Periodic solutions of quaternionic-valued ordinary differential equations

Abstract

This paper uses topological degree methods to prove the existence of periodic solutions of some quaternionic-valued ordinary differential equations.

Determinantal representations of the Moore–Penrose inverse over the quaternion skew field and corresponding Cramer's rules

“…If a is constant, then t t0 adτ = a (t − t 0 ), and Φ l (t) = Φ r (t) = e a(t−t0) . The similar result has been obtained in [9] as well. Moreover, in [9], the following nonhomogeneous differential equations corresponding to (2) has been considered,…”

“…Leo and Ducati [8] solved some simple second order quaternionic differential equations. Campos and Mawhin [9] studied the existence of periodic solutions for the quaternionic Riccati equation.…”

Quaternionic-valued ordinary differential equations. The Riccati equation

“…has no periodic solutions (see [2,Corollary 7.2]). In all the cases l = {q = q 0 + q 1 i + q 2 j + q 3 k ∈ H: …”

“…(2). To prove that a set W ⊂ R × M is an isolating segment for ϕ it is enough to check the behaviour of the vector field (1, v) on the boundary of W .…”

“…Campos and Mawhin [2] have initiated a study of the T -periodic solutions of quaternionic-valued first order differential equationsq …”