The authors consider the following multiple-sets split feasibility problem: Find a vector \[ x^*\in C\equiv\bigcap^t_{i=1} C_i\quad\text{such that}\quad x^*\in Q\equiv\bigcap^r_{j=1} Q_j, \] where \(A\) is a given \((M\times N)\)-matrix, \(C_i\), \(\forall i= 1,\dots, t\) are non-empty closed convex sets in \(\mathbb{R}^N\), and \(Q_j\), \(\forall j= 1,\dots, r\) are non-empty closed convex sets in \(\mathbb{R}^M\).
A new method for solving this problem is proposed, which uses variable step-sizes unlike to other methods known from the literature, whcih use a fixed step-size. The m ethod is an extension and application of the method published by B. S. He, H. Yang, Q. Meng and D. R. Han [J. Optimization Theory Appl. 112, No. 1, 129–143 (2002; Zbl 0998.65066)] to problems with weaker conditions.