Behavior of the solutions of a real second-order differential equation, not solved with respect to the leading derivative (Q1061286)

The authors consider the equation (1) \(w(t)x''(t)=ax'(t)+x^ 3f(t,x,x',x''),\) where \(w(t)\in C^ 1(0,t_ 0)\), \(w>0\), \(w'>0\), \(w(+0)=w'(+0)\), \(a=const.>0\), \(\int_{0}dt/tw(t)=\infty\) \(w(t)[w'(t)]^{-1}=o(1),\) \(w^{-k}(t)\exp [\int^{t}_{t_ 0}ds/w(s)]=o(1)\) as \(t\to 0\) and \(k>0\) arbitrary and f(t,x,y,z) is a continuous function on \(D=\{0\leq t\leq t_ 0,| x| \leq R,| y| \leq R_ 1,| z| \leq R_ 2\}\) and has there continuous derivatives with respect to x,y,z. They investigate the existence and asymptotic behaviour of the nontrivial solutions of (1) which have the properties: \(x^{(i)}(t)\to 0\) as \(t\to 0\) \(i=0,1,2\). The main result is the following: For every number c there exists at least one solution x(t) of (1) such that for sufficiently small t some given estimates hold. The proof is made via the Schauder fixed point theorem.