Stability and boundedness of solutions of nonlinear fourth order differential equations with bounded delay

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SAKARYA ÜNİVERSİTESİ FEN BİLİMLERİ ENSTİTÜSÜ DERGİSİ SAKARYA UNIVERSITY JOURNAL OF SCIENCE

e-ISSN: 2147-835X Dergi sayfası:http://dergipark.gov.tr/saufenbilder Geliş/Received 24-04-2017 Kabul/Accepted 01-08-2017 Doi 10.16984/saufenbilder.308097

Stability and boundedness of solutions of nonlinear fourth order differential

equations with bounded delay

Erdal Korkmaz *1

ABSTRACT

In this paper, we determine sufficient conditions for the boundedness, uniformly asymtotically stability of the solutions to a certain fourth-order non-autonomous differential equations with bounded delay by considering second method of Lyapunov. The results obtain essentially improve, include and complement the consequences in the current literature.

Keywords: Stability, Boundedness, Lyapunov functional, Delay differential equations, Fourth order.

Dördüncü mertebeden sınırlı gecikmeli nonlineer diferansiyel denklemlerin

çözümlerinin kararlılığı ve sınırlılığı

ÖZ

Bu makalede Lyapunov’un ikinci metodu kullanılarak dördüncü mertebeden otonom olmayan değişken gecikmeli diferansiyel denklemlerin çözümlerinin düzgün asimptotik kararlılığı ve sınırlılığı için yeterli şartları veririz. Elde edilen sonuçlar literatürdeki sonuçları tamamlar, kapsar ve geliştirir.

Anahtar Kelimeler: Kararlılık, Sınırlılık, Lyapunov fonksiyonu, Gecikmeli diferansiyel denklemler, Dördüncü mertebe.

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1. INTRODUCTION

Differential equations with higher-order have been widely used in mechanics, vibration theory, electromechanical systems of physics and engineering. Solutions of the boundedness and stability problem assocaited to differential equation in fourth-order is one of the most prominent issue and it has been found hihgly remarkable for many authors. Very interesting results related to the solutions have been obtained. Particularly, majority of these results were obtained using the second method to the Lyapunov, which is thought as the most result-oriented and secured methods (see, Lyapunov [13] and Yoshizawa [28]). However, [4,5,16] include such a useful content about the qualitative behaviors of differential equations without or with delay. To gain much better perspective on the boundedness and stability, see the papers of Ezeilo [6,7], Hara [8], Harrow [9,10], Tunç [22,23,24,25,26], Remili et al. [15,17,18], Wu and Xiong [27] and others and theirs references. As motive from references, we obtain some new consequences on the uniformly asymtotically stability and boundedness of the solutions by means of the Lyapunov's functional approach. Our results differ from that obtained in the literature (see, [1]-[28] and the references therein). By this way, this paper enrich to the current literature and contribute future studies by presenting useful information for the solutions of higher-order functional differential equation’s qualitative behaviors. In view of all the mentioned information, it can be checked the novelty and originality of the current paper.

In this paper, we seek sufficient condition to obtain the uniformly asymptotically stability of the solutions for p(t,x,x′,x′′,x ′′′)≡0 and boundedness of solutions to the fourth order nonlinear differential equation with bounded veriable delay

(

)

(

)

(

)

). , , , , ( ))) ( ( ( ) ( ) ( )) ( ( ) ( ) ( )) ( ( ) ( ) ( )) ( ( ) ( ) ( )) ( ( x x x x t p t r t x h t d t x t x f t c t x t x q t b t x t x k t a t x t x g ′′′ ′′ ′ = − + ′ + ′ + ′′ + ′′ ′ ′ ′′ (1)

For convenience, we let

) ( )) ( ( )) ( ( ) ( ), ( )) ( ( )) ( ( ) ( ₂ ₂ ₂ 1 x t t x g t x k t t x t x g t x g t = ′ ′ θ = ′ ′ θ and ). ( )) ( ( )) ( ( ) ( ), ( )) ( ( )) ( ( ) ( 2 4 ₂ 3 x t t x g t x f t t x t x g t x q t = ′ ′ θ = ′ ′ θ

We write (1) in the system form

(

)

