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On the steepest descent approximation method for the zeros of generalized accretive operators. (English) Zbl 1220.47089
Summary: We present certain characteristic conditions for the convergence of the generalized steepest descent approximation process to a zero of a generalized strongly accretive operator, defined on a uniformly smooth Banach space. Our study is based on an important result of S. Reich [Nonlinear Anal., Theory, Methods Appl. 2, 85–92 (1978; Zbl 0375.47032)] and the given results extend and improve some earlier results which involve the steepest descent approximation method.

MSC:
47J25 Iterative procedures involving nonlinear operators
47J05 Equations involving nonlinear operators (general)
47H06 Nonlinear accretive operators, dissipative operators, etc.
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[1] Browder, F.E., Nonlinear mappings of non-expansive and accretive type in Banach spaces, Bull. amer. math. soc., 73, 875-882, (1967) · Zbl 0176.45302
[2] Chidume, C.E., Steepest descent approximations for accretive operator equations, Nonlinear anal., 26, 299-311, (1996) · Zbl 0941.47039
[3] 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
[4] Ćirić, Lj.B.; Ume, J.S., Ishikawa iterative process for strongly pseudo-contractive operators in arbitrary Banach spaces, Math. commun., 8, 1, 43-48, (2003) · Zbl 1052.47062
[5] Ćirić, Lj.B.; Ume, J.S., Iterative process with errors for approximation solutions to nonlinear equations of generalized monotone type in arbitrary Banach spaces, Bull. austral. math. soc., 69, 177-189, (2004) · Zbl 1082.47507
[6] Deimling, K., Zeros of accretive operators, Manuscripta math., 13, 283-288, (1974)
[7] Ishikawa, S., Fixed points by a new iteration method, Proc. amer. math. soc., 73, 65-71, (1976) · Zbl 0352.47024
[8] Kato, T., Nonlinear semi-groups and evolution equations, J. math. soc. Japan, 18-19, 508-520, (1967) · Zbl 0163.38303
[9] Kartsatos, A.G., Zeros of demicontinuous accretive operators in Banach spaces, J. integral equations, 8, 175-184, (1985) · Zbl 0584.47054
[10] Liu, Z.Q.; Kang, S.M., Convergence theorems for \(\phi\)-strongly accretive and \(\phi\)-hemi-contractive operators, J. math. anal. appl., 253, 35-49, (2001) · Zbl 0999.47039
[11] Mann, W.R., Mean value methods in iteration, Proc. amer. math. soc., 4, 506-510, (1953) · Zbl 0050.11603
[12] Morales, C.H., Surjectivity theorems for multi-valued mappings of accretive types, Comment. math. univ. carol., 26, 2, 397-413, (1985) · Zbl 0595.47041
[13] Morales, C.H.; Chidume, C.E., Convergence of the steepest descent method for accretive operators, Proc. amer. math. soc., 127, 12, 3677-3683, (1999) · Zbl 0937.47057
[14] Osilike, M.O., Iterative solution of nonlinear equations of the \(\phi\)-strongly accretive type, J. math. anal. appl., 200, 2, 259-271, (1996) · Zbl 0860.65039
[15] Reich, S., An iterative procedure for constructing zeros of accretive sets in Banach spaces, Nonlinear anal., 2, 85-92, (1978) · Zbl 0375.47032
[16] Rhoades, B.E., Comments on two fixed point iteration methods, J. math. anal. appl., 56, 741-750, (1976) · Zbl 0353.47029
[17] Tan, K.K.; Hu, H.K., Iterative solutions to nonlinear equations of strongly accretive operators in Banach spaces, J. math. anal. appl., 178, 9-21, (1993) · Zbl 0834.47048
[18] Vainberg, M.M., On the convergence of the method of steepest descent for nonlinear equations, Sibirsk mat. zb., 201-220, (1961) · Zbl 0206.14201
[19] Xu, Z.B.; Roach, G.F., Characteristic inequalities in uniformly convex and uniformly smooth Banach spaces, J. math. anal. appl., 157, 189-210, (1991) · Zbl 0757.46034
[20] Zarantonello, E.H., The closure of the numerical range contains the spectrum, Bull. amer. math. soc., 70, 781-783, (1964) · Zbl 0137.32501
[21] Zeidler, E., Nonlinear functional analysis and its applications. part II. monotone operators, (1985), Springer-Verlag New York, Berlin
[22] Zhou, H.Y.; Guo, J., A characteristic condition for convergence of generalized steepest descent approximation to accretive equations, Indian J. pure appl. math., 32, 2, 277-284, (2001) · Zbl 1018.47049
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