Nagumo viability theorem. Revisited

• Published in: Nonlinear analysis Vol. 64; no. 9; pp. 2043 - 2052
• Main Author: Vrabie, Ioan I.
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01.05.2006; Elsevier
• Subjects: b-compact function; Exact sciences and technology; Lipschitz function; Mathematical analysis; Mathematics; Operator theory; Ordinary differential equations; Partial differential equations; Physics; Pseudoparabolic equation; Sciences and techniques of general use; Slow fluids; Statistical physics, thermodynamics, and nonlinear dynamical systems; Tangency condition; Thermodynamics; Viable set; Tangency condition; Slow fluids; Viable set; Pseudoparabolic equation; secondary 35K70; b-compact function; Lipschitz function; 80A20; Primary 34G20; 47J35; b compact function; Slow fluid; Viability; 80A20 Viable set
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2005.07.037

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 ( τ , ξ ) ∈ I × K there exists T > τ such that the equation above has at least one solution u : [ τ , T ] → K is the tangency condition below lim inf s ↓ 0 1 s d ( ξ + s [ f ( τ , ξ ) + h ( τ , ξ ) ] ; K ) = 0 for each ( τ , ξ ) ∈ I × K . As an application, we deduce the existence of positive solutions for a class of pseudoparabolic semilinear equations.