International Journal of Analysis and Applications

Volume 16, Number 2 (2018), 162-177

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-16-2018-162

PERIODIC AND NONNEGATIVE PERIODIC SOLUTIONS OF NONLINEAR

NEUTRAL DYNAMIC EQUATIONS ON A TIME SCALE

MANEL GOUASMIA1, ABDELOUAHEB ARDJOUNI1,2,∗ AND AHCENE DJOUDI1

1Applied Mathematics Lab, Faculty of Sciences, Department of Mathematics, Univ Annaba, P.O. Box 12,

Annaba 23000, Algeria

2Faculty of Sciences and Technology, Department of Mathematics and Informatics, Univ Souk Ahras, P.O.

Box 1553, Souk Ahras, 41000, Algeria

∗Corresponding author: abd ardjouni@yahoo.fr

Abstract. Let T be a periodic time scale. We use Krasnoselskii–Burton’s fixed point theorem to show

new results on the existence of periodic and nonnegative periodic solutions of nonlinear neutral dynamic

equation with variable delay of the form

x∆(t) = −a(t)h(xσ(t)) + Q(t,x(t− τ(t)))∆ + G(t,x(t),x(t− τ(t))), t ∈ T.

We invert the given equation to obtain an equivalent integral equation from which we define a fixed point

mapping written as a sum of a large contraction and a completely continuous map. The Caratheodory

condition is used for the functions Q and G. The results obtained here extend the work of Mesmouli,

Ardjouni and Djoudi [16].

1. Introduction

In 1988, Stephan Hilger [11] introduced the theory of time scales (measure chains) as a means of unifying

discrete and continuum calculi. Since Hilger’s initial work there has been significant growth in the theory

2010 Mathematics Subject Classification. 34A37, 34A12, 35B09, 35B10, 45J05.

Key words and phrases. Krasnoselskii-Burton’s theorem; large contraction; neutral differential equation; integral equation;

periodic solution; time scales.

c©2018 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

162

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-16-2018-162


Int. J. Anal. Appl. 16 (2) (2018) 163

of dynamic equations on time scales, covering a variety of different problems; see [8, 9, 14] and references

therein.

Let T be a periodic time scale such that 0 ∈ T. In this article, we are interested in the analysis of

qualitative theory of periodic and positive periodic solutions of neutral dynamic equations. Motivated by

the papers [1]- [7], [10], [12], [13], [15], [16] and the references therein, we consider the following nonlinear

neutral dynamic equation

x∆(t) = −a(t)h(xσ(t)) + Q(t,x(t− τ(t)))∆ + G(t,x(t),x(t− τ(t))), t ∈ T. (1.1)

Throughout this paper we assume that a and τ are positive rd-continuous functions, id − τ : T → T is

increasing so that the function x (t− τ (t)) is well defined over T. The function h is continuous, Q and

G satisfying the Caratheodory condition. To reach our desired end we have to transform (1.1) into an

integral equation written as a sum of two mapping, one is a contraction and the other is continuous and

compact. After that, we use Krasnoselskii-Burton’s fixed point theorem, to show the existence of periodic

and nonnegative periodic solutions.

The organization of this paper is as follows. In Section 2, we introduce some notations and definitions,

and state some preliminary material needed in later sections. We will state some facts about the exponential

function on a time scale as well as the fixed point theorems. For details on fixed point theorems we refer

the reader to [10, 17]. In Section 3, we establish the existence of periodic solutions. In Section 4, we give

sufficient conditions to ensure the existence of nonnegative periodic solutions. The results presented in this

paper extend the main results in [16].

2. Preliminaries

In this section, we consider some advanced topics in the theory of dynamic equations on a time scales.

Again, we remind that for a review of this topic we direct the reader to the monographs of Bohner and

Peterson [8] and [9].

A time scale T is a closed nonempty subset of R. For t ∈ T the forward jump operator σ, and the backward

jump operator ρ, respectively, are defined as σ(t) = inf {s ∈ T : s > t} and ρ(t) = sup{s ∈ T : s < t}. These

operators allow elements in the time scale to be classified as follows. We say t is right scattered if σ(t) > t and

right dense if σ(t) = t. We say t is left scattered if ρ(t) < t and left dense if ρ(t) = t. The graininess function

µ : T → [0,∞), is defined by µ(t) = σ(t) − t and gives the distance between an element and its successor.

We set inf ∅ = sup T and sup ∅ = inf T. If T has a left scattered maximum M, we define Tk = T\{M}.

Otherwise, we define Tk = T. If T has a right scattered minimum m, we define Tk = T\{m}. Otherwise,

we define Tk = T.



