In this paper, by combining the shrinking projection method with a modified inertial Siteration process, we introduce a new inertial hybrid iteration for two asymptotically Gnonexpansive mappings and a new inertial hybrid iteration for two G-nonexpansive mappings in Hilbert spaces with graphs. We establish a sufficient condition for the closedness and convexity of the set of fixed points of asymptotically G-nonexpansive mappings in Hilbert spaces with graphs. We then prove a strong convergence theorem for finding a common fixed point of two asymptotically G-nonexpansive mappings in Hilbert spaces with graphs. By this theorem, we obtain a strong convergence result for two G-nonexpansive mappings in Hilbert spaces with graphs. These results are generalizations and extensions of some convergence results in the literature, where the convexity of the set of edges of a graph is replaced by coordinate-convexity. In addition, we provide a numerical example to illustrate the convergence of the proposed iteration processes.
Trang 1Tập 17, Số 6 (2020): 1137-1149 Vol 17, No 6 (2020): 1137-1149 ISSN:
1859-3100 Website: http://journal.hcmue.edu.vn
Research Article *
STRONG CONVERGENCE OF INERTIAL HYBRID ITERATION
FOR TWO ASYMPTOTICALLY G-NONEXPANSIVE MAPPINGS
IN HILBERT SPACE WITH GRAPHS
Nguyen Trung Hieu * , Cao Pham Cam Tu
Faculty of Mathematics Teacher Education, Dong Thap University, Cao Lanh City, Viet Nam
* Corresponding author: Nguyen Trung Hieu – Email: ngtrunghieu@dthu.edu.vn
Received: April 07, 2020; Revised: May 08, 2020; Accepted: June 24, 2020
ABSTRACT
In this paper, by combining the shrinking projection method with a modified inertial S-iteration process, we introduce a new inertial hybrid S-iteration for two asymptotically G-nonexpansive mappings and a new inertial hybrid iteration for two G-G-nonexpansive mappings in Hilbert spaces with graphs We establish a sufficient condition for the closedness and convexity
of the set of fixed points of asymptotically G-nonexpansive mappings in Hilbert spaces with graphs We then prove a strong convergence theorem for finding a common fixed point of two asymptotically G-nonexpansive mappings in Hilbert spaces with graphs By this theorem, we obtain a strong convergence result for two G-nonexpansive mappings in Hilbert spaces with graphs These results are generalizations and extensions of some convergence results in the literature, where the convexity of the set of edges of a graph is replaced by coordinate-convexity
In addition, we provide a numerical example to illustrate the convergence of the proposed iteration processes
Keywords: asymptotically G-nonexpansive mapping; Hilbert space with graphs; inertial
hybrid iteration
1 Introduction and preliminaries
In 2012, by using the combination concepts between the fixed point theory and the
graph theory, Aleomraninejad, Rezapour, and Shahzad (2012) introduced the notions of G-contractive mapping and G-nonexpansive mapping in a metric space with directed graphs
and stated the convergence for these mappings After that, there were many convergence
results for G-nonexpansive mappings by some iteration processes established in Hilbert
spaces and Banach spaces with graphs In 2018, Sangago, Hunde, and Hailu (2018)
introduced the notion of an asymptotically G-nonexpansive mapping and proved the weak
and strong convergence of a modified Noor iteration process to common fixed points of a
finite family of asymptotically G-nonexpansive mappings in Banach spaces with graphs After that some authors proposed a two-step iteration process for two asymptotically
G-nonexpansive mappings T T 1, 2 : (Wattanataweekul, 2018) and a three-step iteration
(Wattanataweekul, 2019) as follows:
Cite this article as: Nguyen Trung Hieu, & Cao Pham Cam Tu (2020) Strong convergence of inertial hybrid
iteration for two asymptotically G-nonexpansive mappings in Hilbert space with graphs Ho Chi Minh City University of Education Journal of Science, 17(6), 1137-1149
Trang 21 ,
(1 )
n
n n n n n
n
n n n n n
u
3 2
(1 ) (1 )
n
n n n n n
n
n n n n n
n
n n n n n
(1.2)
where { },{ },{ }a n b n c n [0,1]. Furthermore, the authors also established the weak and strong convergence results of the iteration process (1.1) and the iteration process (1.2) to common fixed
points of asymptotically G-nonexpansive mappings in Banach spaces with graphs
Currently, there were many methods to construct new iteration processes which generalize some previous iteration processes In 2008, Mainge proposed the inertial Mann iteration by combining the Mann iteration and the inertial term n(u n u n1). In 2018, by combining the CQ-algorithm and the inertial term, Dong, Yuan, Cho, and Rassias (2018) studied an inertial CQ-algorithm for a non-expansive mapping as follows:
u u H
1
1
(1 )
,
n n
n n n n n
n n n n n
n C Q
where { }a n [0,1], { } n [ , ] for some , , T H: H is a nonexpansive mapping,
n n
C Q
P u is the metric projection of u1onto C n Q n.
In 2019, by combining a modified S-iteration process with the inertial extrapolation,
Phon-on, Makaje, Sama-Ae, and Khongraphan (2019) introduced an inertial S-iteration
process for two nonexpansive mappings such as:
u u H
1 1
(1 )
n n n n n
n n n n n
n n n n n
where { },{ }a n b n [0,1], { } n [ , ] for some , , and T T H1, 2 : H are two nonexpansive mappings Recently, by combining the shrinking projection method with a
modified S-iteration process, Hammad, Cholamjiak, Yambangwai, and Dutta (2019) introduced the following hybrid iteration for two G-nonexpansive mappings
u
1
1
1
(1 ) (1 )
,
n
n n n n n
n n n n n
n
(1.3)
where { },{ }a n b n [0,1],T T 1, 2 : are two G-nonexpansive mappings, and
1 1
n
P u
metric projection of u1onto n1.
Trang 3Motivated by these works, we introduce an iteration process for two G-nonexpansive
mappings T T H1, 2 : H such as:
u u H H
1
1 1
1
(1 ) (1 )
,
n
n n n n n
n n n n n
n n n n n
n
(1.4)
and an iteration process for two asymptotically G-nonexpansive mappings T T H1, 2 : H such as:
u u H H
1
1 1
1
(1 ) (1 )
n
n n n n n
n
n n n n n
n n n n n
n
(1.5)
where { },{ }a n b n [0,1], { } n [ , ] for some , , H is a real Hilbert space,
1 1
n
P u
the metric projection of u1onto n1, and nis defined in Theorem 2.2 in Section 2 Then, under some conditions, we prove that the sequence { }u n generated by (1.5) strongly converges to the projection of the initial point u1 onto the set of all common fixed points of T1 and T2 in Hilbert
spaces with graphs By this theorem, we obtain a strong convergence result for two G-nonexpansive mappings by the iteration process (1.4) in Hilbert spaces with graphs In addition,
we give a numerical example for supporting obtained results
We now recall some notions and lemmas as follows:
Throughout this paper, let G ( ( ), ( ))V G E G be a directed graph, where the set all vertices and edges denoted by V G( ) and E G( ),respectively We assume that all directed graphs are reflexive, that is, ( , )u u E G( ) for each u V G ( ), and Ghas no parallel edges
A directed graph G ( ( ), ( ))V G E G is said to be transitive if for any u v w V G, , ( )such that
( , )u v and ( , )v w are in E G( ),then ( , )u w E G( ).
Definition 1.1
Tiammee, Kaewkhao, & Suantai (2015, p.4): Let Xbe a normed space, be a nonempty subset of X, and G ( ( ), ( ))V G E G be a directed graph such that V G ( ) Then
is said to have property ( )G if for any sequence{ }u n in such that ( ,u u n n1) E G( )for all n and { }u n weakly converging to u , then there exists a subsequence {u n k( )} of
{ }u n such that (u n k( ), )u E G( ) for all k
Definition 1.2
( ( ), ( ))
G V G E G be a directed graph such that E G( ) X X The set of edges E G( ) is said
to be coordinate-convex if for all ( , ),( , ),( , ),( , )p u p v u p v p E G( ) and for all t [0,1], then
( , ) (1 )( , ) ( )
t p u t p v E G and t u p( , ) (1 t v p)( , ) E G( ).
Trang 4Definition 1.3
Tripak (2016) - Definition 2.1 and Sangago et al (2018)- Definition 3.1: Let Xbe a normed space, G ( ( ), ( ))V G E G be a directed graph such that V G ( ) X,and
: ( ) ( )
T V G V G be a mapping Then
(1) T is said to be G-nonexpansive if
(a) T is edge-preserving, that is, for all ( , )u v E G( ),we have ( ,Tu Tv) E G( ).
(b) ||Tu Tv || || uv||, whenever ( , )u v E G( ) for any u v V G, ( ).
