Existence And Asymptotic Behavior Of Solutions For Several Classes Of Fractional Differential Equations

In this paper,we mainly study three kinds of problems:the existence of solutions and asymptotic estimates of fractional differential equations with initial conditions,the existence of solutions and asymptotic estimates of differential equations with weakly singular kernel as well as the existence of solutions of fractional differential equations coupled system.First,under some suitable conditions,using boundary layer function method,we construct a formal asymptotic solution for the fractional differential equation with singular perturbation given by(?),t ∈[0,T],x(0,ε)= x0.where 0<α<1.The uniformly validity of asymptotic solution is obtained via upper and lower solutions method.Also an example is given to illustrate our main results.Second,we consider the initial value problem of Volterra integral equation with weakly singular kernel(?),t ≥ 0,y(0)= y0{ε),where ε is a small parameter,0<ε<<1.The integral kernel K is assumed indepen-dent of ε,which has a weak singularity at the initial point.Under some conditions,we construct the formal asymptotic solution for the initial value problem by boundary layer function method and prove the uniformly validity of the asymptotic solution by upper and lower solutions method,which extends the previous results in[61]Finally,we study the fractional differential equation coupled system(?)where 1<α,β<2,0 ≤p,q,γ,η<1,α-γ-1 ≥ 0,α-p-1 ≥ 0,α-q-1 ≥0,β-γ-1 ≥0,β-p-1 ≥0,β-g-1 ≥0,aηα-γ-<1,aηβ-γ-1<1,D is the standard Riemann-Liouville fractional Derivative and f,g:[0,1]× R × R × R →R R are given continuous functions.By Schauder’s fixed point theory,the existence of solutions to the coupled system is proved and an example is given to illustrate the validity of our main results.