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.

We investigate the Hyers-Ulam stability of functional equations involving a single variable in fuzzy Banach spaces using a new direct method. This method imposes no restrictions on the domain or range of functions and is shown to be simpler and more effective for various functional equations. Furthermore, we establish a fuzzy version of the generalized Hyers-Ulam stability for a Pexider-type functional inequality and a linear functional equation with multiple coefficients in a fuzzy Banach linear space. For both equations, we obtain the existence and uniqueness of approximating solutions. To validate the theoretical results, numerical experiments are conducted using the Monte Carlo random sampling method. The results show that the mean ratio of the true error to the theoretical upper bound is only 0.0027, and the 95th percentile is 0.0051, indicating that the derived error bound is both reliable and tight. The proposed method enriches the proof techniques for stability problems of functional equations in fuzzy spaces, and the findings can serve as a reference for theoretical research and practical applications in related fields.