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Positive periodic solutions in shifts for a nonlinear first-order functional dynamic equation on time scales
Advances in Difference Equations volume 2014, Article number: 76 (2014)
Let be a periodic time scale in shifts with period . We consider the existence of positive periodic solutions in shifts for the nonlinear functional dynamic equation of the form
using the cone theory techniques. We extend and unify periodic differential, difference, h-difference and q-difference equations and more by a new periodicity concept on time scales.
MSC:34N05, 39A12, 35B10.
Functional differential equations include many mathematical ecological and population models, such as the Lasota-Wazewska model [1–6], Nicholson’s blowflies model [1, 4, 7–10], the model for blood cell production [1, 4, 9, 11]etc. Particularly, since the periodic variation of the environment plays an important role in many biological and ecological systems, many researchers have been interested in studying the existence of periodic solutions of the above models. Chow , Freedman and Wy , Hadeler and Tomiuk , Kuang , Wang , Weng and Sun  and many others studied the existence of at least one and at least two positive periodic solutions of nonlinear first-order differential equations using the fixed point theorem of cone expansion and the cone compression method, the upper and lower solution method and iterative technique . On the other hand, it has been observed that very few papers exist in the literature on the existence of at least three and the nonexistence of a nonnegative periodic solution for first-order differential equations. For example, see [1, 15, 18].
In fact, both continuous and discrete systems are very important in implementation and application. Therefore, the study of dynamic equations on time scales, which unifies differential, difference, h-difference, q-differences equations and more, has received much attention; see [19–23]. The theory of dynamic equations on times-scales was introduced by Stefan Hilger in 1988 . There are only a few results concerning periodic solutions of dynamic equations on time scales such as in [20, 25]. In these papers, all periodic time scales must be unbounded above and below, but there are many time scales that do not satisfy this condition such as and . Adıvar introduced a new periodicity concept in  with the aid of shift operators . With the new periodicity concept, the time scale need not be closed under the operation for a fixed . There are only few existence results related with the new periodicity; see .
Let be a periodic time scale in shifts with period and . We are concerned with the existence, multiplicity and nonexistence of periodic solutions in shifts for the nonlinear dynamic equation
where is Δ-periodic in shifts with period T and , is a positive parameter, is Δ-periodic in shifts with period T, , is periodic in shifts with period T and is periodic in shifts with period T with respect to the first variable and .
Hereafter, we use the notation to indicate the time scale interval . The intervals , and are similarly defined.
In this study, we shall show that the number of positive periodic solutions in shifts of (1) can be determined by the asymptotic behaviors of the quotient of at zero and infinity. We shall organize this paper as follows. In Section 2, we state some facts about exponential function on time scales, the new periodicity concept for time scales and some important theorems which will be needed to show the existence and nonexistence of periodic solutions in shifts . Besides these, in Section 2, we give some lemmas about the exponential function and the graininess function with shift operators. We also present some lemmas to be used later. Finally, we state our main results and give their proofs in Section 3 by using the Krasnosel’skiĭ fixed point theorem.
In this section, we mention some definitions, lemmas and theorems from calculus on time scales which can be found in [18, 27]. Next, we state some definitions, lemmas and theorems about the shift operators and the new periodicity concept for time scales which can be found in .
Definition 2.1 
A function is said to be regressive provided for all , where . The set of all regressive rd-continuous functions is denoted by ℛ, while the set is given by .
Let and for all . The exponential function on is defined by
where is the cylinder transformation given by
Also, the exponential function is the solution to the initial value problem , . Other properties of the exponential function are given in the following lemma (, Theorem 2.36).
Lemma 2.1 Let . Then
, where ;
The following definitions, lemmas, corollaries and examples are about the shift operators and the new periodicity concept for time scales which can be found in .
Definition 2.2 
Let be a nonempty subset of the time scale including a fixed number such that there exist operators satisfying the following properties:
(P.1) The functions are strictly increasing with respect to their second arguments, i.e., if
(P.2) If with , then , and if with , then ;
(P.3) If , then and . Moreover, if , then and holds;
(P.4) If , then and , respectively;
(P.5) If and , then and , respectively.
Then the operators and associated with (called the initial point) are said to be backward and forward shift operators on the set , respectively. The variable in is called the shift size. The value 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 2.3  (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 shift if there exists such that for all . Furthermore, if
then P is called the period of the time scale .
Example 2.1 
The following time scales are periodic in the sense of shift operators given in Definition 2.3.
, , .
, , , .
, , , .
Notice that the time scale in Example 2.1 is bounded above and below and .
