Stable Perturbed Algorithms for a New Class of Generalized Nonlinear Implicit Quasi Variational Inclusions in Banach Spaces

Salahuddin Salahuddin, Mohammad Kalimuddin Ahmad

Department of Mathematics, Aligarh Muslim University, Aligarh, India.

DOI: 10.4236/apm.2012.23021                

Abstract

In this work, a new class of variational inclusion involving T-accretive operators in Banach spaces is introduced and studied. New iterative algorithms for stability for their class of variational inclusions and its convergence results are established.

Keywords

T-Accretive Operators; Variational Inclusions; Iterative Algorithms; Stability Conditions; Convergence; Strong Accretivity; Banach Spaces

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	S. Adly, “Perturbed Algorithms and Sensitivity Analysis for a General Class of Variational Inclusions,” Journal of Mathematical Analysis and Applications, Vol. 201, No. 2, 1996, pp. 609-630. doi:10.1006/jmaa.1996.0277
[2]	Ya. Alber, “Metric and Generalized Projection Operators in Banach Spaces: Properties and Applications,” In: A. Kartsatos, Ed., Theory and Applications of Nonlinear Operators of Monotone and Accretive Type, Marcel Dekker, New York, 1996, pp. 15-50.
[3]	H. Attouch, “Variational Convergence for Functions and Operators,” Applicable Mathematics Series, Pitman, Massachusetts, 1984.
[4]	C. Baiocchi and A. Capelo, “Variational and Quasi-Variational Inequalities Application to Free Boundary Problems,” Wiley, New York, 1984.
[5]	H. Y. Lan and R. U. Verma, “Iterative Algorithms for Nonlinear Fuzzy Variational Inclusions Systems with (A, η)-accretive Mappings in Banach Spaces,” Advances in Nonlinear Variational Inequalities, Vol. 11, No. 1, 2006, pp. 15-30.
[6]	R. U. Verma, “Generalized System for Relaxed Coercive Variational Inequalities and Projection Methods,” Journal of Optimization Theory and Applications, Vol. 121, No. 1, 2004, pp. 203-210. doi:10.1023/B:JOTA.0000026271.19947.05
[7]	R. U. Verma, “Approximation Solvability of a New Class of Nonlinear Set-valued Variational Inclusions Involving (A, η)-Monotone Mappings,” Journal of Mathematical Analysis and Applications, Vol. 337, No. 2, 2008, pp. 969- 975. doi:10.1016/j.jmaa.2007.01.114
[8]	Z. B. Xu and G. F. Roach, “Characteristic Inequalities Uniformly Convex and Uniformly Smooth Banach Spaces,” Journal of Mathematical Analysis and Applications, Vol. 157, No. 1, 1991, pp. 189-210. doi:10.1016/0022-247X(91)90144-O
[9]	Z. B. Xu, “Inequalities in Banach Spaces with Applications,” Nonlinear Analysis, Vol. 16, No. 12, 1991, pp. 1127-1138. doi:10.1016/0362-546X(91)90200-K
[10]	W. Peng, “Set Valued Variational Inclusions with T- Accretive Operators in Banach Spaces,” Applied Mathematics Letters, Vol. 19, No. 3, 2004, pp. 273-282. doi:10.1016/j.aml.2005.04.009
[11]	Y. P. Fang and N. J. Huang, “H-accretive Operators and Resolvent Operator Technique for Solving Variational Inclusions in Banach Spaces,” Applied Mathematics Letters, Vol. 17, No. 6, 2004, pp. 647-653. doi:10.1016/S0893-9659(04)90099-7
[12]	X. P. Ding, “Generalized Implicit Quasivariational Inclusions with Fuzzy Set Valued Mappings,” Computer Mathematics with Applications, Vol. 38, No. 1, 1999, pp. 71-79.
[13]	Y. P. Fang, N. J. Huang, J. M. Kang and Y. J. Cho, “Generalized Nonlinear Implicit Quasivariational Inclusions,” Journal Inequalities and Applications, Vol. 3, 2005, pp. 261-275.
[14]	N. J. Huang, “Mann and Ishikawa Type Perturbed Algorithms for Generalized Nonlinear Implicit Quasivariational Inclusions,” Journal of Mathematical Analysis and Application, Vol. 210, No. 1, 1997, pp. 88-101.
[15]	X. P. Ding, “Perturbed Proximal Point Algorithms for Generalized Quasivariational Inclusions,” Journal of Mathematical Analysis and Applications, Vol. 210, No. 1, 1997, pp. 88-101. doi:10.1006/jmaa.1997.5370
[16]	S. H. Shim, S. M. Kang, N. J. Huang and Y. J. Cho, “Perturbed Iterative Algorithms with Errors for Com- pletely Generalized Strongly Nonlinear Implicit Quasivaria- tional Inclusions,” Journal of Inequalities and Applications, Vol. 5, No. 4, 2000, pp. 381-395.
[17]	S. S. Chang, “On Chidume’s Open Questions and Approximate Solution of Multivalued Strong Mapping Equations in Banach Spaces,” Journal of Mathematical Analysis and Applications, Vol. 216, No. 1, 1997, pp. 94-111. doi:10.1006/jmaa.1997.5661
[18]	N. J. Huang, M. R. Bai, Y. J. Cho and S. M. Kang, “Gen- eralized Nonlinear Mixed Quasivariational Inequalities,” Computer Mathematics with Applications, Vol. 40, No. 2-3, 2000, pp. 205-215. doi:10.1016/S0898-1221(00)00154-1
[19]	M. O. Osilike, “Stable Iteration Procedures for Strong Pseudo-Contractions and Nonlinear Operators of the Accretive Type,” Journal of Mathematical Analysis and Applications, Vol. 204, No. 3, 1996, pp. 677-692. doi:10.1006/jmaa.1996.0461
[20]	A. M. Harder and T. L. Hicks, “Stability Results for Fixed Point Iteration Procedures,” Mathematics Journal, Vol. 33, 1988, pp. 693-706.
[21]	N. J. Huang and Y. P. Fang, “A Stable Perturbed Proxi- mal Point Algorithm for a New Class of Generalized Strongly Nonlinear Quasivariational Like Inclusions,” in Press.
[22]	W. R. Mann, “Mean Value Methods in Iteration,” Proceeding of American Mathematical Society, Vol. 4, 1953, pp. 506-510.
[23]	Ishikawa, “Fixed Points and Iteration of a Non-Expansive Mapping in Banach Spaces,” Proceeding of American Mathematical Society, Vol. 59, No. 1, 1976, pp. 65-71. doi:10.1090/S0002-9939-1976-0412909-X
[24]	N. J. Huang, “Mann and Ishikawa Type Perturbed Iterative Algorithms for Generalized Nonlinear Implicit Quasivariational Inclusions,” Computer Mathematics with Applications, Vol. 35, No. 10, 1998, pp. 1-7.