Differential Hemivariational Inequalities Research Articles

Improved order in Hilfer fractional differential systems: Solvability and optimal control problem for hemivariational inequalities

The main objective of this study is to analyze the optimal control results and solvability concerning Hilfer fractional differential hemivariational inequalities within the range of (1<ϱ<2). Initially, we explore the solvability of a mild solutions for the Hilfer fractional hemivariational inequality. Subsequently, we investigate the optimal control strategies for the given problems by employing cost functionals, cosine operators, mild solutions, the fixed point method for multivalued functions, and a generalized Clarke subdifferential approach. To illustrate our findings, we provide a practical example and a filter diagram.

Fuzzy fractional delay differential inclusions driven by hemivariational inequalities in Banach spaces

This paper investigates a novel class of nonlinear dynamical fuzzy systems referred to as fuzzy fractional delay differential hemivariational inequalities (FFDDHVIs) in Banach spaces. These systems integrate fractional fuzzy differential inclusions with delay and hemivariational inequalities. The existence theorem for the HVIs is established based on the KKM theorem. Moreover, specific propositions are proved, covering the superpositional measurability and the upper semicontinuity for the HVIs. Next, by using the fixed points theorem, we establish the existence and compactness of mild solution sets for the FFDDHVIs under certain mild conditions. Finally, As an illustrative application, we investigate a frictional quasistatic contact problem for viscoelastic materials, in which the friction and contact conditions are described by the Clarke generalized gradient of nonconvex and nonsmooth functionals.

Optimal control results for fractional differential hemivariational inequalities of order r (1, 2)

ABSTRACT In this paper, we investigate the optimal control results for fractional differential hemivariational inequalities of order r ∈ ( 1 , 2 ) . Multivalued maps, fractional calculus, and the fixed point theorem are being applied to evaluate the main findings of this study. To begin, a well-known fixed point theorems of multivalued maps and the properties of generalized Clarke subdifferential are used to establish and prove the existence of mild solutions. Furthermore, we obtained the conclusion that the presented control system has optimal control under some mild conditions. Finally, an example is given to illustrate the theory.

Periodic solutions to history-dependent differential hemivariational inequalities with applications

In this article, we investigate a hybrid model combined by a parabolic differential equation and a parabolic hemivariational inequality (so-called differential hemivariational inequality of parabolic–parabolic type) in general infinite dimensional spaces which includes the history-dependent operator. The solvability of initial value problems as well as the periodic problems of the hemivariational inequality and the differential hemivariational inequality have been proved. In application, we study a contact problem with normal compliance driven by a history-dependent dynamical system.

Sensitivity analysis for optimal control problems governed by Hilfer fractional differential hemivariational inequalities

Sensitivity analysis for optimal control problems governed by Hilfer fractional differential hemivariational inequalities

A class of differential hemivariational inequalities constrained on nonconvex star-shaped sets

The purpose of this paper is to investigate a class of nonconvex-constrained differential hemivariational inequalities consisting of nonlinear evolution equations and evolutionary hemivariational inequalities. The admissible set of constraints is closed and star-shaped with respect to a certain ball in a reflexive Banach space. We construct an auxiliary inclusion problem and obtain the existence results by applying a surjectivity theorem for multivalued pseudomonotone operators and the properties of Clarke subgradient operator. Moreover, the existence of a solution to the original problem is established by hemivariational inequality approach and a penalization method in which a small parameter does not have to tend to zero. Finally, an application of the main results is provided.

Solvability and pullback attractor for a class of differential hemivariational inequalities with its applications

In this paper, we consider an abstract system which consists of a nonlinear differential inclusion and a parabolic hemivariational inequality (DPHVI) in Banach spaces. The objective of this paper is four fold. The first target is to deal with the existence of solutions and the properties which involve the boundedness and continuous dependence results of the solution set to parabolic hemivariational inequality. The second aim is to investigate the existence of mild solutions to DPHVI by means of a fixed point technique. The third one is to study the existence of a pullback attractor for the multivalued processes governed by DPHVI. Finally, the fourth goal is to demonstrate a concrete application of our main results arising from the dynamic thermoviscoelasticity problems.

