On the stability of finding approximate fixed points by simplicial methods

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R E S E A R C H

Open Access

On the stability of ﬁnding approximate

ﬁxed points by simplicial methods

Qi-Qing Song

1,2

_{, Hui Yang}

1*

_{, Guang-Hui Yang}

1

_{and Chong-Yi Zhong}

1

*_{Correspondence:}

[email protected] 1_{College of Science, Guizhou}

University, Guiyang, 550025, China Full list of author information is available at the end of the article

Abstract

This paper reports some new results in relation to simplicial algorithms considering continuities of approximate fixed point sets. The upper semi-continuity of a set-valued mapping of approximate fixed points using vector-valued simplicial methods is proved, and thus one obtains the existence of finite essential connected components in approximate fixed point sets by vector-valued labels; examples are given to show that this is very different from the property for integer-valued labeling simplicial methods. The existence of essential sets is also proved focusing on both perturbations of domains and functions.

Keywords: stability; approximate ﬁxed point; simplex

1 Introduction

Fixed point theorems have important effects in mathematical and economic sciences. The famous Brouwer fixed point theorem [] plays a key role in many existence problems and also has prompted a wave of finding many kinds of equilibria and other applications, such as the Nash equilibrium[], the general equilibrium [], network problems [–], approx-imation theory [], computer science [],etc.

Naturally, designing algorithms to compute a Brouwer fixed point is also important field. It is well known that Sperner’s lemma became a simple tool for the proof of the existence of Brouwer fixed points. Based on Sperner’s lemma, simplicial algorithms continue to spring up after the excellent work by Scarf [], such as Kuhn’s algorithm [, ], the restart al-gorithms [–], variable dimension alal-gorithms [], and homotopy alal-gorithms [, ]. For simplicial algorithms, one frequently finds a complete labeled sub-simplex (full labeled sub-simplex) in a simplex to approximate a fixed point. Two common labels are integer-valued and vector-integer-valued. Given a fixed grid size, it is well known that the approximate degree of a complete vector-valued sub-simplex to fixed points is better than the other. Is there any difference between the stability of these algorithms? Is a complete labeled sub-simplex able to resist the perturbation of functions or simplices? This paper will focus on these problems.

The stability of fixed points has attracted much attention. After the seminal work for es-sential fixed points of continuous functions (Brouwer type fixed points) in [], eses-sential components and essential sets of fixed points were introduced [, ]. From the view point of stability, as the analogs of singletons, minimal essential sets seem to be good choices []. Essential stabilities (which are related to lower semi-continuity) were used

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to analyze many problems, such as coincidence points [, ], ﬁxed points[], KKM points [, ], game equilibrium points [–], maximal elements [], and variational relation problems [–],etc.

In fact, when a simplicial algorithm is in order, we must face approximate fixed points as the grid size shrinks. By employing the essential stabilities, this concerns the stabil-ity of approximate fixed point sets using simplicial methods under the perturbation of the corresponding functions and domains. We show that there is a significant difference between vector-valued simplicial methods and integer-valued methods. The upper semi-continuity of a set-valued mapping for approximate fixed points using vector-valued la-beling is proved. The existence of finite essential connected components of approximate fixed point sets is also proved for vector-labeled simplicial methods. These results are new.

2 Preliminaries and motivations

LetSbe ann-simplex inRn+ with verticesv,v, . . . ,vn+,C(S) the space of continuous functionsf onSwith uniform metric andIk={, , . . . ,k}. Theith unit vector ofRn+is

denoted bye(i),i= , , . . . ,n, and the (n+ )-vector (, , . . . , )T_{is represented by}_e_.

We recall some deﬁnitions involving simplicial ﬁxed point algorithms. Given the grid size _q, the standard triangulation ofSis the collection of all sub-simplicesσ(y_,_{π) with}

verticesy, . . . ,yn+_in_S_{such that:}

(i) each component ofy_{is a multiple of} 

q;

(ii) π= (π,π, . . . ,πn)is a permutation of the elements ofIn;

(iii) yi+₌_yi₊v(πi+)–v(πi)

q , wherev(i) =vi,∀i∈In.

Note that the mesh of the standard triangulation ofSwith the grid size 

q is √

n+

q or √

n q

ifnis odd or even, respectively. For a functionf ∈C(S), a pointxinSis labeled an integer l(x)∈In+wherel(x) =iif

i=min

jfj(x) –xj= min h∈In+

fh(x) –xh

.

