A note on upper and lower solutions for singular initial value problems (Q2707982)

The authors study the existence of solutions to the first-order singular initial value problem \(y'(t) = q(t) f(t,y(t))\), \(0 < t < T\); \(y(0)=0\), where \(T>0\) is fixed, the nonlinearity \(f\) is allowed to change sign and \(f\) may be nonsingular at \(y=0\). NEWLINENEWLINENEWLINEAssuming, among others, the existence of a pair of lower and upper solutions to this problem, they prove the existence of a solution over the lower one. In the proof they construct a sequence of truncated approximate nonsingular problems that, by the theory of lower and upper solutions for nonsingular problems admits at least one solution lying between both functions. The solution to the singular problem is given as the limit of the solutions to the nonsingular ones. NEWLINENEWLINENEWLINEMoreover, they prove different existence results in this direction and present some examples to illustrate the obtained results.