On the periodic solutions of certain class of seventh-order differential equations (Q1203433)

Consider the constant-coefficient differential equation (1) \(x^{(7)}+ a_ 2 x^{(6)}+ a_ 2 x^{(5)}+ \dots+ a_ 6 \dot x+a_ 7 x=p(t)\) where \(p\) is continuous and \(\omega\)-periodic in \(t\) for some \(w\not\equiv 0\). It is pointed out that if (i) \(a_ 1\leq 0\), \(a_ 3\geq 0\), \(a_ 5\leq 0\) and \(a_ 7>0\), or if (ii) \(a_ 1\geq 0\), \(a_ 3\leq 0\), \(a_ 5>0\) and \(a_ 7<0\), then the equation (1) with \(p\equiv 0\) has no non-trivial periodic solutions while the equation (1) in the case \(p\not\equiv 0\) has an \(\omega\)-periodic solution. The main object of the paper is to obtain some extensions of these results to some nonlinear differential equations (1) in which \(a_ 5\), \(a_ 6\), \(a_ 7\) are not constants and \(p\equiv p(t,x,\dot x,\dots, x^{(6)})\) is bounded and satisfies \(p(t+w,x_ 1,x_ 2,\dots,x_ 7)=p(t,x_ 1,x_ 2,\dots,x_ 7)\) for all \(t\), \(x_ 1,\dots,x_ 7\). It is to be pointed out, however, that the author uses the stronger restrictions \(a_ 1<0\) and \(a_ 3>0\) or \(a_ 1>0\) and \(a_ 3<0\) on \(a_ 1\) and \(a_ 3\) in establishing the existence of an \(\omega\)-periodic solution for the nonlinear equations with \(p\not\equiv 0\).