Fixed Point of a Countable Family of Uniformly Totally Quasi- *Ø* -Asymptotically Nonexpansive Multi-Valued Mappings in Reflexive Banach Spaces with Applications

Author(s)
Yi Li

ABSTRACT

The purpose of this article is to discuss a modified
Halpern-type iteration algorithm for a countable family of uniformly totally
quasi- *?* -asymptotically nonexpansive multi-valued mappings and
establish some strong convergence theorems under certain conditions. We utilize
the theorems to study a modified Halpern-type iterative algorithm for a system
of equilibrium problems. The results improve and extend the corresponding
results of Chang *et al*. (Applied
Mathematics and Computation, 218, 6489-6497).

Cite this paper

Y. Li, "Fixed Point of a Countable Family of Uniformly Totally Quasi-*Ø* -Asymptotically Nonexpansive Multi-Valued Mappings in Reflexive Banach Spaces with Applications," *Applied Mathematics*, Vol. 4 No. 9, 2013, pp. 6-12. doi: 10.4236/am.2013.49A002.

Y. Li, "Fixed Point of a Countable Family of Uniformly Totally Quasi-

References

[1] I. Cioranescu, “Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems,” Kluwer Academic, Dordrecht, 1990. doi:10.1007/978-94-009-2121-4

[2] W. R. Mann, “Mean Value Methods in Iteration,” Proceedings of the American Mathematical Society, Vol. 4, No. 3, 1953, pp. 506-510. doi:10.1090/S0002-9939-1953-0054846-3

[3] A. Genel and J. Lindenstrauss, “An Example Concerning Fixed Points,” Israel Journal of Mathematics, Vol. 22, No. 1, 1975, pp. 81-86.

[4] B. Halpren, “Fixed Points of Nonexpansive Maps,” Bulletin of the American Mathematical Society, Vol. 73, 1967, pp. 957-961. doi:10.1090/S0002-9904-1967-11864-0

[5] K. Nakajo and W. Takahashi, “Strong Convergence Theorems for Nonexpansive Mappings and Nonexpansive Semigroups,” Journal of Mathematical Analysis and Applications, Vol. 279, No. 2, 2003, pp. 372-379. doi:10.1016/S0022-247X(02)00458-4

[6] K. Aoyama and Y. Kimura, “Strong Convergence Theorems for Strongly Nonexpansive Sequences,” Applied Mathematics and Computation, Vol. 217, No.19 , 2011, pp. 7537-7545.

doi:10.1016/j.amc.2011.01.092

[7] S. S. Chang, H. W. J. Lee, C. K. Chan and W. B. Zhang, “A Modified Halpern-Type Iteration Algorithm for Totally Quasi- -Asymptotically Nonexpansive Mappings with Applications,” Applied Mathematics and Computation, Vol. 218, No. 11, 2012, pp. 6489-6497. doi:10.1016/j.amc.2011.12.019

[8] S. S. Chang, L. Yang and J. A. Liu, “Strong Convergence Theorem for Nonexpansive Semi-Groups in Banach Space,” Applied Mathematics and Mechanics, Vol. 28, No. 10, 2007, pp. 1287-1297. doi:10.1007/s10483-007-1002-x

[9] C. E. Chidume and E. U. Ofoedu, “Approximation of Common Fixed Points for Finite Families of Total Asymptotically Nonexpansive Mappings,” Journal of Mathematical Analysis and Applications, Vol. 333, No. 1, 2007, pp. 128-141. doi:10.1016/j.jmaa.2006.09.023

[10] S. Matsushita and W. Takahashi, “Weak and Strong Convergence Theorems for Relatively Nonexpansive Mappings in a Banach Space,” Fixed Point Theory and Applications, Vol. 2004, 2004, pp. 37-47. doi:10.1155/S1687182004310089

[11] S. Matsushita and W. Takahashi, “An Iterative Algorithm for Relatively Nonexpansive Mappings by Hybrid Method and Applications,” Proceedings of the 3rd International Conference on Nonlinear Analysis and Convex Analysis, Tokyo, 2004, pp. 305-313.

[12] S. Matsushita and W. Takahashi, “A Strong Convergence Theorem for Relatively Nonexpansive Mappings in a Banach Space,” The Journal of Approximation Theory, Vol. 134, No. 2, 2005, pp. 257-266. doi:10.1016/j.jat.2005.02.007

[13] X. L. Qin, Y. J. Cho, S. M. Kang and H. Y. Zhou, “Convergence of a Modified Halpern-Type Iterative Algorithm for Quasi- -Nonexpansive Mappings,” Applied Mathematics Letters, Vol. 22, No. 7, 2009, pp. 1051-1055. doi:10.1016/j.aml.2009.01.015

[14] Y. Song, “New Strong Convergence Theorems for Nonexpansive Nonself-Mappings without Boundary Conditions,” Computers and Mathematics with Applications, Vol. 56, No. 6, 2008, pp. 1473-1478. doi:10.1016/j.camwa.2008.03.004

[15] Z. M. Wang, Y. F. Su, D. X. Wang and Y. C. Dong, “A Modified Halpern-Type Iteration Algorithm for a Family of Hemi-Relative Nonexpansive Mappings and Systems of Equilibrium Problems in Banach Spaces,” The Journal of Computational and Applied Mathematics, Vol. 235, No. 8, 2011, pp. 2364-2371. doi:10.1016/j.cam.2010.10.036

[16] Y. I. Alber, “Metric and Generalized Projection Operators in Banach Spaces: Properties and Applications,” In: A. G. Kartosator, Ed., Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Dekker, New York, 1996, pp. 15-50.

