International Journal of Analysis and Applications
ISSN 2291-8639
Volume 3, Number 1 (2013), 60-67
http://www.etamaths.com

FIXED POINTS OF EXPANSIVE TYPE MAPPINGS IN

2-BANACH SPACES

PRABHA CHOUHAN1,∗, NEERAJ MALVIYA2

Abstract. In present paper, we define expansive mappings in 2-Banach space

and prove some common unique fixed point theorems which are the extension

of results of Wang et al. [12] and Rhoades [9] in 2-Banach space.

1. Introduction

The research about fixed points of expansive mapping was initiated by Machuca
(see [6]). Later Jungck discussed fixed points for other forms of expansive mapping
(see [5]). In 1982, Wang et al. (see [12]) presented some interesting work on expan-
sive mappings in metric spaces which correspond to some contractive mapping in
[10]. Also, Zhang has done considerable work in this field. In order to generalize
the results about fixed point theory, Zhang (See [14]) published his work Fixed
Point Theory and Its Applications, in which the fixed point problem for expansive
mapping is systematically presented in a chapter. As applications, he also inves-
tigated the existence of solutions of equations for locally condensing mapping and
locally accretive mapping. On the other hand Gahler ([2],[3]) investigated the idea
of 2-metric and 2-Banach spaces and proved same results. Subsequently several
authors including Iseki [4], Rhoades [8], White [13], Panja and Baisnab [7] and
Saha et al [11] studied various aspects of the fixed point theory and proved fixed
point theorems in 2-metric spaces and 2-Banach spaces. Recently, the study about
fixed point theorem for expansive mapping is deeply explored and has extended too
many others directions. Motivated and inspired by the above work, in this paper we
investigate fixed point for expansive mapping in 2-Banach spaces. The presented
theorems extend, generalize and improve many existing results in the literature [9]
[12].

2. Preliminaries

Definition 2.1. Let X be a real linear space and ‖ ., . ‖ be a non-negative real
valued function defined on X ×X satisfying the following conditions:
i) ‖x,y‖ = 0 if and only if x and y are linearly dependent in X,
ii) ‖x,y‖ = ‖y,x‖ for all x,y ∈ X
iii) ‖x,ay‖ =| a | ‖x,y‖ a being real, x,y ∈ X
iv) ‖x,y + z‖ = ‖x,y‖ + ‖x,z‖ for all x,y,z ∈ X

2010 Mathematics Subject Classification. 47H10, 54H25.

Key words and phrases. 2-normed space, 2-Banach space, Expansive mapping, Fixed point.

c©2013 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

60



FIXED POINTS OF EXPANSIVE TYPE MAPPINGS 61

Then ‖ ., . ‖ is called a 2-norm and the pair (X,‖ ., . ‖) is called a linear 2-
normed space.

Some of the basic properties of 2-norms are that they are non negative satisfying
‖x,y + ax‖ = ‖x,y‖, for all x,y ∈ X and all real numbers a.

Definition 2.2. A sequence {xn} in a linear 2-normed space (X,‖ ., . ‖) is called
Cauchy sequence if

lim
m,n→∞

‖ xm −xn,y ‖= 0 for all y in X.

Definition 2.3. A sequence {xn} in a linear 2-normed space (X,‖ ., . ‖) is said
to be convergent if there is a point x in X such that

lim
n→∞

‖ xn −x,y ‖= 0 for all y in X.

If {xn} converges to x, we write {xn}→ x as n →∞.
Definition 2.4. A linear 2-normed space X is said to be complete if every Cauchy
sequence is convergent to an element of X. We then call X to be a 2- Banach space.

Definition 2.5. Let X be a 2-Banach space and T be a self-mapping of X. T is
said to be continuous at x if for every sequence {xn} in X, {xn} → x as n → ∞
implies {T(xn)}→ T(x) as n →∞ .

Example 2.6. Let X is R3 and consider the following 2-norm on X as

‖ x, y ‖ =

∣∣∣∣∣∣det

 i j kx1 x2 x3

y1 y2 y3



∣∣∣∣∣∣ ,

where x = (x1,x2,x3) and y = (y1,y2,y3). Then (X,‖ ., . ‖) is a 2-Banach space.

Example 2.7. Let Pn denotes the set of all real polynomials of degree ≤ n, on
the interval [0,1]. By considering usual addition and scalar multiplication, Pn is a
linear vector space over the reals. Let {xo,x1, ..........,x2n} be distinct fixed points
in [0,1] and define the following 2-norm on Pn:‖ f,g ‖=

∑2n
k=0 | f(xk) g(xk) |,

whenever f and g are linearly independent and ‖ f,g ‖= 0, if f, g are linearly
dependent. Then (Pn,‖ ., . ‖) is a 2-Banach space.

