Positive solutions of second-order singular initial value problem in Banach space. (English) Zbl 1012.34056

The author establishes conditions for the existence of a positive solution to the following singular initial value problem in Banach space \(E\): \(x''(t)=f(t,x(t),x'(t))\), \(t \in (0,T]\); \(x(0)=x'(0)=\theta\), where \(\theta\) denotes the zero element of \(E\) and the nonlinear term \(f(t,x,y)\) may be singular at \(t=0\), \(x=\theta\), and \(y=\theta\). The case when \(f\) does not depend on \(x'(t)\) is considered separately.


34G20 Nonlinear differential equations in abstract spaces
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations