- Email: [email protected]
Fixed Point Theory for Generalized Contractions on Spaces with Two Metrics
Journal of Mathematical Analysis and Applications 248, 402᎐414 Ž2000. doi:10.1006rjmaa.2000.6914, available online at http:rrwww.idealibrary.com on
Fixed Point Theory for Generalized Contractions on Spaces with Two Metrics Ravi P. Agarwal Department of Mathematics, National Uni¨ ersity of Singapore, 10 Kent Ridge Crescent, Singapore 199260
and Donal O’Regan Department of Mathematics, National Uni¨ ersity of Ireland, Galway, Ireland Submitted by William F. Ames Received April 11, 2000 DEDICATED TO THE MEMORY OF R. KANNAN
We present new fixed point results for generalized contractions on spaces with two metrics. In addition generalized contractive homotopies will also be discussed in detail. 䊚 2000 Academic Press
1. INTRODUCTION This paper presents fixed point theorems for generalized contractions on spaces with two metrics. The results are motivated from the study of differential equations in abstract spaces; see, for example, O’Regan and Precup w6, Chap. 2x. This paper will be divided into two main sections. Section 2 presents new local and global fixed point results for generalized contractions. The theorems in Section 2 extend results of Hardy and Rogers w2x, Kannan w3x, Precup w7x, and Reich w8x. Section 3 presents general continuation type theorems for generalized contractive homotopies on spaces with two metrics. The theorems in this section extend previous results of Granas w1x, O’Regan w5x, and Precup w7x. 402 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
403
FIXED POINT THEORY FOR CONTRACTIONS
2. FIXED POINT THEORY This section presents fixed point results for generalized contractions on spaces with two metrics. The theorems extend the fixed point results in w2᎐4, 7, 8x. Throughout this section Ž X, dX . will be a complete metric space and d will be another metric on X. If x 0 g X and r ) 0 let B Ž x 0 , r . s x g X : d Ž x, x 0 . - r 4 , and we let B Ž x 0 , r .
d
X
denote the dX-closure of B Ž x 0 , r ..
THEOREM 2.1. Let Ž X, dX . be a complete metric space, d another metric X on X, x 0 g X, r ) 0, and F : B Ž x 0 , r . d ª X. Suppose there exists q g Ž0, 1. such that for x, y g B Ž x 0 , r .
d
X
we ha¨ e
d Ž Fx , Fy . F q max d Ž x, y . , d Ž x, Fx . , d Ž y, Fy . , 1 2
d Ž x, Fy . q d Ž y, Fx .
4.
In addition assume the following three properties hold: d Ž x 0 , Fx 0 . - Ž 1 y q . r
Ž 2.1.
X
if d h d assume F is uniformly continuous from Ž B Ž x 0 , r . , d . into Ž X , dX . ,
Ž 2.2. and X
if d / dX assume F is continuous from B Ž x 0 , r . , dX into Ž X , dX . .
ž
d
/
Then F has a fixed point. That is, there exists x g B Ž x 0 , r .
d
X
with x s Fx.
Before we prove Theorem 2.1 it is worth stating the special case of Theorem 2.1 when d s dX . COROLLARY 2.2. Let Ž X, d . be a complete metric space, x 0 g X, r ) 0, d and F: B Ž x 0 , r . ª X. Suppose there exists q g Ž0, 1. such that for x, y d
g B Ž x 0 , r . we ha¨ e d Ž Fx , Fy . F q max d Ž x, y . , d Ž x, Fx . , d Ž y, Fy . , 1 2
d Ž x, Fy . q d Ž y, Fx .
and d Ž x 0 , Fx 0 . - Ž 1 y q . r . d
Then there exists x g B Ž x 0 , r . with x s Fx.
4
404
AGARWAL AND O’REGAN
Proof of Theorem 2.1. Let x 1 s Fx 0 . Now from Ž2.1. we have dŽ x 1 , x 0 . - Ž1 y q . r F r so x 1 g B Ž x 0 , r .. Next let x 2 s Fx 1 and note that d Ž x 1 , x 2 . s d Ž Fx 0 , Fx 1 . F q max d Ž x 0 , x 1 . , d Ž x 0 , Fx 0 . , d Ž x 1 , Fx 1 . , 1 2
d Ž x 0 , Fx 1 . q d Ž x 1 , Fx 0 .
4
F q max d Ž x 0 , x 1 . , d Ž x 0 , x 1 . , d Ž x 1 , x 2 . , 12 d Ž x 0 , x 2 . 4 F q max d Ž x 0 , x 1 . , d Ž x 1 , x 2 . ,
1 2
d Ž x 0 , x1 . q d Ž x1 , x 2 .
4
F qd Ž x 0 , x 1 . . To see the last inequality suppose the maximum on the right hand side of the above displayed equation is 12 w dŽ x 0 , x 1 . q dŽ x 1 , x 2 .x. Then dŽ x 1 , x 2 . F 2q w dŽ x 0 , x 1 . q dŽ x 1 , x 2 .x and so d Ž x1 , x 2 . F
q 2yq
d Ž x 0 , x 1 . F qd Ž x 0 , x 1 . .
The other cases are easier. Thus d Ž x 1 , x 2 . F qd Ž x 0 , x 1 . - q Ž 1 y q . r . Notice x 2 g B Ž x 0 , r . since d Ž x 0 , x 2 . F Ž 1 q q . d Ž x 0 , x1 . - Ž 1 q q . Ž 1 y q . r F Ž 1 y q . r 1 q q q q 2 q ⭈⭈⭈ s r . Proceed inductively to obtain x n s Fx ny1 , n s 3, 4, . . . with d Ž x nq 1 , x n . F qd Ž x n , x ny1 . F q nd Ž x 0 , x 1 . - q n Ž 1 y q . r and x nq 1 g B Ž x 0 , r .. Now since q g Ž0, 1. we have that Ž x n . is a Cauchy sequence with respect to d. We now claim that
Ž x n . is a Cauchy sequence with respect to dX .
Ž 2.4.
If d G dX this is trivial. Next suppose d h dX . Let ⑀ ) 0 be given. Now Ž2.2. guarantees that there exists ␦ ) 0 such that dX Ž Fx , Fy . - ⑀
whenever x, y g B Ž x 0 , r . and d Ž x, y . - ␦ . Ž 2.5.
405
FIXED POINT THEORY FOR CONTRACTIONS
From above we know that there exists N g 1, 2, . . . 4 with dŽ xn , xm . - ␦
whenever n, m G N.
Ž 2.6.
Now Ž2.5. and Ž2.6. imply dX Ž x nq 1 , x mq1 . s dX Ž Fx n , Fx m . - ⑀
whenever n, m G N,
and as a result Ž2.4. holds. Now since Ž X, dX . is complete there exists x X d g B Ž x 0 , r . with dX Ž x n , x . ª 0 as n ª ⬁. We claim that x s Fx.
Ž 2.7.
