Convergence of SP-iteration for generalized nonexpansive mapping in Banach spaces
UDC 517.9 Phuengrattana and Suantai [J. Comput. and Appl. Math., 235, 3006 – 3014 (2011)] introduced an iteration scheme and they named this iteration as SP-iteration. In this paper, we study the convergence behaviour of SP-iteration scheme for the class of generalized nonexpansive mappings. One wea...
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| author | Ali, J. Uddin, I. Ali, J. Uddin, I. Ali, J. Uddin, I. |
| author_facet | Ali, J. Uddin, I. Ali, J. Uddin, I. Ali, J. Uddin, I. |
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| description | UDC 517.9
Phuengrattana and Suantai [J. Comput. and Appl. Math., 235, 3006 – 3014 (2011)] introduced an iteration scheme and they named this iteration as SP-iteration. In this paper, we study the convergence behaviour of SP-iteration scheme for the class of generalized nonexpansive mappings. One weak convergence theorem and two strong convergence theorems in uniformly convex Banach spaces are obtained. We also furnish a numerical example in support of our main result. In process, our results generalize and improve many existing results in the literature. |
| doi_str_mv | 10.37863/umzh.v73i6.350 |
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DOI: 10.37863/umzh.v73i6.350
UDC 517.9
J. Ali (Dep. Math., Aligarh Muslim Univ., India),
I. Uddin (Dep. Math., Jamia Millia Islamia, New Delhi, India)
CONVERGENCE OF SP-ITERATION FOR GENERALIZED
NONEXPANSIVE MAPPING IN BANACH SPACES
ЗБIЖНIСТЬ SP-IТЕРАЦIЙ ДЛЯ УЗАГАЛЬНЕНИХ
НЕРОЗШИРЮЮЧИХ ВIДОБРАЖЕНЬ У БАНАХОВИХ ПРОСТОРАХ
Phuengrattana and Suantai [J. Comput. and Appl. Math., 235, 3006 – 3014 (2011)] introduced an iteration scheme and they
named this iteration as SP-iteration. In this paper, we study the convergence behavior of SP-iteration scheme for the class
of generalized nonexpansive mappings. One weak convergence theorem and two strong convergence theorems in uniformly
convex Banach spaces are obtained. We also furnish a numerical example in support of our main result. In process, our
results generalize and improve many existing results in the literature.
У роботi Phuengrattana i Suantai [J. Comput. and Appl. Math., 235, 3006 – 3014 (2011)] запропоновано iтерацiйну схему
iз назвою SP-iтерацiя. Нашу статтю присвячено вивченню збiжностi цiєї схеми SP-iтерацiй для класу узагальнених
нерозширюючих вiдображень. Доведено одну теорему про слабку збiжнiсть та двi теореми про сильну збiжнiсть
у рiвномiрно опуклих банахових просторах. З метою iлюстрацiї основного результату наведено числовий приклад.
Отриманi результати узагальнюють та удосконалюють багато iнших вiдомих результатiв.
1. Introduction. Let X be an arbitrary nonempty set and T : X \rightarrow X. A point x \in X is said
to be fixed point of mapping T if Tx = x. Fixed point theorems play a very important role in
many fields so that discussions and studies on its concept provide wide applications in various areas
not only in mathematics but also in other allied subjects. For example, in mathematics, fixed point
theorems are vital for the existence of a solution to boundary-value problems and integral equations.
In economics, fixed point results are incredibly useful when it comes to prove the existence of a
solution for various types of Nash equilibria. Moreover, there are some applications in chemistry,
biology, computer science and engineering. The classical contraction mapping principle of Banach
is one of the most powerful theorems in fixed point theory. A number of articles in the fixed point
theory have been dedicated to the improvement and generalization of this pioneer theorem. It is also
well-known that different iteration processes for contraction and nonexpansive mappings have been
successfully used to develop efficient and powerful numerical methods for solving various nonlinear
equations and variational problems, often of great importance for applications in various areas of
pure and applied sciences. By now, there exists an extensive literature on the iterative fixed points for
various classes of mappings. For an up-to date literature on this theme, one can refer to Berinde [1].
Let K be a nonempty subset of Banach space X. A mapping T : K \rightarrow K is said to be nonex-
pansive if
\| Tx - Ty\| \leq \| x - y\| for all x, y \in K.
