Báo cáo hóa học: "Research Article Some Extensions of Banach’s Contraction Principle in Complete Cone Metric Spaces" ppt

We generalize some definitions such as c-nonexpansive and c, λ-uniformly locally contractive functions f-closure, c-isometric in cone metric spaces, and certain fixed point theorems will

Volume 2008, Article ID 768294, 11 pages

doi:10.1155/2008/768294

Research Article

Some Extensions of Banach’s Contraction Principle

in Complete Cone Metric Spaces

P Raja and S M Vaezpour

Department of Mathematics and Computer Sciences, Amirkabir University of Technology,

P.O Box 15914, Hafez Avenue, Tehran, Iran

Correspondence should be addressed to S M Vaezpour,vaez@aut.ac.ir

Received 10 December 2007; Revised 2 June 2008; Accepted 23 June 2008

Recommended by Billy Rhoades

In this paper we consider complete cone metric spaces We generalize some definitions such as

c-nonexpansive and c, λ-uniformly locally contractive functions f-closure, c-isometric in cone

metric spaces, and certain fixed point theorems will be proved in those spaces Among other results, we prove some interesting applications for the fixed point theorems in cone metric spaces Copyrightq 2008 P Raja and S M Vaezpour 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

1 Introduction

The study of fixed points of functions satisfying certain contractive conditions has been at the center of vigorous research activity, for example see 1 5 and it has a wide range of applications in different areas such as nonlinear and adaptive control systems, parameterize estimation problems, fractal image decoding, computing magnetostatic fields in a nonlinear medium, and convergence of recurrent networks, see 6 10 Recently, Huang and Zhang generalized the concept of a metric space, replacing the set of real numbers by an ordered Banach space and obtained some fixed point theorems for mapping satisfying different contractive conditions 11 The study of fixed point theorems in such spaces is followed

by some other mathematicians, see 12–15 The aim of this paper is to generalize some

definitions such as c-nonexpansive and c, λ-uniformly locally contractive functions in these

spaces and by using these definitions, certain fixed point theorems will be proved

Let E be a real Banach space A subset P of E is called a cone if and only if the following

hold:

i P is closed, nonempty, and P / {0},

ii a, b ∈ R, a, b  0, and x, y ∈ P imply that ax  by ∈ P,

iii x ∈ P and −x ∈ P imply that x  0.

Given a cone P ⊂ E, we define a partial ordering  with respect to P by x  y if and only if y − x ∈ P We will write x < y to indicate that x  y but x / y, while x  y will stand for y − x ∈ int P, where int P denotes the interior of P.

The cone P is called normal if there is a number K > 0 such that 0  x  y implies

||x||  K||y||, for every x, y ∈ E The least positive number satisfying above is called the normal constant of P

There are non-normal cones

R0, 1 with the norm ||f||  ||f||∞ ||f||∞, and consider the cone

P  {f ∈ E : f  0} For each K  1, put fx  x and gx  x 2K Then, 0 g  f, ||f||  2,

and||g||  2K  1 Since K||f|| < ||g||, K is not normal constant of P 16

In the following, we always suppose E is a real Banach space, P is a cone in E with int P /  ∅, and  is partial ordering with respect to P.

Let X be a nonempty set As it has been defined in11, a function d : X × X → E is called a cone metric on X if it satisfies the following conditions:

i dx, y  0, for every x, y ∈ X, and dx, y  0 if and only if x  y,

ii dx, y  dy, x, for every x, y ∈ X,

iii dx, y  dx, z  dy, z, for every x, y, z ∈ X.

ThenX, d is called a cone metric space.

the normal constant of P is equal to 116

The sequence{x n } in X is called to be convergent to x ∈ X if for every c ∈ E with 0  c, there is n0 ∈ N such that dx n , x   c, for every n  n0, and is called a Cauchy sequence if

for every c ∈ E with 0  c, there is n0 ∈ N such that dx m , x n   c, for every m, n  n0 A cone metric spaceX, d is said to be a complete cone metric space if every Cauchy sequence

in X is convergent to a point of X A self-mapT on X is said to be continuous if lim n→∞x n  x

implies that limn→∞T x n   Tx, for every sequence {x n } in X The following lemmas are

useful for us to prove our main results

Lemma 1.3 see 11, Lemma 1 Let X, d be a cone metric space, P be a normal cone with normal

Lemma 1.4 see 11, Lemma 3 Let X, d be a cone metric space, {xn } be a sequence in X If {x n}

is convergent, then it is a Cauchy sequence, too.

