Globally Asymptotic Stability Of Solutions Of A Certain Fourth-order Functional Differential Equation

In this paper,we consider globally asymptotic stability of zero solutions of a certainfourth-order functional differential Equation:x⁽⁴⁾+f((x|¨)(t))x⁽³⁾(t)+g((x|¨)(t-(?)))+h((?)(t-(?)))+α₄x(t)+β₄(t-(?))=0 (1．1)whereα₄,β₄,(?) are constants and (?)>0,Symbois".","¨","(3)"and"(4)"arederivatives and higher order derivatives for t,functions f,g,h are continuous differentiableand provides existence uniqueness of solutions．Equivalent system of Equation(1．1) as following:where a₄=α₄+β₄Our main result as following:Theorem．In addition to the basic assumptions for f,g,h,assume that there exists positiveconstants a₁,a₂,a₃,δ,M,N such that as following conditions holds:(i)for anyξ≠0,we have f(ξ)≥a₁>0,h(ξ)/ξ≥a₃>0 and for allξ,z,(a₁a₂-h(ξ))a₃-a₁a₄f(z)≥δ；(ii)h(0)=0,for anyη,03≤h'(η)≤M．and for anyξ≠0,0≤h'-h(ξ)/ξ≤δ₁<(2a₄δ)/(a₁a₃²)ï¼›(iii)for any z≠0,1/z integral from n=0 to z f(ξ)dξ-f(z)≤δ₂<(2δ)/(a₁²a₃)ï¼›(iv)g(0)=0,for anyη,02≤g'(η)≤Nï¼›and for everyξ≠0,0≤g(ξ)/ξ-a₂≤δ₃<(ε₀a₃³)/(a₄²)whereε₀=ε=minï½›a₄/a₃,a₃/(4a₂(a₁a₄+a₃))((2a₄δ)/(a₁a₃²)-δ₁),a₁a₄/4a₂(a₁a₄+a₃)(2δ/a₁²a₃-δ₂)｝(v)let d₁=ε+1/a₁,d₂=ε+a₄/a₃,β=max{β₄,M,N},d=max{1,d₁,d₂},ρ=min{1/3a₃(ε-ε₀),1/3a₁ε,δ/(6a₁a₃)｝βd(?)<ρ(3．3)then the zero solution of is uniformly asymptotically stable and globlly asymptotically sta-ble。...