Investigations into $(n,q)$-symmetry in nonlinear multimappings acting on a metric space

In light of recent advances, this paper introduces a novel class of multimappings within the framework of metric spaces that aims to deepen the understanding of the structure and interplay of symmetric commuting tuples of mappings. A central focus of our study is the analysis of the product of two symmetric commuting tuples of mappings (c.t.m): Specifically, we consider an $(m,q)$-symmetric c.t.m $\mathbf{N}=({\mathcal{N}}_1,\dots ,{\mathcal{N}}_p)$ and an $(n,q)$-symmetric c.t.m $\mathbf{W}=({\mathcal{W}}_1,\dots ,{\mathcal{W}}_r)$, and establish the precise conditions under which their product $\mathbf{N}\odot \mathbf{W}$ yields an $(m+n-1,q)$-symmetric c.t.m. In addition, we investigate the behavior of $(m,q)$-isometric commuting tuples under composition $\mathbf{N}\cdot \mathbf{W}$, deriving new results that generalize existing theorems in the literature, most notably extending of Theorem 2.14 in Bermúdez et al., J. Operat. Theor., 72(2) (2014), 313–329 to a broader setting. The analysis of mappings in metric spaces is inherently complex, as these spaces often lack algebraic structures such as vector addition or scalar multiplication, making the study of nonlinear and multimappings particularly challenging. Our findings contribute to this area by elucidating the subtle relationships among symmetry, isometry, and commutativity in mapping tuples, thereby enabling more structured approaches to higher-dimensional problems within this framework.