), , , , , ( ) ( )) ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( , , ) ( 1 , ) ( 3 2 2 1 w z y x t p d y x h t d x h t d y x f t c t x g t b z t x g t a x g x q t b t x k t a w x g x k t a w w z z x g y y x t t r t + ′ + − + −       ₋ ₋ + − = ′ = ′ = ′ = ′

∫

−

η

θ

(2)

where r(t) is a bounded delay, 0 ≤ r(t)≤Ω, λ

≤ ′ )(t

r , 0<λ <1, λ and Ω some positive constants, Ω which will be determined later, the functions a,b,c,d are continuously differentiable functions and the functions f,h,g,q,k and p are continuous functions depending only on the

arguments shown. Also derivatives

) ( ), ( ), ( ), ( ), (x g x k x q x f x

g′ ′′ ′ ′ ′ and h′(x) exist and

are continuous. The continuity of the functions , , , , , , , , ,b c d g g g k k a ′ ′′ ′ q, ′q , f ,p and h guarantees the existence of the solutions of equation (1). If the right-hand side of the system (2) satisfies a Lipchitz condition in

) ( ), ( ), ( ), (t y t z t w t

x and x(t−r(t)) and exists of solutions of system (2), then it is unique solution of system (2). Assuming , , , , , , , , , , , , , , ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₁ ₁ ₁ ₁ ₁ ₁ 0 b c d f g q k a b c d f g a , , , , ₁ 1 k m M

q and

δ

are constants then, following assumptions hold: (A1) 0<a₀ ≤a(t)≤a₁; ; ) ( 0<b₀ ≤b t ≤b₁ 0<c₀ ≤c(t)≤c₁; 1 0 ( ) 0<d ≤d t ≤d for t≥0. (A2) 0< f₀ ≤ f(x)≤ f₁; ; ) ( 0 ; ) ( 0< g₀ ≤g x ≤g₁ <k₀ ≤k x ≤k₁ 1 0 ( ) 0<q ≤q x ≤q forx ∈R and

{

, , ,1

}

, min 0<m < f₀ k₀ g₀

{

, , ,1

}

. max f₁ g₁ k₁ M >

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(A3) h(_xx) ≥

δ

>0 for x ≠ 0, h(0)=0. (A4) p(t,x,y,z,w) ≤ e(t).

2. PRELIMINARIES

We also consider the functional differential equation . 0 , 0 ), ( ) ( ), , ( . ≥ ≤ ≤ − + = = f t x x xt r t x _t _t

θ

(3)

where f : IxC_H →Rn is a continuous mapping, , 0 ) 0 , (t = f C_H :={

φ

∈(C[−r,0],Rn):

φ

≤H}, and for H <₁ H, there exists L(H₁)>0, with

) ( ) , (t L H₁ f

φ

< when

φ

<H₁.

Theorem 2.1. Let V(t,φ): IxCH →R_{be a} continuous functional satisfying a local Lipchitz condition, V(t,0)= 0, and wedges W_i such that :

1) W₁(

φ

)≤V(t,

φ

)≤W₂(

φ

). 2) V₍′₃₎(t,

φ

)≤−W₃(

φ

).

Then, it implies that the equation (3) is uniformly asymptotically stable for the zero solution (Burton [4]). 3. MAIN RESULTS Lemma 3.1. Let h(0)=0, xh(x)>0 ( ≠x 0) and δ(t)−h′(x)≥0, (δ(t)> 0), then ), ( ) ( ) ( 2

δ

_t _H _x _≥_h2 _x where H(x)=_∫₀xh(s)ds (Hara [8])

Theorem 3.1. Besides to the fundamental assumptions imposed on the functions a,b,c,d,

f q k

g, , , and h let we suppose that there exists non-negative constants h₀,

δ

₀,

υ

₁,

υ

₂,

η

₁,

η

₂,

η

₃ and

4

η

so that the following statements are hold:

i. M h d m a m h x h ₂0 1 0 0 0 − δ ≤ ′₍ ₎≤ _, 4 ) ( ' x <

η

g for R x ∈ .

ii. b0q0 >max

{

υ

1,

υ

2

}

where

(

)

       + + + − = − + + = + − + 0 1 2 1 0 3 0 0 1 1 0 0 0 0 0 1 2 0 0 1 3 3 0 2 1 0 1 ) 2 ( 2 2 2 1 1 ) 1 ( 2 2 1 0 ) ( 1 ) 1 ( h d mM M c c m c M h d a a M c M m M c a h d m a c M m c M d h a M m a a

υ

δ iii. ∫₀∞

(

a′(t) + b′(t) + c′(t) + d′(t)

)

dt<

η

₁. iv. _∫₋+∞_∞

(

g′(s)+ k′(s) +q′(s) + f′(s)

)

ds<

η

₂. v. _∫₀∞ e(t)dt<

η

₃.

