1. Introduction
Let
T be a selfmap on a subset
W of a Banach space
. Subsequently,
T is called contraction map on
W if for each pair of elements
, there is some real constant
, such that
If (
1) holds at
, then
T is called non-expansive. When a point
exists with the property
, then
g is called a fixed point of
T. The fixed point set of
T we often denote by the notation
. In 1922, Banach [
1] proved that any self contraction map of a closed subset
W of a Banach space has a unique fixed point. Later, the Banach result [
1] was extended by Caccioppoli [
2] in complete metric spaces. In 1965, Browder [
3] and Gohde [
4] proved that any self non-expansive map of a convex bounded closed subset
W of a uniformly convex Banach space
U always admits a fixed point. The Browder-Gohde result was proved by Kirk [
5] in the context of reflexive Banach spaces. We know that the class of non-expansive maps is important as an application point of view. Thus, it is very natural to consider larger classes of non-expansive maps. One of the larger class of non-expansive maps was introduced by Suzuki [
6] in 2008. A selfmap
T on a subset
W of a Banach space is said to be Suzuki map (or said to satisfy
-condition), if for each pair of elements
, it follows that
Suzuki also proved that, if a map T satisfies -condition, then for all , holds.
Inspired by Suzuki
-condition, Garcia-Falset et al. [
7] introduced
-condition, as follows: a selfmap
T on a subset
W of a Banach space is said be Garcia-Falset map (or said to satisfy
-condition), if for each pair of elements
, there is some real constant
, such that
We see that any map
T with
-condition satisfies
-condition with real constant
. Nevertheless, an example in the
Section 4 shows that there exists maps in the class of Garcia-Falset maps which does not belong to the class of Suzuki maps. Hence, the class of Garica-Falset maps properly includes the class of Suzuki maps. Garcia-Falset et al. [
7] also proved some existence theorems of fixed points for maps satisfying
-condition. Recently, Usurelu et al. [
8] studied some fixed point results for this class of maps and using an example, they studied the visualization of convergence behaviors of some iterative processes. In this paper, we use the three-step iterative process, which is different from the iterative process used in [
8] for approximating fixed points of maps of this class. We also present a new example of maps satisfying
-condition, and prove that its under the consideration three-step iterative process is more efficient than the other well-known three-step iterative processes. In the last section, we shall apply our results for finding solutions of split feasibility problems.
However, once the existence of fixed point for an operator is established, then the finding of this fixed point is not easy work. One of the simplest iterative method for finding fixed points is the Picard iterative method, which is,
. The Banach–Caccioppoli result states that the unique fixed point of contractions can be obtained by using the Picard iterative method. Nevertheless, the Picard iterative method does not always work properly in the finding of fixed points of non-expansive maps. For finding fixed points of non-expansive maps and to obtain relatively better convergence speed, one deals with the different iterative methods, e.g., Mann [
9], Ishikawa [
10], Agarwal [
11], Noor [
12], Abbas [
13], and others. Among the other things, Thakur et al. [
14] introduced the following three-step iterative process for finding fixed points of non-expansive maps in Banach spaces, as follows:
where
.
In [
14], Thakur et al. proved some important strong and weak convergence theorems of the iterative process (
2) for the class of non-expansive maps in the context of uniformly convex Banach spaces. Recently in 2020, Maniu [
15] extended the results of Thakur et al. [
14] to the setting of Suzuki maps. The purpose of this research is to extend the results of Maniu [
15] to the more general setting of Garcia–Falset maps. We also study the rate of convergence of the iterative process (
2) with the some well-known three-step iterative processes in the setting of Garcia-Falset maps, under different initial points and set of parameters. At the end of the paper, we shall apply our results to find the solution of split feasibility problems.
2. Preliminaries
In this section, we shall deal with some basic definitions and early results. Let
W be any nonempty subset of a Banach space
U. Fix
and assume that
is bounded. Define
by
We denote the asymptotic radius of
with respect to
W by
and define, as follows:
We denote the asymptotic center of
with respect to
W by
and define, as follows:
The asymptotic center of
with respect to
W is nonempty and convex whenever
W is convex weakly compact, (see, e.g., [
16,
17] and others). One of the well known property of the set
is the singletoness property in the frame work of uniformly convex Banach spaces [
18].
Recall that a Banach space
U is said to have Opial’s property [
19], if, for any weakly convergent sequence
in
U with a weak limit
, follows the following strict inequality
The following result shows that the class of Suzuki maps is a sub-class of Garcia–Falset maps.
Lemma 1. [7] Let W be a nonempty subset of a Banach space and let satisfies -condition. Subsequently, T satisfies -condition with . Lemma 2. [7] Let W be a nonempty subset of a Banach space and let satisfies -condition. Subsequently, for all and , we have . Lemma 3. [7] Let T be a selfmap on a subset W of a Banach space having Opial property. Let T satisfy the -condition. If is weakly convergent to g and , then . The following characterization is due to Schu [
20].
Lemma 4. Let U be a uniformly convex Banach space, for every natural number and be some real constant. If and are any two sequences in U, such that , and , then .
