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Positive solutions for singular discrete boundary value problems. (English) Zbl 1070.39006

The paper deals with the second-order difference equation (1) \(\Delta(a_n\Phi_p (\Delta x_n)) = g(n, x_{n+1})\), where \(\Delta x_n = x_{n+1} - x_n,\) \(\{ a_n \}\) is a positive real sequence, \(g\) is a positive continuous function on \(\mathbb{N} \times(0,u_0)\), \(0<u_0\leq\infty\), and \(\Phi_p(u)=| u|^{p-2}u\) with \(p>1\). The function \(g\) can be unbounded with respect to the second variable in a right neighborhood of zero. The authors give conditions for the existence of decaying solutions of (1), i.e. positive solutions \(\{ x_n \}\) with \(\Delta x_n < 0\) and \(\lim_n x_n=0\).

MSC:

39A11 Stability of difference equations (MSC2000)
39A12 Discrete version of topics in analysis
34B16 Singular nonlinear boundary value problems for ordinary differential equations
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