A modified viscosity implicit rule for a variational inequality problem and a uniformly L-Lipschitzian asymptotically pseudocontractive mapping in a Banach space

Poonam Mishra; Shailesh Dhar Diwan; Manoj Kumar

doi:10.26637/MJM0704/0008

Abstract

In this paper, we propose a modified implicit rule for the uniformly L-Lipschitzian and asymptotically pseudocontractive mapping in a Banach space.Related strong convergence theorems are established under the assumptions on certain parameters.Furthermore, it also provides solution to an appropriate variational inequality problem. Our main result improves and extends many known results of the recent literature.

Keywords:

Variational inequality, , asymptotically pseudocontractive mapping, viscosity implicit rule, Banach space

Mathematics Subject Classification:

Mathematics

Authors Imprint References Funding Related Articles Downloads

Poonam Mishra Department of Applied Mathematics, Amity University Chhattisgarh, Raipur-492001, India. https://orcid.org/0000-0002-3444-5353
Shailesh Dhar Diwan Department of Applied Mathematics, Amity University Chhattisgarh, Raipur-492001, India
Manoj Kumar Department of Applied Mathematics, Amity University Chhattisgarh, Raipur-492001, India

Pages: 659-668
Date Published: 01-10-2019
Vol. 7 No. 04 (2019): Malaya Journal of Matematik (MJM)

Moudafi, A. Viscosity approximation methods for fixed points problems, J. Math. Anal.. 241(2000), 46-55.

$mathrm{Xu}, mathrm{H} . mathrm{K}$. , Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298(2004), 279291.

Ke,Y., Ma, C., The generalized viscosity implicit rules of nonexpansive mappings in Hilbert spaces, Fixed Point Theory and Applications, 2015:190.

Xu, H.K., Alghamdi, M.Ali, Shahzad, N., The viscosity technique for the implicit midpoint rule of nonexpansive mappings in Hilbert spaces, Fixed Point Theory and Applications, 2015:41.

Lou, J. , Zhang, L., He, Z., Viscosity approximation methods for asymptotically nonexpansive mappings, Appl. Math. Comput., 203(2008), 171-177.

Alghamadi, M.A., Alghamdi, M.Ali., Shahzad, N., Xu, H.K. The implicit midpoint rule for nonexpansive mappings, Fixed Point Theory and Applications, 2014:96.

Luo, Ping. , Cai, Gang, Shehu, Yekini. The viscosity iterative algorithms for the implicit midpoint rule of nonexpansive mappings in uniformly smooth Banach spaces, Journal of Inequalities and Applications, $2017: 154$.

Bader, G., Deufhard, P, A semi-implicit mid-point rule for stiff systems of ordinary differential equations, Numer. Math., 41(1983), 373-398.

Deuflhard, P, Recent progress in extrapolation methods for ordinary differential equations, SIAM Rev., $27(4)(1985), 505-535$.

Yan,Q., Hu. Shaotao, Strong convergence theorems for the generalized viscosity implicit rules of asymptotically nonexpansive mappings in Hilbert spaces, J. Computational Analysis and Applications, 24(3)(2018), 486-496.

Wang, Y.H., Xia, Y.H., Strong Convergence for asymptotically pseudocontractions with the demiclosedness principle in Banach spaces, Fixed Point Theory Appl., 2012:8.

Yao, Y.H,Shahzad, N.,Liou,Y.,C., Modified semi-implicit midpoint rule for nonexpansive mappings, Fixed Point Theory Appl., 2015:15.

F. E. Browder; W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20(1967), 197-228.

M.A. Krasnoselskij, Two remarks on the method of successive approximations, USPCHI Mat. Nauk, 10(1955), $123-127$.

F. E. Browder, Nonlinear mappings of nonexpansive and accretive in Banach spaces, Bull. Amer. Math. Soc., 73(1967), 875-882.

F. E. Browder: Nonlinear operators and nonlinear equation of evolution in Banach spaces, Proc. Sympus. Pure Math., 18(2)(1976), 112-128.

W. A. Kirk; R. Schoneberg, Some results on pseudocontractive mappings, Pacific J. Math., 71(1977), 89-100.

T. Kato : Accretive operators and nonlinear evolution equations in Banach spaces, Nonlinear Functional Analysis, Proc. Symp. Pure Math., Vol. 18 Part I, ed. Browder, F. E. Amer. Math. Soc. (1970), 138-161.

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

Y.Wang and C.Pan, The modified viscosity implicit rules for uniformly L-Lipschitzian asymptotically pseudocontractive mappings in Banach spaces, J.Nonlinear Sci. Appl., 10(2017), 1582-1592.

Y.-H. Yao, Y.-C. Liou, R.-D. Chen, Strong convergence of an iterative algorithm for pseudocontractive mapping in Banach spaces, Nonlinear Anal., 67 (2007), 3311-3317.

L.-C. Ceng, S. Al-Homidan, Q. H. Ansari, J.-C. Yao, An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings, $J$. Comput. Appl. Math., 223(2009), 967-974.