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Zbl 1197.39006
Ma, Ruyun; Ma, Huili
Positive solutions for nonlinear discrete periodic boundary value problems.
(English)
[J] Comput. Math. Appl. 59, No. 1, 136-141 (2010). ISSN 0898-1221

The article deals with the following boundary value problem $$-\Delta[p(t - 1)\Delta u(t - 1)] + q(t)u(t) = rg(t)f(u(t)), \quad t \in [1,T]_{\Bbb Z},$$ $$u(0) = u(T), \quad p(0)\Delta u(0) = p(T)\Delta u(T),$$ where $r$ is a positive parameter, $T > 2$, $f \in C({\Bbb R},{\Bbb R})$, $sf(s) > 0$ for $s \ne 0$ and there exist the limits $$f_0 = \lim_{|s| \to 0} \ \frac{f(s)}{s}, \qquad f_\infty = \lim_{|s| \to \infty} \ \frac{f(s)}{s},$$ $p, g: \ {\Bbb Z} \to (0,\infty)$ and $q: \ {\Bbb Z} \to [0,\infty)$, $q \not\equiv 0$, 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 $$-\Delta[p(t - 1)\Delta u(t - 1)] + q(t)u(t) = rg(t)u(t), \quad t \in [1,T]_{\Bbb Z},$$ $$u(0) = u(T), \quad p(0)\Delta u(0) = p(T)\Delta u(T).$$
[Peter Zabreiko (Minsk)]
MSC 2000:
*39A23
39A12 Discrete version of topics in analysis
34B15 Nonlinear boundary value problems of ODE
34C25 Periodic solutions of ODE
39A22
34L05 General spectral theory for ODE

Keywords: difference equations; periodic boundary value problem; positive solutions; Green function; periodic solutions; eigenvalue

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