Research Article
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Convergence and Stability Analysis of a New Four-Step Fixed-Point Algorithm

Year 2022, , 57 - 70, 30.06.2022

Yunus Atalan , Esra Kılıç

https://doi.org/10.29002/asujse.1096163

Abstract

The concept of stability is studied on many different types of mathematical structures. This concept can be thought of as the small changes that will be applied in the structure studied should not disrupt the functioning of this structure. In this context, we performed the convergence and stability analysis of the new four-step iteration algorithm that we defined in this study, under appropriate conditions. In addition, we execute a speed comparison with existing algorithms to prove that the new algorithm is effective and useful, and we gave a numerical example to support our result.

Keywords

Stability Convergence Fixed-point algorithm Convergence speed

References

  • [1] I.K. Argyros, S. Hilout, Computational Methods in Nonlinear Analysis: Efficient algorithms, fixed point theory and applications. (World Scientific, London, 2013).
  • [2] J. Borwein, B. Sims, The Douglas–Rachford algorithm in the absence of convexity, H.H. Bauschke Burachik, R.S. Combettes, P.L. Elser, V. Luke, D.R. ve Wolkowicz, H., ed. Fixed-Point Algorithms for Inverse Problems in Science and Engineering (Springer, New York, 2011).
  • [3] L.C. Ceng, Q.H. Ansari, J. C. Yao, Some iterative methods for finding fixed points and for solving constrained convex minimization problems. Nonlinear Analysis: Theory, Methods & Applications 74(16) (2011) 5286-5302.
  • [4] M. Chen, W. Lu, Q. Chen, K.J. Ruchala, G. H. Olivera, A simple fixed-point approach to invert a deformation field. Medical Physics 35(1) (2008) 81-88.
  • [5] E. A. Ok, Real analysis with economic applications. (Princeton University Press, Princeton, 2007).
  • [6] N. Radde, Fixed point characterization of biological networks with complex graph topology. Bioinformatics 26(22) (2010) 2874-2880.
  • [7] C. Somarakis, J.S. Baras, Fixed point theory approach to exponential convergence in LTV continuous time consensus dynamics with delays. Proceedings of the Conference on Control and Its Applications (2013) 129-136.
  • [8] L.E.J. Brouwer, Uber abbildung der mannigfaltigkeiten. Mathematische Annalen 71(1) (1912) 97-115.
  • [9] J. Schauder, Der fixpunktsatz in functional raumen. Studia Math. 2 (1930) 171-180.
  • [10] S. Banach, Sur les operations dans les ensembles abstraits et leur applications aux equations integrales. Fund. Math. 3(1) (1922) 133-181.
  • [11] S. Maldar, F. Gürsoy, Y. Atalan, M. Abbas, On a three-step iteration process for multivalued Reich-Suzuki type α-nonexpansive and contractive mappings. J. Appl. Math. Comput. 68(2) (2022) 863-883.
  • [12] E. Hacıoğlu, F. Gürsoy, S. Maldar, Y. Atalan, G. V. Milovanović, Iterative approximation of fixed points and applications to two-point second-order boundary value problems and to machine learning. Applied Numerical Mathematics 167 (2021) 143-172.
