(ω,c)-Periodic Solutions to Fractional Differential Equations with Impulses

and

JinRong Wang

Department of Mathematics, Guizhou University, Guiyang 550025, China

Author to whom correspondence should be addressed.

Axioms2022,11(3), 83;https://doi.org/10.3390/axioms11030083

Submission received: 19 January 2022 /Revised: 18 February 2022 /Accepted: 21 February 2022 /Published: 22 February 2022

(This article belongs to the Special IssueMathematical Analysis Is the Theoretical Basis of a Number of Mathematical Disciplines)

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Abstract

This paper deals with the

(ω, c)

-periodic solutions to impulsive fractional differential equations with Caputo fractional derivative with a fixed lower limit. Firstly, a necessary and sufficient condition of the existence of

(ω, c)

-periodic solutions to linear problem is given. Secondly, the existence and uniqueness of

(ω, c)

-periodic solutions to semilinear problem are proven. Lastly, two examples are given to demonstrate our results.

Keywords:

(ω,c)-periodic solutions;fractional differential equation;impulse

MSC:

26A33

1. Introduction

Alvarez et al. [1] introduced a new concept of

(ω, c)

-periodic functions: a continuous function

f : R \to X

, whereX is a complex Banach space, is

(ω, c)

-periodic if

f (t + ω) = c f (t)

holds for all

t \in R

, where

ω > 0, c \ C \in {0}

. Then, Alvarez et al. [2] proved the existence and uniqueness of

(N, λ)

-periodic solutions to a a class of Volterra difference equations. For more research on

(ω, c)

-period systems, we refer the readers to [3,4,5,6].

In recent years, impulsive fractional differential equations have attracted more and more scholars’ attentions. For the existence of solutions and control problems, we refer to [7,8,9,10,11]. Recently, Fečkan et al. [12] proved the existence of the periodic solutions of impulsive fractional differential equations. However, to our knowledge, the existence of

(ω, c)

-periodic solutions of impulsive fractional differential equations has not been studied. Motivated by [1,7,12,13,14], we study the following impulsive fractional differential equations with fixed lower limits

\{\begin{matrix} ^{c} D_{t_{0}}^{q} u (t) = f (t, u (t)), q \in (0, 1), t \neq t_{k}, t \in [t_{0}, \infty), \\ u (t_{k}^{+}) = u (t_{k}^{-}) + Δ_{k}, k \in N, \end{matrix}

where

^{c} D_{t_{0}}^{q} u (t)

is the Caputo fractional derivative with the lower time at

t_{0}

, and for any

k \in N

t_{k} < t_{k + 1}

{lim}_{k \to \infty} t_{k} = \infty

In this paper, we deal with the existence of

(ω, c)

-periodic solutions impulsive fractional differential equations with fixed lower limit. We first study the existence of

(ω, c)

-periodic solutions to the linear problem, i.e.,

f (t, u) = ρ u

. Then, we prove the existence of

(ω, c)

-periodic solutions to the semilinear problem. Finally, we give two examples to illustrate our results.

2. Preliminaries

We introduce a Banach space

P C (R, R^{n}) = {x : R \to R^{n} : x \in C ((t_{k}, t_{k + 1}], R^{n}), and x (t_{k}^{-}) = x (t_{k}), x (t_{k}^{+}) exists \forall k \in N}

endowed with the norm

{∥ x ∥}_{\infty} = {sup}_{t \in R} ∥ x (t) ∥

Definition 1.

(see [15]) Let

n \in N^{+}

and u be a n time differentiable function. The Caputo fractional derivative of order

α > 0

with the lower limit zero for u is given by

^{c} D_{0}^{α} u (t) = \frac{1}{Γ (n - α)} \int_{0}^{t} {(t - s)}^{n - α - 1} u^{(n)} (s) d s, n - 1 < α \leq n .

Lemma 1.

Assume that

f : R \times R^{n}

is continuous. A solution

u \in P C (R, R^{n})

of the following impulsive fractional differential equations with fixed lower limit

\{\begin{matrix} ^{c} D_{t_{0}}^{q} u (t) = f (t, u (t)), q \in (0, 1), t \neq t_{k}, t \in [t_{0}, \infty), \\ u (t_{k}^{+}) = u (t_{k}^{-}) + Δ_{k}, k \in N, \\ u (t_{0}) = u_{t_{0}}, \end{matrix}

(1)

is given by

u (t) = u (t_{0}) + \frac{1}{Γ (q)} \int_{t_{0}}^{t} {(t - τ)}^{q - 1} f (τ, u (τ)) d τ + \sum_{t_{0} < t_{i} < t} Δ_{i}, t \geq t_{0} .

(2)

Proof.

