Araştırma Makalesi
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An Examination of Data Dependence For Jungck-Type Iteration Method

Yıl 2020, Cilt: 36 Sayı: 3, 374 - 384, 31.12.2020

Öz

Iteration methods are an important field of study in fixed point theory and have an extensive literature. Different types of iteration methods were defined in many spaces by researchers, and the results such as convergence, rate of convergence, stability and data dependence of these methods were examined. In this study, a new iteration method of Jungck Type was defined and the convergence of this method for a certain mapping class was investigated. Then, using this mapping class for this iteration method, the results of stability and data dependence were obtained. Additionally, the rate of convergence of the newly defined iteration method with Jungck CR iteration method was compared under suitable conditions and an example supporting this result was given.

Kaynakça

  • [1] Mann, W.R. 1953. Mean value methods in iteration. Proceedings of the American Mathematical Society, 4, 506–510.
  • [2] Ishikawa, S. 1974. Fixed points by a new iteration method. Proceedings of the American Mathematical Society, 44(1), 147–150.
  • [3] Noor, M.A. 2000. New approximation schemes for general variational inequalities. Journal of Mathematical Analysis and Applications, 251(1), 217–229.
  • [4] Gürsoy F, Karakaya V, Rhoades B.E. 2013. Data dependence results of new multistep and S-iterative schemes for contractive-like operators. Fixed Point Theory and Applications, 2013: 1-12.
  • [5] Dogan, K. 2019. A Comparative Study On Some Recent Iterative Schemes. Journal Of Nonlinear And Convex Analysis, 20(11), 2411-2423.
  • [6] Atalan, Y. 2019. On a New Fixed Point Iterative Algorithm For General Variational Inequalities. Journal Of Nonlınear And Convex Analysis, 20 (11), 2371-2386.
  • [7] Jungck, G. 1976. Commuting mappings and fixed points. American Mathematical Monthly, 83(4), 261-263.
  • [8] Razani, A., Bagherboum, M. 2013. Convergence and stability of Jungck-type iterative procedures in convex b-metric spaces. Fixed Point Theory and Applications, 2013(1), 1-17.
  • [9] Okeke, G. A., & Kim, J. K. 2016. Convergence and (S, T)-stability almost surely for random Jungck-type iteration processes with applications. Cogent Mathematics, 3(1), 1258768.
  • [10] Khan, A. R., Gürsoy, F., Karakaya, V., 2016. Jungck-Khan iterative scheme and higher convergence rate. International Journal of Computer Mathematics, 93(12), 1029-2105.
  • [11] Singh, S.L., Bhatnagar, C. Mishra, S.N. 2005. Stability of Jungck-type iterative procedures. International Journal of Mathematics and Mathematical Sciences, 19, 3035–3043.
  • [12] Olatinwo, M.O. 2008. Some stability and strong convergence results for the Jungck-Ishikawa iteration process. Creative Mathematics and Informatics, 17, 33–42.
  • [13] Olatinwo, M.O. 2008. A generalization of some convergence results using a Jungck-Noor three-step iteration process in arbitrary Banach space. Polytechnica Posnaniensis, 40, 37–43.
  • [14] Chugh, R., Kumar, V. 2011. Strong Convergence and Stability results for Jungck-SP iterative scheme. International Journal of Computer Applications, 36(12), 40-46.
  • [15] Hussain, N., Kumar, V. Kutbi, M. A. 2013. On rate of convergence of Jungck-type iterative schemes, Abstract and Applied Analysis, 2013, 1-15.
  • [16] Agarwal R. P., O’Regan D., Sahu D. R., 2007, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, Journal of Nonlinear and Convex Analysis, vol. 8, no. 1, pp. 61–79.
  • [17] Khan, A. R., Kumar, V., Hussain, N. 2014. Analytical and numerical treatment of Jungck-type iterative schemes. Applied Mathematics and Computation, 231, 521-535.
  • [18] Akewe, H. 2016. The stability of a modified Jungck-Mann hybrid fixed point iteration procedure. International Journal of Mathematical Analysis and Optimization: Theory and Applications, (2016), 95-104.
  • [19] Atalan, Y., Karakaya, V. 2019. Obtaining New Fixed Point Theorems By Using Generalized Banach-Contraction Principle. Journal of Institue Of Science and Technology, 35(3). 34-45.
  • [20] Jungck, G., Hussain, N. 2007. Compatible Maps and Invariant approximations. J. Math. Anal. Appl., 325 (2), 1003-1012.
  • [21] Weng X, 1991. Fixed Point Iteration for Local Strictly Pseudo-Contractive Mapping. Proceedings of the American Mathematical Society, 113 (3): 727-731.
  • [22] Soltuz S.M, Grosan T. 2008. Data dependence for Ishikawa iteration when dealing with contractive like operators. Fixed Point Theory. (2008), 1-7.
  • [23] Knopp K, 1931. Theory and Application of Infinite Series. Berlin.

