Subdifferentials of convex functions on time scales

Małgorzata Wyrwas; Dorota Mozyrska; Ewa Girejko

doi:10.3934/dcds.2011.29.671

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2011, Volume 29, Issue 2: 671-691. Doi: 10.3934/dcds.2011.29.671

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Subdifferentials of convex functions on time scales

1.
Faculty of Computer Science, Białystok University of Technology, ul. Wiejska 45a, 15-351 Bia lystok
2.
Faculty of Computer Science, Białystok University of Technology, ul. Wiejska 45a, 15-351 Białystok

Received: July 31, 2009

Revised: February 28, 2010

Published: May 2011

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Abstract

The paper studies the notion of subdifferentials of functions defined on a time scale. The subdifferential of a given function $f$ is defined as the set of certain extended functions. Since the convexity of the given function guarantees its subdifferentiability, properties of convex functions on time scales are presented. We show that the convexity of a function is the necessary and sufficient condition for its subdifferentiability. The relations between the delta, nabla, diamond-$\alpha$ derivatives and subdifferentials of convex functions are given.

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Mathematics Subject Classification: Primary: 26D15, 39A13; Secondary: 52A40.

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