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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.

MSC:

39A12 Discrete version of topics in analysis
34B15 Nonlinear boundary value problems for ordinary differential equations
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References:

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