Traveling Wavefronts on Reaction Diffusion Systems with Spatio Temporal Delays

(1)

Traveling Wavefronts on Reaction Diffusion

Systems with Spatio-Temporal Delays

Xinli Han1, Lijun Pan2

1_{College of Science, Nanjing University of Posts and Telecommunications, Nanjing, China} 2_{Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, China}

Email: [email protected], [email protected]

Received November 1, 2012; revised January 1, 2013; accepted January 9, 2013

Copyright © 2013 Xinli Han, Lijun Pan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

By using Schauder’s Fixed Point Theorem, we study the existence of traveling wave fronts for reaction-diffusion sys-tems with spatio-temporal delays. In our results, we reduce the existence of traveling wave fronts to the existence of an admissible pair of upper solution and lower solution which are much easier to construct in practice.

Keywords: Schauder’s Fixed Point Theorem; Traveling Wave Fronts; Reaction-Diffusion; Spatio-Temporal Delays

1. Introduction

Traveling wave solutions, usually characterized as solu-tions invariant with respect to translation in space, have attracted much attention due to their significant nature in science and engineering [1-18]. In which, the theory of wave fronts of reaction diffusion systems is an important part, and its history traces back to the so-called Fisher- KPP equation, the celebrated mathematical works by P. A. Fisher and by Kolmogorov, Petrovskii and Piscunov. Since then, lots of papers are devoted to the study of traveling wave solutions of reaction diffusion systems, and various research methods come forth.

The present paper is mainly devoted to tackle the ex-istence of traveling wave front solutions of the following reaction diffusion system with spatial-temporal delays and with some zero-diffusive coefficients,

 



 



 





2

1

, ,

, , , m ,

U t x U t x

D

t x

F g U t x g U t x

 



 

   

(1.1)

where t0, x; Ddiag



d₁, , d_n



, 2 1

0 n

i i

d 





,

0

i

d 

F f

, U t x

 

, 



u t x1

 

, , , u t xn

 

,



,



_{, ,}_f



T



2_, n



    

T

1 n ,



g_jU t x



 

, 

_{ }

_{ }t g t s x y U s y y s_j



 , 

  

, d d , 1, , .j  m

Here the kernels of convolutions g U jj



1, , m



,

satisfy

 

0

, d d 1,

, 0, ,

j j

g s y y s

g t x t x

  

 

  

 

 . (1.2)

And the kernels used frequently in the reference are as follows

1) g t xj

 

, 

   

t  x ; 2) g t x_j

 

, 

   

t p x_j ; 3) g t xj

 

, 



t j



 

x ; 4) g t xj

 

, q tj

   

 x ; 5) g t x_j

 

, 



t_j



p x_j

 

.

The remaining part of this paper is organized as fol-lows. In the next section, some preliminaries are given. In Section 3, we state and prove the main result of this paper.

2. Preliminaries

A traveling wave solution of (1.1) is a special translation invariant solution of the form U t x

 

,   



x ct



, where _{ }_{C R R}2



_, n



_{is the profile of the wave that}

propagates through the one-dimensional spatial domain at a constant velocity c > 0. If is monotone and satis- 

fies the asymptotic boundary conditions lim

 

s s U

  

and lim

 

,

s s U



(2)

1279

wave front of (1.1), where U



u1, ,un



T,

_ _ 

,

UU





T

1, ,

n n

U u  u  , and U−_,_U+_are

equi-libria of system (1.1). If Y < Z, we also denote



,



:





, n



:

 

,



BC Y Z   C   Y  t Z t Let  be the supremum norm in _n_and_{C a b}



 

_{, ;}_n



_,

and











 













 









 



0

; : su

; : s

: max sup

i

n n

t

n n

t

i d

C C

t





  

  



 













2

; : p

; : up

; : , , ; sup and ; , here, 0

t

n n

i

t t

BC t

BC t e

BC BC C d

 

 



 

  

 

        

 

     

where  will be given in the next section. Obviously,

, and are

Ba-nach spaces respectively with the norms





BC _{ }; n _BC



_; n

  



BC2



 ; n



 





0: sup , ; ,

n t

t BC



    



 

 





: sup e t, ; n ,

t t BC



 





