We provide some results on the convergence of the sequence of fixed points for some different sequences of contraction mappings or fuzzy metrics in G-complete fuzzy metric spaces with the H-type t-norms. We improve the
corresponding conclusions in the literature.

1. Introduction

In 1940, Ulam [1] proposed the following stability problem: “When is it true that a function which satisfies some functional equation approximately must be close to one satisfying the equation exactly?” This problem posed by Ulam has stimulated a long lasting interest in a particular stability of various types of equations and inequalities. That issue is sometimes called Ulam's type stability. In the recent past, several Ulam stability results concerning the various functional equations were determined in [2–4], respectively, in the fuzzy and intuitionistic fuzzy normed spaces.

The concept of fuzzy sets was introduced initially by Zadeh [5] in 1965. After that, to use this concept in topology and analysis many authors have expansively developed the theory of fuzzy sets and applications. In the theory of fuzzy topological spaces, one of the main problems is to obtain an appropriate and consistent notion of a fuzzy metric space. This problem was investigated by many authors from different points of view [6–16]. Kramosil and Michalek [7] gave a notion of fuzzy metric space which could be considered as a reformulation, in the fuzzy context, of the notion of probabilistic metric space due to Menger [17]. Later, George and Veeramani [18] introduced and studied a notion of fuzzy metric space which constitutes a modification of the one due to Kramosil and Michalek.

These fuzzy metric spaces have been widely accepted as an appropriate notion of metric fuzziness in the sense that it provides rich topological structures which can be obtained, in many cases, from classical theorems. Further, these fuzzy metric spaces have very important applications in studying fixed point theorems [17, 19–28]. In [29], Shen et al. have shown that the convergence of the sequence of fixed points to some sequences of contraction mappings or fuzzy metrics satisfies certain conditions in fuzzy metric spaces. However, they have given the most of results in [29] only for the two specific t-norms, minimum t-norm and product t-norm.

In this paper, we will provide some improved results on convergence of fixed points in G-complete fuzzy metric spaces. Specifically, after redefining some basic concepts of [29] in Section 2, we will generalize the results in [29] to the G-complete fuzzy metric spaces with an H-type t-norm in Section 3. Since the H-type t-norms are very important and widely used in fuzzy fixed point theory [17, 20, 24, 30, 31], our results improve the corresponding conclusions in the literature.

2. Preliminaries

For the sake of completeness, we briefly recall some notions from the theory of fuzzy metric spaces used in this paper. Let N denote the set of all positive integers.

Definition 1 (see [<xref ref-type="bibr" rid="B18">32</xref>]).

A triangular norm (t-norm for short) is a binary operation on the unit interval [0,1], that is, a function *:[0,1]2→[0,1], such that for all a,b,c,d∈[0,1] the following four axioms are satisfied:

a*1=a (boundary condition).

a*b≤c*d whenever a≤c and b≤d (monotonicity).

a*b=b*a (commutativity).

a*(b*c)=(a*b)*c (associativity).

A t-norm * is said to be continuous if it is a continuous function in [0,1]2. Examples of t-norms are a*Pb=ab and a*Mb=min{a,b}.

Definition 2 (see [<xref ref-type="bibr" rid="B21">30</xref>]).

A t-norm * is of H-type if the family (x(n))n∈N is equicontinuous at the point x=1, where x(n) is defined by
(1)x(1)=x,x(n)=x(n-1)*x,n≥2,x∈[0,1].
It is obvious that *M is an H-type t-norm. In fact, there are innumerable H-type t-norms [30].

Lemma 3 (see [<xref ref-type="bibr" rid="B29">31</xref>]).

Let * be a t-norm. If * is continuous and of H-type, then there exists a strictly increasing sequence (bn)n∈N from the interval [0,1) such that limn→∞bn=1 and bn*bn=bn.

In [29], Shen et al. redefined the notion of fuzzy metric space by appending the following condition (FM-6) based on the one in the sense of George and Veeramani [18].

Definition 4 (see [<xref ref-type="bibr" rid="B32">33</xref>]).

