Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 43, pp. 1–11.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

µ PSEUDO ROTATING-PERIODIC SOLUTIONS FOR

DIFFERENTIAL EQUATIONS

DANDAN LI, JIAYIN DU

Abstract. In this article, we combine rotating periodic functions with µ er-
godic functions to obtain a new class of functions called µ pseudo rotating

periodic functions. Then we study the existence and uniqueness of µ pseudo

rotating periodic solutions for linear systems, semi-linear systems, and non-
linear systems by exponential dichotomy.

1. Introduction

Researchers pay a lot of attention to the existence and uniqueness of all kinds of
periodic solutions. Newton [16] was the first person who mentioned the existence
of periodic solutions. Then, Bohl and Esclangon gave the concept of quasi-periodic
functions. In 1931, Bohr [5] obtained the concept of almost periodic functions.
Hale [12] studied their properties. In 1955, Bochner [2, 3] gave the concept of
almost autonomous periodic functions. His student Veech [18] researched the group
properties of this kind of functions. Shen and Yi [17], Liang and Xiao [10, 11, 13, 21],
Campos and Tarallo [6], Wang and Li [19] also achieved a series of results in the field.
Recently, many researchers have studied rotating periodic functions and obtained
a series of results; see [14, 15, 20, 22, 23, 24, 25].

In the past, because of the development of ergodic perturbation theory, the study
of systems with perturbations gradually became one of the focus in the field of
dynamics. Soon after, Frechét gave the definition of asymptotically almost periodic
function.

Definition 1.1. A function f ∈ C(R+) is called asymptotically almost periodic if

f = g + φ, (1.1)

where g ∈ AP(R+), and φ ∈ C0(R+) = {h ∈ C(R+) : limt→+∞ h(t) = 0}.

However, there were some shortcomings about asymptotically almost periodic
functions. For example, the indefinite integral of asymptotically almost periodic
function may not be asymptotically almost periodic function. Hence, Zhang [26]
put forward the concept of pseudo almost periodic functions.

Definition 1.2. A function f ∈ C(R)(C(Ω×R)) is called pseudo almost periodic if

f = g + φ, (1.2)

2010 Mathematics Subject Classification. 34D09, 34K14.

Key words and phrases. µ pseudo rotating periodic functions; exponential dichotomy.
c©2020 Texas State University.

Submitted November 24, 2019. Published May 16, 2020.

1



2 D. LI, J. DU EJDE-2020/43

where g ∈ AP(R)(AP(Ω× R)), and φ ∈ C1(R)(C1(Ω× R)). Here,

C1(R) = {h ∈ C(R), lim
r→∞

1

2r

∫ r

−r
‖h(t)‖dt = 0},

C1(Ω× R) = {h ∈ C(Ω× R) : lim
r→∞

1

2r

∫ r

−r
‖h(t, x)‖dt = 0},

(1.3)

where g and φ are called the almost periodic component and the ergodic perturba-
tion, respectively, of the function of f .

Moreover, the completeness and translation invariance of pseudo almost peri-
odic function were also proved in [26]. Then, Ait [1] gave the concept of generalized
pseudo almost periodic functions by extending the ergodic perturbation part of the
function f to an unbounded function. And he proved the existence and unique-
ness of generalized pseudo almost periodic solutions for differential equations with
exponential dichotomy.

In 2006, Diagana [9] introduced a new class of functions called weighted pseudo
almost periodic functions, which generalize the classical pseudo almost periodic
functions.

Definition 1.3. A function f ∈ BC(R,X) is called weighted pseudo almost periodic
if it can be expressed as f = g + φ, where g ∈ AP(X) and φ ∈ C2(R, ρ). Here,

C2(R, ρ) = {h ∈ BC(R,X) : lim
r→∞

1

m(r, ρ)

∫ r

−r
‖h(t)‖ρ(t)dt = 0}. (1.4)

Similarly, he obtained the completeness and translation invariance of weighted
pseudo almost periodic function.

