近算子方程的等价关系,由这个等价关系得到求解邻近算子方程的迭代算法,该算法收敛于上述混合拟变分包含问题的解.     

On the notion of derivo-periodicity

A De Blasi-like differentiable multivalued function is shown to have a periodic derivative (i.e., to be derivo-periodic) if and only if it is a sum of a function of a continuous (single-valued) periodic function, linear function and a bounded interval (a multivalued constant). At the same time, the single-valued part is derivo-periodic a.e. in the usual sense. In the single-valued case, a characterization of a more general class of derivo-periodic ACG_∗-functions is given. Derivo-periodicity in terms of the Clarke subdifferentials and an impossibility of an almost-periodic analogy are also discussed. The obtained results are finally applied to differential equations and inclusions.     

Properties of the set of “trajectory-control” pairs of a control systemwith subdifferential operators

We consider a control system described by an evolution equation with control constraint which is a multivalued mapping of a phase variable with closed nonconvex values. One of the evolution operators of the system is the subdifferential of a time-dependent proper, convex, and lower semicontinuous function. The other operator, acting on the derivative of the required functions, is the subdifferential of a convex continuous function. We also consider systems with the following control constraints: multivalued mappings whose values are the closed convex hulls of the values of the original constraint and multivalued mapping whose values are the extreme points of the convexified constraint that belong to the original one. We study topological properties of the sets of admissible “trajectory–control” pairs of the system with various control constraints and clarify the relations between them. An example of a parabolic system with hysteresis and diffusion phenomena is considered in detail. Bibliography: 19 titles.     

On solutions of differential inclusions in homogeneous spaces of functions

In this paper we study the solvability of some classes of differential inclusions with multivalued linear operators in homogeneous spaces of functions. These spaces include a large number of functional spaces like periodic functions and Bohr and Stepanov almost periodic functions. As an application, we consider some existence results for feedback control systems governed by degenerate differential equations of Sobolev type in a Banach space.     

带有多值右端项的发展方程之解集的稳定性

In this paper we consider the stability of fixed points of the multivalued Nemitsky operator and, further, obtain stability of solution sets of evolution inclusions. Our work extends the stability results of Lim and Constantin.     

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.     

Penalty schemes with inertial effects for monotone inclusion problems

We introduce a penalty term-based splitting algorithm with inertial effects designed for solving monotone inclusion problems involving the sum of maximally monotone operators and the convex normal cone to the (nonempty) set of zeros of a monotone and Lipschitz continuous operator. We show weak ergodic convergence of the generated sequence of iterates to a solution of the monotone inclusion problem, provided a condition expressed via the Fitzpatrick function of the operator describing the underlying set of the normal cone is verified. Under strong monotonicity assumptions we can even show strong nonergodic convergence of the iterates. This approach constitutes the starting point for investigating from a similar perspective monotone inclusion problems involving linear compositions of parallel-sum operators and, further, for the minimization of a complexly structured convex objective function subject to the set of minima of another convex and differentiable function.     

含m—耗散算子的随机微分包含

证明含m-耗散算子的微分包含随机积分解的存在性，统一和推广Avgerinos和Papaeorgiou，Kravvaritis的结果．讨论半线性微分包含的随机适度解问题．     

On semilinear differential inclusions in Banach spaces with nondensely defined operators

We consider a semilinear differential inclusion in a Banach space assuming that its linear part is a nondensely defined Hille–Yosida operator whereas Carathèodory-type multivalued nonlinearity satisfies a regularity condition expressed in terms of the Hausdorff measure of noncompactness. We apply the theory of integrated semigroups and the fixed point theory of condensing multivalued maps to obtain local and global existence results and to prove the continuous dependence of the solutions set on initial data. An application to an optimization problem for a feedback control system is given.     

Second order Hamilton–Jacobi–Bellman equations with an unbounded operator

This work is devoted to the study of a class of Hamilton–Jacobi–Bellman equations associated to an optimal control problem where the state equation is a stochastic differential inclusion with a maximal monotone operator. We show that the value function minimizing a Bolza-type cost functional is a viscosity solution of the HJB equation. The proof is based on the perturbation of the initial problem by approximating the unbounded operator. Finally, by providing a comparison principle we are able to show that the solution of the equation is unique.     

Banach空间中含$(H,\eta)$单调算子变分包含组解的迭代法

In this paper, we introduce and study a new system of variational inclusions involving （H, η）-monotone operators in Banach space. Using the resolveut operator associated with （H, η）- monotone operators, we prove the existence and uniqueness of solutions for this new system of variational inclusions. We also construct a new algorithm for approximating the solution of this system and discuss the convergence of the iterative sequence generated by the algorithm.