# 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)
Generalized nonlinear mixed quasi-variational inequalities. (English) Zbl 0960.47036
The problem is studied to find in a Hilbert space $u$, $x\in Su$, $y\in Tu$, and $z\in Gu$ such that the inclusion $0\in N(x,y)+M(p(u),z)$ holds. Here, $M(\cdot,z)$ is assumed to be maximal monotone where the resolvent $(I+\rho M(\cdot,z))^{-1}$ is Lipschitz with respect to $z$; $S,T,G$ are Lipschitz with respect to the Hausdorff distance, and for $p$ and $N$ Lipschitz and monotonicity conditions are assumed (always with appropriate constants). An iterative algorithm is suggested whose convergence to a solution is proved. Moreover, for single-valued $S,T$ and $G=I$ also stability of a perturbed algorithm is proved.

##### MSC:
 47J20 Inequalities involving nonlinear operators 47J25 Iterative procedures (nonlinear operator equations) 47H05 Monotone operators (with respect to duality) and generalizations 49J40 Variational methods including variational inequalities
Full Text:
##### References:
 [1] Adly, S.: Perturbed algorithm and sensitivity analysis for a general class of variational inclusions. J. math. Anal. appl. 201, 609-630 (1996) · Zbl 0856.65077 [2] Attouch, H.: Variational convergence for functions and operators. Appl. math. Ser. (1974) · Zbl 0561.49012 [3] Baiocchi, C.; Capelo, A.: Variational and quasi-variational inequalities, application to free boundary problems. (1984) · Zbl 0551.49007 [4] Bensoussan, A.: Stochastic control by functional analysis method. (1982) · Zbl 0474.93002 [5] Bensoussan, A.; Lions, J. L.: Impulse control and quasi-variational inequalities. (1984) · Zbl 0324.49005 [6] Chang, S. -S.: Variational inequality and complementarity problem theory with applications. (1991) [7] Chang, S. -S.; Huang, N. -J.: Generalized strongly nonlinear quasi-complementarity problems in Hilbert spaces. J. math. Anal. appl. 158, 194-202 (1991) · Zbl 0739.90067 [8] Chang, S. -S.; Huang, N. -J.: Generalized multivalued implicit complementarity problems in Hilbert spaces. Math. japonica 36, 1093-1100 (1991) · Zbl 0748.49006 [9] Cottle, R. W.; Pang, J. P.; Stone, R. E.: The linear complementarity problem. (1992) · Zbl 0757.90078 [10] Crank, J.: Free and moving boundary problems. (1984) · Zbl 0547.35001 [11] Demyanov, V. F.; Stavroulakis, G. E.; Polyakova, L. N.; Panagiotopoulos, P. D.: Quasidifferentiability and nonsmooth modeling in mechanics, engineering and economics. (1996) · Zbl 1076.49500 [12] Ding, X. -P.: Perturbed proximal point for generalized quasi-variational inclusions. J. math. Anal. appl. 210, 88-101 (1997) · Zbl 0902.49010 [13] Ding, X. -P.: Generalized strongly nonlinear quasi-variational inequalities. J. math. Anal. appl. 173, 577-587 (1993) · Zbl 0779.49010 [14] Duvaut, G.; Lions, J. L.: Inequalities in mechanics and physics. (1976) · Zbl 0331.35002 [15] Giannessi, F.; Maugeri, A.: Variational inequalities and network equilibrium problems. (1995) · Zbl 0834.00044 [16] Harker, P. T.; Pang, J. S.: Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications. Math. programming 48, 161-220 (1990) · Zbl 0734.90098 [17] Hassouni, A.; Moudafi, A.: A perturbed algorithm for variational inclusions. J. math. Anal. appl. 185, 706-712 (1994) · Zbl 0809.49008 [18] Huang, N. -J.: On the generalized implicit quasi-variational inequalities. J. math. Anal. appl. 216, 197-210 (1997) · Zbl 0886.49007 [19] Huang, N. -J.: Mann and Ishikawa type perturbed iterative algorithms for generalized nonlinear implicit quasi-variational inclusions. Computers math. Applic. 35, No. 10, 1-7 (1998) · Zbl 0999.47057 [20] Huang, N. -J.: Generalized nonlinear variational inclusions with noncompact valued mapping. Appl. math. Lett. 9, No. 3, 25-29 (1996) · Zbl 0851.49009 [21] Huang, N. -J.: A new method for a class of nonlinear set-valued variational inequalities. Z. angw. Math. mech. 78, 427-430 (1998) · Zbl 0901.49009 [22] Huang, N. -J.: A new completely general class of variational inclusions with noncompact valued mappings. Computers math. Applic. 