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Lyapunov theorems for measure functional differential equations via Kurzweil‐equations
Author(s) -
Federson Márcia,
Mesquita Jaqueline G.,
Toon Eduard
Publication year - 2015
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201300219
Subject(s) - mathematics , measure (data warehouse) , lyapunov function , stability (learning theory) , ordinary differential equation , differential equation , mathematical analysis , distribution (mathematics) , integrable system , pure mathematics , nonlinear system , physics , quantum mechanics , database , machine learning , computer science
We consider measure functional differential equations (we write measure FDEs) of the form D x = f ( x t , t ) D g , where f is Perron–Stieltjes integrable, x t is given byx t ( θ ) = x ( t + θ ) , θ ∈ [ − r , 0 ] , with r > 0 , and D x and D g are the distributional derivatives in the sense of the distribution of L. Schwartz, with respect to functions x : [ t 0 , ∞ ) → R nand g : [ t 0 , ∞ ) → R ,t 0 ∈ R , and we present new concepts of stability of the trivial solution, when it exists, of this equation. The new stability concepts generalize, for instance, the variational stability introduced by Š. Schwabik and M. Federson for FDEs and yet we are able to establish a Lyapunov‐type theorem for measure FDEs via theory of generalized ordinary differential equations (also known as Kurzweil equations).