A shrinking projection method with extended allowable ranges for common fixed point problems

Authors

Takanori Ibaraki Yokohama National University

Keywords:

shrinking projection method, allowable range, common fixed point, common attractive point

Abstract

In this paper, we propose a shrinking projection method with extended allowable ranges for approximating a common fixed point of a family of nonlinear mappings in a Banach space. Our approach combines and improves ideas from Kimura (Proc. 10th Int. Conf. Fixed Point Theory Appl., 2012, pp. 157–164), who allowed nonsummable errors, and Takeuchi (J. Nonlinear Anal. Optim., 10 (2019), pp. 83–94), who introduced allowable ranges into the shrinking projection method. Using our method, we prove strong convergence theorems for approximating a common fixed point of a family of nonlinear mappings of quasi-nonexpansive type in Banach spaces.

References

[1] Ya. I. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Dekker, New York, 1996, 15–50.

[2] K. Aoyama, F. Kohsaka and W. Takahashi, Three generalizations of firmly nonexpansive mappings: Their relations and continuity properties, J. Nonlinear Convex Anal., 10 (2009), 131–147.

[3] D. Butnariu, S. Reich, and A. J. Zaslavski, Asymptotic behavior of relatively nonexpansive operators in Banach spaces, J. Appl. Anal., 7 (2001), 151–174.

[4] T. Ibaraki, Y. Kimura and W. Takahashi, Convergence Theorems for Generalized Projections and Maximal Monotone Operators in Banach Spaces, Abstr. Appl. Anal., 2003 (2003), 621–629.

[5] T. Ibaraki and S. Saejung, On shrinking projection method for cutter type mappings with nonsummable errors, J. Inequal. Appl., 2023:92 (2023), 20 pages.

[6] T. Ibaraki and Y. Takeuchi, New convergence theorems for common fixed points of a wide range of nonlinear mappings, J. Nonlinear Anal. Optim., 9 (2018), 95–114.

[7] F. Kohsaka and W. Takahashi, Existence and approximation of fixed points of firmly nonexpansive type mappings in Banach spaces, SIAM J. Optim., 19 (2008), 824–835.

[8] F. Kohsaka and W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch. Math. (Basel), 91 (2008), 166–177.

[9] Y. Kimura, Approximation of a fixed point of nonexpansive mapping with nonsummable errors in a geodesic space, Proceedings of the 10th International Conference on Fixed Point Theory and Its Applications, 2012, 157–164.

[10] Y. Kimura, Approximation of a fixed point of nonlinear mappings with nonsummable errors in a Banach space, Proceedings of the International Symposium on Banach and Function Spaces IV (Kitakyushu, Japan), (L. Maligranda, M. Kato, and T. Suzuki eds.), 2014, 303–311.

[11] Y. Kimura, Approximation of a common fixed point of a finite family of nonexpansive mappings with nonsummable errors in a Hilbert space, J. Nonlinear Convex Anal., 15 (2014), 429–436. [12] Y. Kimura and W. Takahashi, On a hybrid method for a family of relatively nonexpansive mappings in a Banach space, J. Math. Anal. Appl., 357 (2009), 356–363.

[13] L. -J. Lin and W. Takahashi, Attractive point theorems for generalized nonspreading mappings in Banach Spaces, J. Convex Anal., 20 (2013), 265–284.

[14] S. Matsushita and W. Takahashi, Weak and strong convergence theorems for relatively nonexpansive mappings in Banach space, Fixed Point Theory Appl., 2004 (2004), 37–47.

[15] S. Matsushita and W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in Banach space, J. Approx. Theory, 134 (2005), 257–266.

[16] U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. in Math., 3 (1969), 510–585.

[17] S. Reich, A weak convergence theorem for the alternating method with Bregman distances Theory and applications of nonlinear operators of accretive and monotone type, Lecture Notes in Pure and Appl. Math., 178, Dekker, New York, 1996, 313–318.

[18] W. Takahashi, Nonlinear Functional Analysis - Fixed Point Theory and Its Applications, Yokohama Publishers, 2000.

[19] W. Takahashi and Y. Takeuchi, Nonlinear ergodic theorem without convexity for generalized hybrid mappings in a Hilbert space, J. Nonlinear Convex Anal., 12 (2011), 399–406.

[20] W. Takahashi, Y Takeuchi and R. Kubota, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 341 (2008), 276–286.

[21] Y. Takeuchi, Shrinking projection method with allowable ranges, J. Nonlinear Anal. Optim., 10 (2019), 83–94.

[22] M. Tsukada, Convergence of best approximations in a smooth Banach space, J. Approx. Theory, 40 (1984), 301–309.