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Zbl 1080.34001
Dibl\'ik, Josef; Ruužičková, Miroslava
Existence of positive solutions of a singular initial problem for a nonlinear system of differential equations.
(English)
[J] Rocky Mt. J. Math. 34, No. 3, 923-944 (2004). ISSN 0035-7596

This work considers initial value problems (IVPs) defined by a diagonally implicit ODE of the form $g(x)y'=A(x)\alpha(y) - \omega(x)$, together with $y(0^+)=0;$ here $y \in \Bbb{R}^n$, $y'=dy/dx$, $g(x)=\text{diag}(g_1(x), \ldots, g_n(x))$, $A(x)=(a_{ij}(x))$ is a non-singular matrix with non-vanishing diagonal entries, $\alpha(y)=(\alpha_1(y_1), \ldots, \alpha_n(y_n)),$ and $\omega(x)=(\omega_1(x), \ldots, \omega_n(x))$; $g$ is continuous, and $A$, $\alpha$, $\omega$ are $C^1$. The condition $g_i(0^+)=0$ for some $i$ defines a singular Cauchy problem. Under componentwise positivity hypotheses on $g(x)$, $\alpha(y)$, $\alpha'(y)$, $\Omega(x) \equiv A^{-1}(x)\omega(x)$ and $\Omega'(x)$, and assuming additionally $\alpha(0^+)=\Omega(0^+)=0$, $\alpha_i(y) \leq M\alpha'_i(y)$, the authors first show that there exists a well-defined, positive, $C^1$ mapping $z=\varphi(x)=\alpha^{-1}(\Omega(x))$ with $\varphi(0^+)=0$. This allows them to construct a funnel'', that is, a domain which will contain the graph of a solution to the singular IVP; specifically, under additional conditions on $a_{ij},$ $\omega_i,$ $g_i$ and $\Omega_i$, they prove using a topological retract method that there exists a family of positive solutions to the IVP. Some generalizations, simplification for linear cases (in which $\alpha_i = \text{id}$), and examples are discussed.
MSC 2000:
*34A09 Implicit equations
34A12 Initial value problems for ODE

Keywords: singular ordinary differential equation; Cauchy problem; positive solution; topological retract

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