The 3-tuple (X,M,*) is said to be a fuzzy metric space if X is an arbitrary nonempty set, * is a continuous t-norm, and M is a fuzzy set on X2×(0,∞) satisfying the following conditions, for all x,y,z∈X,t,s>0:

M(x,y,t)>0,

M(x,y,t)=1 if and only if x=y,

M(x,y,t)=M(y,x,t),

M(x,z,t+s)≥M(x,y,t)*M(y,z,s),

M(x,y,·):(0,∞)→(0,1] is continuous,

limt→∞M(x,y,t)=1.

Definition 5 (see [<xref ref-type="bibr" rid="B10">7</xref>, <xref ref-type="bibr" rid="B34">29</xref>]).

Let (X,M,*) be a fuzzy metric space. Then

a sequence (xn) in X is said to converge to x in X if and only if limn→∞M(xn,x,t)=1 for all t>0; that is, for each r∈(0,1) and t>0, there exists an n0∈N such that M(xn,x,t)>1-r for all n≥n0;

a sequence (xn) in X is a G-Cauchy sequence if and only if limn→∞M(xn,xn+p,t)=1 for any p∈N and t>0;

the fuzzy metric space (X,M,*) is called G-complete if every G-Cauchy sequence is convergent.

Definition 6 (see [<xref ref-type="bibr" rid="B10">7</xref>]).

Let (X,M,*) be a fuzzy metric space. A mapping T:X→X is called a contraction mapping if there exists k∈(0,1) such that
(2)M(Tx,Ty,kt)≥M(x,y,t)
for every x,y∈X and t>0.

Lemma 7.

Let (X,M,*) be a G-complete fuzzy metric space. If T:X→X is a contraction mapping, then T has a unique fixed point.

Definition 8.

Let (X,M,*) be a fuzzy metric space and let (Tn) be a sequence of self-mappings of X. T0:X→X is a given mapping. The sequence (Tn) is said to converge pointwise to T0 if Tnx converge to T0x for each x∈X; that is, for any x∈X, r∈(0,1), and t>0, there exists an n0∈N such that
(3)M(Tnx,T0x,t)>1-r
for all n≥n0.

Definition 9 (see [<xref ref-type="bibr" rid="B34">29</xref>]).

Let (X,M,*) be a fuzzy metric space and let (Tn) be a sequence of self-mappings of X. T0:X→X is a given mapping. The sequence (Tn) is said to converge uniformly to T0 if for each r∈(0,1) and t>0, there exists an n0∈N such that
(4)M(Tnx,T0x,t)>1-r
for all n≥n0 and x∈X.

Definition 10.

Let (X,M,*) be a fuzzy metric space. A sequence of self-mappings (Tn) is uniformly equicontinuous if for each r∈(0,1) and t>0, there exists an ε∈(0,1) such that M(Tnx,Tny,t)>1-r whenever M(x,y,t)>1-ε for all x,y∈X and n∈N.

Definition 11 (see [<xref ref-type="bibr" rid="B7">18</xref>]).

Let (X,M,*) be a fuzzy metric space. The open ball BM(x,r,t) and the closed ball BM[x,r,t] with center x∈X and radius r∈(0,1) and t>0, respectively, are defined as follows:
(5)BM(x,r,t)={y∈X:M(x,y,t)>1-r},BM[x,r,t]={y∈X:M(x,y,t)≥1-r}.

Lemma 12 (see [<xref ref-type="bibr" rid="B7">18</xref>]).

Every open (closed) ball is an open (closed) set.

Definition 13.

A fuzzy metric space (X,M,*) is a compact space if (X,τM) is a compact topological space, where τM is a topology induced by the fuzzy metric M.

Lemma 14 (see [<xref ref-type="bibr" rid="B32">33</xref>]).

Every closed subset A of a compact fuzzy metric space (X,M,*) is compact.

Definition 15 (see [<xref ref-type="bibr" rid="B32">33</xref>]).

A fuzzy metric space (X,M,*) in which every point has a compact neighborhood is called locally compact.

Definition 16 (see [<xref ref-type="bibr" rid="B26">34</xref>, <xref ref-type="bibr" rid="B36">35</xref>]).

Let * be a t-norm and a,b∈[0,1]. Define the pseudodifference a⊖b by
(6)a⊖b=sup{c:b*c≤a}.

Definition 17.

Let (X,M0,*) be a fuzzy metric space and let (Mn) be a sequence of fuzzy metrics with respect to the same t-norm * on X. The sequence (Mn) is said to upper semiconverge uniformly to M0 if for each r∈(0,1) and t>0, there exists an n0∈N such that
(7)Mn(x,y,t)≥M0(x,y,t),M0(x,y,t)⊖Mn(x,y,t)>1-r
for all n≥n0,x,y∈X.

