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Periodic Solutions in Shifts for a Nonlinear Dynamic Equation on Time Scales
Let be a periodic time scale in shifts . We use a fixed point theorem due to Krasnosel'skiĭ to show that nonlinear delay in dynamic equations of the form , has a periodic solution in shifts . We extend and unify periodic differential, difference, -difference, and -difference equations and more by a new periodicity concept on time scales.
The time scales approach unifies differential, difference, -difference, and -differences equations and more under dynamic equations on time scales. The theory of dynamic equations on time scales was introduced by Hilger in this Ph.D. thesis in 1988 . The existence problem of periodic solutions is an important topic in qualitative analysis of ordinary differential equations. There are only a few results concerning periodic solutions of dynamic equations on time scales such as in [2, 3]. In these papers, authors considered the existence of periodic solutions for dynamic equations on time scales satisfying the condition 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 condition (1.1). 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 . He defined a new periodicity concept with the aid of shift operators which are first defined in  and then generalized in .
Let be a periodic time scale in shifts with period and is nonnegative and fixed. We are concerned with the existence of periodic solutions in shifts for the nonlinear dynamic equation with a delay function : where is fixed if and if is periodic in shifts with period .
Kaufmann and Raffoul in  used Krasnosel'skiĭ fixed point theorem and showed the existence of a periodic solution of (1.2) and used the contraction mapping principle to show that the periodic solution is unique when satisfies condition (1.1). Similar results were obtained concerning (1.2) in [6, 7] in the case , , respectively. Currently, Adıvar and Raffoul used Lyapunov's direct method to obtain inequalities that lead to stability and instability of delay dynamic equations of (1.2) when on a time scale having a delay function in  and also using the topological degree method and Schaefers fixed point theorem, they deduce the existence of periodic solutions of nonlinear system of integrodynamic equations on periodic time scales in .
Hereafter, we use the notation to indicate the time scale interval . The intervals , and are similarly defined.
In Section 2, we will state some facts about the exponential function on time scales, the new periodicity concept for time scales, and some important theorems which will be needed to show the existence of a periodic solution in shifts . In Section 3, we will give some lemmas about the exponential function and the graininess function with shift operators. Finally, we present our main result in Section 4 by using Krasnosel'skiĭ fixed point theorem and give an example.
In this section, we mention some definitions, lemmas, and theorems from calculus on time scales which can be found in [10, 11]. 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 (see ). A function is said to be regressive provided for all , where . The set of all regressive rd-continuous functions is denoted by 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 [10, Theorem 2.36].
Lemma 2.2 (see ). Let . Then(i) and ;(ii); (iii), where, ;(iv); (v); (vi); (vii); (viii).
The following definitions, lemmas, corollaries, and examples are about the shift operators and new periodicity concept for time scales which can be found in .
Definition 2.3 (see ). Let be a nonempty subset of the time scale including a fixed number such that there exist operators satisfying the following properties. The function are strictly increasing with respect to their second arguments, that is, if
If with , then , and if with , then . If , then and . Moreover, if , then and holds. If , then and , respectively. 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 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.
Example 2.4 (see ). (i), , , , and .(ii), , , , and .(iii), , , , and .(iv), , , , and .
Definition 2.5 (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 a such that for all . Furthermore, if then is called the period of the time scale .
Example 2.6 (see ). The following time scales are not periodic in the sense of condition (1.1) but periodic with respect to the notion of shift operators given in Definition 2.5:(i),, ,,(ii), , ,,(iii), , ,,(iv),
Notice that the time scale in Example 2.6 is bounded above and below and is constant and .
Remark 2.7 (see ). Let be a time scale, that is, periodic in shifts with the period . Thus, by of Definition 2.3 the mapping defined by is surjective. On the other hand, by of Definition 2.3 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 . The operators are commutative with the forward jump operator given by . That is, for all .
Lemma 2.8 (see ). The mapping preserves the structure of the points in . That is,
Corollary 2.9 (see ). and for all .
Definition 2.10 (periodic function in shift ). Let be a time scale that is periodic in shifts with the period . We say that a real value function defined on is periodic in shifts if there exists a such that where . The smallest number such that (5) holds is called the period of .
Definition 2.11 (periodic function in shifts ). Let be a time scale that is periodic in shifts with the period . We say that a real value function defined on is periodic in shifts if there exists a such that where . The smallest number such that (2.9) hold is called the period of .
Now, we give two theorems concerning the composition of two functions. The first theorem is the chain rule on time scales [10, Theorem 1.93].
Theorem 2.12 (chain rule ). Assume that is strictly increasing and is a time scale. Let . If and exist for , then
Let be a time scale that is periodic in shifts . If one takes , then one has and .
The second theorem is the substitution rule on periodic time scales in shifts which can be found in .
Theorem 2.13 (see ). Let be a time scale that is periodic in shifts with period and a periodic function in shifts with the period . Suppose that , then
This work is mainly based on the following theorem .
Theorem 2.14 (Krasnosel'skiĭ). Let be a closed convex nonempty subset of a Banach space . Suppose that and map into such that (i) imply ,(ii) is completely continuous, (iii) is a contraction mapping.
