## Taiwanese Journal of Mathematics

### CODERIVATIVE AND MONOTONICITY OF CONTINUOUS MAPPINGS

#### Abstract

Sufficient conditions for a norm-to-weak$^*$ continuous mapping $f: X \rightarrow X^*$ being monotone or submonotone are established by its Fréchet and normal coderivatives, where $X$ is an Asplund space with its dual space $X^*$. Under some additional assumptions, they are also necessary conditions. Among other things, we obtain a criterion for the monotonicity of continuous mappings which extends the following classical result: a differentiable mapping $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is monotone if and only if for each $x \in \mathbb{R}^n$ the Jacobian matrix $\nabla F(x)$ is positive semi-definite; see [22, Proposition 12.3]. As a by-product, sufficient conditions for a function being convex or approximately convex are given.

#### Article information

Source
Taiwanese J. Math., Volume 16, Number 1 (2012), 353-365.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500406545

Digital Object Identifier
doi:10.11650/twjm/1500406545

Mathematical Reviews number (MathSciNet)
MR2887869

Zentralblatt MATH identifier
1238.49039

#### Citation

Chieu, N. H.; Trang, N. T. Q. CODERIVATIVE AND MONOTONICITY OF CONTINUOUS MAPPINGS. Taiwanese J. Math. 16 (2012), no. 1, 353--365. doi:10.11650/twjm/1500406545. https://projecteuclid.org/euclid.twjm/1500406545

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