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Some results on asymptotically quasiϕnonexpansive mappings in the intermediate sense and equilibrium problems
Journal of Inequalities and Applications volume 2014, Article number: 202 (2014)
Abstract
In this paper, we investigate a common fixed point problem of a finite family of asymptotically quasiϕnonexpansive mappings in the intermediate sense and an equilibrium problem. Strong convergence theorems of common solutions are established in a uniformly smooth and strictly convex Banach space which also enjoys the KadecKlee property.
MSC:47H09, 47J25, 90C33.
1 Introductionpreliminaries
Let E be a real Banach space. Recall that E is said to be strictly convex if \parallel \frac{x+y}{2}\parallel <1 for all x,y\in E with \parallel x\parallel =\parallel y\parallel =1 and x\ne y. It is said to be uniformly convex if {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{y}_{n}\parallel =0 for any two sequences \{{x}_{n}\} and \{{y}_{n}\} in E such that \parallel {x}_{n}\parallel =\parallel {y}_{n}\parallel =1 and {lim}_{n\to \mathrm{\infty}}\parallel \frac{{x}_{n}+{y}_{n}}{2}\parallel =1. Let {U}_{E}=\{x\in E:\parallel x\parallel =1\} be the unit sphere of E. Then the Banach space E is said to be smooth if
exists for each x,y\in {U}_{E}. It is said to be uniformly smooth if the above limit is attained uniformly for x,y\in {U}_{E}.
Recall that E has KadecKlee property if for any sequence \{{x}_{n}\}\subset E, and x\in E with {x}_{n}\rightharpoonup x, and \parallel {x}_{n}\parallel \to \parallel x\parallel, then \parallel {x}_{n}x\parallel \to 0 as n\to \mathrm{\infty}. For more details of the KadecKlee property, the readers can refer to [1] and the references therein. It is well known that if E is a uniformly convex Banach space, then E enjoys the KadecKlee property.
Recall that the normalized duality mapping J from E to {2}^{{E}^{\ast}} is defined by
where \u3008\cdot ,\cdot \u3009 denotes the generalized duality pairing. It is well known that if E is uniformly smooth, then J is uniformly normtonorm continuous on each bounded subset of E. It is also well known that if E is uniformly smooth if and only if {E}^{\ast} is uniformly convex.
Next, we assume that E is a smooth Banach space. Consider the functional defined by
Observe that, in a Hilbert space H, the equality is reduced to \varphi (x,y)={\parallel xy\parallel}^{2}, x,y\in H. As we all know if C is a nonempty closed convex subset of a Hilbert space H and {P}_{C}:H\to C is the metric projection of H onto C, then {P}_{C} is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. In this connection, Alber [2] recently introduced a generalized projection operator {\mathrm{\Pi}}_{C} in a Banach space E which is an analog of the metric projection {P}_{C} in Hilbert spaces. Recall that the generalized projection {\mathrm{\Pi}}_{C}:E\to C is a map that assigns to an arbitrary point x\in E the minimum point of the functional \varphi (x,y), that is, {\mathrm{\Pi}}_{C}x=\overline{x}, where \overline{x} is the solution to the minimization problem
The existence and uniqueness of the operator {\mathrm{\Pi}}_{C} follow from the properties of the functional \varphi (x,y) and strict monotonicity of the mapping J; see, for example, [1, 2]. In Hilbert spaces, {\mathrm{\Pi}}_{C}={P}_{C}. It is obvious from the definition of the function ϕ that
and
Remark 1.1 If E is a reflexive, strictly convex, and smooth Banach space, then \varphi (x,y)=0 if and only if x=y; for more details, see [1, 2] and the references therein.
Let C be a nonempty subset of E and let T:C\to C be a mapping. In this paper, we use F(T) to denote the fixed point set of T. T is said to be asymptotically regular on C if for any bounded subset K of C,
T is said to be closed if for any sequence \{{x}_{n}\}\subset C such that {lim}_{n\to \mathrm{\infty}}{x}_{n}={x}_{0} and {lim}_{n\to \mathrm{\infty}}T{x}_{n}={y}_{0}, then T{x}_{0}={y}_{0}. In this paper, we use → and ⇀ to denote the strong convergence and weak convergence, respectively.
