Fixed points of a sequence of locally contractive multivalued maps

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Computers and Mathematics with Applications

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Fixed points of a sequence of locally contractive multivalued maps

Akbar Azam

a,∗

, Muhammad Arshad

b

a_{Department of Mathematics, F.G. Postgraduate College, H-8, Islamabad, Pakistan} b_{Department of Mathematics, International Islamic University, H-10, Islamabad, Pakistan}

a r t i c l e i n f o

Article history:

Received 15 March 2008

Received in revised form 5 August 2008 Accepted 10 September 2008

Keywords:

Common fixed point Multivalued mapping Locally contractive mappings Locally expansive mappings

a b s t r a c t

We prove the existence of common fixed points of a sequence of multivalued mappings satisfying Edelstein type contractive condition. As an application, common fixed points of a sequence of single valued expansive type mappings have been obtained.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction and preliminaries

Edelstein [1] introduced the concept of locally contractive mappings and established a generalization of the Banach contraction principle in

ε

-chainable metric spaces. Subsequently, many authors [2–13] studied locally contractive type mappings and obtained important results. In this paper, common fixed points of a sequence of locally contractive multivalued mappings are obtained in

ε

-chainable metric spaces, and some former corresponding results of [1,3,5,8,10,

14,15] are extended and improved. Recently a fixed point theorem in a chainable Menger probabilistic metric space was established by Razani and Fouladgar [16], which motivated us to attempt this work. This area was not in the focus of recent research for a while (since Waters [13]). So we thought of building on the basic work of Nadler [10]. We believe that our work will highlight this area and research focus will again tilt back into fixed points of locally contractive mappings in

ε

-chainable metric spaces.

Let

(

X

,

d

)

be a metric space,

ε >

0, 0

≤

λ <

1 and x

,

y

∈

X . An

ε

-chain from x to y is a finite set of points x1

,

x2

,

x3

, . . . ,

xn

such that x

=

x1

,

xn

=

y, and d

(

xj−1

,

xj

) < ε

for all j

=

1

,

2

,

3

, . . . ,

n. A metric space

(

X

,

d

)

is said to be

ε

-chainable if and

only if given x

,

y

∈

X , there exists an

ε

-chain from x to y. A mapping T

:

X

→

X is called a globally contractive mapping

if for all x

,

y

∈

X , d

(

Tx

,

Ty

) ≤ λ

d

(

x

,

y

)

. A mapping T

:

X

→

X is called an

(ε, λ)

uniformly locally contractive mapping

if 0

<

d

(

x

,

y

) < ε

implies d

(

Tx

,

Ty

) ≤ λ

d

(

x

,

y

)

. For

η >

1, T is called an

(ε, η)

uniformly locally expansive mapping if

0

<

d

(

x

,

y

) < ε

implies d

(

Tx

,

Ty

) ≥ η

d

(

x

,

y

)

. We remark that a globally contractive mapping can be regarded as an

(∞, λ)

uniformly locally contractive mapping and for some special spaces every locally contractive mapping is globally contractive. For details see [1,12,13]. Let

CB

(

X

) = {

A

:

A is a nonempty closed and bounded subset of X

}

.

C

(

X

) = {

A

:

A is a nonempty compact subset of X

}

.

For A

,

B

∈

CB

(

X

)

and e

>

0 the sets N

(

e

,

A

)

and EA,Bare defined as follows:

N

(

e

,

A

) = {

x

∈

X

:

d

(

x

,

A

) <

e

}

,

∗_{Corresponding author.}

E-mail addresses:akbarazam@yahoo.com(A. Azam),marshad_zia@yahoo.com(M. Arshad). 0898-1221/$ – see front matter©2008 Elsevier Ltd. All rights reserved.

EA,B

= {

e

:

A

⊆

N

(

e

,

B

),

B

⊆

N

(

e

,

A

)},

where

d

(

x

,

A

) =

inf

{

d

(

x

,

y

) :

y

∈

A

}

.

The distance function H on CB

(

X

)

defined by H

(

A

,

B

) =

inf

{

e

:

e

∈

EA,B

}

, is known as the Hausdorff metric on X .

A mapping T

:

X

→

C

(

X

)

is called a globally nonexpansive multivalued mapping if for all x

,

y

∈

X

,

H

(

Tx

,

Ty

) ≤

d

(

x

,

y

)

.

T

:

X

→

C

(

X

)

is called an

ε

-nonexpansive multivalued mapping if x

,

y

∈

X 0

<

d

(

x

,

y

) < ε

implies H

(

Tx

,

Ty

) ≤

d

(

x

,

y

)

. A mapping K

:

(

0

, ε) → [

0

,

1

)

is said to have property (p) if for t

∈

(

0

, ε)

there exists

δ(

t

) >

0

,

s

(

t

) <

1 such that 0

≤

r

−

t

< δ(

t

)

implies K

(

r

) ≤

s

(

t

) <

1

(

cf

.