Then there exists with .
3. Some Lemmas
In this section, we show some interesting properties of the exponential functions and shift operators on time scales.
Lemma 3.1. Let be a time scale that is periodic in shifts with the period and the shift is differentiable on where . Then the graininess function satisfies
Lemma 3.2. Let be a time scale, that is, periodic in shifts with the period and the shift is differentiable on , where . Suppose that is periodic in shifts with the period . Then,
Lemma 3.3. Let be a time scale, that is, periodic in shifts with the period and the shift is differentiable on where . Suppose that is periodic in shifts with the period . Then
4. Main Result
We will state and prove our main result in this section. We define where is the space of all real valued continuous functions. Endowed with the norm is a Banach space.
Lemma 4.1. Let . Then exists and .
Proof. Since , then , and by Corollary 2.9, we have . For all ,. Hence . Since , there exists such that . If is left scattered, then . And so, . Thus, we have . If is dense, and .
Assume that is left dense and right scattered. Note that if then we work . Fix and consider a sequence such that . Note that for all . By the continuity of , there exists such that for all , . This implies that . Since was arbitrary, then and the proof is complete.
In this paper we assume that is a continuous function with for all and where is continuous. We further assume that is continuous and periodic with in and Lipschitz continuous in and . That is, and there are some positive constants and such that
Proof. Let be a solution of (1.2). We can rewrite (1.2) as
Multiply both sides of the above equation by and then integrate from to to obtain
We arrive at
Dividing both sides of the above equation by and using and Lemma 2.2, we have
Now, we consider the first term of the integral on the right-hand side of (4.11) Using integration by parts from rule  we obtain Since and , the above equality reduces to
Substituting (4.15) into (4.11) we get
Thus the proof is complete.
Define the mapping by To apply Theorem 2.14 we need to construct two mappings: one map is a contraction and the other map is compact and continuous. We express (4.17) as where , are given and is defined in (4.7).
Proof. We show that . Evaluate (4.20) at ,
Now, since (4.3) and Corollary 2.9 hold, then we have
That is, is periodic in with period . Using the periodicity of , and Lemma 3.2 we get
That is, inside the integral of (4.21) is periodic in with period . By Theorem 2.13 and Lemma 3.2 we have
That is, .
To see that is continuous, we let with and and define
Given that , take such that . By making use of the Lipschitz inequality (4.5) in (4.20), we get where are given by (4.5) and . This proves that is continuous.
We need to show that is compact. Consider the sequence of periodic functions in and assume that the sequence is uniformly bounded. Let be such that , for all . In view of (4.5) we arrive at where . Hence, Thus, the sequence is uniformly bounded. If we find the derivative of , we have
Consequently, for all . That is, , for some positive constant . Thus the sequence is uniformly bounded and equicontinuous. The Arzela-Ascoli theorem implies that uniformly converges to a continuous periodic function in . Thus is compact.
Lemma 4.4. Let be defined by (4.19) and
Then is a contraction.
Proof. Trivially, . For , we have
Hence defines a contraction mapping with contraction constant .
Theorem 4.5. Let . Let , and be given by (4.39). Suppose that (4.3)–(4.5) and (4.31) hold and that there is a positive constant such that all solutions of (1.2), , satisfy , the inequality holds. Then (1.2) has a periodic solution in .
Proof. Define . Then Lemma 4.3 implies that is compact and continuous. Also, from Lemma 4.4, the mapping is contraction.
We need to show that if , we have . Let with . From (4.19) and (4.20) and the fact that , we have We see that all the conditions of Krasnosel'skiĭ theorem are satisfied on the set . Thus there exists a fixed point in such that . By Lemma 4.2, this fixed point is a solution of (2) has a periodic solution in .
Proof. Let the mapping be given by (4.17). For we have This completes the proof.
Example 4.7. Let be a periodic time scale in shift with period . We consider the dynamic equation (1.2) with , and .
The operators and are backward and forward shift operators for . Here , the initial point and for . If we consider conditions (4.3)-(4.4) we find . Then , satisfy condition (4.3), and satisfies the condition (4.4) for all . Also, is Lipschitz continuous in and for . Since , then the condition (4.31) holds.
If we compute , and , we have If we take , then inequality (4.33) satisfies.
Let . We show that . Integrate (1.2) from 1 to 4, we get
Since , then and so after integration by parts (23) becomes
Claim 1. There exist such that .
Suppose that the claim is false. Define . Then there exists such that for all . So, That is, , a contradiction.
As a consequence of the claim, we have where .
So, , which implies . Since for all , we have
This implies that
Taking the norm in (1.2) yields Substitution of (4.47) into (4.46) yields that for all . Then by Theorem 4.5, (1.2) has a periodic solution in shifts .
In this example, if we take , we have
So, all the conditions of Theorem 4.6 are satisfied. Therefore, (1.2) has a unique periodic solution in shifts .
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Copyright © 2012 Erbil Çetin and F. Serap Topal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.