# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 Inequalities involving nonlinear operators 47J25 Iterative procedures (nonlinear operator equations) 49J40 Variational methods including variational inequalities
Full Text:
##### 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