Int. J. Anal. Appl. 16 (2) (2018) 164

Let t ∈ Tk and let f : T → R. The delta derivative of f(t), denoted f∆ (t), is defined to be the number

(when it exists), with the property that, for each ε > 0, there is a neighborhood U of t such that

∣∣f (σ (t)) −f (s) −f∆ (t) [σ (t) −s]∣∣ ≤ ε |σ (t) −s| ,
for all s ∈ U. If T = R then f∆ (t) = f′ (t) is the usual derivative. If T = Z then f∆ (t) = ∆f(t) =

f (t + 1) −f (t) is the forward difference of f at t.

A function is right dense continuous (rd-continuous), f ∈ Crd = Crd(T,R), if it is continuous at every

right dense point t ∈ T and its left-hand limits exist at each left dense point t ∈ T. The function f : T → R

is differentiable on Tk provided f∆ (t) exists for all t ∈ Tk.

We are now ready to state some properties of the delta-derivative of f. Note fσ(t) = f(σ(t)).

Theorem 2.1 ( [8, Theorem 1.20]). Assume f,g : T → R are differentiable at t ∈ Tk and let α be a scalar.

(i) (f + g)
∆

(t) = f∆(t) + g∆(t).

(ii) (αf)
∆

(t) = αf∆(t).

(iii) The product rules

(fg)
∆

(t) = f∆ (t) g (t) + fσ (t) g∆ (t) ,

(fg)
∆

(t) = f (t) g∆ (t) + f∆ (t) gσ (t) .

(iv) If g (t) gσ (t) 6= 0 then (
f

g

)∆
(t) =

f∆ (t) g (t) −f (t) g∆ (t)
g (t) gσ (t)

.

Definition 2.1 ( [12]). We say that a time scale T is periodic if there exist a w > 0 such that if t ∈ T then

t±w ∈ T. For T 6= R, the smallest positive w is called the period of the time scale.

Definition 2.2 ( [12]). Let T 6= R be a periodic time scales with the period w. We say that the function

f : T → R is periodic with period T if there exists a natural number n such that T = nw, f(t±T) = f(t) for

all t ∈ T and T is the smallest number such that f(t±T) = f(t). If T = R, we say that f is periodic with

period T > 0 if T is the smallest positive number such that f(t±T) = f(t) for all t ∈ T.

Remark 2.1 ( [12]). If T is a periodic time scale with period w, then σ(t±nw) = σ(t)±nw. Consequently,

the graininess function µ satisfies µ(t±nw) = σ(t±nw) − (t±nw) = σ(t) − t = µ(t) and so, is a periodic

function with period w.

The next theorem is the chain rule on time scales ( [8, Theorem 1.93]).

Theorem 2.2 (Chain Rule). Assume v : T → R is strictly increasing and T̃ := v(T) is a time scale. Let

w : T̃ → R. If v∆(t) and w∆̃ (v(t)) exist for t ∈ Tk, then

(w ◦v)∆ = (w∆̃ ◦v)v∆.



Int. J. Anal. Appl. 16 (2) (2018) 165

In the sequel we will need to differentiate and integrate functions of the form f(t−τ(t)) = f (v (t)) where,

v(t) := t− τ(t). Our next theorem is the substitution rule ( [8, Theorem 1.98]).

Theorem 2.3. Assume v : T → R is strictly increasing and T̃ := v(T) is a time scale. If f : T → R is

rd-continuous function and v is differentiable with rd-continuous derivative, then for a,b ∈ T ,∫ b
a

f(t)v∆(t)∆t =

∫ v(b)
v(a)

(
f ◦v−1

)
(s) ∆̃s.

A function p : T → R is said to be regressive provided 1 + µ (t) p (t) 6= 0 for all t ∈ Tk. The set of all

regressive rd-continuous function f : T → R is denoted by R. The set of all positively regressive functions

R+, is given by R+ = {f ∈R : 1 + µ (t) f (t) > 0 for all t ∈ T}.

Let p ∈R and µ (t) 6= 0 for all t ∈ T. The exponential function on T is defined by

ep(t,s) = exp

(∫ t
s

1

µ (z)
log (1 + µ (z) p (z)) ∆z

)
.

It is well known that if p ∈R+ , then ep(t,s) > 0 for all t ∈ T. Also, the exponential function y(t) = ep(t,s)

is the solution to the initial value problem y∆ = p (t) y, y (s) = 1. Other properties of the exponential

function are given by the following lemma.

Lemma 2.1 ( [8, Theorem 2.36]). Let p,q ∈R. Then

(i) e0 (t,s) = 1 and ep (t,t) = 1,

(ii) ep (σ (t) ,s) = (1 + µ (t) p (t)) ep (t,s),

(iii) 1
ep(t,s)

= e	p (t,s), where e	p (t,s) = −
p(t)

1+µ(t)p(t)
,

(iv) ep (t,s) =
1

ep(s,t)
= e	p (t,s),

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

(vi) e∆p (.,s) = pep (.,s) and
(

1
ep(.,s)

)∆
= − p(t)

eσp (.,s)
.

