Existence of a positive solution to a nonlocal semipositone boundary value problem on a time scale

Goodrich, Christopher S.

Commentationes Mathematicae Universitatis Carolinae, Tome 54 (2013) no. 4, pp. 509-525

Résumé

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.

Détail

Classification : 26E70, 34B10, 34B15, 34B18, 34N05, 39A10, 47H07
Keywords: time scales; integral boundary condition; second-order boundary value problem; cone; positive solution