[6,9,16]

).

Lemma 1 ([6]). If A

,

B

∈

CB

(

X

)

with H

(

A

,

B

) < ε

, then for each a

∈

A, there exists an element b

∈

B such that d

(

a

,

b

) < ε

.

Lemma 2 ([17]). Let

{

An

}

be a sequence of sets in CB

(

X

)

and limn→∞H

(

An

,

A

) =

0 for A

∈

CB

(

X

)

. If yn

∈

An

(

n

=

1

,

2

, . . .)

and d

(

yn

,

y

) →

0, then y

∈

A.

Lemma 3 ([2]). Let A

,

B

∈

CB

(

X

)

, then for a

∈

A d

(

a

,

B

) ≤

H

(

A

,

B

).

2. Main results

Theorem 1 (Edelstein Contraction Principle [1]). Let

(

X

,

d

)

be a complete

ε

-chainable metric space. If T

:

X

→

X is an

(ε, λ)

uniformly locally contractive mapping, then T has a unique fixed point.

Nadler [10] extended the above theorem to multivalued mappings as follows:

Theorem 2. Let

(

X

,

d

)

be a complete

ε

-chainable metric space and T

:

X

→

C

(

X

)

be a mapping satisfying the following condition:

x

,

y

∈

X and 0

<

d

(

x

,

y

) < ε

implies H

(

Tx

,

Ty

) ≤ λ

d

(

x

,

y

).

Then T has a fixed point.

As an improvement and generalization of the above theorem we have the following result:

Theorem 3. Let

(

X

,

d

)

be a complete

ε

-chainable metric space and

{

Tn

}

∞n=1be a sequence of mappings from X to CB(X) satisfying the following condition:

x

,

y

∈

X and 0

<

d

(

x

,

y

) < ε

implies H

(

Tnx

,

Tmy

) ≤

K

(

d

(

x

,

y

))

d

(

x

,

y

)

(1)

for n

,

m

=

1

,

2

, . . .

where K

:

(

0

, ε) → [

0

,

1

)

is a function having property (p). Then there exists a point y∗

∈

X such that y∗

_{∈ ∩}

∞

n=1Tny∗

.

Proof. Let y0be an arbitrary, but fixed element of X . We shall construct a sequence

{

yn

}

of points of X as follows.

Let y1

∈

X be such that y1

∈

T1y0. Also let y0

=

x(1,0)

,

x(1,1)

,

x(1,2)

, . . . ,

x(1,m)

=

y1

∈

T1y0

be an arbitrary

ε

-chain from y0to y1. Rename y1as x(2,0).

Since x₍2,0)

∈

T1x(1,0),

H

(

T1x(1,0)

,

T2x(1,1)

) ≤

K

(

d

(

x(1,0)

,

x(1,1)

))

d

(

x(1,0)

,

x(1,1)

)

< p[

K

(

d

(

x₍1,0)

,

x(1,1)

))]

d

(

x(1,0)

,

x(1,1)

)

<

d

(

x₍1,0)

,

x(1,1)

)

< ε.

UsingLemma 1. we obtain x₍2,1)

∈

T2x(1,1)such that

d

(

x₍2,0)

,

x(2,1)

) < p[

K

(

d

(

x(1,0)

,

x(1,1)

))]

d

(

x(1,0)

,

x(1,1)

)

<

d

(

x₍1,0)

,

x(1,1)

)

< ε.

(2) Since x₍2,1)

∈

T2x(1,1) H

(

T2x(1,1),T2x(1,2)

) ≤

K

(

d

(

x(1,1)

,

x(1,2)

))

d

(

x(1,1)

,

x(1,2)

)

< p[

K

(

d

(

x₍1,1)

,

x(1,2)

))]

d

(

x(1,1)

,

x(1,2)

)

<

d

(

x₍1,1)

,

x(1,2)

)

< ε.

We may choose an element x₍2,2)

∈

T2x(1,2)such that d

(

x₍2,1)

,

x(2,2)

) < p[

K

(

d

(

x(1,1)

,

x(1,2)

))]

d

(

x(1,1)

,

x(1,2)

)

<

d

(

x₍1,1)

,

x(1,2)

) < ε.

Thus we obtain a finite set of points x₍2,0)

,

x(2,1)

,

x(2,2)

, . . . ,

x(2,m)such that

x₍2,0)

∈

T1x(1,0) and x(2,j)

∈

T2x(1,j)

,

for j

=

1

,

2

,

3

, . . . ,

m

,

with

d

(

x₍2,j)

,

x(2,j+1)

) < p[

K

(

d

(

x(1,j)

,

x(1,j+1)

))]

d

(

x(1,j)

,

x(1,j+1)

)

<

d

(

x₍1,j)

,

x(1,j+1)

)

< ε,

for j

=

0

,

1

,

2

, . . . ,

m

−

1

.

