# Iterative algorithm for strongly continuous semigroup of Lipschitz pseudocontraction mappings

Volume 9, Issue 4, pp 1424--1431
• 1511 Views

### Authors

Liping Yang - School of Applied Mathematics, Guangdong University of Technology, Guangzhou 510520, Guangdong, China.

### Abstract

In this paper, an implicit iterative process is considered for strongly continuous semigroup of Lipschitz pseudocontraction mappings. Weak and strong convergence theorems for common fixed points of strongly continuous semigroup of Lipschitz pseudocontraction mappings are established in a real Banach space.

### Share and Cite

##### ISRP Style

Liping Yang, Iterative algorithm for strongly continuous semigroup of Lipschitz pseudocontraction mappings, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 4, 1424--1431

##### AMA Style

Yang Liping, Iterative algorithm for strongly continuous semigroup of Lipschitz pseudocontraction mappings. J. Nonlinear Sci. Appl. (2016); 9(4):1424--1431

##### Chicago/Turabian Style

Yang, Liping. "Iterative algorithm for strongly continuous semigroup of Lipschitz pseudocontraction mappings." Journal of Nonlinear Sciences and Applications, 9, no. 4 (2016): 1424--1431

### Keywords

• Semigroup of pseudocontraction mappings
• uniformly convex Banach spaces
• Opial's condition
• variational inequality.

•  47H10
•  47J25
•  49J40
•  65K10

### References

• [1] V. Barbu, T. Precupanu, Convexity and optimization in Banach spaces, Editura Academiei, Bucharest (1978)

• [2] C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103-120.

• [3] C. E. Chidume, Global iteration schemes for strongly pseudo-contractive maps, Proc. Amer. Math. Soc., 126 (1998), 2641-2649.

• [4] A. Cochocki, R. Unbehauen , Neural Networks for Optimization and Signal Processing, Wiley, New York (1993)

• [5] K. Deimling, Zeros of accretive operators, Manuscripta Math., 13 (1974), 365-374.

• [6] R. Dewangan, B. S. Thakur, M. Postolache , Strong convergence of asymptotically pseudocontractive semigroup by viscosity iteration, Appl. Math. Comput., 248 (2014), 160-168.

• [7] F. Facchinei, J. Pang, Finite-Dimensional Variational Inequalities and Nonlinear Complementarity Problems, Springer-Verlag, New York (2003)

• [8] P. Harker, J. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications, Math. Program, 48 (1990), 161-220.

• [9] P. Hartman, G. Stampacchia, On some nonlinear elliptic differential functional equations, Acta Math., 115 (1996), 271-310.

• [10] G. Marino, H. X. Xu , A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 318 (2006), 43-52.

• [11] R. H. Martin, Differential equations on closed subsets of a Banach space, Trans. Amer. Math. Soc., 179 (1973), 399-414.

• [12] C. Morales, Pseudocontractive mappings and Leray Schauder boundary condition, Comment. Math. Univ. Carolin., 20 (1979), 745-746.

• [13] C. H. Morales, J. S. Jung, Convergence of paths for pseudocontractive mappings in Banach spaces , Proc. Amer. Math. Soc., 128 (2000), 3411-3419.

• [14] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 591-597.

• [15] M. O. Osilike, Iterative solutions of nonlinear $\phi$-strongly accretive operator equations in arbitrary Banach spaces, Nonlinear Anal. Ser. A: Theory Methods, 36 (1999), 1-9.

• [16] W. Takahashi , Nonlinear Functional Analysis-Fixed Point Theory and Its Applications, Yokohama Publishers Inc., Yokohama (2000)

• [17] B. B. Thakur, R. Dewangan, M. Postolache, Strong convergence of new iteration process for a strongly continuous semigroup of asymptotically pseudocontractive mappings, Numer. Funct. Anal. Optim., 34 (2013), 1418-1431.

• [18] B. S. Thakur, R. Dewangan, M. Postolache , General composite implicit iteration process for a finite family of asymptotically pseudocontractive mappings, Fixed Point Theory Appl., 2014 (2014), 15 pages.

• [19] H. K. Xu, R. G. Ori , An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal. Optim., 22 (2001), 767-773.

• [20] Y. Yao, M. Postolache, S. M. Kang , Strong convergence of approximated iterations for asymptotically pseudocon-tractive mappings, Fixed Point Theory Appl., 2014 (2014), 13 pages.

• [21] D. Youla , On deterministic convergence of iterations of related projection operators, J. Vis. Commun. Image Represent., 1 (1990), 12-20.

• [22] S. S. Zhang, Convergence theorem of common fixed points for Lipschitzian pseudo-contraction semigroups in Banach spaces, Appl. Math. Mech., 30 (2009), 145-152.