Abstract
In this paper, based on the theory of calculus on time scales, by using a multiple fixed point theorem in cones, some criteria are established for the existence and multiplicity of positive periodic solutions in shifts for an impulsive functional dynamic equation on time scales of the following form: , , , , where be a periodic time scale in shifts with period and is nonnegative and fixed. Finally, some numerical examples are presented to illustrate the feasibility and effectiveness of the results.
MSC:34K13, 34K45, 34N05.
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1 Introduction
The time scales approach unifies differential, difference, h-difference, and q-differences equations and more under dynamic equations on time scales. The theory of dynamic equations on time scales was introduced by Hilger in his PhD thesis in 1988 [1]. The existence problem of periodic solutions is an important topic in qualitative analysis of functional dynamic equations. Up to now, there are only a few results concerning periodic solutions of dynamic equations on time scales; see, for example, [2, 3]. In these papers, authors considered the existence of periodic solutions for dynamic equations on time scales satisfying the condition ‘there exists a such that , ’. Under this condition all periodic time scales are unbounded above and below. However, there are many time scales such as and which do not satisfy the condition. Adıvar and Raffoul introduced a new periodicity concept on time scales which does not oblige the time scale to be closed under the operation for a fixed . They defined a new periodicity concept with the aid of shift operators which are first defined in [4] and then generalized in [5].
Recently, based on a fixed point theorem in cones, Çetin et al. studied the existence of positive periodic solutions in shifts for some nonlinear first-order functional dynamic equation on time scales; see [6, 7].
However, to the best of our knowledge, there are few papers published on the existence of positive periodic solutions in shifts for a functional dynamic equation with impulses. As we know, impulsive functional dynamic equation on time scales plays an important role in applications; see, for example, [8, 9].
Motivated by the above, in the present paper, we consider the following system:
where be a periodic time scale in shifts with period and is nonnegative and fixed; are Δ-periodic in shifts with period ω and ; is periodic in shifts with period ω with respect to the first variable; is periodic in shifts with period ω; and represent the right and the left limit of in the sense of time scales, in addition, if is right-scattered, then , whereas, if is left-scattered, then ; , . Assume that there exists a positive constant q such that , , . For each interval of ℝ, we denote , without loss of generality, set .
The main purpose of this paper is to establish some sufficient conditions for the existence of at least three positive periodic solutions in shifts of system (1.1) using a multiple fixed point theorem (Avery-Peterson fixed point theorem) in cones.
The organization of this paper is as follows. In Section 2, we introduce some notations and definitions and state some preliminary results needed in later sections; then we give the Green’s function of system (1.1), which plays an important role in this paper. In Section 3, we establish our main results for positive periodic solutions in shifts by applying Avery-Peterson fixed point theorem. In Section 4, some numerical examples are presented to illustrate that our results are feasible and more general.
2 Preliminaries
Let be a nonempty closed subset (time scale) of ℝ. The forward and backward jump operators and the graininess are defined, respectively, by
A point is called left-dense if and , left-scattered if , right-dense if and , and right-scattered if . If has a left-scattered maximum m, then ; otherwise . If has a right-scattered minimum m, then ; otherwise .
A function is right-dense continuous provided it is continuous at right-dense point in and its left-side limits exist at left-dense points in . If f is continuous at each right-dense point and each left-dense point, then f is said to be a continuous function on . The set of continuous functions will be denoted by .
For the basic theories of calculus on time scales, see [10].
A function is called regressive provided for all . The set of all regressive and rd-continuous functions will be denoted by . Define the set .
If r is a regressive function, then the generalized exponential function is defined by
for all , with the cylinder transformation
Let be two regressive functions, define
Lemma 2.1 [10]
Assume that be two regressive functions, then
-
(i)
and ;
-
(ii)
;
-
(iii)
;
-
(iv)
;
-
(v)
.