It is known that in general, sequence of Picard’s iterates defined as (for any x1 \in K )
xn+1 = Tnx1, n \in \BbbN ,
c\bigcirc J. ALI, I. UDDIN, 2021
738 ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
CONVERGENCE OF SP-ITERATION FOR GENERALIZED NONEXPANSIVE MAPPING . . . 739
does not converge for a nonexpansive mapping, e.g., Picard’s iterates of nonexpansive mapping T :
[ - 1, 1] \rightarrow [ - 1, 1] defined by Tx = - x does not converge for any nonzero x \in [ - 1, 1] even T has
a fixed point.
In an attempt to construct a convergent sequence of iterates in respect of a nonexpansive mapping,
Mann [2] defined an iteration method as follows:
x1 \in K,
xn+1 = (1 - \alpha n)xn + \alpha nTyn, n \in \BbbN ,
where \{ \alpha n\} \subset (0, 1).
In 1974, Ishikawa [3] introduced a new two step iteration procedure as follows:
x1 \in K,
yn = (1 - \alpha n)xn + \alpha nTxn,
xn+1 = (1 - \beta n)xn + \beta nTyn, n \in \BbbN ,
where \{ \alpha n\} and \{ \beta n\} \subset (0, 1).
Mann and Ishikawa iteration procedures are two basic and most utilized iteration schemes. For a
comparison of two iterative schemes in the one-dimensional case, one may refer Rhoades [4] wherein
it is shown that under suitable conditions (see part (a) of Theorem 3) rate of convergence of Ishikawa
iteration is better than that of Mann iteration. Iterative techniques for approximating fixed points of
nonexpansive single-valued mappings have been investigated by various authors (c.f. [5 – 9]).
In 2007, Xu and Noor [10] introduced a three step iteration scheme which is a genuine extension
of Mann and Ishikawa schemes and described as follows: (for x1 \in K )
yn = (1 - \gamma n)xn + \gamma nTxn,
zn = (1 - \beta n)xn + \beta nTyn,
xn+1 = (1 - \alpha n)xn + \alpha nTzn,
where \{ \alpha n\} , \{ \beta n\} and \{ \gamma n\} \subset (0, 1).
Thianwan [11] introduced the following two step iteration scheme:
yn = (1 - \beta n)xn + \beta nTxn,
xn+1 = (1 - \alpha n)yn + \alpha nTyn, (1.1)
where \{ \alpha n\} , \{ \beta n\} \subset (0, 1).
Recently, Phuengrattana and Suantai [12] defined SP-iteration as follows: (for x1 \in K )
yn = (1 - \gamma n)xn + \gamma nTxn,
zn = (1 - \beta n)yn + \beta nTyn,
xn+1 = (1 - \alpha n)zn + \alpha nTzn,
(1.2)
where \{ \alpha n\} , \{ \beta n\} and \gamma n \subset (0, 1).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
740 J. ALI, I. UDDIN
Phuengrattana and Suantai [12] proved some convergence theorems for SP-iteration. They also
proved that the rate of convergence of iterative schemes due to Mann [2], Ishikawa [5], Xu and Noor
[10] and SP-iteration [12] is equivalent for nonexpansive mapping but SP-iteration converges better
than others for the class of continuous and nondecreasing functions.
On the other hand, Suzuki [13] introduced a new class of mappings which is larger than the
class of nonexpansive mappings and named the defining class as condition (C) which also referred as
generalized nonexpansive mapping and proved some existence and convergence theorems for Mann
iteration.
In this paper, we prove weak as well as strong convergence theorems for SP-iteration (1.2) for
generalized nonexpansive mapping. In process, our results generalize several corresponding results
contained in [10, 12, 14].
2. Basic definitions and relevant results. In this section, we collect some basic definitions and
needed results. We start with the following definition due to Opial [15].
Definition 2.1. A Banach space X is said to satisfy Opial’s condition if for any sequence \{ xn\}
in X with xn \rightharpoonup x (\rightharpoonup denotes weak convergence) implies that
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
\| xn - x\| < \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
\| xn - y\|
for all y \in X with y \not = x.
Examples of Banach spaces satisfying Opial’s condition are Hilbert spaces and all lp, 1 < p < \infty ,
spaces. On the other hand, Lp[a, b] with 1 < p \not = 2 fail to satisfy Opial’s condition.
Now, we recall the definition of mapping which satisfies condition (C).
Definition 2.2 [13]. A mapping T defined on a subset K of a Banach space X is said to satisfy
condition (C) if ( for all x, y \in K)
1
2
\| x - Tx\| \leq \| x - y\| \Rightarrow \| Tx - Ty\| \leq \| x - y\| .
