Approximation of Common Fixed Points for Families of Mappings in ...
JOURNAL OF COMPUTERS, VOL. 7, NO. 6, JUNE 2012
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Approximation of Common Fixed Points for Families of Mappings in Banach Space Zhiming Cheng School of Mathematics and Computer, Yangtze Normal University, Chongqing, China Email: [email protected]
Abstract—In this paper. We introduce a general iterative method for the family of mappings and prove the strong convergence of the new iterative scheme in Banach space. The new iterative method includes the iterative scheme of Khan and Domlo and Fukhar-ud-din [Common fixed points Noor iteration for a finite family of asymptotically quasinonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 341 (2008) 1–11]. The results generalize the corresponding results. Index Terms—strong convergence, common fixed point, generalized asymptotically quasi-nonexpansive mapping, viscosity iteration sequence, modified W-mapping
1 ≤ i ≤ k . Then we consider the following mapping of C into itself:
U1n = (1 − α1n ) I + α1n S1n , U 2 n = (1 − α 2 n ) I + α 2 n S 2nU1n , (1.2)
U ( k −1) n = (1 − α ( k −1) n ) I + α ( k −1) n S (nk −1)U ( k − 2 ) n , Wn = U kn = (1 − α kn ) I + α kn S knU ( k −1) n . Such a mapping Wn is called the modified W-mapping
I. INTRODUCTION Let C be a nonempty subset of a real Banach space E and T a self-mapping of C . The set of fixed points of T denote by F (T ) and we assume that F (T ) ≠ φ . The mapping T is said to be generalized asymptotically quasi-nonexpansive [1] if there exists two sequences {un } , {hn } in [0, + ∞) with lim un = 0 and n →∞
lim hn = 0 such that
n→∞
T n x − p ≤ (1 + un ) x − p + hn ,
∀ x ∈ C , p ∈ F (T ) ,
(1.1)
where n = 1, 2 , … . If hn = 0 for all n ≥ 1 , then T
generated by S1 , S 2 ,… , S k and
α1n ,α 2 n ,…,α kn
(See
[4]). In 2008, Khan, Domlo and Fukhar-ud-din [2] introduced the following iteration process for a family of asymptotically quasi-nonexpansive mappings, for an arbitrary x1 ∈ C :
⎧ y = (1 − α ) x + α S n y , 1n n 1n 1 0 n ⎪ 1n ⎪ y2 n = (1 − α 2 n ) xn + α 2 n S 2n y1n , ⎪ (1.3) ⎨ ⎪y n ⎪ ( k −1) n = (1 − α ( k −1) n ) xn + α ( k −1) n Sk −1 y( k − 2 ) n , ⎪ x = (1 − α ) x + α S n y kn n kn k ( k −1) n , ⎩ n +1
becomes asymptotically quasi-nonexpansive mapping; if un = 0 and hn = 0 for all n ≥ 1 , then T becomes
where
quasi-nonexpansive mapping. It is known that an asymptotically nonexpansive mapping is an asymptotically quasi-nonexpansive. Let C be a nonempty closed convex subset of a real Banach E , {S n } a family of generalized asymptotically
fixed point of the family of mappings if and only if lim inf n→∞ d ( xn , F ) = 0 , where d ( xn , F ) =
nonexpansive mappings of C into itself and let {α in : n, i ∈ N , 1 ≤ i ≤ k} be a sequence of real
y0 n = xn , α in ⊂ [0, 1] (i = 1, 2, …, k ) , n = 1, 2, … and proved that the iterative sequence {xn } defined by (1.3), converges strongly to a common
inf x − p . (1.3) may denote by p∈F
xn +1 = Wn xn ,
numbers such that 0 ≤ ain ≤ 1 for every n, i ∈ N ,
{Si }ik=1 is a family of asymptotically quasinonexpansive mappings of C into itself. where
Manuscript received May 10, 2011; revised June 28, 2011, accepted July 15, 2011.
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(1.4)
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Recently, Chang, lee, chan and kim [3] introduced the following iteration process of asymptotically nonexpansive mappings in Banach space:
⎧⎪ xn = λn f ( xn ) + (1 − λn ) S n yn , ⎨ ⎪⎩ yn = β n xn + (1 − β n ) S n xn ,
common fixed points of T . Then, Tn is said to satisfy:
T if for y ∈ F (T ) , there exists ∞ ∞ two sequences {δ n }n =1 and {γ n }n =1 in [0, ∞) with
(a) condition (I) with (1.5)
where {λn } , {β n } ⊂ [0, 1] ， f is a fixed contractive mapping, and gave the sufficient and necessary condition for the iterative sequence {xn } converges to the fixed points of S . For a family of mappings, it is quite significant to devise a general iteration scheme which extends the iteration (1.3) and the iteration (1.5), simultaneously. Thereby, to achieve this goal, we introduce a new iteration process for a family of mappings as follows: Let C be a nonempty closed convex subset of a real Banach E , {Si : C → C , i = 1, 2, … , k} a family of generalized asymptotically quasi-nonexpansive mappings and f : C → C a fixed contractive mapping with contractive coefficient α ∈ [0, 1] . For a given x1 ∈ C , the iteration scheme is defined as follows:
xn +1 = λn f ( xn ) + (1 − λn )Wn xn ,
of all fixed points of Tn and F ( T ) is the set of all
(1.6)
where {λn } ∈ [0, 1] . Further, let {Tn : C → C , n ∈ N }
Σ ∞n =1δ n < +∞ , Σ ∞n =1γ n < +∞ such that Tn x − y ≤ (1 + δ n ) x − y + γ n , ∀ x ∈ C , n ∈ N ; (b) condition (II) with T if condition (I) is satisfied and
F (T ) = ∩∞n =1 F (Tn ) . Lemma 2.2 Let C be a nonempty closed convex subset of a real Banach E , Tn and T two families of mappings of C into itself such that
∩ F (Tn ) . Suppose that Tn satisfies condition (I) with T . Let {λn } be a sequence of real numbers with 0 ≤ λn ≤ 1 for all n ∈ N , Σ ∞n =1λn < ∞ . Let f : C → C be a contraction with 0 < α < 1 . The sequence {xn } defined by (1.7), then (1) there exists a sequence {ξ n } in [0, ∞) with Σ ∞n =1ξ n < +∞ such that xn +1 − p ≤ (1 + δ n ) xn − p + ξ n ,
be a family of mappings. We propose the following iteration scheme:
xn+1 = λn f ( xn ) + (1 − λn )Tn xn ,
(1.7)
where {λn } ∈ [0, 1] .
∀ p ∈ F (T ), ∀ n ∈ N ; (2) there exists a constant M 1 > 0 , such that ∞
xn + m − p ≤ M 1 xn − p + M 1 ∑ ξ j ,
The purpose of this paper is to study the convergence problem of the iterative sequences {xn } defined by (1.6)
j =n
∀ p ∈ F (T ), ∀ n, m ∈ N .
and (1.7). The results extend the results of [2].
