Oscillation For Two Classes Of Fractional Differential Equations

In the self-similar porous structure, viscoelastic, control theory, analytical chemistry, physical chemistry, economics dynamics process, many mathematical models correspond-ing to the practical problems are the fractional differential equations. In the last few decades, many authors have paid much attention to various aspects of fractional differ-ential equations, for example, the existence, uniqueness, stability, asymptotic of solutions for fractional differential equations, the numerical solution methods for fractional differ-ential equations, and so on. In these investigations, we find that very little attention is paid to the oscillatory of fractional differential equations, and no one paid any attention to the oscillation of fractional neutral differential equations.We mainly studied the oscillation of two classes of fractional differential equations, where, in the second chapter, we considered the oscillation of the fractional differential equations with Riemann-Liouville derivative Dαx(t)+b(t)f(x(t))= 0, t≥0, where, Dαx denotes the Riemann-Liouville fractional derivative of order a of x and a ∈ (n - 1,n), n is a positive even. In the third chapter, we considered the oscillation of the fractional differential equations with neutral type where, z(t)= y(t)+py(t - τ), Dα_z denotes the Liouville right-sided fractional derivative of order a of z and α ∈ (n - 1, n), n is a positive odd.These two kinds of equations are both higher order fractional differential equations, thus, the second kind of equations is the fractional differential equations with neutral type. By using transformation and some analysis methods such as inequality and in-tegral, several sufficient conditions for the oscillation of the two equations are obtained respectively through strict proof, and some applications are presented for the established results.