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Keywords:
singular ordinary differential equation of the second order; lower and upper functions; nonlinear boundary conditions; time singularities; phase singularity
singular ordinary differential equation of the second order; lower and upper functions; nonlinear boundary conditions; time singularities; phase singularity
Summary:
The paper deals with the singular nonlinear problem \[ u^{\prime \prime }(t) + f(t,u(t),u^{\prime }(t)) = 0,\quad u(0) = 0,\quad u^{\prime }(T) = \psi (u(T)), \] where $f \in \mathop {\mathit{Car}}((0,T)\times D)$, $D = (0,\infty )\times $. We prove the existence of a solution to this problem which is positive on $(0,T]$ under the assumption that the function $f(t,x,y)$ is nonnegative and can have time singularities at $t = 0$, $t = T$ and space singularity at $x = 0$. The proof is based on the Schauder fixed point theorem and on the method of a priori estimates.
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