## Advances in Differential Equations

### Semipositone boundary value problems with nonlocal, nonlinear boundary conditions

Christopher S. Goodrich

#### Abstract

We demonstrate the existence of at least one positive solution to $$\begin{split} -y''(t)& =\lambda f(t,y(t))\text{, }t\in(0,1)\\ y(0)& =H(\varphi(y))\text{, }y(1)=0,\notag \end{split}$$ where $H : \mathbb{R}\rightarrow\mathbb{R}$ is a continuous function and $\varphi : \mathcal{C}([0,1])\rightarrow\mathbb{R}$ is a linear functional so that the boundary condition at $t=0$ may be both nonlocal and nonlinear. Since the continuous function $f : [0,1]\times\mathbb{R}\rightarrow\mathbb{R}$ may assume negative values, our results apply to semipositone problems. The classical Leray-Schauder degree is utilized to derive the existence result, which we obtain in the case where $\lambda$ is small and which permits $f$ to be negative on its entire domain. The result is illustrated by an example.

#### Article information

Source
Adv. Differential Equations Volume 20, Number 1/2 (2015), 117-142.

Dates
First available in Project Euclid: 11 December 2014

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