Positive periodic solutions in shifts for a nonlinear first-order functional dynamic equation on time scales
© Çetin; licensee Springer. 2014
Received: 15 October 2013
Accepted: 13 February 2014
Published: 26 February 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 .
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 .
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).
, 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 
(P.1) The functions are strictly increasing with respect to their second arguments, i.e., ifthen
(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)
then P is called the period of the time scale .
Example 2.1 
, , .
, , , .
, , , .
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 
Corollary 2.1 
and for all .
Definition 2.4  (Periodic function in shift )
where . The smallest number such that (4) holds is called the period of f.
Definition 2.5  (Δ-periodic function in shifts )
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 .
We give some interesting properties of the exponential functions and shift operators on time scales which can be found in .
then is a Banach space.
Lemma 2.5 
Let . Then exists and .
where is the Green’s function.
Thus, the proof is complete. □
Lemma 2.7 and is compact and continuous.
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.
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
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)
for , for , or
for , for
holds. Then A has a fixed point in .
and so for all .
and so for all .
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. □
Proof It can be proved similarly to the second part of Theorem 3.2 and Theorem 3.3. □
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