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Journal of Mathematics
Volume 2013 (2013), Article ID 434205, 7 pages
Fixed Point Theorems for a Class of Weakly C-Contractive Mappings in a Setting of 2-Banach Space
1Department of Mathematics, The University of Burdwan, Burdwan, West Bengal, 713104, India
2Burdwan Railway Balika Vidyapith High School, Khalasipara, Burdwan, West Bengal, 713101, India
Received 26 September 2012; Accepted 1 December 2012
Academic Editor: Baoding Liu
Copyright © 2013 M. Saha and Anamika Ganguly. 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.
In this paper, the authors have proved some existence theorems of fixed points for a class of weakly C-contractive mappings in a setting of 2-Banach space. The authors have supported the results with the help of suitable examples.
The theory of 2-Banach spaces was investigated by Gähler in  and Iseki in  who proved some fixed point theorems in such spaces and these new spaces have subsequently been studied by many authors. For example, we refer to [3, 4] where authors have dealt with mappings that are of contractive nature. Our findings as presented here have added further findings in fixed point theory in a 2-Banach space by working with a class of mappings so general that take care of those as found in [5, 6].
We now state some definitions before presenting our main results.
Definition 1. Let be a real linear space and a non-negative real valued function defined on satisfying the following conditions:(i) if and only if and are linearly dependent,(ii) for all ,(iii), being real for all ,(iv) for all .Then is called a 2-norm and the pair is called a linear 2-normed space. So a 2-norm always satisfies for all and all scalars .
Definition 2. A sequence in a linear 2-normed space is said to be a Cauchy sequence if for all in .
Definition 3. A sequence in a linear 2-normed space is said to be convergent in if there exists a point in such that for all in . If converges to , we write as .
Definition 4. A linear 2-normed space is said to be complete if every Cauchy sequence is convergent to an element of . A complete linear 2-normed space is called a 2-Banach space.
The main purpose of our paper is to find out the existence of fixed point for a class of weakly C-contractive results due to Choudhury  in a setting of 2-Banach space. We have also established a common fixed point theorem for such type of mappings with a supporting example. Some allied theorems have also been presented here.
3. Main Results
Definition 5. A mapping is said to be weakly -contractive or a weak -contraction if for all in , where is a continuous mapping such that if and only if .
Theorem 6. Let be a 2-Banach space and let be a weak C-contraction. Then has a unique fixed point in .
Proof. Let . Let , .
If , then has a fixed point .
Assume that . Now for any , Therefore, So is a monotone decreasing sequence of real numbers and hence is convergent and let (say) as
We now show that . If not, then Taking , we get implying that
Also from (2) and using continuity of , we get showing that , a contradiction, unless . So, as , that is We shall now show that is Cauchy sequence in .
If not, then there exists a and for each positive integer , there exist integers and with such that and . Then Also, So by using (7), we obtain Now Also Therefore,
Similarly, it can be shown that So from (8), we get implying that , a contradiction.
Hence is a Cauchy sequence and therefore is a convergent in and let (say) as . Now for any , So by the continuity of and taking , we get implying that .
We shall now show that the fixed point is unique. If possible let be another fixed point of , then for any which by property of is a contradiction unless for all that is .
This completes the proof.
Example 7. Let be and consider the following 2-norm on as
where , . Then is a 2-Banach space (see [8, 9]).
is defined by for all . Also let .
Now setting is a continuous mapping such that if and only if . Now and . Take .
Now Also, All the conditions of Theorem 6 are satisfied.
Hence has a unique fixed point in .
Theorem 8. Let and be two self-mappings of a 2-Banach space . Suppose that for any , where is a continuous mapping such that if and only if . Then and have a unique common fixed point in .
Proof. For , define by . So for any ,
Thus is a monotone decreasing sequence of real numbers and hence is convergent and
We now claim that .
If not, So, from (24) taking implying that , a contradiction.
Now for any integer
So is Cauchy in and hence by completeness of , let Now for all , By routine check up, we can see that for sufficiently large values of , Hence, .
Similarly we can show that .
We shall now show that is unique.
If not let be another fixed point of and . Then for any
So , showing that . So implies .
Example 9. Let be and consider the following 2-norm on as
where , . Then is a 2-Banach space (see [8, 9]).
Define by for all and for all .
Now setting is a continuous mapping such that if and only if . Take .
Now By routine calculation, it can be seen that the condition in Theorem 8 is satisfied. Also and have a unique common fixed point in .
Theorem 10. Let be a 2-Banach space with and let be a sequence of mappings such that(i) for all and where is a continuous mapping such that if and only if ,(ii) for each . Then has a unique fixed point in such that where is the fixed point of .
As is continuous,
So from (40) taking limit as , we obtain
Hence by Theorem 6, has a unique fixed point in .
If possible, let Now for let Now Now So from (46), (47), (48), and (49), we get
Hence, showing that which is possible only when
Hence , implying that .
Theorem 11. Let be a 2-Banach space, a sequence of mappings of onto itself with fixed points . Let be a mapping of into itself satisfying the condition of Theorem 6 with fixed point , such that uniformly on . Then .
Proof. Fix . From the uniform convergence of on , there exists an integer such that for all , and for all , for all .
Now If possible let that is there exists , Now for all , So for satisfying (52) Therefore As is arbitrary, Again Hence As is arbitrary, Therefore Hence Taking in (53), we get , showing that which possible only when , a contradiction to (52).
Hence , implying that .
Theorem 12. Let be a 2-Banach space and for any , a sequence of mappings of onto itself. Suppose there exists a sequence of nonnegative integers such that for all , and every pair with satisfying where is a continuous mappings such that if and only if . Then have a unique common fixed point in .
Proof. Define .
Pick and define .
Now By routine calculation, we can see that Implying that is a monotone decreasing sequence of real numbers and hence is convergent and let We shall now show that .
If not, Taking , Hence Taking in (64), we get , showing that which is possible only when .
Now for any integer Therefore is Cauchy in and hence by completeness of , let
Now for any integer , Hence Therefore implying that for all .
Hence has a common fixed point .
If is another fixed point of , then Implying that . Hence implying that .
Now for each , showing that is the fixed point of .
Also and . Hence for all .
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