Fixed Point Theorems for a Demicontractive Mapping and Equilibrium Problems in Hilbert Spaces

Wongvisarut Khuangsatung, Sarawut Suwannaut

Abstract


In this research, we introduce some properties of demicontractive mapping and the combination of equilibrium problem. Then, we prove a strong convergence for the iterative sequence converging to a common element of fixed point set of demicontractive mapping and a common solution of equilibrium problems. Finally, we give a numerical example for the main theorem to support our results.


Keywords


The combination of equilibrium problem; Fixed point; Demicontractive mapping

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References


E. Blum and W. Oettli, From optimization and variational inequilities to equilibrium problems, Mathematics Students 63(14) (1994), 123 – 145.

A. Bnouhachem, A hybrid iterative method for a combination of equilibria problem, a combination of variational inequality problems and a hierarchical fixed point problem. Fixed Point Theory and Applications 2014 (2014), Article number 163, DOI: 10.1186/1687-1812-2014-163.

P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, Journal of Nonlinear and Convex Analysis 6(1) (2005), 117 – 136, https://www.ljll.math.upmc.fr/plc/jnca1.pdf.

S. Iemoto and W. Takahashi, Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space, Nonlinear Analysis: Theory, Methods & Applications 71 (2009), 2082 – 2089, DOI: 10.1016/j.na.2009.03.064.

A. Kangtunyakarn, A new iterative scheme for fixed point problems of infinite family of (kappa_i)-pseudo contractive mappings, equilibrium problem, variational inequality problems, Journal of Global Optimization 56, 1543 – 1562, DOI: 10.1007/s10898-012-9925-0.

A. Kangtunyakarn, Convergence theorem of (kappa)-strictly pseudo-contractive mapping and a modification of generalized equilibrium problems, Fixed Point Theory and Applications 2012 (2012), Article number 89, DOI: 10.1186/1687-1812-2012-89.

W. Khuangsatung and A. Kangtunyakarn, Algorithm of a new variational inclusion problem and strictly pseudo nonspreading mapping with application, Fixed Point Theory and Applications 2014 (2014), Article number 209, DOI: 10.1186/1687-1812-2014-209.

F. Kohsaka and W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Archiv der Mathematik 91 (2008), 166 – 177, DOI: 10.1007/s00013-008-2545-8.

S. Maruster, Strong convergence of the mann iteration of demicontractive mappings, Applied Mathematical Sciences 9(42) (2015), 2061 – 2068, DOI: 10.12988/ams.2015.5166.

S. Maruster, The solution by iteration of nonlinear equations in hilbert spaces, Proceedings of the American Mathematical Society 63(1) (1977), 69 – 73, DOI: 10.2307/2041067.

C. Mongkolkeha, Y. J. Cho and P. Kumam, Convergence theorems for k-demicontractive mappings in Hilbert spaces, Mathematical Inequalities and Applications 16(4) (2013), 1065 – 1082, DOI: 10.7153/mia-16-83.

Z. Opial, Weak convergence of the sequence of successive approximation of nonexpansive mappings, Bulletin of the American Mathematical Society 73(1967), 591 – 597, https://projecteuclid.org/download/pdf1/euclid.bams/1183528964.

M. O. Osilike and F. O. Isiogugu,Weak and strong convergence theorems for nonspreading-type mappings in Hilbert spaces, Nonlinear Analysis 74 (2011), 1814 – 1822, DOI: 10.1016/j.na.2010.10.054.

S. Suwannaut and A. Kangtunyakarn, Convergence analysis for the equilibrium problems with numerical results, Fixed Point Theory and Applications 2014 (2014), Article number 167, DOI: 10.1186/1687-1812-2014-167.

S. Suwannaut and A. Kangtunyakarn, The combination of the set of solutions of equilibrium problem for convergence theorem of the set of fixed points of strictly pseudo-contractive mappings and variational inequalities problem, Fixed point Theory and Applications 2013 (2013), Article number 291, DOI: 10.1186/1687-1812-2013-291.

W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama (2000), http://www.ybook.co.jp/nonlinear.htm.

H. K. Xu, An iterative approach to quadric optimization, Journal of Optimization Theory and Applications 116 (2003), 659 – 678, DOI: 10.1023/A:1023073621589.




DOI: http://dx.doi.org/10.26713%2Fcma.v11i2.1237

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