Proof. (1) Let p ∈ F (T ) , by (1.7), we have
II. PRELIMINARIES Lemma 2.1 (see [5]) Let {an } ， {δ n } and {γ n } be
xn+1 − p
sequences of nonnegative real sequences satisfying the following conditions:
≤ λn f ( xn ) − p + (1 − λn ) Tn xn − p ≤ λnα xn − p + λn f ( p ) − p
an +1 = (1 + δ n )an + γ n , ∀ n ∈ N ,
+ (1 − λn ){(1 + δ n ) xn − p + γ n }
Σ ∞n =1δ n < ∞ and Σ ∞n =1γ n < ∞ , then limn → ∞ an exists. Moreover, if in addition, lim inf n → ∞ an = 0 , then lim n → ∞ an = 0 . where
= {λnα + (1 − λn )(1 + δ n )} xn − p + (1 − λn )γ n + λn f ( p) − p ≤ {λnα + (1 − λn )(1 + δ n )} xn − p + γ n + λn f ( p ) − p
Next, we introduce two new conditions. Let C be a nonempty closed convex subset of a real Banach E . Let Tn and T be families of mappings of C into itself such that
φ ≠ F (T ) ⊂ ∩∞n =1 F (Tn ) , where F (Tn )
is the set
≤ (1 + δ n ) xn − p + ξ n , where
(2.1)
ξ n = γ n + λn f ( p) − p , Σ ∞n =1ξ n < +∞ . This
completes the proof of (1).
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φ ≠ F (T ) ⊂
∞ n =1
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(2) If t ≥ 0 , then 1 + t by (2.1), we obtain
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≤ et . And for all integer m ≥ 1 ,
Let
xn+m − p ≤ (1 + δ n+m−1 ) xn+m−1 − p + ξ n+m−1
Σ u < +∞ for each i , therefore Σ ∞n =1ν n < +∞ . For all p ∈ ∩ i =1 F ( Si ) and x ∈ C , it follows from (1.2) k
+ ξ n+m−2 } + ξ n+m−1 ≤
∏ (1 +δ
j
j = n + m −2
that
U1n x − p ≤ (1 − α1n ) x − p + α1n S1n x − p
) xn + m − 2 − p
≤ (1 − α1n ) x − p + α1n{(1 + u1n ) x − p + h1n }
+ (1 + δ n+m−1 )(ξ n+ m−2 + ξ n+ m−1 )
≤
n + m−1
≤ (1 + u1n ) x − p + h1n
j =n + m −2
≤ (1 +ν n ) x − p + h1n .
∏ (1 +δ j )[(1 + δ n+m−3 ) xn+m−3 − p
+ ξ n+m−3 ] + (1 + δ n+m−1 )(ξ n+ m−2 + ξ n+m−1 ) ≤
n + m −1
∏ (1 +δ
j = n + m −3
+
Assume that
j ) x n + m −3 − p
n + m −1
∏ (1 +δ
j =n + m− 2
≤
ν n = max1≤i ≤ k uin
∞ n =1 in
≤ (1 + δ n+m−1 ){(1 + δ n+m−2 ) xn+m−2 − p n + m−1
∩ ki=1 F ( Si ) ⊂ F (Wn ) . , for all n ∈ N . Since
Proof. From (1.2) we obtain that
n + m −1
∏ (1 +δ
j
j
U jn x − p ≤ (1 + ν n ) j x − p + (1 + ν n ) j −1 Σ ij=1hin ,
n + m−1
)
∑ξ
holds for some 1 ≤ j ≤ k − 1 . Then
j
j = n + m −3
U ( j +1) n x − p ≤ (1 − α ( j +1) n ) x − p
) xn − p
+ α ( j +1) n S nj+1U jn x − p
j =n
+
n + m −1
∏ (1 +δ j )
j = n +1
⎛ ≤ exp⎜⎜ ⎝
n + m −1
∑ξ
≤ (1 − α ( j +1) n ) x − p + α ( j +1) n
j
j =n
× {(1 + u( j +1) n ) U jn x − p + h( j +1) n }
⎞ δ j ⎟⎟ xn − p ∑ j =n ⎠ n + m − 1 ⎛ ⎞n + m −1 + exp⎜⎜ ∑ δ j ⎟⎟ ∑ ξ j ⎝ j = n +1 ⎠ j = n n + m −1
≤ (1 − α ( j +1) n ) x − p + α ( j +1) n × (1 + u( j +1) n ){(1 +ν n ) j x − p + (1 + ν n ) j −1 ∑i =1 hin } + α ( j +1) n h( j +1) n j
≤ {(1 − α ( j +1) n ) + α ( j +1) n (1 + ν n ) j +1}
∞
≤ M 1 xn − p + M 1 ∑ ξ j ,
× x − p + (1 + ν n ) j ∑i=1 hin + h( j +1) n j
j =n
where M 1 = exp(Σ
j +1
≤ (1 + ν n ) j +1 x − p + (1 +ν n ) j ∑ hin
δ j ) . This completes the proof.
∞ j =1
i =1
Lemma 2.3 Let C be a nonempty closed convex subset of a real Banach E . Let Si (i = 1, 2, … , k ) be k generalized asymptotically quasi-nonexpansive selfmappings of C with uin , hin ⊂ [0, ∞) such that
Σ ∞n =1uin < +∞ and
Σ ∞n =1hin < +∞ for all i ∈ N
,
1 ≤ i ≤ k . Suppose ∩ ki=1 F ( Si ) ≠ φ and {α in } ⊂
[0, 1] (i = 1, 2, … , k ) for all n ∈ N . Let Wn be the modified W-mapping generated by S1 , S 2 , … , S k and α1n , α 2 n , … , α kn for all n ∈ N . Then Wn satisfies condition (I) with {Si }i =1 . k
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Thus, by induction, we have
U jn x − p ≤ (1 + ν n ) j x − p + (1 + ν n ) j −1 ∑i =1 hin j
⎧ ⎫ j! j ≤ ⎨1 + ∑r =1 ν nr ⎬ x − p r!( j − r )! ⎭ ⎩
+ eν n ( j −1) ∑i =1 hin j
= (1 + δ jn ) x − p + γ jn ,
j = 1, 2, … , k ,
(2.2)
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where,
δ jn = ∑ r =1 j
M j = sup eν n ( j −1)
j j! ν nr , γ jn = M j ∑i =1 hin , r!( j − r )!
.
Since
n
Σ ∞n =1ν n < +∞
and
Σ∞n =1hin < +∞ for all 1 ≤ i ≤ k , therefore Σ ∞n =1δ jn < +∞ and Σ ∞n =1γ jn < +∞ (1 ≤ j ≤ k ) . Hence,
Wn x − p = U kn x − p ≤ (1 + δ kn ) x − p + γ kn . k Thus, Wn satisfies condition (I) with {Si }i =1 .
Lemma 2.1 in [2] can be easily elicited from lemma 2.2 and lemma 2.3. III. STRONG CONVERGENCE THEOREMS FOR THE FAMILY OF MAPPINGS
Theorem 3.1 If all the condition of lemma 2.2 are conformed, and plus one another that F ( T ) be a closed
E . Then the sequence {xn } defined by (1.7)
subset of
∩∞n =1 F (Tn ) if and lim inf n →∞ d ( xn , F (T )) = 0 , where
converges strongly to p ∈ F (T ) ⊂ only
if
d ( xn , F (T )) = inf
y∈F ( T )
xn − y .
