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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.

MSC:
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)
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[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
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