If Ž2.7. is true then we are finished. First consider the case when d / dX . Notice dX Ž x, Fx . F dX Ž x, x n . q dX Ž x n , Fx . s dX Ž x, x n . q dX Ž Fx ny1 , Fx . . Let n ª ⬁ and use Ž2.3. to obtain dX Ž x, Fx . s 0, so Ž2.7. is true in this case. Next suppose d s dX . Then d Ž x, Fx . F d Ž x, x n . q d Ž Fx ny1 , Fx . F d Ž x, x n . q q max d Ž x, x ny1 . , d Ž x, Fx . , d Ž x ny1 , Fx ny1 . , 1 2
d Ž x, Fx ny 1 . q d Ž x ny1 , Fx .
4
F d Ž x, x n . q q max d Ž x, x ny1 . , d Ž x, Fx . , d Ž x ny1 , x n . , 1 2
d Ž x, x n . q d Ž x ny1 , x . q d Ž x, Fx .
4.
Letting n ª ⬁ gives d Ž x, Fx . F qd Ž x, Fx . . That is, dŽ x, Fx . s 0, so Ž2.7. holds. The following global result can easily be deduced from Theorem 2.1. THEOREM 2.3. Let Ž X, dX . be a complete metric space, d another metric on X, and F : X ª X. Suppose there exists q g Ž0, 1. such that for x, y g X we ha¨ e d Ž Fx , Fy . F q max d Ž x, y . , d Ž x, Fx . , d Ž y, Fy . , 1 2
d Ž x, Fy . q d Ž y, Fx .
4.
406
AGARWAL AND O’REGAN
In addition assume the following two properties hold: if d h dX assume F is uniformly continuous from Ž X , d . into Ž X , dX . Ž 2.8. and if d / dX assume F is continuous from Ž X , dX . into Ž X , dX . .
Ž 2.9.
Then F has a fixed point. Proof. Fix x 0 g X. Choose r ) 0 so that d Ž x 0 , Fx 0 . - Ž 1 y q . r . d
X
Now Theorem 2.1 guarantees that there exists x g B Ž x 0 , r . with x s Fx. Theorem 2.3 immediately yields the following result of Hardy and Rogers w2x. COROLLARY 2.4. Let Ž X, d . be a complete metric space and F : X ª X. Suppose there exists q g Ž0, 1. such that for x, y g X we ha¨ e d Ž Fx , Fy . F q max d Ž x, y . , d Ž x, Fx . , d Ž y, Fy . , 1 2
d Ž x, Fy . q d Ž y, Fx .
4.
Then F has a fixed point.
3. HOMOTOPY RESULTS In this section we present homotopy results for the maps discussed in Section 2. Our theorems improve the results in w1, 5, 7x. THEOREM 3.1. Let Ž X, dX . be a complete metric space and let d be another metric on X. Let Q : X be dX-closed and let U : X be d-open and U : Q. Suppose H : Q = w0, 1x ª X satisfies the following fi¨ e properties: Ži. x / H Ž x, . for x g Q_U and g w0, 1x; Žii. there exists q g Ž0, 1. such that for all g w0, 1x and x, y g Q we ha¨ e dŽ H Ž x, ., H Ž y, .. F q max dŽ x, y ., dŽ x, H Ž x, .., dŽ y, H Ž y, .., 1 w Ž Ž .. Ž Ž ..x4; 2 d x, H y, q d y, H x, Žiii. H Ž x, . is continuous in with respect to d, uniformly for x g Q; Živ. if d h dX assume H is uniformly continuous from U = w0, 1x endowed with the metric d on U into Ž X, dX .; and Žv. if d / dX assume H is continuous from Q = w0, 1x endowed with the metric dX on Q into Ž X, dX .. In addition assume H0 has a fixed point. Then for each g w0, 1x we ha¨ e that H has a fixed point x g U Ž here HŽ . . s H Ž . , ...
FIXED POINT THEORY FOR CONTRACTIONS
407
Before we prove Theorem 2.1 we state the special case of Theorem 3.1 when d s dX . COROLLARY 3.2. Let Ž X, d . be a complete metric space. Let Q : X be d-closed and let U : X be d-open and U : Q. Suppose H : Q = w0, 1x ª X satisfies the following three properties: Ži. x / H Ž x, . for x g Q_U and g w0, 1x; Žii. there exists q g Ž0, 1. such that for all g w0, 1x and x, y g Q we ha¨ e dŽ H Ž x, ., H Ž y, .. F q max dŽ x, y ., dŽ x, H Ž x, .., dŽ y, H Ž y, .., 1 w Ž Ž .. Ž Ž ..x4; and 2 d x, H y, q d y, H x, Žiii. H Ž x, . is continuous in with respect to d, uniformly for x g Q. In addition assume H0 has a fixed point. Then for each g w0, 1x we ha¨ e that H has a fixed point x g U Ž here HŽ . . s H Ž . , ... Remark 3.1. Usually in Corollary 3.2 we have Q s U d Žthe d-closure of U in X .. Notice in this case that Ži. becomes x / H Ž x, . for all x g ⭸ U Žthe boundary of U in X . and g w0, 1x. Proof of Theorem 3.1. Let A s g w 0, 1 x : H Ž x, . s x for some x g U 4 . Now since H0 has a fixed point Žand Ži. holds. we have that 0 g A, so A is nonempty. We will show A is both closed and open in w0, 1x, and so by the connectedness of w0, 1x we are finished since A s w0, 1x. First we show A is closed in w0, 1x. Let Ž k . be a sequence in A with k ª g w0, 1x as k ª ⬁. By definition for each k, there exists x k g U with x k s H Ž x k , k .. Now we have d Ž x k , x j . s d Ž H Ž x k , k . , H Ž x j , j . . F d Ž H Ž x k , k . , H Ž x k , . . q d Ž H Ž x k , . , H Ž x j , . . q d Ž H Ž x j , . , H Ž x j , j . . F d Ž H Ž x k , k . , H Ž x k , . . q q max d Ž x k , x j . , d x k , H Ž x k , . , d Ž x j , H Ž x j , . . ,
½
ž
1 2
d Ž x k , H Ž x j , . . q d Ž x j , H Ž x k , . .
5
408
AGARWAL AND O’REGAN
q d Ž H Ž x j , . , H Ž x j , j . . F d Ž H Ž x k , k . , H Ž x k , . . q q max d Ž x k , x j . , d Ž H Ž x k , k . , H Ž x k , . . ,
½
d Ž H Ž x j , j . , H Ž x j , . . , 1 2
2 d Ž x k , x j . q d Ž H Ž x j , j . , H Ž x j , . . qd Ž H Ž x k , k . , H Ž x k , . .
5
q d Ž H Ž x j , . , H Ž x j , j . . F
Ž2 q q. d Ž H Ž x k , k . , H Ž x k , . . 2Ž 1 y q . qd Ž H Ž x j , . , H Ž x j , j . . .
To see the last inequality suppose the maximum on the right hand side of the above displayed equation is 1 2
2 d Ž x k , x j . q d Ž H Ž x j , j . , H Ž x j , . . q d Ž H Ž x k , k . , H Ž x k , . . .
Then d Ž x i , x j . F d Ž H Ž x k , k . , H Ž x k , . . q
q 2
2 d Ž x k , x j . q d Ž H Ž x j , j . , H Ž x j , . . qd Ž H Ž x k , k . , H Ž x k , . .
q d Ž H Ž x j , . , H Ž x j , j . . and so dŽ xk , x j . F
Ž2 q q. d Ž H Ž x k , k . , H Ž x k , . . 2Ž 1 y q . qd Ž H Ž x j , . , H Ž x j , j . . .