Lemma 1.5 see 11, Lemma 4 Let X, d be a cone metric space, P be a normal cone with

limm,n→∞d x m , x n   0.

The following example is a cone metric space, see11

d x, y  |x − y|, α|x − y|, where α  0 is a constant Then X, d is a cone metric space.

2 Certain nonexpansive mappings

Then f is said to be c-nonexpansive, for 0  c, if

for every x, y ∈ X with dx, y  c If we have

for every x, y ∈ X with x / y and dx, y  c, then f is called c-contractive.

to belong to the f-closure of Y and is denoted by y ∈ Y f , if fY ⊆ Y and there are a point

x ∈ Y and an increasing sequence {n i} ⊆ N such that limi→∞f n i x  y.

said to be a c-isometric sequence if



 dx m k , x n k

for all k, m ∈ N with dx m , x n  < c A point x ∈ X is said to generate a c-isometric sequence under the function f : X → X, if {f n x} is a c-isometric sequence.

Theorem 2.4 Let X, d be a cone metric space, where P is a normal cone with normal constant K.

{m j } ⊆ N such that lim j→∞f m j x  x.

f m y  x, for some m ∈ N Put m j  n j − mn j > m , is a sequence as desired, then {m j} is

a sequence with desired property Otherwise, for  > 0, fix δ, 0 < δ <  Choose c ∈ E with

0 c and K||c|| < δ Then there is i  ic such that

for every j ∈ N ∪ {0} So by c-nonexpansivity of f and putting j  0, we have

f n i k −n i x, f n i k y< c

for every k∈ N Therefore,

f n i y, f n i k y dx, f n i y dx, f n i k y c

for every k∈ N Hence,

x, f n i1−n i x dx, f n i y df n i y, f n i1y df n i1y, f n i1−n i x< c

4 c

2 c

4  c,

2.7

which implies

d

Put m1  n i1− n i and suppose that m1< m2< · · · < m j−1chosen such that

d

x, f m i y  1

2m 1, ,mmini−1

d

for i  2, 3, , j −1 We put m j  n l1−n l , where l is chosen so as to satisfy dx, f l j y  c/4, with δ replaced by

min



2m 1, ,mmini−1

d

It is easily seen that the sequence{m j} that is defined in the above satisfies the requirements

of the theorem The proof is complete

Theorem 2.5 Let X, d be a cone metric space, where P is a normal cone with normal constant K.

p  df m x, f n x − df m k x, f n k x / 0 By the assumption, p ∈ P and

0 < p  df m x, f n x− df m l x, f n l x, 2.11

for l  k, l ∈ N It means that

d

f m x, f n x− df m k x, f n k x, 2.12

for l  k, l ∈ N Also by the assumption andTheorem 2.4,

lim

j→∞f n j

f l x lim

i∈ N such that

f m n j x, f m x c



f n n j x, f n x c

for every j  i However,

f m x, f n x df m x, f m n j x df m n j x, f n n j x

 df n n j x, f n x c  df m n j x, f n n j xc

2.

2.15 So

f m x, f n x− df m n j x, f n n j x c. 2.16

It means that

d

f m x, f n x− df m n j x, f n n j x

that is a contradiction by 2.12, for n j  max{n i , k } Therefore, p  0 and the proof is

complete

The following corollary implies immediately

Corollary 2.6 Let X, d be a cone metric space, where P is a normal cone with normal constant K.

3 Extended contraction principle

We have the following generalized form of Banach’s contraction for cone metric spaces

Theorem 3.1 see 11, Theorem 1 Let X, d be a complete cone metric space, P be a normal cone

x ∈ X, the sequence {T n x} converges to the fixed point.

It is natural to ask whether the mentioned theorem could be modified if3.1 holds for just sufficiently close points To be more specific, we introduce the following definitions

contractive, if for every x ∈ X there is c ∈ X with 0  c and 0  λ < 1 such that

for every p, q ∈ {y ∈ X : dx, y  c} A function f : X → X is said to be c, λ-uniformly locally contractive if it is locally contractive and both c and λ do not depend on x.

