Existence of periodic solutions in totally nonlinear delay dynamic equations

By means of a fixed point theorem we offer sufficient conditions f or the existence of periodic solutions of totally nonlinear delay dynamic equations, where the solution maps a periodic time scale into another time scale.


Introduction
We begin with asking the following questions: • When does a nonlinear delay difference equation have an integer valued periodic solution?
EJQTDE Spec.Ed.I, 2009 No. 1 Time scale theory has given mathematicians a general perspective of the understanding on how to combine and unify the theories of difference and differential equations under the umbrella of dynamic equations on time scales.Hence, it is natural to ask the following question which is much more general than the ones above: • When does a totally nonlinear delay dynamic equation have a non-zero periodic solution which maps a periodic time scale T 1 into another time scale T 2 ?
The expression totally nonlinear implies that the functions h and G of (3) are nonlinear in x.In earlier time scale papers (e.g., [2], [3], [7]) concerning the existence of periodic solutions of dynamic equations on a time scale T, sufficient conditions are given only for the existence of real valued periodic solutions in C(T, R).By doing so, existence is shown but the existence of positive periodic solutions is handled in a totally different manner.On the other hand, most of the studies of difference equations show the existence of real valued solutions x : Z → R.However, when we study some problem from biology, physics or any other applicable science described by difference equations, we should have integer valued solutions (see for instance [8], [9]).This, in return, requires showing the existence of integer valued solutions.The advantage of handling a problem on two time scales T 1 and T 2 instead of on a periodic time scale T and R not only fills this gap but also helps us to understand positivity of solutions.By this approach it is enough to set the problem on T 1 = T 2 = Z to obtain existence of integer valued solutions of a difference equation and it is easy to obtain positivity of solutions by taking the positive part of the time scale T 2 as the range of functions.It is worth mentioning that existence of periodic solutions of the equation (3) has not been studied before even for the particular case when T 1 is a periodic time scale and T 2 = R.
For clarity, we restate the following definitions, introductory examples, and lemmas which can be found in [3] and [7].Definition 1.1 A time scale T is said to be periodic if there exists P > 0 such that t ± P ∈ T for all t ∈ T. If T = R, the smallest positive P is called the period of the time scale.
Remark 1.1 All periodic time scales are unbounded above and below.Definition 1.2 Let T = R be a periodic time scale with period P .We say that the function f : T → R is periodic with period T if there exists a natural number n such that T = nP , f (t ± T ) = f (t) for all t ∈ T and T is the smallest number such that Let T 1 be a periodic time scale and T 2 a time scale that is closed under addition, i.e., u + v ∈ T 2 for all u, v ∈ T 2 .In this paper, using the concept of large contraction, we study existence of periodic solutions x : where a : In the following, we give some particular time scales with corresponding delay functions.

