Studies On Oscillation For One Class Superlinear Second-order Differential Equations

The oscillation theory of differential equation is one of important branch of differential equations. In the field of modern applied mathematics, it has made considerable headway in recent years, because all the structures of its emergence have deep physical background and realistic mathematical models. Many scholars take on the research of this field, they have achieved many good results. With the increasing development of science and technology, there are many problems relating to differential equation derived from lots of real applications and practice, such as whether differential equation has a oscillating solution or not, and whether all of its solutions are oscillatory or not. In very resent years, great changes of this field have taken place. Especially, the second order superlinear differential equation has been paid more attentions and investigated in various classes by using different methods(see [1]-[35]).The present paper employs a generalized Riccati transformation, integral average technique and the monotone of functions to investigate the oscillation criteria for some class of superlinear differential equations, the results of which generalized and improved some known oscillation criteria.The thesis is divided into three sections according to contents.In Chapter 1, Preface, we introduce the main contents of this paper.In chapter 2, The chapter is divided into three sections to investigate the oscillation criteria for some class of superlinear second order damped differential equation. We state the main results as follows: First, we are concerned with the second-order superlinear damped differential equation,(a(t)y'(t))' + p(t)y'(t) + q(t)f(y(t)) = 0, t ≥ t₀ . (2.1.1)In this section, we mainly employed a generalized Riccati transformation and integral average technique in the study of oscillatory properties of more general second-order superlinear differential equation (2.1.1). We shall further the investigation and improve the main results of Yu [17]. We obtained several new oscillation criteria at the end of this section.Second, we employ a integral operator A_a^b of second-order superlinear differential equation (2.1.1) in this section. And, we obtained several new oscillation criteria at the end of this paper.Third, we consider the oscillatory behavior of the second order superlinear differential equation(a(t)y'(t))'+ q(t)f(y(t)) = 0, t≥t₀. (2.3.1)In this section, several new oscillation criteria are established under quite general assumptions of second-order superlinear differential equation (2.3.1). Our methodology is somewhat different from that of previous authors. We obtained several new oscillation criteria at the end of this section.In chapter 3, we study the interval oscillation of second order nonlinear differential equations with delayed argument,x"(t)+q(t)f(x(Î¥(t))) =0. (3.1.1)In this chapter, our results aslo including that the order of the function f(x) don't influence the oscillation of equation (3.1.1). We shall further the investigation and improve the main results of D. Cakmak and A. Tiryaki [31] and obtained several new oscillation criteria in the subinterval of [t₀, ∞) at the end of this chapter.