SOME STRONG CONVERGENCE RESULTS OF RANDOM ITERATIVE ALGORITHMS WITH ERRORS IN BANACH SPACES

- Journal title : Communications of the Korean Mathematical Society
- Volume 31, Issue 1, 2016, pp.147-161
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/CKMS.2016.31.1.147

Title & Authors

SOME STRONG CONVERGENCE RESULTS OF RANDOM ITERATIVE ALGORITHMS WITH ERRORS IN BANACH SPACES

Chugh, Renu; Kumar, Vivek; Narwal, Satish;

Chugh, Renu; Kumar, Vivek; Narwal, Satish;

Abstract

In this paper, we study the strong convergence and stability of a new two step random iterative scheme with errors for accretive Lipschitzian mapping in real Banach spaces. The new iterative scheme is more acceptable because of much better convergence rate and less restrictions on parameters as compared to random Ishikawa iterative scheme with errors. We support our analytic proofs by providing numerical examples. Applications of random iterative schemes with errors to variational inequality are also given. Our results improve and establish random generalization of results obtained by Chang [4], Zhang [31] and many others.

Keywords

random iterative schemes;stability;accretive operator;variational inequality;

Language

English

Cited by

References

1.

I. Beg and M. Abbas, Equivalence and stability of random fixed point iterative procedures, J. Appl. Math. Stoch. Anal. 2006 (2006), Art. ID 23297, 19 pp.

2.

I. Beg and M. Abbas, Iterative procedures for solution of random equations in Banach spaces, J. Math. Anal. Appl. 315 (2006), 181-201.

3.

T. Bharucha-Reid, Fixed point theorems in probabilistic analysis, Bull. Amer. Math. Soc. 82 (1976), no. 5, 611-657.

4.

S. S. Chang, The Mann and Ishikawa iterative approximation of solutions to variational inclusions with accretive type mappings, Comput. Math. Appl. 37 (1999), no. 9, 17-24.

6.

B. S. Choudhury and M. Ray, Convergence of an iteration leading to a solution of a random operator equation, J. Appl. Math. Stoch. Anal. 12 (1999), no. 2, 161-168.

7.

B. S. Choudhury and A. Upadhyay, An iteration leading to random solutions and fixed points of operators, Soochow J. Math. 25 (1999), no. 4, 395-400.

8.

R. Chugh and V. Kumar, Convergence of SP iterative scheme with mixed errors for accretive Lipschitzian and strongly accretive Lipschitzian operators in Banach spaces, Int. J. Comput. Math. 90 (2013), no. 9, 1865-1880.

9.

R. Chugh, S. Narwal, and V. Kumar, Convergence of random SP iterative scheme, Appl. Math. Sci. Vol.7 (2013), no. 46, 2283-2293.

10.

I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer Academic Publishers, Dordrecht, 1990.

11.

L. B. Ciric, A. Rafiq, and N. Cakic, On Picard iterations for strongly accretive and strongly pseudo-contractive Lipschitz mappings, Nonlinear Anal. 70 (2009), no. 12, 4332-4337.

12.

L. B. Ciric and J. S. Ume, Ishikawa iterative process for strongly pseudocontractive operators in arbitrary Banach spaces, Math. Commun. 8 (2003), no. 1, 43-48.

13.

L. B. Ciric and J. S. Ume, Ishikawa iterative process with errors for nonlinear equations of generalized monotone type in Banach spaces, Math. Nachr. 278 (2005), no. 10, 1137-1146.

14.

L. B. Ciric, J. S. Ume, and S. N. Jesic, On random coincidence and fixed points for a pair of multivalued and single-valued mappings, J. Inequal. Appl. 2006 (2006), Art. ID 81045, 12 pp.

15.

L. B. Ciric, J. S. Ume, S. N. Jesic, Arandjelovic-Milovanovic, and M. Marina, Modified Ishikawa iteration process for nonlinear Lipschitz generalized strongly pseudocontractive operators in arbitrary Banach spaces, Numer. Funct. Anal. Optim. 28 (2007), no. 11-12, 1231-1243.

16.

X. P. Ding, Generalized strongly nonlinear quasivariational inequalities, J. Math. Anal. Appl. 173 (1993), no. 2, 577-587.

17.

X. P. Ding, Perturbed proximal point algorithms for generalized quasivariational inclusions, J. Math. Anal. Appl. 210 (1997), no. 1, 88-101.

18.

A. Hassouni and A. Moudafi, A perturbed algorithms for variational inclusions, J. Math. Anal. Appl. 185 (1994), no. 3, 706-721.

19.

J. Himmelberg, Measurable relations, Fund. Math. 87 (1975), 53-72.

21.

K. R. Kazmi, Mann and Ishikawa type perturbed iterative algorithms for generalized quasivariational inclusions, J. Math. Anal. Appl. 209 (1997), no. 2, 572-584.

22.

A. R. Khan, F. Akbar, and N. Sultana, Random coincidence points of subcompatible multivalued maps with applications, Carpathian J. Math. 24 (2008), no. 2, 63-71.

23.

A. R. Khan, A. B. Thaheem, and N. Hussain, Random fixed points and random approximations in nonconvex domains, J. Appl. Math. Stoch. Anal. 15 (2002), no. 3, 263-270.

24.

A. R. Khan, A. B. Thaheem, and N. Hussain, Random fixed points and random approximations, Southeast Asian Bull. Math. 27 (2003), no. 2, 289-294.

25.

M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000), no. 1, 217-229.

26.

B. E. Rhoades, Iteration to obtain random solutions and fixed points of operators in uniformly convex Banach spaces, Soochow J. Math. 27 (2001), no. 4, 401-404.

27.

A. H. Siddiqi and Q. H. Ansari, General strongly nonlinear variational inequalities, J. Math. Anal. Appl. 166 (1992), no. 2, 386-392.

28.

H. Siddiqi, Q. H. Ansari, and K. R. Kazmi, On nonlinear variational inequalities, Indian J. Pure Appl. Math. 25 (1994), no. 9, 969-973.

29.

E. Zeidler, Nonlinear Functional Analysis and its Applications. Part II: Monotone Operators, Springer-Verlag, New York, 1985.

30.

L. C. Zeng, Iterative algorithms for finding approximate solutions for general strongly nonlinear variational inequalities, J. Math. Anal. Appl. 187 (1994), no. 2, 352-360.

31.

S. S. Zhang, Existence and approximation of solutions to variational inclusions with accretive mappings in Banach spaces, Appl. Math. Mech. 22 (2001), no. 9, 997-1003.