TY - JOUR AU - Cho, Ilwoo PY - 2022 TI - Certain nonlinear functions acting on the vector space \(\mathbb{H}^{n}\) over the Quaternions \(\mathbb{H}\) JO - Journal of Nonlinear Sciences and Applications SP - 14--40 VL - 15 IS - 1 AB - In this paper, we consider a certain type of nonlinear functions acting on a finite-dimensional vector space \(\mathbb{H}^{n}\) over the ring \(\mathbb{H}\) of all quaternions, for \(n\) \(\in\) \(\mathbb{N}.\) Our main results show that: (i) every quaternion \( {q\in\mathbb{H}}\) is classified by its spectrum of the realization under a canonical representation on \(\mathbb{C}^{2}\); (ii) each vector of \(\mathbb{H}^{n}\) is classified by \(\mathbb{C}^{n}\) in an extended set-up of (i); and (iii) the (usual linear) spectral analysis on the matricial ring \( {M_{n}\left(\mathbb{C}\right)}\) of all \(\left(n\times n\right)\)-matrices (over \(\mathbb{C}\)) affects some fixed point theorems for our nonlinear functions on \(\mathbb{H}^{n}\). In conclusion, we study the connections between the ``linear'' spectral theory over the complex numbers \(\mathbb{C}\), and fixed point theorems for ``nonlinear'' functions over \(\mathbb{H}\). SN - ISSN 2008-1901 UR - http://dx.doi.org/10.22436/jnsa.015.01.02 DO - 10.22436/jnsa.015.01.02 ID - Cho2022 ER -