Variational inequalities for \((\eta,\theta)\)-pseudomonotone operators in nonreflexive Banach spaces
The paper concerns a variational problem: Find \(x_0\in K\) such that, for all \(x\in K\), \(\langle Tx_0,\eta(x,x_0)\rangle\geq 0\). Here \(X\) is a nonreflexive Banach space, \(K\subseteq X^{**}\) is a convex set, \(T:K\to X^{*}\) is a hemicontinuous operator, and \(\eta:K\times K\to X^{**}\) is an appropriate substitute for the standard operator \((x,x_0)\to x-x_0\). The main result asserts solvability in case \(T\) is \((\eta,\theta)\)-pseudomonotone for some \(\theta:K\times K\to X^{**}\), which resembles \(\eta\). This generalized monotonicity is a property which, for some \(r>0\), renders the problem above equivalent to the problem below: find \(x_0\in K\) such that, for all \(x\in K\), \(\langle Tx,\eta(x,x_0)\rangle\geq r\|\theta(x,x_0)\|^2\).
scientific article; zbMATH DE number 1369320