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Fixed points of mixed monotone operators withapplicationsDajun Guo aa Department of Mathematics, Shandong University, People's Republic, Jinan, Shandong, ChinaVersion of record first published: 02 May 2007.

To cite this article: Dajun Guo (1988): Fixed points of mixed monotone operators with applications, Applicable Analysis: AnInternational Journal, 31:3, 215-224

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Applicable Analysis, Vol. 31. pp. 215-224 Reprints available directly from the publisher Photocopying permitted by license only 0 1988 Gordon and Breach, Science Publishers, Inc Pr~nted in Great Britain

Fixed Points of Mixed Monotone Operators with Applications Communicated by R . P. Gilbert

DAJUN GUO Department of Mathematics, Shandong Univers i ty , J inan , Shandong, People ' s Republic of China.

AMS(M0S): 47H

Abstract Existence and uniqueness theorems of f ixed p o i n t s f o r some mixed monotone ope ra to r s a r e inves- t i g a t e d and a p p l i c a t i o n s t o ord inary d i f f e r e n t i a l equat ions a r e given.

KEY WORDS: Cone and partial ordering, mixed monotone operators, coupled fixed points and fixed points.

(Received for Publication 1 August 1988)

INTRODUCTION

This paper i s cont inua t ion of uuo & ~akshmikantham' . Let t he r e a l L3anach space E be p a r t i a l l y ordered by a cone P

of E, i.e. x s y i f f y-x€P. Let DCE. Operator A:DXD+E is s a i d t o be mixed monotone i f A(x,y) i s nondecreasing

i n x and nonincreesing i n y. Poin t (x* ,yY ~ ~ D X D is c a l l e d

a coupled f i xed po in t of A i f A(x*,yi) = x* and A(y*,x*) = y*. Element x * ~ D i s c a l l e d a f i xed po in t of A i f

A(x*,x*) = x*.

Recal l t h a t cone P is s a i d t o be s o l i d i f t he i n t e r -

i o r $ of P i s nonempty, and P is s a i d t o be normal i f t h e r e e x i s t s a p o s i t i v e cons tan t ti such t h a t O S x S y

imp l i e s ~ B ~ ( c N Iwll ( see Guo & I,akshmikantham2). I f y-x€p, we wr i t e x a y .

In t h i s paper , we f i r s t g ive ex i s t ence and unique- nes s theorems of f ixed p o i n t s f o r some mixed monotone ope ra to r s , and then o f f e r a p p l i c a t i o n s t o t he i n i t i a l value problems of ord inary d i f f e r e n t i a l equat ions.

MA I N THEOREMS

215

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216 DAJUN GUO

Theorem 1 Le t P be s o l i d and normal, and A : B x L fi be a mixed monotone o p e r a t o r . Suppose t h a t t h e r e e x i s t s O b a

1 such t h a t

Then A h a s e x a c t l y one f i x e d p o i n t x i n and, c o n s t r u c t - i n g s u c c e s s i v e l y sequences

xn= A ( X , , ~ ,Y, ,~ 1, Y,= A ( Y , - ~ , x ~ - ~ ("=I ,2, ( 2 )

f o r any i n i t i a l we have

with convergence r a t e

where 0 < r < 1 and r depends on (xo, yo). Moreover, f o r any

coupled f i x e d p o i n t (.ji,y)e6xP of A , i t must be 37=y=x*.

Proof From h y p o t h e s i s ( 1 ) we know f i r s t

and s o

L e t zoek be a r b i t r a r i l y given. S ince A(zo ,z0) th , we can choose 0 4 to< 1 s u f f i c i e n t l y s m a l l such t h a t

Le t uo= t t z o , vo= t,fzo an.

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FIXED POINTS OF OPERATORS 217

C l e a r 1 y,

u 0 , v o ~ ~ , U O < V O , u0= t0vo (6 1

and, by v i r t u e o f ( 1 ) , ( 5 ) and t h e mixed monotone proper ty o f A , we have

Now, i t is e a s y t o show by i n d u c t i o n t h a t

an n ~f u r, to vn, then vn s tGa u, and n

a -a n n+ 1 u ,+~= A(un,vn) & A ( t o v n , t o u n ) A(vnyun)

hence, by (8) and i n d u c t i o n , we g e t n

u,zt: vn ( n = O , l , . - . ) .

