id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ma-53	Argyros, Ioannis K.; George, Santhosh ; Argyros, Christopher I.	On the Ostrowski Method for Solving Equations	2021	17	.pdf	application/pdf	5305	395	83	For each x, y ∈ Ω0 ‖F ′(x0)−1(F ′(y)− F ′(x))‖ ≤ K‖y − x‖, ‖F ′(x0)−1([y , x ;F ]− F ′(x0))‖ ≤ K1(‖y ′(x∗)−1(A0 − F ′(y0))‖‖A−10 F ′(x∗)‖‖F ′(x∗)−1F (y0)‖ ≤ L‖y0 − x∗‖2 2(1− L0‖x0 − x∗‖ + (L2 + L3)‖y0 − x0‖L4‖y0 − x∗‖ (1− L0‖y0 − x∗‖)(1− p(‖x0 − x∗‖)) ≤ ϕ2(‖x0 − x∗‖)‖x0 − x∗‖ ≤ ‖x0 − x∗‖ < r, so x1 ∈ U(x∗, r), where we also used ‖F ′(x∗)−1(A0 − F ′(y0))‖ ≤ ‖F ′(x∗)−1([y0, x0;F ]− F ′(x0))‖ +‖F ′(x∗)−1([y0, x0;F ]− F ′(y0))‖ ≤ (L2 + L3)‖y0 − x0‖ ≤ (L2 + L3)(‖y0 − x∗‖+ ‖x0 − x∗‖) ≤ (L2 + L3)(1 + ϕ1(‖x0 − x∗‖))‖x0 − x∗‖, and ‖F ′(x∗)−1F (y0)‖ = ‖ ∫ 1 0 F ′(x∗)	cache/ma-53.pdf	txt/ma-53.txt