International Journal of Analysis and Applications
ISSN 2291-8639
Volume 2, Number 1 (2013), 54-61
http://www.etamaths.com

CONVERGENCE THEOREMS OF AN IMPLICIT ITERATES WITH
ERRORS FOR TOTAL ASYMPTOTICALLY PSEUDO-CONTRACTIVE

MAPPINGS

G. S. SALUJA

Abstract. The goal of this paper is to establish weak and strong convergence
theorems of an implicit iteration process with errors to converge to common fixed
points for a finite family of uniformly L-Lipschitzian total asymptotically pseudo-
contractive mappings in the framework of Banach spaces. Our results extend the
corresponding result of [2, 5, 8, 10] and many others.

1. Introduction and Preliminaries

In recent years, the implicit iteration scheme for approximating fixed point of
nonlinear mappings has been introduced and studied by various authors (see, e.g.,
[1, 4, 7, 10, 11]).

In 2001, Xu and Ori [11] 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 =

⋂N
i=1 F(Ti) , ∅,

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(1.1)

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 (1.1).

2010 Mathematics Subject Classification. 47H09, 47H10.
Key words and phrases. Total asymptotically pseudo-contractive mapping, common fixed point,

implicit iteration with errors, strong convergence, weak convergence, Banach space.

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.

54



CONVERGENCE THEOREMS OF AN IMPLICIT ITERATES 55

In 2002, Zhou and Chang [12] introduced the following implicit iteration scheme
for common fixed points of a finite family of asymptotically nonexpansive map-
pings {Ti}Ni=1 in Banach space:

xn = αnxn−1 + (1 −αn)T
n
n (mod N)xn, n ≥ 1(1.2)

By this implicit iteration scheme, Zhou and Chang proved some weak and strong
convergence theorems in Banach spaces for a finite family of asymptotically non-
expansive mappings.

In 2003, Sun [10] modified the implicit iteration process of Xu and Ori [11]
and applied the modified averaging iteration process for the approximation of
fixed points of asymptotically quasi-nonexpansive mappings. Sun introduced the
following implicit iteration process for common fixed points of a finite family of
asymptotically quasi-nonexpansive mappings {Ti}Ni=1 in Banach spaces:

xn = αnxn−1 + (1 −αn)T
k
i xn, n ≥ 1,(1.3)

where n = (k − 1)N + i, i ∈ I = {1, 2, . . . , N}.

In this paper, we propose the following implicit iteration process with errors
for a finite family of total asymptotically pseudo-contractive mappings {Ti}Ni=1 and
prove some strong convergence theorems for said mappings and iteration scheme
in Banach spaces. The results presented in this paper extend the corresponding
results of Chang [2], Miao et al. [5], Sun [10], Osilike and Akuchu [8] and many
others. The proposed implicit iteration scheme is as follows: x1 ∈ C and

xn = αnxn−1 + (1 −αn)T
n
n xn + un, ∀n ≥ 1,(1.4)

where C is a closed convex subset of a Banach space E with C+C ⊂ C, Tnn = T
n
n (mod N)

and {un} is a bounded sequence in C.

Definition 1.1. ([5]) A mapping T : C → C is said to be total asymptotically pseudo
contractive if there exists a nonnegative real sequence {µn}, n ≥ 1, with µn → 0 as
n →∞ and there exists a strictly function φ: R+ → R+ with φ(0) = 0 such that for
all x, y ∈ C,

〈Tnx − Tn y, j(x − y)〉 ≤
∥∥∥x − y∥∥∥2 + µnφ(∥∥∥x − y∥∥∥).(1.5)

Remark 1.2. If φ(λ) = λ2, then (1.5) reduces to

〈Tnx − Tn y, j(x − y)〉 ≤ (1 + µn)
∥∥∥x − y∥∥∥2 .(1.6)

The total asymptotically pseudo contractive mappings coincide with asymptot-
ically pseudo contractive mappings. If µn = 0 for all n ≥ 1, we obtain from (1.5)
the class of mappings that includes the class of pseudo contractive mappings.

