ABSTRACT

In this paper, we generalize H(.,.) accretive operator introduced by Zou and Huang [1] and we call it*H(.,.)- φ - η -* accretive operator. We define the resolvent operator associated with *H(.,.)- φ - η -* accretive operator and prove its Lipschitz continuity. By using these concepts an iterative algorithm is suggested to solve a generalized variational-like inclusion problem. Some examples are given to justify the definition of *H(.,.)- φ - η -* accretive operator.

In this paper, we generalize H(.,.) accretive operator introduced by Zou and Huang [1] and we call it

KEYWORDS

*H*(.,
.)*- φ - η -* Accretive Operator,
Variational-Like Inclusion,
Resolvent Operator,
Algorithm,
Convergence

Cite this paper

nullR. Ahmad and M. Dilshad, "*H*(.,.)*- φ - η -* Accretive Operators and Generalized Variational-Like Inclusions," *American Journal of Operations Research*, Vol. 1 No. 4, 2011, pp. 305-311. doi: 10.4236/ajor.2011.14035.

nullR. Ahmad and M. Dilshad, "

References

[1] Y. Z. Zou and N. J. Huang, “ -Accretive Operator with an Application for Solving Variational Inclusions in Banach Spaces,” Applied Mathematics and Computation, Vol. 204, No. 2, 2008, pp. 809-816. doi:10.1016/j.amc.2008.07.024

[2] N. J. Huang and Y. P. Fang, “Generalized -Accretive Mappings in Banach Spaces,” Journal of Sichuan University, Vol. 38, No.4, 2001, pp. 591-592.

[3] R. P. Agarwal, Y. J. Cho and N. J. Huang, “Sensitivity Analysis for Strongly Nonlinear Quasi-Variational Inclusions,” Applied Mathematics Letters, Vol. 13, No.6, 2000, pp. 19-24. doi:10.1016/S0893-9659(00)00048-3

[4] R. Ahmad and Q. H. Ansari, “An Iterative Algorithm for Generalized Nonlinear Variational Inclusions,” Applied Mathematics Letters, Vol. 13, No.5, 2000, pp. 23-26. doi:10.1016/S0893-9659(00)00028-8

[5] X. P. Ding and C. L. Luo, “Perturbed Proximal Point Algorithms for General Quai-Variational-Like Inclusions,” Journal of Computational and Applied Mathematics, Vol. 210, No. 1-2, 2000, pp. 153-165. doi:10.1016/S0377-0427(99)00250-2

[6] Y. P. Fang and N. J. Huang, “ -Monotone Operator and Resolvent Operator Technique for Variational Inclusions,” Applied mathematics and Computation, Vol. 145, No. 2-3, 2003, pp. 795-803. doi:10.1016/S0096-3003(03)00275-3

[7] Y. P. Fang and N. J. Huang, “Approximate Solutions for Nonlinear Operator Inclusions with -Monotone Operator,” Research Report, Sichuan University, 2003.

[8] Y. P. Fang and N. J. Huang, “ -Accretive Operator 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

[9] Y. P. Fang, Y. J. Cho and J. K. Kim, “ -Accretive Operator and Approximating Solutions for Systems of Variational Inclusions in Banach Spaces,” Applied Ma- thematics Letters, 2011, in press.

[10] Y. P. Fang, N. J. Huang and H. P. Thompson, “A New Systems of Variational Inclusions with -Monotone Operators in Hilbert Spaces,” Computers and Mathematics with Applications, Vol. 49, No. 2-3, 2005, pp. 365-374. doi:10.1016/j.camwa.2004.04.037

[11] N. J. Huang and Y. P. Fang, “Generalized -Accretive Mappings in Banach Spaces,” Journal of Sichuan University, Vol. 38, No. 4, 2001, pp. 591-592.

[12] N. J. Huang and Y. P. Fang, “A New Class of General Variational Inclusions Involving Maximal -Monotone Mappings,” Publicationes Mathematicae Debrecen, Vol. 62, No. 1-2, 2003, pp. 83-98.

[13] H. Y. Lan, Y. J. Cho and R. U. Verma, “Nonlinear Relaxed Cocoercive Variational Inclusions Involving - Accretive Mappings in Banach Spaces,” Computers and Mathematics with Applications, Vol. 51, No. 9-10, 2006, pp. 1529-1538. doi:10.1016/j.camwa.2005.11.036

[14] J. H. Sun, S. W. Zhang and L. W. Zhang, “An Algorithm Based on Resolvent Operators for Solving Positively Semi-Definite Variational Inequalities,” Fixed Point Theory and Applications, Vol. 2007, 2007, Article ID 76040. doi:10.1155/2007/76040

[15] R. U. Verma, “A Monotonicity and Applications to Non- linear Variational Inclusions,” Journal of Applied Mathematics and Stochastic Analysis, Vol. 2004, No. 2, 2004, pp. 193-195. doi:10.1155/S1048953304403013

[16] Y. Z. Zou, K. Ding and N. J. Huang, “New Global Set- valued Projected Dynamical Systems,” Impulsive Dynamical Systems and Applications, Vol. 4, 2006, pp. 233- 237.