In 2012, Blot [4] gave the definition of µ pseudo almost periodic functions, which
further expanded the weighted pseudo almost periodic functions.

Definition 1.4. A function f ∈ BC(R,X) is called µ pseudo almost periodic if it
can be expressed as f = g + φ, where g ∈ AP(X), and φ ∈ C3(R, µ). Here,

C3(R, µ) = {h ∈ BC(R,X, µ) : lim
r→∞

1

µ[−r, r]

∫ r

−r
‖h(t)‖dµ(t) = 0}, (1.5)

where µ represents a measure function defined on R.

Blot et al, transformed the density function ρ of weighted pseudo almost pe-
riodic function into measure function by measure theory. They also proved the
completeness and translation invariance of µ pseudo almost periodic function by
introducing measure equivalence. In this article, we introduce a kind of pseudo
rotating-periodic functions and prove the existence uniqueness of pseudo rotating
periodic solutions.

This article is organized as follows: in section 2, we show the concept of µ pseudo
rotating periodic function and exponential dichotomy. The basic properties of the
space of µ pseudo rotating periodic function are shown in section 3. In section 4, we
prove the existence and uniqueness of µ pseudo rotating periodic solution for linear
systems, semi-linear systems, and non-linear systems. Finally, section 5 verifies the
flexibility of our conditions by an example.



EJDE-2020/43 µ PSEUDO ROTATING-PERIODIC SOLUTIONS 3

2. µ pseudo rotating periodic function

Firstly, we give some notation. B denotes the Lebesgue σ-field of R. M is the set
that consist of all positive measures µ on B satisfying µ(R) = +∞ and µ([a, b]) <
+∞, for all a, b ∈ R (a ≤ b). We denote the space of all n-order orthogonal matrices
by O(n). BC(R,Rn) denotes the Banach space of all bounded continuous functions
from R to Rn, equipped with the supremum norm ‖f‖ = supt∈R ‖f(t)‖.

Definition 2.1. A function g ∈ BC(R,Rn) is said to be (T,Q) rotating periodic if
there exist T > 0 and Q ∈ O(n) such that

g(t+ T ) = Qg(t) ∀t ∈ R.

Definition 2.2. A function g ∈ BC(R × Rn,Rn) is said to be (T,Q) rotating
periodic if the exist T > 0 and Q ∈ O(n) such that

g(t+ T, x) = Qg(t, Q−1x) ∀t ∈ R.
We denote the space of rotating periodic functions byRP(R,Rn) (RP(R×Rn,Rn)),
equipped with the supremum norm.

Remark 2.3. If Q = id or Q = − id, then a (T,Q) rotating periodic function is the
usual T -periodic function or T -anti-periodic one, respectively. If for some positive
integer k0 such that Qk0 = id, a (T,Q) rotating periodic function is a subharmonic
one. If for all positive integer k1 such that Qk1 6= id, then a (T,Q) rotating periodic
function is a quasi-periodic function.

Definition 2.4. Let µ ∈M. A function φ ∈ BC(R,Rn) is called µ-ergodic if

lim
r→∞

1

µ([−r, r])

∫ r

−r
‖φ(t)‖dµ(t) = 0.

Definition 2.5. Let µ ∈M. A function φ ∈ BC(R×Rn,Rn) is called µ-ergodic if

lim
r→∞

1

µ([−r, r])

∫ r

−r
‖φ(t, x)‖dµ(t) = 0. (2.1)

We denote the space of all such functions by E(R,Rn, µ), E(R×Rn,Rn, µ), equipped
with the supremum norm.

Remark 2.6. If µ is the Lebesgue measure, then the µ ergodic function φ is the
same as the function given by Zhang [26]. If the Radon-Nikodym derivative of µ
with respect to the Lebesgue measure on R is ρ, the µ-ergodic function is a Weighted
ergodic one by Diagana [9].

Definition 2.7. Let µ ∈ M. A function f ∈ BC(R,Rn) is said to be µ pseudo
rotating periodic if f is written in the form

f = g + φ,

where g ∈ RP(R,Rn), and φ ∈ E(R,Rn, µ).

