We establish adequate conditions for the existence and uniqueness of solutions to systems of two matrix equations when the unknown matrices are raised to a power $k\in [-1,1]\setminus \left\{0\right\}$. The findings from coupled fixed points for ordered pairs of maps are used. Numerical examples are provided to illustrate the results shown. Some of the known results are the consequence of our acquisitions.

Using the deep result of Ran & Reunrings, we generalize existing results for tripled fixed points. In contrast to the previously known results for tripled fixed points of maps with or without the mixed monotone property in partially ordered complete metric spaces, we demonstrate that it is possible to obtain results for the existence and uniqueness of such points for arbitrary maps with a type of monotonicity, with the partial ordering in the Cartesian product arising from the maps itself. We prove theorems that ensure the existence and uniqueness for tripled fixed points for maps with different types of monotone properties. We obtain sufficient conditions for the existence and uniqueness of systems of three nonlinear matrix equations. The obtained results are illustrated by solving systems of matrix equations.

Geometric constructions are widely used in computer graphics and engineering drawing. A right generalized cylinder is a ruled surface whose base curve is a plane curve perpendicular to the rulings. The paper discusses relations between the base curve and the non-planar space curve on the right generalized cylinder. Based on these relations, a method for obtaining a new space curve from a given plane curve parameterized about an arbitrary parameter is presented. At first, we define a non-planar space curve on the right generalized cylinder whose base curve is the considered plane curve, parameterized about an arbitrary parameter. Later on, we examine the focal curve of the obtained cylindrical curve which is also a non-planar curve. The Frenet-Seret system of the cylindrical curve and its focal curve are expressed in terms of the signed curvature of the abovementioned plane curve and its derivatives. Finally, we obtained the parametric representation of orthogonal projection of the focal curve onto the Euclidean plane via before mentioned plane curve and its derivatives. That curve is called generalized focal curve of a plane curve. The proposed method is demonstrated for several closed plane curves used in engineering practice. These curves include: epicycloid, hypocycloid and a curve that is orthogonal projection of toroidal helix onto the Euclidean plane.

In this paper, we establish relations between Frenet frames and differential-geometric invariants of a non-helical space curve in a three-dimensional Euclidean space and its closely related curve in a four-dimensional Euclidean space. The focal curve of the constructed four-dimensional curve is used to obtain a parametric representation of the generalized focal curve of a spherical curve.