Abstract
In this paper, we consider S-iteration to find fixed points of continuous mappings on an arbitrary
interval. We give some necessary and sufficient conditions for the convergence of this iteration. Also, we proved
that the rate of convergence of S-iteration is better than some other iterations for continuous and nondecreasing
mappings. It is also noted that the method of proof of Lemma 3 using S-iteration is slightly different from that using
the iteration schemes like Mann.
Keywords
Computational cost continuous function convergence theorems fixed point rate of convergence
References
- Agarwal RP, O’Regan D, Sahu DR, 2007. Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex. Anal., 8(1): 61–79.
- Mann WR, 1953. Mean value methods in iteration. Proc. Amer. Math. Soc., 4: 506–510.
- Ishikawa S, 1974. Fixed points by a new iteration method. Proc. Amer. Math. Soc., 44: 147-150.
- Rhoades BE, 1974. Fixed point iterations using in finite matrices. Trans. Amer. Math. Soc., 196: 161-176.
- Borwein D, Borwein J, 1991. Fixed point iterations for real functions. J. Math. Anal. Appl., 157(1): 112-126.
- Qing Y, Qihou L, 2006. The necessary and sufficient condition for the convergence of Ishikawa iteration on an arbitrary interval. J. Math. Anal. Appl., 323 (2): 1383-1386.
- Rhoades BE, 1976. Comments on two fixed point iteration methods. J. Math. Anal. Appl., 56: 741–750.
- Phuengrattana W, Suantai S, 2011. On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval. J. Comput. Appl. Math., 235: 3006-3014
- Noor MA, 2000. New approximation schemes for general variational inequalities. J. Math. Anal. Appl., 251:217–229.
- Dong QL, He S, Liu X, 2013. Rate of convergence of Mann, Ishikawa and Noor iterations for continuous functions on an arbitrary interval. J. Ineq. Appl., 2013.1: doi: 10.1186/1029-242X-2013-269.
- Khan SH, 2013. A Picard-Mann hybrid iterative process. Fixed Point Theory and Appl., doi:10.1186/1687-1812-2013-69.
- Sahu DR, 2011. Applications of S iteration process to constrained minimization problems and split feasibility problems. Fixed Point Theory, 12(1): 187-204.
- Karahan I, Ozdemir M, 2013. Fixed point problems of Picard-Mann hybrid iterative process for continuous functions on an arbitrary interval. Fixed Point Theory and Appl., 2013:244, doi: 10.1186/1687-1812-2013-244.
Abstract
References
- Agarwal RP, O’Regan D, Sahu DR, 2007. Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex. Anal., 8(1): 61–79.
- Mann WR, 1953. Mean value methods in iteration. Proc. Amer. Math. Soc., 4: 506–510.
- Ishikawa S, 1974. Fixed points by a new iteration method. Proc. Amer. Math. Soc., 44: 147-150.
- Rhoades BE, 1974. Fixed point iterations using in finite matrices. Trans. Amer. Math. Soc., 196: 161-176.
- Borwein D, Borwein J, 1991. Fixed point iterations for real functions. J. Math. Anal. Appl., 157(1): 112-126.
- Qing Y, Qihou L, 2006. The necessary and sufficient condition for the convergence of Ishikawa iteration on an arbitrary interval. J. Math. Anal. Appl., 323 (2): 1383-1386.
- Rhoades BE, 1976. Comments on two fixed point iteration methods. J. Math. Anal. Appl., 56: 741–750.
- Phuengrattana W, Suantai S, 2011. On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval. J. Comput. Appl. Math., 235: 3006-3014
- Noor MA, 2000. New approximation schemes for general variational inequalities. J. Math. Anal. Appl., 251:217–229.
- Dong QL, He S, Liu X, 2013. Rate of convergence of Mann, Ishikawa and Noor iterations for continuous functions on an arbitrary interval. J. Ineq. Appl., 2013.1: doi: 10.1186/1029-242X-2013-269.
- Khan SH, 2013. A Picard-Mann hybrid iterative process. Fixed Point Theory and Appl., doi:10.1186/1687-1812-2013-69.
- Sahu DR, 2011. Applications of S iteration process to constrained minimization problems and split feasibility problems. Fixed Point Theory, 12(1): 187-204.
- Karahan I, Ozdemir M, 2013. Fixed point problems of Picard-Mann hybrid iterative process for continuous functions on an arbitrary interval. Fixed Point Theory and Appl., 2013:244, doi: 10.1186/1687-1812-2013-244.
Details
| Primary Language | Turkish |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Matematik / Mathematics |
| Authors | |
| Publication Date | June 30, 2018 |
| Submission Date | December 25, 2017 |
| Acceptance Date | February 21, 2018 |
| Published in Issue | Year 2018 Volume: 8 Issue: 2 |