@ Appl. Gen. Topol. 20, no. 1 (2019), 43-55doi:10.4995/agt.2019.9168
c© AGT, UPV, 2019

Approximation of common fixed points in

2-Banach spaces with applications

D. Ramesh Kumar
a
and M. Pitchaimani

b

a
Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology,

Vellore - 632014, India. (rameshkumard14@gmail.com)
b

Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai -

600005, India. (mpitchaimani@yahoo.com)

Communicated by E. A. Sánchez-Pérez

Abstract

The purpose of this paper is to establish the existence and uniqueness

of common fixed points of a family of self-mappings satisfying gener-

alized rational contractive condition in 2-Banach spaces. An example

is included to justify our results. We approximate the common fixed

point by Mann and Picard type iteration schemes. Further, an applica-

tion to well-posedness of the common fixed point problem is given. The

presented results generalize many known results on 2-Banach spaces.

2010 MSC: 47H10; 54H25.

Keywords: common fixed point; Mann iteration; Picard iteration; well-

posedness; 2-Banach space.

1. Introduction

The Banach Contraction Principle [5] is one of the pivotal results in nonlinear
analysis which guarantees the existence and uniqueness of fixed points of map-
pings. There is a great number of extensions of the Banach contraction princi-
ple using different forms of contractive conditions in various generalized metric
spaces, we refer the reader to [1, 2, 6, 17, 13, 14, 20, 23, 24, 25, 27, 28, 29, 30].
Some of such generalizations are obtained through rational contractive condi-
tions. Das and Gupta [8] studied the extention of Banach contraction in term

Received 09 January 2018 – Accepted 26 September 2018

http://dx.doi.org/10.4995/agt.2019.9168


D. Ramesh Kumar, M. Pitchaimani

of rational inequality and derived a fixed point result for a self mapping which
runs as follows:

Theorem 1.1. Let f be a mapping of a complete metric space X into itself

such that

(i)

d(f(x), f(y)) ≤ α
d(y, f(y))[1 + d(x, f(x))]

1 + d(x, y)
+ βd(x, y),

for all x, y ∈ X, α > 0, β > 0, α + β < 1 and
(ii) for some x0 ∈ X, the sequence of iterates {f

n(x0)} has a subsequence
{fnk (x0)} with ξ = lim

n→∞
fnk (x0).

Then ξ is a unique fixed point of f.

Meanwhile Pachpatte [19] extended it to a pair of self mappings. Recently,
Azam et al. [4] obtained fixed point theorems for a pair of contractive mappings
using an analogical rational inequality of [8] in a complex-valued metric space
setting. Also, in [18], Nashine et al. studied some common fixed point theorems
for a pair of mappings under certain rational contractions in complex valued
metric spaces. In [33], Shahkoohi and Razani established the existence of fixed
point of a self mapping satisfying rational Geraghty contractive mappings in
partially ordered b-metric spaces. For related study [3, 32].

In the 1960’s, Gähler [10, 11, 12] generalized the idea of metric space and
introduced a new theory of 2-metric space. On the other hand, White [35]
started the investigation on the concept of 2-Banach spaces. Since then, many
authors have focused on these spaces and presented papers that dealt with
fixed point theory for single-valued and multi-valued operators in 2-Banach
spaces (see [7, 15]). Recently, Pitchaimani and Ramesh Kumar [21] obtained
common fixed points under implicit relation in 2-Banach spaces and proved
some common and coincidence fixed point theorems for asymptotically regular
mappings in [22].

Inspired by the concept of 2-Banach spaces and notion of rational type con-
ditions, an attempt has been made in this paper to prove the existence and
uniqueness of common fixed point of a family of self mappings satisfying the
generalized rational contractive condition in 2-Banach spaces with an example
which illustrates our result. Further, the approximation of the common fixed
point by means of Mann and Picard iteration method is given. Finally, we
prove the well-posedness of the common fixed point problem.

2. Preliminaries

In this section, we recall the notions which will be required in the sequel.
Throughout this paper, N denotes the set of all positive integers and R denotes
the set of all real numbers.

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:

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Approximation of common fixed points in 2-Banach spaces

(i) ‖x, y‖ = 0 if and only if x and y are linearly dependent;
(ii) ‖x, y‖ = ‖y, x‖, for all x, y ∈ X;
(iii) ‖x, ay‖ = |a|‖x, y‖, for all x, y ∈ X and a ∈ R;
(iv) ‖x, y + z‖ ≤ ‖x, y‖ + ‖x, z‖, for all x, y, z ∈ X;

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 sat-
isfying ‖x, y + ax‖ = ‖x, y‖, for all x, y ∈ X and a ∈ R.

