# Common solutions of fixed point and generalized equilibrium problems using asymptotically nonexpansive mapping

## Main Article Content

## Abstract

In this paper, a common solution of fixed point and generalized equilibrium problems using an asymptotically nonexpansive mapping is determined via iterative approach. Further, an application and numerical example of the main result are given.

## Article Details

*Tamkang Journal of Mathematics*,

*55*(3), 287–306. https://doi.org/10.5556/j.tkjm.55.2024.5172

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

## References

R. P. Agarwal, D. O’Regan and D. R. Sahu, Fixed point theory for Lipschitzian-type mappings with applications, 1st 9 edition, Springer, New York, 2009.

M. O. Aibinu, S. C. Thakur and S. Moyo, The implicit midpoint procedures for asymptotically nonexpansive mappings, Journal of Mathematics, 2020 (2020).

E. Blum, From optimization and variational inequalities to equilibrium problems, Math. student, 63 (1994), 123-145.

F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, Journal of Mathematical Analysis and Applications, 20(2) (1967), 197-228.

Y. J. Cho, H. Zhou and G. Guo, Weak and strong convergence theorems for three step iterations with errors for asymptotically nonexpansive mappings, Computers and Mathematics with Applications, 47(4-5) (2004), 707-717.

F. Cianciaruso, G. Marino, L. Muglia and Y. Yao, A hybrid projection algorithm for finding solutions of mixed equilibrium problem and variational inequality problem, Fixed Point Theory and Applications, 2010 (2009), 1-19.

P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal, 6(1) (2005), 117-136.

K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proceedings of the American Mathematical Society, 35(1) (1972), 171-174.

K. R. Kazmi and S. H. Rizvi, Iterative approximation of a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem, Journal of the Egyptian Mathematical society, 21(1) (2013), 44-51.

T. H. Meche, M. G. Sangago and H. Zegeye, Approximating a common solution of a finite family of generalized equilibrium and fixed point problems, SINET: Ethiopian Journal of Science, 38(1) (2015), 17-28.

A. Moudafi and M. Thera, Proximal and dynamical approaches to equilibrium problems, In Ill-posed variational problems and regularization techniques, Springer, Berlin, Heidelberg, (1999), 187-201.

R. Osward, S. Kumar and M. G. Sangago, Approximation of common solutions for a fixed point problem of asymptotically nonexpansive mapping and a generalized equilibrium problem in Hilbert space, Journal of the Egyptian Mathematical Society,

27(1) (2019), 1-16.

S. Takahashi and W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space, Nonlinear Analysis: Theory, Methods and Applications, 69(3) (2008), 1025-1033.

H. K. Xu, Iterative algorithms for nonlinear operators, Journal of the London Mathematical Society, 66(1) (2002), 240-256.

H. K. Xu, Another control condition in an iterative method for nonexpansive mappings, Bulletin of the Australian Mathematical Society, 65(1) (2002), 109-113.

H. Zegeye, T. H. Meche and M. G. Sangago, Algorithms of common solutions for a fixed point of hemicontractive-type mapping and a generalized equilibrium problem, Inter. J. Adv. Math. Sci, 5 (2017), 20-26.