Algorithms for solving generalized equilibrium problems and fixed points of nonexpansive semigroups in Hilbert spaces

References

• Atsushiba , S and Takahashi , W . 1997 . Approximation common fixed points of nonexpansive semigroups by the Mann iteration process . Ann. Univ. Mariae Curie-Sklodowska , 51 : 1 – 16 .
   Google Scholar
• Blum , E and Oettli , W . 1994 . From optimization and variational inequalities to equilibrium problems . Math. Stud. , 63 : 123 – 145 .
   Google Scholar
• Browder , FE . 1965 . Nonexpansive nonlinear operators in a Banach space . Proc. Nat. Acad. Sci. USA , 54 : 1041 – 1044 .
   PubMed Web of Science ®Google Scholar
• Bruck , RE . 1981 . On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces . Israel J. Math. , 38 : 304 – 314 .
   Web of Science ®Google Scholar
• Chen , R and He , H . 2007 . Viscosity approximation of common fixed points of nonexpansive semigroups in Banach space . Appl. Math. Lett. , 20 : 751 – 757 .
   Web of Science ®Google Scholar
• Chen , R and Song , Y . 2007 . Convergence to common fixed point of nonexpansive semigroups . J. Comput. Appl. Math. , 200 : 566 – 575 .
   Web of Science ®Google Scholar
• Cholamjiak , P . 2010 . Strong convergence of projection algorithms for a family of relatively quasi-nonexpansive mappings and an equilibrium problem in Banach spaces . Optimization , 60 : 495 – 507 .
   Web of Science ®Google Scholar
• Cholamjiak , P and Suantai , S . 2010 . An iterative method for equilibrium problems and a finite family of relatively nonexpansive mappings in a Banach space . Appl. Math. Comput. , 217 : 3825 – 3831 .
   Web of Science ®Google Scholar
• Cholamjiak , P and Suantai , S . 2010 . Convergence analysis for a system of equilibrium problems and a countable family of relatively quasi-nonexpansive mappings in Banach spaces . Abstr. Appl. Anal. , 2010 Article ID 141376, p. 17
   Web of Science ®Google Scholar
• Cholamjiak , P and Suantai , S . 2011 . Viscosity approximation methods for a nonexpansive semigroup in Banach spaces with gauge functions . J. Glob. Optim. , doi: 10.1007/s10898-011-9756-4. Available at http://www.springerlink.com/content/r63467l887862u51/
   Web of Science ®Google Scholar
• Cholamjiak , P and Suantai , S . 2011 . A hybrid method for a family of relatively quasi-nonexpansive mappings and an equilibrium problem in Banach spaces . J. Glob. Optim. , doi: 10.1007/s10898-011-9743-9. Available at http://www.springerlink.com/content/r272275823256158/
   Web of Science ®Google Scholar
• Combettes , PL and Hirstoaga , SA . 2005 . Equilibrium programming in Hilbert spaces . J. Nonlinear Convex Anal. , 6 : 117 – 136 .
   Web of Science ®Google Scholar
• Day , MM . 1957 . Amenable semigroup . Illinois J. Math. , 1 : 509 – 544 .
   Google Scholar
• Goebel , K and Kirk , WA . 1990 . “ Topics in Metric Fixed Point Theory ” . In Cambridge Studies in Advanced Mathematics , Vol. 28 , Cambridge , , UK : Cambridge University Press .
   Google Scholar
• Halpern , B . 1967 . Fixed points of nonexpanding maps . Bull. Am. Math. Soc. , 73 : 957 – 961 .
   Web of Science ®Google Scholar
• Kumam , P and Katchang , K . 2009 . A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings . Nonlinear Anal. Hybrid Syst. , 3 : 475 – 486 .
   Web of Science ®Google Scholar
• Lau , AT-M . 1973 . Invariant means on almost periodic functions and fixed point properties . Rocky Mountain J. , 3 : 69 – 76 .
   Google Scholar
• Lau , AT-M . 2008 . Invariant means and fixed point properties of semigroup of nonexpansive mappings . Taiwanese J. Math. , 12 : 1525 – 1542 .
   Web of Science ®Google Scholar
• Lau , AT-M , Miyake , H and Takahashi , W . 2007 . Approximation of fixed points for amenable semigroups of nonexpansive mappings in Banach spaces . Nonlinear Anal. TMA , 67 : 1211 – 1225 .
   Web of Science ®Google Scholar
• Lau , AT-M , Shioji , N and Takahashi , W . 1999 . Existence of nonexpansive retractions for amenable semigroups of nonexpansive mappings and nonlinear ergodic theorems in Banach spaces . J. Funct. Anal. , 161 : 62 – 75 .
   Web of Science ®Google Scholar
• Lau , AT-M and Takahashi , W . 1995 . Invariant means and fixed point properties for nonexpansive representations of topological semigroups . Topol. Methods Nonlinear Anal. , 5 : 39 – 57 .
   Google Scholar
• Lim , TC . 2001 . On characterizations of Meir–Keeler contractive maps . Nonlinear Anal. TMA , 46 : 113 – 120 .
   Web of Science ®Google Scholar
• Lin , Q . 2009 . Viscosity approximation to common fixed points of a nonexpansive semigroup with a generalized contraction mappings . Nonlinear Anal. TMA , 71 : 5451 – 5457 .
