Comparison theorems for third- and fourth-order linear equations

Comparison Theorems for Fourth Order Differential Equations Garret

This paper establishes an apparently overlooked relationship between the iv yiV p'air of fourth order line,r equations y p(x)y 0 and + p(x)y 0, where p is , positive, continuous function defined on [O,m). It is shown that if all solutions of the first equation are nonoscillatory, then all solutions of the second equation must be nonoscillatory as well. An oscillation criterion for these equatio...

Comparison Theorems for Half-linear Differential Equations of the Fourth Order

An identity of the Picone type for fourth-order half-linear ordinary differential operators of the form lα[x] ≡ (pφ(x′′))′′ − (rφ(x′))′ + qφ(x) and Lα[y] ≡ (Pφ(y′′))′′ − (Rφ(y′))′ +Qφ(y). where φ(u) := |u|α−1u, α > 0, u ∈ R, and p, q, r, P,Q and R are continuous functions on a given interval I is derived and then Sturmian comparison theory for the corresponding fourth-order equations lα[x] = 0 ...

THIRD-ORDER AND FOURTH-ORDER ITERATIVE METHODS FREE FROM SECOND DERIVATIVE FOR FINDING MULTIPLE ROOTS OF NONLINEAR EQUATIONS

In this paper, we present two new families of third-order and fourth-order methods for finding multiple roots of nonlinear equations. Each of them requires one evaluation of the function and two of its first derivative per iteration. Several numerical examples are given to illustrate the performance of the presented methods.    

New family of Two-Parameters Iterative Methods for Non-Linear Equations with Fourth-Order Convergence

Oscillation Theorems for Third-Order Half-linear Delay Dynamic Equations with Damping

This article is concerned with oscillatory behavior of a class of third order half-linear delay dynamic equation with damping ( r(x 2 ) ) (t)+ p(t) ( x 2 )γ (t)+ q(t)x (τ (t)) = 0, on an arbitrary time scaleTwith supT = ∞ and lim t→∞ τ(t) = ∞.We establish some new sufficient conditions which ensure that every solution oscillates or converges to zero. Example is given to illustrate our main resu...