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.


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


Zbl 0462.34007
Full Text: DOI