Studies On Oscillation For Some Nonlinear Differential Equations

Oscillation theory is one of the most important qualitative theories for dif-ferential equations, which has profound physical background, and there are many mathematical models involve oscillation theory. As we know, the comparison and separation theories of zeros distribution for second order homogeneous linear differ-ential equations established by G. Sturm lay a foundation of oscillation theory for differential equations. During one and a half century, oscillation theory of differ-ential equations has developed quickly and played an important role in qualitative theories and theory of boundary value problems. There are many mathematicians are major in this subject and they obtained many useful results.The oscillation of solutions is one of important properties of solutions for dif-ferential equations. With the development of natural sciences and technology im-provements, there are many problems dealt with oscillation and non-oscillation of solutions. Particularly, there is a rapid development in this field among the past decades. The oscillation for second order differential equations is obtained partic-ular attention due to its extensive applications in mechanics and other fields, both the categories of equations and methods to these questions are improved rapidly.The present paper employs the generalized variational principles, the Riccati transformation, and the monotone of functions and inequalities to investigate the oscillation of several kind of nonlinear second order differential equations, the results we obtained generalize and improve some known oscillation criteria in literature.The thesis is divided into four chapters according to contents.In Chapter 1, we introduce the main contents of this paper.In Chapter 2, we are concerned with oscillation for the second order forced quasi-linear differential equation of the form where p,q,e∈C([t0,∞),R) with p(t)> 0 and 0<α≤βare constants.In this Chapter, we use Riccati transformation to give a new oscillation criterion for the quasi-linear differential equation (0.0.1), this oscillation criterion is closely related to the generalized Leighton's variational principle, which generalizes and improves the results mention above.In Chapter 3, we consider the oscillatory behavior of the nonlinear nonhomo-geneous differential equation of the form where a is a positive constant, p,q,e∈C([t0,∞), R) with p(t)> 0,Ψ∈C(R, (0,∞)), f∈C(R,R) satisfying uf(u)> 0 for u≠0.The purpose of this chapter is to use the generalized Leighton variational prin-ciple, the Riccati transformation, and Young's inequalities to study oscillation, which improve and generalize the results in [36] of Zhaowen Zheng and S. Cheng.In Chapter 4, we are concerned with the oscillation for the second-order forced quasi-linear differential equation, for t≥to> 0, where(I1) 0<α<βare constants;(I2) r,q,e∈C([t0,∞),R) with r(t)> 0;(I3) F:[t0,∞)×R×R×R×R→R is a continuous function;(I4)τ:[to,00)→(0,00) is a continuous function and limt→∞τ(t)= 00.A.Tiryaki [42] obtained some new interval criteria for the equations in the form for t≥t0> 0.In 2007, Zhaowen Zheng and Fanwei Meng [40] studied the oscillation of the equation where p, q, e∈C([t0,∞),R) with p(t)> 0, and 0< a≤βare constants.Motivated by the ideas in [39]-[47], the purpose of this chapter is to further their investigation of a kind of non-linear second order difference equations using two different methodologies. Our methodology is somewhat different from that of previous authors. We shall establish several new interval criteria for oscillation of equation, which the function F(t,x,u, v,w) satisfies three different conditions. Our conditions do not require the signs of p′(t) andτ′(t) in the first two cases. In above mentioned papers, oscillation criteria are obtained separately in retard caseτ(t)≤t and advanced caseτ(t)≥t. In this chapter, we note that the delayτ(t)does not affect our oscillation criteria.