Particularly, iff(x) =xandx= , then assign the label ofxas the ﬁrst indexisuch that

xi> ,i∈In+. We calll:S→In+a standard integer-valued labeling function. Given the

mesh of a standard triangulation onS, from Sperner’s lemma, there exists at least one sub-simplex with complete integer labels (completely labeled simplex, with the meaning of vertices of the sub-simplex with totally diﬀerent labels).

If a pointxinSreceives the (n+ )-vectorL(x), where

L(x) = –f(x) +x+e,

in this case, we call L:S→Rn+ _{a standard vector-valued labeling function. For a}

tri-angulation of S, a sub-simplex σ(y_{, . . . ,}_yn+_{) with vector-valued labels is complete if}

n+

i=λiL(yi) =ehas a solutionλ∗= (λ,λ, . . . ,λn+) withλ∗∈Rn++.

Given a grid size _q, for eachf ∈C(S), denote byF(f,q) (F(f,q)) the collection of all sub-simplices with complete integer-valued (vector-valued) labels inS, then we deﬁne a set-valued mapping fromC(S) toSwithF(F) :C(S)→S_{, in addition, for convenience}

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be decomposed as_i_∈CiwithCi∩Cj=∅for anyi=j, andCi,∀i∈is a connected

component.

There are signiﬁcant diﬀerences in relation to semi-continuities betweenFandF, and the following example shows that the set-valued mappingFis not upper semi-continuous onC(S); further results will be demonstrated in Section .

Example . LetSbe standard simplex inR_{. A map}_f_∈_C₍_S_{) is the identity, that is,}_f₍_x_{) =}

x,∀x∈S. Given the grid size_q with_q=_, then, for the integer labels of the sub-simplices of the triangulation with the grid size _q, we have

l(x) = ⎧ ⎪ ⎨ ⎪ ⎩

, x= (, ), , x= (, ), , x= (_,_).

It can be checked thatF(f,q) ={(x,  –x)|≤x≤_}. For eachn= , , . . . , deﬁnefn∈

C(S) satisfying

fn(x,x) =

(x)

n

+n_{,  – (}_x_₎+nn_.

Then the corresponding integer labels usingfnfor eachn= , , . . . is the same as

l(x) = ⎧ ⎪ ⎨ ⎪ ⎩

, x= (, ), , x= (, ), , x= (_,_).

We can calculate thatF(fn_,_q_{) =}_{₍_x

,  –x)|_≤x≤}. Obviously, for small enough open

setUwithF(f,q)⊂U, however closefn_{is to}_f_{, we have}_F₍_fn_,_q₎_⊂_U_{. Therefore,}_F_{is not}

upper semi-continuous onC(S), hence, the graph ofFis not closed. In fact, clearly,Fis also not lower semi-continuous.

For eachf ∈C(S), denote byFix(f) the ﬁxed point set off onS. Note the fact of Exam-ple . aboutF, the following deﬁnitions consider a kind of description for stability ofF and subsets ofFix(f).

Deﬁnition . Given the grid size _q, for eachf ∈C(S), a closed subset e(f) ofF(f,q) is called an essential set with respect toC(S) if for any open setUwithU⊃e(f), there exists an open neighborhoodO(f) off inC(S) such thatF(f,q)∩U=∅,∀f∈O(f). If a connected componentC⊂F(f,q) is an essential set,Cis called an essential connected component ofF(f,q) with respect toC(S).

Deﬁnition . Letf∈C(S),e(f) be a closed subset ofFix(f). We calle(f) an approximate essential set if for eachεneighborhoodB(e(f),ε) ofe(f), there exists aδ>  such that, for eachf∈C(S) withf–f<δ, we can ﬁnd a numberZ, such thatF(f,q)∩B(e(f),ε)=∅,

∀q>Z.

Lemma .(see []) Let Y be a metric space,E be a Baire space,and F:E→Y be an

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3 Stability results under function perturbations

Theorem . Given a triangulation of S with vertices v,v, . . . ,vn+with a grid size_q,the graph of the set-valued mapping F,

GrF=(f,x)|f ∈C(S),x∈F(f,q), is closed.

