## 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
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