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CUBO A Mathematical Journal
Vol.13, No¯ 01, (137–147). March 2011

Strong convergence of an implicit iteration process for a
finite family of strictly asymptotically pseudocontractive

mappings

Gurucharan Singh Saluja

Department of Mathematics & Information Technology,

Govt. Nagarjuna P.G. College of Science, Raipur (C.G.).

email: saluja 1963@rediffmail.com

and

Hemant Kumar Nashine

Department of Mathematics,

Disha Institute of Management and Technology,

Satya Vihar, Vidhansabha-Chandrakhuri Marg

Mandir Hasaud, Raipur-492101(Chhattisgarh), India.

email: hnashine@rediffmail.com, nashine 09@rediffmail.com

ABSTRACT

In this paper, we establish the strong convergence theorems for a finite family of k-

strictly asymptotically pseudo-contractive mappings in the framework of Hilbert spaces.

Our results improve and extend the corresponding results of Liu [5] and many others.

RESUMEN

En este trabajo, hemos establecido los teoremas de convergencia para una familia finita

de asignaciones de k-estrictamente asintticamente pseudo-contraccin en el marco de los

espacios de Hilbert. Nuestros resultados mejoran y amplan los resultados correspondi-

entes de Liu [5] y muchos otros.



138 Gurucharan Singh Saluja and Hemant Kumar Nashine CUBO
13, 1 (2011)

Keywords: Strictly asymptotically pseudo-contractive mapping, implicit iteration scheme, com-

mon fixed point, strong convergence, Hilbert space.

AMS Subject Classification: 47H09, 47H10.

1 Introduction

Let H be a real Hilbert space with the scalar product and norm denoted by the symbols 〈., .〉 and

‖ . ‖ respectively, and C be a closed convex subset of H. Let T be a (possibly) nonlinear mapping

from C into C. We now consider the following classes:

(1) T is contractive, i.e., there exists a constant k < 1 such that

‖T x − T y‖ ≤ k ‖x − y‖ , (1.1)

for all x, y ∈ C.

(2) T is nonexpansive, i.e.,

‖T x − T y‖ ≤ ‖x − y‖ , (1.2)

for all x, y ∈ C.

(3) T is uniformly L-Lipschitzian, i.e., if there exists a constant L > 0 such that

‖T nx − T ny‖ ≤ L ‖x − y‖ , (1.3)

for all x, y ∈ C and n ∈ N.

(4) T is pseudo-contractive, i.e.,

〈T x − T y, j(x − y)〉 ≤ ‖x − y‖
2
, (1.4)

for all x, y ∈ C.

(5) T is strictly pseudo-contractive, i.e., there exists a constant k ∈ [0, 1) such that

‖T x − T y‖
2

≤ ‖x − y‖
2

+ k ‖(x − T x) − (y − T y)‖
2
, (1.5)



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Strong convergence of an implicit iteration process for a finite
family of strictly asymptotically pseudocontractive mappings 139

for all x, y ∈ C.

(6) T is asymptotically nonexpansive [3], i.e., if there exists a sequence {rn} ⊂ [0, ∞) with

limn→∞ rn = 0 such that

‖T nx − T ny‖ ≤ (1 + rn) ‖x − y‖ , (1.6)

for all x, y ∈ C and n ∈ N.

(7) T is k-strictly asymptotically pseudo-contractive [6], i.e., if there exists a sequence

{rn} ⊂ [0, ∞) with limn→∞ rn = 0 such that

‖T nx − T ny‖
2

≤ (1 + rn)
2 ‖x − y‖

2

+k ‖(x − T nx) − (y − T ny)‖
2

(1.7)

for some k ∈ [0, 1) for all x, y ∈ C and n ∈ N.

Remark 1.1 [6]: If T is k-strictly asymptotically pseudo-contractive mapping, then it is uni-

formly L-Lipschitzian, but the converse does not hold.