From Lemma 3.2 in [7], a solutionu of Equation (1) is given by

u (t) = u (t_{0}) + \frac{1}{Γ (q)} \int_{t_{0}}^{t} {(t - τ)}^{q - 1} f (τ, u (τ)) d τ + \sum_{i = 1}^{k} Δ_{i}, t \in (t_{k}, t_{k + 1}] .

(3)

Using

\sum_{i = 1}^{k} Δ_{i} = \sum_{t_{0} < t_{i} < t} Δ_{i}, \forall t \in (t_{k}, t_{k + 1}],

we get that (3) is equivalent to

u (t) = u (t_{0}) + \frac{1}{Γ (q)} \int_{t_{0}}^{t} {(t - τ)}^{q - 1} f (τ, u (τ)) d τ + \sum_{t_{0} < t_{i} < t} Δ_{i}

(4)

(t_{k}, t_{k + 1}]

. Using the arbitrariness ofk, we obtain that (4) holds on

⋃_{k = 1}^{\infty} (t_{k}, t_{k + 1}]

. Since (4) is independent ofk, we obtain that (2) holds on

[t_{0}, \infty)

. □

Definition 2.

(see [16], Theorem 2.4) A solution

u \in P C (R, R^{n})

of following linear impulsive fractional differential equations with fixed lower limit

\{\begin{matrix} ^{c} D_{t_{0}}^{q} u (t) = ρ u (t), ρ \in R, q \in (0, 1), t \neq t_{k}, t \in [t_{0}, \infty), \\ u (t_{k}^{+}) = (1 + α_{k}) u (t_{k}^{-}), k \in N, \\ u (t_{0}) = u_{t_{0}}, \end{matrix}

is given by

u (t) = \{\begin{matrix} u_{t_{0}} E_{q} (ρ {(t - t_{0})}^{q}), t \in [t_{0}, t_{1}] \\ u_{t_{0}} \prod_{i = 1}^{k} (1 + α_{i} E_{q} (ρ {(t_{i} - t_{0})}^{q})) E_{q} (ρ {(t - t_{0})}^{q}), t \in (t_{k}, t_{k + 1}], k \in N, \end{matrix}

where

E_{q} (\cdot)

is the Mittag–Leffler function.

Definition 3.

(see [1]) Let

c \in C \ {0}

ω > 0

, X denote a complex Banach space with norm

∥ \cdot ∥

. A continuous function

f : R \to X

is said to be

(ω, c)

-periodic if

f (t + ω) = c f (t)

for all

t \in R

Lemma 2.

(see [3], Lemma 2.2) Set

Φ_{ω, c} : = {u : u \in P C (R, R^{n})} and u (\cdot + ω) = c u (\cdot)}

. Then,

u \in Φ_{ω, c}

if, and only if, it holds

\begin{matrix} u (ω) = c u (0) . \end{matrix}

(5)

3. ( $(ω, c)$ )-Periodic Solutions to Linear Problem

Set

t_{0} = 0

, we consider the following linear impulsive fractional differential equation with fixed lower limit

\{\begin{matrix} ^{c} D_{0}^{q} u (t) = ρ u (t), ρ \in R, q \in (0, 1), t \neq t_{k}, t \in [0, \infty), \\ u (t_{k}^{+}) = (1 + α_{k}) u (t_{k}^{-}), k \in N, \\ u (0) = u_{0} . \end{matrix}

(6)

Theorem 1.

Assume that there exists a constant

N \in N

such that

ω = t_{N}, t_{k + N} = t_{k} + ω, \forall k \in N, and α_{i + N} = α_{i}, \forall i \in N .

Then, the linear impulsive fractional differential Equation (6)has a

(ω, c)

-periodic solution

u \in Φ_{ω, c}

if, and only if

u_{0} (c - \prod_{i = 1}^{N} (1 + α_{i} E_{q} (ρ t_{i}^{q})) E_{q} (ρ ω^{q})) = 0 .

(7)

Proof.

“⇒” If (6) has a

(ω, c)

-periodic solution

u \in Φ_{ω, c}

, i.e.,

u (\cdot + ω) = c u (\cdot)

, then

u (ω) = c u (0)

, i.e.,

u_{0} \prod_{i = 1}^{N} (1 + α_{i} E_{q} (ρ t_{i}^{q})) E_{q} (ρ ω^{q}) = c u_{0}

which implies that (7) holds.

