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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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##### References:
 [1] Shkarin, Trans. Matematicheskie Zametki 51 pp 128– (1992) [2] Phelps, Convex functions, monotone operators and differentiability 1364 (1989) · Zbl 0658.46035 · doi:10.1007/978-3-662-21569-2_5 [3] Haddad, Calcul sous-differentiel et solutions de viscosité des équations de Hamilton-Jacobi (1994) [4] DOI: 10.2307/2155218 · Zbl 0849.49016 · doi:10.2307/2155218 [5] Deville, Serdica 21 pp 59– (1995) [6] Cohn, Measure theory (1980) · doi:10.1007/978-1-4899-0399-0 [7] Azagra, J. Math. Anal. Appl. [8] Deville, Smoothness and renormings in Banach spaces 64 (1993) · Zbl 0782.46019
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