Existence of positive solutions of \(p\)-Laplacian difference equations. (English) Zbl 1125.39007

Li, Yongkun; Lu, Linghong

Appl. Math. Lett. 19, No. 10, 1019-1023 (2006).

Using a fixed point theorem, the authors give conditions for the existence of at least two positive solutions for boundary value problem \(\Delta u(0) = u(T+2) = 0\) of the \(p\)-Laplacian difference equation
\[ \Delta [\phi_p(\Delta u(t-1))] + a(t)f(u(t)) = 0, \quad t \in \{1,\dots, T+1\}, \]
where \(\phi_p(s) = | s| ^{p-2}s, \;p > 1,\) the function \(f: \mathbb R^+ \to\mathbb R^+\) is continuous and \(a(t)\) is a positive valued function defined on \(\{1,\dots, T+1\}.\)

Reviewer: Victor I. Tkachenko (Kyïv)

MSC:

39A11	Stability of difference equations (MSC2000)
39A12	Discrete version of topics in analysis
34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations

Keywords:

\(p\)-Laplacian difference equation; boundary value problem; positive solution

References:

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