The following definitions and lemmas about the shift operators and the new periodicity concept for time scales can be found in [7, 11].
Let be a nonempty subset of the time scale and be a fixed number, define operators . The operators and associated with (called the initial point) are said to be forward and backward shift operators on the set , respectively. The variable in is called the shift size. The values and in indicate s units translation of the term to the right and left, respectively. The sets
are the domains of the shift operator , respectively. Hereafter, is the largest subset of the time scale such that the shift operators exist.
Definition 2.1 (Periodicity in shifts [11])
Let be a time scale with the shift operators associated with the initial point . The time scale is said to be periodic in shifts if there exists such that for all . Furthermore, if
then P is called the period of the time scale .
Definition 2.2 (Periodic function in shifts [11])
Let be a time scale that is periodic in shifts with the period P. We say that a real-valued function f defined on is periodic in shifts if there exists such that and for all , where . The smallest number is called the period of f.
Definition 2.3 (Δ-periodic function in shifts [11])
Let be a time scale that is periodic in shifts with the period P. We say that a real-valued function f defined on is Δ-periodic in shifts if there exists such that for all , the shifts are Δ-differentiable with rd-continuous derivatives and for all , where . The smallest number is called the period of f.
Lemma 2.2 [11]
and for all .
Lemma 2.3 [7]
Let be a time scale that is periodic in shifts with the period P. Suppose that the shifts are Δ-differentiable on where and is Δ-periodic in shifts with the period ω. Then
-
(i)
for ;
-
(ii)
for .
Lemma 2.4 [11]
Let be a time scale that is periodic in shifts with the period P, and let f be a Δ-periodic function in shifts with the period . Suppose that , then
Lemma 2.5 [10]
Suppose that r is regressive and is rd-continuous. Let , , then the unique solution of the initial value problem
is given by
Define
Set
with the norm , then X is a Banach space.
Lemma 2.6 is an ω-periodic solution in shifts of system (1.1) if and only if is an ω-periodic solution in shifts of
where
Proof If is an ω-periodic solution in shifts of system (1.1). For any , there exists such that is the first impulsive point after t. By using Lemma 2.5, for , we have
then
Again using Lemma 2.5 and (2.2), for , then
Repeating the above process for , we have
Let in the above equality, we have
Noticing that , , by Lemma 2.1, then x satisfies (2.1).
Let x be an ω-periodic solution in shifts of (2.1). If , , then by (2.1) and Lemma 2.2, we have
If , , then by (2.1), we have
So, x is an ω-periodic solution in shifts of system (1.1). This completes the proof. □
It is easy to verify that the Green’s function satisfies the property
where . By Lemma 2.3, we have
In order to obtain the existence of periodic solutions in shifts of system (1.1), we need the following concepts and Avery-Peterson fixed point theorem.
Let X be a Banach space and K be a cone in X, define . A map α is said to be a nonnegative continuous concave functional on K if is continuous and
Let γ and θ be nonnegative continuous convex functionals on K, α be a nonnegative continuous concave functional on K, and ψ be a nonnegative continuous functional on K. Then for positive real numbers a, b, c, and d, we define the following convex sets:
and a closed set .
Lemma 2.7 (Avery-Peterson fixed point theorem [12])
Let γ and θ be nonnegative continuous convex functionals on K, α be a nonnegative continuous concave functional on K, and ψ be a nonnegative continuous functional on K satisfying for , such that for some positive numbers E and d,
for all . Suppose is completely continuous and there exist positive numbers a, b, and c with such that:
-
(1)
and for ,
-
(2)
, for with ,
-
(3)
and for with .
Then H has at least three fixed points such that
Define K, a cone in X, by
and an operator by
For convenience, we denote
In the following, we shall give some lemmas concerning K and H defined by (2.6) and (2.7), respectively.
Lemma 2.8 is well defined.
Proof For any , it is clear that . In view of (2.7), by Lemma 2.4 and (2.4), for , we have
that is, .
