More general viscosity implicit midpoint rule for nonexpansive mapping with applications
-
2111
Downloads
-
2927
Views
Authors
Hui-Ying Hu
- Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China.
Abstract
The more general viscosity implicit midpoint rule of fixed point of nonexpansive mapping in Hilbert space is established.
The strong convergence of this rule is proved under certain assumptions imposed on the sequence of parameters, which, in
addition, is the unique solution of the variational inequality problem. Applications to variational inequalities, hierarchical
minimization problems, Fredholm integral equations, and nonlinear evolution equations are included. The results presented in
this work may be treated as an improvement, extension and refinement of some corresponding ones in the literature.
Share and Cite
ISRP Style
Hui-Ying Hu, More general viscosity implicit midpoint rule for nonexpansive mapping with applications, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 5, 2743--2756
AMA Style
Hu Hui-Ying, More general viscosity implicit midpoint rule for nonexpansive mapping with applications. J. Nonlinear Sci. Appl. (2017); 10(5):2743--2756
Chicago/Turabian Style
Hu, Hui-Ying. "More general viscosity implicit midpoint rule for nonexpansive mapping with applications." Journal of Nonlinear Sciences and Applications, 10, no. 5 (2017): 2743--2756
Keywords
- More general viscosity implicit midpoint rule
- nonexpansive mapping
- fixed point problem
- iterative scheme
- variational inequality.
MSC
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]
H. Attouch, Viscosity solutions of minimization problems, SIAM J. Optim., 6 (1996), 769–806.
-
[3]
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.
-
[4]
G. Bader, P. Deuflhard, A semi-implicit mid-point rule for stiff systems of ordinary differential equations, Numer. Math., 41 (1983), 373–398.
-
[5]
F. E. Browder, Existence of periodic solutions for nonlinear equations of evolution, Proc. Nat. Acad. Sci. U.S.A., 53 (1965), 1100–1103.
-
[6]
F. E. Browder, Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces, Arch. Rational Mech. Anal., 24 (1967), 82–90.
-
[7]
A. Cabot, Proximal point algorithm controlled by a slowly vanishing term: applications to hierarchical minimization, SIAM J. Optim., 15 (2005), 555–572.
-
[8]
P. Deuflhard, Recent progress in extrapolation methods for ordinary differential equations, SIAM Rev., 27 (1985), 505–535.
-
[9]
B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc., 73 (1967), 957–961.
-
[10]
P.-L. Lions, Approximation de points fixes de contractions, (French) C. R. Acad. Sci. Paris Sér. A-B, 284 (1977), A1357– A1359.
-
[11]
G. Marino, H.-K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 318 (2006), 43–52.
-
[12]
A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46–55.
-
[13]
S. H. Rizvi, General viscosity implicit midpoint rule for nonexpansive mapping, ArVix, 2016 (2016), 14 pages.
-
[14]
C. Schneider, Analysis of the linearly implicit mid-point rule for differential-algebraic equations, Electron. Trans. Numer. Anal., 1 (1993), 1–10.
-
[15]
S. Somalia, Implicit midpoint rule to the nonlinear degenerate boundary value problems, Int. J. Comput. Math., 79 (2002), 327–332.
-
[16]
M. van Veldhuizen, Asymptotic expansions of the global error for the implicit midpoint rule (stiff case) , Computing, 33 (1984), 185–192.
-
[17]
H.-K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004), 279–291.
-
[18]
H.-K. Xu, Averaged mappings and the gradient-projection algorithm, J. Optim. Theory Appl., 150 (2011), 360–378.
-
[19]
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.
-
[20]
I. Yamada, The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, Inherently parallel algorithms in feasibility and optimization and their applications, Haifa, (2000), Stud. Comput. Math., North-Holland, Amsterdam, 8 (2001), 473–504.