A New Kind of Lax-Oleinik Type Operator with Parameters for Time-Periodic Positive Definite Lagrangian Systems

In this paper we introduce a new kind of Lax-Oleinik type operator with parameters associated with positive definite Lagrangian systems for both the time-periodic case and the time-independent case. On one hand, the family of new Lax-Oleinik type operators with an arbitrary u is an element of C(M, R-1) as initial condition converges to a backward weak KAM solution in the time-periodic case, while it was shown by Fathi and Mather that there is no such convergence of the Lax-Oleinik semigroup. On the other hand, the family of new Lax-Oleinik type operators with an arbitrary u is an element of C(M, R-1) as initial condition converges to a backward weak KAM solution faster than the Lax-Oleinik semigroup in the time-independent case.