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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:
47J20Inequalities involving nonlinear operators
47J25Iterative procedures (nonlinear operator equations)
49J40Variational methods including variational inequalities
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References:
[1] Noor, M. A.: An implicit method for mixed variational inequalities. Appl. math. Lett. 11, No. 4, 109-113 (1998) · Zbl 0941.49005
[2] Noor, M. A.: An extragradient method for general monotone variational inequalities. Adv. nonlinear var. Inequal. 2, No. 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, No. 8, 55-62 (2000) · Zbl 0988.47041
[4] Verma, R. U.: Nonlinear variational and constrained hemivariational inequalities involving relaxed operators. Zamm 77, No. 5, 387-391 (1997) · Zbl 0886.49006
[5] Baiocchi, C.; Capelo, A.: Variational and quasivariational inequalities. (1984) · 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, No. 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, No. 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) · 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) · Zbl 0583.47050
[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, No. 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, No. 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