Remark 2.1 
Let be a time scale that is periodic in shifts with the period P. Thus, by (P.4) of Definition 2.2, the mapping defined by is surjective. On the other hand, by (P.1) of Definition 2.2, shift operators are strictly increasing in their second arguments. That is, the mapping is injective. Hence, is an invertible mapping with the inverse defined by .
We assume that is a periodic time scale in shift with period P. The operators are commutative with the forward jump operator given by . That is, for all .
Lemma 2.2 
The mapping preserves the structure of the points in . That is,
Corollary 2.1 
and for all .
Definition 2.4  (Periodic function in shift )
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
where . The smallest number such that (4) holds is called the period of f.
Definition 2.5  (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
where . The smallest number such that (5)-(7) hold is called the period of f.
Notice that Definition 2.4 and Definition 2.5 give the classic periodicity definition on time scales whenever are the shifts satisfying the assumptions of Definition 2.4 and Definition 2.5.
Now, we give a theorem which is the substitution rule on periodic time scales in shifts which can be found in .
Theorem 2.1 Let be a time scale that is periodic in shifts with period , and let f be a Δ-periodic function in shifts with the period . Suppose that , then
We give some interesting properties of the exponential functions and shift operators on time scales which can be found in .
Lemma 2.3 Let be a time scale that is periodic in shifts with the period P. Suppose that the shifts are Δ-differentiable on where . Then the graininess function satisfies
Lemma 2.4 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 T. Then
where is the space of all real-valued continuous functions endowed with the norm
then is a Banach space.
Lemma 2.5 
Let . Then exists and .
Lemma 2.6 is a solution of (1) if and only if
where is the Green’s function.
Proof Let be a solution of (1). We can rewrite equation (1) as
Multiply both sides of the above equation by and then integrate from t to to obtain
We arrive at
Dividing both sides of the above equation by and using Lemma 2.1, we have
Thus, the proof is complete. □
It is easy to verify that the Green’s function satisfies the property
Define K, a cone in , by
and an operator by
Lemma 2.7 and is compact and continuous.
Proof By using Theorem 2.1, for , we have
One can show that for , we have
Therefore, . We will prove that is continuous and compact. Firstly, we will consider the continuity of . Let and as , then and as for any . Because of the continuity of f, for any and , we have
where . Thus is continuous on K.
Next, we prove that is a compact operator. It is equal to proving that maps bounded sets into relatively compact sets.
Let be an arbitrary bounded set in K, then there exists a number such that for any . We prove that is compact. In fact, for any and , we have
which imply that and are uniformly bounded on . There exists a subsequence of converging uniformly on , namely, is compact. The proof is complete. □
Lemma 2.8 The existence of positive periodic solutions in shifts of (1) is equivalent to the existence of fixed point problem of in K.
The proof of Lemma 2.8 is straightforward and hence omitted.
3 Main result
In this section, we consider the existence of one or two positive T-periodic solutions in of (1). Let us define
To prove the results, we will use the following theorem which can be found in Krasnosel’skiĭ’s book .
Theorem 3.1 (Guo-Krasnoselskiĭ fixed point theorem)
Let X be a Banach space, be a cone, and suppose that and are open, bounded subsets of X with and . Suppose further that is a completely continuous operator such that either
for , for , or
for , for
holds. Then A has a fixed point in .
Theorem 3.2 If either , or , holds, then equation (1) has a positive T-periodic solution x in shifts for
Proof At first, in view of uniformly on , there exists such that for , , where . We define and if , then and for all t. We get
and so for all .
Next we use the assumption uniformly on . We can choose large enough such that
where . Then if is the ball in K centered at the origin with radius R and if , then we have
and so for all . Consequently, Theorem 3.1 yields the existence of a positive T-periodic solution of (1) in shifts , that is,
Next, let , hold. In view of , there exists such that
where . We define and if , then and for all t. We get
and so for all .
Next, since , there exists such that
where . Let and it follows that for and , where . For , we have
and again we have for . It follows from part (ii) of Theorem 3.1 that has a fixed point in and this implies that our given equation (1) has a positive T-periodic solution x in shifts , that is,
Theorem 3.3 Let hold. Further, assume that there is a constant such that
Then equation (1) has two positive T-periodic solutions in shifts for
Proof At first, in view of , there exists such that
where . Set . Then, for , we have
Next, since , then for any there exists such that for . Set . For , since , , we have
Finally, let . For , then from (14), we have
Hence, since and (16), (17) and (18), it follows from Theorem 3.1 that has a fixed point in and a fixed point in . Both are positive T-periodic solutions in shifts of equation (1) and . The proof is therefore complete. □
Theorem 3.4 Let hold. There exists a constant such that
Then equation (1) has two positive T-periodic solutions in shifts for
Proof It can be proved similarly to the second part of Theorem 3.2 and Theorem 3.3. □
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The author declares that they have no competing interests.