Ratio Mathematica Volume 47, 2023

Fixed point theorems in uniformly convex
Banach spaces

Jahir Hussain Rasheed*
Manoj Karuppasamy†

Abstract

In this article, we establish a concept of fixed point result in Uni-
formly convex Banach space. Our main finding uses the Ishikawa
iteration technique in uniformly convex Banach space to demonstrate
strong convergence. Additionally, we use our primary result to demon-
strate some corollaries.
Keywords: Fixed point; Mann iteration; Ishikawa iteration.
2020 AMS subject classifications: 55M20, 46B80, 52A21.1

*Jamal Mohamed College (Autonomous), Affiliated to Bharathidasan University,
Tiruchirapplli-620020, Tamilnadu India ; hssn jhr@yahoo.com.

†Jamal Mohamed College(Autonomous), Affiliated to Bharathidasan University,
Tiruchirapplli-620020, Tamilnadu India ; manojguru542@gmail.com.
1Received on August 25, 2023. Accepted on February 27, 2023. Published on April 4, 2023. doi:
10.23755/rm.v39i0.835. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is
published under the CC-BY licence agreement.

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R. Jahir Hussain, K. Manoj

1 Introduction
Mann [Mann [1953]] defined mean value methods in an iterative scheme in

1953, and Ishikawa [Ishikawa [1974]] established fixed points using a new it-
eration method technique in 1954. Takahashi [Takahashi [1970] ] introduced the
idea of convexity in metric spaces and non-expansive mappings in 1970. Then
Machado [Machado [1973]] went on to discuss about a classification of convex
subsets of normed spaces. After that, Luis Bernal-Gonzalez [Bernal-Gonzalez
[1996]] discussed convex domain in uniformly Banach spaces. Berinde [Berinde
[2004]] investigates iterative scheme to finding fixed points using quasi contrac-
tive mappings in uniformly convex Banach spaces(stands for CBS), extended to
uniformly convex Banach spaces(CBS).

Throughout this paper, we use strong convergence of ishikawa iterations to
prove such fixed point results in uniformly CBS using different type of contrac-
tions.

2 Preliminaries
Definition 2.1. Let (X, d) be a metric space and I = [0, 1]. A mapping W : X ×
X ×I → X is said to be a convex structure on X if for each (x, y, λ) ∈ X ×X ×I
and u ∈ X

d (u, W(x, y, λ)) ≤ λd(u, x) + (1 − λ)d(u, y).
A metric space (X, d) together with a convex structure W is called a convex metric
space,which is denoted by (X, d, W).

Definition 2.2. Let (X, d, W) be a convex metric space. A nonempty subset C of
X is said to be convex if W(x, y, λ) ∈ C whenever (x, y, λ) ∈ C × C × I.
Definition 2.3. Let f : X → X. A point x ∈ X is called a fixed point of f if
f(x) = x.

Definition 2.4. Let E be a uniformly Banach space and T : E → E a map for
which there is a real constant k1 ∈ (0, 1/5) such that each pair u, v ∈ X,

∥Tu − Tv∥ ≤ k1{∥u − v∥ + ∥u − Tu∥ + ∥v − Tv∥ + ∥u − Tv∥ + ∥v − Tu∥}.

Then, T has a fixed point by the approximation of Picard.

Definition 2.5. Let E be a uniformly Banach space and T : E → E a map for
which there is a real constant k2 ∈ (0, 1/3) such that for each pair u, v ∈ X,

∥Tu − Tv∥ ≤ k2{∥u − v∥ +
∥u − Tu∥ + ∥v − Tv∥

2
+

∥u − Tv∥ + ∥v − Tu∥
2

}.

Then, T has a fixed point by the approximation of Picard.

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Fixed point theorems in uniformly convex Banach spaces

Definition 2.6. Let E be a uniformly Banach space and T : E → E a given
operator. Let u0 ∈ E be arbitrary and {αn} ⊂ [0, 1] a sequence of real numbers.
The sequence { un} ⊂ E defined by

un+1 = (1 − αn)un + αnTun, n = 0, 1, 2, · · · (1)

is called the Mann iteration.