409
FIXED POINT THEORY FOR CONTRACTIONS
The other cases are easier Žnote as well that - 2ŽŽ21qyqq.. .. Now dŽ xk , x j . F
1 1y q
-
Ž2 q q . 2 Ž1 y q .
and 1 q q
Ž2 q q. d Ž H Ž x k , k . , H Ž x k , . . 2Ž 1 y q . qd Ž H Ž x j , . , H Ž x j , j . .
and Žiii. guarantees that Ž x k . is a Cauchy sequence with respect to d. We now claim that
Ž x i . is a Cauchy sequence with respect to dX . X
Ž 3.1.
X
If d G d this is trivial. If d h d then dX Ž x k , x j . s dX Ž H Ž x k , k . , H Ž x j , j . . and Živ. guarantee that Ž3.1. is true Žnote as well that Ž x k . is a Cauchy sequence with respect to d and Ž k . is a Cauchy sequence in w0, 1x.. Now since Ž X, dX . is complete there exists an x g Q with dX Ž x k , x . ª 0 as k ª ⬁. We next claim that x s H Ž x, . . Ž 3.2. We consider first the case d / dX . Now dX Ž x, H Ž x, . . F dX Ž x, x k . q dX Ž x k , H Ž x, . . s dX Ž x, x k . q dX Ž H Ž x k , k . , H Ž x, . . together with Žv. guarantees that dX Ž x, H Ž x, .. s 0, so Ž3.2. holds. Next suppose that d s dX . Now d Ž x, H Ž x, . . F d Ž x, x k . q d Ž H Ž x k , k . , H Ž x, . . F d Ž x, x k . q d Ž H Ž x k , k . , H Ž x, k . . q d Ž H Ž x, k . , H Ž x, . . F d Ž x, x k . q q max d Ž x k , x . , d Ž x k , H Ž x k , k . , d Ž x, H Ž x, k . . , 1 2
d Ž x k , H Ž x, k . . q d Ž x, H Ž x k , k . .
q d Ž H Ž x, k . , H Ž x, . . F d Ž x, x k . q q max d Ž x k , x . , 0, d Ž x, H Ž x, . . qd Ž H Ž x, . , H Ž x, k . . , 1 2
d Ž x k , x . q d Ž x, H Ž x, . . qd Ž H Ž x, . , H Ž x, k . . q d Ž x, x k .
q d Ž H Ž x, k . , H Ž x, . . .
4
4
410
AGARWAL AND O’REGAN
Let k ª ⬁ and use Žiii. to obtain d Ž x, H Ž x, . . F qd Ž x, H Ž x, . . . That is, dŽ x, H Ž x, .. s 0, so Ž3.2. holds when d s dX . Now from Ž3.2. and Ži. we have x g U. Consequently g A so A is closed in w0, 1x. Next we show A is open in w0, 1x. Let 0 g A and x 0 g U with x 0 s H Ž x 0 , 0 .. Since U is d-open there exists a d-ball B Ž x 0 , ␦ . s x g X : dŽ x, x 0 . - ␦ 4 , ␦ ) 0, with B Ž x 0 , ␦ . : U. Now Žiii. guarantees that there exists s Ž ␦ . ) 0 with d Ž x 0 , H Ž x 0 , . . s d Ž H Ž x 0 , 0 . , H Ž x 0 , . . - Ž 1 y q . ␦ for g w 0, 1 x and < y 0 < F . Now Žii., Živ., and Žv. together with Theorem 2.1 Žin this case r s ␦ and X d F s H . guarantee that there exists x g B Ž x 0 , ␦ . : Q with x s HŽ x . for g w0, 1x and < y 0 < F . Consequently A is open in w0, 1x. We now discuss a special case of Theorem 3.1 which is particularly useful in the study of second order differential equations in Banach spaces Žsee w6, Chap. 2x.. Let I s w0, 1x and E s Ž E, < . <. be a real Banach space. C Ž I; E . is the space of continuous functions y : I ª E endowed with the norm 5 y 5 ⬁ s sup t g I < y Ž t .<. For every integer k G 1, C k Ž I; E . is the space of functions y : I ª E, such that for each j s 1, 2, . . . , k, y Ž j. exists and is continuous, endowed with the norm 5 y 5 k , ⬁ s max 5 y 5 ⬁ , 5 yX 5 ⬁ , . . . , 5 y Ž k . 5 ⬁ 4 . Also for 1 F p F ⬁, we let L p Ž I; E . be the Banach space of all measurable functions u : I ª E such that < u < p is Lebesgue integrable on I, with the norm 5 u5 p s
1
žH
uŽ t .
p
0
1rp
dt
/
if p - ⬁
and 5 u 5 ⬁ s inf M G 0 : u Ž t . F M for a.e. t g I 4 . For 1 F p F ⬁ we define the Sobolev space W m, p Ž I; E . inductively as follows. A function u belongs to W 1, p Ž I; E . if it is continuous and there exists a ¨ g L p Ž I; E . with uŽ t . s uŽ o . q
1
H0 ¨ Ž s . ds,
t g I;
411
FIXED POINT THEORY FOR CONTRACTIONS
here by H01 ¨ Ž s . ds we mean the Bochner integral. For any integer m ) 1, u g W m, p Ž I; E . if u, uX g W my1, p Ž I; E .. The space W m, p Ž I; E . is a Banach space with norm 5 u 5 m , p s max uŽ j.
p
: j s 0, 1, . . . , m4 .
THEOREM 3.3. Let R ) 0, 1 - p F ⬁, DR s u g C 1 Ž I; E . : 5 u 5 1, ⬁ F R4 , and T : DR ª W 2, p Ž I; E .. Assume the following conditions are satisfied: Ža. T Ž DR . is bounded in Ž C 1 Ž I; E ., 5 . 5 1, ⬁ . and there exists R 0 ) 0 with < yY Ž t .< F R 0 for a.e. t g I and any y g T Ž DR .; Žb. there exists a metric d on C 1 Ž I; E . equi¨ alent to the metric induced by 5 . 5 1, p , which satisfies d Ž u, ¨ . F c 0 5 u y ¨ 5 1 , p for all u, ¨ g C 1 Ž I; E . for some constant c 0 ) 0, such that 5 Tu y T¨ 5 1, ⬁ F c1 d Ž u, ¨ . for some constant c1 ) 0, and there exists q g Ž0, 1. with d Ž Tu, T¨ . F q max d Ž u, ¨ . , d Ž u, Tu . , d Ž ¨ , T¨ . , 1 2
d Ž u, T¨ . q d Ž ¨ , Tu .