Concerning the convergence problem of iterative sequences for strictly pseudocontractive map-

pings has been studied by several authors (see, e.g., [2, 4, 7, 11, 12]). Concerning the class of

strictly asymptotically pseudocontractive mappings, Liu [5] proved the following result in Hilbert

space:

Theorem 1.1(Liu [5]): Let H be a real Hilbert space, let C be a nonempty closed convex

and bounded subset of H, and let T : C → C be a completely continuous uniformly L-Lipschitzian

(λ, {kn})-strictly asymptotically pseudocontractive mapping such that
∑∞

n=1(k
2
n − 1) < ∞. Let

{αn} ⊂ (0, 1) be a sequence satisfying the following condition:

0 < ǫ ≤ αn ≤ 1 − λ − ǫ ∀ n ≥ 1 and some ǫ > 0.

Then, the sequence {xn} generated from an arbitrary x1 ∈ C by

xn+1 = (1 − αn)xn + αnT
n
xn, ∀ n ≥ 1 (1.8)

converges strongly to a fixed point of T .



140 Gurucharan Singh Saluja and Hemant Kumar Nashine CUBO
13, 1 (2011)

In 2001, Xu and Ori [12] have introduced an implicit iteration process for a finite family of

nonexpansive mappings in a Hilbert space H. Let C be a nonempty subset of H. Let T1, T2, . . . , TN

be self-mappings of C and suppose that F = ∩Ni=1F (Ti) 6= ∅, the set of common fixed points of

Ti, i = 1, 2, . . . , N . An implicit iteration process for a finite family of nonexpansive mappings is

defined as follows, with {tn} a real sequence in (0, 1), x0 ∈ C:

x1 = t1x0 + (1 − t1)T1x1,

x2 = t2x1 + (1 − t2)T2x2,

...

xN = tN xN−1 + (1 − tN )TN xN ,

xN+1 = tN+1xN + (1 − tN+1)T1xN+1,

...

which can be written in the following compact form:

xn = tnxn−1 + (1 − tn)Tnxn, n ≥ 1 (1.9)

where Tk = Tk mod N . (Here the mod N function takes values in {1, 2, . . . , N}). And they proved

the weak convergence of the process (1.9).

Very recently, Acedo and Xu [1] still in the framework of Hilbert spaces introduced the fol-

lowing cyclic algorithm.

Let C be a closed convex subset of a Hilbert space H and let {Ti}
N−1
i=0 be N k-strict pseudo-

contractions on C such that F =
⋂N−1

i=0 F (Ti) 6= ∅. Let x0 ∈ C and let {αn} be a sequence in

(0, 1). The cyclic algorithm generates a sequence {xn}
∞
n=1 in the following way:

x1 = α0x0 + (1 − α0)T0x0,

x2 = α1x1 + (1 − α1)T1x1,

...

xN = αN−1xN−1 + (1 − αN−1)TN−1xN−1,

xN+1 = αN xN + (1 − αN )T0xN ,

...

In general, {xn+1} is defined by

xn+1 = αnxn + (1 − αn)T[n]xn, (1.10)



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Strong convergence of an implicit iteration process for a finite
family of strictly asymptotically pseudocontractive mappings 141

where T[n] = Ti with i = n (mod N ), 0 ≤ i ≤ N −1. They also proved a weak convergence theorem

for k-strict pseudo-contractions in Hilbert spaces by cyclic algorithm (1.10). More precisely, they

obtained the following theorem:

Theorem AX [1]: Let C be a closed convex subset of a Hilbert space H. Let N ≥ 1 be

an integer. Let for each 0 ≤ i ≤ N − 1, Ti : C → C be a ki-strict pseudo-contraction for some

0 ≤ ki < 1. Let k = max{ki : 1 ≤ i ≤ N}. Assume the common fixed point the set
⋂N−1

i=0 F (Ti)

of {Ti}
N−1
i=0 is nonempty. Given x0 ∈ C, let {xn}

∞
n=0 be the sequence generated by the cyclic

algorithm (1.10). Assume that the control sequence {αn} is chosen so that k + ǫ < αn < 1 − ǫ for

all n and for some ǫ ∈ (0, 1). Then {xn} converges weakly to a common fixed point of the family

{Ti}
N−1
i=0 .

