  We apply methods of the fixed point theory to a Lambda policy iteration with
a randomization algorithm for weak contractions mappings. This type of mappings
covers a broader range than the strong contractions typically considered in the
literature, such as \'Ciri\'c contraction. Specifically, we explore the
characteristics of reinforcement learning procedures developed for feedback
control within the context of fixed point theory. Under relatively general
assumptions, we identify the sufficient conditions for convergence with a
probability of one in infinite-dimensional policy spaces.