“⇐” It follows from Definition 2 that Equation (7) has a solutionu given by

u (t) = \{\begin{matrix} u_{0} E_{q} (ρ t^{q}), t \in [0, t_{1}] \\ u_{0} \prod_{i = 1}^{k} (1 + α_{i} E_{q} (ρ t_{i}^{q})) E_{q} (ρ t^{q}), t \in (t_{k}, t_{k + 1}], k \in N . \end{matrix}

If (7) holds, we obtain

u (t_{N}) = u (ω) = c u_{0}

. Now, we prove that the solution

u \in Φ_{ω, c}

Case 1: For

t \in (0, t_{1}]

, we have

t + ω \in (t_{N}, t_{N + 1}]

, then

\begin{matrix} u (t + ω) & = & u_{t_{N}} E_{q} (ρ {(t + ω - t_{N})}^{q}) \\ = & u_{t_{N}} E_{q} (ρ t^{q}) = c u_{0} E_{q} (ρ t^{q}) = c u (t) . \end{matrix}

Case 2: For

t \in (t_{k}, t_{k + 1}]

k \in N

, we have

t + ω \in (t_{k + N}, t_{k + N + 1}]

, then

\begin{matrix} u (t + ω) & = & u_{t_{N}} \prod_{i = 1}^{k} (1 + α_{i + N} E_{q} (ρ {(t_{i + N} - t_{N})}^{q})) E_{q} (ρ {(t + ω - t_{N})}^{q}) \\ = & u_{t_{N}} \prod_{i = 1}^{k} (1 + α_{i} E_{q} (ρ t_{i}^{q})) E_{q} (ρ t^{q}) \\ = & c u_{0} \prod_{i = 1}^{k} (1 + α_{i} E_{q} (ρ t_{i}^{q})) E_{q} (ρ t^{q}) \\ = & c u (t) . \end{matrix}

So, we obtain that (6) has a

(ω, c)

-periodic solution

u \in Φ_{ω, c}

. □

4. $(ω, c)$ -Periodic Solutions to Semilinear Problem

Set

t_{0} = 0

, we consider the

(ω, c)

-periodic solutions of following impulsive fractional differential equations with fixed lower limit

\{\begin{matrix} ^{c} D_{0}^{q} u (t) = f (t, u (t)), q \in (0, 1), t \neq t_{k}, t \in [0, \infty), \\ u (t_{k}^{+}) = u (t_{k}^{-}) + Δ_{k}, k \in N, \\ u (0) = u_{0} . \end{matrix}

(8)

We assume the following conditions:

(I): $f : R \times R^{n} \to R^{n}$ is continuous and
$f (t + ω, c u) = c f (t, u), \forall t \in R, \forall u \in R^{n} .$
(II): There exists a constant $A > 0$ such that
$∥ f (t, u) - f (t, v) ∥ \leq A ∥ u - v ∥, \forall t \in R, \forall u, v \in R^{n} .$
(III): There exist constant $B > 0$ , $P > 0$ such that
$∥ f (t, u) ∥ \leq B ∥ u ∥ + P, \forall t \in R, \forall u \in R^{n} .$
(IV): $Δ_{k} \in R^{n}$ and there exists a constant $M \in N$ such that $ω = t_{M}$ , $t_{k + M} = t_{k} + ω$ and $Δ_{k + M} = Δ_{k}$ hold for any $k \in N$ .

Lemma 3.

Suppose that conditions

(I)

(I V)

hold and

c \neq 1

. Then, the solution

u \in Ψ : = P C ([0, ω], R^{n})

of Equation (8)satisfying (5)is given by

\begin{matrix} u (t) & = & {(c - 1)}^{- 1} \frac{1}{Γ (q)} \int_{0}^{ω} {(ω - τ)}^{q - 1} f (τ, u (τ)) d τ + \frac{1}{Γ (q)} \int_{0}^{t} {(t - τ)}^{q - 1} f (τ, u (τ)) d τ \\ + {(c - 1)}^{- 1} \sum_{k = 1}^{M} Δ_{k} + \sum_{0 < t_{k} < t} Δ_{k} t \in [0, ω] . \end{matrix}

Proof.

It follows from (2) that the solution

u \in P C ([0, ω], R^{n})

is given by

u (t) = u (0) + \frac{1}{Γ (q)} \int_{0}^{t} {(t - τ)}^{q - 1} f (τ, u (τ)) d τ + \sum_{t_{0} < t_{k} < t} Δ_{k}, t \in [0, ω] .

(9)

So we get

u (ω) = u (0) + \frac{1}{Γ (q)} \int_{0}^{ω} {(ω - τ)}^{q - 1} f (τ, u (τ)) d τ + \sum_{t_{0} < t_{k} < ω} Δ_{k} = c u_{0}

which is equivalent to

u_{0} = {(c - 1)}^{- 1} (\frac{1}{Γ (q)} \int_{0}^{ω} {(ω - τ)}^{q - 1} f (τ, u (τ)) d τ + \sum_{t_{0} < t_{k} < ω} Δ_{k}) .

