Existence and uniqueness of solutions for a general nonlinear variational inequality
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.
general nonlinear variational inequality
scientific article; zbMATH DE number 737516
scientific article