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

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.


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


Zbl 1124.34318
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