Abstract

Let be a periodic time scale in shifts with period and is nonnegative and fixed. By using a multiple fixed point theorem in cones, some criteria are established for the existence and multiplicity of positive solutions in shifts for a class of higher-dimensional functional dynamic equations with impulses on time scales of the following form: where is a nonsingular matrix with continuous real-valued functions as its elements. Finally, numerical examples are presented to illustrate the feasibility and effectiveness of the results.

1. Introduction

As is known to all, both continuous and discrete systems are very important in implementation and application. The study of dynamic equations on time scales, which unifies differential, difference, -difference, and -differences equations and more, has received much attention; see, for example, [116] and the references therein. The theory of dynamic equations on time scales was introduced by Hilger in his PhD thesis in 1988 [5]. 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, [69]. 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 this condition. Adivar 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 [10] and then generalized in [11].

Recently, based on a fixed-point theorem in cones, Çetin and Serap Topal studied the existence of positive periodic solutions in shifts for some nonlinear first-order functional dynamic equation on time scales; see [12, 13]. However, to the best of our knowledge, there are few papers published on the existence of positive periodic solutions in shifts for higher-dimensional functional dynamic equations with impulses, especially systems with the coefficient matrix being an arbitrary nonsingular matrix.

Motivated by the above, in the present paper, we consider the following system: where is a periodic time scale in shifts with period and is nonnegative and fixed; is a nonsingular matrix with continuous real-valued functions as its elements, , and is -periodic in shifts with period ; and is -periodic in shifts with period ; and is periodic in shifts with period with respect to the first variable; is periodic in shifts with period ; 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 ; and . Assume that there exists a positive constant such that , , . For each interval of , we denote ; without loss of generality, set .

In [14], Li and Hu studied the existence of positive periodic solutions of system (1) on a periodic time scale with . The time scale considered in [14] is unbounded above and below. Moreover, the condition in [14] is too strict so that it cannot be satisfied even if the coefficient matrix is a diagonal matrix. Therefore, the results in [14] are less applicable.

The main purpose of this paper is to study the existence and multiplicity of positive periodic solutions in shifts of system (1) under more general assumptions. By using Leggett-Williams fixed point theorem, sufficient conditions for the existence of at least three positive periodic solutions in shifts of system (1) will be established. The results presented in this paper improve and generalize the results in [14].

In this paper, for each , the norm of is defined as , where and when it comes to the fact that is continuous, delta derivative, delta integrable, and so forth; we mean that each element is continuous, delta derivative, delta integrable, and so forth.

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. Besides, in Section 2, we give some lemmas about the exponential function with shift operators, and Green’s function of system (1). In Section 3, we establish our main results for positive periodic solutions in shifts by applying Leggett-Williams fixed point theorem. In Section 4, 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 . 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 , then ; otherwise, . If has a right-scattered minimum , then ; otherwise, .

A function is right-dense continuous provided that it is continuous at right-dense point in and its left-side limits exist at left-dense points in . If is continuous at each right-dense point and each left-dense point, then 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 [15].

Definition 1 (see [15]). An -matrix-valued function on a time scale is called regressive (with respect to ) provided that is invertible for all . The set of all regressive and rd-continuous functions will be denoted by .

Definition 2 (see [15]). Let and assume that is a regressive -matrix-valued function. The unique matrix-valued solution of the IVP is where denotes as usual the -identity matrix, is called the matrix exponential function (at ), and is denoted by .

Lemma 3 (see [15]). If is a regressive -matrix-valued function on , then(i) and ;(ii);(iii);(iv).

Lemma 4 (see [15)). Let be a regressive -matrix-valued function on and suppose that is rd-continuous. Let and has a unique solution . Moreover, the solution is given by

The following definitions and lemmas about the shift operators and the new periodicity concept for time scales can be found in [16].

Let be a nonempty subset of the time scale and let 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 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 5 (see [16], periodicity in shifts ). 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 is called the period of the time scale .

Definition 6 (see [16], periodic function in shifts ). Let be a time scale that is periodic in shifts with the period . We say that a real-valued function defined on is periodic in shifts if there exists such that and for all , where . The smallest number is called the period of .

Definition 7 (see [16], -periodic function in shifts ). Let be a time scale that is periodic in shifts with the period . We say that a real-valued function defined on is -periodic in shifts if there exists such that for all and the shifts are -differentiable with rd-continuous derivatives and for all , where . The smallest number is called the period of .

Lemma 8 (see [16]). Consider and for all .

Lemma 9 (see [16]). Let be a time scale that is periodic in shifts with the period , and let be a -periodic function in shifts with the period . Suppose that , then

Let be a time scale that is periodic in shifts . If one takes , then one has and .

Now, we prove two properties of the exponential functions and shift operators on time scales.

Lemma 10. Let be a time scale that is periodic in shifts with the period . Suppose that the shifts are -differentiable on , where and is -periodic in shifts with the period . Then

Proof. Let , where , then and . Hence, solves the IVP, which has exactly one solution according to Lemma 4, and therefore we have This completes the proof.

Lemma 11. Let be a time scale that is periodic in shifts with the period . Suppose that the shifts are -differentiable on , where and is -periodic in shifts with the period . Then

Proof. From Lemma 8, we know . By Lemmas 10 and 3, we can obtain This completes the proof.

Define

Set with the norm defined by , where ; then is a Banach space.

