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

\[ y^{(\ell)}(x_k)= 0,\quad \ell= \alpha_1,\dots, \alpha_{n_k}, \] 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 \(\forall j= 1,\dots, n_k\).

Reviewer: Mithat Idemen (İstanbul)

### MSC:

34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |

34B27 | Green’s functions for ordinary differential equations |

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\textit{P. W. Eloe} and \textit{L. Zhang}, J. Math. Anal. Appl. 246, No. 1, 296--307 (2000; Zbl 0964.34012)

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