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

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.

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.

Cite this paper

S. Salahuddin and M. Ahmad, "Stable Perturbed Algorithms for a New Class of Generalized Nonlinear Implicit Quasi Variational Inclusions in Banach Spaces,"*Advances in Pure Mathematics*, Vol. 2 No. 3, 2012, pp. 139-148. doi: 10.4236/apm.2012.23021.

S. Salahuddin and M. Ahmad, "Stable Perturbed Algorithms for a New Class of Generalized Nonlinear Implicit Quasi Variational Inclusions in Banach Spaces,"

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.

[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.