Existence-uniqueness of nonlinear fractional differential equation

• 1、Department of Mathematics, School of Science, Wuhan University of Technology, Wuhan 430073

Abstract：We prove the existence-uniqueness of the solution to the nonlinearfractional differential equation in the Banach space $C([0, T]),$$$ egin{array}{lll} left {egin{array}{lll}D^{ lpha} u (t) = f (t, D^{ eta_{i}} u(t), 0leq ileqm), hspace{0.3cm} t in (0, T ),xin mathbf{R}, \ left . rac{d ^{k} u(t)}{d t ^k} ight |_{t=0} =eta_k hspace{0.3cm} (0leq k leq {lpha }-1),end{array} ight . end{array} $$ where $ lpha > eta _i>eta_0= 0, 1leq ileqm, { lpha }$ is the minimum integer no less than $lpha,f (t, v_i(t), 0leq ileq m)in C([0, T ) imes C([0, T )) imes C([0, T )) imescdots imes C([0, T ))), $$$| f ( t, u_i(t), 0leq ileq m)- f ( t, v_i(t),0leq ileq m)|leq sumlimits_{i=0}^m |a_i(t) (u_i(t)-v_i(t))| $$where $a_iin C([0, T )),max limits_{tin [0,T]} max limits_{0leq i,jleq m}int_0^t|a_i(s)|(t-s)^{lpha-eta_j-1} ds <+infty ( 0leq ileq m) $ and $ t in[0, T ] $.

Keywords： Differential equation, Fractional order, Existence-uniqueness theorem, Laplace transform