Stability of iterative procedures with errors for approximating common fixed points of a couple of q-contractive-like mappings in Banach spaces

Abstract

Recently, Agarwal, Cho, Li and Huang [R.P. Agarwal, Y.J. Cho, J. Li, N.J. Huang, Stability of iterative procedures with errors approximating common fixed points for a couple of quasi-contractive mappings in q-uniformly smooth Banach spaces, J. Math. Anal. Appl. 272 (2002) 435–447] introduced the new iterative procedures with errors for approximating the common fixed point of a couple of quasi-contractive mappings and showed the stability of these iterative procedures with errors in Banach spaces. In this paper, we introduce a new concept of a couple of q-contractive-like mappings ( q > 1 ) in a Banach space and apply these iterative procedures with errors for approximating the common fixed point of the couple of q-contractive-like mappings. The results established in this paper improve, extend and unify the corresponding ones of Agarwal, Cho, Li and Huang [R.P. Agarwal, Y.J. Cho, J. Li, N.J. Huang, Stability of iterative procedures with errors approximating common fixed points for a couple of quasi-contractive mappings in q-uniformly smooth Banach spaces, J. Math. Anal. Appl. 272 (2002) 435–447], Chidume [C.E. Chidume, Approximation of fixed points of quasi-contractive mappings in L p spaces, Indian J. Pure Appl. Math. 22 (1991) 273–386], Chidume and Osilike [C.E. Chidume, M.O. Osilike, Fixed points iterations for quasi-contractive maps in uniformly smooth Banach spaces, Bull. Korean Math. Soc. 30 (1993) 201–212], Liu [Q.H. Liu, On Naimpally and Singh's open questions, J. Math. Anal. Appl. 124 (1987) 157–164; Q.H. Liu, A convergence theorem of the sequence of Ishikawa iterates for quasi-contractive mappings, J. Math. Anal. Appl. 146 (1990) 301–305], Osilike [M.O. Osilike, A stable iteration procedure for quasi-contractive maps, Indian J. Pure Appl. Math. 27 (1996) 25–34; M.O. Osilike, Stability of the Ishikawa iteration method for quasi-contractive maps, Indian J. Pure Appl. Math. 28 (1997) 1251–1265] and many others in the literature.