A new class of fractional differential hemivariational inequalities with application to an incompressible Navier-Stokes system coupled with a fractional diffusion equation

This paper is devoted to the study of a new and complicated dynamical system, called a fractional differential hemivariational inequality, which consists of a quasilinear evolution equation involving the fractional Caputo derivative operator and a coupled generalized parabolic hemivariational inequality. Under certain general assumptions, existence and regularity of a mild solution to the dynamical system are established by employing a surjectivity result for weakly-weakly upper semicontinuous multivalued mappings, and a feedback iterative technique together with a temporally semi-discrete approach through the backward Euler difference scheme with quasi-uniform time-steps. To illustrate the applicability of the abstract results, we consider a nonstationary and incompressible Navier-Stokes system supplemented by a fractional reaction-diffusion equation, which is studied as a fractional hemivariational inequality. Bibliography: 57 titles.

A new class of fractional differential hemivariational inequalities with application to an incompressible Navier-Stokes system coupled with a fractional diffusion equation

This paper is devoted to the study of a new and complicated dynamical system, called a fractional differential hemivariational inequality, which consists of a quasilinear evolution equation involving the fractional Caputo derivative operator and a coupled generalized parabolic hemivariational inequality. Under certain general assumptions, existence and regularity of a mild solution to the dynamical system are established by employing a surjectivity result for weakly-weakly upper semicontinuous multivalued mappings, and a feedback iterative technique together with a temporally semi-discrete approach through the backward Euler difference scheme with quasi-uniform time-steps. To illustrate the applicability of the abstract results, we consider a nonstationary and incompressible Navier-Stokes system supplemented by a fractional reaction-diffusion equation, which is studied as a fractional hemivariational inequality.

Analysis and simulation of an adhesive contact problem governed by fractional differential hemivariational inequalities with history-dependent operator

The main idea of this manuscript is to study an adhesive contact problem with long memory which is governed by a hemivariational inequality and a fractional differential equation. We first prove the existence of a unique solution of the fractional differential hemivariational inequality system. Subsequently, we consider a fully discrete scheme of this system and then focus on deriving error estimates for numerical solutions. To the tail of this manuscript, we present two numerical simulation examples for the adhesive contact problem, which provide numerical evidence to support our theoretical predictions.

Coupled system of fractional hemivariational inequalities with applications

ABSTRACT The aim of the paper is to study a coupled system which consists of a fractional differential hemivariational inequality (FDHVI, for short) combined with a fractional hemivariational inequality (FHVI, for short). By utilizing Rothe method and surjectivity result, the existence of coupled system is established. In addition, we apply the results to a new quasistatic contact problem, in which the constitutive equations are described by the fractional Kelvin-Voigt laws.

Decay mild solutions of fractional differential hemivariational inequalities

Decay mild solutions of fractional differential hemivariational inequalities

A new class of fractional impulsive differential hemivariational inequalities with an application

We consider a new fractional impulsive differential hemivariational inequality, which captures the required characteristics of both the hemivariational inequality and the fractional impulsive differential equation within the same framework. By utilizing a surjectivity theorem and a fixed point theorem we establish an existence and uniqueness theorem for such a problem. Moreover, we investigate the perturbation problem of the fractional impulsive differential hemivariational inequality to prove a convergence result, which describes the stability of the solution in relation to perturbation data. Finally, our main results are applied to obtain some new results for a frictional contact problem with the surface traction driven by the fractional impulsive differential equation.

