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Multiple solutions for higher-order difference equations. (English) Zbl 0941.39003

Let \(T\in\{1,2, \dots\}\), \(I_0=\{0,1, \dots,T\}\) and \(y:I_n= \{0,1, \dots, T+n\}\to\mathbb{R}\). The authors discuss the \(n\)th \((n\geq 2)\) order discrete conjugate problem \[ (-1)^{n-p} \Delta^n y(k)=f \bigl(k,y(k)\bigr),\;k\in I_0, \]
\[ \Delta^iy(0)=0,\;0\leq i\leq p-1\quad \text{(here }1\leq p\leq n-1), \] and the \(n\)th \((n\geq 2)\) order discrete \((n,p)\) problem \[ \begin{aligned} & \Delta^n y(k)+f\bigl(k,y(k)\bigr) =0,\;k\in i_0,\\ & \Delta^iy(0)=0,\;0\leq i\leq n-2,\\ & \Delta^p y(T+n-p)=0,\;0\leq p\leq n-1\;\text{ is fixed}.\end{aligned} \] The technique employed is based on existence principles, lower type inequalities and Krasnoselskij’s fixed point theorem in a cone.

MSC:

39A10 Additive difference equations
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References:

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