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In this paper, we investigate a generalized nonlinear quasi-hemivariational inequality (QHI) involving a multivalued map in a Banach space. Under general assumptions, by using a fixed point theorem...

We consider a history-dependent variational–hemivariational inequality with unilateral constraints in a reflexive Banach space. The unique solvability of the inequality follows from an existence and uniqueness result obtained in Sofonea and Migórski (2016, 2018). In this current paper we introduce and study a generalized penalty method associated to the inequality. To this end we consider a sequence of generalized penalty problems, governed by a parameter and an operator . We prove the unique solvability of the penalty problems as well as the convergence of corresponding solutions sequence to the solution of original problem. These results extend the previous results in Sofonea etal. (2018) and Xiao and Sofonea (2019). Finally, we illustrate them in the study of a history-dependent problem with unilateral boundary conditions which describes the quasistatic evolution of a rod–spring system under the action of given applied force.

The goal of the present paper is to investigate an abstract system, called fractional differential variational inequality, which consists of a mixed variational inequality combined with a fractional evolution equation in the framework of Banach spaces. Using discrete approximation approach, an existence theorem of solutions for the inequality is established under some suitable assumptions.

The aim the paper is to study a large class of variational-hemivariational inequalities involving constraints in a Banach space. First, we establish a general existence theorem for this class. Second, we introduce a sequence of penalized problems without constraints. Under the suitable assumptions, we prove that the Kuratowski upper limit with respect to the weak topology of the sets of solutions to penalized problems, is nonempty and is contained in the set of solutions to original inequality problem. Also, we prove the identity, when operator A satisfies -property. Finally, we illustrate the applicability of the theoretical results and we explore two complicated partial differential systems of elliptic type, which are an elliptic mixed boundary value problem involving a nonlinear nonhomogeneous differential operator with an obstacle effect, and a nonlinear elastic contact problem in mechanics with unilateral constraints, respectively.

The primary objective is to investigate a class of noncoercive variational–hemivariational inequalities on a Banach space. We start with several new existence results for the abstract inequalities in which our approach is based on arguments of recession analysis and the theory of pseudomonotone operators. A nonsmooth elastic contact problem is considered as an illustrative application.

We consider a hemivariational inequality of elliptic type (HVI, for short) in a reflexive Banach space, prove its solvability and the compactness of its set of solutions. To this end we employ a surjectivity theorem for multivalued mappings that we use for the sum of a maximal monotone operator and a bounded pseudomonotone operator. Next, we introduce the concepts of strongly and weakly well-posedness in the generalized sense for the HVI and provide two characterizations for the strongly well-posedness, under different assumptions on the data. These characterizations are formulated in terms of the metric properties of the approximating sets. We also provide sufficient conditions which guarantee the weakly and strongly well-posedness in the generalized sense of the HVI. Finally, we consider two perturbations of the HVI for which we obtain convergence results in the sense of Kuratowski.

A system which couples an abstract hemivariational inequality of hyperbolic type and an evolution equation in a Banach space is studied. The global existence of the system is established by exploiting the Rothe method. An application to a dynamic adhesive viscoelastic contact problem with friction is provided for which results on existence and regularity of weak solutions are proved.

In this paper, we firstly introduce a complicated system obtained by mixing a nonlinear evolutionary partial differential equation and a mixed variational inequality in infinite dimensional Banach spaces in the case where the set of constraints is not necessarily bounded and the problem is driven by nonlocal boundary conditions, which is called partial differential variational inequality ((PDVI), for short). Then, we show that the solution set of the mixed variational inequality involved in problem (PDVI) is nonempty, bounded, closed and convex. Moreover, the upper semicontinuity and measurability properties for set-valued mapping U : [ 0 , T ] × E 2 → C b v ( E 1 ) (see (3.7) , below) are also established. Finally, several existence results for (PDVI) are obtained by using a fixed point theorem for condensing set-valued operators and theory of measure of noncompactness.

The goal of this paper is to study a comprehensive system called differential variational–hemivariational inequality which is composed of a nonlinear evolution equation and a time-dependent variational–hemivariational inequality in Banach spaces. Under the general functional framework, a generalized existence theorem for differential variational–hemivariational inequality is established by employing KKM principle, Minty’s technique, theory of multivalued analysis, the properties of Clarke’s subgradient. Furthermore, we explore a well-posedness result for the system, including the existence, uniqueness, and stability of the solution in mild sense. Finally, using penalty methods to the inequality, we consider a penalized problem-associated differential variational–hemivariational inequality, and examine the convergence result that the solution to the original problem can be approached, as a parameter converges to zero, by the solution of the penalized problem.

This paper is devoted to studying an inverse problem of parameter identification in a nonlinear quasi-hemivariational inequality posed in a Banach space. We employ the Kluge's fixed point theorem for the set-valued selection map, use the Minty approach and some properties of the Clarke subgradient to prove that the quasi-hemivariational inequality associated to the inverse problem has a nonempty, bounded, and weakly compact solution set. We develop a general regularization framework to provide an existence result for the inverse problem. As an illustrative application, we study an identification inverse problem in a complicated mixed elliptic boundary value problem with p-Laplace operator and an implicit obstacle.

The primary objective of this paper is to explore a complicated differential variational-hemivariational inequality involving a history-dependent operator in Banach spaces. A well-posedness result for the inequality, including the existence, uniqueness, and continuous dependence on the initial data of the solution is established by using a fixed point principle for history-dependent operators. Moreover, to illustrate the applicability of the theoretical results, an elastic contact problem with wear and long time dependent effort is explored.

