A SHRINKING PROJECTION METHOD FOR SOLVING THE SPLIT COMMON FIXED POINT PROBLEM IN HILBERT SPACES

Abstract

We study the split common fixed point problem in two Hilbert spaes. Let H₁ and H₂ be two real Hilbert spaces. Let S₁ : H₁ → H₁, and S₂ : H₂ → H₂, be two nonexpansive mappings on H₁ and H₂, respectively. Consider the following problem: find an element x† ∈ H₁ such that

x† ∈ Ω := Fix(S1) ∩ T−1( Fix(S2)) ≠ ∅,

where T : H₁ → H₂ is a given bounded linear operator from H₁ to H₂.

Using the shrinking projection method, we propose a new algorithm for solving this problem and establish a strong convergence theorem for that algorithm.