It is easy to find cone metric spaces which admit locally contractive which are not globally contractive



x, y | x  cos t, y  sin t; 0  t  3π

2



and P  {x, y ∈ E | x, y  0}, and d : X × X → E such that dx, y  |x − y|, α|x − y|, where α  0 is a constant It is easily checked that X, d is a cone metric space Suppose that

f cos t, sin t  cost/2, sint/2 It is not hard to see that f is locally contractive but not

globally contractive

Note that every locally contractive function is c-nonexpansive for some c 0

there is a finite set of points a  x0, x1, , x n  b, n depends on both a and b, such that

d x i−1, x i  < c, for i, 1  i  n.

c-chainable

Theorem 3.6 Let X, d be a complete c-chainable cone metric space, P be a normal cone with normal

i1

x i−1, x i



We have

x i−1

 βdx i−1, x i

for every 1 i  n, and by induction

x i−1

x i



x i−1

x i



< · · · < β m c, 3.1

for every m∈ N Hence

f m x, f m1xn

i1

x i−1

for every m ∈ N Now, for m, p ∈ N with m < p, we have

f m x, f p x

p−1



i m

f i x, f i1x< nc

β m  · · ·  β p−1

m

1− β . 3.3

It means that

d

for m, p ∈ N with m < p Since k ∈ 0, 1, then lim m,p→∞ m x, f p

limm,p→∞d f m x, f p x  0, and by Lemma 1.5, {f m x} is a Cauchy sequence Since X

is complete, then limm→∞f m x  z, for some z ∈ X From the continuity of f it follows that

f z  z To complete the proof it is enough to show that z is the unique point with this property To do this, suppose that there is z∈ X such that fz  z Let z x0, x1, , x t  z

be a c-chain By3.1, we obtain

f z, fz

 df l z, f l

z

t

i1

f l

x i−1

, f l

x i



It means that

d

Corollary 3.7 Let X, d be a complete c-chainable cone metric space, P be a normal cone with normal

then f has a unique fixed point.

Proof It is an immediate consequence of the fact that for the inverse function all assumptions

of theTheorem 3.6are satisfied

In the following theorem we investigate a kind of functions which are not necessarily contractions but have a unique fixed point First, we will prove the following lemma which will be used later

Lemma 3.8 Let X, d be a complete cone metric space, P be a normal cone with normal constant K,

such that

there is s ∈ N ∪ {0} such that snx < n  s  1nx and we have

f n x, x df n x

f n −nx x, f n x x df n x x, x

 βdf n −nx x, x df n x x, x

 df n x x, x βd

f n −nx x, f n x x df n x x, x

 df n x x, x ββd

f n −2nx x, x df n x x, x

 · · ·  df n x x, x1 β  β2 · · ·  β s

.

3.8

It means that

d

f n x, x  K 1

1− βd

f n x x, x  K 1

Hence rx is finite and the proof is complete.

Theorem 3.9 Let X, d be a complete cone metric space, P be a normal cone with normal constant

K, β ∈ 0, 1, and f : X → X be a continuous function such that for every x ∈ X, there is an

n x ∈ N such that

f m i x i , where m i  nx i  We show that {x n} is a Cauchy sequence We have

x n1, x n

 df m n−1

, f m n−1

 βdf m n

x n−1

, x n−1

 · · ·  β n d

x0

, x0

,

3.11

for every n∈ N So byLemma 3.8, n1, x n n r x0, for every n ∈ N Now, suppose that m, n ∈ N with m < n, we have

d

x n , x m   K n−1

i m

d

x i1, x i   K β n

1− β r



Since limn→∞β n / 1 − β  0, then lim m,n→∞ n , x m Lemma 1.5,{x n} is a

Cauchy sequence Completeness of X implies that lim n→∞x n  u, for some u ∈ X Now, we

show that f u  u By contradiction, suppose that fu / u We claim that there are c, d ∈ E

such that 0 c, 0  d and B c u and B d fuhave no intersection, where B e x  {y ∈ X :

d x, y  e}, for every x ∈ X and 0  e If not, then suppose that  > 0, and choose c ∈ E

It means that

f u  u, a contradiction Therefore, assume that c, d ∈ E with 0  c, 0  d are such that