Definition 2.8. Let µ ∈M. A function f ∈ BC(R×Rn,Rn) is said to be µ pseudo
rotating periodic if f is written in the form

f = g + φ,

where g ∈ RP(R× Rn,Rn), and φ ∈ E(R× Rn,Rn, µ).
We denote the space of such functions by PRP(R,Rn, µ) (PRP(R×Rn,Rn, µ)),

equipped with the new norm ‖f‖1 = ‖g‖+ ‖φ‖.



4 D. LI, J. DU EJDE-2020/43

Consider the system
ẋ = A(t)x, (2.2)

where A(t) is a continuous matrix on R and X(t) is a fundamental matrix of system
(2.2) satisfying X(0) = id.

Definition 2.9 ([8]). The system of differential equations (2.2) is said to possess
exponential dichotomy on R, if there exist a projection matrix P and constants
K > 1, α > 0, such that

‖X(t)PX−1(s)‖ ≤ K exp(−α(t− s)), t ≥ s,
‖X(t)(I − P )X−1(s)‖ ≤ K exp(−α(s− t)), t ≤ s.

We define the Green’s function as follows

U(t, s) =

{
X(t)PX−1(s) t ≥ s,
−X(t)(I − P )X−1(s) t ≤ s.

(2.3)

3. Basic properties

For µ ∈ M, and τ ∈ R, we define µτ (A) = µ({a+ τ : a ∈ A}) for A ∈ B. Then
we make the following hypothesis.

(H1) Let µ ∈M, there exist β > 0 and a bounded interval I such that µτ (A) ≤
βµ(A), when A ∈ B satisfies A ∩ I = ∅.

Lemma 3.1. If µ satisfies (H1), then (PRP(R,Rn, µ), ‖ · ‖1) is a Banach space.

Proof. Let {fn} be a Cauchy sequence in PRP(R,Rn, µ). For all ε > 0, there exist
N > 0 such that

‖fn(t)− fm(t)‖1 < ε (m,n > N, ∀t ∈ R),

and

fn(t) = gn(t) + φn(t) ∀t ∈ R,
fm(t) = gm(t) + φm(t) ∀t ∈ R.

where gn, gm ∈ RP(R,Rn), and φn, φm ∈ E(R,Rn, µ). So, we have

‖fn(t)− fm(t)‖1 = ‖gn(t) + φn(t)− gm(t)− φm(t)‖1
= ‖gn(t)− gm(t)‖+ ‖φn(t)− φm(t)‖
= sup

t∈R
|gn(t)− gm(t)|+ sup

t∈R
|φn(t)− φm(t)| < ε.

We deduce that

sup
t∈R
|gn(t)− gm(t)| ≤ ε ∀t ∈ R,

sup
t∈R
|φn(t)− φm(t)| ≤ ε ∀t ∈ R.

According to the definition of Cauchy sequence, we can conclude that {gn} and
{φn} are the Cauchy sequence of RP(R,Rn) and E(R,Rn, µ), respectively. Then
we use that RP(R,Rn) and E(R,Rn, µ) are Banach space [7, 4]. Hence there exist
g0(t) ∈ RP(R,Rn) and φ0(t) ∈ E(R,Rn, µ) such that

lim
t→∞

‖gn(t)− g0(t)‖ = 0, lim
t→∞

‖φn(t)− φ0(t)‖ = 0.

Let f0(t) = g0(t) + φ0(t). We deduce that f0 ∈ PRP(R,Rn, µ). So we can prove
that (PRP(R,Rn, µ), ‖ · ‖1) is a Banach space. �



EJDE-2020/43 µ PSEUDO ROTATING-PERIODIC SOLUTIONS 5

Lemma 3.2. If µ satisfies (H1), then (PRP(R × Rn,Rn, µ), ‖ · ‖1) is a Banach
space.

The proof of the above lemma is similar to that of Lemma 3.1.

