International Journal of Analysis and Applications
ISSN 2291-8639
Volume 11, Number 2 (2016), 110-123
http://www.etamaths.com

STABILITY IN TOTALLY NONLINEAR NEUTRAL DYNAMIC EQUATIONS ON

TIME SCALES

MALIK BELAID2, ABDELOUAHEB ARDJOUNI1,2,∗ AND AHCENE DJOUDI2

Abstract. Let T be a time scale which is unbounded above and below and such that 0 ∈ T. Let
id− τ : [0,∞) ∩T → T be such that (id− τ) ([0,∞) ∩T) is a time scale. We use the Krasnoselskii-
Burton’s fixed point theorem to obtain stability results about the zero solution for the following
totally nonlinear neutral dynamic equation with variable delay

x4 (t) = −a (t) h (xσ (t)) + c (t) x4̃ (t− τ (t)) + b (t) G (x (t) ,x (t− τ (t))) , t ∈ [0,∞) ∩T,

where f4 is the 4-derivative on T and f4̃ is the 4-derivative on (id− τ) (T). The results obtained
here extend the work of Ardjouni, Derrardjia and Djoudi [2].

1. Introduction

The concept of time scales analysis is a fairly new idea. In 1988, it was introduced by the German
mathematician Stefan Hilger in his Ph.D. thesis [12]. It combines the traditional areas of continuous
and discrete analysis into one theory. After the publication of two textbooks in this area by Bohner
and Peterson [6] and [7], more and more researchers were getting involved in this fast-growing field of
mathematics.

The study of dynamic equations brings together the traditional research areas of differential and
difference equations. It allows one to handle these two research areas at the same time, hence shedding
light on the reasons for their seeming discrepancies. In fact, many new results for the continuous and
discrete cases have been obtained by studying the more general time scales case (see [1, 3, 4, 13] and
the references therein).

There is no doubt that the Lyapunov method have been used successfully to investigate stability
properties of wide variety of ordinary, functional and partial equations. Nevertheless, the application
of this method to problem of stability in differential equations with delay has encountered serious
difficulties if the delay is unbounded or if the equation has unbounded term. It has been noticed that
some of theses difficulties vanish by using the fixed point technic. Other advantages of fixed point
theory over Lyapunov’s method is that the conditions of the former are average while those of the
latter are pointwise (see [2, 5, 8, 9, 10, 11] and references therein).

In paper, we consider the following neutral nonlinear dynamic equations with variable delay given
by

(1.1) x4 (t) = −a (t) h (xσ (t)) + c (t) x4̃ (t− τ (t)) + b (t) G (x (t) ,x (t− τ (t))) , t ∈ [0,∞) ∩T,

with an assumed initial function

x (t) = ψ (t) , t ∈ [m0, 0] ∩T,

where T is an unbounded above and below time scale and such that 0 ∈ T, ψ : [m0, 0] ∩ T → R
is rd-continuous and m0 = inf {t− τ (t) : t ∈ [0,∞) ∩T}. Throughout this paper, we assume that
a,b : [0,∞) ∩ T → R are rd-continuous, h : R → R is continuous with h (0) = 0 and c : [0,∞) ∩
T → R is continuously delta-differentiable. In order for the function x (t− τ (t)) to be well-defined and

2010 Mathematics Subject Classification. 34K20, 34K30, 34k40.

Key words and phrases. Fixed points, neutral dynamic equations, stability, time scales.

c©2016 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

110



STABILITY IN TOTALLY NONLINEAR NEUTRAL DYNAMIC EQUATIONS 111

differentiable over [0,∞)∩T, we assume that τ : [0,∞)∩T → T is positive and twice continuously delta-
differentiable, and that id−τ : [0,∞)∩T → T is an increasing mapping such that (id− τ) ([0,∞) ∩T)
is closed where id is the identity function.

Our purpose here is to use a modification of Krasnoselskii’s fixed point theorem due to Burton (see
[8] Theorem 3) to show the asymptotic stability and the stability of the zero solution for equation (1.1).
Clearly, the present problem is totally nonlinear so that the variation of parameters can not be applied
directly. Then, we resort to the idea of adding and subtracting a linear term. As noted by Burton
in [8], the added term destroys a contraction already present in part of the equation but it replaces
it with the so called a large contraction mapping which is suitable for fixed point theory. During the
process we have to transform (1.1) into an integral equation written as a sum of two mapping; one is
a large contraction and the other is compact. After that, we use a variant of Krasnoselskii fixed point
theorem, to show the asymptotic stability and the stability of the zero for equation (1.1). In the special
case T = R, Ardjouni, Derrardjia and Djoudi [2] show the zero solution of (1.1) is asymptotically stable
by using Krasnoselskii-Burton’s fixed point theorem.

In Section 2, we present some preliminary material that we will need through the remainder of
the paper. We will state some facts about the exponential function on a time scale as well as the
modification of Krasnoselskii’s fixed point theorem established by Burton (see ([8] Theorem 3) and
[10]). For details on Krasnoselskii’s theorem we refer the reader to [14]. We present our main results
on stability in Section 3 and 4. The results presented in this paper extend the main results in [2].

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 [6] and [7].

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{t ∈ 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.

Let t ∈ Tk and let f : T → R. The delta derivative of f (t), denoted f4 (t), is defined to be the
number (when it exists), with the property that, for each � > 0, there is a neighborhood U of t such
that ∣∣f (σ (t)) −f (s) −f4 (t) [σ (t) −s]∣∣ ≤ � |σ (t) −s| ,
for all s ∈ U. If T = R then f4 (t) = f′ (t) is the usual derivative. If T = Z then f4 (t) = 4f (t) =
f (t + 1) −f (t) is the forward difference of f at t.

