Nagumo viability theorem. Revisited

We consider the nonlinear ordinary differential equation u′(t)=f(t,u(t))+h(t,u(t)),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→Xf:I×K→X a compact function, and h:I×K→Xh:I×K→X continuous on I×KI×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(τ,ξ)∈I×K there exists T>τT>τ such that the equation above has at least one solution u:[τ,T]→Ku:[τ,T]→K is the tangency condition belowliminfs↓01sd(ξ+s[f(τ,ξ)+h(τ,ξ)];K)=0for each (τ,ξ)∈I×K(τ,ξ)∈I×K. As an application, we deduce the existence of positive solutions for a class of pseudoparabolic semilinear equations.