Theorem 2.4 ( [7, Theorem 2.1]). Let T be a periodic time scale with period w > 0. If p ∈ Crd(T) is a

periodic function with the period T = nw, then∫ b+T
a+T

p(u)∆u =

∫ b
a

p(u)∆u, ep(b + T,a + T) = ep(b,a) if p ∈R,

and ep(t + T,t) is independent of t ∈ T whenever p ∈R.

Lemma 2.2 ( [1]). If p ∈R+, then

0 < ep (t,s) ≤ exp
(∫ t

s

p (u) ∆u

)
, ∀t ∈ T.

Corollary 2.1 ( [1]). If p ∈R+ and p (t) < 0 for all t ∈ T, then for all s ∈ T with s ≤ t we have

0 < ep (t,s) ≤ exp
(∫ t

s

p (u) ∆u

)
< 1.



Int. J. Anal. Appl. 16 (2) (2018) 166

Now, we give some definitions which we are going to use in what follows.

Definition 2.3. A function f : [0,T]×Rn → R is an L1∆−Caratheodory function if it satisfies the following

conditions

(i) For each z ∈ Rn, the mapping t 7→ f(t,z) is ∆−measurable.

(ii) For almost all t ∈ [0,T], the mapping z 7→ f(t,z) is continuous on Rn.

(iii) For each r > 0, there exists αr ∈ L1∆ ([0,T] ,R) such that for almost all t ∈ [0,T] and for all z such

that |z| < r, we have |f(t,z)| ≤ αr(t).

Burton observed that Krasnoselskii’s result (see [17]) can be more attractive in applications with certain

changes and formulated Theorem 2.5 below (see [10] for the proof).

Definition 2.4. Let (M,d) be a metric space and assume that B : M 7→ M. B is said to be a large

contraction, if for ϕ,ψ ∈M, with ϕ 6= ψ, we have d(Bϕ,Bψ) < d(ϕ,ψ), and if ∀ε > 0, ∃δ < 1 such that

[ϕ,ψ ∈M, d(ϕ,ψ) ≥ ε] ⇒ d(Bϕ,Bψ) < δd(ϕ,ψ).

It is proved in [10] that a large contraction defined on a closed bounded and complete metric space has a

unique fixed point.

Theorem 2.5 (Krasnoselskii-Burton). Let M be a closed bounded convex nonempty subset of a Banach

space (B,‖.‖). Suppose that A and B map M into M such that

(i) A is completely continuous,

(ii) B is large contraction,

(iii) x,y ∈M, implies Ax + By ∈M.

Then there exists z ∈M with z = Az + Bz.

3. Existence of periodic solutions

Let T > 0, T ∈ T be fixed and if T 6= R, T = nw for some n ∈ N. By the notation [a,b] we mean

[a,b] = {t ∈ T : a ≤ t ≤ b}, unless otherwise specified. The intervals [a,b), (a,b] and (a,b) are defined

similarly. Define

PT = {φ ∈ C(T,R), φ(t + T) = φ(t)} ,

where C(T,R) is the space of all real valued rd-continuous functions. Then (PT ,‖.‖) is a Banach space when

it is endowed with the supremum norm

‖φ‖ = sup
t∈[0,T]

|φ(t)| .

We will need the following lemma whose proof can be found in [12].



Int. J. Anal. Appl. 16 (2) (2018) 167

Lemma 3.1. Let x ∈ PT . Then ‖xσ‖ = ‖x◦σ‖ exists and ‖xσ‖ = ‖x‖.

In this paper we assume that h is continuous, a ∈R+ is rd-continuous and

a(t−T) = a(t), τ(t−T) = τ(t), τ(t) ≥ τ∗ > 0, (3.1)

with τ continuously and τ∗ is positive constant, a is positive function and

1 −e�a(t,t−T) ≡
1

η
6= 0. (3.2)

The functions Q(t,x) and G(t,x,y) are periodic in t of period T . That is

Q(t−T,x) = Q(t,x), G(t−T,x,y) = G(t,x,y). (3.3)

The following lemma is fundamental to our results.

Lemma 3.2. Suppose (3.1)–(3.3) hold. If x ∈ PT , then x is a solution of equation (1.1) if and only if

x(t) = η

∫ t
t−T

k(t,u)a(u)[xσ(u) −h(xσ(u))]∆u + Q(t,x(t− τ(t)))

+ η

∫ t
t−T

k(t,u) [−a(u)Qσ(u,x(u− τ(u))) + G(u,x(u),x(u− τ(u)))] ∆u, (3.4)

where

k(t,u) = e�a(t,u). (3.5)

Proof. Let x ∈ PT be a solution of (1.1). Rewrite the equation (1.1) as

(x(t) −Q(t,x(t− τ(t))))∆ + a(t)[xσ(t) −Qσ(t,x(t− τ(t)))]

= a(t) [xσ(t) −h(xσ(t))] −a(t)Qσ(t,x(t− τ(t))) + G(t,x(t),x(t− τ(t))).

