In this paper we propose a modification of the von Neumann method of alternating projection x
k+1=P
A
P
B
x
k
where A,B are closed and convex subsets of a real Hilbert space ℋ. If Fix P
A
P
B
≠ ∅ then any sequence generated by the classical method converges weakly to a fixed point of the operator T=P
A
P
B
. If the distance δ=inf
x∈A,y∈B
∥
x-y
∥ is known then one can efficiently apply a modification of the von Neumann method, which has the form x
k+1=P
A
(x
k
+λ
k
(P
A
P
B
x
k
-x
k
)) for λ
k
>0 depending on x
k
(for details see: Cegielski and Suchocka, SIAM J. Optim. 19:1093–1106, 2008). Our paper contains a generalization of this modification, where we do not suppose that we know the value δ. Instead of δ we apply its approximation which is updated in each iteration.
