A counterexample in the theory of linear singularly perturbed systems

In this note we compare the bounded solutions of the linear singularly perturbed system ε X' = A (t) x + f (t), with the solutions of the algebraic system A (t) x + f (t) = 0.  Here A and  f  are bounded C1 functions with bounded derivatives. We assume that the eigenvalues of A (t) satisfy | R e λ (t )| ≥ y and gt; 0. It is known that for small ε, the following estimate is valid hasta ||kε (f) + A-1 f || ≤  εL || f ||1, where kε(f) denotes the bounded solution of ε X'  = A(t)x +f (t), || f || = sup R| f(t)|, || f ||1 : = || f || +|| f´ ||  and L is a constant. We prove that this estimate cannot be replaced by ||kε (f) + A-1 f || ≤  εL || f ||. Futhermore, if, instead of the condition that A be C1, we require that the function be bounded and Lipschitz continuous, we show that the same estimate, ||kε (f) + A-1 || ≤  εL || f ||1, can be obtained.

• Autores: Naulin, Raúl
• Tipo de recurso: Article of journal
• Fecha de publicación: 1991
• Institución: Universidad Nacional de Colombia
• Repositorio: Universidad Nacional de Colombia
• Idioma: spa
• OAI Identifier: oai:repositorio.unal.edu.co:unal/43433
• Acceso en línea: https://repositorio.unal.edu.co/handle/unal/43433; http://bdigital.unal.edu.co/33531/
• Palabra clave: 51 Matemáticas / Mathematics; Bounded solutions; system linear algebraic system; bounded functions; Lipschitz function; Soluciones acotadas; sistema lineal; sistema algebráico; funciones acotadas; función de Lipschitz
• Rights: openAccess
• License: Atribución-NoComercial 4.0 Internacional