3. Convergence Results in Banach Spaces
This section contains some weak and strong convergence results of the iterative process (
2) for operators satisfying
-condition. Throughout the section,
U will stand for uniformly convex Banach space.
Lemma 5. Let W be a nonempty convex closed subset of U and be a map satisfying -condition with . If is generated by (2), then exists for every . Proof. Let
. By Lemma 2, we have
and
While using the above inequilities, we have
Thus, is bounded and non-increasing, which implies that exists for each . □
Now, we establish the following result which will be used throughout in the upcoming theorems.
Theorem 1. Let W be a nonempty closed convex subset of U and let be a map satisfying -condition. Let be the sequence defined by (2). Subsequently, if and only if is bounded and . Proof. Let
be bounded and
. Let
. We shall prove that
. Since
T satisfies
-condition, we have
It follows that . Since is singleton set, we have . Hence, .
Conversely, we assume that
and
. We shall prove that
is bounded and
. By Lemma 5,
exists and
is bounded. Put
From the proof of Lemma 5, it follows that
Again, from the proof of Lemma 5,
Accordingly, we can get
.
Applying Lemma 4, we obtain
□
Using compactness of the domain
W, we establish the following strong convergence of
generated by (
2) for maps satisfying
-condition.
Theorem 2. Let W be a nonempty convex compact subset of U and let T and be as in Theorem 1 and . Subsequently, converges strongly to a fixed point of T.
Proof. By compactness of
W we can construct a subsequence
of
such that
, for some
. Because the map
T satisfies
-condition, one can find some real constant
, such that
In the view of Theorem 1,
. Now, using
and
, we have from (
8),
. Now, the uniqueness of limits in Banach space follows that
. Hence,
u is the fixed point of
T. By Lemma 5,
exists. Hence,
u is the strong limit of
. □
Theorem 3. Let W be a nonempty closed convex subset of U and let T and be as in Theorem 1. If and . Subsequently, converges strongly to a fixed point of T.
Proof. The proof is elementary and, hence, omitted. □
The next theorem requires condition
I of Sentor and Dotson [
21]. The detail definition is given below.
Definition 1. [21] Let W be a nonempty subset of U. A selfmap T of W is said to satisfy condition I if there is a nondecreasing function ξ with the properties if and only if , for every and for each . Theorem 4. Let W be a nonempty closed convex subset of U and let T and be as in Theorem 1 and . If T satisfies condition I, then converges strongly to a fixed point of T.
Proof. From Theorem 1, it follows that
From the definition of condition
I, we have
The function
is non-decreasing and satisfy
,
for every
. Hence
By Theorem 3, we conclude that T converges to some fixed point of T. □
Using Opial’s property, we obtain the weak convergence of for maps with -condition.
Theorem 5. Let W a nonempty closed convex subset of U having Opial property and let T and be as in Theorem 1 and . Subsequently, converges weakly to a fixed point of T.
Proof. By Theorem 1, the sequence
is bounded and
.
U is reflexive because
U is uniform convex. Now, by reflexivity of
U, we can construct a weakly convergent subsequence
of
with some weak limit
. By Lemma 3, we conclude that
. We claim that
converges weakly to
. Assume that
is not the weak limit of
. Accordingly, we choose another weakly convergent subsequence
of
with some weak limit
and assume that
. Again, by Lemma 3,
. Using Lemma 5 and Opial condition, we have
This is a contradiction. Hence is the weak limit of and fixed point of T. □
5. Application
In this section, we are interested in finding of the solution of a split feasibility problem (in short SFP) by using the three-step iterative method (
2). To do this, we assume that
and
are any two real Hilbert spaces,
and
be convex and closed. Assume that
be a linear and bounded. Subsequently, the SFP mathematically described as finding an element
, such that
Next we assume that the solution set
associated with the SFP (
10) is nonempty and let
We see that the set
is nonempty convex as well as closed. Censor and Elfving [
22] solved the class of inverse problems with the help of SFP. In the year 2002, Byrne [
23] proposed the remarkable
-algorithm for solving the SFP. If
,
and
represent the projections onto
C and
Q respectively and
is the adjoint of
L, then the sequence
of
-algorithm is generated iteratively, as follows:
The following facts are in [
24].
Lemma 6. If , for , then T is non-expansive.
By assumption, the set
that is associated with a SFP is nonempty, one can see that the element
is the solution of SFP if and only if it solves the following fixed point equation:
Hence, the solution set
coincides with the fixed point set of the operator
T, that is,
. For details, one can refer [
25,
26].
Now, we present our main results.
Theorem 6. Let and be a sequence defined by the iterative process (2), then converges weakly to the some solution of a SFP (10). Proof. By Lemma 6, the operator T is non-expansive. In the view of Lemma 1, T is Garcia–Falset operator. The conclusions follows from Theorem 5. □
Theorem 7. Let and be a sequence defined by the iterative process (2), then converges strongly to the solution of SFP (10), provided that . Proof. Proof follows from Theorem 3. □