Definition 2.7. Let E be a uniformly Banach space and T : E → E a given
operator. Let u0 ∈ E be arbitrary, {αn} and {βn} ⊂ [0, 1] a sequence of real
numbers. The sequence { un} ⊂ E defined by

un+1 = (1 − αn)un + αnTvn, n = 0, 1, 2, · · · (2)

vn = (1 − βn)un + βnTun, n = 0, 1, 2, · · · . (3)
Then {un} is called Ishikawa iteration.

Result 2.1. Berinde [2004] The condition of Mann and Ishikawa iteration for
strong convergence are given below

(a) Let K be a closed convex subset of a uniformly Banach space E and T : K →
K as an operator satisfying contraction. Let {un} be defined by Definition
2.6 and x0 ∈ K, with {αn} ∈ [0, 1] satisfying

∞∑
n=0

αn = ∞. (4)

Then, {un} converges strongly to a fixed point.

(b) Let K be a closed convex subset of a uniformly Banach space E and T :
K → K as an operator satisfying the contraction. Let {un} be defined by
Definition 2.7 and u0 ∈ K, with {αn}, {βn} ∈ [0, 1] satisfying

∞∑
n=0

αn(1 − αn) = ∞. (5)

Then, {un} strongly converges to a fixed point.2

3 Main Results
Theorem 3.1. Let K be a closed convex subset of a uniformly Banach space E and
T : K → K an operator satisfying equation

∥Tu−Tv∥ ≤ k1{∥u−v∥+∥u−Tu∥+∥v −Tv∥+∥u−Tv∥+∥v −Tu∥}. (6)

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R. Jahir Hussain, K. Manoj

Let {un} be the Ishikawa iteration and u0 ∈ K, where {αn}, {βn} ⊂ [0, 1] with
{αn} satisfying equation

∑∞
n=0 αn = ∞. Then {un} converges strongly to a fixed

point of T .

Proof. Suppose T has a fixed point p in K. Consider u, v ∈ K and T is an
operator satisfies above equation,

∥Tu − Tv∥ ≤ k1{∥u − v∥ + ∥u − Tu∥ + ∥v − Tv∥ + ∥u − Tv∥
+ ∥v − Tu∥}
= k1{2∥u − v∥ + 2∥u − Tv∥ + ∥u − Tu∥ + ∥v − Tu∥}
≤ k1{2∥u − v∥ + 2[∥u − Tu∥ + ∥Tu − Tv∥] + ∥u − Tu∥
+ ∥v − Tu∥}

(1 − 2k1)∥Tu − Tv∥ ≤ 2k1∥u − v∥ + 3k1∥u − Tu∥ + k1∥v − Tu∥

∥Tu − Tv∥ ≤
2k1

1 − 2k1
∥u − v∥ +

3k1
1 − 2k1

∥u − Tu∥ +
k1

1 − 2k1
∥v − Tu∥.

Take δ = k1
1−2k1

, then we have δ ∈ [0, 1), it result that the inequality

∥Tu − Tv∥ ≤ 2δ∥u − v∥ + 3δ∥u − Tu∥ + δ∥v − Tu∥∀u, v ∈ K. (7)

Now let {un}∞n=0 be Ishikawa iteration defined on Definition 2.7 and u0 ∈ K
arbitrary then

∥un+1 − p∥ = ∥(1 − αn)un + αnTvn − (1 − αn + αn)p∥
= ∥(1 − αn)(un − p) + αn(Tvn − p)∥

∥un+1 − p∥ ≤ (1 − αn)∥un − p∥ + αn∥Tvn − p∥. (8)

In equation (7), put u = p and v = vn, we have

∥Tvn − p∥ ≤ 2δ∥p − vn∥ + 3δ∥p − Tp∥ + δ∥vn − Tp∥
= 2δ∥p − vn∥ + δ∥vn − p∥

∥Tvn − p∥ ≤ 3δ∥vn − p∥. (9)

Furthermore, ∥vn − p∥ = ∥(1 − βn)un + βnTun − (1 − βn + βn)p∥

= ∥(1 − βn)(un − p) + βn(Tun − p)∥

∥vn − p∥ ≤ (1 − βn)∥un − p∥ + βn∥Tun − p∥. (10)

Again in equation (7), put u = p and v = un, we get

∥Tun − p∥ ≤ 2δ∥p − un∥ + 3δ∥p − Tp∥ + δ∥un − p∥
= (2δ + δ)∥un − p∥

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Fixed point theorems in uniformly convex Banach spaces