3. Main ResultsTheorem 18.

Let (X,M,*) be a fuzzy metric space and let (Tn) be a sequence of self-mappings of X. T0:X→X is a contraction mapping; that is, there exists a k∈(0,1) such that M(T0x,T0y,kt)≥M(x,y,t) for every x,y∈X and t>0. A is a compact subset of X. If (Tn) converges pointwise to T0 in A and it is a uniformly equicontinuous sequence, then the sequence (Tn) converges uniformly to T0 in A.

Proof.

For each r-∈(0,1), we may choose an appropriate r∈(0,1) such that
(8)(1-r)*(1-r)*(1-r)>1-r-.
At the same time, for any fixed t>0, since (Tn) is uniformly equicontinuous, there exists ε∈(0,r) such that
(9)M(Tnx,Tny,t2+k)>1-r,
whenever M(x,y,t/(2+k))>1-ε, for all x,y∈X and n∈N. Now define the open covering C of A as
(10)C={BM(x,ε,t2+k):x∈A}.
Since A is compact, there exist xi∈A, i=1,2,…,m such that
(11)A⊆⋃i=1mBM(xi,ε,t2+k).
Since Tn converges pointwise to T0 in A, for each xi, there exists ni∈N such that
(12)M(Tnxi,T0xi,t2+k)>1-r
for all n≥ni. Set l=max{ni,i=1,2,…,m}. For any x∈A, there exists an xi0 such that x∈B(xi0,ε,t/(2+k)). Now for all n≥l, we have that
(13)M(Tnx,T0x,t)≥M(Tnx,Tnxi0,t2+k)*M(Tnxi0,T0xi0,t2+k)*M(T0xi0,T0x,kt2+k)≥(1-r)*(1-r)*M(xi0,x,t2+k)≥(1-r)*(1-r)*(1-ε)≥(1-r)*(1-r)*(1-r)≥1-r-.
By the arbitrariness of r- and t, we get that the sequence (Tn) converges uniformly to T0 in A.

Theorem 19.

Let (X,M,*) be a G-complete fuzzy metric space and let (Tn) be a sequence of self-mappings of X, where the t-norm * is of H-type. T0:X→X is a contraction mapping; that is, there exists a k∈(0,1) such that M(T0x,T0y,kt)≥M(x,y,t) for every x,y∈X and t>0, and then T0 has a unique fixed point x0∈X. If there exists at least one fixed point xn for each Tn and the sequence (Tn) converges uniformly to T0, then xn converges to x0.

Proof.

Since * is of H-type, by Lemma 3 there exists a strictly increasing sequence (ri) from the interval [0,1) such that limi→∞ri=1 and ri*ri=ri.

Suppose that xn does not converge to x0. Thus, without loss of generality, we can suppose that there exists t0>0 and 1-r′∈(ri) such that
(14)1-r′>M(xn,x0,t0)
for all n∈N. Let α be a given number in (k,1). According to the condition (FM-6) of Definition 4, for any xn, we can find a p(n)∈N such that
(15)M(xn,x0,t0(ak)p(n))>1-r′.
In addition, since the sequence (Tn) converges uniformly to T0, there exists an n0∈N such that
(16)M(Tnx,T0x,(1-a)t0)>1-r′
for all x∈X and n≥n0. Now for n≥n0, we get that
(17)1-r′>M(xn,x0,t0)=M(Tnxn,T0x0,t0)≥M(Tnxn,T0xn,(1-a)t0)*M(T0xn,T0x0,at0)≥M(Tnxn,T0xn,(1-a)t0)*M(xn,x0,akt0)≥M(Tnxn,T0xn,(1-a)t0)*M(Tnxn,T0xn,ak(1-a)t0)*M(xn,x0,(ak)2t0)≥M(Tnxn,T0xn,(1-a)t0)*M(Tnxn,T0xn,ak(1-a)t0)*⋯*M(Tnxn,T0xn,(ak)p(n)-1(1-a)t0)*M(xn,x0,(ak)p(n)t0).
Since
(18)(ak)i(1-a)t0≥(1-a)t0,i=1,2,…,p(n)-1,
by the monotonicity of fuzzy metric M with respect to t, we have that
(19)M(Tnxn,T0xn,(ak)i(1-a)t0)≥M(Tnxn,T0xn,(1-a)t0)>1-r′,i=1,2,…,p(n)-1.
Thus from inequality (15) and inequality (17), we get
(20)1-r′>(1-r′)*(1-r′)*⋯*(1-r′).
But since * is of H-type and 1-r′∈(rn), inequality (20) is a contradiction. Thence, we get that xn converges to x0.

