Research On Oscillation For Several Classes Of Differential Equations And Difference Equations

This thesis is concerned with oscillatory and nonoscillatory criteria for several classes of differential equations and difference equations,thus extending and improving a number of results in the literature.The main results are listed as f'ollows:In chapter 1,we introduce the background and progress of the differential equations and difference equations,and also introduce the main results in this thesis.In chapter 2,we study the oscillation of second order delay dynamic equations by using Riccati transformations,mean value theorem and inequality techniques.Some known results in the literature are improved.We study the following equationwhere ?>0 is a quotient of two positive integers,and satisfies the following conditions:In this chapter,we assume the following conditions hold:Theorem 0.0.1.Assume the conditions(2),(3)and ? ? 1 hold.If there exists some positive constant k ?(0,1)such that holds,then equation(1)is oscillatory on[t0,?)T·Corollary 0.0.1.Assume the conditions(2),(3)and ? ? 1 hold.If there exists some constant k ?(0,1)sch that holds,then eqzuation(1)is oscillatory on[t,?)T·Theorem 0.0.2.Assume conditions(2)and(3)hold,and there exists a function?(t)?t,such that as ? ? 1,then equation(1)is oscillatory on[t0,?)T·We use the following notations:Theorem 0.0.3.Assume conditions(2)and(3)hold,and To? ? ?oT.If there exists a positive constant k ?(0,1)and a function ?(t)? Crd([t0,?)T,R)such that where then equation(1)is oscillatory on[t0,?)T.Theorem 0.0.4.Asscume conditions(2)and(3)hold,and ?o? = ?o?.Assume there exist functions H,h ? Crd(D,R),D ?{(t,s)}?T2:t ?s? t0} and a function?(t)? Crd([t0,?)T,R),in the mean time,H has a nonpositive partial derivative H?s(t,s)with respect to s and satisfies and where then equation(1)is oscillatory on[t0,?)T.In chapter 3,we consider the oscillation criteria for a class of neutral delay differential equations by using Riccati transformation,thus complementing and improving those results in the literature.By comparing the sizes of ? and ?,we construct different weight functions to deduce oscillation criteria.We consider the following equation where z(t)= x(t)+p(t)x(?(t)),f(t,x)sgnx ? q(t)?x??,x ?0,? ? 1 is a quotient of two positive odd integers.In this chapter,we assume the following conditions hold:Theorem 0.0.5.Suppose 0<? ? ?,then there exists a function ?? C([t0,+?))such that then equation(4)oscillates.Theorem 0.0.6.If ?>?>0,and there exists a function ??C([t0,+?))such that then equation(4)oscillates.In chapter 4,we study oscillation for difference equations in two parts.By applying the definitions of difference,formulas and inequality techniques,we study the oscillation of second order difference equations and two dimensional difference systems.New oscillation criteria are established,thus extending and improving the results in literature.Illustrating examples are also given.We study the following equation:In this chapter we use the following notations:Theorem 0.0.7.Let q ? 1/4.If there exists a constant ??[0,1)such that then eqzuatiorn(5)is oscillatory.Theorem 0.0.8.Let p*(0)? 1/4 and q<1/4.If there exists a constant ?? G[M2,1)such that then equation(5)is oscillatory.In this chapter we always assume that where n0?N = {1,2,…},{pn},{qn} are real sequences,pn?0,qn?0,andTheorem 0.0.9.Assume that g*(2)?1/4 and are true,and there exists??[2,+?),such that then the system of equations(6)oscillateCorollary 0.0.2.Let the condition then the system of eqzuations(6)oscillates.In chapter 5,we establish several sufficient conditions for nonoscillation of fourth order difference equations by employing equality properties,thus enriching the present nonoscillation theory for difference equations.We study the following equation Assume z(n)= ?y(n),then the above equation is equivalent with ie.Assume that we have the following conditionsTheorem 0.0.10.If P1(n),pa(n)and p3(n)satisfy one of the following conditions,then eqzuation(8)is nonoscillatory.Remark 0.0.1.SubstitZuting the relatiofns p1(n)= a(n + 1)+ b(n ?1)-3,p2(n)=3-2a(n + 1)-b(n+1)+c(n + 1),p3(n)=a(n + 1)-1 into conditions from(A1)to(A12),we can deduce the ineqzualities of a(n),b(n),c(n),which become nonoscillatory criteria for eqzuation(7).In chapter 6,the significance,innovation and the future work are introduced.