Well-posedness Of Cauchy Problems In Fr(?)chet Spaces

In this paper we are interested in the well-posedness for the Cauchy problems in Fr(?)chet space. Since this kind of equation has important and wide applications in practice, it has received many attentions, such as Lobanov studied the existence and uniqueness of the solution for the Cauchy problems x’= f(t, x,λ),x(θ)= u on I in Fr(?)chet space. This article is based on his results and studies the continuous dependence on parameters λ for the Cauchy problems x’= f(t, x, λ),x(θ)= u. Next, we generalize the Cauchy problems x’= f(t, x,λ),x(θ)= u of that Lobanov studied the existence and uniqueness. We study the existence and uniqueness of solution for the Cauchy problem x’= f(t, x,Φ(x)),,x(θ)= u on I in Fr(?)chet space when the parameter y=Φ(x) is a state connected with x by using the Picard iteration in Fr(?)chet spaces.