分数阶微分方程初值问题解的存在唯一性

The existence and uniqueness of the solutions for initial value problems of fractional differential equations

文章研究了具有两个阻尼项分数阶微分方程的动力系统解的存在性和唯一性.该文考虑以下具有两个阻尼项的分数阶微分方程{^(c)D^(α)_(0+)x(t)-A^(c)D^(β)_(0+)x(t)-B^(c)D^(γ)_(0+)x(t)=f(t,x(t)),t∈J:=[0,T],x(0)=x_(0),x′(0)=x′_(0),x″(0)=x″_(0),其中0<γ≤1<β≤2<α≤3,0<T<∞,x∈R^(n),A和B是R^(n×n)矩阵,f:J×R^(n)→R^(n)是连续函数.运用Arzelà-Ascoli定理,Banach压缩映射原理和Leray-Schauder度理论得出以上方程解的存在性和唯一性结果.

The existence and uniqueness of the solutions of fractional dynamical systems with two damping terms are considered,{^(c)D^(α)_(0+)x(t)-A^(c)D^(β)_(0+)x(t)-B^(c)D^(γ)_(0+)x(t)=f(t,x(t)),t∈J:=[0,T],x(0)=x_(0),x′(0)=x′_(0),x″(0)=x″_(0),where 0<γ≤1<β≤2<α≤3,0<T<∞,x∈R^(n),A and B are R^(n×n) matrix and f:J×R^(n)→R^(n) is jointly continuous.The existence and uniqueness of solutions of the above equations are obtained by using Arzel-Ascoli theorem,Banach contraction mapping principle and Leray Schauder degree theory.