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| Convexity and Uniform Monotone Approximation of Differentiable Function in Banach Spaces |
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Citation： |
Shaoqiang SHANG.Convexity and Uniform Monotone Approximation of Differentiable Function in Banach Spaces[J].Chinese Annals of Mathematics B,2025,46(2):271~286 |
| Page view： 453
Net amount： 214 |
Authors： |
Shaoqiang SHANG; |
Foundation： |
the National Natural Science Foundation of China (No. 12271121) |
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| Abstract： |
In this paper, the author proves that if the dual X? of X is weakly locally
uniformly convex and the convex function f is continuous on X, then there exist two
sequences {fn}∞ n=1 and {gn}∞ n=1 of continuous functions on X?? such that (1) fn(x) ≤
fn+1(x) ≤ f(x) ≤ gn+1(x) ≤ gn(x) whenever x ∈ X; (2) the two convex functions fn and
gn are G?ateaux differentiable on X; (3) fn → f and gn → f uniformly on X. Moreover,
if the function f is coercive on X, then (1) fn and gn are two w?-lower semicontinuous
convex functions on X??; (2) epifn = epi fn ∩ (X × R)w? and epi gn = epi gn ∩ (X × R)w? |
Keywords： |
Uniform monotone approximation Gˆateaux differentiable Weakly locally
uniformly convex space w∗-Lower semicontinuous convex functions |
Classification： |
46B20 |
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