4
for all u, ¨ g DR and g w0, 1x; Žc. if u g DR sol¨ es u s Tu for some g w0, 1x then 5 u 5 1, ⬁ - R. Then T has a fixed point in DR . Remark 3.2. In the proof of Theorem 3.3 we will use a standard result from the literature Žsee w6, Chap. 2x.. For convenience we state it here. Suppose g W 1, ⬁Ž I; E . is such that < Ž t .< G a ) 0 for some t g I and < X Ž s .< F M for a.e. s g I. Then 1
H0
Ž s . ds G min
½
a 3a2 , . 2 8M
5
Proof. Denote by dX the metric induced by 5 . 5 1, ⬁ on C 1 Ž I; E .. Let K s co Ž 0 4 j T Ž DR . . . Notice K : C 1 Ž I; E .. Denote by X the dX-closure of K in C 1 Ž I; E . and let Q s X l DR . Notice Q is dX-closed in X. Define H : Q = w0, 1x ª X by H Ž x, . s Tx and let U be the d-interior of Q in X. Notice condition Žii. of Theorem 3.1 follows immediately from Žb. since Q : DR . Also since
412
AGARWAL AND O’REGAN
T Ž DR . is bounded in Ž C 1 Ž I; E ., 5 . 5 1, ⬁ . there exists r 0 ) 0 Ždepending only on R . with H Ž u, . y H Ž ¨ , .
1, ⬁
F H Ž u, . y H Ž ¨ , .
1, ⬁
q H Ž ¨ , . y H Ž ¨ , .
1, ⬁
F 5 Tu y T¨ 5 1, ⬁ q r 0 < y < F c1 d Ž u, ¨ . q r 0 < y < for all u, ¨ g Q and , g w0, 1x. Thus condition Živ. of Theorem 3.1 is true. In addition 5 . 5 1, p F 5 . 5 1, ⬁ and dŽ u, ¨ . F c 0 5 u y ¨ 5 1, p , yields dX Ž H Ž u, . , H Ž ¨ , . . s H Ž u, . y H Ž ¨ , .
1, ⬁
F c1 d Ž u, ¨ . q r 0 < y <
F c1 c 0 5 u y ¨ 5 1, p q r 0 < y < F c1 c 0 5 u y ¨ 5 1, ⬁ q r 0 < y < s c1 c 0 dX Ž u, ¨ . q r 0 < y < , so condition Žv. of Theorem 3.1 is true. Also notice that Žb. together with 5 H Ž u, . y H Ž u, .5 1, ⬁ F r 0 < y < yields d Ž H Ž u, . , H Ž u, . . F c 0 H Ž u, . y H Ž u, .
1, p
F c 0 H Ž u, . y H Ž u, .
1, ⬁
F c0 r 0 < y < ,
so condition Žiii. of Theorem 3.1 holds. The result of the theorem will follow once we show condition Ži. of Theorem 3.1 holds. To show this we will show that ugQ
and
5 u 5 1, ⬁ - R implies u g U.
Ž 3.3.
Notice Ž3.3. together with condition Žc. guarantees that condition Ži. of Theorem 3.1 holds. It remains to show Ž3.3.. Let u g Q and 5 u 5 1, ⬁ - R. If we show there exists r ) 0 such that ¨ gX
and
5 ¨ y u 5 1, p - r implies ¨ g DR ,
Ž 3.4.
then of course Ž3.3. holds Ži.e., Ž3.4. implies ¨ g Q for all ¨ g X with 5 ¨ y u 5 1, p - r and now since dŽ u, ¨ . F c 0 5 ¨ y u 5 1, p for all u, ¨ g C 1 Ž I; E . we have u g U .. Suppose Ž3.4. is false. Then there exists a sequence Ž u k . : X with 5 u k y u 5 1, p - 1k and u k f DR . Note < u k Ž t .< ) R or < uXk Ž t .< ) R for some t g I. Let R⬁ s 5 u 5 1, ⬁ . Now R⬁ - R and < uŽ t .< F R⬁ ,
413
FIXED POINT THEORY FOR CONTRACTIONS
< uX Ž t .< F R⬁ for all t g I. Thus, for each k, there is at least one t such that either < u k Ž t . y uŽ t .< G < u k Ž t .< y < uŽ t .< G < u k Ž t .< y R⬁ ) R y R⬁ or < uXk Ž t . y uX Ž t .< G < uXk Ž t .< y < uX Ž t .< ) R y R⬁
Ž1. Ž2. holds.
Case Ži.. Suppose that < uXk Ž t . y uX Ž t .< F R y R⬁ for all t g I, and infinitely many values of k. Then, passing to a subsequence if necessary, we may assume that for each k we have uXk Ž t . y uX Ž t . F R y R⬁
for all t g I
and u k Ž t . y u Ž t . ) R y R⬁
for at least one t g I.
Then by Remark 3.2 we have 1
H0
u k Ž s . y u Ž s . ds G
3 Ž R y R⬁ . 8
)0
for all k.
This implies 5 u k y u 5 1, p ¢ 0 as k ª ⬁, a contradiction. Case Žii.. We assume the opposite to Case Ži., so we may suppose without loss of generality that for any k, we have < uXk Ž t . y uX Ž t .< ) R y R⬁ for at least one t g I. Let ⑀ ) 0 be chosen appropriately Žsee below.. Since u, u k g X there exists u, ˜ u˜k g K with u ˜Xk Ž t . y u˜X Ž t . ) R y R⬁
for at least one t g I,
with 1
uXk Ž s . y u ˜Xk Ž s . ds F
H0
⑀
and
2
1
H0
uX Ž s . y u ˜X Ž s . ds F
⑀ 2
Now since u, ˜ u˜k g K we have from condition Ža. that u ˜Yk Ž s . y u˜Y Ž s . F 2 R 0
for all s g I.
Then by Remark 3.2 we have 1
H0
R y R⬁ 3 Ž R y R⬁ . , ˜ Ž s . y u˜ Ž s . ds G min 2 16 R 0
uXk
X
½
2
5
' ) 0.
.
414
AGARWAL AND O’REGAN
Consequently
F
1
H0
uXk Ž s . y uX Ž s . ds q ⑀ F ⑀ q 5 u k y u 5 1, p .
Thus 5 u k y u 5 1, p G y ⑀ for all k. Choose ⑀ - and so 5 u k y u 5 1, p ¢ 0 as k ª ⬁, a contradiction.
REFERENCES 1. A. Granas, Continuation methods for contractive maps, Topol. Methods Nonlinear Anal. 3 Ž1994., 375᎐379. 2. G. E. Hardy and T. G. Rogers, A generalization of a fixed point theorem of Reich, Canad. Math. Bull. 16 Ž1973., 201᎐206. 3. R. Kannan, Some remarks on fixed points, Bull. Calcutta Math. Soc. 60 Ž1960., 71᎐76. 4. M. G. Maia, Un’obsservazione sulle contrazioni metriche, Rend. Sem. Mat. Uni¨ . Pado¨ a 40 Ž1968., 139᎐143. 5. D. O’Regan, Fixed point theorems for nonlinear operators, J. Math. Anal. Appl. 202 Ž1996., 413᎐432. 6. D. O’Regan and R. Precup, ‘‘Theorems of Leray᎐Schauder Type and Applications,’’ Gordon & Breach, New York, in press. 7. R. Precup, Discrete continuation method for boundary value problems on bounded sets in Banach spaces, J. Comput. Appl. Math. 113 Ž2000., 267᎐281. 8. S. Reich, Kannan’s fixed point theorem, Bull. Uni¨ . Mat. Italiana 4 Ž1971., 1᎐11.
