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Two-step algorithms and their applications to variational problems. (English) Zbl 1050.49006
Summary: The two-step algorithm is introduced and applied to the approximation solvability of a system of nonlinear variational inequalities in a Hilbert space setting. Let $H$ be a real Hilbert space and $K$ be a nonempty closed convex subset of $H$. For arbitrarily chosen initial points $x^0$, $y^0\in K$, compute sequences $\{x^k\}$ and $\{y^k\}$ such that $$x^{k+1}= P_K \biggl[a^kx^k+ (1-a^k)P_K\bigl[y^k- \rho T(y^k)\bigr]\biggr] \quad\text{for }\rho>0$$ $$y^k= P_K\biggl[b^k x^k+ (1-b^k)P_K\bigl[x^k-\eta T(x^k)\bigr]\biggr]\quad\text{for } \eta >0,$$ where $T:K\to H$ is a nonlinear pseudococoercive mapping on $K$, $P_K$ is the projection of $H$ onto $K$, and $0\le a^k,b^k\le 1$ for $k\ge 0$. Based on the two-step algorithm, we explore the approximation solvability of a system of nonlinear variational inequality problems: determine elements $x^*,y^*\in K$ such that $$\bigl\langle\rho T(y^*)+x^*-y^*,x-x^*\bigr\rangle\ge 0\ \forall x\in K$$ $$\bigl\langle \eta T(x^*)+y^*-x^*,x-y^*\bigr\rangle\ge 0\ \forall x\in K,$$ where $\rho$, $\eta>0$.

49J40Variational methods including variational inequalities
47J25Iterative procedures (nonlinear operator equations)