Strong convergence theorems for asymptotically nonexpansive

Guo et al. Fixed Point Theory and Applications (2015) 2015:212 DOI 10.1186/s13663-015-0463-6

RESEARCH

Open Access

Strong convergence theorems for asymptotically nonexpansive nonself-mappings with applications Weiping Guo1 , Afrah AN Abdou2* , Liaqat A Khan2 and Yeol Je Cho2,3* *

Correspondence: [email protected]; [email protected] 2 Department of Mathematics, King Abdulaziz University, Jeddah, 21589, Saudi Arabia 3 Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju, 660-701, Korea Full list of author information is available at the end of the article

Abstract In this paper, first, we introduce the condition (BP) which is weaker than the completely continuous mapping in Banach spaces. Second, we consider a simple iteration and prove some strong convergence theorems of the proposed iteration for an asymptotically nonexpansive nonself-mapping with the condition (BP). Finally, we give two examples to illustrate the main result in this paper. Our results improve and extend the corresponding results given by some authors. Keywords: asymptotically nonexpansive nonself-mapping; strong convergence; fixed point; uniformly convex Banach space

1 Introduction In , Goebel and Kirk [] introduced the class of asymptotically nonexpansive selfmappings. Let C be a nonempty subset of a real normed linear space E. A mapping T : C → C is said to be asymptotically nonexpansive if there exists a sequence {kn } ⊂ [, ∞) with limn→∞ kn =  such that n T x – T n y ≤ kn x – y for all x, y ∈ C and n ≥ , and one proved the following. Theorem GK If C is a nonempty closed convex subset of a real uniformly convex Banach space E and T : C → C is an asymptotically nonexpansive self-mapping, then T has a fixed point in C. On the other hand, in , Schu [] introduced the modified Mann process to approximate fixed points of an asymptotically nonexpansive self-mapping defined on a nonempty closed convex and bounded subset C of a Hilbert space H as follows: Theorem JS Let H be a Hilbert space and C be a nonempty closed convex bounded subset of H. Let T : C → C be a completely continuous and asymptotically nonexpansive mapping  with the sequence {kn } ⊂ [, ∞) with limn→∞ kn =  and ∞ n= (kn – ) < ∞. Let {αn } be a real sequence in [, ] satisfying the condition ε ≤ αn ≤  – ε for all n ≥  and for some ε > . © 2015 Guo et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Guo et al. Fixed Point Theory and Applications (2015) 2015:212

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Then the sequence {xn } generated by ⎧ ⎨x ∈ C arbitrarily,  ⎩xn+ = ( – αn )xn + αn T n xn

(JS)

for each n ≥  converges strongly to some fixed point of T. In Theorem JS, the mapping T remains a self-mapping of a nonempty closed convex subset C of a Hilbert space H. If, however, the domain D(T) of T is a proper subset of H and T : D(T) → H is a mapping, the modified iteration {xn } defined by (JS) may fail to be well defined. To overcome this situation, in , Chidume et al. [] introduced the concept of asymptotically nonexpansive nonself-mappings. Let E be a real Banach space. A subset C of E is called a retract of E if there exists a continuous mapping P : E → K such that Px = x for all x ∈ C. Every closed convex subset of a uniformly convex Banach space is a retract. A mapping P : E → E is called a retraction if P = P. It follows that, if a mapping P is a retraction, then Py = y for all y in the range of P (see []). Definition . Let C be a nonempty subset of a real normed linear space E. Let P : E → C be a nonexpansive retraction of E onto C. A nonself-mapping T : C → E is said to be asymptotically nonexpansive if there exists a sequence {kn } ⊂ [, ∞) with limn→∞ kn =  such that T(PT)n– x – T(PT)n– y ≤ kn x – y

(NSM)

for all x, y ∈ C and n ≥ . Also, Chidume et al. [] introduced the following iteration scheme: x ∈ C arbitrarily, xn+ = P(( – αn )xn + αn T(PT)n– xn )

(.)

