zbMATH — the first resource for mathematics

On Picard iterations for strongly accretive and strongly pseudo-contractive Lipschitz mappings. (English) Zbl 1225.47092
Summary: We speed up the convergence of the Picard sequence of iterations for strongly accretive and strongly pseudo-contractive mappings. Our results improve the results of C. E. Chidume [“Picard iterations for strongly accretive and strongly pseudocontractive Lipschitz maps” (ICTP Preprint No. IC2000098) (2000); “Iterative algorithms for non-expansive mappings and some of their generalizations”, in: Ravi P. Agarwal (ed.) et al., Nonlinear analysis and applications: To V. Lakshmikantham on his 80th birthday, Vol. 1 (Dordrecht: Kluwer), 383–429 (2003; Zbl 1057.47003)], L.-W. Liu [Proc. Am. Math. Soc. 125, No. 5, 1363–1366 (1997; Zbl 0870.47039)], and some other known results. The technique of the proof presented in this paper is different from the technique used by Chidume.

47J25 Iterative procedures involving nonlinear operators
47H06 Nonlinear accretive operators, dissipative operators, etc.
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J05 Equations involving nonlinear operators (general)
65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx)
Full Text: DOI
[1] Browder, F.E., Nonlinear mappings of the non-expansive and accretive type in Banach spaces, Bull. amer. math. soc., 73, 875-882, (1967) · Zbl 0176.45302
[2] Chen, R.; Song, Y.Y., Convergence to common fixed point of non-expansive semigroups, J. comput. appl. math., 200, 2, 566-575, (2007) · Zbl 1204.47076
[3] Chidume, C.E., Iterative approximation of fixed points of Lipschitz strictly pseudo-contractive mappings, Proc. amer. math. soc., 99, 2, 283-288, (1987) · Zbl 0646.47037
[4] Chidume, C.E., An iterative process for nonlinear Lipschitz strongly accretive mappings in \(L_p\) spaces, J. math. anal. appl., 151, 453-461, (1990) · Zbl 0724.65058
[5] C.E. Chidume, Picard iteration for strongly accretive and strongly pseudo-contractive Lipschitz maps, ICTP Preprint no. IC2000098
[6] Chidume, C.E., (), 383-429
[7] Ćirić, Lj.B.; Ume, J.S., Ishikawa process with errors for nonlinear equations of generalized monotone type in Banach spaces, Math. nachr, 10, 1137-1146, (2005) · Zbl 1092.47054
[8] Ćirić, L.B.; Ume, J.S., Convergence theorems for the Ishikawa iterative process associated with a pair of strongly pseudo-contractive operators, Italian J. pure appl. math., 22, 139-148, (2007) · Zbl 1145.47049
[9] Ćirić, L.B.; Ješić, S.N.; Milovanović, M.M.; Ume, J.S., On the steepest descent approximation method for the zeros of generalized accretive operators, Nonlinear anal. theor., 69, 763-769, (2008) · Zbl 1220.47089
[10] Ćirić, L.; Rafiq, A.; Cakić, N.; Ume, J.S., Implicit Mann fixed point iterations for pseudo-contractive mappings, Appl. math. lett., (2008)
[11] Deimling, K., Nonlinear functional analysis, (1980), Springer-Verlag Berlin, Heidelberg, New York, Tokyo
[12] Deng, L., On chidume’s open problem, J. math. anal. appl., 174, 2, 441-449, (1991) · Zbl 0784.47051
[13] Deng, L., An iterative process for nonlinear Lipschitz and strongly accretive mappings in uniformly convex and uniformly smooth Banach spaces, Acta appl. math., 32, 183-196, (1993) · Zbl 0801.47040
[14] Deng, L., Iteration processes for nonlinear Lipschitzian strongly accretive mappings in \(L_p\) spaces, J. math. anal. appl., 188, 128-140, (1994) · Zbl 0828.47042
[15] Deng, L.; Ding, X.P., Iterative approximation of Lipschitz strictly pseudo-contractive mappings in uniformly smooth spaces, Nonlinear analysis, 24, 7, 981-987, (1995) · Zbl 0827.47041
[16] Ishikawa, S., Fixed point by a new iteration method, Proc. amer. math. soc., 44, 147-150, (1974) · Zbl 0286.47036
[17] Kato, T., Nonlinear semigroups and evolution equations, J. math. soc. Japan, 19, 508-520, (1967) · Zbl 0163.38303
[18] Liu, L., Approximation of fixed points of a strictly pseudo-contractive mapping, Proc. amer. math. soc., 125, 2, 1363-1366, (1997) · Zbl 0870.47039
[19] Liu, Q.H., The convergence theorems of the sequence of Ishikawa iterates for hemi-contractive mappings, J. math. anal. appl., 148, 55-62, (1990) · Zbl 0729.47052
[20] Mann, W.R., Mean value methods in iteration, Proc. amer. math. soc., 4, 506-510, (1953) · Zbl 0050.11603
[21] Sastry, W.R.; Babu, G.V.R., Approximation of fixed points of strictly pseudo-contractive mappings on arbitrary closed, convex sets in a Banach space, Proc. amer. math. soc., 128, 2907-2909, (2000) · Zbl 0956.47040
[22] Shahzad, N.; Udomene, A., Fixed point solutions of variational inequalities for asymptotically non-expansive mappings in Banach spaces, Nonlinear anal. theor., 64, 3, 558-567, (2006) · Zbl 1102.47056
[23] Weng, X.L., Fixed point iteration for local strictly pseudo-contractive mapping, Proc. amer. math. soc., 113, 727-731, (1991) · Zbl 0734.47042
[24] Yao, Y.; Liou, Y.C.; Chen, R., Strong convergence of an iterative algorithm for pseudo-contractive mapping in Banach spaces, Nonlinear anal. theor., 67, 12, 3311-3317, (2007) · Zbl 1129.47059
[25] Yao, Y.; Chen, R., Convergence to common fixed points of averaged mappings without commutativity assumption in Hilbert spaces, Nonlinear analysis:TMA, 67, 6 (A, 1758-1763, (2007) · Zbl 1134.47053
[26] Yao, Y.; Noor, M.A., Convergence of three-step iterations for asymptotically nonexpansive mappings, Appl. math. comput., 187, 883-892, (2007) · Zbl 1136.65060
[27] Zhou, H., Convergence theorems of common fixed points for a finite family of Lipschitz pseudo-contractions in Banach spaces, Nonlinear anal. theor., 68, 10, 2977-2983, (2008) · Zbl 1145.47055
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.