Existence results for positive solutions of non-homogeneous BVPs for second order difference equations with one-dimensional \(p\)-Laplacian. (English) Zbl 1191.39008

Liu, Yuji

J. Korean Math. Soc. 47, No. 1, 135-163 (2010).

This paper is concerned with boundary value problems of the form
\[ \Delta(\varphi(\Delta x(n)))+ f(n,x(n+1),\Delta x(n),\Delta x(n+1))=0,\quad n=0,1,\dots,N, \]
\[ x(0) -\sum_{i=1}^m \alpha_i x(n_i)=A, \qquad x(N+2)- \sum_{i=1}^m \beta _i x(n_i)=B, \]
where \(\Delta \) is the forward difference operator, \(\varphi(x)=|x|^{p-2}x\) for \(x\neq 0\) and \(\varphi(0)=0\) with \(p>1\), \(f\) is positive and continuous; while \(\alpha_1,\dots, \alpha_m\), \(\beta_1,\dots,\beta_m\), \(A,B\geq 0\) and \(0<n_1<\cdots <n_m<N+2\). By means of a fixed point theorem for operators defined on Banach spaces with cones [stated by Z. Bai and W. Ge, Acta Math. Sin., Chin. Ser. 49, No. 5, 1045–1052 (2006; Zbl 1124.34318)], existence of three positive solutions is established.

Reviewer: Sui Sun Cheng (Hsinchu)

MSC:

39A12	Discrete version of topics in analysis
34B15	Nonlinear boundary value problems for ordinary differential equations
39A22	Growth, boundedness, comparison of solutions to difference equations

Keywords:

Laplacian difference equation; multi-point boundary condition; positive solution

Citations:

Zbl 1124.34318