Positive solutions of a second-order difference equation with summation boundary value problem (Q2873063)

This paper studies the existence of positive solutions to the second-order difference equation NEWLINE\[NEWLINE\Delta^2u(t-1)+a(t)f(u)=0,\;\;t\in\{1,2,\dots, T\},NEWLINE\]NEWLINE with the three-point summation boundary condition NEWLINE\[NEWLINEu(0)=0, \;\;u(T+1)=\alpha\sum_{s=1}^{\eta}u(s),NEWLINE\]NEWLINE where \(f\) is continuous, \(T\geq 3\) is a fixed integer and \(\eta\in\{1,2,\dots,T-1\}\).NEWLINENEWLINEThe authors first define the superlinear and sublinear cases for \(f\). Let NEWLINE\[NEWLINEf_0=\lim_{u\rightarrow 0^{+}}\frac{f(u)}{u},\;\;\;f_{\infty}=\lim_{u\rightarrow \infty}\frac{f(u)}{u}.NEWLINE\]NEWLINE Then \(f_0=0\) and \(f_\infty=\infty\) correspond to the superlinear case, and \(f_0=\infty\) and \(f_\infty=0\) correspond to the sublinear case.NEWLINENEWLINEIn the second section, several preliminary results are obtained that are about the difference equation \(\Delta^2u(t-1)+y(t)=0\), \(t\in \mathbb{N}_{1,T}\), with the same boundary condition given above.NEWLINENEWLINEThe main results are obtained in the third section. It is stated that if \(0<\alpha<\frac{2T+2}{\eta(\eta+1)}\), and the following conditions hold:NEWLINENEWLINE(H1) \(f\in C([0,\infty),[0,\infty))\),NEWLINENEWLINE(H2) \(a\in C(\mathbb{N}_{T+1},[0,\infty))\) and there exists \(t_0\in\mathbb{N}_{\eta,T+1}\) such that \(a(t_0)>0\),NEWLINENEWLINEthen the main boundary value problem has at least one positive solution when \(f\) is either superlinear or sublinear. Krasnoselskii's fixed point theorem plays an important role in the proof.NEWLINENEWLINEAt last, two examples are discussed in order to illustrate the main result.