Abstract
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.
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