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Several recent oscillation criteria are obtained for nonlinear delay impulsive differential equations by relating them to linear delay impulsive differential equations or inequalities, and then comparison and oscillation criteria for the latter are applied. However, not all nonlinear delay impulsive differential equations can be directly related to linear delay impulsive differential equations or inequalities. Moreover, standard oscillation criteria for linear equations cannot be applied directly...

We study the exact multiplicity and bifurcation curves of positive solutions of generalized logistic problems$$\left\{\begin{array}{cc}-{\left[\phi \left(u^{\text{'}}\right)\right]}^{\text{'}}=\lambda u^p\left(1-{\displaystyle \frac{u}{N}}\right)\hfill & \text{in}(-L,L),\hfill \\ u(-L)=u\left(L\right)=0,\hfill \end{array}\right.$$ where $p>1$, $N>0$, $\lambda >0$ is a bifurcation parameter, $L>0$ is an evolution parameter, and $\phi \left(u\right)$ is either $\phi \left(u\right)=u$ or $\phi \left(u\right)=u/\sqrt{1-u^2}$. We prove that the corresponding bifurcation curve is $\subset$-shape. Thus, the exact multiplicity of positive solutions can be obtained.