Fractional partial differential variational inequality

In this present paper, we introduce and study a dynamical systems involving fractional derivative operator and nonlocal condition, which is constituted of a fractional evolution equation and a time-dependent variational inequality, and is named as fractional partial differential variational inequality (FPDVI, for short). By employing the estimates involving the one-and two-parameter Mittag-Leffler functions, fixed-point theory for set-value mappings, and non-compactness measure theory, we develop a general framework to establish the existence of smooth solutions to (FPDVI).Comment: 12 page

Differential variational-hemivariational inequalities: existence, uniqueness, stability, and convergence

The goal of this paper is to study a comprehensive systemcalled differential variational–hemivariational inequality which is com-posed of a nonlinear evolution equation and a time-dependentvariational–hemivariational inequality in Banach spaces. Under the gen-eral functional framework, a generalized existence theorem for differ-ential variational–hemivariational inequality is established by employ-ing 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, andstability of the solution in mild sense. Finally, using penalty methods tothe inequality, we consider a penalized problem-associated differentialvariational–hemivariational inequality, and examine the convergence re-sult that the solution to the original problem can be approached, as aparameter converges to zero, by the solution of the penalized problem

Convergence of a generalized penalty and regularization method for quasi-variational-hemivariational inequalities

In the paper an elliptic quasi–variational–hemivariational inequality with constraints in a Banach space is studied. First, we apply the Minty technique, the KKM principle and the theory of nonsmooth analysis to establish the solvability of the inequality problem. Then, we employ a generalized penalty and regularization method for the inequality and introduce a family of penalized and regularized problems with no constraints and with Gâteaux differentiable potentials. Through a limit procedure, we prove that the Kuratowski upper limit with respect to the weak topology of the solution sets to penalized and regularized problems, is a nonempty subset of the solution set to the original inequality problem. Next, if a set-valued operator in the inequality has (S)+-property, then the Kuratowski upper limits with respect to the weak and strong topologies for the solution sets coincide. Finally, we illustrate our results by examining a nonlinear elliptic inclusion with the subgradient term of a locally Lipschitz function, mixed boundary conditions and an obstacle unilateral constraint which appears in a semipermeability problem

Existence of projected solutions for quasi-variational hemivariational inequality

In this short article, we prove the existence of projected solutions to a class of quasi-variational hemivariational inequalities with non-self-constrained mapping, which generalizes the results of Allevi et al. (Quasi-variational problems with non-self map on Banach spaces: Existence and applications, Nonlinear Anal. Real World Appl. 67 (2022), 103641, DOI: https://doi.org/10.1016/j.nonrwa.2022.103641.

On the well-posedness of differential quasi-variational-hemivariational inequalities

The goal of this paper is to discuss the well-posedness and the generalized well-posedness of a new kind of differential quasi-variational-hemivariational inequality (DQHVI) in Hilbert spaces. Employing these concepts, we explore the essential relation between metric characterizations and the well-posedness of DQHVI. Moreover, the compactness of the set of solutions for DQHVI is delivered, when problem DQHVI is well-posed in the generalized sense

Multiplicity results of solutions to the double phase anisotropic variational problems involving variable exponent

Aim of this paper is to discuss the existence of multiple so-lutions to double phase anisotropic variational problems for the case of a combined effect of concave-convex nonlinearities. Especially the super -linear (convex) term to the given problem substantially fulfills a weaker condition as well as Ambrosetti-Rabinowitz condition. To achieve these results, we apply the variational methods such as the famous mountain pass theorem and Ekeland's type variational principle when an energy functional corresponding to our problem satisfies the compactness con-dition of the Palais-Smale type. In particular, we establish several ex-istence results of a sequence of infinitely many solutions by employing the Cerami compactness condition. The key tools for obtaining these results are the fountain theorem and the dual fountain theorem