Recall that a point p in C is said to be an asymptotic fixed point of T [3] iff C contains a sequence \{{x}_{n}\} which converges weakly to p such that {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}T{x}_{n}\parallel =0. The set of asymptotic fixed points of T will be denoted by \tilde{F}(T).
A mapping T is said to be relatively nonexpansive iff
A mapping T is said to be relatively asymptotically nonexpansive iff
where \{{\mu}_{n}\}\subset [0,\mathrm{\infty}) is a sequence such that {\mu}_{n}\to 0 as n\to \mathrm{\infty}.
Remark 1.2 The class of relatively asymptotically nonexpansive mappings were first considered in [4]; see also, [5] and the references therein.
Recall that a mapping T is said to be quasiϕnonexpansive iff
Recall that a mapping T is said to be asymptotically quasiϕnonexpansive iff there exists a sequence \{{\mu}_{n}\}\subset [0,\mathrm{\infty}) with {\mu}_{n}\to 0 as n\to \mathrm{\infty} such that
Remark 1.3 The class of quasiϕnonexpansive mappings was considered in [6]. The class of asymptotically quasiϕnonexpansive mappings which was investigated in [7] and [8] includes the class of quasiϕnonexpansive mappings as a special case.
Remark 1.4 The class of quasiϕnonexpansive mappings and the class of asymptotically quasiϕnonexpansive mappings are more general than the class of relatively nonexpansive mappings and the class of relatively asymptotically nonexpansive mappings. Quasiϕnonexpansive mappings and asymptotically quasiϕnonexpansive do not require the restriction F(T)=\tilde{F}(T).
Remark 1.5 The class of quasiϕnonexpansive mappings and the class of asymptotically quasiϕnonexpansive mappings are generalizations of the class of quasinonexpansive mappings and the class of asymptotically quasinonexpansive mappings in Banach spaces.
Recall that T is said to be asymptotically quasiϕnonexpansive in the intermediate sense iff F(T)\ne \mathrm{\varnothing} and the following inequality holds:
Putting
it follows that {\xi}_{n}\to 0 as n\to \mathrm{\infty}. Then (1.3) is reduced to the following:
Remark 1.6 The class of asymptotically quasiϕnonexpansive mappings in the intermediate sense was first considered by Qin and Wang [9].
Remark 1.7 The class of asymptotically quasiϕnonexpansive mappings in the intermediate sense is a generalization of the class of asymptotically quasinonexpansive mappings in the intermediate sense, which was considered by Kirk [10], in the framework of Banach spaces.
Let f be a bifunction from C\times C to ℝ, where ℝ denotes the set of real numbers. Recall the following equilibrium problem. Find p\in C such that
We use EP(f) to denote the solution set of the equilibrium problem (1.5). That is,
We remark here that the equilibrium problem was first introduced by Fan [11]. Given a mapping Q:C\to {E}^{\ast}, let
Then p\in EP(f) if and only if p is a solution of the following variational inequality. Find p such that
To study the equilibrium problems (1.5), we may assume that F satisfies the following conditions:
(A1) F(x,x)=0 for all x\in C;
(A2) F is monotone, i.e., F(x,y)+F(y,x)\le 0 for all x,y\in C;
(A3) for each x,y,z\in C,
(A4) for each x\in C, y\mapsto F(x,y) is convex and weakly lower semicontinuous.
Numerous problems in physics, optimization, and economics reduce to find a solution of (1.5). Recently, many authors have investigated common solutions of fixed point and equilibrium problems in Banach spaces; see, for example, [12–33] and the references therein.
In this paper, we consider a projection algorithm for treating the equilibrium problem and fixed point problems of asymptotically quasiϕnonexpansive mappings in the intermediate sense.
In order to prove our main results, we need the following lemmas.
