Periodic Solutions for Non-Linear Systems of Boundary Value Problems

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Periodic Solutions for Non-Linear Systems of Boundary Value Problems

Raad N. Butris¹ and Suham Kh. Shammo²

1,2Department of Mathematics Faculty of Science, University of Zakho

1E-mail: raad.khlka@yahoo.com

2E-mail: Suhammath55@yahoo.com (Received: 3-1-14 / Accepted: 17-2-14)

Abstract

In this work, we investigate the periodic solutions for new non-linear system of boundary value problems by using the numerical analytic method, which was introduced by Samoilenko. These investigations lead us to improving and extending the results of Samoilenko.

Keywords: Numerical-analytic methods, Existence of periodic solutions, Nonlinear system, Boundary value problem.

I Introduction

Many results about the existence and approximation of periodic solutions for system of non–linear differential equations have been obtained by the numerical analytic methods that were proposed by Samoilenko [6,7] which had been later applied in many studies [1, 2, 3, 4, 5].

Samoilenko [6, 7] has used the numerical-analytic methods of periodic solutions for ordinary differential equation with boundary and boundary integral conditions which has the form:

= ,

= , ∈ ,

Where ∈ , is closed and bounded subset of , the vector function.

, is de"ined on the domain:

, ∈ ^%× = −∞, ∞ × , Which is continuous in and and periodic in of period T.

Our work, we investigate the periodic solution for new non-linear system of differential equations with boundary integral conditions which has the form:

)

= * + , , ,, -

. = _%

, = /, + 0 , , ,, -

. , = ₁

- = 2- + ℎ , , ,, -

. - = ₄

56 66 66 7 66 66 68

⋯ BVP

Where ∈ ⊂ , , ∈ _% ⊂ ^> and - ∈ ₁ ⊂ ^?.The domains , _% and ₁ are closed and bounded.

Let the vector functions , , ,, - , 0 , , ,, - and ℎ , , ,, - are defined and continuous on the domain:

, , ,, - ∈ ^%× × _%× ₁ ⋯ 1 And periodic in t of period T, and * = A*_BCD, *_% = A*_%BCD, *₁ = A*_1BCD,

/ = A/_BCD, /_% = A/_%BCD, /₁ = A/_1BCD, 2 = A2_BCD, 2_% = A2_%BCD, 2₁ = A2_1BCD are

E × E non-negative matrices, also F_% = F_%%, F_%1 , … , F₁ = F_1%, F₁₁ , … , F₄ = F_4%, F₄₁, … are positive constant vectors.

Suppose that the functions , 0 and ℎ satisfy the following inequalities:

‖ , , ,, - ‖ ≤ J_%, ‖0 , , ,, - ‖ ≤ J₁, ‖ℎ , , ,, - ‖ ≤ J₄ ⋯ 2

‖ , _%, ,_%, -_% − , ₁, ,₁, -₁ ‖ ≤ L_%‖ _%− ₁‖ + L₁‖,_%− ,₁‖ +

+ L₄‖-_%) − )-₁‖ ⋯ 3

‖0 , %, ,_%, -_% − 0 , ₁, ,₁, -₁ ‖ ≤ N%‖ _% − ₁‖ + N1‖,%− ,₁‖ +

+N₄ ‖-_%− -₁‖ ⋯ 4

‖ ℎ , %, ,^%, -_% − ℎ , ₁, ,₁, -₁ ‖ ≤ P_%‖ _%− ₁‖ + P1‖,%− ,₁‖ +

+ P₄ ‖-_%− -₁‖ ⋯ 5 for all ∈ ^%, , _% , ₁ ∈ , ,, ,_%, ,₁ ∈ _%, -, -_%, -₁ ∈ ₁ .

WherF J_%, J₁, J₁, L_%, L₁, L₄, N_%, N₁, N₄, P_%, P₁, P₄ are positive constants.

Provided that:

Se^{T UVW} S ≤γ_%

λ_%, Se^{Z UVW}S ≤γ₁

λ₁ , Se^{[ UVW} S ≤γ₄

λ₄ ⋯ 6

Where γ_%, γ₁ , γ₄ , λ_%, λ₁, λ₄ are positive conistants and ‖. ‖ = max_{_∈` , a}|. | .