Theorem 20.

Let (X,M,*) be a G-complete fuzzy metric space. If T:X→X is a self-mapping of X such that the iteration mapping Tm is a contraction mapping for a certain positive integer m, then T has a unique fixed point.

Proof.

Without loss of generality, we suppose m>1. Since Tm is a contraction mapping, by Lemma 7, Tm has a unique fixed point x. Thus Tmx=x and T(Tmx)=Tx which implies Tm(Tx)=Tx. It follows that Tx=x; that is, x is a fixed point of T. If Ty=y for some y∈X, then we can get that Tmy=y which implies x=y. Thus T has a unique fixed point.

Theorem 21.

Let (X,M,*) be a G-complete fuzzy metric space and let (Tn) be a sequence of self-mappings of X, where the t-norm * is of H-type. Suppose that T0:X→X is a self-mapping such that T0m is a contraction mapping for a certain positive integer m. If there exists at least one fixed point xn for each Tn and the sequence (Tn) converges uniformly to T0, then xn converges to x0.

Proof.

It is a corollary of Theorems 19 and 20.

Theorem 22.

Let (X,M,*) be a locally compact fuzzy metric space and let (Tn) be a sequence of self-mappings of X, where the t-norm * is of H-type. T0:X→X is a contraction mapping; that is, there exists a k∈(0,1) such that M(T0x,T0y,kt)≥M(x,y,t) for every x,y∈X and t>0. If the following conditions are satisfied:

Tnm is a contraction mapping for a certain number m=m(n),

(Tn) converges pointwise to T0 and it is a uniformly equicontinuous sequence,

Tnxn=xn,n=0,1,2,…,

then xn converges to x0.
Proof.

Since * is of H-type, by Lemma 3 there exists a strictly increasing sequence (ri) from the interval [0,1) such that limi→∞ri=1 and ri*ri=ri. Since X is a locally compact space, for the given point x0, we can choose a t0>0 and an r′ such that 1-r′∈(ri) and the closed ball BM[x0,r′,t0] is a compact set of X. Since Tn is uniformly equicontinuous and pointwise convergent on BM[x0,r′,t0], by Theorem 18, Tn converges uniformly to T0 on BM[x0,r′,t0]. Thus there exists n0∈N such that
(21)M(Tnx,T0x,(1-k)t0)>1-r′
for all x∈BM[x0,r′,t0] and n≥n0. Consequently, for all x∈BM[x0,r′,t0] and n≥n0, we have that
(22)M(Tnx,x0,t0)=M(Tnx,T0x0,t0)≥M(Tnx,T0x,(1-k)t0)*M(T0x,T0x0,kt0)≥M(Tnx,T0x,(1-k)t0)*M(x,x0,t0)≥1-r′*1-r′=1-r′.
Then we have Tnx∈BM[x0,r′,t0], and then by inductive reasoning Tnix∈BM[x0,r′,t0] for all i∈N whenever x∈BM[x0,r′,t0] and n≥n0. Especially, BM[x0,r′,t0] is an invariant set for Tnm(n) whenever n≥n0. It follows that the fixed point xn of Tn is contained in the set BM[x0,r′,t0] whenever n≥n0. Now by Theorem 19, we get that xn converges to x0.

Theorem 23.

Let (X,M0,*) be a G-complete fuzzy metric space and let A be a compact subset of X. If (Mn) is a sequence of fuzzy metrics of X with respect to * and (Tn) is a sequence of self-mappings of X satisfying the following conditions:

(Mn) upper semiconverges uniformly to M0,

Tn is a contraction mapping with respect to the fuzzy metric Mn for n=0,1,2,…,

(Tn) converges pointwise to T0 with respect to M0,

then Tn converges uniformly to T0 in A with respect to the fuzzy metric M0.
Proof.

