On monotone nonlinear variational inequality problems. (English) Zbl 0937.47060

Verma, Ram U.

Commentat. Math. Univ. Carol. 39, No. 1, 91-98 (1998).

Variational inequalities of the type \[ u \in K;\;(Su - Tu - w, v -u) + f(v) - f(u) \geq 0 \text{ for all } v \in K \] are studied where \(K\) is a closed convex subset in a reflexive real Banach space \(X\), \(0 \in K\), \(S,\;T:X \to X^*\) are nonlinear operators, \(f:X \to (-\infty , +\infty ]\) is a convex lower semicontinuous functional, \(f \not \equiv \infty ,\;w \in X^*\) is a given element. It is supposed that \(S,\;T\) are hemicontinuous, \(S\) is \(p\)-monotone (i.e. \((Su-Sv,u-v) \geq r\|u-v\|^p)\), \(T\) is \(p\)-Lipschitz continuous (i.e. \((Su-Sv,u-v) \leq k\|u-v\|^p)\). The equivalence with the problem \[ u \in K;\;(Sv - Tv - w, v - u) + f(v) - f(u) \geq c\|v-u\|^p \text{ for all } v \in K \] with \(c=r-k\) is shown and the existence and unicity of a solution for any \(w \in X^*\) is proved.

Reviewer: M.Kučera (Praha)

MSC:

47J05

Equations involving nonlinear operators (general)

Keywords:

nonlinear variational inequality problems; \(p\)-monotone operators; \(p\)-Lipschitzian operators; KKM mappings