(2) T is call asymptotically G -nonexpansive mapping if
(a) Tis edge-preserving
(b) There exists a sequence { } n [1, ) with
1
( n 1)
n
n
T u T v u v for all n , whenever ( , )u v E G( ) for any u v V G, ( ), where
{ } n is said to be an asymptotic coefficient sequence
Remark 1.4
Every G-nonexpansive mapping is an asymptotically G-nonexpansive mapping with
the asymptotic coefficients n 1 for all n
Lemma 1.5
Sangago et al (2018) - Theorem 3.3: Let be a nonempty closed, convex subset of a real Banach space X, have Property ( ),G G ( ( ), ( ))V G E G be a directed graph such that
( ) ,
V G T : be an asymptotically G-nonexpansive mapping, { }u n be a sequence in
converging weakly to u , ( ,u u n n1) E G( ) and lim || n n || 0.
n Tu u
Let H be a real Hilbert space with inner product ,. and norm || ||, be a nonempty, closed and convex subset of a Hilbert space H. Now, we recall some basic notions of Hilbert spaces which we will use in the next section
The nearest point projection of H onto is denoted by P, that is, for all uH, we have
||uP u || inf{|| uv||:v }. Then P is called the metric projection of Honto It is known that for each u H,pP u is equivalent to u p p, v 0 for allv .
Lemma 1.6
Alber (1996, p.5): Let H be a real Hilbert space, be a nonempty, closed and convex subset of H, and P is the metric projection of H onto Then for all u H and v ,we have ||vP u ||2 ||uP u || ||2 uv|| 2
Lemma 1.7
Bauschke and Combettes (2011)- Corollary 2.14: Let H be a real Hilbert space Then for all [0,1] and u v, H, we have
|| u (1 ) ||v || ||u (1 ) || ||v (1 ) ||uv||
Lemma 1.8
Martinez-Yanes and Xu (2006) – Lemma 13: Let H be a real Hilbert space and be
a nonempty, closed and convex subset of H. Then for x y z, , H and a , the following set is convex and closed: {w :||yw|| || 2 xw|| 2 z w, a}.
Trang 5The following result will be used in the next section The proof of this lemma is easy and is omitted
Lemma 1.9
Let H be a real Hilbert space Then for all u v w, , H, we have
||uv|| || uw || ||wv|| 2u w w, v
2 Main results
First, we denote by F T( ){uH Tu: u}the set of fixed points of the mapping
T H H The following result is a sufficient condition for the closedness and convexity
of the set F T( ) in real Hilbert spaces, where T is an asymptotically G-nonexpansive
mapping
Proposition 2.1
Let H be a real Hilbert space, G ( ( ), ( ))V G E G be a directed graph such that
V G H T H: H be an asymptotically G-nonexpansive mapping with an asymptotic
coefficient sequence { } n [1, ) satisfying
1
( n 1) ,
n
and F T( ) F T( ) E G( ). Then
(1) If H have property ( ),G then F T( ) is closed
(2) If the graph G is transitive, E G( ) is coordinate-convex, then F T( ) is convex
Proof
(1) Suppose that F T ( ) Let { }p n be a sequence in F T( )such that lim || n || 0
n p p
some pH Since F T( ) F T( ) E G( ),we have ( ,p p n n1)E G( ) By combining this with
property ( )G of H, we conclude that there exists a subsequence {p n k( )}of { }p n such that
( )
(p n k, )p E G( ) for k Since T is an asymptotically G-nonexpansive mapping, we obtain
||p Tp || || pp n k ||||Tp n k Tp|| (1 ) ||pp n k ||
It follows from the above inequality and lim || n || 0
n p p
that Tp p, that is, p F T( ).
