Research Article

Common Solution for Nonlinear Operators in Banach Spaces

Volume: 33 Number: 3 September 1, 2020
EN

Common Solution for Nonlinear Operators in Banach Spaces

Abstract

This paper formulates a hybrid approximation process involving inertial component and demonstrates a convergence results for it. The formulated scheme converges faster and finds a common solution for some nonlinear operators in Banach spaces. The method of our proof and results obtained is well involved and significant.

Keywords

References

  1. [1] S. Reich, S. Sabach, “A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces”, Journal of Nonlinear Convex Analysis, 10(3), 471-485, 2009.[2] D. Butnariu, A. N. Iusem, “Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization”, vol. 40, Kluwer Academic, Dordrecht, 2000.[3] D.P. Bertsekas, “Convex optimization theory”, Anthena Scientific, Belmont, Massachusetts.[4] D. Butnariu, E. Resmerita, “Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces”, Abstr. Appl. Anal., 1-39, Art ID 84919, 2006.[5] E. Naraghirad, J. C. Yao, “Bregman weak relatively nonexpansive mappings in Banach spaces”, Fixed Point Theory and Applications, 2013(141), 2013.[6] B.T. Polyak, “Some methods of speeding up the convergence of iteration methods”, USSR Comput. Math. and Math. Phys., 4(5), 1–17, 1964. [7] Y. I. Alber, “Metric and generalized projection operators in Banach Spaces: Properties and Applications”, Lecture Notes in Pure and Applied Mathematics, 15–50, 1996.[8] R. T. Rockafellar, “Convex Analysis”, Princeto University Press, Princeton, 1970.[9] C. Zalinescu, “Convex Analysis in General Vector Spaces”, World Scientific, River Edge, NJ, USA, 2002.[10] J. F. Bonnas, A. Shapiro, “Perturbation Analysis of Optimization Problems”, Springer, New York, 2000. [11] H. H. Bauschke, J. M. Borwein, P. L. Combettes, “Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces”, Communications in Contemporary Mathematics, 3(4), 615–647, 2001. [12] R. P. Phelps, “Convex Functions, Monotone Operators and Differentiability”, Lecture notes in Mathematics, Vol. 1364, Springer, Berlin, 1993. [13] Q. L. Dong, Y, J. Cho, Th. M. Rassias, “General Inertial Mann Algorithms and their convergence analysis for nonexpansive mappings”, Applications of Nonlinear Analysis, Springer Optimization and its Applications, 134, 2018, https://doi.org/10.1007/978-3-319-89815-5_7. [14] M. A. Alghamdi, N. Shahzad, H. Zegeye, “Fixed points of Bregman relatively nonexpansive mappings and solutions of variational inequality problems”, Journal of Nonlinear Science and Appl. 9, 2541-2552, 2016. [15] C.E. Chidume, S.I. Ikechukwu, A. Adamu, “Inertial algorithm for approximatinga common fixed point for a countable family of relatively nonexpansive maps”, Fixed Point Theory and Applications, 2018(9), 2018, https://doi.org/10.1186/s13663-018-0634-3. [16] S. Reich, S. Sabach, “Existence and approximation of fixed points of Bregman firmly nonexpansive mappings in reflexive Banach spaces”, Fixed-point Algorithms for Inverse Problems in Science and Engineering, 49, 301–316, 2011.[17] G. Inoue, W. Takahashi, K. Zembayashi, “Strong convergence theorems by hybrid method for maximal monotone operators and relatively nonexpansive mappings in Banach spaces”, J. Convex Anal., 16, 791-806, 2009.[18] E. Naraghirad, “Halpern’s iteration for Bregman relatively nonexpansive mappings in Banach spaces”, Numerical Functional Analysis and optimization, 34(10), 1129-1155, 2013, DOI:10.1080/01630563.2013.767269.

Details

Primary Language

English

Subjects

Engineering

Journal Section

Research Article

Publication Date

September 1, 2020

Submission Date

July 10, 2019

Acceptance Date

April 3, 2020

Published in Issue

Year 2020 Volume: 33 Number: 3

APA
Ekuma-okereke, E., & Okoro, F. (2020). Common Solution for Nonlinear Operators in Banach Spaces. Gazi University Journal of Science, 33(3), 737-749. https://doi.org/10.35378/gujs.590435
AMA
1.Ekuma-okereke E, Okoro F. Common Solution for Nonlinear Operators in Banach Spaces. Gazi University Journal of Science. 2020;33(3):737-749. doi:10.35378/gujs.590435
Chicago
Ekuma-okereke, Enyinnaya, and Felix Okoro. 2020. “Common Solution for Nonlinear Operators in Banach Spaces”. Gazi University Journal of Science 33 (3): 737-49. https://doi.org/10.35378/gujs.590435.
EndNote
Ekuma-okereke E, Okoro F (September 1, 2020) Common Solution for Nonlinear Operators in Banach Spaces. Gazi University Journal of Science 33 3 737–749.
IEEE
[1]E. Ekuma-okereke and F. Okoro, “Common Solution for Nonlinear Operators in Banach Spaces”, Gazi University Journal of Science, vol. 33, no. 3, pp. 737–749, Sept. 2020, doi: 10.35378/gujs.590435.
ISNAD
Ekuma-okereke, Enyinnaya - Okoro, Felix. “Common Solution for Nonlinear Operators in Banach Spaces”. Gazi University Journal of Science 33/3 (September 1, 2020): 737-749. https://doi.org/10.35378/gujs.590435.
JAMA
1.Ekuma-okereke E, Okoro F. Common Solution for Nonlinear Operators in Banach Spaces. Gazi University Journal of Science. 2020;33:737–749.
MLA
Ekuma-okereke, Enyinnaya, and Felix Okoro. “Common Solution for Nonlinear Operators in Banach Spaces”. Gazi University Journal of Science, vol. 33, no. 3, Sept. 2020, pp. 737-49, doi:10.35378/gujs.590435.
Vancouver
1.Enyinnaya Ekuma-okereke, Felix Okoro. Common Solution for Nonlinear Operators in Banach Spaces. Gazi University Journal of Science. 2020 Sep. 1;33(3):737-49. doi:10.35378/gujs.590435

Cited By

A Hybrid Algorithm for Solving Nonlinear Problems

Faculty of Natural and Applied Sciences Journal of Mathematical Modeling and Numerical Simulation

https://doi.org/10.63561/jmns.v2i3.860