Set-Valued Non-Linear Random Implicit Quasivariational Inclusions

ABSTRACT

In this paper, we propose iterative algorithms for set valued nonlinear random implicit quasivariational inclusions. We define the related random implicit proximal operator equations and establish an equivalence between them. Finally, we prove the existence and convergence of random iterative sequences generated by random iterative algorithms.

In this paper, we propose iterative algorithms for set valued nonlinear random implicit quasivariational inclusions. We define the related random implicit proximal operator equations and establish an equivalence between them. Finally, we prove the existence and convergence of random iterative sequences generated by random iterative algorithms.

Cite this paper

S. and M. Ahmad, "Set-Valued Non-Linear Random Implicit Quasivariational Inclusions,"*Applied Mathematics*, Vol. 4 No. 3, 2013, pp. 421-428. doi: 10.4236/am.2013.43063.

S. and M. Ahmad, "Set-Valued Non-Linear Random Implicit Quasivariational Inclusions,"

References

[1] C. Baiocchi and A. Capelo, “Variational and Quasi-Variational Inequalities,” John Wiley and Sons, New York, 1984.

[2] H. Brezis, “Operateurs Maximmaux Monotones,” North Holland, Amsterdam, 1973.

[3] F. Giannessi and A. Maugeri, “Variational Inequalities and Network Equilibrium Problems,” Plenum Press, New York, 1995.

[4] C. P. Toskos and V. J. Padgett, “Random Integral Equations with Applications to Stochastic Systems,” Springer-Verlag, Berlin and New York, 1971.

[5] C. P. Toskos and V. J. Padgett, “Random Integral Equations with Applications in Life Sciences and Engineering,” Academic Press, New York, 1974.

[6] M. A. Noor, “Some Recent Advances in Variational Inequalities II,” New Zealand Journal of Mathematics, Vol. 26, No. 2, 1997, pp. 229-255.

[7] M. A. Noor, “Generalized Set-Valued Variational Inclusions and Resolvent Equations,” Journal of Mathematical Analysis and Application, Vol. 228, No. 1, 1998, pp. 206-220. doi:10.1006/jmaa.1998.6127

[8] S. H. Shim, S. M. Kang, N. J. Huang and Y. J. Cho, “Ge-neralized Set-Valued Strongly Nonlinear Quasi-Variational Inclusions,” Indian Journal of Pure Applied Mathematics, Vol. 31, No. 9, 2000, pp. 1113-1122.

[9] S. S. Chang and N. J. Huang, “Generalized Random Multivalued Quasi Complementarity Problems,” Indian Journal of Mathematics, Vol. 35, No. 3, 1993, pp. 305-320.

[10] N. J. Huang, X. Long and Y. J. Cho, “Random Generalized Nonlinear Variational Inclusions,” Bulletin of Korean Mathematical Society, Vol. 34, No. 4, 1997, pp. 603-615.

[11] T. Hussain, E. Tarafdar and X. Z. Yuan, “Some Results on Random Generalized Games and Random Quasi-Variational Inequalities,” Far East Journal Mathematical Society, Vol. 2, No. 1, 1994, pp. 35-55.

[12] N. X. Tan, “Random Quasi-Variational Inequalities,” Mathematische Nachrichten, Vol. 125, 1986, pp. 319-328.

[13] R. U. Verma, “On Generalized Variational Inequalities Involving Relaxed Lipschitz and Relaxed Monotone Operators,” Journal of Mathematical Analysis and Application, Vol. 23, 1997, pp. 389-392.

[14] X. Z. Yuan, “Non Compact Random Generalized Games and Random Quasi-Variational Inequalities,” Journal of Applied Stochastic Analysis, Vol. 7, No. 4, 1994, pp. 467-486. doi:10.1155/S1048953394000377

[15] X. P. Ding and Y. J. Park, “A New Class of Generalized Nonlinear Implicit Quasi-Variational Inclusions with Fuzzy Mappings,” Journal of Computer and Applied Mathematics, Vol. 138, No. 2, 2002, pp. 243-257. doi:10.1016/S0377-0427(01)00379-X

[16] A. Hassouni and A. Moudafi, “A Perturbed Algorithm for Variational Inclusions,” Journal of Mathematical Analysis and Application, Vol. 185, No. 3, 1994, pp. 706-712. doi:10.1006/jmaa.1994.1277

[17] N. J. Huang, “Generalized Nonlinear Variational Inclusions with Noncompact Valued Mappings,” Applied Mathematics Letter, Vol. 9, No. 3, 1996, pp. 25-29. doi:10.1016/0893-9659(96)00026-2

[18] L. U. Uko, “Strongly Nonlinear Generalized Equations,” Journal of Mathematical Analysis and Application, Vol. 220, No. 2, 1998, pp. 65-76. doi:10.1006/jmaa.1997.5796

[19] Salahuddin, “Some Aspects of Variational Inequalities,” Ph.D. Thesis, Aligarh Muslim University, Aligarh, 2000.