Example 2.8. Let X is Q3, the field of rational number and consider the fol-
lowing 2-norm on X as:

‖ x, y ‖ =

∣∣∣∣∣∣det

 i j kx1 x2 x3

y1 y2 y3



∣∣∣∣∣∣ ,

where x = (x1,x2,x3) and y = (y1,y2,y3). Then (X,‖ ., . ‖) is not a 2-Banach
space but a 2-normed space.



62 CHOUHAN AND MALVIYA

3. Main Results

Definition 3.1. Let (X,‖ ., . ‖) be a 2-Banach space with 2-norm ‖ ., . ‖ . A
mapping T of X into itself is said to be expansive if there exists a constant h > 1
such that

‖ Tx−Ty,a ‖≥ h ‖ x−y,a ‖ for all x,y ∈ X

Example 3.2. Let X = R2 and consider the following 2-norm on X as
‖ x,y ‖ = | x1y2 −x2y1 | where x = (x1,x2) and y = (y1,y2). Then (X,‖ ., . ‖) is a
2-Banach space. Define a self map T on X as follows Tx = βx where β > 1 for all
x ∈ X, clearly T is an expansive mapping.

Theorem 3.3. Let (X,‖ ., . ‖) be a 2-Banach space and S, T be surjective mapping
of X into itself satisfying.

‖ Sx−Ty,a ‖ + k
[
‖ x−Ty,a ‖ + ‖ y −Sx,a ‖

]
≥ a1 ‖ x−Sx,a ‖

+ b1 ‖ y −Ty,a ‖ +c1 ‖ x−y,a ‖(3.3.1)

for all x,y,a ∈ X, x 6= y where a1,b1,c1,k ≥ 0, satisfy a1 < k + 1, b1 < k + 1,c1 >
2k + 1. Then S and T have a common unique fixed point in X.

Proof: We define a sequence {xn} as follows for n = 0, 1, 2, 3, ...

(3.3.2) x2n = Sx2n+1, x2n+1 = Tx2n+2

If x2n = x2n+1 = x2n+2 for some n then we see that x2n is a fixed point of S and
T. Therefore, we suppose that no two consecutive terms of sequence {xn} are equal.

Now we put x = x2n+1 and y = x2n+2 in (3.3.1) we get

‖ Sx2n+1 −Tx2n+2,a ‖ +k
[
‖ x2n+1 −Tx2n+2,a ‖ + ‖ x2n+2 −Sx2n+1,a ‖

]
≥ a1 ‖ x2n+1 −Sx2n+1,a ‖ +b1 ‖ x2n+2 −Tx2n+2,a ‖ +c1 ‖ x2n+1 −x2n+2,a ‖

⇒‖ x2n −x2n+1,a ‖ + k
[
‖ x2n+1 −x2n+1,a ‖ + ‖ x2n+2 −x2n,a ‖

]
≥ a1 ‖ x2n+1 −x2n,a ‖

+ b1 ‖ x2n+2 −x2n+1,a ‖ +c1 ‖ x2n+1 −x2n+2,a ‖

⇒‖ x2n −x2n+1,a ‖ + k
[
‖ x2n+2 −x2n,a ‖

]
≥ a1 ‖ x2n+1 −x2n,a ‖ +b1 ‖ x2n+2 −x2n+1,a ‖

+ c1 ‖ x2n+1 −x2n+2,a ‖

⇒‖ x2n −x2n+1,a ‖ + k
[
‖ x2n+2 −x2n+1,a ‖ + ‖ x2n+1 −x2n,a ‖

]
≥ a1 ‖ x2n+1 −x2n,a ‖

+ b1 ‖ x2n+2 −x2n+1,a ‖ +c1 ‖ x2n+1 −x2n+2,a ‖

⇒ (1 + k −a1) ‖ x2n −x2n+1,a ‖≥ (b1 + c1 −k) ‖ x2n+1 −x2n+2,a ‖

⇒‖ x2n+1 −x2n+2,a ‖≤
(1+k−a1)
(b1+c1−k)

‖ x2n −x2n+1,a ‖

⇒‖ x2n+1 −x2n+2,a ‖≤ k1 ‖ x2n −x2n+1,a ‖
where k1 =

(1+k−a1)
(b1+c1−k)

< 1 (As a1 + b1 + c1 > 1 + 2k)



FIXED POINTS OF EXPANSIVE TYPE MAPPINGS 63

Similarly, we can calculate

‖ x2n+2 −x2n+3,a ‖≤ k2 ‖ x2n+1 −x2n+2,a ‖ for n = 0, 1, 2, ...
where k2 =

(1+k−a1)
(b1+c1−k)