Fixed Point Theory for Generalized Contractions on Spaces with Two Metrics Ravi P. Agarwal Department of Mathematics, National Uni¨ ersity of Singapore, 10 Kent Ridge Crescent, Singapore 199260
and Donal O’Regan Department of Mathematics, National Uni¨ ersity of Ireland, Galway, Ireland Submitted by William F. Ames Received April 11, 2000 DEDICATED TO THE MEMORY OF R. KANNAN
We present new fixed point results for generalized contractions on spaces with two metrics. In addition generalized contractive homotopies will also be discussed in detail. 䊚 2000 Academic Press
1. INTRODUCTION This paper presents fixed point theorems for generalized contractions on spaces with two metrics. The results are motivated from the study of differential equations in abstract spaces; see, for example, O’Regan and Precup w6, Chap. 2x. This paper will be divided into two main sections. Section 2 presents new local and global fixed point results for generalized contractions. The theorems in Section 2 extend results of Hardy and Rogers w2x, Kannan w3x, Precup w7x, and Reich w8x. Section 3 presents general continuation type theorems for generalized contractive homotopies on spaces with two metrics. The theorems in this section extend previous results of Granas w1x, O’Regan w5x, and Precup w7x. 402 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
403
FIXED POINT THEORY FOR CONTRACTIONS
2. FIXED POINT THEORY This section presents fixed point results for generalized contractions on spaces with two metrics. The theorems extend the fixed point results in w2᎐4, 7, 8x. Throughout this section Ž X, dX . will be a complete metric space and d will be another metric on X. If x 0 g X and r ) 0 let B Ž x 0 , r . s x g X : d Ž x, x 0 . - r 4 , and we let B Ž x 0 , r .
d
X
denote the dX-closure of B Ž x 0 , r ..
THEOREM 2.1. Let Ž X, dX . be a complete metric space, d another metric X on X, x 0 g X, r ) 0, and F : B Ž x 0 , r . d ª X. Suppose there exists q g Ž0, 1. such that for x, y g B Ž x 0 , r .
d
X
we ha¨ e
d Ž Fx , Fy . F q max d Ž x, y . , d Ž x, Fx . , d Ž y, Fy . , 1 2
d Ž x, Fy . q d Ž y, Fx .
4.
In addition assume the following three properties hold: d Ž x 0 , Fx 0 . - Ž 1 y q . r
Ž 2.1.
X
if d h d assume F is uniformly continuous from Ž B Ž x 0 , r . , d . into Ž X , dX . ,
Ž 2.2. and X
if d / dX assume F is continuous from B Ž x 0 , r . , dX into Ž X , dX . .
ž
d
/
Then F has a fixed point. That is, there exists x g B Ž x 0 , r .
d
X
with x s Fx.
Before we prove Theorem 2.1 it is worth stating the special case of Theorem 2.1 when d s dX . COROLLARY 2.2. Let Ž X, d . be a complete metric space, x 0 g X, r ) 0, d and F: B Ž x 0 , r . ª X. Suppose there exists q g Ž0, 1. such that for x, y d
g B Ž x 0 , r . we ha¨ e d Ž Fx , Fy . F q max d Ž x, y . , d Ž x, Fx . , d Ž y, Fy . , 1 2
d Ž x, Fy . q d Ž y, Fx .
and d Ž x 0 , Fx 0 . - Ž 1 y q . r . d
Then there exists x g B Ž x 0 , r . with x s Fx.
4
404
AGARWAL AND O’REGAN
Proof of Theorem 2.1. Let x 1 s Fx 0 . Now from Ž2.1. we have dŽ x 1 , x 0 . - Ž1 y q . r F r so x 1 g B Ž x 0 , r .. Next let x 2 s Fx 1 and note that d Ž x 1 , x 2 . s d Ž Fx 0 , Fx 1 . F q max d Ž x 0 , x 1 . , d Ž x 0 , Fx 0 . , d Ž x 1 , Fx 1 . , 1 2
d Ž x 0 , Fx 1 . q d Ž x 1 , Fx 0 .
4
F q max d Ž x 0 , x 1 . , d Ž x 0 , x 1 . , d Ž x 1 , x 2 . , 12 d Ž x 0 , x 2 . 4 F q max d Ž x 0 , x 1 . , d Ž x 1 , x 2 . ,
1 2
d Ž x 0 , x1 . q d Ž x1 , x 2 .
4
F qd Ž x 0 , x 1 . . To see the last inequality suppose the maximum on the right hand side of the above displayed equation is 12 w dŽ x 0 , x 1 . q dŽ x 1 , x 2 .x. Then dŽ x 1 , x 2 . F 2q w dŽ x 0 , x 1 . q dŽ x 1 , x 2 .x and so d Ž x1 , x 2 . F
q 2yq
d Ž x 0 , x 1 . F qd Ž x 0 , x 1 . .
The other cases are easier. Thus d Ž x 1 , x 2 . F qd Ž x 0 , x 1 . - q Ž 1 y q . r . Notice x 2 g B Ž x 0 , r . since d Ž x 0 , x 2 . F Ž 1 q q . d Ž x 0 , x1 . - Ž 1 q q . Ž 1 y q . r F Ž 1 y q . r 1 q q q q 2 q ⭈⭈⭈ s r . Proceed inductively to obtain x n s Fx ny1 , n s 3, 4, . . . with d Ž x nq 1 , x n . F qd Ž x n , x ny1 . F q nd Ž x 0 , x 1 . - q n Ž 1 y q . r and x nq 1 g B Ž x 0 , r .. Now since q g Ž0, 1. we have that Ž x n . is a Cauchy sequence with respect to d. We now claim that
Ž x n . is a Cauchy sequence with respect to dX .
Ž 2.4.
If d G dX this is trivial. Next suppose d h dX . Let ⑀ ) 0 be given. Now Ž2.2. guarantees that there exists ␦ ) 0 such that dX Ž Fx , Fy . - ⑀
whenever x, y g B Ž x 0 , r . and d Ž x, y . - ␦ . Ž 2.5.
405
FIXED POINT THEORY FOR CONTRACTIONS
From above we know that there exists N g 1, 2, . . . 4 with dŽ xn , xm . - ␦
whenever n, m G N.
Ž 2.6.
Now Ž2.5. and Ž2.6. imply dX Ž x nq 1 , x mq1 . s dX Ž Fx n , Fx m . - ⑀
whenever n, m G N,
and as a result Ž2.4. holds. Now since Ž X, dX . is complete there exists x X d g B Ž x 0 , r . with dX Ž x n , x . ª 0 as n ª ⬁. We claim that x s Fx.
Ž 2.7.
If Ž2.7. is true then we are finished. First consider the case when d / dX . Notice dX Ž x, Fx . F dX Ž x, x n . q dX Ž x n , Fx . s dX Ž x, x n . q dX Ž Fx ny1 , Fx . . Let n ª ⬁ and use Ž2.3. to obtain dX Ž x, Fx . s 0, so Ž2.7. is true in this case. Next suppose d s dX . Then d Ž x, Fx . F d Ž x, x n . q d Ž Fx ny1 , Fx . F d Ž x, x n . q q max d Ž x, x ny1 . , d Ž x, Fx . , d Ž x ny1 , Fx ny1 . , 1 2
d Ž x, Fx ny 1 . q d Ž x ny1 , Fx .