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

m,n→∞
‖xm − xn, y‖ = 0 for all y ∈ X.

Definition 2.3. A sequence {xn} in a linear 2-normed space (X, ‖·, ·‖) is said
to converge to a point x ∈ X if lim

n→∞
‖xn − x, y‖ = 0 for all y ∈ X.

Definition 2.4. A linear 2-normed space (X, ‖·, ·‖) in which every Cauchy
sequence is convergent is called a 2-Banach space.

Example 2.5. Let X = R3 and a 2-norm ‖·, ·‖ be defined as follows:

‖x, y‖ = |x × y| = det





~i ~j ~k

x1 x2 x3
y1 y2 y3





where x = (x1, x2, x3), y = (y1, y2, y3) ∈ X and~i,~j,~k are the unit vectors along
the axes. Note that (X, ‖·, ·‖) is a 2-Banach space.

Definition 2.6. A sequence {xn} in a 2-Banach space X is said to be asymp-
totically T - regular if lim

n→∞
‖xn − T xn, y‖ = 0 for all y ∈ X.

Definition 2.7. Let X be a nonempty set and S, T : X → X be self mappings.
Then

(i) an element x ∈ X is said to be a fixed point of T if x = T x.
(ii) If x = Sx = T x then x is called a common fixed point of S and T .

Let C be a nonempty convex subset of a 2-Banach space (X, ‖·, ·‖) and
T : X → X be a mapping then

(i) the sequence {xn} defined by

x0 ∈ C, xn+1 = (1 − βn)xn + βnT (xn), ∀n ≥ 0,

where {βn} satisfies 0 < βn ≤ 1, ∀n and
∞
∑

n=0

βn = ∞, is called the

Mann iteration scheme.
(ii) the sequence {xn} defined by

x0 ∈ C, xn+1 = T (xn), ∀n ≥ 0,

is called the Picard iteration scheme which is particular case of the
Mann iteration scheme.

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D. Ramesh Kumar, M. Pitchaimani

3. Main Results

In this section, we prove the existence and uniqueness of common fixed point
of a family of self mappings.

Theorem 3.1. Let (X, ‖·, ·‖) be a 2-Banach space and S, T : X → X be two
self mappings such that

‖Sx − T y, u‖ ≤ a
‖y − T y, u‖

[

1 + ‖x − Sx, u‖
]

1 + ‖x − y, u‖
+ b

‖x − Sx, u‖
[

1 + ‖y − T y, u‖
]

1 + ‖x − y, u‖

+c
‖x − T y, u‖

[

1 + ‖y − Sx, u‖
]

1 + ‖x − y, u‖
+ d

‖y − Sx, u‖
[

1 + ‖x − T y, u‖
]

1 + ‖x − y, u‖

+e‖x − y, u‖,(3.1)

for all x, y, u ∈ X, x 6= y, where a, b, c, d, e are non negative constants such that
a + b + c + d + e < 1. Then S and T have a unique common fixed point in X.

Proof. For x0 ∈ X, define a sequence as follows:

x2n+1 = Sx2n, x2n+2 = T x2n+1, n = 0, 1, 2, . . .

Now for all u ∈ X, using (3.1), we get

‖x2n+1 − x2n, u‖ = ‖Sx2n − T x2n−1, u‖

≤ a
‖x2n−1 − x2n, u‖[1 + ‖x2n − x2n+1, u‖]

1 + ‖x2n − x2n−1, u‖

+b
‖x2n − x2n+1, u‖[1 + ‖x2n − x2n−1, u‖]

1 + ‖x2n − x2n−1, u‖

+c
‖x2n − x2n, u‖[1 + ‖x2n−1 − x2n+1, u‖]

1 + ‖x2n − x2n−1, u‖

+d
‖x2n−1 − x2n+1, u‖[1 + ‖x2n − x2n, u‖]

1 + ‖x2n − x2n−1, u‖

+e‖x2n − x2n−1, u‖,

which gives
‖x2n+1 − x2n, u‖ ≤ kn‖x2n − x2n−1, u‖,

where kn =
(a + d + e) + e‖x2n − x2n−1, u‖

(1 − b − d) + (1 − a − b)‖x2n − x2n−1, u‖
.

