Algorithms for Computing of Optimization Problems with a Common Minimum-Norm Fixed Point with Applications
This research is aimed to study a two-step iteration
process defined over a finite family of σ-asymptotically
quasi-nonexpansive nonself-mappings. The strong convergence
is guaranteed under the framework of Banach spaces with some
additional structural properties including strict and uniform
convexity, reflexivity, and smoothness assumptions. With similar
projection technique for nonself-mapping in Hilbert spaces, we
hereby use the generalized projection to construct a point within
the corresponding domain. Moreover, we have to introduce the use
of duality mapping and its inverse to overcome the unavailability
of duality representation that is exploit by Hilbert space theorists.
We then apply our results for σ-asymptotically quasi-nonexpansive
nonself-mappings to solve for ideal efficiency of vector optimization
problems composed of finitely many objective functions. We also
showed that the obtained solution from our process is the closest to
the origin. Moreover, we also give an illustrative numerical example
to support our results.
[1] K. Goebel, W. A. Kirk, A fixed point theorem for asymptotically
nonexpansive mapping, Proc. Amer. Math. Soc. 35(1972), 171-174.
[2] C. E. Chidume, E. U. Ofoedu, H. Zegeye, Strong and weak convergence
theorems for asymptotically nonexpansive mappings, J. Math. Anal. Appl.
280(2003), 364–374.
[3] L. Wang, Strong and weak convergence theorems for common fixed points
of nonself asymptotically nonexpansive mappings, J. Math. Anal. Appl.
323(2006), 550-557.
[4] W. Guo, W. Guo, Weak convergence theorems for asymptotically
nonexpansive nonself-mappings, Appl. Math. Lett. 24(2011), 2181-2185.
[5] W. Guo, Y. J. Cho, W. Guo, Convergence theorems for mixed
type asymptotically nonexpansive mappings, Fixed Point Theory and
Applications. 2012 2012:224.
[6] H. Zegeye, N. Shahzed, Approximation of the common minimum-norm
fixed point of a finite family of asymptotically nonexpansive mappings,
Fixed Point Theory Appl., 2013 2013:1, 12 pages.
[7] J. Schu, Iteration construction of fixed points of asymptotically
nonexpansive mappings, J. Math. Anal. Appl. 158(1991), 407-413.
[8] M. O. Osilike, S. C. Aniagbosor, Weak and strong convergence theorems
for fixed points of asymptotically nonexpansive mappings, Math. Comput.
Modelling. 256(2001), 431-445.
[9] Y.I. Alber, Metric and generalized projection opeators in Banach spaces:
properties and applications, in: Theory and Applications of Nonlinear
Operators of Accretive and Monotone Type, Lecture Notes in Pure and
Applied Mathematics, Dekker, New York, 1996, pp. 1550.
[10] C. Byrne, Iterative oblique projection onto convex subset and the split
feasibility problem, Inverse Problems. 18 (2002), 441-453.
[11] H. K. Pathak, V. K. Sahu, Y. J. Cho, Approximation of a common
minimum-norm fixed point of a finite family of σ-asymptotically
quasi-nonexpansive mappings with applications, J. Nonlinear Sci. Appl.
9 (2016), 3240-3254.
[12] Y. Censor, T. Bortfeld, B. Martin, A. Trofimov, A unified approach for
inversion problem in intensity-modulated radiation therapy, Phys. Med.
Biol. 51(2006), 2353-2365.
[13] Y. Censor, T. Elfving, A multiprojection algorithm using Bregman
projection in a product space, Numer. Algoritms. 8 (1994), 221-239.
[14] X. Yang, Y. C. Liou, Y. You, Finding minimum-norm fixed point of
nonexpansive mapping and applications, Math. Problem in Engin. Article
ID 106450, (2011), 13 pp.
[15] Y. Hao, S. Y. Cho, X. Qin, Some weak convergence theorems for a
family of asymptotically nonexpansive nonself mappings, Fixed Point
Theo. Appl. 2010, Article ID 218573, 11 pp.
[16] S. Kamimura, W. Takahashi, Strong convergence of a proximal-type
algorithm in a Banach space, SIAM J. Optim. 13 (2002) 938-945.
[17] J. G. Ohara, P. Pillay, H. K. Xu, Iteration approaches to convex
feasibility problem in Banach spaces, Nonlinear Anal. 64 (2006),
2022-2042.
[18] P. E. Manige, Strong convergence of projected subgradient methods for
nonsmooth and nonstrictly convex minization, Set-Valued Anal, 16 (2008),
899-912.
[19] H. Zegeye, E. U. Ofoedu, N. Shahzad, Convergence theorems for
equilibrium problem, variational inequality problem and countably
infinite relatively quasi-nonexpansive mappings, Appl. Math. Comput.
216 (2010), 3439-3449
[1] K. Goebel, W. A. Kirk, A fixed point theorem for asymptotically
nonexpansive mapping, Proc. Amer. Math. Soc. 35(1972), 171-174.