Note. The idea of Definition 1.1 is to unify various definitions of classes of map-
pings associated with the class of asymptotically pseudo contractive mappings
and which are extensions of pseudo contractive mappings.

Observe that if C is a nonempty closed convex subset of a real Banach space E
with C + C ⊂ C and {Ti}Ni=1 : C → C be N uniformly Li-Lipschitzian total asymp-
totically pseudo-contractive mappings. If (1 − αn)L < 1, where L = max{Li : i =



56 SALUJA

1, 2, . . . , N}, then for given xn ∈ C, the mapping Wn : C → C defined by

Wn(x) = αnxn−1 + (1 −αn)T
n
n x + un, ∀n ≥ 1,(1.7)

is a contraction mapping. In fact, the following are observed

‖Wnx − Wn y‖ =‖αnxn−1 + (1 −αn)T
n
n x

+ un − (αnxn−1 + (1 −αn)T
n
n y + un)‖

=(1 −αn)
∥∥∥Tnn x − Tnn y∥∥∥

≤(1 −αn)L
∥∥∥x − y∥∥∥ , ∀x, y ∈ C.(1.8)

Since (1 −αn)L < 1 for all n ≥ 1, hence Wn : C → C is a contraction mapping. By
Banach contraction mapping principle, there exists a unique fixed point xn ∈ C
such that

xn = αnxn−1 + (1 −αn)T
n
n xn + un, ∀n ≥ 1.(1.9)

Therefore, if (1 − αn)L < 1 for all n ≥ 1, then the iterative sequence (1.4) can be
employed for the approximation of common fixed points for a finite family of uni-
formly Li-Lipschitzian total asymptotically pseudo-contractive mappings.

Recall that a Banach space E satisfies the Opial’s condition [6] if for each sequence
{xn} in E weakly convergent to a point x and for all y , x

lim inf
n→∞

‖xn − x‖ < lim inf
n→∞

∥∥∥xn − y∥∥∥ .
The examples of Banach spaces which satisfy the Opial’s condition are Hilbert

spaces and all Lp[0, 2π] with 1 < p , 2 fail to satisfy Opial’s condition [6].

Let C be a nonempty closed convex subset of a Banach space E. Then I − T is
demiclosed at zero if, for any sequence {xn} in C, condition xn → x weakly and
limn→∞ ‖xn − Txn‖ = 0 implies (I − T)x = 0.

Recall that a family {Ti}Ni=1 : C → C with F = ∩
N
i=1F(Ti) , ∅ is said to satisfy

Condition (B) [3] on C if there is a nondecreasing function f : [0,∞) → [0,∞) with
f (0) = 0, f (r) > 0 for all r ∈ (0,∞) such that for all x ∈ C

max
1≤i≤N

{
‖x − Tix‖

}
≥ f (d(x,F )).

In the sequel we need the following lemma to prove our main results.

Lemma 1.3. (see [9]) Let {an}∞n=1, {bn}
∞

n=1 and {cn}
∞

n=1 be sequences of nonnegative real
numbers satisfying the inequality

an+1 ≤ (1 + bn)an + cn, n ≥ 1.

If
∑
∞

n=1 bn < ∞ and
∑
∞

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

n=1 has a
subsequence which converges strongly to zero, then limn→∞ an = 0.



CONVERGENCE THEOREMS OF AN IMPLICIT ITERATES 57

2. Main Results

Theorem 2.1. Let E be a real Banach space. Let C be a closed convex subset of E with
C + C ⊂ C and {Ti}Ni=1 be a finite family of uniformly Li-Lipschitzian total asymptotically
pseudo contractive self mappings of C into itself such that F = ∩Ni=1F(Ti) , ∅ is closed. Let
L = max{Li : i = 1, 2, . . . , N},

∑
∞

n=1 ‖un‖ < ∞,
∑
∞

n=1 µn < ∞, and suppose that there exist
Ki > 0 such that φi(λi) ≤ Kiλi, i = 1, 2, . . . , N. Given x1 ∈ C, let {xn}∞n=1 be the sequence
generated by an implicit iteration scheme (1.4). If {αn} is chosen such that αn ∈ (0, 1)
with 0 < τ < αn < 1, where τ is some constant, then {xn} strongly to a common fixed
point of the family {Ti}Ni=1 if and only if lim infn→∞ d(xn,F ) = 0, where d(x,F ) denotes
the distance between x and the set F .