Note that kn < 1, as a+b+c+d+e < 1. Repeating the same argument several
times, we obtain

‖x2n+1 − x2n, u‖ ≤ (kn)
2n‖x1 − x0, u‖.

Now, for n > m, we have

‖xn − xm, u‖ ≤ ‖xn − xn−1, u‖ + ‖xn−1 − xn−2, u‖ + · · · · · · + ‖xm+1 − xm, u‖

≤ ((kn)
n−1 + (kn)

n−2 + · · · · · · + (kn)
m)‖x1 − x0, u‖

≤
(kn)

m

1 − kn
‖x1 − x0, u‖.

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Approximation of common fixed points in 2-Banach spaces

Thus ‖xn −xm, u‖ → 0 as m, n → ∞, since
(kn)

m

1 − kn
→ 0 as m → ∞. This shows

that {xn} is a Cauchy sequence in X. Hence, there exists a point w ∈ X such
that xn → w as n → ∞. Further, we get

‖w − T w, u‖ ≤ ‖w − x2n+1, u‖ + ‖x2n+1 − T w, u‖

= ‖w − x2n+1, u‖ + ‖Sx2n − T w, u‖

≤ ‖w − x2n+1, u‖ + a
‖w − T w, u‖[1 + ‖x2n − x2n+1, u‖]

1 + ‖x2n − w, u‖

+b
‖x2n − x2n+1, u‖[1 + ‖w − T w, u‖]

1 + ‖x2n − w, u‖

+c
‖x2n − T w, u‖[1 + ‖w − x2n+1‖]

1 + ‖x2n − w, u‖

+d
‖w − x2n+1, u‖[1 + ‖x2n − T w, u‖]

1 + ‖x2n − w, u‖

+e‖x2n − w, u‖.

Letting n → ∞, we have

‖w − T w, u‖ ≤ (a + c)‖w − T w, u‖,

which implies that T w = w, as (a + c) < 1 for all u ∈ X. Similarly, we get
w = Sw. Thus w is a common fixed point of S and T.

For the uniqueness, assume that v ∈ X is another common fixed point of S
and T , that is, v = Sv = T v. Then

‖w − v, u‖ = ‖Sw − T v, u‖

≤ a
‖v − T v, u‖[1 + ‖w − Sw, u‖]

1 + ‖w − v, u‖

+b
‖w − Sw, u‖[1 + ‖v − T v, u‖]

1 + ‖w − v, u‖

+c
‖w − T v, u‖[1 + ‖v − Sw, u‖]

1 + ‖w − v, u‖

+d
‖v − Sw, u‖[1 + ‖w − T v, u‖]

1 + ‖w − v, u‖

+e‖w − v, u‖,

which yields

‖w − v, u‖ + ‖w − v, u‖2 ≤ (c + d + e)‖w − v, u‖ + (c + d + e)‖w − v, u‖2,

which is a contradiction, since c + d + e < 1. Hence w is the unique common
fixed point of S and T . �

Taking S = T in Theorem 3.1, we obtain the following result.

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D. Ramesh Kumar, M. Pitchaimani

Corollary 3.2. Let (X, ‖·, ·‖) be a 2-Banach space and T : X → X be a self
mapping such that

‖T x − T y, u‖ ≤ a
‖y − T y, u‖

[

1 + ‖x − T x, u‖
]

1 + ‖x − y, u‖
+ b

‖x − T x, u‖
[

1 + ‖y − T y, u‖
]

1 + ‖x − y, u‖

+c
‖x − T y, u‖

[

1 + ‖y − T x, u‖
]

1 + ‖x − y, u‖
+ d

‖y − T x, u‖
[

1 + ‖x − T y, u‖
]

1 + ‖x − y, u‖

+e‖x − y, u‖,(3.2)

for all x, y, u ∈ X, x 6= y, where a, b, c, d, e are non negative constants and
a + b + c + d + e < 1. Then T have a unique fixed point in X.

Proof. The proof follows from Theorem 3.1. �

Remark 3.3. In the case b = c = d = 0, Corollary 3.2 reduces to Theorem 1.1.

Now we extend the Theorem 3.1 to the case of pair of mappings Sp and T q

where p and q are some positive integers.

