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Stability results for Jungck-type iterative processes in convex metric spaces. (English) Zbl 1293.54033
Let \((E, d)\) be a complete convex metric space in the sense of W. Takahashi [Kodai Math. Semin. Rep. 22, 142–149 (1970; Zbl 0268.54048)] and \(Y\) a nonempty closed convex subset of \(E\). Following the reviewer, C. Bhatnagar and S. N. Mishra [Int. J. Math. Math. Sci. 2005, No. 19, 3035–3043 (2005; Zbl 1117.26005)], the authors obtain stability results for Jungck-Mann and Jungck-Ishikawa iterations for a pair of nonself mappings \(S, T: Y \to E\) satisfying the conditions \(T(Y)\subseteq S(Y)\), \(S(Y)\) is a complete subspace of \(E\), and \(d(Tx, Ty)\leq \delta d(Sx,Sy) + Lu(x, y)\), where \(0\leq\delta<1\), \(L\geq 0\) and \(u(x, y) = \min\{d(Sx,Tx), d(Sy, Ty), d(Sx, Ty), d(Sy, Tx),\frac{1}{2}[d(Sx,Tx)+d(Sy,Ty)], \frac{1}{2}[d(Sx,Ty)+d(Sy,Tx)] \}.\)

MSC:
54H25 Fixed-point and coincidence theorems (topological aspects)
47J25 Iterative procedures involving nonlinear operators
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[1] Agarwal, R.P.; O’Regan, D.; Sahu, D.R., Fixed point theory for Lipschitzian-type mappings with applications, Topological fixed point theory, vol. 6, (2009), Springer Science+Business Media · Zbl 1176.47037
[2] Banach, S., Sur LES operations dans LES ensembles abstraits et leur applications aux equations integrales, Funda. math., 3, 133-181, (1922) · JFM 48.0201.01
[3] Beg, I., Structure of the set of fixed points of nonexpansive mappings on convex metric spaces, Ann. univ. marie curie – sklodowska sec. A LII, 7-14, (1998) · Zbl 1004.54031
[4] Beg, I., Inequalities in metric spaces with applications, Topol. methods nonlinear anal., 17, 183-190, (2001) · Zbl 0998.47040
[5] Berinde, V., On the stability of some fixed point procedures, Bull. stiint. univ. baia mare, ser. B, matemat. inform., XVIII, 1, 7-14, (2002) · Zbl 1031.47030
[6] Berinde, V., Iterative approximation of fixed points, (2007), Springer-Verlag Berlin Heidelberg · Zbl 1165.47047
[7] Berinde, V., Some remarks on a fixed point theorem for ciric-type almost contractions, Carpathian J. math., 25, 2, 157-162, (2009) · Zbl 1249.54078
[8] Bosede, A.O.; Rhoades, B.E., Stability of Picard and Mann iteration for a general class of functions, J. adv. math. stud., 3, 2, 22-25, (2010) · Zbl 1210.47093
[9] Chatterjea, S.K., Fixed-point theorems, C.R. acad. bulgare sci., 10, 727-730, (1972) · Zbl 0274.54033
[10] Ciric, L., On some discontinuous fixed point theorems on convex metric spaces, Czechoslovak math. J., 43, 319-326, (1993) · Zbl 0814.47065
[11] Guay, M.D.; Singh, K.L.; Whitfield, J.H.M., Fixed point theorems for nonexpansive mappings in convex metric spaces, (), 179-189 · Zbl 0501.54030
[12] Harder, A.M.; Hicks, T.L., Stability results for fixed point iteration procedures, Math. japonica, 33, 5, 693-706, (1988) · Zbl 0655.47045
[13] Imoru, C.O.; Olatinwo, M.O., On the stability of Picard and Mann iteration processes, Carpathian J. math., 19, 2, 155-160, (2003) · Zbl 1086.47512
[14] Ishikawa, S., Fixed point by a new iteration method, Proc. amer. math. soc., 44, 1, 147-150, (1974) · Zbl 0286.47036
[15] Jachymski, J.R., An extension of A. ostrowski’s theorem on the round-off stability of iterations, Aequationes math., 53, 242-253, (1997) · Zbl 0885.47023
[16] Jungck, G., Commuting mappings and fixed points, Amer. math. mon., 83, 4, 261-263, (1976) · Zbl 0321.54025
[17] Kannan, R., Some results on fixed points, Bull. Calcutta math. soc., 10, 71-76, (1968) · Zbl 0209.27104
[18] Mann, W.R., Mean value methods in iteration, Proc. amer. math. soc., 44, 506-510, (1953) · Zbl 0050.11603
[19] Olatinwo, M.O., Some stability and strong convergence results for the jungck – ishikawa iteration process, Creative math. inf., 17, 33-42, (2008) · Zbl 1199.47282
[20] Osilike, M.O., Stability results for fixed point iteration procedures, J. Nigerian math. soc., 14/15, 17-29, (1995/96) · Zbl 0847.47043
[21] Osilike, M.O.; Udomene, A., Short proofs of stability results for fixed point iteration procedures for a class of contractive-type mappings, Indian J. pure appl. math., 30, 12, 1229-1234, (1999) · Zbl 0955.47038
[22] Ostrowski, A.M., The round-off stability of iterations, Z. angew. math. mech., 47, 77-81, (1967) · Zbl 0149.36601
[23] Popescu, O., Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators, Math. commun., 12, 195-202, (2007) · Zbl 1153.47055
[24] Rhoades, B.E., Fixed point theorems and stability results for fixed point iteration procedures, Indian J. pure appl. math., 21, 1, 1-9, (1990) · Zbl 0692.54027
[25] Rhoades, B.E., Fixed point theorems and stability results for fixed point iteration procedures II, Indian J. pure appl. math., 24, 11, 691-703, (1993) · Zbl 0794.54048
[26] Rus, I.A., Generalized contractions and applications, (2001), Cluj University Press Cluj Napoca · Zbl 0968.54029
[27] I.A. Rus, A. Petrusel, G. Petrusel, Fixed point theory, 1950-2000, Romanian Contributions, House of the Book of Science, Cluj Napoca, 2002. · Zbl 1171.54034
[28] Shimizu, T.; Takahashi, W., Fixed point theorems in certain convex metric spaces, Math. japon., 37, 855-859, (1992) · Zbl 0764.47030
[29] Singh, S.L.; Bhatnagar, C.; Mishra, S.N., Stability of Jungck-type iterative procedures, Int. J. math. math. sci., 19, 3035-3043, (2005) · Zbl 1117.26005
[30] Takahashi, W., A convexity in metric spaces and nonexpansive mapping I, Kodai math. sem. rep., 22, 142-149, (1970) · Zbl 0268.54048
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