We define the non-empty sets as follows:

)

c = −d

2γ_%

λ_%J_%−γ_%

λ_%e d

f = _%−d 2γ₁

λ₁J₁−γ₁

λ₁g , d

h = ₁−d 2γ₄

λ₄J₄−γ₄

λ₄F - d

566 7 66 8

⋯ 7

Where

e = _%j A F^Tl− E − _%+ . N , , ,, - nj , g , = ₁j,

B F^Zl− E − ₁+ . N0 , , ,, - j , F , = ₄j-

C F^[l− E − ₄+ . Nℎ , , ,, - j ,

% = j A¹

F^Tl− TAE − E j, ¹ = j B¹

F^Zl− TBE − E j , ⁴

= j C¹

F^[l− TCE − E j

Furthermore, we suppose that the largest eigen- value of the matrix

q = r ss st

u_% v_%d

2 L^%w_% u_% v_%d

2 L¹w_% u_% v_%d

2 L⁴w_% u₁

v₁d

2 N^%w₁ u₁ v₁d

2 N¹w₁ u₁ v₁d

2 N⁴w₁ d2u₄

v₄P_%w₄ d 2u₄

v₄P₁w₄ d 2u₄

v₄P₄w₄x yy yz

Does not exceed unity

}_% 3 +}_~

6 +2}₁

3 +2}_%¹

3 < 1 , ⋯ 8 where }_% = d

2 L^%w_%u_%

v_%+ N₁w₁u₁

v₁+ P₄w₄u₄ v₄ }₁ = d¹

2 u_% v_%u₁

v₁w_%w₁ L₁N_%− L_%N₁ +u_% v_%u₄

v₄w_%w₄ L₄P_%− L_%P₄ + u₁

v₁u₄

v₄w₁w₄ N₄P₁− N₁P₄ , }₄ = d⁴

2 u_% v_%u₁

v₁u₄

v₄w_%w₁w₄ L_% N₁P₄− N₄P₁ + L₁ N₄P_%− N_%P₄ + L₄ N_%P₁− N₁P_% ,

}_~ = 8}_%⁴+ 180}₄+ 36}_%}₁+ 12 81}₄¹+ 12}₄}_%⁴+ 54}_%}₁}₄− 3}_%¹}₁¹− 12}₁^{4 %/1 %/4} ,

Also w_% = 1 +u_%

v_%d¹ _% , w₁ = 1 +u₁

v₁d¹ ₁ , w₄ = 1 +u₄

v₄d¹ ₄ . Define a sequence of functions:

„ _> , , , , - , ,> , , , , - , -_> , , , , - …_>† ^‡ by

>ˆ% , , , , - = F^TU+ . F^{_ ‰ _VŠ} ` n, _> n, , , , - . ,_> n, , , , - , -_> n, , , , - −

F^TlA− E . F^{‰ VŠ} n, _> n, , , , - , ,_> n, , , , - ,)

-_> n, , , , - n − ‹a n ⋯ 9 with , , , , - = F^TU

where

‹ = A¹

F^Tl− TAE − E • A F^Tl− E − _%+ . N , , ,, - Ž ; F A ≠ 0 and F F^Tl− TAE − E ≠ 0.

and N , , ,, - = . F^{‰ _VŠ} ` n, , ,, - − A

F^Tl− E . F^{‰ VŠ} n, , ,, - na n And

,_>ˆ% , , , , - = , F^‘_ + . F^{_ ‘ _VŠ} `0) n, _> n, , , , - ,) ,_> n, , , , - , -_> n, , , , - −

B

F^Zl− E . F^{‘ VŠ} )0 n, _> n, , , , - , ,_> n, , , , - ,))

-_> n, , , , - n − ’a n ⋯ 10

with, , , , , - = , F^‘_

where

’ = B¹

F^Zl− TBE − E •,

B F^Zl− E − ₁+ . N0 , , ,, - nŽ ; F B ≠ 0, F F^Zl− TBE − E ≠ 0.

N0 , , ,, - = . F^{_ ‘ _VŠ} `0 n, , ,, - − B

F^Zl− E ). F^{‘ VŠ} 0 n, , ,, - n)a n Also

-_>ˆ% , , , , - = -_“F^”_ + . F^{_ ” _VŠ} `ℎ n, _> n, , , , - ),) ,_> n, , , , - , -_> n, , , , - − C

F^[l− E . F^{” VŠ} ℎ n, _> n, , , , - ,),_> n, , , , - ,

-_> n, , , , - n − •a n ⋯ 11 with- , , , , - = - F^”_

Where

• = C¹

F^[l− TCE − E •-

C F^[l− E − ₄+ . Nℎ , , ,, -, Ž ; F C ≠ 0, F F^[l− TCE − E ≠ 0

Nℎ , , ,, - = . F^{_ ” _VŠ} `ℎ n, , ,, - −

_A–—˜^[_V™D F^{” VŠ} ℎ n, , ,, - na n, m=0, 1, 2, …

By using lemma3.1 [6], we can state and proof the following lemma:

Lemma 1: Suppose that the functions , 0 šE be vectors which are defined in the interval `0, da, then the following inequality holds:

›‖œ_% , , , , - ‖

‖œ₁ , , , , - ‖

‖œ₄ , , , , - ‖• ≤ r ss

tž_% u_% v_%J_% ž₁ u₁

v₁J₁ ž₄ u₄

v₄J₄x yy

z , … 12

for 0 ≤ ≤ d , ž_% ≤ d

2 , ž¹ ≤d

2 , ž⁴ ≤ d 2 ,

Where

œ_% , , , , - = . F^{_ ‰ _VŠ} ` n, , , , - ) −

− A

F^Tl− E . F^{‰ VŠ} ) n, ,, ,- na n œ₁ , , , , - = . F^{_ ‘ _VŠ} `0 n, , , , - ) −

− B

F^Zl− E . F^{‘ VŠ} )0 n, ,, ,- na n œ₄ , , , , - = . F^{_ ” _VŠ} `ℎ n, , , , - ) − Ÿ −

− C

F^[l− E . F^{” VŠ} )ℎ n, ,, ,- n a n And

ž_% = A2F^{‖‰‖ V_} − F^‖‰‖ − ‖ ‖D + dAF^‖‰‖ − F^{‖‰‖ V_} D F^‖‰‖ − ‖ ‖

ž₁ = A2F^{‖‘‖ V_} − F^‖‘‖ − ‖ ‖D + dAF^‖‘‖ − F^{‖‘‖ V_} D F^‖‘‖ − ‖ ‖

ž₄ = A2F^{‖”‖ V_} − F^‖”‖ − ‖ ‖D + dAF^‖”‖ − F^{‖”‖ V_} D F^‖”‖ − ‖ ‖

Proof:

‖œ_% , , , , - ‖ ≤

¡‖ ‖ − F^‖‰‖ − F^{‖‰‖ V_}

F^‖T‖l− ‖ ‖ ¢ . SF^{_ ‰ _VŠ} S‖ n, , , , - ‖ n + + ¡F^‖‰‖ − F^{‖‰‖ V_}

F^‖T‖l− ‖ ‖ ¢ . SF^{‰ _VŠ} S

_ ‖ n, , , , - ‖ n

≤ ž_% u_%

v_%J_% … 13

And similarly

‖œ1 , , , , - ‖ ≤ ž₁ u₁

v₁J₁ … 14

‖œ₄ , , , , - ‖ ≤ ž₄ u₄

v₄J₄ … 15

from 13 , 14 and 15 we conclude that the inequality 12 holds. ⧠

II Approximation of Periodic Solution for (BVP)

The investigation of approximate solution of BVP will be introduced by the following theorem:

Theorem 1: Let the vector functions , 0 šE ℎ are defined and continuous on the domain (1) and periodic in t of period T. Suppose that these functions satisfy the inequalities (2), (3), (4), (5) and the conditions (6), (7) and (8), then there exist a sequences of functions (9), (10) and (11), converges uniformly on the domain:

, , , , - ∈ `0, da × _c × _f × _h ⋯ 16

To the limit function › , , , , - , , , , , -

- , , , , - • which is continuous in the domain (16) and periodic in t of period T and satisfies the following vector form:

› , , , , - , , , , , - - , , , , - • =

r ss ss

t F^TU+ . F^{_ ‰ VŠ} • n, , ,, - − A

F^Tl− E . F^{‰ VŠ} n, , ,, - n − ‹Ž n ,_“F^‘_+ . F^{_ ‘ _VŠ} •0 n, , ,, - − B

F^Zl− E . F^{‘ VŠ} 0 n, , ,, - n − ’Ž n -_“F^”_ + . F^{_ ” _VŠ} •ℎ n, , ,, - − C

F^[l− E . F^{” VŠ} ℎ n, , ,, - n − •Ž n x yy yy z

⋯ 17

And it is a unique solution of (BVP) which satisfies the following inequality:

›‖ , , , , - − _> , , , , - ‖

‖, , , , , - − ,_> , , , , - ‖

‖- , , , , - − -> , , , , - ‖• ≤ q^> − q ^V%¥_%

where ¥_% = r ss st d 2u_%

v_%J_%+u_%

v_%e d d

2u₁

v₁J₁+u₁

v₁g , d d

2γ₄

λ₄M₄ +γ₄

λ₄F - dx yy yz ,

for all t ∈ [0,T] and ∈ _c , , ∈ _f , - ∈ _h .