A function f 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 f4 (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 1 ([6, Theorem 1.20]). Assume f,g : T → R are differentiable at t ∈ Tk and let α be a
scalar.

(i) (f + g)
4

(t) = g4 (t) + f4 (t).

(ii) (αf)
4

(t) = αf4 (t).
(iii) The product rules

(fg)
4

(t) = f4 (t) g (t) + fσ (t) g4 (t) ,

(fg)
4

(t) = f (t) g4 (t) + f4 (t) gσ (t) .



112 BELAID, ARDJOUNI AND DJOUDI

(iv) If g (t) gσ (t) 6= 0 then (
f

g

)4
(t) =

f4 (t) g (t) −f (t) g4 (t)
g (t) gσ (t)

.

The next theorem is the chain rule on time scales ([6, Theorem 1.93], Theorem 1.93).

Theorem 2 (Chain Rule). Assume ν : T → R is strictly increasing and T̃ := ν (T) is a time scale.
Let ω : T̃ → R. If ν4 (t) and ω4̃ (ν (t)) exist for t ∈ Tk, then (ω ◦ν)4 =

(
ω4̃ ◦ν

)
ν4.

In the sequel we will need to differentiate and integrate functions of the form f (t− τ (t)) = f (ν (t))
where, ν (t) := t−τ (t). Our next theorem is the substitution rule ([6, Theorem 1.98], Theorem 1.98).

Theorem 3 (Substitution). Assume ν : T → R is strictly increasing and T̃ := ν (T) is a time scale.
If f : T → R is rd-continuous function and ν is differentiable with rd-continuous derivative, then for
a,b ∈ T , ∫ b

a

f (t) ν4 (t)4t =
∫ ν(b)
ν(a)

(
f ◦ν−1

)
(s)4̃s.

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

4 = p (t) y, y (s) = 1. Other properties of
the exponential function are given by the following lemma.

Lemma 1 ([6, 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 	p (t) = −
p(t)

1+µ(t)p(t)
,

(iv) ep (t,s) =
1

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

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

(vi) e4p (.,s) = pep (.,s) and
(

1
ep(.,s)

)4
= − p(t)

eσp(.,s)
.

Lemma 2 ([1]). If p ∈R+, then

0 < ep (t,s) ≤ exp
(∫ t

s

p (u)4u
)
, ∀t ∈ T.

Corollary 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)4u
)
< 1.

3. The inversion and the fixed point theorem

In addition to the conditions mentioned in Section 1, we assume that a ∈ R+ is rd-continuous
and a (t) > 0 for all t ∈ [0,∞) ∩T, c is continuously delta-differentiable and τ is twice continuously
delta-differentiable with

(3.1) τ4 (t) 6= 1, t ∈ [0,∞) ∩T.
We also assume that G (x,y) is locally Lipschitz continuous in x and y. That is, there are positive

constants N1 and N2 so that if |x| , |y| ≤
√

3/3, then

(3.2) |G (x,y) −G (z,w)| ≤ N1 ‖x−z‖ + N2 ‖y −w‖ and G (0, 0) = 0.



STABILITY IN TOTALLY NONLINEAR NEUTRAL DYNAMIC EQUATIONS 113

One crucial step in the investigation of an equation using fixed point theory involves the construction
of a suitable fixed point mapping. For that end we must invert (1.1) to obtain an equivalent integral
equation from which we derive the needed mapping. During the process, an integration by parts has

to be performed on the neutral term x4̃ (t− τ (t)). Unfortunately, when doing this, a derivative τ4 (t)
of the delay appears on the way, and so we have to support it.

Lemma 3. Suppose (3.1) holds. x is a solution of equation (1.1) if and only if

x (t) =

[
ψ (0) −

c (0)

1 − τ4 (0)
ψ (−τ (0))

]
e	a(t, 0) +

∫ t
0

a (s) (Hx) (s) e	a (t,s) ∆s

+
c (t)

1 − τ4 (t)
x (t− τ (t)) −

∫ t
0

µ (s) xσ (s− τ (s)) e	a (t,s) ∆s

+

∫ t
0

b (s) G (x (s) ,x (s− τ (s))) e	a (t,s) ∆s,(3.3)

where

(3.4) µ (t) =

(
c4 (t) + a (t) cσ (t)

)(
1 − τ4 (t)

)
+ c (t) τ44 (t)

(1 − τ4 (t)) (1 − τ4 (σ (t)))
,

and

(3.5) (Hx) (t) = xσ (t) −h (xσ (t)) .

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

x4 (t) + a (t) xσ (t)

= a (t) xσ (t) −a (t) h (xσ (t)) + c (t) x4̃ (t− τ (t)) + b (t) G (t,x (t) ,x (t− τ (t))) .

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

x (t) = ψ (0) e	a (t, 0)

+

∫ t
0

a (s) (Hx) (s) e	a (t,s) ∆s +

∫ t
0

c (s) x4̃ (s− τ (s)) e	a (t,s) ∆s

+

∫ t
0

b (s) G (x (s) ,x (s− τ (s))) e	a (t,s) ∆s,(3.6)

letting ∫ t
0

c (s) x4̃ (s− τ (s)) e	a (t,s) ∆s

=

∫ t
0

c (s)

(1 − τ4 (s))
(
1 − τ4 (s)

)
x4̃ (s− τ (s)) e	a (t,s) ∆s.

By performing an integration by parts, we obtain∫ t
0

c (s) x4̃ (s− τ (s)) e	a (t,s) ∆s

=
c (t)

1 − τ4 (t)
x (t− τ (t)) −

c (0)

1 − τ4 (0)
ψ (−τ (0)) e	a (t, 0)

−
∫ t
0

µ (s) xσ (s− τ (s)) e	a (t,s) ∆s,(3.7)

where µ (s) is given by (3.4). We obtain (3.3) by substituting (3.7) in (3.6). Since each step is reversible,
the converse follows easily. This completes the proof. �

Burton [8] observed that Krasnoselskii result can be more interesting in applications with certain
changes and formulated in Theorem 5 below (see [8] for the proof).