Jungck-Tipi İterasyon Yöntemi İçin Veri Bağlılığının İncelenmesi

Yıl 2020, Cilt: 36 Sayı: 3, 374 - 384, 31.12.2020

Öz

İterasyon yöntemleri sabit nokta teorisinde önemli bir çalışma alanı olup geniş bir literatüre sahiptir. Araştırmacılar tarafından birçok uzayda farklı türden iterasyon yöntemleri tanımlanarak, bu yöntemlerin yakınsaklığı, yakınsaklık hızları, kararlılığı ve veri bağlılığı gibi sonuçlar irdelenmiştir. Bu çalışmada, Jungck tipi yeni bir iterasyon yöntemi tanımlanarak bu yöntemin belirli bir dönüşüm sınıfı için yakınsaklığı incelenmiştir. Daha sonra bu iterasyon yöntemi için söz konusu dönüşüm sınıfı kullanılarak kararlılık ve veri bağlılığı sonuçları elde edilmiştir. Ayrıca yeni tanımlanan iterasyon yönteminin uygun şartlar altında yakınsaklık hızı Jungck CR iterasyon yöntemiyle karşılaştırılarak bu sonucu destekleyen örnek verilmiştir.

Kaynakça

  • [1] Mann, W.R. 1953. Mean value methods in iteration. Proceedings of the American Mathematical Society, 4, 506–510.
  • [2] Ishikawa, S. 1974. Fixed points by a new iteration method. Proceedings of the American Mathematical Society, 44(1), 147–150.
  • [3] Noor, M.A. 2000. New approximation schemes for general variational inequalities. Journal of Mathematical Analysis and Applications, 251(1), 217–229.
  • [4] Gürsoy F, Karakaya V, Rhoades B.E. 2013. Data dependence results of new multistep and S-iterative schemes for contractive-like operators. Fixed Point Theory and Applications, 2013: 1-12.
  • [5] Dogan, K. 2019. A Comparative Study On Some Recent Iterative Schemes. Journal Of Nonlinear And Convex Analysis, 20(11), 2411-2423.
  • [6] Atalan, Y. 2019. On a New Fixed Point Iterative Algorithm For General Variational Inequalities. Journal Of Nonlınear And Convex Analysis, 20 (11), 2371-2386.
  • [7] Jungck, G. 1976. Commuting mappings and fixed points. American Mathematical Monthly, 83(4), 261-263.
  • [8] Razani, A., Bagherboum, M. 2013. Convergence and stability of Jungck-type iterative procedures in convex b-metric spaces. Fixed Point Theory and Applications, 2013(1), 1-17.
  • [9] Okeke, G. A., & Kim, J. K. 2016. Convergence and (S, T)-stability almost surely for random Jungck-type iteration processes with applications. Cogent Mathematics, 3(1), 1258768.
  • [10] Khan, A. R., Gürsoy, F., Karakaya, V., 2016. Jungck-Khan iterative scheme and higher convergence rate. International Journal of Computer Mathematics, 93(12), 1029-2105.
  • [11] Singh, S.L., Bhatnagar, C. Mishra, S.N. 2005. Stability of Jungck-type iterative procedures. International Journal of Mathematics and Mathematical Sciences, 19, 3035–3043.
  • [12] Olatinwo, M.O. 2008. Some stability and strong convergence results for the Jungck-Ishikawa iteration process. Creative Mathematics and Informatics, 17, 33–42.
  • [13] Olatinwo, M.O. 2008. A generalization of some convergence results using a Jungck-Noor three-step iteration process in arbitrary Banach space. Polytechnica Posnaniensis, 40, 37–43.
  • [14] Chugh, R., Kumar, V. 2011. Strong Convergence and Stability results for Jungck-SP iterative scheme. International Journal of Computer Applications, 36(12), 40-46.