Provided that:

›‖ , , , , - − ‖

‖, , , , , - − , ‖

‖- , , , , - − - ‖• ≤ r st¹

§¨

©¨J_%+^§_©^¨_¨e d

1§ª

©ªJ₁+_©^§ª_ªg , d

1«¬

-¬J₄+^«_-^¬_¬F - dx yz

⋯ 18

Proof: Setting m=0 in (9), (10) and 11 , we have

‖ _% , , , , - − ‖

≤ j. `F^{_ ‰ _VŠ} ) n, ,, ,- − A

F^Tl− E ). F^{‰ VŠ} n, , , , - na nj + . SF^{_ ‰ _VŠ} Sj) A¹

F^Tl− TAE − E ) )®A F^Tl− E ))) − _%+ )+). N , ,, ,- Žj n

By using Lemma1 we get

‖ _% , , , , - − ‖ ≤ ž_% u_%

v_%J_%+u_%

v_%e d.

Hence _% , , , , - ∈ _c for all t ∈ [0,T],

‖,% , , , , - − , ‖ ≤ ž_{1 ©}^§ª

ªJ₁+ _©^§ª

ªg , d Hence ,_% , , , , - ∈ _f for all t ∈ [0,T],

‖-_% , , , , - − - ‖ ≤ ž_{4 ©}^§¬_¬J₄+ _©^§¬_¬F - d Hence -_% , , , , - ∈ _h for all t ∈ [0,T]

Now by mathematical induction, we can prove the following inequalities for

¯ = 0, 1, 2,…,

‖ _> , , , , - − ‖ ≤ ž_% u_%

v_%J_%+u_%

v_%e d, ⋯ 19

‖,> , , , , - − , ‖ ≤ ž₁ u₁

v₁J₁+ u₁

v₁g , d , ⋯ 20

‖-_> , , , , - − - ‖ ≤ ž_{4 ©}^§¬

¬J₄+ _©^§¬

¬F - d. ⋯ 21

That is _> , , , , - ∈ _c ,_> , , , , - ∈ _f and -_> , , , , - ∈ _h for all t ∈ [0,T]

Next, we shall prove that the sequence of functions (9), (10) and (11) are convergent uniformly on the domain (16).Then by mathematical induction we can prove the following inequalities:

‖ _>ˆ% , , , , - − _> , , , , - ‖

≤ ž_% u_%

v_%L_%w_%‖ _> , , , , - ) − ) _>V% , , , , - ‖ + +ž_% u_%

v_%L₁w_%‖,_> , , , , - − ,_>V% , , , , - ‖ + +ž_% u_%

v_%L₄w_%‖-_> , , , , - − -_>V% , , , , - ‖ , ⋯ 22

‖,_>ˆ% , , , , - − ,_> , , , , - ‖ ≤ ž₁ u₁

v₁N_%w₁‖ _> , , , , - − _>V% , , , , - ‖ + +ž₁ u₁

v₁N₁w₁‖,_> , , , , - − ,_>V% , , , , - ‖ + +ž₁ u₁

v₁N₄w₁‖-> , , , , - − -_>V% , , , , - ‖ , ⋯ 23

‖-_>ˆ% , , , , - − -_> , , , , - ‖

≤ ž₄ u₄

v₄P_%w₄‖ _> , , , , - − _>V% , , , , - ‖ + +ž₄ u₄

v₄P₁w₄‖,_> , , , , - − ,_>V% , , , , - ‖ + +ž₄ ^§_©^¬_¬P₄w₄‖-_> , , , , - − -_>V% , , , , - ‖ ⋯ 24

Rewrite (22), (23) and (24) in a vector form i. e.