114 BELAID, ARDJOUNI AND DJOUDI

Definition 1. Let (M,d) be a metric space and F : M → M. F is said to be a large contraction if
ϕ,ψ ∈ M with ϕ 6= ψ, then d (Fϕ,Fψ) < d (ϕ,ψ), and if for all � > 0, there exists η < 1 such that

[ϕ,ψ ∈ M, d (ϕ,ψ) ≥ �] =⇒ d (Fϕ,Fψ) ≤ ηd (ϕ,ψ) .

Theorem 4 (Burton). Let (M,d) be a complete metric space and F be a large contraction. Suppose
there is x ∈ M and ρ > 0 such that d (x,Fnx) ≤ ρ for all n ≥ 1. Then F has a unique fixed point in
M.

Below, we state Krasnoselskii-Burton’s hybrid fixed point theorem which enables us to establish a
stability result of the trivial solution of (1.1). For more details on Krasnoselskii’s captivating theorem
we refer to Smart [14] or [10].

Theorem 5 (Krasnoselskii-Burton). Let M be a closed bounded convex non-empty subset of a Banach
space (S,‖.‖). Suppose that A, B map M into M and that

(i) for all x,y ∈ M =⇒ Ax + By ∈ M,
(ii) A is continuous and AM is contained in a compact subset of M,
(iii) B is a large contraction.

Then there is z ∈ M with z = Az + Bz.

Here we manipulate function spaces defined on infinite t-intervals. So for compactness, we need an
extension of Arzela-Ascoli theorem. This extension is taken from [10, Theorem 1.2.2, p. 20] and is as
follows.

Theorem 6. Let q : [0,∞) ∩T −→ R+ be a rd-continuous function such that q (t) −→ 0 as t −→∞.
If {ϕn (t)} is an equicontinuous sequence of Rm-valued functions on [0,∞)∩T with |ϕn (t)| ≤ q (t) for
t ∈ [0,∞) ∩T, then there is a subsequence that converges uniformly on [0,∞) ∩T to a rd-continuous
function ϕ (t) with |ϕ (t)| ≤ q (t) for t ∈ [0,∞) ∩T, where |.| denotes the Euclidean norm on Rm.

4. Stability by Krasnoselskii-Burton’s theorem

From the existence theory which can be found in [10], we conclude that for each continuous initial
function ψ : [m0, 0] ∩T → R, there exists a continuous solution x (t, 0,ψ) which satisfies (1.1) on an
interval [0,σ) ∩T for some σ > 0 and x (t, 0,ψ) = ψ (t), t ∈ [m0, 0] ∩T.

We need the following stability definitions taken from [10].

Definition 2. The zero solution of (1.1) is said to be stable at t = 0 if for each � > 0, there exists
δ > 0 such that ψ : [m0, 0] ∩T → (−δ,δ) implies that |x (t)| < � for t ≥ m0.

Definition 3. The zero solution of (1.1) is said to be asymptotically stable if it is stable at t = 0 and
δ > 0 exists such that for any rd-continuous function ψ : [m0, 0] ∩T → (−δ,δ), the solution x with
x (t) = ψ (t) on [m0, 0] ∩T tends to zero as t →∞.

To apply Theorem 5, we have to choose carefully a Banach space depending on the initial function
ψ and construct two mappings, a large contraction and a compact operator which obey the conditions
of the theorem. So let S be the Banach space of rd-continuous bounded functions ϕ : [m0,∞)∩T → R
with the supremum norm ‖.‖. Let L > 0 and define the set

Sψ = {ϕ ∈ S | ϕ is Lipschitzian |ϕ (t)| ≤ L, t ∈ [m0,∞) ∩T,
ϕ (t) = ψ (t) if t ∈ [m0, 0] ∩T and ϕ (t) → 0 as t →∞} .

Clearly, if {ϕn} is a sequence of k-Lipschitzian functions converging to a function ϕ, then

|ϕ (u) −ϕ (v)| ≤ |ϕ (u) −ϕn (u)| + |ϕn (u) −ϕn (v)| + |ϕn (v) −ϕ (v)|
≤ ‖ϕ−ϕn‖ + k |u−v| + ‖ϕ−ϕn‖ .

Consequently, as n → ∞, we see that ϕ is k-Lipschitzian. It is clear that Sψ is convex, bounded and
complete endowed with ‖.‖.



STABILITY IN TOTALLY NONLINEAR NEUTRAL DYNAMIC EQUATIONS 115

For ϕ ∈ Sψ and t ∈ [0,∞) ∩T, define the maps A, B and C on Sψ as follows

(Aϕ) (t) :=
c (t)

1 − τ4 (t)
ϕ (t− τ (t)) +

∫ t
0

b (s) G (ϕ (s) ,ϕ (s− τ (t))) e	a (t,s) ∆s

−
∫ t
0

µ (s) ϕσ (s− τ (s)) e	a (t,s) ∆s,(4.1)

(Bϕ) (t) :=

[
ψ (0) −

c (0)

1 − τ4 (0)
ψ (−τ (0))

]
e	a (t, 0)

+

∫ t
0

a (s) (Hx) (s) e	a(t,s)∆s,(4.2)

and

(4.3) (Cϕ) (t) := (Aϕ) (t) + (Bϕ) (t) .

If we are able to prove that C possesses a fixed point ϕ on the set Sψ, then x (t, 0,ψ) = ϕ (t) for
t ∈ [0,∞) ∩T, x (t, 0,ψ) = ψ (t) on [m0, 0] ∩T, x (t, 0,ψ) satisfies (1.1) when its derivative exists and
x (t, 0,ψ) → 0 as t →∞.