Proof Let (fm_,_xm₎_∈_Gr_F_{with (}_fm_,_xm₎_→₍_f_,_x_),_m_{= , , . . . . It is clear that (}_f_,_x₎_∈

C(S)×S. We need to show thatx_{is a point of a complete sub-simplex}_σ

fwith

vector-valued labels in S. For each m= , , . . . , since (fm_,_xm₎ _∈_Gr_F_{, there exists a}

com-plete labeled sub-simplex σfm such thatx m _∈_σ

fm ⊂F(fm,q)⊂S, hence, denoteσ_fm as

σfm(y

m,ym, . . . ,ymn+) =σfm(y m,πm).

Since{π_m}belongs in the ﬁnite set In+,{πm} has a convergence subsequence{πmk },

such thatπ

mi=πmj for large enoughiandjwithi=j. For{πmk }, we can also ﬁnd such

a convergence subsequence which is denoted {π

mk} for convenience of notation. Then,

following this method, we will get a convergence subsequence{π_mki }of{πi

m}which can be

uniﬁed as one{πi}withπi=πj,∀i=j, that is,σ_fmk(y_mk,π_mk) =σ_fmk(y_mk,π). Since{y_mk} ⊂

X, there is a sequence, being its convergence subsequence, that, without loss of generality, we also denote it by{y

mk}withymk→y(k→ ∞). So far, by choosing some real numbers

pi

mk,i∈In+, we can writeσfmk(y_mk,π) as

y_mk=p_mk,p_mk, . . . ,pn_mk+/q

and

yi_mk+=yi_mk+vπi+–vπi/q, ∀i∈In.

Then we have a point yi_ such thatyi_mk →yi_∈S for each i∈In+. That is,σ(y,π) =

σ(y_,y_, . . . ,yn_+) is deﬁnitely a sub-simplex in the triangulation ofSgiven the grid size_q. Note that (fmk_,_xmk₎_∈ _Gr_F _{with (}_fmk_,_xmk₎ _→₍_f_,_x_{) as} _k _{tends to inﬁnity. Since}

σ(y

mk,ymk, . . . ,ynmk+) is a complete sub-simplex with vector-valued labels, there exists a

nonnegtive vector (λ

mk,λmk, . . . ,λnmk+) such that

n+

i=

λi_mk–fmkyi_mk+yi_mk+e=e. ()

There is a convergence subsequences{λi_mkj}of{λi_mk}withλi_mkj→λi_≥ (j→ ∞),∀i∈ In+. Then, by substitutingmkwithmkjin equation (), asj→ ∞, we have

n+

i=

λi_–fyi_+yi_+e=e. ()

Therefore,σ_f=σ(y_,y_, . . . ,yn_+) is a complete sub-simplex with vector-valued labels.

Next, we havex∈σ_f. Sincexmkj∈σ_fmkj, there existsβ_mkji ≥ such that

xmkj=

n+

i=

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withn_i_=+β_mkji = . Without loss of generality, we can assume thatβ_mkji is convergent with the limitβ_i, that is,β_mkji →β_i (j→ ∞). Then, asj→ ∞, for equation (), we havex₌

n+

i=βiyi∈σf.

From Theorem ., we have the following direct corollary.

Corollary . Given a triangulation of S with a grid size _q,the set-valued mapping Fis

upper semi-continuous on C(S).

The following example shows thatFdoes not possess the property of being lower semi-continuous onC(S).

Example . LetS,f∈C(S) be the same as Example .. With the grid size _q with_q=_, for the vector-valued labels of the sub-simplices of the triangulation withf, we have for each grid pointx= (/, /), (/, /),L(x) = (, ). Then the sub-simplexσ={(x,x)∈

S: /≤x≤/}is complete andσ⊂F(f,q). We take a pointx¯= (/, /)∈σ. For each

n= , , . . . , we deﬁnefn_∈_C₍_S_{) such that}

fn₍_x

,x) =

(x)

n+

n _{,  – (}_x )

n+

n _.

Then, for eachn= , , . . . , the vector-valued labels for the sub-simplexσusingfn_is

L(x) =

( – (

 )

n+

n , ( )

n+

n +

) = (a,c), x= (/, /),

(_– (_)nn+_{, (} )

n+

n ₊

) = (b,d), x= (/, /),

hence, the right-hand side of the equation

a b c d

–

 

=  ad–bc

d–b a–c

is the solution (λ∗_,λ∗_) of the equationsλL(/, /) +λL(/, /) =e, thenλ∗=ada––cbc. By

a straightforward calculation, for eachn= , , . . . , we havea–c=



n–

·n > , whilead–bc= (–n)n–

·n < . Thenλ ∗

< ,∀n= , , . . . . Thus, by labeling the sub-simplexσusingfn, one

sees thatσis not complete. Therefore, for a small enough open neighborhoodUofx¯, we haveF(fn,q)∩U=∅, for eachn= , , . . . , that is,Fis not lower semi-continuous onC(S).

