Fixed point approximations of noncommutative generic 2-generalized Bregman nonspreading mappings with equilibriums
Volume 13, Issue 6, pp 303--316
http://dx.doi.org/10.22436/jnsa.013.06.01
Publication Date: March 29, 2020
Submission Date: December 18, 2019
Revision Date: December 18, 2019
Accteptance Date: February 10, 2020
-
1469
Downloads
-
3079
Views
Authors
Bashir Ali
- Department of Mathematical Sciences, Bayero University, Kano, Nigeria.
Lawal Yusuf Haruna
- Department of Mathematical Sciences, Kaduna State University, Kaduna, Nigeria.
Abstract
In this paper, a Halpern type iterative scheme for finding a common element in the set of fixed points of generic 2-generalized Bregman nonspreading mappings and the solution set of equilibrium problem have been proposed. We also prove that the sequence generated by the scheme converges strongly to the element in a real reflexive Banach space. Our results improve and generalize some announced results in the literature.
Share and Cite
ISRP Style
Bashir Ali, Lawal Yusuf Haruna, Fixed point approximations of noncommutative generic 2-generalized Bregman nonspreading mappings with equilibriums, Journal of Nonlinear Sciences and Applications, 13 (2020), no. 6, 303--316
AMA Style
Ali Bashir, Haruna Lawal Yusuf, Fixed point approximations of noncommutative generic 2-generalized Bregman nonspreading mappings with equilibriums. J. Nonlinear Sci. Appl. (2020); 13(6):303--316
Chicago/Turabian Style
Ali, Bashir, Haruna, Lawal Yusuf. "Fixed point approximations of noncommutative generic 2-generalized Bregman nonspreading mappings with equilibriums." Journal of Nonlinear Sciences and Applications, 13, no. 6 (2020): 303--316
Keywords
- invex set
- normally 2-generalized hybrid mapping
- fixed point
- generic 2-generalized Bregman nonspreading mapping
- equilibrium problem
MSC
References
-
[1]
Y. I. Alber, The Young-Fenchel Transformation and some new characteristics of Banach Spaces, In: Functional Spaces, 2007 (2007), 1--19
-
[2]
B. Ali, M. H. Harbau, L. H. Yusuf, Existence Theorems for Attractive Points of Semigroups of Bregman Generalized Nonspreading Mappings in Banach Spaces, Adv. Oper. Theory, 2 (2017), 257--268
-
[3]
B. Ali, L. Y. Haruna, Iterative Approximations of Attractive Point of a New Generalized Bregman Nonspreading Mapping in Banach Spaces, Bull. Iranian Math. Soc., 46 (2020), 331--354
-
[4]
B. Ali, L. H. Yusuf, M. H. Harbau, Common Attractive Points Approximation for Family of Generalized Bregman Nonspreading Mappings in Banach Space, J. Nonlinear Convex Anal., (), Submitted
-
[5]
S. Alizadeh, F. Moradlou, Weak Convergence Theorems for $2$-Generalized Hybrid Mappings and Equilibrium Problems, Commun. Korean Math. Soc., 31 (2016), 765--777
-
[6]
E. Asplund, R. T. Rockafellar, Gradient of Convex Function, Trans. Amer. Math. Soc., 139 (1969), 443--467
-
[7]
H. H. Bauschke, J. M. Borwein, Legendre Function and the Method of Bregman Projections, J. Convex Anal., 4 (1997), 27--67
-
[8]
H. H. Bauschke, P. L. Combettes, J. M. Borwein, Essential Smoothness, Essential Strict Convexity, and Legendre functions in Banach Spaces, Commun. Contemp. Math., 3 (2001), 615--647
-
[9]
E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123--145
-
[10]
L. M. Bregman, The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming, USSR Comput. Math. Phys., 7 (1967), 200--217