Let x₍2,m)

=

y2, then the set of points

y1

=

x(2,0)

,

x(2,1)

,

x(2,2)

, . . . ,

x(2,m)

=

y2

∈

T2y1

is an

ε

-chain from y1to y2.

Rename y2as x(3,0), then by the same procedure we obtain an

ε

-chain y2

=

x(3,0)

,

x(3,1)

,

x(3,2)

, . . . . ,

x(3,m)

=

y3

∈

T3y2 from y2to y3. Inductively, we obtain yn

=

x(n+1,0)

,

x(n+1,1)

,

x(n+2,2)

, . . . ,

x(n+1,m)

=

yn+1

∈

Tn+1yn with d

(

x₍n+1,j)

,

x(n+1,j+1)

) < p[

K

(

d

(

x(n,j)

,

x(n,j+1)

))]

d

(

x(n,j)

,

x(n,j+1)

)

<

d

(

x₍n,j)

,

x(n,j+1)

)

< ε,

for j

=

0

,

1

,

2

, . . . ,

m

−

1 and n

=

0

,

1

,

2

, . . . .

(3) Consequently, we obtain a sequence

{

yn

}

of points of X with

y1

=

x(1,m)

=

x(2,0)

∈

T1y0 y2

=

x(2,m)

=

x(3,0)

∈

T2y1 y3

=

x(3,m)

=

x(4,0)

∈

T3y2

. . . .

. . . .

. . . .

That is, yn+1

=

x(n+1,m)

=

x(n+2,0)

∈

Tn+1yn

,

for n

=

0

,

1

,

2

, . . . .

Then from(3)we see that

{

d

(

x₍n,j)

,

x(n,j+1)

) :

n

≥

0

}

is a decreasing sequence of non-negative real numbers and therefore

tends to a limit t

≥

0. We claim t

=

0. For if t

>

0

,

inequality(3)yields t

< ε

. Then by the property (p) of K there exists

δ(

t

) >

0

,

s

(

t

) <

1 such that 0

≤

r

−

t

< δ(

t

)

implies K

(

r

) ≤

s

(

t

) <

1

.

For this

δ(

t

) >

0 there exists an integer N such that

0

≤

d

(

x₍n,j)

,

x(n,j+1)

) −

t

< δ(

t

)

for n

≥

N

.

Hence, K

(

d

(

x₍n,j)

,

x(n,j+1)

)) ≤

s

(

t

) <

1 whenever n

≥

N

.

Then d

(

x₍n,j)

,

x(n,j+1)

) < p[

K

(

d

(

x(n−1,j)

,

x(n−1,j+1)

))]

d

(

x(n−1,j)

,

x(n−1,j+1)

)

<

max



N max i=1

p

[

K

(

d

(

x₍i,j)

,

x(i,j+1)

))],

s

(

t

)



d

(

x₍n−1,j)

,

x(n−1,j+1)

)

<

max

(r

N max i=1

[

K

(

d

(

x₍i,j)

,

x(i,j+1)

))],

s

(

t

)

)!

n d

(

x₍0,j)

,

x(0,j+1)

).

On letting n

→ ∞

and in the view of max



_q

maxN

i=1

[

K

(

d

(

x(i,j)

,

x(i,j+1)

))],

s

(

t

)



<

1, the above inequality yields

d

(

x₍n,j)

,

x(n,j+1)

) →

0

,

a contraction. Hence t

=

0

.

Hence,

d

(

yn−1,yn

) =

d

(

x(n,0)

,

x(n,m)

) ≤

m−1

X

j=0

d

(

x₍n,j)

,

x(n,j+1)

) →

0

.

Now we prove that

{

yn

}

is a Cauchy sequence. Assume that

{

yn

}

is not a Cauchy sequence. Then there exist a number t

>

0

(we may assume t

< ε

without loss of generality) and two sequences

{

nj

}

, {

mj

}

of natural numbers with nj

<

mjsuch that

d

(

xnj

,

xmj

) ≥

t

,

d

(

xnj

,

xmj−1

) <

t for j

=

1

,

2

,

3

, . . . .

Then, t

≤

d

(

xnj

,

xmj

) ≤

d

(

xnj

,

xmj−1

) +

d

(

xmj−1

,

xmj

)

<

t

+

d

(

xmj−1

,

xmj

).

It follows that lim j→∞d

(

xnj

,

xmj

) =

t

∈

(

0

, ε).

For this t

>

0, by property (p) of K, we can find

δ(

t

) >

0

,

s

(

t

) <

1 such that 0

≤

r

−

t

< δ(

t

)

implies K

(

r

) ≤

s

(

t

) <

1

.

