Problema cauchy si functii continue

[Cezar Lupu]

Fie \( a(.), b(.):[0, \infty)\to\mathbb{R} \) continue, \( b(x)\leq 0, \forall x\geq 0 \) si \( f:\mathbb{R}^{2}\to\mathbb{R} \) astfel incat \( f(x, y)\leq a(x)y+b(x), \forall (x,y)\in\mathbb{R}^{2} \). Sa se arate ca problema Cauchy

\( \left{\begin{array}{c}
y\prime=f(x,y), x\geq 0\\
y(0)=0\end{array} \)

are solutie \( y:[0, \infty)\to [0, \infty) \) daca si numai daca \( b(x)=0 \) si \( f(x, 0)=0, \forall x\geq 0 \).