Motivated by Xu and Ori [12], Acedo and Xu [1] and some others we introduce and study

the following:

Let C be a closed convex subset of a Hilbert space H and let {Ti}
N−1
i=0 be N k-strictly asymp-

totically pseudo-contractions on C such that F =
⋂N−1

i=0 F (Ti) 6= ∅. Let x0 ∈ C and let {αn} be a

sequence in (0, 1). The implicit iteration scheme generates a sequence {xn}
∞
n=0 in the following way:

x1 = α0x0 + (1 − α0)T0x0,

x2 = α1x1 + (1 − α1)T1x1,

...

xN = αN−1xN−1 + (1 − αN−1)TN−1xN−1,

xN+1 = αN xN + (1 − αN )T
2
0 x0,

...

x2N = α2N−1x2N−1 + (1 − α2N−1)T
2
N−1x2N−1,

x2N+1 = α2N x2N + (1 − α2N )T
3
0 x0,

...

In general, {xn} is defined by

xn+1 = αnxn + (1 − αn)T
s
[n]xn, (1.11)

where T s
[n]

= T s
n (mod N)

= T si with n = (s − 1)N + i and i ∈ I = {0, 1, . . . , N − 1}.

The aim of this paper is to establish strong convergence theorems of implicit iteration pro-

cess (1.11) for a finite family of k-strictly asymptotically pseudo-contraction mappings in Hilbert



142 Gurucharan Singh Saluja and Hemant Kumar Nashine CUBO
13, 1 (2011)

spaces. Our results extend the corresponding results of Liu [5] and many others.

In the sequel, we will need the following lemmas.

Lemma 1.1: Let H be a real Hilbert space. There holds the following identities:

(i) ‖x − y‖
2

= ‖x‖
2
− ‖y‖

2
− 2〈x − y, y〉 ∀ x, y ∈ H.

(ii) ‖tx + (1 − t)y‖
2

= t ‖x‖
2

+ (1 − t) ‖y‖
2
− t(1 − t) ‖x − y‖

2
,

∀ t ∈ [0, 1], ∀ x, y ∈ H.

(iii) If {xn} be a sequence in H weakly converges to z, then

lim sup
n→∞

‖xn − y‖
2

= lim sup
n→∞

‖xn − z‖
2

+ ‖z − y‖
2

∀y ∈ H.

Lemma 1.2 [9]: Let {an}
∞
n=1, {βn}

∞
n=1 and {rn}

∞
n=1 be sequences of nonnegative real num-

bers satisfying the inequality

an+1 ≤ (1 + rn)an + βn, n ≥ 1.

If
∑∞

n=1 rn < ∞ and
∑∞

n=1 βn < ∞, then limn→∞ an exists. If in addition {an}
∞
n=1 has a

subsequence which converges strongly to zero, then limn→∞ an = 0.

2 Main Results

Theorem 2.1: Let C be a closed convex subset of a Hilbert space H. Let N ≥ 1 be an inte-

ger. Let for each 0 ≤ i ≤ N − 1, Ti : C → C be N ki-strictly asymptotically pseudo-contraction

mappings for some 0 ≤ ki < 1 and
∑∞

n=1 rn < ∞. Let k = max{ki : 0 ≤ i ≤ N − 1} and

rn = max{rni : 0 ≤ i ≤ N − 1}. Assume that F =
⋂N−1

i=0 F (Ti) 6= ∅. Given x0 ∈ C, let {xn}
∞
n=0 be

the sequence generated by an implicit iteration scheme (1.11). Assume that the control sequence

{αn} is chosen so that k < αn < 1 for all n and
∑∞

n=0(αn − k)(1 − αn) = ∞. Then the iterative

sequence {xn} has the following properties:



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Strong convergence of an implicit iteration process for a finite
family of strictly asymptotically pseudocontractive mappings 143

(1) limn→∞ ‖xn − p‖ exists for each p ∈ F ,

(2) limn→∞ d(xn, F ) exists,

(3) lim infn→∞

∥

∥

∥
xn − T

s
[n]

xn

∥

∥

∥
= 0,

(4) the sequence {xn}
∞
n=0 converges strongly to a common fixed point p ∈ F if and only if

lim inf
n→∞

d(xn, F ) = 0.

Proof: We divide the proof of Theorem 2.1 into three steps.

(I) First, we proof the conclusions (1)and (2).

