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

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)
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