×

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
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Blum, E.; Oettli, W., From optimization and variational inequalities to equilibrium problems, Math. Student, 63, 123-145 (1994) · Zbl 0888.49007
[2] Flam, S. D.; Antipin, A. S., Equilibrium programming using proximal-link algorithms, Math. Program., 78, 29-41 (1997) · Zbl 0890.90150
[3] Moudafi, A.; Thera, M., Proximal and dynamical approaches to equilibrium problems, (Lecture Note in Economics and Mathematical Systems, vol. 477 (1999), Springer-Verlag: Springer-Verlag New York), 187-201 · Zbl 0944.65080
[4] Tada, A.; Takahashi, W., Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem, J. Optim. Theory Appl., 133, 359-370 (2007) · Zbl 1147.47052
[5] Su, Yongfu; Shang, Meijuan; Qin, Xiaolong, An iterative method of solution for equilibrium and optimization problems, Nonlinear Anal., 69, 2709-2719 (2008) · Zbl 1170.47047
[6] Takahashi, S.; Takahashi, W., Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., 331, 506-515 (2007) · Zbl 1122.47056
[7] Combettes, P. L.; Hirstoaga, S. A., Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6, 117-136 (2005) · Zbl 1109.90079
[8] Ceng, L. C.; Yao, J. C., Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings, Appl. Math. Comput., 198, 729-741 (2008) · Zbl 1151.65058
[9] Ceng, L. C.; Yao, J. C., A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math., 214, 186-201 (2008) · Zbl 1143.65049
[10] Qin, Xiaolong; Cho, Yeol Je; Kang, Shin Min, Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces, J. Comput. Appl. Math., 225, 20-30 (2009) · Zbl 1165.65027
[11] Combettes, P. L.; Hirstoaga, S. A., Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6, 117-136 (2005) · Zbl 1109.90079
[12] Kumam, P.; Wattanawitoon, K., Convergence theorems of a hybrid algorithm for equilibrium problems, Nonlinear Anal. Hybrid Syst. (2009) · Zbl 1184.47056
[13] Takahashi, K.; Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal., 70, 45-57 (2009) · Zbl 1170.47049
[14] Takahashi, W.; Zembayashi, K., Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings, Fixed Point Theory Appl. (2008) · Zbl 1187.47054
[15] Xu, H. K., An iterative approach to quadratic optimization, J. Optim. Theory Appl., 116, 659-678 (2003) · Zbl 1043.90063
[16] Suzuki, T., Strong convergence theorems for infinite families of nonexpansive mapping in general Banach spaces, Fixed Point Theory Appl., 1, 103-123 (2005) · Zbl 1123.47308
[17] Qin, Xiaolong; Cho, Yeol Je; Kang, Shin Min; Zhou, Haiyun, Convergence of a modified Halpern-type iteration algorithm for quasi-\( \phi \)-nonexpansive mappings, Appl. Math. Lett., 22, 1051-1055 (2009) · Zbl 1179.65061
[18] Su, Yongfu; Wang, Ziming; Xu, Hongkun, Strong convergence theorems for a common fixed point of two hemi-relatively nonexpansive mappings, Nonlinear Anal. (2009) · Zbl 1206.47088
[19] Kimura, Yasunori; Takahashi, Wataru, On a hybrid method for a family of relatively nonexpansive mappings in Banach space, J. Math. Anal. Appl., 357, 356-363 (2009) · Zbl 1166.47058
[20] Takahashi, W., Convex Analysis and Approximation Fixed points (2000), Yokohama-Publishers, (in Japanese)
[21] Matsushita, S.; Takahashi, W., Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl., 2004, 37-47 (2004) · Zbl 1088.47054
[22] Matsushita, S. Y.; Takahashi, W., A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory, 134, 257-266 (2005) · Zbl 1071.47063
[23] Kamimura, S.; Takahashi, W., Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim., 13, 938-945 (2002) · Zbl 1101.90083
[24] Su, Yongfu; Wang, Dongxing; Shang, Meijuan, Strong convergence of monotone hybrid algorithm for hemi-relatively nonexpansive mappings, Fixed Point Theory Appl., 2008 (2008), 8 pp. Article ID 284613 · Zbl 1203.47078
[25] Mosco, U., Convergence of convex sets and of solutions of variational inequalities, Adv. Math., 3, 510-585 (1969) · Zbl 0192.49101
[26] Ibaraki, T.; Kimura, Y.; Takahashi, W., Convergence theorems for generalized projections and maximal monotone operators in Banach spaces, Abstr. Appl. Anal., 621-629 (2003) · Zbl 1045.47041
[27] Matsushita, Shin-ya; Takahashi, Wataru, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory, 134, 257-266 (2005) · Zbl 1071.47063
[28] Colao, Vittorio; Acedo, Genaro López; Marino, Giuseppe, An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings, Nonlinear Anal., 71, 2708-2715 (2009) · Zbl 1175.47058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.