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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.

MSC:
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
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[1] Merdivenci Atici, F.; Guseinov, G.Sh., On Green’s functions and positive solutions for boundary value problems on time scales, J. comput. appl. math., 141, 1-2, 75-99, (2002) · Zbl 1007.34025
[2] Atici, F. Merdivenci; Topal, S. Gulsan, Nonlinear three point boundary value problems on time scales, Dynam. systems appl., 13, 327-337, (2004) · Zbl 1069.39018
[3] Bohner, M.; Peterson, A., Dynamic equations on time scales, an introduction with applications, (2001), Birkhäuser · Zbl 0978.39001
[4] Bohner, M.; Peterson, A., Advances in dynamic equations on time scales, (2003), Birkhäuser Boston · Zbl 1025.34001
[5] Davis, J.M.; Eloe, P.W.; Henderson, J., Triple positive solutions and dependence on higher order derivatives, J. math. anal. appl., 237, 710-720, (1999) · Zbl 0935.34020
[6] Deimling, K., Nonlinear functional analysis, (1985), Springer New York · Zbl 0559.47040
[7] Eloe, P.W.; Henderson, J., Comparison of eigenvalues for a class of two point boundary value problems, Appl. anal., 34, 25-34, (1989) · Zbl 0662.34026
[8] Henderson, J.; Prasad, K.R., Comparison of eigenvalues for lidstone boundary value problems on a measure chain, Comput. math. appl., 38, 55-62, (1999) · Zbl 1010.34079
[9] Krasnosel’skii, M., Positive solutions of operator equations, (1964), Noordhoff Groningen · Zbl 0121.10604
[10] Wong, P.J.Y.; Agarwal, R.P., Eigenvalues of lidstone boundary value problems, Appl. math. comput., 104, 15-31, (1999) · Zbl 0933.65089
[11] Yao, Q., Monotone iterative technique and positive solutions of lidstone boundary value problems, Appl. math. comput., 138, 1-9, (2003) · Zbl 1049.34028
[12] Yao, Q., On the positive solutions of lidstone boundary value problems, Appl. math. comput., 137, 477-485, (2003) · Zbl 1093.34515
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