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Multiple solutions for \(2m\)th order Sturm-Liouville boundary value problems on a measure chain. (English) Zbl 0965.39008

Existence criteria for boundary value problem of the form \[ (-1)^m u^{(2m)}= f(t,u(t)),\quad t\in[0,1], \] \(\alpha_{i+1} u^{(2i)}(0)- \beta_{i+1} u^{(2i+ 1)}(0)= 0= \gamma_{i+1} u^{(2i)}(1)+ \delta_{i+1} u^{(2i+1)}(1)\), \(0\leq i\leq m+1\), have been investigated to some extent. In this paper, similar boundary value problems of the form \[ (-1)^m u^{\Delta^{2m}}= f(t, u(\sigma(t))),\quad t\in [0, 1], \]
\[ \alpha_{i+1} u^{\Delta^{2i}}(0)- \beta_{i+1} u^{\Delta^{2i+1}}(0)= 0= \gamma_{i+1} u^{\Delta^{2i}}(\sigma(1))+ \delta_{i+1} u^{\Delta^{2i+1}}(\sigma(1)),\quad 0\leq i\leq m+1, \] are studied, where \(\sigma(t)= \inf\{\tau\in T\mid\tau> t\}\) and \(u^\Delta(t)= (x(\sigma(t))- x(t))/(\sigma(t)- t)\), and the so-called measure chain \(T\) is a nonempty closed subset of \(\mathbb{R}\) containing \(0\) and \(1\). By means of Krasnosel’skii fixed point theorem and various properties of the Green’s function associated to the corresponding homogeneous problem, existence of positive solutions are established under several standard assumptions on \(f\).

MSC:

39A12 Discrete version of topics in analysis
34B15 Nonlinear boundary value problems for ordinary differential equations
34B24 Sturm-Liouville theory
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