35, No. 10, 9-14 (1998) · Zbl 0999.47056 [23] Huang, N. -J.; Cao, S. Y.: Generalized set-valued strongly nonlinear quasi-variational inequalities. J. xinjiang univ. 10, No. 4, 42-47 (1993) · Zbl 1056.49502 [24] Huang, N. -J.; Hu, X. Q.: Generalized multi-valued nonlinear quasi-complementarity problems in Hilbert spaces. J. sichuan univ. 31, 306-310 (1994) · Zbl 0808.47030 [25] Huang, N. -J.; Wu, D. P.: Boundedness and perturbed iterative algorithm of solutions of the generalized nonlinear implicit quasi complementarity problems. J. sichuan univ. 33, 490-493 (1996) · Zbl 0869.90070 [26] Isac, G.: Complementarity problems. Lecture notes in math. 1528 (1992) · Zbl 0795.90072 [27] Kazmi, K. R.: Mann and Ishikawa type perturbed iterative algorithms for generalized quasi-variational inclusions. J. math. Anal. appl. 209, 572-584 (1997) · Zbl 0898.49007 [28] Mosco, U.: Implicit variational problems and quasi-variational inequalities. Lecture notes in math. 543 (1976) · Zbl 0346.49003 [29] Nagurney, A.; Siokos, S.: Variational inequalities for international general financial equilibrium modeling and computation. Mathl. comput. Modelling 25, No. 1, 31-49 (1997) · Zbl 0881.90014 [30] Noor, M. A.: On the nonlinear complementarity problem. J. math. Anal. appl. 123, 455-460 (1987) · Zbl 0624.41041 [31] Noor, M. A.: The quasi-complementarity problem. J. math. Anal. appl. 130, 344-353 (1988) · Zbl 0645.90086 [32] Noor, M. A.; Noor, K. I.: Multivalued variational inequalities and resolvent equations. Mathl. comput. Modelling 26, No. 7, 109-121 (1997) · Zbl 0893.49005 [33] Noor, M. A.; Noor, K. I.; Rassias, T. M.: Set-valued resolvent equations and mixed variational inequalities. J. math. Anal. appl. 220, 741-759 (1998) · Zbl 1021.49002 [34] Noor, M. A.; Noor, K. I.; Rassias, T. M.: Some aspects of variational inequalities. J. comput. Appl. math. 47, 285-312 (1993) · Zbl 0788.65074 [35] Panagiotopoulos, P. D.; Stavroulakis, G. E.: New types of variational principles based on the notion of quasidifferentiability. Acta. mech. 94, 171-194 (1992) · Zbl 0756.73096 [36] Robinson, S. M.: Generalized equation and their solutions. Part I. Basic theory. Math. programming stud. 10, 128-141 (1979) · Zbl 0404.90093 [37] Siddiqi, A. H.; Ansari, Q. H.: Strongly nonlinear quasi-variational inequalities. J. math. Anal. appl. 149, 444-450 (1990) · Zbl 0712.49009 [38] Siddiqi, A. H.; Ansari, Q. H.: General strongly nonlinear variational inequalities. J. math. Anal. appl. 166, 386-392 (1992) · Zbl 0770.49006 [39] Uko, L. U.: Strongly nonlinear generalized equations. J. math. Anal. appl. 220, 65-76 (1998) · Zbl 0918.49007 [40] Yao, J. C.: The generalized quasi variational inequality problem with applications. J. math. Anal. appl. 158, 139-160 (1991) · Zbl 0739.49010 [41] Zeng, L. -C.: Completely generalized strongly nonlinear quasicomplementarity problems in Hilbert spaces. J. math. Anal. appl. 193, 706-714 (1995) · Zbl 0832.47053 [42] Zhu, D. L.; Marcotte, P.: Scheme for solving variational inequalities. SIAM J. Control. optim. 6, No. 3, 714-726 (1996) · Zbl 0855.47043 [43] Minty, G. J.: On the monotonicity of the gradient of a convex function. Pacific J. Math. 14, 243-247 (1964) · Zbl 0123.10601 [44] Rockaffellar, R.: Convex analysis. (1970) [45] Jr., S. B. Nadler: Multi-valued contraction mappings. Pacific J. Math. 30, 475-488 (1969) · Zbl 0187.45002 [46] Harder, A. M.; Hicks, T. L.: Stability results for fixed point iteration procedures. Math. japonica 33, 693-706 (1988) · Zbl 0655.47045 [47] Osilike, M. O.: Stable iteration procedures for strong pseudo-contractions and nonlinear operator equations of the accretive type. J. math. Anal. appl. 204, 677-692 (1996) · Zbl 0882.47030 [48] Chang, S. -S.: On chidume’s open questions and approximate solution of multivalued strongly accretive mapping equations in Banach spaces. J. math. Anal. appl. 216, 94-111 (1997) · Zbl 0909.47049