Approximation of fixed points for Garcia-Falset mappings in a uniformly convex Banach space

Year 2021, Volume: 4 Issue: 4, 200 - 206, 31.12.2021

Tanapat Chalarux Khuanchanok Chaichana

https://doi.org/10.53006/rna.960564

Abstract

The aim of this research is to introduce a novel iterative technique termed CC-iteration for identifying the fixed points of Garcia-Falset mappings. In uniformly convex Banach spaces, we establish both weak and strong convergence characteristics. Additionally, numerical examples of the iterative approach are presented in the form of a signal recovery application in a compressed sensing issue.

Keywords

Garcia-Falset mapping, Iteration, Convergence, Signal recovery

Supporting Institution

Chiang Mai University

Project Number

R000027527

Thanks

CMU Junior Research Fellowship Program

References

• [1] P. L. Combettes, V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul. 4 (2005) 1168-1200.
• [2] J. Duchi, S. Shalev-Shwartz, Y. Singer, T. Chandra, E?cient projections onto the l 1 -ball for learning in high dimensions, In: Proceedings of the 25th International Conference on Machine Learning, Helsinki, Finland. (2008) 272-279.
• [3] J. Garci­a-Falset, E. Llorens-Fuster, T. Suzuki, Fixed point theory for a class of generalized nonexpansive mappings, J.Math. Anal. Appl. 375 (2011) 185-195.
• [4] H. Iiduka, W. Takahashi, Strong convergence theorems for nonexpansive nonself-mappings and inverse-strongly-monotone mappings, J. Convex Anal. 11 (2004) 69-79.
• [5] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 4(1) (1974) 147-150.
• [6] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953) 506-510.
• [7] M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251(1) (2000) 217-229.
• [8] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math.Soc. 73(4) (1967) 591-598.
• [9] J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Aust. Math. Soc.43(1) (1991) 153-159.
• [10] H. F. Senter, W. G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44 (1974)375-380.
• [11] S. Suantai, Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings, J. Math.Anal. Appl. 311(2) (2005) 506-517.
• [12] R. Suparatulatorn, P. Charoensawan, K. Poochinapan, Inertial self-adaptive algorithm for solving split feasible problems with applications to image restoration, Math. Methods Appl. Sci. 42(18) (2019) 7268-7284.
• [13] R. Suparatulatorn, A. Khemphet, Tseng type methods for inclusion and fixed point problems with applications, Mathematics. 7(12) (2019) 1175.
• [14] R. Suparatulatorn, A. Khemphet, P. Charoensawan, S. Suantai, N. Phudolsitthiphat, Generalized self-adaptive algorithm for solving split common fixed point problem and its application to image restoration problem, Int. J. Comput. Math.97(7) (2020) 1431-1443.
• [15] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal.Appl. 340(2) (2008) 1088-1095.
• [16] B. S. Thakur, D. Thakur, M. Postolache, A new iteration scheme for approximating fixed points of nonexpansive mappings, Filomat. 30(10) (2016) 2711-2720.
• [17] G. I. Usurelu, A. Bejenaru, M. Postolache, Operators with property (E) as concerns numerical analysis and visualization, Numer. Funct. Anal. Optim. 41(11) (2020) 1398-1419.
• [18] P. Cholamjiak, W. Cholamjiak, Fixed point theorems for hybrid multivalued mappings in Hilbert spaces, J. Fixed Point Theory Appl. 18 (2016) 673-688.
• [19] P. Cholamjiak, A. A. Abdou, Y. J. Cho, Proximal point algorithms involving fixed points of nonexpansive mappings in CAT(0) spaces, Fixed Point Theory Appl. 2015 (2015) 227.
• [20] S. Kesornprom, N. Pholasa, P. Cholamjiak, On the convergence analysis of the gradient-CQ algorithms for the split feasibility problem, Numerical Algorithms. 84(3) (2020) 997-1017.
• [21] S. Kesornprom, P. Cholamjiak, Proximal type algorithms involving linesearch and inertial technique for split variational inclusion problem in hilbert spaces with applications, Optimization. 68(12) (2019) 2369-2395.
• [22] S. Suantai, S. Kesornprom, P. Cholamjiak, A new hybrid CQ algorithm for the split feasibility problem in Hilbert spaces and its applications to compressed sensing, Mathematics. 7(9) (2019) 789.
• [23] R. Suparatulatorn, P. Charoensawan, K. Poochinapan, S. Dangskul, An algorithm for the split feasible problem and image restoration, RACSAM. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 115 (2021). https://doi.org/10.1007/s13398-020-00942-z.
• [24] R. Suparatulatorn, P. Charoensawan, A. Khemphet, An inertial subgradient extragradient method of variational inequality problems involving quasi-nonexpansive operators with applications, Math. Methods Appl. Sci. (2021).https://doi.org/10.1002/mma.7576.