## On the existence of positive solutions of fourth-order difference equations.(English)Zbl 1068.39008

Let $$T\geq 1$$ fixed, $$f\in C(\mathbb{R}^+,\mathbb{R}^+)$$, $$a:\{1,2,\dots, T+1\}\subset\mathbb{Z}\to \mathbb{R}^+$$ and $$\lim_{x\to 0,x> 0}\,x^{-1} f(x)$$ and $$\lim_{x\to+\infty}\, x^{-1} f(x)$$ be limits that exist.
The authors determine eigenvalues $$\lambda$$ for which there exist positive solutions $$u$$ of the fourth-order difference equation $\Delta^4 u(t- 2)- \lambda a(t) f(u(t))= 0,\quad t\in \{2,3,\dots, T+ 2\}\subset\mathbb{Z}$ satisfying the boundary conditions $u(0)= \Delta^2 u(0)= u(T+ 2)= \Delta^2 u(T)= 0$ or $u(0)= \Delta^2 u(0)= \Delta u(t+ 1)= \Delta^3 u(T- 1)= 0$ by means of Krasnosel’skij’s fixed point theorem.
Reviewer: D. M. Bors (Iaşi)

### MSC:

 39A11 Stability of difference equations (MSC2000)
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### References:

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