  • [13] S. Maldar, Y. Atalan, K. Dogan, Comparison rate of convergence and data dependence for a new iteration method. Tbilisi Mathematical Journal 13(4) (2020) 65-79.
  • [14] S. Maldar, V. Karakaya, Convergence of Jungck-Kirk type iteration method with applications. Punjab University Journal of Mathematics 54(2) (2022) 75-87.
  • [15] A. R. Khan, F. Gürsoy, V. Kumar, Stability and data dependence results for the Jungck--Khan iterative scheme. Turkish J. Math. 40(3) (2016) 631-640.
  • [16] Magnus, K., Development of the stability concept in mechanics. Naturwissenschaften 46 (1959) 590-595.
  • [17] M. Urabe, Convergence of numerical iteration in solution of equations. J. Sci. Hiroshima Univ. Ser. A 19(3) (1956) 479-489.
  • [18] A. Ostrowski, The round-off stability of iterations. ZAMM Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 47(2) (1967) 77-81.
  • [19] A. M. Harder, T. L. Hicks, Stability results for fixed point iteration procedures. Math. Japonica 33 (5) (1988) 693-706.
  • [20] Y. Atalan, V. Karakaya, Stability of nonlinear Volterra-Fredholm integro differential equation: A fixed point approach. Creat. Math. Inform. 26(3) (2017) 247-254.
  • [21] K. Doğan, V. Karakaya, On the convergence and stability results for a new general iterative process. The Scientific World Journal (2014) 852475.
  • [22] B. S. Thakur, D. Thakur, M. Postolache, A new iterative scheme for numerical reckoning fixed points of Suzuki’s generalized nonexpansive mappings. App. Math. Comp. 275 (2016) 147–155.
  • [23] K. Ullah, M. Arshad, New three-step iteration process and fixed point approximation in Banach spaces. Journal of Linear and Topological Algebra 7(02) (2018) 87-100.
  • [24] N. Hussain, K. Ullah, M. Arshad, Fixed point approximatıon of Suzuki generalized nonexpansive mappings via new faster iteration process, J. Nonlinear Convex Anal. 19(8) (2018) 1383-1393.
  • [25] V. Karakaya, Y. Atalan, K. Doğan, NEH. Bouzara, Daha hızlı yeni bir iterasyon metodu için yakınsaklık analizi. İstanbul Ticaret Üniversitesi Fen Bilimleri Dergisi 15(30) (2016) 35-53.
  • [26] S.M. Şoltuz, T. Grosan, Data dependence for Ishikawa iteration when dealing with contractive-like operators. Fixed Point Theory and Applications 2008 (2008) 242916.
  • [27] X. Weng, Fixed point iteration for local strictly pseudocontractive mapping. Proc. Amer. Math. Soc. 113 (1991) 727-731.
  • [28] V. Berinde, On a family of first order difference inequalities used in the iterative approximation of fixed points. Creat. Math. Inform. 18 (2009) 110-122.
  • [29] W. Phuengrattana, S. Suantai, Comparison of the rate of convergence of various iterative methods for the class of weak contractions in Banach spaces. Thai Journal of Mathematics 11 (1) (2012) 217-226.
  • [30] V. Berinde, On the stability of some fixed point procedures. Bul. Stiințific Univ. Baia Mare Ser. B Fascicola Matematica-Informatica 18(1) (2002) 7-14.