for all n ≥ , where {αn } ⊂ (, ), and proved some strong and weak convergence theorems for asymptotically nonexpansive nonself-mappings. We denote the set of fixed points of T by F(T) = {x ∈ C : Tx = x}. Remark . If a mapping T : C → C is a self-mapping, then P becomes the identity mapping and so we have the following: () The nonself-mapping with (NSM) coincides with an asymptotically nonexpansive self-mapping. () The iteration defined by (.) coincides with the iteration defined by (JS). Since the results of Chidume et al., some authors proved weak and strong convergence theorems for asymptotically nonexpansive nonself-mappings in Banach spaces (see [–]). Let E be a real Banach space, C be a nonempty closed convex subset of E and P : E → C be a nonexpansive retraction of E onto E. Let T : C → E be an asymptotically nonexpan-


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sive nonself-mapping. Now, we define the iterative scheme {xn } as follows: ⎧ ⎪ ⎨x ∈ C arbitrarily, xn+ = P(( – αn )xn + αn T(PT)n– yn ), ⎪ ⎩ yn = P(( – βn )xn + βn T(PT)n– xn )

(.)

for all n ≥ , where {αn } ⊂ (, ) and {βn } ⊂ [, ]. In this paper, first, we introduce the condition (BP) which is weaker that the completely continuous mapping. Second, we introduce a new iteration (.) and prove some strong convergence theorems of the proposed iteration for an asymptotically nonexpansive nonself-mapping T : C → E with the condition (BP). Finally, we give two examples to illustrate the main result in this paper. Our results improve and extend the corresponding results given by some authors.

2 Some lemmas For our main results, we need the following lemmas. Lemma . ([]) Let p >  and R >  be two fixed numbers and X be a Banach space. Then X is uniformly convex if and only if there exists a continuous, strictly increasing and convex function g : [, ∞) → [, ∞) with g() =  such that

λx + ( – λ)yp ≤ λxp + ( – λ)yp – ωp (λ)g x – y for all x, y ∈ B(, R) = {x ∈ E : x ≤ R} and λ ∈ [, ], where ωp (λ) = λ( – λ)p + λp ( – λ). Letting S = S = I, where I denotes the identity mapping, and T = T = T in Lemma . of [], we have the following. Lemma . Let E be a real uniformly convex Banach space and C be a nonempty closed convex subset of E. Let T : C → E be an asymptotically nonexpansive nonself-mapping with {hn } ⊂ [, ∞) such that ∞ n= (hn – ) < ∞ and F(T) = ∅. Let {xn } be the sequence defined by (.), where {αn } ⊂ (, ) and {βn } ⊂ [, ). Then: () xn+ – p ≤ hn xn – p for all p ∈ F(T); () limn→∞ xn – p exists for all p ∈ F(T). Lemma . Let E be a real uniformly convex Banach space and C be a nonempty closed convex subset of E. Let T : C → E be an asymptotically nonexpansive nonself-mapping with {hn } ⊂ [, ∞) such that ∞ n= (hn – ) < ∞ and F(T) = ∅. Let {xn } be the sequence defined by (.), where  < lim inf αn , n→∞

lim sup αn < , n→∞

and lim sup βn < . n→∞

Then limn→∞ xn – Txn = .


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Proof By Lemma ., we know that limn→∞ xn – p exists for any p ∈ F(T). It follows that {xn – p}, {yn – p}, {T(PT)n– xn – p}, {T(PT)n– yn – p} are all bounded, so there exists a real number R >  such that xn – p, yn – p, T(PT)n– xn – p, T(PT)n– yn – p ⊂ B(, R) for all n ≥ . It follows from (.) and Lemma . that

 yn – p ≤ ( – βn )(xn – p) + βn T(PT)n– xn – p  ≤ ( – βn )xn – p + βn T(PT)n– xn – p

– βn ( – βn )g xn – T(PT)n– xn ≤ ( – βn )hn xn – p + βn hn xn – p = hn xn – p and so

 xn+ – p ≤ ( – αn )(xn – p) + αn T(PT)n– yn – p  ≤ ( – αn )xn – p + αn T(PT)n– yn – p

