Consider the second order Hamiltonian system
where is a symmetric matrix valued function and . A nonzero solution of (HS) is said to be homoclinic (to 0) if as .
The authors prove that problem (1) has infinitely many homoclinic orbits under the following conditions:
is a symmetric and positively definite matrix for all and there exists a continuous function such that for all and
, where is a continuous function such that
and is a constant.
In fact, in Theorem 1.2 the condition respect to is not sufficient and the condition that is positive is also used in the proof of the Lemma 3.1.
Theorem 1.2 in this paper generalizes the result in [Z. Zhang and R. Yuan, Nonlinear Anal., Theory Methods Appl. 71, No. 9, A, 4125–4130 (2009; Zbl 1173.34330)], in which is a positive continuous function such that