Time scale Delay function
Throughout the paper we suppose that the functions a, h, and G are continuous in their respective domains and that for at least T > 0 To avoid obtaining the zero solution we also suppose that G(t, 0) = a(t)h(0) for some t ∈ T 1 .
In the analysis, we employ a fixed point theorem in which the notion of a large contraction is required as one of the sufficient conditions.First, we give the following definition which can be found in [6].B is said to be a large contraction if φ, ϕ ∈ M, with φ = ϕ then d(Bφ, Bϕ) ≤ d(φ, ϕ) and if for all ε > 0, there exists a δ < 1 such that The next theorem, which constitutes a basis for our main result, is a reformulated version of Krasosel'kii's fixed point theorem.
Theorem 1.1 [6] Let M be a bounded convex nonempty subset of a Banach space (B, • ).Suppose that A and B map M into B such that i. x, y ∈ M, implies Ax + By ∈ M; ii.A is compact and continuous; iii.B is a large contraction mapping.
Then there exists z ∈ M with z = Az + Bz.
Define the forward jump operator σ by The set of all regressive rd-continuous functions p : T → R is denoted by R while the set R + is given by R + = {p ∈ R : 1 + µ(t)p(t) > 0 for all t ∈ T}.
Let p ∈ R and µ(t) > 0 for all t ∈ T. The exponential function on T is defined by The exponential function y(t) = e p (t, s) is the solution of the initial value problem y ∆ = p(t)y, y(s) = 1.Other properties of the exponential function are given by the following.
and contains no right scattered elements of T, and f is differentiable at each t ∈ D.
We will resort to the next theorem at several occasions in our further work.
Theorem 1.3 [4, Theorem 1.67-Corollary 1.68] Let f and g be real-valued functions defined on T, both pre-differentiable with D ⊂ T. Then 2. If U is a compact interval with endpoints r, s ∈ T, then 2 Existence of periodic solution Suppose that T 1 is a periodic time scale and that T 2 is an arbitrary time scale that is closed under addition.The space P T (T 1 , T 2 ), given by is a Banach space when it is endowed with the supremum norm Determine α ∈ (0, ∞) to be a fixed real number such that We ask for the condition (10) since we need to guarantee that the set M α given by (which will be shown to include a solution of ( 4) Moreover, M α is a closed, bounded, and convex subset of the Banach space P T (T 1 , T 2 ).Hereafter, we use the notation γ = −a and assume that γ ∈ R + .
Lemma 2.1 If x ∈ P T (T 1 , T 2 ), then x is a solution of equation ( 4) if, and only if, where Proof.Let x ∈ P T (T 1 , T 2 ) be a solution of (4).The equation ( 4) can be expressed as Multiplying both sides of ( 14) by e ⊖γ (σ (t) , t 0 ) we get Taking the integral from t − T to t, we arrive at It follows from ( 7) that e p (t, s) > 0. On the other hand, since we have exp(u) ≥ 1 + u, and therefore, u ≥ log(1 + u) for all u ∈ ( − 1, ∞), we find This completes the proof.We derive the next result from (15).
Corollary 2.1 If p ∈ R + and p(t) < 0 for all t ∈ T, then for all s ∈ T with s ≤ t we have As a consequence of Lemma 2.2 we note that for Let the maps A and B be defined by respectively.It is clear from (6) that the maps A and B are T periodic.To make sure A : M α → P T (T 1 , T 2 ) and B : M α → P T (T 1 , T 2 ) we also need to ask for the following condition: In the proof of the next result, we use a time scale version of the Lebesgue dominated convergence theorem.For a detailed study on ∆-Riemann and Lebesgue integrals on time scales we refer the reader to [5].
Lemma 2.3 Suppose that there exists a positive valued function ξ : Then the mapping A, defined by (18), is continuous on M α .
Proof.To see that A is a continuous mapping, let {ϕ i } i∈N be a sequence of functions in M α such that ϕ i → ϕ as i → ∞.Since (21) holds, the continuity of G, and the dominated convergence theorem yield where K is defined as in (17).This shows continuity of the mapping A. The proof is complete.
One may illustrate with the following example what kind of functions ξ, satisfying (21), can be chosen to show the continuity of A.
Example 2.1 Assume that G(t, x) satisfies a Lipschitz condition in x; i.e., there is a positive constant k such that Then for ϕ ∈ M α , In this case we may choose ξ as EJQTDE Spec.Ed.I, 2009 No. 1 Another possible ξ satisfying (21) is the following where n is a positive integer and g and p are continuous functions on T 1 , and y ∈ M α .
Remark 2.1 Condition ( 22) is strong since it requires the function G to be globally Lipschitz.A lesser condition is (21) in which ξ can be directly chosen as in ( 23) or (24).
In next two results we assume that for all t ∈ T 1 and ψ ∈ M α , where ξ is defined by (21).This shows A(M α ) is uniformly bounded.It is left to show that A(M α ) is equicontinuous.Since ξ is continuous and T -periodic, by (21) and differentiation of (18) with respect to t ∈ T 1 (for the differentiation rule see [1, Lemma 1]) we arrive at where L is a positive constant.Thus, the estimation on |(Aϕ i ) ∆ (t)| and (9) imply that A(M α ) is equicontinuous.Then the Arzela-Ascoli theorem yields compactness of the mapping A. The proof is complete.T 2 is closed under addition and so (20) implies Proof.Let A and B be defined by ( 18) and (19), respectively.By Lemma 2.4, the mapping A is compact and continuous.Then using (25), (26), and the periodicity of A and B, we have Hence an application of Theorem 1.1 implies the existence of a periodic solution in M α .This completes the proof.
The next result gives a relationship between the mappings H and B in the sense of large contraction.Taking the supremum over the set [0, T ] ∩ T 1 , we get that Bx − By ≤ ||x − y||.One may also show in a similar way that Bx − By ≤ δ||x − y|| holds if we know the existence of a 0 < δ < 1 such that for all ε > 0 The proof is complete.
From Theorem 2.1 and Lemma 2.5, we deduce the following result.

Classification and applications
We derive the next result by making use of Theorem 1.3.
Lemma 3.1 Suppose g : T → R is pre-differentiable with D. Suppose U is a compact interval with endpoints r, s ∈ T and g ∆ (t) ≥ 0 for all t ∈ U κ ∩ D. Then we have Proof.Let the function f : T → R be defined by Evidently, f is pre-differentiable with D and From ( 8), we derive as desired.The proof is complete.
Corollary 3.1 Suppose g : T → R is pre-differentiable with D. Suppose U is a compact interval with endpoints r, s ∈ T. g ∆ (t) ≥ 0 for all t ∈ U κ ∩ D if and only if g is non-decreasing on U.
Proof.If g ∆ (t) ≥ 0 for all t ∈ U κ ∩ D, then from (27), we have for s, r ∈ U with s ≥ r.Conversely, let g be non-decreasing on U.For a t ∈ U κ ∩ D, there are two possible cases: If µ(t) = 0, then from [4, Theorem 1.16, (iii), Exercise 1.17] we find This completes the proof.Corollary 2.2 shows that having a large contraction on a class of periodic functions plays a substantial role in proving existence of periodic solutions.We deduce by the next theorem that are the conditions implying that the mapping H in ( 13) is a large contraction on the set M α .Theorem 3.1 Let h : T 2 → T 2 be a function satisfying (H.1-H.3).Then the mapping H is a large contraction on the set M α .
To see this, choose a fixed ε ∈ (0, 1) and assume that φ and ϕ are two functions in Since h is continuous and strictly increasing, the function h u + ε 2 − h(u) attains its minimum on the closed and bounded interval [−α, α].Thus, if ε 2 < |φ(t) − ϕ(t)| for some t ∈ D(φ, ϕ), then from (31) and (H.3) we conclude that and therefore, where Consequently, it follows from (33) and (34) that where The proof is complete.If T 2 is a time scale such that the interval U α = [−α, α] ∩ T 2 contains negative reals, then functions of type h 1 (t) = ∆ .For the particular time scale T = R, it is easy to see by the chain rule, But for an arbitrary time scale T, it is not that easy since the rule for {f n+1 (t)} ∆ is changed to (see [4,Exercise 1.23]).Throughout the discussion below we shall always assume that f : T → R is a nondecreasing differentiable function.If f (t)f (σ(t)) = 0 for some t ∈ T, then (35) implies and therefore, On the other hand, If f (t)f (σ(t)) = 0, there are three possibilities: (i) 0 < f (t) ≤ f (σ(t)), (ii) f (t) ≤ f (σ(t)) < 0, and (iii) f (t) < 0 < f (σ(t)).Let us separate these cases by defining the sets The next lemma gives the relationship between [f 2n+1 (t)] ∆ and (2n+1)f 2n (t)f ∆ (t).

Proof.
We use the formula (35).(37) follows from (35) and the fact that On the other hand, for all t ∈ S − we have and hence, The proof is complete.From ( 36)-(39) we derive the following result.The proof is complete.
Then the mapping H defined by (13) defines a large contraction on the set M α .
By choosing T 2 = Z in the next result one obtains sufficient conditions for the existence of integer valued periodic solution of the nonlinear difference equation (1).Corollary 3.5 Let T 1 be a periodic time scale.For a fixed n ∈ Z + define the function h by (52) and set G(t, x(δ(t))) = b(t) (2n + 1)(α + 1) 2n x 2n+1 (δ(t)) + c(t).
Let the mappings A and B be given by ( 18) and (19), respectively.Suppose that the time scale T 2 is closed under addition and that the condition (20) holds.If a is positive valued T periodic function and c = 0 ∈ P T (T 1 , R), then (α + 1) It is worth mentioning that Theorem 3.5 is given for arbitrary time scales T 1 and T 2 , where T 1 is assumed to be periodic and T 2 is a time scale, closed under addition, such that (20) holds.One may apply this theorem to the particular time scales including the following cases:

Example 1 . 1
The following time scales are periodic.i. T = Z has period P = 1; ii.T = hZ has period P = h; iii.T = R; EJQTDE Spec.Ed.I, 2009 No. 1

Definition 1 . 3 (
Large Contraction) Let (M,d) be a metric space and B: M → M.

Corollary 2 . 2
In addition to the assumptions of Theorem 2.1, suppose also that a is a positive valued function.If H is a large contraction on M α , then (4) has a periodic solution in M α .EJQTDE Spec.Ed.I, 2009 No. 1 EJQTDE Spec.Ed.I, 2009 No. 1 EJQTDE Spec.Ed.I, 2009 No. 1

1 Lemma 3 . 2
EJQTDE Spec.Ed.I, 2009 No. Let f : T → R be a differentiable function in D. If f is a non decreasing function, then Spec.Ed.I, 2009 No. 1

1 .
T 1 = T 2 = R; EJQTDE Spec.Ed.I, 2009 No. 1 {s > t : s ∈ T} , and the graininess function µ by µ(t) = σ(t) − t.A point t in a time scale is called right scattered if σ(t) > t.Hereafter, we denote by x σ the composite function x • σ.Note that in a periodic time scale T with period P the inequality 0 ≤ µ(t) ≤ P holds for all t ∈ T.