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Keywords:
time scales; integral boundary condition; second-order boundary value problem; cone; positive solution
Summary:
We consider the existence of at least one positive solution to the dynamic boundary value problem \begin{align*} -y^{\Delta\Delta}(t) & = \lambda f(t,y(t))\text{, }t\in [0,T]_{\mathbb{T}} y(0) & = \int_{\tau_1}^{\tau_2}F_1(s,y(s)) \Delta s y\left(\sigma^2(T)\right) & = \int_{\tau_3}^{\tau_4}F_2(s,y(s)) \Delta s, \end{align*} where $\mathbb{T}$ is an arbitrary time scale with $0<\tau_1<\tau_2<\sigma^2(T)$ and $0<\tau_3<\tau_4<\sigma^2(T)$ satisfying $\tau_1$, $\tau_2$, $\tau_3$, $\tau_4\in \mathbb{T}$, and where the boundary conditions at $t=0$ and $t=\sigma^2(T)$ can be both nonlinear and nonlocal. This extends some recent results on second-order semipositone dynamic boundary value problems, and we illustrate these extensions with some examples.
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