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 .