On Data Dependency and Solutions of Nonlinear Fredholm Integral Equations with the Three-Step Iteration Method

Yıl 2023, Cilt: 5 Sayı: 2, 53 - 64, 20.11.2023

Lale Cona , Kadir Şengül

https://doi.org/10.54286/ikjm.1303219

Cited By: 1

Öz

In this study, the solution of the second type of homogeneous nonlinear Fredholm integral equations is investigated using a three-step iteration algorithm. In other words, it has been shown that the sequences obtained from this algorithm converge to the solution of the mentioned equations. Also, data dependency is obtained for the second type of homogeneous nonlinear Fredholm integral equations and this result is supported by an example.

Anahtar Kelimeler

Banach Fixed Point Theorem, Nonlinear Fredholm Integral Equations, Three-Step Iteration Method, Data Dependency

Kaynakça

• 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, 8(1): 61-79.
• Akbulut, A. (2007) Application of fixed point theorems to Cauchy problem and integral equations. M.Sc. thesis, Gazi University, Ankara, Turkey.
• Atalan, Y. (2017) Solutions of some differential and integral equations with fixed point approach. Ph.D. thesis, Yıldız Technical University, İstanbul, Turkey.
• Atalan, Y., Gürsoy, F., Khan, A.R. (2021) Convergence of S-iterative method to a solution of Fredholm integral equation and data dependency. Facta Universitiy, 4(36): 685-694.
• Atalan, Y. (2019) Examination of the solution of a class of functional-integral equation under iterative approach. Journal of the Institute of Science and Technology, 9(3): 1622-1632.
• Banach, S. (1922) Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundamenta Mathematicae, 3(1): 133-181.
• Brouwer, L. E. J. (1911) Über abbildung von mannigfaltigkeiten. Mathematische Annalen, 71(1): 97-115.
• Chugh, R., Kumar, V., Kumar, S. (2012) Strong convergence of a new three step iterative scheme in Banach spaces. American Journal of Computational Mathematics, 2(4): 345-357.
• Doğan, K. (2016) Some geometrical properties and new fixed point iteration procedures. Ph.D. thesis, Yıldız Technical University, İstanbul, Turkey.
• Gürsoy, F. (2014) Investigation of convergences and stabilities of some new fixed point iteration procedures. Ph.D. thesis, Yıldız Technical University, İstanbul, Turkey.
• Gürsoy F. (2016) A Picard-S iterative method for approximating fixed point of weak-contraction mappings. Filomat, 30(10): 2829-2845.
• Hussain, N., Chugh, R., Kumar, V., Rafiq, A. (2012) On the rate of convergence of Kirk-type iterative schemes. Journal of Applied Mathematics, 2012.
• Ishikawa, S. (1974) Fixed points by a new iteration method. Proceedings of the American Mathematical Society, 44(1): 147-150.
• Karakaya, V., Atalan, Y., Doğan, K., Bouzara, N. E. H. (2017) Some fixed point results for a new three steps iteration process in Banach spaces. Fixed Point Theory, 18(2): 625-640.
• Khan, S. H. (2013) A Picard-Mann hybrid iterative process. Fixed Point Theory and Applications, 2013(69): 1-10.
• Kirk, W. A. (1971) On successive approximations for nonexpansive mappings in Banach spaces. Glasgow Mathematical Journal, 12(1): 6-9.
• Krasnosel’skii, M. A. (1955) Two comments on the method of successive approximations. Usp. Math. Nauk, 10(1): 123-127.
• Mann, W. R. (1953) Mean value methods in iteration. Proceedings of the American Mathematical Society, 4(3): 506-510.
• Noor, M. A. (2000) New approximation schemes for general variational inequalities. Journal of Mathematical Analysis and applications, 251(1): 217-229.
• Olatinwo, M. O. (2009) Some stability results for two hybrid fixed point iterative algorithms in normed linear space. Matematički Vesnik, 61(4): 247-256.
• Phuengrattana, W., Suantai, S. (2011) 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(9): 3006-3014.
• Picard, E. (1890) Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives. Journal de Mathématiques Pures et Appliquées, 6: 145-210.
• Rhoades, B. E., Şoltuz, Ş. M. (2004) The equivalence between Mann–Ishikawa iterations and multistep iteration. Nonlinear Analysis: Theory, Methods & Applications, 58(1-2): 219-228.
• Soltuz, Ş. M., Grosan, T. (2008) Data dependence for Ishikawa iteration when dealing with contractive-like operators. Fixed Point Theory and Applications, 2008(1): 1-7.
• Weng, X. (1991) Fixed point iteration for local strictly pseudo-contractive mapping. Proceedings of the American Mathematical Society, 113(3): 727-731.