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In this paper we introduce a new kind of Lax-Oleinik type operator withparameters associated with positive definite Lagrangian systems for both thetime-periodic case and the time-independent case. On one hand, the new familyof Lax-Oleinik type operators with an arbitrary $u\in C(M,\mathbb{R}^1)$ asinitial condition converges to a backward weak KAM solution in thetime-periodic case, while it was shown by Fathi and Mather that there is nosuch convergence of the Lax-Oleinik semigroup. On the other hand, the newfamily of Lax-Oleinik type operators with an arbitrary $u\in C(M,\mathbb{R}^1)$as initial condition converges to a backward weak KAM solution faster than theLax-Oleinik semigroup in the time-independent case.

A new kind of Lax-Oleinik type operator with parameters for time-periodic positive definite Lagrangian systems