Introduce gradual Banach lattices and Hyers stability

The main objective of this article is to introduce gradual normed Riesz spaces and their properties. In fact, by adding a property to gradual normed spaces as discussed in reference [20], we have defined the gradual Banach lattices. Next, we focus on the stability of the supremum-preserving functional equation in these spaces, utilizing the fixed-point theorem to examine the conditions ensuring stability. Also, we can verify the stability of various functional equations in these spaces.