Multiply both sides of the above equation by ea(t, 0) and then integrate from t−T to t to obtain

∫ t
t−T

[(x(u) −Q(u,x(u− τ(u))))ea(u, 0)]
∆

∆u

=

∫ t
t−T

a(u)[xσ(u) −h(xσ(u))]ea(u, 0)∆u

+

∫ t
t−T

[−a(u)Qσ(u,x(u− τ(u))) + G(u,x(u),x(u− τ(u)))]ea(u, 0)∆u.



Int. J. Anal. Appl. 16 (2) (2018) 168

As a consequence, we arrive at

(x(t) −Q(t,x(t− τ(t))))ea(t, 0)

− (x(t−T) −Q(t−T,x(t−T − τ(t−T))))ea(t−T, 0)

=

∫ t
t−T

a(u)[xσ(u) −h(xσ(u))]ea(u, 0)∆u

+

∫ t
t−T

[−a(u)Qσ(u,x(u− τ(u))) + G(u,x(u),x(u− τ(u)))]ea(u, 0)∆u.

By dividing both sides of the above equation by ea(t, 0) and using the fact that x(t) = x(t−T), we obtain

x(t) −Q(t,x(t− τ(t)))

= η

∫ t
t−T

a(u)[xσ(u) −h(xσ(u))]e�a(t,u)∆u

+ η

∫ t
t−T

[−a(u)Qσ(u,x(u− τ(u))) + G(u,x(u),x(u− τ(u)))]e�a(t,u)∆u. (3.6)

The converse implication is easily obtained and the proof is complete. �

To apply Theorem 2.5, we need to define a Banach space B, a closed bounded convex subset M of B

and construct two mappings; one is a completely continuous and the other is large contraction. So, we let

(B,‖.‖) = (PT ,‖.‖) and

M = {ϕ ∈ PT , ‖ϕ‖≤ L} , (3.7)

with L ∈ (0, 1]. For x ∈M, let the mapping H be defined by

H(x) = xσ −h(xσ), (3.8)

and by (3.4), define the mapping S : PT → PT by

(Sϕ) (t)

= η

∫ t
t−T

k(t,u)a(u)H(ϕ(u))∆u + Q(t,ϕ(t− τ(t)))

+ η

∫ t
t−T

k(t,u)[−a(u)Qσ(u,ϕ(u− τ(u))) + G(u,ϕ(u),ϕ(u− τ(u)))]∆u. (3.9)

Therefore, we express the above equation as

(Sϕ) (t) = (Aϕ) (t) + (Bϕ) (t),



Int. J. Anal. Appl. 16 (2) (2018) 169

where A,B : PT → PT are given by

(Aϕ) (t)

= Q(t,ϕ(t− τ(t)))

+ η

∫ t
t−T

k(t,u) [−a(u)Qσ(u,ϕ(u− τ(u))) + G(u,ϕ(u),ϕ(u− τ(u)))] ∆u. (3.10)

and

(Bϕ) (t) = η

∫ t
t−T

k(t,u)a(u)H(ϕ(u))∆u. (3.11)

We will assume that the following conditions hold.

(H1) a ∈ L1∆ [0,T] is bounded.

(H2) Q and G satisfy Caratheodory conditions with respect to L1∆ [0,T].

(H3) There exist periodic functions q1,q2 ∈ L1∆ [0,T], with period T, such that

|Q(t,x)| ≤ q1(t) |x| + q2(t).

(H4) There exist periodic functions g1,g2,g3 ∈ L1∆ [0,T], with period T , such that

|G(t,x,y)| ≤ g1(t) |x| + g2(t) |y| + g3(t).

Now, we need the following assumptions

q1(t)L + q2(t) ≤
γ1
2
L, (3.12)

g1(t)L + g2(t)L + g3(t) ≤ γ2La(t), (3.13)

and

J (γ1 + γ2) ≤ 1, (3.14)

where γ1, γ2 and J are positive constants with J ≥ 3.

Lemma 3.3. For A defined in (3.10), suppose that (3.1)–(3.3), (3.12)–(3.14 ) and (H1)–(H4) hold. Then

A : M→M.

Proof. Let A be defined by (3.10). Obviously, Aϕ is rd-continuous. First by (3.1) and (3.3), a change of

variable in (3.10) shows that (Aϕ)(t + T) = (Aϕ)(t). That is, if ϕ ∈ PT then Aϕ is periodic with period T.



Int. J. Anal. Appl. 16 (2) (2018) 170

Next, let ϕ ∈M, by (3.12)–(3.14) and (H1)–(H4) we have

|(Aϕ) (t)|

≤ |Q(t,ϕ(t− τ(t)))|

+ η

∫ t
t−T

k(t,u) [a(u) |Qσ(u,ϕ(u− τ(u)))| + |G(u,ϕ(u),ϕ(u− τ(u)))|] ∆u

≤ q1(t) |ϕ(t− τ(t))| + q2(t)

+ η

∫ t
t−T

k(t,u)a(u)[q1(u) |ϕ(u− τ(u))| + q2(u)]∆u

+ η

∫ t
t−T

k(t,u)[g1(u) |ϕ(u)| + g2(u) |ϕ(u− τ(u))| + g3(u)]∆u

≤ γ1L + γ2L ≤
L

J
≤ L.

