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

Christopher S. Goodrich

Commentationes Mathematicae Universitatis Carolinae (2013)

  • Volume: 54, Issue: 4, page 509-525
  • ISSN: 0010-2628

Abstract

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We consider the existence of at least one positive solution to the dynamic boundary value problem - y Δ Δ ( t ) = λ f ( t , y ( t ) ) , t ∈ [ 0 , T ] 𝕋 y ( 0 ) = ∫ τ 1 τ 2 F 1 ( s , y ( s ) ) Δ s y σ 2 ( T ) = ∫ τ 3 τ 4 F 2 ( s , y ( s ) ) Δ s , where 𝕋 is an arbitrary time scale with 0 < τ 1 < τ 2 < σ 2 ( T ) and 0 < τ 3 < τ 4 < σ 2 ( T ) satisfying τ 1 , τ 2 , τ 3 , τ 4 ∈ 𝕋 , and where the boundary conditions at t = 0 and t = σ 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.

How to cite

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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 54.4 (2013): 509-525. <http://eudml.org/doc/260748>.

@article{Goodrich2013,
abstract = {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.},
author = {Goodrich, Christopher S.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {time scales; integral boundary condition; second-order boundary value problem; cone; positive solution; second-order boundary value problem; time scale; integral boundary condition; positive solution},
language = {eng},
number = {4},
pages = {509-525},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Existence of a positive solution to a nonlocal semipositone boundary value problem on a time scale},
url = {http://eudml.org/doc/260748},
volume = {54},
year = {2013},
}

TY - JOUR
AU - Goodrich, Christopher S.
TI - Existence of a positive solution to a nonlocal semipositone boundary value problem on a time scale
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2013
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 54
IS - 4
SP - 509
EP - 525
AB - 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.
LA - eng
KW - time scales; integral boundary condition; second-order boundary value problem; cone; positive solution; second-order boundary value problem; time scale; integral boundary condition; positive solution
UR - http://eudml.org/doc/260748
ER -

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