On the monotonicity property for a certain class of second order differential equations
Consider the equations (1) \((p(x)x'(t))'=q(t)x(t)\) and (2) \((p(t)x'(t))'=q(t)f(x(t)),\) where p,q: [0,\(\infty)\to R\) and f: \(R\to R\) are continuous, \(p(t)>0\), \(q(t)>0\), and \(uf(u)>0\) for \(u\neq 0\). Let \(I_ 1=\int^{\infty}_{0}1/p(t)\int^{s}_{0}q(r)drds,\quad I_ 2=\int^{\infty}_{0}q(s)\int^{s}_{0}1/p(r)drds,\) \(B=\{x\), solution of (1) [(2)]: x(t)x\({}'(t)<0\) for \(t>0[t>\alpha_ x]\}\). In section 1, the authors give necessary and sufficient conditions related to the quantities of \(I_ 1\) and \(I_ 2\) for the asymptotic behavior of solutions of (1) in B. In section 2, they give sufficient and necessary conditions for the existence of a solution x of (2) respectively in B and in B with \(x(+\infty)=0\) or \(x(+\infty)\neq 0\).
scientific article; zbMATH DE number 4136383
scientific article