In this paper a class of generalized differential variational inequalities with constraints involving history-dependent operators in Banach spaces is investigated. The unique solvability and regularity results are obtained via surjectivity of multivalued pseudomonotone operators combined with a fixed point principle. From abstract results, a theorem concerning existence, uniqueness and regularity of weak solution to a frictional viscoelastic contact problem with adhesion and history-dependent operator is established. Further, a theoretical analysis of a penalty method for history-dependent differential variational inequality is provided. The unique solvability of a penalized problem is shown, as well as the convergence of its solution to the solution of the original history-dependent differential variational inequality, as a penalty parameter tends to zero. Finally, results on a penalty method are applied to another contact problem, history-dependent frictional viscoelastic contact problem with a generalized normal compliance condition instead of a generalized Signorini contact condition.

This paper is devoted to introduce and study a new class of generalized vector complementarity problems ((GVCP), for short) and generalized vector variational inequalities ((GVVI), for short) in fuzzy environment. Under suitable conditions, we prove the equivalence between (GVCP) and (GVVI) in Banach spaces. Then, without any monotonicity assumption, we apply KKM-technique to establish an existence theorem of solutions for (GVVI). Finally, we show that the solution set of (GVCP) is nonempty and closed.

In this paper, we study a class of generalized differential hemivariational inequalities of parabolic type involving the time fractional order derivative operator in Banach spaces. We use the Rothe method combined with surjectivity of multivalued pseudomonotone operators and properties of the Clarke generalized gradient to establish existence of solution to the abstract inequality. As an illustrative application, a frictional quasistatic contact problem for viscoelastic materials with adhesion is investigated, in which the friction and contact conditions are described by the Clarke generalized gradient of nonconvex and nonsmooth functionals, and the constitutive relation is modeled by the fractional Kelvin–Voigt law.

In this paper we investigate an abstract system which consists of a hemivariational inequality of parabolic type combined with a nonlinear evolution equation in the framework of an evolution triple of spaces which is called a differential hemivariational inequality [(DHVI), for short]. A hybrid iterative system corresponding to (DHVI) is introduced by using a temporally semi-discrete method based on the backward Euler difference scheme, i.e., the Rothe method, and a feedback iterative technique. We apply a surjectivity result for pseudomonotone operators and properties of the Clarke subgradient operator to establish existence and a priori estimates for solutions to an approximate problem. Finally, through a limiting procedure for solutions of the hybrid iterative system, the solvability of (DHVI) is proved without imposing any convexity condition on the nonlinear function $$u\mapsto f(t,x,u)$$ and compactness of $$C_0$$ -semigroup $$e^{A(t)}$$ .

In this paper we investigate the system obtained by mixing a nonlinear evolutionary equation and a mixed variational inequality ((EEVI), for short) on Banach spaces in the case where the set of constraints is not necessarily compact and the problem is driven by a ϕ -pseudomonotone operator which is not necessarily monotone. In this way, we extend the recent results in Liu–Zeng–Motranu, (2016). First, it is shown that the solution set for the mixed variational inequality associated to problem (EEVI) is nonempty, closed, convex and bounded. Upper semicontinuity and measurability properties are also established. Then, relying on these results, we prove the existence of solutions for problem (EEVI) as well as a compactness property for the solution set. Finally, as an application, we study a new class of partial differential complementarity problems.

In this paper, we consider a new differential variational inequality (DVI, for short) which is composed of an evolution equation and a variational inequality in infinite Banach spaces. This kind of problems may be regarded as a special feedback control problem. Based on the Browder's theorem and the optimal control theory, we show the existence of solutions to the mentioned problem.

The primary objective of this research is to investigate an inverse problem of parameter identification in nonlinear mixed quasi-variational inequalities posed in a Banach space setting. By using a fixed point theorem, we explore properties of the solution set of the considered quasi-variational inequality. We develop a general regularization framework to give an existence result for the inverse problem. Finally, we apply the abstract framework to a concrete inverse problem of identifying the material parameter in an implicit obstacle problem given by an operator of $p$-Laplacian type.

We investigate a generalized Lagrange multiplier system in a Banach space, called a mixed variational–hemivariational inequality (MVHVI, for short), which contains a hemivariational inequality and a variational inequality. First, we employ the Minty technique and a monotonicity argument to establish an equivalence theorem, which provides three different equivalent formulations of the inequality problem. Without compactness for one of operators in the problem, a general existence theorem for (MVHVI) is proved by using the Fan–Knaster–Kuratowski–Mazurkiewicz principle combined with methods of nonsmooth analysis. Furthermore, we demonstrate several crucial properties of the solution set to (MVHVI) which include boundedness, convexity, weak closedness, and continuity. Finally, a uniqueness result with respect to the first component of the solution for the inequality problem is proved by using the Ladyzhenskaya–Babuska–Brezzi (LBB) condition. All results are obtained in a general functional framework in reflexive Banach spaces.

The purpose of this paper is to investigate a class of differential variational inequalities involving a constraint set in Banach spaces. A well-posedness result for the inequality is obtained, including the existence, uniqueness, and stability of the solution in mild sense. Further, we introduce a penalized problem without constraints and prove that the solution to the original inequality can be approached, as a parameter converges to zero, by the solution of the penalized problem. Finally, an application to a comprehensive obstacle parabolic-elliptic system delineates the abstract results.