Summary: Let be real Hilbert spaces, let , be two nonempty closed convex level sets, let , be two bounded linear operators. Our interest is in solving the following new convex feasibility problem
which allows asymmetric and partial relations between the variables and . In this paper, we present and study the convergence of a relaxed alternating CQ-algorithm (RACQA) and show that the sequences generated by such an algorithm weakly converge to a solution of the above problem. The interest of RACQA is that we just need projections onto half-spaces, thus making the relaxed CQ-algorithm implementable. Note that by taking we recover the split convex feasibility problem originally introduced by Y. Censor and J. Elfving [Numer. Algorithms 8, No. 2–4, 221–239 (1994; Zbl 0828.65065)] and used later in intensity-modulated radiation therapy [Y. Censor et al., “A unified approach for inversion problems in intensity-modulated radiation therapy”, Physics in Medicine and Biology 51, 2353–2365 (2006)]. We also recover the relaxed CQ-algorithm introduced by Q. Yang [Inverse Probl. 20, No. 4, 1261–1266 (2004; Zbl 1066.65047)] by particularizing both and a given parameter.