¥_>ˆ% ≤ q ¥_> ⋯ 25

¥_>ˆ%= ›‖ _>ˆ% , , , , - − _> , , , , - ‖

‖,_>ˆ% , , , , - − ,_> , , , , - ‖

‖-_>ˆ% , , , , - − -_> , , , , - ‖•

¥_> = ›‖ _> , , , , - − _>V% , , , , - ‖

‖,_> , , , , - − ,_>V% , , , , - ‖

‖-_> , , , , - − -_>V% , , , , - ‖• And

q =

r ss

tž_% u_%

v_%L_%w_% ž_% u_%

v_%L₁w_% ž_% u_% v_%L₄w_% ž₁ u₁

v₁N_%w₁ ž₁ u₁

v₁N₁w₁ ž₁ u₁ v₁N₄w₁ ž₄ u₄

v₄P_%w₄ ž₄ u₄

v₄P₁w₄ ž₄ u₄

v₄P₄w₄x yy z

Now, we take the maximum value for the both sides of the inequalities (25) we get

¥_>ˆ%≤ q¥_> ⋯ 26 Where q = max_{_∈` , a}q

q = r ss st

u_% v_%d

2 L^%w_% u_% v_%d

2 L¹w_% u_% v_%d

2 L⁴w_% u₁

v₁d

2 N^%w₁ u₁ v₁d

2 N¹w₁ u₁ v₁d

2 N⁴w₁ d

2u₄

v₄P_%w₄ d 2u₄

v₄P₁w₄ d 2u₄

v₄P₄w₄x yy yz

And by repetition (26) we find ¥_>ˆ% ≤ q^>¥_% and also we get

° ¥_B

>

B†%

≤ ° q^BV%

>

B†%

¥_% . ⋯ 27

By condition (8) then the sequence (27) is uniformly convergent that is

>→‡lim ° q^BV%

>

B†%

¥_% = ° q^BV%

‡ B†%

¥_% = − q ^V%¥_% ⋯ 28

Let

>→‡lim › ^> , , , , - ,_> , , , , -

-_> , , , , - • = › , , , , - , , , , , -

- , , , , - • ⋯ 29

Since the sequence of functions (3), (4) and (5) is defined and continuous in the domain (1) then the limiting function › , , , , -

, , , , , -

- , , , , - • is also defined and continuous in the same domain.

Moreover, by using Lemma1, the relation (29) and proceeding (9), (10) and (11) to the limit › , , , , -

, , , , , -

- , , , , - • when m→∞ , the equality (28) satisfied for all ¯ ≥ 0 , and this show that the limiting function › , , , , -

, , , , , -

- , , , , - • is the solution of (BVP).

Finally, we have to show that › , , , , - , , , , , -

- , , , , - • is a unique solution of (BVP).

Let › , , , , - , , , , , -

- , , , , - • be another solution of (BVP) where

, , , , - = F^TU+ . F^{_ ‰ _VŠ} ` n, n, , , , - , , n, , , , - ), , - n, , , , - − A

F^Tl− E . F^{‰ VŠ} ) n, n, ,, ,- ,, n, ,, ,- ,) , - n, , , , - n − A¹

F^Tl− TAE − E ® A F^Tl− E ) − _%+ )+ N , ,,,- ³a n,

, , , , , - = , F^ZU+ . F^{_ ‘ _VŠ} `0 n, n, , , , - , , n, , , , - ), , - n, , , , - − B

F^Zl− E . F^{‘ VŠ} 0 n, n, , , , - , , n, , , , - ,) , - n, , , , - n − B¹

F^Zl− TBE − E ®,

B F^Zl− E ) − ₁ + )+. N0 , ,,,- Ža n,

- , , , , - = - F^”_+ . F^{_ ” _VŠ} `ℎ n, n, , , , - , , n, , , , - ),

, - n, , , , - − C

F^[l− E . F^{” VŠ} )ℎ n, n, ,, ,- ,, n, ,, ,- ,) - n, , , , - n − C¹

F^[l− TCE − E ®-

C F^[l− E ) − ₄+ + ). Nℎ , , ,, - Ža n

∴ ‖ , , , , - ) − ) , , , , - ‖

≤ ž_% u_%

v_%L_%w_%‖ , , , , - ) − ) , , , , - ‖ + +ž_% u_%

v_%L₁w_%‖, , , , , - ) − ), , , , , - ‖ + +ž_% u_%

v_%L₄w_%‖- , , , , - ) − )- , , , , - ‖ ⋯ 30 Similarly

‖, , , , , - − , , , , , - ‖

≤ ž₁ u₁

v₁N_%w₁‖ , , , , - ) − ) , , , , - ‖ + +ž₁ u₁

v₁N₁w₁‖, , , , , - ) − ), , , , , - ‖ + ž₁ u₁

v₁N₄w₁‖- , , , , - ) − )- , , , , - ‖ ⋯ 31

‖- , , , , - − - , , , , - ‖

≤ ž₄ u₄

v₄P_%w₄‖ , , , , - ) − ) , , , , - ‖ + +ž₄ u₄

v₄P₁w₄‖, , , , , - ) − ), , , , , - ‖ + +ž₄ u₄

v₄P₄w₄‖- , , , , - ) − )- , , , , - ‖ ⋯ 32

Then we can rewrite the inequalities 30 , 31 and 32 by the vector form:

›‖ , , , , - − , , , , - ‖

‖, , , , , - − , , , , , - ‖

‖- , , , , - ) − )- , , , , - ‖•

≤ q ›‖ , , , , - − , , , , - ‖

‖, , , , , - − , , , , , - ‖

‖- , , , , - ) − )- , , , , - ‖• ⋯ 33

Now by the condition (8), we get

›‖ , , , , - − , , , , - ‖

‖, , , , , - − , , , , , - ‖

‖- , , , , - ) − )- , , , , - ‖• → µ0 00¶

∴ › , , , , - , , , , , -

- , , , , - • = › , , , , - , , , , , - - , , , , - •.

This proves that the solution is a unique and this completes the proof. ⧠

III Existence of Periodic Solution for (BVP) [7]

The problem of the existence solution for (BVP) is uniquely connected with existence of zero of the functions

∆_% , , , - , ∆₁ , , , - and ∆₄ , , , - , which defined by:

∆_% , , , - = A¹

F^Tl− TAE − E ® A F^Tl− E ) − _%+

)+. N , ,, ,- Ž + A

F^Tl− E . F^{‰ VŠ} ` n, n, , , , - ), , , n, , , , - , - n, , , , - a n ⋯ 34

∆_%: _c× _f × _h→

∆₁ , , , - = B¹

F^Zl− TBE − E ®,

B F^Zl− E ) − ₁ + + ). N0 , , , , - Ž +

+ B

F^Zl− E . F^{‘ VŠ} `0 n, n, , , , - , , n, , , , - ),

, - n, , , , - a n ⋯ 35

∆₁: _c× _f × _h → And

∆₄ , , , - = C¹

F^[l− TCE − E ®-

C F^[l− E ) − ₄+ + ). Nℎ , , , , - Ž +

+ C

F^[l− E . F^{” VŠ} `ℎ n, n, , , , - , , n, , , , - )

- n, , , , - ⋯ 36

∆₄: _c× _f × _h →

Since the functions are approximately determined from the sequence of functions ∆_% , , , - , ∆₁ , , , - and ∆₄ , , , - :

∆_%> , , , - = A¹

F^Tl− TAE − E ® A F^Tl− E ) − _%+ + ). N , _>, ,_>, -_> Ž +

+ A

F^Tl− E . F^{‰ VŠ} n, _> n, , , , - , ,_> n, , , , - ), , -_> n, , , , - n ⋯ 37

∆_%>: _c× _f × _h →

∆_1> , , , - = B¹

F^Zl− TBE − E ®,

B F^Zl− E ) − ₁+ + ). N0 , _>, ,_>-_> Ž +

+ B

F^Zl− E . F^{‘ VŠ} 0 n, _> n, , , , - , ,_> n, , , , - ,)

, -_> n, , , , - n ⋯ 38

∆_1>: _c × _f × _h →

∆_4> , , , - = C¹

F^[l− TCE − E ®-

C F^[l− E ) − ₄+ + ). Nℎ , _>, ,_>, -_> Ž +

+ C

F^[l− E . F^{” VŠ} ℎ n, _> n, , , , - , ,_> n, , , , - ) , -_> n, , , , - n ⋯ 39

∆_4>: _c × _f × _h →

Theorem 2: Let all assumptions and conditions of Theorem1 were given, then the following inequality holds:

