Multi-valued Bregman nonspreading mapping in Banach spaces

Subject Areas : Statistics

Roushanak Lotfikar ¹ , Gholamreza Zamani Eskandani ² , Massoumeh Raeisi ³

1 - Faculty of Basic Science, Ilam University, P. O. Box 69315-516, Ilam, Iran,
2 - Department of Pure Mathematics, University of Tabriz,Tabriz, Iran
3 - Department of Pure Mathematics, University of Tabriz, Tabriz, Iran

Received: 2021-11-10 Accepted : 2022-12-05 Published : 2024-09-25

Keywords: فاصله برگمن, متر هاسدورف, نقطه ثابت, نگاشت های غیر‌پخشی,

Abstract :

In 2008, Kohsaka and Takahashi introduced a nonlinear mapping called nonspreading mapping in a smooth, strictly convex, and reflexive Banach space[1]. Since then, some fixed point theorems of such mapping has been studied by many researchers. We note that the concept of nonspreading mapping is very important because of useful applications. In this paper, we introduce and investigate the concept of multi-valued nonspreading mapping in Banach spaces which is called multi-valued Bregman nonspreading mapping. For this purpose, we introduce the Bregman Hausdorff distance on closed and bounded subsets of a Banach space X. We prove some properties of multi-valued Bregman nonspreading mapping. Furthermore, a weak convergence theorem for approximating a common fixed point of a finite family of multi-valued Bregman nonspreading mappings is established in Banach spaces. Also, we prove a theorem of the existence of a fixed point for the single-valued Bregman nonspreading mappings. The theorems proved improve and complete a host of important recent results.

References:

[1] F. Kohsaka, and W. Takahashi, “Proximal point algorithm with Bregman functions in Banach spaces”, J. Nonlinear Convex Anal. 6, pp. 505-523, 2005.

[2] S. Iemoto, and W. Takahashi, “Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space”, Nonlinear Anal. 71, pp. 2080-2089, 2009.

[3] W. Cholamjiaky, S. Suantai, and Y.J. Cho, “Fixed points for nonspreading-type multi-valued mappings: existence and convergence results”, Ann. Acad. Rom. Sci. Ser. Math. Appl. Vol.10, No. 2, 2018.

[4] S. Suantai, P. Cholamjiak, Y. J. Cho, and W. Cholamjiak, “On solving split equilibrium problems and fixed point problems of nonspreading multi-valued mappings in Hilbert spaces”, Fixed Point TheoryAppl. 2016.

[5] L. M. Bregman, “A relaxation method for finding the common point of convex sets and its application to the solution of problems in convex programming, USSR Comput”, Math. Math. Phys. 7, pp. 200-217, 1967.

[6] H. H. Bauschke, J. M. Borwein, and P. L. Combettes, “Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces”, Commun. Contemp. Math. 3, 615-647, 2001.

[7] H. H, Bauschke, J. M. Borwein, and P. L. Combettes, “Bregman monotone optimization algorithms”, SIAM J. Control Optim. 42, pp. 596-636, 2003.

[8] D. Butnariu, and A. N. Iusem, “Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization”, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.

[9] D. Butnariu, A. N. Iusem, and C. Zalinescu, “On uniform convexity, total convexity and convergence of the proximal point and outer Bregman projection algorithms in Banach spaces”, J. Convex Anal. 10, pp. 35-61, 2003.

[10] D, Butnariu, and A. N. Iusem, “Bregman distances, totally convex functions and a methodfor solving operator equations in Banach spaces”, Abstr. Appl. Anal. Pp. 1-39, 2006, Art.

[11] G. Z. Eskandani, M. Raeisi, and Th. M. Rassias, “A hybrid extragradient method for solving pseudomonotone equilibrium problems using Bregman distance”, J. Fixed Point Theory Appl, pp. 20:132, 2018.

[12] M. Raeisi, G. Z. Eskandani, and M. Eslamian, “A general algorithm for multiple-sets split feasibility problem involving resolvents and Bregman mappings”, Optimization. 68. Pp. 309-327, 2018.

[13] S. Reich, and S. Sabach, “A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces”, J. Nonlinear Convex Anal. 10, pp.471-485, 2009.

[14] S. Reich, and S. Sabach, “Two strong convergence theorems for a proximal method in reflexive Banach spaces”, Numer. Funct. Anal. Optim. 31, pp.22-44, 2010.

[15] S. Reich, and S. Sabach, “Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces”, Nonlinear Anal. 73, pp.122-135, 2010.

[16] S. Reich, and S. Sabach, “Existence and approximation of fixed points of Bregman firmly nonexpansive operators in reflexive Banach spaces. In: Fixed-point algorithms for inverse problems in science and engineering”, optimization and its applications. NewYork (NY): Springer; pp. 301-316, 2011.

[17] J. F, Bonnans, and A. Shapiro, “Perturbation Analysis of Optimization Problems”, Springer Verlag, New York, 2000.

[18] S. Sabach, “Products of finitely many resolvents of maximal monotone mappings in reflexive Banach spaces”, SIAM J. Optim. 21, pp.1289-1308, 2011.

[19] Y. I. Alber, “Metric and generalized projection operators in Banach spaces: Properties and applications. in: Kartsatos A.G., Theory and Applications of Nonlinear Operator of Accretive and Monotone Type”, Marcel Dekker, New York, pp.15-50, 1996.

[20] Y. Censor, and A. Lent, “An iterative row-action method for interval convex programming”, J. Optim. Theory Appl. 34, pp.321-353, 1981.

[21] R. P. Phelps, “Convex Functions, Monotone Operators, and Differentiability, second ed.in: Lecture Notes in Mathematics”, vol. 1364, Springer Verlag, Berlin, 1993.

[22] E. Naraghirad, and J. C. Yao, “Bregman weak relatively nonexpansive mappings in Banach spaces”, Fixed Point Theory Appl. doi: 10.1186/1687-1812-2013-141, 2013.

[23] C. Zalinescu, “Convex analysis in general vector spaces”, World Scientific Publishing, Singapore, 2002.

[24] F. Schopfer, T. Schuster, and A. K. Louis, “An iterative regularization method for the solution of the split feasibility problem in Banach spaces”, Inverse Problems 24, 2008.

[25] D, Butnariu, A. N. Iusem, and E. Resmerita, “Total convexity for powers of the norm in uniformly convex Banach spaces”, J. Convex Anal. 7, pp.319-334, 2000.

[26] A. Ambrosetti, and G. Prodi, “A Primer of Nonlinear Analysis”, Cambridge University Press, Cambridge, 1993.

[27] W. Takahashi, “Nonlinear Functional Analysis, Fixed Point Theory and its Applications”, Yokohama Publishers, Yokohama, 2000 (in Japanese).