For each r∈(0,1), we can choose an ε∈(0,1) such that
(23)(1-ε)*(1-ε)>1-r.
Since Mn upper semiconverges uniformly to M0, for ε∈(0,1) and t>0, there exists n(ε,t) such that
(24)Mn(x,y,t)≥M0(x,y,t),M0(x,y,t)⊖Mn(x,y,t)>1-ε
for all x,y∈X and n≥n(ε,t). It should be noted that
(25)M0(x,y,t)≥(M0(x,y,t)⊖Mn(x,y,t))*Mn(x,y,t)
because of the continuity of *.

Now for any x,y∈X satisfying M0(x,y,t)>1-ε and n≥n(ε,t), we get that
(26)M0(Tnx,Tny,t)≥(M0(Tnx,Tny,t)⊖Mn(Tnx,Tny,t))*Mn(Tnx,Tny,t)≥(1-ε)*Mn(Tnx,Tny,t)≥(1-ε)*Mn(x,y,tkn),
where kn∈(0,1) is the constant in Definition 6 for Tn. By the monotonicity of fuzzy metric M with respect to t, we have that
(27)Mn(x,y,tkn)≥Mn(x,y,t)>M0(x,y,t)>1-ε.
Now from inequality (26), for n≥n(ε,t), we get that
(28)M0(Tnx,Tny,t)≥(1-ε)*(1-ε)>1-r,
whenever M0(x,y,t)>1-ε. Thus Tn is uniformly equicontinuous in X with respect to the fuzzy metric M0. Since A is a compact subset of X and Tn converges pointwise to T0 with respect to M0, it follows from Theorem 18 that Tn converges uniformly to T0 in A with respect to the fuzzy metric M0.

Theorem 24.

Let (X,M0,*) be a locally compact fuzzy metric space where the t-norm * is of H-type. T0:X→X is a contraction mapping; that is, there exists a k∈(0,1) such that M0(T0x,T0y,kt)≥M0(x,y,t) for every x,y∈X and t>0. If (Mn) is a sequence of fuzzy metrics of X with respect to * and (Tn) is a sequence of self-mappings of X satisfying the following conditions:

(Mn) upper semiconverges uniformly to M0;

Tn is a contraction mapping with respect to the fuzzy metric Mn and Tnxn=xn for n=0,1,2,…;

(Tn) converges pointwise to T0 with respect to M0.

Then the sequence of fixed points (xn) converges to x0.

Proof.

Since * is of H-type, by Lemma 3 there exists a strictly increasing sequence (ri) from the interval [0,1) such that limi→∞ri=1 and ri*ri=ri. Since X is a locally compact space, for the given point x0, we can choose a t0>0 and an r′ such that 1-r′∈(ri) and the closed ball BM0[x0,r′,t0] is a compact set of X.

It follows form Theorem 23 that Tn converges uniformly to T0 in BM[x0,r′,t0] with respect to the fuzzy metric M0. Thus there exists n0∈N such that
(29)M0(Tnx,T0x,(1-k)t0)>1-r′
for all x∈BM[x0,r′,t0] and n≥n0.

Meantime, since Mn upper semiconverges uniformly to M0, for r′∈(0,1) and t0>0, there exists n1 such that
(30)Mn(x,y,t0)≥M0(x,y,t0),M0(x,y,t0)⊖Mn(x,y,t0)>1-r′
for all x,y∈X and n≥n1. Let n′=max{n0,n1}. If n≥n′ and x∈BMn[x0,r′,t0], then we have that Mn(x,x0,t0)≥1-r′ and M0(x,x0,t)⊖Mn(x,x0,t)>1-r′. This implies that M0(x,x0,t)≥1-r′ because * is of H-type and
(31)M0(x,x0,t0)≥(M0(x,x0,t0)⊖Mn(x,x0,t0))*Mn(x,x0,t0).
Thus, we have
(32)BMn[x0,r′,t0]⊆BM0[x0,r′,t0],
for all n≥n′. But for n≥n′, it is easy to see that
(33)BMn[x0,r′,t0]⊇BM0[x0,r′,t0],
because
(34)Mn(x,y,t0)≥M0(x,y,t0)≥1-r′.
Therefore, we get BMn[x0,r′,t0]=BM0[x0,r′,t0], for all n≥n′. Now for x∈BM0[x0,r′,t0] and n≥n′, we have
(35)Mn(Tnx,x0,t0)≥M0(Tnx,x0,t0)=M0(Tnx,T0x0,t0)≥M0(Tnx,T0x,(1-k)t0)*M0(T0x,T0x0,kt0)≥M0(Tnx,T0x,(1-k)t0)*M0(x,x0,t0)≥1-r′*1-r′=1-r′.
Then we have Tnx∈BMn[x0,r′,t0]=BM0[x0,r′,t0], which implies that BM0[x0,r′,t0] is an invariant set in X with respect to Mn for n≥n′. It follows that the fixed point xn of Tn is contained in the set BM0[x0,r′,t0] whenever n≥n′. Now by Theorem 19, we get that xn converges to x0.