(10)

By (9) and (10), we obtain

\begin{matrix} u (t) & = & {(c - 1)}^{- 1} \frac{1}{Γ (q)} \int_{0}^{ω} {(ω - τ)}^{q - 1} f (τ, u (τ)) d τ + \frac{1}{Γ (q)} \int_{0}^{t} {(t - τ)}^{q - 1} f (τ, u (τ)) d τ \\ + {(c - 1)}^{- 1} \sum_{k = 1}^{M} Δ_{k} + \sum_{0 < t_{k} < t} Δ_{k} . \end{matrix}

The proof is finished. □

Theorem 2.

Suppose that conditions

(I)

(I I)

(I V)

hold and

c \neq 1

. If

0 < \frac{A ω^{q} {(| c - 1 |}^{- 1} + 1)}{Γ (q + 1)} < 1

, then the impulsive fractional differential Equation (8)has a unique

(ω, c)

-periodic solution

u \in Φ_{ω, c}

. Furthermore, we have

\begin{matrix} {∥ u ∥}_{\infty} \leq \frac{μ ω^{q} {(| c - 1 |}^{- 1} {+ 1) + Γ (q + 1) (| c - 1 |}^{- 1} + 1) \sum_{k = 1}^{M} ∥ Δ_{k} ∥}{Γ (q + 1) - A ω^{q} {(| c - 1 |}^{- 1} + 1)}, \end{matrix}

where

μ = {sup}_{t \in [0, ω]} ∥ f (t, 0) ∥

Proof.

It follows from

(I)

that for any

u \in Φ_{ω, c}

, we have

f (t + ω, u (t + ω)) = f (t + ω, c u (t)) = c f (t, u (t)), \forall t \in R

which implies that

f (\cdot, u (\cdot)) \in Φ_{ω, c}

Define the operator

F : Ψ \to Ψ

\begin{matrix} (F u) (t) & = & {(c - 1)}^{- 1} \frac{1}{Γ (q)} \int_{0}^{ω} {(ω - τ)}^{q - 1} f (τ, u (τ)) d τ + \frac{1}{Γ (q)} \int_{0}^{t} {(t - τ)}^{q - 1} f (τ, u (τ)) d τ \\ + {(c - 1)}^{- 1} \sum_{k = 1}^{M} Δ_{k} + \sum_{0 < t_{k} < t} Δ_{k} . \end{matrix}

(11)

From Lemmas 2 and 3, we obtain that the fixed points ofF determine the

(ω, c)

-periodic solutions of Equation (8). It is easy to see that

F (Ψ) \subseteq Ψ

. For any

u, v \in Ψ

, we have

\begin{matrix} ∥ (F u) (t) - (F v) (t) ∥ \\ = & ∥ {(c - 1)}^{- 1} \frac{1}{Γ (q)} \int_{0}^{ω} {(ω - τ)}^{q - 1} f (τ, u (τ)) d τ + \frac{1}{Γ (q)} \int_{0}^{t} {(t - τ)}^{q - 1} f (τ, u (τ)) d τ \\ - {(c - 1)}^{- 1} \frac{1}{Γ (q)} \int_{0}^{ω} {(ω - τ)}^{q - 1} f (τ, v (τ)) d τ - \frac{1}{Γ (q)} \int_{0}^{t} {(t - τ)}^{q - 1} f (τ, v (τ)) d τ ∥ \\ \leq & {| c - 1 |}^{- 1} \frac{1}{Γ (q)} \int_{0}^{ω} {(t - τ)}^{q - 1} ∥ f (τ, u (τ)) - f (τ, v (τ)) ∥ d τ \\ + \frac{1}{Γ (q)} \int_{0}^{t} {(t - τ)}^{q - 1} ∥ f (τ, u (τ)) - f (τ, v (τ)) ∥ d τ \\ \leq & {| c - 1 |}^{- 1} \frac{A}{Γ (q)} \int_{0}^{ω} {(ω - τ)}^{q - 1} ∥ u (τ) - v (τ) ∥ d τ \\ + \frac{A}{Γ (q)} \int_{0}^{t} {(t - τ)}^{q - 1} ∥ u (τ) - v (τ) ∥ d τ \\ \leq & \frac{A}{Γ (q)} {∥ u - v ∥}_{\infty} ({| c - 1 |}^{- 1} \int_{0}^{ω} {(ω - τ)}^{q - 1} d τ + \int_{0}^{t} {(t - τ)}^{q - 1} d τ) \\ \leq & \frac{A ω^{q} {(| c - 1 |}^{- 1} + 1)}{Γ (q + 1)} {∥ u - v ∥}_{\infty} \end{matrix}

which implies that

{∥ F u - F v ∥}_{\infty} \leq \frac{A ω^{q} {(| c - 1 |}^{- 1} + 1)}{Γ (q + 1)} {∥ u - v ∥}_{\infty} .

