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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.

MSC:

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