    



 

 









2





0 0

2: max₀ , sup , ; .

i

n i

d  _t_ t BC

     



 

Substituting into (1.1) and

denot-ing still by t the traveling coordinate

 



,

U t x   x ct



x ct , we obtain the corresponding wave equations

 



1

 

, ,



 



, ,

m

c t D t F g t g t

t

 

     





 (2.1)

where c > 0 is velocity, 2 , ,

1

0 n

i i

d 





d_i0 i1, , n;

and



gj



 

t ₀ g s yj

  

, t y cs



d d ,

  

 

_{ }

   y s



(2.2)

Without loss of generality, we assume





T

0, ,0 ,

U_{ }

0  the

as-ymptotic boundary conditions are replaced by



T 1, , n ,

U _ _ K _ K

K

 

lim 0, lim .

t t  t t K (2.3)

In the following, we list the basic assumptions of this paper:

(A1) F



0, , 0



F



K, , K



0.

(A2) There exist positive constants j1 and Lj,

such that for all Y Zj, j



0,K



, j1,, ,m



1





1



1

, , , , m j.

m m j j

j

F Y Y F Z Z L Y Z 



 





  _j (2.4)

(A3) There exists an constant  0 such that for

and 1, , ,

j  m  BC



K K,



,

  



0 , d d

t j

g s y t y cs y s  _ 

 

     

 

(A4) One of the following two cases holds. 1

4

(A ) ₀ g t x tj

 

, d





is uniformly convergent for



,



x a a , where a0, i.e., for given 0, there

exists 0 s.t. _j

b



g t

 

,x td  for all x 



a a,



, 1, ,

j  m.

2 4

(A ) g t x xj

 

, d

 



is uniformly convergent for

 

0,

t b , where b > 0, i.e., for given 0, there exists , s.t.

0



a _a g t xj

 

, dx 





 and



_a g t x xj

 

, d 

for all t

 

0, ,b j1,,m. (A5) There exists a matrix

s.t.



1



diag , , n , i 0,

      



 



 





 



 



 





 

1 1

, , , ,

m m

F g t g t t

F g t g t

 

   

      



 t (2.5)

where t and _{  }, _C



_{ }, n



satisfy ˆ_{  } ˆ

0  K, here and in the sequel, denote the

constant vector function on , taking the value .

ˆ

U





n

 



1, , T

U u un

At the end of this section, we give the following two useful lemmas.

 

Lemma 2.1. [8]Let be a differentiable

function. If

0

:

x  

 

lim sup

t

 

lim inf ,

t x t   x t

1

n

there are

se-quences

 

sn



 and

 

tn n1



 with

s.t.

lim

 n lim n

n s nt  

 

lim lim inf and 0,

lim lim sup and 0.

n n

n t

n n

n t

x s x t x s

x t x t x t

 



   



 _ _ _ _ _



Lemma 2.2. [3,8] Let , and

be a differentiable function. If li

a

t





: ,

x a  





mx t

 exists (finite)

and the derivative function x t

 

is uniformly

continu-e . _{ous on}

_

_a_,_

_

_then_lim

(3)

3. Main Theorem

First, we introduce the definition an upper-lower solution

of wave Equations (2.1)

Definition 3.1. A continuous function 



1, , n



n



  is called an upper solution of (2.1), if _i

 

t

and _i

 

t (if d_i0) exist almost everyw e essentially bounded, and

here and they

ar _i satisfies almost

whe 

every-re on

 



1

 

, ,



 



i i i i m

c t dt  f g  t  g  t (3.1)

A lower solution 



₁, , _n



of (2.1) can be given in a similar way by reversing the inequality in (3.1)

ns (2.1), we h .

For wave equatio ave the following

re-sults.

Proposition 3.1. Assume (A ) holds. If wave e2

qua-tions (2.1) have a monotone solution _{ }_C



_{ }; n



satisfying lim

 

t t V and tlim

 

t V, where

0V VK, V V, n then



 



 



1 , , m



F g  t  g  t

is uniformly continuous in

ow is uniformly

.