Proof: We will only prove the sufficiency; the necessity is obvious. From Lemma2.2 (1), we have
d ( xn +1 , F (T )) ≤ (1 + δ n )d ( xn , F (T )) + ξ n , ∀ p ∈ F (T ), n > 1 . By Lemma 2.1 and lim inf n →∞ d ( xn , F (T )) = 0 ,
F (T ) ) = 0 . Next, we
we get that lim n →∞ d ( xn ,
Now,
lim n→∞ d ( xn , F (T )) = 0
since
Σ ξ j < +∞ , given ε > 0 , there exists an integer
N1 > 0 such that for all n > N1 , d ( xn , F (T ))
N , d ( xn , F (T )) < and xn − q < . In 2 2
Hence,
particular, we have d ( xN , F (T )) = inf p∈F ( T ) xN − p
Є , i.e., there exists p ∈ F (T ) , such that 2 є xN − p < , hence 2
N1 , m ≥ 1 .
of a real Banach
∞
ε
2( M 1 + 2) 2( M 1 + 2) j =n integers n > N1 , m ≥ 1 , we obtain from (3.2) that n
prove that {xn } is a Cauchy sequence. From Lemma 2.2
xn + m − p ≤ M 1 xn − p + M 1 ∑ ξ j ,
and
∞ j =1
(3.2)
STRONG CONVERGENCE THEOREMS FOR GENERALIZED ASYMPTOTICALLY QUASI-NONEXPANSIVE MAPPING
Theorem 4.1 If all the condition of lemma 2.3 are conformed. Let
∩∞n =1 F ( Sn ) be a closed subset of E .
Let {λn } be a sequence of real numbers with
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0 ≤ λn ≤ 1 for all n ∈ N , Σ ∞n =1λn < ∞ . Let f : C → C be a contraction with 0 < α < 1 . Starting from arbitrary x1 ∈ C , define the sequence {xn } defined by (1.6), then the sequence {xn } converges strongly to
p ∈ ∩ ki=1 F ( Si )
if
and
only
if
lim inf n →∞
d ( xn , ∩ i =1 F ( Si )) = 0 . k
Proof: By Lemma 2.3,
Wn satisfies condition (I) with
{Si }ik=1 , we obtain from Theorem 3.1 that {xn }
p ∈ ∩ ki=1 F ( Si ) . This completes
converges strongly to
the proof. Using Theorem 4.1, we also obtain the following theorem which was proved by Khan, Domlo, and Fukharud-din [2]. Theorem 4.2 Let C be a nonempty closed convex subset of a real Banach E . Let Si (i = 1, 2, … , k ) be k asymptotically quasi-nonexpansive self-mappings of C with
uin ⊂ [0, ∞) such that Σ ∞n =1uin < +∞ for all
i∈N
1 ≤ i ≤ k . Suppose ∩ ki=1 F ( Si ) ≠ φ and
,
{α in } ⊂ [0, 1] (i = 1, 2, … , k ) for all n ∈ N . For any given x1 ∈ C , define the sequence {xn } by the recursion (1.3). Then {xn } converges strongly to
p ∈ ∩ ki=1 F ( Si )
if
and
only
if
lim inf n →∞
d ( xn , ∩ i =1 F ( Si )) = 0 . k
conclusion of Theorem 4.2 can be obtained from Theorem 4.1 immediately. Theorem 4.3 Let C be a nonempty closed convex subset of a real Banach E . Let S be a asymptotically
un ⊂ [0, ∞) ,
S n x − S n y ≤ (1 + un ) x − y for all x, y ∈ C ,
F ( S ) ≠ φ . Let {λn } be a sequence of real numbers with 0 ≤ λn ≤ 1 such that ∞
where Σ n =1un < +∞ , suppose
Σ ∞n =1λn < ∞ for all n ∈ N , and f : C → C a contraction with 0 < α < 1 . Starting from arbitrary x1 ∈ C , define the sequence {xn } defined by (1.5). Then the sequence {xn } converges strongly to p ∈ F ( S ) if and only if lim inf n →∞ d ( xn , F ( S )) = 0 .
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α kn = 1 , 1 − α ( k −1) n
(i = 1, 2, …, k ) for all n ∈ N , (1.6) is reduced to (1.5). Therefore the conclusion of Theorem 4.3 can be obtained from Theorem 4.1 immediately. g: Let G = { g : [0, ∞ ) → [0, ∞ ) : g (0) = 0, continuous; strictly increasing; convex}. We have the following lemma for a uniformly convex Banach space. Lemma 4.1 (see [6]) E is a uniformly convex Banach space if and only if for every bounded subset B of E , there exists g ∈ G such that 2
2
λx + (1 − λ ) y ≤ λ x + (1 − λ ) y
2
− λ (1 − λ ) g ( x − y ) For all
x, y ∈ B and λ ∈ [0, 1] .
Lemma 4.2 Let E be a real uniformly convex Banach space and C a nonempty bounded closed convex subset of E . Let Si : C → C (i = 1, 2, … , k ) be k uniformly L-Lipschitzian, generalized asymptotically quasi-nonexpansive mappings with
{uin }, {hin } ⊂ [0, ∞) such that Σ ∞n =1uin < +∞ and
Σ ∞n =1hin < +∞ for all i ∈ N , 1 ≤ i ≤ k . Let {α in } ( i, n ∈ N , 1 ≤ i ≤ k ) be a sequence of real numbers with 0 < a ≤ α kn ≤ 1 , 0 < b ≤ α in ≤ c < 1 ( 1 ≤ i ≤ k − 1 ) for all n ∈ N and Wn be a modified Wk
∩ ki=1 F ( Si ) is closed. In (1.6), taking , λn = 0 and hin = 0 (i = 1, 2,…, k ) for all n ∈ N , (1.6) is reduced to (1.3). Therefore the
i.e.,
S1 = S 2 , = = S k −2 = I , = β n , S k = S K −1 = S , hin = 0
Proof: In (1.6), taking
mapping generated by {Si }i =1 and {α in }i =1 . Suppose
Proof: Easy to show that
nonexpansive self-mappings of C with
1471
k
∩ ki=1 F ( Si ) ≠ φ , then ∩ ki=1 F ( Si ) = ∩∞n =1 F (Wn ) . Proof:
∩ ki=1 F ( Si )
⊂ ∩∞n =1 F (Wn ) is obvious. Let
p ∈ ∩∞n =1 F (Wn ) , we obtain from (1.2) that Wn p = (1 − α kn ) p + α kn S knU ( k −1) n p . Since 0 < a ≤ α kn ≤ 1
z∈
for all n ∈ N , therefore p = S k U ( k −1) n p . Let n
∩
k n =1
F ( Sn ) , we have 2
{
p − z ≤ p − SinU ( i −1) n p + SinU ( i −1) n p − z
{ p − z }+ S U
}
2
= p − SinU ( i −1) n p p − SinU ( i −1) n p + 2 SinU ( i −1) n
n i
( i −1) n
≤ M 2 p − SinU ( i −1) n p
{
+ (1 + uin ) U ( i −1) n p − z + hin
}
2
p−z
2
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= (1 + uin ) 2 U (i −1) n p − z
2
lim g ( S kn−1U ( k − 2) n p − p ) = 0 ,
n →∞
+ 2(1 + uin )hin U ( i −1) n p − z + M 2 p − SinU ( i −1) n p + hin
(4.1) 2
hence, lim n → ∞ S k −1U ( k − 2 ) n p − p = 0 . By the same n
token, we have
lim S mnU ( m −1) n p − p = 0, m = 2, …, k .
