Affine-periodic solutions by asymptotic and homotopy equivalence

This paper studies the existence of affine-periodic solutions which have the form of x(t+T)=Qx(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x(t+T)=Qx(t)$\end{document} 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.


Introduction
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]. Besides periodicity, many systems may also have other symmetric structures. The antiperiodic system together with the existence of anti-periodic solutions, for example, is paid high attention to; see [1-5, 20, 25, 27].
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. For example, one may ask whether the system under a spiral external force has a spiral form solution. Recently, the concept of affine periodicity, including the spiral symmetry was introduced. Some problems and methods concerning affine-periodic solutions, such as Levinson's problem, Lyapunov function type theorems, the dissipative second order rotating periodic systems, LaSalle type theorems, Hamiltonian systems and the averaging method of higher order perturbed systems were given; see [8,16,18,19,23,24,28].
Consider the following (Q, T)-affine-periodic system: where f (t, x) : R × R n → R n is continuous and f (t + T, x) = Qf (t, Q -1 x) for every (t, x) ∈ R × R n , Q ∈ GL(n) (all nonsingular n × n matrices). We want to find the solution of (1.1) with Such a solution is called a (Q, T)-affine-periodic solution.
According to the structure of Q, it will be seen that (i) x(t) is a T-periodic solution if Q = I (the identity matrix), and an anti-periodic one if Q = -I; (ii) If Q ∈ O(n), that is, Q is an orthogonal matrix, x(t) is a special quasi-periodic solution corresponding to the rotation of a rigid body; (iii) A (Q, T)-affine-periodic solution x(t) can be unbounded and x(t) |x(t)| is quasi-periodic, like a helical line, for example, x(t) = e at (cos ωt, sin ωt). For periodic systems, 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. They proved that, if each solution starting from the boundary of a bounded region will not return to the initial point during a periodic time and the topological degree of f (0, ·) is not equal to zero, then the system will have a periodic solution. In this paper, we give Krasnosel'skii-Perov type results for affine-periodic systems. When I -Q is not invertible, we give a general result, which is comparable with Krasnosel'skii and Perov's theorem in the periodic case. It is well known that sometimes the conditions of the Krasnosel'skii-Perov type theorem are difficult to verify, but we give a more flexible condition. When I -Q is invertible, we find that the existence of affine-periodic solutions can also be obtained without calculating the topological degree of f (0, ·).
It is also meaningful to find the relationship for the existence of periodic solutions between asymptotically equivalent equations. It is well known that the existence of periodic solutions of a system is rather related to that of another asymptotically equivalent one; see [15,17]. In the asymptotically linear case, more results have been obtained; for example see [6,10]. In this paper, we use the method of asymptotically equivalent equation to study the existence of affine-periodic solutions. In our results, the asymptotically equivalent equation can be nonlinear, and the conditions are much easier to verify in linear case.
In the study of periodic solutions for differential equations, the alternative is an interesting phenomenon. Krasnosel'skii and Perov's theorem is a kind of alternative theorem. Another alternative result is achieved by the homotopy method. If the solutions of the auxiliary equations starting at the boundary are not periodic, then the system will have at least one periodic solution in the interior of the region; for example, see [11,21,22]. In this paper, we give a method to study the existence of affine-periodic solutions by using homotopy approach.
If there is a linear coordinate transformation u = Bx with B ∈ GL(n), then system (1.1) becomes where g(t, u) = Bf (t, B -1 u). According to the symmetry of f (t, x), we have This means the affine periodicity is invariant under linear transformations.
In Sect. 2 and Sect. 4 of the paper, we assume Q has the following Jordan normal form: where O denotes the zero matrix, 0 ≤ m ≤ n, and I (n-m)×(n-m) -C (n-m)×(n-m) is invertible. For example, Q is a symmetric or orthogonal matrix. Meanwhile in Sect. 3, we only need Q ∈ GL(n). The paper is organized as follows. In Sect. 2, we give some Krasnosel'skii-Perov type results. In Sect. 3, we study the existence of affine-periodic solutions by asymptotic equivalence. In Sect. 4, we give an existence theorem through the homotopy method. In Sect. 5, some examples are given to illustrate the characteristics of (Q, T)-affine-periodic systems and to show the effectiveness of the theorems.

