LOCAL STABLE MANIFOLDS THROUGH PERIODIC SOLUTIONS TO EVOLUTION EQUATIONS

Tóm tắt

The matter of finding the relevant conditions for the existences of the integral
manifolds to evolution equations is a great interest of many Mathematic researchers.
The first studies on the existence of integral manifolds on differential equations in
case finite-dimensional phase spaces were given by Hadamard [5], Perron [12], and
Bogoliubov and Mitropolsky [1, 2]. Next, Daleckii and Krein [3] extended these
results to the case of bounded coefficients acting on Banach spaces. Later, Henry
[10], Sell and You [13] proved the case of unbounded coefficients, and the papers by
N.T.Huy [6,7] in which nonlinear part satisfies the  -Lipschitz condition. However,
the existences of the integral manifolds near periodic solution of evolution equations
with Nemytskii’s nonlinear operator have been studied in no researches before.
Therefore, in this paper, we would like to prove the existence of local stable
manifolds near periodic solutions to evolution equationse form u A t u g u t   ( ) ( )( ) 
where the operator-valued function t A t  ( ) is T -periodic, and the Nemytskii's
operator g x t ( )( ) is T -periodic with respect to t for each fixed x and satisfies
locally Lipschitz condition.