< 1 (As a1 + b1 + c1 > 1 + 2k)

and so on

So, in general

⇒‖ xn −xn+1,a ‖≤ k ‖ xn−1 −xn,a ‖ for n = 1, 2, 3...
where k = max{k1,k2} then k < 1

(3.3.3) ⇒‖ xn −xn+1,a ‖≤ kn ‖ x0 −x1,a ‖
We can prove that {xn} is a Cauchy sequence (using (3.3.3)). So there exists a
point x in X such that

(3.3.4) {xn}→ x as n →∞
Existence of fixed point: Since S and T are surjective maps, so there exist

two points y and y′ in X such that

(3.3.5) x = Sy and x = Ty′

Consider
‖ x2n −x,a ‖=‖ Sx2n+1 −Ty′,a ‖

≥−k
[
‖ x2n+1 −Ty′,a ‖ + ‖ y′ −Sx2n+1,a ‖

]
+ a1 ‖ x2n+1 −Sx2n+1,a ‖ +b1 ‖ y′ −Ty′,a ‖

+ c1 ‖ x2n+1 −y′,a ‖

⇒‖ x2n −x,a ‖≥−k
[
‖ x2n+1 −Ty′,a ‖ + ‖ y′ −x2n,a ‖

]
+ a1 ‖ x2n+1 −x2n,a ‖

+ b1 ‖ y′ −Ty′,a ‖ +c1 ‖ x2n+1 −y′,a ‖

As {x2n}, {x2n+1} are subsequences of {xn} as n → ∞, {x2n} → x, {x2n+1} → x
(Using 3.3.4)

Therefore

‖ x−x,a ‖≥−k
[
‖ x−x,a ‖ + ‖ y′ −x,a ‖

]
+ a1 ‖ x−x,a ‖ +b1 ‖ y′ −x,a ‖

+ c1 ‖ x−y′,a ‖

⇒ 0 ≥ (b1 + c1 −k) ‖ x−y′,a ‖

⇒‖ x−y′,a ‖= 0 (As 2k + 1 < a1 +b1 +c1 < k + 1 +b1 +c1, so that k < b1 +c1)
(3.3.6) ⇒ x = y′

In an exactly similar way (Using b1 < k + 1) we can prove that,

(3.3.7) x = y

The fact (3.3.5) along with (3.3.6) and (3.3.7) shows that x is a common fixed point
of S and T .



64 CHOUHAN AND MALVIYA

Uniqueness: Let z be another common fixed point of S and T , that is

Sz = z and Tz = z

‖ x−z,a ‖ = ‖ Sx−Tz,a ‖

≥−k
[
‖ x−Tz,a ‖ + ‖ z −Sx,a ‖

]
+ a1 ‖ x−Sx,a ‖ +b1 ‖ z −Tz,a ‖

+ c1 ‖ x−z,a ‖

⇒‖ x−z,a ‖≥−k
[
‖ x−z,a ‖ + ‖ z −x,a ‖

]
+ a1 ‖ x−x,a ‖ +b1 ‖ z −z,a ‖ +c1 ‖ x−z,a ‖

⇒‖ x−z,a ‖≥ (−2k + c1) ‖ x−z,a ‖
⇒ (1 + 2k − c1) ‖ x−z,a ‖≥ 0

⇒‖ x−z,a ‖= 0 (As c1 > 2k + 1)

⇒ x = z

This completes the proof of the theorem 3.3

Corollary 3.4. Let (X,‖ ., . ‖) be a 2-Banach space and S and T be two
surjective mappings of X into itself such that for every x,y,a ∈ X

‖ Sx−Ty,a ‖≥ a1 ‖ x−y,a ‖(3.4.1)

where a1 > 1. Then S and T have a common unique fixed point in X.

Proof: If we put k,a1,b1 = 0 and c1 = a1 in theorem 3.3 then we get above corol-
lary 3.4.

Corollary 3.5. Let (X,‖ ., . ‖) be a 2-Banach space and T be a surjective
mapping of X into itself such that for every x,y,a ∈ X

‖ Tx−Ty,a ‖≥ a1 ‖ x−y,a ‖(3.5.1)

where a1 > 1. Then T has a unique fixed point in X.

Proof: If we put S = T in corollary 3.4 then we get above corollary 3.5 which is
an extension of theorem 1 of Wang et al. [12] in 2-Banach space.

Corollary 3.6. Let (X,‖ ., . ‖) be a 2-Banach space and S, T be two surjective
mappings of X into itself satisfying.

‖ Sx−Ty,a ‖≥ a1 ‖ x−Sx,a ‖ +a2 ‖ y −Ty,a ‖ +a3 ‖ x−y,a ‖(3.6.1)

for each x,y ∈ X, with x 6= y where a1,a2,a3 ≥ 0 and a3 > 1. Then S and T have
a common unique fixed point in X.

