Higher order boundary value problems on time scales. (English) Zbl 1124.34009

This paper deals with the existence of positive solutions to the Lidstone boundary value problem \[ (-1)^n y^{\Delta^{2n}}(t) = f(t, y(\sigma(t))), \quad t \in [0,1], \]
\[ y^{\Delta^{2i}}(0) = y^{\Delta^{2i}}(\sigma(1)) = 0, \quad 0 \leq i \leq n-1, \] where \(n \geq 1\) and \(f: [0,\sigma(1)] \times \mathbb{R} \to \mathbb{R}\) is continuous. The problem is considered on a time scale with the right-dense \(\sigma(1)\). The paper develops as follows.
The introductory Section 1 provides an overview of related recent results and estimates on the Green function associated with the problem. Section 2 contains an existence result under the assumption of a bounded inhomogeneous term, which follows from the Schauder fixed point theorem. Another existence result is based on the assumption that lower and upper solutions \(u\), \(v\) exist and are in the well order, that is, \(u \leq v\). Sections 3 is devoted to the existence of a positive solutions. The assumptions made here are the sub- and super-linearity of \(f \geq 0\). The results of the last section follow from the Krasnosel’skiĭ fixed point theorem of cone compression-expansion type.


34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B60 Applications of boundary value problems involving ordinary differential equations
39A10 Additive difference equations
Full Text: DOI


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