Now for this

δ(

t

) >

0, there exists an integer N such that

j

≥

N implies 0

≤

d

(

xnj

,

xmj

) −

t

< δ(

t

)

and hence

K

(

d

(

xnj

,

xmj

)) <

s

(

t

)

if j

≥

N

.

(4)

Thus

d

(

xnj

,

xmj

) ≤

d

(

xnj

,

xnj+1

) +

d

(

xnj+1

,

xmj+1

) +

d

(

xmj+1

,

xmj

)

≤

d

(

xnj

,

xnj+1

) +

K

(

d

(

xnj

,

xmj

))

d

(

xnj

,

xmj

) +

d

(

xmj+1

,

xmj

).

This in the view of inequality(4)implies that t

≤

s

(

t

)

t

<

t, a contradiction. It follows that

{

yn

}

is a Cauchy sequence.

Since X is complete, yn

→

y∗

∈

X . Hence there exists an integer M

>

0 such that n

>

M implies d

(

yn

,

y∗

) < ε

. This in the

view of inequality(1)implies

H

(

Tn+1yn

,

Tjy∗

) ≤

K

(

d

(

yn

,

y∗

)

d

(

yn

,

y∗

)).

Consequently,

H

(

Tn+1yn

,

Tjy∗

) →

0

.

Since yn+1

∈

Tn+1ynwith d

(

yn+1

,

y∗

) →

0, nowLemma 2implies that y∗

∈

Tjy∗

,

therefore, y∗

∈ ∩

∞n=1Tny∗. This completes

the proof. 

Corollary 4. Let

(

X

,

d

)

be a complete

ε

-chainable metric space and

{

Tn

}

∞n=1be a sequence of mappings from X to CB(X) satisfying the following condition:

x

,

y

∈

X and 0

<

d

(

x

,

y

) < ε

implies H

(

Tnx

,

Tmy

) ≤ λ(

d

(

x

,

y

))

for n

,

m

=

1

,

2

,

3

. . .

. Then there exists a point y∗

∈

X such that y∗

∈ ∩

∞

n=1Tny∗.

3. An application to single valued expansive mappings

Nadler [10] usedTheorem 2 and obtained some results regarding fixed points of a single valued (not necessarily one to one) uniformly locally expansive mapping T

:

dom

(

T

) →

X by placing some conditions on the inverse of T

(e.g., T−1x

∈

C

(

domT

)

and T−1is

ε

-nonexpansive). We useCorollary 4to improve and generalize corresponding results of [10] as follows:

Theorem 4. Let

(

X

,

d

)

be a complete

ε

-chainable metric space,

η ≥

1,

φ 6=

A

⊆

X and

{

Tn

}

∞n=1be a sequence of mappings from A onto X satisfying the following condition:

x

,

y

∈

X and 0

<

d

(

x

,

y

) < ε

implies d

(

Tnx

,

Tmy

) ≥ η(

d

(

x

,

y

))

, for n

,

m

=

1

,

2

, . . .

. If for each n

=

1

,

2

, . . .

and x

∈

X ,

T−1

n x

∈

CB

(

A

)

and 0

<

d

(

x

,

y

) < ε

implies H

(

T

−1

n x

,

T

−1

m y

) < ε

, for n

,

m

=

1

,

2

, . . .

, then there exists a point y

∗

_∈

_{X such that}

y∗

₌

_T

Proof. Let x

,

y

∈

X such that 0

<

d

(

x

,

y

) < ε

and choose

β >

0

.

Let u

∈

T_n−1x. Since H

(

T_n−1x

,

T_m−1y

) < ε

, byLemma 1there exists a point

v ∈

T−1

m y such that d

(

u

, v) < ε.

Therefore

d

(

Tnu

,

Tm

v) ≥ η

d

(

u

, v).

That is, d

(

u

, v) <



1

η

+

β



d

(

x

,

y

).

It follows that d

(

u

,

T_m−1y

) <



1

η

+

β



d

(

x

,

y

).

This further implies that

u

∈

N



1

η

+

β



d

(

x

,

y

),

T_m−1y



.

This proves that

T_n−1x

⊆

N



1

η

+

β



d

(

x

,

y

),

T_m−1y



.

Similarly it can be shown that

T−1 m y

⊆

N

h

1 η

+

β

i

d

(

x

,

y

),

T−1

n x

 .

Since

β

is arbitrary, it now follows that H

(

T

−1

n x

,

T

−1

m y

) ≤

1

ηd

(

x

,

y

).

We may now apply

Corollary 4to conclude that there is a point y∗

∈

X such that y∗

∈ ∩

∞_n=1T_n−1y∗. Clearly

y∗

=

Tny∗ for each n

=

1

,

2

, . . . .



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