On Stability of Nonlinear Differential System via Cone Perturbing Liapunov Function Method

(1)

http://dx.doi.org/10.4236/am.2015.610157

On Stability of Nonlinear Differential

System via Cone-Perturbing

Liapunov Function Method

A. A. Soliman, W. F. Seyam

Department Mathematics, Faculty of Sciences, Benha University, Benha, Egypt Email: [email protected]

Received 8 August 2015; accepted 21 September 2015; published 24 September 2015

This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Abstract

Totally equistable, totally φ0-equistable, practically equistable, and practically φ0-equistable of system of differential equations are studied. Cone valued perturbing Liapunov functions method and comparison methods are used. Some results of these properties are given.

Keywords

Totally Equistable, Totally φ0-Equistable, Practically Equistable, Practically φ0-Equistable

1. Introduction

Consider the non linear system of ordinary differential equations

( )

, ,

( )

0 0

x′ = f t x x t =x (1.1) and the perturbed system

( )

,

( )

, ,

( )

0 0.

x′ = f t x +R t x x t =x (1.2) Let Rn be Euclidean n-dimensional real space with any convenient norm . , and scalar product

( )

.,. ≤ . . . Let for some

ρ

>0

{

n,

}

Sρ = x∈R x <ρ

(2)

Consider the scalar differential equations with an initial condition

( )

1 , , 0 0

u′ =g t u u t =u , (1.3)

(

)

( )

2 , , 0 0

g t t

ω′ = ω ω =ω (1.4)

and the perturbing equations

( )

1 , 1 , 0 0

u′ =g t u +ϕ t u t =u (1.5)

(

)

( )

2 , 2 , 0 0

g t t t

ω′ = ω +ϕ ω =ω (1.6)

where g g₁, ₂∈C J

[

×R R,

]

, ϕ ϕ₁, ₂∈C J R

[

,

]

respectively.

Other mathematicians have been interested in properties of qualitative theory of nonlinear systems of diffe-rential equations. In last decade, in [1], some different concepts of stability of system of ordinary diffediffe-rential Equations (1.1) are considered namely, say totally stability, practically stability of (1.1), and (1.2); and in [2], methods of perturbing Liapunov function are used to discuss stability of (1.1). The authors in [3] discussed some stability of system of ordinary differential equations, and in [4] [5] the authors discussed totally and totally φ0-stability of system of ordinary differential Equations (1.1) using Liapunov function method that was played essential role for determine stability of system of differential equations. In [6] the authors discussed practically stability for system of functional differential equations.

In [7], and [8], the authors discussed new concept namely, φ0-equitable of the zero solution of system of or-dinary differential equations using cone-valued Liapunov function method. In [4], the author discussed and im-proved some concepts stability and discussed concept mix between totally stability from one side and φ0- stability on the other side.

In this paper, we will discuss and improve the concept of totally stability, practically stability of the system of ordinary differential Equations (1.1) with Liapunov function method, and comparison technique. Furthermore, we will discuss and improve the concept of totally φ0-stability, and practically φ0-stability of the system of ordi-nary differential Equations (1.1). These concepts are mix and lie somewhere between totally stability and prac-tically stability from one side and φ0-stability on the other side. Our technique depends on cone-valued Liapunov function method, and comparison technique. Also we give some results of these concepts of the zero solution of differential equations.

The following definitions [8] will be needed in the sequal.

Definition 1.1. A proper subset K of n

R is called a cone if

( )

0

( )

( ) { }

i λK⊂K,λ≥0. ii K+K⊂K, iii K=K, iv K ≠ ∅, v K −K = 0

where K and K0 denote the closure and interior of K respectively and ∂K denotes the boundary of K.

Definition 1.2. The set *

{

(

)

}

, , 0,

n

K = φ∈R φ x ≥ x∈K is called the adjoint cone if it satisfies the properties of the definition 3.1.

( )

*

{ }

0 0

if , 0 for some , / 0 .

x∈∂K φ x = φ∈K K =K

Definition 1.3. A function g D: →K D, ⊂Rn is called quasimonotone relative to the cone K if

, ,

x y∈D y− ∈∂x K

then there exists * 0 K0

φ ∈ such that

(

φ0,y−x

)

=0 and

(

φ0,g y

( )

−g x

( )

)

>0.