From the condition

0 < \frac{A ω^{q} {(| c - 1 |}^{- 1} + 1)}{Γ (q + 1)} < 1

, we obtain thatF is a contraction mapping. So, there exists a unique fixed pointu of (11) satisfying

u (ω) = c u (0)

. It follows from Lemma 2 that

u \in Φ_{ω, c}

. Then, we obtain that Equation (8) has a unique

(ω, c)

-periodic solution

u \in Φ_{ω, c}

Furthermore, we have

\begin{matrix} ∥ u (t) ∥ & \leq & {| c - 1 |}^{- 1} \frac{1}{Γ (q)} \int_{0}^{ω} {(ω - τ)}^{q - 1} ∥ f (τ, u (τ)) - f (τ, 0) ∥ d τ \\ {+ | c - 1 |}^{- 1} \frac{1}{Γ (q)} \int_{0}^{ω} {(ω - τ)}^{q - 1} ∥ f (τ, 0) ∥ d τ \\ + \frac{1}{Γ (q)} \int_{0}^{t} {(t - τ)}^{q - 1} ∥ f (τ, u (τ)) - f (τ, 0) ∥ d τ \\ + \frac{1}{Γ (q)} \int_{0}^{t} {(t - τ)}^{q - 1} ∥ f (τ, 0) ∥ d τ \\ {+ | c - 1 |}^{- 1} \sum_{k = 1}^{M} ∥ Δ_{k} ∥ + \sum_{t_{0} < t_{k} < t} ∥ Δ_{k} ∥ \\ \leq & {| c - 1 |}^{- 1} \frac{A}{Γ (q)} \int_{0}^{ω} {(ω - τ)}^{q - 1} {∥ u (τ) ∥ d τ + | c - 1 |}^{- 1} \frac{μ}{Γ (q)} \int_{0}^{ω} {(ω - τ)}^{q - 1} d τ \\ + \frac{A}{Γ (q)} \int_{0}^{t} {(t - τ)}^{q - 1} ∥ u (τ) ∥ d τ + \frac{μ}{Γ (q)} \int_{0}^{t} {(t - τ)}^{q - 1} d τ \\ + ({| c - 1 |}^{- 1} + 1) \sum_{k = 1}^{M} ∥ Δ_{k} ∥ \\ \leq & \frac{A ω^{q} {(| c - 1 |}^{- 1} + 1)}{Γ (q + 1)} {∥ u ∥}_{\infty} + \frac{μ ω^{q} {(| c - 1 |}^{- 1} + 1)}{Γ (q + 1)} + ({| c - 1 |}^{- 1} + 1) \sum_{k = 1}^{M} ∥ Δ_{k} ∥, \end{matrix}

which implies that

\begin{matrix} {∥ u ∥}_{\infty} \leq \frac{μ ω^{q} {(| c - 1 |}^{- 1} {+ 1) + Γ (q + 1) (| c - 1 |}^{- 1} + 1) \sum_{k = 1}^{M} ∥ Δ_{k} ∥}{Γ (q + 1) - A ω^{q} {(| c - 1 |}^{- 1} + 1)} . \end{matrix}

The proof is completed. □

Theorem 3.

Suppose that conditions

(I)

(I I I)

(I V)

hold and

c \neq 1

. If

B ω^{q} {(| c - 1 |}^{- 1} + 1) < Γ (q + 1)

, then the impulsive fractional differential Equation (8)has at least one

(ω, c)

-periodic solution

u \in Φ_{ω, c}

Proof.

Let

B_{r} = {{u \in Ψ : ∥ u ∥}_{\infty} \leq r}

, where

r \geq \frac{P ω^{q} {(| c - 1 |}^{- 1} {+ 1) + Γ (q + 1) (| c - 1 |}^{- 1} + 1) \sum_{k = 1}^{M} ∥ Δ_{k} ∥}{Γ (q + 1) - B ω^{q} {(| c - 1 |}^{- 1} + 1)} .