Furthermore, for any , , we have
that is, . This completes the proof. □
Lemma 2.9 is completely continuous.
Proof Firstly, we show that H is continuous. Because of the continuity of f and , , for any and , there exists such that
imply
and
Therefore, if with , , , then
which yields , that is, H is continuous.
Next, we show that H maps any bounded sets in K into relatively compact sets. We first prove that f maps bounded sets into bounded sets. Indeed, let , for any , there exists such that imply
and
Choose a positive integer N such that . Let and define , . If , then
Thus
for all , and
and these yield
and
It follows from (2.7), (2.8), and (2.9) that for ,
Furthermore, for , we have
and
To sum up, is a family of uniformly bounded and equicontinuous functionals on . By a theorem of Arzela-Ascoli, the functional H is completely continuous. This completes the proof. □
3 Main result
Now, we fix , , and let the nonnegative continuous concave functional α, the nonnegative continuous convex functionals θ, γ, and the nonnegative continuous functional ψ be defined on the cone K by
respectively, where , .
The functionals defined above satisfy the following relations:
Lemma 3.1 For , there exists a constant such that
Proof For , we have
where . Setting . This completes the proof. □
Moreover, for each ,
and , , for all . It follows from (3.2) and (3.3) that condition (2.5) in Lemma 2.7 is satisfied.
For convenience in the following discussion, we introduce the following notations:
To present our main result, we assume that there exist constants with such that:
(S1) , for , ;
(S2) , for , ;
(S3) , for , .
Theorem 3.1 Under assumptions (S1)-(S3), system (1.1) has at least three positive ω-periodic solutions , , and in shifts satisfying
Proof For , then and . From Lemma 3.1, we have , that is, , for . Then, by Lemma 2.8 and assumption (S1), for , we have , and
then
Therefore, .
To check condition (1) in Lemma 2.7, we take . It is easy to verify that , and , and so .
For , we have
that is, , , for .
Then, by assumption (S2), we have
that is, for all . This shows that condition (1) in Lemma 2.7 is satisfied.
Secondly, by (3.1) and the cone K we defined in (2.6), we can get for all with . Thus condition (2) in Lemma 2.7 is satisfied.
Finally, we show that condition (3) in Lemma 2.7 also holds. Clearly, as , we have . Suppose that with , this implies that for , there is , . Hence,
By assumption (S3), we have
So, condition (3) in Lemma 2.7 is satisfied.
To sum up, all conditions in Lemma 2.7 are satisfied. Hence, H has at least three fixed points, that is, system (1.1) has at least three positive ω-periodic solutions in shifts . This completes the proof. □
4 Numerical examples
Consider the following system with impulses:
Example 1 Let
in system (4.1), then
Case I: Let , , then . It is easy to verify , , satisfy
and .
By a direct calculation, we can get
Choose , , , , then
According to Theorem 3.1, when , for system (4.1) there exist at least three positive periodic solutions , , in shifts with period , and
Case II: Let , , then . It is easy to verify , , satisfy
and .
By a direct calculation, we can get
Choose , , , , then
According to Theorem 3.1, when , for system (4.1) there exist at least three positive periodic solutions , , in shifts with period , and
Example 2 Let
in system (4.1), where , then
Let , , then . It is easy to verify , , satisfy
and .
By a direct calculation, we can get
Choose , , , , then
According to Theorem 3.1, when , for system (4.1) there exist at least three positive periodic solutions , , in shifts with period , and
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Acknowledgements
This work is supported by the Basic and Frontier Technology Research Project of Henan Province (Grant No. 142300410113).
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Hu, M., Wang, L. Multiple periodic solutions in shifts for an impulsive functional dynamic equation on time scales. Adv Differ Equ 2014, 152 (2014). https://doi.org/10.1186/1687-1847-2014-152
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DOI: https://doi.org/10.1186/1687-1847-2014-152