Definition 1.4. A function a r

( )

is said to belong to the class  if a∈_R+,R+_,a

( )

0 =0 and a r

( )

is strictly monotone increasing in r.

2. Totally Equistable

(3)

functions method and Comparison principle method.

We define for V∈C J ×Sρ,Rn, the function D V t x+

( )

, by

( )

_1.2

(

( )

)

( )

)

0 1

, lim sup , , , , .

h

D V t x V t h x h f t x R t x V t x

h

+

→

= + + + −

The following definition [1] will be needed in the sequal.

Definition 2.1. The zero solution of the system (1.1) is said to be T₁-totally equistable (stable with respect to permanent perturbations), if for every >0,t₀∈J there exist two positive numbers δ₁=δ₁

( )

 >0 and

( )

2 2 0

δ =δ  > such that for every solution of perturbed Equation (1.2), the inequality

(

, ,0 0

)

0

x t t x < for t≥t

holds, provided that x₀ <δ₁ and R t x

( )

, <δ₂.

Definition 2.2. The zero solution of the Equation (1.3) is said to be T₁-totally equistable (stable with respect to permanent perturbations), if for every >0,t₀∈J , there exist two positive numbers * *

( )

1 1 0

δ =δ  > and

( )

* *

2 2 0

δ =δ  > such that for every solution of perturbed Equation (1.5). The inequality

(

, ,0 0

)

, 0 u t t u < t≥t holds, provided that *

0 1

u <δ and

( )

*

1 t 2

ϕ <δ .

Theorem 2.1. Suppose that there exist two functions

[

]

1, 2 ,

g g ∈C J×R R with g t₁

( )

, 0 =g₂

( )

t, 0 =0

and there exist two Liapunov functions

1 ,

n

V ∈C J ×Sρ R  and 2 ,

C n

Vη∈C J ×SρSη R  with V t1

( )

, 0 =V2η

( )

t, 0 =0

where Sη =

{

x∈Rn, x <η

}

for

η

>0 and SηC denotes the complement of Sη satisfying the following conditions:

(H1) V t x1

( )

, is locally Lipschitzian in x.

( )

(

( )

)

( )

1 , 1 , 1 , , , .

D V t x+ ≤g t V t x ∀ t x ∈ ×J Sρ

(H2) V2η

( )

t x, is locally Lipschitzian in x.

( )

2

( )

,

( )

,

( )

,

C

b x ≤V_η t x ≤a x ∀ t x ∈ ×J S_ρS_η

where a b, ∈ are increasing functions. (H3)

( )

(

( )

)

( )

1 , 2 , 2 , 1 , 2 , , , .

C

D V t x+ +D V+ _η t x ≤g t V t x +V_η t x ∀ t x ∈ ×J S_ρS_η

(H4) If the zero solution of (1.3) is equistable, and the zero solution of (1.4) is totally equistable. Then the zero solution of (1.1) is totally equistable.

Proof. Since the zero solution of the system (1.4) is totally equistable, given b

( )

 >0, there exist two positive numbers δ₁*=δ₁*

( )

 >0 and δ₂*=δ₂*

( )

 >0 such that for every solution ω

(

t t, ,₀ ω₀

)

of perturbed equation (1.6) the inequality

(

t t, ,0 0

)

, t t0

ω ω < ≥ (2.1)

holds, provided that *

0 1

ω <δ and ϕ₂

( )

t <δ₂*.

Since the zero solution of (1.3) is equistable given 0

( )

2 δ 

and t₀∈J, there exists δ δ=

(

t₀,

)

>0 such that

(

)

0

( )

0 0 , ,

2

(4)

holds, provided that u₀≤δ.

From the condition (H2) we can find δ1=δ1

( )

 >0 such that

( )

0 *

1 1

2 .

a δ +δ <δ (2.3)

To show that the zero solution of (1.1) is T₁-totally equistable, it must show that for every >0,t₀∈J there exist two positive numbers δ₁=δ₁

( )

 >0 and δ₂=δ₂

( )

 >0 such that for every solution x t t x

(

, ,₀ ₀

)

of per-turbed Equation (1.2). The inequality

(

, ,0 0

)

for 0 x t t x < t≥t

( )

, <δ₂.

