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Öğe A New Note on Convergence and Data Dependence Concept for a Volterra Integral Equation by Fixed Point Iterative Algorithm(Islamic Azad University, 2023) Atalan, Yunus; Maldar, S.In this paper, we prove that a three-step iterative algorithm converges strongly to the solution of a functional Volterra integral equation of the second kind. We also show the solution presented in the convergence theorem can obtain by removing a strong restriction imposed on the control sequence. Finally, we analyse the data-dependence result by using this iterative algorithm, and to support obtained result we give an example.Öğe An example of data dependence result for the class of almost contraction mappings(University of Maragheh, 2020) Atalan, Yunus; Karakaya, VatanIn the present paper, we show that S* iteration method can be used to approximate fixed point of almost contraction mappings. Furthermore, we prove that this iteration method is equivalent to CR iteration method and it produces a slow convergence rate compared to the CR iteration method for the class of almost contraction mappings. We also present table and graphic to support this result. Finally, we obtain a data dependence result for almost contraction mappings by using S* iteration method and in order to show validity of this result we give an example.Öğe Analyzing stability and data dependence notions by a novel Jungck-Type iteration method(Naim ÇAĞMAN, 2023) Atalan, Yunus; Erbaş, EsraFinding the ideal circumstances for a mapping to have a fixed point is the fundamental goal of fixed point theory. These criteria can also be used for the structure under investigation. One of this theory’s most well-known theorems, Banach’s fixed point theorem, has been expanded adopting various methods, making it possible to conduct numerous research studies. Thanks to the Jungck-Contraction Theorem, which has been proven through commutative mappings, many fixed point theorems have been obtained using classical fixed point iteration methods and newly defined methods. This study aims to investigate the convergence, stability, convergence rate, and data dependency of the new four-step fixed-point iteration method. Nontrivial examples are also included to support some of the results herein.Öğe Common fixed point theorems for complex-valued mappings with applications(Kangwon-Kyungki Mathematical Society, 2022) Maldar, Samet; Atalan, YunusThe aim of this paper is to obtain some results which belong to fixed point theory such as strong convergence, rate of convergence, stability, and data dependence by using the new Jungck-type iteration method for a mapping defined in complex-valued Banach spaces. In addition, some of these results are supported by nontrivial numerical examples. Finally, it is shown that the sequence obtained from the new iteration method converges to the solution of the functional integral equation in complex-valued Banach spaces. The results obtained in this paper may be interpreted as a generalization and improvement of the previously known results.Öğe Comparison rate of convergence and data dependence for a new iteration method(Tbilisi Mathematical, 2020) Maldar, Samet; Atalan, Yunus; Doğan, KadriIn this paper, we have defined hyperbolic type of some iteration methods. The new iteration has been investigated convergence for mappings satisfying certain condition in hyperbolic spaces. It has been proved that this iteration is equivalent in terms of convergence with another iteration method in the same spaces. The rate of convergence of these two iteration methods have been compared. We have investigated data dependence result using hyperbolic type iteration. Finally, we have given numerical examples about rate of convergence and data dependence.Öğe Investigation of some fixed point theorems in hyperbolic spaces for a three step iteration process(Korean Mathematical Society, 2019) Atalan, Yunus; Karakaya, VatanIn the present paper, we investigate the convergence, equivalence of convergence, rate of convergence and data dependence results using a three step iteration process for mappings satisfying certain contractive condition in hyperbolic spaces. Also we give non-trivial examples for the rate of convergence and data dependence results to show effciency of three step iteration process. The results obtained in this paper may be interpreted as a refinement and improvement of the previously known results.