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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 \(\forall j= 1,\dots, n_k\).

MSC:

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

[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
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