Nagumo viability theorem. Revisited

✍ Scribed by Ioan I. Vrabie

• Publisher: Elsevier Science
• Year: 2006
• Tongue: English
• Weight: 132 KB
• Volume: 64
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2005.07.037

✦ Synopsis

We consider the nonlinear ordinary differential equation u (t) = f (t, u(t)) + h(t, u(t)), where X is a real Banach space, I is a nonempty and open interval, K a nonempty and locally closed subset in X, f : I × K → X a compact function, and h : I × K → X continuous on I × K and locally Lipschitz with respect to its last argument. We prove that a necessary and sufficient condition in order that for each ( , ) ∈ I × K there exists T > such that the equation above has at least one solution u : [ , T ] → K is the tangency condition below lim inf s↓0 1 s d( + s[f ( , ) + h( , )]; K) = 0 for each ( , ) ∈ I × K. As an application, we deduce the existence of positive solutions for a class of pseudoparabolic semilinear equations.