Lemma 1.8 [2]
Let E be a reflexive, strictly convex and smooth Banach space. Let C be a nonempty closed convex subset of E and let x\in E. Then
Lemma 1.9 [2]
Let C be a nonempty closed convex subset of a smooth Banach space E and let x\in E. Then {x}_{0}={\mathrm{\Pi}}_{C}x if and only if
Lemma 1.10 Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space E. Let f be a bifunction from C\times C to ℝ satisfying (A1)(A4). Let r>0 and x\in E. Then

(a)
[34]There exists z\in C such that
f(z,y)+\frac{1}{r}\u3008yz,JzJx\u3009\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}y\in C. 
(b)
[6, 24]Define a mapping {T}_{r}:E\to C by
{S}_{r}x=\{z\in C:f(z,y)+\frac{1}{r}\u3008yz,JzJx\u3009,\mathrm{\forall}y\in C\}.
Then the following conclusions hold:

(1)
{S}_{r} is singlevalued;

(2)
{S}_{r} is a firmly nonexpansivetype mapping, i.e., for all x,y\in E,
\u3008{S}_{r}x{S}_{r}y,J{S}_{r}xJ{S}_{r}y\u3009\le \u3008{S}_{r}x{S}_{r}y,JxJy\u3009 
(3)
F({S}_{r})=EP(f);

(4)
{S}_{r} is quasiϕnonexpansive;

(5)
\varphi (q,{S}_{r}x)+\varphi ({S}_{r}x,x)\le \varphi (q,x), \mathrm{\forall}q\in F({S}_{r});

(6)
EP(f) is closed and convex.
Lemma 1.11 [35]
Let E be a smooth and uniformly convex Banach space and let r>0. Then there exists a strictly increasing, continuous and convex function g:[0,2r]\to R such that g(0)=0 and
for all x,y\in {B}_{r}=\{x\in E:\parallel x\parallel \le r\} and t\in [0,1].
2 Main results
Theorem 2.1 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the KadecKlee property and let C be a nonempty closed and convex subset of E. Let f be a bifunction from C\times C to ℝ satisfying (A1)(A4) and let N be some positive integer. Let {T}_{i}:C\to C an asymptotically quasiϕnonexpansive mapping in the intermediate sense for every 1\le i\le N. Assume that {T}_{i} is closed asymptotically regular on C and {\bigcap}_{i=1}^{N}F({T}_{i})\cap EF(f) is nonempty and bounded. Let \{{x}_{n}\} be a sequence generated in the following manner:
where {\xi}_{n}=max\{0,{sup}_{p\in F({T}_{i}),x\in C}(\varphi (p,{T}_{i}^{n}x)\varphi (p,x))\}, \{{\alpha}_{n,i}\} is a real number sequence in (0,1) for every 1\le i\le N, \{{r}_{n}\} is a real number sequence in [k,\mathrm{\infty}), where k is some positive real number. Assume that {\sum}_{i=0}^{N}{\alpha}_{n,i}=1 and {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n,0}{\alpha}_{n,i}>0 for every 1\le i\le N. Then the sequence \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{{\bigcap}_{i=1}^{N}F({T}_{i})\cap EF(f)}{x}_{1}, where {\mathrm{\Pi}}_{{\bigcap}_{i=1}^{N}F({T}_{i})\cap EF(f)} is the generalized projection from E onto {\bigcap}_{i=1}^{N}F({T}_{i})\cap EF(f).
