## 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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