≤ (1 + uin ) 2 (1 − α (i −1) n )( p − z )
{
+ α (i −1) n Sin−1U (i − 2 ) n p − z
}
n→∞
2
Since each Si is uniformly L-Lipschitzian, we conclude 2
+ M 3hin + M 2 p − SinU (i −1) n p + hin , where,
that
p − Smn p ≤ p − SmnU ( m −1) n p + S mnU ( m −1) n p − S mn p ≤ p − S mnU ( m −1) n p + L U ( m −1) n p − p
M 2 = sup{ p − SinU (i −1) n p + 2 SinU ( i −1) n p − z } n∈ N
= p − S mnU ( m −1) n p
M 3 = sup{2(1 + uin ) U ( i −1) n p − z } . n∈ N
+ Lα ( m −1) n p − S mn −1U ( m − 2 ) n p ,
By Lemma 4.1 and (4.1), we get
{
2
p − z ≤ (1 + uin ) 2 (1 − α ( i −1) n ) p − z + α (i −1) n Sin−1U ( i − 2) n p − z − (1 − α (i −1) n )α ( i −1) n
hence,
2
}
n i
→ 0,
2
p = Sm p , (4.2)
2
+ M 2 p − S U ( i −1) n p + M 3hin + hin . By (2.2) and (4.2), we have
p − z ≤ (1 + uin ) (1 − α (i −1) n ) p − z 2
∞
Σ ∞n =1hin < +∞ for all
i ∈ N , 1 ≤ i ≤ k . Suppose ∩ ki=1 F ( Si ) ≠ φ is closed.
× [(1 + δ ( i − 2 ) n p − z + γ ( i − 2 ) n ] + h( i −1) n }
2
− (1 + uin ) (1 − α ( i −1) n )
Let {α in } ( i, n ∈ N , 1 ≤ i ≤ k ) be a sequence of real
2
(4.3)
× α ( i −1) n g ( S U (i − 2 ) n p − p ) n i −1
2
+ M 2 p − SinU ( i −1) n p + M 3hin + hin . If i = k , there is lim p − Si U ( i −1) n p = lim uin = n
n→∞
lim δ ( i − 2 ) n = lim γ ( i − 2 ) n = lim hin = lim u( i −1) n = 0 , n→∞
0 < b ≤ α ( i −1) n ≤ c < 1 , taking limit in (4.3), we have © 2012 ACADEMY PUBLISHER
Theorem 4.4 Let E be a real uniformly convex Banach space, C a nonempty bounded closed convex subset of E and Si : C → C (i = 1, 2, … , k ) be k uniformly L-
such that Σ n =1uin < +∞ and
+ (1 + uin ) 2 α ( i −1) n {(1 + u( i −1) n )
n →∞
= ∩∞n =1 F (Wn ) . This
∩ ki=1 F ( Si )
Lipschitzian, generalized asymptotically quasinonexpansive mappings with {uin }, {hin } ⊂ [0, ∞)
2
n →∞
m = 1, 2, …, k , i.e.,
completes the proof.
× α ( i −1) n g ( Sin−1U ( i − 2) n p − p ) n i
n → ∞.
So, we obtain p − S m p = 0 for
+ h( i −1) n ]2 }− (1 + uin ) 2 (1 − α ( i −1) n )
n →∞
all
≤ L p − S mn p + S mn +1 p − p
2
+ α ( i −1) n [(1 + u( i −1) n ) U ( i − 2) n p − z
n→∞
for
p − S m p ≤ S m p − S mn +1 p + S mn +1 p − p
≤ (1 + uin ) 2 (1 − α ( i −1) n ) p − z
2
lim n → ∞ p − S mn p = 0
lim n → ∞ p − S1n p = 0 . Since
+ M 2 p − S U ( i −1) n p + M 3hin + hin
{
get
m = 2, …, k . Repeat the above process, we get
2
× g ( S U (i − 2) n p − p ) n i −1
we
numbers with 0 < a ≤ α kn ≤ 1 , 0 < b ≤ α in ≤ c < 1 ( 1 ≤ i ≤ k − 1 ) for all n ∈ N and Wn be a modified W-mapping generated by {Si }i =1 and {α in }i =1 , {λn } a k
k
sequence of real numbers with 0 ≤ λn ≤ 1 for all
n ∈ N , Σ ∞n =1λn < +∞ and f : C → C a contraction with 0 < α < 1 . For any given x1 ∈ C , the sequence
{xn } defined by the recursion (1.6). Then {xn }
JOURNAL OF COMPUTERS, VOL. 7, NO. 6, JUNE 2012
∞
converges strongly to p ∈ ∩ n =1 F (Wn ) if and only if
lim inf n →∞ d ( xn , ∩ i =1 F ( Si )) = 0 . k
Proof: By Lemma 4.2, there exists {Si }i =1 such that k
∩ ki=1 F ( Si ) = ∩∞n =1 F (Wn ) , using Theorem 4.1, we obtain Theorem 4.4. REFERENCES [1] N. Shahzad, H. Zegeye, “Strong convergence of an implicit iteration process for a finite family of generalized asymptotically quasi-nonexpansive maps,” Appl. Math. Comp., vol. 189, pp. 1058–1065, June 2007. [2] A. R. Khan, A-A. Domlo, and H. Fukhar-ud-din, “Common fixed points Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces,” J. Math. Anal. Appl., vol. 341, pp. 1–11, 2008. [3] S. S. Chang, H. W. J. Lee, Chi Kin Chan, and J. K. Kim, “Approximating solutions of variational inequalities for asymptotically nonexpansive mappings,” Appl. Math. Comput., vol. 212, pp. 51-59, June 2009. [4] K. Nakajo, K. Shimoji, and W. Takahashi, “On strong convergence by the hybrid method for families of mappings in Hilbert spaces,” Nonlinear Anal. TMA., vol. 71, pp. 112–119, July 2009. [5] H. K. Xu, “Approximating fixed points of nonexpansive mappings by the Ishikawa iterative process,” J. Math. Anal. Appl., vol. 178, pp. 301–308, 1993. [6] K. Nakajo, K. Shimoji, and W. Takahashi, “Strong convergence theorems by the hybrid method for families of mappings in Banach spaces,” Nonlinear Anal. TMA., vol. 71, pp. 812–818, August 2009. [7] S. Temir, “On the convergence theorems of implicit iteration process for a finite family of I-asymptotically nonexpansive mappings,” J. Comp. Appl. Math., vol. 225 pp. 398–405, March 2009. [8] T. C. Lim, H. K. Xu, “Fixed point theorems for asymptotically nonexpansive mappings,” Nonlinear Anal. TMA., vol. 22, pp. 1345-1355, June 1994.
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[9] C. E. Chidume, E. U. Ofoedub, “Approximation of common fixed points for finite families of total asymptotically nonexpansive mappings,” J. Math. Anal. Appl., vol. 333, pp. 128-141, September 2007. [10] Z. h. Sun, “Strong convergence of an implicit iteration process for a finite family of asymptotically quasinonexpansive mappings,” J. Math. Anal. Appl., vol. 286, pp. 351–358, October 2003. [11] Z. M. Cheng, L. Deng, “Approximation of common fixed points for finite families of generalized asymptotically quasi-nonexpansive mappings,” Journal of Southwest University (Science Edition), Chongqing, vol. 32, pp. 143– 147, June 2010. [12] Y. Yao, J. C. Yao, and H. Zhou, “Approximation methods for common fixed points of infinite countable family of nonexpansive mappings,” Comp. Math. Appl., vol. 53, pp. 1380–1389, May 2007. [13] V. Colao, G. Marino, “Common fixed points of strict pseudocontractions by iterative algorithms,” J. Math. Anal. Appl., vol. 382, pp. 631–644, October 2011. [14] W. Q. deng, “A modified Mann iteration process for common fixed points of an infinite family of nonexpansive mappings in Banach spaces,” Appl. Math. Sciences, vol. 4, pp. 1521–1526, 2011. [15] A. R. Khan, M.A. Ahmed, “Convergence of a general iterative scheme for a finite family of asymptotically quasinonexpansive mappings in convex metric spaces and applications,” Comp. Math. Appl., vol. 59, pp. 2990–2995, April 2010. [16] A. Kettapun, A. Kananthai, and S. Suantai, “A new approximation method for common fixed points of a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces,” Comp. Math. Appl., vol. 60, pp. 1430– 1439, September 2010. [17] C. Wang, J. Zhu, and L. An, “New iterative schemes for asymptotically quasi-nonexpansive nonself-mappings in Banach spaces,” J. Comp. Appl. Math., vol. 233, pp. 2948– 2955, April 2010. [18] H. Zhou, G. Gao, and B. Tan, “Convergence theorems of a modified hybrid algorithm for a family of quasi-φasymptotically nonexpansive mappings,” J. Appl. Math. Comput., vol. 32., pp. 453–464, 2010.