From the definition, it is easy to see that every nonexpansive mapping satisfies condition (C).
If a mapping T satisfies condition (C) and has a fixed point, then T is a quasinonexpansive mapping.
But converse need not be true in general. The following examples justify these facts.
Example 2.1 [13]. Define a self mapping T on [0, 3] \subset \BbbR by
Tx =
\Biggl\{
0, when x \not = 3,
1, when x = 3.
Then T satisfies condition (C) but T is not a nonexpansive mapping.
Example 2.2 [13]. Define a self mapping T on [0, 3] \subset \BbbR by
Tx =
\Biggl\{
0, when x \not = 3,
2, when x = 3.
Then F (T ) \not = \varnothing and T is a quasinonexpansive mapping but does not satisfy condition (C).
Suzuki [13] also proved the following existence theorem for generalized nonexpansive mappings,
i.e., condition (C).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
CONVERGENCE OF SP-ITERATION FOR GENERALIZED NONEXPANSIVE MAPPING . . . 741
Theorem 2.1 [13]. Let T be a mapping defined on a convex subset K of a Banach space X
which enjoys condition (C). Also, assume that either of the following holds:
(i) K is compact
or
(ii) K is weakly compact and X has the Opial property.
Then T has a fixed point in K.
The following theorem is also very important which characterizes the fixed point set of genera-
lized nonexpansive mapping.
Theorem 2.2 [13]. Let T be a mapping defined on a closed subset K of a Banach space X.
Assume that T satisfies condition (C). Then F (T ) is closed. Moreover, if X is strictly convex and
K is convex, then F(T) is also convex.
The following results are useful and will be used repeatedly.
Lemma 2.1 [13]. Let K be a subset of a Banach space X and T : K \rightarrow K be a mapping
which satisfies condition (C), then for all x, y \in K following holds:
\| x - Ty\| \leq 3\| x - Tx\| + \| x - y\| .
The following theorems due Xu [16] and Sun et al. [17] are crucial to prove our results.
Theorem 2.3 [16]. Let X be a Banach space. Then X is uniformly convex if and only if, for
any p, 1 < p < \infty , and r > 0, there exists a continuous strictly increasing convex function gr :
\BbbR + \rightarrow \BbbR + such that gr(0) = 0 and
\| tx+ (1 - t)y\| p \leq t\| x\| p + (1 - t)\| y\| p - t(1 - t)gr(\| x - y\| )
for all x, y \in Br[0] and t \in [0, 1].
Theorem 2.4 [17]. Let g : \BbbR + \rightarrow \BbbR + be a continuous strictly increasing map with g(0) = 0. If
a sequence \{ xn\} in [0,\infty ) satisfies \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty g(xn) = 0, then \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty xn = 0.
In 1974, Senter and Dotson [18] introduced condition (I) as follows.
Definition 2.3 [18]. A mapping T : K \rightarrow K is said to satisfy condition (I) if there exists a
nondecreasing function f : [0,\infty ) \rightarrow [0,\infty ) with f(0) = 0 and f(r) > 0 for all r \in (0,\infty ) such
that d(x, Tx) \geq f(d(x, F (T ))) for all x \in K.
3. Main results. Firstly, we prove the following auxiliary lemma.
Lemma 3.1. Let K be a nonempty closed convex subset of a uniformly convex Banach space
X and T : K \rightarrow K be generalized nonexpansive mapping with F (T ) \not = \varnothing . Let \{ \alpha n\} , \{ \beta n\} be
sequences in [0, 1] and \{ \gamma n\} be a sequence in [\varepsilon , 1 - \varepsilon ] for some \varepsilon \in (0, 1). If \{ xn\} is described as
in (1.2), then
(i) \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \| xn - p\| exists for all p \in F (T ),
(ii) \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \| xn - Txn\| = 0.
Proof. (i) Let p \in F (T ). Since,
1
2
\| p - Tp\| = 0 \leq \| xn - p\| ,
which due to condition (C) gives rise \| Txn - Tp\| \leq \| xn - p\| .
