Systems-Disconjugacy of a Fourth-Order Differential Equation

Systems-conjugate and focal points of fourth order nonselfadjoint differential equations

“…Whyburn [1] showed that the general real selfadjoint fourth order equation (3) /[«] = (p2(t)u")" -0,(0«')' + Po(t)u = 0 can be written in the form (1) with b(t) = -l/p2(i) > 0 and a(f) = d(t). Later Kreith [2] also showed the general real linear fourth order equation (4) /[«] a (p2(t)u" -q2(t)u')" -(p^u' -qi(t)u)' + p0(t)u = 0 can be reduced to the form (1), with the nonselfadjointness reflected by the inequality of a(t) and d(t).…”

“…This paper is then concerned primarily with the existence of ^(a) and ¿4 (a) which Barrett [3] originally defined as the systems-conjugate and systems-focal points of a with respect to (2). Actually systems of the form (2) had been considered earlier by Bliss and Schoenberg [4], Hartman and Wintner [5] and others in connection with the calculus of variations which naturally requires the coefficient matrix in (2) to be symmetric.…”

Oscillation theorems for fourth‐order delay differential equations with a negative middle term

“…Consider the fourth-order linear delay differential equation y (4) (t) − p(t)y ′ (t) + q(t)y( (t)) = 0, t ≥ t 0 ,…”

“…In particular, there exists a very large body of literature devoted to the corresponding 2-term equation and its various generalizations. For a broader view, we refer the reader to some classical papers [2][3][4][5][6] and the references cited therein. Recently, investigation of higher-order differential equations with a middle term y (n) (t) + p(t)y (n−2) (t) + q(t)y( (t)) = 0 has received a considerable portion of interest (see, for instance Agarwal [7][8][9][10][11][12][13][14][15][16][17] ).…”

“…However, a little attention has been paid to a case when a difference in the derivative order between the first and the second terms of the studied equation is greater than 2. Some attempts in this direction have been made by Hou and Cheng 18 and Liang 19 who studied asymptotic and oscillatory properties of the equation y (4) (t) + p(t)y ′ (t) + q(t)y( (t)) = 0 and established criteria for all nonoscillatory solutions to tend to zero near infinity. These results have been recently extended and generalized by present authors, see Džurina et al [20][21][22] The present paper is a continuation of the above-mentioned recent works [18][19][20][21][22] in case when the middle term is negative.…”