Proof. It is not difficult to sh

continuous in , then we k

 

t



 now that   0 sufficiently

small





1

max

, j

j l

mL L



 

 L , there is a c stant on  , s.t.

for t₁,t2 and t1t2  ,







 



1

1 2 ,

t y cs t y cs  mL 

     

where

 

0

s , y,

1

min_{j l} j

 

 

 , and

n in 2). In dition, we can obtain



1, ,



j j m

  

are give (A ad



gi

  

t1  gi

 

t2



gi

  

t1  gi

 

t2 



0,K j



, 1, , . m Then by (A2), we have



 



 







 



 



  







1 1 1 1 2 2

1 2

0 1

, , , ,

, d j

m m

m

j j

j

d

F g t g t F g t g t

L  _ g s y t y cs t y cs y s  



    

 

 _        _ 

 

  

 

hus, ,

T F g



1

 

t , ,



gm

 

t



in. This completes the pro

is uniformly

con-position 3.2. Assume (A2) and (A4) hold. If wave

eq

tinuous of of the

proposi-tion.

Pro

uations (2.1) have a monotone solution _{ }_C



_{ }; n



satisfying lim

 

t t V and tlim

 

t V , where

0__V __V , n,_the

 

,

K V V

    n



, ,







0

F V V F V,V  .

Proof. We only give the proof under th ase

an

e c 1

4

(A )

d the case 2

4

(A ) is similar. Firstly, we show



 



 



1





lim , , _m ,



.

tF g  t  g  t F V V

For fixed let , then

by Hence

s.t.

1, , ,

j  m

know



_

1

0 ,

j m

mL L

 



 

₀

 

, d

j j

h x 

_

g s x s



dx1, 1, , .j  m

, there is a constant

(1.2), we



,





 

  A0

j

h x

maxLj 

 









 









1

0 1

0

d 8 ,

1, , .

d 8 ,

j A

A j

h x x mL

j m

h x x mL







 

 _ _

 _



  





 (3.2)

where

 

1minj m j .

 

 



constant B

By 1 , for the above A, there

4

(A )

exists a 0 s.t.

 





1





0

, d 16 , 1, , .

i

B g t x t  mL A j m



 (3.3)

holds uniformly for

 

 



,



.

x A A Since lim

 

,

t t V

ts

for the above constan ,  , m and L, there exists an constant T 0 s.t.

 





1

4, .

t V  mL  t T

      (3.4)

Obviously,



gj



 

t ,



g Vj 



 

t 



0,K



, t,

1, ,

j  m, then by (3.2)-( 4) and (A3. 2), we have



 



 









  



  



  



  



1 ₀

1 0

, , , , , d d

, d d , d d

, d d ,

.

j

m

m j i

j

m _A _B _A

j _A j _{A B} j

j

j A

F g t g t F V V L g s y t y cs V y s

L g s y t y cs V s y g s y t y cs V s y g s y t y cs V s y

t T A





 

  _ 





 



 



 

    _     _

 



 _         



     _ 

  

  

 

(4)

1281

Therefore,



 



 









1

lim , ,

, , ,

m

t F g t g t

F V V



 

 





similarly,



 



 









1

lim , ,

, , .

m

t F g t g t

V V



 

  



F 

For the i that di > 0, we denote Bi: limsup_t__ i

 

t > bi. We claim Bi =

.1 we know there with

 

: liminf , 1, , ,

i _t i

b  t i

 

  

Bi > bi, then by

are sequences

 

s_{N N}_₁ and

lim N lim N

Ns Nt   s.t.

n then Bi

Lemma 2

bi, otherwise

 

t_{N N}_₁

 

lim liminf and

and 0.

i N i

N t

i i N

s t

t t

 

   

   



 _ _

  

Substituting

 

s_{N N}_₁ and

 

t_{N N}_₁ the ith equation of

(2.1), we have f V_i



, 



cb_i f_i



ness o

 

0,

lim limsup

i N i N

N t

s t

 

 



 







as

tonicity and bounded f exists

(finite). From the ith equat have

i

cB

This contradicts with ,V ,

Bi

, , ,

V V

   

> bi, by the mono-

.

N 

 

, i t



ion of

 

lim i 0

t t 

(2.1), we also

 

1



V_ ,V_



exists (fi

lim _i lim _i

t t

i i

c

t t

d d

 

      nite).