Then any solution x(t) equation (1) are bounded and trival solution of equation (1) for

0 ) , , , , (t x x′ x′′ x ′′′ ≡ p is uniformly asymtotically stability, if . ) ( ) ( , ) 1 ( , 1 ) 2 ( min ) 1 ( 2 2 1 1 2 1 0 0 2 0 0 1 1    + − −    − + − + − < Ω Mm mM c a M q b m M m a m c h d

ε

υ

λ

α

ε

λ

β

α

ε

λ

Proof We take a Lyapunov functional for the usage of basic tool for the proof,

( , , , , ) 0 ( ) , 1 V e w z y x t W W sds t γ η∫ = = − (4) where , ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 4 3 2 1 t t t t t d t c t b t a t

θ

γ

+ + + + ′ + ′ + ′ + ′ = and

[

]

ds d y zw yw yz x f t c z x h t d y x h x g t d w z x g y t d h x q t b yz x g x k t a z x g x k t a z x g x q t b y x f x g t c x H t d V t s t t r

γ

σ

β

α

β

α

β

α

β

) ( 2 2 ) ( ) ( 2 ) ( ) ( 2 ) ( ) ( ) ( 2 ) ( 1 ) ( ) ( ) ( ) ( ) ( ) ( 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 2 2 0 ) ( 2 2 2 0 2 2 2

∫

− + + + + + + + + − − + + + + + = with H(x)=_∫₀xh(s)ds, , 0

ε

α

= _aM_m+ , 0 0 1 ε β =d_ch_m +

and η are non-negative constants to be described later. We can rewrite it in the form

V 2 as

(4)

(

)

, 3 2 1 2 0 ) ( 2 2 2 ) ( ) ( ) ( 2 ) ( ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( 2 L L L ds d y x H t d x f t c x h t d y x g x f t c z y x f t c x h t d x f t c y x g z x k t a w x k t a V t s t t r + + + + +       + − +       + + +       + + =

∫

− + γ γ σ ε α β where , 1 ) ( 1 ) ( ) ( ) ( ) ( ) ( 1 ) ( ) ( ) ( , ) ( ) ( ) ( ) ( 2 ) ( ) ( 2 2 2 2 0 0 1 0 1 z x g x k t a x f t c x g x g x q t b L ds s h x f t c t d m c h d s h t d L x          − +    − − =       ′ − =

_∫

α β α

(

)

]

. ) ( 1 1 2 ) ( ) ( 1 2 ) ( 3 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 2 2 2 2 0 3 yw x g w x k t a y x g x g x f t c x g x k t a t d h x q t b L       − +       − + + − −    − − = β α β α β Let

(

)

      + − < ) ( , , min 1 1 2 1 0 0 2 0 0 1 0 M a mMc q b m m c h d m a M

υ

ε

(5) then 2 , 2 . 0 0 1 0 0 1 0 0 c m h d m c h d m a M m a M < < < <α β (6)

Considering conditions (A1)-(A3), (i)-(ii) and inequalities (5), (6) we have

(

)

2 0 2 0 1 1 0 1 0 0 1 0 0 2 0 0 0 1 1 1 1 , 0 ) ( 2 ) ( ) ( 4 z m M a z M M m a a m c m a M m a m c h d M q b L ds s h M h s h m c d t d L x       ₋ +    − −           + −       + − ≥ ≥     _′ − ≥

_∫

α β ε ε α

(

)

(

)

(

)

(

)

(

)

0, , 1 2 1 1 1 0 0 2 1 1 0 1 0 2 1 2 0 1 0 1 0 0 ≥ + − − ≥    + −    − − − − ≥ z mM c a M q b m Mm z mM c a m M M m a a m a M c m c a h d M q b ε υ α ε α and

