As generalizations of Yoshizawa’s theorem, it is proved that a dissipative affine-periodic system admits affine-periodic solutions. This result reveals some oscillation mechanism in nonlinear systems.

1. Introduction

Consider the system

()

where f : ℝ¹ × ℝⁿ → ℝⁿ is continuous and ensures the uniqueness of solutions with respect to initial values. Fix T > 0. The system (1) is said to be T-periodic if f(t + T, x) = f(t, x) for all (t, x) ∈ ℝ¹ × ℝⁿ. For this T-periodic system, a major problem is to seek the existence of T-periodic solutions. Actually, some physical systems also admit the certain affine-periodic invariance. For example, let Q ∈ GL(n), and

()

This affine-periodic invariance exhibits two characters: periodicity in time and symmetry in space. Obviously, when Q = id, the invariance is just the usual periodicity; when Q = −id, the invariance implies the usual antisymmetry in space. When Q ∈ SO(n), the invariance shows the rotating symmetry in space. Hence, the invariance also reflects some properties of solutions in geometry. Now, (2) is said to possess the affine-periodic structure. For this affine-periodic system, we are concerned with the existence of affine-periodic solutions x(t) with

()

In the qualitative theory, it is a basic result that the dissipative periodic systems admit the existence of periodic solutions. The related topics had ever captured the main field in periodic solutions theory from the 1960s to the 1990s. For some litratures, see, for example, [1–12].

In the present paper, we will see whether (1) admits affine-periodic solutions or not if (1) is affine-dissipative. Here, (1) is said to be affine-dissipative if Q^−mx(t + mT) are ultimately bounded. Our main result is the following.

Theorem 1. Let Q ∈ GL(n). If the system (1) is Q-affine-periodic, that is,

()

and affine-dissipative, then it admits a Q-affine-periodic solution x_*(t); that is,

()

The paper is organized as follows. In Section 2, we use the asymptotic fixed-point theorem, for example, Horn’s fixed-point theorem to prove Theorem 1. Section 3 deals with the case of functional differential equations, where an anagolous version is given and the proof is sketched. Finally, in Section 4, we illustrate some applications.

2. Proof of Theorem 1

In order to prove Theorem 1, we first recall some preliminaries.

Lemma 2 (Horn’s fixed-point theorem [13]). Let X be a Banach space, and let S₀ ⊂ S₁ ⊂ S₂ ⊂ X be convex sets, where S₀ is compact, S₁ relatively open with respect to S₂, and S₂ closed. Assume that P : S₀ → X is continuous and satisfies

()

Then, P has a fixed point in S₀.

The following is a usual definition.

Definition 3. The system (1) is said to be dissipative or ultimately bounded, if there is B₀ > 0 and for any B > 0, there are M = M(B) > 0 and L = L(B) > 0 such that for |x₀ | ≤ B,

()

where x(t, x₀) denotes the solution of (1) with the initial value x(0) = x₀.

For the affine-periodic system (1), we have the following.

Definition 4. The system (1) is said to be Q-affine-dissipative, if there is B₀ > 0 and for any B > 0, there are M = M(B) > 0 and L = L(B) > 0 such that

()

whenever |x₀ | ≤ B.

Proof of Theorem 1. Define the map P : ℝⁿ → ℝⁿ by

()

and set

()

where

()

By uniqueness and the affine periodicity of f(t, x), Q^−mx(t + mT, x₀) is still the solution of (1) for each

. Therefore,

()

It follows from (8) that

()

Thus, Horn’s fixed-point theorem implies that P has a fixed point

in S₀; that is,

. Also, uniqueness yields

()

This completes the proof of Theorem 1.

3. A Version to Functional Differential Equations

Consider the functional differential equation (FDE)

()

where F : ℝ¹ × ℂ → ℝⁿ is continuous, takes any bounded set in ℂ to a bounded set in ℝⁿ, and ensures the uniqueness of solutions with respect to initial values, where ℂ = ℂ([−r, 0], ℝⁿ), x_t(s) = x(t + s), and s ∈ [−r, 0]. Moreover, F is Q-affine-periodic; that is,

()

Definition 5. The system (15) is said to be Q-affine-dissipative; if there is B₀ > 0 and for any B > 0, there are M = M(B) > 0 and L = L(B) > 0 such that

()

whenever ∥φ∥ = max _[−r,0] |φ(s)| ≤ B; here, x(t, φ) denotes the solution of (15) at initial value x₀ = φ.