Lemma 3.3. Let µ satisfies (H1). If y ∈ PRP(R,Rn, µ), f ∈ PRP(R×Rn,Rn, µ),
and f satisfies the Lipschitz condition in x, then the convolution product

(f ∗ U)(t) =

∫ +∞

−∞
U(t, s)f(s, y(s))ds (3.1)

is also a µ pseudo rotating periodic function.

Proof. Since y ∈ PRP(R,Rn, µ) and f ∈ PRP(R× Rn,Rn, µ), we have

f(t, y(t)) = f1(t, y1(t)) + f(t, y(t))− f(t, y1(t)) + f2(t, y1(t))

=: f1(t, y1(t)) + f̃(t, y(t), y1(t)),

where y1 ∈ RP(R,Rn), y2 ∈ E(R,Rn, µ), f1 ∈ RP(R × Rn,Rn), f2 ∈ E(R ×
Rn,Rn, µ). Then

(f ∗ U)(t) =

∫ +∞

−∞
U(t, s)f1(s, y1(s))ds+

∫ +∞

−∞
U(t, s)f̃(s, y(s), y1(s))ds

=: (f ∗ U)1(t) + (f ∗ U)2(t).

For f1 ∈ RP(R× Rn,Rn), it is easy to prove that (f ∗ U)1(t) ∈ RP(R,Rn).
Next we prove that (f ∗ U)2(t) is a µ ergodic function. i.e.

lim
r→∞

1

µ([−r, r])

∫ r

−r
‖(f ∗ U)2(t)‖dµ(t) = 0.

Since f satisfies the Lipschitz condition, we have a constant N such that

‖f(s, y(s))− f(s, y1(s))‖ ≤ ‖f(s, y(s))− f(s, y1(s))‖1
≤ N‖y(s)− y1(s)‖1
= N‖y2(t)‖.

Hence, we deduce that∫ r

−r
‖
∫ +∞

−∞
U(t, s)f̃(s, y(s), y1(s))ds‖dµ(t)

≤ N
∫ r

−r

∫ +∞

−∞
‖U(t, s)‖‖y2(s)‖ ds dµ(t)

+

∫ r

−r

∫ +∞

−∞
‖U(t, s)‖‖f2(s, y1(s))‖ ds dµ(t)

=: Z1(t) + Z2(t).

According to (2.3) and definition 2.5, we conclude that

Z1(t) ≤
∫ r

−r

∫ t

−∞
KN exp(−α(t− s))‖y2(s)‖ ds dµ(t)

+

∫ r

−r

∫ +∞

t

KN exp(α(t− s))‖y2(s)‖ ds dµ(t)

≤
∫ 0

−∞

∫ r

−r
KN‖y2(s+ t)‖dµ(t)ds



6 D. LI, J. DU EJDE-2020/43

+

∫ +∞

0

∫ r

−r
KN‖y2(s+ t)‖dµ(t)ds

= KN

∫ +∞

−∞

∫ r

−r
‖y2(s+ t)‖dµ(t)ds.

For y2 ∈ E(R,Rn, µ), Joël Blot [4] proved y2(t+ s) ∈ E(R,Rn, µ). Using Lebesgue’s
dominated convergence theorem, we deduce that

lim
r→∞

Z1(t)

µ([−r, r])
≤

∫ +∞

−∞
lim
r→∞

KN

µ([−r, r])

∫ r

−r
‖y2(s+ t)‖dµ(t)ds = 0.

Similarly, limr→∞
Z2(t)

µ([−r,r]) = 0. Therefore, we conclude that (f ∗ U)2(t) is a µ

ergodic function. Finally, we obtain that (f ∗ U)(t) ∈ PRP(R,Rn, µ). �

4. Existence and uniqueness

Theorem 4.1. Consider the linear system

ẋ = A(t)x+ f(t), (4.1)

where A(t) ∈ PR(R,Rn), f(t) ∈ PRP(R,Rn, µ). If the linear homogeneous system
ẋ = A(t)x satisfies exponential dichotomy, then (4.1) has a unique µ pseudo rotating
periodic solution.