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Approximation of Common Fixed Points for Families of Mappings in Banach Space Zhiming Cheng School of Mathematics and Computer, Yangtze Normal University, Chongqing, China Email: [email protected]
Abstract—In this paper. We introduce a general iterative method for the family of mappings and prove the strong convergence of the new iterative scheme in Banach space. The new iterative method includes the iterative scheme of Khan and Domlo and Fukhar-ud-din [Common fixed points Noor iteration for a finite family of asymptotically quasinonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 341 (2008) 1–11]. The results generalize the corresponding results. Index Terms—strong convergence, common fixed point, generalized asymptotically quasi-nonexpansive mapping, viscosity iteration sequence, modified W-mapping
1 ≤ i ≤ k . Then we consider the following mapping of C into itself:
U1n = (1 − α1n ) I + α1n S1n , U 2 n = (1 − α 2 n ) I + α 2 n S 2nU1n , (1.2)
U ( k −1) n = (1 − α ( k −1) n ) I + α ( k −1) n S (nk −1)U ( k − 2 ) n , Wn = U kn = (1 − α kn ) I + α kn S knU ( k −1) n . Such a mapping Wn is called the modified W-mapping
I. INTRODUCTION Let C be a nonempty subset of a real Banach space E and T a self-mapping of C . The set of fixed points of T denote by F (T ) and we assume that F (T ) ≠ φ . The mapping T is said to be generalized asymptotically quasi-nonexpansive [1] if there exists two sequences {un } , {hn } in [0, + ∞) with lim un = 0 and n →∞
lim hn = 0 such that
n→∞
T n x − p ≤ (1 + un ) x − p + hn ,
∀ x ∈ C , p ∈ F (T ) ,
(1.1)
where n = 1, 2 , … . If hn = 0 for all n ≥ 1 , then T
generated by S1 , S 2 ,… , S k and
α1n ,α 2 n ,…,α kn
(See
[4]). In 2008, Khan, Domlo and Fukhar-ud-din [2] introduced the following iteration process for a family of asymptotically quasi-nonexpansive mappings, for an arbitrary x1 ∈ C :
⎧ y = (1 − α ) x + α S n y , 1n n 1n 1 0 n ⎪ 1n ⎪ y2 n = (1 − α 2 n ) xn + α 2 n S 2n y1n , ⎪ (1.3) ⎨ ⎪y n ⎪ ( k −1) n = (1 − α ( k −1) n ) xn + α ( k −1) n Sk −1 y( k − 2 ) n , ⎪ x = (1 − α ) x + α S n y kn n kn k ( k −1) n , ⎩ n +1
becomes asymptotically quasi-nonexpansive mapping; if un = 0 and hn = 0 for all n ≥ 1 , then T becomes
where
quasi-nonexpansive mapping. It is known that an asymptotically nonexpansive mapping is an asymptotically quasi-nonexpansive. Let C be a nonempty closed convex subset of a real Banach E , {S n } a family of generalized asymptotically
fixed point of the family of mappings if and only if lim inf n→∞ d ( xn , F ) = 0 , where d ( xn , F ) =
nonexpansive mappings of C into itself and let {α in : n, i ∈ N , 1 ≤ i ≤ k} be a sequence of real
y0 n = xn , α in ⊂ [0, 1] (i = 1, 2, …, k ) , n = 1, 2, … and proved that the iterative sequence {xn } defined by (1.3), converges strongly to a common
inf x − p . (1.3) may denote by p∈F
xn +1 = Wn xn ,
numbers such that 0 ≤ ain ≤ 1 for every n, i ∈ N ,
{Si }ik=1 is a family of asymptotically quasinonexpansive mappings of C into itself. where
Manuscript received May 10, 2011; revised June 28, 2011, accepted July 15, 2011.
© 2012 ACADEMY PUBLISHER doi:10.4304/jcp.7.6.1467-1473
(1.4)
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Recently, Chang, lee, chan and kim [3] introduced the following iteration process of asymptotically nonexpansive mappings in Banach space:
⎧⎪ xn = λn f ( xn ) + (1 − λn ) S n yn , ⎨ ⎪⎩ yn = β n xn + (1 − β n ) S n xn ,
common fixed points of T . Then, Tn is said to satisfy:
T if for y ∈ F (T ) , there exists ∞ ∞ two sequences {δ n }n =1 and {γ n }n =1 in [0, ∞) with
(a) condition (I) with (1.5)
where {λn } , {β n } ⊂ [0, 1] ， f is a fixed contractive mapping, and gave the sufficient and necessary condition for the iterative sequence {xn } converges to the fixed points of S . For a family of mappings, it is quite significant to devise a general iteration scheme which extends the iteration (1.3) and the iteration (1.5), simultaneously. Thereby, to achieve this goal, we introduce a new iteration process for a family of mappings as follows: Let C be a nonempty closed convex subset of a real Banach E , {Si : C → C , i = 1, 2, … , k} a family of generalized asymptotically quasi-nonexpansive mappings and f : C → C a fixed contractive mapping with contractive coefficient α ∈ [0, 1] . For a given x1 ∈ C , the iteration scheme is defined as follows:
xn +1 = λn f ( xn ) + (1 − λn )Wn xn ,
of all fixed points of Tn and F ( T ) is the set of all
(1.6)
where {λn } ∈ [0, 1] . Further, let {Tn : C → C , n ∈ N }
Σ ∞n =1δ n < +∞ , Σ ∞n =1γ n < +∞ such that Tn x − y ≤ (1 + δ n ) x − y + γ n , ∀ x ∈ C , n ∈ N ; (b) condition (II) with T if condition (I) is satisfied and
F (T ) = ∩∞n =1 F (Tn ) . Lemma 2.2 Let C be a nonempty closed convex subset of a real Banach E , Tn and T two families of mappings of C into itself such that
∩ F (Tn ) . Suppose that Tn satisfies condition (I) with T . Let {λn } be a sequence of real numbers with 0 ≤ λn ≤ 1 for all n ∈ N , Σ ∞n =1λn < ∞ . Let f : C → C be a contraction with 0 < α < 1 . The sequence {xn } defined by (1.7), then (1) there exists a sequence {ξ n } in [0, ∞) with Σ ∞n =1ξ n < +∞ such that xn +1 − p ≤ (1 + δ n ) xn − p + ξ n ,
be a family of mappings. We propose the following iteration scheme:
xn+1 = λn f ( xn ) + (1 − λn )Tn xn ,
(1.7)
where {λn } ∈ [0, 1] .
