Banach Uzaylarında Zenginleştirilmiş Daralmalarla İlişkilendirilen Bir İteratif Algoritma Üzerine Bazı Sonuçlar
Abstract
Bu çalışmada, Banach uzaylarında zenginleştirilmiş daralmalar vasıtasıyla tanımlanan, ortalama dönüşüm sınıfları ile üretilen Picard-S algoritması ele alınmıştır. Bu dönüşüm sınıfları kullanılarak Picard-S algoritmasından elde edilen iteratif dizinin, zenginleştirilmiş daralmanın sabit noktasına yakınsaklığı kontrol dizileri üzerinde herhangi ek şartlar olmaksızın elde edilmiştir. Bu dönüşüm sınıfları ile Picard-S ve CR algoritmalarından elde edilen iteratif dizilerin sabit noktaya yakınsaklıklarının denk olduğu gösterilmiş ve aynı dönüşüm sınıfları için Picard-S algoritmasının veri bağlılığı üzerine bir sonuç elde edilmiştir. Elde edilen tüm sonuçlar sonsuz boyutlu Banach uzaylarında sayısal örneklerle desteklenmiştir.
References
- É. Picard, Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives, Journal de Mathématiques pures et appliquées. 6 (1890), 145-210.
- W.R. Mann, Mean value methods in iteration, Proceedings of the American Mathematical Society. 4 (1953), 506-510.
- S. Ishikawa, Fixed points by a new iteration method, Proceedings of the American Mathematical Society. 44(1) (1974), 147-150.
- R. Chugh, V. Kumar, S. Kumar, Strong convergence of a new three step iterative scheme in Banach spaces, American Journal of Computational Mathematics. 2 (4) (2012), 345-357.
- F. Gürsoy, V. Karakaya, A Picard-S hybrid type iteration method for solving a differential equation with retarded argument, (2014), arXiv preprint arXiv:1403.2546.
- F. Gürsoy, A Picard-S iterative method for approximating fixed point of weak-contraction mappings, Filomat. 30 (10) (2016), 2829-2845.
- M. Ertürk, F. Gürsoy, Some convergence, stability and data dependency results for a Picard-S iteration method of quasi-strictly contractive operators, Mathematica Bohemica. 144 (1) (2019), 69-83.
- V. Berinde, M. Păcurar, Approximating fixed points of enriched contractions in Banach spaces, Journal of Fixed Point Theory and Applications. 22 (2020), 1-10.
- M. Abbas, R. Anjum, V. Berinde, Equivalence of certain iteration processes obtained by two new classes of operators, Mathematics. 9 (18) (2021), 2292.
- R. Anjum, N. Ismail, A. Bartwal, Implication between certain iterative processes via some enriched mappings, The Journal of Analysis. (2023), 1-14.
- L. Qihou, A convergence theorem of the sequence of Ishikawa iterates for quasi-contractive mappings, Journal of Mathematical Analysis and Applications. 146 (2) (1990), 301-305.
- V. Berinde, Iterative Approximation of Fixed Points, Springer, Berlin, 2007.
- R.E. Megginson, An Introduction to Banach Space Theory, Springer, New York, 1998.
Abstract
In this study, the Picard-S algorithm, which is generated by the average mapping classes defined by the enriched contractions in Banach spaces, is considered. Using these mapping classes, the convergence of the iterative sequence obtained from the Picard-S algorithm to the fixed point of the enriched contraction has been obtained without any additional conditions on the control sequences. It has been shown that the convergence of the iterative sequences generated by Picard-S and the CR algorithms with these mapping classes to the fixed point is equivalent, and a result regarding the data dependency of the Picard-S algorithm for the same mapping classes has been obtained. All results obtained are supported by numerical examples in infinite-dimensional Banach spaces.
References
- É. Picard, Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives, Journal de Mathématiques pures et appliquées. 6 (1890), 145-210.
- W.R. Mann, Mean value methods in iteration, Proceedings of the American Mathematical Society. 4 (1953), 506-510.
- S. Ishikawa, Fixed points by a new iteration method, Proceedings of the American Mathematical Society. 44(1) (1974), 147-150.
- R. Chugh, V. Kumar, S. Kumar, Strong convergence of a new three step iterative scheme in Banach spaces, American Journal of Computational Mathematics. 2 (4) (2012), 345-357.
- F. Gürsoy, V. Karakaya, A Picard-S hybrid type iteration method for solving a differential equation with retarded argument, (2014), arXiv preprint arXiv:1403.2546.
- F. Gürsoy, A Picard-S iterative method for approximating fixed point of weak-contraction mappings, Filomat. 30 (10) (2016), 2829-2845.
- M. Ertürk, F. Gürsoy, Some convergence, stability and data dependency results for a Picard-S iteration method of quasi-strictly contractive operators, Mathematica Bohemica. 144 (1) (2019), 69-83.
- V. Berinde, M. Păcurar, Approximating fixed points of enriched contractions in Banach spaces, Journal of Fixed Point Theory and Applications. 22 (2020), 1-10.
- M. Abbas, R. Anjum, V. Berinde, Equivalence of certain iteration processes obtained by two new classes of operators, Mathematics. 9 (18) (2021), 2292.
- R. Anjum, N. Ismail, A. Bartwal, Implication between certain iterative processes via some enriched mappings, The Journal of Analysis. (2023), 1-14.
- L. Qihou, A convergence theorem of the sequence of Ishikawa iterates for quasi-contractive mappings, Journal of Mathematical Analysis and Applications. 146 (2) (1990), 301-305.
- V. Berinde, Iterative Approximation of Fixed Points, Springer, Berlin, 2007.
- R.E. Megginson, An Introduction to Banach Space Theory, Springer, New York, 1998.
Details
| Primary Language | English |
|---|---|
| Subjects | Operator Algebras and Functional Analysis |
| Journal Section | Articles |
| Authors | |
| Early Pub Date | December 18, 2023 |
| Publication Date | December 31, 2023 |
| Submission Date | October 23, 2023 |
| Acceptance Date | November 30, 2023 |
| Published in Issue | Year 2023 Volume: 5 Issue: 2 |
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