STRONG CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEMS AND FIXED POINT PROBLEMS OF STRICT PSEUDO-CONTRACTION MAPPINGS

Volume 2, Issue 2, pp 78-91 http://dx.doi.org/10.22436/jnsa.002.02.02

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1796 Downloads
• 3056 Views

Authors

LIANG CAI ZHAO - Department of Mathematics, Yibin University, Yibin, Sichuan 644000, China. SHIH-SEN CHANG - Department of Mathematics, Sichuan University, Chengdu 610064, P. R. China.

Abstract

The purpose of this paper is to introduce an iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a k−strict pseudo-contraction non-self mapping in Hilbert space. By the viscosity approximation algorithms, under suitable conditions , some strong convergence theorems for approximating to this common elements are proved. The results presented in the paper extend and improve some recent results of Marino and Xu [G.Marino,H.K.Xu, Weak and strong convergence theorems for k−strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007) 336–349], Zhou [H.Zhou, Convergence theorems of fixed Points for k−strict pseudo-contractions in Hilbert spaces, Nonlinear Anal. 69 (2008) 456–462], Takahashi and Takahashi [S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506– 515], Ceng,Homidan,etc [L. C. Ceng, S.A.Homidan, Q.H.Ansari, J. C. Yao, An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings, J. Comput. Appl. Math. 223 (2009) 967–974].

Share and Cite

Keywords

• Equilibrium problem
• strict pseudo-contraction mapping
• fixed point
• strong convergence theorem.

MSC

•  47H05
•  47H09
•  47H10

References

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


• [2] F. E. Browder, Fixed point theorems for noncompact mappings in Hilbert spaces, Proc. Natl. Acad. Sci. USA, 53 (1965), 1272–1276.


• [3] F. E. Browder, W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal. Appl. , 20 (1967), 197–228.


• [4] L. C. Ceng, S. A.Homidan, Q. H. Ansari, J. C. Yao, An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings, J. Comput. Appl. Math., 223 (2009), 967–974.


• [5] S. S. Chang, Some problems and results in the study of nonlinear analysis, Nonlinear Anal., 30 (1997), 4197–4208.


• [6] P. L. Combettes, A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117–136.


• [7] S. D. Flam, A. S. Antipin, Equilibrium programming using proximal-like algorithms, Math. Program. , 78 (1997), 29–41.


• [8] F. Flores-Bazan, Existence theory for finite-dimensional pseudomonotone equilibrium problems, Acta. Appl. Math. , 77 (2003), 249–297.


• [9] N. Hadjisavvas, S. Schaible, From scalar to vector equilibrium problems in the quasimonotone case, J. Optim. Theory Appl., 96 (1998), 297–309.


• [10] H. Iiduka, W. Takahashi , Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal., 61 (2005), 341–350.


• [11] T. H. Kim, H. K. Xu, Strong convergence of modified Mann iteration, Nonlinear Anal., 61 (2005), 51–60.


• [12] W. R. Mann, Mean value methods in iterations, Proc. Amer. Math. Soc., 4 (1953), 506–510.


• [13] G. Marino, H. K. Xu, Weak and strong convergence theorems for k−strict pseudocontractions in Hilbert spaces, J. Math. Anal. Appl., 329 (2007), 336–349.


• [14] A. Moudafi, Viscosity approximation methods for fixed points problems, J. Math. Anal. Appl. , 241 (2000), 46–55.


• [15] K. Nakajo, W. Takahashi , Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups , J. Math. Anal. Appl., 279 (2003), 372–379.


• [16] Z. Opial , Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 595–597.


• [17] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. , 67 (1979), 274–276.


• [18] B. E. Rhoades, Fixed point iterations using infinite matrices, Trans. Amer. Math. Soc. , 196 (1974), 162–176.


• [19] T. Suzuki, Strong convergence of Krasnoselskii and Mann’s type sequences for oneparameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl. , 305 (2005), 227- 239.


• [20] A. Tada, W. Takahashi, Strong convergence theorem for an equilibrium problem and a nonexpansive mapping, in: W. Takahashi, T. Tanaka(Eds.), Nonlinear Analysis and Convex Analysis, 609–617, Yokohama Publishers, Yokohama (2006)


• [21] S. Takahashi, W. Takahashi , Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., 331 (2007), 506–515.


• [22] W. Takahashi, M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417–428.


• [23] R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math., 58 (1992), 486–491.


• [24] H. K. Xu , Iterative algorithms for nonlinear operators , J. London Math. Soc. , 66 (2002), 240–256.


• [25] H. Zhou , Convergence theorems of fixed Points for k−strict pseudo-contractions in Hilbert spaces, Nonlinear Anal., 69 (2008), 456–462.