4
F d Ž x, x n . q q max d Ž x, x ny1 . , d Ž x, Fx . , d Ž x ny1 , x n . , 1 2
d Ž x, x n . q d Ž x ny1 , x . q d Ž x, Fx .
4.
Letting n ª ⬁ gives d Ž x, Fx . F qd Ž x, Fx . . That is, dŽ x, Fx . s 0, so Ž2.7. holds. The following global result can easily be deduced from Theorem 2.1. THEOREM 2.3. Let Ž X, dX . be a complete metric space, d another metric on X, and F : X ª X. Suppose there exists q g Ž0, 1. such that for x, y g X we ha¨ e d Ž Fx , Fy . F q max d Ž x, y . , d Ž x, Fx . , d Ž y, Fy . , 1 2
d Ž x, Fy . q d Ž y, Fx .
4.
406
AGARWAL AND O’REGAN
In addition assume the following two properties hold: if d h dX assume F is uniformly continuous from Ž X , d . into Ž X , dX . Ž 2.8. and if d / dX assume F is continuous from Ž X , dX . into Ž X , dX . .
Ž 2.9.
Then F has a fixed point. Proof. Fix x 0 g X. Choose r ) 0 so that d Ž x 0 , Fx 0 . - Ž 1 y q . r . d
X
Now Theorem 2.1 guarantees that there exists x g B Ž x 0 , r . with x s Fx. Theorem 2.3 immediately yields the following result of Hardy and Rogers w2x. COROLLARY 2.4. Let Ž X, d . be a complete metric space and F : X ª X. Suppose there exists q g Ž0, 1. such that for x, y g X we ha¨ e d Ž Fx , Fy . F q max d Ž x, y . , d Ž x, Fx . , d Ž y, Fy . , 1 2
d Ž x, Fy . q d Ž y, Fx .
4.
Then F has a fixed point.
3. HOMOTOPY RESULTS In this section we present homotopy results for the maps discussed in Section 2. Our theorems improve the results in w1, 5, 7x. THEOREM 3.1. Let Ž X, dX . be a complete metric space and let d be another metric on X. Let Q : X be dX-closed and let U : X be d-open and U : Q. Suppose H : Q = w0, 1x ª X satisfies the following fi¨ e properties: Ži. x / H Ž x, . for x g Q_U and g w0, 1x; Žii. there exists q g Ž0, 1. such that for all g w0, 1x and x, y g Q we ha¨ e dŽ H Ž x, ., H Ž y, .. F q max dŽ x, y ., dŽ x, H Ž x, .., dŽ y, H Ž y, .., 1 w Ž Ž .. Ž Ž ..x4; 2 d x, H y, q d y, H x, Žiii. H Ž x, . is continuous in with respect to d, uniformly for x g Q; Živ. if d h dX assume H is uniformly continuous from U = w0, 1x endowed with the metric d on U into Ž X, dX .; and Žv. if d / dX assume H is continuous from Q = w0, 1x endowed with the metric dX on Q into Ž X, dX .. In addition assume H0 has a fixed point. Then for each g w0, 1x we ha¨ e that H has a fixed point x g U Ž here HŽ . . s H Ž . , ...
FIXED POINT THEORY FOR CONTRACTIONS
407
Before we prove Theorem 2.1 we state the special case of Theorem 3.1 when d s dX . COROLLARY 3.2. Let Ž X, d . be a complete metric space. Let Q : X be d-closed and let U : X be d-open and U : Q. Suppose H : Q = w0, 1x ª X satisfies the following three properties: Ži. x / H Ž x, . for x g Q_U and g w0, 1x; Žii. there exists q g Ž0, 1. such that for all g w0, 1x and x, y g Q we ha¨ e dŽ H Ž x, ., H Ž y, .. F q max dŽ x, y ., dŽ x, H Ž x, .., dŽ y, H Ž y, .., 1 w Ž Ž .. Ž Ž ..x4; and 2 d x, H y, q d y, H x, Žiii. H Ž x, . is continuous in with respect to d, uniformly for x g Q. In addition assume H0 has a fixed point. Then for each g w0, 1x we ha¨ e that H has a fixed point x g U Ž here HŽ . . s H Ž . , ... Remark 3.1. Usually in Corollary 3.2 we have Q s U d Žthe d-closure of U in X .. Notice in this case that Ži. becomes x / H Ž x, . for all x g ⭸ U Žthe boundary of U in X . and g w0, 1x. Proof of Theorem 3.1. Let A s g w 0, 1 x : H Ž x, . s x for some x g U 4 . Now since H0 has a fixed point Žand Ži. holds. we have that 0 g A, so A is nonempty. We will show A is both closed and open in w0, 1x, and so by the connectedness of w0, 1x we are finished since A s w0, 1x. First we show A is closed in w0, 1x. Let Ž k . be a sequence in A with k ª g w0, 1x as k ª ⬁. By definition for each k, there exists x k g U with x k s H Ž x k , k .. Now we have d Ž x k , x j . s d Ž H Ž x k , k . , H Ž x j , j . . F d Ž H Ž x k , k . , H Ž x k , . . q d Ž H Ž x k , . , H Ž x j , . . q d Ž H Ž x j , . , H Ž x j , j . . F d Ž H Ž x k , k . , H Ž x k , . . q q max d Ž x k , x j . , d x k , H Ž x k , . , d Ž x j , H Ž x j , . . ,
½
ž
1 2
d Ž x k , H Ž x j , . . q d Ž x j , H Ž x k , . .
5
408
AGARWAL AND O’REGAN
q d Ž H Ž x j , . , H Ž x j , j . . F d Ž H Ž x k , k . , H Ž x k , . . q q max d Ž x k , x j . , d Ž H Ž x k , k . , H Ž x k , . . ,
½
d Ž H Ž x j , j . , H Ž x j , . . , 1 2
2 d Ž x k , x j . q d Ž H Ž x j , j . , H Ž x j , . . qd Ž H Ž x k , k . , H Ž x k , . .
5
q d Ž H Ž x j , . , H Ž x j , j . . F
Ž2 q q. d Ž H Ž x k , k . , H Ž x k , . . 2Ž 1 y q . qd Ž H Ž x j , . , H Ž x j , j . . .
To see the last inequality suppose the maximum on the right hand side of the above displayed equation is 1 2
2 d Ž x k , x j . q d Ž H Ž x j , j . , H Ž x j , . . q d Ž H Ž x k , k . , H Ž x k , . . .
Then d Ž x i , x j . F d Ž H Ž x k , k . , H Ž x k , . . q
q 2
2 d Ž x k , x j . q d Ž H Ž x j , j . , H Ž x j , . . qd Ž H Ž x k , k . , H Ž x k , . .
q d Ž H Ž x j , . , H Ž x j , j . . and so dŽ xk , x j . F
Ž2 q q. d Ž H Ž x k , k . , H Ž x k , . . 2Ž 1 y q . qd Ž H Ž x j , . , H Ž x j , j . . .
