Convergence, stability and data dependence results for contraction and nonexpansive mappings by a new four step algorithm
الموضوعات :
1 - Department of Mathematics, Akwa Ibom state University, Ikot Akpaden, Mkpat Enin, Nigeria
2 - Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Nigeria
الکلمات المفتاحية: Stability, Nonexpansive mapping, Banach space, contraction mapping, Data Dependence,
ملخص المقالة :
Here we show that the UI-iteration scheme (Udofia and Igbokwe, [24]) can be used to approximate the fixed points of contraction and nonexpansive mappings. we prove a strong and weak convergence of the iteration scheme to the fixed point of contraction and nonexpansive mappings. We also prove that the scheme is Γ-stable and data dependent. Analytically and with numerical example we show that the UI-iteration scheme has a faster rate of convergence for contraction and nonexpansive mappings than some well known existing iteration schemes in literature. Finally, we apply the UI-iteration scheme to find the solution of constrained convex minimization problem.
[1] M. Abbas, T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Mat. Vesnik. 66 (2) (2014), 223-234.
[2] R. P. Agarwal, D. ORegan, D. R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal. 8 (1) (2007), 61-79.
[3] S. Banach, Fixed point iteration for local strictly pseudocontractive mapping, Proc. Amer. Math. Soc. 113 (1991), 727-731.
[4] A. Bejenaru, M. Postolache, Partially projective algorithm for the split feasibility problem with visualization of the solution set, Symmetry. 12 (4) (2020), 12:608.
[5] V. Berinde, Picard iteration converges faster than Mann iteration for a class of quasicontractive operators, Fixed Point Theory Appl. 2 (2004), 97-105.
[6] G. Cai, Y. Shehu, An iterative algorithm for fixed point problem and convex minimization problem with applications, Fixed Point Theory Appl. (2015), 2015:7.
[7] C. D. Enyi, M. E. Soh, Modified gradient-projection algorithm for solving convex minimization problem in Hilbert spaces, Inter. J. Appl. Math. 44 (2014), 44:3.
[8] C. Garodia, I. Uddin, A new iterative method for solving split feasibility problem, J. Appl. Anal. Compt. 10 (3) (2020), 986-1004.
[9] K. Goebel, W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge, 1990.
[10] K. Goebel, S. Reich, Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings, Marcel Dekker Inc, New York, 1984.
[11] M. A. Harder, Fixed point theory and stability results for fixed point iteration procedures, PhD thesis, University of Missouri-Rolla, Missouri, United States, 1987.
[12] S. He, Z. Zhao, Strong convergence of a relaxed CQ algorithm for the split feasibility problem, J. Inequal. Appl. (2013), 2013:197.
[13] S. Ishikawa, Fixed point by a new iteration method, Proc. Amer. Math. Soc. 4 (1) (1974), 147-150.
[14] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510.
[15] A. A. Mebawondu, O. T. Mewomo, Fixed point results for a new three steps iteration process, Annals. Uni. Craiova. Math. Comput. Sci. Ser. 46 (2) (2019), 298-319.
[16] M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000), 217-229.
[17] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591-597.
[18] M. O. Osilike, D. I. Igbokwe, Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations, Comput. Math. Appl. 40 (2000), 559-567.
[19] H. Piri, B. Daraby, S. Rahrovi, M. Ghasemi, Approximating fixed points of generalized α-nonexpansive mappings in Banach spaces by new faster iteration process, Numer. Algor. 81 (2019), 1129-1148.
[20] Y. Shehu, O. S. Iyiola, C. D. Enyi, Iterative approximation of solutions for constrained convex minimization problem, Arab J. Math. 2 (2013), 393-402.
[21] S. M. Soltuz, T. Grosan, Data dependence for Ishikawa iteration when dealing with contractive like operators, Fixed Point Theory Appl. (2008), 2008:242916.
[22] K. K. Tan, H. K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. 178 (1993), 301-308.
[23] D. Thakur, B. S. Thakur, M. Postolache, A new iterative scheme for numerical reckoning fixed points of Suzuki’s generalized nonexpansive mappings, Appl. Math. Comput. 275 (2016), 147-155.
[24] U. E. Udofia, D. I. Igbokwe, Convergence theorems for monotone generalized α-nonexpansive mappings in ordered Banach space by a new four-step iteration process with application, Commun. Nonlinear Anal. 1 (2021), 1-18.
[25] K. Ullah, M. Arshad, New iteration process and numerical reckoning fixed points in Banach spaces, Scientific Bull. (Series A). 79 (2017), 113-122.
[26] K. Ullah, M. Arshad, Numerical reckoning fixed points for Suzuki generalized nonexpansive mappings via new iteration process, Filomat. 32 (2018), 187-196.
[27] X. Weng, Fixed point iteration for local strictly pseudocontractive mapping, Proc. Amer. Math. Soc. 113 (1991), 727-731.
[28] H. K. Xu, Inequality in Banach spaces with applications, Nonlinear Anal. 16 (1991), 1127-1138.