That is Aϕ ∈M. �

Lemma 3.4. For A : M → M defined in (3.10), suppose that (3.1)–(3.3), (3.12)–(3.14) and (H1)–(H4)

hold. Then A is completely continuous.

Proof. We show that A is continuous in the supremum norm, Let ϕn ∈M where n is a positive integer such

that ϕn → ϕ as n →∞. Then

|(Aϕn) (t) − (Aϕ) (t)|

≤ |Q(t,ϕn(t− τ(t)) −Q(t,ϕ(t− τ(t))|

+ η

∫ t
t−T

k(t,u)a(u) |Qσ(u,ϕn(u− τ(u))) −Qσ(u,ϕ(u− τ(u)))|∆u

+ η

∫ t
t−T

k(t,u) |G(u,ϕn(u),ϕn(u− τ(u))) −G(u,ϕ(u),ϕ(u− τ(u)))|∆u.

By the Dominated Convergence Theorem, limn→∞ |(Aϕn) (t) − (Aϕ) (t)| = 0. Then A is continuous.

We next show that A is completely continuous. Let ϕ ∈M, then, by Lemma 3.3, we see that

‖Aϕ‖≤ L.



Int. J. Anal. Appl. 16 (2) (2018) 171

And so the family of functions Aϕ is uniformly bounded. Again, let ϕ ∈M. Without loss of generality, we

can pick ω < t such that t−ω < T . Then

|(Aϕ) (t) − (Aϕ) (ω)|

≤ |Q(t,ϕ(t− τ(t))) −Q(ω,ϕ(ω − τ(ω)))|

+ η

∣∣∣∣
∫ t
t−T

k(t,u)a(u)Qσ(u,ϕ(u− τ(u)))∆u

−
∫ ω
ω−T

k(ω,u)a(u)Qσ(u,ϕ(u− τ(u)))∆u
∣∣∣∣

+ η

∣∣∣∣
∫ t
t−T

k(t,u)G(u,ϕ(u),ϕ(u− τ(u)))∆u

−
∫ ω
ω−T

k(ω,u)G(u,ϕ(u),ϕ(u− τ(u)))∆u
∣∣∣∣

≤ |Q(t,ϕ(t− τ(t))) −Q(ω,ϕ(ω − τ(ω)))|

+ 2ηk0

∫ t−T
ω−T

[
a(u)qL(u) + g√2L(u)

]
∆u

+ η

∫ ω
ω−T
|k(t,u) −k(ω,u)|

[
a(u)qL(u) + g√2L(u)

]
∆u

≤ |Q(t,ϕ(t− τ(t))) −Q(ω,ϕ(ω − τ(ω)))|

+ 2ηk0

∫ t
ω

[
a(u)qL(u) + g√2L(u)

]
∆u

+ η

∫ T
0

|k(t,u) −k(ω,u)|
[
a(u)qL(u) + g√2L(u)

]
∆u,

where k0 = max
u∈[t−T,t]

{k(t,u)}, then by the Dominated Convergence Theorem |(Aϕ)(t) − (Aϕ)(ω)| → 0 as

t−ω → 0 independently of ϕ ∈ M. Thus (Aϕ) is equicontinuous. Hence by Ascoli-Arzela’s theorem A is

completely continuous. �

Now, we state an important result see [1] and for convenience we present below its proof, we deduce by

this theorem that the following are sufficient conditions implying that the mapping H given by (3.8) is a

large contraction on the set M.

(H5) h : R → R is continuous on [−L,L] and differentiable on (−L,L),

(H6) the function h is strictly increasing on [−L,L],

(H7) sup
t∈(−L,L)

h′(t) ≤ 1.

Theorem 3.1. Let h : R → R be a function satisfying (H5)–(H7). Then the mapping H in (3.8) is a large

contraction on the set M.



Int. J. Anal. Appl. 16 (2) (2018) 172

Proof. Let ϕσ,ψσ ∈M with ϕσ 6= ψσ. Then ϕσ(t) 6= ψσ(t) for some t ∈ T. Let us denote the set of all such

t by D(ϕ,ψ), i.e.,

D(ϕ,ψ) = {t ∈ T : ϕσ(t) 6= ψσ(t)} .

