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Zbl 1078.47034
Cai, Tao; Liu, Zeqing; Kang, Shin Min
Existence of solutions for a system of generalized nonlinear implicit variational inequalities. (English)
[J] Math. Sci. Res. J. 8, No. 6, 176-183 (2004). ISSN 1537-5978

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.
[Mikhail Yu. Kokurin (Yoshkar-Ola)]

MSC 2000:: *47J20 Inequalities involving nonlinear operators
49J40 Variational methods including variational inequalities

Keywords: Hilbert space; iterative algorithm; strong convergence; monotonicity; continuity

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