Theorem 25.

Let (X,M,*) be a compact fuzzy metric space where the t-norm * is of H-type. (Mn) is a sequence of fuzzy metrics of X with respect to * and (Tn) is a sequence of self-mappings of X satisfying the following conditions:

(Mn) upper semiconverges uniformly to M0;

Tn is a contraction mapping with respect to the fuzzy metric Mn for n=0,1,2,…;

(Tn) converges pointwise to T0 with respect to M0.

If xn is the fixed point of Tn and there is a subsequence (xnk) of (xn) which converges to x0∈X, then T0x0=x0.

Proof.

Since * is of H-type, by Lemma 3, there exists a strictly increasing sequence (ri) from the interval [0,1) such that limi→∞ri=1 and ri*ri=ri.

Let K be the closure of the set {xn:n∈N}. By Lemma 14, we can easily know that K is a compact set. According to Theorem 23, it follows that the subsequence (Tnk) converges uniformly to T0 in K with respect to M0; that is, for each r∈(ri) and t>0, there exists an k0∈N such that M(Tnx,T0x,t/2)>r for all x∈K and k≥k0. Now for k≥k0, we have that
(36)M(Tnkxnk,T0x0,t)≥M(Tnkxnk,T0xnk,t2)*M(T0xnk,T0x0,t2)≥r*M(xnk,x0,t2k0)≥r*r=r,
where k0∈(0,1) is the constant in Definition 6 for T0. Thus Tnkxnk=xnk converges to T0x0. Then T0x0=x0.

Example 26.

Let (1-1/2n)n∈N be a strictly increasing sequence from the interval [0,1), and let (αn,βn)=(1-1/2n,1-1/2n+1), for all n∈N. Since limn→∞1-1/2n=1, by Lemma 3 in [36], the ordinal sum *=(〈(αn,βn),*P〉)n∈N is an H-type t-norm. Let X=[-1/2,1/2]. Define a fuzzy set M on X2×(0,∞) by
(37)M(x,y,t)=tt+|x-y|,
for all x,y∈X and t>0. Then we can get that (X,M,*) is a G-complete fuzzy metric space. Let T0 be a mapping defined by T0(x)=x2/4 for all x∈X, and let (Tn) be a sequence of self-mappings of X defined by
(38)Tn(x)=(14+14n)x2-14n,
for all x∈X and n∈N. By some simple calculations, we can get that T0 is a contraction mapping and (Tn) converges uniformly to T0. Thus by Theorem 19, we obtain that the sequence (xn) of fixed point of (Tn) converges to the fixed point x0 of T0. In fact, we have that x0=0 and
(39)xn=1-1+(1/4n)(1+(1/4n-1))(1/2)+(1/22n-1),
for all n∈N.

4. Conclusions

In this paper, we provide some results on the convergence of the sequence of fixed points for some different sequences of contraction mappings or fuzzy metrics in G-complete fuzzy metric spaces with the H-type t-norms. We improve the corresponding conclusions in the literature. The first solution to the Ulam problem was obtained through a classical approach known from the fixed point theory, by iteration of a simple operator. Later it has been shown that some fixed point theorems can be directly applied in investigations of Ulam's type stability. Thus we hope our results would provide a background to ongoing work in the problems of those related fields.

Conflict of Interests

The authors have no conflicts of interests regarding this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 11201512), the Natural Science Foundation Project of CQ CSTC (cstc2012jjA00001), and the Science and Technology Project of Chongqing Municipal Education Committee of China (Grant no. KJ120520, KJ1400426).

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