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

Ma, Ruyun; Ma, Huili

Comput. Math. Appl. 59, No. 1, 136-141 (2010).

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 \[ -\Delta[p(t - 1)\Delta u(t - 1)] + q(t)u(t) = rg(t)u(t), \quad t \in [1,T]_{\mathbb Z}, \]
\[ u(0) = u(T), \quad p(0)\Delta u(0) = p(T)\Delta u(T). \]

Reviewer: Peter Zabreiko (Minsk)

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

Keywords:

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

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