Positive solutions to certain classes of singular nonlinear second order boundary value problems. (English) Zbl 0808.34022

The author continues to investigate the existence of solutions to the nonlinear singular boundary value problem \((1/p(t)) (p(t)y')' + q(t) f(t,y,p(t)y') = 0\), \(0 < t < 1\), with \(y\) satisfying either \(y(0) = y(1)\), \(y(1) = 0\); or \(y(1) = 0\), \(\lim_{t \to 0^ +} p(t)y'(t) = 0\). The nonlinearity \(qf\) is allowed to be singular at \(t=0\), \(t=1\), \(y=0\) and/or \(py'=0\). The main idea is to consider the corresponding approximate problems with \(y(1) = 1/n\) and/or \(\lim_{t \to 0^ +} p(t)y'(t) = - 1/n\) \((n \in \mathbb{N}^*)\) when \(qf\) is singular at \(y = 0\) and/or at \(py' = 0\), and then, letting \(n \to \infty\), to obtain the solution via the Arzela-Ascoli theorem.


34B15 Nonlinear boundary value problems for ordinary differential equations
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