Suppose that this is false, then there exists a solution x t t x

(

, ,₀ ₀

)

of (1.2) with t₁>t₀ such that

(

0, ,0 0

)

1,

(

1, ,0 0

)

x t t x =δ x t t x = (2.4)

(

)

[

]

1 x t t x, ,0 0 fort t t0,1 .

δ ≤ ≤ ∈

Let δ η₁= and setting m t x

( )

, =V t x₁

( )

, +V₂η

( )

t x, .

Since V t x₁

( )

, and V₂η

( )

t x, are Lipschitzian in x for constants M1 and M2 respectively. Then

( )

1 , 1.2 2 , 1.2 1 , 1.1 2 , 1.1 ,

D V t x+ +D V+ η t x ≤D V t x+ +D V+ η t x +M R t x

where M =M₁+M₂ From the condition (H3) we obtain the differential inequality

( )

(

( )

)

( )

1 , 2 , 2 , 1 , 2 , ,

D V t x+ +D V+ η t x ≤g t V t x +Vη t x +M R t x

for t∈

[

t t₀,₁

]

Then we have

( )

, 2

(

,

( )

,

)

( )

, .

D m t x+ ≤g t m t x +M R t x

Let ω₀ =m t

(

₀,x₀

)

=V t₁

(

₀,x₀

)

+V₂_η

(

t₀,x₀

)

.

Applying the comparison Theorem (1.4.1) of [1], it yields

( )

, 2

(

, ,0 0

)

for

[

0,1

]

m t x ≤r t t ω t∈ t t

where r t t₂

(

, ,₀ ω₀

)

is the maximal solution of the perturbed Equation (1.6). Define ϕ₂

( )

t =M R t x

( )

, .

To prove that

(

)

( )

2 , ,0 0 .

r t t ω <b 

It must be show that

*

0 1

ω <δ and ϕ₂

( )

t <δ₂*.

Choose u₀=V t x₁

(

₀, ₀

)

. From the condition (H1) and applying the comparison Theorem of [1], it yields

( )

(

)

1 , 1 , ,0 0 V t x ≤r t t u

where r t t u₁

(

, ,₀ ₀

)

is the maximal solution of (1.3). From (2.2) at t=t₀

(

)

(

)

0

( )

1 0, 0 1 0, ,0 0 . 2

(5)

From the condition (H2) and (2.4), at t=t0

(

)

( )

2 0, 0 0 1 .

V_η t x ≤a x ≤a δ (2.6)

From (2.3), we get

(

)

(

)

0

( )

*

0 1 0, 0 2 0, 0 1 1.

2

V t x V_η t x δ a

ω = + ≤  + δ <δ

Since

( )

*

2 t M R t x, M 2 2.

ϕ = ≤ δ =δ

From (2.1), we get

( )

, 2

(

, 0, 0

)

( )

.

m t x ≤r t t ω <b  (2.7) Then from the condition (H2), (2.4) and (2.7) we get t=t1

( )

(

( )

1

)

2

(

1,

( )

1

)

(

1,

( )

1

)

2

(

1, ,0 0

)

( )

.

b  =b x t ≤Vη t x t <m t x t ≤r t t ω <b 

This is a contradiction, then it must be

(

, ,0 0

)

for 0 x t t x < t≥t

( )

, <δ₂. Therefore the zero solution of (1.1) is totally equistable.

3. Totally

φ

0

-Equistable

In this section we discuss the concept of Totally φ0-equistable of the zero solution of (1.1) using cone valued perturbing Liapunov functions method and Comparison principle method.

Definition 3.1. The zero solution of the system (1.1) is said to be totally φ0-equistable (φ0-equistable with re-spect to permanent perturbations), if for every >0, t₀∈J and *

0 K0

φ ∈ , there exist two positive numbers

( )

1 1 0

δ =δ  > and δ₂=δ₂

( )

 >0 such that the inequality

(

)

(

φ0,x t t x, ,0 0

)

< fort≥t0

holds, provided that

(

φ₀,x₀

)

<δ₁ and R t x

( )

, <δ₂ where x t t x

(

, ,₀ ₀

)

is the maximal solution of perturbed Equation (1.2).