For any p ∈ F , it follows from (1.11) and Lemma 1.1(ii), we note that

‖xn+1 − p‖
2

=
∥

∥

∥
αnxn + (1 − αn)T

s
[n]xn − p

∥

∥

∥
(2.1)

=
∥

∥

∥
αn(xn − p) + (1 − αn)(T

s
[n]xn − p)

∥

∥

∥

≤ αn ‖xn − p‖
2

+ (1 − αn)
∥

∥

∥
T

s
[n]xn − p

∥

∥

∥

2

−αn(1 − αn)
∥

∥

∥
xn − T

s
[n]xn

∥

∥

∥

2

≤ αn ‖xn − p‖
2

+ (1 − αn)[(1 + rn)
2 ‖xn − p‖

2

+k
∥

∥

∥
xn − T

s
[n]xn

∥

∥

∥

2

] − αn(1 − αn)
∥

∥

∥
xn − T

s
[n]xn

∥

∥

∥

2

≤ [αn(1 + rn)
2 + (1 − αn)(1 + rn)

2] ‖xn − p‖
2

−(αn − k)(1 − αn)
∥

∥

∥
xn − T

s
[n]xn

∥

∥

∥

2

≤ (1 + rn)
2 ‖xn − p‖

2
− (αn − k)(1 − αn)

∥

∥

∥
xn − T

s
[n]xn

∥

∥

∥

2

≤ (1 + dn) ‖xn − p‖
2
− (αn − k)(1 − αn)

∥

∥

∥
xn − T

s
[n]xn

∥

∥

∥

2

where dn = r
2
n + 2rn, since

∑∞
n=1 rn < ∞ thus

∑∞
n=1 dn < ∞ and since k < αn < 1, we get

‖xn+1 − p‖
2

≤ (1 + dn) ‖xn − p‖
2

(2.2)

and therefore



144 Gurucharan Singh Saluja and Hemant Kumar Nashine CUBO
13, 1 (2011)

‖xn+1 − p‖ ≤ (1 + dn)
1/2 ‖xn − p‖ . (2.3)

Since
∑∞

n=1 dn < ∞, it follows from Lemma 1.2, we know that limn→∞ ‖xn − p‖ exists for each

p ∈ F . So that there exists K > 0 such that ‖xn − p‖ ≤ K for all n ≥ 1. Consequently, we obtain

from (2.3) that

‖xn+1 − p‖ ≤ (1 + dn)
1/2 ‖xn − p‖

≤ (1 + dn) ‖xn − p‖

≤ ‖xn − p‖ + Kdn. (2.4)

It follows from (2.4) that

d(xn+1, F ) ≤ (1 + dn)d(xn, F ), ∀ n ≥ 1 (2.5)

so that it again follows from Lemma 1.2 that limn→∞ d(xn, F ) exists.

The conclusions (1)and (2) are proved.

(II) The proof of conclusion (3).

It follows from (2.1) that

‖xn+1 − p‖
2

≤ (1 + dn) ‖xn − p‖
2

(2.6)

−(αn − k)(1 − αn)
∥

∥

∥
xn − T

s
[n]xn

∥

∥

∥

2

where dn = r
2
n + 2rn, since

∑∞
n=1 rn < ∞ thus

∑∞
n=1 dn < ∞ and since k < αn < 1, we get

‖xn+1 − p‖
2

≤ (1 + dn) ‖xn − p‖
2

(2.7)

that means the sequence {‖xn − p‖} is decreasing. Now, since
∑∞

n=1 dn < ∞ it follows that
∏∞

i=1(1 + di) < ∞, from (2.6), we have

∞
∑

n=0

(αn − k)(1 − αn)
∥

∥

∥
xn − T

s
[n]xn

∥

∥

∥

2

≤
∞
∏

i=1

(1 + di) ‖x0 − p‖
2

(2.8)

< ∞.



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Strong convergence of an implicit iteration process for a finite
family of strictly asymptotically pseudocontractive mappings 145

Since
∑∞

n=0(αn − k)(1 − αn) = ∞, (2.8) implies that

lim inf
n→∞

∥

∥

∥
xn − T

s
[n]xn

∥

∥

∥
= 0. (2.9)

(IV) Next, we prove the conclusion (4).

