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Implicit relaxed and hybrid methods with regularization for minimization problems and asymptotically strict pseudocontractive mappings in the intermediate sense. (English) Zbl 1273.47107
Summary: We first introduce an implicit relaxed method with regularization for finding a common element of the set of fixed points of an asymptotically strict pseudocontractive mapping $S$ in the intermediate sense and the set of solutions of the minimization problem (MP) for a convex and continuously Fréchet differentiable functional in the setting of Hilbert spaces. The implicit relaxed method with regularization is based on three well-known methods: the extragradient method, the viscosity approximation method, and the gradient projection algorithm with regularization. We derive a weak convergence theorem for two sequences generated by this method. On the other hand, we also prove a new strong convergence theorem by an implicit hybrid method with regularization for the MP and the mapping $S$. The implicit hybrid method with regularization is based on four well-known methods: the CQ method, the extragradient method, the viscosity approximation method, and the gradient projection algorithm with regularization.
MSC:
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
65J20Improperly posed problems; regularization (numerical methods in abstract spaces)
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References:
[1] K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 35, pp. 171-174, 1972. · Zbl 0256.47045 · doi:10.2307/2038462
[2] R. Bruck, T. Kuczumow, and S. Reich, “Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property,” Colloquium Mathematicum, vol. 65, no. 2, pp. 169-179, 1993. · Zbl 0849.47030 · eudml:210212
[3] T.-H. Kim and H.-K. Xu, “Convergence of the modified Mann’s iteration method for asymptotically strict pseudo-contractions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 68, no. 9, pp. 2828-2836, 2008. · Zbl 1220.47100 · doi:10.1016/j.na.2007.02.029
[4] D. R. Sahu, H.-K. Xu, and J.-C. Yao, “Asymptotically strict pseudocontractive mappings in the intermediate sense,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 10, pp. 3502-3511, 2009. · Zbl 1221.47122 · doi:10.1016/j.na.2008.07.007
[5] L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “Some iterative methods for finding fixed points and for solving constrained convex minimization problems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 16, pp. 5286-5302, 2011. · Zbl 05930925 · doi:10.1016/j.na.2011.05.005
[6] G. M. Korpelevi\vc, “An extragradient method for finding saddle points and for other problems,” Èkonomika i Matematicheskie Metody, vol. 12, no. 4, pp. 747-756, 1976. · Zbl 0342.90044
[7] R. P. Agarwal, D. O’Regan, and D. R. Sahu, “Iterative construction of fixed points of nearly asymptotically nonexpansive mappings,” Journal of Nonlinear and Convex Analysis, vol. 8, no. 1, pp. 61-79, 2007. · Zbl 1134.47047
[8] Q. H. Ansari, C. S. Lalitha, and M. Mehta, Generalized Convexity, Nonsmooth Variational Inequalities and Nonsmooth Optimization, Taylor & Francis, Boca Raton, Fla, USA, 2013.
[9] A. Auslender and M. Teboulle, Asymptotic Cones and Functions in Optimization and Variational Inequalities, Springer Monographs in Mathematics, Springer, New York, NY, USA, 2003. · Zbl 1017.49001
[10] F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. 1-2 of Springer Series in Operations Research, Springer, New York, NY, USA, 2003. · Zbl 1062.90002
[11] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer Series in Computational Physics, Springer, New York, NY, USA, 1984. · Zbl 0536.65054
[12] R. Glowinski, J.-L. Lions, and R. Trémolières, Numerical Analysis of Variational Inequalities, vol. 8 of Studies in Mathematics and its Applications, North-Holland Publishing, Amsterdam, The Netherlands, 1981.
[13] I. V. Konnov, Equilibrium Models and Variational Inequalities, vol. 210 of Mathematics in Science and Engineering, Elsevier B. V., Amsterdam, The Netherlands, 2007. · Zbl 1140.91056
[14] J. Reinermann, “Über fixpunkte kontrahierender abbildungen und schwach konvergente Toeplitz-verfahren,” Archiv der Mathematik, vol. 20, pp. 59-64, 1969. · Zbl 0174.19401 · doi:10.1007/BF01898992
[15] M. O. Osilike, S. C. Aniagbosor, and B. G. Akuchu, “Fixed points of asymptotically demicontractive mappings in arbitrary Banach spaces,” Panamerican Mathematical Journal, vol. 12, no. 2, pp. 77-88, 2002. · Zbl 1018.47047
[16] K.-K. Tan and H. K. Xu, “Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process,” Journal of Mathematical Analysis and Applications, vol. 178, no. 2, pp. 301-308, 1993. · Zbl 0895.47048 · doi:10.1006/jmaa.1993.1309
[17] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990. · Zbl 0708.47031 · doi:10.1017/CBO9780511526152
[18] Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 591-597, 1967. · Zbl 0179.19902 · doi:10.1090/S0002-9904-1967-11761-0
[19] R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,” Transactions of the American Mathematical Society, vol. 149, pp. 75-88, 1970. · Zbl 0222.47017 · doi:10.2307/1995660
[20] L.-C. Ceng and J.-C. Yao, “Strong convergence theorems for variational inequalities and fixed point problems of asymptotically strict pseudocontractive mappings in the intermediate sense,” Acta Applicandae Mathematicae, vol. 115, no. 2, pp. 167-191, 2011. · Zbl 1220.47086 · doi:10.1007/s10440-011-9614-x