Proof. The necessity is obvious and so it is omitted. Now, we prove the sufficiency.
For any p ∈F = ∩Ni=1F(Ti), from (1.4) and (1.5), we have∥∥∥xn − p∥∥∥2 = ∥∥∥αnxn−1 + (1 −αn)Tnn xn + un − p∥∥∥2

= αn〈xn−1 − p, j(xn − p)〉 + (1 −αn)〈T
n
n xn − p, j(xn − p)〉

+〈un, j(xn − p)〉

≤ αn
∥∥∥xn−1 − p∥∥∥ ∥∥∥xn − p∥∥∥ + (1 −αn)[∥∥∥xn − p∥∥∥2

+µnφ(
∥∥∥xn − p∥∥∥)] + ‖un‖∥∥∥xn − p∥∥∥

≤

∥∥∥xn−1 − p∥∥∥ ∥∥∥xn − p∥∥∥ + (1 −αn)
αn

µnφ(
∥∥∥xn − p∥∥∥)

+
1
αn
‖un‖

∥∥∥xn − p∥∥∥
≤

∥∥∥xn−1 − p∥∥∥ ∥∥∥xn − p∥∥∥ + (1 −τ)
τ

µnφ(
∥∥∥xn − p∥∥∥)

+
1
τ
‖un‖

∥∥∥xn − p∥∥∥ .(2.1)
Simplify both the sides of above inequality, we get

∥∥∥xn − p∥∥∥ ≤ ∥∥∥xn−1 − p∥∥∥ + (1 −τ)
τ

µn
φ(

∥∥∥xn − p∥∥∥)∥∥∥xn − p∥∥∥ + ‖un‖τ
≤

∥∥∥xn−1 − p∥∥∥ + θn,(2.2)
where

θn =
(1 −τ)
τ

µn
φ(

∥∥∥xn − p∥∥∥)∥∥∥xn − p∥∥∥ + ‖un‖τ .
Since φ is an strictly increasing continuous function, by hypothesis, there exists

K such that
φ(‖xn−p‖)

‖xn−p‖
≤ K and by the assumptions of the theorem we know that∑

∞

n=1 µn < ∞ and
∑
∞

n=1 ‖un‖ < ∞, it follows that
∑
∞

n=1 θn < ∞. Then from (2.2), we
have

d(xn,F ) ≤ d(xn−1,F ) + θn.(2.3)



58 SALUJA

By Lemma 1.3, we know that limn→∞ d(xn,F ) exists. Because lim infn→∞ d(xn,F ) =
0, then

lim
n→∞

d(xn,F ) = 0.(2.4)

Next we prove that {xn} is a Cauchy sequence in C. It follows from (2.2) that for
any m ≥ 1, for all n ≥ n0 and for any p ∈F , we have∥∥∥xn+m − p∥∥∥ ≤ ∥∥∥xn+m−1 − p∥∥∥ + θn+m

≤

∥∥∥xn+m−2 − p∥∥∥ + [θn+m + θn+m−1]
≤ . . .

≤ . . .

≤

∥∥∥xn − p∥∥∥ + n+m∑
k=n+1

θk.(2.5)

So we have

‖xn+m − xn‖ ≤
∥∥∥xn+m − p∥∥∥ + ∥∥∥xn − p∥∥∥

≤ 2
∥∥∥xn − p∥∥∥ + n+m∑

k=n+1

θk

≤ 2
∥∥∥xn − p∥∥∥ + ∞∑

k=n

θk(2.6)

Then, we have

‖xn+m − xn‖ ≤ 2d(xn,F ) +
∞∑

k=n

θk, ∀n ≥ n0.(2.7)

For any given ε > 0, there exists a positive integer n1 ≥ n0 such that for any n ≥ n1,

d(xn,F ) <
ε
4

and
∞∑

k=n

θk <
ε
2
.(2.8)

