## 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

### Keywords:

second-order difference equation; decaying solution
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