   Web of Science ®Google Scholar
• Meir , A and Keeler , E . 1969 . A theorem on contraction mappings . J. Math. Anal. Appl. , 28 : 326 – 329 .
   Web of Science ®Google Scholar
• Moudafi , A . 2000 . Viscosity approximation methods for fixed point problems . J. Math. Anal. Appl. , 241 : 46 – 55 .
   Web of Science ®Google Scholar
• Moudafi , A and Théra , M . 1999 . “ Proximal and Dynamical Approaches to Equilibrium Problems ” . In Lecture Notes in Economics and Mathematical Systems , Vol. 477 , 187 – 201 . Berlin : Springer .
   Google Scholar
• Petruşel , A and Yao , J-C . 2008 . Viscosity approximation to common fixed points of families of nonexpansive mappings with generalized contractions mappings . Nonlinear Anal. TMA , 69 : 1100 – 1111 .
   Web of Science ®Google Scholar
• Qin , X , Chang , SS and Cho , YJ . 2010 . Iterative methods for generalized equilibrium problems and fixed point problems with applications . Nonlinear Anal. RWA , 11 : 2963 – 2972 .
   Web of Science ®Google Scholar
• Qin , X , Cho , YJ and Kang , SM . 2009 . Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces . J. Comput. Appl. Math. , 225 : 20 – 30 .
   Web of Science ®Google Scholar
• Reich , S . 1972 . Fixed point of contractive functions . Boll. Unione Mat. Ital. , 5 : 26 – 42 .
   Google Scholar
• Saeidi , S . 2008 . Existence of ergodic retractions for semigroups in Banach spaces . Nonlinear Anal. TMA , 69 : 3417 – 3422 .
   Web of Science ®Google Scholar
• Saeidi , S . 2009 . Iterative algorithms for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of families and semigroups of nonexpansive mappings . Nonlinear Anal. TMA , 70 : 4195 – 4208 .
   Web of Science ®Google Scholar
• Saewan , S and Kumam , P . 2010 . A hybrid iterative scheme for a maximal monotone operator and two countable families of relatively quasi-nonexpansive mappings for generalized mixed equilibrium and variational inequality problems . Abstr. Appl. Anal. , 2010 Article ID 123027, p. 31
   Google Scholar
• Shehu , Y . 2011 . Strong convergence theorems for an infinite family of quasi-nonexpansive mappings and generalized equilibrium problems and variational inequality problems . Comput. Math. Appl. , 61 : 357 – 366 .
   Web of Science ®Google Scholar
• Shioji , N and Takahashi , W . 1999 . Strong convergence of average approximants for asymptotically nonexpansive mappings in Banach spaces . J. Approx. Theor. , 97 : 53 – 64 .
   Web of Science ®Google Scholar
• Song , Y and Xu , S . 2008 . Strong convergence theorems for nonexpansive semigroup in Banach spaces . J. Math. Anal. Appl. , 338 : 152 – 161 .
   Web of Science ®Google Scholar
• Suzuki , T . 2005 . Strong convergence of Krasnoselskii and Mann's type sequences for one parameter nonexpansive semigroups without Bochner integrals . J. Math. Anal. Appl. , 305 : 227 – 239 .
   Web of Science ®Google Scholar
• Suzuki , T . 2007 . Moudafi's viscosity approximations with Meir–Keeler contractions . J. Math. Anal. Appl. , 325 : 342 – 352 .
   Web of Science ®Google Scholar
• Tada , A and Takahashi , W . 2007 . Weak and strong convergence theorems for nonexpansive mappings and an equilibrium problem . J. Optim. Theor. Appl. , 133 : 359 – 370 .
   Web of Science ®Google Scholar
• Takahashi , W . 1981 . A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space . Proc. Am. Math. Soc. , 81 : 253 – 256 .
   Web of Science ®Google Scholar
• Takahashi , W . 2000 . Nonlinear Function Analysis , Yokohama : Yokohama Publishers .
   Google Scholar
• Takahashi , W . 2009 . Viscosity approximation methods for countable families of nonexpansive mappings in Banach spaces . Nonlinear Anal. TMA , 70 : 719 – 734 .
   Web of Science ®Google Scholar
• Takahashi , S and Takahashi , W . 2007 . Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces . J. Math. Anal. Appl. , 331 : 506 – 515 .
   Web of Science ®Google Scholar
• Takahashi , S and Takahashi , W . 2008 . Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space . Nonlinear Anal. TMA , 69 : 1025 – 1033 .
   Web of Science ®Google Scholar
• Xu , HK . 2002 . Iterative algorithms for nonlinear operators . J. Lond. Math. Soc. , 66 : 240 – 256 .
   Web of Science ®Google Scholar
• Xu , HK . 2004 . Viscosity approximation methods for nonexpansive mappings . J. Math. Anal. Appl. , 298 : 279 – 291 .
   Web of Science ®Google Scholar
• Xu , HK . 2005 . A strong convergence theorem for contraction semigroups in Banach spaces . Bull. Aust. Math. Soc. , 72 : 371 – 379 .
   Web of Science ®Google Scholar
• Yao , Y , Liou , Y-C and Yao , J-C . 2007 . Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings . Fixed Point Theory Appl. , 2007 Article ID 64363, p. 12
   Google Scholar