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Existence of solutions for a system of generalized nonlinear implicit variational inequalities. (English) Zbl 1078.47034
Let $H$ be a real Hilbert space with inner product $\langle \cdot, \cdot \rangle$. Let $\xi, \eta$ be elements of $H$, $\rho$ and $\beta$ be positive constants, and $K$ a nonempty closed convex subset of $H$. Let $f, g, N: H \to H$ and $M: H \times H \to H$ be nonlinear mappings with $K \subset g(H)$. The paper is concerned with the following problem: find $x, y \in H$ such that $f(x), g(y) \in K$ and $\langle \rho(M(x, g(y))-\xi)+f(x)-g(y), v-f(x) \rangle \geq 0$, $\langle \beta(N(x)-\eta)+g(y)-f(x), v-g(y) \rangle \geq 0$, $\forall v \in K$. The authors construct an iterative algorithm for the problem and establish strong convergence of the algorithm under appropriate monotonicity and continuity conditions on the operators.

47J20Inequalities involving nonlinear operators
49J40Variational methods including variational inequalities