Proof First, we show that {\bigcap}_{i=1}^{N}F({T}_{i})\cap EF(f) is closed and convex. From [9], we find that {\bigcap}_{i=1}^{N}F({T}_{i}) is closed and convex, which combines with Lemma 1.10 shows that {\bigcap}_{i=1}^{N}F({T}_{i})\cap EF(f) is closed and convex. Next, we show that {C}_{n} is closed and convex. It is obvious that {C}_{1}=C is closed and convex. Suppose that {C}_{h} is closed and convex for some positive integer h. For z\in {C}_{h}, we see that \varphi (z,{u}_{h})\le \varphi (z,{x}_{h})+{\xi}_{h} is equivalent to
It is to see that {C}_{h+1} is closed and convex. This proves that {C}_{n} is closed and convex. This in turn shows that {\mathrm{\Pi}}_{{C}_{n+1}}{x}_{1} is well defined. Putting {u}_{n}={S}_{{r}_{n}}{y}_{n}, we from Lemma 1.10 see that {S}_{{r}_{n}} is quasiϕnonexpansive. Now, we are in a position to prove that {\bigcap}_{i=1}^{N}F({T}_{i})\cap EF(f)\subset {C}_{n}. Indeed, {\bigcap}_{i=1}^{N}F({T}_{i})\cap EF(f)\subset {C}_{1}=C is obvious. Assume that {\bigcap}_{i=1}^{N}F({T}_{i})\cap EF(f)\subset {C}_{h} for some positive integer h. Then, for \mathrm{\forall}w\in {\bigcap}_{i=1}^{N}F({T}_{i})\cap EF(f)\subset {C}_{h}, we have
which shows that w\in {C}_{h+1}. This implies that {\bigcap}_{i=1}^{N}F({T}_{i})\cap EF(f)\subset {C}_{n}.
Next, we prove that the sequence \{{x}_{n}\} is bounded. Notice that {x}_{n}={\mathrm{\Pi}}_{{C}_{n}}{x}_{1}. We find from Lemma 1.9 that \u3008{x}_{n}z,J{x}_{1}J{x}_{n}\u3009\ge 0, for any z\in {C}_{n}. Since {\bigcap}_{i=1}^{N}F({T}_{i})\cap EF(f)\subset {C}_{n}, we find that
It follows from Lemma 1.8 that
This implies that the sequence \{\varphi ({x}_{n},{x}_{1})\} is bounded. It follows from (1.1) that the sequence \{{x}_{n}\} is also bounded. Since the space is reflexive, we may assume, without loss of generality, that {x}_{n}\rightharpoonup \overline{x}. Next, we prove that \overline{x}\in {\bigcap}_{i=1}^{N}F({T}_{i})\cap EF(f). Since {C}_{n} is closed and convex, we find that \overline{x}\in {C}_{n}. This implies from {x}_{n}={\mathrm{\Pi}}_{{C}_{n}}{x}_{1} that \varphi ({x}_{n},{x}_{1})\le \varphi (\overline{x},{x}_{1}). On the other hand, we see from the weakly lower semicontinuity of \parallel \cdot \parallel that
which implies that {lim}_{n\to \mathrm{\infty}}\varphi ({x}_{n},{x}_{1})=\varphi (\overline{x},{x}_{1}). Hence, we have {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}\parallel =\parallel \overline{x}\parallel. In view of the KadecKlee property of E, we find that {x}_{n}\to \overline{x} as n\to \mathrm{\infty}. Since {x}_{n}={\mathrm{\Pi}}_{{C}_{n}}{x}_{1}, and {x}_{n+1}={\mathrm{\Pi}}_{{C}_{n+1}}{x}_{1}\in {C}_{n+1}\subset {C}_{n}, we find that \varphi ({x}_{n},{x}_{1})\le \varphi ({x}_{n+1},{x}_{1}). This shows that \{\varphi ({x}_{n},{x}_{1})\} is nondecreasing. We find from its boundedness that {lim}_{n\to \mathrm{\infty}}\varphi ({x}_{n},{x}_{1}) exists. It follows that
This implies that
In the light of {x}_{n+1}={\mathrm{\Pi}}_{{C}_{n+1}}{x}_{1}\in {C}_{n+1}, we find that
It follows from (2.3) that
In view of (1.1), we see that {lim}_{n\to \mathrm{\infty}}(\parallel {x}_{n+1}\parallel \parallel {u}_{n}\parallel )=0. This implies that {lim}_{n\to \mathrm{\infty}}\parallel {u}_{n}\parallel =\parallel \overline{x}\parallel. That is,
This implies that \{J{u}_{n}\} is bounded. Note that both E and {E}^{\ast} are reflexive. We may assume, without loss of generality, that J{u}_{n}\rightharpoonup {u}^{\ast}\in {E}^{\ast}. In view of the reflexivity of E, we see that J(E)={E}^{\ast}. This shows that there exists an element u\in E such that Ju={u}^{\ast}. It follows that
