Taiwanese Journal of Mathematics

CODERIVATIVE AND MONOTONICITY OF CONTINUOUS MAPPINGS

N. H. Chieu and N. T. Q. Trang

Full-text: Open access

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

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406545

Digital Object Identifier
doi:10.11650/twjm/1500406545

Mathematical Reviews number (MathSciNet)
MR2887869

Zentralblatt MATH identifier
1238.49039

Subjects
Primary: 49K40: Sensitivity, stability, well-posedness [See also 90C31] 49J40: Variational methods including variational inequalities [See also 47J20] 49J52: Nonsmooth analysis [See also 46G05, 58C50, 90C56] 49J53: Set-valued and variational analysis [See also 28B20, 47H04, 54C60, 58C06]

Keywords
monotonicity submonotonicity convexity coderivative positive semi-definiteness

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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