We considerF defined in (11) on

B_{r}

. For any

t \in [0, ω]

and any

u \in B_{r}

\begin{matrix} ∥ F (u) (t) ∥ & \leq & {| c - 1 |}^{- 1} \frac{B}{Γ (q)} \int_{0}^{ω} {(ω - τ)}^{q - 1} {∥ u (τ) ∥ d τ + | c - 1 |}^{- 1} \frac{P}{Γ (q)} \int_{0}^{ω} {(ω - τ)}^{q - 1} d τ \\ + \frac{B}{Γ (q)} \int_{0}^{t} {(t - τ)}^{q - 1} ∥ u (τ) ∥ d τ + \frac{P}{Γ (q)} \int_{0}^{t} {(t - τ)}^{q - 1} d τ \\ {+ | c - 1 |}^{- 1} \sum_{k = 1}^{M} ∥ Δ_{k} ∥ + \sum_{0 < t_{k} < t} ∥ Δ_{k} ∥ \\ \leq & \frac{B ω^{q} {(| c - 1 |}^{- 1} + 1)}{Γ (q + 1)} {∥ u ∥}_{\infty} + \frac{P ω^{q} {(| c - 1 |}^{- 1} + 1)}{Γ (q + 1)} + ({| c - 1 |}^{- 1} + 1) \sum_{k = 1}^{M} ∥ Δ_{k} ∥ \\ \leq & r, \end{matrix}

(12)

which implies

{∥ F u ∥}_{\infty} \leq r

. So,

F (B_{r}) \subseteq B_{r}

We prove thatF is continuous on

B_{r}

Let

{u_{i}}_{i \geq 1} \subseteq B_{r}

and

u_{i} \to u

B_{r}

i \to \infty

. By the continuity off, we get

f (τ, u_{i} (τ)) \to f (τ, u (τ))

i \to \infty

. Thus, we have

\begin{matrix} {(ω - τ)}^{q - 1} f (τ, u_{i} (τ)) & \to & {(ω - τ)}^{q - 1} f (τ, u (τ)) as i \to \infty, \\ {(t - τ)}^{q - 1} f (τ, u_{i} (τ)) & \to & {(t - τ)}^{q - 1} f (τ, u (τ)) as i \to \infty . \end{matrix}

Using condition

(I I I)

, we obtain that for any

0 \leq τ \leq t \leq ω

\begin{matrix} \int_{0}^{ω} ∥ {(ω - τ)}^{q - 1} f (τ, u_{i} (τ)) - {(ω - τ)}^{q - 1} f (τ, u (τ)) ∥ d τ \\ \leq 2 (B r + P) \int_{0}^{ω} {(ω - τ)}^{q - 1} d τ \leq 2 (B r + P) q^{- 1} ω^{q} < \infty, \end{matrix}

and

\begin{matrix} \int_{0}^{t} ∥ {(t - τ)}^{q - 1} f (τ, u_{i} (τ)) - {(t - τ)}^{q - 1} f (τ, u (τ)) ∥ d τ \\ \leq 2 (B r + P) \int_{0}^{t} {(t - τ)}^{q - 1} d τ \leq 2 (B r + P) q^{- 1} ω^{q} < \infty . \end{matrix}

Then, by Lebesgue dominated convergence theorem, we get

\int_{0}^{ω} ∥ {(ω - τ)}^{q - 1} f (τ, u_{i} (τ)) - {(ω - τ)}^{q - 1} f (τ, u (τ)) ∥ d τ \to 0 as i \to \infty,

and

\int_{0}^{t} ∥ {(t - τ)}^{q - 1} f (τ, u_{i} (τ)) - {(t - τ)}^{q - 1} f (τ, u (τ)) ∥ d τ \to 0 as i \to \infty .

So, for any

t \in [0, ω]

, it holds

\begin{matrix} ∥ (F u_{i}) (t) - (F u) (t) ∥ \\ \leq & {(c - 1)}^{- 1} \frac{1}{Γ (q)} \int_{0}^{ω} ∥ {(ω - τ)}^{q - 1} f (τ, u_{i} (τ)) - {(ω - τ)}^{q - 1} f (τ, u (τ)) ∥ d τ \\ + \frac{1}{Γ (q)} \int_{0}^{t} ∥ {(t - τ)}^{q - 1} f (τ, u_{i} (τ)) - {(t - τ)}^{q - 1} f (τ, u (τ)) ∥ d τ \to 0 as i \to \infty . \end{matrix}

Then,F is continuous on

B_{r}

We prove thatF is pre-compact.

For any

t_{i} < t \leq s \leq t_{i + 1}

i \in N_{0}

, we have

\begin{matrix} ∥ \sum_{0 < t_{k} < t} Δ_{k} - \sum_{0 < t_{k} < s} Δ_{k} ∥ = ∥ \sum_{k = 1}^{i} Δ_{k} - \sum_{k = 1}^{i} Δ_{k} ∥ = 0 \end{matrix}

which implies that

∥ \sum_{0 < t_{k} < t} Δ_{k} - \sum_{0 < t_{k} < s} Δ_{k} ∥ \to 0, as t \to s .

