Geodesic
Existence of a positive solution to a nonlocal semipositone boundary value problem on a time scale
Commentationes Mathematicae Universitatis Carolinae, Tome 54 (2013) no. 4, pp. 509-525 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 26E70, 34B10, 34B15, 34B18, 34N05, 39A10, 47H07
Keywords: time scales; integral boundary condition; second-order boundary value problem; cone; positive solution 
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     author = {Goodrich, Christopher S.},
     title = {Existence of a positive solution to a nonlocal semipositone boundary value problem on a time scale},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {509--525},
     year = {2013},
     volume = {54},
     number = {4},
     mrnumber = {3125073},
     zbl = {06373981},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2013_54_4_a4/}
}
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Goodrich, Christopher S. Existence of a positive solution to a nonlocal semipositone boundary value problem on a time scale. Commentationes Mathematicae Universitatis Carolinae, Tome 54 (2013) no. 4, pp. 509-525. http://geodesic.mathdoc.fr/item/CMUC_2013_54_4_a4/