Abstract
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.
1. Introduction
The theory of 2-Banach spaces was investigated by Gähler in [1] and Iseki in [2] 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.
2. Preliminaries
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 [7] 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 ,
Hence
Similarly
Thus is a monotone decreasing sequence of real numbers and hence is convergent and
Let
We now claim that .
If not,
So,
from (24) taking
implying that , a contradiction.
Therefore, .
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 .
Proof.
Now
Similarly
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 .
Then
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 .