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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
\[ f(x,y) \geq 0 \quad \text{for all} \quad y \in C\tag{1} \]
where \(C\) is a nonempty closed convex subset of a real Banach space with the dual \(E^*\), \(f:\;C \times C \to {\mathbb R}\). 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:
\[ \begin{cases} x_1 \in C, \\ C_1 = C, \\ y_{n,\lambda} = J^{-1}(\alpha_nJx_1 + (1 - \alpha_n)JS_\lambda x_n), \\ C_{n+1} = \{z \in C_n: \;\sup\limits_{\lambda \in \Lambda} \phi(z,y_{n,\lambda}) \leq \alpha_n\phi(z,x_1) + (1 - \alpha_n)\phi(z,x_n)], \\ x_{n+1} = \Pi_{C_{n+1}}x; \end{cases} \]
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
\[ \begin{cases} x_1 \in C, \\ C_1 = C, \\ f_\lambda(u_n,y) + \frac1r\langle y - u_n,Ju_n - Jx_n \rangle \geq 0, \;\;\text{for all} \;y \in C, \;r > 0, \\ y_{n,\lambda} = J^{-1}(\alpha_nJx_1 + (1 - \alpha_n)Ju_{n,\lambda}), \\ C_{n+1} = \{z \in C_n:\;\sup\limits_{\lambda \in \Lambda} \phi(z,y_{n,\lambda}) \leq \alpha_n\phi(z,x_1) + (1 - \alpha_n)\phi(z,x_n)\}, \\ x_{n+1} = \Pi_{C_{n+1}}x\end{cases} \]
is considered too; in this case it is assumed that \(f(x,y)\) satisfies the following properties: (A1) \(f(x,x) = 0\); (2) \(f(x,y) + f(y,x) \leq 0\); (3) \(\lim\limits_{t \to 0} f(tz + (1 - t)x,y) \leq f(x,y)\); (4) the functions \(y \to f(x,y)\), \(x \in C\), are convex and lower semicontinuous.

MSC:
65J15 Numerical solutions to equations with nonlinear operators
47H05 Monotone operators and generalizations
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
47J20 Variational and other types of inequalities involving nonlinear operators (general)
47J25 Iterative procedures involving nonlinear operators
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