New Faster Four Step Iterative Algorithm for Suzuki Generalized Nonexpansive Mappings With an Application

Year 2021, Volume: 5 Issue: 4, 482 - 506, 30.12.2021

Austine Ofem , Donatus Igbokwe

Abstract

The focus of this paper is to introduce a four step iterative algorithm, called A* iterative method, for approximating the fixed points of Suzuki generalized nonexpansive mappings. We prove analytically and numerically that our new iterative algorithm converges faster than some leading iterative algorithm in the literature for almost contraction mappings and Suzuki generalized nonexapansive mapping. Furthermore, we prove weak and strong convergence theorems of our new iterative method for Suzuki generalized nonexpansive mappings in uniformly convex Banach spaces. Again, we show analytically and numerically that our new iterative algorithm is G-stable and data dependent. Finally, to illustrate the applicability of our new iterative method, we will find the unique solution of a functional Volterra Fredholm integral equation with a deviating argument via our new iterative method. Hence, our results generalize and improve several well known results in the existing literature.

Keywords

Banach space, fixed point, stability, almost contraction map, Suzuki generalized nonexpansive mapping, data dependence, convergence, iterative scheme, Volterra Fredholm integral equation

References

• M. Abbas and T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems. Mat Vesn, 66(2014), 223--234.
• R. P. Agarwal, D. O. Regan, D. R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J Nonlinear Convex Anal, 8(2007), 61-79.
• V. Berinde, Picard iteration converges faster than Mann iteration for a class of quasicontractive operators, Fixed Point Theory Appl. , 2 (2004), 97-105.
• V. Berinde, On the approximation of fixed points of weak contractive mapping, Carpath J Math, 19(2003), 7-22.
• A. Bielecki, Une remarque sur l'application de la methode de Banach{Cocciopoli-Tichonov dans la thorie de l'equation s = f(x; y; z; p; q), Bull. Pol. Acad. Sci. Math. 4(1956), 265-357.
• S.K. Chatterjea, Fixed point theorems, C R Acad Bulg Sci. 25(1972), 727-730.
• R. Chugh R, V. Kumar and S. Kumar, Strong convergence of a new three step iterative scheme in Banach spaces, American J Comp Math, 2(2012), 345-357.
• C. Garodia and I. Uddin, A new xed point algorithm for nding the solution of a delay differential equation, AIMS Mathematics, 5(4), 3182-3200. DOI:10.3934/math.2020205.
• C. Garodia and I. Uddin, Solution of a nonlinear integral equation via new fixed point iteration process, arXiv:1809.03771v1 [math.FA] 11 Sep 2018.
• F. Gursoy, Applications of normal S-iterative method to a nonlinear integral equation, Scientic World Journal Volume 2014, Article ID 943127, 5 pages http://dx.doi.org/10.1155/2014/943127.
• F. Gursoy and V Karakaya, A Picard-S hybrid type iteration method for solving a differential equation with retarded argument, (2014), arXiv:1403.2546v2.
• G. Hmmerlin and K. H. Hoffmann, Numerical Mathematics. Springer, Berlin (1991).
• M. A. Harder, Fixed point theory and stability results for fixed point iteration procedures. PhD thesis, University of Missouri-Rolla, Missouri.
• I. Karahan and M. Ozdemir, A general iterative method for approximation of fixed points and their applications, Adv Fixed Point Theory, 3(2013), 510-526.
• S. Ishikawa, Fixed points by a new iteration method. Proc Am Math Soc, 44(1974), 147-150.
• N. Lungu and I. A. RUS, On a functional volterra fredholm integral equation, via picard operators, Journal of Mathematical Inequalities 3(4) (2009), 519-527.
• W. R. Mann, Mean value methods in iteration, Proc. Am. Math. Soc. 4(1953), 506-510.
• K. Maleknejad and P. Torabi, Application of xed point method for solving Volterra-Hammerstein integral equation, 74(1) (2012), U.P.B. Sci. Bull., Series A.
• R. Kannan, Some results on xed point. Bull Calcutta Math Soc. 10(1968), 71-76.
• K. Maleknejad and M. Hadizadeh, A New computational method for Volterra-Fredholm integral equations, Comput. Math. Appl., 37 (1999) 18.
• M. A. Noor, New approximation schemes for general variational inequalities, J Math Anal Appl. 251(2000), 217-229.
• W. Phuengrattana and S. Suantai, On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval, J. Comput. Appl. Math. 235(2011), 3006-3014.
• D. R. Sahu and A. Petrusel, Strong convergence of iterative methods by strictly pseudo- contractive mappings in Banach spaces. Nonlinear Anal Theory Methods Appl, 74(2011), 6012-6023.
• J. Schu, Weak and strong convergence to xed points of asymptotically nonexpansive mappings, B. Aust. Math. Soc., 43(1991), 153-159.
• H. F. Senter and W. G. Dotson, Approximating xed points of nonexpansive mapping, Proc. Amer. Math. Soc., 44(1974), 375-380.
• S. M. Soltuz and T. Grosan, Data dependence for Ishikawa iteration when dealing with contractive like operators. Fixed Point Theory Appl. (2008). doi:10.1155/2008/242916.
• T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. Math. 340(2008), 1088-10995.
• S. Thianwan, Common xed points of new iterations for two asymptotically nonexpansive nonself-mappings in a Banach space. Journal of Computational and Applied Mathematics, 224(2009), 688-695.
• D. Thakur, B. S. Thakur, M. Postolache, A new iterative scheme for numerical reckoning fixed points of Suzuki generalized nonexpansive mappings, Appl. Math. Comput., 275 (2016), 147-155.
• B. S. Thakur, D. Thakur, M. Postolache, A new iterative scheme for numerical reckoning fixed points of Suzuki's generalized nonexpansive mappings, Appl Math Comput, 275(2016), 147-155.
• K. Ullah and M. Arshad, New iteration process and numerical reckoning xed points in Banach spaces, University Politehnica of Bucharest Scientic Bulletin Series A, 79 (2017), 113-122.
• K. Ullah and M. Arshad, Numerical Reckoning Fixed Points for Suzukis Generalized Nonexpansive Mappings via New Iteration Process, Filomat, 32(2018), 187-196.
• K. Ullah and M. Arshad, New three-step iteration process and fixed point approximation in Banach spaces, Journal of Linear and Topological Algebra 7(2),(2018), 87- 100.
• A.M. Wazwaz, A reliable treatment for mixed Volterra-Fredholm integral equations, Appl. Math. Comput., 127 (2002), 405-414.
• X.Weng, Fixed point iteration for local strictly pseudocontractive mapping. Proc Am MathSoc 113(1991), 727-731.
• T. Zamferescu, Fixed point theorems in metric spaces, Arch. Math. (Basel). 23 (1972), 292-298.