Vol 2, No 12 (2011)

SOME RESULTS OF FIXED POINT THEOREMS IN L–SPACE

Rajesh Shrivastava*, Animesh Gupta*, R. N. Yadava*** and Dilip Jaiswal**

*Department of Mathematics, Govt. Benazir Science & Commerce College, Bhopal, (M.P.), India

**Department of Mathematics, Moti Lal Vigyan Mahavidhyalaya (MVM), Bhopal, (M.P.), India

***Ex. Director Gr. Scientist & Head Resource Development Center, Regional Research Laboratory Bhopal (M.P, India)

E-mail: [email protected]

(Received on: 23-10-11; Accepted on: 09-12-11)

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ABSTRACT

T

here are several Theorems are prove in L- Space, using various type of mappings .In this paper, we prove some fixed point theorem and common fixed point Theorems, in L- space using different, symmetric rational mappings.

Keywords: Fixed point, Commmon Fixed point, L-Space, Continuous Mapping, Self Mapping, Weakly Compatible Mappings.

2000 Mathematics Subject Classification: 47H10, 54H25.

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1.INTRODUCTION:

It was shown by S. Kasahara [13] in 1976, that several known generalization of the Banach Contraction Theorem can be derived easily from a Fixed Point Theorem in an L-Space. Iseki [10] has used the fundamental idea of Kasahara to investigate the generalization of some known Fixed Point Theorem in L-Space.

Let be the set of natural numbers and be a nonempty set. Then L-Space is defined to be the pair of the set and a subset of the set , satisfying the following conditions;

!_" #

$ %s&'( )& * !

" In what follows instead of writing , we write

and read converges to x. Further we give some definitions regarding L-Space.

Definition: 1 Let be an L-Space. It is said to be ‘separated’ if each sequence in converges to at most one point of +

Definition: 2 A mapping f on into an L-Space is said to be continuous f: implies

for some subsequence _"

Definition: 3 Let d- be a non negative extended real valued function on , - . / % . for all % The L-Space is said to be d- complete if each sequence in with 0 /_{"34 " "12} 5 converges to the atmost one point of +

In this context Kasahara S. proved a lemma, which as follows,

Lemma (S. Kasahara): Let be an L- space which is d- complete for a non negative real valued function d on

+ if is separated then, / % / % - 67 ( % % During the past few years many great mathematicians Yeh[19], Singh[18], Pathak, and Dubey[14], Sharma, and Agrawa[17], Patel,Sahu, and Sao[15], Patel and Patel[15], worked for L- Space. In this chapter, we similar investigation for the study of Fixed Point

/ 3 4 5

Theorems in L- Space are worked out. We find some more Fixed Point Theorem and Common Fixed Point Theorem in L- Sapce.

2. MAIN RESULT:

89:;<:=, + Let be a separated L-space, which is d- complete for a non negative real valued function d on with / - , for each . Let > ? / @ be three continuous self mapping of into itself, satisfying the following condition;

A*2, > B @ / ? B @ >@ @> ?@ @?+

A*C, / > ?% . D E FG FH +IE FG JH 1 E FH KG L_{E FG KG 1E FH JH}

For all x, y in X, where non negative D such that - . D 5 A, with @ M @% + Then > ? @ have unique common fixed point.

Proof: Let ₄ ( * > B @ we can choose a point ₂ , such that @ ₂ > ₄ also ? B @ we can choose _C such that _C ? ₂ In general we can choose the point;

@ C 12 > C (1.1) @ C 1C ? C 12 (1.2)

Now consider,

/ @ C 12@ C 1C / > C ? C 12

FromA*C

/ > C ? C 12 . D/ @ C @_{/ @}C 12 + I/ @ C ? C 12 N / @ C 12> C L C > C N / @ C 12 ? C 12

/ @ C 12@ C 1C . D / @ C @_{/ @}C 12 + I/ @ C @ C 1C N / @ C 12 @ C 12 L C @ C 12 N / @ C 12@ C 1C

/ @ C 12@ C 1C . D / @ C @ C 12

For A O P Q Q Q+

Whether, / @ _{C 12} @ _{C 1C}

-Similarly, we have

/ @ C 12 @ C 1C . D + / @ 4 @ 2

For every positive integer n , this means that,

0 / @"34 C"12@ C"1C 5

Thus the d- completeness of the space implies that, the sequence @ ₄ converges to some & in. so by (1.1) and (1.2); > ₄ / ? ₄ also converges to the some point& respectively.

Since > ? @ are continuous, there is a subsequence of @ ₄ such that,

>R@ S > & @R> S @ & ,

TR@ S ? & , @R? S @ &, ,

By (AA *2 we have,

> & ? & @ & (1.3)

Thus,

@ @& @ >& > @& > >& > ?& @ ?& ? @& ? >& ? ?& (1.4)

/ 3 4 5

6 3 ! ! % 7

If > & M ? >&

/ >& ? >& . D E FU F KU +IE FU J KU 1 E F KU KU L_{E FU KU 1E F KU J KU}

/ >& ? >& . - Which contradiction

Hence; >& ? >& (1.5)

From (1.4) and (1.5) we have

>& ? >& @ >& > >&

Hence >& is a common fixed point of > ? / @+

Uniqueness:

Let $ is another fixed point of > ? / @ different from & then by A*C we have,

/ & $ / >& ?$

/ >& ?$ . D / @& @$ + I/ @& ?$ N / @$ >& L_{/ @& >& N / @$ ?$}

/ >& ?$ .

-Which contradiction.

Therefore &is unique fixed point of > ? / @ in X.

