Vol 3, No 1 (2012)

Some Fixed Point Theorems for

-maps

M. Akram* and Z. Afzal

1

1

Department of Mathematics, GC University, Katchery Road Lahore 54000, Pakistan

Email: [email protected], [email protected], [email protected]

(Received on: 19-11-11; Accepted on: 10-12-11)

________________________________________________________________________________________________ ABSTRACT

I

n this paper, we introduce the idea of -mapping for -distance and proved some results for fixed point and common fixed point for -mappings. These results, generalized and improved the corresponding results of A.Latif, Suzuki and others.

AMS (2000) Mathematics Subject Classification: 47H10, 54H25.

Key Words: -maps, Fixed points, Multi-valued maps, -distance.

________________________________________________________________________________________________

1 INTRODUCTION:

In [6], Kada et al., have introduced a notion of -distance on a metric space and improved several results replacing the involved metric by a generalized distance. Using this generalized distance, Suzuki and Takahashi in [12] introduced notions of single-valued and multi-valued weakly contractive maps and proved some fixed point results for such maps. Consequently, they generalized the Banach contraction principle and Nadler's (see [10]) fixed point results . While, in [13] Suzuki generalized Kannan's [7] fixed point result in -distance. Further fixed point results concerning -distance can be found in [6,7,11]. In [2], M. Akram, et al., introduced the general class of contraction called A-contraction and proved that the class of A-contraction is a proper super class of Kannan's contractions including several other famous contractions like Bianchini's [4] and Reich's [11] contractions. In [1] M. Akram et al., using the idea of A-contractions, proved several fixed point theorems for single valued maps on generalized complete metric spaces. In [3], M. Akram, et al., introduced general multi-valued -maps and proved some fixed point theorems for these maps. In this paper, using the concept of w-distance, we define -maps and proved some fixed point and common fixed point results for these maps. These results generalizes the results of [9] and others, for general class of maps called -maps.

2 PRELIMINARIES:

Throughout this paper, is a metric space with metric . Let denote a set of all nonempty subsets of X, a family of all nonempty closed subsets of . Consider a multi-valued map . An element is said to be a fixed point of if . Let denote the set of all non-negative real numbers and stands for the set of all functions satisfying

1. is continuous on the of all triplets of non-negative reals (with respect to the Euclidean metric on ); 2. is non-decreasing in each coordinate variables;

3. for some , whenever or or for all .

Definition: 2.1 A map is called Lower Semi-Continuous if for any sequence ! _"# $ with _" imply that %&%'(" " .

Kada et all. [6], introduced a concept of -distance as follows:

Definition: 2.2 A function ) is called -distance on if it satisfies the following conditions, for any

* + :

1. + * , * +

2. the map - is lower semi-continuous.

3. for any . / , there exist a 0 / such that + 0 and + * 0 imply that * ..

. / + ) / ) 0 The metric on is a simple example of -distance on . Many other examples of -distance can be found in [8, 10, 13].

Definition: 2.3 A multi-valued map is -map if there exists a -distance function such that for any

1 2 there exist 3 * for all * 1, and

2 3 * 2 * 3

for some .

In particular, if we take 4 , then -map becomes an -map as defined in [3].

Definition: 2.4 A multi-valued map is said to be 5 -map if there exists a nonnegative number 6 and a -distance function such that for any 1 2 there exist 3 * for all * 1, satisfies the inequality 2 3 6 2 , * 3 -

The following Lemma concerning -distance is very important.

Lemma: 2.5 [8] Let be metric space with metric and let be the -distance on . Let ! _"# and !*_"# be sequences in . Let ! "# and !7"# be a sequences in converging to 0, and let * + , then the following

hold:

(a) If _" * _" and _" + 7_" for any ' 8, then * 4 +; in particular, if * 4 and + 4 -Then * 4 +;

(b) If _" *" " and " + 7" for any ' 8, then !*"# converges to +; (c) If _{" 9 "} for any ' & 8 with & / ', then ! _"# is Cauchy sequence; (d) If * " " for any ' 8, then ! "# be Cauchy sequence.

