Existence of periodic solutions for a class of second-order superquadratic delay differential equations (Q2927688)

Consider the second-order delay differential equation NEWLINE\[NEWLINE x''(t)=-f(t,x(t-\tau)), NEWLINE\]NEWLINE where \(f(t,x)\) is periodic with respect to the first variable with period \(2\tau\) and there exists \(F(t,x)\) such that \(\nabla _xF=f\). Using the crititcal point theory, the authors establish two results on the existence of a nontrivial \(2\tau\)-periodic solution. The results can be applied to the case where \(f\) does not satisfy the Ambrosetti-Rabinowitz growth condition. However, from the view of the reviewer, the authors should give some background of the problem, especially the importance of the existence of \(2\tau\)-periodic solution.NEWLINENEWLINEIt seems that there is a typo in (\(F_2\)), that is, \(F(t,x)\) should be \(2\tau\)-periodic in \(t\). If this is the case, then (\(F_1\)) is redundant.