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On approximately convex functions

Author(s): Zsolt Páles
Journal: Proc. Amer. Math. Soc. 131 (2003), 243-252.
MSC (2000): Primary 26A51, 26B25
Posted: June 5, 2002
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Abstract: A real valued function $f$ defined on a real interval $I$ is called $(\varepsilon,\delta)$-convex if it satisfies

\begin{displaymath}f(tx+(1-t)y)\le tf(x)+(1-t)f(y) + \varepsilon t(1-t)\vert x-y\vert + \delta \quad \text{for} x,y\in I,\, t\in[0,1]. \end{displaymath}

The main results of the paper offer various characterizations for $(\varepsilon,\delta)$-convexity. One of the main results states that $f$is $(\varepsilon,\delta)$-convex for some positive $\varepsilon$ and $\delta$ if and only if $f$ can be decomposed into the sum of a convex function, a function with bounded supremum norm, and a function with bounded Lipschitz-modulus. In the special case $\varepsilon=0$, the results reduce to that of Hyers, Ulam, and Green obtained in 1952 concerning the so-called $\delta$-convexity.

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Additional Information:

Zsolt Páles
Affiliation: Institute of Mathematics and Informatics, University of Debrecen, H-4010 Debrecen, Pf. 12, Hungary
Email: pales@math.klte.hu

DOI: 10.1090/S0002-9939-02-06552-8
PII: S 0002-9939(02)06552-8
Keywords: Convexity, $(\varepsilon,\delta)$-convexity, stability of convexity, $(\varepsilon,\delta)$-subgradient, $(\varepsilon,\delta)$-subdifferentiability
Received by editor(s): April 2, 2001
Received by editor(s) in revised form: September 4, 2001
Posted: June 5, 2002
Additional Notes: This research was supported by the Hungarian Scientific Research Fund (OTKA) Grant T-038072 and by the Higher Education, Research and Development Fund (FKFP) Grant 0215/2001.
Communicated by: Jonathan M. Borwein
Copyright of article: Copyright 2002, American Mathematical Society

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