Therefore, F T( )is closed
(2) Let p p1, 2 F T( ).For t [0,1], we put ptp1 (1 t p) 2
Since F T( )F T( )E G( ) and p p1, 2F T( ), we get ( , ),( , ),( , ),( , )p p1 1 p p1 2 p p2 1 p p2 2 E G( )
( , ) (1 )( , ) ( , ) ( ),
t p p t p p p p E G t p p( , ) (11 1 t p p)( , ) ( , )2 1 p p1 E G( ) and
t p p t p p p p E G Due to the fact that T is an asymptotically
G-nonexpansive mapping, for each i 1,2, we get
Furthermore, by using Lemma 1.9, we get
||p T p n || || p p || ||p T p n || 2 p p p T p, n (2.2)
and
||p T p n || || p p|| ||p T p n || 2 p p p T p, n (2.3)
It follows from (2.1) and (2.2) that
Trang 62 2 2
n
p T p p p p p p T p (2.4) Also, we conclude from (2.1) and (2.3) that
n
p T p p p p p p T p (2.5)
By multiplying ton the both sides of (2.4), and multiplying (1t)on the both sides
of (2.5), we get
2t p p p T p, n 2(1 t p) p p T p, n
Since
1
n
n
Therefore, from (2.6), we find that
1
( ,p T p n )E G( ). Then, by the transitive property of G and ( , ),( ,p p1 p T p1 n )E G( ),we get
( ,p T p n )E G( ). Due to asymptotically G-nonexpansiveness of T,we obtain
1
||Tpp|| || Tp T np|| ||T npp|| ||p T p n || ||T npp|| (2.8) Taking the limit in (2.8) as n and using (2.7), we find that Tpp, that is, ( )
p F T Therefore, F T( )is convex
Let T T H1, 2: H be two asymptotically G-nonexpansive mappings with asymptotic
coefficient sequences { },{ } n n [1, )such that
1
( n 1)
n
and
1
n
max{ , },
1
( n 1)
n
( , )u v E G( ) and for each i 1, 2, we have || n n || || ||
T u T v uv In the following theorem, we also assume that F F T( )1 F T( )2 is nonempty and bounded in H,that is, there exists a positive number such that F {uH :|| ||u }.The following result shows the strong convergence of iteration process (1.5) to common fixed points of two asymptotically
G-nonexpansive mappings in Hilbert spaces with directed graphs
Theorem 2.2
Let H be a real Hilbert space, H have property ( ),G G ( ( ), ( ))V G E G be a directed transitive graph such that V G( ) H,E G( )be coordinate-convex, T T H1, 2 : H be two asymptotically G -nonexpansive mappings such that F T( )i F T( )i E G( ) for all i 1,2, { }u n be a sequence generated by (1.5) where { },{ }a n b n are sequences in [0,1] such that
0 lim inf n lim sup n 1,
n a n a
n b n b
Trang 7such that ( , ),( , ),( , )u p p u n n z p n E G( ) for all p F ; n ( n2 1)(1 b n n 2 )(|| ||z n ) 2 Then the sequence { }u n strongly converges to P u F 1.
Proof
The proof of Theorem 2.2 is divided into six steps
Step 1 We show that P u F 1is well-defined Indeed, by Proposition 2.1, we conclude that
1
( )
F T and F T( )2 are closed and convex Therefore, F F T( )1 F T( )2 is closed and convex Note that Fis nonempty by the assumption This fact ensures that P u F 1is well-defined
Step 2 We show that
1 1
n
P u
is well-defined We first prove by a mathematical induction
that nis closed and convex for n Obviously, 1 H is closed and convex Now we suppose that nis closed and convex Then by the definition of n1and Lemma 1.8, we conclude that n1is closed and convex Therefore, nis closed and convex for n
Next, we show that F n1for all n Indeed, for p F ,we have T p1 T p2 p.
Since( , )z p n E G( )and T1nis edge-preserving, we obtain ( 1n , ) ( ).
n
T z p E G Due to the
v p b z p b T z p E G It follows from Lemma 1.7 and asymptotically G-nonexpansiveness of T T1, 2 that
w p a T v p a T v p
n v n p a n a n T v n T v n
n v n p
(2.9)
and
1
v p b z p b T z p
1
1
[1 b n( n 1)] ||z n p||
(2.10)
By substituting (2.10) into (2.9), we obtain
||w n p|| n[1 b n( n 1)] ||z n p||
||z n p|| ( n 1)(1 b n n )(||z n || || ||)p
||z n p|| ( n 1)(1 b n n )(||z n || )
2
||z n p|| n.
(2.11)
It follows from (2.11) that p n1and hence F n1 for all n Since F ,
we have n1 for all n Therefore, we find that
1 1
n
P u
Trang 8Step 3 We show that lim || n 1||
n u u
n
n
u P u , we have
||u n u || || xu || for all x n. (2.12) Since
1
n
by taking x u n1 in (2.12), we obtain
||u n u || || u n u ||
Since Fis nonempty, closed and convex subset of H,there exists a unique q P u F 1
||u n u || || qu || By the above, we conclude that the sequence { ||u n u1|| } is bounded and nondecreasing Therefore, lim || n 1||
n u u
Step 4 We show that lim n
n u u
for some uH. Indeed, it follows from 1
n
n
u P u
and Lemma 1.6, we get
||vu n || ||u u n || || vu || for all v n. (2.13)
m
u P u By taking vu min (2.13), we have
||u mu n|| ||u u n || || u mu || This implies that ||u mu n || ||2 u m u1||2 ||u nu1|| 2
n u u
,
lim || m n || 0
m n u u
n u u
1
lim || n n || 0.
n u u
Step 5 We show that uF. Indeed, since u n1 n, by the definition of n1, we get
||w n u n || || z n u n || n (2.15)
It follows from ||z n u n || | n | ||u n u n1|| and (2.14) that
lim || n n || 0.
