*(English)*Zbl 1256.49044

Summary: Let ${H}_{1},{H}_{2},{H}_{3}$ be real Hilbert spaces, let $C\subset {H}_{1}$, $Q\subset {H}_{2}$ be two nonempty closed convex level sets, let $A:{H}_{1}\to {H}_{3}$, $B:{H}_{2}\to {H}_{3}$ 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 $x$ and $y$. 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 $B=I$ 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 $B$ and a given parameter.