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Iterative processes with errors for nonlinear equations

Published online by Cambridge University Press:  17 April 2009

Łjubomir Ćirić
Affiliation:
Faculty of Mechanical Engineering, University of Blegrade, Belgrade, 27. marta 80, Yugoslavia e-mail: [email protected]
Jeong Sheok Ume
Affiliation:
Depatment of Applied Mathematics, Changwon National University, Changwon 641-773, Korea e-mail: [email protected]
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In this paper we introduce and consider a class of multi-valued and single-valued operators of generalised monotone type. We proved a new general lemma on the convergence of real sequences and some new convergence theorems for the Ishikawa and Mann iteration processes with errors to the unique fixed point of such operators, which are not necessarily Lipschitz operators. Our results generalise, improve, and extend several recent results.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Browder, F.E., ‘Nonlinear monotone and accreative opeators in Banach spaces’, Proc. Nat. Acad. Sci. U.S.A. 61 (1968), 388393.CrossRefGoogle ScholarPubMed
[2]Chang, S.S. and Tan, K.K., ‘Iteration processes for approximating fixed points of operators of monotone type’, Bull. Austral. Math. Soc. 57 (1998), 433445.CrossRefGoogle Scholar
[3]Chang, S.S., Cho, Y.J., Lee, B.S., Jung, J.S. and Kang, S.M., ‘Iterative approximations of fixed points and solutions for strongly accretive and strongly pseudo-contractive mappings in Banach spaces’, J. Math. Anal. Appl. 224 (1998), 149165.CrossRefGoogle Scholar
[4]Chang, S.S., ‘Some problems and results in the study of nonlinear analysis’, Nonlinear Anal. 30 (1997), 41974208.CrossRefGoogle Scholar
[5]Chidume, C.E., ‘Iterative construction of fixed points for multi-valued operators of the monotone type’, Appl. Anal. 23 (1986), 209218.CrossRefGoogle Scholar
[6]Chidume, C.E., ‘Iterative solution of nonlinear equations of the monotone type in Banach spaces’, Bull. Austral. Math. Soc. 42 (1990), 2131.CrossRefGoogle Scholar
[7]Chidume, C.E., ‘Approximation of fixed points of strongly pseudo-contractive mappings’, Proc. Amer. Math. Soc. 120 (1994), 545551.CrossRefGoogle Scholar
[8]Chidume, C.E., ‘Iterative solution of nonlinear equations with strongly accretive operators’, J. Math. Anal. Appl. 192 (1995), 502518.CrossRefGoogle Scholar
[9]Chidume, C.E., ‘Steepest descent approximations for accretive operator equations’, Nonlinear Anal. 26 (1996), 299311.CrossRefGoogle Scholar
[10]Deng, L. and Ding, X. P., ‘Iterative approximation of Lipschitz strictly pseudo-contractive mappings in uniformly smooth Banach spaces’, Nonlinear Anal. 24 (1995), 981987.CrossRefGoogle Scholar
[11]Dunn, J.C., ‘Iterative construction of fixed points for multi-valued operators of the monotone type’, J. Funct. Anal. 27 (1978), 3850.CrossRefGoogle Scholar
[12]Gu, F.Iteration processes with errors approximating fixed points of operators of monotone type, Proc. Amer. Math. Soc. 129 (2001), 22932300. (electronic).Google Scholar
[13]Liu, L.S., ‘Ishikawa and Man iterative process with errors for nonlinear strongly accretive mappings in Banach spaces’, J. Math. Anal. Appl. 194 (1995), 114125.CrossRefGoogle Scholar
[14]Martin, R.H. Jr., ‘A global existence theorem for autonomous differential equations in Banach spaces’, Proc. Amer. Math. Soc. 26 (1970), 307314.CrossRefGoogle Scholar
[15]Osilike, M.O., ‘Ishikawa and Mann iteration methods for nonlinear strongly accretive mappings’, Bull. Austral. Math. Soc. 46 (1992), 411422.CrossRefGoogle Scholar
[16]Osilike, M.O., ‘Stable iteration procedures for strong pseudo-contractions and nonlinear operator equations of the accretive type’, J. Math. Anal. Appl. 204 (1996), 677692.CrossRefGoogle Scholar
[17]Osilike, M.O., ‘Iterative solution of nonlinear equations of the φ-strongly accretive type’, J. Math. Anal. Appl. 200 (1996), 259271.CrossRefGoogle Scholar
[18]Tan, K.K. and Xu, H.K., ‘Iterative solutions to nonlineear equations of strongly accretive operators in Banach spaces’, Math. Anal. Appl. 178 (1993), 921.CrossRefGoogle Scholar
[19]Zeidler, E., Nonlinear functional analysis and its applications. Part II. Monotone operators (Springer-Verlag, New York, Berlin, Heidelberg, 1985).Google Scholar