Modified multistep iteration for approximating a general class of functions in Locally Convex Spaces
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Abstract
In this paper, we study the convergence of modied multistep iteration and use the scheme to approximate the fixed point of a general class of functions introduced by Bosede and Rhoades [5] in a complete metrisable locally convex space. As corollaries, the convergence results for SP and Mann iterations are also established. Moreover, most convergence results in Banach spaces are generalized to complete metrisable locally convex spaces. Our convergence results generalize and extend the results of Berinde [2], Olaleru [11], Phuengrattana and Suantai [13], Raq [14] among others.
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Akewe, H.
(2014).
Modified multistep iteration for approximating a general class of functions in Locally Convex Spaces.
Acta Mathematica Universitatis Comenianae, 83(1), 39-45.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/74/42
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References
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[2] V. Berinde, A convergence theorem for Mann iteration in the class of Zamrescu operators, Seria Mathematica Informatica, XLV (1)(2007), 33-41.
[3] A. O. Bosede, Strong convergence of Noor iteration for a general class of functions. Bulletin of Mathematical Analysis and Applications, Vol.(2011), 140-145.
[4] A. O. Bosede and B. E. Rhoades, Stability of Picard and Mann iteration for a general class of functions. Journal of Advanced Mathematical Studies, 3 (2)(2010), 1-3.
[5] S. K. Chatterjea, Fixed point theorems, Comptes Rendus de l'Academie Bulgare des Sciences, 25(6)(1972), 727-730.
[6] C. O. Imoru and M. O. Olatinwo, On the stability of Picard and Mann iteration, Carpathian Journal of Mathematics, 19(2003), 155-160.
[7] S. Ishikawa, Fixed points by a new iteration method,Proceedings of the American Mathematical Society, 44(1974), 147-150.
[8] R. Kannan, Some results on fixed points, Bulletin Calcutta Mathematical Society, 10(1968), 71-76.
[9] W. R. Mann, Mean value methods in iterations, Proceedings of the American Mathematical Society, 44(1953), 506-510.
[10] M. A. Noor, New approximation schemes for general variational inequalities, Journal of Mathematical Analysis and Applications, 251(2000), 217-229.
[11] J. O. Olaleru, On the convergence of the Mann iteration in locally convex spaces, Carpathian Journal of Mathematics, vol. 22, no. 1-2 (2006), pp. 115-120.
[12] M. O. Osilike, Stability results for Ishikawa fixed point iteration procedure, Indian Journal of Pure and Applied Mathematics, 26 (10)(1995/96), 937-941.
[13] W. Phuengrattana and S. Suantai, On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval, Journal of Computational and Applied Mathematics, 235(2011), 3006-3014.
[14] A. Raq, A convergence theorem for Mann fixed point iteration procedure, Applied Mathematics E-Notes,
6(2006), 289-293.
[15] A. Raq, On the convergence of the three-step iteration process in the class of quasi-contractive operators, Acta Mathematica Academiae
Paedagogicae Nyregyhziensis, 22(2006 ), 305-309.
[16] B. E. Rhoades, Fixed point iteration using finnite matrices, Transactions of the American Mathematical Society, 196 (1974), 161-176.
[17] B. E. Rhoades, A comparison of various denition of contractive mapping, Transactions of the American Mathematical Society, 226(1977), 257-290.
[18] B. E. Rhoades, Fixed point theorems and stability results for fixed point iteration procedure II, Indian Journal of Pure and Applied Mathematics, 24 (11) (1993), 691-703.
[19] B. E. Rhoades and S. M. Soltuz, The equivalence between Mann-Ishikawa iterations and multi-step iteration, Nonlinear Analysis, 58(2004), 219-228.
[20] H. H. Schaer, Topological Vector Spaces, Springer-Verlag, 1999.
[21] X. Weng, Fixed point iteration for local strictly pseudo-contractive mapping, Proceedings of the American Mathematical Society, 113 (1991) no.3, 727-731.
[22] T. Zamrescu, Fixed point theorems in metric spaces, Archiv der Mathematik (Basel), 23(1972), 292-298.
[2] V. Berinde, A convergence theorem for Mann iteration in the class of Zamrescu operators, Seria Mathematica Informatica, XLV (1)(2007), 33-41.
[3] A. O. Bosede, Strong convergence of Noor iteration for a general class of functions. Bulletin of Mathematical Analysis and Applications, Vol.(2011), 140-145.
[4] A. O. Bosede and B. E. Rhoades, Stability of Picard and Mann iteration for a general class of functions. Journal of Advanced Mathematical Studies, 3 (2)(2010), 1-3.
[5] S. K. Chatterjea, Fixed point theorems, Comptes Rendus de l'Academie Bulgare des Sciences, 25(6)(1972), 727-730.
[6] C. O. Imoru and M. O. Olatinwo, On the stability of Picard and Mann iteration, Carpathian Journal of Mathematics, 19(2003), 155-160.
[7] S. Ishikawa, Fixed points by a new iteration method,Proceedings of the American Mathematical Society, 44(1974), 147-150.
[8] R. Kannan, Some results on fixed points, Bulletin Calcutta Mathematical Society, 10(1968), 71-76.
[9] W. R. Mann, Mean value methods in iterations, Proceedings of the American Mathematical Society, 44(1953), 506-510.
[10] M. A. Noor, New approximation schemes for general variational inequalities, Journal of Mathematical Analysis and Applications, 251(2000), 217-229.
[11] J. O. Olaleru, On the convergence of the Mann iteration in locally convex spaces, Carpathian Journal of Mathematics, vol. 22, no. 1-2 (2006), pp. 115-120.
[12] M. O. Osilike, Stability results for Ishikawa fixed point iteration procedure, Indian Journal of Pure and Applied Mathematics, 26 (10)(1995/96), 937-941.
[13] W. Phuengrattana and S. Suantai, On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval, Journal of Computational and Applied Mathematics, 235(2011), 3006-3014.
[14] A. Raq, A convergence theorem for Mann fixed point iteration procedure, Applied Mathematics E-Notes,
6(2006), 289-293.
[15] A. Raq, On the convergence of the three-step iteration process in the class of quasi-contractive operators, Acta Mathematica Academiae
Paedagogicae Nyregyhziensis, 22(2006 ), 305-309.
[16] B. E. Rhoades, Fixed point iteration using finnite matrices, Transactions of the American Mathematical Society, 196 (1974), 161-176.
[17] B. E. Rhoades, A comparison of various denition of contractive mapping, Transactions of the American Mathematical Society, 226(1977), 257-290.
[18] B. E. Rhoades, Fixed point theorems and stability results for fixed point iteration procedure II, Indian Journal of Pure and Applied Mathematics, 24 (11) (1993), 691-703.
[19] B. E. Rhoades and S. M. Soltuz, The equivalence between Mann-Ishikawa iterations and multi-step iteration, Nonlinear Analysis, 58(2004), 219-228.
[20] H. H. Schaer, Topological Vector Spaces, Springer-Verlag, 1999.
[21] X. Weng, Fixed point iteration for local strictly pseudo-contractive mapping, Proceedings of the American Mathematical Society, 113 (1991) no.3, 727-731.
[22] T. Zamrescu, Fixed point theorems in metric spaces, Archiv der Mathematik (Basel), 23(1972), 292-298.