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Existence of positive solution for nonlinear fourth-order difference equations. (English) Zbl 1220.39008
Let $T \geq 5$ be an integer $\mathbb{T}_{0} = \{0,\dots,T+2 \}$, $\mathbb{T}_{2} = \{2,\dots,T \}$ and let $f: \mathbb{T}_{2} \times [0, \infty) \to [0, \infty)$ be a continuous function. The author gives some sufficient conditions under which the difference problem $$\Delta^{4}u(t - 2) - \lambda f(t, u(t)) = 0,\quad T \in \mathbb{T}_{2} \tag1$$ $$u(1) = u(T + 1) = \Delta^{2} u(0) = \Delta^{2} u(T) = 0, \tag2$$ 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 $\{u_{n} \}$ of positive solutions of (1), (2) for which $$ \|u_{n}\|:= \max \{|u_{n}(j)|: j \in \mathbb{T}_{0} \} \to \infty. $$

39A12Discrete version of topics in analysis
39A22Growth, boundedness, comparison of solutions (difference equations)
39A10Additive difference equations
34B15Nonlinear boundary value problems for ODE
Full Text: DOI
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