Taking {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}} on both sides of the equality aboven yields
That is, \overline{x}=u, which in turn implies that {u}^{\ast}=J\overline{x}. It follows that J{u}_{n}\rightharpoonup J\overline{x}\in {E}^{\ast}. Since {E}^{\ast} enjoys the KadecKlee property, we obtain from (2.5) that {lim}_{n\to \mathrm{\infty}}J{u}_{n}=J\overline{x}. Since {J}^{1}:{E}^{\ast}\to E is demicontinuous and E enjoys the KadecKlee property, we obtain {u}_{n}\to \overline{x}, as n\to \mathrm{\infty}. Note that
It follows that
Since J is uniformly normtonorm continuous on any bounded sets, we have
On the other hand, we have
We, therefore, find that
Since E is uniformly smooth, we know that {E}^{\ast} is uniformly convex. In view of Lemma 1.11, we find that
It follows that
In view of the restriction on the sequences, we find from (2.8) that {lim}_{n\to \mathrm{\infty}}g(\parallel J{x}_{n}J{T}_{1}^{n}{x}_{n}\parallel )=0. It follows that
In the same way, we obtain
Notice that \parallel J{T}_{i}^{n}{x}_{n}J\overline{x}\parallel \le \parallel J{T}_{i}^{n}{x}_{n}J{x}_{n}\parallel +\parallel J{x}_{n}J\overline{x}\parallel. It follows that
The demicontinuity of {J}^{1}:{E}^{\ast}\to E implies that {T}_{i}^{n}{x}_{n}\rightharpoonup \overline{x}. Note that
This implies from (2.9) that {lim}_{n\to \mathrm{\infty}}\parallel {T}_{i}^{n}{x}_{n}\parallel =\parallel \overline{x}\parallel. Since E has the KadecKlee property, we obtain {lim}_{n\to \mathrm{\infty}}\parallel {T}_{i}^{n}{x}_{n}\overline{x}\parallel =0. On the other hand, we have
It follows from the uniformly asymptotic regularity of {T}_{i} that
That is, {T}_{i}{T}_{i}^{n}{x}_{n}\to \overline{x}. From the closedness of {T}_{i}, we find \overline{x}={T}_{i}\overline{x} for every 1\le i\le N. This proves \overline{x}\in {\bigcap}_{i=1}^{N}F({T}_{i}).
Next, we show that \overline{x}\in EF(f). In view of Lemma 1.8, we find that
It follows from (2.8) that {lim}_{n\to \mathrm{\infty}}\varphi ({u}_{n},{y}_{n})=0. This implies that {lim}_{n\to \mathrm{\infty}}(\parallel {u}_{n}\parallel \parallel {y}_{n}\parallel )=0. It follows from (2.6) that
It follows that
This shows that \{J{y}_{n}\} is bounded. Since {E}^{\ast} is reflexive, we may assume that J{y}_{n}\rightharpoonup {v}^{\ast}\in {E}^{\ast}. In view of J(E)={E}^{\ast}, we see that there exists v\in E such that Jv={v}^{\ast}. It follows that
Taking {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}} the both sides of equality above yields that
That is, \overline{x}=v, which in turn implies that {v}^{\ast}=J\overline{x}. It follows that J{y}_{n}\rightharpoonup J\overline{x}\in {E}^{\ast}. Since {E}^{\ast} enjoys the KadecKlee property, we obtain J{y}_{n}J\overline{x}\to 0 as n\to \mathrm{\infty}. Note that {J}^{1}:{E}^{\ast}\to E is demicontinuous. It follows that {y}_{n}\rightharpoonup \overline{x}. Since E enjoys the KadecKlee property, we obtain {y}_{n}\to \overline{x} as n\to \mathrm{\infty}. Note that
This implies that {lim}_{n\to \mathrm{\infty}}\parallel {u}_{n}{y}_{n}\parallel =0. Since J is uniformly normtonorm continuous on any bounded sets, we have {lim}_{n\to \mathrm{\infty}}\parallel J{u}_{n}J{y}_{n}\parallel =0. From the assumption {r}_{n}\ge k, we see that
Since {u}_{n}={S}_{{r}_{n}}{y}_{n}, we find that
It follows from (A2) that
In view of (A4), we find from (2.10) that
For 0<t<1 and y\in C, define {y}_{t}=ty+(1t)\overline{x}. It follows that {y}_{t}\in C, which yields f({y}_{t},\overline{x})\le 0. It follows from (A1) and (A4) that
That is,
Letting t\downarrow 0, we obtain from (A3) that f(\overline{x},y)\ge 0, \mathrm{\forall}y\in C. This implies that \overline{x}\in EP(f).