Similarly, we have \| Tyn - Tp\| \leq \| yn - p\| and \| Tzn - Tp\| \leq \| zn - p\| . By (1.2), we get
\| yn - p\| = \| (1 - \gamma n)xn + \gamma nTxn - p\| \leq
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
742 J. ALI, I. UDDIN
\leq (1 - \gamma n)\| xn - p\| + \gamma n\| Txn - Tp\| \leq
\leq (1 - \gamma n)\| xn - p\| + \gamma n\| xn - p\| = \| xn - p\| . (3.1)
Also, we obtain
\| zn - p\| = \| (1 - \beta n)yn + \beta nTyn - p\| \leq
\leq (1 - \beta n)\| yn - p\| + \beta n\| Tyn - Tp\| \leq
\leq (1 - \beta n)\| yn - p\| + \beta n\| yn - p\| \leq \| yn - p\| . (3.2)
From (3.1) and (3.2), we have
\| zn - p\| \leq \| xn - p\| . (3.3)
Now, consider
\| xn+1 - p\| = \| (1 - \alpha n)zn + \alpha nTzn - p\| \leq
\leq (1 - \alpha n)\| zn - p\| + \alpha n\| Tzn - Tp\| \leq
\leq (1 - \alpha n)\| zn - p\| + \alpha n\| zn - p\| \leq \| zn - p\| . (3.4)
Combining (3.3) and (3.4), we get
\| xn+1 - p\| \leq \| xn - p\|
which shows that \{ \| xn - p\| \} is a decreasing sequence of nonnegative reals. Thus, sequence \{ \| xn -
- p\| \} is bounded below and decreasing and, hence, it is convergent.
(ii) From part (i), \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \| xn - p\| exists for all p \in F (T ). Let us write
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\| xn - p\| = c. (3.5)
From (3.4) and (3.5), we obtain
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
\| zn - p\| \geq c,
but using (3.3) and (3.4), we get
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
\| zn - p\| \leq c
and, hence,
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\| zn - p\| = c. (3.6)
Also, by (3.2) and (3.6), we have
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
\| yn - p\| \geq c,
while (3.1) implies
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
\| yn - p\| \leq c.
Hence,
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
CONVERGENCE OF SP-ITERATION FOR GENERALIZED NONEXPANSIVE MAPPING . . . 743
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\| yn - p\| = c. (3.7)
Now, in view of Theorem 2.3, there exists a continuous strictly increasing convex function g :
\BbbR + \rightarrow \BbbR + with g(0) = 0 such that
\| yn - p\| 2 = \| (1 - \gamma n)xn + \gamma nTxn - p\| 2 \leq
\leq (1 - \gamma n)\| xn - p\| 2 + \gamma n\| Txn - Tp\| 2 - \gamma n(1 - \gamma n)g(\| xn - Txn\| ) \leq
\leq (1 - \gamma n)\| xn - p\| 2 + \gamma n\| xn - p\| 2 - \gamma n(1 - \gamma n)g(\| xn - Txn\| ) \leq
\leq \| xn - p\| 2 - \gamma n(1 - \gamma n)g(\| xn - Txn\| )
yielding there by
\gamma n(1 - \gamma n)g(\| xn - Txn\| ) \leq \| xn - p\| 2 - \| yn - p\| 2,
or
g(\| xn - Txn\| ) \leq
1
\gamma n(1 - \gamma n)
\bigl[
\| xn - p\| 2 - \| yn - p\| 2
\bigr]
.
As \{ \gamma n\} \in [\varepsilon , 1 - \varepsilon ] for some \varepsilon \in (0, 1), then
g(\| xn - Txn\| ) \leq
1
\varepsilon 2
\bigl[
\| xn - p\| 2 - \| yn - p\| 2
\bigr]
.
In view of (3.5) and (3.7), \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty g(\| xn - Txn\| ) = 0 and, hence, owing to Theorem 2.4, we have
\mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \| xn - Txn\| = 0.
Lemma 3.1 is proved.
Now, we prove the following weak convergence theorem for SP-iteration scheme.
Theorem 3.1. Let K be a nonempty closed convex subset of a uniformly convex Banach space
X which satisfies Opial’s condition and T : K \rightarrow K be a generalized nonexpansive mapping with
F (T ) \not = \varnothing . Let \{ \alpha n\} , \{ \beta n\} be sequences in [0, 1] and \{ \gamma n\} be a sequence in [\varepsilon , 1 - \varepsilon ] for some
\varepsilon \in (0, 1). If \{ xn\} is described as in (1.2), then the sequence \{ xn\} weakly converges to a fixed point
of T.
Proof. From Lemma 3.1, we have \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \| xn - p\| exists for each p \in F (T ) so that the
sequence \{ xn\} is bounded and \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \| xn - Txn\| = 0. As X is uniformly convex, there exists
a subsequence \{ xnk
\} of \{ xn\} such that xnk
\rightharpoonup q for some q \in K. Now, we show that q \in F (T ).
