×

An inverse mapping theorem for set-valued maps. (English) Zbl 0804.49021

Let \((X,\rho)\) be a complete metric space, and \((Y,d)\) any linear space with a translation-invariant metric. Let \(f: X\to Y\) be strictly stationary at some point \(x_ 0\in X\), i.e., for every \(\varepsilon>0\) there exists \(\delta>0\) satisfying: \(d(f (x_ 1), f(x_ 2))\leq \varepsilon\rho (x_ 1, x_ 2)\), whenever \(\rho(x_ i, x_ 0)<\delta\), \(i=1,2\). Let \(F: X\to {\mathcal P}(Y)\) be any set-valued map, and \((x_ 0, y_ 0)\) be any point in graph \(F\). In the paper, four Lipschitz conditions are considered for \(F^{-1}\):
(L1): \(F^{-1}\) has a closed-valued pseudo-Lipschitz selection around \((y_ 0, x_ 0)\).
(L2): \(F^{-1}\) is locally closed-valued and pseudo-Lipschitz around \((y_ 0, x_ 0)\).
(L3): \(F^{-1}\) has a Lipschitz selection around \((y_ 0, x_ 0)\).
(L4): \(F^{-1}\) is locally single-valued and Lipschitz around \((y_ 0, x_ 0)\).
The main theorem says that, for \(i=1,\dots, 4\), the following are equivalent:
1) \(F^{-1}\) enjoys property (Li) around \((y_ 0, x_ 0)\).
2) \((f+F)^{-1}\) enjoys property (Li) around \((y_ 0+ f(x_ 0), x_ 0)\).
To prove the theorem, a fixed-point lemma is stated for closed-valued maps \(\phi: X\to {\mathcal P} (X)\), asserting that \(\phi\) has a fixed point provided that there exist a point \(\xi_ 0\in X\), and real numbers \(r\), \(\lambda\), \(0\leq\lambda <1\), such that a) \(\text{dist} (\xi_ 0, \phi(\xi_ 0))< r(1-\lambda)\), b) \(e(\phi (x_ 1)\cap B_ r (\xi_ 0), \phi(x_ 2))\leq \lambda\rho (x_ 1, x_ 2)\) for all \(x_ 1, x_ 2\in B_ r (\xi_ 0)\), where \(B_ r(\xi_ 0)\) is the \(r\)-ball centered at \(\xi_ 0\), and \(e(A,B)\) is the Hausdorff excess from \(B\) to \(A\).
Among the consequences of the theorem, it is worthwhile mentioning a characterization of the strong regularity for the variational inequality \(y\in \phi(x)+ \partial \Omega(x)\), where \(X\) is a Banach space, \(\phi: X\to X^*\), \(\Omega\) is any set in \(X\), and \(\partial\Omega(x)\) is the normal cone to \(\Omega\) at \(x\).
Reviewer: C.Vinti (Perugia)

MSC:

49K40 Sensitivity, stability, well-posedness
26E25 Set-valued functions
47H04 Set-valued operators
49J52 Nonsmooth analysis
90C31 Sensitivity, stability, parametric optimization
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Jean-Pierre Aubin, Lipschitz behavior of solutions to convex minimization problems, Math. Oper. Res. 9 (1984), no. 1, 87 – 111. · Zbl 0539.90085
[2] Jean-Pierre Aubin and Hélène Frankowska, Set-valued analysis, Systems & Control: Foundations & Applications, vol. 2, Birkhäuser Boston, Inc., Boston, MA, 1990. · Zbl 0713.49021
[3] J. M. Borwein, Stability and regular points of inequality systems, J. Optim. Theory Appl. 48 (1986), no. 1, 9 – 52. · Zbl 0557.49020
[4] J. M. Borwein and D. M. Zhuang, Verifiable necessary and sufficient conditions for openness and regularity of set-valued and single-valued maps, J. Math. Anal. Appl. 134 (1988), no. 2, 441 – 459. · Zbl 0654.49004
[5] Frank H. Clarke, Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. · Zbl 0582.49001
[6] Asen L. Dontchev and William W. Hager, Lipschitzian stability in nonlinear control and optimization, SIAM J. Control Optim. 31 (1993), no. 3, 569 – 603. · Zbl 0779.49032
[7] Asen L. Dontchev and William W. Hager, Implicit functions, Lipschitz maps, and stability in optimization, Math. Oper. Res. 19 (1994), no. 3, 753 – 768. · Zbl 0835.49019
[8] Lawrence M. Graves, Some mapping theorems, Duke Math. J. 17 (1950), 111 – 114. · Zbl 0037.20401
[9] A. D. Ioffe and V. M. Tihomirov, Theorie der Extremalaufgaben, VEB Deutscher Verlag der Wissenschaften, Berlin, 1979 (German). Translated from the Russian by Bernd Luderer. A. D. Ioffe and V. M. Tihomirov, Theory of extremal problems, Studies in Mathematics and its Applications, vol. 6, North-Holland Publishing Co., Amsterdam-New York, 1979. Translated from the Russian by Karol Makowski.
[10] Alexander D. Ioffe, Regular points of Lipschitz functions, Trans. Amer. Math. Soc. 251 (1979), 61 – 69. · Zbl 0427.58008
[11] A. D. Ioffe, Global surjection and global inverse mapping theorems in Banach spaces, Reports from the Moscow refusnik seminar, Ann. New York Acad. Sci., vol. 491, New York Acad. Sci., New York, 1987, pp. 181 – 188. · Zbl 0709.47054
[12] E. B. Leach, A note on inverse function theorems, Proc. Amer. Math. Soc. 12 (1961), 694 – 697. · Zbl 0191.15003
[13] Boris Mordukhovich, Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions, Trans. Amer. Math. Soc. 340 (1993), no. 1, 1 – 35. · Zbl 0791.49018
[14] -, Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis, IMA Preprints Series 994, 1992.
[15] Sam B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475 – 488. · Zbl 0187.45002
[16] Albert Nijenhuis, Strong derivatives and inverse mappings, Amer. Math. Monthly 81 (1974), 969 – 980. · Zbl 0296.58002
[17] Jean-Paul Penot, Metric regularity, openness and Lipschitzian behavior of multifunctions, Nonlinear Anal. 13 (1989), no. 6, 629 – 643. · Zbl 0687.54015
[18] Stephen M. Robinson, An inverse-function theorem for a class of multivalued functions, Proc. Amer. Math. Soc. 41 (1973), 211 – 218. · Zbl 0293.58003
[19] Stephen M. Robinson, Regularity and stability for convex multivalued functions, Math. Oper. Res. 1 (1976), no. 2, 130 – 143. · Zbl 0418.52005
[20] Stephen M. Robinson, Stability theory for systems of inequalities. II. Differentiable nonlinear systems, SIAM J. Numer. Anal. 13 (1976), no. 4, 497 – 513. · Zbl 0347.90050
[21] Stephen M. Robinson, Strongly regular generalized equations, Math. Oper. Res. 5 (1980), no. 1, 43 – 62. · Zbl 0437.90094
[22] Stephen M. Robinson, An implicit-function theorem for a class of nonsmooth functions, Math. Oper. Res. 16 (1991), no. 2, 292 – 309. · Zbl 0746.46039
[23] R. Tyrrell Rockafellar, Lipschitzian properties of multifunctions, Nonlinear Anal. 9 (1985), no. 8, 867 – 885. · Zbl 0573.54011
[24] Corneliu Ursescu, Multifunctions with convex closed graph, Czechoslovak Math. J. 25(100) (1975), no. 3, 438 – 441. · Zbl 0318.46006
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.