Let for some

ρ

>0

(

)

{

}

* *

0 0 0

, , , .

n

Sρ = x∈R φ x <ρ φ ∈K

[

]

1, 2 ,

( )

, 0 =g₂

( )

t, 0 =0

and let there exist two cone valued Liapunov functions

*

1 ,

V ∈C J ×Sρ K and 2 * * ,

C

Vη∈C J ×SρSη K with V t1

( )

, 0 =V2η

( )

t, 0 =0

where *

{

(

)

*

}

0 0 0

, , ,

Sη = x∈K φ x <η φ ∈K for

η

>0 and Sη*C denotes the complement of Sη* satisfying the following conditions:

(h1) V t x1

( )

, is locally Lipschitzian in x and

( )

(

)

(

( )

)

( )

*

0, 1 , 1 , 1 , for , .

D+ φ V t x ≤g t V t x t x ∈ ×J S_ρ

(h2) V₂η

( )

t x, is locally Lipschitzian in x and

(

)

(

( )

)

(

)

(

)

* *

0, 0, 2 , 0, for ,

C t

(6)

where a b, ∈ are increasing functions.

(h3)

(

0, 1

( )

,

)

(

0, 2

( )

,

)

2

(

, 1

( )

, 2

( )

,

)

for

( )

, * * .

C

D+ φ V t x +D+ φ Vη t x ≤g t V t x +Vη t x t x ∈ ×J SρSη

(h4) If the zero solution of (1.3) is φ0-equistable, and the zero solution of (1.4) is totally φ0-equistable. Then the zero solution of (1.1) is totally φ0-equistable.

Proof. Since the zero solution of (1.4) is totally φ0-equistable, given, given b

( )

 >0 there exist two positive numbers * *

( )

1 1 0

δ =δ  > and * *

( )

2 2 0

δ =δ  > such that the inequality

(

)

(

φ0,r t t2 , ,0 ω0

)

<, t≥t0 (3.1) holds, provided that

(

φ ω₀, ₀

)

<δ₁* and ϕ₂

( )

t <δ₂*. where r t t₂

(

, ,₀ ω₀

)

Since the zero solution of the system (1.3) is φ0-equistable, given

( )

0 2 δ 

and t₀∈J there exists

(

t0,

)

0

δ δ=  > such that

(

)

(

)

0

( )

0,1 , ,0 0 2 r t t u δ

φ <  (3.2)

(

φ₀,u₀

)

≤δ where r t t u₁

(

, ,₀ ₀

)

is the maximal solution of (1.3). From the condition (h2) we can choose δ1=δ1

( )

 >0 such that

( )

0 *

1 1

2 .

a δ +δ <δ (3.3)

To show that the zero solution of (1.1) is T1-totally φ0-equistable, it must be prove that for every >0,t0∈J

and *

0 K0

φ ∈ there exist two positive numbers δ₁=δ₁

( )

 >0 and δ₂=δ₂

( )

 >0 such that the inequality

(

)

(

φ0,x t t x, ,0 0

)

< fort≥t0

(

φ₀,x₀

)

<δ₁ and R t x

( )

(

, ,₀ ₀

)

Suppose that is false, then there exists a solution x t t x

(

, ,₀ ₀

)

(

)

(

φ0,x t t x0, ,0 0

)

=δ1,

(

φ0,x t t x

(

1, ,0 0

)

= (3.4)

(

)

(

)

[

]

1 0,x t t x, ,0 0 fort t t0,1 .

δ ≤ φ ≤ ∈

Let δ η₁= and setting m t x

( )

, =V t x₁

( )

, +V₂η

( )

t x, .