From ( 9 ) and (10) we f i n d n n

O C U ~ + ~ - u,+vn- u n + ( l - t ; ) v n d ( l - t ; )Vo,

and consequent ly n

I I U , + ~ - ~416 N ( 1 -t: ) I!v& which i m p l i e s t h a t {un] converges ( i n norm) t o some u*e E,

S i m i l a r l y , we can prove t h a t {vnj a l s o converges t o some V*&E and, by (y ) ,

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21 8 DAJUN GUO

Hence u*,v*ch. Now, ( 1 1 ),(9) and (10) imply

and the re fo re u*= v*. Let x*= u = v . On account of (11 ),

Taking l i m i t a s n+clo, we ge t

hence, A(x*,x*) = x*, i .e. x* is a f ixed poin t of A.

For any coupled f ixed poin t ( T E , J ) C ~ & of A , l e t t l= s u p l o c t c l l t x * ~ ~ ~ t - ~ x " , t x * ~ ~ ~ t - l x * j . c l e a r l y , 0 < t l + 1 and t l x * ~ ~ ~ t ~ l x * , t lx*+ ~ l t ; ' x * . II O c t l < I ,

then by v i r t u e of ( 1 ) and ( 5 ) , we have

s i m i l a r l y , we g e t

(13) and (14) con t r ad i c t the d e f i n i t i o n of t l , s ince t y > t , . Hence t l = 1 and = 7 = x* . This a$ t he same time proves the uniqueness of f ixed poin t of A i n $.

We remain t o show t h a t (3) and (4 ) hold. Let (xo,yo) 0 P$ be given. We can choose t 0 ( o < t o < I ) s o small t h a t

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and, s i m i l a r l y , y n z u n , yngvn . Hence, by induct ion ,

Now, from

i t fol lows t h a t

I n the same way, we g e t

F ina l ly , (16) and (17) imply ( 4 ) with r = to, and there- fo re (3) holds. The proof i s complete.

Theorem 2 Let P be s o l i d and normal, and A:fix8+; be a mixed monotone opera tor . Suppose t h a t t h e r e e x i s t s O S a t 1 such t h a t ( 1 ) holds. Let x t be t he unique s o l u t i o n i n $ of the equat ion

Then x: i s continuous with r e spec t t o t , i.e. UX~-X~,II-+O

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220 DAJUN GUO

a s t + t o ( t o > 0 ) . I f , i n add i t i on , 0 6 a c * , then x; i s s t rong ly decreas ing with r e spec t t o t , i.e.

(19)

and

1 x * = 0 t384 = +" . t++m t (20)

Proof Since opera tor t" A s a t i s f i e s a l l condi t ions of - Theorem 1 , equat ion (18) has exac t ly one s o l u t i o n x; i n 8 . Given t2 p t l 2 0 a r b i t r a r i l y and l e t so= s u p i s ;. 0 I X;~%SX; , X;~&SX; 3 . Clear ly , 0 < soc+oo and

2 1

x ; ~ * s x* t2 , xi2& . (21

It i s easy t o see from (21) t h a t so 5 1 i s impossible. Hence 0 < so<l. ~3y (1 ) and (21 ), we f ind

=t

Consequently

Observing the d e f i n i t i o n of so and t2ty1 st > s o , we con- clude t t;'st s so, and so

so & ( t l / t 2 ) l / ( l - a ) . (23)

I t fol lows from (21 ) and (23) t h a t

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FIXED POINTS OF OPERATORS 22 1

I n e q u a l i t i e s (24) and (251, t oge the r with t he normal i ty

of cone P, imply

Hence, t he c o n t i n u i t y of x: with r e s p e c t t o t ( t > 0 ) i s

proved.