›‖∆_% , , , - − ∆_%> , , , - ‖

‖∆₁ , , , - − ∆_1> , , , - ‖

‖∆₄ , , , - − ∆_4> , , , - ‖•

≤ r ss t¸_%u_%

v_%L_% ¸_%u_%

v_%L₁ ¸_%u_% v_%L₄

¸₁u₁

v₁N_% ¸₁u₁

v₁N₁ ¸₁u₁ v₁N₄

¸₄u₄

v₄P_% ¸₄u₄

v₄P₁ ¸₄u₄ v₄P₄x

yy

z q^> − q ^V%¥_% ⋯ 40

Where ¸_% = ‖A‖T

F^‖T‖l− ‖E‖ + ž^{% %}d, ¸₁ = ‖B‖T

F^‖Z‖l− ‖E‖ + ž^{1 1}d ¸₄ = ‖C‖T

F^‖[‖l− ‖E‖ + ž^{4 4}d Proof: By the equations (34) and (37), we have

‖∆_% , , , - − ∆_%> , , , - ‖

≤ ‖A‖T

F^‖T‖l− ‖E‖ u_%

v_%L_%‖ , , , , - ) − ) _> , , , , - ‖ +

+ ‖A‖T

F^‖T‖l− ‖E‖ u_%

v_%L₁‖, , , , , - ) − ),> , , , , - ‖ +

+ ‖A‖T

F^‖T‖l− ‖E‖ u_%

v_%L₄‖- , , , , - ) − )-> , , , , - ‖ + +u_%

v_%ž_{% %}dL_%‖ , , , , - ) − ) _> , , , , - ‖ + +u_%

v_%ž_{% %}dL₁‖, , , , , - ) − ),_> , , , , - ‖ + +u_%

v_%ž_{% %}dL₄‖- , , , , - ) − )-_> , , , , - ‖

∴ ‖∆_% , , , - − ∆_%> , , , - ‖

≤ 〈 ¸_%u_%

v_%L_% ¸_%u_%

v_%L₁ ¸_%u_%

v_%L₄ , q^> − q ^V%¥_%〉 = »_> ⋯ 41 And from the equation (35) and (38), we have

‖∆₁ , , , - − ∆_1> , , , - , - ‖

≤ 〈 ¸₁u₁

v₁N_% ¸₁u₁

v₁N₁ ¸₁u₁

v₁N₄ , q^> − q ^V%¥_% 〉 = ¼_> ⋯ 42 Also from the equation (36) and (39), we have

‖∆₄ , , , - − ∆_4> , , , - , - ‖ ≤ 〈 ¸₄u₄

v₄P_% ¸₄u₄

v₄P₁ ¸₄u₄

v₄P₄ , q^> − q ^V%¥_% 〉 = ½_> ⋯ 43 Then we rewrite (41), (42) and 43 by the vector form, then we get (40). ⧠

Now, we prove the following theorem taking into account that the inequality (41), (42) and 43 will be satisfied for all ¯ ≥ 0.

Theorem 3[6] Let (BVP) be defined in the `š, ea, `g, ašE ` ¾, ¿a ÀE ^%, šE ÁFw¾À ¾g ¾E À ÁFw¾À d. Suppose that for ¯ ≥ 0 the sequences of functions ∆_%> , , , - , ∆_1> , , , - šE ∆_4> , , , - which are defined in (37), (38) and (39) satisfy the inequalities:

)

¯¾E∆_%> , , , - ≤ −»_>

∈ Â_%, , ∈ Â₁ , - ∈ Â₄

¯š ∆_%> , , , - ≥ »_>

∈ Â_%, , ∈ Â₁ , - ∈ Â₄

56 7 68

⋯ 44

)

¯¾E∆_1> , , , - ≤ −¼_>

∈ Â_%, , ∈ Â₁ , - ∈ Â₄

¯š ∆_1> , , , - ≥ ¼_>

∈ Â_%, , ∈ Â₁ , - ∈ Â₄

56 7 68

⋯ 45

)

¯¾E∆_4> , , , - ≤ −½_>

∈ Â_%, , ∈ Â₁ , - ∈ Â₄

¯š ∆_4> , , , - ≥ ½_>

∈ Â_%, , ∈ Â₁ , - ∈ Â₄

56 7 68

⋯ 46

Then the system (BVP) has a periodic solution = , , , , - , , = , , , , , - and - = - , , , , - such that

∈ Â_% = ® š +_1©^§¨

¨J_% +^§_©^¨

¨e d , e −_1©^§¨

¨J_%−^§_©^¨

¨e d³ , , ∈ Â₁ = ® g +_1©^§ª

ªJ₁+_©^§ª

ªg , d , −_1©^§ª

ªJ₁−_©^§ª

ªg , d³ and - ∈ Â₄ = ` ¾ +₁^§_©^¬_¬J₄+^§_©^¬_¬F - d, ¿ −₁^§_©^¬_¬J₄−^§_©^¬_¬F - d]

Proof: Let _%, ₁ be any points in the interval Â_%, ,_%, ,₁ be any points in the interval Â₁, and -_%, -₁ beany points in the Interval Â₄, then:

)

∆_{%> %}, ,_%, -_% = ¯¾E∆_%> , , , -

∈ Â_%, , ∈ Â₁ , - ∈ Â₄

∆_{%> 1}, ,₁, -₁ = ¯š ∆_%> , , , -

∈ Â_%, , ∈ Â₁ , - ∈ Â₄

56 7 68

⋯ 47

)