[2] C. E. Chidume, E. U. Ofoedu, H. Zegeye, Strong and weak convergence
theorems for asymptotically nonexpansive mappings, J. Math. Anal. Appl.
280(2003), 364–374.
[3] L. Wang, Strong and weak convergence theorems for common fixed points
of nonself asymptotically nonexpansive mappings, J. Math. Anal. Appl.
323(2006), 550-557.
[4] W. Guo, W. Guo, Weak convergence theorems for asymptotically
nonexpansive nonself-mappings, Appl. Math. Lett. 24(2011), 2181-2185.
[5] W. Guo, Y. J. Cho, W. Guo, Convergence theorems for mixed
type asymptotically nonexpansive mappings, Fixed Point Theory and
Applications. 2012 2012:224.
[6] H. Zegeye, N. Shahzed, Approximation of the common minimum-norm
fixed point of a finite family of asymptotically nonexpansive mappings,
Fixed Point Theory Appl., 2013 2013:1, 12 pages.
[7] J. Schu, Iteration construction of fixed points of asymptotically
nonexpansive mappings, J. Math. Anal. Appl. 158(1991), 407-413.
[8] M. O. Osilike, S. C. Aniagbosor, Weak and strong convergence theorems
for fixed points of asymptotically nonexpansive mappings, Math. Comput.
Modelling. 256(2001), 431-445.
[9] Y.I. Alber, Metric and generalized projection opeators in Banach spaces:
properties and applications, in: Theory and Applications of Nonlinear
Operators of Accretive and Monotone Type, Lecture Notes in Pure and
Applied Mathematics, Dekker, New York, 1996, pp. 1550.
[10] C. Byrne, Iterative oblique projection onto convex subset and the split
feasibility problem, Inverse Problems. 18 (2002), 441-453.
[11] H. K. Pathak, V. K. Sahu, Y. J. Cho, Approximation of a common
minimum-norm fixed point of a finite family of σ-asymptotically
quasi-nonexpansive mappings with applications, J. Nonlinear Sci. Appl.
9 (2016), 3240-3254.
[12] Y. Censor, T. Bortfeld, B. Martin, A. Trofimov, A unified approach for
inversion problem in intensity-modulated radiation therapy, Phys. Med.
Biol. 51(2006), 2353-2365.
[13] Y. Censor, T. Elfving, A multiprojection algorithm using Bregman
projection in a product space, Numer. Algoritms. 8 (1994), 221-239.
[14] X. Yang, Y. C. Liou, Y. You, Finding minimum-norm fixed point of
nonexpansive mapping and applications, Math. Problem in Engin. Article
ID 106450, (2011), 13 pp.
[15] Y. Hao, S. Y. Cho, X. Qin, Some weak convergence theorems for a
family of asymptotically nonexpansive nonself mappings, Fixed Point
Theo. Appl. 2010, Article ID 218573, 11 pp.
[16] S. Kamimura, W. Takahashi, Strong convergence of a proximal-type
algorithm in a Banach space, SIAM J. Optim. 13 (2002) 938-945.
[17] J. G. Ohara, P. Pillay, H. K. Xu, Iteration approaches to convex
feasibility problem in Banach spaces, Nonlinear Anal. 64 (2006),
2022-2042.
[18] P. E. Manige, Strong convergence of projected subgradient methods for
nonsmooth and nonstrictly convex minization, Set-Valued Anal, 16 (2008),
899-912.
[19] H. Zegeye, E. U. Ofoedu, N. Shahzad, Convergence theorems for
equilibrium problem, variational inequality problem and countably
infinite relatively quasi-nonexpansive mappings, Appl. Math. Comput.
216 (2010), 3439-3449
@article{"International Journal of Engineering, Mathematical and Physical Sciences:74647", author = "Apirak Sombat and Teerapol Saleewong and Poom Kumam and Parin Chaipunya and Wiyada Kumam and Anantachai Padcharoen and Yeol Je Cho and Thana Sutthibutpong", title = "Algorithms for Computing of Optimization Problems with a Common Minimum-Norm Fixed Point with Applications", abstract = "This research is aimed to study a two-step iteration
process defined over a finite family of σ-asymptotically
quasi-nonexpansive nonself-mappings. The strong convergence
is guaranteed under the framework of Banach spaces with some
additional structural properties including strict and uniform
convexity, reflexivity, and smoothness assumptions. With similar
projection technique for nonself-mapping in Hilbert spaces, we
hereby use the generalized projection to construct a point within
the corresponding domain. Moreover, we have to introduce the use
of duality mapping and its inverse to overcome the unavailability
of duality representation that is exploit by Hilbert space theorists.
We then apply our results for σ-asymptotically quasi-nonexpansive
nonself-mappings to solve for ideal efficiency of vector optimization
problems composed of finitely many objective functions. We also
showed that the obtained solution from our process is the closest to
the origin. Moreover, we also give an illustrative numerical example
to support our results.", keywords = "σ-asymptotically quasi-nonexpansive nonselfmapping,
strong convergence, fixed point, uniformly convex and
uniformly smooth Banach space.", volume = "10", number = "12", pages = "688-9", }
