In the paper, we present a necessary and sufficient condition for the existence of a sequence of measurable functions with finite values, which converge to any given essential bounded function in the topology of essential supremum in a Banach space. A new convergence method is proposed, which allows for the discovery of an essential bounded function $F$ that is valued in a Banach space. Generally speaking, there exists a Banach-valued essential bounded function $F$ which $F_n$ can't converge to $F$ in the topology of essential supremum for any sequence of finite-valued measurable function.

In this paper, we consider the variational inequality problem and the split common fixed point problem. Considering the common fixed points of an infinite family of nonexpansive mappings, instead of just the fixed point of one nonexpansive mapping, we generalize the results of Tian and Jiang. By removing a projection operator, we improve the efficiency of our algorithm. Finally, we propose a very simple modification to the extragradient method, which gives our algorithm strong convergence properties. We also provide some numerical examples to illustrate our main results.

Self-adaptive algorithms are presented for solving the split common fixed point problem of quasi-pseudocontractive operators in Hilbert spaces. Weak and strong convergence theorems are given under some mild assumptions.