We are in position to state another main result.

Theorem 6. If the system (15) is Q-affine-periodic-dissipative, then it admits a Q-affine-periodic solution x(t); that is,

()

Proof. Define the map P : ℂ → ℂ by

()

and set

()

where

()

where N = [L(B₁) + r] + 2. Then, (17) and the constructions imply that

()

Hence, P has a fixed point φ_* ∈ S₀ via Horn’s theorem. The uniqueness implies that x(t, φ_*) is the desired affine-periodic solution of (15). The proof is complete.

4. Some Applications

First, we observe a simple example to show the meanings of affine-periodic solutions.

Example 7. Consider the equation

()

Put f(t, x) = −2x + e^−t. The general solution of (24) is

()

Obviously, for given τ > 0,

()

and any solution x(t) satisfies

()

which implies that (24) is e^−τ-periodic-dissipative. By Theorem 1, (24) has an e^−τ-affine-periodic solution. This solution is just x(t) = e^−t and different from the usual periodic solutions!

As usual, Lyapunov’s method is flexible in studying the existence of affine-periodic solutions. The following results illustrate applications in this aspect.

Theorem 8. Assume that there exists a Lyapunov’s function such that

(i)
V(t, x) is of ℂ¹;
(ii)
V′(t, x)≤−W(t, x), |x | ≥ M > 0, where W(t, x) is continuous in , and W(t, x) ≥ α > 0, |x | ≥ M;
(iii)
Uniformly in t,
()

Then, the system (1) has a Q-affine-periodic solution.

Proof. Let x(t, x₀) denote the solution of (1) with the initial value x(0) = x₀. Put

()

By assumption (iii), G is bounded and closed. In the following, we will prove that for each B > 0, there are M = M(B) > 0 and N = N(B) > 0 such that

()

whenever |x₀ | ≤ B.

In fact, given that x₀ ∈ ℝⁿ, |x₀ | > M implies on the maximal interval [0, L) that |x(t, x₀)| > M; we have

()

This shows that there is t₁ ∈ (0, ∞) such that

()

Note that

()

which together with the construction of G yields

()

If |x₀ | < M, and there is a

such that

()

then we also have

()

Of course, in case of |x₀ | = M, we have

()

Taking these cases into account, we choose

()

Now, the existence of affine-periodic solutions is an immediate consequence. The proof is complete.

Theorem 9. Assume that

()

where

satisfies

()

Then, (1) has an affine-periodic solution.

Proof. Let

()

Then,

()

By assumption,

, there is t₁ ∈ (0, ∞) such that

()

Thus,

()

By Theorem 1, (1) has an affine-periodic solution. This finishes the proof.

Example 10. Consider the system

where β ≥ 0; x ∈ ℂⁿ; Θ = (θ₁, θ₂, …, θ_n) ^T, θ_i > 0, i = 1,2, …, n. Let

()

Then

()

In the following, we only consider the case (*)₋. Otherwise, set t → −t for (*)₊. Take V(t, x) = (1/2) | x|². Notice that for

()

Hence, by Theorem 8, (*)₋ has a Q-affine T-periodic solution. Now, if leting p/q be a reduced fraction and θ_iT = p/q, i = 1,2, …, n, then the Q-affine T-periodic solutions are just q-subharmonic ones; if ΘT ∈ ℚⁿ (the set of rational vectors), then there is a K such that these affine T-periodic solutions are K-periodic ones; if ΘT ∈ ℝⁿ∖ℚⁿ, then these solutions are quasiperiodic ones with frequency ΘT.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grants nos. 11171132 and 11201173).