Theorem 3.4. Let (X, ‖·, ·‖) be a 2-Banach space and S, T : X → X be two
self mappings such that

‖Spx − T qy, u‖ ≤ a
‖y − T qy, u‖

[

1 + ‖x − Spx, u‖
]

1 + ‖x − y, u‖
+ b

‖x − Spx, u‖
[

1 + ‖y − T qy, u‖
]

1 + ‖x − y, u‖

+c
‖x − T qy, u‖

[

1 + ‖y − Spx, u‖
]

1 + ‖x − y, u‖
+ d

‖y − Spx, u‖
[

1 + ‖x − T qy, u‖
]

1 + ‖x − y, u‖

+e‖x − y, u‖,(3.3)

for all x, y, u ∈ X, x 6= y, where p and q are some positive integers and
a, b, c, d, e are non negative constants with a + b + c + d + e < 1. Then S and T
have a unique common fixed point in X.

Proof. Note that Sp and T q satisfy the conditions of Theorem 3.1, so Sp and
T q have a unique common fixed point. Let w be the common fixed point. Then

Spw = w ⇒ S(Spw) = Sw,
Sp(Sw) = Sw.

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Approximation of common fixed points in 2-Banach spaces

If Sw = x0 then S
p(x0) = x0. So, Sw is a fixed point of S

p. Similarly, T q(T w) =
T w. Now, we have

‖w − T w, u‖ = ‖Spw − T q(T w), u‖

≤ a
‖T w − T q(T w), u‖[1 + ‖w − Spw, u‖]

1 + ‖w − T w, u‖

+b
‖w − Spw, u‖[1 + ‖T w − T q(T w), u‖]

1 + ‖w − T w, u‖

+c
‖w − T q(T w), u‖[1 + ‖T w − Spw, u‖]

1 + ‖w − T w, u‖

+d
‖T w − Spw, u‖[1 + ‖w − T q(T w), u‖]

1 + ‖w − T w, u‖

+e‖w − T w, u‖,

which implies that

‖w −T w, u‖+‖w −T w, u‖2 ≤ (c+d+e)‖w −T w, u‖+(c+d+e)‖w −Tw, u‖2,

which is a contradiction, as we are having c + d + e < 1. Thus, w = T w for all
u ∈ X. With the similar arguments, we obtain w = Sw.

Finally, in order to prove the uniqueness of v, let w 6= v be another common
fixed point of S and T . Then clearly w is also a common fixed point of Sp

and T q which implies w = v. Therefore, S and T have a unique common fixed
point. �

Hence we have proved that if x0 is a unique common fixed point of S
p and

T q, where p, q are some positive integers then x0 is a unique common fixed
point of S and T . Next we extend Theorem 3.1 to a case of family of mappings
satisfying the condition (3.1).

Theorem 3.5. Let (X, ‖·, ·‖) be a 2-Banach space and {Fα} be a family of self
mappings on X such that

‖Fαx − Fβy, u‖ ≤ a
‖y − Fβy, u‖

[

1 + ‖x − Fαx, u‖
]

1 + ‖x − y, u‖
+ b

‖x − Fαx, u‖
[

1 + ‖y − Fβy, u‖
]

1 + ‖x − y, u‖

+c
‖x − Fβy, u‖

[

1 + ‖y − Fαx, u‖
]

1 + ‖x − y, u‖
+ d

‖y − Fαx, u‖
[

1 + ‖x − Fβy, u‖
]

1 + ‖x − y, u‖

+e‖x − y, u‖,

for all α, β ∈ Λ with α 6= β and x, y, u ∈ X with x 6= y, where a, b, c, d, e are
non negative constants such that a + b + c + d + e < 1. Then there exists a
unique w ∈ X satisfying Fαw = w for all α ∈ Λ.

Proof. Let us take Fα and Fβ in place of S and T respectively in Theorem 3.1,
an application of which gives a unique w ∈ X to satisfy Fαw = Fβw = w. For
any other member Fγ, uniqueness of w gives Fγw = w and this completes the
proof. �

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 49



D. Ramesh Kumar, M. Pitchaimani

Remark 3.6. Theorem 3.5 generalizes and improves the results of [4, 8, 18, 19,
32] in 2-Banach space setting.

Example 3.7. Let X = R2 and a 2-norm ‖·, ·‖ be defined by ‖x, y‖ = |x1y2 −
x2y1|. Note that (X, ‖·, ·‖) is a 2-Banach space. Let S, T : X → X be two self
mappings defined as follows:

S(x, y) =

(

x + y

2
,
x + y

2

)

and

T (u, v) =

(

u

2
,
u

2

)

.

For a, b, c, d, e ∈ [1
2
, 1), it can be easily seen that (3.1) is satisfied. Hence, by

Theorem 3.1, S and T have a unique common fixed point in X. Here,

(

1

2
,
1

2

)

is the unique common fixed point of S and T .