So, for any

0 \leq s_{1} < s_{2} \leq ω

, and any

u \in B_{r}

, it holds

\begin{matrix} ∥ (F u) (s_{1}) - (F u) (s_{2}) ∥ \\ \leq & ∥ \frac{1}{Γ (q)} \int_{0}^{s_{1}} {(s_{1} - τ)}^{q - 1} f (τ, u (τ)) d τ - \frac{1}{Γ (q)} \int_{0}^{s_{2}} {(s_{2} - τ)}^{q - 1} f (τ, u (τ)) d τ ∥ \\ + ∥ \sum_{0 < t_{k} < s_{1}} Δ_{k} - \sum_{0 < t_{k} < s_{2}} Δ_{k} ∥ \\ \leq & \frac{1}{Γ (q)} \int_{0}^{s_{1}} ({(s_{1} - τ)}^{q - 1} - {(s_{2} - τ)}^{q - 1}) ∥ f (τ, u (τ)) ∥ d τ \\ + \frac{1}{Γ (q)} \int_{s_{1}}^{s_{2}} {(s_{2} - τ)}^{q - 1} ∥ f (τ, u (τ)) ∥ d τ + ∥ \sum_{0 < t_{k} < s_{1}} Δ_{k} - \sum_{0 < t_{k} < s_{2}} Δ_{k} ∥ \\ \leq & \frac{B r + P}{Γ (q)} \int_{0}^{s_{1}} ({(s_{1} - τ)}^{q - 1} - {(s_{2} - τ)}^{q - 1}) d τ + \frac{B r + P}{Γ (q)} \int_{s_{1}}^{s_{2}} {(s_{2} - τ)}^{q - 1} d τ \\ + ∥ \sum_{0 < t_{k} < s_{1}} Δ_{k} - \sum_{0 < t_{k} < s_{2}} Δ_{k} ∥ \\ \leq & \frac{B r + P}{Γ (q + 1)} ((s_{2}^{q} - s_{1}^{q}) + 2 {(s_{2} - s_{1})}^{q}) + ∥ \sum_{0 < t_{k} < s_{1}} Δ_{k} - \sum_{0 < t_{k} < s_{2}} Δ_{k} ∥ \to 0 as s_{1} \to s_{2} . \end{matrix}

So,

F (B_{r})

is equicontinuous. By (12), we obtain that

F (B_{r})

is uniformly bounded. Using Arzelà-Ascoli theorem, we obtain that

F (B_{r})

is pre-compact.

It follows from Schauder’s fixed point theorem that the impulsive fractional differential Equation (8) has at least one

(ω, c)

periodic solution

u \in Φ_{ω, c}

. The proof is finished. □

Remark 1.

c = 1

(ω, c)

-periodic solution is standard ω-periodic solution. If

c = - 1

(ω, c)

-periodic solution is ω-antiperiodic solution. Moreover, all results obtained in this paper are based on the fixed lower limit of Caputo fractional derivative.

5. Examples

Example 1.

We consider the following impulsive fractional differential equation:

\{\begin{matrix} ^{c} D_{0}^{\frac{1}{2}} u (t) = λ cos 2 t sin u (t), t \neq t_{k}, t \in [0, \infty), \\ u (t_{k}^{+}) = u (t_{k}^{-}) + cos k π, k = 1, 2, 3, \dots, \end{matrix}

(13)

where

λ \in R

t_{k} = \frac{k π}{2}

Δ_{k} = cos k π

f (t, u) = λ cos 2 t sin u (t)

. Set

ω = π

c = - 1

. It is easy to see that for any

k \in N

t_{k + 2} = t_{k} + π

Δ_{k + 2} = Δ_{k}

. So, we obtain

M = 2

, and

(I V)

holds. For any

t \in R

and any

u \in R

, we have

f (t + ω, c u) = f (t + π, - u) = - λ cos 2 t sin u (t) = - f (t, u) = c f (t, u)

which implies that

(I)

holds. For any

t \in R

and any

u, v \in R

, we have

| f (t, u) - f (t, v) | \leq | λ | | u - v |

which implies that

A = | λ |

and

(I I)

holds. Note that

\frac{A ω^{q} {(| c - 1 |}^{- 1} + 1)}{Γ (q + 1)} = \frac{3 | λ | \sqrt{π}}{Γ (\frac{1}{2})}

. Letting

0 < | λ | < \frac{Γ (\frac{1}{2})}{3 \sqrt{π}}

, we obtain

0 < \frac{A ω^{q} {(| c - 1 |}^{- 1} + 1)}{Γ (q + 1)} < 1

. Then, all assumptions in Theorem 2 hold for Equation (13).

