Abstract
In this paper, we consider the multiplicity of homoclinic solutions for the following damped vibration problems x ¨ ( t ) + B x ˙ ( t ) − A ( t ) x ( t ) + H x ( t , x ( t ) ) = 0 , where A ( t ) ∈ ( R , R N ) is a symmetric matrix for all t ∈ R , B = [ b i j ] is an antisymmetric N × N constant matrix, and H ( t , x ) ∈ C 1 ( R × B δ , R ) is only locally defined near the origin in x for some δ > 0 . With the nonlinearity H ( t , x ) being partially sub-quadratic at zero, we obtain infinitely many homoclinic solutions near the origin by using a Clark's theorem.
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