A generalized nonlinear iterative algorithm for the explicit midpoint rule of nonexpansive semigroup

Yıl 2020, Cilt: 69 Sayı: 1, 613 - 628, 30.06.2020

Hamid Reza Sahebi Mahdi Azhını Masume Cheraghi

https://doi.org/10.31801/cfsuasmas.484452

Öz

In this paper, we introduce a new iterative midpoint rule for finding a solution of xed point problem for a nonexpansive semigroup in real Hilbert spaces. We establish a strong convergence theorem for the sequences generated by our proposed iterative scheme. Furthermore, we provide application to Fredholm integral equations. A
numerical example is presented to illustrate the convergence result. Our results improve and extend the corresponding results in the literature.

Anahtar Kelimeler

Nonexpansive semigroup, equilibrium problem, midpoint method, strongly positive linear bounded operator

Kaynakça

• Alghamdi, M.A., Shahzad, N. and Xu, H.K., The implicit midpoint rule for nonexpansive mappings, Fixed Point Theory Appl., 96 (2014), 9 pages.
• Auzinger, W. and Frank, R., 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.
• Bader, G. and Deuflhard, P., A semi-implicit mid-point rule for stiff systems of ordinary differential equations, Numer. Math., 41 (1983) 373-398.
• Bagherboum, M. and Razani, A. A., modified Mann iterative scheme for a sequence of nonexpansive mappings and a monotone mapping with applications, Bulletin of the Iranian Mathematical Society, 40 (2014), No. 4, 823--849
• Bayreuth, A. The implicit midpoint rule applied to discontinuous differential equations, Computing, 49 (1992) 45-62.
• Chang, S. S., Lee, J. and Chan, H. W., An new method for solving equilibrium problem, fixed point problem and variational inequality problem with application to optimization, Nonlinear Analysis, 70(2009)3307-3319.
• Chen, R. and Song, Y., Convergence to common fixed point of nonexpansive semigroups, J. Comput. Appl. Math. 200, 566-575 (2007).
• Crombez, G., A hicrarchical presentation of operators with fixed points on Hilbert spaces, Numer. Funct. Anal. Optim. 27(2006)259-277.
• Deuflhard, P., Recent progress in extrapolation methods for ordinary differential equations, SIAM Rev., 27(4) (1985) 505-535.
• Hofer, E., 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.
• Homaeipour, S. and Razani, A., Convergence of an iterative method for relatively nonexpansive multi-valued mappings and equilibrium problems in Banach spaces, Optimization Letters, 8 (2014), no. 1, 211-225.
• Kang, J., Su, Y. and Zhang, X., Genaral iterative algorithm for nonexpansive semigroups and variational inequalitis in Hilbert space, Journal of Inequalities and Applications ( 2010) Article ID.264052, 10 pages.
• Lions, P.L., Approximation de points fixes de contractions, C.R. Acad. Sci., Ser. A-B, Paris, 284 (1977) 1357-1359.
• Mahdioui, H. and Chadli, O., On a system of generalized mixed equilibrium problem involving variational-like inequalities in Banach spaces: existence and algorithmic aspects, Advances in Operations Research. 2012(2012)843-486.
• Marino, G. and Xu, H.K., A general iterative method for nonexpansive mappings in Hilbert spaces, Math. Appl. 318(2006)43-52.
• Moradi, R. and Razani, A., Nonlinear iterative algorithms for quasi contraction mapping in modular space, Georgian Mathematics Journal, 23 (2016), No. 1, 105--119.
• Opial, Z., Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Am. Math. Soc. 73(4)(1967)595-597.
• Plubtieng, S. and Punpaeng, R., Fixed point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces, Math. Comput. Model. 48(2008) 279-286.
• Razani, A. and Bagherboum, M., Convergence and stability of Jungck-type iterative procedures in convex b-metric spaces, Fixed Point Theory and Applications, 2013:331 (2013),17 pages.
• Razani, A. and Yazdi, M., A new iterative method for nonexpansive mappings in Hilbert Spaces, Journal of Nonlinear and Optimization: Theory and Applications, 3 (2012), No.1, 85--92.
• Razani, A. and Yazdi, M., Viscosity approximation methods for a countable family of quasi-nonexpansive mappings, World Applied Sciences Journal, 17(12) (2012), 1618--1622.
• Razani, A. and Yazdi, M., A new iterative method for generalized equilibrium and fixed point problems of nonexpansive mappings, Bull. Malays. Math. Sci. Soc. (2) 35(4) (2012), 1049--1061.
• Razani, A. and Yazdi, M., A new iterative method for a family of nonexpansive mappings, Mathematical Reports, 16(66), 1 (2014), 7--23.
• Rizvi, S. H., General Viscosity Implicit Midpoint Rule For Nonexpansive Mapping, 2016.
• Schneider, C., Analysis of the linearly implicit mid-point rule for differential-algebra equations, Electron. Trans. Numer. Anal., 1 (1993) 1-10.
• Shimizu, T. and Takahashi, W., Strong convergence to common fixed points of families of nonexpansive mappings, J. Math. Anal. Appl. 211(1997)71-83.
• Somalia, S. and Davulcua, S., Implicit midpoint rule and extrapolation to singularly perturbed boundary value problems, Int. J. Comput. Math., 75(1) (2000) 117-127.
• Somalia, S., Implicit midpoint rule to the nonlinear degenerate boundary value problems, Int. J. Comput. Math., 79(3) (2002) 327-332.
• Xu, H. K., Viscosity approximation method for nonexpansive semigroups, J. Math. Anal. Appl. 298(2004)279-291.
• Xu, H.K., Alghamdi, M.A. and Shahzad, N., The viscosity technique for the implicit mid point rule of nonexpansive mappings in Hilbert spaces, Fixed point Theory Appl., 41 (2015), 12 pages.