Control and Cybernetics a Generalization of the Opial's Theorem

Opial presented in 1967 a theorem, which can be applied in order to prove the weak convergence of sequences (xk) in a Hilbert space, generated by iterative schemes of the form xk+1 = Uxk for a nonexpansive and asymptotically regular operator U with nonempty FixU . Several iterative schemes have, however, the form xk+1 = Ukxk, where (Uk) is a sequence of operators with a common fixed point. We show that under some conditions on the sequence (Uk) the sequence (xk) converges weakly to a common fixed point of operators Uk. We show also that the Opial’s theorem and the Krasnoselskii–Mann theorem are the corollaries descending from the obtained results. Finally, we present some applications of the results to the convex feasibility problems.