A global description of the positive solutions of sublinear second-order discrete boundary value problems. (English) Zbl 1177.39006

Summary: Let \(T\in\mathbb N\) be an integer with \(T>1\), \(\mathbb T:=\{1,\dots,T\}\), \(\widehat{\mathbb T}:=\{0,1,\dots,T+1\}\). We consider boundary value problems of nonlinear second-order difference equations of the form \[ \Delta^2u(t-1)+\lambda a(t)f(u(t))=0,\quad t\in\mathbb T,\;u(0)=u(T+1)=0, \]
where \(a:\mathbb T\to\mathbb R^+\), \(f\in C([0,\infty),[0,\infty))\) and, \(f(s)>0\) for \(s>0\), and \(f_0=f_\infty=0\), \(f_0=\lim_{s\to 0^+}f(s)/s\), \(f_\infty=\lim_{s\to+\infty}f(s)/s\). We investigate the global structure of positive solutions by using the Rabinowitz’s global bifurcation theorem.


39A12 Discrete version of topics in analysis
34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text: DOI EuDML


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