International Journal of Nonlinear Analysis and Applications

Common fixed points for hybrid pair of generalized non-expensive mappings by a three-step iterative scheme

Document Type : Research Paper

Authors

1 Department of Mathematics, Farhangian University, Tehran, Iran

2 Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India

Abstract

In this paper, we introduce a three-step iterative scheme, called the MF-iteration process to approximate a common fixed point for a hybrid pair $\{\tau, T\}$  of single-valued and multi-valued maps satisfying a generalized contractive condition defined on uniformly convex Banach spaces. We establish the strong convergence theorem for the proposed process under some basic boundary conditions. We give a numerical example to prove our results' convergence rate. Further, we compare the convergence speed of Sokhuma and Kaewkhao [29] and MF-iterations. we show numerically that the considered iterative scheme converges faster than Sokhuma and Kaewkhao [29] for single-valued and multi-valued non-expansive mappings. Our newly proven results generalize several relevant results in the literature.

Keywords

[1] M. Abbas and T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Mater. Vesn. 66 (2014), 223–234.
[2] R.P. Agarwal, D. O’ Regan, and V. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal. 8 (2007), 61–79.
[3] F.E. Browder, Non-expansive nonlinear operators in a Banach space, Proc. Natl. Acad. Sci. USA 54 (1965), 1041–1044.
[4] F.E. Browder, Semicontractive and semiaccretive nonlinear mappings in Banach spaces, Bull. Amer. Math. Soc. 74 (1968), 660–665.
[5] G. Das and J.P. Debata, Fixed points of quasi-nonexpansive mappings, Indian J. Pure Appl. Math. 17 (1986), 1263–1269.
[6] S. Dhompongsa, A. Kaewcharoen and A. Kaewkhao, The Domenguez-Lorenzo condition and multivalued nonexpansive mappings, Nonlinear Anal. Theory Meth. Appl. 64 (2006), no. 5, 958–970.
[7] W.G. Dotson, On the Mann iterative process, Trans. Amer. Math. Soc. 149 (1970) 65–73.
[8] J. Garcia-Falset, E. Llorens-Fuster, and T. Suzuki, Fixed point theory for a class of generalized non-expansive mappings, J. Math. Anal. Appl. 375 (2011), 185–195.
[9] P. Gholamjiak, W. Gholamjiak, Y.J. Cho and S. Suantai, Weak and strong convergence to common fixed points of a countable family of multi-valued mappings in Banach spaces, Thai j. Math. Appl. 9 (2011), 505–520.
[10] W. Gholamjiak and S. Suantai, Approximation of common fixed points of two quasi non-expansive multi-valued maps in Banach spaces, Comput. Math. Appl. 61 (2011), no. 4, 941–949.
[11] L. Gorniewicz, Topological Fixed Point Theory of Multivalued Mappings, Kluwer, Dordrecht, 1999.
[12] F. Gursoy and V. Karakaya, A Picard S-hybrid type iteration method for solving differential equation with retarded argument, Filomat 30 (2016), no. 10, 2829–2845.
[13] T. Hu, J.C. Huang and B.E. Rhoades, A general principle for Ishikawa iterations for multivalued mappings, Indian J. Pure Appl. Math. 28 (1997), no. 8, 1091–1098.
[14] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147–150.
[15] S. Ishikawa, Fixed point and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc. 59 (1976), 65–71.
[16] T. Kaczynski, Multivalued maps as a tool in modeling and rigorous numerics, J. Fixed Point Theory Appl. 4 (2008), 151–176.
[17] S.H. Khan, A Picard-Mann hybrid iterative process, Fixed Point Theory Appl. 2013 (2013), 69.
[18] S.H. Khan and W. Takahashi, Approximating common fixed points of two asymptotically nonexpansive mappings, Sci. Math. Jpn. 53 (2001), no. 1, 143–148.
[19] T.C. Lim, A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach spaces, Bull. Am. Math. Soc. 80 (1974), 1123–1126.
[20] W.R. Mann, Mean value methods in iterations, Proc. Amer. Math. Soc. 4 (1953), 506–510.
[21] J.T. Markin, Continuous dependence of fixed point sets, Proc. Amer. Math. Soc. 38 (1973), 545–547.
[22] S.B. Nadler, Multivalued contraction mappings, Pac. J. Math. 30 (1969), 475–488.
[23] M.A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000), 217–229.
[24] B. Panyanak, Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces, Comput. Math. Appl. 54 (2007), no. 6, 872–877.
[25] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67 (1979), 274–276.
[26] D.R. Sahu and A. Petrusel, Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces, Nonlinear Anal.: Theory Meth. Appl. 74 (2011), no. 17, 6012–6023.
[27] K.P.R. Sastry and G.V.R. Babu, Convergence of Ishikawa iterates for a multi-valued mapping with a fixed point, Czechoslovak Math. J. 55 (2005), no. 4, 817–826.
[28] N. Shahzad and H. Zegeye, On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces, Nonlinear Anal. Theory Math. Appl. 71 (2009), no. 3-4, 838–844.
[29] K. Sokhuma and A. Kaewkhao, Ishikawa iterative process for a pair of single-valued and multivalued nonexpansive mappings in Banach spaces, Fixed Point Theory Appl. 2010 (2010), 1–9.
[30] Y. Song and H. Wang, Convergence of iterative algorithms for multivalued mappings in Banach spaces, Nonlinear Anal. Theory Math. Appl. 70 (2009), no. 4, 1547–1556.
[31] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008), 1088–1095.
[32] W. Takahashi, Iterative methods for approximation of fixed points and their applications, J. Oper. Res. Soc. Jpn. 43 (2000), no. 1, 87–108.
[33] W. Takahashi and T. Tamura, Convergence theorems for a pair of nonexpansive mappings, J. Convex Anal. 5 (1998), no. 1, 45–58.
[34] K.K. Tan and H.K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. 178 (1993), 301–308.
[35] B.S. Thakur, D. Thakur, M. Postolache, A new iterative scheme for numerical reckoning fixed points of Suzuki’s generalized nonexpansive mappings, Appl. Math. Comp. 275 (2016), 147–155.
International Journal of Nonlinear Analysis and Applications
Volume 15, Issue 3
March 2024
Pages 91-102
  • Receive Date: 17 April 2022
  • Accept Date: 14 September 2022