– αn ( – αn )g xn – T(PT)n– yn

≤ ( – αn )hn xn – p + αn hn yn – p – αn ( – αn )g xn – T(PT)n– yn

≤ hn xn – p – αn ( – αn )g xn – T(PT)n– yn ,

where g : [, ∞) → [, ∞) is a continuous strictly increasing and convex function with g() = . By the conditions  < lim infn→∞ αn and lim supn→∞ αn < , we know that there exist a positive integer n and two real numbers a, b ∈ (, ) such that a ≤ αn ≤ b for all n ≥ n , thus

a( – b)g xn – T(PT)n– yn ≤ hn xn – p – xn+ – p ,

∀n ≥ n .

Since limn→∞ xn – q exists, it follows from limn→∞ hn =  that

lim g xn – T(PT)n– yn = .

n→∞

Using the properties of g, we have lim xn – T(PT)n– yn = .

n→∞

(.)

Since we have the conditions lim supn→∞ βn <  and limn→∞ hn = , there exist a positive integer m and a real number s ∈ (, ) such that βn hn ≤ s for all n ≥ m and yn – xn ≤ βn T(PT)n– xn – xn ≤ βn T(PT)n– xn – T(PT)n– yn + βn xn – T(PT)n– yn ≤ βn hn xn – yn + xn – T(PT)n– yn .


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Hence ( – s)yn – xn ≤ ( – βn hn )yn – xn ≤ xn – T(PT)n– yn for all n ≥ m . From (.), we have lim yn – xn = .

n→∞

(.)

Furthermore, from (.), (.), and xn – T(PT)n– xn ≤ xn – T(PT)n– yn + T(PT)n– yn – T(PT)n– xn ≤ xn – T(PT)n– yn + hn yn – xn , it follows that lim xn – T(PT)n– xn = .

n→∞

(.)

It follows from (.) and xn+ – T(PT)n– yn ≤ xn+ – xn + xn – T(PT)n– yn ≤ (αn + )xn – T(PT)n– yn that lim xn+ – T(PT)n– yn = .

n→∞

(.)

Since xn+ – yn ≤ xn+ – T(PT)n– yn + T(PT)n– yn – xn + xn – yn , we have lim xn+ – yn = 

n→∞

(.)

by (.), (.) and (.). It follows from (.), (.), (.), and xn – Txn ≤ xn – T(PT)n– xn + T(PT)n– xn – Txn ≤ xn – T(PT)n– xn + T(PT)n– xn – T(PT)n– yn– + T(PT)n– yn– – Txn ≤ xn – T(PT)n– xn + hn xn – yn– + h T(PT)n– yn– – xn that limn→∞ xn – Txn = . This completes the proof.

3 The condition (BP) Let E be a Banach space and T : E → E be a bounded linear operator. In , Browder and Petryshyn [] considered the existence of a solution of the equation u – Tu = f

(EQ)

by the iteration of Picard-Poincaré-Neumann, ⎧ ⎨x ∈ E arbitrarily,  ⎩xn+ = Txn + f ,

(BP)