From Theorem ., the set-valued mapping F with F :C(S)→ S _{is upper}

semi-continuous. If the set-valued mappingFis lower semi-continuous at a pointf_{, then, given}

a grid size 

q, clearly, each point inF(f

_,_q_{) is essential. Thus, by Fort’s lemma (Lemma .)}

and Deﬁnition ., we can obtain the following generic stability result.

Corollary . Given a grid size_q,there exists a dense residual set Q in C(S)such that for each f ∈C(S),each point in F(f,q)is essential with respect to C(S).

Theorem . Given a triangulation of S with a grid size 

q,for each f ∈C(S),there exist

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Proof From Theorem ., the set-valued mappingFis upper semi-continuous onS. Then the setF(f,q) itself is an essential set with respect toC(S). Let denote the collection of all essential sets inF(f,q). Note that each decreasing chain in with the set inclusion order has its intersection as a lower bound. Therefore, there exists a minimal elemente(f) in , which is an essential set inF(f,q). Hence, it is clear that each connected componentC withC⊃e(f) is an essential connected component by Deﬁnition .. Then the remaining problem is to show that eache(f) is connected.

If not, lete(f) =D_∪_D_{. Nonessential closed sets}_D _and_D _{can be separated by two}

open setsU_and_U_with_Di_⊂_Ui_,_i_{= , . For each}_i_{= ,  and}_ε_{> , there exists an open}

setWi_and_fi_∈_C₍_S_{) with}_Di_⊂_Wi_⊂_Wi_⊂_Ui_{such that}_f_–_fi_<ε

butF(fi,q)∩Wi=∅;

meanwhile, for anyf∈C(S) withf–f<ε, we haveF(f,q)∩(W_∪_W₎₌_∅_{. Construct}

a specialf∈C(S) by deﬁning

f(x) =λ(x)f(x) + –λ(x)f(x), ∀x∈S,

whereλ(x) =d(x,W)/(d(x,W) +d(x,W)). Routinely, we can check thatf–f<ε, this means that there is at least a pointxsuch thatx∈F(f,q)∩(W_∪_W_{). For each}_i_{= , ,}

ifx∈Wi_{, such that}_f₍_x_{) =}_fi₍_x_{), then the labels for the sub-simplices’ vertices in}_Wi_using

fi_or_f _{are no diﬀerent. Therefore,}_F₍_fi_,_q₎_∩_Wi₌_F₍_f_,_q₎_∩_Wi_{, from which one deduces}

the fact thatx∈/F(f,q), a contradiction.

Finally, from the ﬁniteness of the complete labeled simplex inS, the result follows.

The following result shows that essential connected components under the grid size _q can be close to an approximate ﬁxed point set asqtends to inﬁnity.

Theorem . Given a continuous function f ∈C(S),for each grid size _q,let Cq_⊂_F₍_f_,_q₎

be an essential connected component with respect to C(S),there exists a subsequence{Cqk}

of{Cq_}_{with C}qk→h _C_{and C}_{is an approximate essential connected set in}_Fix(_f_),_where

h is the Hausdorﬀ metric induced by the Euclidean metric on Rn+_.

Proof Since{Cq}is a sequence inK(S), whereK(S) is the collection of nonempty compact subsets ofS, from the compactness ofS, there is a subsequence{Cqk}_of{_Cq}_{with the limit}

C∈K(S). We denote the subsequence just as{Cq}for convenience. For eachx∈C, there is a sequence{xq}withxq∈Cqandxq→x. For eachε> , sincef is continuous, there exists a numberNsuch that, for each sub-simplexσf in the triangulation ofSunder

the grid size _N, we have

max

i∈In+

fi(x) –fi(y)<

 N <

ε

√n+ , ∀x,y∈σf.