[17] S. S. Chang, C. K. Chan and H. W. J. Lee, “Modified Block Iterative Algorithm for Quasi- -Asymptotically Nonexpansive Mappings and Equilibrium Problem in Banach Spaces,” Applied Mathematics and Computation, Vol. 217, No. 18, 2011, pp. 7520-7530. doi:10.1016/j.amc.2011.02.060

[18] E. Blum and W. Oettli, “From Optimization and Variational Inequalities to Equilibrium Problems,” Mathematical Studies, Vol. 63, No. 1/4, 1994, pp. 123-145.

[1] I. Cioranescu, “Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems,” Kluwer Academic, Dordrecht, 1990. doi:10.1007/978-94-009-2121-4

[2] W. R. Mann, “Mean Value Methods in Iteration,” Proceedings of the American Mathematical Society, Vol. 4, No. 3, 1953, pp. 506-510. doi:10.1090/S0002-9939-1953-0054846-3

[3] A. Genel and J. Lindenstrauss, “An Example Concerning Fixed Points,” Israel Journal of Mathematics, Vol. 22, No. 1, 1975, pp. 81-86.

[4] B. Halpren, “Fixed Points of Nonexpansive Maps,” Bulletin of the American Mathematical Society, Vol. 73, 1967, pp. 957-961. doi:10.1090/S0002-9904-1967-11864-0

[5] K. Nakajo and W. Takahashi, “Strong Convergence Theorems for Nonexpansive Mappings and Nonexpansive Semigroups,” Journal of Mathematical Analysis and Applications, Vol. 279, No. 2, 2003, pp. 372-379. doi:10.1016/S0022-247X(02)00458-4

[6] K. Aoyama and Y. Kimura, “Strong Convergence Theorems for Strongly Nonexpansive Sequences,” Applied Mathematics and Computation, Vol. 217, No.19 , 2011, pp. 7537-7545.

doi:10.1016/j.amc.2011.01.092

[7] S. S. Chang, H. W. J. Lee, C. K. Chan and W. B. Zhang, “A Modified Halpern-Type Iteration Algorithm for Totally Quasi- -Asymptotically Nonexpansive Mappings with Applications,” Applied Mathematics and Computation, Vol. 218, No. 11, 2012, pp. 6489-6497. doi:10.1016/j.amc.2011.12.019

[8] S. S. Chang, L. Yang and J. A. Liu, “Strong Convergence Theorem for Nonexpansive Semi-Groups in Banach Space,” Applied Mathematics and Mechanics, Vol. 28, No. 10, 2007, pp. 1287-1297. doi:10.1007/s10483-007-1002-x

[9] C. E. Chidume and E. U. Ofoedu, “Approximation of Common Fixed Points for Finite Families of Total Asymptotically Nonexpansive Mappings,” Journal of Mathematical Analysis and Applications, Vol. 333, No. 1, 2007, pp. 128-141. doi:10.1016/j.jmaa.2006.09.023

[10] S. Matsushita and W. Takahashi, “Weak and Strong Convergence Theorems for Relatively Nonexpansive Mappings in a Banach Space,” Fixed Point Theory and Applications, Vol. 2004, 2004, pp. 37-47. doi:10.1155/S1687182004310089

[11] S. Matsushita and W. Takahashi, “An Iterative Algorithm for Relatively Nonexpansive Mappings by Hybrid Method and Applications,” Proceedings of the 3rd International Conference on Nonlinear Analysis and Convex Analysis, Tokyo, 2004, pp. 305-313.

[12] S. Matsushita and W. Takahashi, “A Strong Convergence Theorem for Relatively Nonexpansive Mappings in a Banach Space,” The Journal of Approximation Theory, Vol. 134, No. 2, 2005, pp. 257-266. doi:10.1016/j.jat.2005.02.007

[13] X. L. Qin, Y. J. Cho, S. M. Kang and H. Y. Zhou, “Convergence of a Modified Halpern-Type Iterative Algorithm for Quasi- -Nonexpansive Mappings,” Applied Mathematics Letters, Vol. 22, No. 7, 2009, pp. 1051-1055. doi:10.1016/j.aml.2009.01.015

[14] Y. Song, “New Strong Convergence Theorems for Nonexpansive Nonself-Mappings without Boundary Conditions,” Computers and Mathematics with Applications, Vol. 56, No. 6, 2008, pp. 1473-1478. doi:10.1016/j.camwa.2008.03.004

[15] Z. M. Wang, Y. F. Su, D. X. Wang and Y. C. Dong, “A Modified Halpern-Type Iteration Algorithm for a Family of Hemi-Relative Nonexpansive Mappings and Systems of Equilibrium Problems in Banach Spaces,” The Journal of Computational and Applied Mathematics, Vol. 235, No. 8, 2011, pp. 2364-2371. doi:10.1016/j.cam.2010.10.036

[16] Y. I. Alber, “Metric and Generalized Projection Operators in Banach Spaces: Properties and Applications,” In: A. G. Kartosator, Ed., Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Dekker, New York, 1996, pp. 15-50.

[17] S. S. Chang, C. K. Chan and H. W. J. Lee, “Modified Block Iterative Algorithm for Quasi- -Asymptotically Nonexpansive Mappings and Equilibrium Problem in Banach Spaces,” Applied Mathematics and Computation, Vol. 217, No. 18, 2011, pp. 7520-7530. doi:10.1016/j.amc.2011.02.060

[18] E. Blum and W. Oettli, “From Optimization and Variational Inequalities to Equilibrium Problems,” Mathematical Studies, Vol. 63, No. 1/4, 1994, pp. 123-145.