Green's functions and disconjugacy

Focal points for a linear differential equation whose coefficients are of constant signs

“…Clealry, a < s0 < f {a). To complete the proof, we shall show that s0 = f (a) and since ¿> < f (a), the solution^ will satisfy inequalities (6). Suppose on the contrary that s0 < f (a).…”

“…The necessity part of Theorem 2 is a particular case of the necessity part of Theorem 1, since we may choose/as the solution of (1) which satisfies (6). And conversely, the sufficiency of Theorem 2 implies that of Theorem 1, since a solution which satisfies (6), satisfies (7) trivially.…”

“…It turns out that the {k, n -A:)-focal points and the associated solutions are useful tools for the study of the disconjugacy of (2) [6]. The concept of {k, n -&)-focal point was generalized by Keener and Travis [5] for the equation where Piix) >0, i = 0,...,k-l, {-\y-kPi{x)>0, i = k,...,n-\.…”

Conjugate points for fractional differential equations

“…Disconjugacy of ordinary differential equations is well-studied and has enjoyed a rich history; we refer the reader to the monograph of Coppel [7] or landmark papers of Hartman [20], Levin [25] or Nehari [28]. Very little progress has been made to develop analogous theories for linear fractional differential equations, although a Fite type result [1] has recently been obtained for an iterated Riemann-Liouville fractional differential operator.…”

“…The classical result is obtained by elementary methods, although we refer the reader to [7] or [28]. Schmitt and Smith [30] applied the theory of cones to extend this theory to second order, m−dimensional systems of Brought to you by | MIT Libraries Authenticated Download Date | 5/12/18 9:59 PM two-point conjugate BVPs.…”