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Zbl 1068.39008
He, Zhimin; Yu, Jianshe
On the existence of positive solutions of fourth-order difference equations.
(English)
[J] Appl. Math. Comput. 161, No. 1, 139-148 (2005). ISSN 0096-3003

Let $T\ge 1$ fixed, $f\in C(\bbfR^+,\bbfR^+)$, $a:\{1,2,\dots, T+1\}\subset\bbfZ\to \bbfR^+$ and $\lim_{x\to 0,x> 0}\,x^{-1} f(x)$ and $\lim_{x\to+\infty}\, x^{-1} f(x)$ be limits that exist.\par The authors determine eigenvalues $\lambda$ for which there exist positive solutions $u$ of the fourth-order difference equation $$\Delta^4 u(t- 2)- \lambda a(t) f(u(t))= 0,\quad t\in \{2,3,\dots, T+ 2\}\subset\bbfZ$$ 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.
[D. M. Bors (Iaşi)]
MSC 2000:
*39A11 Stability of difference equations

Keywords: Positive solution; Krasnosel'skii's fixed point theorem; boundary value problem; fourth-order difference equation

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