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

Abstract

<jats:title>Abstract</jats:title> <jats:p>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.</jats:p>

Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis

“…We sharpened traditional results by showing that a larger class of problems admit a unique solution and achieved this by drawing on the fixed-point theory in an interesting and alternative way via an application of Rus's contraction mapping theorem. Our results add to the recent literature on BVPs [54,[59][60][61] and collectively move towards a more complete understanding of their underlying theory and application.…”

On a Class of Differential Variational Inequalities in Infinite-Dimensional Spaces

“…The paper by Liu et al [15] was devoted to discuss the wellposedness and the generalized well-posedness of a differential mixed quasi-variational inequality and to provide criteria of well-posedness in the generalized sense of the inequality. For solving the problems or phenomena described by nonconvex superpotential functions, which are locally Lipschitz, Cen et al [16] extended the results derived in Liu et al [15]. Migórski and Bai [17] studied a class of evolution subdifferential inclusions involving history-dependent operators.…”

On a class of differential quasi-variational-hemivariational inequalities in infinite-dimensional Banach spaces

“…c(t) ∈ S(K, g(t, s(t)) + Q(•), J, Ψ), a.e. t ∈ [0, L], s(0) = s 0 , where S(K, g(t, s(t)) + Q(•), J, Ψ) stands for the solution set of the following quasivariational-hemivariational inequality: find c : [0, L] → K such that g(t, s(t))+Q(c(t)),c−c(t) +J 0 (c(t);c − c(t))+Ψ(c)−Ψ(c(t)) ≥ 0, ∀c ∈ K. (2) According to Pang and Stewart [20], Pazy [22] and Liu et al [12], the solutions of evolutionary problem (DQVHI) are understood in the following mild sense: Differential variational inequalities represent an important mathematical tool of variational and nonlinear analysis for studying real-life problems coming from natural sciences, operations research, engineering, and physical sciences. In finitedimensional spaces, several contributions related to differential variational inequalities have been established so far (see, for instance, [3]- [9], [15], [17], [21], [25] and references therein).…”

“…In this regard, the paper Liu et al [14] studied the well-posedness and the generalized well-posedness associated with a class of differential mixed quasi-variational inequalities. To investigate some problems or phenomena governed by locally Lipschitz nonconvex superpotential functions, Cen et al [2] extended the results included in Liu et al [14]. Moreover, Migórski and Bai [18] introduced and studied a class of evolution subdifferential inclusions involving history-dependent operators.…”