Si oof of t rmly c

,

i

f 

milar to the pr he unifo ontinuity of 

 

t

in Proposition 3.1, we can obtain that _i

 

t is uniformly

 

t t

continuous in



0,



. Combining lim V_

 

 

t

  en  

 

t

and

Lemma 2.2, lim _i 0,

t th  erefore,

by the ith equation of (2.1), we have

 

t 0. lim

t i 0. Th



, ,



lim

 

i _t i i i

f V_ V_ c t d



   

 _  _



For the i that d_i 0,



i1, , n



, by Propositio we know

n 3.1,

 



1

 

, ,



 



i t fi g t gm  t c.

ering

    

Is uniformly continuous in . Consid lim

 

t t V

0

 . Hence,

is finite, by Lemma 2.2, we ve ha lim_t_i

 

t







 



, ,



lim

 

0.

f_i g V₁  g_mV t c _i t

t

  _



, ,



0

F V V



Then   . Similarly, F V



, ,V



0.

ted.

The pro 2 is comple

Define

of of Proposition 3.





T









1, , : 0, ;

n n

H  H  H BC K BC_ 





 



 



T

1

0, , , , n

BC K H   H   H sat-

,



s.t. isfying

  

  

 

,

H t



 



 



1 , ,



.

m

F g t t

t



    



 

)

g t



(3.5

Then we have the following lemma.

Lemma 3.1. Assume (A1) and (A5) hold, for all ,



0,



BC K

  , the operator H defined by (3.5) satisfi

1)

es

  

0H  t K t, .

 

,H H .

     

2) If

3) If 

 

t is nondecreasing in , H

 

 is also nondecreasi

Proof. 1) and 2) can be given directly by (A1) and

(A5).

3) For

ng in .

let 

 

   



t 



0,

  then













 

, 1, , .

j j

g t g

j m



      

t

By the monotonicity of 

 

t we know ˆ0    

ˆ

K, by 2),

 



  

  ,

,

H t H t H t

t



      



and this complete the proof of Lemma 3.1.

Without loss of generality, we assume  _i 0 in (A5),

and denote





 





 

2 1

2 2

4 2

if 0

4 2

if 0.

i i i i

i

i i i i

i

i i

c c d d

d

c c d d

d c

 

 

 _{ } _ 

 _{  }



 _ _ _ _

 _



       

(3.6)

Defining the integral operator P on BC



0,K



,



0,



BC K

  , t,

is given by

  



1

  

, , n

  



P  t  P  t  P  t T

  

 

_{  }

 

_{  }

_

_

 

_{  }

1 2

1

2 1

e d e d ] if

e d if

i i

i

t _{t s} _{t s}

i _t i i i i

i

t _{t s}

i i

H s s H s s d d

P t

H s s c d

 



 



 



 

 _ _ _ _ _ 

 

 

  _{ }

 _ _





0

0 i



(3.7)

(5)

w defined by (3.5

tion 3.3. Assume (A2), (A4) and (A5) hold

The integral operator P defined by (3.7) maps

here H ₁ ,H_n

 

 T is ).

Then we have the following two propositions.

 



 



,

H

   

Proposi .



0,



BC K

, and _P

 

_{ }_BC2



_{ }_; n



,



0,



BC K

  .

Proof. We on

and the case 2

4

(A

ly give the proof under th case is similar.

into BC



0,K



e 1

4

(A ), )

For  BC



0,K



, by Lemma 3.1 1), we have

  

,

H t K t

0   ,  then, if di 0

  

1  2 

_

_

2 1

0 t ei t s d + ei t s d

i i i _t i i i i

P t    K s     K s d   K



 

  __

_

__  _i  _i

If d_i0, ₀

  

t _ei1 t s_d

i i

P t   s  K c K



  

_

 _i  _i. on, similar to the pr

Therefore, 0ˆP

 

 Kˆ. In additi

oof of Proposition 3.2 we can obtain,   0



m



, i ts a onstant A > 0 s.t. for j1, , m,

L  

there ex s c

 





1

d 8

j

A h x x  L K,



 _m  

 





1

d 8

A j

h x x  mL  K



 



_.

n B > 0

And for this A, there is a consta t s.t. for

j1, , , m

 





1





, d 16 , ,

j

B g t x t  mL K x A A



 



where

 

1 in . 1min j m j , min1 j m j , m i

For fixed

i m K

 

L L K

     

, we can find T > 0 s.t. t 



T T,



.

t

Since  BC



0,K



,

 

t is uniformly continuous in



T A cB 1 ,



T A 1

     

 

 , hence there is 0  1

s.t.





