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Existence results for an even-order boundary value problem on time scales. (English) Zbl 1159.34019
Summary: Let \(\mathbb T\) be a time scale with \(t_1,t_2,t_3\in\mathbb T\). We investigate the existence of solutions to the nonlinear even-order three-point boundary value problem
\[ \begin{aligned} &(-1)^n y^{\Delta^{2n}}(t)= f(t,y(\sigma(t))), \quad t\in[t_1,t_3]\subset\mathbb T,\\ &y^{\Delta^{2i+1}}(t_1)=0, \quad \alpha y^{\Delta^{2i}} (\sigma(t_3))+ \beta y^{\Delta^{2i+1}} (\sigma(t_3))= y^{\Delta^{2i+1}}(t_2), \qquad 0\leq i\leq n-1, \end{aligned} \]
where \(n\in\mathbb N\), \(t_2\in(t_1,\sigma(t_3))\), \(\alpha>0\) and \(\beta>1\) are given constants.

MSC:
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
39A10 Additive difference equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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