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Zbl 1025.39006
Zhang, Binggen; Kong, Lingju; Sun, Yijun; Deng, Xinghua
Existence of positive solutions for BVPs of fourth-order difference equations.
(English)
[J] Appl. Math. Comput. 131, No.2-3, 583-591 (2002). ISSN 0096-3003

The authors consider the boundary value problem (BVP) $$\Delta^4x(t-2)= \lambda a(t)f\bigl(t,x(t) \bigr),$$ $$x(0)=x(T+2)= \Delta^2x(0)= \Delta^2x(T)= 0,\ 2\le t\le T\text{ and }T\ge 6.$$ Under the conditions $f$ increasing in $x$ and continuous in both arguments, $f(t,0)>0$ and $a(t)\ge 0$ there exists a $\lambda_0\ge 0$ such that the BVP has at least one positive solution for $0\le\lambda \le\lambda_0$. If, moreover, $f(t,x)\ge dx$ for some $d>0$ then there is no solution for $\lambda> \lambda_0$.\par Reviewer's remark: The case where $a(t)$ vanishes identically must be excluded in the last assertion.
[Lothar Berg (Rostock)]
MSC 2000:
*39A11 Stability of difference equations

Keywords: positive solutions; fourth-order difference equations; boundary value problem

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