 

1

1 _,

4

t t t

mL





       _{ } _

   t  ,





,

t _ T A cB T  A_.









Obviously, gj* t t ,



gj*



 

t 



0,K



,

t j1, , , m then by (A2) we obtain



















 



 



  







 





 

0 , 2 d d

A j

A

 

1 1

0 1

1 0

1

0

* , , * , , *

, d d

, 4 d d , 2 d d

j

m m

m

j j

j

m _A _B _A

j _A j _{A B} j

j

F

 

, 2 d d



.

*

g t t g t t t g t

L g s y t t y cs t y cs y s

L g s y mL y s g s y K y s

g





   



 

  



      

 

 _          _

 

 

  

 

 



F g

 

g s y K y s  



_{ }

 

_{ }

s y K y s  

Hence, F g



1*

 

t , ,



gm*

 

t



is continuous in



, then 

 



1*

 

, ,



m*

 



 

H  t F g  t  g  t   t

is continuous in . In addition, by calculating directly we can obtain the following, if di0,

   

1 

_{  }

2 

_{  }

_

_

1 ei d 2 ei d

t _{t s} _{t s}

i i i i _t i i i

P t    H s s     H s s d



  

2 i1

 

 

 _    _ _  _







   

2 1 

_{  }

_

_{ }

_{ }

2 2 

_{  }

_

_

1 ei d 1 2 2 ei d

t _{t s} _{t s}

i i i i i i i _t i i i

P t    H s s   H     H s s d ₂ _i₁



  _{ } _

 __

_

    t 

_

 __ _  _.

If di0,

   

t _ei t s

_{  }

_d

_{  }

i i i i

P   t _ _   H  s s H  t _ c





 .

Therefore, _P

 

_{ }_BC

 

_0,_K __BC2



_{ }_; 2



_.

3.3 is completed.

The proof of Proposition

Proposition 3.4. For  BC



0,K



,  BC2



 ; n



,

 

t P

  

t

   iff

 

  

,

D  t c  t  t H t

       

.



t Especially,  BC



0,K



only if

is a solution of wave

if and a fixed point of P.

f. We only prove t

case can be g

Equation (2.1)  is

Proo he case di 0,and the proof

for the iven similarly.

If

0,

i

d 

 

  

t _ei t s

_{  }

_d _c_,

i t Pi t Hi s s



 



  

_

 

let i  i c, We obtain ci

 

t  i i

 

Hi

  

 t . bove argument

t

, On the other hand, by the a

So we have



i i



i

 

cP_i 

  

 t  P

 

 t H 

 

t .

   

0.

i i i i i

c_ P  _ t  _ P  _ t 

Then

 

  

eit

i t Pi t 

(6)

1283



0,



BC K

 

By Proposition 3.3, we know ,

 



0,



P  BC

 

  

i t Pi t

   K is bou. Since





2 _; n

BC   , d in , hence

 

nde  0, then

And th proof of

By Lemma 3.1, we ca following

lemma on the monotonicity o or P.

Lemma 3.2. Assume (A1

1) If

 

  

.

i t Pi t

  

Proposition 3.4.

is completes the

n easily obtain the f the integral operat ) and (A5) hold, then





, BC 0,K

   , and   , P

 

 P

 

 . 2) If BC



0,K



is nondecreasing in , P

 



is also .

On the continuity of the i , we have

5. Ass

nondecreasing in 

ntegral operator P

2)-(A4) hold. Then

the following.

Proposition 3. ume (A









: 0, 0,

P BC K BC K

the norm

is continuous with respect to

u

 in BC



;



, where

,

2

u  



1, 2,



0  min  _i _i _i and   i1, i2, i are

defined by (3.5) is c

given by

oof. We first claim H ontinuous

in (3.6).

