Abstract
In this paper, we introduce and study a new class of general strongly nonlinear quasivariational inequalities and construct a general iterative algorithm by using the projection method. We establish the existence of a unique solution for general strongly nonlinear quasivariational inequalities involving relaxed Lipschitz, relaxed monotone, and strongly monotone mappings; we obtain the convergence and stability of the iterative sequences generated by the algorithm. Our results extend, improve, and unify many known results due to Bose, Noor, Siddiqi-Ansari, Verma, Yao, Zeng, and others.
Similar content being viewed by others
References
BOSE, R. K., On a General Nonlinear Variational Inequality, Bulletin of the Australian Mathematical Society, Vol. 42, pp. 399–406, 1990.
NOOR, M. A., An Iterative Scheme for a Class of Quasivariational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 110, pp. 463–468, 1985.
NOOR, M. A., General Nonlinear Variational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 126, pp. 78–84, 1987.
NOOR, M. A., On a Class of Variational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 128, pp. 138–155, 1987.
SIDDIQI, A. H., and ANSARI, Q. H., An Algorithm for a Class of Quasivariational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 145, pp. 413–418, 1990.
VERMA, R. U., Iteratie Algorithms for Variational Inequalities and Associated Nonlinear Equations Inolûing Relaxed Lipschitz Operators, Applied Mathematics Letters, Vol. 9, pp. 61–63, 1996.
VERMA, R. U., Generalized Variational Inequalities and Associated Nonlinear Equations, Czechoslovak Mathematical Journal, Vol. 48, pp. 413–418, 1998.
VERMA, R. U., The Solvability of a Class of Generalized Nonlinear Variational Inequalities Based on an Iterative Algorithm, Applied Mathematics Letters, Vol. 12, pp. 51–53, 1999.
ZENG, L. C., A Note on a General Algorithm for Variational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 223, pp. 354–363, 1998.
ZENG, L. C., On a General Projection Algorithm for Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 97, pp. 229–235, 1998.
NOOR, M. A., New Approximation Schemes for General Variational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 251, pp. 217–229, 2000.
SIDDIQI, A. H., and ANSARI, Q. H., Strongly Nonlinear Quasivariational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 149, pp. 444–450, 1990.
YAO, J. C., Applications of Variational Inequalities to Nonlinear Analysis, Applied Mathematics Letters, Vol. 4, pp. 89–92, 1991.
LIU, L. S., Ishikawa and Mann Iterative Process with Errors for Nonlinear Strongly Accretive Mappings in Banach Spaces, Journal of Mathematical Analysis and Applications, Vol. 194, pp. 114–125, 1995.
MOSCO, U., Implicit Variational Problems and Quasivariational Inequalities, Lecture Notes in Mathematics, Springer Verlag, Berlin, Germany, Vol. 543, 1976.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Liu, Z., Ume, J. & Kang, S. General Strongly Nonlinear Quasivariational Inequalities with Relaxed Lipschitz and Relaxed Monotone Mappings. Journal of Optimization Theory and Applications 114, 639–656 (2002). https://doi.org/10.1023/A:1016079130417
Issue Date:
DOI: https://doi.org/10.1023/A:1016079130417