# 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 & 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., 20 (1967), 197. Google Scholar [2] R. Chen, P. K. Lin and Y. Dong, An approximation method for strictly pseudocontractive mappings,, Nonlinear Analysis (TMA), 64 (2006), 2527. doi: 10.1016/j.na.2005.08.031. Google Scholar [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. doi: 10.1016/S0898-1221(01)00138-9. Google Scholar [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. doi: 10.1016/j.jmaa.2004.01.038. Google Scholar [5] H. K. Xu, Inequality in Banach spaces with applications,, Nonlinear Anal., 16 (1991), 1127. doi: 10.1016/0362-546X(91)90200-K. Google Scholar [6] H. K. Xu and R. G. Ori, An implicit iteration process for nonexpansive mappings,, Numer. Funct. Anal., 22 (2001), 767. doi: 10.1081/NFA-100105317. Google Scholar

show all references

##### References:
 [1] F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space,, J. Math. Anal., 20 (1967), 197. Google Scholar [2] R. Chen, P. K. Lin and Y. Dong, An approximation method for strictly pseudocontractive mappings,, Nonlinear Analysis (TMA), 64 (2006), 2527. doi: 10.1016/j.na.2005.08.031. Google Scholar [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. doi: 10.1016/S0898-1221(01)00138-9. Google Scholar [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. doi: 10.1016/j.jmaa.2004.01.038. Google Scholar [5] H. K. Xu, Inequality in Banach spaces with applications,, Nonlinear Anal., 16 (1991), 1127. doi: 10.1016/0362-546X(91)90200-K. Google Scholar [6] H. K. Xu and R. G. Ori, An implicit iteration process for nonexpansive mappings,, Numer. Funct. Anal., 22 (2001), 767. doi: 10.1081/NFA-100105317. Google Scholar
 [1] Teck-Cheong Lim. On the largest common fixed point of a commuting family of isotone maps. Conference Publications, 2005, 2005 (Special) : 621-623. doi: 10.3934/proc.2005.2005.621 [2] Dalila Azzam-Laouir, Fatiha Selamnia. On state-dependent sweeping process in Banach spaces. Evolution Equations & Control Theory, 2018, 7 (2) : 183-196. doi: 10.3934/eect.2018009 [3] Samir Adly, Ba Khiet Le. Unbounded state-dependent sweeping processes with perturbations in uniformly convex and q-uniformly smooth Banach spaces. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 81-95. doi: 10.3934/naco.2018005 [4] Cleon S. Barroso. The approximate fixed point property in Hausdorff topological vector spaces and applications. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 467-479. doi: 10.3934/dcds.2009.25.467 [5] Byung-Soo Lee. A convergence theorem of common fixed points of a countably infinite family of asymptotically quasi-$f_i$-expansive mappings in convex metric spaces. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 557-565. doi: 10.3934/naco.2013.3.557 [6] Sonja Cox, Arnulf Jentzen, Ryan Kurniawan, Primož Pušnik. On the mild Itô formula in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2217-2243. doi: 10.3934/dcdsb.2018232 [7] Nicholas Long. Fixed point shifts of inert involutions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1297-1317. doi: 10.3934/dcds.2009.25.1297 [8] Goro Akagi, Mitsuharu Ôtani. Evolution equations and subdifferentials in Banach spaces. Conference Publications, 2003, 2003 (Special) : 11-20. doi: 10.3934/proc.2003.2003.11 [9] Yuri Latushkin, Valerian Yurov. Stability estimates for semigroups on Banach spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5203-5216. doi: 10.3934/dcds.2013.33.5203 [10] Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409 [11] Yakov Krasnov, Alexander Kononovich, Grigory Osharovich. On a structure of the fixed point set of homogeneous maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1017-1027. doi: 10.3934/dcdss.2013.6.1017 [12] Jorge Groisman. Expansive and fixed point free homeomorphisms of the plane. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1709-1721. doi: 10.3934/dcds.2012.32.1709 [13] Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692 [14] Luis Hernández-Corbato, Francisco R. Ruiz del Portal. Fixed point indices of planar continuous maps. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2979-2995. doi: 10.3934/dcds.2015.35.2979 [15] Antonio Garcia. Transition tori near an elliptic-fixed point. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 381-392. doi: 10.3934/dcds.2000.6.381 [16] Yong Ji, Ercai Chen, Yunping Wang, Cao Zhao. Bowen entropy for fixed-point free flows. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6231-6239. doi: 10.3934/dcds.2019271 [17] Óscar Vega-Amaya, Joaquín López-Borbón. A perturbation approach to a class of discounted approximate value iteration algorithms with borel spaces. Journal of Dynamics & Games, 2016, 3 (3) : 261-278. doi: 10.3934/jdg.2016014 [18] Fatihcan M. Atay, Lavinia Roncoroni. Lumpability of linear evolution Equations in Banach spaces. Evolution Equations & Control Theory, 2017, 6 (1) : 15-34. doi: 10.3934/eect.2017002 [19] Viviane Baladi, Sébastien Gouëzel. Banach spaces for piecewise cone-hyperbolic maps. Journal of Modern Dynamics, 2010, 4 (1) : 91-137. doi: 10.3934/jmd.2010.4.91 [20] Cecilia González-Tokman, Anthony Quas. A concise proof of the multiplicative ergodic theorem on Banach spaces. Journal of Modern Dynamics, 2015, 9: 237-255. doi: 10.3934/jmd.2015.9.237

Impact Factor: