AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 707319 10.1155/2012/707319 707319 Research Article Periodic Solutions in Shifts δ± for a Nonlinear Dynamic Equation on Time Scales Çetin Erbil Topal F. Serap Braverman Elena Department of Mathematics Ege University Bornova, 35100 Izmir Turkey ege.edu.tr 2012 13 8 2012 2012 16 03 2012 27 06 2012 2012 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.

Let 𝕋 be a periodic time scale in shifts δ±. We use a fixed point theorem due to Krasnosel'skiĭ to show that nonlinear delay in dynamic equations of the form xΔ(t)=-a(t)xσ(t)+b(t)xΔ(δ-(k,t))δ-Δ(k,t)+q(t,x(t),x(δ-(k,t))),t𝕋, has a periodic solution in shifts δ±. We extend and unify periodic differential, difference, h-difference, and q-difference equations and more by a new periodicity concept on time scales.

1. Introduction

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 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 (1.1)“thereexistsa  ω>0    suchthat    t±ωTtT. Under this condition all periodic time scales are unbounded above and below. However, there are many time scales such as q¯={qn:n}{0} and ={n:n} 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 t±ω for a fixed ω>0. He defined a new periodicity concept with the aid of shift operators δ± which are first defined in  and then generalized in .

Let 𝕋 be a periodic time scale in shifts δ± with period P(t0,)𝕋 and t0𝕋 is nonnegative and fixed. We are concerned with the existence of periodic solutions in shifts δ± for the nonlinear dynamic equation with a delay function δ-(k,t): (1.2)xΔ(t)=-a(t)xσ(t)+b(t)xΔ(δ-(k,t))δ-Δ(k,t)+q(t,x(t),x(δ-(k,t))),tT, where k is fixed if 𝕋= and k[P,)𝕋 if 𝕋 is periodic in shifts δ± with period P.

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 𝕋 satisfies condition (1.1). Similar results were obtained concerning (1.2) in [6, 7] in the case 𝕋=, 𝕋=, 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 q=0 on a time scale having a delay function δ- 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 [a,b]𝕋 to indicate the time scale interval [a,b]𝕋. The intervals [a,b)𝕋, (a,b]𝕋 and (a,b)𝕋 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 δ±. 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.

2. Preliminaries

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 [<xref ref-type="bibr" rid="B4">10</xref>]).

A function p:𝕋 is said to be regressive provided 1+μ(t)p(t)0 for all t𝕋κ, where μ(t)=σ(t)-t. The set of all regressive rd-continuous functions φ:𝕋 is denoted by while the set + is given by +={φ:1+μ(t)φ(t)>0  forall  t𝕋}.

Let φ and μ(t)>0 for all t𝕋. The exponential function on 𝕋 is defined by (2.1)eφ(t,s)=exp(stζμ(r)(φ(r))Δr), where ζμ(s) is the cylinder transformation given by (2.2)ζμ(r)(φ(r)):={1μ(r)Log(1+μ(r)φ(r)),  if    μ(r)>0,φ(r),  if    μ(r)=0.   Also, the exponential function y(t)=ep(t,s) is the solution to the initial value problem yΔ=p(t)y,y(s)=1. Other properties of the exponential function are given in the following lemma [10, Theorem 2.36].

Lemma 2.2 (see [<xref ref-type="bibr" rid="B4">10</xref>]).

Let p,q. Then

e0(t,s)1 and ep(t,t)1;

ep(σ(t),s)=(1+μ(t)p(t))ep(t,s);

1/ep(t,s)=e(t,s), where, p(t)=-(p(t))/(1+μ(t)p(t));

ep(t,s)=1/ep(s,t)=ep(s,t);

ep(t,s)ep(s,r)=ep(t,r);

ep(t,s)eq(t,s)=epq(t,s);

ep(t,s)/eq(t,s)=epq(t,s);

(1/ep(·,s))Δ=-p(t)/epσ(·,s).

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 [<xref ref-type="bibr" rid="B3">12</xref>]).

Let 𝕋* be a nonempty subset of the time scale 𝕋 including a fixed number t0𝕋* such that there exist operators δ±:[t0,)𝕋×𝕋*𝕋* satisfying the following properties.

The function δ± are strictly increasing with respect to their second arguments, that is, if (2.3)(T0,t),(T0,u)D±:={(s,t)[t0,)T×T*:δ(s,t)T*}, then (2.4)T0t<u    implies    δ±(T0,t)<δ±(T0,u).

If (T1,u),(T2,u)𝒟 with T1<T2, then δ-(T1,u)>δ-(T2,u), and if (T1,u),(T2,u)D+ with T1<T2, then δ+(T1,u)<δ+(T2,u).

If t[t0,)𝕋, then (t,t0)D+ and δ+(t,t0)=t. Moreover, if t𝕋*, then (t0,t)D+ and δ+(t0,t)=t holds.

If (s,t)D±, then (s,δ±(s,t))D and δ(s,δ±(s,t))=t, respectively.

If (s,t)D± and (u,δ±(s,t))D±, then (s,δ(u,t))D± and δ(u,δ±(s,t))=δ±(s,δ(u,t)), respectively.

Then the operators δ- and δ+ associated with t0𝕋* (called the initial point) are said to be backward and forward shift operators on the set 𝕋*, respectively. The variable s[t0,)𝕋 in δ±(s,t) is called the shift size. The values δ+(s,t) and δ-(s,t) in 𝕋* indicate s units translation of the term t𝕋* to the right and left, respectively. The sets 𝒟± are the domains of the shift operator δ±, respectively. Hereafter, 𝕋* is the largest subset of the time scale 𝕋 such that the shift operators δ±:[t0,)𝕋×𝕋*𝕋* exist.

Example 2.4 (see [<xref ref-type="bibr" rid="B3">12</xref>]).

𝕋 = , t0=0, 𝕋*=, δ-(s,t)=t-s, and δ+(s,t)=t+s.

𝕋 = , t0=0, 𝕋*=, δ-(s,t)=t-s, and δ+(s,t)=t+s.

𝕋 = q { 0 } , t0=1, 𝕋*=q, δ-(s,t)=t/s, and δ+(s,t)=ts.

𝕋 = 1 / 2 , t0=0, 𝕋*=1/2, δ-(s,t)=t2-s2, and δ+(s,t)=t2+s2.

Definition 2.5 (periodicity in shifts [<xref ref-type="bibr" rid="B3">12</xref>]).

Let 𝕋 be a time scale with the shift operators δ± associated with the initial point t0𝕋*. The time scale 𝕋 is said to be periodic in shift δ± if there exists a p(t0,)𝕋* such that (p,t)D± for all t𝕋*. Furthermore, if (2.5)P:=inf  {p(t0,)T*:(p,t)D±,tT*}t0, then P is called the period of the time scale 𝕋.

Example 2.6 (see [<xref ref-type="bibr" rid="B3">12</xref>]).

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:

𝕋1={±n2:n},δ±(P,t)={(t±P)2,t>0;±P,t=0;-(-t±P)2,t<0;, P=1,t0=0,

𝕋2=q¯, δ±(P,t)=P±1t, P=q,t0=1,

𝕋3=n[22n,22n+1]¯,  δ±(P,t)=P±1t,  P=4,t0=1,

𝕋4={qn/(1+qn):q>1    is  constant  and  n}{0,1}, (2.6)δ±(P,t)=q((ln(t/(1-t))±ln(P/(1-P)))/lnq)1+q((ln(t/(1-t))±ln(P/(1-P)))/lnq),P=q1-q.

Notice that the time scale 𝕋4 in Example 2.6 is bounded above and below and 𝕋4*={qn/(1+qn):q>1 is constant and n}.

Remark 2.7 (see [<xref ref-type="bibr" rid="B3">12</xref>]).

Let 𝕋 be a time scale, that is, periodic in shifts with the period P. Thus, by (P.4) of Definition 2.3 the mapping δ+P:𝕋*𝕋* defined by δ+P(t)=δ+(P,t) is surjective. On the other hand, by (P.1) of Definition 2.3 shift operators δ± are strictly increasing in their second arguments. That is, the mapping δ+P(t)=δ+(P,t) is injective. Hence, δ+P is an invertible mapping with the inverse (δ+P)-1=δ-P defined by δ-P(t):=δ-(P,t).

We assume that 𝕋 is a periodic time scale in shift δ± with period P. The operators δ±P:𝕋*𝕋* are commutative with the forward jump operator σ:𝕋𝕋 given by σ(t):=inf{s𝕋:s>t}. That is, (δ±Pσ)(t)=(σδ±P)(t) for all t𝕋*.

Lemma 2.8 (see [<xref ref-type="bibr" rid="B3">12</xref>]).

The mapping δ+P:𝕋*𝕋* preserves the structure of the points in 𝕋*. That is, (2.7)σ(t)=timpliesσ(δ+(P,t))=δ+(P,t)andσ(t)>timpliesσ(δ+(P,t))>δ+(P,t).

Corollary 2.9 (see [<xref ref-type="bibr" rid="B3">12</xref>]).

δ + ( P , σ ( t ) ) = σ ( δ + ( P , t ) ) and     δ-(P,σ(t))=σ(δ-(P,t)) for all t𝕋*.

Definition 2.10 (periodic function in shift <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M189"><mml:mrow><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mo>±</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> [<xref ref-type="bibr" rid="B3">12</xref>]).

Let 𝕋 be a time scale that is periodic in shifts δ± with the period P. We say that a real value function f defined on 𝕋* is periodic in shifts δ± if there exists a T[P,)𝕋* such that (2.8)(T,t)D±,  f(δ±T(t))=f(t)tT*, where δ±T:=δ±(T,t). The smallest number T[P,)𝕋* such that (5) holds is called the period of f.

Definition 2.11 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M201"><mml:mi mathvariant="normal">Δ</mml:mi><mml:mtext>-</mml:mtext></mml:math></inline-formula>periodic function in shifts <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M202"><mml:mrow><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mo>±</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> [<xref ref-type="bibr" rid="B3">12</xref>]).

Let 𝕋 be a time scale that is periodic in shifts δ± with the period P. We say that a real value function f defined on 𝕋* is Δ-periodic in shifts δ± if there exists a T[P,)𝕋* such that (2.9)(T,t)D±tT*,      theshiftsδ±TareΔ-differentiablewithrd-continuousderivatives,f(δ±T(t))δ±ΔT=f(t)tT*, where δ±T:=δ±(T,t). The smallest number T[P,)𝕋* such that (2.9) hold is called the period of f.

Notice that Definitions 2.10 and 2.11 give the classic periodicity definition on time scales whenever δ±T:=t±T are the shifts satisfying the assumptions of Definitions 2.10 and 2.11.

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 [<xref ref-type="bibr" rid="B4">10</xref>]).

Assume that υ:𝕋 is strictly increasing and 𝕋~:=υ(𝕋) is a time scale. Let w:𝕋~. If νΔ(t) and wΔ~ exist for t𝕋κ, then (2.10)(wv)Δ=(wΔ~ν)νΔ.

Let 𝕋 be a time scale that is periodic in shifts δ±. If one takes ν(t)=δ±(T,t), then one has ν(𝕋)=𝕋 and [f(ν(t))]Δ=(fΔν(t))νΔ(t).

The second theorem is the substitution rule on periodic time scales in shifts δ± which can be found in .

Theorem 2.13 (see [<xref ref-type="bibr" rid="B3">12</xref>]).

Let 𝕋 be a time scale that is periodic in shifts δ± with period P[t0,)𝕋* and f a Δ-periodic function in shifts δ± with the period T[P,)𝕋*. Suppose that f𝒞rd(𝕋), then (2.11)t0tf(s)Δs=δ±T(t0)δ±T(t)f(s)Δs.

This work is mainly based on the following theorem .

Theorem 2.14 (Krasnosel'skiĭ).

Let M be a closed convex nonempty subset of a Banach space (𝔹,.). Suppose that A and B map M into 𝔹 such that

x,yM imply Ax+ByM,

A is completely continuous,

B is a contraction mapping.

Then there exists zM with z=Az+Bz.

3. Some Lemmas

In this section, we show some interesting properties of the exponential functions ep(t,t0) and shift operators on time scales.

Lemma 3.1.

Let 𝕋 be a time scale that is periodic in shifts δ± with the period P and the shift δ±T is Δ- differentiable on t𝕋* where T[P,)𝕋*. Then the graininess function μ:𝕋[0,) satisfies (3.1)μ(δ±T(t))=δ±ΔT(t)μ(t).

Proof.

Since δ±T is Δ-differentiable at t we can use Theorem 1.16 (iv) in . Then we have (3.2)μ(t)δ±ΔT(t)=δ±T(σ(t))-δ±T(t).

Then by using Corollary 2.9 we have (3.3)μ(t)δ±ΔT(t)=σ(δ±T(t))-δ±T(t)=μ(δ±T(t)).

Thus, the proof is complete.

Lemma 3.2.

Let 𝕋 be a time scale, that is, periodic in shifts δ± with the period P and the shift δ±T is Δ-differentiable on t𝕋*, where T[P,)𝕋*. Suppose that p is Δ-periodic in shifts δ± with the period T[P,)𝕋*. Then, (3.4)ep(δ±T(t),δ±T(t0))=ep(t,t0)fort,t0T*.

Proof.

Assume that μ(τ)0. Set f(τ)=(1/μ(τ))Log(1+p(τ)μ(τ)). Using Lemma 3.1 and Δ-periodicity of p in shifts δ± we get (3.5)f(δ±T(τ))δ±ΔT(τ)=δ±ΔT(τ)μ(δ±T(τ))Log(1+p(δ±T(τ))μ(δ±T(τ)))=δ±ΔT(τ)μ(δ±T(τ))Log(1+p(δ±T(τ))δ±ΔT1δ±ΔTμ(δ±T(τ)))=1μ(τ)Log(1+p(τ)μ(τ))=f(τ).

Thus, f is Δ-periodic in shifts δ± with the period T. By using Theorem 2.13 we have (3.6)ep(δ±T(t),δ±T(t0))={exp(δ±T(t0)δ±T(t)1μ(τ)Log(1+p(τ)μ(τ))Δτ),forμ(τ)0,exp(δ±T(t0)δ±T(t)p(τ)Δτ),forμ(τ)=0,={exp(t0t1μ(τ)Log(1+p(τ)μ(τ))Δτ),for    μ(τ)0,exp(t0tp(τ)Δτ),for    μ(τ)=0,=ep(t,t0).

The proof is complete.

Lemma 3.3.

Let 𝕋 be a time scale, that is, periodic in shifts δ± with the period P and the shift δ±T is Δ-differentiable on t𝕋* where T[P,)𝕋*. Suppose that p is Δ-periodic in shifts δ± with the period T[P,)𝕋*. Then (3.7)ep(δ±T(t),σ(δ±T(s)))=ep(t,σ(s))=ep(t,s)1+μ(t)p(t) fort,sT*.

Proof.

From Corollary 2.9, we know σ(δ±T(s))=δ±T(σ(s)). By Lemmas 3.2 and 2.2 we obtain (3.8)ep(δ±T(t),σ(δ±T(s)))=ep(δ±T(t),δ±T(σ(s)))=ep(t,σ(s))=ep(t,s)1+μ(t)p(t).

The proof is complete.

4. Main Result

We will state and prove our main result in this section. We define (4.1)PT={xC(T,R):x(δ+T(t))=x(t)}, where 𝒞(𝕋,) is the space of all real valued continuous functions. Endowed with the norm (4.2)x=maxt[t0,δ+T(t0)]T|x(t)|,PT is a Banach space.

Lemma 4.1.

Let xPT. Then xσ exists and xσ=x.

Proof.

Since xPT, then x(δ+T(t0))=x(t0), and by Corollary 2.9, we have x(σ(δ+T(t0)))=x(σ(t0)). For all t[t0,δ+T(t0)]𝕋,|x(σ(t))|x. Hence xσx. Since x𝒞(𝕋,), there exists t1[t0,δ+T(t0)] such that x=|x(t1)|. If t1 is left scattered, then σ(ρ(t1))=t1. And so, xσ|xσ(ρ(t1))|=x(t1)=x. Thus, we have xσ=x. If t1 is dense, σ(t1)=t1 and xσ=x.

Assume that t1 is left dense and right scattered. Note that if t1=t0 then we work t1=δ+T(t0). Fix ϵ>0 and consider a sequence {an} such that ant1. Note that σ(an)t1 for all n. By the continuity of x, there exists N such that for all n>N, |x(t1)-xσ(an)|<ϵ. This implies that x-ϵxσ. Since ϵ>0 was arbitrary, then x=xσ and the proof is complete.

In this paper we assume that a(t)+ is a continuous function with a(t)>0 for all t𝕋 and (4.3)a(δ+T(t))δ+ΔT(t)=a(t),      b(δ+T(t))=b(t), where bΔ(t) is continuous. We further assume that q(t,x,y) is continuous and periodic with δ± in t and Lipschitz continuous in x and y. That is, (4.4)q(δ+T(t),x,y)δ+ΔT(t)=q(t,x,y), and there are some positive constants L and E such that (4.5)|q(t,x,y)-q(t,z,w)|Lx-z+Ey-w.

Lemma 4.2.

Suppose that (4.3)–(4.5) hold. If x(t)PT, then x(t) is a solution of (1.2) if and only if (4.6)  x(t)=b(t)x(δ-(k,t))+11-ea(t)(t,δ-T(t))×δ-T(t)t[-r(s)xσ(δ-(k,s))+q(s,x(s),x(δ-(k,s)))]ea(s)(t,s)Δs, where (4.7)r(s)=a(s)bσ(s)+bΔ(s).

Proof.

Let x(t)PT be a solution of (1.2). We can rewrite (1.2) as (4.8)xΔ(t)+a(t)xσ(t)=b(t)xΔ(δ-(k,t))δ-Δ(k,t)+q(t,x(t),x(δ-(k,t))).         Multiply both sides of the above equation by ea(t)(t,t0) and then integrate from δ-T(t) to t to obtain (4.9)δ-T(t)t[x(s)ea(s)(s,t0)]ΔΔs=δ-T(t)t[b(s)xΔ(δ-(k,s))δ-Δ(k,s)+q(s,x(s),x(δ-(k,s)))]ea(s)(s,t0)Δs. We arrive at (4.10)x(t)[ea(t)(t,t0)-ea(t)(δ-T(t),t0)]=δ-T(t)t[b(s)xΔ(δ-(k,s))δ-Δ(k,s)+q(s,x(s),x(δ-(k,s)))]ea(s)(s,t0)Δs. Dividing both sides of the above equation by ea(t)(t,t0) and using x(δ+T(t))=x(t) and Lemma 2.2, we have (4.11)x(t)(1-ea(t)(δ-T(t),t))=δ-T(t)t[b(s)xΔ(δ-(k,s))δ-Δ(k,s)+q(s,x(s),x(δ-(k,s)))]ea(s)(s,t)Δs.

Now, we consider the first term of the integral on the right-hand side of (4.11) (4.12)δ-T(t)tb(s)xΔ(δ-(k,s))δ-Δ(k,s)ea(s)(s,t)Δs. Using integration by parts from rule  we obtain (4.13)δ-T(t)tb(s)xΔ(δ-(k,s))δ-Δ(k,s)ea(s)(s,t)Δs=δ-T(t)t[b(s)ea(s)(s,t)x(δ-(k,s))]ΔΔs-δ-T(t)t[b(s)ea(s)(s,t)]sΔxσ(δ-(k,s))Δs(4.14)δ-T(t)tb(s)xΔ(δ-(k,s))δ-Δ(k,s)ea(s)(s,t)Δs=b(t)ea(t)(t,t)x(δ-(k,t))-b(δ-T(t))ea(s)(δ-T(t),t)x(δ-(k,δ-T(t)))-δ-T(t)t[bσ(s)a(s)ea(s)(s,t)+bΔ(s)ea(s)(s,t)]xσ(δ-(k,s))Δs.                     Since b(δ-T(t))=b(t) and x(δ-T(t))=x(t), the above equality reduces to (4.15)δ-T(t)tb(s)xΔ(δ-(k,s))δ-Δ(k,s)ea(s)(s,t)Δs=b(t)x(δ-(k,t))(1-ea(s)(δ-T(t),t))-δ-T(t)t[a(s)bσ(s)+bΔ(s)]xσ(δ-(k,s))ea(s)(s,t)Δs.

Substituting (4.15) into (4.11) we get (4.16)x(t)=b(t)x(δ-(k,t))+11-ea(t)(t,δ-T(t))        ×δ-T(t)t[-r(s)xσ(δ-(k,s))+q(s,x(s),x(δ-(k,s)))]ea(s)(t,s)Δs.

Thus the proof is complete.

Define the mapping H:PTPT by (4.17)Hx(t):=b(t)x(δ-(k,t))+1(1-ea(t)(t,δ-T(t)))×δ-T(t)t[-r(s)xσ(δ-(k,s))+q(s,x(s),x(δ-(k,s)))]ea(s)(t,s)Δs. 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 (4.18)Hx(t)=Bx(t)+Ax(t), where A, B are given (4.19)Bx(t)=b(t)x(δ-(k,t)),(4.20)Ax(t)=11-ea(t)(t,δ-T(t))×δ-T(t)t[-r(s)xσ(δ-(k,s))+q(s,x(s),x(δ-(k,s)))]ea(s)(t,s)Δs, and r(s) is defined in (4.7).

Lemma 4.3.

Suppose that (4.3)–(4.5) hold. Then A:PTPT, as defined by (4.20), is compact and continuous.

Proof.

We show that A:PTPT. Evaluate (4.20) at δ+T(t), (4.21)        Ax(δ+T(t))=11-ea(t)(δ+T(t),δ-T(δ+T(t)))×δ-T(δ+T(t))δ+T(t)[-r(s)xσ(δ-(k,s))+q(s,x(s),x(δ-(k,s)))]ea(s)(δ+T(t),s)Δs=11-ep(t)(δ+T(t),t)×δ+T(δ-T(t))δ+T(t)[-r(s)xσ(δ-(k,s))+q(s,x(s),x(δ-(k,s)))]ea(s)(δ+T(t),s)Δs. Now, since (4.3) and Corollary 2.9 hold, then we have (4.22)r(δ+T(s))δ+TΔ(s)=a(δ+T(s))δ+TΔ(s)bσ(δ+T(s))+bΔ(δ+T(s))δ+TΔ(s)=a(s)bσ(s)+bΔ(s)=r(s). That is, r(s) is Δ-periodic in δ± with period T. Using the periodicity of r,  x,q, and Lemma 3.2 we get (4.23)[-r(δ+T(s))xσ(δ-(k,δ+T(s)))+q(δ+T(s),x(δ+T(s)),x(δ-(k,δ+T(s))))]δ+TΔ(s)×ea(s)(δ+T(t),δ+T(s))=[-r(s)xσ(δ-(k,s))+q(s,x(s),x(δ-(k,s)))]ea(s)(t,s). That is, inside the integral of (4.21) is Δ-periodic in δ± with period T. By Theorem 2.13 and Lemma 3.2 we have (4.24)Ax(δ+T(t))=11-ea(t)(t,δ-T(t))×δ-T(t)t[-r(s)xσ(δ-(k,s))+q(s,x(s),x(δ-(k,s)))]ea(s)(t,s)Δs=Ax(t).

That is, A:PTPT.

To see that A is continuous, we let φ,  ψPT with φC and ψC and define (4.25)η:=maxt[t0,δ+T(t0)]T|(1-ea(t)(t,δ-T(t)))-1|,  γ:=maxu[δ-T(t),t]Tea(t)(t,u),  β:=maxt[t0,δ+T(t0)]T|r(t)|.

Given that ϵ>0, take δ=ϵ/M such that φ-ψ<δ. By making use of the Lipschitz inequality (4.5) in (4.20), we get (4.26)Aφ-Aψγηδ-T(t)tβφ-ψ+Lφ-ψ+Eφ-ψΔs=ηγ[β+L+E](t0-δ-T(t0))φ-ψMφ-ψ<ϵ, where L,E are given by (4.5) and M=ηγ[β+L+E](t0-δ-T(t0)). This proves that A is continuous.

We need to show that A is compact. Consider the sequence of periodic functions in δ±{φn}PT and assume that the sequence is uniformly bounded. Let R>0 be such that φnR, for all n. In view of (4.5) we arrive at (4.27)|q(t,x,y)|=|q(t,x,y)-q(t,0,0)+q(t,0,0)||q(t,x,y)-q(t,0,0)|+|q(t,0,0)|Lx+Ey+α, where α:=maxt[t0,δ+T(t0)]𝕋|q(t,0,0)|. Hence, (4.28)|Aφn|=|11-ea(t)(t,δ-T(t))δ-T(t)t[-r(s)φnσ(δ-(k,s))+q(s,φn(s),φn(δ-(k,s)))]ea(s)(t,s)Δs|ηγ[(β+L+E)φn+α](t0-δ-T(t0))ηγ[(β+L+E)R+α](t0-δ-T(t0)):=D. Thus, the sequence {Aφn} is uniformly bounded. If we find the derivative of Aφn, we have (4.29)(Aφn)Δ(t)=a(t)Aφn(t)+-a(t)+a(t)1-ea(t)(σ(t),δ-T(σ(t)))×δ-T(t)t[-r(s)φnσ(δ-(k,s))+q(s,φn(s),φn(δ-(k,s)))]ea(s)(t,s)Δs+11+μ(t)a(t)[-r(t)φσ(δ-(k,s))+q(s,φn(s),φn(δ-(k,s)))].

Consequently, (4.30)|(Aφn)Δ(t)|Da+[(β+E+L)R+α][2aγη(t0-δ-T(t0))]:=F. for all n. That is, |(Aφn)Δ(t)|F, for some positive constant F. Thus the sequence {Aφn} is uniformly bounded and equicontinuous. The Arzela-Ascoli theorem implies that {Aφnk} uniformly converges to a continuous T-periodic function φ* in δ±. Thus A is compact.

Lemma 4.4.

Let B be defined by (4.19) and (4.31)b(t)ξ<1.

Then B:PTPT is a contraction.

Proof.

Trivially, B:PTPT. For φ,ψPT, we have (4.32)Bφ-Bψ=maxt[t0,δ+T(t0)]T|Bφ(t)-Bψ(t)|=maxt[t0,δ+T(t0)]T{|b(t)||φ(δ-(k,s))-ψ(δ-(k,s))|}ξφ-ψ.

Hence B defines a contraction mapping with contraction constant ξ.

Theorem 4.5.

Let α:=maxt[t0,δ+T(t0)]𝕋|q(t,0,0)|. Let β,η, and γ be given by (4.39). Suppose that (4.3)–(4.5) and (4.31) hold and that there is a positive constant G such that all solutions x(t) of (1.2), xPT, satisfy |x(t)|G, the inequality (4.33){ξ+γη(β+L+E)(t0-δ-T(t0))}G+γηα(t0-δ-T(t0))G holds. Then (1.2) has a T-periodic solution in δ±.

Proof.

Define M:={xPT:xG}. Then Lemma 4.3 implies that A:PTPT is compact and continuous. Also, from Lemma 4.4, the mapping B:PTPT is contraction.

We need to show that if φ,ψM, we have Aφ-BψG. Let φ,ψM with φ,  ψG. From (4.19) and (4.20) and the fact that |q(t,x,y)|Lx+Ey+α, we have (4.34)Aφ+Bψγηδ-T(t)t[Lφ+Eφ+βφ+α]Δs+ξψ{ξ+γη(β+L+E)(t0-δ-T(t0))}G+γηα(t0-δ-T(t0))G. We see that all the conditions of Krasnosel'skiĭ theorem are satisfied on the set M. Thus there exists a fixed point z in M such that z=Bz+Az. By Lemma 4.2, this fixed point is a solution of (2) has a T-periodic solution in δ±.

Theorem 4.6.

Suppose that (4.3)–(4.5) and (4.31) hold. Let β,η, and γ be given by (4.39). If (4.35)ξ+γη(β+L+E)(t0-δ-T(t0))1, then (1.2) has a unique T-periodic solution in δ±.

Proof.

Let the mapping H be given by (4.17). For φ,ψPT we have (4.36)Hφ-Hψξφ-ψ+γηδ-T(t)t[Lφ-ψ+Eφ-ψ+βφ-ψ]Δs[ξ+γη(β+L+E)(t0-δ-T(t0))]φ-ψ. This completes the proof.

Example 4.7.

Let 𝕋={2n}nN0{1/4,1/2} be a periodic time scale in shift δ±(P,t)=P±1t with period P=2. We consider the dynamic equation (1.2) with a(t)=1/5t, b(t)=(1/500)(-1)lnt/lnq and q(t,x,y)=(sin x+arctan x+1)/1000t.

The operators δ-(s,t)=t/s and δ+(s,t)=st are backward and forward shift operators for (s,t)D±. Here 𝕋*=𝕋, the initial point t0=1 and δ-(k,t)=t/k for k[2,)𝕋. If we consider conditions (4.3)-(4.4) we find T=4. Then a(t), b(t) satisfy condition (4.3), a(t)+ and q(t,x,y) satisfies the condition (4.4) for all t𝕋. Also, q(t,x,y) is Lipschitz continuous in x and y for L=E=1/250. Since b(t)=(1/500)(-1)lnt/lnq=(1/500)=ξ<1, then the condition (4.31) holds.

If we compute η,γ, and β, we have (4.37)η=maxt[t0,δ+T(t0)]T|(1-ea(t)(t,δ-T(t)))-1|3,45,α=maxt[t0,δ+T(t0)]T|q(t,0,0)|=1250(4.38)γ=maxu[δ-T(t),t]Tea(t)(t,u)1,5,β=maxt[t0,δ+T(t0)]T|r(t)|=112500. If we take G=1, then inequality (4.33) satisfies.

Let x(t)PT. We show that x(t)G=1. Integrate (1.2) from 1 to 4, we get (4.39)x(4)-x(1)=14[-a(t)xσ(t)+b(t)xΔ(tk)1k+q(t,x(t),x(tk))]Δt.

Since x(t)PT, then x(4)=x(1) and so after integration by parts (23) becomes (4.40)14a(t)xσ(t)Δt=14q(t,x(t),x(tk))-bΔ(t)x(tk)Δt.

Claim 1.

There exist t*[1,4]𝕋 such that 3a(t*)xσ(t*)14a(t)xσ(t)Δt.

Suppose that the claim is false. Define S:=14a(t)xσ(t)Δt. Then there exists ϵ>0 such that (4.41)3a(t)xσ(t)>S+ϵ, for all t[1,4]𝕋. So, (4.42)S=14a(t)xσ(t)Δt>1314(S+ϵ)ΔT=S+ϵ. That is, S>S+ϵ, a contradiction.

As a consequence of the claim, we have (4.43)3|a(t*)||xσ(t*)|14|q(t,x(t),x(tk))|+|bΔ(t)x(tk)|Δt14[(L+E)x+α+δx]Δt=3[(2250+1250)x+1250]=3[3250x+1250], where δ=max[1,4]𝕋|bΔ(t)|=1/250.

So, |a(t*)||xσ(t*)|(3/250)x+(1/250), which implies |xσ(t*)|20[(3/250)x+(1/250)]. Since for all t[1,4]𝕋, (4.44)xσ(t)=xσ(t*)+t*txΔ(σ(s))Δs, we have (4.45)|xσ(t)||xσ(t*)|+1t|xΔ(σ(s))|Δs20[3250x+1250]+3xΔ.

This implies that (4.46)x219+150019xΔ.

Taking the norm in (1.2) yields (4.47)xΔ(a+L+E)x+α1-b=(1/5+2/250)x+1/2501-1/500=104x+2250.499. Substitution of (4.47) into (4.46) yields that for all x(t)PT,  x(t)G=1. Then by Theorem 4.5, (1.2) has a 4-periodic solution in shifts δ±.

In this example, if we take q(t,x,y)=(sinx+arctanx)/1000t, we have (4.48)ξ+γη(β+L+E)(t0-δ-T(t0))=1500+962552.106<1.

So, all the conditions of Theorem 4.6 are satisfied. Therefore, (1.2) has a unique 4-periodic solution in shifts δ±.

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