Funções convexas em escalas temporais

Abstract

In this work we study some results of the theory of time scales, which are closed nonempty subsets of the real numbers. The time scales represent a powerful tool to describe models which involve evolution of time, where R and Z are considered special cases, called continuous and discrete time respectively. The theory and applications of the derivation (delta, nabla and α-diamond) and the Riemann’s integration in time scales have recently received considerable attention. The main objective of this work is to study convex functions on time scales and to present some properties such as: the convexity of a function is a necessary and sufficient condition for its sub-differentiability. The subdifferential of a function ƒ is given as a set of certain extended functions. Using the convexity of a function we prove a generalized version of Jensen’s inequality on time scales via the delta integral. In addition, we present some corollaries and an application in variational calculus.