Suppose q \not = Tq. Then owing to Lemma 2.1, we have
\mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow \infty
\| xnk
- Tq\| \leq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
k\rightarrow \infty
\{ 3\| xnk
- Txnk
\| + \| Txnk
- Tq\| \} \leq
\leq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
k\rightarrow \infty
\{ 3\| xnk
- Txnk
\| + \| xnk
- q\| \} .
On making k \rightarrow \infty , we get xnk
\rightharpoonup Tq which is a contradiction to uniqueness of limit of convergent
sequence and hence Tq = q. Now, we prove that \{ xn\} has unique weak subsequential limit in F (T ).
To show this, let \{ xni\} and \{ xnj\} be subsequences of \{ xn\} such that xni \rightharpoonup q1 and xnj \rightharpoonup q2. If
q1 \not = q2, then owing to Opial’s condition
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\| xn - q1\| = \mathrm{l}\mathrm{i}\mathrm{m}
j\rightarrow \infty
\| xnj - q1\| < \mathrm{l}\mathrm{i}\mathrm{m}
j\rightarrow \infty
\| xnj - q2\| = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\| xn - q2\| =
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
744 J. ALI, I. UDDIN
= \mathrm{l}\mathrm{i}\mathrm{m}
i\rightarrow \infty
\| xni - q2\| < \mathrm{l}\mathrm{i}\mathrm{m}
i\rightarrow \infty
\| xni - q1\| = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\| xn - q1\| ,
which is a contradiction and hence \{ xn\} converges weakly to a fixed point of T.
Theorem 3.1 is proved.
By setting \beta n = 0, SP-iteration reduces to two step iteration scheme, i.e., (1.1). Now, from above
theorem we can draw the following corollary.
Corollary 3.1. Let K be a nonempty closed convex subset of a uniformly convex Banach space
X which satisfies Opial’s condition and T : K \rightarrow K be a generalized nonexpansive mapping with
F (T ) \not = \varnothing . Let \{ \alpha n\} be sequence in [0, 1] and \{ \gamma n\} be a sequence in [\varepsilon , 1 - \varepsilon ] for some \varepsilon \in (0, 1).
If \{ xn\} is described as in (1.1), then the sequence \{ xn\} weakly converges to a fixed point of T.
Now, we prove following strong convergence theorem.
Theorem 3.2. Let K be a nonempty closed convex subset of a uniformly convex Banach space
X and T : K \rightarrow K be a generalized nonexpansive mapping with F (T ) \not = \varnothing . Let \{ \alpha n\} , \{ \beta n\}
be sequences in [0, 1] and \{ \gamma n\} be a sequence in [\varepsilon , 1 - \varepsilon ] for some \varepsilon \in (0, 1). If \{ xn\} is
described as in (1.2), then the sequence \{ xn\} converges to a fixed point of T if and only if
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}n\rightarrow \infty d(xn, F (T )) = 0, where d(z, F (T )) = \mathrm{i}\mathrm{n}\mathrm{f}\{ \| z - p\| : p \in F (T )\} .
Proof. If \{ xn\} converges to a fixed point p of T, then
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
\| xn - p\| = 0
and, hence,
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
d(xn, F (T )) = 0.
For converse part, let \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}n\rightarrow \infty d(xn, F (T )) = 0. In view of (3.4) \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{l}\mathrm{l} p \in F (T ), we have
\| xn+1 - p\| \leq \| xn - p\| ,
which implies
\mathrm{i}\mathrm{n}\mathrm{f}
p\in F (T )
\| xn+1 - p\| \leq \mathrm{i}\mathrm{n}\mathrm{f}
p\in F (T )
\| xn - p\| .
Therefore, we get
d(xn+1, F (T )) \leq d(xn, F (T )).
Hence, \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty d(xn, F (T )) exists and thus by assumption we have \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty d(xn, F (T )) = 0.
Therefore, for any \varepsilon > 0, there exists a positive integer k such that (\mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{l}\mathrm{l} n \geq k)
d(xk, F (T )) <
\varepsilon
4
or
\mathrm{i}\mathrm{n}\mathrm{f}\{ \| xk - p\| : p \in F (T )\} <
\varepsilon
4
,
so that there exists p \in F (T ) such that
\| xk - p\| <
\varepsilon
2
.