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equivalently, ⎧ ⎨x ∈ E arbitrarily,  ⎩xn = T n x + (f + Tf + · · · + T n– f ), for each n ≥  and f ∈ E. In fact, in , Browder [] proved the following. Theorem B Let E be a reflexive Banach space. Then a solution u of the equation u – Tu = f exists for a given point f ∈ E and an operator T which is asymptotically bounded (i.e., there exists M ≥  such that T n x ≤ M for all x ∈ E and n ≥ ) if and only if the sequence {xn } defined by (BP) is bounded for any fixed x ∈ E. But, without any assumption of the reflexivity on E, under a slight sharper condition on T, Browder and Petryshyn proved the following: Theorem BP Let E be a Banach space and T : E → E be a bounded linear operator which is asymptotically convergent, i.e., {T n x} converges in E for all x ∈ E. Then we have the following: () If f ∈ R(I – T), the sequence {xn } defined by (BP) converges to a solution u of the equation u + Tu = f . () If any subsequence {xni } of the sequence {xn } converges to an element y ∈ E, then y is a solution of the equation y – Ty = f . () If E is a reflexive Banach space and the sequence {xn } is bounded, then the sequence {xn } converges to a solution of the equation u + Tu = f . Motivated by Theorem BP, we have the concept of the condition (BP) as follows: Let E be a real normed linear space, C be a nonempty subset of E and T : C → E be a mapping. Definition . The pair (T, C) is said to satisfy the condition (BP) if, for any bounded closed subset G of C, {z : z = x – Tx, x ∈ G} is a closed subset of E. Let E and F be Banach spaces. Recall that a mapping T : E → F is completely continuous if it is continuous and compact (i.e., C is bounded implies that T(C) is relatively compact, i.e., T(C) is compact) or a weakly convergent sequence (xn → x weakly) implies a strongly convergent sequence (Txn → Tx). We give some relations between the condition (BP) and a completely continuous mapping as follows. Proposition . Let E be a real normed linear space, C be a nonempty subset of E and T : C → E be a completely continuous mapping. Then the pair (T, C) satisfies the condition (BP). Proof For any bounded closed subset G of C, we denote M = {z : z = x – Tx, x ∈ G}. For any zn ∈ M with zn → z, there exists xn ∈ G such that zn = xn – Txn . Since T is completely continuous and the sequence {xn } is bounded, there exists a subsequence {xnk } of {xn } such


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that {Txnk } is convergent. Letting Txnk → x , it follows that xnk = znk + Txnk → z + x as k → ∞. Since G is closed, it follows that z + x ∈ G. Furthermore, since T is a completely continuous mapping, we have z = lim znk = lim (xnk – Txnk ) = (z + x ) – T(z + x ). k→∞

k→∞

It follows from z + x ∈ G that z ∈ M and so M is closed. This shows that the pair (T, C) satisfies the condition (BP) and so Proposition . holds. This completes the proof. Remark . If the pair (T, C) satisfies the condition (BP), then T is not completely continuous in general. Example . Let E = (–∞, ∞) with the usual norm | · | and C = {, ,  , . . .}. Define a mapping T : C → E by , if x = , Tx = n, if x = n , n ≥ . Then, for any bounded closed subset G of C, the set {z : z = x – Tx, x ∈ G}d = ∅, where Ad denotes the set of accumulation points of A, and so {z : z = x – Tx, x ∈ G} is a closed subset of E. This shows that the pair (T, C) satisfies the condition (BP), but the set T(C) = {, , , . . . , n, . . .} is unbounded. So, T is not completely continuous.

4 Strong convergence theorems Now, we prove strong convergence theorems for asymptotically nonexpansive nonselfmappings with the condition (BP) in real uniformly convex Banach spaces. Theorem . Let E be a real uniformly convex Banach space and C be a nonempty closed convex subset of E. Let T : C → E be an asymptotically nonexpansive nonself-mapping with {hn } ⊂ [, ∞) such that ∞ n= (hn – ) < ∞ and F(T) = ∅. Let {xn } be the sequence defined by ⎧ ⎪ ⎨x ∈ C arbitrarily, xn+ = P(( – αn )xn + αn T(PT)n– yn ), ⎪ ⎩ yn = P(( – βn )xn + βn T(PT)n– xn )

(.)

for all n ≥ , where  < lim inf αn , n→∞



If the pair (T, C) satisfies the condition (BP), then the sequence {xn } converges strongly to a fixed point of T. Proof Letting G = {xn }, where {xn } denotes the closure of {xn }, since the sequence {xn } is bounded in C by Lemma .() and so G is a bounded closed subset of C. Since the


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pair (T, C) satisfies the condition (BP), it follows that M = {z = x – Tx : x ∈ G} is closed. From {xn – Txn } ⊂ M and xn – Txn →  as n → ∞ by Lemma ., we know that the zero vector  ∈ M and so there exists a q ∈ G such that q = Tq. This shows that q is a fixed point of T. Since q ∈ G, there exists a positive integer n such that xn = q or there exists a subsequence {xnk } of {xn } such that xnk → q as k → ∞. If xn = q, then it follows from Lemma .() that xn = q for all n ≥ n and so xn → q as n → ∞. If xnk → q, then, since limn→∞ xn – q exists, we have xn → q as n → ∞. This completes the proof. Using Theorem . and Proposition ., we have the following. Corollary . Let E be a real uniformly convex Banach space and C be a nonempty closed convex subset of E. Let T : C → E be an asymptotically nonexpansive nonself-mapping with {hn } ⊂ [, ∞) such that ∞ n= (hn – ) < ∞ and F(T) = ∅. Let {xn } be the sequence defined by (.), where  < lim inf αn , n→∞