Krasnosel'skii-Perov type results
Now we give our first main result.  Proof Suppose x(t) is a solution of system (1.1) with boundary condition x(T) = Qx(0).
For t ∈ R, let where m is an integer such that t -mT ∈ [0, T], the solution x(t) can be extended to the whole real line. So to prove the existence of (Q, T)-affine-periodic solutions of system (1.1), we just need to prove the existence of solutions of (1.1) with boundary condition We claim that, for each zero y of Φ, x(t, y) is a solution of (1.1) with boundary condition (2.1).
In fact, if y is a zero of Φ, we have and hence x(T, y) = Qy.
Consider the homotopy operator H : V × (0, 1] → R n : When λ = 0, denote It is easy to prove the operator is continuous, we omit the proof. Now we prove that Suppose on the contrary that there exists (ỹ,λ) ∈ ∂V × [0, 1], such that That is, and Pf (0,ỹ) = 0.
By (2.5), we getỹ ∈ ∂V ∩ Ker(I -Q), which contradicts assumption (iii). (II): Whenλ ∈ (0, 1], we have Then This contradicts assumption (ii). By (I) and (II), we obtain Without loss of generality, we assume that Q(0) has the form (1.2). Let Pf (0, y) be twice continuously differentiable and satisfy It is easy to see that H(y * , 0) = 0 if and only if y * ∈ Ker(I -Q) and Pf (0, y * ) = 0. Moreover, When Pf (0, y) is only continuous, the same result can be obtained by selecting suitable twice continuously differentiable functions to approximate it.
From the homotopy invariance of topological degree, we have Then there exists a y * ∈ V such that Φ y * = 0, and x(t, y * ) is a solution of equation (1.1) with boundary condition (2.1). Thus the existence of (Q, T)-affine-periodic solutions of system (1.1) is obtained.
When Q = I, Theorem 2.1 is consistent with Krasnosel'skii and Perov's theorem. For a perturbed system, we give the following corollary.
(ii) Let V be an open bounded subset of R n , and P : R n → Ker(I -Q) the orthogonal projection, and denote Pf i (t, x) = ((Pf i ) 1 (t, x), . . . , (Pf i ) n (t, x)) . Suppose that, for every point p ∈ ∂V ∩ Ker(I -Q), there exists a neighborhood U p of p, a constant σ p > 0 that are both independent of ε, and an integer 1 ≤ j ≤ n, such that Then system (2.9) has a (Q, T)-affine-periodic solution for |ε| > 0 small enough.
Proof It is easy to prove that there exist positive constants r and σ , such that, for every p ∈ ∂V ∩ Ker(I -Q) and every y ∈ B r (p), one has (2.14) At the same time, one has If x(ω, y) = Qy, then x(t, y) + ε k+1 r t, x(t, y), ε dt = 0.
When y ∈ and x(ω, y) = Qy, one has

Asymptotic equivalence
In this section, we study the relationship for the existence of affine-periodic solutions between system (1.1) and the following system: where A(t, x) : R × R n → R n is continuous, and for every t ∈ R, A(t, x) is continuously differentiable in the variable x. Moreover, A(t + T, x) = QA(t, Q -1 x) for every (t, x) ∈ R × R n , Let and define the norm as x = sup t∈[0,T] |x(t)|. It is easy to see that C Q,T is a Banach space with the norm · .
Theorem 3.1 Consider the system (1.1) and system (3.1). Assume the following conditions hold: (iii) There exists a constant σ > 0 such that, for every ϕ ∈ C Q,T , the solution y(t) with y = 1 of the system satisfies |y(T) -Qy(0)| ≥ σ , where A 1 (t, ϕ(t)) = Proof For each ϕ ∈ C Q,T and λ ∈ [0, 1], consider the following equation: A(t, 0) . (3.3) It is easy to see the system (3.3) is also a (Q, T)-affine-periodic system, and by assumption (iii) it has a unique (Q, T)-affine-periodic solution x ϕ,λ (t). Consider the homotopy operator H : Denote Ω p = {x ∈ C Q,T , x ≤ p}. Now we prove that there exists a constant p 0 > 0 large enough, such that, for every ϕ ∈ ∂Ω p 0 and λ ∈ [0, 1], Let Φ ϕ (t) be a fundamental matrix solution of (3.2) such that Φ ϕ (0) = I. Then for every ϕ ∈ C Q,T and t ∈ [0, T], where tr(A) denote the trace of matrix A. Thus, we have Denote by y ϕ (t) the solution of (3.2) with initial value y ϕ (0) = x ϕ (0). Then there exists a constant p 1 > 0, such that for every ϕ ∈ Ω p with p ≥ p 1 . If not, there would exist ϕ k ∈ Ω k , k = 1, 2, . . . , such that y ϕ k > k 2 . By the variation of constants formula, we get By assumption (i), we get which contradicts assumption (iii). Also by the variation of constant formula, for each t ∈ [0, T] and λ ∈ [0, 1], we have By assumption (i), there exists a p 2 > 0, such that Take p 0 = max{p 1 , p 2 }, then Next we prove H : Ω p 0 × [0, 1] → C Q,T is compact and continuous. By (3.3) and (3.4), it is easy to see that there exists a constant M 0 > 0, such that By Arzela-Ascoli's theorem, H is compact.
Let w(t) = xφ ,λ (t)u(t). Then w(t) is a solution of (3.2) and w(T) = Qw(0), this contradicts assumption (iii). Now by the homotopy invariance of topological degree, we get Then there exists a ϕ ∈ Ω p 0 , such that which can be extended to a (Q, T)-affine-periodic solution of system (1.1).
We give the asymptotically linear case as a corollary.
Consider the system

Homotopy method
To investigate the affine-periodic solutions of system (1.1), in this section we consider the following auxiliary equation: with λ ∈ [0, 1]. Then there exists aỹ ∈ V , such that which implies x(T,ỹ) = Qỹ.

Examples
Using the results in this paper, we can obtain periodic, anti-periodic, quasi-periodic or general affine-periodic solutions. In this section we give some examples to show this.
Example 5.3 Consider the following system in R n : where α > 0 is a constant, e : R → R n×n and h : R → R n are continuous. Moreover, with Q ∈ O(n). Since -|x| 2α x + e(t)x + h(t), x < 0 for (t, x) ∈ [0, T] × ∂B R with R > 0 large enough, we see that the vector field is inward to B R . By Theorem 2.3, the system has a (Q, T)-affine-periodic solution.
Denote by Φ(t) the fundamental matrix solution of y = A(t)y, (5.2) such that Φ(0) = I. We claim that if Φ(T) + I is invertible, the system (5.1) would have a T-anti-periodic solution. In fact, the system (5.1) is a (-I, T)-affine-periodic system. Since Φ(T) + I is invertible, the system (5.2) has only a trivial T-anti-periodic solution. It is easy to see By Corollary 3.1, the system has a T-anti-periodic solution.