Nonlinear Evolution Inclusions Research Articles

Optimal control, well-posedness and sensitivity analysis for a class of generalized evolutionary systems

In this paper, we are concerned with a generalized evolution dynamical system, called fractional differential variational-hemivariational inequality (FDVHVI, for short), which is composed of a nonlinear fractional evolution inclusion and a time-dependent mixed variational-hemivariational inequality in the framework of Banach spaces. The objective of this paper is four fold. The first one is to investigate the nonemptiness as well as the compactness of the mild solutions set to the FDVHVI. The second aim is to study the optimal control problems described by the FDVHVI. The third goal is to establish the well-posedness results of the FDVHVI, including the existence, uniqueness, and stability. Furthermore, the sensitivity analysis of a perturbed problem associated to the FDVHVI with respect to the initial state and the two parameters is also obtained. Finally, a comprehensive fractional model is given to illustrate the validity of our main results.

Well-posedness of a fully nonlinear evolution inclusion of second order

Well-posedness of a fully nonlinear evolution inclusion of second order

Second order doubly nonlinear evolution inclusions –quasi-variational approach–

Second order doubly nonlinear evolution inclusions –quasi-variational approach–

Abstract nonlinear evolution inclusions of second order with applications in visco-elasto-plasticity

Existence of strong solutions of an abstract Cauchy problem for a class of doubly nonlinear evolution inclusion of second order is established via a semi-implicit time discretization method. The principal parts of the operators acting on u and u′ are multi-valued subdifferential operators and are discretized implicitly. A non-variational and non-monotone perturbation acting nonlinearly on u and u′ is allowed and discretized explicitly in time. The convergence of a variational approximation scheme is established using methods from convex analysis. In addition, it is proven that the solution satisfies an energy-dissipation equality. Applications of the abstract theory to various examples, e.g., a model in visco-elastic-plasticity, are provided.

Dynamic viscoelastic unilateral constrained contact problems with thermal effects

A new model that describes a dynamic frictional contact between a viscoelastic body and an obstacle is investigated in this paper. We consider a nonlinear viscoelastic constitutive law which involves a convex subdifferential inclusion term and thermal effects. The contact condition is modeled with unilateral constraint condition for a version of normal velocity. The boundary conditions that describe the contact, friction and heat flux are govern by the generalized Clarke multivalued subdifferential. We derive a coupled system of two nonlinear first order evolution inclusions problems, which consists of a parabolic variational-hemivariational inequality for the displacement and a hemivariational inequality of parabolic type for the temperature. Then, the unique weak solvability of the contact problem is obtained by virtue of a fixed point theorem and the surjectivity result of multivalued maps. Finally, we deliver a continuous dependence result on a coupled system when the data are subjected to perturbations.

Global solvability for nonlinear nonautonomous evolution inclusions of Volterra-type and its applications

We study a nonlinear nonautonomous evolution inclusion of Volterra-type with time delay involving a multivalued nonlinearity. The family of operators (possibly unbounded) in the principal part of the evolution inclusion generates a noncompact evolution family. Global solvability for local and nonlocal initial value problems corresponding to the evolution inclusion is considered. The investigation in the present paper is based on two topological fixed point theorems. We impose a condition of Hausdorff-measure of noncompactness on the nonlinearity, which allows us to overcome the lack of compactness of the evolution family. Finally, we present an example to illustrate the abstract results.

Nonlinear Volterra delay evolution inclusions subjected to nonlocal initial conditions

This paper deals with a nonlinear Volterra delay evolution inclusion subjected to a nonlocal implicit initial condition. The evolution inclusion involves an $m$-dissipative operator (possibly multivalued and/or nonlinear) and a noncompact interval. We first consider the evolution inclusion subjected to a local initial condition and prove an existence result for bounded $C^0$-solutions. Then, using a fixed point theorem for upper semicontinuous multifunctions with contractible values, we obtain a global solvability result for the original problem. Finally, we present an example to illustrate the abstract result.

Sensitivity Analysis of Optimal Control Problems Governed by Nonlinear Hilfer Fractional Evolution Inclusions

This article studies sensitivity properties of optimal control problems that are governed by nonlinear Hilfer fractional evolution inclusions (NHFEIs) in Hilbert spaces, where the initial state $$\xi $$ is not the classical Cauchy, but is the Riemann–Liouville integral. First, we obtain the nonemptiness and the compactness properties of mild solution sets $$\mathbb {S}(\xi )$$ for NHFEIs, and also establish an extension Filippov’s theorem. Then we obtain the continuity and upper semicontinuity of optimal control problems connected with NHFEIs depending on a initial state $$\xi $$ as well as a parameter $$\lambda $$ . Finally, An illustrating example is given.

A parabolic–elliptic chemotaxis system with nonlinear diffusion approached from a Cahn–Hilliard-type system

This paper deals with a parabolic–elliptic chemotaxis system with nonlinear diffusion. It was proved that there exists a solution of a Cahn–Hilliard system as an approximation of a nonlinear diffusion equation by applying an abstract theory by Colli–Visintin (Commun Partial Differ Equ 15:737–756, 1990) for a doubly nonlinear evolution inclusion with some bounded monotone operator and subdifferential operator of a proper lower semicontinuous convex function (cf. Colli–Fukao in J Math Anal Appl 429, 2015:1190–1213). Moreover, Colli–Fukao (J Differ Equ 260:6930–6959, 2016) established existence of solutions to the nonlinear diffusion equation by passing to the limit in the Cahn–Hilliard equation. However, Cahn–Hilliard approaches to chemotaxis systems with nonlinear diffusions seem to be not studied yet. This paper will try to derive existence of solutions to a parabolic–elliptic chemotaxis system with nonlinear diffusion by passing to the limit in a Cahn–Hilliard-type chemotaxis system. Although in this paper, we employ a time discretization scheme to prove existence for the Cahn–Hilliard-type chemotaxis system in reference to Colli–Kurima (Nonlinear Anal 190:111613, 2020), please note that this reference does not deal with chemotaxis terms.

