Existence of positive solutions for a semipositone boundary value problem on the half-line
The paper discusses the existence of single and twin positive solutions to the following boundary value problem \[ (P)\quad\begin{cases} x''(t)-k^2(t)x(t)+\lambda m(t)f(t,x(t))=0,\qquad t\in(0,+\infty),\\ x(0)=0,\qquad x(+\infty)=0, \end{cases} \] where \(k: [0,\infty)\longrightarrow(0,\infty)\) is continuous, bounded, \(f: [0,\infty)\longrightarrow\mathbb{R}\) is continuous and semi-positone, \(m\) is continuous and positive, and \(\lambda\) is a positive parameter. After defining a suitable Green's function, the authors formulate Problem (P) as a fixed point problem for a compact nonlinear mapping. Under some growth assumptions and some restrictions on \(\lambda\), the authors then prove the existence of a single or twin solutions in a positive cone of a Banach space via the classical Krasnosel'skii's fixed point theorem of cone compression and expansion. Existence theorems extend some known results for the constant case \(k>0\) (see e.g. [\textit{S. Djebali, K. Mebarki}, Comput. Math. Appl. 55, No.~12, 2940--2952 (2008; Zbl 1142.34316)] and the references therein).
scientific article; zbMATH DE number 5727369
scientific article