Hostname: page-component-7b7567c8c-kwt9d Total loading time: 0 Render date: 2026-11-14T21:03:39.417Z Has data issue: false hasContentIssue false
  • English
  • Français

An Inequality Implicit Function Theorem

Part of:Topological linear spaces and related structuresFunctions of several variables

Published online by Cambridge University Press:  09 April 2009

Kung-Fu Ng
Kung-Fu Ng
Affiliation:
Department of MathematicsThe Chinese University of Hong Kong, Hong Kong
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

Let f be a continuous function, and u a continuous linear function, from a Banach space into an ordered Banach space, such that f − u satisfies a Lipschitz condition and u satisfies an inequality implicit-function condition. Then f also satisfles an inequality implicit-function condition. This extends some results of Flett, Craven and S. M. Robinson.

MSC classification

Secondary:46A30: Open mapping and closed graph theorems; completeness (including $B$-, $B_r$-completeness)26B10: Implicit function theorems, Jacobians, transformations with several variables46A40: Ordered topological linear spaces, vector lattices

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 44 , Issue 2 , April 1988, pp. 146 - 150
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Bender, P. J., ‘Nonlinear programming in normed linear spaces’, J. Optimization Theory Appl. 24 (1978), 263–285.CrossRefGoogle Scholar
[2]Craven, B. D., ‘A generalization of Lagrange multipliers’, Bull. Austral. Math. Soc. 3 (1970), 353–362.CrossRefGoogle Scholar
[3]Craven, B. D., Mathematical programming and control theory, (Chapman and Hall, 1978).CrossRefGoogle Scholar
[4]Flett, T. M., ‘On differentiation in normed vector spaces’, J. London Math. Soc. 42 (1967), 523–533.CrossRefGoogle Scholar
[5]Jameson, G., Ordered linear spaces, (Springer-Verlag, 1970).CrossRefGoogle Scholar
[6]Luenberger, D., Optimization by vector space method, (John Wiley, 1968).Google Scholar
[7]Nadler, S. B. Jr, ‘Multi-valued contraction mappings’, Pacific J. Math. 30 (1969), 475–488.CrossRefGoogle Scholar
[8]Ng, K. F., ‘An open mapping theorem’, Math. Proc. Cambridge Philos. Soc. 74 (1973), 61–66.CrossRefGoogle Scholar
[9]Ng, K. F. and Yost, D., ‘Quasi-regularity in optimization’, J. Austral. Math. Soc. Ser. A 41 (1986), 188–192.CrossRefGoogle Scholar
[10]Robinson, S. M., ‘Normed convex processes’, Trans. Amer. Math. Soc. 174 (1972), 127–140.CrossRefGoogle Scholar
[11]Robinson, S. M., ‘An inverse-function theorem for a class of multivalued functions’, Proc. Amer. Math. Soc. 41 (1973), 211–218.CrossRefGoogle Scholar
[12]Robinson, S. M., ‘Stability theory for systems of inequalities, Part II: differentiable nonlinear systems’, SIAM J. Numer. Anal. 13 (1976), 497–513.CrossRefGoogle Scholar
[13]Rockafellar, R. T., ‘Monotone processes of convex and concave type’, Mem. Amer. Math. Soc. 77 (1967).Google Scholar
[14]Zowe, J. and Kurcyusz, S., ‘Regularity and stability for the mathematical programming problem in Banach spaces’, Appl. Math. Optim. 5 (1979), 49–62.CrossRefGoogle Scholar