id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejpam-5591	Almutairi, Alanoud	Fourth-Order Differential Equations: Asymptotic andOscillatory Behaviors of Solutions	2025	11	.pdf	application/pdf	3977	274	79	By Lemma 4, we see (11) holds and from Lemma 1 with U = σ (η) , V = (p− 1) εη2/ ( 2 (κ (η)µ (η))1/(p−1) ) and η = ζ, we get ζ ′ (η) ≤ −ϱ (η) + ( 2 εη2 )(p−1) κ (η)µ (η) (σ (η))(p−1)+1 pp . Lemma 1 is used with U = σ∗ (η) , V = 1/ς (η) , p = 2 and η = w, We obtainThe formula is ζ ′ (η) ≤ −ϱ∗ (η) + 1 4 ς (η) (σ∗ (η))2 .	cache/ejpam-5591.pdf	txt/ejpam-5591.txt