(

)

(

)

, ) ( 1 1 2 1 1 1 ) 1 ( 2 ) ( 1 1 2 1 2 2 2 ) ( 1 1 2 1 2 ) ( 2 0 2 2 0 0 0 1 2 0 2 0 1 2 1 0 3 0 0 1 1 0 0 0 0 2 0 2 2 1 2 1 1 0 0 0 3 yw x g w m a M y M m M c a h d yw x g w m a M y h d mM M c c m c M h d a a Mc q b yw x g w m a M y M M c x g M a d h q b L       − +       − +       ₋ − ≥       − +       − +    + −    − − ≥       − +       − +    + −    − − ≥

β

α

β

and by calculating the discriminant, we obtain 2 0 0 1 2 2 2 1 1 ) ( 1 1       ₋ −       − = ∆ M m m c h d x g

β

. 0 1 1 2 1 1 2 2 0 0 1 2 0 0 1 ₌               ₋ −       ₋ ≤ ∆ M m m c h d M m m c h d β Thus 0 3 ≥ L .

From the above inequalities, there exists non-negative constant D₀ so that

2V ≥ D0(y2 + z2 + w2 + H(x)). (7) Considering Lemma 3.1, (A3) and (i), we find a positive constant D₁ such that

(5)

In this way V is positive definite. In consideration of (A1)-(A3), we can have a positive constant U₁ such that

V ≤U1(x2 + y2 +z2 +w2). (9) Considering the condition (iv), we write

(

)

(

)

∞ < < ′ + ′ + ′ + ′ ≤ ′ + ′ + ′ + ′ =

∫

∑

∫

∞ + ∞ − ∞ + ∞ − = 2 2 2 2 2 ) ( ) ( 2 ) ( ) ( 4 1 0 ) ( ) ( 1 ) ( ) ( 1 ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 1 2 1 m du u f u q m du u k u g m du u g u f u q du u g u k u g ds s t t t t i i t η θ α α α α (10)

where

α

₁(t =) min

{

x(0),x(t)

}

and

{

(0), ( )

}

. max ) ( 2 t = x x t

α

From inequalities (5), (9)

and (10), it follows that

W ≥D₂(x2 + y2 +z2 +w2) (11) where 2 . 2 1 1 1 2 2       ₊ − = m e D D η η η

Also, it is easy to see that there is a positive constant U ₂ such that

W ≤U2(x2 + y2 +z2 +w2) (12) for all x,y,z,w and all t≥0.

Now our goal is to show that .

W is negative definite function. For the function V taking derivative with respect to t yields to obtain following statement along any solution

(

x(t),y(t),z(t),w(t)

)

_{of the system (2)} ) , , , , ( ) ( 2 ) ( ) ( 2 2 9 8 7 6 5 4 2 ) 2 ( . w z y x t p w z y L L L L L L y x f t c V

α

β

ε

+ + + + + + + + + − = where , ) ( ) ( ) ( 2 ) ( ) ( ) ( ) ( ) ( 2 0 2 0 0 1 4 yz x h x g h t d y x h x g t d x f t c m c h d L       _′ − −       ′ − − =

α

η

σ

η

β

η

α

β

α

d y t r t ry d y x h t z t d d y x h t y t d d y x h w t d L w x g x k t a L z x g x k t a x g x f t c x g x q t b L t t r t t t r t t t r t t t r t ) ( )) ( 1 ( ) ( ) ( )) ( ( ) ( ) ( 2 ) ( )) ( ( ) ( ) ( 2 ) ( )) ( ( ) ( 2 , 1 ) ( ) ( ) ( 2 , ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 2 ) ( 2 ) ( ) ( ) ( 7 2 6 2 2 5

∫

− − − − ′ − − + ′ + ′ + ′ =       − − =       − − − =

(

)

(

)

(

)

(

( ) ( ) 2 ( ) ( )

)