A class of history-dependent systems of evolution inclusions with applications

This paper is devoted to studying a system of coupled nonlinear first order history-dependent evolution inclusions in the framework of evolution triples of spaces. The multivalued terms are of the Clarke subgradient or of the convex subdifferential form. Using a surjectivity result for multivalued maps and a fixed point argument for a history-dependent operator, we prove that the system has a unique solution. We conclude with two examples of an evolutionary differential variational–hemivariational inequality and of a dynamic frictional contact problem in mechanics, which illustrate the abstract results.

Evolutionary variational\u2013hemivariational inequalities with applications to dynamic viscoelastic contact mechanics

The purpose of this work is to introduce and investigate a complicated variational–hemivariational inequality of parabolic type with history-dependent operators. First, we establish an existence and uniqueness theorem for a first-order nonlinear evolution inclusion problem, which is driven by a convex subdifferential operator for a proper convex function and a generalized Clarke subdifferential operator for a locally Lipschitz superpotential. Then, we employ the fixed point principle for history-dependent operators to deliver the unique solvability of the parabolic variational–hemivariational inequality. Finally, a dynamic viscoelastic contact problem with the nonlinear constitutive law involving a convex subdifferential inclusion is considered as an illustrative application, where normal contact and friction are described, respectively, by two nonconvex and nonsmooth multi-valued terms.

Approximate controllability of fractional delay evolution inclusions with noncompact semigroups

ABSTRACTThis paper is devoted to investigate the control systems governed by nonlinear fractional delay evolution inclusions (FDEIs) with noncompact semigroups. Under suitable assumptions, sufficient conditions for approximate controllability of FDEIs are established. The approximate controllability results are further extended to FDEIs with nonlocal conditions, where the compactness or Lipschitz continuity on the nonlocal functions is not required. Finally, two examples are given to illustrate the effectiveness of our conclusions.

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.

Periodic solutions for implicit evolution inclusions

We consider a nonlinear implicit evolution inclusion driven by a nonlinear, nonmonotone, time-varying set-valued map and defined in the framework of an evolution triple of Hilbert spaces. Using an approximation technique and a surjectivity result for parabolic operators of monotone type, we show the existence of a periodic solution.

Sensitivity analysis for optimal control problems described by nonlinear fractional evolution inclusions

Abstract In this paper, a sensitivity analysis of optimal control problem for a class of systems described by nonlinear fractional evolution inclusions (NFEIs, for short) on Banach spaces is investigated. Firstly, the nonemptiness as well as the compactness of the mild solutions set S(ζ) (ζ being the initial condition) for the NFEIs are obtained, and we also present an extension Filippov’s theorem and whose proof differs from previous work only in some technical details. Finally, the optimal control problems described by NFEIs depending on the initial condition ζ and the parameter η are considered and the sensitivity properties of the optimal control problem are also established.

Solvability and optimal controls of impulsive Hilfer fractional delay evolution inclusions with Clarke subdifferential

We study the solvability and optimal controls of an impulsive nonlinear Hilfer fractional delay evolution inclusion in Banach spaces. For the main results, we use fractional calculus, fixed point technique, semigroup theory and multivalued analysis. We introduce the notion of Clarke delay subdifferential. To continue and extend previous works in the field, we prove an existence result of optimal control pair for Lagrange problem L P under appropriate set of sufficient conditions. Finally, as application, we give an impulsive control partial differential inclusion with Hilfer fractional derivative.

Anti-periodic solutions for nonlinear evolution inclusions

We consider an anti-periodic evolution inclusion defined on an evolution triple of spaces, driven by an operator of monotone-type and with a multivalued reaction term F(t, x). We prove existence theorem for the “convex” problem (that is, F is convex-valued) and for the “nonconvex” problem (that is, F is nonconvex-valued) and we also show the existence of extremal trajectories (that is, when F is replaced by mathrm {ext},F). Finally, we prove a “strong relaxation” theorem, showing that the extremal trajectories are dense in the set of solutions of the convex problems.

Periodic solutions for nonlinear nonmonotone evolution inclusions

We study periodic problems for nonlinear evolution inclusions defined in the framework of an evolution triple $(X,H,X^*)$ of spaces. The operator $A(t,x)$ representing the spatial differential operator is not in general monotone. The reaction (source) term $F(t,x)$ is defined on $[0,b]× X$ with values in $2^{X^*}\setminus\{\emptyset\}$. Using elliptic regularization, we approximate the problem, solve the approximation problem and pass to the limit. We also present some applications to periodic parabolic inclusions.

Nonlinear second order evolution inclusions with noncoercive viscosity term

In this paper we deal with a second order nonlinear evolution inclusion, with a nonmonotone, noncoercive viscosity term. Using a parabolic regularization (approximation) of the problem and a priori bounds that permit passing to the limit, we prove that the problem has a solution.

The Anti-periodic Solutions Set for a Class of Nonlinear Evolution Inclusions in RN

The Anti-periodic Solutions Set for a Class of Nonlinear Evolution Inclusions in RN