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Zbl 1213.65082
Wang, Ziming; Su, Yongfu; Wang, Dongxing; Dong, Yucai
A modified Halpern-type iteration algorithm for a family of hemi-relatively nonexpansive mappings and systems of equilibrium problems in Banach spaces.
(English)
[J] J. Comput. Appl. Math. 235, No. 8, 2364-2371 (2011). ISSN 0377-0427

The article deals with the equilibrium problem $$f(x,y) \ge 0 \quad \text{for all} \quad y \in C\tag1$$ where $C$ is a nonempty closed convex subset of a real Banach space with the dual $E^*$, $f:\ C \times C \to {\Bbb R}$. The special case of this problem is the known problem of solving a variational inequality; in this problem $f(x,y) = \langle Tx,y - x \rangle$, $T: \ C (\subset E) \to E^*$. It is assumed that $E$ is a strictly convex reflexive Banach space with the Kadec--Klee property and Fréchet differentiable norm, $\{S_\lambda:\ \lambda \in \Lambda\}$ a family of closed hemi-relatively nonexpansive mappings of $C$ into itself with a common fixed point, $\{\alpha_n\}$ a sequence in $[0,1]$ converging to zero. The following iterative scheme is studied: $$\cases x_1 \in C, \\ C_1 = C, \\ y_{n,\lambda} = J^{-1}(\alpha_nJx_1 + (1 - \alpha_n)JS_\lambda x_n), \\ C_{n+1} = \{z \in C_n: \ \sup\limits_{\lambda \in \Lambda} \phi(z,y_{n,\lambda}) \le \alpha_n\phi(z,x_1) + (1 - \alpha_n)\phi(z,x_n)], \\ x_{n+1} = \Pi_{C_{n+1}}x; \endcases$$ here $J$ is the duality mapping, $\phi(x,y) = \|x\|^2 - 2\langle x,Jy \rangle + \|y\|^2$, $\Pi_{C_n}$ is the generalized projection from $C$ onto $C_n$. It is proved that $\{x_n\}$ converges strongly to $\Pi_Fx$, where $\Pi_F$ is the generalized projection from $C$ onto $F$, $F = \bigcap\limits_{\lambda \in \Lambda} F(S_\lambda)$ is the set of common fixed points of $\{S_\lambda\}$. The following modified iterative scheme $$\cases x_1 \in C, \\ C_1 = C, \\ f_\lambda(u_n,y) + \frac1r\langle y - u_n,Ju_n - Jx_n \rangle \ge 0, \ \ \text{for all} \ y \in C, \ r > 0, \\ y_{n,\lambda} = J^{-1}(\alpha_nJx_1 + (1 - \alpha_n)Ju_{n,\lambda}), \\ C_{n+1} = \{z \in C_n:\ \sup\limits_{\lambda \in \Lambda} \phi(z,y_{n,\lambda}) \le \alpha_n\phi(z,x_1) + (1 - \alpha_n)\phi(z,x_n)\}, \\ x_{n+1} = \Pi_{C_{n+1}}x\endcases$$ is considered too; in this case it is assumed that $f(x,y)$ satisfies the following properties: (A1) $f(x,x) = 0$; (2) $f(x,y) + f(y,x) \le 0$; (3) $\lim\limits_{t \to 0} f(tz + (1 - t)x,y) \le f(x,y)$; (4) the functions $y \to f(x,y)$, $x \in C$, are convex and lower semicontinuous.
[Peter Zabreiko (Minsk)]
MSC 2000:
*65J15 Equations with nonlinear operators (numerical methods)
47H05 Monotone operators (with respect to duality)
47H09 Mappings defined by "shrinking" properties
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47J20 Inequalities involving nonlinear operators
47J25 Methods for solving nonlinear operator equations (general)

Keywords: hemi-relatively nonexpansive mappings; hybrid algorithm; equilibrium problem; variational inequalities; Halpern-type iteration algorithm; Banach space

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