Proof. Since f(t) ∈ PRP(R,Rn, µ), we have a bounded solution (see [8])

x(t) =

∫ +∞

−∞
U(t, s)g(s)ds+

∫ +∞

−∞
U(t, s)φ(s)ds =: x1(t) + x2(t),

where g ∈ RP(R,Rn), φ ∈ E(R,Rn, µ). It follows from g(s) ∈ RP(R,Rn) that
x1(t) ∈ RP(R,Rn).

Then we will prove that x2(t) is a µ ergodic function. By (2.3) and definition
2.5, we have∫ r

−r
‖x2(t)‖dµ(t)

≤
∫ r

−r

∫ t

−∞
K exp(−α(t− s))‖φ(s)‖ ds dµ(t)

+

∫ r

−r

∫ +∞

t

K exp(−α(s− t))‖φ(s)‖ ds dµ(t)

≤ K
∫ r

−r

∫ 0

−∞
‖φ(s+ t)‖ ds dµ(t) +K

∫ r

−r

∫ +∞

0

‖φ(s+ t)‖ ds dµ(t)

= K

∫ 0

−∞

∫ r

−r
‖φ(s+ t)‖dµ(t)ds+K

∫ +∞

0

∫ r

−r
‖φ(s+ t)‖dµ(t)ds,

by Fubini’s theorem.
The translation invariance of E(R,Rn, µ) was proved in [4], i.e. φ(s + t) ∈

E(R,Rn, µ). So we can obtain that limr→+∞
x2(t)

µ([−r,r]) = 0 by Lebesgue’s domi-

nated convergence theorem. Then system (4.1) has a unique µ pseudo rotating
periodic solution. �



EJDE-2020/43 µ PSEUDO ROTATING-PERIODIC SOLUTIONS 7

Theorem 4.2. Consider the semi-linear system

ẋ = A(t)x+ f(t, x), (4.2)

where A(t) ∈ RP(R,Rn), f(t, x) ∈ PRP(R× Rn,Rn, µ), and it satisfies the Lips-
chitz condition in x. If the linear homogeneous system ẋ = A(t)x satisfies exponen-
tial dichotomy and there is a Lipschitz constant N > 0, such that 2KN

α < 1, then
system (4.2) has a unique µ pseudo rotating periodic solution.

Proof. Firstly, for all y(t) ∈ PRP(R,Rn, µ), we consider the system

ẋ = A(t)x+ f(t, y(t)),

which has a bounded solution

x(t) =

∫ +∞

−∞
U(t, s)f(s, y(s))ds

and x(t) ∈ PRP(R,Rn, µ) by lemma 3.3. Then we define the mapping H(y)(t) :
PRP(R,Rn, µ)→ PRP(R,Rn, µ) by

H(y)(t) =

∫ +∞

−∞
U(t, s)f(s, y(s))ds. (4.3)

For all y1(t), y2(t) ∈ PRP(R,Rn, µ), we deduce that

‖H(y1)(t)−H(y2)(t)‖1 ≤
∫ +∞

−∞
‖U(t, s)‖1N‖y1(s)− y2(s)‖1ds

≤ KN
∫ t

−∞
exp(−α(t− s))‖y1(s)− y2(s)‖1ds

+KN

∫ +∞

t

exp(α(t− s))‖y2(s)− y1(s)‖1ds

≤ 2KN

α
‖y1(s)− y2(s)‖1.

Since 2KN
α < 1, the mapping H(y)(t) is a strict contraction. Using Banach fixed

point theorem, there exists a unique y(t) ∈ PRP(R,Rn, µ) such that H(y)(t) =
y(t). So system (4.2) has a unique µ pseudo rotating periodic solution. �

In the following, we replace the Lipschitz condition by the linear growth condi-
tion. Then the existence of µ pseudo rotating wave solutions for semi-linear systems
is also going to be proved.

Corollary 4.3. Consider system (4.2), where A(t) is an rotating periodic ma-
trix, f(t, x) ∈ PRP(R × Rn,Rn, µ), and f is uniformly continuous function in
x. Suppose that the homogeneous linear system ẋ = A(t)x satisfies an exponential
dichotomy, and the function f satisfies linear growth condition, i. e.

