# An iterative algorithm based on the generalized viscosity explicit methods for an infinite family of accretive operators

## Keywords:

Proximal-point algorithm, Generalized viscosity explicit methods, Accretive operators, Common zeros## Abstract

In this paper, we introduce and study a new iterative method based on the generalized viscosity explicit methods (GVEM) for solving the inclusion problem with an infinite family of multivalued accretive operators in real Banach spaces. Applications to equilibrium and to convex minimization problems involving an infinite family of semi-continuous and convex functions are included. Our results improve important recent results.

## References

W. Auzinge and 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), pp. 469-499.

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

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

F.E. Browder, Convergenge theorem for sequence of nonlinear operator in Banach spaces, Math. Z., 100 (1967), pp. 201-225.

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud., 63 (1994), pp. 123-145.

R.E. Bruck, Jr., A strongly convergent iterative solution of $0 in U (x)$ for a maximal monotone operator U in Hilbert spaces, J. Math. Anal. Appl., 48 (1974), pp. 114-126.

P.L. Combettes and S. Ahirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), pp. 117-136.

I. Cioranescu, Geometry of Banach space, duality mapping and nonlinear problems, Kluwer, Dordrecht (1990).

S. Chang, J.K. Kim and X.R. Wang, Modified block iterative algorithm for solving convex feasibility problems in Banach spaces, J. Inequal. Appl., (2010), pp. 1-14.

A. Genel and J. Lindenstrauss, An example concerning fixed points, Israel J. Math., 22 (1975), pp. 81-86.

K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York (1984).

E. Hairer, S.P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd edn. Springer Series in Computational Mathematics. Springer, Berlin (1993).

Y. Ke and Ch. Ma, The generalized viscosity implicit rules of nonexpansive mappings in Hilbert spaces, J. Fixed Point Theory Appl., (2015), pp. 1-21.

N. Lehdili and A. Moudafi, Combining The Proximal Algorithm And Tikhonov Regularization, Optimization, (1996), pp. 239-252.

T.C. Lim and H.K. Xu, Fixed point theorems for assymptoticaly nonexpansive mapping, Nonliear Anal., 22 (1994), pp. 1345-1355.

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

W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), pp. 506-510.

G. Marino, B. Scardamaglia and R. Zaccone, A general viscosity explicit midpoint rule for quasi-nonexpansive mappings, J. Nonlinear Convex Anal., 48 (2017), pp. 137-148.

G.J. Minty, Monotone (nonlinear) operator in Hilbert space, Duke Math, 29 (1962), pp. 341-346.

I. Miyadera, Nonlinear semigroups, Translations of Mathematical Monographs, American Mathematical Society, Providence, RI (1992).

Z. Opial, Weak convergence of sequence of succecive approximation of nonexpansive mapping, Bull. Amer. Math. Soc., 73 (1967), pp. 591-597.

R.T. Rockafellar, Operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), pp. 877-898.

S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 67 (1979 ), pp. 274-276.

M.V. Solodov and B.F. Svaiter, Forcing strong convergence of proximal point iterations in a Hilbert space, Math. Program. Ser. A, 87 (2000), pp. 189-202.

S. Takahashi, W. Takahashi and M. Toyoda, Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces, J. Optim. Theory Appl., 147 (2010), pp. 27-41.

H.K. Xu, A regularization method for the proximal point algorithm, J. Global. Optim., 36 (2006), pp. 115-125.

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

H.K. Xu, M.A. Alghamdi and N. Shahzad, The viscosity technique for the implicit midpoint rule of nonexpansive mappings in Hilbert spaces, J. Fixed Point Theory Appl., (2015), pp. 1-41.

## Downloads

## Published

## How to Cite

*Armenian Journal of Mathematics*,

*12*(9). Retrieved from http://armjmath.sci.am/index.php/ajm/article/view/315

## Issue

## Section

## License

Copyright (c) 2020 Armenian Journal of Mathematics

This work is licensed under a Creative Commons Attribution 4.0 International License.