References

1 Arino O. A., Burton T. A., and Haddock J. R., Periodic solutions to functional differential equations, Proceedings of the Royal Society of Edinburgh A. (1985) 101, 253–271, https://doi.org/10.1017/S0308210500020813, ZBL0582.34077.
10.1017/S0308210500020813
Web of Science® Google Scholar
2 Burton T. A. and Zhang S., Unified boundedness, periodicity, and stability in ordinary and functional differential equations, Annali di Matematica Pura ed Applicata. (1986) 145, no. 1, 129–158, 2-s2.0-0001013306, https://doi.org/10.1007/BF01790540, ZBL0626.34038.
10.1007/BF01790540
Web of Science® Google Scholar
3 Chu J., Sun J., and Wong P. J. Y., Existence for singular periodic problems: a survey of recent reaults, Abstract and Applied Analysis. (2013) 17, 420835, https://doi.org/10.1155/2013/420835.
10.1155/2013/420835
Google Scholar
4 Ergoren H. and Tunc C., Anti-periodic solutions for a class of fourth-order nonlinear differential equations with multiple deviating arguments, Azerbaijan Journal of Mathematics. (2013) 3, no. 1, 111–126, ZBL06165017.
Google Scholar
5 Gao H., Li Y., and Zhang B., A fixed point theorem of Krasnoselskii-Schaefer type and its applications in control and periodicity of integral equations, Fixed Point Theory. (2011) 12, no. 1, 91–112, 2-s2.0-79952551528, ZBL05871149.
Web of Science® Google Scholar
6 Hale J. K. and Lopes O., Fixed point theorems and dissipative processes, Journal of Differential Equations. (1973) 13, no. 2, 391–402, 2-s2.0-0000266374, https://doi.org/10.1016/0022-0396(73)90025-9, ZBL0256.34069.
10.1016/0022-0396(73)90025-9
Web of Science® Google Scholar
7 Küpper T., Li Y., and Zhang B., Periodic solutions for dissipative-repulsive systems, Tohoku Mathematical Journal. (2000) 52, no. 3, 321–329, 2-s2.0-0034259098, https://doi.org/10.2748/tmj/1178207816, ZBL0971.34061.
10.2748/tmj/1178207816
Web of Science® Google Scholar
8 Li Y., Zhou Q. D., and Lu X. R., Periodic solutions and equilibrium states for functional differential inclusions with nonconvex right-hand side, Quarterly of Applied Mathematics. (1997) 55, no. 1, 57–68, 2-s2.0-0031103712, ZBL0870.34019.
10.1090/qam/1433752
Web of Science® Google Scholar
9 Makay G., Periodic solutions of dissipative functional differential equations, Tohoku Mathematical Journal. (1994) 46, 417–426.
10.2748/tmj/1178225721
Web of Science® Google Scholar
10 Seidman T. I. and Klein O., Periodic solutions of isotone hybrid systems, Discrete Contin, Dynamic Systems Service B. (2013) 18, no. 2, 483–493.
Google Scholar
11 Yoshizawa T., Stability Theory by Liapunov′s Second Method, 1966, The Mathematical Society of Japan, Tokyo, Japan.
Google Scholar
12 Yuan Y. and Guo Z., On the existence and stability of periodic solutions for a nonlinear neutral functional differential equation, Abstract and Applied Analysis. (2013) 2013, 8, 175479, https://doi.org/10.1155/2013/175479, ZBL06209193.
10.1155/2013/175479
Google Scholar
13 Horn W. A., Some fixed point theorems for compact maps and flows in Banach spaces, Transactions of the American Mathematical Society. (1970) 149, 391–404, ZBL0201.46203.
10.2307/1995402
Web of Science® Google Scholar

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Affine-Periodic Solutions for Dissipative Systems

Abstract

1. Introduction

2. Proof of Theorem 1

3. A Version to Functional Differential Equations

4. Some Applications

Conflict of Interests

Acknowledgment

References

References

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Affine-Periodic Solutions for Dissipative Systems

Abstract

1. Introduction

2. Proof of Theorem 1

3. A Version to Functional Differential Equations

4. Some Applications

Conflict of Interests

Acknowledgment

References

References

Related

Information