Approaching the split common solution problem for nonlinear demicontractive mappings by means of averaged iterative algorithms

We consider new iterative algorithms for solving split common solution problems in the class of demicontractive mappings. These algorithms are obtained by inserting an averaged term into the algorithms previously used in [He, Z. and Du, W-S., Nonlinear algorithms approach to split common solution problems, Fixed Point Theory Appl. 2012, 2012:130, 14 pp] for the case of quasi-nonexpansive mappings. In this way, we are able to solve the split common solution problem in the larger class of demicontractive mappings, which strictly includes the class of quasi-nonexpansive mappings. Our investigation is based on the embedding of demicontractive operators in the class of quasi-nonexpansive operators by means of averaged mappings. For the considered algorithms we prove weak and strong convergence theorems in the setting of a real Hilbert space.