×

An implicit-function theorem for a class of monotone generalized equations. (English) Zbl 0792.49005

Summary: We prove an implicit function theorem for a class of generalized equations defined by a monotone set-valued mapping in Hilbert spaces. We give applications to variational inequalities, single-valued functions and a class of nonsmooth functions.

MSC:

49J40 Variational inequalities
26B10 Implicit function theorems, Jacobians, transformations with several variables
54C60 Set-valued maps in general topology
26E25 Set-valued functions
PDFBibTeX XMLCite
Full Text: EuDML Link

References:

[1] W. Alt: The Lagrange-Newton method for infinite-dimensional optimization problems. Numer. Funct. Anal. Optim. 11 (1990), 201-224. · Zbl 0694.49022 · doi:10.1080/01630569008816371
[2] W. Alt: Parametric programming with applications to optimal control and sequential quadratic programming. Bayreuther Mathematische Schriften 35 (1991), 1-37. · Zbl 0734.90094
[3] J.-P. Aubin, I. Ekeland: Applied Nonlinear Analysis. J. Wiley, New York 1984. · Zbl 0641.47066
[4] J.-P. Aubin, H. Frankowska: Set-valued Analysis. Birkhauser, Boston 1990. · Zbl 0713.49021
[5] F. E. Browder: Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces. Amer. Math. Soc, Providence, Rhode Island 1976. · Zbl 0327.47022
[6] S. Dafermos: Sensitivity analysis in variational inequalities. Math. Oper. Res. IS (1988), 421-434. · Zbl 0674.49007 · doi:10.1287/moor.13.3.421
[7] A. V. Fiacco: Sensitivity analysis for nonlinear programming using penalty methods. Math. Programming 10 (1976), 287-311. · Zbl 0357.90064 · doi:10.1007/BF01580677
[8] A. V. Fiacco: Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. Academic Press, New York - London 1983. · Zbl 0543.90075
[9] A. D. Ioffe, V. M. Tihomirov: Theory of Extremal Problems. North-Holland, Amsterdam - New York - Oxford 1979. · Zbl 0407.90051
[10] K. Ito, K. Kunisch: Sensitivity analysis of solutions to optimization problems in Hilbert spaces with applications to optimal control and estimation. Preprint, 1989. · Zbl 0790.49028 · doi:10.1016/0022-0396(92)90133-8
[11] G. Kassay, I. Kolumban: Implicit-function theorems for monotone mappings. Research Seminar on Mathematical Analysis, Babes-Bolyai University, Preprint Nr. 6 (1988), 7-24. · Zbl 0664.46045
[12] G. Kassay, I. Kolumban: Implicit functions and variational inequalities for monotone mappings. Research Seminar on Mathematical Analysis, Babes-Bolyai University, Preprint Nr.7 (1989), 79-92. · Zbl 0724.47027
[13] K. Malanowski: Second-order conditions and constraint qualifications in stability and sensitivity analysis of solutions to optimization problems in Hilbert spaces. Appl. Math. Optim. · Zbl 0756.90093 · doi:10.1007/BF01184156
[14] G. Minty: Monotone (nonlinear) operators in Hilbert spaces. Duke Math. J. 29 (1962), 341-346. · Zbl 0111.31202 · doi:10.1215/S0012-7094-62-02933-2
[15] S. M. Robinson: Strongly regular generalized equations. Math. Oper. Res. 5 (1980), 43-62. · Zbl 0437.90094 · doi:10.1287/moor.5.1.43
[16] S. M. Robinson: Generalized equations. Mathematical Programming - The State of the Art (A. Bachem, M. Grotschel, B. Korte, Springer-Verlag, Berlin 1983, pp. 346-368. · Zbl 0554.34007
[17] S. M. Robinson: An implicit-function theorem for a class of nonsmooth functions. Math. Oper. Res. 16 (1991), 292-309. · Zbl 0746.46039 · doi:10.1287/moor.16.2.292
[18] R. T. Rockafellar: On the maximality of sums of nonlinear monotone operators. Trans. Amer. Math. Soc. 149 (1970), 75-88. · Zbl 0222.47017 · doi:10.2307/1995660
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.