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Extension of the hybrid steepest descent method to a class of variational inequalities and fixed point problems with nonself-mappings. (English) Zbl 1159.65005
Consider $$H$$ a Hilbert space, $$D\subset H$$ a closed convex set, $$F,T:D\rightarrow H$$ with $$\text{Fix}(T):=\{x\in D\mid Tx=x\}\neq\emptyset.$$ The aim of the paper is to establish an algorithm to solve the problem VIP$$F,\text{Fix}(T)$$: find $$x_{\ast}\in Fix(T)$$ such that $$\left\langle F(x_{\ast}),v-x_{\ast}\right\rangle \geq0$$ for all $$v\in \text{Fix}(T).$$ For this one considers the iteration $$x_{n+1}:=P_{D}T_{w_{n}} P_{D}x_{n}-\alpha_{n}F(P_{D}T_{w_{n}}P_{D}x_{n}),$$ where $$(\alpha_{n} )\subset[0,1],$$ $$(w_{n})\subset(0,\infty),$$ $$T_{w_{n}}:=(1-w_{n} )I+w_{n}T,$$ $$I$$ being the identity mapping on $$D$$ and $$P_{D}$$ being the metric projection from $$H$$ to $$D.$$ Under some additional assumptions on $$F,T,$$ $$(\alpha_{n})$$ and $$(w_{n})$$ it is shown that the sequence $$(x_{n})$$ converges strongly to the (unique) solution VIP$$F,\text{Fix}(T)$$.

##### MSC:
 65C20 Probabilistic models, generic numerical methods in probability and statistics 90C29 Multi-objective and goal programming 90C26 Nonconvex programming, global optimization
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