Multiple periodic solutions in shifts for an impulsive functional dynamic equation on time scales
© Hu and Wang; licensee Springer. 2014
Received: 22 March 2014
Accepted: 12 May 2014
Published: 22 May 2014
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.
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 . 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  and then generalized in .
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].
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.
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 .
A function is called regressive provided for all . The set of all regressive and rd-continuous functions will be denoted by . Define the set .
Lemma 2.1 
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 )
then P is called the period of the time scale .
Definition 2.2 (Periodic function in shifts )
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 )
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 
and for all .
Lemma 2.3 
Lemma 2.4 
Lemma 2.5 
with the norm , then X is a Banach space.
Noticing that , , by Lemma 2.1, then x satisfies (2.1).
So, x is an ω-periodic solution in shifts of system (1.1). This completes the proof. □
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.
and a closed set .
Lemma 2.7 (Avery-Peterson fixed point theorem )
and for ,
, for with ,
and for with .
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.
that is, .
that is, . This completes the proof. □
Lemma 2.9 is completely continuous.
which yields , that is, H is continuous.
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
respectively, where , .
where . Setting . This completes the proof. □
and , , for all . It follows from (3.2) and (3.3) that condition (2.5) in Lemma 2.7 is satisfied.
To present our main result, we assume that there exist constants with such that:
(S1) , for , ;
(S2) , for , ;
(S3) , for , .
To check condition (1) in Lemma 2.7, we take . It is easy to verify that , and , and so .
that is, , , for .
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.
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
This work is supported by the Basic and Frontier Technology Research Project of Henan Province (Grant No. 142300410113).
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