Let α (t) =
c(t)

1−τ4(t) and assume that there are constants k1,k2,k3 > 0 such that for 0 ≤ t1 < t2,

(4.4)

∣∣∣∣
∫ t2
t1

a (u) ∆u

∣∣∣∣ ≤ k1 |t2 − t1| ,
(4.5) |τ (t2) − τ (t1)| ≤ k2 |t2 − t1| ,

and

(4.6) |α (t2) −α (t1)| ≤ k3 |t2 − t1| .

Suppose that for t ∈ [0,∞) ∩T,

(4.7) |µ (t)| ≤ δa (t) ,

(4.8) (N1 + N2) |b (t)| ≤ βa (t) ,

(4.9) sup
t≥0
|α (t)| = α0,

and that

(4.10) J (α0 + β + δ) < 1,

(4.11) max (|H (−L)| , |H (L)|) ≤
2L

J
,

where α0, β, δ and J are constants with J > 3.
Choose γ > 0 small enough and such that

(4.12)

(
1 +

∣∣∣∣ c (0)1 − τ4 (0)
∣∣∣∣
)
γ +

3L

J
≤ L.

The chosen γ in the relation (4.12) is used below in Lemma 5 to show that if � = L and if ‖ψ‖ < γ,
then the solutions satisfy |x (t, 0,ψ)| < �.

Assume further that

(4.13) t− τ (t) →∞ as t →∞ and
∫ t
0

a (u) ∆u →∞ as t →∞,

(4.14) α (t) → 0 as t →∞,

(4.15)
µ (t)

a (t)
→ 0 as t →∞,



116 BELAID, ARDJOUNI AND DJOUDI

and

(4.16)
b (t)

a (t)
→ 0 as t →∞.

We begin with the following theorem (see [1]) and for convenience we present its proof below. In
the next theorem, we prove that for a well chosen function h, the mapping H given by (3.5) is a large
contraction on the set Sψ. So let us make the following assumptions on the function h : R → R.

(H1) h : R → R is continuous on [−L,L] and differentiable on (−L,L),
(H2) the function h is strictly increasing on [−L,L],
(H3) sup

t∈(−L,L)
h′ (t) ≤ 1.

Theorem 7. Let h : R → R be a function satisfying (H1) − (H3). Then the mapping H in (3.5) is a
large contraction on the set Sψ.

Proof. Let φ,ϕ ∈ Sψ with φσ 6= ϕσ. Then φσ (t) 6= ϕσ (t) for some t ∈ T. Let us denote the set of all
such t by D (φ,ϕ), i.e.,

D (φ,ϕ) = {t ∈ T : φσ (t) 6= ϕσ (t)} .
For all t ∈ D (φ,ϕ), we have

|(Hφ) (t) − (Hϕ) (t)| = |φσ (t) −h (φσ (t)) −ϕσ (t) + h (ϕσ (t))|

= |φσ (t) −ϕσ (t)|
∣∣∣∣1 −

(
h (φσ (t)) −h (ϕσ (t))

φσ (t) −ϕσ (t)

)∣∣∣∣ .(4.17)
Since h is a strictly increasing function, we have

(4.18)
h (φσ (t)) −h (ϕσ (t))

φσ (t) −ϕσ (t)
> 0 for all t ∈ D (φ,ϕ) .

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

Ut =

{
(ϕσ (t) ,φσ (t)) if φσ (t) > ϕσ (t) ,
(φσ (t) ,ϕσ (t)) if ϕσ (t) > φσ (t) .

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

h (φσ (t)) −h (ϕσ (t))
φσ (t) −ϕσ (t)

= h′ (ct) .

By (H2) and (H3), we have

(4.19) 0 ≤ inf
u∈(−L,L)

h′ (u) ≤ inf
u∈Ut

h′ (u) ≤ h′ (ct) ≤ sup
u∈Ut

h′ (u) ≤ sup
u∈(−L,L)

h′ (u) ≤ 1.

Hence, by (4.17)-(4.19), we obtain

(4.20) |(Hφ) (t) − (Hϕ) (t)| ≤
∣∣∣∣1 − inf

u∈(−L,L)
h′ (u)

∣∣∣∣ |φσ (t) −ϕσ (t)| ,
for all t ∈ D (φ,ϕ). Then by (H3), we have

‖Hφ−Hϕ‖≤‖φ−ϕ‖ .
Now, choose a fixed � ∈ (0, 1) and assume that φ and ϕ are two functions in Sψ satisfying

� ≤ sup
t∈D(φ,ϕ)

|φ (t) −ϕ (t)| = ‖φ−ϕ‖ .

If |φσ (t) −ϕσ (t)| ≤ �
2

for some t ∈ D (φ,ϕ), then by (4.19) and (4.20), we get

(4.21) |(Hφ) (t) − (Hϕ) (t)| ≤ |φσ (t) −ϕσ (t)| ≤
1

2
‖φ−ϕ‖ .

Since h is continuous and strictly increasing, the function h
(
u + �

2

)
− h (u) attains its minimum on

the closed and bounded interval [−L,L]. Thus, if �
2
≤ |φσ (t) −ϕσ (t)| for some t ∈ D (φ,ϕ), then by

(H2) and (H3), we conclude that

1 ≥
h (φσ (t)) −h (ϕσ (t))

φσ (t) −ϕσ (t)
> λ,



STABILITY IN TOTALLY NONLINEAR NEUTRAL DYNAMIC EQUATIONS 117

where

λ :=
1

2L
min

{
h
(
u +

�

2

)
−h (t) , u ∈ [−L,L]

}
> 0.

