Oscillation Criteria For Some Nonlinear Fractional Differential Equations

With the development of science, the oscillation of the differential equations as the fundament of dynamics has attracted more attention. Mathematics began to pay much attention to the field of fractional differential equations due to their wide application in practical questions. As an important part of qualitative theory of differential equations, the oscillation of the fractional differential equations still need further investigation, especially for nonlinear fractional differential equations system.In recent years, the research on oscillation of nonlinear fractional differential equa-tions has become a hot topic. A lot of elegant results were givenBased on the previous research, we studied the oscillatory criteria of three classes of nonlinear fractional differential equations using generalized Riccati transformation and some integral inequalities.This paper includes the following four parts:Chapter 1 Preference, we mainly introduced some basic definition and properties about fractional derivative and integral and some lemmas.Chapter 2 Motivated by references [16-25], considered the fractional differential equations In this Chapter, to illustrate the main results, some oscillatory criteria are given for the following two equationsChapter 3 Under the condition of reference [25], the oscillation of equations ef-fected by the added the damping term was discussed. some sufficient conditions to guarantee the equation’s oscillatory were obtained By means of the main results, we considered the following two equationsIn Chapter 2 and Chapter 3, α∈(0,1), Dαx(t) means the Liouville right-sided fractional derivative of order a of x.Chapter 4 By means of the properties of Riemann-Liouville fractional derivative, we studied the oscillation of following fractional differential equations where Dax is the Riemann-Liouville fractional derivative of order α∈(0,1) of x. Some oscillatory criteria for the following equations are given.