On the general variational inequality (Q2367814)

Given \(f,g: R^ n\to R^ n\) nonlinear mappings and \(K\), a closed convex set in \(R^ n\), the author considers the problem of finding \(u\in K\) such that (1) \(\langle f(u), g(v)- g(u)\rangle\geq 0\) for all \(v\in K\), which is known as a general variational inequality problem, being a special case of the problem as formulated by the reviewer [Appl. Math. Lett. 1, No. 2, 119-122 (1988; Zbl 0655.49005)]. Some existence results are derived using fixed point theory. It should be remarked that to discuss the uniqueness of the solution of (1) in addition to the hypotheses of Theorem 3.3, one has to assume that the mapping \(g^{-1}\) exists.