In this section, we prove strong convergence theorem which is our main result.
Theorem 3.1.
Let
be a nonempty, closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space
, let
be an
-inverse-strongly monotone mapping of
into
with
for all
and
. Let
and
be two finite families of closed relatively weak quasi-nonexpansive mappings from
into itself with
, where
. Assume that
and
are uniformly continuous for all
. Let
be a sequence generated by the following algolithm:
where
, and
is the normalized duality mapping on
. Assume that
,
and
are the sequences in
satisfying the restrictions:
(C1)
;
(C2)
for some
with
, where
is the 2-uniformly convexity constant of
;
(C3)
and if one of the following conditions is satisfied
(a)
and
and
(b)
and
.
Then
converges strongly to
, where
is the generalized projection from
onto
.
Proof.
By the same method as in the proof of Cai and Hu [22], we can show that
is closed and convex. Next, we show
for all
. In fact,
is obvious. Suppose
for some
. Then, for all
, we know from Lemma 2.5 that
Since
and
is
-inverse-strongly monotone, we have
Therefore, from Lemma 2.1 and the assumption that
for all
and
, we obtain that
Substituting (3.3) and (3.4) into (3.2) and using the condition that
, we get
Using (3.5) and the convexity of
, for each
, we obtain
It follows from (3.6) that
So,
. Then by induction,
for all
and hence the sequence
generated by (3.1) is well defined. Next, we show that
is a convergent sequence in
. From
, we have
It follows from
for all
that
From Lemma 2.4, we have
for each
and for all
. Therefore, the sequence
is bounded. Furthermore, since
=
and
=
, we have
This implies that
is nondecreasing and hence
exists. Similarly, by Lemma 2.4, we have, for any positive integer
, that
The existence of
implies that
as
. From Lemma 2.2, we have
Hence,
is a Cauchy sequence. Therefore, there exists a point
such that
as 
Now, we will show that
.
-
(I)
We first show that
. Indeed, taking
in (3.12), we have
It follows from Lemma 2.2 that
This implies that
The property of the function
implies that
Since
, we obtain
It follows from the condition (3.14) and (3.17) that
From Lemma 2.2, we have
Combining (3.15) and (3.20), we have
Since
is uniformly norm-to-norm continuous on any bounded sets, we have
On the other hand, noticing
Since
is uniformly norm-to-norm continuous on any bounded sets, we have
Using (3.15), (3.20), and (3.24) that
Taking the constant
, we have, from Lemma 2.6, that there exists a continuous strictly increasing convex function
satisfying the inequality (2.7) and
.
Case 1.
Assume that (a) holds. Applying (2.7) and (3.5), we can calculate
This implies that
We observe that
It follows from (3.15), (3.22), (3.23) and (3.25) that
From
and (3.27), we get
By the property of function
, we obtain that
Since
is uniformly norm-to-norm continuous on any bounded sets, we have
From (3.15) and (3.32), we have
Noticing that
for all
. By the uniformly continuity of
, (3.16) and (3.33), we obtain
Thus
From the closeness of
, we get
. Therefore
. In the same manner, we can apply the condition
to conclude that
Again, by (C2) and (3.26), we have
It follows from (3.29) and
that
Since
, we have
From Lemmas 2.4, 2.5, and (3.4), we have
It follows from (3.40) that
Lemma 2.2 implies that
Since
is uniformly norm-to-norm continuous on any bounded sets, we have
Combining (3.37) and (3.43), we also obtain
Moreover
By (3.43), (3.15), we have
This implies that
Noticing that
for all
. Since
is uniformly continuous, we can show that
. From the closeness of
, we get
. Therefore
. Hence
.
Case 2.
Assume that (b) holds. Using the inequalities (2.7) and (3.5), we obtain
This implies that
It follows from (3.21), (3.24) and the condition
that
By the property of function
, we obtain that
Since
is uniformly norm-to-norm continuous on any bounded sets, we have
On the other hand, we can calculate
Observe that
It follows from (3.53) and (3.54) that
Applying
and (3.57) and the fact that
is bounded to (3.55), we obtain
From Lemma 2.2, one obtains
We observe that
This together with (3.25) and (3.59), we obtain
Noticing that
for all
. By the uniformly continuity of
, (3.16) and (3.61), we obtain
Thus
From the closeness of
, we get
. Therefore
. By the same proof as in Case 1, we obtain that
Hence
as
for each
and
Combining (3.54), (3.61), and (3.65), we also have
Moreover
By (3.43), (3.15), we have
This implies that
Noticing that
for all
. Since
is uniformly continuous, we can show that
. From the closeness of
, we get
. Therefore
. Hence
.