409
FIXED POINT THEORY FOR CONTRACTIONS
The other cases are easier Žnote as well that - 2ŽŽ21qyqq.. .. Now dŽ xk , x j . F
1 1y q
-
Ž2 q q . 2 Ž1 y q .
and 1 q q
Ž2 q q. d Ž H Ž x k , k . , H Ž x k , . . 2Ž 1 y q . qd Ž H Ž x j , . , H Ž x j , j . .
and Žiii. guarantees that Ž x k . is a Cauchy sequence with respect to d. We now claim that
Ž x i . is a Cauchy sequence with respect to dX . X
Ž 3.1.
X
If d G d this is trivial. If d h d then dX Ž x k , x j . s dX Ž H Ž x k , k . , H Ž x j , j . . and Živ. guarantee that Ž3.1. is true Žnote as well that Ž x k . is a Cauchy sequence with respect to d and Ž k . is a Cauchy sequence in w0, 1x.. Now since Ž X, dX . is complete there exists an x g Q with dX Ž x k , x . ª 0 as k ª ⬁. We next claim that x s H Ž x, . . Ž 3.2. We consider first the case d / dX . Now dX Ž x, H Ž x, . . F dX Ž x, x k . q dX Ž x k , H Ž x, . . s dX Ž x, x k . q dX Ž H Ž x k , k . , H Ž x, . . together with Žv. guarantees that dX Ž x, H Ž x, .. s 0, so Ž3.2. holds. Next suppose that d s dX . Now d Ž x, H Ž x, . . F d Ž x, x k . q d Ž H Ž x k , k . , H Ž x, . . F d Ž x, x k . q d Ž H Ž x k , k . , H Ž x, k . . q d Ž H Ž x, k . , H Ž x, . . F d Ž x, x k . q q max d Ž x k , x . , d Ž x k , H Ž x k , k . , d Ž x, H Ž x, k . . , 1 2
d Ž x k , H Ž x, k . . q d Ž x, H Ž x k , k . .
q d Ž H Ž x, k . , H Ž x, . . F d Ž x, x k . q q max d Ž x k , x . , 0, d Ž x, H Ž x, . . qd Ž H Ž x, . , H Ž x, k . . , 1 2
d Ž x k , x . q d Ž x, H Ž x, . . qd Ž H Ž x, . , H Ž x, k . . q d Ž x, x k .
q d Ž H Ž x, k . , H Ž x, . . .
4
4
410
AGARWAL AND O’REGAN
Let k ª ⬁ and use Žiii. to obtain d Ž x, H Ž x, . . F qd Ž x, H Ž x, . . . That is, dŽ x, H Ž x, .. s 0, so Ž3.2. holds when d s dX . Now from Ž3.2. and Ži. we have x g U. Consequently g A so A is closed in w0, 1x. Next we show A is open in w0, 1x. Let 0 g A and x 0 g U with x 0 s H Ž x 0 , 0 .. Since U is d-open there exists a d-ball B Ž x 0 , ␦ . s x g X : dŽ x, x 0 . - ␦ 4 , ␦ ) 0, with B Ž x 0 , ␦ . : U. Now Žiii. guarantees that there exists s Ž ␦ . ) 0 with d Ž x 0 , H Ž x 0 , . . s d Ž H Ž x 0 , 0 . , H Ž x 0 , . . - Ž 1 y q . ␦ for g w 0, 1 x and < y 0 < F . Now Žii., Živ., and Žv. together with Theorem 2.1 Žin this case r s ␦ and X d F s H . guarantee that there exists x g B Ž x 0 , ␦ . : Q with x s HŽ x . for g w0, 1x and < y 0 < F . Consequently A is open in w0, 1x. We now discuss a special case of Theorem 3.1 which is particularly useful in the study of second order differential equations in Banach spaces Žsee w6, Chap. 2x.. Let I s w0, 1x and E s Ž E, < . <. be a real Banach space. C Ž I; E . is the space of continuous functions y : I ª E endowed with the norm 5 y 5 ⬁ s sup t g I < y Ž t .<. For every integer k G 1, C k Ž I; E . is the space of functions y : I ª E, such that for each j s 1, 2, . . . , k, y Ž j. exists and is continuous, endowed with the norm 5 y 5 k , ⬁ s max 5 y 5 ⬁ , 5 yX 5 ⬁ , . . . , 5 y Ž k . 5 ⬁ 4 . Also for 1 F p F ⬁, we let L p Ž I; E . be the Banach space of all measurable functions u : I ª E such that < u < p is Lebesgue integrable on I, with the norm 5 u5 p s
1
žH
uŽ t .
p
0
1rp
dt
/
if p - ⬁
and 5 u 5 ⬁ s inf M G 0 : u Ž t . F M for a.e. t g I 4 . For 1 F p F ⬁ we define the Sobolev space W m, p Ž I; E . inductively as follows. A function u belongs to W 1, p Ž I; E . if it is continuous and there exists a ¨ g L p Ž I; E . with uŽ t . s uŽ o . q
1
H0 ¨ Ž s . ds,
t g I;
411
FIXED POINT THEORY FOR CONTRACTIONS
here by H01 ¨ Ž s . ds we mean the Bochner integral. For any integer m ) 1, u g W m, p Ž I; E . if u, uX g W my1, p Ž I; E .. The space W m, p Ž I; E . is a Banach space with norm 5 u 5 m , p s max uŽ j.
p
: j s 0, 1, . . . , m4 .
THEOREM 3.3. Let R ) 0, 1 - p F ⬁, DR s u g C 1 Ž I; E . : 5 u 5 1, ⬁ F R4 , and T : DR ª W 2, p Ž I; E .. Assume the following conditions are satisfied: Ža. T Ž DR . is bounded in Ž C 1 Ž I; E ., 5 . 5 1, ⬁ . and there exists R 0 ) 0 with < yY Ž t .< F R 0 for a.e. t g I and any y g T Ž DR .; Žb. there exists a metric d on C 1 Ž I; E . equi¨ alent to the metric induced by 5 . 5 1, p , which satisfies d Ž u, ¨ . F c 0 5 u y ¨ 5 1 , p for all u, ¨ g C 1 Ž I; E . for some constant c 0 ) 0, such that 5 Tu y T¨ 5 1, ⬁ F c1 d Ž u, ¨ . for some constant c1 ) 0, and there exists q g Ž0, 1. with d Ž Tu, T¨ . F q max d Ž u, ¨ . , d Ž u, Tu . , d Ž ¨ , T¨ . , 1 2
d Ž u, T¨ . q d Ž ¨ , Tu .
