Directional asymptotics of Fej\'er monotone sequences

The notion of Fej'er monotonicity is instrumental in unifying the convergence proofs of many iterative methods, such as the Krasnoselskii-Mann iteration, the proximal point method, the Douglas-Rachford splitting algorithm, and many others. In this paper, we present directionally asymptotical results of strongly convergent subsequences of Fej'er monotone sequences. We also provide examples to show that the sets of directionally asymptotic cluster points can be large and that weak convergence is needed in infinite-dimensional spaces.