# 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)
Existence and stability of iterative algorithms for the system of nonlinear quasi-mixed equilibrium problems. (English) Zbl 1220.47125

Let $ℋ$ be a real Hilbert space and the maps ${{\Phi }}_{1},{{\Phi }}_{2}:ℋ×ℋ\to ℋ$ satisfy ${{\Phi }}_{i}\left(x,x\right)=0,$ $i=1,2$; let ${T}_{1},{T}_{2}:ℋ×ℋ\to ℋ$ be nonlinear maps, and ${C}_{1},{C}_{2}:ℋ⊸ℋ$ be multimaps with nonempty convex closed values. The authors consider the problem of finding $\left({x}^{*},{y}^{*}\right)\in ℋ×ℋ$ such that ${x}^{*}\in {C}_{1}\left({x}^{*}\right),$ ${y}^{*}\in {C}_{2}\left({y}^{*}\right)$ and

$\left\{\begin{array}{cc}{{\Phi }}_{1}\left({x}^{*},z\right)+\left({T}_{1}\left({x}^{*},{y}^{*}\right),z-{x}^{*}\right)\ge 0,\hfill & \forall z\in {C}_{1}\left({x}^{*}\right),\hfill \\ {{\Phi }}_{2}\left({y}^{*},z\right)+\left({T}_{2}\left({x}^{*},{y}^{*}\right),z-{y}^{*}\right)\ge 0,\hfill & \forall z\in {C}_{2}\left({y}^{*}\right)·\hfill \end{array}\right\$

They prove existence and uniqueness results and describe convergence and stability of a Mann type iterative algorithm.

##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47J20 Inequalities involving nonlinear operators 47H04 Set-valued operators 47H05 Monotone operators (with respect to duality) and generalizations 47H10 Fixed point theorems for nonlinear operators on topological linear spaces
##### References:
 [1] Noor, M. A.: Generalized set-valued mixed nonlinear quasi variational inequalities, Korean J. Comput. appl. Math. 5, No. 1, 73-89 (1998) · Zbl 0917.49009 [2] Ding, K.; Yan, W. Y.; Huang, N. J.: A new system of generalized nonlinear relaxed cocoercive variational inequalities, J. inequal. Appl., 1-14 (2006) · Zbl 1090.49008 · doi:10.1155/JIA/2006/40591 [3] Cho, Y. J.; Petrot, N.: On the system of nonlinear mixed implicit equilibrium problems in Hilbert spaces, J. inequal. Appl., 1-12 (2010) · Zbl 1184.49003 · doi:10.1155/2010/437976 [4] Agarwal, R. P.; Cho, Y. J.; Li, J.; Huang, N. J.: Stability of iterative procedures with errors approximating common fixed points for a couple of quasi-contractive mappings in q-uniformly smooth Banach spaces, J. math. Anal. appl. 272, No. 2, 435-447 (2002) · Zbl 1012.65051 · doi:10.1016/S0022-247X(02)00150-6 [5] Chang, S. S.; Lee, H. W. Joseph; Chan, C. K.: Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces, Appl. math. Lett. 20, 329-334 (2007) · Zbl 1114.49008 · doi:10.1016/j.aml.2006.04.017 [6] Cho, Y. J.; Fang, Y. P.; Huang, N. J.; Hwang, H. J.: Algorithms for systems of nonlinear variational inequalities, J. korean math. Soc. 41, No. 3, 489-499 (2004) · Zbl 1057.49010 · doi:10.4134/JKMS.2004.41.3.489 [7] He, Z.; Gu, F.: Generalized system for relaxed cocoercive mixed variational inequalities in Hilbert spaces, Appl. math. Comput. 214, 26-30 (2009) · Zbl 1166.49011 · doi:10.1016/j.amc.2009.03.056 [8] Verma, R. U.: Projection methods, algorithms, and a new system of nonlinear variational inequalities, Comput. math. Appl. 41, 1025-1031 (2001) · Zbl 0995.47042 · doi:10.1016/S0898-1221(00)00336-9 [9] Verma, R. U.: Generalized system for relaxed cocoercive variational inequalities and its projection methods, J. optim. Theory appl. 121, 203-210 (2004) · Zbl 1056.49017 · doi:10.1023/B:JOTA.0000026271.19947.05 [10] Verma, R. U.: General convergence analysis for two-step projection methods and application to variational problems, Appl. math. Lett. 18, 1286-1292 (2005) · Zbl 1099.47054 · doi:10.1016/j.aml.2005.02.026 [11] Blum, E.; Oettli, W.: From optimization and variational inequalities to equilibrium problems, Math. student 63, 123-145 (1994) · Zbl 0888.49007 [12] Combettes, P. L.; Hirstoaga, S. A.: Equilibrium programming in Hilbert spaces, J. nonlinear convex anal. 6, 117-136 (2005) · Zbl 1109.90079 [13] Harder, A. M.; Hick, T. L.: Stability results for fixed-point iteration procedures, Math. japonica 33, No. 5, 693-706 (1998) · Zbl 0655.47045 [14] Kazmi, K. R.; Khan, F. A.: Iterative approximation of a unique solution of a system of variational-like inclusions in real q-uniformly smooth Banach spaces, Nonlinear anal. 67, No. 3, 917-929 (2007) · Zbl 1121.49008 · doi:10.1016/j.na.2006.06.049