Proof: If we put k = 0 and b1 = a2, c1 = a3 in theorem 3.3 then we get above
corollary 3.6.



FIXED POINTS OF EXPANSIVE TYPE MAPPINGS 65

Corollary 3.7. Let (X,‖ ., . ‖) be a 2-Banach space and T be a surjective
mapping of X into itself satisfying.

‖ Tx−Ty,a ‖≥ a1 ‖ x−Tx,a ‖ +a2 ‖ y −Ty,a ‖ +a3 ‖ x−y,a ‖(3.7.1)

for each x,y ∈ X with x 6= y where a1,a2,a3 ≥ 0 and a3 > 1. Then T has a unique
fixed point in X.

Proof: If we put S = T in corollary 3.6 then we get above corollary 3.7 which is
an extension of theorem 2 of Wang et al. [12] in 2-Banach space.

The following theorem 3.8 is an example in which the fixed point need not be
unique and continuity of self maps are required and it extend the theorem 3 of
Rhoades [9] in 2-Banach spaces.

Theorem 3.8. Let (X,‖ ., . ‖) be a 2-Banach space and S, T be two continuous
mappings of X into itself satisfying.

‖ Sx−Ty,a ‖≥ a1 min{‖ x−y,a ‖, ‖ x−Sx,a ‖, ‖ y −Ty,a ‖}(3.8.1)

for every x,y ∈ X, x 6= y Where a1 > 1. Then S and T have a common fixed point
in X.

Proof: We define a sequence {xn} as follows for n = 0, 1, 2, 3, ...

x2n = Sx2n+1, x2n+1 = Tx2n+2

Now we put x = x2n+1 and y = x2n+2 in (3.8.1) we get

‖Sx2n+1 −Tx2n+2,a‖≥ a1 min
{
‖x2n+1 −x2n+2,a‖ ,‖x2n+1 −Sx2n+1,a‖ ,‖x2n+2 −Tx2n+2,a‖

}
= a1 min

{
‖x2n+1 −x2n+2,a‖ ,‖x2n+1 −x2n,a‖ ,‖x2n+2 −x2n+1,a‖

}
⇒‖x2n −x2n+1,a‖≥ a1 min

{
‖x2n+1 −x2n+2,a‖ ,‖x2n −x2n+1,a‖

}

Case I

‖ x2n −x2n+1,a ‖≥ a1 ‖ x2n+1 −x2n,a ‖

⇒ 1 ≥ a1

Which is contradiction.

Case II
‖ x2n+1 −x2n+2,a ‖≤ 1a1 ‖ x2n −x2n+1,a ‖

‖ x2n+1 −x2n+2,a ‖≤ k ‖ x2n −x2n+1,a ‖



66 CHOUHAN AND MALVIYA

where k = 1
a1
< 1 (As a1 > 1)

So, in general

⇒‖ xn −xn+1,a ‖≤ k ‖ xn−1 −xn,a ‖ for n = 1, 2, 3...

(3.8.2) ⇒‖ xn −xn+1,a ‖≤ kn ‖ x0 −x1,a ‖
We can prove that {xn} is a Cauchy sequence (using (3.8.2). So there exists a point
x in X such that

(3.8.3) {xn}→ x as n →∞
Existence of Fixed Point: If S and T are continuous then existence part follows
very easily. As shown below

x = lim
n→∞

x2n = lim
n→∞

Sx2n+1 = S lim
n→∞

x2n+1 = Sx (as n →∞{x2n+1}→ x)

Similarly

x = lim
n→∞

x2n+1 = lim
n→∞

Tx2n+2 = T lim
n→∞

x2n+2 = Tx (as n →∞{x2n+2}→ x)

This completes the proof of the theorem 3.8

Corollary 3.9. Let (X,‖ ., . ‖) be a 2-Banach space and T be a continuous
mapping of X into itself satisfying.

‖ Tx−Ty,a ‖≥ a1 min{‖ x−y,a ‖, ‖ x−Tx,a ‖, ‖ y −Ty,a ‖}(3.9.1)
for every x,y ∈ X, x 6= y Where a1 > 1. Then T has a fixed point in X.

Proof: If we put S = T in theorem 3.8 then we get above corollary 3.9 which is
an extension of theorem 3 of Wang et al. [12] in 2-Banach space.

Remark 1: If mappings are continuous in theorem 3.3 then existence of fixed
point follows very easily as proved in theorem 3.8.

Remark 2 : In corollary 3.9, we proved the fixed point is unique by using only
a3 > 1 and there is no need of a1 < 1 and a2 < 1, so it extend and unify the
theorem 2 of Wang et al. [12].

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1Scope College of Engineering Bhopal (M.P.), India

2NRI Institute of Information Science and Technology, Bhopal (M.P.) India

∗Corresponding author