Now, for all m, n \geq k, we have
\| xm - xn\| \leq \| xm - p\| + \| p - xn\| \leq 2\| xk - p\| < 2
\Bigl( \varepsilon
2
\Bigr)
= \varepsilon .
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CONVERGENCE OF SP-ITERATION FOR GENERALIZED NONEXPANSIVE MAPPING . . . 745
Hence, \{ xn\} is a Cauchy sequence in K and converges to some x in K. As \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty d(xn, F (T )) =
= 0 which implies that d(x, F (T )) = 0. In view of Theorem 2.2, F (T ) is closed and thus x \in F (T ).
Again, by setting \beta n = 0, SP-iteration reduces to two step iteration scheme, i.e., (1.1) and we
can draw the following corollary.
Corollary 3.2. Let K be a nonempty closed convex subset of a uniformly convex Banach space
X and T : K \rightarrow K be a generalized nonexpansive mapping with F (T ) \not = \varnothing . Let \{ \alpha n\} be sequence
in [0, 1] and \{ \gamma n\} be a sequence in [\varepsilon , 1 - \varepsilon ] for some \varepsilon \in (0, 1). If \{ xn\} is described as in (1.1),
then the sequence \{ xn\} converges to a fixed point of T if and only if \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}n\rightarrow \infty d(xn, F (T )) = 0.
Now, we prove the following strong convergence theorem using condition (I).
Theorem 3.3. Let K be a nonempty closed convex subset of a uniformly Banach space X and
T : K \rightarrow K be a generalized nonexpansive mapping which satisfies condition (I) with F (T ) \not = \varnothing .
Let \{ \alpha n\} , \{ \beta n\} be sequences in [0, 1] and \{ \gamma n\} be a sequence in [\varepsilon , 1 - \varepsilon ] for some \varepsilon \in (0, 1). If
\{ xn\} is described as in (1.2), then \{ xn\} converges to a fixed point of T.
Proof. By Lemma 3.1, \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty d(xn, p) exists for all p \in F (T ) and let us take to be c. If
c = 0, then there is nothing to prove. If c > 0, then as argued in Theorem 3.2, \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty d(xn, F (T ))
exists. Owing to condition (I) there exists a nondecreasing function f such that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
f(d(xn, F (T ))) \leq \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
d(xn, Txn) = 0
so that \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty f(d(xn, F (T ))) = 0. Since, f is a nondecreasing function and f(0) = 0, therefore,
\mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty d(xn, F (T )) = 0. Now, in view of Theorem 3.2, we are through.
Theorem 3.3 is proved.
Corollary 3.3. Let K be a nonempty closed convex subset of a uniformly Banach space X and
T : K \rightarrow K be a generalized nonexpansive mapping which satisfies condition (I) with F (T ) \not = \varnothing .
Let \{ \alpha n\} be sequence in [0, 1] and \{ \gamma n\} be a sequence in [\varepsilon , 1 - \varepsilon ] for some \varepsilon \in (0, 1). If \{ xn\} is
described as in (1.1), then \{ xn\} converges to a fixed point of T.
Remark 3.1. Theorems 3.1 – 3.3 and Corollaries 3.1 – 3.3 generalize and extend the relevant re-
sults from [11, 12, 14].
4. Numerical example. To illustrate the genuineness of our results and the convergence behavior
of SP-iteration scheme, we furnish following example of Suzuki’s generalized nonexpansive mapping
which is not a nonexpansive mapping.
Example 4.1. Define a self mapping T on [0, 1] by
T (x) =
\left\{
1 - x, if x \in
\biggl[
0,
1
33
\biggr)
,
x+ 32
33
, if x \in
\biggl[
1
33
, 1
\biggr]
.
Here T is a Suzuki’s generalized nonexpansive mapping, but T is not a nonexpansive.
Verification. Take x =
3
100
and y =
1
33
, then
\| x - y\| =
\bigm\| \bigm\| \bigm\| \bigm\| 3
100
- 1
33
\bigm\| \bigm\| \bigm\| \bigm\| =
1
3300
and
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
746 J. ALI, I. UDDIN
\| Tx - Ty\| =
\bigm\| \bigm\| \bigm\| \bigm\| 1 - 3
100
- 1057
1089
\bigm\| \bigm\| \bigm\| \bigm\| =
67
108900
>
1
3300
= \| x - y\| .
Hence, T is not a nonexpansive mapping.
Claim: T is a Suzuki’s generalized nonexpansive mapping.