Hence, (4.17) implies

(4.22) |(Hφ) (t) − (Hϕ) (t)| ≤ (1 −λ)‖φ−ϕ‖ .
Consequently, combining (4.21) and (4.22), we obtain

|(Hφ) (t) − (Hϕ) (t)| ≤ η‖φ−ϕ‖ ,
where

η = max

{
1

2
, 1 −λ

}
< 1.

The proof is complete. �

By step we will prove the fulfillment of (i), (ii) and (iii) in Theorem 5.

Lemma 4. Suppose that (3.1), (3.2), (4.7)-(4.10) and (4.13) are true. For A defined in (4.1), if
ϕ ∈ Sψ, then |(Aϕ) (t)| ≤ L/J < L. Moreover, (Aϕ) (t) → 0 as t →∞.

Proof. Using the conditions (4.7)-(4.10) and the expression (4.1) of the map A, we get

|(Aϕ) (t)| ≤
∣∣∣∣ c (t)1 − τ4 (t)ϕ (t− τ (t))

∣∣∣∣ +
∫ t
0

|b (t)| |G (ϕ (s) ,ϕ (s− τ (s)))|e	a (t,s) ∆s

+

∫ t
0

|µ (s)| |ϕσ (s− τ (s))|e	a (t,s) ∆s

≤ α0L +
∫ t
0

(N1 + N2) |b (t)|Le	a (t,s) ∆s + L
∫ t
0

|µ (s)|e	a (t,s) ∆s

≤ L
{
α0 +

∫ t
0

βa (s) e	a (t,s) ∆s +

∫ t
0

δa (s) e	a (t,s) ∆s

}
≤ L (α0 + β + δ) ≤

L

J
< L.

So ASψ is bounded by L as required.
Let ϕ ∈ Sψ be fixed. We will prove that (Aϕ) (t) → 0 as t → 0. Due to the conditions t−τ (t) →∞

as t → ∞ in (4.13) and (4.9), it is obvious that the first term on the right hand side of A tends to 0
as t →∞. That is ∣∣∣∣ c (t)1 − τ4 (t)ϕ (t− τ (t))

∣∣∣∣ ≤ α0 |ϕ (t− τ (t))|→ 0 as t →∞.
It is left to show that the two remaining integral terms of A go to zero as t →∞. Let � > 0 be given.
Find T such that |ϕσ (t− τ (t))| < � for t ≥ T. Then we have∣∣∣∣

∫ t
0

µ (s) ϕσ (s− τ (s)) e	a (t,s) ∆s
∣∣∣∣

≤
∫ T
0

|µ (s) ϕσ (s− τ (s))|e	a (t,s) ∆s +
∫ t
T

|µ (s)| |ϕσ (s− τ (s))|e	a (t,s) ∆s

≤ Le	a (t,T)
∫ T
0

|µ (s)|e	a (T,s) ∆s + �
∫ t
T

|µ (s)|e	a (t,s) ∆s

≤ Lδe	a (t,T) + �δ.
The term Lδe	a (t,T) is arbitrarily small as t → ∞, because of (4.13). The remaining integral term
in A goes to zero by just a similar argument. This ends the proof. �

Lemma 5. Let (H1) − (H3), (3.1), (3.2), (4.7)-(4.11) and (4.13) hold. For A, B defined in (4.1)
and (4.2), if φ,ϕ ∈ Sψ are arbitrary, then

‖Bϕ + Aφ‖≤ L.
Moreover, B is a large contraction on Sψ with a unique fixed point in Sψ and Bϕ (t) → 0 as t →∞.



118 BELAID, ARDJOUNI AND DJOUDI

Proof. Using the definitions (4.1), (4.2) of A and B and applying (4.7)-(4.11), we obtain

|(Bϕ) (t) + (Aφ) (t)|

≤
(

1 +

∣∣∣∣ c (0)1 − τ4 (0)
∣∣∣∣
)
‖ψ‖e	a (t, 0) + α0L + L

∫ t
T

|µ (s)|e	a (t,s) ∆s

+

∫ t
0

(N1 + N2) |b (s)|Le	a (t,s) ∆s +
2L

J

∫ t
0

a (s) e	a (t,s) ∆s

≤
(

1 +

∣∣∣∣ c (0)1 − τ4 (0)
∣∣∣∣
)
‖ψ‖ + (α0 + β + δ) L +

2L

J

≤
(

1 +

∣∣∣∣ c (0)1 − τ4 (0)
∣∣∣∣
)
‖ψ‖ +

L

J
+

2L

J
,

by the monotonicity of the mapping H. So from the above inequality, by choosing the initial function
ψ having small norm, say ‖ψ‖ < γ, then, and referring to (4.12), we obtain

|(Bϕ) (t) + (Aφ) (t)| ≤
(

1 +

∣∣∣∣ c (0)1 − τ4 (0)
∣∣∣∣
)
γ +

3L

J
≤ L.

Since 0 ∈ Sψ, we have also proved that |(Bϕ) (t)| ≤ L. The proof that Bϕ is Lipschitzian is similar to
that of the map Aϕ below. To see that B is a large contraction on Sψ with a unique fixed point, we
know from Theorem 7 that H (ϕ) = ϕσ −h (ϕσ) is a large contraction within the integrand. Thus, for
any �, from the proof of that Theorem 7, we have found η < 1 such that

|(Bϕ) (t) − (Aφ) (t)| ≤
∫ t
0

a (s) |(Hφ) (s) − (Hϕ) (s)|e	a (t,s) ∆s

≤ η
∫ t
0

a (s)‖ϕ−φ‖e	a (t,s) ∆s

≤ η‖ϕ−φ‖ .

To prove that (Bϕ) (t) → 0 as t → ∞, we use (4.13) for the first term, and for the second term, we
argue as above for the map A. �

Lemma 6. Suppose (3.1), (3.2), (4.7)-(4.10) hold. Then the mapping A is continuous on Sψ.