Since V t x₁

( )

, and V₂η

( )

t x, are Lipschitzian in x for constants M1 and M2 respectively. Then

( )

(

)

(

( )

)

( )

(

)

(

( )

)

( )

0 1 _1.2 0 2 _1.2

0 1 _1.1 0 2 _1.1

, , , ,

, , , , ,

D V t x D V t x

D V t x D V t x MR t x

η

φ φ

+ +

+

≤ + +

where M =M₁+M₂ From the condition (h3) we obtain the differential inequality

( )

(

0, 1 ,

)

(

0, 2

( )

,

)

2

(

, 1

( )

, 2

( )

,

)

( )

,

D+ φ V t x +D+ φ V_η t x ≤g t V t x +V_η t x +M R t x

for t∈

[

t t₀,₁

]

Then we have

( )

(

0, ,

)

2

(

,

( )

,

)

( )

, .

D+ φ m t x ≤g t m t x +M R t x

Let ω₀=m t x

(

₀, ₀

)

=V t x₁

(

₀, ₀

)

+V₂η

(

t x₀, ₀

)

. Applying the comparison Theorem of [1], yields

( )

(

φ0,m t x,

)

≤

(

φ0,r t t2

(

, ,0 ω0

)

fort∈

[

t t0,1

]

.

(7)

To prove that

(

)

(

φ0,r t t2 , ,0ω0

)

<b

( )

 .

It must be shown that

(

)

*

( )

*

0, 0 1 and 2 t 2. φ ω <δ ϕ <δ

Choose u₀ =V t x₁

(

₀, ₀

)

. From the condition (h1) and applying the comparison Theorem [1], it yields

( )

(

φ0,V t x1 ,

)

≤

(

φ0,r t t u1

(

, ,0 0

)

.

From (3.2) at t=t₀

(

)

(

)

(

)

0

( )

0, 1 0, 0 0,1 0, ,0 0 .

2

V t x r t t u δ

φ ≤ φ <  (3.5)

From the condition (h2) and (3.4), at t=t0

(

)

(

φ0,V2η t x0, 0

)

≤a

(

φ0,x0

)

≤a

( )

δ1 . (3.6) From (3.3), we get

(

)

(

)

(

)

0

( )

*

0, 0 0, 1 0, 0 0, 2 0, 0 1 1.

2

V t x V_η t x δ a

φ ω = φ + φ ≤  + δ <δ

Since

( )

*

2 t M R t x, M 2 2.

ϕ = ≤ δ =δ

From (3.1), we get

( )

(

φ0,m t x,

)

≤

(

φ0,r t t2

(

, ,0 ω0

)

<b

( )

 . (3.7)

Then from the condition (h2), (3.4) and (3.7) we get at t=t1

( )

(

0,

( )

1

)

(

0, 2

(

1,

( )

1

)

(

0,

(

1,

( )

1

)

(

0, 2

(

1, ,0 0

)

( )

.

b  =b φ x t ≤ φ Vη t x t < φ m t x t ≤ φ r t t ω <b 

This is a contradiction, then

(

)

(

φ0,x t t x, ,0 0

)

< fort≥t0

provided that

(

φ₀,x₀

)

<δ₁ and R t x

( )

(

, ,₀ ₀

)

is the maximal solution of perturbed equation (1.2). Therefore the zero solution of (1.1) is totally φ0-equistable.

4. Practically Equistable

In this section, we discuss the concept of practically equistable of the zero solution of (1.1) using perturbing Liapunov functions method and Comparison principle method.

Definition 4.1. Let 0< <

λ

A be given. The system (1.1) is said to be practically equistable if for t₀∈J such that the inequality

(

, ,0 0

)

for 0

x t t x <A t≥t (4.1)

holds, provided that x₀ <λ where x t t x

(

, ,₀ ₀

)

is any solution of (1.1).

In case of uniformly practically equistable, the inequality (4.1) holds for any t₀. We define

( )

{

n: , 0

}

S A = x∈R x ≤A A> .

(8)

[

]

1, 2 ,

( )

, 0 =g₂

( )

t, 0 =0

and there exist two Liapunov functions

( )

1 ,

n

V ∈C J_ ×S A R _ and 2

( )

,

C _n

V_η∈C J_ ×S A S B R _ with V t₁

( )

, 0 =V₂_B

( )

t, 0 =0

where S B

( )

=

{

x∈Rn,x<B, 0< <B A

}

and S B

( )

C denotes the complement of S B

( )

satisfying the fol-lowing conditions:

(I) V t x₁

( )

, is locally Lipschitzian in x.

