×

zbMATH — the first resource for mathematics

Data dependence results of new multi-step and S-iterative schemes for contractive-like operators. (English) Zbl 1423.47041
Summary: In this paper, we prove that the convergence of a new iteration and S-iteration can be used to approximate the fixed points of contractive-like operators. We also prove some data dependence results for these new iteration and S-iteration schemes for contractive-like operators. Our results extend and improve some known results in the literature.

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] Rhoades, BE, A comparison of various definitions of contractive mappings, Trans. Am. Math. Soc, 226, 257-290, (1977) · Zbl 0365.54023
[2] Mann, WR, Mean value methods in iterations, Proc. Am. Math. Soc, 4, 506-510, (1953) · Zbl 0050.11603
[3] Ishikawa, S, Fixed points by a new iteration method, Proc. Am. Math. Soc, 44, 147-150, (1974) · Zbl 0286.47036
[4] Noor, MA, New approximation schemes for general variational inequalities, J. Math. Anal. Appl, 251, 217-229, (2000) · Zbl 0964.49007
[5] Rhoades, BE; Şoltuz, SM, The equivalence between Mann-Ishikawa iterations and multistep iteration, Nonlinear Anal, 58, 219-228, (2004) · Zbl 1064.47070
[6] Thianwan, S, Common fixed points of new iterations for two asymptotically nonexpansive nonself mappings in a Banach space, (2008)
[7] 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
[8] Glowinski R, Le Tallec P: Augmented Langrangian and Operator Splitting Methods in Nonlinear Mechanics. SIAM, Philadelphia; 1989. · Zbl 0698.73001
[9] Xu, B; Noor, MA, Ishikawa and Mann iteration process with errors for nonlinear strongly accretive operator equations, J. Math. Anal. Appl, 224, 91-101, (1998) · Zbl 0936.47041
[10] Takahashi, W, Iterative methods for approximation of fixed points and their applications, J. Oper. Res. Soc. Jpn, 43, 87-108, (2000) · Zbl 1004.65069
[11] Das, G; Debata, JP, Fixed points of quasi-nonexpansive mappings, Indian J. Pure Appl. Math, 17, 1263-1269, (1986) · Zbl 0605.47054
[12] Agarwal, RP; O’Regan, D; Sahu, DR, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal, 8, 61-79, (2007) · Zbl 1134.47047
[13] Berinde, V, On the convergence of the Ishikawa iteration in the class of quasi contractive operators, Acta Math. Univ. Comen, 73, 119-126, (2004) · Zbl 1100.47054
[14] Chidume, CE; Chidume, CO, Convergence theorem for fixed points of uniformly continuous generalized phihemicontractive mappings, J. Math. Anal. Appl, 303, 545-554, (2005) · Zbl 1070.47055
[15] Chidume, CE; Chidume, CO, Iterative approximation of fixed points of nonexpansive mappings, J. Math. Anal. Appl, 318, 288-295, (2006) · Zbl 1095.47034
[16] Suantai, S, Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings, J. Math. Anal. Appl, 311, 506-517, (2005) · Zbl 1086.47057
[17] Sahu, DR, Applications of the S-iteration process to constrained minimization problems and split feasibility problems, Fixed Point Theory Appl, 12, 187-204, (2011) · Zbl 1281.47053
[18] Yıldırım, İ; Özdemir, M; Kızıltunç, H, On the convergence of a new two-step iteration in the class of quasi-contractive operators, Int. J. Math. Anal, 3, 1881-1892, (2009) · Zbl 1236.47081
[19] Chugh, R; Kumar, V, Strong convergence of SP iterative scheme for quasi-contractive operators in Banach spaces, Int. J. Comput. Appl, 31, 21-27, (2011)
[20] Osilike, MO; 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, 1229-1234, (1999) · Zbl 0955.47038
[21] Rafiq, A, On the convergence of the three step iteration process in the class of quasi-contractive operators, Acta Math. Acad. Paedagog. Nyházi, 22, 305-309, (2006) · Zbl 1120.47314
[22] Hussain, N; Rafiq, A; Damjanović, B; Lazović, R, On rate of convergence of various iterative schemes, No. 2011, (2011) · Zbl 1315.47065
[23] Rus, IA; Muresan, S, Data dependence of the fixed points set of weakly Picard operators, Stud. Univ. Babeş-Bolyai, Math, 43, 79-83, (1998) · Zbl 1005.54036
[24] Rus, IA; Petruşel, A; Sîntamarian, A, Data dependence of the fixed points set of multivalued weakly Picard operators, Stud. Univ. Babeş-Bolyai, Math, 46, 111-121, (2001) · Zbl 1027.47053
[25] Rus, IA; Petruşel, A; Sîntamarian, A, Data dependence of the fixed point set of some multivalued weakly Picard operators, Nonlinear Anal., Theory Methods Appl, 52, 1947-1959, (2003) · Zbl 1055.47047
[26] Berinde, V, On the approximation of fixed points of weak contractive mappings, Carpath. J. Math, 19, 7-22, (2003) · Zbl 1114.47045
[27] Espínola, R; Petruşel, A, Existence and data dependence of fixed points for multivalued operators on gauge spaces, J. Math. Anal. Appl, 309, 420-432, (2005) · Zbl 1070.47046
[28] Markin, JT, Continuous dependence of fixed point sets, Proc. Am. Math. Soc, 38, 545-547, (1973) · Zbl 0278.47036
[29] Chifu, C; Petruşel, G, Existence and data dependence of fixed points and strict fixed points for contractive-type multivalued operators, (2007) · Zbl 1155.54337
[30] Olatinwo, MO, Some results on the continuous dependence of the fixed points in normed linear space, Fixed Point Theory Appl, 10, 151-157, (2009) · Zbl 1186.47072
[31] Olatinwo, MO, On the continuous dependence of the fixed points for [inlineequation not available: see fulltext.]-contractive-type operators, Kragujev. J. Math, 34, 91-102, (2010) · Zbl 1258.47084
[32] Şoltuz, SM, Data dependence for Mann iteration, Octogon Math. Mag, 9, 825-828, (2001)
[33] Şoltuz, SM, Data dependence for Ishikawa iteration, Lect. Mat, 25, 149-155, (2004) · Zbl 1381.47042
[34] Şoltuz, SM; Grosan, T, Data dependence for Ishikawa iteration when dealing with contractive like operators, No. 2008, (2008) · Zbl 1205.47059
[35] Chugh, R; Kumar, V, Data dependence of Noor and SP iterative schemes when dealing with quasi-contractive operators, Int. J. Comput. Appl, 40, 41-46, (2011)
[36] Akewe, H, Strong convergence and stability of Jungck-multistep-SP iteration for generalized contractive-like inequality operators, Adv. Nat. Sci, 5, 21-27, (2012)
[37] Zamfirescu, T, Fix point theorems in metric spaces, Arch. Math, 23, 292-298, (1972) · Zbl 0239.54030
[38] Imoru, CO; Olantiwo, MO, On the stability of Picard and Mann iteration processes, Carpath. J. Math, 19, 155-160, (2003) · Zbl 1086.47512
[39] Agarwal RP, O’Regan D, Sahu DR: Fixed Point Theory for Lipschitzian Type-Mappings with Applications. Springer, New York; 2009. · Zbl 1176.47037
[40] Berinde V: Iterative Approximation of Fixed Points. Springer, Berlin; 2007. · Zbl 1165.47047
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.