B c u ∩ B d fu  ∅ Since f is continuous, then there is n0 ∈ N such that x n ∈ B c u and

f x n  ∈ B d fu, for every n ∈ N and n  n0 Then

, x n

 df m n−1

, f m n−1

≤ βdf

x n−1

, x n−1

 · · ·  β n d

, x0

,

3.14

limn→∞d fx n , x n   0, a contradiction Thus fu  u The uniqueness of the fixed point

follows immediately from the hypothesis

Now, suppose that x0∈ X is arbitrary To show that lim n→∞f n x0  u, set

r0 maxd

x0

, u: m  0, 1, , nu − 1

If n is sufficiently large, then n  rnu  q, for r > 0 and 0  q < nu, and we have

 df rn uq

, f n u u βdf r−1nuq

≤ · · ·  β r d

.

3.16

It means that

d

, u   Kβ rd

function if for every x, y ∈ X, x  y, implies that ϕx  ϕy, ϕx  x, and

limn→∞||ϕ n x||  0, for every x ∈ X.

with ϕx, y  ax, ay, for some a ∈ 0, 1 is a comparison function Also if ϕ1, ϕ2are two comparison functions overR, then ϕx, y  ϕ1x, ϕ2y is also a comparison function over E.

Recall that for a cone metric spaceX, d, where P is a cone with normal constant K, since for every x

Theorem 3.12 Let X, d be a complete cone metric space, where P is a normal cone with normal

such that

Proof Let x0 ∈ X be arbitrary We have

, f n1

 ϕd

f n−1

 ϕ2

, f n−1

 · · ·  ϕ n

,

3.19

for every n ∈ N Since limn→∞ n dx0, f x0

n∈ N such that

d

x0

, f n1

for every n ≥ n0and c ∈ P with



Kϕ c  ϕdf n

, f n1

For n ≥ n0, we have

, f n2

 df n

, f n1

 df n1

, f n2

So df n

, f n2

x0   Kdf n

, f n1

x0   Kdf n1

, f n2

K

 K2ϕ

x0

, f n1

x0

 .

3.23

Now, for every n  n0, we have



, f n3

 df n

, f n1

 df n1

x0

, f n3

Since K≥ 1, then we have

d

, f n3

x0   Kdf n

, f n1

x0   Kdf n1

, f n3

K

 K2ϕ

, f n2

x0

 .

3.25

Lemma 1.5, we have{f n x0} is a Cauchy sequence in X, d So lim n→∞f n x0  x∗, for some

x∗∈ X Now, we will prove fx∗  x∗ Since limn→∞f n x0  x∗, for every c 0, there exists

n c ∈ N such that for every n  n c , we have df n x0, x∗ < c Therefore,

 dx∗, f n1

 df

 dx∗, f n1

 ϕd

< d

x∗, f n1

 df n

, x∗

< 2c,

3.26

for every c 0 So fx∗  x∗ For the uniqueness of the fixed point, suppose that there exists

y∗∈ X such that fy∗  y∗ Hence

x∗, y∗

 df n

x∗

 ϕ n

So

d

x∗, y∗  Kϕ n

Since limn→∞ n dy∗, x∗ ∗ y∗and the proof is complete

4 Applications

Theorem 4.1 Consider the integral equation

x t 

b

a

Suppose that

i k : a, b × a, b × R n→ Rn and g : a, b → R n ;

ii kt, s, · : R n→ Rn is increasing for every t, s ∈ a, b;

iii there exists a continuous function p : a, b × a, b → R and a comparison function

iv supt ∈a,bb

a pt, s, αpt, sds  1.

g ∞, for every f, g ∈ X Then it is easily seen that X, d is a cone metric space Define

Ax t :

b

a

For every x, y ∈ X, we have

Ax t − Ayt , α Ax t − Ayt



b

a



b

a



a

b

a

α k t, s, xs − kt, s, y s ds



b

a



p t, s, αpt, sϕ x s − ys , α x s − ys ds

a



p t, s, αpt, sds

.

4.3

Hence dAx, Ay  ϕdx, y, for every x, y ∈ X The conclusion follows now from

Theorem 3.12

iii there exists a continuous function p : a, b × a, b → R...

g ∞, for every f, g ∈ X Then it is easily seen that X, d is a cone metric space Define

Ax t :

b

a