, 2 ) ( ) ( ) ( 2 ) ( ) ( 2 ) ( ) ( ) ( ) ( 2 ) ( ) ( ) ( 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 2 3 4 2 2 2 2 3 2 2 2 2 2 2 1 8 yz x g t c y x g t c zw z x g t a yz x g y x g zw x g z x g t b zw x k t a y x h x g t d z y x g x f t c z x q t b z x k t a L α θ α θ β α α θ α β α θ + + + − + + + − + + + + − =

[

]

[

]

. ) ( ) ( 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 ) ( ) ( ) ( ) ( 2 ) ( ) ( 2 ) ( 2 ) ( 2 2 2 2 2 0 9       + ′ +       + ′ + + ′ + + + − ′ = yz x g x k z x g x k t a y x q z x g x q t b yz x f y x f x g t c z x h y x h x g y h x H t d L

β

α

β

By regarding conditions (A1), (A2), (i), (ii) and inequalty (6), (7) we have the following

. ) ( ) ( 2 2 2 ) ( ) ( ) ( 2 ) ( ) ( ) ( 2 ) ( ) ( ) ( ) ( 2 2 0 2 2 2 0 0 2 0 4 z x h m h t d m z m z m y x h x g h m t d yz x h x g h t d y x h x g h x g t d L     ₋ _′ ≤               −       +       ′ − ≤       ′ − −       ′ − − ≤ α α α α In that case,           + − − ≤ + m M c m a M M q b L L4 5 2 0 0

ε

1

(6)

(

)

(

)

(

)

[

]

0, 2 2 4 2 1 1 2 1 0 0 2 2 2 1 1 2 0 0 2 3 0 1 0 1 1 2 0 2 0 0 2 0 0 2 2 1 0 0 1 ≤ + − − − ≤          ₊ − −    − − − ≤    −       + − z m c a M q b m Mm z c m a m M m a M m c M a h d c m a M M q b z a m m M a m c h d

ε

υ

ε

δ

α

ε

and . 0 2 1 2 0 2 0 2 6  =− ≤      − − ≤ w M m a w M m a L α ε By taking 1 max

{

,₂

}

, 0 1 0 0 0 M h d m a m h h = − δ we get

[

d h

]

y s ds y t r z y w t r h d L t t r t ( ) ) 1 ( ) 1 ( ) ( ) )( ( 2 ) ( 1 1 2 2 2 2 1 1 7

∫

− − − + + + + + + ≤ λ σ β α σ β α If we choose = ₍₁11₎( + +1), −

α

β

σ

dh_λ we get

[

]

. ) 1 ) 2 ( ( ) 1 ( 1 ) ( 2 2 2 1 1 7 z y w t r h d L + + − + + − − ≤ λ β α α λ λ

Thus, there exists a positive constant D₃ such that

). ( 2 ) ( ) ( 2 2 2 2 3 7 6 5 4 2 w z y D L L L L y x f t c + + − ≤ + + + + − ε

From (7), and the Cauchy Schwartz inequality, we obtain

(

)

(

)

(

)

(

)

(

)

(

)

(

)

, 2 ) ( ) )( ( ) ( ) )( ( ) ( ) ( ) ( ) ( ) ( ) )( ( ) ( ) ( ) ( ) )( ( ) ( ) ) ( )( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 4 3 2 1 0 1 2 2 2 4 3 2 1 1 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 3 4 2 2 2 2 2 2 2 2 2 2 1 8 V D x H w z y z y x g y x g w z x g z x g t b w z z x g t a z y x g t c y x g t c w z x k t a y x h x g t d z y x g x f t c z x q t b z x k t a L θ θ θ θ λ θ θ θ θ λ β α α θ α θ α θ α β α θ + + + ≤ + + + + + + ≤ + + + + + + + + + + + + + + + + + + + ≤ where

{

}

. ) ( ), ( ), ( , max 1 1 1 2 1 1 1 1 0 1 1 M c c d M a b a M M M M M b M h d + + + + + + + + = α α α β β α α λ

Using condition (i) and Lemma 3.1, we can write

[

(

)

]

(

)

[

]

(

)

[

]

[

](

)

[

( ) ( ) ( ) ( )

]

, 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( ) ( ) ( ) ) ( )( ( ) ( 2 ) ( 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0 9 V t d t c t b t a D x H w z y t d t c t b t a z y z x g x k t a y z x g x q t b z y y x g x f t c z x h y x h x g y h x H t d L ′ + ′ + ′ + ′ ≤ + + + ′ + ′ + ′ + ′ ≤ + + ′ +       + ′ + + + ′ + + + + + + ′ ≤

