Strong convergence of perturbed Mann iteration for systems of variational inequality problems over the set of common fixed points of a finite family of demicontractive mappings in Banach spaces
Volume 17, Issue 1, pp 70--81
https://dx.doi.org/10.22436/jnsa.017.01.04
Publication Date: February 14, 2024
Submission Date: October 27, 2023
Revision Date: December 01, 2023
Accteptance Date: December 05, 2023
Authors
T. M. M. Sow
- Universite Amadou Mahtar Mbow, Senegal.
Abstract
In this paper, we propose an iterative algorithm, which is based on the Mann iterative method for solving simultaneously common fixed point problem with a finite family of demicontractive mappings and systems of variational inequalities involving an infinite family of strongly accretive operators. Under suitable assumptions, we prove the strong convergence of this algorithm in Banach spaces. Application to systems of constrained convex minimization problem is provided to support our main results. The results of this paper improve and extend results of [M. Eslamian, C. R. Math. Acad. Sci. Paris, \(\bf 355\) (2017), 1168--1177], and of many others.
Share and Cite
ISRP Style
T. M. M. Sow, Strong convergence of perturbed Mann iteration for systems of variational inequality problems over the set of common fixed points of a finite family of demicontractive mappings in Banach spaces, Journal of Nonlinear Sciences and Applications, 17 (2024), no. 1, 70--81
AMA Style
Sow T. M. M., Strong convergence of perturbed Mann iteration for systems of variational inequality problems over the set of common fixed points of a finite family of demicontractive mappings in Banach spaces. J. Nonlinear Sci. Appl. (2024); 17(1):70--81
Chicago/Turabian Style
Sow, T. M. M.. "Strong convergence of perturbed Mann iteration for systems of variational inequality problems over the set of common fixed points of a finite family of demicontractive mappings in Banach spaces." Journal of Nonlinear Sciences and Applications, 17, no. 1 (2024): 70--81
Keywords
- Perturbed Mann iteration
- systems of variational inequalities
- demicontractive operators
- strongly accretive operators
MSC
References
-
[1]
T. O. Alakoya, L. O. Jolaoso, O. T. Mewomo, A general iterative method for finding common fixed point of finite family of demicontractive mappings with accretive variational inequality problems in Banach spaces, Nonlinear Stud., 27 (2020), 213–236
-
[2]
Y. I. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, In: Theory and applications of nonlinear operators of accretive and monotone type, Dekker, New York, 178 (1996), 15–50
-
[3]
K. O. Aremu, L. O. Jolaoso, C. Izuchukwu, O. T. Mewomo, Approximation of common solution of finite family of monotone inclusion and fixed point problems for demicontractive multivalued mappings in CAT(0) spaces, Ric. Mat., 69 (2020), 13–34
-
[4]
J. B. Baillon, G. Haddad, Quelques propri´et´es des op´erateurs angle-born´es etn-cycliquement monotones, Isr. J. Math., 26 (1977), 137–150
-
[5]
O. A. Boikanyo, G. Moros¸anu, On the method of alternating resolvents, Nonlinear Anal., 74 (2011), 5147–5160
-
[6]
F. E. Browder, Convergenge theorem for sequence of nonlinear operator in Banach spaces, Math. Z., 100 (1967), 201–225
-
[7]
Y. Censor, A. N. Iusem, S. A. Zenios, An interior point method with Bregman functions for the variational inequality problem with paramonotone operators, Math. Programming, 81 (1998), 373–400
-
[8]
S. Chang, J. K. Kim, X. R. Wang, Modified block iterative algorithm for solving convex feasibility problems in Banach spaces, J. Inequal. Appl., 2010 (2010), 14 pages
-
[9]
P. Charoensawan, R. A Suparatulatorn, A modified Mann algorithm for the general split problem of demicontractive operators, Results Nonlinear Anal., 5 (2022), 213–221
-
[10]
C. E. Chidume, The solution by iteration of nonlinear equations in certain Banach spaces, J. Nigerian Math. Soc., 3 (1984), 57–62
-
[11]
C. E. Chidume, Geometric properties of Banach spaces and nonlinear iterations, Springer-Verlag, London (2009)
-
[12]
I. Cioranescu, Geometry of Banach space, duality mapping and nonlinear problems, Kluwer Academic Publishers Group, Dordrecht (1990)
-
[13]
M. Eslamian, Common solutions to a system of variational inequalities over the set of common fixed points of demicontractive operators, C. R. Math. Acad. Sci. Paris, 335 (2017), 1168–1177
-
[14]
B. Halpern, Fixed points of nonexpansive maps, Bull. Amer. Math. Soc., 73 (1967), 957–961
-
[15]
W. Khuangsatung, A. Kangtunyakarn, A Method for Solving the Variational Inequality Problem and Fixed Point Problems in Banach Spaces, Tamkang J. Math., 53 (2022), 23–36
-
[16]
T.-C. Lim, H. K. Xu, Fixed point theorems for assymptoticaly nonexpansive mapping, Nonliear Anal., 22 (1994), 1345–1355
-
[17]
P.-E. Maing´e, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899–912
-
[18]
W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506–510
-
[19]
Y. Song, Y. J. Cho, Some notes on Ishikawa iteration for multivalued mappings, Bull. Korean Math. Soc., 48 (2011), 575–584
-
[20]
T. M. M. Sow, N. Djitte, C. E. Chidume, A path convergence theorem and construction of fixed points for nonexpansive mappings in certain Banach spaces, Carpathian J. Math., 32 (2016), 241–250
-
[21]
D.-J.Wen, Modified Krasnoselski-Mann type iterative algorithm with strong convergence for hierarchical fixed point problem and split monotone variational inclusions, J. Comput. Appl. Math., 393 (2021), 13 pages
-
[22]
H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. (2), 66 (2002), 240–256
-
[23]
I. Yamada, The hybrid steepest-descent method for variational inequality problems over the intersection of the fixed point sets of nonexpansive mappings, In: Inherently parallel algorithms in feasibility and optimization and their applications, Stud. Comput. Math., 8 (2001), 473–504
-
[24]
J. C. Yao, Variational inequalities with generalized monotone operators, Math. Oper. Res., 19 (1994), 691–705