For all t ∈ D(ϕ,ψ), we have

|(Hϕ)(t) − (Hψ)(t)|

≤ |ϕσ(t) −ψσ(t) −h(ϕσ(t)) + h(ψσ(t))|

≤ |ϕσ(t) −ψσ(t)|
∣∣∣∣1 − h(ϕσ(t)) −h(ψσ(t))ϕσ(t) −ψσ(t)

∣∣∣∣ . (3.15)
Since h is a strictly increasing function we have

h(ϕσ(t)) −h(ψσ(t))
ϕσ(t) −ψσ(t)

> 0 for all t ∈ D(ϕ,ψ). (3.16)

For each fixed t ∈ D(ϕ,ψ) define the interval It ⊂ [−L,L] by

It =


 (ϕ

σ(t),ψσ(t)) if ϕσ(t) < ψσ(t),

(ψσ(t),ϕσ(t)) if ψσ(t) < ϕσ(t).

The Mean Value Theorem implies that for each fixed t ∈ D(ϕ,ψ) there exists a real number ct ∈ It such

that

h(ϕσ(t)) −h(ψσ(t))
ϕσ(t) −ψσ(t)

= h′(ct).

By (H6) and (H7) we have

0 ≤ inf
u∈(−L,L)

h′(u) ≤ inf
u∈It

h′(u) ≤ h′(ct) ≤ sup
u∈It

h′(u) ≤ sup h′
u∈(−L,L)

(u) ≤ 1. (3.17)

Hence, by (3.15)–(3.17) we obtain

|(Hϕ)(t) − (Hψ)(t)| ≤ |ϕσ(t) −ψσ(t)|
∣∣∣∣1 − inf

u∈(−L,L)
h′(u)

∣∣∣∣ , (3.18)
for all t ∈ D(ϕ,ψ). This implies a large contraction in the supremum norm. To see this, choose a fixed

ε ∈ (0, 1) and assume that ϕ and ψ are two functions in M satisfying

ε ≤ sup
t∈(−L,L)

|ϕ(t) −ψ(t)| = ‖ϕ−ψ‖ .

If |ϕσ(t) −ψσ(t)| ≤ ε
2

for some t ∈ D(ϕ,ψ), then we get by (3.17) and (3.18) that

|(Hϕ)(t) − (Hψ)(t)| ≤ |ϕσ(t) −ψσ(t)| ≤
1

2
‖ϕ−ψ‖ . (3.19)

Since h is continuous and strictly increasing, the function h(u + ε
2
)−h(u) attains its minimum on the closed

and bounded interval [−L,L]. Thus, if ε
2
≤ |ϕσ(t) −ψσ(t)| for some t ∈ D(ϕ,ψ), then by (H6) and (H7) we

conclude that

1 ≥
h(ϕσ(t)) −h(ψσ(t))

ϕσ(t) −ψσ(t)
> λ,



Int. J. Anal. Appl. 16 (2) (2018) 173

where

λ :=
1

2L
min

{
h(u +

ε

2
) −h(u) : u ∈ [−L,L]

}
> 0.

Hence, (3.15) implies

|(Hϕ)(t) − (Hψ)(t)| ≤ (1 −λ)‖ϕ−ψ‖ . (3.20)

Consequently, combining (3.19) and (3.20) we obtain

|(Hϕ)(t) − (Hψ)(t)| ≤ δ‖ϕ−ψ‖ , (3.21)

where

δ = max

{
1

2
, 1 −λ

}
.

The proof is complete. �

The next result shows the relationship between the mappings H and B in the sense of large contractions.

Assume that

max{|H(−L)| , |H(L)|}≤
2L

J
. (3.22)

Lemma 3.5. Let B be defined by (3.11), suppose (H5)–(H7) hold. Then B : M→M is a large contraction.

Proof. Let B be defined by (3.11). Obviously, Bϕ is continuous and it is easy to show that (Bϕ)(t + T) =

(Bϕ)(t). Let ϕ ∈M

|(Bϕ)(t)| ≤ η
∫ t
t−T

k(t,u)a(u) max{|H(−L)| , |H(L)|}∆u

≤
2L

J
< L,

which implies B : M→M.

By Theorem 3.1, H is large contraction on M, then for any ϕ,ψ ∈ M, with ϕ 6= ψ and for any ε > 0,

from the proof of that Theorem, we have found a δ < 1, such that

|(Bϕ)(t) − (Bψ)(t)| =
∣∣∣∣η
∫ t
t−T

k(t,u)a(u)[H(ϕ(u)) −H(ψ(u))]∆u
∣∣∣∣

≤ δ‖ϕ−ψ‖η
∫ t
t−T

k(t,u)a(u)∆u ≤ δ‖ϕ−ψ‖ .

The proof is complete. �

Theorem 3.2. Suppose the hypothesis of Lemmas 3.3, 3.4 and 3.5 hold. Let M defined by (3.7). Then the

equation (1.1) has a T -periodic solution in M.



Int. J. Anal. Appl. 16 (2) (2018) 174

Proof. By Lemma 3.3, 3.4, A is continuous and A(M) is contained in a compact set. Also, from Lemma

3.5, the mapping B is a large contraction. Next, we show that if ϕ,ψ ∈ M, we have ‖Aψ + Bϕ‖ ≤ L. Let

ϕ,ψ ∈M with ‖ϕ‖ ,‖ψ‖≤ L. By (3.12)–(3.14)

‖Aψ + Bϕ‖≤ (γ1 + γ2)L +
2L

J

≤
L

J
+

2L

J
≤ L.