Impulsive hemivariational inequality for a class of history-dependent quasistatic frictional contact problems

&lt;p style='text-indent:20px;'&gt;This paper deals with a class of history-dependent frictional contact problem with the surface traction affected by the impulsive differential equation. The weak formulation of the contact problem is a history-dependent hemivariational inequality with the impulsive differential equation. By virtue of the surjectivity of multivalued pseudomonotone operator theorem and the Rothe method, existence and uniqueness results on the abstract impulsive differential hemivariational inequalities is established. In addition, we consider the stability of the solution to impulsive differential hemivariational inequalities in relation to perturbation data. Finally, the existence and uniqueness of weak solution to the contact problem is proved by means of abstract results.&lt;/p&gt;

A General Class of Differential Hemivariational Inequalities Systems in Reflexive Banach Spaces

We consider an abstract system consisting of the parabolic-type system of hemivariational inequalities (SHVI) along with the nonlinear system of evolution equations in the frame of the evolution triple of product spaces, which is called a system of differential hemivariational inequalities (SDHVI). A hybrid iterative system is proposed via the temporality semidiscrete technique on the basis of the Rothe rule and feedback iteration approach. Using the surjective theorem for pseudomonotonicity mappings and properties of the partial Clarke’s generalized subgradient mappings, we establish the existence and priori estimations for solutions to the approximate problem. Whenever studying the parabolic-type SHVI, the surjective theorem for pseudomonotonicity mappings, instead of the KKM theorems exploited by other authors in recent literature for a SHVI, guarantees the successful continuation of our demonstration. This overcomes the drawback of the KKM-based approach. Finally, via the limitation process for solutions to the hybrid iterative system, we derive the solvability of the SDHVI with no convexity of functions u↦fl(t,x,u),l=1,2 and no compact property of C0-semigroups eAl(t),l=1,2.

History-dependent fractional hemivariational inequality with time-delay system for a class of new frictionless quasistatic contact problems

We study a new frictionless quasistatic contact problem for viscoelastic materials, in which contact conditions are described by the fractional Clarke generalized gradient of nonconvex and nonsmooth functions and a time-delay system. In addition, our constitutive relation is modeled using the fractional Kelvin–Voigt law with long memory. The existence of mild solutions for new history-dependent fractional differential hemivariational inequalities with a time-delay system are obtained by the Rothe method, properties of the Clarke generalized gradient, and a fixed-point theorem.

Rothe method and numerical analysis for a new class of fractional differential hemivariational inequality with an application

The goal of this paper is to introduce and study a new class of fractional differential hemivariational inequality formulated by an evolutionary hemivariational inequality and a fractional differential equation in Banach spaces. By employing the Rothe method and the surjectivity result, we derive the existence of unique solution for such a problem under some mild conditions. Moreover, we use the fully discrete scheme to approximate the fractional differential hemivariational inequality and provide an error estimate for the approximation. Finally, the main results are applied to obtain the unique solvability as well as the numerical analysis for a viscoelastic frictional contact problem with adhesion.

Penalty Method for a Class of Differential Hemivariational Inequalities with Application

In this paper, we consider a penalty method for a class of differential hemivariational inequalities in reflexive Banach spaces, governed by a set of constraints. First, we recall the existence and uniqueness result of the differential hemivariational inequality. Further, we introduce a penalized problem without constraints and we prove that, as a penalty parameter tends to zero, the solution of the original inequality can be approached to the solution of the penalized problem. Finally, we illustrate our results by an example of a contact problem for which our abstract results can be applied.

Solvability and optimal control of fractional differential hemivariational inequalities

ABSTRACT This paper investigates the solvability and the optimal control of fractional differential hemivariational inequalities (FDHVIs, for short). Firstly, we establish several sufficient assumptions for FDHVIs to transform the solution mappings into differential inclusions. Then, we verify the non-emptiness and the compactness of the set of all the mild solutions for FDHVIs. Furthermore, we discuss the optimal control problems of FDHVIs via some appropriate hypotheses. Finally, a typical example is presented to illustrate our main results.

Existence of a global attractor for fractional differential hemivariational inequalities

In this paper, we introduce and study a new class of fractional differential hemivariational inequalities ((FDHVIs), for short) formulated by an initial-value fractional evolution inclusion and a hemivariational inequality in infinite Banach spaces. First, by applying measure of noncompactness, a fixed point theorem of a condensing multivalued map, we obtain the nonemptiness and compactness of the mild solution set for (FDHVIs). Further, we apply the obtained results to establish an existence theorem of the mild solution of a global attractor for the semiflow governed by a fractional differential hemivariational inequality ((FDHVI), for short). Finally, we provide an example to demonstrate the main results.