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Subdifferential Rolle’s and mean value inequality theorems. (English) Zbl 0894.49008
Let \(X\) be a Banach space and let \({\mathcal U}\) be an open convex subset of \(X\). A function \(f:{\mathcal U}\to\mathbb{R}\) is said to be Gâteaux subdifferentiable at \(x\) if there exists \(p\in X^*\) such that for every \(h\in X\) \[ \liminf_{t\to 0} {f(x+ th)- f(x)- \langle p,th\rangle\over\| th\|}\geq 0, \] and the Gâteaux subdifferential set of \(f\) at the point \(x\) is defined by \[ D^-_Gf(x)= \Biggl\{p\in X^* \Biggl| \forall h\in S_X,\;\liminf_{| t|\to 0} {f(x+ th)- f(x)-\langle p,th\rangle\over\| th\|}\geq 0\Biggr\}. \] This paper proves that if for every \(x\in{\mathcal U}\) there exists \(p\in D^-_Gf(x)\) such that \(\| p\|\leq M\), then \(| f(x)- f(y)|\leq M\| x-y\|\) for all \(x,y\in{\mathcal U}\). Moreover, it is proved that if \(x,y\in{\mathcal U}\) and \(M\geq 0\) such that for every \(t\in[0,1]\) there exists \(p\in D^-_Gf(tx+ (1-t)y)\) with \(\| p\|\leq M\), then \(| f(x)- f(y)|\leq M\| x-y\|\). These are two mean value inequality theorems.
This paper also gives a subdifferential approximate Rolle’s theorem as follows. Let \({\mathcal U}\) be an open connected bounded set in a Banach space \(X\) that has a Gâteaux differentiable Lipschitz bump function. Let \(f:\overline{\mathcal U}\to \mathbb{R}\) be continuous and bounded Gâteaux subdifferentiable in \({\mathcal U}\). Let \(R>0\) and \(x_0\in{\mathcal U}\) be such that \(\text{dist}(x_0,\partial{\mathcal U})= R\). Suppose that \(f(\partial{\mathcal U})\subset[-\varepsilon, \varepsilon]\). Then there exist \(x_\varepsilon\in{\mathcal U}\) and \(p\in D^-_Gf(x_\varepsilon)\) such that \(\| p\|\leq 2\varepsilon/R\). For showing this theorem, one needs Ekeland’s variational principle. There is also a Fréchet-subdifferential approximate Rolle’s theorem, but it has to use Deville-Godefroy-Zizler smooth variational principle.
Reviewer: S.Shih (Tianjin)

MSC:
49J52 Nonsmooth analysis
26D15 Inequalities for sums, series and integrals
49J50 Fréchet and Gateaux differentiability in optimization
46G05 Derivatives of functions in infinite-dimensional spaces
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