Approximating the fixed points of Suzuki's generalized non-expansive map via an efficient iterative scheme with an application
Main Article Content
Abstract
This paper is aimed at proving the efficiency of a faster iterative scheme called $PC^*$-iterative scheme to approximate the fixed points for the class of Suzuki's Generalized non-expansive mapping in a uniformly convex Banach space. We will prove some weak and strong convergence results. It is justified numerically that the $PC^*$-iterative scheme converges faster than many other remarkable iterative schemes. We will also provide numerical illustrations with graphical representations to prove the efficiency of $PC^*$ iterative scheme. As an application of the solution of a fractional differential equation is obtained by using $PC^*$ iterative scheme.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
References
bibitem{1}Ali, D., Hussain, A., Karapinar, E. and Cholamjiak, P.(2022). Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces. Open Mathematics, 20(1), 1753-1769.
bibitem{2} Ali, J. and Ali, F.(2020). A new iterative scheme to approximating fixed points and the solution of a delay differential equation. Journal of Nonlinear and Convex Analysis, 21(9), 2151-2163.
bibitem{3} Bae, J.S.(1984). Fixed point theorems of generalized nonexpansive maps. Journal of the Korean mathematical society, 21(2), 233-248.
bibitem{4} Banach, S.(1922). Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fund. math, 3(1), 133-181.
bibitem{5} Berinde, V.(2004). Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators. Fixed Point Theory and Applications, 2004(2), 1-9.
bibitem{6} Berinde, V.(2007). A convergence theorem for Mann iteration in the class of Zamfirescu operators. Seria Matematica Informatica XLV, 1, 33-41.
bibitem{7} Bhutia, J.D. and Tiwary, K.(2019). New iteration process for approximating fixed points in Banach spaces. Journal of Linear and Topological Algebra, 8(04),237-250.
bibitem{8} Browder, F.E. (1965). Nonexpansive nonlinear operators in a Banach space. Proceedings of the National Academy of Sciences, 54(4), 1041-1044.
bibitem{9} Gautam, P. and Kaur, C.(2022). Fixed points of interpolative Matkowski type contraction and its application in solving non-linear matrix equations. Rendiconti del Circolo Matematico di Palermo Series 2, 1-18.
bibitem{10} Gautam, P., Kaur, C.(2023). A novel iterative scheme to approximate the fixed points of Zamfirescu operator and Generalized non-expansive map with an application. Lobachevskii Journal of Mathematics.(Accepted)
bibitem{11} Gautam, P., Kumar, S., Verma, S. and Gupta, G.(2021). Nonunique Fixed Point Results via Kannan-Contraction on Quasi-Partial-Metric Space. Journal of Function Spaces , Article ID 2163108.
bibitem{12} Gautam, P., Sánchez Ruiz, L.M. and Verma, S.(2020). Fixed point of interpolative Rus–Reich–Ćirić contraction mapping on rectangular quasi-partial b-metric space. Symmetry, 13(1), 32.
bibitem{13} Goebel, K. and Kirk, W.A.(1990).Topics in metric fixed point theory (No. 28). Cambridge university press.
bibitem{14} Göhde, D.(1965). Zum prinzip der kontraktiven abbildung. Mathematische Nachrichten, 30(3‐4), 251-258.
bibitem{15} Hardy, G.E. and Rogers, T.D.(1973). A generalization of a fixed point theorem of Reich. Canadian Mathematical Bulletin, 16(2), 201-206.
bibitem{16} Hassan, S., De la Sen, M., Agarwal, P., Ali, Q. and Hussain, A.(2020). A new faster iterative scheme for numerical fixed points estimation of Suzuki’s generalized nonexpansive mappings. Mathematical Problems in Engineering.
bibitem{17} Hussain, N., Ullah, K. and Arshad, M. (2018). Fixed point approximation of Suzuki generalized nonexpansive mappings via new faster iteration process. arXiv preprint arXiv:1802.09888.
bibitem{18} Ishikawa, S.(1974). Fixed points by a new iteration method. Proceedings of the American Mathematical Society, 44(1), 147-150.
bibitem{19} Karapinar, E. and K. Tas,(2011). Generalized (C)-conditions and related fixed point theorems, Comput. Math. Appl. 61 ,
bibitem{20} Kirk, W.A.(1965). A fixed point theorem for mappings which do not increase distances. The American mathematical monthly, 72(9), 1004-1006.
bibitem{21} Krasnosel'skii, M.A.(1955). Two comments on the method of successive approximations. Usp. Math. Nauk, 10, 123-127.
bibitem{22} Mann, W.R.(1953). Mean value methods in iteration. Proceedings of the American Mathematical Society, 4(3), 506-510.
no. 11, 3370–3380.
bibitem{23} Opial, Z.(1967). Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bulletin of the American Mathematical Society, 73(4), 591-597.
bibitem{24} Ostrowski, A.M.(1967). The Round‐off Stability of Iterations. ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 47(2), 77-81.
bibitem{25} 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.
bibitem{26} Qing, Y. and Rhoades, B.E.(2008). Comments on the rate of convergence between Mann and Ishikawa iterations applied to Zamfirescu operators. Fixed Point Theory and Applications,1-3.
bibitem{27} Rhoades, B.E. and Soltuz, S.M.(2004). The equivalence between Mann–Ishikawa iterations and multistep iterations. Nonlinear Analysis: Theory, Methods and Applications, 58(1-2), 219-228.
bibitem{28} Sahu, D.R.(1974).Senter, H.F.; Dotson, W.G. Approximating fixed points of non-expansive mappings. Proc. Am. Math. Soc.
bibitem{29} Schu, J.(1991). Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bulletin of the Australian Mathematical Society, 43(1), 153-159.
bibitem{30} Şoltuz, Ş.M. and Otrocol, D. (2007). Classical results via Mann-Ishikawa iteration. Revue d'analyse numérique et de théorie de l'approximation, 36(2), 193-197.
bibitem{31} Suzuki, T. (2008). Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. Journal of mathematical analysis and applications, 340(2), 1088-1095.
bibitem{32} Ullah, K. and Arshad, M.(2018). Numerical reckoning fixed points for Suzuki’s generalized nonexpansive mappings via new iteration process. Filomat, 32(1),187-196.
bibitem{33} Ullah, K., Ahmad, J. and Sen, M.D.L.(2020). On generalized nonexpansive maps in Banach spaces. Computation, 8(3), 61.