Oscillation Of Nonlinear Impulsive Delayed Partial Differential Equations And Fractional Differential Equations

Oscillation is a common movement in the form of substance and one of the main dynamical properties of the system. Oscillation theories of differential equations have a wide range of ap-plications in control engineering, mechanical vibration, mechanics and others. As we know, the comparison and separation theories of zeros distribution for second order homogeneous linear d-ifferential equations established by G. Sturm lay a foundation of oscillation theory for differential equations. During one and a half century, oscillation theory of differential equations has devel-oped quickly and played an important role in qualitative theories and theory of boundary value problems. There are many mathematician in this subject and they obtained many useful results. Among these theories, oscillation theories for delayed ordinary (partial) differential equations and impulsive ordinary (partial) differential equations are the most important qualitative theories for these equations. The existence of delay and impulse makes the system reflects the variations of things more accurately, but it makes the analysis of oscillation of the system more difficult. The oscillation theory of impulsive delayed ordinary (partial) differential equations has become one of new research areas in recent years, which leads to fundamental theoretical interest.In addition, of course, to the theories of differential, integral and integro-differential equation-s, and special functions of mathematical physics as well as their extensions and generalizations in one or more variables, some of the areas of present day applications of fractional calculus include fluid flow, rheology, dynamical processes in self-similar and porous structures, diffusive transport akin to diffusion, probability and statistics, control theory of dynamical systems, viscoelasticity, electrochemistry of corrosion, optics and signal processing and so on.The thesis is concerned with the oscillation of delayed differential equations with impulses and fractional differential equations, forced oscillation of fractional partial differential equations. The results obtained extend and improve many results reported in the literature. The main results are described as follows:In chapter 1, we summarize the historical background and recent development of oscillation of delayed differential equations with impulses, fractional differential equations. The main results of this thesis are also briefly introduced.In chapter 2, by using generalized Riccati transformation and integral average, the oscilla-tion of nonlinear delayed hyperbolic differential equations (systems) with impulses is discussed. Several sufficient conditions are presented by reducing the oscillation of delayed partial differ- ential equations to delayed differential inequalities with impulses having no eventually positive solutions, and the new oscillation criteria are obtained.In chapter 3, by considering the function H(t, s) that may not have a nonpositive partial derivative on s and employing generalized Riccati transformation as well as integral averaging technique, oscillation criteria for nonlinear delayed impulsive partial differential equations with damping term are established. These criteria complement and improve the results in the literature.In chapter 4, some definitions for fractional calculus are first introduced. By using the prop-erties of fractional derivatives and fractional integrals, oscillation of a fractional ordinary differen-tial equation and forced oscillation of a fractional partial differential equation are discussed, some oscillation criteria are obtained which can be considered as improvements for known results of fractional differential equations.In chapter 5, the main research contents are summarized, and the future research tendency is prospected.