×

zbMATH — the first resource for mathematics

Projection methods, algorithms, and a new system of nonlinear variational inequalities. (English) Zbl 0995.47042
Let \(H\) be a real Hilbert space and \(T: K \to H\) a strongly monotone and Lipschitz continuous mapping from a closed convex subset \(K \subset H\) into \(H\). The paper is concerned with the problem of finding elements \(x^*, y^* \in K\) such that \(x^*=P_K [y^*-\rho T(y^*)], y^*=P_K [x^*-\gamma T(x^*)]\), where \(\rho, \gamma >0\) and \(P_K\) is the projection of \(H\) onto \(K\). To solve the problem the author proposes and studies the following iterative algorithm: \(x^{k+1}=(1-a^k)x^k+ a^k P_K [y^k-\rho T(y^k)], y^k=P_K [x^k-\gamma T(x^k)]\), where \(0 \leq a^k <1\) and \(\sum_{k=0}^{\infty} a^k =\infty\). The strong convergence of \(\{ x^k\}\) to \(x^*\) is established provided that \(\rho\) and \(\gamma\) are sufficiently small.

MSC:
47J20 Variational and other types of inequalities involving nonlinear operators (general)
47J25 Iterative procedures involving nonlinear operators
49J40 Variational inequalities
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Noor, M.A., An implicit method for mixed variational inequalities, Appl. math. lett., 11, 4, 109-113, (1998) · Zbl 0941.49005
[2] Noor, M.A., An extragradient method for general monotone variational inequalities, Adv. nonlinear var. inequal., 2, 1, 25-31, (1999) · Zbl 1007.49507
[3] Verma, R.U., A class of projection-contraction methods applied to monotone variational inequalities, Appl. math. lett., 13, 8, 55-62, (2000) · Zbl 0988.47041
[4] Verma, R.U., Nonlinear variational and constrained hemivariational inequalities involving relaxed operators, Zamm, 77, 5, 387-391, (1997) · Zbl 0886.49006
[5] Baiocchi, C.; Capelo, A., Variational and quasivariational inequalities, (1984), Wiley & Sons New York · Zbl 0551.49007
[6] Chan, D.; Pang, J.S., Iterative methods for variational and complementarity problems, Math. programming, 24, 284-313, (1982) · Zbl 0499.90074
[7] Ding, X.P., A new class of generalized strongly nonlinear quasivariational inequalities and quasicomplementarity problems, Indian J. pure appl. math., 25, 11, 1115-1128, (1994) · Zbl 0821.49013
[8] Dunn, J.C., Convexity, monotonicity and gradient processes in Hilbert spaces, J. math. anal. appl., 53, 145-158, (1976) · Zbl 0321.49025
[9] Guo, J.S.; Yao, J.C., Extension of strongly nonlinear quasivariational inequalities, Appl. math. lett., 5, 3, 35-38, (1992) · Zbl 0778.49009
[10] He, B.S., A projection and contraction method for a class of linear complementarity problems and its applications, Applied math. optim., 25, 247-262, (1992) · Zbl 0767.90086
[11] He, B.S., A new method for a class of linear variational inequalities, Math. programming, 66, 137-144, (1994) · Zbl 0813.49009
[12] He, B.S., Solving a class of linear projection equations, Numer. math., 68, 71-80, (1994) · Zbl 0822.65040
[13] He, B.S., A class of projection and contraction methods for monotone variational inequalities, Applied math. optim., 35, 69-76, (1997) · Zbl 0865.90119
[14] Kinderlehrer, D.; Stampacchia, G., An introduction to variational inequalities, (1980), Academic Press New York · Zbl 0457.35001
[15] Korpelevich, G.M., The extragradient method for finding saddle points and other problems, Matecon, 12, 747-756, (1976) · Zbl 0342.90044
[16] Zeidler, E., Nonlinear functional analysis and its applications I, (1986), Springer-Verlag New York
[17] Marcotte, P.; Wu, J.H., On the convergence of projection methods, J. optim. theory appl., 85, 347-362, (1995) · Zbl 0831.90104
[18] R.U. Verma, A class of iterative algorithms and solvability of nonlinear inequalities involving multivalued mappings, J. of Computational Analysis and Applications (to appear). · Zbl 1094.49503
[19] R.U. Verma, An extension of a class of iterative procedures for nonlinear variational inequalities, J. of Computational Analysis and Applications (to appear). · Zbl 1033.65052
[20] Verma, R.U., A class of quasivariational inequalities involving cocoercive mappings, Adv. nonlinear var. inequal., 2, 2, 1-12, (1999) · Zbl 1007.49512
[21] Verma, R.U., An extension of a class of nonlinear quasivariational inequality problems based on a projection method, Math. sci. res. hot-line, 3, 5, 1-10, (1999) · Zbl 0954.49008
[22] Wittmann, R., Approximation of fixed points of nonexpansive mappings, Arch. math. (basel), 58, 486-491, (1992) · Zbl 0797.47036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.