2022
DOI: 10.1007/s10957-021-01995-9
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Coupled Variational Inequalities: Existence, Stability and Optimal Control

Abstract: In this paper, we introduce and investigate a new kind of coupled systems, called coupled variational inequalities, which consist of two elliptic mixed variational inequalities on Banach spaces. Under general assumptions, by employing Kakutani-Ky Fan fixed point theorem combined with Minty technique, we prove that the set of solutions for the coupled variational inequality (CVI, for short) under consideration is nonempty and weak compact. Then, two uniqueness theorems are delivered via using the monotonicity a… Show more

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Cited by 6 publications
(5 citation statements)
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“…The main feature of the system is a strong coupling which appears in the nonlinear operators and the generalized directional derivatives. Our results concern the existence and compactness of the solution set to the system, and generalize the results obtained recently in [1] by using a different method.…”
Section: Introductionsupporting
confidence: 83%
See 1 more Smart Citation
“…The main feature of the system is a strong coupling which appears in the nonlinear operators and the generalized directional derivatives. Our results concern the existence and compactness of the solution set to the system, and generalize the results obtained recently in [1] by using a different method.…”
Section: Introductionsupporting
confidence: 83%
“…for all v ∈ C, and B(u, w), z − w E + θ (z) − θ (w) ≥ l, z − w E (1.8) for all z ∈ D. This system was considered and investigated in [1] in which the authors applied the Kakutani-Ky Fan fixed point theorem for multivalued operators to prove the existence of the solutions of system (1.7)- (1.8). In this paper, in contrast to [1], we give a new proof which is based on a multivalued version of the Tychonoff fixed point principle in a Banach space combined with the theory of nonsmoth analysis, generalized monotonicity arguments, and the Minty approach.…”
Section: Introductionmentioning
confidence: 99%
“…This leads to a contradiction. Thus, we conclude that Γ(γ, ζ) is bounded in X × Y, allowing us to find a constant M > 0 satisfying (13).…”
Section: Andmentioning
confidence: 69%
“…Results on the existence and convergence for these problems are investigated. For more details on these topics, the reader is referred to [8, 24, 35, 36, 38, 44, 45] and references therein. Also, there is no paper studying the convergence of solutions in the sense of Painlevnormalé$\rm \acute{e}$–Kuratowski (P.K.)…”
Section: Introductionmentioning
confidence: 99%