In this section, we prove strong convergence theorems for the iterative scheme given in (2.1) in uniformly convex Banach spaces.

**Lemma 3.1**
*Let*
*E*
*be a real uniformly convex Banach space and*
*K*
*be a nonempty closed convex subset of*
*E*.

*Let*
*be two asymptotically nonexpansive self*-

*mappings with*
*and*
*be two asymptotically nonexpansive nonself*-

*mappings with*
*such that*
*and*
*for*
,

*respectively*,

*and*
.

*Let*
*be the sequence defined by* (2.1),

*where*
*and*
*are two real sequences in*
.

*Then*
- (1)
*exists for any*
;

- (2)
*exists*.

*Proof* (1) Set

. For any

, it follows from (2.1) that

Since
and
for
, we have
. It follows from Lemma 2.1 that
exists.

(2) Taking the infimum over all

in (3.2), we have

for each
. It follows from
and Lemma 2.1 that the conclusion (2) holds. This completes the proof. □

**Lemma 3.2**
*Let*
*E*
*be a real uniformly convex Banach space and*
*K*
*be a nonempty closed convex subset of*
*E*.

*Let*
*be two asymptotically nonexpansive self*-

*mappings with*
*and*
*be two asymptotically nonexpansive nonself*-

*mappings with*
*such that*
*and*
*for*
,

*respectively*,

*and*
.

*Let*
*be the sequence defined by* (2.1)

*and the following conditions hold*:

- (a)

- (b)

*Then*
*for*
.

*Proof* Set

. For any given

,

exists by Lemma 3.1. Now, we assume that

. It follows from (3.2) and

that

Taking lim sup on both sides in (3.1), we obtain

and so

By the condition (b), it follows that

and so, from (3.3), we have

Taking lim inf on both sides in the inequality above, we have

It follows from Lemma 2.2 that

Indeed, since

by the condition (b). It follows from (3.5) that

Since

and

is a nonexpansive retraction of

*E* onto

*K*, we have

Thus it follows from (3.5), (3.6) and (3.7) that

Since

by the condition (b) and

Using (3.3) and (3.8), we have

Using (3.3) and (3.11), we obtain that

Thus, using (3.9), (3.10) and the inequality

we have

. It follows from (3.6) and the inequality

from (3.8), (3.11) and (3.13), it follows that

Again, since

,

for

and

,

are two asymptotically nonexpansive nonself-mappings, we have

for

. It follows from (3.12), (3.14) and (3.15) that

for

. Moreover, we have

Using (3.4), (3.8) and (3.12), we have

for

. Thus it follows from (3.6), (3.10), (3.16) and (3.17) that

In fact, by the condition (b), we have

for

. Thus it follows from (3.5), (3.6), (3.9) and (3.10) that

This completes the proof. □

Now, we find two mappings,
and
, satisfying the condition (b) in Lemma 3.2 as follows.

**Example 3.1**[20]

Let ℝ be the real line with the usual norm

and let

. Define two mappings

by

Now, we show that

*T* is nonexpansive. In fact, if

or

, then we have

If

and

or

and

, then we have

This implies that *T* is nonexpansive and so *T* is an asymptotically nonexpansive mapping with
for each
. Similarly, we can show that *S* is an asymptotically nonexpansive mapping with
for each
.

Next, we show that two mappings *S*, *T* satisfy the condition (b) in Lemma 3.2. For this, we consider the following cases:

Case 1. Let

. Then we have

Case 2. Let

. Then we have

Case 3. Let

and

. Then we have

Case 4. Let

and

. Then we have

Therefore, the condition (b) in Lemma 3.2 is satisfied.

**Theorem 3.1**
*Under the assumptions of Lemma *3.2, *if one of*
,
,
*and*
*is completely continuous*, *then the sequence*
*defined by* (2.1) *converges strongly to a common fixed point of*
,
,
*and*
.

*Proof* Without loss of generality, we can assume that

is completely continuous. Since

is bounded by Lemma 3.1, there exists a subsequence

of

such that

converges strongly to some

. Moreover, we know that

by Lemma 3.2, which imply that

as

and so

. Thus, by the continuity of

,

,

and

, we have

for
. Thus it follows that
. Furthermore, since
exists by Lemma 3.1, we have
. This completes the proof. □

**Theorem 3.2**
*Under the assumptions of Lemma *3.2, *if one of*
,
,
*and*
*is semi*-*compact*, *then the sequence*
*defined by* (2.1) *converges strongly to a common fixed point of*
,
,
*and*
.

*Proof* Since
for
by Lemma 3.2 and one of
,
,
and
is semi-compact, there exists a subsequence
of
such that
converges strongly to some
. Moreover, by the continuity of
,
,
and
, we have
and
for
. Thus it follows that
. Since
exists by Lemma 3.1, we have
. This completes the proof. □

*for all*
, *where*
, *then the sequence*
*defined by* (2.1) *converges strongly to a common fixed point of*
,
,
*and*
.

*Proof* Since
for
by Lemma 3.2, we have
. Since
is a nondecreasing function satisfying
,
for all
and
exists by Lemma 3.1, we have
.

Now, we show that

is a Cauchy sequence in

*K*. In fact, from (3.2), we have

for each

, where

and

. For any

*m*,

*n*,

, we have

where

. Thus, for any

, we have

Taking the infimum over all

, we obtain

Thus it follows from
that
is a Cauchy sequence. Since *K* is a closed subset of *E*, the sequence
converges strongly to some
. It is easy to prove that
,
,
and
are all closed and so *F* is a closed subset of *K*. Since
,
, the sequence
converges strongly to a common fixed point of
,
,
and
. This completes the proof. □