4. Approximation by Mann and Picard Type Iteration

In this section, we approximate the common fixed point of S and T by a
Mann and Picard type iteration schemes.

Theorem 4.1. Let C be a nonempty closed convex subset of a 2-Banach space

X and S, T : C → C be two self mappings such that (1−k)S(C)+kT (C) ⊂ S(C)
for 0 < k ≤ 1 and S(C) is closed. Suppose that S and T satisfy all the
conditions of Theorem 3.1, then S and T have a unique common fixed point.
Moreover, if, for arbitrary y0 ∈ C, the sequence {yn} defined by

(4.1) S(yn+1) = (1 − βn)S(yn) + βnT (yn), ∀n ≥ 0,

where {βn} satisfies 0 < βn ≤ 1, ∀n and
∞
∑

n=0

βn = ∞, is asymptotically T -

regular, then it converges to the unique common fixed point of S and T , with a

rate estimated by

‖(S(yn+1) − w, u‖ ≤ λ
n+1L,

where λ ∈ [0, 1) and L ≥ 0 are some constants.

Proof. From Theorem 3.1, S and T have a unique common fixed point w ∈ X.
Let y0 ∈ C and the sequence {yn} be defined by (4.1). Then, for all u ∈ X and
n ≥ 0, we have the following

‖S(yn+1) − w, u‖ = ‖(1 − βn)S(yn) + βnT (yn) − w, u‖

≤ (1 − βn)‖S(yn) − w, u‖ + βn‖T (yn) − w, u‖(4.2)

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Approximation of common fixed points in 2-Banach spaces

From (3.1), we get

‖T (yn) − w, u‖ = ‖T (yn) − Sw, u‖

≤ a
‖w − Sw, u‖[1 + ‖yn − T yn, u‖]

1 + ‖yn − w, u‖

+b
‖yn − T yn, u‖[1 + ‖w − Sw, u‖]

1 + ‖yn − w, u‖

+c
‖yn − Sw, u‖[1 + ‖w − T yn, u‖]

1 + ‖yn − w, u‖

+d
‖w − T yn, u‖[1 + ‖yn − Sw, u‖]

1 + ‖yn − w, u‖

+e‖yn − w, u‖.

Since {yn} is asymptotically T -regular and Sw = T w = w, letting n → ∞, we
have

‖T yn−w, u‖+‖T yn−w, u‖
2 ≤ (c+d+e)‖T yn−w, u‖+(c+d+e)‖T yn−w, u‖

2,

which is not possible as c + d + e < 1. Hence lim
n→∞

‖T (yn) − w, u‖ = 0.

Now it follows from (4.2) that

‖S(yn+1) − w, u‖ ≤ (1 − βn)‖S(yn) − w, u‖.

Continuing in this way, we have

(4.3) ‖(S(yn+1) − w, u‖ ≤ λ
n+1L,

where λ = (1 − βn) ∈ [0, 1) and L = ‖Sy0 − w, u‖ ≥ 0 are some constants.
Since λ ∈ [0, 1), from (4.3) we get

lim
n→∞

‖(S(yn+1) − w, u‖ → 0.

This completes the proof. �

Corollary 4.2. Let C be a nonempty closed convex subset of a 2-Banach space

X and T : C → C be a mapping satisfying all the conditions of Corollary 3.2.
Then T has a unique fixed point w ∈ X. In addition, if for arbitrary x0 ∈ C,
the sequence {xn} defined by

(4.4) xn+1 = (1 − βn)xn + βnT (xn, . . . , xn), ∀n ≥ 0,

where {βn} satisfies 0 < βn ≤ 1, ∀n and
∞
∑

n=0

βn = ∞, is asymptotically T -

regular, then it converges to the unique fixed point of T , with a rate estimated

by

‖xn − w, u‖ ≤ λ
nL,

where λ ∈ [0, 1) and L ≥ 0 are some constants.

Remark 4.3. From Corollary 4.2, we obtain an approximation of fixed point of
a self mapping by the Mann iteration scheme in 2-Banach spaces. Note that
the result holds even if {xn} is asymptotically S-regular.