[20] Salahuddin and M. K. Ahmad, “On Generalized Multivalued Random Variational Like Inclusions,” Applied Mathematics, Vol. 2, No. 8, 2011, pp. 1011-1018. doi:10.4236/am.2011.28140

[1] C. Baiocchi and A. Capelo, “Variational and Quasi-Variational Inequalities,” John Wiley and Sons, New York, 1984.

[2] H. Brezis, “Operateurs Maximmaux Monotones,” North Holland, Amsterdam, 1973.

[3] F. Giannessi and A. Maugeri, “Variational Inequalities and Network Equilibrium Problems,” Plenum Press, New York, 1995.

[4] C. P. Toskos and V. J. Padgett, “Random Integral Equations with Applications to Stochastic Systems,” Springer-Verlag, Berlin and New York, 1971.

[5] C. P. Toskos and V. J. Padgett, “Random Integral Equations with Applications in Life Sciences and Engineering,” Academic Press, New York, 1974.

[6] M. A. Noor, “Some Recent Advances in Variational Inequalities II,” New Zealand Journal of Mathematics, Vol. 26, No. 2, 1997, pp. 229-255.

[7] M. A. Noor, “Generalized Set-Valued Variational Inclusions and Resolvent Equations,” Journal of Mathematical Analysis and Application, Vol. 228, No. 1, 1998, pp. 206-220. doi:10.1006/jmaa.1998.6127

[8] S. H. Shim, S. M. Kang, N. J. Huang and Y. J. Cho, “Ge-neralized Set-Valued Strongly Nonlinear Quasi-Variational Inclusions,” Indian Journal of Pure Applied Mathematics, Vol. 31, No. 9, 2000, pp. 1113-1122.

[9] S. S. Chang and N. J. Huang, “Generalized Random Multivalued Quasi Complementarity Problems,” Indian Journal of Mathematics, Vol. 35, No. 3, 1993, pp. 305-320.

[10] N. J. Huang, X. Long and Y. J. Cho, “Random Generalized Nonlinear Variational Inclusions,” Bulletin of Korean Mathematical Society, Vol. 34, No. 4, 1997, pp. 603-615.

[11] T. Hussain, E. Tarafdar and X. Z. Yuan, “Some Results on Random Generalized Games and Random Quasi-Variational Inequalities,” Far East Journal Mathematical Society, Vol. 2, No. 1, 1994, pp. 35-55.

[12] N. X. Tan, “Random Quasi-Variational Inequalities,” Mathematische Nachrichten, Vol. 125, 1986, pp. 319-328.

[13] R. U. Verma, “On Generalized Variational Inequalities Involving Relaxed Lipschitz and Relaxed Monotone Operators,” Journal of Mathematical Analysis and Application, Vol. 23, 1997, pp. 389-392.

[14] X. Z. Yuan, “Non Compact Random Generalized Games and Random Quasi-Variational Inequalities,” Journal of Applied Stochastic Analysis, Vol. 7, No. 4, 1994, pp. 467-486. doi:10.1155/S1048953394000377

[15] X. P. Ding and Y. J. Park, “A New Class of Generalized Nonlinear Implicit Quasi-Variational Inclusions with Fuzzy Mappings,” Journal of Computer and Applied Mathematics, Vol. 138, No. 2, 2002, pp. 243-257. doi:10.1016/S0377-0427(01)00379-X

[16] A. Hassouni and A. Moudafi, “A Perturbed Algorithm for Variational Inclusions,” Journal of Mathematical Analysis and Application, Vol. 185, No. 3, 1994, pp. 706-712. doi:10.1006/jmaa.1994.1277

[17] N. J. Huang, “Generalized Nonlinear Variational Inclusions with Noncompact Valued Mappings,” Applied Mathematics Letter, Vol. 9, No. 3, 1996, pp. 25-29. doi:10.1016/0893-9659(96)00026-2

[18] L. U. Uko, “Strongly Nonlinear Generalized Equations,” Journal of Mathematical Analysis and Application, Vol. 220, No. 2, 1998, pp. 65-76. doi:10.1006/jmaa.1997.5796

[19] Salahuddin, “Some Aspects of Variational Inequalities,” Ph.D. Thesis, Aligarh Muslim University, Aligarh, 2000.

[20] Salahuddin and M. K. Ahmad, “On Generalized Multivalued Random Variational Like Inclusions,” Applied Mathematics, Vol. 2, No. 8, 2011, pp. 1011-1018. doi:10.4236/am.2011.28140