∀ p ∈ F (T ), ∀ n ∈ N ; (2) there exists a constant M 1 > 0 , such that ∞
xn + m − p ≤ M 1 xn − p + M 1 ∑ ξ j ,
The purpose of this paper is to study the convergence problem of the iterative sequences {xn } defined by (1.6)
j =n
∀ p ∈ F (T ), ∀ n, m ∈ N .
and (1.7). The results extend the results of [2].
Proof. (1) Let p ∈ F (T ) , by (1.7), we have
II. PRELIMINARIES Lemma 2.1 (see [5]) Let {an } ， {δ n } and {γ n } be
xn+1 − p
sequences of nonnegative real sequences satisfying the following conditions:
≤ λn f ( xn ) − p + (1 − λn ) Tn xn − p ≤ λnα xn − p + λn f ( p ) − p
an +1 = (1 + δ n )an + γ n , ∀ n ∈ N ,
+ (1 − λn ){(1 + δ n ) xn − p + γ n }
Σ ∞n =1δ n < ∞ and Σ ∞n =1γ n < ∞ , then limn → ∞ an exists. Moreover, if in addition, lim inf n → ∞ an = 0 , then lim n → ∞ an = 0 . where
= {λnα + (1 − λn )(1 + δ n )} xn − p + (1 − λn )γ n + λn f ( p) − p ≤ {λnα + (1 − λn )(1 + δ n )} xn − p + γ n + λn f ( p ) − p
Next, we introduce two new conditions. Let C be a nonempty closed convex subset of a real Banach E . Let Tn and T be families of mappings of C into itself such that
φ ≠ F (T ) ⊂ ∩∞n =1 F (Tn ) , where F (Tn )
is the set
≤ (1 + δ n ) xn − p + ξ n , where
(2.1)
ξ n = γ n + λn f ( p) − p , Σ ∞n =1ξ n < +∞ . This
completes the proof of (1).
© 2012 ACADEMY PUBLISHER
φ ≠ F (T ) ⊂
∞ n =1
JOURNAL OF COMPUTERS, VOL. 7, NO. 6, JUNE 2012
(2) If t ≥ 0 , then 1 + t by (2.1), we obtain
1469
≤ et . And for all integer m ≥ 1 ,
Let
xn+m − p ≤ (1 + δ n+m−1 ) xn+m−1 − p + ξ n+m−1
Σ u < +∞ for each i , therefore Σ ∞n =1ν n < +∞ . For all p ∈ ∩ i =1 F ( Si ) and x ∈ C , it follows from (1.2) k
+ ξ n+m−2 } + ξ n+m−1 ≤
∏ (1 +δ
j
j = n + m −2
that
U1n x − p ≤ (1 − α1n ) x − p + α1n S1n x − p
) xn + m − 2 − p
≤ (1 − α1n ) x − p + α1n{(1 + u1n ) x − p + h1n }
+ (1 + δ n+m−1 )(ξ n+ m−2 + ξ n+ m−1 )
≤
n + m−1
≤ (1 + u1n ) x − p + h1n
j =n + m −2
≤ (1 +ν n ) x − p + h1n .
∏ (1 +δ j )[(1 + δ n+m−3 ) xn+m−3 − p
+ ξ n+m−3 ] + (1 + δ n+m−1 )(ξ n+ m−2 + ξ n+m−1 ) ≤
n + m −1
∏ (1 +δ
j = n + m −3
+
Assume that
j ) x n + m −3 − p
n + m −1
∏ (1 +δ
j =n + m− 2
≤
ν n = max1≤i ≤ k uin
∞ n =1 in
≤ (1 + δ n+m−1 ){(1 + δ n+m−2 ) xn+m−2 − p n + m−1
∩ ki=1 F ( Si ) ⊂ F (Wn ) . , for all n ∈ N . Since
Proof. From (1.2) we obtain that
n + m −1
∏ (1 +δ
j
j
U jn x − p ≤ (1 + ν n ) j x − p + (1 + ν n ) j −1 Σ ij=1hin ,
n + m−1
)
∑ξ
holds for some 1 ≤ j ≤ k − 1 . Then
j
j = n + m −3
U ( j +1) n x − p ≤ (1 − α ( j +1) n ) x − p
) xn − p
+ α ( j +1) n S nj+1U jn x − p
j =n
+
n + m −1
∏ (1 +δ j )
j = n +1
⎛ ≤ exp⎜⎜ ⎝
n + m −1
∑ξ
≤ (1 − α ( j +1) n ) x − p + α ( j +1) n
j
j =n
× {(1 + u( j +1) n ) U jn x − p + h( j +1) n }
⎞ δ j ⎟⎟ xn − p ∑ j =n ⎠ n + m − 1 ⎛ ⎞n + m −1 + exp⎜⎜ ∑ δ j ⎟⎟ ∑ ξ j ⎝ j = n +1 ⎠ j = n n + m −1
≤ (1 − α ( j +1) n ) x − p + α ( j +1) n × (1 + u( j +1) n ){(1 +ν n ) j x − p + (1 + ν n ) j −1 ∑i =1 hin } + α ( j +1) n h( j +1) n j
≤ {(1 − α ( j +1) n ) + α ( j +1) n (1 + ν n ) j +1}
∞
≤ M 1 xn − p + M 1 ∑ ξ j ,
× x − p + (1 + ν n ) j ∑i=1 hin + h( j +1) n j
j =n
where M 1 = exp(Σ
j +1
≤ (1 + ν n ) j +1 x − p + (1 +ν n ) j ∑ hin
δ j ) . This completes the proof.
∞ j =1
i =1
Lemma 2.3 Let C be a nonempty closed convex subset of a real Banach E . Let Si (i = 1, 2, … , k ) be k generalized asymptotically quasi-nonexpansive selfmappings of C with uin , hin ⊂ [0, ∞) such that
Σ ∞n =1uin < +∞ and
Σ ∞n =1hin < +∞ for all i ∈ N
,
1 ≤ i ≤ k . Suppose ∩ ki=1 F ( Si ) ≠ φ and {α in } ⊂
[0, 1] (i = 1, 2, … , k ) for all n ∈ N . Let Wn be the modified W-mapping generated by S1 , S 2 , … , S k and α1n , α 2 n , … , α kn for all n ∈ N . Then Wn satisfies condition (I) with {Si }i =1 . k
© 2012 ACADEMY PUBLISHER
Thus, by induction, we have
U jn x − p ≤ (1 + ν n ) j x − p + (1 + ν n ) j −1 ∑i =1 hin j
⎧ ⎫ j! j ≤ ⎨1 + ∑r =1 ν nr ⎬ x − p r!( j − r )! ⎭ ⎩
+ eν n ( j −1) ∑i =1 hin j
= (1 + δ jn ) x − p + γ jn ,
j = 1, 2, … , k ,
(2.2)
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where,
δ jn = ∑ r =1 j
M j = sup eν n ( j −1)
j j! ν nr , γ jn = M j ∑i =1 hin , r!( j − r )!
.
Since
n
Σ ∞n =1ν n < +∞
and
Σ∞n =1hin < +∞ for all 1 ≤ i ≤ k , therefore Σ ∞n =1δ jn < +∞ and Σ ∞n =1γ jn < +∞ (1 ≤ j ≤ k ) . Hence,
Wn x − p = U kn x − p ≤ (1 + δ kn ) x − p + γ kn . k Thus, Wn satisfies condition (I) with {Si }i =1 .