Proof. Let ϕ,φ ∈ Sψ, then

|(Aϕ) (t) − (Aφ) (t)|
≤ {α0 |ϕ (t− τ (t)) −φ (t− τ (t))|

+

∣∣∣∣
∫ t
0

b (s) [G (ϕ (s) ,ϕ (s− τ (s))) −G(φ(s),φ(s− τ(s)))] e	a (t,s) ∆s
∣∣∣∣

+

∣∣∣∣
∫ t
0

µ (s) [ϕσ (s− τ (s)) −φσ (s− τ (s))] e	a (t,s) ∆s
∣∣∣∣
}

≤ α0 ‖ϕ−φ‖ +
∫ t
0

(N1 + N2) |b (s)|‖ϕ−φ‖e	a (t,s) ∆s

+ ‖ϕ−φ‖
∫ t
0

|µ (s)|e	a (t,s) ∆s

≤ (α0 + β + δ)‖ϕ−φ‖
∫ t
0

a (s) e	a (t,s) ∆s

≤ (α0 + β + δ)‖ϕ−φ‖≤ (1/J)‖ϕ−φ‖ .

Let � > 0 be arbitrary. Define η = �J. Then for ‖ϕ−φ‖≤ η, we obtain

‖Aϕ−Aφ‖≤
1

J
‖ϕ−φ‖≤ �.

Therefore, A is continuous. �



STABILITY IN TOTALLY NONLINEAR NEUTRAL DYNAMIC EQUATIONS 119

Lemma 7. Let (3.1), (3.2), (4.4)-(4.9) and (4.14)-(4.16) hold. The function Aϕ is Lipschitzian and
the operator A maps Sψ into a compact subset of Sψ

Proof. Let ϕ ∈ Sψ and let 0 ≤ t1 < t2. Then

|(Aϕ) (t2) − (Aϕ) (t1)|

≤
∣∣∣∣ c (t2)1 − τ4 (t2)ϕ (t2 − τ (t2)) − c (t1)1 − τ4 (t1)ϕ (t1 − τ (t1))

∣∣∣∣
+

∣∣∣∣
∫ t2
0

µ (s) ϕσ (s− τ (s)) e	a (t2,s) ∆s−
∫ t1
0

µ (s) ϕσ (s− τ (s)) e	a (t1,s) ∆s
∣∣∣∣

+

∣∣∣∣
∫ t2
0

b (s) G (ϕ (s) ,ϕ (s− τ (s))) e	a (t2,s) ∆s

−
∫ t1
0

b (s) G (ϕ (s) ,ϕ (s− τ (s))) e	a (t1,s) ∆s
∣∣∣∣ .(4.23)

By hypotheses (4.5)-(4.6), we have

|α (t2) ϕ (t2 − τ (t2)) −α (t1) ϕ (t1 − τ (t1))|
≤ |α (t2)| |ϕ (t2 − τ (t2)) −ϕ (t1 − τ (t1))| + |ϕ (t1 − τ (t1))| |α (t2) −α (t1)|
≤ α0k |(t2 − t1) − (τ (t2) − τ (t1))| + Lk3 |t2 − t1|
≤ (α0k + α0kk2 + Lk3) |t2 − t1| ,(4.24)

where k is the Lipschitz constant of ϕ. By hypotheses (4.4) and (4.7), we have

∣∣∣∣
∫ t2
0

µ (s) ϕσ (s− τ (s)) e	a (t2,s) ∆s−
∫ t1
0

µ (s) ϕσ (s− τ (s)) e	a (t1,s) ∆s
∣∣∣∣

=

∣∣∣∣
∫ t1
0

µ (s) ϕσ (s− τ (s)) e	a (t1,s) (e	a (t2, t1) − 1) ∆s

+

∫ t2
t1

µ (s) ϕσ (s− τ (s)) e	a (t2,s) ∆s
∣∣∣∣

≤ L |e	a (t2, t1) − 1|
∫ t1
0

δa (s) e	a (t1,s) ∆s + L

∫ t2
t1

|µ (s)|e	a (t2,s) ∆s

≤ Lδ
∫ t2
t1

a (s) ∆s + L

∫ t2
t1

e	a (t2,s)

(∫ s
t1

|µ (v)|∆v
)4
4s

≤ Lδ
∫ t2
t1

a (s) ∆s + L

{[
e	a (t2,s)

∫ s
t1

|µ (v)|∆v
]t2
t1

+

∫ t2
t1

a (s) e	a (t2,s)

∫ s
t1

|µ (v)|∆v∆s
}

≤ Lδ
∫ t2
t1

a (s) ∆s + L

∫ t2
t1

|µ (s)|∆s
(

1 +

∫ t2
t1

a (s) e	a (t2,s) ∆s

)
≤ Lδ

∫ t2
t1

a (s) ∆s + 2L

∫ t2
t1

|µ (s)|∆s

≤ Lδ
∫ t2
t1

a (s) ∆s + 2Lδ

∫ t2
t1

a (s) ∆s

≤ 3Lδk1 |t2 − t1| .(4.25)



120 BELAID, ARDJOUNI AND DJOUDI

Similarly, by (4.4) and (4.8), we deduce∣∣∣∣
∫ t2
0

b (s) G (ϕ (s) ,ϕ (s− τ (s))) e	a (t2,s) ∆s

−
∫ t1
0

b (s) G (ϕ (s) ,ϕ (s− τ (s))) e	a (t1,s) ∆s
∣∣∣∣

=

∣∣∣∣
∫ t1
0

b (s) G (ϕ (s) ,ϕ (s− τ (s))) e	a (t1,s) (e	a (t2, t1) − 1) ∆s

+

∫ t2
t1

b (s) G (ϕ (s) ,ϕ (s− τ (s))) e	a (t2,s) ∆s

∣∣∣∣∣
≤ L |e	a (t2, t1) − 1|

∫ t1
0

βa (s) e	a (t1,s) ∆s + (N1 + N2) L

∫ t2
t1

|b (s)|e	a (t2,s) ∆s

≤ Lβ
∫ t2
t1

a (u) ∆u + (N1 + N2) L

∫ t2
t1

e	a (t2,s)

