Structure of positive solution sets of sub-linear semi-positone operator equations
In this paper, the authors consider the nonlinear operator equation \(x=G(x,\lambda)\) in a real Banach space \(E\). Here, \(\lambda\in(0,+\infty)\) and \(x\in P\), where \(P\) is an order cone. Moreover, \(G(x,\lambda):=\lambda KFx+e_0\), where \(K : P\to P\) is a linear, completely continuous operator, \(F : P\to E\) is a bounded nonlinear continuous operator, and \(e_0\in(-P)\). Due to this latter fact (i.e., that \(e_0\in(-P)\)), it follows that the authors study the so-called semipositone problem. The authors employ classical degree theory as well as Krein-Rutman theory. After studying the aforementioned, more abstract problem, the authors provide some results for the boundary value problem \(\left(p(t)u'\right)'+\lambda f(t,u)+q(t)=0\) subject to the boundary conditions \(au(0)-bp(0)u'(0)=0\) and \(cu(1)+dp(1)u'(1) =0\). By imposing standard hypotheses on \(f\) (e.g., \(\frac{f(t,u)}{u}\to 0\), uniformly for \(t\in[0,1]\), as \(u\to+\infty\)), the authors deduce the existence of at least one positive solution to the above boundary value problem for certain values of the parameter \(\lambda\). The paper is well written and concludes with an example of a boundary value problem to which their theory is applied. Unfortunately, no additional examples are provided to explicate the more abstract theory.
scientific article; zbMATH DE number 6121311