A new modified semi-implicit midpoint rule for nonexpansive mappings and 2-generalized hybrid mappings

Volume 9, Issue 12, pp 6348--6363 http://dx.doi.org/10.22436/jnsa.009.12.35

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Authors

Yanlai Song - College of Science, Zhongyuan University of Technology, 450007 Zhengzhou, China. Yonggang Pei - Engineering Laboratory for Big Data Statistical Analysis and Optimal Control, College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China.

Abstract

In this paper, we present a new modified semi-implicit midpoint rule with the viscosity technique for finding a common fixed point of nonexpansive mappings and 2-generalized hybrid mappings in a real Hilbert space. The proposed algorithm is based on implicit midpoint rule and viscosity approximation method. Under some mild conditions, the strong convergence of the iteration sequences generated by the proposed algorithm is derived.

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Keywords

• Hilbert space
• nonexpansive mapping
• invex set
• fixed point
• semi-implicit midpoint rule.

MSC

•  47H09
•  47H10

References

• [1] M. A. Alghamdi, M. A. Alghamdi, N. Shahzad, H.-K. Xu, The implicit midpoint rule for nonexpansive mappings, Fixed Point Theory Appl., 2014 (2014), 9 pages


• [2] W. Auzinger, R. Frank, Asymptotic error expansions for stiff equations: an analysis for the implicit midpoint and trapezoidal rules in the strongly stiff case, Numer. Math., 56 (1989), 469--499


• [3] G. Bader, P. Deu hard, A semi-implicit mid-point rule for stiff systems of ordinary differential equations, Numer. Math., 41 (1983), 373--398


• [4] C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Problems, 18 (2002), 441--453


• [5] Y. Censor, T. Bortfeld, B. Martin, A. Trofimov, A unified approach for inversion problems in intensity-modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353--2365


• [6] Y. Censor, T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221--239


• [7] Y. Censor, T. Elfving, N. Kopf, T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems, 21 (2005), 2071--2084


• [8] Y. Censor, A. Motova, A. Segal, Perturbed projections and subgradient projections for the multiple-sets split feasibility problem, J. Math. Anal. Appl., 327 (2007), 1244--1256


• [9] P. Deu hard, Recent progress in extrapolation methods for ordinary differential equations, SIAM Rev., 27 (1985), 505--535


• [10] K. Goebel, W. A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (1990)


• [11] E. Hofer, A partially implicit method for large stiff systems of ODEs with only few equations introducing small time-constants, SIAM J. Numer. Anal., 13 (1976), 645--663


• [12] M. Hojo, W. Takahashi, I. Termwuttipong, Strong convergence theorems for 2-generalized hybrid mappings in Hilbert spaces, Nonlinear Anal., 75 (2012), 2166--2176


• [13] J. Hrabalová, P. Tomášek, On stability regions of the modified midpoint method for a linear delay differential equation, Adv. Difference Equ., 2013 (2013), 10 pages


• [14] A. E. Kastner-Maresch, The implicit midpoint rule applied to discontinuous differential equations, Computing, 49 (1992), 45--62


• [15] P. Kocourek, W. Takahashi, J.-C. Yao, Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces, Taiwanese J. Math., 14 (2010), 2497--2511


• [16] P. E. Maingé, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899--912


• [17] G. Marino, H.-K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl., 329 (2007), 336--346


• [18] E. Naraghirad, L.-J. Lin, Strong convergence theorems for generalized nonexpansive mappings on star-shaped set with applications, Fixed Point Theory Appl., 2014 (2014), 24 pages


• [19] X.-L. Qin, Y. J. Cho, J. I. Kang, S. M. Kang, Strong convergence theorems for an infinite family of nonexpansive mappings in Banach spaces, J. Comput. Anal. Appl., 230 (2009), 121--127


• [20] C. Schneider, Analysis of the linearly implicit mid-point rule for differential-algebraic equations, Electron. Trans. Numer. Anal., 1 (1993), 1--10


• [21] S. Somali, Implicit midpoint rule to the nonlinear degenerate boundary value problems, Int. J. Comput. Math., 79 (2002), 327--332


• [22] S. Somali, S. Davulcu, Implicit midpoint rule and extrapolation to singularly perturbed boundary value problems, Int. J. Comput. Math.,, 75 (2000), 117--127


• [23] H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240--256


• [24] H.-K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004), 279--291


• [25] H.-K. Xu, , , 26 (2010), 17 pages., Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Problems, 26 (2010), 17 pages


• [26] H.-K. Xu, M. A. Alghamdi, N. Shahzad, The viscosity technique for the implicit midpoint rule of nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl., 2015 (2015), 12 pages


• [27] Y.-H. Yao, Y. J. Cho, Y.-C. Liou, Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems, European J. Oper. Res., 212 (2011), 242--250


• [28] Y.-H. Yao, M. A. Noor, On viscosity iterative methods for variational inequalities, J. Math. Anal. Appl., 325 (2007), 776--787


• [29] Y.-H. Yao, N. Shahzad, Y.-C. Liou, Modified semi-implicit midpoint rule for nonexpansive mappings, Fixed Point Theory Appl., 2015 (2015), 15 pages


• [30] Z.-T. Yu, L.-J. Lin, C.-S. Chuang, Mathematical programming with multiple sets split monotone variational inclusion constraints, Fixed Point Theory Appl., 2014 (2014), 27 pages


• [31] Y.-L. Yu, C.-F. Wen, A modified iterative algorithm for nonexpansive mappings, J. Nonlinear Sci. Appl., 9 (2016), 3719--3726


• [32] H. Zegeye, N. Shahzad, Convergence of Mann's type iteration method for generalized asymptotically nonexpansive mappings, Comput. Math. Appl., 62 (2011), 4007--4014