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Existence of positive solution for nonlinear fourth-order difference equations. (English) Zbl 1220.39008

Let $T\ge 5$ be an integer ${𝕋}_{0}=\left\{0,\cdots ,T+2\right\}$, ${𝕋}_{2}=\left\{2,\cdots ,T\right\}$ and let $f:{𝕋}_{2}×\left[0,\infty \right)\to \left[0,\infty \right)$ be a continuous function. The author gives some sufficient conditions under which the difference problem

${{\Delta }}^{4}u\left(t-2\right)-\lambda f\left(t,u\left(t\right)\right)=0,\phantom{\rule{1.em}{0ex}}T\in {𝕋}_{2}\phantom{\rule{2.em}{0ex}}\left(1\right)$
$u\left(1\right)=u\left(T+1\right)={{\Delta }}^{2}u\left(0\right)={{\Delta }}^{2}u\left(T\right)=0,\phantom{\rule{2.em}{0ex}}\left(2\right)$

where $\lambda >0$ is a parameter, has at least two positive solutions.

Moreover, the author presents two theorems that describe conditions such that there exists a sequence $\left\{{u}_{n}\right\}$ of positive solutions of (1), (2) for which

$\parallel {u}_{n}\parallel :=max\left\{|{u}_{n}\left(j\right)|:j\in {𝕋}_{0}\right\}\to \infty ·$

##### MSC:
 39A12 Discrete version of topics in analysis 39A22 Growth, boundedness, comparison of solutions (difference equations) 39A10 Additive difference equations 34B15 Nonlinear boundary value problems for ODE
##### References:
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