Convergence Theorems for k-Strictly Pseudononspreading Multivalued in Hilbert Spaces

We introduce a k-strictly pseudononspreading multivalued in Hilbert spaces more general than the class of nonspreading multivalued. We establish some weak convergence theorems of the sequences generated by our iterative process. Some new iterative sequences for finding a common element of the set of solutions for equilibrium problem was introduced. The results improve and extend the corresponding results of Osilike Isiogugu [1] (Nonlinear Anal.74 (2011)) and others.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Liu, H. and Li, Y. (2014) Convergence Theorems for k-Strictly Pseudononspreading Multivalued in Hilbert Spaces. Advances in Pure Mathematics, 4, 317-323. doi: 10.4236/apm.2014.47042.

 [1] Osilike, M.O. and Isiogugu, F.O. (2011) Weak and Strong Convergence Theorems for Nonspreading-Type Mappings in Hilbert Spaces. Nonlinear Analysis: Theory, Methods & Applications, 74, 1814-1822. http://dx.doi.org/10.1016/j.na.2010.10.054 [2] Song, Y. and Wang, H. (2009) Convergence of Iterative Algorithms for Multivalued Mappings in Banach Spaces. Nonlinear Analysis, 70, 1547-1556. http://dx.doi.org/10.1016/j.na.2008.02.034 [3] Shahzad, N. and Zegeye, H. (2009) On Mann and Ishikawa Iteration Schemes for Multivalued Maps in Banach Space. Nonlinear Analysis, 71, 838-844. http://dx.doi.org/10.1016/j.na.2008.10.112 [4] Eslamian, M. and Abkar, A. (2011) One-Step Iterative Process for a Finite Family of Multivalued Mappings. Mathematical and Computer Modelling, 54, 105-111. http://dx.doi.org/10.1016/j.mcm.2011.01.040 [5] Takahashi, W. and Toyoda, M. (2003) Weak Convergence Theorems for Nonexpansive Mappings and Monotone Mappings. Journal of Optimization Theory and Applications, 118, 417-428. http://dx.doi.org/10.1023/A:1025407607560 [6] Kohsaka, F. and Takahashi, W. (2008) Fixed Point Theorems for a Class of Nonlinear Mappings Relate to Maximal Monotone Operators in Banach Spaces. Archiv der Mathematik (Basel), 91, 166-177. http://dx.doi.org/10.1007/s00013-008-2545-8 [7] Kohsaka, F. and Takahashi, W. (2008) Existence and Approximation of Fixed Points of Firmly Nonexpansive-Type Mappings in Banach Spaces. SIAM Journal on Optimization, 19, 824-835. http://dx.doi.org/10.1137/070688717 [8] Iemoto, S. and Takahashi, W. (2009) Approximating Commom Fixed Points of Nonexpansive Mappings and Nonspreading Mappings in a Hilbert Space. Nonlinear Analysis, 71, 2082-2089. http://dx.doi.org/10.1016/j.na.2009.03.064 [9] Blum, E. and Oettli, W. (1994) From Optimization and Variational Inequalities to Equilibrium Problems. The Mathematics Student, 63, 123-145. [10] Combettes, P.L. and Hirstoaga, S.A. (2005) Equilibrium Programming in Hilbert Spaces. Journal of Nonlinear and Convex Analysis, 6, 117-136. [11] Li, X.B. and Li, S.J. (2010) Existence of Solutions for Generalized Vector Quasi-Equilibrium Problems. Optimization Letters, 4, 17-28. http://dx.doi.org/10.1007/s11590-009-0142-9 [12] Giannessi, F., Maugeri, G. and Pardalos, P.M. (2001) Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models. Kluwer Academics Publishers, Dordrecht. [13] Moudafi, A. and Thera, M. (1999) Proximal and Dynamical Approaches to Equilibrium Problems, In: Lecture Note in Economics and Mathematical Systems, Vol. 477, Springer-Verlag, New York, 187-201. [14] Pardalos, P.M., Rassias, T.M. and Khan, A.A. (2010) Nonlinear Analysis and Variational Problems. Springer, Berlin. http://dx.doi.org/10.1007/978-1-4419-0158-3 [15] Ceng, L.C., Al-Homidan, S., Ansari, Q.H. and Yao, J.C. (2009) An Iterative Scheme for Equilibrium Problems and Fixed Point Problems of Strict Pseudo-Contraction Mappings. Journal of Computational and Applied Mathematics, 223, 967-974. http://dx.doi.org/10.1016/j.cam.2008.03.032