∆_{1> %}, ,_%, -_% = ¯¾E∆_1> , , , - ∈ Â_%, , ∈ Â₁ , - ∈ Â₄

∆_{1> 1}, ,₁, -₁ = ¯š ∆_1> , , , -

∈ Â_%, , ∈ Â₁ , - ∈ Â₄

56 7 68

⋯ 48

)

∆_{4> %}, ,_%, -_% = ¯¾E∆_4> , , , - ∈ Â_%, , ∈ Â₁ , - ∈ Â₄

∆_{4> 1}, ,₁, -₁ = ¯š ∆_4> , , , -

∈ Â_%, , ∈ Â₁ , - ∈ Â₄

56 7 68

⋯ 49

By using the inequalities 44 , 45 , (46), (47), (48) and (49) we have )∆^{% %}, ,_%, -_% = ∆_{%> %}, ,_%, -_% + A∆_{% %}, ,_%, -_% − ∆_{%> %}, ,_%, -_% D < 0

∆_{% 1}, ,₁, -₁ = ∆_{%> 1}, ,₁, -₁ + A∆_{% 1}, ,₁, -₁ − ∆_{%> 1}, ,₁, -₁ D > 0Å ⋯ 50 )∆^{1 %}, ,_%, -_% = ∆_{1> %}, ,_%, -_% + A∆_{1 %}, ,_%, -_% − ∆_{1> %}, ,_%, -_% D < 0

∆_{1 1}, ,₁, -₁ = ∆_{1> 1}, ,₁, -₁ + A∆_{1 1}, ,₁, -₁ − ∆_{1> 1}, ,₁, -₁ D > 0Å ⋯ 51 )∆^{4 %}, ,_%, -_% = ∆_{4> %}, ,_%, -_% + A∆_{4 %}, ,_%, -_% − ∆_{4> %}, ,_%, -_% D < 0

∆_{4 1}, ,₁, -₁ = ∆_{4> 1}, ,₁, -₁ + ∆_{4 1}, ,₁, -₁ − ∆_{4> 1}, ,₁, -₁ > 0Å

⋯ 52

From the continuity of the functions ∆_{% %}, ,_%, -_% , ∆_{1 1}, ,₁, -₁ and

∆_{4 1}, ,₁, -₁ and the inequalities (50), (51) and (52), then there exist an isolated points , , , - = , , , - and ∈ ` <sub>%</sub> , <sub>1</sub>a, , ∈ `,_% , ,₁a - ∈ `-_% , -₁a where ∆_% , , , - = ∆₁ , , , - = ∆₄ , , , - =0

This means that (17) is a periodic solution

= , , , , - , , = , , , , , - and - = - , , , , - . ⧠

Theorem 4: Suppose that the vector functions ∆_% , , , - ,, ∆₁ , , , - šE

∆₄ , , , - be defined by (34), (35) and (36), and then the following inequality holds:

›‖∆% , , , - ‖

‖∆₁ , , , - ‖

‖∆₄ , , , - ‖• ≤ r ss tÆ_%u_%

v_%J_%+ e Æ₁u₁

v₁J₁+ g , Æ₄u₄

v₄J₄ + F - x yy

z ⋯ 53

Where Æ_% = ‖A‖T

F^‖T‖l− ‖E‖ , Æ₁ = ‖B‖T F^‖Z‖l− ‖E‖ , Æ₄ = ‖C‖T

F^‖[‖l− ‖E‖

Proof: From the properties of the functions , , , , - , , , , , , - and - , , , , - are "ixative intheorem 1.

Then the functions ∆_% , , , - , ∆₁ , , , - and ∆₄ , , , - are continuous, bounded in the domain (1.)

By using (34), we get

‖∆_% , , , - ‖ = ‖ A¹

F^Tl− TAE − E ) ®A F^Tl− E − _%) +

+ ). N , , , , - Ž + A

F^Tl− E . F^{‰ VŠ} ) n, n, ,, , - ,)) ),, n, ,, ,- ,- n, ,, ,- n‖

∴ ‖∆_% , , , - ‖ ≤ Æ_%u_%

v_%J_% + e ⋯ 54 Similarly by using (35), (36) we get

‖∆₁ , , , - ‖ ≤ Æ₁u₁

v₁J₁+ g , ⋯ 55

‖∆4 , , , - ‖ ≤ Æ₄u₄

v₄J₄+ F - ⋯ 56 Then we rewrite (54), (55) and (56) by the vector form we get (53). ⧠

References

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