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Gâteaux differentiability and uniform monotone approximation of convex functions in Banach spaces
Acta Mathematica Hungarica ( IF 0.6 ) Pub Date : 2021-05-06 , DOI: 10.1007/s10474-021-01146-6
S. Shang

We prove that if \(X^{*}\) is strictly convex, a convex function \(f\) is coercive and b-Lipschitzian iff there exist two convex function sequences \(\{f_{n}\}_{n=1}^{\infty}\) and \(\{g_{n}\}_{n=1}^{\infty}\) such that (1) \(f_{n}\leq f_{n+1}\leq f\) and \(f\leq g_{n+1}\leq g_{n}\) for all integers \(n \geq 1\); (2) \(f_{n}\) and \(g_{n}\) are continuous and Gâteaux differentiable on \(X\); (3) \(f_n \to f\) and \({g_n \to f}\) uniformly on \(X\); (4) \(f_n\) and \(g_n\) are coercive and b-Lipschitzian. Moreover, we also prove that if \(X^{*}\) is strictly convex, then a convex function f is Lipschitzian iff conditions (1)-(3) are true and there exists \(m>0\) such that \(|f_{n}(x)-f_{n}(y)|\leq m\|x-y\|\) and \(|g_{n}(x)-g_{n}(y)|\leq m\|x-y\|\) whenever \(x, y \in X\) and \(n\in N\).



中文翻译:

Banach空间中凸函数的Géteaux可微性和一致单调逼近

我们证明如果\(X ^ {*} \)是严格凸的,则凸函数\(f \)是强制的,并且如果存在两个凸函数序列 \(\ {f_ {n} \} _ { n = 1} ^ {\ infty} \)\(\ {g_ {n} \} _ {n = 1} ^ {\ infty} \)这样(1)\(f_ {n} \ leq f_ { n + 1} \ leq f \)\(f \ leq g_ {n + 1} \ leq g_ {n} \)对于所有整数\(n \ geq 1 \) ; (2)\(f_ {n} \)\(g_ {n} \)是连续的,并且Gâteaux在\(X \)上是可区分的;(3)\(f_n \ to f \)\({g_n \ to f} \)统一放在\(X \)上;(4)\(f_n \)\(g_n \)是强制性的,是b-Lipschitzian。此外,我们还证明,如果\(X ^ {*} \)是严格凸的,则凸函数f为Lipschitzian iff条件(1)-(3)为真,并且存在\(m> 0 \)使得\(| f_ {n}(x)-f_ {n}(y)| \ leq m \ | xy \ | \)\(| g_ {n}(x)-g_ {n}(y)| \每次\(x,y \ in X \)\(n \ in N \)都应使用leq m \ | xy \ | \\)

更新日期:2021-05-06
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