Lemma 2.1 in [2] can be easily elicited from lemma 2.2 and lemma 2.3. III. STRONG CONVERGENCE THEOREMS FOR THE FAMILY OF MAPPINGS
Theorem 3.1 If all the condition of lemma 2.2 are conformed, and plus one another that F ( T ) be a closed
E . Then the sequence {xn } defined by (1.7)
subset of
∩∞n =1 F (Tn ) if and lim inf n →∞ d ( xn , F (T )) = 0 , where
converges strongly to p ∈ F (T ) ⊂ only
if
d ( xn , F (T )) = inf
y∈F ( T )
xn − y .
Proof: We will only prove the sufficiency; the necessity is obvious. From Lemma2.2 (1), we have
d ( xn +1 , F (T )) ≤ (1 + δ n )d ( xn , F (T )) + ξ n , ∀ p ∈ F (T ), n > 1 . By Lemma 2.1 and lim inf n →∞ d ( xn , F (T )) = 0 ,
F (T ) ) = 0 . Next, we
we get that lim n →∞ d ( xn ,
Now,
lim n→∞ d ( xn , F (T )) = 0
since
Σ ξ j < +∞ , given ε > 0 , there exists an integer
N1 > 0 such that for all n > N1 , d ( xn , F (T ))
N , d ( xn , F (T )) < and xn − q < . In 2 2
Hence,
particular, we have d ( xN , F (T )) = inf p∈F ( T ) xN − p
Є , i.e., there exists p ∈ F (T ) , such that 2 є xN − p < , hence 2
N1 , m ≥ 1 .
of a real Banach
∞
ε
2( M 1 + 2) 2( M 1 + 2) j =n integers n > N1 , m ≥ 1 , we obtain from (3.2) that n
prove that {xn } is a Cauchy sequence. From Lemma 2.2
xn + m − p ≤ M 1 xn − p + M 1 ∑ ξ j ,
and
∞ j =1
(3.2)
STRONG CONVERGENCE THEOREMS FOR GENERALIZED ASYMPTOTICALLY QUASI-NONEXPANSIVE MAPPING
Theorem 4.1 If all the condition of lemma 2.3 are conformed. Let
∩∞n =1 F ( Sn ) be a closed subset of E .
Let {λn } be a sequence of real numbers with
JOURNAL OF COMPUTERS, VOL. 7, NO. 6, JUNE 2012
0 ≤ λn ≤ 1 for all n ∈ N , Σ ∞n =1λn < ∞ . Let f : C → C be a contraction with 0 < α < 1 . Starting from arbitrary x1 ∈ C , define the sequence {xn } defined by (1.6), then the sequence {xn } converges strongly to
p ∈ ∩ ki=1 F ( Si )
if
and
only
if
lim inf n →∞
d ( xn , ∩ i =1 F ( Si )) = 0 . k
Proof: By Lemma 2.3,
Wn satisfies condition (I) with
{Si }ik=1 , we obtain from Theorem 3.1 that {xn }
p ∈ ∩ ki=1 F ( Si ) . This completes
converges strongly to
the proof. Using Theorem 4.1, we also obtain the following theorem which was proved by Khan, Domlo, and Fukharud-din [2]. Theorem 4.2 Let C be a nonempty closed convex subset of a real Banach E . Let Si (i = 1, 2, … , k ) be k asymptotically quasi-nonexpansive self-mappings of C with
uin ⊂ [0, ∞) such that Σ ∞n =1uin < +∞ for all
i∈N
1 ≤ i ≤ k . Suppose ∩ ki=1 F ( Si ) ≠ φ and
,
{α in } ⊂ [0, 1] (i = 1, 2, … , k ) for all n ∈ N . For any given x1 ∈ C , define the sequence {xn } by the recursion (1.3). Then {xn } converges strongly to
p ∈ ∩ ki=1 F ( Si )
if
and
only
if
lim inf n →∞
d ( xn , ∩ i =1 F ( Si )) = 0 . k
conclusion of Theorem 4.2 can be obtained from Theorem 4.1 immediately. Theorem 4.3 Let C be a nonempty closed convex subset of a real Banach E . Let S be a asymptotically
un ⊂ [0, ∞) ,
S n x − S n y ≤ (1 + un ) x − y for all x, y ∈ C ,
F ( S ) ≠ φ . Let {λn } be a sequence of real numbers with 0 ≤ λn ≤ 1 such that ∞
where Σ n =1un < +∞ , suppose
Σ ∞n =1λn < ∞ for all n ∈ N , and f : C → C a contraction with 0 < α < 1 . Starting from arbitrary x1 ∈ C , define the sequence {xn } defined by (1.5). Then the sequence {xn } converges strongly to p ∈ F ( S ) if and only if lim inf n →∞ d ( xn , F ( S )) = 0 .
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α kn = 1 , 1 − α ( k −1) n
(i = 1, 2, …, k ) for all n ∈ N , (1.6) is reduced to (1.5). Therefore the conclusion of Theorem 4.3 can be obtained from Theorem 4.1 immediately. g: Let G = { g : [0, ∞ ) → [0, ∞ ) : g (0) = 0, continuous; strictly increasing; convex}. We have the following lemma for a uniformly convex Banach space. Lemma 4.1 (see [6]) E is a uniformly convex Banach space if and only if for every bounded subset B of E , there exists g ∈ G such that 2
2
λx + (1 − λ ) y ≤ λ x + (1 − λ ) y
2
− λ (1 − λ ) g ( x − y ) For all
x, y ∈ B and λ ∈ [0, 1] .
Lemma 4.2 Let E be a real uniformly convex Banach space and C a nonempty bounded closed convex subset of E . Let Si : C → C (i = 1, 2, … , k ) be k uniformly L-Lipschitzian, generalized asymptotically quasi-nonexpansive mappings with
{uin }, {hin } ⊂ [0, ∞) such that Σ ∞n =1uin < +∞ and
Σ ∞n =1hin < +∞ for all i ∈ N , 1 ≤ i ≤ k . Let {α in } ( i, n ∈ N , 1 ≤ i ≤ k ) be a sequence of real numbers with 0 < a ≤ α kn ≤ 1 , 0 < b ≤ α in ≤ c < 1 ( 1 ≤ i ≤ k − 1 ) for all n ∈ N and Wn be a modified Wk
∩ ki=1 F ( Si ) is closed. In (1.6), taking , λn = 0 and hin = 0 (i = 1, 2,…, k ) for all n ∈ N , (1.6) is reduced to (1.3). Therefore the
i.e.,
S1 = S 2 , = = S k −2 = I , = β n , S k = S K −1 = S , hin = 0
Proof: In (1.6), taking
mapping generated by {Si }i =1 and {α in }i =1 . Suppose
Proof: Easy to show that
nonexpansive self-mappings of C with
1471
k
∩ ki=1 F ( Si ) ≠ φ , then ∩ ki=1 F ( Si ) = ∩∞n =1 F (Wn ) . Proof:
∩ ki=1 F ( Si )
⊂ ∩∞n =1 F (Wn ) is obvious. Let
p ∈ ∩∞n =1 F (Wn ) , we obtain from (1.2) that Wn p = (1 − α kn ) p + α kn S knU ( k −1) n p . Since 0 < a ≤ α kn ≤ 1
z∈
for all n ∈ N , therefore p = S k U ( k −1) n p . Let n
∩
k n =1
F ( Sn ) , we have 2
{
p − z ≤ p − SinU ( i −1) n p + SinU ( i −1) n p − z
{ p − z }+ S U
}
2
= p − SinU ( i −1) n p p − SinU ( i −1) n p + 2 SinU ( i −1) n
n i
( i −1) n
≤ M 2 p − SinU ( i −1) n p
{
+ (1 + uin ) U ( i −1) n p − z + hin
}
2
p−z
2
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= (1 + uin ) 2 U (i −1) n p − z
2
lim g ( S kn−1U ( k − 2) n p − p ) = 0 ,
n →∞
+ 2(1 + uin )hin U ( i −1) n p − z + M 2 p − SinU ( i −1) n p + hin
(4.1) 2
hence, lim n → ∞ S k −1U ( k − 2 ) n p − p = 0 . By the same n
token, we have
lim S mnU ( m −1) n p − p = 0, m = 2, …, k .
