Document Type: Research Paper

**Author**

Gaston Berger University, Saint Louis, Senegal.

**Abstract**

In this paper, we introduce two iterative schemes by a modified Krasnoselskii-Mann algorithm for finding a common element of the set of solutions of equilibrium problems and the set of fixed points of multivalued nonexpansive mappings in Hilbert space. We prove that the sequence generated by the proposed method converges strongly to a common element of the set of solutions of equilibruim problems and the set of fixed points of multivalued nonexpansive mappings which is also the minimum-norm element of the above two sets. Finally, some applications of our results to optimization problems with constraint and the split feasibility problem are given. No compactness assumption is made. The methods in the paper are novel and different from those in early and recent literature.

**Keywords**

[1] E. Blum and W. Oettli, *From optimization and variational inequalities to equilibrium problems*, Math. Student, 63 (1994), pp. 123-145.

[2] C. Bryne, *Iterative oblique projection onto convex set and the split feasiblity problem*, Inverse Problems, 18 (2002), pp. 441-453.

[3] J. Caristi, *Fixed points theorems and selections of set-valued contraction*, J. Math. Anal., 227 (1988), pp. 55-67.

[4] Y. Censor and T. Elfving, *A multiprojection algorithm using Bregman projections in a product space*, Numerical Algorithms, 8 (1994), pp. 221-239.

[5] S. Chang, Y.Tang, L. Wang, Y. Xu, Y. Zhao, and G. Wang, *Convergence theorems for some multivalued generalized nonexpansive mappings*, Fixed Point Theory Appl., 33 (2014), pp. 1-11.

[6] C.E. Chidume, *Geometric Properties of Banach spaces and Nonlinear Iterations*, Springer Verlag, 2009.

[7] C.E. Chidume, C.O. Chidume, N. Djitte, and M. S. Minjibir, *Convergence Theorems for Fixed Points of Multivalued Strictly Pseudocontractive Mappings in Hilbert Spaces*, Abstract and Applied Analysis, 2013 (2013), pp. 1-10.

[8] D. Downing and W.A. Kirk, *Fixed point theorems for set-valued mappings in metric and Banach spaces*, Mathematica Japonica, 22 (1977), pp. 99-112.

[9] K. Fan, *A minimax inequality and applications, in Inequalities III, (O. Shisha, ed.)*, Academic Press, New York, 1972.

[10] Q-W. Fan and Z. Yao, *Strong convergence theorems for nonexpansive mapping and its applications for solving the split feasibility problem*, J. Nonlinear. Sci. App., 10 (2017), pp. 1470-1477.

[11] J. Geanakoplos, *Nash and Walras equilibrium via Brouwer*, Economic Theory, 21 (2003), pp. 585-603.

[12] L. Gorniewicz, *Topological Fixed Point Theory of Multivalued Mappings*, Dordrecht: Springer, 1999.

[13] S. Kakutani, *A generalization of Brouwer’s fixed point theorem*, Duke Mathematical Journal, 8 (1941), pp. 457-459.

[14] T.C. Lim and H.K. Xu, *Fixed point theorems for assymptoticaly nonexpansive mapping*, Nonlinear Analysis: Theory, Methods & Applications, 22 (1994), pp. 1345-1355.

[15] J.T. Markin, *Continuous dependence of fixed point sets*, Proc. Am. Math. Soc., 38 (1973), pp. 545-547.

[16] Jr. S. B. Nadler, *Multivalued contraction mappings*, Pacific J. Math., 30 (1969), pp. 475-488.

[17] J.F. Nash, *Equilibrium points in $n$-person games*, Proceedings of the National Academy of Sciences of the United States of America, 36 (1950), pp. 48-49.

[18] J.F. Nash, *Non-coperative games, Annals of Mathematics*, Second series, 54 (1951), pp. 286-295.

[19] B. Panyanak, *Mann and Ishikawa iteration processes for multivalued mappings in Banach Spaces*, Comput. Math. Appl. 54 (2007), pp. 872-877.

[20] N. Petrot, K. Wattanawitoon, and P. Kumam, *A hybrid projection method for generalized mixed equilibrium problems and fixed point problems in Banach spaces*, Nonlinear Analysis. Hybrid Systems, 4 (2010), pp 631-643.

[21] X. Qin, S.Y. Cho, and S.M. Kang, *Convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications*, Journal of Computational and Applied Mathematics, 233 (2009), pp 231-240.

[22] X. Qin, Y.J. Cho, S.M. Kang, and H. Zhou, *Convergence of a modified Halpern-type iteration algorithm for quasi-$phi$-nonexpansive mappings*, Applied Mathematics Letters, 22 (2009), pp. 1051-1055.

[23] T.M.M. Sow, N. Djitte, and C.E. Chidume, *A path convergence theorem and construction of fixed points for nonexpansive mappings in certain Banach spaces*, Carpathian J.Math, 32 (2016), pp. 217-226.

[24] S. Shoham, *Iterative methods for solving optimization problems*, Technion-Isreal Institute of Technology, Haifa, 2012.

[25] H.K. Xu, *A variable Krasnoselskii-Mann algorithm and the multiple set split feasiblity problem*, Inverse Problem, 26 (2006), pp. 2021-2034.

[26] H.K. Xu, *Iterative algorithms for nonlinear operators*, J. London Math. Soc., 66 (2002), pp. 240-256.

[27] H.K. Xu, *Iterative methods for the split feasiblity problem in infinite-dimensional Hilbert spaces*, Inverse Problem, 26 (2010), pp. 1-17.