3. MAIN RESULTS:

In the sequel, we take complete and 1 a non-empty closed subset of .

Theorem 3.1 Let 1 1 be a multi-valued -map such that for any 1, there exists 2 , there exist

3 * for all * 1 and

:;<! 2 , == # /

for every 2 with 2 2 . Then has a fixed point.

Proof. Let 2_> be an element of 1, and 2_? 2_> be fixed. Since is -map, there exist 2_@ 2_? s.t

2? 2@ 2> 2? 2> 2? 2? 2@

== 2_>2_?

where . By repeating this process, we have

2" 2" ? "== 2>2? -Now, for & / ', we have

29 2" 2" 2" ? , 2" ? 2" @ ,- - - , 29A? 29 ==================================== "_{== 2}

>2? , " ?== 2>2? ,- - - , 9A?== 2>2? ==================================== "_, " ?_{,- - - ,} 9A? _{== 2}

> 2? ==================================== ! " _B 9 _{C B #== 2}

> 2? == D=='

-Hence, by lemma(1.4), !2_"# is a Cauchy sequence. From the completeness of , we get that !2_"# converges to some 3> . Since 1 being closed, we have 3> 1. Let ' 8 be fixed. Since !29# converges to some 3> and 2"E is lower semi-continuous, we get

2" 3> F:G:;<₉ 2" 29 ! " B 9 C B #== 2> 2?

So, as ' , we have 2" 3> . Assume that 3> 3> . Then by hypothesis, we have

H %'(! 2 3> , == 2 2 2 #

=== :;<! " _B 9 _{C B == 2}

> 2? ,== " 2> 2 ' 8# 4

-Which is impossible and hence 3> 3> .

Note that Theorem 3.1 of [9] becomes the corollary of our Theorem 3.1 as below.

Corollary: 3.2 Let 1 1 be a multi-valued 5 -map such that

:;<! 2 , # / ,

for every 2 with 2 2 . Then has a fixed point.

Proof: Define by 2 3 4 6 2 , 3 , where H 6 H C . One can easily see that . For any

1 2 there exist a 3 * for all * 1, If we take 2 4 2 3 4 * 3 and 4 * , in the definition of 5 -map, we have

2 3 IJ 2 , * 3 K 4 I 2 , 3 4 2 3

-This gives 2 3 * 2 * 3 . That is is a -map. Hence by Theorem 3.1, we have the conclusion.

Theorem 3.3 Each -map 1 1 has a fixed point, provided that for any iterative sequence !2_"# in 1 with 2_" 3_> 1, the sequence of real numbers ! 3_>2_" # converges to zero.

Proof. In a similar way, as in Theorem 3.1, there exists a convergent iterative sequence !2"# such that 2" 3> 1 with

2" 3> F:G:;<₉ 2" 29 ! " B 9 C B #== 2> 2? and 2" 2" ? "== 2> 2?

-Note that 2_" 3_> as ' . Further since 2_" 2_"A? and is a -map therefore there is 3_" 3_> such that

2" 3" 2"A? 3> 2"A? 2" 3> 3"

-For this we use definition of lower semi-continuous

3> _"F:G3" F:G:;<_" 2" 3" _"F:G 2" 3"

-By using this expression given above, we get

F:G

" 2" 3" "F:G 2"A? 3> 2"A? 2" 3>3" F:G

" 2"A? 3> "F:G 2"A? 2" "F:G 3> 3" ========================= 3> _"F:G3"

========================= F:G

" 2" 3" ========================= 4

-Thus 2_" 3_" as ' . Then by Lemma 2.5, we get that 3_" 3_>- As 3_" 3_> , and 3_> is closed, 3> 3> .

Theorem 3.2 of [9] becomes the special case of our Theorem 3.3 as bellow.

Corollary: 3.4 Each 5 -map 1 1 has a fixed point, provided that for any iterative sequence !2"# in 1

with 2_" 3_> 1, the sequence of real numbers ! 3_>2_" # converges to zero.

