## 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:

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