CUBO A Mathematical Journal
Vol.15, No¯ 02, (105–110). June 2013

An iterative method for finite family of hemi contractions in
Hilbert space

Balwant Singh Thakur

School of Studies in Mathematics,

Pt.Ravishankar Shukla University, Raipur,492010, India.

balwantst@gmail.com

ABSTRACT

We consider the problem of finding a common fixed point of N hemicontractions defined

on a compact convex subset of a Hilbert space, an algorithm for solving this problem

will be studied. We will prove strong convergence theorem for this algorithm.

RESUMEN

Consideramos el problema de búsqueda de un punto fijo común de N hemicontracciones

definida sobre un subconjunto convexo compacto de un espacio de Hilbert. Se estudiará

un algoritmo para resolver este problema. Probaremos el teorema de convergencia

fuerte para este algoritmo.

Keywords and Phrases: Hemicontraction, Mann iteration, implicit iteration, common fixed

point.

2010 AMS Mathematics Subject Classification: 47H09,47H10.



106 Balwant Singh Thakur CUBO
15, 2 (2013)

1 Introduction

Let H be a Hilbert space and let K be a nonempty subset of H. A map T : K → K is called

nonexpansive if

‖Tx − Ty‖ ≤ ‖x − y‖ , ∀x, y ∈ K .

An important generalization of nonexpansive mapping is pseudocontractive mapping. A mapping

T : K → K is said to be pseudocontractive if, ∀x, y ∈ K ,

‖Tx − Ty‖
2
≤ ‖x − y‖

2
+ ‖(I − T)x − (I − T)y‖

2

holds. T is said to be strongly pseudocontractive if, there exists k ∈ (0, 1) such that,

‖Tx − Ty‖
2
≤ ‖x − y‖

2
+ k ‖(I − T)x − (I − T)y‖

2
, ∀x, y ∈ K.

For importance of fixed points of pseudocontractive mappings one may refer [1].

Iterative methods for approximating fixed points of nonexpansive mappings have been exten-

sively studied (see e.g. [2, 3, 4, 7]), but, iterative methods for approximating pseudocontractive

mappings are far less developed than those of nonexpansive mappings. However, on the other-

hand pseudocontractions have more powerful applications than nonexpansive mappings in solving

nonlinear inverse problems. In recent years many authors have studied iterative approximation of

fixed point of strongly pseudocontractive mappings. Most of them used Mann’s iteration process

[6]. But in the case of pseudocontractive mapping, it is well known that Mann’s iteration fails

to converge to fixed point of Lipschitz pseudocontractive mappings in a compact convex subset

of a Hilbert space. In 1974, Ishikawa [5] introduced an iteration process which converges to a

fixed point of Lipschitz pseudocontractive mapping in a compact convex subset of a Hilbert space.

Qihou [8], extended result of Ishikawa to slightly more general class of Lipschitz hemicontractive

mappings. A mapping T : K → K is said to be hemicontractive if F(T) 6= ∅ and

‖Tx − p‖
2
≤ ‖x − p‖

2
+ ‖x − Tx‖

2
, ∀x ∈ K, p ∈ F(T)

where F(T) := {x ∈ K : Tx = x} is the fixed point set of T.

It is easy to see that the class of pseudocontractive mappings with fixed points is a subclass

of the class of hemicontractive mappings.

More recently, Rafiq [9], proposed Mann type implicit iteration process to approximate fixed

points of hemicontractive mapping defined in a compact convex subset of a Hilbert space.

For arbitrary chosen x0 ∈ K the iteration process is given by

xn = αnxn−1 + (1 − αn)Txn



CUBO
15, 2 (2013)

An iterative method for finite family of hemi contractions . . . 107

where {αn} is a real sequence in [0, 1] satisfying some appropriate conditions.

The purpose of this paper to study the problem of finding a point x such that

x ∈
N
⋂

i=1

Fix(Ti)

where N ≥ 1 is a positive integer and {Ti}
N
i=1 are N hemicontractive mappings defined on a compact

convex subset K of a Hilbert space H. We study the strong convergence of the algorithm which

generates a sequence {xn} in the following way:

xn = αnxn−1 + (1 − αn)

N∑

i=1

λ
(n)
i

Tixn , (1)

where the sequence of weights {λ
(n)
i

}Ni=1 satisfies appropriate assumptions.

2 Preliminaries

Following well known identity holds in a Hilbert space H :

‖(1 − λ)x + λy‖
2
= (1 − λ) ‖x‖

2
+ λ ‖y‖

2
− λ(1 − λ) ‖x − y‖

2

for all x, y ∈ H and λ ∈ [0, 1].

We shall use the following lemma to prove our main result:

Lemma 2.1. [10] Suppose {ρn} and {σn} are two sequences of nonnegative numbers such that for

some real number N0 ≥ 1,

ρn+1 ≤ ρn + σn ∀n ≥ N0 .

(a) If
∑

σn < ∞ then, lim ρn exists.

(b) If
∑

ρn < ∞ and {ρn} has a subsequence converging to zero, then lim ρn = 0.

Given an integer N ≥ 1, for each 1 ≤ i ≤ N, assume that Ti : K → K is a hemicontractive mapping.

Then the family {Ti}
N
i=1 is said to satisfy condition B if

N
⋂

i=1

Fix(Ti) 6= ∅ , and Fix

(

N∑

i=1

λiTi

)

=

N
⋂

i=1

Fix(Ti)

where {λi} is a positive sequence such that
∑N

i=1
λi = 1.