λ

β

α

β

such that

{

}

. ) 1 ( ), 1 ( ), 1 ( 2 max 0 0 2 + + + + + + + = β α α α β λ α m M M h M M h By taking 1 1 max

{

₁, ₂

}

, 0

λ

η = D we have

(

)

(

)

(

)

. ) ( ) ( ) ( ) ( 1 ) , , , , ( 4 3 2 1 2 2 2 3 ) 2 ( . V t d t c t b t a w z y x t p w z y w z y D V θ θ θ θ η α β + + + + ′ + ′ + ′ + ′ + + + + + + − ≤ (13)

From (A4), (iii), (iv), (10), (11), (13) and the Cauchy Schwartz inequality, we get

(

)

(

)

(

)

(

)

(

)

) ( 1 3 ) ( 3 ) ( ) , , , , ( ) , , , , ( ) ( 1 2 4 2 2 2 4 4 ) ( 2 2 2 3 ) ( ) 2 ( . ) 2 ( . 0 1 0 1 t e W D D t e w z y D t e w z y D w z y x t p w z y e w z y x t p w z y w z y D e V t V W ds s ds s t t       + ≤ + + + ≤ + + ≤ + + ≤ ∫ + + + + + − ≤ ∫       ₋ = − −

α

β

α

β

γ

η

γ γ η η (14) 3 ( ) ( ) 2 4 4 W e t D D t e D + ≤ (15)

(7)

where D₄ =max

{

α

,

β

,1

}

. Integrating (15) from 0 to t and using the condition (v) and the Gronwall inequality, we have

(

)

(

)

(

)

(

)

es ds t t D D e D w z y x W ds s e s w s z s y s x s W D D D w z y x W W ) ( 3 4 0 2 4 3 4 0 2 4 3 ) 0 ( ), 0 ( ), 0 ( ), 0 ( , 0 ) ( ) ( ), ( ), ( ), ( , 3 ) 0 ( ), 0 ( ), 0 ( ), 0 ( , 0 ∫ + ≤ + + ≤

∫

η η

(

)

(

)

∞ < = + ≤ 1 3 4 3 2 4 3 ) 0 ( ), 0 ( ), 0 ( ), 0 ( , 0 K e D w z y x W D D _η η ₍₁₆₎

Because of inequalities (11) and (16), we write

(

)

1 ₂, 2 2 2 2 2 K W D w z y x + + + ≤ ≤ (17) where . 2 1 2 D K

K = Clearly (17) imlies that

. 0 all for ) ( , ) ( , ) ( , ) ( 2 2 2 2 ≥ ≤ ≤ ≤ ≤ t K t w K t z K t y K t x

Thus, by using conditions (A2), (i) and (17) with the system (2) we have

. 0 1 ) ( ) ( ) ( ) ( ' ) ( ) ( 1 ) ( '' ' , 1 ) ( ) ( 1 ) ( ' ) ( '' , ) ( ' , ) ( 2 2 0 4 2 0 2 2 0 2 2 ≥ + ≤ + ≤ ≤ = ≤ ≤ ≤ t all for K g K g t z t y x g x g t w x g t x K g t z x g t y t x K t x K t x

η

(18)

In this case x(t),x'(t),x ''(t) and x' ''(t) are bounded.

By taking p(t,x,y,z,w)≡ 0 in the inequality (14) obtained ), ( ) ( ) ( 1 2 2 2 ) ( 2 2 2 3 ) ( ) 2 ( . . ) 2 ( 0 1 0 1 w z y e w z y D e V t V W ds s ds s t t + + − ≤ ∫ + + − ≤ ∫       − = − − µ γ η γ γ η η where . 2 1 3 η η η

µ =De− + It can also be observed that the only solution of system (2) for which

0 ) , , , , ( . ) 2 ( t x y z w = W is the solution = = = =

equation (1) is uniformly asymptotically stable and are bounded solutions of equation (1).

ACKNOWLEDGMENTS

The author would like to present his sincere thanks to the referee(s) for the detailed read and valuable input on the manuscript.

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