Clearly, all the hypotheses of the Krasnoselskii-Burton’s theorem are satisfied. Thus there exists a fixed

point z ∈M such that z = Az + Bz. By Lemma 3.2 this fixed point is a solution of (1.1). Hence (1.1) has

a T-periodic solution. �

4. Existence of nonnegative periodic solutions

In this section we obtain the existence of a nonnegative periodic solution of (1.1). By applying Theorem

2.5, we need to define a closed, convex, and bounded subset M of PT . So, let

M = {φ ∈ PT : 0 ≤ φ ≤ K} , (4.1)

where K is positive constant. To simplify notation, we let

m = min
u∈[t−T,t]

e�a(t,u), M = max
u∈[t−T,t]

e�a(t,u). (4.2)

It is easy to see that for all (t,u) ∈ [0, 2T]2,

m ≤ k(t,u) ≤ M. (4.3)

Then we obtain the existence of a nonnegative periodic solution of (1.1) by considering the two cases;

1) Q(t,y) ≥ 0, ∀t ∈ [0,T] , y ∈ M.

2) Q(t,y) ≤ 0, ∀t ∈ [0,T] , y ∈ M.

In the case one, we assume for all t ∈ [0,T], x,y ∈ M, that there exists a positive constant c1 such that

0 ≤ Q(t,y) ≤ c1y, (4.4)

c1 < 1, (4.5)

0 ≤−a(t)Qσ(t,y) + G(t,x,y), (4.6)

a(t)H(ϕ(t)) −a (t) Qσ (t,y) + G(t,x,y) ≤
K (1 − c1)
MηT

. (4.7)

Lemma 4.1. Let A and B given by (3.10) and (3.11) respectively, assume that (4.4)–(4.7) hold. Then

A,B : M → M.



Int. J. Anal. Appl. 16 (2) (2018) 175

Proof. Let A defined by (3.11). So, for any ϕ ∈ M, we have

0 ≤ (Aϕ) (t) ≤ Q(t,ϕ(t− τ(t)))

+ η

∫ t
t−T

k(t,u) [−a(u)Qσ(u,ϕ(u− τ(u))) + G(u,ϕ(u),ϕ(u− τ(u)))] ∆u

≤ η
∫ t
t−T

M
K (1 − c1)
MηT

∆u + c1K = K,

That is Aϕ ∈ M.

Now, let B defined by (3.11). So, for any ϕ ∈ M, we have

0 ≤ (Bϕ)(t) ≤ η
∫ t
t−T

M
K (1 − c1)
MηT

∆u ≤ MηT
K

MηT
= K.

That is Bϕ ∈ M. �

Theorem 4.1. Suppose the hypothesis of Lemmas 3.4, 3.5 and 4.1 hold. Then equation (1.1) has a non-

negative T -periodic solution x in the subset M.

Proof. By Lemma 3.4, A is completely continuous. Also, from Lemma 3.5, the mapping B is a large

contraction. By Lemma 4.1, A,B : M → M. Next, we show that if ϕ,ψ ∈ M, we have 0 ≤ Aψ + Bϕ ≤ K.

Let ϕ,ψ ∈ M with 0 ≤ ϕ,ψ ≤ K. By (4.4)–(4.7)

(Aψ)(t) + (Bϕ)(t)

= η

∫ t
t−T

k(t,u)a(u)H(ϕ(u))∆u + Q(t,ψ(t− τ(t)))

+ η

∫ t
t−T

k(t,u) [−a(u)Qσ(u,ψ(u− τ(u))) + G(u,ψ(u),ψ(u− τ(u)))] ∆u

≤ η
∫ t
t−T

k(t,u)
K (1 − c1)
MηT

∆u + c1K

≤ η
∫ t
t−T

M
K (1 − c1)
MηT

∆u + c1K = K.

On the other hand,

(Aψ)(t) + (Bϕ)(t) ≥ 0.

Clearly, all the hypotheses of the Krasnoselskii-Burton’s theorem are satisfied. Thus there exists a fixed

point z ∈ M such that z = Az + Bz. By Lemma 1 this fixed point is a solution of (1.1) and the proof is

complete. �

In the case two, we substitute conditions (4.4)–(4.7) with the following conditions respectively. We assume

that there exist a negative constant c2 such that

c2y ≤ Q(t,y) ≤ 0, (4.8)



Int. J. Anal. Appl. 16 (2) (2018) 176

− c2 < 1, (4.9)

−c2K
MηT

≤ a(t)H(ϕ(t)) −a(t)Q(t,y) + G(t,x,y), (4.10)

a(t)H(ϕ(t)) −a (t) Q (t,y) + G(t,x,y) ≤
K

MηT
. (4.11)

Theorem 4.2. Suppose (4.8)–(4.11) and the hypothesis of Lemmas 3.3, 3.4 and 3.5 hold. Then equation

(1.1) has a nonnegative T -periodic solution x in the subset M.