If T is completely continuous, then the sequence {xn } converges strongly to a fixed point of T. Letting βn =  for all n ≥  in Theorem ., we have the following. Theorem . Let E be a real uniformly convex Banach space and C be a nonempty closed convex subset of E. Let T : C → E be an asymptotically nonexpansive nonself-mapping with {hn } ⊂ [, ∞) such that ∞ n= (hn – ) < ∞ and F(T) = ∅. Let {xn } be the sequence defined by (.), where  < lim inf αn , n→∞

lim sup αn < . n→∞

If the pair (T, C) satisfies the condition (BP), then the sequence {xn } converges strongly to a fixed point of T. Using Theorem . and Proposition ., we obtain the following. Corollary . Let E be a real uniformly convex Banach space and C be a nonempty closed convex subset of E. Let T : C → E be an asymptotically nonexpansive nonself-mapping with {hn } ⊂ [, ∞) such that ∞ n= (hn – ) < ∞ and F(T) = ∅. Let {xn } be the sequence defined by (.), where  < lim inf αn , n→∞

lim sup αn < . n→∞

If T is completely continuous, then the sequence {xn } converges strongly to a fixed point of T.


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Remark . Theorem . and Corollary . improve and extend Theorem . in [] and extend Theorem . in [], Theorem . in [] and the corresponding results in [] to the setting of Banach spaces and the more general class of nonself-mappings.

5 Examples Now, we give two examples to illustrate Theorem . as follows. Proposition . ([]) Let X be a Hilbert space and C = {x ∈ X : x ≤ r}, where r > . Let P : X → C be a mapping defined by Px =

x, rx , x

if x ∈ C, if x ∈ X – C.

Then P is a nonexpansive retraction of X onto C. Example . Let X = Rn with the inner product x, y = ni= xi yi and the usual norm x = n   ( i= xi )  , then X is a Hilbert space. Let C = {x ∈ X : x ≤ }. Define the mapping P : X → C by Px =

x, x , x

if x ∈ C, if x ∈ X – C.

Then P is nonexpansive retraction of X onto C by Proposition .. Define a mapping T : C → X by Tx = ( – x , , . . . , ) for all x = (x , x , . . . , xn ) ∈ C. Then we have Tx – Ty = (y – x , , . . . , ) ≤ x – y for all x = (x , x , . . . , xn ), y = (y , y , . . . , yn ) ∈ C, which shows that T is a nonexpansive nonself-mapping and so T(PT)– x – T(PT)– y ≤ P(Tx) – P(Ty) ≤ Tx – Ty ≤ x – y. Suppose that T(PT)k– x – T(PT)k– y ≤ x – y for n = k. Then, when n = k + , we have T(PT)(k+)– x – T(PT)(k+)– y ≤ (PT)(k+)– x – (PT)(k+)– y

= P T(PT)k– x – P T(PT)k– y ≤ T(PT)k– x – T(PT)k– y ≤ x – y.