Case I. Let x \in
\biggl[
0,
1
33
\biggr)
. Then
1
2
\| x - Tx\| =
1
2
\| 2x - 1\| . For
1
2
\| x - Tx\| \leq \| x - y\| , we must
have
1 - 2x
2
\leq y - x =\Rightarrow y \geq 1
2
and, hence, y \in
\biggl[
1
2
, 1
\biggr]
. Now, we have
\| Tx - Ty\| =
\bigm\| \bigm\| \bigm\| \bigm\| 1 - x - y + 32
33
\bigm\| \bigm\| \bigm\| \bigm\| =
\bigm\| \bigm\| \bigm\| \bigm\| y + 33x - 1
33
\bigm\| \bigm\| \bigm\| \bigm\| <
1
33
and
\| x - y\| = | x - y| > 31
66
.
Hence,
1
2
\| x - Tx\| \leq \| x - y\| =\Rightarrow \| Tx - Ty\| \leq \| x - y\| .
Case II. Let x \in
\biggl[
1
33
, 1
\biggr]
. Then
1
2
\| x - Tx\| =
1
2
\bigm\| \bigm\| \bigm\| x+ 32
33
- x
\bigm\| \bigm\| \bigm\| =
\bigm\| \bigm\| \bigm\| 16 - 16x
33
\bigm\| \bigm\| \bigm\| .
For
1
2
\| x - Tx\| \leq \| x - y\| , we must have
16 - 16x
33
\leq | x - y| , which gives two possibilities:
(A) Let x < y, then
16 - 16x
33
\leq y - x, i.e.,
17x+ 16
33
\leq y =\Rightarrow y \in
\biggl[
545
1089
, 1
\biggr]
\subset
\biggl[
1
33
, 1
\biggr]
.
So,
\| Tx - Ty\| =
1
33
\| x - y\| \leq \| x - y\| .
Hence,
1
2
\| x - Tx\| \leq \| x - y\| =\Rightarrow \| Tx - Ty\| \leq \| x - y\| .
(B) Let x > y, then
16 - 16x
33
\leq x - y, i.e., y \leq 49x - 16
33
, so y \in [0, 1]. Also,
33y + 16
49
\leq x
which gives x \in
\biggl[
16
49
, 1
\biggr]
. For x \in
\biggl[
16
49
, 1
\biggr]
and y \in
\biggl[
1
33
, 1
\biggr]
case II(A) can be used. Therefore,
consider x \in
\biggl[
16
49
, 1
\biggr]
and y \in
\biggl[
0,
1
33
\biggr)
. Then
\| Tx - Ty\| =
\bigm\| \bigm\| \bigm\| \bigm\| x+ 32
33
- 1 + y
\bigm\| \bigm\| \bigm\| \bigm\| =
\bigm\| \bigm\| \bigm\| \bigm\| x+ 33y - 1
33
\bigm\| \bigm\| \bigm\| \bigm\| <
1
33
and
\| x - y\| = | x - y| > 479
1617
>
1
33
.
Hence,
1
2
\| x - Tx\| \leq \| x - y\| =\Rightarrow \| Tx - Ty\| \leq \| x - y\| .
Thus, T is Suzuki’s generalized nonexpansive mapping. It is easy to see that x = 1 is fixed point
of T. If we choose \gamma n =
1
n+ 68
, \beta n =
\sqrt{}
n
n+ 68
and \alpha n =
\sqrt{}
n+ 67
n+ 68
, then we have the following
table which illustrate that convergence behavior of Mann, Ishikawa and SP-iteration.
Thus, Table 1 and Fig. 1 show that SP-iteration (1.2) converges faster than the Mann and Ishikawa
iteration even for Suzuki’s generalized nonexpansive mapping as claimed in Theorem 3.7 of Phuen-
grattana and Suantai [12]. Also, rate of convergence of SP-iteration is better than Noor iteration and
in our case mapping is discontinuous.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
CONVERGENCE OF SP-ITERATION FOR GENERALIZED NONEXPANSIVE MAPPING . . . 747
Table 1
Step Mann Ishikawa Noor SP
1 0.02 0.02 0.02 0.02
2 0.03391304347826 0.1344537965569 0.9668347163656 0.9681242789516
3 0.04729606625259 0.2763857687951 0.9989280684102 0.9990208877563
4 0.06030781438742 0.4206836352069 0.9999666029046 0.9999712693742
5 0.07296360139904 0.5531477763904 0.999998992079 0.9999991896535
6 0.08527790806373 0.6665975873401 0.9999999704407 0.9999999779427
7 0.09726443834299 0.7586939227074 0.9999999991557 0.9999999994189
8 0.1089361708573 0.8302090841836 0.9999999999765 0.9999999999851
Fig. 1
References
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4. B. E. Rhoades, Comments on two fixed point iteration methods, J. Math. Anal. and Appl., 56, 741 – 750 (1976).
