Browse > Article
http://dx.doi.org/10.22771/nfaa.2021.26.05.10

INERTIAL PICARD NORMAL S-ITERATION PROCESS  

Dashputre, Samir (Department of Mathematics Govt. College)
Padmavati, Padmavati (Department of Mathematics Govt. V.Y.T. Autonomous P.G. College)
Sakure, Kavita (Department of Mathematics Govt. Digvijay Autonomous P.G. College)
Publication Information
Nonlinear Functional Analysis and Applications / v.26, no.5, 2021 , pp. 995-1009 More about this Journal
Abstract
Many iterative algorithms like that Picard, Mann, Ishikawa and S-iteration are very useful to elucidate the fixed point problems of a nonlinear operators in various topological spaces. The recent trend for elucidate the fixed point via inertial iterative algorithm, in which next iterative depends on more than one previous terms. The purpose of the paper is to establish convergence theorems of new inertial Picard normal S-iteration algorithm for nonexpansive mapping in Hilbert spaces. The comparison of convergence of InerNSP and InerPNSP is done with InerSP (introduced by Phon-on et al. [25]) and MSP (introduced by Suparatulatorn et al. [27]) via numerical example.
Keywords
Picard normal S-iteration process; normal S-itetation process; inertial extrapolation; weak convergence; strong convergence;
Citations & Related Records
연도 인용수 순위
  • Reference
1 J. Franklin, Methods of mathematical economics, Springer Verlag, New York, 1980.
2 N. Kadioglu and I. Yildirim, Approximating fixed points of nonexpansive mappings by faster iteration proess, J. Adv. Math. Stud., 8(2) (2015), 257-264.
3 S.H. Khan, A Picard-Mann hybrid iterative process, Fixed Point Theory Appl., 1 (2013), 1-10.
4 W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506-610.   DOI
5 P.E. Mainge, Convergence theorems for inertial KM-type algorithms, J. Comput. Appl. Math., 219 (2008), 223-236.   DOI
6 K. Nakajo and W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl., 279 (2003), 372-379.   DOI
7 Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 591-597.   DOI
8 D.R. Sahu, Application of the S-iteration process to constrained minimization problems and feasibility problems, Fixed Point Theory, 12 (2011), 187-204.
9 L.U. Uko, Remarks on the generalized Newton method, Math. Program., 59 (1993), 404-412.
10 L.U. Uko, Generalized equations and the generalized Newton method, Math. Program., 73 (1996), 251-268.   DOI
11 O. Ege and I. Karaca, Banach fixed point theorem for digital images, J. Nonlinear Sci. Appl., 8 (2015), 237-245.   DOI
12 V. Berinde, Iterative approximation of fixed points, Editura Dfemeride, Baia Mare, 2002.
13 B. Mercier, Mechanics and Variational Inequalities, Lecture Notes, Orsay Centre of Paris University, 1980.
14 R. Suparatulotorn, W. Cholamjiak and S. Suantai, Modified S-iteration process for G-nonexpansive mappings in Banach spaces with graphs, Numer. Algo., 77(2) (2018), 479-490.   DOI
15 H.H. Bauschke and P.L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, Springer, Berlin, 2011.
16 C.E. Chidume and S. Mutangadura, An example on the Mann iteration method for Lipschitzian pseudocontractions, Proc. Amer. Math. Soc., 129 (2001), 2359-2363.   DOI
17 S. Dashputre, Padmavati and K. Sakure, Convergence results for proximal point algorithm in complete CAT(0) space for multivalued mappings, J. Indone. Math. Soc., 27(1) (2021), 29-47.
18 W.G. Jr. Dotson, On the Mann iterative process, Trans. Amer. Math. Soc., 149 (1970), 65-73.   DOI
19 S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44 (1974), 147-150.   DOI
20 J.L. Lions and G. Stampacchia, Variational inequalities, Commun. Pure Appl. Math., 20 (1967), 493-519.   DOI
21 A. Phon-on and N. Makaje, A. Sama-Ae and K. Khongraphan, An inertial S-iteration process, Fixed Point Theory Appl., 2019, 2019:4.
22 W. Takahashi, Y. Takeuchi and R. Kubota, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 341(1) (2008), 276-286.   DOI
23 C.L. Byrne, A unified treatment of some iterative algorithm in signal processing and image reconstruction, Inverse Problems, 18 (2004), 441-453.   DOI
24 R.P. Agarwal, D. O'Regan and D.R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal., 8(1) (2007), 61-79.
25 F. Alvarez, Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space, SIAM J. Optim., 14(3) (2004), 773-782.   DOI
26 H.H. Bauschke and J.M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Rev., 38 (1996), 367-426.   DOI
27 S. Dashputre, Padmavati and K. Sakure, On approximation of fixed point in Busemann space via generalized Picard normal S-iteration process, Malaya J. Math., 8(3) (2020), 1055-1062.   DOI
28 S. Dashputre, Padmavati and K. Sakure, Strong and ∆-convergence results for generalized nonexpansive mapping in hyperbolic space, Commu. Math. Appl., 11(3) (2020), 389-401.
29 Q.L. Dong and Y.Y. Lu, A new hybrid algorithm for nonexpansive mapping, Fixed Point Theory Appl., 2015, 2015:37.
30 Q.L. Dong, H.B. Yuan, Y.J. Cho and T.M. Rassias, Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings, Optim. Lett., 12(1) (2018), 87-102.   DOI