Hybrid Proximal-Type and Hybrid Shrinking Projection Algorithms for Equilibrium Problems, Maximal Monotone Operators, and Relatively Nonexpansive Mappings

REFERENCES

• Ya. I. Alber ( 1996 ). Metric and generalized projection operators in Banach spaces: properties and applications . In: Theory and Applications of Nonlinear Operators of Monotonicand Accretive Type . ( A.G. Kartsatos , ed.), New York , Marcel Dekker , pp. 15 – 50 .
   Google Scholar
• Ya. I. Alber and S. Guerre-Delabriere ( 2001 ). On the projection methods for fixed point problems . Analysis 21 : 17 – 39 .
   Google Scholar
• Ya. I. Alber and S. Reich ( 1994 ). An iterative method for solving a class of nonlinear operator equations in Banach spaces . Panamer. Math. J. 4 ( 2 ): 39 – 54 .
   Google Scholar
• Q.H. Ansari , N.C. Wong , and J.C. Yao ( 1999 ). The existence of nonlinear inequalities . Appl. Math. Lett. 12 ( 5 ): 89 – 92 .
   Web of Science ®Google Scholar
• M. Bianchi and S. Schaible ( 2004 ). Equilibrium problems under generalized convexity and generalized monotonicity . J. Global Optim. 30 : 121 – 134 .
   Web of Science ®Google Scholar
• E. Blum and W. Oettli ( 1994 ). From optimization and variational inequalities to equilibrium problems . Math. Student 63 : 123 – 145 .
   Google Scholar
• L.C. Ceng , T.C. Lai , and J.C. Yao ( 2008 ). Approximate proximal algorithms for generalized variational inequalities with paramonotonicity and pseudomonotonicity . Comput. Math. Appl. 55 : 1262 – 1269 .
   Web of Science ®Google Scholar
• L.C. Ceng and J.C. Yao ( 2008 ). Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinite many nonexpansive mappings . Appl. Math. Computat. 198 : 729 – 741 .
   Web of Science ®Google Scholar
• L.C. Ceng and J.C. Yao ( 2008 ). Approximate proximal algorithms for generalized variational inequalities with pseudomonotone multifunctions . J. Comput. Appl. Math. 213 : 423 – 438 .
   Web of Science ®Google Scholar
• L.C. Ceng and J.C. Yao ( 2008 ). A hybrid iterative scheme for mixed equilibrium problems and fixed point problems . J. Comput. Appl. Math. 214 : 186 – 201 .
   Web of Science ®Google Scholar
• I. Cioranescu ( 1990 ). Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems . Kluwer Academic Publishers , Dordrecht , Germany .
   Google Scholar
• P.L. Combettes and S.A. Hirstoaga ( 2005 ). Equilibrium programming in Hilbert spaces . J. Nonlinear Convex Anal. 6 : 117 – 136 .
   Web of Science ®Google Scholar
• F. Flores-Bazán ( 2000 ). Existence theorems for generalized noncoercive equilibrium problems: the quasi-convex case . SIAM J. Optm. 11 : 675 – 690 .
   Web of Science ®Google Scholar
• F. Flores-Bazán ( 2003 ). Existence theory for finite-dimensional pseudomonotone equilibrium problems . Acta Appl. Math. 77 : 249 – 297 .
   Web of Science ®Google Scholar
• O. Güler ( 1991 ). On the convergence of the proximal point algorithm for convex minimization . SIAM J. Control Optim. 29 : 403 – 419 .
   Web of Science ®Google Scholar
• S. Kamimura and W. Takahashi ( 2000 ). Approximating solutions of maximal monotone operators in Hilbert spaces . J. Approx. Theory 106 : 226 – 240 .
   Web of Science ®Google Scholar
• S. Kamimura and W. Takahashi ( 2003 ). Strong convergence of a proximal-type algorithm in a Banach space . SIAM J. Optim. 13 : 938 – 945 .
   Web of Science ®Google Scholar
• A. Moudafi ( 2003 ). Second-order differential proximal methods for equilibrium problems . J. Inequal. Pure Appl. Math. 4 : Article 18 .
   Google Scholar
• B. Martinet ( 1970 ). Regularisation d'inequations variationnelles par approximations successives . Rev. Franc. Inform. Rech. Oper. 4 : 154 – 159 .
   Google Scholar
• S. Matsushita and W. Takahashi ( 2005 ). A strong convergence theorem for relatively nonexpansive mappings in a Banach space . J. Approx. Theory 134 : 257 – 266 .
   Web of Science ®Google Scholar
• S. Reich ( 1996 ). A weak convergence theorem for the alternating method with Bergman distance . In: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type . ( A.G. Kartsatos , ed.), Marcel Dekker , New York , pp. 313 – 318 .
   Google Scholar
• R.T. Rockafellar (1976). Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14:877–898.
   Web of Science ®Google Scholar
• M.V. Solodov and B.F. Svaiter ( 2000 ). Forcing strong convergence of proximal point iterations in a Hilbert space . Math. Prog. 87 : 189 – 202 .
   Web of Science ®Google Scholar
• A. Tada and W. Takahashi ( 2007 ). Strong convergence theorem for an equilibrium problem and a nonexpansive mapping . In: Nonlinear Analysis and Convex Analysis . ( W. Takahashi and T. Tanaka , eds.), Yokohama Publishers , Yokohama , Japan , pp. 609 – 617 .
   Google Scholar
• A. Tada and W. Takahashi ( 2007 ). Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem . J. Optim. Theory Appl. 133 : 359 – 370 .
   Web of Science ®Google Scholar
• S. Takahashi and W. Takahashi ( 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
• W. Takahashi , Y. Takeuchi , and R. Kubota ( 2008 ). Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces . J. Math. Anal. Appl. 341 : 276 – 286 .
   Web of Science ®Google Scholar
• W. Takahashi and K. Zembayashi ( 2008 ). Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings . Fixed Point Theory Appl. Article ID 528476, 11 pages .
   Google Scholar
• W. Takahashi and K. Zembayashi ( 2008 ). Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces . Nonlinear Anal. doi: doi:10.1016/j.na.2007.11.031 .
   Google Scholar
• H.K. Xu ( 1991 ). Inequalities in Banach spaces with applications . Nonlinear Anal. 16 : 1127 – 1138 .
   Web of Science ®Google Scholar
• L.C. Zeng and J.C. Yao ( 2007 ). An inexact proximal-type algorithm in Banach spaces . J. Optim. Theory Appl. 135 ( 1 ): 145 – 161 .
   Google Scholar
• X. Qin and Y. Su ( 2007 ). Strong convergence theorems for relatively nonexpansive mappings in a Banach space . Nonlinear Anal. 67 : 1958 – 1965 .
   Web of Science ®Google Scholar