Advances in Differential Equations

Semipositone boundary value problems with nonlocal, nonlinear boundary conditions

Christopher S. Goodrich

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We demonstrate the existence of at least one positive solution to \begin{equation} \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} \end{equation} 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

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

First available in Project Euclid: 11 December 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34B09: Boundary eigenvalue problems 34B10: Nonlocal and multipoint boundary value problems 34B18: Positive solutions of nonlinear boundary value problems 47H07: Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces 47H30: Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) [See also 45Gxx, 45P05] 47B40: Spectral operators, decomposable operators, well-bounded operators, etc. 47G10: Integral operators [See also 45P05] 47H11: Degree theory [See also 55M25, 58C30]


Goodrich, Christopher S. Semipositone boundary value problems with nonlocal, nonlinear boundary conditions. Adv. Differential Equations 20 (2015), no. 1/2, 117--142.

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