Pr



0,



C K

B with respect to the norm _u.

For  , BC



0,K



, we know obviously



gj



 

t ,



g_j 



 

t BC



0,K



, ,t j1, , . m

(A

By (A2) and 3), we obtain

 





 





 









1

sup * e

e

, Suppose

j

m

u t

j j j

t j

m j j

L g t g t

t t

L



1

max sup t

i i

i n t



1

sup m j e t je t

t j

L   

*



1

H H

 

 

 



 



 





 _ _ 

  __    __



 



 



_  _       

 







re

 

  _

 





  

 

 _ _

 _    _   











  

 



whe

 

1min j m j 1, 1 j n i

    max

 



 



 . Thus   0,

choose

1 1 1

min m j ,1

j j

L M

 

  





_ _ _ _ 

_ _ 

 _ _  _ _

  

_



 _ 

then we have



 

 

H  H  _ for    _ ,

i.e., H is continuous in BC



0,K



with respect to the norm _u.

Now, we show P BC:



0,K



BC



0,K



_is

continu-ous with respect to the norm u. For ,  BC



0,K



, sim_{ilar to the method in Ma [4], we obtain ther}e is

G > 0 s.t.

an constant

 

P  P  _ G H H _

, we ow P is continuous with

respect tothe norm

   

By the continuity of H kn

u

 . The proof is completed.

In the following, we state and prove the main theorem of this paper.

Theorem3.1. Assume (A1)-(A5) hold. Suppose wave

equations (2.1) has a pair of upper and lower solution





, BC 0,K

  satisfying

1) sup

 



or

 

inf

 



, .

s t

s t s t t s t

   



    

2)



, ,



0,



0,inf

 

sup

 

, .

t _t

F V V V  t  t K

 _

 



  _ _

  

 

 _

Then (2.1) and (2.3 have a monotone solution, i.e., front so tion.

we define the following profile set

)

(1.1) has a traveling wave lu

To prove Theorem 3.1,





 

1) is nondecreasing in .

, : : 2) , 0, .

3) , , .

BC K

s t M s t s t

   

 _ 

 

 

_ _  _      _

 

     



 

 

where





1 i n i i

max ,

M  K c we first prove two lemmas.

hold, then for

 

Lemma 3.3. If the conditions of Theorem 3.1 



; n



C

    with    , we have

 

.

P

  

Proof. For t, we denote

 



 



 



  

 



  

 



, ,

n

t w t w t

W t  t



  





1

T

1 1 , ,

n

P  t  t  P t

T

W

(3.8)

then we have _W__BC



_{ }; n



_{. In order to obtain}

 

,

P   it suffices to prove W t

 

0. 0,

i

d 

By Proposition 3.4, we know if

   

  

,

i i i i

cP   t P  t H  t t

From the definition of lower solution, we know

 

, . . .

i i i i

c t   t H  t a e t 



  and by Lemma 3.1 2), we obtain

Considering

  

 

  

 

0,

. . .

i i i i i

c P t t P t t

a e t

   

        

   

 

Let

 

t

 

, .

i i i i

r t cw  w t  t  ) Then

(3.9

 

0.

i

r t 

riant of c

By the continuity of and

for-mula of va onstants, we have

d , (3.10)

 

i

w t

 

_eit t _ei t s

_{ }

i i

w t   

_

_   r s s

where  is a constant. By Lemma 3.1, we know Pi

 

 ,

 

Pi   are both bounded, It follows by (3.8) that w ti

 

,

 

i

(7)

 

i

r t is also essentially bounded in , so  0 in (3.10), and Pi

 

 i,

for

 di 0



. In a similar way, we can obtain Pi

 

 i,



for di 0



therefore P

 

 

 . Similarly, we can prove P

 

 . The proof is completed.