≤ (1 + uin ) 2 (1 − α (i −1) n )( p − z )
{
+ α (i −1) n Sin−1U (i − 2 ) n p − z
}
n→∞
2
Since each Si is uniformly L-Lipschitzian, we conclude 2
+ M 3hin + M 2 p − SinU (i −1) n p + hin , where,
that
p − Smn p ≤ p − SmnU ( m −1) n p + S mnU ( m −1) n p − S mn p ≤ p − S mnU ( m −1) n p + L U ( m −1) n p − p
M 2 = sup{ p − SinU (i −1) n p + 2 SinU ( i −1) n p − z } n∈ N
= p − S mnU ( m −1) n p
M 3 = sup{2(1 + uin ) U ( i −1) n p − z } . n∈ N
+ Lα ( m −1) n p − S mn −1U ( m − 2 ) n p ,
By Lemma 4.1 and (4.1), we get
{
2
p − z ≤ (1 + uin ) 2 (1 − α ( i −1) n ) p − z + α (i −1) n Sin−1U ( i − 2) n p − z − (1 − α (i −1) n )α ( i −1) n
hence,
2
}
n i
→ 0,
2
p = Sm p , (4.2)
2
+ M 2 p − S U ( i −1) n p + M 3hin + hin . By (2.2) and (4.2), we have
p − z ≤ (1 + uin ) (1 − α (i −1) n ) p − z 2
∞
Σ ∞n =1hin < +∞ for all
i ∈ N , 1 ≤ i ≤ k . Suppose ∩ ki=1 F ( Si ) ≠ φ is closed.
× [(1 + δ ( i − 2 ) n p − z + γ ( i − 2 ) n ] + h( i −1) n }
2
− (1 + uin ) (1 − α ( i −1) n )
Let {α in } ( i, n ∈ N , 1 ≤ i ≤ k ) be a sequence of real
2
(4.3)
× α ( i −1) n g ( S U (i − 2 ) n p − p ) n i −1
2
+ M 2 p − SinU ( i −1) n p + M 3hin + hin . If i = k , there is lim p − Si U ( i −1) n p = lim uin = n
n→∞
lim δ ( i − 2 ) n = lim γ ( i − 2 ) n = lim hin = lim u( i −1) n = 0 , n→∞
0 < b ≤ α ( i −1) n ≤ c < 1 , taking limit in (4.3), we have © 2012 ACADEMY PUBLISHER
Theorem 4.4 Let E be a real uniformly convex Banach space, C a nonempty bounded closed convex subset of E and Si : C → C (i = 1, 2, … , k ) be k uniformly L-
such that Σ n =1uin < +∞ and
+ (1 + uin ) 2 α ( i −1) n {(1 + u( i −1) n )
n →∞
= ∩∞n =1 F (Wn ) . This
∩ ki=1 F ( Si )
Lipschitzian, generalized asymptotically quasinonexpansive mappings with {uin }, {hin } ⊂ [0, ∞)
2
n →∞
m = 1, 2, …, k , i.e.,
completes the proof.
× α ( i −1) n g ( Sin−1U ( i − 2) n p − p ) n i
n → ∞.
So, we obtain p − S m p = 0 for
+ h( i −1) n ]2 }− (1 + uin ) 2 (1 − α ( i −1) n )
n →∞
all
≤ L p − S mn p + S mn +1 p − p
2
+ α ( i −1) n [(1 + u( i −1) n ) U ( i − 2) n p − z
n→∞
for
p − S m p ≤ S m p − S mn +1 p + S mn +1 p − p
≤ (1 + uin ) 2 (1 − α ( i −1) n ) p − z
2
lim n → ∞ p − S mn p = 0
lim n → ∞ p − S1n p = 0 . Since
+ M 2 p − S U ( i −1) n p + M 3hin + hin
{
get
m = 2, …, k . Repeat the above process, we get
2
× g ( S U (i − 2) n p − p ) n i −1
we
numbers with 0 < a ≤ α kn ≤ 1 , 0 < b ≤ α in ≤ c < 1 ( 1 ≤ i ≤ k − 1 ) for all n ∈ N and Wn be a modified W-mapping generated by {Si }i =1 and {α in }i =1 , {λn } a k
k
sequence of real numbers with 0 ≤ λn ≤ 1 for all
n ∈ N , Σ ∞n =1λn < +∞ and f : C → C a contraction with 0 < α < 1 . For any given x1 ∈ C , the sequence
{xn } defined by the recursion (1.6). Then {xn }
JOURNAL OF COMPUTERS, VOL. 7, NO. 6, JUNE 2012
∞
converges strongly to p ∈ ∩ n =1 F (Wn ) if and only if
lim inf n →∞ d ( xn , ∩ i =1 F ( Si )) = 0 . k
Proof: By Lemma 4.2, there exists {Si }i =1 such that k
∩ ki=1 F ( Si ) = ∩∞n =1 F (Wn ) , using Theorem 4.1, we obtain Theorem 4.4. REFERENCES [1] N. Shahzad, H. Zegeye, “Strong convergence of an implicit iteration process for a finite family of generalized asymptotically quasi-nonexpansive maps,” Appl. Math. Comp., vol. 189, pp. 1058–1065, June 2007. [2] A. R. Khan, A-A. Domlo, and H. Fukhar-ud-din, “Common fixed points Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces,” J. Math. Anal. Appl., vol. 341, pp. 1–11, 2008. [3] S. S. Chang, H. W. J. Lee, Chi Kin Chan, and J. K. Kim, “Approximating solutions of variational inequalities for asymptotically nonexpansive mappings,” Appl. Math. Comput., vol. 212, pp. 51-59, June 2009. [4] K. Nakajo, K. Shimoji, and W. Takahashi, “On strong convergence by the hybrid method for families of mappings in Hilbert spaces,” Nonlinear Anal. TMA., vol. 71, pp. 112–119, July 2009. [5] H. K. Xu, “Approximating fixed points of nonexpansive mappings by the Ishikawa iterative process,” J. Math. Anal. Appl., vol. 178, pp. 301–308, 1993. [6] K. Nakajo, K. Shimoji, and W. Takahashi, “Strong convergence theorems by the hybrid method for families of mappings in Banach spaces,” Nonlinear Anal. TMA., vol. 71, pp. 812–818, August 2009. [7] S. Temir, “On the convergence theorems of implicit iteration process for a finite family of I-asymptotically nonexpansive mappings,” J. Comp. Appl. Math., vol. 225 pp. 398–405, March 2009. [8] T. C. Lim, H. K. Xu, “Fixed point theorems for asymptotically nonexpansive mappings,” Nonlinear Anal. TMA., vol. 22, pp. 1345-1355, June 1994.
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