Positive solutions of nonlocal fractional boundary value problems
John R. Graef Lingju Kong Qingkai Kong Min Wang
Conference Publications 2013, 2013(special): 283-290 doi: 10.3934/proc.2013.2013.283
The authors study a type of nonlinear fractional boundary value problem with nonlocal boundary conditions. An associated Green's function is constructed. Then a criterion for the existence of at least one positive solution is obtained by using fixed point theory on cones.
keywords: fractional calculus. positive solution Green's function
Optimal control of a vector-host epidemics model
Qingkai Kong Zhipeng Qiu Zi Sang Yun Zou
Mathematical Control & Related Fields 2011, 1(4): 493-508 doi: 10.3934/mcrf.2011.1.493
In this paper, a vector-host epidemic model with control measures is considered to assess the impact of control measures on the prevalence of the vector-host diseases. We incorporated mosquito-reduction strategy and host medical treatment into the model. For the basic vector-host model, we provide sufficient conditions for the local stability of the disease free equilibrium (DFE) and the sensitivity analysis for the reproduction number with respect to the model parameters. Using the optimal control theory, the optimal levels of the two controls are characterized, and then the existence and uniqueness for the optimal control pair are established. Numerical simulations are further conducted to confirm and extend the analytical results. Numerical results suggest that optimal multi-control strategy is a more beneficial choice in fighting the outbreak of vector-host diseases. For the vector-host epidemics, vector control measures should be taken prior to other measures.
keywords: Epidemic model sensitivity indices reproduction number. optimal control
Existence of nodal solutions of multi-point boundary value problems
Lingju Kong Qingkai Kong
Conference Publications 2009, 2009(Special): 457-465 doi: 10.3934/proc.2009.2009.457
We study the nonlinear boundary value problem consisting of the equation $y^{''}+ w(t)f(y)=0$ on $[a,b]$ and a multi-point boundary condition. By relating it to the eigenvalues of a linear Sturm-Liouville problem with a two-point separated boundary condition, we obtain results on the existence and nonexistence of nodal solutions of this problem. We also discuss the changes of the existence of different types of nodal solutions as the problem changes.
keywords: multi-point boundary value problems Nodal solutions eigenvalues Sturm-Liouville problems

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