Lemma 12. The function is an -periodic solution in shifts of system (1) if and only if is an -periodic solution in shifts of where

Proof. If is an -periodic solution in shifts of system (1), for any , there exists such that is the first impulsive point after . By using Lemma 4, for , we have Then Again, using Lemma 4 and (21), for , then Repeating the above process for , we have Let in the above equality; we have Noticing that and , by Lemma 3, then satisfies (18).
Let be an -periodic solution in shifts of (18). If , , then, by (18) and Lemma 8, we have If , , then, by (18), we have So, is an -periodic solution in shifts of system (1). This completes the proof.

By using Lemmas 10 and 11, it is easy to verify that Green’s function satisfies

For convenience, we introduce the following notations: Hereafter, we assume that ), ;(), , , .

Let where . Obviously, is a cone in .

Define an operator by that is, for all , , where is defined by (19), and where

In the following, we will give some lemmas concerning and defined by (29) and (31), respectively.

Lemma 13. Assume that hold; then is well defined.

Proof. For any , it is clear that . In view of (31), by Lemma 9 and (27), for , we obtain that is, .
Furthermore, for any , by , we have that is, . This completes the proof.

Define

Lemma 14. Assume that hold; then is completely continuous.

Proof. We first show that is continuous. Because of the continuity of and , for any and , there exists a such that imply that Therefore, if with , , , then for all , which yields that is, is continuous.
Next, we show that maps any bounded sets in into relatively compact sets. We first prove that maps bounded sets into bounded sets. Indeed, let ; for any , there exists such that imply that Choose a positive integer such that . Let and define , . If , then So for all , and and these yield It follows from (32) and (45) that, for ,
Finally, for , we have So where , .
To sum up, is a family of uniformly bounded and equicontinuous functionals on . By a theorem of Arzela-Ascoli, we know that the functional is completely continuous. This completes the proof.

3. Main Results

In this section, we will state and prove our main results about the existence of at least three positive periodic solutions of system (1) via Leggett-Williams fixed point theorem.

Let be a Banach space with cone . A map is said to be a nonnegative continuous concave functional on if is continuous and

Let be two numbers such that and let be a nonnegative continuous concave functional on . We define the following convex sets:

Lemma 15 (see [17] Leggett-Williams fixed point theorem). Let be completely continuous and let be a nonnegative continuous concave functional on such that for all . Suppose that there exist such that (1) and for ;(2) for all ;(3) for all with . Then has at least three fixed points satisfying and .

For convenience, we introduce the following notations:

Theorem 16. Assume that hold, and there exists a number such that the following conditions (i), ,(ii) for , hold. Then system (1) has at least three positive -periodic solutions in shifts .

Proof. By the condition of , one can find that, for there exists a such that where .
Let ; if , , then , and we have
Take ; then the set is a bounded set. According to the fact that is completely continuous, then maps bounded sets into bounded sets and there exists a number such that If , we deduce that is completely continuous. If , then, from (54), we know that for any and hold. Thus we have is completely continuous. Now, take ; then , so is completely continuous.
Denote the positive continuous concave functional as . Firstly, let and take , , , then the set . By , if , then , and we have Hence, condition (1) of Lemma 15 holds.
Secondly, by the condition of (i), one can find that, for there exists a    such that where . If , we have that is, condition (2) of Lemma 15 holds.
Finally, if with , by the definition of the cone , we have which implies that condition (3) of Lemma 15 holds.
To sum up, all conditions in Lemma 15 hold. By Lemma 15, the operator has at least three fixed points in . Therefore, system (1) has at least three positive -periodic solutions in shifts , and This completes the proof.

Corollary 17. Using the following (), , , , instead of in Theorem 16, the conclusion of Theorem 16 remains true.

4. Numerical Examples

Consider the following system with impulses on time scales:

Example 1. Let in system (62), where . Then

Case 1. , and . Let ; then . It is easy to verify that , , and satisfy and . By a direct calculation, we can get Since , . Then From the above, we can see that conditions and hold.

Let ; then(i), ;(ii) for , .

According to Theorem 16, when , system (62) has at least three positive -periodic solutions in shifts .

Case 2. , and . Let ; then . It is easy to verify that , , and satisfy and . By a direct calculation, we can get Since , . Then From the above, we can see that conditions and hold.

Let ; then(i), ;(ii) for , .

According to Theorem 16, when , system (62) has at least three positive -periodic solutions in shifts .

Example 2. Let in system (62), where .

Let , , and ; then . It is easy to verify that , , and satisfy and . By a direct calculation, we can get Since , , then , , . Moreover, we have From the above, we can see that conditions and hold.

Let ; then(i), ;(ii) for , .

According to Theorem 16, when , system (62) has at least three positive -periodic solutions in shifts .

Remark 3. From Examples 1 and 2, we can see that the results obtained in this paper can be applied to systems on more general time scales, and not only time scales are unbounded above and below.

Remark 4. In system (62), if is a diagonal matrix, a similar calculation in Example 2 shows that , , , and the condition in [14] cannot be satisfied. So the main results in [14] cannot ensure the existence of positive periodic solution of system (62), while is a diagonal matrix. Therefore, our main results improve and generalize the results in [14].

Conflict of Interests

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

Acknowledgments

This work is supported by the Basic and Frontier Technology Research Project of Henan Province (Grant no. 142300410113) and the National Science Foundation of People Republic of China (Grant no. 11301008).