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Existence of positive solutions for BVPs of fourth-order difference equations. (English) Zbl 1025.39006

The authors consider the boundary value problem (BVP) \[ \Delta^4x(t-2)= \lambda a(t)f\bigl(t,x(t) \bigr), \]
\[ x(0)=x(T+2)= \Delta^2x(0)= \Delta^2x(T)= 0,\;2\leq t\leq T\text{ and }T\geq 6. \] Under the conditions \(f\) increasing in \(x\) and continuous in both arguments, \(f(t,0)>0\) and \(a(t)\geq 0\) there exists a \(\lambda_0\geq 0\) such that the BVP has at least one positive solution for \(0\leq\lambda \leq\lambda_0\). If, moreover, \(f(t,x)\geq dx\) for some \(d>0\) then there is no solution for \(\lambda> \lambda_0\).
Reviewer’s remark: The case where \(a(t)\) vanishes identically must be excluded in the last assertion.

MSC:

39A11 Stability of difference equations (MSC2000)
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References:

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