∥Tun − p∥ ≤ 3δ∥un − p∥. (11)
Using equation (9),(10),(11) in equation (8), we get

∥un+1 − p∥ ≤ (1 − αn)∥un − p∥ + αn∥Tvn − p∥
≤ (1 − αn)∥un − p∥ + 3αnδ∥vn − p∥
≤ (1 − αn)∥un − p∥ + 3αnδ[(1 − βn)∥un − p∥ + βn∥Tun − p∥]
= (1 − αn)∥un − p∥ + 3αnδ(1 − βn)∥un − p∥ + 3αnδβn∥Tun − p∥
≤ (1 − αn)∥un − p∥ + 3αnδ(1 − βn)∥un − p∥ + 9αnδ2βn∥un − p∥
= [1 − αn + 3αnδ − 3αnβnδ + 9αnδ2βn]∥un − p∥
= 1 − αn(1 − 3δ) − 3αnβnδ(1 − 3δ)]∥un − p∥
= [1 − (1 − 3δ)αn(1 + 3δβn)]∥un − p∥

which by the inequality, 1 − (1 − 3δ)αn(1 + 3δβn) ≤ 1 − (1 − 3δ)2αn

⇒ ∥un+1 − p∥ ≤ [1 − (1 − 3δ)2αn]∥un − p∥, n = 0, 1, 2, ... (12)
by equation (8), we obtain

∥un+1 − p∥ ≤
n∏

k=0

[1 − (1 − 3δ)2αk]∥u0 − p∥. (13)

Where, δ ∈ (0, 1), αk, βn ∈ [0, 1] and
∞∑
n=0

αn = ∞ by result (a), we get

limn→∞

n∏
k=0

[1 − (1 − 3δ)2αk] = 0. By equation (13) which implies

limn→∞ ∥un+1 − p∥ = 0. Therefore, {un}∞n=0 converges strongly to p. 2
Theorem 3.2. Let K be a closed convex subset of a uniformly Banach space E and
T : K → K an operator satisfying equation

∥Tu − Tv∥ ≤ k2{∥u − v∥ +
∥u − Tu∥ + ∥v − Tv∥

2
+

∥u − Tv∥ + ∥v − Tu∥
2

}.

Let {un} be the Ishikawa iteration and u0 ∈ K, where {αn} and {βn} are se-
quences in [0, 1] with {αn} satisfying equation

∑∞
n=0 αn = ∞. Then {un} con-

verges strongly to a fixed point of T .

Proof. Consider T has a fixed point p in K. Consider u, v ∈ K and T is an
operator satisfies equation

∥Tu − Tv∥ ≤ k2{∥u − v∥ +
∥u−Tu∥+∥v−Tv∥

2
+

∥u−Tv∥+∥v−Tu∥
2

}
≤ k2{∥u − v∥ +

∥u−v∥+∥v−Tu∥+∥v−Tv∥
2

+
∥u−Tv∥+∥v−Tu∥

2
}

= k2{32∥u − v∥ + ∥v − Tu∥ +
1
2
∥v − Tv∥ + ∥u−Tv∥

2
}

(1 − k2)∥Tu − Tv∥ ≤ 32k2{∥u − v∥ + ∥v − Tv∥} +
k2
2
∥u − Tv∥

∥Tu − Tv∥ ≤ 3k2
2(1−k2)

{∥u − v∥ + ∥v − Tv∥} + k2
2(1−k2)

∥u − Tv∥

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R. Jahir Hussain, K. Manoj

Take δ =
k2

1 − k2
∈ [0, 1).∥Tu−Tv∥ ≤

3

2
δ{∥u−v∥+∥v −Tv∥}+

δ

2
∥u−Tv∥.