n z u
Therefore, we conclude from (2.14) and (2.16) that
1
lim || n n || 0.
n z u
It follows from (2.17) and the boundedness of the sequence { }u n that { }z n is
bounded Thus, there exists A 1 0 such that
1
0 n ( n 1)(1 b n n )(||z n || ) A( n 1). Taking the limit in the above
inequality as n and using lim n 1,
n
n
with (2.15) and (2.17), we have
1
lim || n n || 0.
n w u
It follows from (2.14) and (2.18) that
lim || n n || 0.
n w u
Then by combining (2.16) and (2.19), we obtain that
Trang 9lim || n n || 0.
n z w
(2.20) Next, for p F , by the same proof of (2.9), (2.10) and (2.11), we get
1
w p b z p b b T z z
1
It follows from (2.19) and the boundedness of the sequence { }u n that { }w n is bounded Moreover, by the boundedness of { }z n and { },w n we conclude that there exists A 2 0 such that ||z n || ||w n || A2 for all n It follows from (2.21) that
b b T z z z p w p
||z n ||2 ||w n ||2 2w n z p n, n
(||z n || || w n ||)(||z n || ||w n ||) 2 || w n z n || || ||p n
A2 ||z n w n || 2 || w n z n || || ||p n. (2.22) Therefore, by combining (2.22) with (2.20) and using lim n 0,
n
lim inf (1n n) 0,
n b b
we get
1
lim || n || 0.
n n
n T z z
Then by ( , ),( , )z p n p u n E G( ) and the transitive property of G, we obtain ( , )z u n n E G( ).
Since T1 is asymptotically G-nonexpansive and ( , )z u n n E G( ), we get
n u n z n T z n z n z n u n
n z n u n T z n z n
It follows from (2.16), (2.23) and (2.24) that
1
lim || n || 0.
n n
n T u u
Next, by using similar argument as in the proof of (2.9), (2.10) and (2.11), we also obtain
By the same proof of (2.22), from (2.26) and lim inf (1n n) 0,
n a a
lim || n n || 0.
n n
n T v T v
n n n n n
lim || n n || 0.
n v z
(2.28) Then by combining (2.16) and (2.28), we have
lim || n n || 0.
n u v
Now, by ( , ),( , )v p n p u n E G( ) and the transitive property of G, we obtain
( , )v u n n E G( ). Since T T1, 2 are asymptotically G-nonexpansive mappings, we get
|| 2n ||
n n
T u u
Trang 102 1 1
n v n u n T v n T v n n v n u n T u n u n
It follows from (2.25), (2.27), (2.29) and (2.30) that
2
lim || n || 0.
n n
n T u u
Now, by combining ( , ),( ,u p p u n n1) E G( ) and the transitive property of G, we conclude that ( ,u u n n1) E G( ). Then, for each i 1,2, due to the fact that T i is an
asymptotically G-nonexpansive mapping, we have
n i n n n n i n i n i n
u T u u u u T u T u T u
n n n i n n n n
1
n u n u n u n T u i n
It follows from (2.14), (2.25), (2.31) and (2.32) that
lim || n || 0.
n i n
n u T u
Since ( ,p u n1) E G( )for p F and T i nis edge-preserving, we have 1
( , n ) ( ).
i n
p T u E G By combining this with (u n1, )p E G( )and using the transitive property
mapping, we have
n i n n i n i n i n
1
Taking the limit in (2.34) as n and using (2.25), (2.31) and (2.33), we find that
lim || i n n || 0.
n T u u
Therefore, by Lemma 1.5, (2.35), we find that T u1 T u2 u and hence uF
Step 6 We show that u q P u F 1. Indeed, since 1,
n
n
u P u we get
for all y n. (2.36) Let p F Since F n, we have p n.Then, by choosing y p in (2.36), we obtain u1 u u n, n p 0. Taking the limit in this inequality as n and using
lim n ,
n u u
we find that u1 u u, p 0. This implies that u P u F 1.
Since every G-nonexpansive mapping is an asymptotically G-nonexpansive mapping
with the asymptotic coefficient n 1 for all n , from Theorem 2.2, we get the following corollary
Corollary 2.3
Let H be a real Hilbert space, H have property ( ),G G ( ( ), ( ))V G E G be a directed transitive graph such that V G( ) H, E G( ) be coordinate-convex, T T 1, 2 : be two G -nonexpansive mappings such that F F T( )1 F T( )2 ( ) ,F T i F T( )i E G( ) for all
1,2,
i { }u n be a sequence generated by (1.4) where { },{ }a n b n are sequences in [0,1] such that 0 lim inf n lim sup n 1,
n a n a
n b n b