n a non linear differential equation of the Riccati form +=p\_(), with an initial condition $\theta(0)=\theta_i$ then, if there exists a constant $c_\Lambda>0$ such that \_i+&lt; \_ \_[0]{}\^e\^[-c\_]{} p\_() d, it follows that a solution $\theta_\Lambda$ does not extend beyond a finite value of the affine parameter $\lambda=\lambda_e$.

The equation (\[u\]) may be converted, by the change of variables y()=-(\_+c\_)e\^[-c\_]{},into the following one =+r(),y(0)=y\_0. Here r()=e\^[-c\_]{}(p\_()+),q()=e\^[-c\_]{},and $y_0=-\theta_i-c_\Lambda$. From the definition of $q(\lambda)$ given in (\[erre\]) it is clear that \_[+]{}\_0\^+. Assume that the initial condition is such that \_[+]{} \^\_0 r()d&gt;-y\_0, and that $y(\lambda)$ extend to the whole interval $[0,\infty)$. This will imply a contradiction which will show that this affirmation is false thus, $y(\lambda)$ does not extend beyond a finite affine parameter value $\lambda=\lambda_e$. The same observation will hold for $\theta_\Lambda$, as it is defined in terms of $y(\lambda)$ by (\[y\]).

In order to show the aforementioned contradiction, note that (\[cudo\]) implies the existence of affine parameter $\lambda_1$ for which $$\int^\lambda_0 r(\xi)d\xi>-y_0, \qquad \textrm{for}\qquad \lambda>\lambda_1.$$ By integrating the equation (\[teor\]) and taking into account the last inequality, it follows that y()=\_0\^ d+\_0\^r() d+y\_0&gt;\_0\^ d. It is convenient to introduce the quantity given by R()=\_0\^ d. As $q(\lambda)$ is positive and $\lambda>0$, it can directly be seen that that $R(\lambda)\geq 0$, the equality holds only for $\lambda=0$. This definition and the inequality (\[teor2\]) shows that &lt;=, for $\lambda>\lambda_1$. From here it is concluded, for every $\lambda_2>\lambda_1$, that $$\int_{\lambda_2}^{\lambda}\frac{d\xi}{q(\xi)}<\int_{\lambda_2}^{\lambda}\frac{1}{R^2}\frac{dR}{d\lambda}d\xi=\frac{1}{R(\lambda_2)}-\frac{1}{R(\lambda)}<\frac{1}{R(\lambda_2)}.$$ However, by condition (\[cudon\]) it follows that the left hand is not bounded when $\lambda