# American Institute of Mathematical Sciences

2014, 4(4): 287-293. doi: 10.3934/naco.2014.4.287

## Strong convergence of an implicit iteration process for a finite family of Lipschitz $\phi -$uniformly pseudocontractive mappings in Banach spaces

 1 Department of Mathematics, Kyungsung University, Busan 608-736, South Korea 2 Department of Mathematics, Lahore Leads University, Lahore, Pakistan

Received  September 2013 Revised  September 2014 Published  December 2014

The purpose of this paper is to establish a strong convergence of an implicit iteration process to a common fixed point for a finite family of Lipschitz $\phi-$uniformly pseudocontractive mappings in real Banach spaces. The results presented here improve and extend the corresponding results in [2, 4, 6] and the consecutive remark explains in details about the facts.
Citation: B. S. Lee, Arif Rafiq. Strong convergence of an implicit iteration process for a finite family of Lipschitz $\phi -$uniformly pseudocontractive mappings in Banach spaces. Numerical Algebra, Control and Optimization, 2014, 4 (4) : 287-293. doi: 10.3934/naco.2014.4.287
##### References:
 [1] F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal., Appl., 20 (1967), 197-228. [2] R. Chen, P. K. Lin and Y. Dong, An approximation method for strictly pseudocontractive mappings, Nonlinear Analysis (TMA), 64 (2006), 2527-2535. doi: 10.1016/j.na.2005.08.031. [3] C. Moore and B. V. C. Nnoli, Iterative solution of nonlinear equations involving set-valued uniformly accretive operators, Computers Math. Applic, 42 (2001), 131-140. doi: 10.1016/S0898-1221(01)00138-9. [4] M. O. Osilike, Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps, J. Math. Anal., 294 (2004), 73-81. doi: 10.1016/j.jmaa.2004.01.038. [5] H. K. Xu, Inequality in Banach spaces with applications, Nonlinear Anal., 16 (1991), 1127-1138. doi: 10.1016/0362-546X(91)90200-K. [6] H. K. Xu and R. G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal., Optim., 22 (2001), 767-773. doi: 10.1081/NFA-100105317.

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##### References:
 [1] F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal., Appl., 20 (1967), 197-228. [2] R. Chen, P. K. Lin and Y. Dong, An approximation method for strictly pseudocontractive mappings, Nonlinear Analysis (TMA), 64 (2006), 2527-2535. doi: 10.1016/j.na.2005.08.031. [3] C. Moore and B. V. C. Nnoli, Iterative solution of nonlinear equations involving set-valued uniformly accretive operators, Computers Math. Applic, 42 (2001), 131-140. doi: 10.1016/S0898-1221(01)00138-9. [4] M. O. Osilike, Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps, J. Math. Anal., 294 (2004), 73-81. doi: 10.1016/j.jmaa.2004.01.038. [5] H. K. Xu, Inequality in Banach spaces with applications, Nonlinear Anal., 16 (1991), 1127-1138. doi: 10.1016/0362-546X(91)90200-K. [6] H. K. Xu and R. G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal., Optim., 22 (2001), 767-773. doi: 10.1081/NFA-100105317.
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