Common Solution for Nonlinear Operators in Banach Spaces

Year 2020, Volume: 33 Issue: 3, 737 - 749, 01.09.2020

Enyinnaya Ekuma-okereke Felix Okoro

https://doi.org/10.35378/gujs.590435

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

Bregman nonexpansive-type operator, Common element, Continuous monotone operator, Inertial component

References

• [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.