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D. Ramesh Kumar, M. Pitchaimani

Corollary 4.4. Let (X, ‖·, ·‖) be a 2-Banach space and S, T : X → X be two
mappings such that T (X) ⊂ S(X) and S(X) is a closed subset of X. Suppose
all the conditions of Theorem 3.1 are satisfied, then S and T have a unique
common fixed point w ∈ X. Further, if for arbitrary x0 ∈ C, the sequence {yn}
defined by

yn = S(xn) = T (xn−1), ∀n ∈ N

is asymptotically T -regular, then it converges to the unique common fixed point

of S and T , with a rate estimated by

‖yn − w, u‖ ≤ λ
n
L,

where λ ∈ [0, 1) and L ≥ 0 are some constants.

Corollary 4.5. Let (X, ‖·, ·‖) be a 2-Banach space and T : X → X be a
mapping such that all the conditions of Corollary 3.2 are satisfied. Then T has
a unique fixed point X. In addition, if for arbitrary x0 ∈ C, the sequence {zn}
defined by

zn = T (xn−1), ∀n ∈ N,

is asymptotically T -regular, then it converges to the unique fixed point of T ,

with a rate estimated by

‖zn − w, u‖ ≤ λ
nL,

where λ ∈ [0, 1) and L ≥ 0 are some constants.

Remark 4.6. From Corollary 4.5, we obtain an approximation of fixed point of
self mapping by the Picard iteration scheme in 2-Banach space.

5. Applications

The notion of well-posedness of a fixed point problem was introduced in
[9] and has generated much interest to several mathematicians, for example
[16, 26, 31]. In this section, we study well-posedness of the common fixed point
obtainined in Theorem 3.1.

Definition 5.1. Let (X, ‖·, ·‖) be a 2-Banach space and T be a self mapping
on X. Then the fixed point problem of T is said to be well-posed if

(i) T has a unique fixed point x0 ∈ X
(ii) for any sequence {xn} ⊂ X and lim

n→∞
‖xn − T xn, u‖ = 0, we have

lim
n→∞

‖xn − x0, u‖ = 0.

Let CFP(S, T, X) denote a common fixed point problem of self mappings T
and f on X and CF(S, T ) denote the set of all common fixed points of T and
f.

Definition 5.2. CFP(S, T, X) is called well- posed if CF(S, T ) is singleton
and for any sequence {xn} in X with

x̂ ∈ CF(S, T ) and lim
n→∞

‖xn − Sxn, u‖ = lim
n→∞

‖xn − T xn, u‖ = 0

implies x̂ = lim
n→∞

xn.

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 52



Approximation of common fixed points in 2-Banach spaces

Theorem 5.3. Let (X, ‖·, ·‖) be a 2-Banach space, S and T be two self map-
pings on X as in Theorem 3.1. Then the common fixed point problem of S and
T is well posed.

Proof. From Theorem 3.1, the mappings S and T have a unique common fixed
point, say w ∈ X. Let {xn} be a sequence in X and lim

n→∞
‖Sxn − xn, u‖ =

lim
n→∞

‖T xn − xn, u‖ = 0. Without loss of generality, assume that w 6= xn for

any non-negative integer n. Using (3.1) and Sw = T w = w, we get

‖w − xn, u‖ ≤ ‖T v − T xn, u‖ + ‖T xn − xn, u‖

= ‖T xn − xn, u‖ + ‖Sw − T xn, u‖

≤ ‖T xn − xn, u‖ + a
‖xn − T xn, u‖[1 + ‖w − Sw, u‖]

1 + ‖w − xn, u‖

+b
‖w − Sw, u‖[1 + ‖xn − T xn, u‖]

1 + ‖w − xn, u‖

+c
‖w − T xn, u‖[1 + ‖xn − Sw, u‖]

1 + ‖w − xn, u‖

+d
‖xn − Sw, u‖[1 + ‖w − T xn, u‖]

1 + ‖w − xn, u‖

+e‖w − xn, u‖.

Taking limit n → ∞, we get

‖w − xn, u‖ + ‖w − xn, u‖
2 ≤ (c + d + e)‖w − xn, u‖ + (c + d + e)‖w − xn, u‖

2,

which gives the contradiction as c + d + e < 1. This completes the proof. �

Corollary 5.4. Let (X, ‖·, ·‖) be a 2-Banach space and T be a self mapping
on X as in Corollary 3.2. Then the fixed point problem of T is well posed.

Remark 5.5. Notice that well-posedness of the common fixed points obtained
in Theorems 3.4 and 3.5 can easily be viewed.

Acknowledgements. The authors thank the reviewers for valuable com-

ments. The first author D. Ramesh Kumar would like to thank the University

Grants Commission, New Delhi, India for providing the financial support in

preparation of this manuscript.

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 53



D. Ramesh Kumar, M. Pitchaimani

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