(14)
Now, let {un} be Ishikawa iteration defined on Definition 2.7 and u0 ∈ K then,
∥un+1 − p∥ = ∥(1 − αn)un + αnTvn − (1 − αn + αn)p∥

∥un+1 − p∥ ≤ (1 − αn)∥un − p∥ + αn∥Tvn − p∥ (15)

In equation (14), put v = p and u = un, ∥Tun − p∥ ≤ 3δ2 ∥un − p∥ +
δ
2
∥un − p∥

∥Tun − p∥ ≤ 2δ∥un − p∥ (16)

In equation (14), put u = vn and v = p, ∥Tvn − p∥ ≤ 3δ2 ∥vn − p∥ +
δ
2
∥vn − p∥

∥Tvn − p∥ ≤ 2δ∥vn − p∥ (17)

∥vn − p∥ = ∥(1 − βn)un + βnTun − (1 − βn + βn)p∥
∥vn − p∥ ≤ (1 − βn)∥un − p∥ + βn∥Tun − p∥ (18)

Using equation (16), (17), (18) in equation (15), we get

∥un+1 − p∥ ≤ (1 − αn) + 2αnδ∥vn − p∥
≤ (1 − αn) + 2δαn[(1 − βn)∥un − p∥ + βn∥Tun − p∥]
≤ (1 − αn) + 2δαn(1 − βn)∥un − p∥ + 4δ2αnβn∥un − p∥
= [1 − αn(1 + 2δβn)(1 − 2δ)]∥un − p∥

Which by inequality, 1 − αn(1 + 2δβn)(1 − 2δ) ≤ 1 − (1 − 2δ)2αn
⇒ ∥un+1 − p∥ ≤ [1 − (1 − 2δ)2αn]∥un − p∥ (19)

By equation (19), we obtain

∥un+1 − p∥ ≤
n∏

k=0

[1 − (1 − 2δ)2αk]∥u0 − p∥ (20)

where δ ∈ (0, 1), αk, βn ∈ [0, 1] and
∞∑
n=0

αn = ∞ by result (a), we get,

limn→∞

n∏
k=0

[1 − (1 − 2δ)2αk] = 0. By equation (20), ⇒ limn→∞ ∥un+1 − p∥ = 0.

Therefore {un}∞n=0 converges strongly to p.2

Corolary 3.1. Let K be a closed convex subset of a uniformly Banach space E and
T : K → K an operator satisfying equation
∥Tu − Tv∥ ≤ k

4
{∥u − Tu∥ + ∥v − Tv∥ + ∥u − Tv∥ + ∥v − Tu∥}.

Let {un} be the Ishikawa iteration and u0 ∈ K, where {αn} and {βn} are se-
quences of positive numbers in [0, 1] with {αn} satisfying equation

∑∞
n=0 αn =

∞. Then {un} converges strongly to a fixed point of T .

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Fixed point theorems in uniformly convex Banach spaces

Corolary 3.2. Let E be an uniformly Banach space, K is a closed convex subset
of E and T : K → K an operator satisfying equation

∥Tu − Tv∥ ≤ k∥u − v∥ (21)

Let {un}∞n=0 be the Ishikawa iteration and u0 ∈ K, where {αn}∞n=0 and {βn}∞n=0
are sequences of positive numbers in [0, 1] with {αn}∞n=0 satisfying

∑∞
n=0 αn =

∞. Then {un}∞n=0 converges strongly to a fixed point of T .

Corolary 3.3. Let E be an uniformly Banach space, K is a closed convex subset
of E and T : K → K an operator satisfying equation

∥Tu − Tv∥ ≤
k

2
{∥u − Tu∥ + ∥v − Tv∥} (22)

Let {un}∞n=0 be the Ishikawa iteration and u0 ∈ K, where {αn}∞n=0 and {βn}∞n=0
are sequences of positive numbers in [0, 1] with {αn}∞n=0 satisfying

∑∞
n=0 αn =

∞. Then {un}∞n=0 converges strongly to a fixed point of T .

4 Conclusions
In this work, we presented the result on strong convergence of fixed point of

T. We developed different kinds of contractive conditions to prove strong con-
vergence using Ishikawa iterative method in uniforly convex Banach space. Our
main result may be vision for other authors using different contraction to prove
several converging fixed point result.

Acknowledgements
The editors and referees are greatly appreciated by the authors for their in-

sightful input, which helped the paper’s presentation.

References
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contractive operators. Acta Math. Univ. Comenianae, LXXIII:1–11, 2004.

L. Bernal-Gonzalez. A schwarz lemma for convex domains in arbitrary banach
spaces. Journal of Mathematical Analysis and Applications, 200:511–517,
1996.

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R. Jahir Hussain, K. Manoj

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