‖f(t, x)‖1 ≤ a‖x‖1 + b (t ∈ R, x ∈ Rn), (4.4)

where a > 0, 2Kaα < 1. Then system (4.2) has a µ pseudo rotating periodic solution.

Proof. Let G := {y ∈ PRP(R,Rn, µ), ‖y‖1 ≤ N1}, where N1 = 2Kb
α−2Ka . We can

deduce that G is a closed convex subset of the space of Banach. For all y ∈ G, we
have

‖H(y)(t)‖1 = ‖
∫ +∞

−∞
U(t, s)f(s, y(s))ds‖1



8 D. LI, J. DU EJDE-2020/43

≤
∫ +∞

−∞
‖U(t, s)‖1‖f(s, y(s))‖1ds

≤ (a‖y‖1 + b)(

∫ t

−∞
K exp(−α(t− s))ds+

∫ +∞

t

K exp(α(t− s))ds)

≤ (aN1 + b)
2K

α
≤ N1.

By using the uniform continuity of f , for all ε > 0, there exist δ > 0 such that
‖y1 − y2‖1 < δ, and we have

‖f(t, y1)− f(t, y2)‖1 < ε.

From the inequality

‖H(y1)(t)−H(y2)(t)‖1

≤ ‖
∫ +∞

−∞
U(t, s)f(s, y1(s))ds−

∫ +∞

−∞
U(t, s)f(s, y2(s))ds‖1

≤
∫ t

−∞
‖X(t)PX−1(s)‖1‖f(s, y1(s))− f(s, y2(s))‖1ds

+

∫ +∞

t

‖X(t)(I − P )X−1(s)‖1‖f(s, y2(s))− f(s, y1(s))‖1ds

≤
∫ t

−∞
K exp(−α(t− s))εds+

∫ +∞

t

K exp(α(t− s))εds

≤ 2Kε

α
,

we deduce that H(y)(t) is uniformly continuous in x and satisfies (4.2). Then we
have

‖Ḣ(y)(t)‖1 = ‖A(t)‖1‖H(y)(t)‖1 + ‖f(t, x)‖1 ≤ (a+ sup
t∈R
|A(t)|)M1 + b.

Therefore, the mapping H(y)(t) is equicontinuous. By using Arzela-Ascoli theorem,
there exist a subsequence of {fk(t)} ⊂ {fn(t)} ∈ G such that

lim
k→∞

fk(t) = y0(t),

where y0(t) ∈ G. We can prove that H(y)(t) has fixed point in the space G by
Schauder fixed point theorem. Then the system (4.2) has a µ pseudo rotating
periodic solution. �

Theorem 4.4. Consider the nonlinear system

ẋ = f(t, x), (4.5)

where f ∈ PRP(R×Rn,Rn, µ), and f is a continuous differentiable function in x.
Suppose

ẋ =

∫ 1

0

∂f

∂x
(θy)dθ · x, y ∈ PRP(R,Rn, µ) (4.6)

satisfies the exponential dichotomy, i.e.

‖Φ(t, y)PyΦ−1(s, y)‖ ≤ K exp(−α(t− s)) t ≥ s,
‖Φ(t, y)(I − Py)Φ−1(s, y)‖ ≤ K exp(−α(s− t)) t ≤ s,



EJDE-2020/43 µ PSEUDO ROTATING-PERIODIC SOLUTIONS 9

where Φ(t, y) is a fundamental matrix of system (4.6), and satisfies Φ(0, x) = id.
Then (4.5) has a unique µ pseudo rotating periodic solution.

Proof. Firstly, we prove uniqueness. Suppose x, y are µ pseudo rotating wave solu-
tions of the system (4.5). Let u = x− y. Then we have

u̇ = ẋ− ẏ = f(t, x)− f(t, y)

=

∫ 1

0

∂f

∂x
(θx+ (1− θ)y)dθ · (x− y)

=

∫ 1

0

∂f

∂x
(y + θu)dθ · u.