For eachq>N, sincexq_∈_Cq_⊂_F₍_f_,_q_{), we have}_f₍_xq_{) –}_xq_<ε

. Then we can ﬁnd a large

enoughqsuch that the following inequality holds:

fx–x≤fx–fxq+fxq–xq+xq–x

<ε +

ε +

ε =ε.

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Assume thatC_{is not connected, then}_C_{can be decomposed as two disjoint compact}

sets likeC₌_C_∪_C_{with two open sets}_W_and_W_{such that}_C_⊂_W_,_C_⊂_W_{, and}

W∩W=∅. By the compactness ofCandC, there are two open setsUandUsuch that C⊂U⊂ ¯U⊂WandC⊂U⊂ ¯U⊂W. SinceCqis connected, we haveCq⊂Uor Cq_⊂_U_as_q_{large enough. Then the limit of}_Cq_{is in}_W_or_W_{, which contradicts the fact}

thatCq h→C∪CandC⊂W,C⊂W, andW∩W=∅. Therefore,Cis connected. Finally, we show thatC_{is an approximate essential set of}_Fix(_f_{). If not, then there exists}

aε¯>  andfj₍_j_{= , , . . .) with}_fj_→_f_{, such that for each number}_q_,_F₍_fj_,_q₎_∩_B₍_C_,_{ε) =}_¯ _∅_,

j= , , . . . . SinceCq_→h _C_{, there is a number}_N _{such that}_Cq_⊂_B₍_C_,_{ε) when}_¯ _q_≥_N_.

BecauseCN _{is essential, for the open set}_B₍_C_,_{ε), there is a}_δ_{>  such that for any}_f_with

f–f<δ, we haveF(f,N)∩B(C_,_ε)₌_∅_{. From the fact that}_fj_→_f_{, for large enough}_j_,

we haveF(fj_,_N₎_∩_B₍_C_,_ε)₌_∅_{, a contradiction.}

4 Stability results under perturbations of simplices and functions

In order to analyze the perturbation of domains, letX⊂Rn+ be andimensional com-pact set, M the collection of all n-simplex in X. For any two S(v,v, . . . ,vn+) and

S(v,v, . . . ,vn+) inM, deﬁne

ρ(S,S) =min_π

n+

k=

vk_–vπk  .

Lemma . ρis a metric on M.

Proof (i) For anyS(v,v, . . . ,vn+),S(v,v, . . . ,vn+)∈M, we haveρ(S,S) =ρ(S,S). Let

¯

π= (π¯,π¯, . . . ,π¯n+) matchρ(S,S). We haveρ(S,S) =

n+

k=vk–v ¯ πk  . Then

ρ(S,S) =min_π

n+

k=

vπ¯k  –v

πk  = n+ k=

vπ¯k

 –vk=ρ(S,S).

(ii) For any S(v,v, . . . ,vn+),S(v,v, . . . ,vn+)∈M, we haveρ(S,S) = ⇔S=S.

From the deﬁnition ofρ, one needs only the proof of the necessity. Letρ(S,S) = , then

there existsπ¯ such thatn_k+_=vk_–vπ¯k

 = , which means thatvk–v ¯ πk

 = ,∀k∈In+.

That is,S=S.

(iii) For any S(v,v, . . . ,vn+),S(v,v, . . . ,vn+),S(v,v, . . . ,vn+)∈M, we have ρ(S,

S)≤ρ(S,S) +ρ(S,S). Letρ(S,S) =

n+

k=vk–vπ¯k. Then we have

ρ(S,S) =min π

n+

k=

vk_–vπk

 ≤min π _n_+ k=

vk_–vπ¯k₊ n+

k=

vπ¯k_–_vπk

 = n+ k=

vk_–vπ¯k₊_min π

n+

k=

vπ¯k_–_vπk

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=ρ(S,S) +min π

k=

vk–vπk



=ρ(S,S) +ρ(S,S).

Concerning a stability analysis of approximate ﬁxed points, we intend to restrain do-mains to avoiding a domain perturbed in a large-scale range. Letbe anndimensional subset of a compact setXinRn+_{. Let}_M_⊂_M_satisfy_M₌_{_S_∈_M_:_⊂_S_⊂_X_}_.

Lemma . The metric space(M,ρ)is complete.

