On the existence and uniquness theorem of the global solutions for UFDES

The uncertain functional differential equation (UFDE) is a type of functional differential equations driven by a canonical uncertain process. Uncertain functional differential equation with infinite delay (IUFDE) have been widely applied in sciences and technology. In this paper, we prove an existence and uniqueness theorem for IUFDE intheinterval $[t_{0},T]$, underuniform Lipschitz condition and weak condition. Also, the novel existence and uniqueness theorem under the linear growth condition and the local Lipschitz condition is proven. In the following, a more general type of UFDE considers, which the future state is determined by entire of the past states rather than some of them. Finally, the existence and uniqueness theorem is considered on theinterval $[t_{0},infty ]$.