## Focal point characterizations and comparisons for right focal differential operators.(English)Zbl 0834.47044

Let $$B> 0$$ and $$k\in \{1,\dots, n- 1\}$$, $$n\in \mathbb{N}$$, be given. Further, let $$a_i(x)\in C[0, B]$$, $$1\leq i\leq n$$ and define the linear differential operator $$L$$ by $Ly= y^{(n)}+ \sum^n_{i= 1} a_i(x) y^{(n- i)},\qquad 0\leq x\leq B.$ Also, put $$\alpha= (\alpha_1,\dots, \alpha_{n- k})$$, where $$0\leq \alpha_1<\cdots< \alpha_{n- k}\leq n- 1$$ and let $$\mu\in \{0,\dots, \alpha_1\}$$ and $$P_i(x)\in C[0, B]$$, $$0\leq i\leq \mu$$. The authors consider the following boundary value problems for each $$b\in [0, B]$$. $Ly= \sum^\mu_{i= 0} p_i(x) y^{(i)}(x),\qquad 0\leq x\leq b,\tag{1}$
$y^{(i)}(0)= 0,\quad 0\leq i\leq k- 1,\quad y^i(b)= 0,\quad i= \alpha_1,\dots, \alpha_{n- k}.\tag{2}$ Also, given $$\alpha$$, the first $$\alpha$$-focal point of (1) corresponding to (2) is defined as $$b_0(\mu, \alpha)= \inf\{0< b\leq B\mid$$ (1)–(2) has a nontrivial solution}. Assuming that $$L$$ is right disfocal on $$[0, B]$$ as defined by J. S. Muldowney [SIAM J. Math. Anal. 12, 368-384 (1981; Zbl 0462.34007)], the authors discuss some properties of $$b_0(\mu, \alpha)$$. The obtained results extend previous ones for the two term operator $$Hy= y^{(n)}+ q(x) y$$ of the authors, U. Elias and Z. Nehari.

### MSC:

 47E05 General theory of ordinary differential operators 34B05 Linear boundary value problems for ordinary differential equations

Zbl 0462.34007
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