Qualitative Analysis Of Several Classes Of Higher-Order Delay Differential Equations

The increasing interest in oscillation and asymptotic behavior of solutions to dynamic equations is motivated by their applications in the engineering and natural sciences. This thesis is concerned with the oscillation and asymptotic properties of several classes of higher-order delay dynamic equations. The results obtained extend and improve a number of results reported in the literature. The main results are described as follows:In Chapter1, we briefly summarize background and development of the dy-namic equations. The main results in this thesis are also introduced.In Chapter2, we study the oscillation of second-order delay dynamic equa-tions by using Riccati transformation and comparison methods. Section2.1is con-cerned with oscillatory behavior of a class of second-order delay dynamic equations on a time scale. Two Philos-type oscillation criteria are presented that improve some known results in the literature. Section2.2is concerned with oscillatory be-havior of a certain class of second-order linear neutral delay dynamic equations on a time scale. Some new results are presented that complement and improve those related results in the literature. Section2.3is concerned with the oscillation of a class of second-order half-linear neutral dynamic equations on a time scale."Four new oscillation criteria are presented that supplement those results given in the literature. Section2.4is concerned with oscillatory behavior of a class of second-order neutral differential equations. New theorems complement and improve a number of results reported in the literature. Section2.5is concerned with oscilla-tion of a class of second-order Emden-Fowler neutral delay differential equations. Some new oscillation criteria are obtained.In Chapter3, we consider the oscillation and asymptotic properties of third-order delay dynamic equations by using Riccati transformation and comparison methods. Section3.1is concerned with the oscillation of a class of third-order non-linear delay differential equations. Some new oscillation criteria are presented that improve related contributions to the subject. Section3.2is concerned with oscil-lation and asymptotic behavior of a class of third-order neutral delay differential equations. Some new criteria are presented which ensure that all solutions to the studied equations are oscillatory or convergent to zero asymptotically. These cri-teria complement and improve those results in the literature. Section3.3presents new Hille and Nehari type asymptotic criteria for a class of third-order delay dy-namic equations on a time scale. The results obtained extend and improve some previous results.In Chapter4, we study the oscillation of fourth-order dynamic equations on time scales by using Riccati transformation and comparison methods. Section4.1is concerned with oscillatory behavior of a certain class of fourth-order nonlinear dynamic equations on a time scale. A new oscillation result is presented. Sec-tion4.2is concerned with oscillation of a class of fourth-order nonlinear delay dynamic equations on a time scale. A new comparison theorem is established that improves related results reported in the literature. Section4.3is concerned with oscillatory behavior of a class of fourth-order half-linear delay dynamic equations with damping on a time scale. Some new oscillation criteria are established.In Chapter5, we consider the oscillation and asymptotic properties of higher-order delay differential equations by using Riccati transformation and comparison methods. Section5.1is concerned with oscillation and asymptotic behavior of a class of higher-order delay differential equations. Some sufficient conditions are es-tablished which ensure that all solutions to the studied equations are oscillatory or convergent to zero asymptotically. The results obtained complement and improve those results reported in the literature. Section5.2is concerned with oscillation and asymptotic properties of a class of higher-order delay damped differential equations with p-Laplacian like operators. Some new criteria are presented that improve the related contributions to the subject. Section5.3is concerned with oscillatory behavior of solutions to a certain class of even-order neutral delay dif-ferential equations. Some new criteria are established that improve a number of related results reported in the literature.In Chapter6, the main research contents and main results are introduced and summarized, and the future research tendency is prospected.