Partial Functional Differential Equations Of Neutral Higher Order Delay Equations Vibration Principle Study

Since 1700's,ordinary functional differential equations and partial functional differential equations with time delays have widely appeared in many fields such as biology, physics, control theory and engineering problems, which made researchers analyze the difficult problems carefully .In various engineering systems time delays are quite populous, the quantity of time delays in automatic control systems is especially great, the forms are regular,the independent variable t represents the time.The systemic theories of this kind of equations consist of the main part of the theories of functional differential equations(including the properties of the solutions such as existence , stableness. boundedness, oscillation, periodicity, asymptotic).Oscillation, as an important property of functional differential equations,is widely used in practical problems.In this article,oscillation of three different kinds of functional differential equations are discussed,they are:(1)nonlinear neutral hyperbolic equation with continuously distributed timedelayboundary condition:Ωis a bounded region with smooth boundary (?)Ωin Rⁿ,R⁺ = (0,∞), R₀⁺ = [0,∞),△is Laplace operator, n is the unit outer normal vector on (?)Ω,the integral in the equation is Stieljies integral.(2)higher order nonlinear delayed differential equation(3)higher order nonlinear neutral delayed differential equationBased on the results of the references, we use Green formula, Riccati transformation , Jensen inequality, method of Philos integral average and Kiguradze Lemma here to deal with the difficulties . obtain the oscillating criterions ,and extend the results of the references.