The findings here stem from a literature review of evidence for low prevalence countries (QAP & UNICEF, 2008) and a comparison with programming principles drawn from research in high prevalence countries (Joint Learning Initiative on Children and HIV and AIDS [JLICA], 2008).
They are defined by means of a parabolic comparison principle with respect to solutions.
We end this section with a comparison principle for system (3.1).
Applying the comparison principle with respect to the initial value of the variational inequality (see [16]), we obtain (3.21).
Hence, by the comparison principle with (h(0)=h_{1}(0)=0), we obtain h varrho ge h_{1} varrho) quad mbox{for } varrhoin[0, varrho_{2}].
By (4.9), (4.12), and (4.13), we have that is a lower solution of (1.1) and (1.3), which with the comparison principle implies that the solutions of (1.1)–(1.3) blow up in finite time.
By constructing two sine functions as the shooting objects and combining with the comparison principle, the author obtained some better results than those via fixed point techniques for the existence of positive solutions to (1.4 - 1.5 1.4 - 1.5
Applying the comparison principle with respect to the initial value of variational inequality (see [12]), we obtain begin{aligned} u x,tau+delta)=hat{u} x,tau geq u x,tau), quad x,tau in Rtimes 0,T- delta]. end{aligned} So, we have begin{aligned} partial _{tau}ugeq0,quad x,tau inOmega_{T}.
For any (tin[0, T)), we know that frac{Y t)}{dt}=bigl[gamma+Rbigl(X t bigr bigr]Y t geq frac{gamma}{2}Y t), which, together with the comparison principle of ordinary differential equation, gives Y t geq Y 0)e^{frac{gamma}{2}t}=F(x_{0})e^{frac{gamma}{2}t}geq frac{Ce^{frac{gamma}{2} t}x_{0}}{2}.
By constructing a quadratic function and a sine function as the shooting objects and combining the integral mean value theorem with the comparison principle, we consider the existence of positive solutions to the BVP respectively under the case 0 < ∑ i = 1 n α i η i ≤ 1 and the case ∑ i = 1 n α i η i > 1.
