## 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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### References:

 [1] Agarwal R.P., Results Math 35 pp 3– (1999) [2] Agarwal, R.P., O’Regan, D. and Wong, P.J.Y. 1999. ”Positive Solutions of Differential Difference and Integral Equations”. Edited by: Dordrecht: Kluwer Academic. [3] Anderson D., Positivity of Green’s functions for an n-point right focal boundary value problem on a measure chain · Zbl 1042.39504 [4] DOI: 10.1080/10236199508808026 · Zbl 0854.39001 [5] Atici F., PanAmer. Math. J 5 pp 71– (1995) [6] Aulback B., Nonlinear Dynamics and Quantum Dynamical Systems, Math, Res 59 (1990) [7] Avery R.I., PanAmer. Math. J 8 pp 39– (1998) [8] Avery R.I., PanAmer.Math.J 8 pp 1– (1998) [9] Chyan C.J., Tamkang J. Math 30 (1999) [10] Chyan C.J., Multiple solutions for 2mth order Sturm-Liouville boundary value problems · Zbl 0958.34018 [11] Davis J.M., Multiplicity of positive solutions for higher order Sturm-Liouville problems · Zbl 0989.34012 [12] Erbe L.H., Diff.Eqns. Dynam. Sys 1 pp 223– (1993) [13] DOI: 10.1006/jmaa.1994.1227 · Zbl 0805.34021 [14] DOI: 10.3934/dcds.2000.6.121 · Zbl 1034.35128 [15] Erbe L.H., Eigenvalue conditions and Positive solutions · Zbl 0949.34015 [16] Erbe L.H., Diff. Eqns. Dynam. Sys 4 pp 313– (1996) [17] DOI: 10.1090/S0002-9939-1994-1204373-9 [18] Hilger S., Resultate Math 18 pp 18– (1990) · Zbl 0722.39001 [19] DOI: 10.1216/rmjm/1181071751 · Zbl 0930.34010 [20] Kaymakcalan, B., Lakshmikantham, V. and Sivasundaram, S. 1996. ”Dynamical Systems on Measure Chains”. Edited by: Boston: Kluwer Academic. · Zbl 0869.34039 [21] Kaymakcalan, B. 1994. ”Positive Solution of Operator Equation”. Edited by: Groningen: Noordhoof. [22] DOI: 10.1512/iumj.1979.28.28046 · Zbl 0421.47033 [23] DOI: 10.1090/S0002-9939-96-03403-X · Zbl 0857.34036 [24] DOI: 10.1006/jdeq.1994.1042 · Zbl 0798.34030
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