4
for all u, ¨ g DR and g w0, 1x; Žc. if u g DR sol¨ es u s Tu for some g w0, 1x then 5 u 5 1, ⬁ - R. Then T has a fixed point in DR . Remark 3.2. In the proof of Theorem 3.3 we will use a standard result from the literature Žsee w6, Chap. 2x.. For convenience we state it here. Suppose g W 1, ⬁Ž I; E . is such that < Ž t .< G a ) 0 for some t g I and < X Ž s .< F M for a.e. s g I. Then 1
H0
Ž s . ds G min
½
a 3a2 , . 2 8M
5
Proof. Denote by dX the metric induced by 5 . 5 1, ⬁ on C 1 Ž I; E .. Let K s co Ž 0 4 j T Ž DR . . . Notice K : C 1 Ž I; E .. Denote by X the dX-closure of K in C 1 Ž I; E . and let Q s X l DR . Notice Q is dX-closed in X. Define H : Q = w0, 1x ª X by H Ž x, . s Tx and let U be the d-interior of Q in X. Notice condition Žii. of Theorem 3.1 follows immediately from Žb. since Q : DR . Also since
412
AGARWAL AND O’REGAN
T Ž DR . is bounded in Ž C 1 Ž I; E ., 5 . 5 1, ⬁ . there exists r 0 ) 0 Ždepending only on R . with H Ž u, . y H Ž ¨ , .
1, ⬁
F H Ž u, . y H Ž ¨ , .
1, ⬁
q H Ž ¨ , . y H Ž ¨ , .
1, ⬁
F 5 Tu y T¨ 5 1, ⬁ q r 0 < y < F c1 d Ž u, ¨ . q r 0 < y < for all u, ¨ g Q and , g w0, 1x. Thus condition Živ. of Theorem 3.1 is true. In addition 5 . 5 1, p F 5 . 5 1, ⬁ and dŽ u, ¨ . F c 0 5 u y ¨ 5 1, p , yields dX Ž H Ž u, . , H Ž ¨ , . . s H Ž u, . y H Ž ¨ , .
1, ⬁
F c1 d Ž u, ¨ . q r 0 < y <
F c1 c 0 5 u y ¨ 5 1, p q r 0 < y < F c1 c 0 5 u y ¨ 5 1, ⬁ q r 0 < y < s c1 c 0 dX Ž u, ¨ . q r 0 < y < , so condition Žv. of Theorem 3.1 is true. Also notice that Žb. together with 5 H Ž u, . y H Ž u, .5 1, ⬁ F r 0 < y < yields d Ž H Ž u, . , H Ž u, . . F c 0 H Ž u, . y H Ž u, .
1, p
F c 0 H Ž u, . y H Ž u, .
1, ⬁
F c0 r 0 < y < ,
so condition Žiii. of Theorem 3.1 holds. The result of the theorem will follow once we show condition Ži. of Theorem 3.1 holds. To show this we will show that ugQ
and
5 u 5 1, ⬁ - R implies u g U.
Ž 3.3.
Notice Ž3.3. together with condition Žc. guarantees that condition Ži. of Theorem 3.1 holds. It remains to show Ž3.3.. Let u g Q and 5 u 5 1, ⬁ - R. If we show there exists r ) 0 such that ¨ gX
and
5 ¨ y u 5 1, p - r implies ¨ g DR ,
Ž 3.4.
then of course Ž3.3. holds Ži.e., Ž3.4. implies ¨ g Q for all ¨ g X with 5 ¨ y u 5 1, p - r and now since dŽ u, ¨ . F c 0 5 ¨ y u 5 1, p for all u, ¨ g C 1 Ž I; E . we have u g U .. Suppose Ž3.4. is false. Then there exists a sequence Ž u k . : X with 5 u k y u 5 1, p - 1k and u k f DR . Note < u k Ž t .< ) R or < uXk Ž t .< ) R for some t g I. Let R⬁ s 5 u 5 1, ⬁ . Now R⬁ - R and < uŽ t .< F R⬁ ,
413
FIXED POINT THEORY FOR CONTRACTIONS
< uX Ž t .< F R⬁ for all t g I. Thus, for each k, there is at least one t such that either < u k Ž t . y uŽ t .< G < u k Ž t .< y < uŽ t .< G < u k Ž t .< y R⬁ ) R y R⬁ or < uXk Ž t . y uX Ž t .< G < uXk Ž t .< y < uX Ž t .< ) R y R⬁
Ž1. Ž2. holds.
Case Ži.. Suppose that < uXk Ž t . y uX Ž t .< F R y R⬁ for all t g I, and infinitely many values of k. Then, passing to a subsequence if necessary, we may assume that for each k we have uXk Ž t . y uX Ž t . F R y R⬁
for all t g I
and u k Ž t . y u Ž t . ) R y R⬁
for at least one t g I.
Then by Remark 3.2 we have 1
H0
u k Ž s . y u Ž s . ds G
3 Ž R y R⬁ . 8
)0
for all k.
This implies 5 u k y u 5 1, p ¢ 0 as k ª ⬁, a contradiction. Case Žii.. We assume the opposite to Case Ži., so we may suppose without loss of generality that for any k, we have < uXk Ž t . y uX Ž t .< ) R y R⬁ for at least one t g I. Let ⑀ ) 0 be chosen appropriately Žsee below.. Since u, u k g X there exists u, ˜ u˜k g K with u ˜Xk Ž t . y u˜X Ž t . ) R y R⬁
for at least one t g I,
with 1
uXk Ž s . y u ˜Xk Ž s . ds F
H0
⑀
and
2
1
H0
uX Ž s . y u ˜X Ž s . ds F
⑀ 2
Now since u, ˜ u˜k g K we have from condition Ža. that u ˜Yk Ž s . y u˜Y Ž s . F 2 R 0
for all s g I.
Then by Remark 3.2 we have 1
H0
R y R⬁ 3 Ž R y R⬁ . , ˜ Ž s . y u˜ Ž s . ds G min 2 16 R 0
uXk
X
½
2
5
' ) 0.
.
414
AGARWAL AND O’REGAN
Consequently
F
1
H0
uXk Ž s . y uX Ž s . ds q ⑀ F ⑀ q 5 u k y u 5 1, p .
Thus 5 u k y u 5 1, p G y ⑀ for all k. Choose ⑀ - and so 5 u k y u 5 1, p ¢ 0 as k ª ⬁, a contradiction.
REFERENCES 1. A. Granas, Continuation methods for contractive maps, Topol. Methods Nonlinear Anal. 3 Ž1994., 375᎐379. 2. G. E. Hardy and T. G. Rogers, A generalization of a fixed point theorem of Reich, Canad. Math. Bull. 16 Ž1973., 201᎐206. 3. R. Kannan, Some remarks on fixed points, Bull. Calcutta Math. Soc. 60 Ž1960., 71᎐76. 4. M. G. Maia, Un’obsservazione sulle contrazioni metriche, Rend. Sem. Mat. Uni¨ . Pado¨ a 40 Ž1968., 139᎐143. 5. D. O’Regan, Fixed point theorems for nonlinear operators, J. Math. Anal. Appl. 202 Ž1996., 413᎐432. 6. D. O’Regan and R. Precup, ‘‘Theorems of Leray᎐Schauder Type and Applications,’’ Gordon & Breach, New York, in press. 7. R. Precup, Discrete continuation method for boundary value problems on bounded sets in Banach spaces, J. Comput. Appl. Math. 113 Ž2000., 267᎐281. 8. S. Reich, Kannan’s fixed point theorem, Bull. Uni¨ . Mat. Italiana 4 Ž1971., 1᎐11.