Several Classes Inequalities On Time Scale And On Oscillation For Higher-order Differential Equations

The development of the theory of time scales was initiated by Hilger [2] in 1988 as a theory capeble to contain both difference and differential calculus in a consistent way. Since then many authors have expounded on various aspects of the theory of dynamic inequalities on time scales and the integral inequalities(see [3]-[11]).The oscillation of ordinary differential equation is one of important branche of differential equations. With the increasing development of science and technol-ogy, there are many problems relating to differential equation derived from lots of practical applications, such as whether differential equation has a oscillating solution or not, and whether all of its solutions are oscillatory or not. Becauce all the structures of its emergence have deep physical background and realistic mathematical models. In the field of modern applied mathematics, it has made considerable headway in recent years. In this dissertation, we employ a gen-eralized Riccati transformation, the estimations of inequalities, integral average technique and monotonicity of a function to investigate the oscillation problems for several classes of higher-order nonlinear differential equations. We obtain some meaningfully new results.The thesis is divided into three chapters according to contents.In Chapter 1, we introduce the background and the main problems of this dissertation.In Chapter 2, we study the five Pachpatte type inequalities on time scales: where m≥1,m≥n≥l>0. Meanwhile, we extend the main results of Li [1].Chapter 3 is divided into two sections to investigate the oscillation criteria for some classes of higher-order nonlinear differential equations. First, new oscillation criteria for higher-order differential equation where z(t)=x(t)+q(t)x(t-τ), a>0 are established under the conditions or by employing functionφ(t,s,l), generalized Riccati transformation and the esti-mations of inequalities. The obtained results extend and improve some known results in literature.Second, we are concerned with the higher-order nonlinear differential equa-tionsBy employing functions H(t, s) and h(t, s) under the four conditions, we obtain new oscillation criteria, which extend and improve the known results.