Existence and uniqueness results for perturbed Neumann boundary value problems (Q1957505)

The authors study existence and uniqueness of positive solution for the following nonlinear perturbed BVP: \[ \pm u''(t)+m^2u(t)=f(t,u(t))+g(t),\quad 0<t<1, \] both with Neumann boundary condition \[ u'(0)=u'(1)=0, \] where \(m\) is a positive constant, \(f: [0,1]\times[0,+\infty)\rightarrow[0,+\infty)\) and \(g: [0,1]\rightarrow[0,+\infty)\) are continuous functions. \(f\) is further increasing in the second argument and satisfies for any \(t\in[0,1)\), \(f(t,a)>0\) where \(a=1/2(\text{ch}\,m+1)\) and the following condition \[ \forall \gamma\in(0,1), x>0,\;\exists\varphi(\gamma)\in(\gamma,1]: f(t,\gamma x)\geq \varphi(\gamma)f(t,x),\;\forall t\in[0,1]. \] Using a fixed point theorem for \(\alpha\)-concave operators in [\textit{C.-B. Zhai, C. Yang} and \textit{C.-M. Guo}, Comput. Math. Appl. 56, No.~12, 3150--3156 (2008; Zbl 1165.47308)], the authors prove existence and uniqueness of solutions in the positive cone of the Banach space of continuous functions. An example is provided.