Proof. By Lemma 3.3, 3.4, A is completely continuous. Also, from Lemma 3.5, the mapping B is a large

contraction. To see that, it is easy to show as in Lemma 4.1, A,B : M → M. Next, we show that if ϕ,ψ ∈ M,

we have 0 ≤ Aψ + Bϕ ≤ K. Let ϕ,ψ ∈ M, with 0 ≤ ϕ,ψ ≤ K. By (4.8)– (4.11)

(Aψ) (t) + (Bϕ) (t)

= η

∫ t
t−T

k(t,u)a(u)H(ϕ(u))∆u + Q(t,ψ(t− τ(t)))

+ η

∫ t
t−T

k(t,u) [−a(u)Qσ(u,ψ(u− τ(u))) + G(u,ψ(u),ψ(u− τ(u)))] ∆u

≤ η
∫ t
t−T

k(t,u)
K

MηT
∆u = η

∫ t
t−T

M
K

MηT
∆u = K.

On the other hand,

(Aψ)(t) + (Bϕ) (t) ≥ η
∫ t
t−T

M
−c2K
MηT

∆u + c2K = 0.

Clearly, all the hypotheses of the Krasnoselskii-Burton’s theorem are satisfied. Thus there exists a fixed

point z ∈ M such that z = Az + Bz. By Lemma 3.2 this fixed point is a solution of (1.1) and the proof is

complete. �

References

[1] M. Adivar and Y. N. Raffoul, Existence of periodic solutions in totally nonlinear delay dynamic equations, Electronic

Journal of Qualitative Theory of Differential Equations, 2009, No. 1, 1–20.

[2] A. Ardjouni and A. Djoudi, Existence of periodic solutions for nonlinear neutral dynamic equations with variable delay on

a time scale. Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 3061–3069.

[3] A. Ardjouni and A. Djoudi, Existence of positive periodic solutions for nonlinear neutral dynamic equations with variable

delay on a time scale, Malaya J. Mat. 2(1) (2013) 60–67.

[4] A. Ardjouni and A. Djoudi, Existence of periodic solutions for nonlinear neutral dynamic equations with functional delay

on a time scale, Acta Univ. Palacki. Olomnc., Fac. rer. nat., Mathematica 52, 1 (2013) 5–19.

[5] A. Ardjouni and A. Djoudi, A. Existence, uniqueness and positivity of solutions for a neutral nonlinear periodic dynamic

equation on a time scale, J. Nonlinear Anal. Optim. 6 (2) (2015), 19–29.

[6] M. Belaid, A. Ardjouni and A.Djoudi, Stability in totally nonlinear neutral dynamic equations on time scales, Int. J. Anal.

Appl. 11 (2) (2016), 110–123.



Int. J. Anal. Appl. 16 (2) (2018) 177

[7] L. Bi, M. Bohner and M. Fan, Periodic solutions of functional dynamic equations with infinite delay, Nonlinear Anal. 68

(2008), 1226–1245.

[8] M. Bohner, A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications, Birkhäuser, Boston,

2001.

[9] M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003.

[10] T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, 2006.

[11] S. Hilger, Ein Masskettenkalkül mit Anwendung auf Zentrumsmanningfaltigkeiten. PhD thesis, Universität Würzburg,

1988.

[12] E. R. Kaufmann and Y. N. Raffoul, Periodic solutions for a neutral nonlinear dynamical equation on a time scale, J. Math.

Anal. Appl. 319 (2006), no. 1, 315–325.

[13] E. R. Kaufmann and Y. N. Raffoul, Periodicity and stability in neutral nonlinear dynamic equations with functional delay

on a time scale, Electron. J. Differential Equations, 2007 (2007), No. 27, 1–12.

[14] V. Lakshmikantham, S. Sivasundaram, B. Kaymarkcalan, Dynamic Systems on Measure Chains, Kluwer Academic Pub-

lishers, Dordrecht, 1996.

[15] M. B. Mesmouli, A. Ardjouni, A. Djoudi, Existence and stability of periodic solutions for nonlinear neutral differential

equations with variable delay using fixed point technique, Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica 54 (1)

(2015), 95–108.

[16] M. B. Mesmouli, A. Ardjouni and A. Djoudi, Study of periodic and nonnegative periodic solutions of nonlinear neutral

functional differential equations via fixed points, Acta Univ. Sapientiae, Mathematica, 8 (2) (2016), 255–270.

[17] D. R. Smart, Fixed point theorems, Cambridge Tracts in Mathematics, No. 66. Cambridge University Press, London-New

York, 1974.


	1. Introduction
	2. Preliminaries
	3. Existence of periodic solutions
	4. Existence of nonnegative periodic solutions
	References