Abstract
This paper studies the existence of affine-periodic solutions which have the form of x(t+T)=Qx(t) with some nonsingular matrix Q. Depending on the structure of Q, they can be periodic, anti-periodic, quasi-periodic or even unbounded. Krasnosel’skii–Perov type existence theorem, asymptotic and homotopy equivalence approaches are given.
Highlights
For more than a century after Poincaré and Lyapunov, the existence theory of periodic solutions for a periodic system has been well developed; for example, see [7, 9, 12, 26, 29]
If a system is subjected to an external force with a certain symmetry structure, a natural question is whether the system has a solution with the same symmetry structure
One may ask whether the system under a spiral external force has a spiral form solution
Summary
For more than a century after Poincaré and Lyapunov, the existence theory of periodic solutions for a periodic system has been well developed; for example, see [7, 9, 12, 26, 29]. Krasnosel’skii and Perov gave an interesting existence theorem of periodic solutions in [13, 14], which is well known today by using the method of topological degree. We give Krasnosel’skii–Perov type results for affine-periodic systems. When I – Q is invertible, we find that the existence of affine-periodic solutions can be obtained without calculating the topological degree of f (0, ·). It is meaningful to find the relationship for the existence of periodic solutions between asymptotically equivalent equations. We use the method of asymptotically equivalent equation to study the existence of affine-periodic solutions. We give a method to study the existence of affine-periodic solutions by using homotopy approach. 3, we study the existence of affine-periodic solutions by asymptotic equivalence.
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