Thus, when n ≥ n1, we have

‖xn+m − xn‖ < 2.
ε
4

+
ε
2

= ε.(2.9)

This implies that {xn} is a Cauchy sequence in C. Thus, the completeness of E
implies that {xn} must be convergent. Assume that limn→∞ xn = p∗. Now, we have
to show that p∗ is a common fixed point of {Ti : i = 1, 2, . . . , N}, that is we have to
show that p∗ ∈ F . Suppose for contradiction that p∗ ∈ F c (where F c denotes the
complement of F ). Since F is a closed subset of E, we have that d(p∗,F ) > 0. But
for all p ∈F , we have ∥∥∥p∗ − p∥∥∥ ≤ ∥∥∥p∗ − xn∥∥∥ + ∥∥∥xn − p∥∥∥ ,(2.10)
which implies that

d(p∗,F ) ≤
∥∥∥xn − p∗∥∥∥ + d(xn,F ),(2.11)

so that, we obtain d(p∗,F ) = 0 as n → ∞, which contradicts d(p∗,F ) > 0. Thus, p∗

is a common fixed point of the mappings {Ti : i = 1, 2, . . . , N}. This completes the
proof. �



CONVERGENCE THEOREMS OF AN IMPLICIT ITERATES 59

Theorem 2.2. Let E be a real Banach space. Let C be a closed convex subset of E with
C + C ⊂ C and {Ti}Ni=1 be a finite family of uniformly Li-Lipschitzian total asymptotically
pseudo contractive self mappings of C into itself such that F = ∩Ni=1F(Ti) , ∅ is closed. Let
L = max{Li : i = 1, 2, . . . , N},

∑
∞

n=1 ‖un‖ < ∞,
∑
∞

n=1 µn < ∞, and suppose that there exist
Ki > 0 such that φi(λi) ≤ Kiλi, i = 1, 2, . . . , N. Given x1 ∈ C, let {xn}∞n=1 be the sequence
generated by an implicit iteration scheme (1.4). If {αn} is chosen such that αn ∈ (0, 1) with
0 < τ < αn < 1, where τ is some constant, then {xn} converges strongly to a common fixed
point p∗ of the family of mappings {Ti}Ni=1 if and only if there exists a subsequence {xn j} of
{xn} which converges to p∗.

Proof. The proof of Theorem 2.2 follows from Lemma 1.3 and Theorem 2.1. This
completes the proof. �

Theorem 2.3. Let E be a real Banach space. Let C be a closed convex subset of E with
C + C ⊂ C and {Ti}Ni=1 be a finite family of uniformly Li-Lipschitzian total asymptotically
pseudo contractive self mappings of C into itself such that F = ∩Ni=1F(Ti) , ∅ is closed. Let
L = max{Li : i = 1, 2, . . . , N},

∑
∞

n=1 ‖un‖ < ∞,
∑
∞

n=1 µn < ∞, and suppose that there exist
Ki > 0 such that φi(λi) ≤ Kiλi, i = 1, 2, . . . , N. Given x1 ∈ C, let {xn}∞n=1 be the sequence
generated by an implicit iteration scheme (1.4). If {αn} is chosen such that αn ∈ (0, 1)
with 0 < τ < αn < 1, where τ is some constant. Assume that limn→∞ ‖xn − Tixn‖ = 0,
for all i ∈ I = {1, 2, . . . , N}. Suppose {Ti : i = 1, 2, . . . , N} satisfies condition (B), then
the sequence {xn} converges strongly to a common fixed point of the mappings {Ti : i =
1, 2, . . . , N}.