Theorem: 3.5 Let ! _"# be a sequence of multi-valued maps of 1 into 1 . For any two maps _{L M} ! _"# and for any * 1 there exists 2 _L and 3 _M * with

2 3 * 2 * 3

-. / + ) / ) 2 Then the sequence ! "# has a common fixed point.

Proof. Let 2> be an arbitrary element of 1 and let 2? ? 2> , then there is 2@ @ 2? such that 2?2@ == 2> 2? - So, there exist a sequence !2"# such that 2" ? " ? 2" for all ' N . Then as ' , we get that !2"# is a Cauchy sequence in . Let P 4 F:GQ RQ in 1. Now we show that P S " P . Let 9 be any arbitrary member of ! _"#. Since 2_{" "} 2_"A? , by hypothesis there is D_{" 9} P such that

2" D" 2"A? P 2"A? 2" P D"

when ' , we have 2"A?P and 2"A? 2" O

F:G

" 2" D" "F:G 2" D" 4

This implies that 2_" D_" as ' and we get the following

2" P F:G:;<= 2" 29 ! " B 9 C B #== 2> 2? '

-This implies that 2_" P as ' - Now, assume that P ₉ P . Then by given hypothesis and for ' / & and & N , we have

H %'(! 2 P , 2 9 2 2 #

==== :;<! 29A? P , 29A? 9 29A? & 8#

:;<! 2_9A? P , 2_9A? 2₉ & 8#

:;<! 9A? B "C B == 2_> 2_? , 9A?== 2_> 2_? & 8# 4

which is impossible and hence P 9 P . But 9 is an arbitrary, hence P is a common fixed point.

Next corollary, is the Theorem 3.3 of [9].

Corollary: 3.6 Let ! _"# be a sequence of multi-valued maps of 1 into 1 . Suppose that there exist a constant

6 H C such that for any two maps _{L M} ! "# and for any 1 2 L , there exists 3 M * for all * 1 with

2 3 6==! 2 , * 3 #

and for each ' N :;<! 2 , _" # /

for any 2 " 2 . Then the sequence ! "# has a common fixed point.

Theorem: 3.7 Let ! "# be a sequence of multi-valued maps of 1 into 1 . For any two maps L M ! "# and for

any * 1 there exists 2 _L and 3 _M * with

2 3 * 2 * 3

-Then ! _"# has a common fixed point, provided that for any iterative sequence !2_"# in 1 with 2_" 3_> 1. The sequence of real numbers ! 3> 2" # converges to .

Proof. This result can be proved in a similar way, as Theorem 3.5 is proved.

Theorem 3.4 of [9] becomes the corollary of our theorem given above as follows.

Corollary 3.8 Let ! _"# be a sequence of multi-valued maps of 1 into 1 . Suppose that there exit a constant

6 H C such that for any two maps _{L M} and for any 1 2 _L . There exists

v

∈

T

_j

( )

y

for all

y

∈

M

with

(

.

)

{

(

,

)

(

,

)

}

.

w u v

≤

r w x u

+

w y v

Then

{ }

T

_n has a common fixed point, provided that for any iterative sequence

{ }

u

_n in

M

with

u

_n

→

v

₀

∈

M

. The

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[2] M. Akram, A.A. Siddique, A. A. Zafar, A general class of contractions: A-contractions, Novi Sad J. Math. Vol. 38, (2008), 25-33.

[3] M. Akram, A. A. Siddique, A. A. Zafar, Some fixed point theorems for multi-valued

A

*

- mappings, Korean J. Math. Science Vol. 10, (2003),7-12.

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[10] S. B. Nadler, Multivalued contraction mappings, Pacific J. Math., 30 (1969), 475-488.

[11] Reich S., Kannan's fixed point theorem, Boll.Un.Math.Ital., Vol.4 (1971), 1-11.

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[13] T. Suzuki, Several fixed point theorems in complete metric spaces,Yokohama Math. J., 44 (1997), 61-72.

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