Hence, if

0 < | λ | < \frac{Γ (\frac{1}{2})}{3 \sqrt{π}}

, (13)has a unique

(π, - 1)

-periodic solution

u \in Φ_{π, - 1}

Furthermore, we have

\begin{matrix} {∥ u ∥}_{\infty} \leq \frac{μ ω^{q} {(| c - 1 |}^{- 1} {+ 1) + Γ (q + 1) (| c - 1 |}^{- 1} + 1) \sum_{k = 1}^{M} ∥ Δ_{k} ∥}{Γ (q + 1) - A ω^{q} {(| c - 1 |}^{- 1} + 1)} = \frac{3 Γ (\frac{1}{2})}{Γ (\frac{1}{2}) - 3 | λ | \sqrt{π}} . \end{matrix}

Example 2.

We consider the following impulsive fractional differential equation:

\{\begin{matrix} ^{c} D_{0}^{\frac{1}{2}} u (t) = λ u (t) sin (3^{- t} u (t)), t \neq t_{k}, t \in [0, \infty), \\ u (t_{k}^{+}) = u (t_{k}^{-}) + 2, k = 1, 2, 3, \dots, \end{matrix}

(14)

where

λ \in R

t_{k} = \frac{k}{2}

Δ_{k} = 2

f (t, u) = λ u sin (3^{- t} u)

. Set

ω = 1

c = 3

. Obviously,

t_{k + 2} = t_{k} + 1

Δ_{k + 2} = Δ_{k}

hold for all

k \in N

. So we obtain

M = 2

, and

(I V)

holds. For any

t \in R

and any

u \in R

, we have

f (t + ω, c u) = f (t + 1, 3 u) = 3 λ u sin (3^{- t} u) = 3 f (t, u) = c f (t, u)

which implies that

(I)

holds. For any

t \in R

and any

u \in R

, we have

| f (t, u) | \leq | λ | | u |

which implies that

B = | λ |

P = 0

and

(I I I)

holds. Note that

B ω^{q} {(| c - 1 |}^{- 1} + 1) = \frac{3}{2} | λ |

. Letting

| λ | < \frac{1}{Γ (\frac{5}{2})}

, we get

B ω^{q} {(| c - 1 |}^{- 1} + 1) < Γ (q + 1)

. Then, all assumptions in Theorem 3 hold for Equation (13).

Therefore, if

| λ | < \frac{1}{Γ (\frac{5}{2})}

, Equation (14)has at least one

(1, 3)

-periodic solution

u \in Φ_{1, 3}

6. Conclusions

In this paper, we mainly study the existence of

(ω, c)

-periodic solutions for impulsive fractional differential equations with fixed lower limits. In future work, we shall study the

(ω, c)

-periodic solutions for impulsive fractional differential equations with varying lower limits.

Author Contributions

The contributions of all authors (L.R. and J.W.) are equal. All the main results were developed together. All authors have read and agreed to the published version of the manuscript.

Funding

This work is partially supported by Foundation of Postgraduate of Guizhou Province (YJSCXJH[2019]031), Training Object of High Level and Innovative Talents of Guizhou Province ((2016)4006), Major Research Project of Innovative Group in Guizhou Education Department ([2018]012), and Guizhou Data Driven Modeling Learning and Optimization Innovation Team ([2020]5016).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

The authors are grateful to the referees for their careful reading of the manuscript and valuable comments. The authors thank the editor too.

Conflicts of Interest

The authors declare no conflict of interest.

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Ren, L.; Wang, J. (ω,c)-Periodic Solutions to Fractional Differential Equations with Impulses.Axioms2022,11, 83. https://doi.org/10.3390/axioms11030083

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Ren L, Wang J. (ω,c)-Periodic Solutions to Fractional Differential Equations with Impulses.Axioms. 2022; 11(3):83. https://doi.org/10.3390/axioms11030083

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Ren, Lulu, and JinRong Wang. 2022. "(ω,c)-Periodic Solutions to Fractional Differential Equations with Impulses"Axioms 11, no. 3: 83. https://doi.org/10.3390/axioms11030083

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Ren, L., & Wang, J. (2022). (ω,c)-Periodic Solutions to Fractional Differential Equations with Impulses.Axioms,11(3), 83. https://doi.org/10.3390/axioms11030083

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(ω,c)-Periodic Solutions to Fractional Differential Equations with Impulses

Abstract

1. Introduction

2. Preliminaries

3. ((ω,c))-Periodic Solutions to Linear Problem

4.(ω,c)-Periodic Solutions to Semilinear Problem

5. Examples

6. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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3. ( $(ω, c)$ )-Periodic Solutions to Linear Problem

4. $(ω, c)$ -Periodic Solutions to Semilinear Problem