108 Balwant Singh Thakur CUBO
15, 2 (2013)

Proposition 2.2. Let N ≥ 1 be a given integer. For each 1 ≤ i ≤ N, assume that Ti : K → K is

a hemicontractive mapping and the family {Ti}
N
i=1 satisfies the condition B. Then

∑N
i=1

λiTi is a

hemicontractive mapping.

Proof. Let us consider the case of N = 2. Set V = (1 − λ)T1 + λT2, where λ ∈ (0, 1), where T1, T2
are hemicontractions. We have to prove that

‖Vx − p‖
2
≤ ‖x − p‖

2
+ ‖(I − V)x‖

2
∀x ∈ K , p ∈ Fix(V) .

We have

‖(I − V)x‖
2
= ‖(I − ((1 − λ)T1 + λT2))x‖

2

= (1 − λ) ‖(I − T1)x‖
2
+ λ ‖(I − T2)x‖

2
− λ(1 − λ) ‖(T1 − T2)x‖

2

so,

‖Vx − p‖
2
= ‖(1 − λ)(T1x − p) + λ(T2x − p)‖

2

= (1 − λ) ‖T1x − p‖
2
+ λ ‖T2x − p‖

2
− λ(1 − λ) ‖T1x − T2x‖

2

≤ (1 − λ)
[

‖x − p‖
2
+ ‖(I − T1)x‖

2
]

+ λ
[

‖x − p‖
2
+ ‖(I − T2)x‖

2
]

− λ(1 − λ) ‖T1x − T2x‖
2

= ‖x − p‖
2
+ ‖(I − V)x‖

2
.

Hence V is a hemicontraction. The general case can be proved by induction.

3 Main result

Theorem 3.1. Let K be a compact convex subset of a Hilbert space H. Let N ≥ 1 be an integer. For

each n ≥ 1, assume that {λ
(n)
i

}Ni=1 is a finite sequence of positive numbers such that
∑N

i=1
λ
(n)
i

= 1

and infn≥1 λ
(n)
i

> 0 for all 1 ≤ i ≤ N. For each 1 ≤ i ≤ N, let Ti : K → K is a hemicontractive

mapping and the family {Ti}
N
i=1 satisfies the condition B. For arbitrary chosen x0 ∈ K, let {xn} be

a sequence generated by the algorithm (1), where the sequence {αn} ⊂ [δ, 1 − δ] for some δ ∈ (0, 1).

Then {xn} converges strongly to a common fixed point of the family {Ti}
N
i=1.

Proof. Write, for each n ≥ 1,

Sn =

N∑

i=1

λ
(n)
i

Ti .

By the Proposition 2.2, each Sn is hemicontractive on K, and the algorithm (1) can be rewritten

as,

xn = αnxn−1 + (1 − αn)Snxn . (2)



CUBO
15, 2 (2013)

An iterative method for finite family of hemi contractions . . . 109

For p ∈ F :=
⋂N

i=1
Fix(Ti), we have

‖xn − p‖
2
= ‖αn(xn−1 − p) + (1 − αn)(Snxn − p)‖

2

= αn ‖xn−1 − p‖
2
+ (1 − αn) ‖Snxn − p‖

2
− αn(1 − αn) ‖xn−1 − Snxn‖

2

≤ αn ‖xn−1 − p‖
2
+ (1 − αn)

[

‖xn − p‖
2
+ ‖xn − Snxn‖

2
]

− αn(1 − αn) ‖xn−1 − Snxn‖
2

(3)

Also,

‖xn − Snxn‖
2
= ‖αnxn−1 + (1 − αn)Snxn − Snxn‖

2

= α2n ‖xn−1 − Snxn‖
2
. (4)

Using (3) and (4), we have

‖xn − p‖
2
≤ ‖xn−1 − p‖

2
− (1 − αn)

2 ‖xn−1 − Snxn‖
2
. (5)

From the condition {αn} ⊂ [δ, 1 − δ] for some δ ∈ (0, 1), we conclude that

‖xn − p‖
2
≤ ‖xn−1 − p‖

2
− δ2 ‖xn−1 − Snxn‖

2
(6)

holds for all p ∈ F. Now,

δ2 ‖xn−1 − Snxn‖
2
≤ ‖xn−1 − p‖

2
− ‖xn − p‖

2

and hence,

δ2
∞∑

j=1

‖xj−1 − Sjxj‖
2
≤

∞∑

j=1

(

‖xj−1 − p‖
2
− ‖xj − p‖

2
)

= ‖x0 − p‖
2

implies,

δ2
∞∑

j=1

‖xj−1 − Sjxj‖
2
< ∞ . (7)

So,

lim
n→∞

‖xn−1 − Snxn‖ = 0 . (8)

From (4), we have

lim
n→∞

‖xn − Snxn‖ = 0 . (9)

Without loss of generality, we may assume that

λ
(nl)
i

→ λi ( as l → ∞), 1 ≤ i ≤ N .

It is easily seen that each λi > 0 and
∑N

i=1
λi = 1. We also have

Snlx → Sx ( as l → ∞), for all x ∈ K ,



110 Balwant Singh Thakur CUBO
15, 2 (2013)

where

S =

N∑

i=1

λiTi .

Since K is compact, there is a subsequence {xnj} of {xn} which converges to a fixed point of S, say

z. Using (6), we have

‖xn − z‖
2
≤ ‖xn−1 − z‖

2
− δ2 ‖xn−1 − Sxn‖

2

In view of Lemma 2.1 and (7), we conclude that

‖xn − z‖ → 0 as n → ∞

i.e. xn → z as n → ∞.

This completes the proof.

Acknowledgements

The author is supported by a project (FN-41-1390/2012) of University Grants Commission of India.

Received: December 2011. Accepted: September 2012.

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