Necessity

If {xn} converges strongly to some point p ∈ F , then from 0 ≤ d(xn, F ) ≤ ‖xn − p‖ → 0 as

n → ∞, we have

lim inf
n→∞

d(xn, F ) = 0. (2.10)

Sufficiency

If lim infn→∞ d(xn, F ) = 0, it follows from the conclusion (2) that limn→∞

d(xn, F ) = 0. Next, we prove that {xn} is a Cauchy sequence in C. In fact, since for any x > 0,

1 + x ≤ exp(x), therefore, for any m, n ≥ 1 and for given p ∈ F , from (2.4), we have

‖xn+m − p‖ ≤ (1 + dn+m−1) ‖xn+m−1 − p‖

≤ edn+m−1 ‖xn+m−1 − p‖

≤ edn+m−1 [edn+m−2 ‖xn+m−2 − p‖]

≤ e{dn+m−1+dn+m−2} ‖xn+m−2 − p‖

≤ . . .

≤ e
∑

n+m−1

j=n
dj ‖xn − p‖

≤ K′ ‖xn − p‖ < ∞ (2.11)

where K′ = e
∑

∞

j=1
dj < ∞. Since

lim
n→∞

d(xn, F ) = 0, (2.12)

for any given ǫ > 0, there exists a positive integer n1 such that



146 Gurucharan Singh Saluja and Hemant Kumar Nashine CUBO
13, 1 (2011)

d(xn, F ) <
ǫ

2(K′ + 1)
, ∀ n ≥ n1. (2.13)

Hence, there exists p1 ∈ F such that

‖xn − p1‖ <
ǫ

(K′ + 1)
∀ n ≥ n1. (2.14)

Consequently, for any n ≥ n1 and m ≥ 1, from (2.11), we have

‖xn+m − xn‖ ≤ ‖xn+m − p1‖ + ‖xn − p1‖

≤ K′ ‖xn − p1‖ + ‖xn − p1‖

≤ (K′ + 1) ‖xn − p1‖

< (K′ + 1).
ǫ

(K′ + 1)
= ǫ.

This implies that {xn} is a Cauchy sequence in C. Let xn → x
∗ ∈ C. Since lim infn→∞ d(xn, F ) =

0, and so d(x∗, F ) = 0. Again since {Ti}
N−1
i=0 is a finite family of k-strictly asymptotically pseudo-

contractive mappings, by Remark 1.1 of [6], it is a finite family of uniformly Lipschitzian mappings.

Hence, the set F of common fixed points of {Ti}
N−1
i=0 is closed and so x

∗ ∈ F . Thus the sequence

{xn} converges strongly to a common fixed point of the family {Ti}
N−1
i=0 . This completes the proof.

Theorem 2.2: Let C be a closed convex compact subset of a Hilbert space H. Let N ≥ 1

be an integer. Let for each 0 ≤ i ≤ N − 1, Ti : C → C be N ki-strictly asymptotically pseudo-

contraction mappings for some 0 ≤ ki < 1 and
∑∞

n=1 rn < ∞. Let k = max{ki : 0 ≤ i ≤ N −1} and

rn = max{rni : 0 ≤ i ≤ N − 1}. Assume that F =
⋂N−1

i=0 F (Ti) 6= ∅. Given x0 ∈ C, let {xn}
∞
n=0 be

the sequence generated by an implicit iteration scheme (1.11). Assume that the control sequence

{αn} is chosen so that k < αn < 1 for all n. Then {xn} converges strongly to a common fixed

point of the family {Ti}
N−1
i=0 .

Proof: We only conclude the difference. By compactness of C this immediately implies that

there is a subsequence {xnj } of {xn} which converges to a common fixed point of {Ti}
N−1
i=0 , say, p.

Combining (2.3) with Lemma 1.2, we have limn→∞ ‖xn − p‖ = 0. Thus {xn} converges strongly

to a common fixed point of the family {Ti}
N−1
i=0 . This completes the proof.

Remark 2.1 Our results extend and improve the corresponding results of Liu [5] and we

also extend the iteration process (1.8) of [5] to an implicit iteration process for a finite family of



CUBO
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Strong convergence of an implicit iteration process for a finite
family of strictly asymptotically pseudocontractive mappings 147

mappings.

Received: June 2009. Revised: November 2009.

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