-
(II)
We next show that
.
Let
be an operator defined by:
By Lemma 2.7,
is maximal monotone and
. Let
, since
, we have
. From
, we get
Since
is
-inverse-strong monotone, we have
On other hand, from
and Lemma 2.3, we have
, and hence
Because
is
constricted, it holds from (3.74) and (3.75) that
for all
. By taking the limit as
in (3.76) and from (3.43) and (3.44), we have
as
. By the maximality of
we obtain
and hence
. Hence we conclude that
Finally, we show that
. Indeed, taking the limit as
in (3.9), we obtain
and hence
by Lemma 2.3. This complete the proof.
Remark 3.2.
Theorem 3.1 improves and extends main results of Iiduka and Takahashi [15], Xu and Ori [19], Qin et al. [21], and Cai and Hu [22] because it can be applied to solving the problem of finding the common element of the set of common fixed points of two families of relatively weak quasi-nonexpansive mappings and the set of solutions of the variational inequality for an inverse-strongly monotone operator.
Strong convergence theorem for approximating a common fixed point of two finite families of closed relatively weak quasi-nonexpansive mappings in Banach spaces may not require that
is 2-uniformly convex. In fact, we have the following theorem.
Corollary 3.3.
Let
be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth Banach space
. Let
and
be two finite families of closed relatively weak quasi-nonexpansive mappings from
into itself with
, where
. Assume that
and
are uniformly continuous for all
. Let
be a sequence generated by the following algolithm:
where
, and
is the normalized duality mapping on
. Assume that
,
and
are the sequences in
satisfying the following restrictions:
(C1)
;
(C2)
and if one of the following conditions is satisfied
(a)
and
and
(b)
and
.
Then
converges strongly to
, where
is the generalized projection from
onto
.
Proof.
Put
in Theorem 3.1. Then, we get that
. Thus, the method of the proof of Theorem 3.1 gives the required assertion without the requirement that
is 2-uniformly convex.
Remark 3.4.
Corollary 3.3 improves Theorem 3.1 of Cai and Hu [22] from a finite family of of relatively weak quasi-nonexpansive mappings to two finite families of relatively weak quasi-nonexpansive mappings.
If
, a Hilbert space, then
is
-uniformly convex (we can choose
) and uniformly smooth real Banach space and closed relatively weak quasi-nonexpansive map reduces to closed weak quasi-nonexpansive map. Furthermore,
, identity operator on
and
, projection mapping from
into
. Thus, the following corollaries hold.
Corollary 3.5.
Let
be a nonempty, closed and convex subset of a Hilbert space
. Let
and
be two finite families of closed weak quasi-nonexpansive mappings from
into itself with
, where
with
for all
and
. Assume that
and
are uniformly continuous for all
. Let
be a sequence generated by the following algorithm:
where
, and
is the normalized duality mapping on
. Assume that
,
,
,
, and
are the sequences in
satisfying the restrictions:
(C1)
;
(C2)
for some
with
, where
is the 2-uniformly convexity constant of
;
(C3)
and if one of the following conditions is satisfied
(a)
and
and
(b)
and
.
Then
converges strongly to
, where
is the metric projection from
onto
.
Let
be a nonempty closed convex cone in
, and let
be an operator from
into
. We define its polar in
to be the set
Then an element
in
is called a solution of the complementarity problem if
The set of all solutions of the complementarity problem is denoted by
. Several problem arising in different fields, such as mathematical programming, game theory, mechanics, and geometry, are to find solutions of the complementarity problems.
Theorem 3.6.
Let
be a nonempty, closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space
, let
be an
-inverse-strongly monotone mapping of
into
with
for all
and
. Let
and
be two finite families of closed relatively weak quasi-nonexpansive mappings from
into itself with
, where
. Assume that
and
are uniformly continuous for all
. Let
be a sequence generated by the following algorithm:
where
=
=
, and
is the normalized duality mapping on
. Assume that
,
and
are the sequences in
satisfying the restrictions:
(C1)
;
(C2)
for some
with
, where
is the 2-uniformly convexity constant of
;
(C3)
and if one of the following conditions is satisfied
(a)
and
and
(b)
and
.
Then
converges strongly to
, where
is the generalized projection from
onto
.
Proof.
From [25, Lemma
], we have
. From Theorem 3.1, we can obtain the desired conclusion easily.