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A modified Halpern-type iteration algorithm for a family of hemi-relatively nonexpansive mappings and systems of equilibrium problems in Banach spaces. (English) Zbl 1213.65082

The article deals with the equilibrium problem

$f\left(x,y\right)\ge 0\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{1.em}{0ex}}y\in C\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $C$ is a nonempty closed convex subset of a real Banach space with the dual ${E}^{*}$, $f:\phantom{\rule{4pt}{0ex}}C×C\to ℝ$. The special case of this problem is the known problem of solving a variational inequality; in this problem $f\left(x,y\right)=〈Tx,y-x〉$, $T:\phantom{\rule{4pt}{0ex}}C\left(\subset E\right)\to {E}^{*}$. It is assumed that $E$ is a strictly convex reflexive Banach space with the Kadec–Klee property and Fréchet differentiable norm, $\left\{{S}_{\lambda }:\phantom{\rule{4pt}{0ex}}\lambda \in {\Lambda }\right\}$ a family of closed hemi-relatively nonexpansive mappings of $C$ into itself with a common fixed point, $\left\{{\alpha }_{n}\right\}$ a sequence in $\left[0,1\right]$ converging to zero. The following iterative scheme is studied:

$\left\{\begin{array}{c}{x}_{1}\in C,\hfill \\ {C}_{1}=C,\hfill \\ {y}_{n,\lambda }={J}^{-1}\left({\alpha }_{n}J{x}_{1}+\left(1-{\alpha }_{n}\right)J{S}_{\lambda }{x}_{n}\right),\hfill \\ {C}_{n+1}=\left\{z\in {C}_{n}:\phantom{\rule{4pt}{0ex}}\underset{\lambda \in {\Lambda }}{sup}\phi \left(z,{y}_{n,\lambda }\right)\le {\alpha }_{n}\phi \left(z,{x}_{1}\right)+\left(1-{\alpha }_{n}\right)\phi \left(z,{x}_{n}\right)\right],\hfill \\ {x}_{n+1}={{\Pi }}_{{C}_{n+1}}x;\hfill \end{array}\right\$

here $J$ is the duality mapping, $\phi \left(x,y\right)={\parallel x\parallel }^{2}-2〈x,Jy〉+{\parallel y\parallel }^{2}$, ${{\Pi }}_{{C}_{n}}$ is the generalized projection from $C$ onto ${C}_{n}$. It is proved that $\left\{{x}_{n}\right\}$ converges strongly to ${{\Pi }}_{F}x$, where ${{\Pi }}_{F}$ is the generalized projection from $C$ onto $F$, $F=\bigcap _{\lambda \in {\Lambda }}F\left({S}_{\lambda }\right)$ is the set of common fixed points of $\left\{{S}_{\lambda }\right\}$. The following modified iterative scheme

$\left\{\begin{array}{c}{x}_{1}\in C,\hfill \\ {C}_{1}=C,\hfill \\ {f}_{\lambda }\left({u}_{n},y\right)+\frac{1}{r}〈y-{u}_{n},J{u}_{n}-J{x}_{n}〉\ge 0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4pt}{0ex}}y\in C,\phantom{\rule{4pt}{0ex}}r>0,\hfill \\ {y}_{n,\lambda }={J}^{-1}\left({\alpha }_{n}J{x}_{1}+\left(1-{\alpha }_{n}\right)J{u}_{n,\lambda }\right),\hfill \\ {C}_{n+1}=\left\{z\in {C}_{n}:\phantom{\rule{4pt}{0ex}}\underset{\lambda \in {\Lambda }}{sup}\phi \left(z,{y}_{n,\lambda }\right)\le {\alpha }_{n}\phi \left(z,{x}_{1}\right)+\left(1-{\alpha }_{n}\right)\phi \left(z,{x}_{n}\right)\right\},\hfill \\ {x}_{n+1}={{\Pi }}_{{C}_{n+1}}x\hfill \end{array}\right\$

is considered too; in this case it is assumed that $f\left(x,y\right)$ satisfies the following properties: (A1) $f\left(x,x\right)=0$; (2) $f\left(x,y\right)+f\left(y,x\right)\le 0$; (3) $\underset{t\to 0}{lim}f\left(tz+\left(1-t\right)x,y\right)\le f\left(x,y\right)$; (4) the functions $y\to f\left(x,y\right)$, $x\in C$, are convex and lower semicontinuous.

MSC:
 65J15 Equations with nonlinear operators (numerical methods) 47H05 Monotone operators (with respect to duality) and generalizations 47H09 Mappings defined by “shrinking” properties 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47J20 Inequalities involving nonlinear operators 47J25 Iterative procedures (nonlinear operator equations)