×

zbMATH — the first resource for mathematics

Convergence and stability theorems for the Picard-Mann hybrid iterative scheme for a general class of contractive-like operators. (English) Zbl 1312.47078
Summary: In this paper we use the general class of contractive-like operators introduced by A. O. Bosede and B. E. Rhoades [J. Adv. Math. Stud. 3, No. 2, 23–25 (2010; Zbl 1210.47093)] to prove strong convergence and stability results for Picard-Mann hybrid iterative schemes considered in a real normed linear space. We establish the strong convergence and stability of the Picard iterative scheme as a corollary. Our results generalize and improve a multitude of results in the literature, including the recent results of C. E. Chidume [Fixed Point Theory Appl. 2014, Article ID 233 (2014; doi:10.1186/1687-1812-2014-233)].

MSC:
47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Hussain, N; Chugh, R; Kumar, V; Rafiq, A, On the rate of convergence of kirk-type iterative schemes, J. Appl. Math., 2012, (2012) · Zbl 1325.47120
[2] Phuengrattana, W; Suantai, S, On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval, J. Comput. Appl. Math., 235, 3006-3014, (2011) · Zbl 1215.65095
[3] Khan, SH, A Picard-Mann hybrid iterative process, Fixed Point Theory Appl., 2013, (2013) · Zbl 1317.47065
[4] Harder, AM; Hicks, TL, A stable iteration procedure for nonexpansive mappings, Math. Jpn., 33, 687-692, (1988) · Zbl 0655.47046
[5] Harder, AM; Hicks, TL, Stability results for fixed point iteration procedures, Math. Jpn., 33, 693-706, (1988) · Zbl 0655.47045
[6] Harder, AM: Fixed point theory and stability results for fixed point iteration procedures. Ph. D. thesis, University of Missouri-Rolla (1987)
[7] Chidume, CE, Strong convergence and stability of Picard iteration sequences for a general class of contractive-type mappings, Fixed Point Theory Appl., 2014, (2014) · Zbl 1346.47037
[8] Rhoades, BE, Fixed point iteration using infinite matrices, Trans. Am. Math. Soc., 196, 161-176, (1974) · Zbl 0285.47038
[9] Mann, WR, Mean value methods in iterations, Proc. Am. Math. Soc., 44, 506-510, (1953) · Zbl 0050.11603
[10] Chugh, R; Kumar, V, Strong convergence of SP iterative scheme for quasi-contractive operators, Int. J. Comput. Appl., 31, 21-27, (2011)
[11] Rhoades, BE; Soltuz, SM, The equivalence between Mann-Ishikawa iterations and multi-step iteration, Nonlinear Anal., 58, 219-228, (2004) · Zbl 1064.47070
[12] Akewe, H; Okeke, GA; Olayiwola, A, Strong convergence and stability of kirk-multistep-type iterative schemes for contractive-type operators, Fixed Point Theory Appl., 2014, (2014) · Zbl 1345.47029
[13] Ishikawa, S, Fixed points by a new iteration method, Proc. Am. Math. Soc., 44, 147-150, (1974) · Zbl 0286.47036
[14] Berinde, V: Iterative Approximation of Fixed Points. Editura Efemeride, Baia Mare (2002) · Zbl 1036.47037
[15] Osilike, MO, Stability results for Ishikawa fixed point iteration procedure, Indian J. Pure Appl. Math., 26, 937-941, (1995) · Zbl 0847.47043
[16] Rhoades, BE, Fixed point theorems and stability results for fixed point iteration procedures, Indian J. Pure Appl. Math., 21, 1-9, (1990) · Zbl 0692.54027
[17] Imoru, CO; Olatinwo, MO, On the stability of Picard and Mann iteration, Carpath. J. Math., 19, 155-160, (2003) · Zbl 1086.47512
[18] Bosede, AO; Rhoades, BE, Stability of Picard and Mann iteration for a general class of functions, J. Adv. Math. Stud., 3, 1-3, (2010)
[19] Chidume, CE; Olaleru, JO, Picard iteration process for a general class of contractive mappings, J. Niger. Math. Soc., 33, 19-23, (2014) · Zbl 1341.47079
[20] Bosede, AO, Noor iterations associated with Zamfirescu mappings in uniformly convex Banach spaces, Fasc. Math., 42, 29-38, (2009) · Zbl 1178.47042
[21] Rhoades, BE, A comparison of various definition of contractive mapping, Trans. Am. Math. Soc., 226, 257-290, (1977) · Zbl 0365.54023
[22] Zamfirescu, T, Fixed point theorems in metric spaces, Arch. Math. (Basel), 23, 292-298, (1972) · Zbl 0239.54030
[23] Berinde, V, On the stability of some fixed point procedures, Bul. Ştiinţ. - Univ. Baia Mare, Ser. B Fasc. Mat.-Inform., 18, 7-14, (2002) · Zbl 1031.47030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.