Proof. By assumption limn→∞ ‖xn − Tixn‖ = 0, for all i ∈ I = {1, 2, . . . , N}. Since
{Ti : i = 1, 2, . . . , N} satisfies condition (B), so condition (B) guarantees that limn→∞
f (d(xn,F )) = 0. Since f is a non-decreasing function and f (0) = 0, it follows that
limn→∞ d(xn,F ) = 0. Therefore, Theorem 2.1 implies that {xn} converges strongly
to a point in F . This completes the proof. �

Theorem 2.4. Let E be a real Banach space satisfying Opial’s condition and C be a
weakly compact subset of E with C + C ⊂ C. Let {Ti}Ni=1 be a finite family of uniformly
Li-Lipschitzian total asymptotically pseudo contractive self mappings of C into itself such
that F = ∩Ni=1F(Ti) , ∅ is closed. Let L = max{Li : i = 1, 2, . . . , N},

∑
∞

n=1 ‖un‖ < ∞,∑
∞

n=1 µn < ∞, and suppose that there exist Ki > 0 such that φi(λi) ≤ Kiλi, i = 1, 2, . . . , N.
Given x1 ∈ C, let {xn}∞n=1 be the sequence generated by an implicit iteration scheme (1.4)
and the sequence {αn} is chosen such that αn ∈ (0, 1) with 0 < τ < αn < 1, where τ is
some constant. Suppose that {Ti : i = 1, 2, . . . , N} has a common fixed point, I − Ti for all
i ∈ I = {1, 2, . . . , N} is demiclosed at zero and {xn} is an approximating common fixed point
sequence for Ti, that is, limn→∞ ‖xn − Tixn‖ = 0, for all i ∈ I = {1, 2, . . . , N}. Then the
sequence {xn} defined by (1.4) converges weakly to a common fixed point of the mappings
{Ti : i = 1, 2, . . . , N}.

Proof. First, we show that ωw(xn) ⊂ F . Let xnk → x weakly. By assumption, we
have limn→∞ ‖xn − Tixn‖ = 0. Since I − Ti, for all i ∈ I = {1, 2, . . . , N} is demiclosed
at zero, x ∈F . By Opial’s condition, {xn} possesses only one weak limit point, that
is, {xn} converges weakly to a common fixed point of {Ti}Ni=1. This completes the
proof. �



60 SALUJA

Remark 2.5. Theorem 2.1 extends the corresponding result of Chang [2] to the
case of more general class of asymptotically nonexpansive mappings and implicit
iteration scheme with errors considered in this paper.

Remark 2.6. Theorem 2.1 also extends the corresponding result of Miao et al. [5] to
the case of implicit iteration scheme with errors considered in this paper.

Remark 2.7. Theorem 2.1 also extends the corresponding result of Sun [10] to the
case of more general class of asymptotically quasi nonexpansive mapping and
implicit iteration scheme with errors considered in this paper.

Remark 2.8. Theorem 2.1 also extends the corresponding result of Osilike and
Akuchu [8] to the case of more general class of asymptotically pseudo contractive
mapping and implicit iteration scheme with errors considered in this paper.

3. Conclusion

The class of total asymptotically pseudo contractive mapping is more general
than the class of pseudo contractive mapping and it also unify various definitions of
classes of mappings associated with the class of asymptotically pseudo contractive
mappings. Thus the results obtained in this paper are good improvement and
generalization of several known results in the existing literature.

References

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spaces, Nonlinear Anal. 67(2007), 2258-2271.

[2] S.S. Chang, On the convergence of implicit iteration process with error for a finite
family of asymptotically nonexpansive mappings, J. Math. Anal. Appl. 313 (2006),
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[3] C.E. Chidume, N. Shahzad, Strong convergence of an implicit iteration process for
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[4] G. Marino and H.K. Xu, Weak and strong convergence theorems for strict pseudo-
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[5] Q. Miao, R. Chen and H. Zhou, Convergence of an implicit iteration process for
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[6] Z. Opial, Weak convergence of the sequence of successive approximations for nonex-
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[7] M.O. Osilike, Implicit iteration process for common fixed points of a finite family of
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[8] M.O. Osilike and B.G. Akuchu, Common fixed points of a finite family of asymp-
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[9] M.O. Osilike, S.C. Aniagbosor and B.G. Akuchu, Fixed points of asymptotically
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[10] Z.H. Sun, Strong convergence of an implicit iteration process for a finite family
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CONVERGENCE THEOREMS OF AN IMPLICIT ITERATES 61

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Departmentof Mathematics, Govt. Nagarjuna P.G. Collegeof Science, Raipur (C.G.), India