Positive solutions for nonlinear discrete periodic boundary value problems.(English)Zbl 1197.39006

The article deals with the following boundary value problem $-\Delta[p(t - 1)\Delta u(t - 1)] + q(t)u(t) = rg(t)f(u(t)), \quad t \in [1,T]_{\mathbb Z},$
$u(0) = u(T), \quad p(0)\Delta u(0) = p(T)\Delta u(T),$ where $$r$$ is a positive parameter, $$T > 2$$, $$f \in C({\mathbb R},{\mathbb R})$$, $$sf(s) > 0$$ for $$s \neq 0$$ and there exist the limits
$f_0 = \lim_{|s| \to 0} \;\frac{f(s)}{s}, \qquad f_\infty = \lim_{|s| \to \infty} \;\frac{f(s)}{s},$
$$p, g: \;{\mathbb Z} \to (0,\infty)$$ and $$q: \;{\mathbb Z} \to [0,\infty)$$, $$q \not\equiv 0$$, are $$T$$-periodic. The main result is the following:
The boundary value problem under consideration has two $$T$$-periodic solutions $$u^+$$ and $$u^-$$, $$u^+(t) > 0$$ and $$u^-(t) < 0$$ for $$t \in (0,T)$$, provided that either $$\lambda_1 / f_\infty < r < \lambda_1 / f_0$$ or $$\lambda_1 / f_0 < r < \lambda_1 / f_\infty$$, where $$\lambda_1$$ is the first eigenvalue of the linear eigenvalue problem $-\Delta[p(t - 1)\Delta u(t - 1)] + q(t)u(t) = rg(t)u(t), \quad t \in [1,T]_{\mathbb Z},$
$u(0) = u(T), \quad p(0)\Delta u(0) = p(T)\Delta u(T).$

MSC:

 39A23 Periodic solutions of difference equations 39A12 Discrete version of topics in analysis 34B15 Nonlinear boundary value problems for ordinary differential equations 34C25 Periodic solutions to ordinary differential equations 39A22 Growth, boundedness, comparison of solutions to difference equations 34L05 General spectral theory of ordinary differential operators
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