×

Positive solutions for singular initial value problems with sign changing nonlinearities depending on \(y'\). (English) Zbl 1155.34001

The authors consider the singular initial value problem
\[ y''(t)= \Phi(t)f(t,y,y'), \quad t\in(0,T], \qquad y(0)= y'(0)=0, \tag{1} \]
where \(f(t,y,y')\) changes sign and may be singular at \(y=0\) and \(y'=0\) and \(f(t,y,y')\) may be superlinear at \(y=+\infty\).
Using the theory of fixed point index on a cone they discuss the existence of positive solutions to (1) when \(f(t,y,y')\) is singular at \(y'=0\) but not \(y=0\) and when \(f\) may change sign. Finally they discuss the existence of positive solutions to (1) when \(f(t,y,y')\) is singular at \(y'=0\) and \(y=0\) and when \(f\) may change sign.

MSC:

34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
PDFBibTeX XMLCite