(∫ s
t1

|b (v)|∆v

)4
4s

≤ Lβ
∫ t2
t1

a (u) ∆u + (N1 + N2) L



[
e	a (t2,s)

∫ s
t1

|b (v)|∆v

]t2
t1

+

∫ t2
t1

a (s) e	a (t2,s)

∫ s
t1

|b (v)|∆v∆s

}

≤ Lβ
∫ t2
t1

a (u) ∆u + (N1 + N2) L

∫ t2
t1

|b (s)|∆s
(

1 +

∫ t2
t1

a (s) e	a (t2,s) ∆s

)
≤ Lβ

∫ t2
t1

a (u) ∆u + 2 (N1 + N2) L

∫ t2
t1

|b (s)|∆s

≤ Lβ
∫ t2
t1

a (u) ∆u + 2Lβ

∫ t2
t1

a (s) ∆s

≤ 3Lβk1 |t2 − t1| .(4.26)

Thus, by substituting (4.24)-(4.26) in (4.23), we obtain

|(Aϕ) (t2) − (Aϕ) (t1)|
≤ (α0k + α0kk2 + Lk3) |t2 − t1| + 3Lδk1 |t2 − t1| + 3Lβk1 |t2 − t1|
≤ K |t2 − t1| ,(4.27)

for a constant K > 0. This shows that Aϕ is Lipschitzian if ϕ is and that ASψ is equicontinuous.
Next, we notice that for arbitrary ϕ ∈ Sψ, we have

|Aϕ (t)| ≤
∣∣∣∣ c (t)1 − τ4 (t)ϕ (t− τ (t))

∣∣∣∣ +
∫ t
0

|b (s)| |G (ϕ (s) ,ϕ (s− τ (s)))|e	a (t,s) ∆s

+

∫ t
0

|µ (s) ϕ (s− τ (s))|e	a (t,s) ∆s

≤ L |α (t)| + (N1 + N2) L
∫ t
0

a (s) [|b (s)|/a (s)] e	a (t,s) ∆s

+ L

∫ t
0

a (s) [|µ (s)|/a (s)] e	a (t,s) ∆s

= q (t) ,

because of (4.14)-(4.16). Using a method like the one used for the map A, we see that q (t) → 0 as
t →∞. By Theorem 6, we conclude that the set ASψ resides in a compact set. �



STABILITY IN TOTALLY NONLINEAR NEUTRAL DYNAMIC EQUATIONS 121

Theorem 8. Let L > 0. Suppose that the conditions (H1)−(H3), (3.1), (3.2) and (4.14)-(4.16) hold.
If ψ is a given initial function which is sufficiently small, then there is a solution x (t, 0,ψ) of (1.1)
with |x (t, 0,ψ)| ≤ L and x (t, 0,ψ) → 0 as t →∞.

Proof. From Lemmas 4 and 7 we have A is bounded by L, Lipschitzian and (Aφ) (t) → 0 as t →∞. So
A maps Sψ into Sψ. From Lemmas 5 and 7 for arbitrary, we have φ,ϕ ∈ Sψ, Bϕ+Aφ ∈ Sψ since both
Aφ and Bϕ are Lipschitzian bounded by L and (Bϕ) (t) → 0 as t →∞. From Lemmas 6 and 7, we have
proved that A is continuous and ASψ resides in a compact set. Thus, all the conditions of Theorem
5 are satisfied. Therefore, there exists a solution of (1.1) with |x (t, 0,ψ)| ≤ L and x (t, 0,ψ) → 0 as
t →∞. �

5. Stability and Compactness

Referring to Burton [10], except for the fixed point method, we know of no other way proving that
solutions of (1.1) converge to zero. Nevertheless, if all we need is stability and not asymptotic stability,
then we can avoid conditions (4.14)-(4.16) and still use Krasnoselskii-Burton’s theorem on a Banach
space endowed with a weighted norm.

Let g : [m0,∞)∩T → [1,∞) be any strictly increasing and rd-continuous function with g (m0) = 1,
g (s) →∞ as s →∞. Let

(
S, |.|g

)
be the Banach space of rd-continuous function ϕ : [m0,∞)∩T → R

for which

|ϕ|g := sup
t≥m0

∣∣∣∣ϕ (t)g (t)
∣∣∣∣ < ∞,

exists. We continue to use ‖.‖ as the supremum norm of any ϕ ∈ S provided ϕ bounded. Also, we use
‖ψ‖ as the bound of the initial function. Further, in a similar way as Theorem 7, we can prove that
the function H (ϕ) = ϕσ −h (ϕσ) is still a large contraction with the norm |.|g.

Theorem 9. If the conditions of Theorem 8 hold, except for (4.14)-(4.16), then the zero solution of
(1.1) is stable.

Proof. We prove the stability starting at t0 = 0. Let � > 0 be given such that 0 < � < L, then for
|x| ≤ � find γ∗ with |xσ −h (xσ)| ≤ γ∗ and choose a number γ such that

(5.1) γ + γ∗ +
�

J
≤ �.

In fact, since xσ −h (xσ) is increasing on (−L,L), we may take γ∗ = 2�
J

. Thus, inequality (5.1) allows
γ > 0. Now, remove the condition ϕ (t) → 0 as t → 0 from Sψ defined previously and consider the set

Mψ = {ϕ ∈ S | ϕ Lipschitzian, |ϕ (t)| ≤ �, t ∈ [m0,∞) ∩T
and ϕ (t) = ψ (t) if t ∈ [m0, 0] ∩T} .