-
[11]
D. Butnariu, A. N. Iusem, Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization, Kluwer Academic Publ., Dordrecht (2000)
-
[12]
D. Butnariu, E. Resmerita, Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces, Abstr. Appl. Anal., 2006 (2006), 39 pages
-
[13]
Y. Censor, A. Lent, An Iterative row-action Method Interval Convex Programming, J. Optim. Theory Appl., 34 (1981), 321--353
-
[14]
P. L. Combettes, S. A. Hirstoaga, Equilibrium Programming in Hilbert Spaces, J. Nonlinear Convex Anal., 6 (2005), 117--136
-
[15]
G. Z. Eskandani, M. Raeisi, J. K. Kim, A Strong Convergence theorem for Bregman quasi-nonexpansive mappings with applications, Revista de la Real Academia de la Ciancias Exactas, Fisicas y, Naturales. Serie A. Mathematics, 113 (2019), 353--366
-
[16]
J.-B. Hiriart-Urruty, C. Lemaréchal, Convex analysis and minimization algorithms. II. Advanced theory and bundle methods, Springer-Verlag, Berlin (1993)
-
[17]
M. Hojo, A. Kondo, W. Takahashi, Weak and Strong Convergence Theorems for Commutative Normally 2-Generalized Hybrid Mappings in Hilbert Spaces, Linear Nonlinear Anal., 4 (2018), 117--134
-
[18]
F. Kohsaka, W. Takahashi, Proximal point algorithms with Bregman functions in Banach spaces, J. Nonlinear Convex Anal., 6 (2005), 505--523
-
[19]
A. Kondo, W. Takahashi, Attractive points and weak convergence theorems for normally N-generalized hybrid mappings in Hilbert spaces, Linear Nonlinear Anal., 3 (2017), 297--310
-
[20]
P.-E. Maingé, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899--912
-
[21]
V. Martín-Márquez, S. Reich, S. Sabach, Iterative Methods for Approximating Fixed Points of Bregman Nonexpansive Operators, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 1043--1063
-
[22]
T. Maruyama, W. Takahashi, M. Yao, Fixed Point and Ergodic Theorems for New Nonlinear Mappings in Hilbert Spaces, J. Nonlinear Convex Anal., 12 (2011), 185--197
-
[23]
E. Naraghirad, N.-C. Wong, J.-C. Yao, Applications of Bregman-Opial Property to Bregman Nonspreading Mappings in Banach Spaces, Abstr. Appl. Anal., 2014 (2014), 14 pages
-
[24]
E. Naraghirad, J.-C. Yao, Bregman weak relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl., 2013 (2013), 43 pages
-
[25]
S. Reich, S. Sabach, Existence and approximation of fixed points of Bregman firmly nonexpansive mappings in reflexive Banach spaces, in: Fixed-point algorithms for inverse problems in science and engineering, 2011 (2011), 301--316
-
[26]
E. Resmerita, On Total Convexity, Bregman Projections and Stability in Banach Spaces, J. Convex Anal., 11 (2004), 1--16
-
[27]
W. Takahashi, Convex Analysis and Approximation of Fixed Points, Yokohama Publ., Yokohama (2000)
-
[28]
S. Takahashi, W. Takahashi, Weak and strong convergence theorems for noncommutative normally $2$-generalized hybrid mappings in Hilbert spaces, J. Nonlinear Convex Anal., 19 (2018), 1427--1441
-
[29]
W. Takahashi, C.-F. Wen, J.-C. Yao, Strong convergence theorems by hybrid methods for noncommutative normally $2$-generalized hybrid mappings in Hilbert spaces, Appl. Anal. Optim., 3 (2019), 43--56
-
[30]
H.-K. Xu, Another Control Condition in an Iterativemethod for Nonexpansive Mappings, Bull. Austral. Math. Soc., 65 (2002), 109--113
-
[31]
C. Zalinescu, Convex Analysis in General Vector Spaces, World Scientific Publishing Co., River Edge (2002)