L

then

emma 3.4. If the co rem

P is equi-continuous.

nditions of Theo 3.1 hold,

Proof. For  BC



0,K



, if di0, by Lemma 3.1

1) and Proposition 3. we have, for 3, t,

   

_

i1

_

t _ei1 t s

_

_

_d i i_, i

2 1

i i i

i i

K P _  t _     K s_{ }



d c

and,

  





   

2 

_

_

2 1

i t s

i

2

i

t e d ,

i i

i i i

i i

K

P t K s

c



  

d

 



 

 



 

then,

   

K ci , this is also

ilar an

i i

P   t  holds for the

the proof is sim mit here.

case di0, d we o

0,



  we choose   M, here





1

max_{i n} i i

M  K c

 

en for

 



 ,

th P  P BC



0,K





, by Lagrange theorem,

if t t1,2, t1t2 ,

  

   





1 2 1 2

1 2

,

, ,

i

t P t P t M t t

t t 

      



then

1 2

i  i t i

P 

  

1

  

2 ,

P  t P  t  i.e., P BC





0,K



f Lemma



is

equi-continuous. This comple 3.4.

Proof of Theorem 3.1. We roof of

into five steps.

tes the proof o divide the p rem 3.1

Step 1, _{ } , _ is a nonempty and convex set.

1) Denote

 

sup

 



or

 



 

s



,t

s t

t s

 t infs t

   

O is c ng in

then by Lem 3.2 1), reasing in

2) By The m 3.1(a)



is nondec know

.

bviously,  _{ontinuous and nondecreasi}

ma ore

,

.

 

P 

, we 0ˆ      Kˆ,

then by lemma 3.2 1) and Lemma 3.3 we have



 

.

P P P

     _ _

nd 2) we al

3) By the above 1) a so know   BC



0,K



_,

similar to the proof of Lemma 3.4 ,we obtain

 

1

 

2 1 i





P  t P  t M t 

2, 1,2 .

t t t

  

Therefore, P

 

_{   }  , _, then _{ } , _ is non-empty. It is obvious that _{ } , _ is convex.

Step 2, _{ } , _ is a closed set in i.e.,

if seque





; n ,

BC_   nces

 







T



1

1 , , ₁ ,

k k k

n

k   _k_  

   _{  } _

converge to with respect to the norm

 

 _, then

, .

 

 

 _{ } _ Since

 





1

max sup e t 0,

k k

i i

i n t t



  

   _

    



for the fixed t



 

e t 0, as .

k _t k 



t k

           (3.11)

0,



  we choose   3 ,M then by

 

1 ,

k

k  





 

  _ _ we know

 





3

k _t k _t _t _

      ,  t  , t, k1, 2,. 1), there is a N s.t. for the above ,t t

By (3.1

 

3

N _t _t _

    ,

and









3.

N _t _t _t _t _

       

Therefore, for fixed t, if  t ,

 





 





N _t _t N _t _t _t

t t t

    









N

N _t _t _t _t _

        

        

i.e., 

 

t t

is continuous in . In addition, we have

1) 1 , t2 , t1t2 , sin

 

1

 

2

k _t k _t

  

ce ,

1,

k 2,, by (3.11),

 

1

 

2

 

1

 

1

 

2

 

2 0,

k k

t t t t t t

           

i.e., 

 

t t

is continuous in .

2)  , since _k

 

_t __

 

_{t k}, _1, 2, ,_ _{by (3.11),}

 

_t _

 

_t

 

_t k

 

_t 0.

       Similarly   , so



0,



BC  

3

.

K

) Since _k

 

_s _{ }k

 

_t __{M s t}_ _{, ,}_{s t}___,

k1, 2,,

 

,

k k

s t

s s M s t t t

  

         

combining with (3.11), we obtain

 

s

 

t M s t.

     Therefore, _{   } , _.

Step 3,

 



,



, .

P _ _  _ _ This can be easily proved followed by Proposition 3.3 and Lemma 3.2 - 3.4.

Step 4, P



  _ , _



is sequentially compact. For _{   } , _. by Proposition 3.3 we know

 



0,



,

P  BC K then _P



_



₀K, _i.e._,_P



_{ }__{ }_, _





3.4 that is uniformly bounded. It follows by Lemma



,



P _{ } _ us. wo

opera-tors

and

satisfying

is equi-continuo We define t





T





1, , : 0,

N N Nn

P  P  P BC K BC



0,K







T













, , Nn : 0, , ; n

Q  Q Q BC K C N N 

1