Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

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

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]