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Nonlinear variational and constrained hemivariational inequalities involving relaxed operators. (English) Zbl 0886.49006

The paper is devoted to study the problem of finding solutions to the so-called generalized nonlinear variational inequality: Find \(x\in K\) such that \[ \langle(I- (S- T))x, v-x\rangle\geq 0,\quad\forall v\in K, \] and to the constrained nonlinear hemivariational inequality: Find \(u\in C\) such that \[ \langle(A- B)u- g,v\rangle\geq 0,\quad\forall v\in T_C(u), \] where \(S, T, A, B: H\to H\) are given operators on a Hilbert space \(H\), \(T_C(u)\) is Clarke’s tangent cone of a star-shaped \(C\subset H\) at \(u\in C\).
The solution of the first problem has been obtained by applying the iterative procedure under the hypotheses that the corresponding operators satisfy some relaxed monotonicity conditions. The constrained problem has been solved by making use of a modified method proposed in [Z. Naniewicz, J. Optimization Theory Appl. 83, No. 4, 97-112 (1994; Zbl 0808.49019)].

MSC:

49J40 Variational inequalities
47J20 Variational and other types of inequalities involving nonlinear operators (general)

Citations:

Zbl 0808.49019
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