Oscillzation For Nonlinear Delay Differential Equations

Oscillation of differential equations is not only an important branch , but also a very active direction of differential equations of qualitative theory. Especially in recent decades, Oscillation of differential equations have very rapid development,and get rich results whether from the linear equations to nonlinear equations, or from the low order to high order equation. Therefore research has also been extended to functional differential equation. differential equation, partial differential equation and functional matrix differential equation by many domestic and foreign scholars. Among them, oscillation of the semi-linear functional differential equation is one of the important aspects.In this paper, we study oscillation of two types of semi-linear differential equations,and get many valuable generalation based in summing oscillation theory of differential equationsThe first part of this paper give an overview and development of oscillation of differential equations.Then we give a brief introduction on mine.The second part introduce basic theory of oscillation of differential equations, and this is closely related to latest study results differential equations with various forms in detail.The third part introduce the basic theory and the developping of oscillation on the semi-linear differential equations with delay. using Riccatti skill and integral averagemethod,we study oscillation of two types of semi-linear differential equations with delay , and get some of new oscillatory criteria.The first equation (E₁):For equation (E₁),let:(ii) (?),f:R→R are continuous ,and (?)(x)=|x|^α-1x,f(x)∈C(-∞,∞),xf(x)>0,x≠0,α>0 is a constant;(iii) g(t)∈C[t₀,∞),t₀∈R⁺,g'(t)>0,when t→∞,g(t)→∞,And for any larget,g(t)≤t.Theorem 1 Assume thatandThere exists an average functionsuch that whent≥t₀,then H(t, t) = 0 , and when (t, s)∈D , H(t, s) > 0.is nonnegative continuous function in D. If C₂:then the equation (E₁)is oscillatory.Corollary 1 If the conditions in (E₁) are replaced by:(ii) (?),f:R→Rare continuous,and (?)'(s)≥0,f(x)∈C(-∞,∞),xf(x)>0,x≠0,α>0 is a constant;(iii) g(t)∈C[t₀,∞),t₀∈R⁺,g'(t)>0, when t→∞,g(t)→∞,And forany large t,g(t)≤t.then the equation(E₁)is oscillatory.Theorem2 Let condition (C₁)holds and let the functions H and h be defined as in Theoreml,such that condition (C₃):is satisfied,then the equation(E₁)is oscillatory.Corollary2 Let condition (C₃) be replaced by (C₄): and (C₅) =then the equation (E₁)is oscillatory.The second type equation (E₂):For equation (E₂) ,let:(i) q: [t₀,∞)→R,q(t)> 0;(ii):f:R→R are continuous,f(x)∈C(-∞,∞),xf(x)>0,x≠0,α>0is a constant;(iii) g(t)∈C[t₀,∞),t₀∈R⁺,g'(t)>0,when t→∞,g(t)→∞,And forany large t,g(t)≤t.Theorem3 Let (C₁):andThere exists an average functionsuch that when t≥t₀,then H(t, t) = 0 , and when (t, s)∈D , H(t, s) > 0.is nonnegative continuous function in D. If (C₂): then the equation (E₂) is oscillatory.