Define A, B on Mψ as before by (4.1), (4.2). We easily check that if ϕ ∈ Mψ, then |(Aϕ) (t)| < �, and
B is a large contraction on Mψ. Also, by choosing ‖ψ‖ < γ and referring to (5.1), we verify that for
ϕ,φ ∈ Mψ |(Bϕ) (t) + (Aφ) (t)| ≤ � and |(Bϕ) (t)| ≤ �. AMψ is an equicontinuous set. According to
[10, Theorem 4.0.1], in the space

(
S, |.|g

)
the set AMψ resides in a compact subset of Mψ. Moreover,



122 BELAID, ARDJOUNI AND DJOUDI

the operator A : Mψ → Mψ is continuous. Indeed, for ϕ,φ ∈ Sψ,

|(Aϕ) (t) − (Aφ) (t)|
g (t)

≤
1

g (t)
{α0 |ϕ (t− τ (t)) −φ (t− τ (t))|

+

∣∣∣∣
∫ t
0

b (s) [G (ϕ (s) ,ϕ (s− τ (s))) −G (φ (s− τ (s)))] e	a (t,s) ∆s
∣∣∣∣

+

∣∣∣∣
∫ t
0

µ(s) [ϕσ (s− τ (s)) −φσ (s− τ (s))] e	a (t,s) ∆s
∣∣∣∣
}

≤ α0 |ϕ−φ|g

+

∫ t
0

|b(s)|
(
N1 |ϕ (s) −φ (s)|

g (t)
+
N2 |ϕ (s− τ (s)) −φ (s− τ (s))|

g (t)

)
e	a (t,s) ∆s

+

∫ t
0

|µ (s)|
|ϕσ (s− τ (s)) −φσ (s− τ (s))|

g (t)
e	a (t,s) ∆s

≤ α0 |ϕ−φ|g +
∫ t
0

|b (s)|
[
N1 |ϕ (s) −φ (s)|

g (s)

g (s)

g (t)

]
e	a (t,s) ∆s

+

∫ t
0

|b (s)|
[
N2 |ϕ (s− τ (s)) −φ (s− τ (s))|

g (s− τ (s))
g (s− τ (s))

g (t)

]
e	a (t,s) ∆s

+

∫ t
0

|µ (s)|
[
|ϕσ (s− τ (s)) −φσ (s− τ (s))|

gσ (s− τ (s))
gσ (s− τ (s))

g (t)

]
e	a (t,s) ∆s

≤ α0 |ϕ−φ|g + |ϕ−φ|g
∫ t
0

|b(s)|
[
N1g(s) + N2g(s− τ(s))

g(t)

]
e	a(t,s)∆s

+ δ |ϕ−φ|g
∫ t
0

a (s)
gσ (s− τ (s))

g (t)
e	a (t,s) ∆s

≤ α0 |ϕ−φ|g + β |ϕ−φ|g
∫ t
0

a (s) e	a (t,s) ∆s + δ |ϕ−φ|g
∫ t
0

a (s) e	a (t,s) ∆s

≤
1

J
|ϕ−φ|g .

The conditions of Theorem 5 are satisfied on Mψ, and so there exists a fixed point lying in Mψ and
solving (1.1). �

References

[1] M. Adıvar, Y. N. Raffoul, Existence of periodic solutions in totally nonlinear delay dynamic equations. Electron. J.

Qual. Theory Differ. Equ., Spec. Ed. 1 (2009), 1–20.
[2] A. Ardjouni, I. Derrardjia and A. Djoudi, Stability in totally nonlinear neutral differential equations with variable

delay, Acta Math. Univ. Comenianae, 83 (2014), 119-134.

[3] A. Ardjouni, 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.
[4] A. Ardjouni, A Djoudi, Stability in neutral nonlinear dynamic equations on time scale with unbounded delay, Stud.

Univ. Babeç-Bolyai Math. 57(2012), No. 4, 481-496.
[5] A. Ardjouni, A Djoudi, Fixed points and stability in linear neutral differential equations with variable delays,

Nonlinear Analysis 74 (2011), 2062-2070.

[6] M. Bohner, A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications, Birkhauser,
Boston, 2001.

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

[8] T. A. Burton, Liapunov functionals, fixed points and stability by Krasnoselskii’s theorem, Nonlinear Stud. 9 (2001),
181–190.

[9] T. A. Burton, Stability by fixed point theory or Liapunov theory: A Comparaison, Fixed Point Theory, 4(2003),

15-32.
[10] T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York,

2006.



STABILITY IN TOTALLY NONLINEAR NEUTRAL DYNAMIC EQUATIONS 123

[11] I. Derrardjia, A. Ardjouni and A. Djoudi, Stability by Krasnoselskii’s theorem in totally nonlinear neutral differential
equations, Opuscula Math. 33(2) (2013), 255-272.

[12] S. Hilger, Ein Maβkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph. D. thesis, Universität

Würzburg, Würzburg, 1988.
[13] E. R. Kaufmann, Y. N. Raffoul, Stability in neutral nonlinear dynamic equations on a time scale with functional

delay, Dynamic Systems and Applications 16 (2007) 561-570.

[14] D. R. Smart, Fixed point theorems, Cambridge Tracts in Mathematics, no. 66, Cambridge University Press, London–
New York, 1974.

1Department of Mathematics and Informatics, University of Souk Ahras, P.O. Box 1553, Souk Ahras,

41000, Algeria

2Applied Mathematics Lab, Faculty of Sciences, Department of Mathematics, University of Annaba, P.O.

Box 12, Annaba 23000, Algeria

∗Corresponding author: abd ardjouni@yahoo.fr