Proof Take a Cauchy sequence{Sm(vm,vm, . . . ,vnm+)} inM. Then, for each ε> , there

exists a numberNsuch thatρ(Ss,St) <εfor anys,t>N. Without loss of generality, we can

assume thatρ(Ss,St) =

n+

k=vks–vtk. Therefore,{vkm}is a Cauchy sequence with the limit

vk

, ∀k∈In+. Denote byS the simplexS(v,v, . . . ,vn+). Then we have ρ(Sm,S)→.

Since⊂∞_m_=Sm⊂X, it follows that⊂S⊂X, henceSis ann-simplex inM.

LetPbe the set of pairs (f,S) such that

P=(f,S)∈C(X)×M:f(x)∈S,∀x∈S.

Deﬁne the metricdbetween twou= (f,S) andu= (f,S) inMas

d(u,u) =max

x∈Xf(x) –f(x)+ρ(S,S).

Given a grid size_q, for eachu= (f,S)∈P, letT(u,q) be the set of all sub-simplices with complete vector-valued labels with the functionf in the triangulation ofSunder the grid size _q, then we deﬁne a set-valued mappingT fromPtoX.

Similar to Deﬁnition ., we consider the essential stability of approximate ﬁxed points under both perturbations of functions and domains.

Deﬁnition . Given the grid size _q, for eachu= (f,S)∈P, we call a closed subsete(f) inT(u,q) an essential set with respect toPif, for any open setUwithU⊃e(f), there is an openO(u) ofuinPsuch thatU∩T(u,q)=∅,∀u∈O(u). A minimal element in the collection (ordered by set inclusion) of essential sets inT(u,q) is called a minimal essential set with respect toP.

Theorem . Given a grid size _q and a continuous function f ∈C(X),the graph of the

set-valued mapping T,GrT={(u,x)|u∈P,x∈T(u,q)},is closed.

Proof Let{(um,xm)} ⊂GrTwith (um,xm)→(u,x), whereum= (fm,Sm),u= (f,S), and

Smis the simplex withvm,vm, . . . ,vmn+as its vertices for eachm= , , . . . . Since (um,xm)∈

GrT, there exists a complete sub-simplexσfm(ym,ym, . . . ,ynm+) with vector-valued labels

such thatxm∈σfm⊂T(um,q)⊂Sm,m= , , . . . .

Denoteσfm(ym,ym, . . . ,ymn+) asσfm(ym,πm). Similar to Theorem ., there exists a

sub-sequence{mk}of{m}and a permutationπsuch thatσfmk(y 

mk,πmk) =σfmk(y 

mk,π). There

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(k→ ∞). So far, for eachmk, by choosing some real numberspi_mk(i∈In+) withpimk→pi

(k→ ∞), the sub-simplexσfmk(y 

mk,π) can be written as

y_mk=p_mk,p_mk, . . . ,pn_mk+/q

and

yi_mk+=yi_mk+vπ_mki+–vπ_mki/q, ∀i∈In.

Sinceum d

→u, which means thatSm

ρ

→S∈M, then, by the deﬁnition ofρ, we have

vπi

mk→vπ i

,∀i∈In+. Noting thatymk →y, we haveσfmk(y 

mk,π)

ρ

→σ(y_,π) ask→ ∞. Clearly,σf(y,π) is a simplex in the triangulation ofS(v,v, . . . ,vn+) with the grid sizeq.

To ﬁnish the proof thatx∈σ(y,π) andσ(y,π) is a complete sub-simplex with

vector-valued labels by functionf, we can adopt the corresponding part of Theorem ..

From Theorem ., T is upper semi-continuous onP. Following the proof of Theo-rem . for the part of the existence of minimal element of essential sets, we obtain the following result.

Theorem . For each u= (f,S)∈P,given a triangulation of S with a grid size 

q,there

exists a minimal essential set in T(u,q)with respect to P.

Remark . Theorems ., . and Theorem . obtain the stability for approximate ﬁxed

points with simplicial methods. From the point of applications, this may facilitate the sta-bility analysis of equilibrium problems, such as Nash equilibria andε-approximate Nash equilibria.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the manuscript.

Author details

1_{College of Science, Guizhou University, Guiyang, 550025, China.}2_{College of Science, Guilin University of Technology,}

Guilin, 541004, China.

Acknowledgements

This project is supported by the National Natural Science Foundation of China (11271098, 11661030), the China Postdoctoral Science Foundation (2016M590905), and the Doctoral Research Fund of Guilin University of Technology.

Received: 24 June 2016 Accepted: 4 September 2016

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