Now, assume Od a c*. By v i r t u e of (22) and (23 ) , we

have

which imp l i e s (19) s i n c e

L e t t i n g t l = 1 and t2= t i n (26 ) , we f ind

and s o

which imp l i e s )(xEll-t 0 a s t+ +co . On the o t h e r hand,

l e t t i n g t l = t and t2= 1 i n (2b ) , we g e t

and t he re fo re

which imp l i e s IIX;~I + +w a s t+ +O. Hence, (20) ho ld s

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222 DAJUN GUO

and our theorem is psoved. Remark I t should be pointed out t h a t i n Theorems 1 and 2 we do no t r equ i r e ope ra to r A t o be continuous,

APPLICATIONS

In t h i s s ec t ion we give a p p l i c a t i o n s of Theorems 1 and 2

t o i n i t i a l value problem

where J = [O,Tl (T>O) , O < r i K 1 , O * s j < l ( i=1,2,*.- ,n; J = 1,2,...,m), xo > 0 , ai ( t ) a r e nonnegative bounded measur- a b l e func t ions (on J ) and b ( t ) a r e nonnegative measurable func t ions such t h a t

j

i n f 2 b j ( t ) > 0 . trJ jat

The s e t of a l l abso lu t e ly continuous func t ions from J i n t o R' i s denoted by ACCJ,R'J . A func t ion x ( t ) on J is s a i d t o be a s o l u t i o n of the i n i t i a l value problem (27) i f x ( t ) O A C I J , R 1 ] and s a t i s f i e s (27).

Theorem 3 Under condi t ions mentioned above, i n i t i a l value problem (27) has e x a c t l y one p o s i t i v e s o l u t i o n x* ( t ) . Moreover, cons t ruc t ing success ive ly sequence of func t ions

1 f o r any i n i t i a l p o s i t i v e funct ion X ~ ( ~ ) ~ C [ J , R ) , t he sequence of func t ions [ x n ( t ) ) converges t o xi( t ) uniformly on J. Proof It is c l e a r , X ( ~ ) € A C ( J , R ~ J i s a p o s i t l v e s o l u t i o n

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FIXED POINTS OF OPERATORS 223

of (27) i f and only i f x ( ~ ) ~ c [ J , H ~ I is a p o s i t i v e s o l u t i o n of the fol lowing i n t e g r a l equat ion

L e t E = C [ J , R ~ I and P = f X C C [ J , R ? ~ ~ ( ~ ) 30,tiJj . hen P

is a normal s o l i d cone i n E and the equat ion (29) can be w r i t t e n i n t he form

where A(x,y) = A l ( x ) + A2(y),

It 1s c l e a r t h a t A l : P+ b i s nondecreasing and A2:;+ P i s nonincreasing, and the re fo re A : h & i s a mixed monotone opera tor . Moreover, f o r x,ycF and O< t < 1 , i t 1s easy t o see

where rO= maxfrl ,- .*,rnj , so= maxfsl ,-..,sm j , LI < ro< 1 ,

0 < so< 1 . And the re fo re

where r = max{ro,so), 0 < r c 1. Hence, by Theorem 1 , we conclude t h a t A has e x a c t l y one f i xed po in t x* i n 8 and, f o r any i n i t l a 1 xoeb,

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224 DAJUN GUO

where

xn = A ( K - ~ , X ~ - , ) (n=1,2,...).

The proof i s complete. Using Theorem 2, we g e t s i m i l a r l y t h e fo l lowing

Theorem 4 Let t h e hypotheses of 'Sheorem 3 be s a t x s f i e d .

Denote by x;( t ) t h e unxque p o s i t i v e s o l u t i o n of t h e i n i - t i a l va lue problem

Then x:(t) converges t o x* ( t ) u n i r o

formly on t€J a s r+ ro

( r O > 0 ) . I f , i n a d d i t i o n , 0 c r i < 3 , O c s . ci (i=1,2,..;n; 3

j=1,2,- .*,m), then

a x x ; ( t ) + 0 a s r + + w , max x;(t)-r+oo a s r - c + U . & J t C J

REFERENCES

1 . Dajun Guo & V.Lakshmikantham, Coupled f i x e d p o i n t s of n o n l i n e a r o p e r a t o r s with a p p l x c a t i o n s , Nonlxnear Anal. TMA, 11 623-632 ( 1967 1.

2. Dajun Guo & V.Lakshmikantham, Nonl inear problems xn a b s t r a c t cones, Academic Press , Inc . , Boston & New Y 0 r k - 7 (m

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