Finally, we turn our attention to proving that \overline{x}={\mathrm{\Pi}}_{{\bigcap}_{i=1}^{N}F({T}_{i})\cap EF(f)}{x}_{1}.
Letting n\to \mathrm{\infty} in (2.2), we obtain
In view of Lemma 1.9, we find that \overline{x}={\mathrm{\Pi}}_{{\bigcap}_{i=1}^{N}F({T}_{i})\cap EF(f)}{x}_{1}. This completes the proof. □
From the definition of quasiϕnonexpansive mappings, we see that every quasiϕnonexpansive mapping is asymptotically quasiϕnonexpansive in the intermediate sense. We also know that every uniformly smooth and uniformly convex space is a uniformly smooth and strictly convex Banach space which also enjoys the KadecKlee property (note that every uniformly convex Banach space enjoys the KadecKlee property).
Remark 2.2 Theorem 2.1 can be viewed an extension of the corresponding results in Qin et al. [6], Kim [12], Qin et al. [22], Takahashi and Zembayashi [24], respectively. The space {L}^{p}, where p>1, satisfies the restriction in Theorem 2.1.
3 Applications
Theorem 3.1 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the KadecKlee property and let C be a nonempty closed and convex subset of E. Let f be a bifunction from C\times C to ℝ satisfying (A1)(A4). Let T:C\to C an asymptotically quasiϕnonexpansive mapping in the intermediate sense. Assume that T is closed asymptotically regular on C and F(T)\cap EF(f) is nonempty and bounded. Let \{{x}_{n}\} be a sequence generated in the following manner:
where {\xi}_{n}=max\{0,{sup}_{p\in F(T),x\in C}(\varphi (p,{T}^{n}x)\varphi (p,x))\}, \{{r}_{n}\} is a real number sequence in [k,\mathrm{\infty}), where k is some positive real number, \{{\alpha}_{n,0}\} and \{{\alpha}_{n}n,1\} are two real number sequence in (0,1). Assume that {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n,0}{\alpha}_{n,1}>0. Then the sequence \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{F(T)\cap EF(f)}{x}_{1}, where {\mathrm{\Pi}}_{F(T)\cap EF(f)} is the generalized projection from E onto F(T)\cap EF(f).
Proof Putting N=1, we draw from Theorem 2.1 the desired conclusion immediately. □
Remark 3.2 If the mapping T in Theorem 3.1 is quasiϕnonexpansive, then the restrictions that T is closed asymptotically regular on C and F(T)\cap EF(f) is bounded will not be required anymore.
If {T}_{i}=I, where I is the identity for every 1\le i\le N, then we find from Theorem 2.1 the following.
Theorem 3.3 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the KadecKlee property and let C be a nonempty closed and convex subset of E. Let f be a bifunction from C\times C to ℝ satisfying (A1)(A4). Assume that EF(f) is nonempty. Let \{{x}_{n}\} be a sequence generated in the following manner:
where \{{r}_{n}\} is a real number sequence in [k,\mathrm{\infty}), where k is some positive real number. Then the sequence \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{EF(f)}{x}_{1}, where {\mathrm{\Pi}}_{EF(f)} is the generalized projection from E onto EF(f).
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Huang, C., Ma, X. Some results on asymptotically quasiϕnonexpansive mappings in the intermediate sense and equilibrium problems. J Inequal Appl 2014, 202 (2014). https://doi.org/10.1186/1029242X2014202
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DOI: https://doi.org/10.1186/1029242X2014202