## 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.

### 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

Zbl 1124.34318
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