( )

(

( )

)

( )

1 , 1 , 1 , , , .

D V t x+ ≤g t V t x ∀ t x ∈ ×J S A

(II) V₂B

( )

t x, is locally Lipschitzian in x.

( )

2

( )

,

( )

,

( )

,

( )

C B

b x ≤V t x ≤a x ∀ t x ∈ ×J S A S B

where a b, ∈ are increasing functions. (III)

( )

(

( )

)

( )

1 , 2 , 2 , 1 , 2 , , , .

C B

D V t x+ +D V+ _η t x ≤g t V t x +V t x ∀ t x ∈ ×J S A S B

(IV) If the zero solution of (1.3) is equistable, and the zero solution of (1.4) is uniformly practically equistable. Then the zero solution of (1.1) is practically equistable.

Proof. Since the zero solution of (1.4) is uniformly practically equistable, given 0<λ₀<A such that for every solution ω

(

t t, ,₀ ω₀

)

of (1.4) the inequality

(

t t, ,0 0

)

b A

( )

ω ω < (4.2)

holds provided ω₀≤λ₀.

Since the zero solution of the system (1.3) is equistable, given 0 2 λ

and t₀∈R+ there exist δ δ=

(

t0,

)

>0 such that for every solution u t t u

(

, ,₀ ₀

)

of (1.3)

(

)

0

0 0 , ,

2

u t t u <λ (4.3)

holds provided that u₀≤δ.

From the condition (II) we can find

λ

>0 such that

( )

0

2 .

a λ +λ ≤λ (4.4)

To show that the zero solution of (1.1) practically equistable, it must be exist 0< <

λ

A such that for any solution x t t x

(

, ,₀ ₀

)

of (1.1) the inequality

(

, ,0 0

)

for 0 x t t x <A t≥t

holds, provided that x₀ <λ.

Suppose that this is false, then there exists a solution x t t x

(

, ,₀ ₀

)

(

0, ,0 0

)

,

(

1, ,0 0

)

x t t x =λ x t t x =A (4.5)

(

, ,0 0

)

for

[

0,1

]

.

x t t x A t t t

λ≤ ≤ ∈

Let

λ

=B and setting

( )

, 1

( )

, 2

( )

, .

(9)

From the condition (III) we obtain the differential inequality for t∈

[

t t₀,₁

]

( )

, 2

(

,

( )

,

)

.

D m t x+ ≤g t m t x

Let ω₀ =m t

(

₀,x₀

)

=V t₁

(

₀,x₀

)

+V₂B

(

t₀,x₀

)

.

Applying the comparison Theorem [8], yields

( )

, 2

(

, ,0 0

)

for

[

0,1

]

m t x ≤r t t ω t∈ t t

where r t t₂

(

, ,₀ ω₀

)

is the maximal solution of (1.4). To prove that

(

)

( )

2 , ,0 0 .

r t t ω <b A

It must be show that ω₀≤λ₀.

(

₀, ₀

)

, from the condition (II) and applying the comparison Theorem of [1], yields

( )

(

)

1 , 1 , ,0 0 V t x ≤r t t u

where r t t u₁

(

, ,₀ ₀

)

is the maximal solution of (1.3). From (4.3) at t=t₀

( )

(

)

0

1 , 1 , ,0 0 .

2

V t x ≤r t t u <λ (4.6)

From the condition (II) and (4.5), at t=t₀

(

)

(

( )

)

( )

2B 0, 0 0 .

V t x ≤a x t ≤a λ (4.7)

From (4.4), (4.6) and (4.7), we get

(

)

(

)

0 V t x1 0, 0 V2B t x0, 0 0.

ω = + ≤λ

From (4.2), we get

( )

, 2

(

, ,0 0

)

( )

.

m t x ≤r t t ω <b A (4.8)

Then from the condition (II), (4.5) and (4.8), we get at t=t₁

( )

(

( )

1

)

2B

(

1, 1

)

(

1,

( )

1

)

2

(

1, ,0 0

)

( )

.

b A =b x t ≤V t x <m t x t ≤r t t ω <b A

This is a contradiction, then

(

, ,0 0

)

for 0 x t t x <A t≥t

provided that x₀ <λ.