Öğe Iterative approximation of fixed points and applications to two-point second-order boundary value problems and to machine learning(Elsevier B.V., 2021) Hacıoğlu, Emirhan; Gürsoy, Faik; Maldar, Samet; Atalan, Yunus; Milovanovic, Gradimir V.In this paper, we revisit two recently published papers on the iterative approximation of fixed points by Kumam et al. (2019) [17] and Maniu (2020) [19] and reproduce convergence, stability, and data dependency results presented in these papers by removing some strong restrictions imposed on parametric control sequences. We confirm the validity and applicability of our results through various non-trivial numerical examples. We suggest a new method based on the iteration algorithm given by Thakur et al. (2014) [28] to solve the two-point second-order boundary value problems. Furthermore, based on the above mentioned iteration algorithm and S-iteration algorithm, we propose two new gradient type projection algorithms and applied them to supervised learning. In both applications, we present some numerical examples to demonstrate the superiority of the newly introduced methods in terms of convergence, accuracy, and computational time against some earlier methods.Öğe Novel algorithms based on forward-backward splitting technique: effective methods for regression and classification(Springer, 2024) Atalan, Yunus; Hacıoğlu, Emirhan; Ertürk, Müzeyyen; Gürsoy, Faik; Milovanovi?, Gradimir V.In this paper, we introduce two novel forward-backward splitting algorithms (FBSAs) for nonsmooth convex minimization. We provide a thorough convergence analysis, emphasizing the new algorithms and contrasting them with existing ones. Our findings are validated through a numerical example. The practical utility of these algorithms in real-world applications, including machine learning for tasks such as classification, regression, and image deblurring reveal that these algorithms consistently approach optimal solutions with fewer iterations, highlighting their efficiency in real-world scenarios.Öğe Obtaining new fixed point theorems by using generalized banach-contraction principle(Erciyes Üniversitesi, 2019) Atalan, Yunus; Karakaya, VatanIn this study, a new three step iterative algorithm was introduced with the help of Jungck-contraction principle which is one of the remerkable generalizations of Banach-contraction principle. Also, the convergence and stability results were obtained for the pair of nonself mappings which satisfy a certain contractive condition by using this iterative algorithm in any Banach space. In addition, it was shown that the new iterative algorithm has a better convergence speed when compared the other Jungck-type iterative algorithms in the current literature, and to support this result, numerical examples were given.Öğe On a three-step iteration process for multivalued Reich-Suzuki type ? -nonexpansive and contractive mappings(Springer Science and Business Media Deutschland GmbH, 2022) Maldar, Samet; Gürsoy, Faik; Atalan, Yunus; Abbas, MujahidThe aim of this paper is twofold: (a) to revisit the results in Iqbal et al. (Numer Algorithms 1–21, 2019. https:doi.org/10.1007/s11075-019-00854-z) and prove some convergence and stability results for the subclass of multivalued contractive operators under some mild conditions (b) to introduce a multivalued Reich-Suzuki type ?-nonexpansive mappings and present some fixed point results for this class of mappings. We considered a more natural notion of stability called weak w2-stability instead of simple stability in Iqbal et al. (Numer Algorithms 1–21, 2019. https:doi.org/10.1007/s11075-019-00854-z). Some illustrative examples to support the results obtained herein are also presented.Öğe The local and semilocal convergence analysis of new Newton-like iteration methods(SCIENTIFIC TECHNICAL RESEARCH COUNCIL TURKEY-TUBITAK, 2018) Karakaya, Vatan; Doğan, Kadri; Atalan, Yunus; Bouzara, Nour El HoudaThe aim of this paper is to find new iterative Newton-like schemes inspired by the modified Newton iterative algorithm and prove that these iterations are faster than the existing ones in the literature. We further investigate their behavior and finally illustrate the results by numerical examples.Öğe Yeni bir iterasyon yöntemi için hemen-hemen büzülme dönüşümleri altında bazı sabit nokta teoremleri(Marmara Üniversitesi, 2018) Atalan, YunusBu makalede (1) ile verilen iterasyon yönteminden daha sade olan yeni bir iterasyon yöntemi tanımlanmıştır. Bu iterasyon yönteminin hemen hemen büzülme dönüşümü şartını sağlayan iki operatörün ortak sabit noktasına yakınsak olduğu ispatlanmıştır. Ayrıca yeni iterasyon yönteminin (1) ile verilen iterasyon yönteminden daha hızlı olduğu gösterilmiştir ve bu sonucu destekleyen bir nümerik örnek verilmiştir. Son olarak, hemen hemen büzülme dönüşümü şartını sağlayan iki operatör için yeni tanımlanan iterasyon kullanılarak veri bağlılığı sonucu elde edilmiştir.