  • [15] Hussain, N., Kumar, V. Kutbi, M. A. 2013. On rate of convergence of Jungck-type iterative schemes, Abstract and Applied Analysis, 2013, 1-15.
  • [16] Agarwal R. P., O’Regan D., Sahu D. R., 2007, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, Journal of Nonlinear and Convex Analysis, vol. 8, no. 1, pp. 61–79.
  • [17] Khan, A. R., Kumar, V., Hussain, N. 2014. Analytical and numerical treatment of Jungck-type iterative schemes. Applied Mathematics and Computation, 231, 521-535.
  • [18] Akewe, H. 2016. The stability of a modified Jungck-Mann hybrid fixed point iteration procedure. International Journal of Mathematical Analysis and Optimization: Theory and Applications, (2016), 95-104.
  • [19] Atalan, Y., Karakaya, V. 2019. Obtaining New Fixed Point Theorems By Using Generalized Banach-Contraction Principle. Journal of Institue Of Science and Technology, 35(3). 34-45.
  • [20] Jungck, G., Hussain, N. 2007. Compatible Maps and Invariant approximations. J. Math. Anal. Appl., 325 (2), 1003-1012.
  • [21] Weng X, 1991. Fixed Point Iteration for Local Strictly Pseudo-Contractive Mapping. Proceedings of the American Mathematical Society, 113 (3): 727-731.
  • [22] Soltuz S.M, Grosan T. 2008. Data dependence for Ishikawa iteration when dealing with contractive like operators. Fixed Point Theory. (2008), 1-7.
  • [23] Knopp K, 1931. Theory and Application of Infinite Series. Berlin.
Toplam 23 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Mühendislik
Bölüm Makale
Yazarlar

Samet Maldar

Yayımlanma Tarihi 31 Aralık 2020
Yayımlandığı Sayı Yıl 2020 Cilt: 36 Sayı: 3

Kaynak Göster

APA Maldar, S. (2020). An Examination of Data Dependence For Jungck-Type Iteration Method. Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi, 36(3), 374-384.
AMA Maldar S. An Examination of Data Dependence For Jungck-Type Iteration Method. Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi. Aralık 2020;36(3):374-384.
Chicago Maldar, Samet. “An Examination of Data Dependence For Jungck-Type Iteration Method”. Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi 36, sy. 3 (Aralık 2020): 374-84.
EndNote Maldar S (01 Aralık 2020) An Examination of Data Dependence For Jungck-Type Iteration Method. Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi 36 3 374–384.
IEEE S. Maldar, “An Examination of Data Dependence For Jungck-Type Iteration Method”, Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi, c. 36, sy. 3, ss. 374–384, 2020.
ISNAD Maldar, Samet. “An Examination of Data Dependence For Jungck-Type Iteration Method”. Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi 36/3 (Aralık 2020), 374-384.
JAMA Maldar S. An Examination of Data Dependence For Jungck-Type Iteration Method. Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi. 2020;36:374–384.
MLA Maldar, Samet. “An Examination of Data Dependence For Jungck-Type Iteration Method”. Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi, c. 36, sy. 3, 2020, ss. 374-8.
Vancouver Maldar S. An Examination of Data Dependence For Jungck-Type Iteration Method. Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi. 2020;36(3):374-8.

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