89:;<:= V + Let be a separated L-space, which is d- complete for a non negative real valued function d on with / -, for each . Let W ? / @ be three self mapping of into itself, satisfying the following condition;

O*2, W X ? Y @ / W@ @W ?W W?+

O*C, / ? @% . DE ZG JG I21E ZH FH L_{21E ZG ZH} N [E ZG ZH IE ZG FH 1E ZH JG L_{E ZG JG 1E ZH FH} N \ / W W%

For all x, y in X, where non negative D [ \ ]uch that D N [ N \ 5 A,

Then W ? @ have unique common fixed point.

Proof: for any arbitrary ₄ in X, we define a sequence of elements of X such that,

W C 12 ? C (2.1)

W _{C 1C} @ _{C 12} (2.2)

For all - A O P Q Q Q+

Now consider,

/ W C 12+ W C 1C / ? C + @ C 12 From O*C

/R? C @ C 12S . D E ZG^_JG_{21E ZG}^_ I21E ZG_{^_ZG^_`a</sub>^_`aFG^_`aL N [ E ZG^_ZG<sub>ERZG</sub>^_`a_{^_}IE ZG_{JG^_}^__{S1E ZG}FG^_`a<sub>^_`a}1E ZG<sub>FG^_`a</sub>^_`aJG^_ L

N \ / W C W C 12

/ W C 12+ W C 1C . D E ZG^_ZG_{21E ZG}^_`aI21E ZG<sub>^_`aZG^_`^</sub>^_`^ZG^_`aL N [E ZG^_ZG<sub>ERZG</sub>^_`a_{^_}IE ZG<sub>ZG^_`a</sub>^_<sub>S1E ZG</sub>ZG^_`^<sub>^_`a</sub>1E ZG<sub>ZG</sub>^_`a<sub>^_`^</sub>ZG^_`aL N \ / W _C W _{C 12}

/ 3 4 5

Where ) D N [ N \ 5 A ; Processing the same way we get,

/ W C 12+ W C 1C . ) / W C W C 12

For A O P Q Q Q+

Whether, / W _{C 12} W _{C 1C}

-Similarly, we have

/ W C 12 W C 1C . ) + / W 4W 2

For every positive integer n, this means that,

0 / W"34 C"12W C"1C 5

Thus the d- completeness of the space implies that, the sequence W ₄ converges to some & in. so by (2.1) and (2.2); ? ₄ / @ ₄ also converges to the some point & respectively.

Since W ? @ are continuous, there is a subsequence of W ₄ such that,

?RW S ? & WR? S W &,

TRW S @ &, WR@ S W & ,

By (O*2 we have,

W & ? & @ & (2.3)

& ( * 66 b / 7 W ? / @+

Uniqueness:

Let us assume that w is another fixed point of W ? / @ different from u & M c then

/ W& Wc / ?& @c from AO*C

?& @c . D E ZU JU I21E Zd Fd L_{21E ZU Zd} N [ E ZU Zd IE ZU Fd 1E Zd JU L_{E ZU JU 1E Zd Fd} N \ / W& Wc

/ W& Wc . \ / W& Wc

Which contradiction.

Hence & ( & )& * 66 b / 7 W ? @+ +

89:;<:= V eLet be a separated L-space, which is d- complete for a non negative real valued function d on with / -, for each . Let > ? / @ be three continuous self mapping of into itself, satisfying the following condition;

P*2, > B @ / ? B @ >@ @> ?@ @?+

P*C, / > ?% . D + / @ > + / @% ?% + / % af

For all x, y in X, where non negative D [ \ such that

- . D N [ N \ 5 A, with @ M @% + Then > ? @ have unique common fixed point .

Proof: Let ₄ ( * > B @ we can choose a point ₂ , such that @ ₂ > ₄ also ? B @ we can choose _C such that @ _C ? ₂ In general we can choose the point;

/ 3 4 5

6 3 ! ! % 9

@ C 1C ? C 12 (3.2)

For every , we have

/ @ C 12@ C 1C / > C ? C 12

From P*C

/ > C ? C 12 . D + / @ C > C + / @ C 12? C 12 + / C C 12 2 g

/ @ C 12 @ C 1C . D + / @ C @ C 12 + / @ C 12 @ C 1C + / C C 12 2 g / @ C 12 @ C 1C . D

f

^ / @ _C @ _{C 12}

Let ) Df^ 5 A

/ @ C 12@ C 1C . ) / @ C @ C 12

For A O P Q Q Q+

Whether, / @ _{C 12} @ _{C 1C}

-Similarly, we have

/ @ C 12@ C 1C . ) + / @ 4 @ 2

For every positive integer n, this means that,

h / @ C"12@ C"1C "34

5

Thus the d- completeness of the space implies that, the sequence @ ₄ converges to some & in. so by (3.1) and (3.2); > ₄ / ? ₄ also converges to the some point& respectively.

Since > ? @ are continuous, there is a subsequence of @ ₄ such that,

>R@ S > & @R> S @ & , ?R@ S ? & , @R? S @ & ,

By (P*2 we have,

> & ? & @ & (3.3)

Thus,

@ @& @ >& > @& > >& > ?& @ ?& ? @& ? >& ? ?& (3.4)

By P*C, (3.3) and (3.4) we have,

If > & M ? >&

/ >& ? >& . D + / @& >& + / @ >& ? >& + / & >& af

/ >& ? >& . -

Which contradiction

Hence; >& ? >& (3.5)

From (3.4) and (3.5), we have

>& ? >& @ >& > >&

/ 3 4 5

Uniqueness:

Let $ is another fixed point of > ? / @ different from & then by P*C we have,

/ & $ / >& ?$

/ >& ?$ . D + / @& >& + / @$ ?$ + / & $ af

/ >& ?$ . - Which contradiction.

Therefore &is unique fixed point of > ? / @ in X.

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