[17] H. K. Xu, “Inequalities in Banach Spaces and Applications,” Nonlinear Analysis, Theory Methods and Aplications, Vol. 16, No. 12, 1991, pp. 1127-1138. doi:10.1016/0362-546X(91)90200-K

[18] N. J. Huang, M. R. Bai, Y. J. Cho and S. M. Kang, “Generalized Nonlinear Mixed Quasi-Variational Inequalities,” Computers and Mathematics with Applications, Vol. 40, No. 2-3, 2000, pp. 205-215. doi:10.1016/S0898-1221(00)00154-1

[19] Z. S. Bi, Z. Han and Y. P. Fang, “Sensitivity Analysis for Nonlinear Variational Inclusions Involving Generalized -Accretive Mappings,” Journal of Sichuan University, Vol. 40, No. 2, 2003, pp. 240-243.

[1] Y. Z. Zou and N. J. Huang, “ -Accretive Operator with an Application for Solving Variational Inclusions in Banach Spaces,” Applied Mathematics and Computation, Vol. 204, No. 2, 2008, pp. 809-816. doi:10.1016/j.amc.2008.07.024

[2] N. J. Huang and Y. P. Fang, “Generalized -Accretive Mappings in Banach Spaces,” Journal of Sichuan University, Vol. 38, No.4, 2001, pp. 591-592.

[3] R. P. Agarwal, Y. J. Cho and N. J. Huang, “Sensitivity Analysis for Strongly Nonlinear Quasi-Variational Inclusions,” Applied Mathematics Letters, Vol. 13, No.6, 2000, pp. 19-24. doi:10.1016/S0893-9659(00)00048-3

[4] R. Ahmad and Q. H. Ansari, “An Iterative Algorithm for Generalized Nonlinear Variational Inclusions,” Applied Mathematics Letters, Vol. 13, No.5, 2000, pp. 23-26. doi:10.1016/S0893-9659(00)00028-8

[5] X. P. Ding and C. L. Luo, “Perturbed Proximal Point Algorithms for General Quai-Variational-Like Inclusions,” Journal of Computational and Applied Mathematics, Vol. 210, No. 1-2, 2000, pp. 153-165. doi:10.1016/S0377-0427(99)00250-2

[6] Y. P. Fang and N. J. Huang, “ -Monotone Operator and Resolvent Operator Technique for Variational Inclusions,” Applied mathematics and Computation, Vol. 145, No. 2-3, 2003, pp. 795-803. doi:10.1016/S0096-3003(03)00275-3

[7] Y. P. Fang and N. J. Huang, “Approximate Solutions for Nonlinear Operator Inclusions with -Monotone Operator,” Research Report, Sichuan University, 2003.

[8] Y. P. Fang and N. J. Huang, “ -Accretive Operator 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

[9] Y. P. Fang, Y. J. Cho and J. K. Kim, “ -Accretive Operator and Approximating Solutions for Systems of Variational Inclusions in Banach Spaces,” Applied Ma- thematics Letters, 2011, in press.

[10] Y. P. Fang, N. J. Huang and H. P. Thompson, “A New Systems of Variational Inclusions with -Monotone Operators in Hilbert Spaces,” Computers and Mathematics with Applications, Vol. 49, No. 2-3, 2005, pp. 365-374. doi:10.1016/j.camwa.2004.04.037

[11] N. J. Huang and Y. P. Fang, “Generalized -Accretive Mappings in Banach Spaces,” Journal of Sichuan University, Vol. 38, No. 4, 2001, pp. 591-592.

[12] N. J. Huang and Y. P. Fang, “A New Class of General Variational Inclusions Involving Maximal -Monotone Mappings,” Publicationes Mathematicae Debrecen, Vol. 62, No. 1-2, 2003, pp. 83-98.

[13] H. Y. Lan, Y. J. Cho and R. U. Verma, “Nonlinear Relaxed Cocoercive Variational Inclusions Involving - Accretive Mappings in Banach Spaces,” Computers and Mathematics with Applications, Vol. 51, No. 9-10, 2006, pp. 1529-1538. doi:10.1016/j.camwa.2005.11.036

[14] J. H. Sun, S. W. Zhang and L. W. Zhang, “An Algorithm Based on Resolvent Operators for Solving Positively Semi-Definite Variational Inequalities,” Fixed Point Theory and Applications, Vol. 2007, 2007, Article ID 76040. doi:10.1155/2007/76040

[15] R. U. Verma, “A Monotonicity and Applications to Non- linear Variational Inclusions,” Journal of Applied Mathematics and Stochastic Analysis, Vol. 2004, No. 2, 2004, pp. 193-195. doi:10.1155/S1048953304403013

[16] Y. Z. Zou, K. Ding and N. J. Huang, “New Global Set- valued Projected Dynamical Systems,” Impulsive Dynamical Systems and Applications, Vol. 4, 2006, pp. 233- 237.

[17] H. K. Xu, “Inequalities in Banach Spaces and Applications,” Nonlinear Analysis, Theory Methods and Aplications, Vol. 16, No. 12, 1991, pp. 1127-1138. doi:10.1016/0362-546X(91)90200-K

[18] N. J. Huang, M. R. Bai, Y. J. Cho and S. M. Kang, “Generalized Nonlinear Mixed Quasi-Variational Inequalities,” Computers and Mathematics with Applications, Vol. 40, No. 2-3, 2000, pp. 205-215. doi:10.1016/S0898-1221(00)00154-1

[19] Z. S. Bi, Z. Han and Y. P. Fang, “Sensitivity Analysis for Nonlinear Variational Inclusions Involving Generalized -Accretive Mappings,” Journal of Sichuan University, Vol. 40, No. 2, 2003, pp. 240-243.