Oscillation criteria for fourth-order functional differential equations

Abstract

ABSTRACT. Some new criteria for the oscillation of all solutions of certain fourth-order functional differential equations are established.

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

“…Corollary 1. Let (5) hold, z(t) ∈  1 be a principal solution of (P 1 ) and v(t) its associated function. Then, (E) has an equivalent binomial representation (E c ) in such way that (E c ) is in canonical form.…”

“…Lemma 5. Let (5) hold, z(t) ∈  1 be a principal solution of (P 1 ) and v(t) its associated function. If y(t) is an eventually positive solution of (E), then…”

“…Theorem 3. Let (5), (13), (15) and (26) hold. Assume that the fourth-order delay differential equation…”

“…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] ).…”

“…Consider again the Equation (Ẽ x ), where a < 2∕(, b > 0, ∈ (0, 1), and t ≥ 1. It is easy to see that conditions(5),(13), and (15) are satisfied. By Corollary 4, (Ẽ x ) is oscillatory provided that (…”