Therefore the zero solution of (1.1) is practically equistable.

5. Practically

φ

0

-Equistable

In this section we discuss the concept of practically φ0-equistable of the zero solution of (1.1) using cone valued perturbing Liapunov functions method and comparison principle method.

The following definitions [6] will be needed in the sequal.

Definition 5.1. Let 0< <

λ

A be given. The system (1.1) is said to be practically φ0-equistable, if for t0∈J

and *

0 K0

φ ∈ such that the inequality

(

)

(10)

In case of uniformly practically φ0-equistable, the inequality (5.1) holds for any t0. We define

( )

{

(

)

}

* *

0 0 0

, , , .

S A = x∈K φ x <Aφ ∈K

[

]

1, 2 ,

( )

, 0 =g₂

( )

t, 0 =0

and let there exist two cone valued Liapunov functions

( )

*

1 ,

V ∈C J_ ×S A K_ and 2 *

( )

*

( )

,

C B

V ∈C J_ ×S A S B K_ with V t₁

( )

, 0 =V₂B

( )

t, 0 =0

where *

( )

{

(

)

*

}

0 0 0 0

, , , 0 ,

S B = x∈K φ x <B < <B Aφ ∈K and *

( )

C

S B denotes the complement of S*

( )

B sa-tisfying the following conditions:

(i) V t x₁

( )

, is locally Lipschitzian in x relative to K.

( )

(

)

(

( )

)

( )

*

( )

0, 1 , 1 , 1 , , , .

D+ φ V t x ≤g t V t x ∀ t x ∈ ×J S A

(ii) V₂_B

( )

t x, is locally Lipschitzian in x relative to K.

(

)

(

( )

)

(

)

( )

*

( )

*

( )

0, 0, 2 , 0, , ,

C B

b φ x ≤ φ V t x ≤a φ x ∀ t x ∈ ×J S A S B

where a b, ∈ are increasing functions.

(iii)

(

( )

)

(

( )

)

(

( )

)

( )

*

( )

*

( )

0, 1 , 0, 2 , 2 , 1 , 2 , , , .

C

B B

D+ φ V t x +D+ φ V t x ≤g t V t x +V t x ∀ t x ∈ ×J S A S B

(iv) If the zero solution of (1.3) is φ0-equistable, and the zero solution of (1.4) is uniformly practically φ0- equistable.

Then the zero solution of (1.1) is practically φ0-equistable.

Proof. Since the zero solution of the system (1.4) is uniformly practically φ0-equistable, given 0<λ0<a B

( )

for a B

( )

>0 such that the inequality

(

)

(

φ0,r t t2 , ,0ω0

)

<a B

( )

(5.2)

holds provided

(

φ ω₀, ₀

)

≤λ₀, where r t t₂

(

, ,₀ ω₀

)

is the maximal solution of (1.4). Since the zero solution of the system (1.3) is φ0-equistable, given 0

2 λ

and t₀∈R+ there exist δ δ=

(

t0,λ0

)

such that the inequality

(

)

(

)

0

0,1 , ,0 0 . 2 r t t u λ

φ < (5.3)

From the condition (ii), assume that

( )

a B ≤b A (5.4) also we can choose λ >₁ 0 such that

( )

0

2 .

a λ +λ ≤λ (5.5)

To show that the zero solution of (1.1) is practically φ0-equistable. It must be show that for 0< <λ A t,0∈J

and *

0 K0

φ ∈ such that the inequality

(

)

(

φ0,x t t x, ,0 0

)

<A fort≥t0

(

φ₀,x₀

)

<λ where x t t x

(

, ,₀ ₀

)

is the maximal solution of (1.1).

(11)

(

)

(

φ0,x t t x1, ,0 0

)

=λ1,

(

φ0,x t t x

(

2, ,0 0

)

=A (5.6)

(

)

(

)

[

]

1 0,x t t x, ,0 0 A fort t t1, 2 .

