A generalization of the Opial's theorem
Andrzej Cegielski · 2007
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.