Existence and uniqueness of solutions for a general nonlinear variational inequality
type: Item
instance of: scholarly article
title: Existence and uniqueness of solutions for a general nonlinear variational inequality
zbMATH Open document ID: 0824.49009
DOI: 10.1016/0893-9659(94)00106-M
published in: Applied Mathematics Letters
publication date: 1995-03-27T00:00:00Z
review text: Let \(B\) be a Banach space with norm \(\|\cdot\|\), \(B^*\) be the dual space of \(B\) and \(K\) be a nonempty closed convex subset of \(B\). We denote by \(\langle u, v\rangle\) the pairing between \(u\in B^*\) and \(v\in B\). Let \(a(u, v)\) be a bilinear coercive continuous form and the functional \(b(\cdot, \cdot): B\times B\to R\) satisfy the following properties: (i) \(b(\cdot, \cdot)\) is linear in the first argument, (ii) for each \(u\in B\), \(b(u, \cdot)\) is a convex functional, (iii) \(b(u, v)\) is continuous (bounded), (iv) for all \(u,v,w\in B\), \(b(u, v)- b(u, w)\leq b(u, v- w)\). For given \(A: B\to B^*\) and \(g: B\to B\) nonlinear operators, consider the problem of finding \(u\in K\) such that \(g(u)\in K\) and \[ a(u, v- u)+ b(u, v)- b(u, u)\geq \langle A(u), g(v)- g(u)\rangle,\quad\text{for all } gw\in K.\tag{1} \] The inequality (1) is called the general nonlinear variational inequality, which was introduced and studied by the reviewer [Appl. Math. Lett. 5, No. 4, 11-14 (1992; Zbl 0757.65077); Panam. Math. J. 4, No. 1, 27-44 (1994)] in the Hilbert space. In this paper, the authors have proved two existence and uniqueness results for solutions of the general nonlinear variational inequalities in the reflexive Banach spaces by applying minimax inequality and fixed point techniques.
reviewed by: Muhammad Aslam Noor
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 90C33
zbMATH DE Number: 737516
zbMATH Keywords: complementarity problems
zbMATH Keywords: general nonlinear variational inequality
zbMATH Keywords: existence and uniqueness results
zbMATH Keywords: minimax inequality
zbMATH Keywords: fixed point techniques
author: Xieping Ding
author: E. Tarafdar
MaRDI profile type: Publication
cites work: Q3721287
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description: scientific article; zbMATH DE number 737516