λ ≤ φ ≤ ∈

Let λ =₁ B and setting

( )

, 1

( )

, 2B

( )

, .

m t x =V t x +V t x From the condition (iii) we obtain the differential inequality

( )

(

0, ,

)

(

0, 2

(

,

( )

,

)

for

[

1, 2

]

.

D+ φ m t x ≤ φ g t m t x t∈ t t

Let ω₀ =m t x t

(

₁,

( )

₁

)

=V t x t₁

(

₁,

( )

₁

)

+V₂_B

(

t x t₁,

( )

₁

)

. Applying the comparison Theorem of [1], yields

( )

(

φ0,m t x,

)

≤

(

φ0,r t t2

(

, ,0 ω0

)

.

To prove that

(

)

(

φ0,r t t2 , ,0 ω0

)

<a B

( )

.

It must be show that

(

φ ω0, 0

)

≤λ0.

(

₀, ₀

)

From the condition (i) and applying the comparison Theorem of [1], yield

( )

(

φ0,V t x1 ,

)

≤

(

φ0,r t t u1

(

, ,0 0

)

.

From (5.3) at t=t₁

( )

(

)

(

)

0

0, 1 , 0,1 , ,0 0 . 2

V t x r t t u λ

φ ≤ φ < (5.8)

From the condition (ii) and (5.6), at t=t₁

( )

(

)

(

φ0,V2B t x t1, 1

)

≤

(

φ0,x t

( )

1

)

≤a

( )

λ1 . (5.9) From (5.5), (5.8) and (5.9), we get

(

φ ω0, 0

)

=

(

φ0,V t x t1

(

1,

( )

1

)

+

(

φ0,V2B

(

t x t1,

( )

1

)

≤λ0.

From (5.2), we get

( )

(

φ0,m t x,

)

≤

(

φ0,r t t2

(

, ,0 ω0

)

<a B

( )

. (5.10)

Then from the condition (ii), (5.4), (5.6) and (5.10), we get at t=t₂

( )

(

0,

( )

2

)

(

0,

(

2,

( )

2

)

(

0, 2

(

2, ,0 0

)

( )

b A =b φ x t ≤ φ m t x t < φ r t t ω <a B ≤a A

which leads to a contradiction, then it must be

(

)

(

φ0,x t t x, ,0 0

)

<A for t≥t0

(

φ₀,x₀

)

<λ. Therefore the zero solution of (1.1) is practically φ0-equistable.

Acknowledgements

(12)

References

[1] Lakshmikantham, V. and Leela, S. (1969) Differential and Integeral Inequalities. Vol. I, Academic Press, New York. [2] Lakshmikantham, V. and Leela, S. (1976) On Perturbing Liapunov Functions. Journal of Systems Theory, 10, 85-90.

http://dx.doi.org/10.1007/BF01683265

[3] El Sheikh, M.M.A., Soliman, A.A. and Abd Alla, M.H. (2000) On Stability of Non Linear Systems of Ordinary Diffe-rential Equations. Applied Mathematics and Computation, 113, 175-198.

http://dx.doi.org/10.1016/S0096-3003(99)00082-X

[4] Soliman, A.A. (2002) On Total φ0-Stability of Non Linear Systems of Differential Equations. Applied Mathematics and Computation, 130, 29-38. http://dx.doi.org/10.1016/S0096-3003(01)00087-X

[5] Soliman, A.A. (2003) On Total Stability of Perturbed System of Differential Equations. Applied Mathematics Letters,

16, 1157-1162. http://dx.doi.org/10.1016/S0893-9659(03)90110-8

[6] Soliman, A.A. (2005) On Practical Stability of Perturbed Differential Systems. Applied Mathematics and Computation,

163, 1055-1060. http://dx.doi.org/10.1016/j.amc.2004.06.073

[7] Akpan, E.P. and Akinyele, O. (1992) On φ0-Stability of Comparison Differential Systems. Journal of Mathematical Analysis and Applications, 164, 307-324. http://dx.doi.org/10.1016/0022-247X(92)90116-U