5. S. Ishikawa, Fixed point and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc., 59,
65 – 71 (1976).
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274 – 276 (1979).
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Received 25.09.18
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
|
| id | umjimathkievua-article-350 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:28Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/1f/5e48dc053daabe4a4cb1531d2bf7451f.pdf |
| spelling | umjimathkievua-article-3502022-03-26T11:03:01Z Convergence of SP-iteration for generalized nonexpansive mapping in Banach spaces Convergence of SP-iteration for generalized nonexpansive mapping in Banach spaces Convergence of SP-iteration for generalized nonexpansive mapping in Banach spaces Ali, J. Uddin, I. Ali, J. Uddin, I. Ali, J. Uddin, I. Banach spaces Fixed point Weak-convergence condition (C) and Opial’s property Banach spaces Fixed point Weak-convergence condition (C) and Opial’s property UDC 517.9 Phuengrattana and Suantai [J. Comput. and Appl. Math., 235, 3006 – 3014 (2011)] introduced an iteration scheme and they named this iteration as SP-iteration. In this paper, we study the convergence behaviour of SP-iteration scheme for the class of generalized nonexpansive mappings. One weak convergence theorem and two strong convergence theorems in uniformly convex Banach spaces are obtained. We also furnish a numerical example in support of our main result. In process, our results generalize and improve many existing results in the literature. UDC 517.9 Збiжнiсть SP-iтерацiй для узагальнених нерозширюючих вiдображень у банахових просторах У роботі Phuengrattana і Suantai [J. Comput. and Appl. Math., {\bf 235}, 3006\,--\,3014 (2011)] запропоновано ітераційну схему із назвою SP-ітерація.&nbsp;Нашу статтю присвячено вивченню збіжності цієї схеми SP-ітерацій для класу узагальнених нерозширюючих відображень.&nbsp;Доведено одну теорему про слабку збіжність та дві теореми про сильну збіжність у рівномірно опуклих банахових просторах.&nbsp;З метою ілюстрації основного результату наведено числовий приклад.&nbsp;Отримані результати узагальнюють та удосконалюють багато інших відомих результатів. Institute of Mathematics, NAS of Ukraine 2021-06-18 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/350 10.37863/umzh.v73i6.350 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 6 (2021); 738 - 748 Український математичний журнал; Том 73 № 6 (2021); 738 - 748 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/350/9023 |
| spellingShingle | Ali, J. Uddin, I. Ali, J. Uddin, I. Ali, J. Uddin, I. Convergence of SP-iteration for generalized nonexpansive mapping in Banach spaces |
| title | Convergence of SP-iteration for generalized nonexpansive mapping in Banach spaces |
| title_alt | Convergence of SP-iteration for generalized nonexpansive mapping in Banach spaces Convergence of SP-iteration for generalized nonexpansive mapping in Banach spaces |
| title_full | Convergence of SP-iteration for generalized nonexpansive mapping in Banach spaces |
| title_fullStr | Convergence of SP-iteration for generalized nonexpansive mapping in Banach spaces |
| title_full_unstemmed | Convergence of SP-iteration for generalized nonexpansive mapping in Banach spaces |
| title_short | Convergence of SP-iteration for generalized nonexpansive mapping in Banach spaces |
| title_sort | convergence of sp-iteration for generalized nonexpansive mapping in banach spaces |
| topic_facet | Banach spaces Fixed point Weak-convergence condition (C) and Opial’s property Banach spaces Fixed point Weak-convergence condition (C) and Opial’s property |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/350 |
| work_keys_str_mv | AT alij convergenceofspiterationforgeneralizednonexpansivemappinginbanachspaces AT uddini convergenceofspiterationforgeneralizednonexpansivemappinginbanachspaces AT alij convergenceofspiterationforgeneralizednonexpansivemappinginbanachspaces AT uddini convergenceofspiterationforgeneralizednonexpansivemappinginbanachspaces AT alij convergenceofspiterationforgeneralizednonexpansivemappinginbanachspaces AT uddini convergenceofspiterationforgeneralizednonexpansivemappinginbanachspaces |