Because (4.6) satisfies the exponential dichotomy, we get a unique solution u = 0,
i.e. x = y.

Next we prove existence. It is easy to obtain

ẋ = f(t, x) =

∫ 1

0

∂f

∂x
(θx)dθ · x+ f(t, 0).

For all y ∈ PRP(R,Rn, µ), we consider the equation

ẋ =

∫ 1

0

∂f

∂x
(θy)dθ · x+ f(t, 0), (4.7)

which has a solution

xy(t) =

∫ t

−∞
Φ(t, y)PyΦ−1(s, y)f(s, 0)ds

−
∫ +∞

t

Φ(t, y)(I − Py)Φ−1(s, y)f(s, 0)ds.

Because f ∈ PRP(R×Rn,Rn, µ), we easily prove that xy ∈ PRP(R×Rn,Rn, µ).
For all s ∈ R, there exists a constant M1 such that ‖f(s, 0)‖1 ≤ M1. Then we
consider the set

D = {h ∈ PRP(R× Rn,Rn, µ), ‖h‖1 ≤M},

where M = 2KM1

α + 1. For all y ∈ D, we define the mapping T[y](t) = xy(t). We
have

‖T[y](t)‖1 ≤
∫ t

−∞
‖Φ(t, y)PyΦ−1(s, y)‖1‖f(s, 0)‖1ds

+

∫ ∞
t

‖Φ(t, y)(I − Py)Φ−1(s, y)‖1‖f(s, 0)‖1ds

≤
∫ t

−∞
K exp(−α(t− s))M1ds+

∫ ∞
t

K exp(α(t− s))M1ds

≤ 2KM1

α
≤M.

Then T[y] : D → D and uniformly bounded with respect to y. Because T[y] satisfies
the system (4.7), we have

‖Ṫ[y](t)‖1 ≤ ‖
∫ 1

0

∂f

∂x
(θy)dθ‖1‖T[y](t)‖1 + ‖f(t, 0)‖1



10 D. LI, J. DU EJDE-2020/43

≤
∫ 1

0

‖∂f
∂x

(θy)‖1dθM +M1.

It follows from f is a continuous differentiable function in x that T[y] is equi-
continuous. Hence, (4.5) has µ pseudo rotating periodic solution by Schauder’s
fixed point theorem. �

5. Example

Consider the equation

ẋ = −3x+ exp(−t) + x2 exp(−t), (5.1)

for all t ∈ R+, x ∈ Ω, where Ω(⊆ R) is a bounded set.
We will prove that (5.1) has a µ pseudo rotating periodic solution. Let

f(t, x) = (−3x+ exp(−t)) + x2 exp(−t) =: g(t, x) + φ(t, x).

For a given τ ∈ R, let

g(t+ τ, x) = −3x+ exp(−(t+ τ))

= exp(−τ)(−3 exp(τ)x+ exp(−t))
= exp(−τ)g(t, exp(τ)x).

Therefore, g(t, x) ∈ PR(R+ × Ω,Ω).
If the Radon-Nikodym derivative ρ is defined by ρ(t) = 1, then

lim
r→+∞

∫ r
0
x2 exp (−t)dµ(t)

µ([0, r])
= lim
r→+∞

∫ r
0
x2 exp (−t)dt∫ r

0
1dt

= lim
r→+∞

x2(1− exp (−r))
r

= 0.

Hence, f ∈ PRP(R+ × Ω,Ω, µ). By theorem 4.4, equation (5.1) has a unique µ
pseudo rotating periodic solution.

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Dandan Li

Department of Mathematics, Northeast Normal University, Changchun 130024, China
Email address: lidandan0104@163.com

Jiayin Du (corresponding author)
Department of Mathematics, Jilin University, Changchun, 130012, China

Email address: dujiayinmath@163.com


	1. Introduction
	2.  pseudo rotating periodic function
	3. Basic properties
	4. Existence and uniqueness
	5. Example
	References