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It follows from the mathematical induction that T is an asymptotically nonexpansive nonself-mapping with the sequence {hn } defined by hn =  for each ≥ and F(T) = {(  , , . . . , )}. Now, we prove that the pair (T, C) satisfies the condition (BP). For any closed subset G of C, we denote M = {z = x – Tx : x ∈ G}. Then M is closed. Indeed, for any zn ∈ M with zn → z, there exists xn ∈ G such that zn = xn – Txn . Since G is bounded closed in C and so G is compact. Therefore, there exists a convergence subsequence {xnk } of {xn }. Letting xnk → x as k → ∞, we have x ∈ G and it follows from the continuous property of T that z = lim znk = lim (xnk – Txnk ) = x – Tx ∈ M. k→∞

k→∞

For any given x ∈ C, define a sequence {xn } by xn+ = P(( – αn )xn + αn T(PT)n– yn ), yn = P(( – βn )xn + βn T(PT)n– xn )  for each n ≥ , {αn } is defined by for all n ≥ , where {αn– } is defined by αn– =  + n   n αn =  + n and {βn } is defined by βn = n+ for each n ≥ . It is easy to prove

lim inf αn = n→∞

 , 

lim sup αn = n→∞

 , 

and  lim sup βn = .  n→∞ Hence all the conditions of Theorem . are satisfied and so {xn } converges strongly to the fixed point (  , , . . . , ) of T. Example . Let X = l with the inner product x, y = ∞ i= xi yi and the norm x =   ( ∞ x ) . Then X is a real infinite dimensional Hilbert space. Let C = {x ∈ X : x ≤ }. i= i Define the mapping P : X → C by Px =

if x ∈ C, if x ∈ X – C.

x, x , x

Then P is a nonexpansive retraction of X onto C. Define a mapping T : C → X by Tx = (–x , –x , . . . , –xi , . . .) for all x = (x , x , . . . , xi , . . .) ∈ C. Then we have Tx – Ty = (y – x , y – x , . . . , yi – xi , . . .) =

∞

 (yi – xi )

i=

= x – y




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for all x = (x , x , . . . , xi , . . .), y = (y , y , . . . , yi , . . .) ∈ C and so T is nonexpansive and, using the same method as given in Example ., we can prove that T is an asymptotically nonexpansive mapping with the sequence {hn } defined by hn =  for each n ≥  and F(T) = {(, , . . .)}. Now, we prove that the pair (T, C) satisfies the condition (BP) and T is not completely continuous. In fact, for any closed subset G of C, we denote M = {z = x – Tx : x ∈ G}. For (n) (n) any z(n) ∈ M with z(n) → z as n → ∞, there exists x(n) = (x(n)  , x , . . . , xi , . . .) ∈ G such that z(n) = x(n) – Tx(n) = x(n) . It follows from z(n) → z that x(n) →  z as n → ∞. Since G is closed in C, it follows that  z ∈ G. From the continuous property of T, it follows that z = lim z

(n)

n→∞

= lim x n→∞

(n)

– Tx

(n)

  = z–T z ∈ M.  

This shows that the pair (T, C) satisfies the condition (BP). Since T(C) = C and the unit ball C in infinite dimensional Hilbert space is not sequential compact, it follows that T is not completely continuous. For any given x() ∈ C, define a sequence {x(n) } by x(n+) = P(( – αn )x(n) + αn T(PT)n– y(n) ), y(n) = P(( – βn )x(n) + βn T(PT)n– x(n) )  for all n ≥ , where {αn– } is defined by αn– =  + n for each n ≥ , {αn } is defined by   n αn =  + n and {βn } is defined by βn = n+ for each n ≥ . It is easy to prove

 lim inf αn = , n→∞ 

lim sup αn = n→∞

 

and lim sup βn = n→∞

 . 

Hence all the conditions of Theorem . are satisfied and so {x(n) } converges strongly to the fixed point (, , . . .) of T. Since T is not completely continuous, the above conclusions cannot be obtained by Theorem . in [].

Competing interests The authors declare that they have no competing interest. Authors’ contributions All authors read and approved the final manuscript. Author details 1 School of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou, Jiangsu 215009, P.R. China. 2 Department of Mathematics, King Abdulaziz University, Jeddah, 21589, Saudi Arabia. 3 Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju, 660-701, Korea. Acknowledgements This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under grant no. (18-130-36-HiCi). The authors, therefore, acknowledge with thanks DSR technical and financial support. Also, this project (WP Guo) was supported by the National Natural Science Foundation of China (Grant Number: 11271282) and the third author (YJ Cho) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning (2014R1A2A2A01002100). Received: 2 May 2015 Accepted: 6 November 2015


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