Year 2022, , 57 - 70, 30.06.2022

Yunus Atalan , Esra Kılıç

https://doi.org/10.29002/asujse.1096163

Abstract

References

  • [1] I.K. Argyros, S. Hilout, Computational Methods in Nonlinear Analysis: Efficient algorithms, fixed point theory and applications. (World Scientific, London, 2013).
  • [2] J. Borwein, B. Sims, The Douglas–Rachford algorithm in the absence of convexity, H.H. Bauschke Burachik, R.S. Combettes, P.L. Elser, V. Luke, D.R. ve Wolkowicz, H., ed. Fixed-Point Algorithms for Inverse Problems in Science and Engineering (Springer, New York, 2011).
  • [3] L.C. Ceng, Q.H. Ansari, J. C. Yao, Some iterative methods for finding fixed points and for solving constrained convex minimization problems. Nonlinear Analysis: Theory, Methods & Applications 74(16) (2011) 5286-5302.
  • [4] M. Chen, W. Lu, Q. Chen, K.J. Ruchala, G. H. Olivera, A simple fixed-point approach to invert a deformation field. Medical Physics 35(1) (2008) 81-88.
  • [5] E. A. Ok, Real analysis with economic applications. (Princeton University Press, Princeton, 2007).
  • [6] N. Radde, Fixed point characterization of biological networks with complex graph topology. Bioinformatics 26(22) (2010) 2874-2880.
  • [7] C. Somarakis, J.S. Baras, Fixed point theory approach to exponential convergence in LTV continuous time consensus dynamics with delays. Proceedings of the Conference on Control and Its Applications (2013) 129-136.
  • [8] L.E.J. Brouwer, Uber abbildung der mannigfaltigkeiten. Mathematische Annalen 71(1) (1912) 97-115.
  • [9] J. Schauder, Der fixpunktsatz in functional raumen. Studia Math. 2 (1930) 171-180.
  • [10] S. Banach, Sur les operations dans les ensembles abstraits et leur applications aux equations integrales. Fund. Math. 3(1) (1922) 133-181.
  • [11] S. Maldar, F. Gürsoy, Y. Atalan, M. Abbas, On a three-step iteration process for multivalued Reich-Suzuki type α-nonexpansive and contractive mappings. J. Appl. Math. Comput. 68(2) (2022) 863-883.
  • [12] E. Hacıoğlu, F. Gürsoy, S. Maldar, Y. Atalan, G. V. Milovanović, Iterative approximation of fixed points and applications to two-point second-order boundary value problems and to machine learning. Applied Numerical Mathematics 167 (2021) 143-172.
  • [13] S. Maldar, Y. Atalan, K. Dogan, Comparison rate of convergence and data dependence for a new iteration method. Tbilisi Mathematical Journal 13(4) (2020) 65-79.
  • [14] S. Maldar, V. Karakaya, Convergence of Jungck-Kirk type iteration method with applications. Punjab University Journal of Mathematics 54(2) (2022) 75-87.
  • [15] A. R. Khan, F. Gürsoy, V. Kumar, Stability and data dependence results for the Jungck--Khan iterative scheme. Turkish J. Math. 40(3) (2016) 631-640.
  • [16] Magnus, K., Development of the stability concept in mechanics. Naturwissenschaften 46 (1959) 590-595.
  • [17] M. Urabe, Convergence of numerical iteration in solution of equations. J. Sci. Hiroshima Univ. Ser. A 19(3) (1956) 479-489.
  • [18] A. Ostrowski, The round-off stability of iterations. ZAMM Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 47(2) (1967) 77-81.
  • [19] A. M. Harder, T. L. Hicks, Stability results for fixed point iteration procedures. Math. Japonica 33 (5) (1988) 693-706.
  • [20] Y. Atalan, V. Karakaya, Stability of nonlinear Volterra-Fredholm integro differential equation: A fixed point approach. Creat. Math. Inform. 26(3) (2017) 247-254.
  • [21] K. Doğan, V. Karakaya, On the convergence and stability results for a new general iterative process. The Scientific World Journal (2014) 852475.
  • [22] B. S. Thakur, D. Thakur, M. Postolache, A new iterative scheme for numerical reckoning fixed points of Suzuki’s generalized nonexpansive mappings. App. Math. Comp. 275 (2016) 147–155.
  • [23] K. Ullah, M. Arshad, New three-step iteration process and fixed point approximation in Banach spaces. Journal of Linear and Topological Algebra 7(02) (2018) 87-100.
  • [24] N. Hussain, K. Ullah, M. Arshad, Fixed point approximatıon of Suzuki generalized nonexpansive mappings via new faster iteration process, J. Nonlinear Convex Anal. 19(8) (2018) 1383-1393.
  • [25] V. Karakaya, Y. Atalan, K. Doğan, NEH. Bouzara, Daha hızlı yeni bir iterasyon metodu için yakınsaklık analizi. İstanbul Ticaret Üniversitesi Fen Bilimleri Dergisi 15(30) (2016) 35-53.
  • [26] S.M. Şoltuz, T. Grosan, Data dependence for Ishikawa iteration when dealing with contractive-like operators. Fixed Point Theory and Applications 2008 (2008) 242916.
  • [27] X. Weng, Fixed point iteration for local strictly pseudocontractive mapping. Proc. Amer. Math. Soc. 113 (1991) 727-731.
  • [28] V. Berinde, On a family of first order difference inequalities used in the iterative approximation of fixed points. Creat. Math. Inform. 18 (2009) 110-122.
  • [29] W. Phuengrattana, S. Suantai, Comparison of the rate of convergence of various iterative methods for the class of weak contractions in Banach spaces. Thai Journal of Mathematics 11 (1) (2012) 217-226.
  • [30] V. Berinde, On the stability of some fixed point procedures. Bul. Stiințific Univ. Baia Mare Ser. B Fascicola Matematica-Informatica 18(1) (2002) 7-14.
There are 30 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

Yunus Atalan 0000-0002-5912-7087

Esra Kılıç 0000-0002-3460-5089

Publication Date June 30, 2022
Submission Date March 30, 2022
Acceptance Date April 27, 2022
Published in Issue Year 2022

Cite

APA Atalan, Y., & Kılıç, E. (2022). Convergence and Stability Analysis of a New Four-Step Fixed-Point Algorithm. Aksaray University Journal of Science and Engineering, 6(1), 57-70. https://doi.org/10.29002/asujse.1096163

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