Comparison of Green’s functions for a family of multipoint boundary value problems. (English) Zbl 0964.34012

The authors consider a multipoint boundary value problem connected with the function \(y(x)\) which is defined on \([0,B]\) and satisfies the homogeneous differential equation \[ Ly\equiv y^{(n)}+ \sum^n_{j=1} a_j(x) y^{(n- 1)}= 0 \] under certain homogeneous conditions of the form \[ y^{(\ell)}(x_j)= 0,\quad \ell= 0,\dots, n-1;\;j=1,\dots, k-1, \]
\[ y^{(\ell)}(x_k)= 0,\quad \ell= \alpha_1,\dots, \alpha_{n_k}, \] with \(n\geq 2\) and \(k\in \{2,\dots, n\}\) while \(x_j\), \(j= 1,\dots, k\), are given numbers such that \(0\leq x_1<\cdots< x_{k-1}< x_k= b\leq B\). The coefficients \(a_j(x)\) are supposed to be continuous on \([0,B]\). As to the numbers \(\alpha_1,\dots, \alpha_{n_k}\), they denote given nonnegative integers such that \(0\leq \alpha_1<\cdots< \alpha_{n_k}\leq n-1\) and \(n_1+\cdots+ n_k= n\).
Their aim is to show that if \(L\) is right disfocal, then the Green function \(G(x,s)\), which is denoted here by \(G(k,\alpha; x,s)\) with \(\alpha= (\alpha_1,\dots, \alpha_{n_k})\), as well as its derivatives with respect to \(x\) meet certain inequalities of the form \[ 0< (-1)^{m_i} G(k, \alpha, b_1; x,s)< (-1)^{m_i} G(k,\widehat\alpha, b_2; x,s). \] Here, one supposes \((x,s)\in (x_i, x_{i+1})\), \(x\in (x_1, b_1)\), \(b_1\leq b_2\) and \(\alpha\leq \widehat\alpha\). The latter means that \(\alpha_j\leq \widehat\alpha_j\) for \(\forall j= 1,\dots, n_k\).


34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B27 Green’s functions for ordinary differential equations
Full Text: DOI


[1] Coppel, W., Disconjugacy, Lecture notes in mathematics, 220, (1971), Springer-Verlag New York/Berlin
[2] Elias, U., Green’s functions for a nondisconjugate differential operator, J. differential equations, 37, 319-350, (1980)
[3] Elias, U., Oscillation theory for two-term differential equations, (1997), Kluwer Boston
[4] Eloe, P.W.; Henderson, J., Multipoint boundary value problems for ordinary differential systems, J. differential equations, 114, 232-242, (1994) · Zbl 0809.34028
[5] Eloe, P.W.; Kaufmann, E.R., A singular boundary value problem for a right disfocal linear differential operator, Dynam. systems appl., 5, 174-182, (1996) · Zbl 0857.34029
[6] Eloe, P.W.; Ridenhour, J., Sign properties of Green’s functions for a family of two-point boundary value problems, Proc. amer. math. soc., 120, 443-452, (1994) · Zbl 0795.34010
[7] Muldowney, J.S., A necessary and sufficient condition for disfocality, Proc. amer. math. soc., 74, 49-55, (1979) · Zbl 0402.34009
[8] Muldowney, J.S., On invertibility of linear ordinary differential boundary value problems, SIAM J. math. anal., 12, 368-384, (1981) · Zbl 0462.34007
[9] Nehari, Z., Disconjugate linear differential operators, Trans. amer. math. soc., 129, 500-516, (1967) · Zbl 0183.09101
[10] Nehari, Z., Green’s functions and disconjugacy, Arch. ration. mech. anal., 62, 53-76, (1976) · Zbl 0339.34036
[11] Peterson, A., Green’s functions for focal type boundary value problems, Rocky mountain J. math., 9, 721-732, (1979) · Zbl 0387.34014
[12] Peterson, A., Focal Green’s functions for fourth-order differential equations, J. math. anal. appl., 75, 602-610, (1980) · Zbl 0439.34026
[13] Peterson, A.; Ridenhour, J., Comparison theorems for Green’s functions for focal boundary value problems, Recent trends in differential equations, World scientific series in applicable analysis, 1, (1992), World Scientific River Edge, p. 493-506 · Zbl 0832.34022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.