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A class of generalized invex functions and vector variationallike inequalities
Journal of Inequalities and Applications volumeÂ 2017, ArticleÂ number:Â 70 (2017)
Abstract
In this paper, a class of generalized invex functions, called \((\alpha,\rho,\eta)\)invex functions, is introduced, and some examples are presented to illustrate their existence. Then we consider the relationships of solutions between two types of vector variationallike inequalities and multiobjective programming problem. Finally, the existence results for the discussed variationallike inequalities are proposed by using the KKMFan theorem.
1 Introduction
As we all know, convexity and generalized convexity of functions play an important role in the field of optimization theory and its application. Among many generalized convexities given in the literature, a meaningful notion is called invex function, which was firstly introduced by Hanson [1]. Then Jeyakumar and Mond [2] popularized it and put forward the concept of Vinvex function, and they studied the optimality conditions for nonlinear programming problem. Ivanov [3] provided the concept of secondorder invex functions and dealt with the optimal solution of nonlinear programming problem. Yu and Liu [4] investigated a class of generalized invex type I functions and presented the optimality conditions of multiobjective programming. One aim of this paper is to present a new kind of generalized invex functions, termed \((\alpha,\rho,\eta)\)invex functions, which is weaker than the above mentioned generalized invexities.
It is well known that vector variationallike inequalities are of importance in the field of applied mathematics. There are several interesting and important topics, for instance, existence results of vector variational inequalities, which ensure the existence of efficient solutions of vector optimization problems and establish the relations between both problems. In order to do so, a great quantity of researchers have been attracted towards this direction, see [5â€“19]. In this note, we shall investigate the relations between vector variationallike inequalities and multiobjective programming problems under the hypothesis of \((\alpha,\rho,\eta)\)invexity. We also define a kind of generalized monotonicity functions, called \((\alpha,\rho,\eta)\)monotone functions. Then, by using the KKMFan theorem, we present the existence theorems for vector variationallike inequalities under the assumption of \((\alpha,\rho,\eta)\)monotonicity.
The content of the present work can be organized into four sections of which this introduction is the first. In SectionÂ 2, some notations and the concepts are recalled. Besides, a new class of generalized invex functions, called \((\alpha,\rho,\eta)\)invex functions, is introduced, and examples are provided in the support of this generalization. In SectionÂ 3, the relationships between Stampacchia and Minty invex vector variationallike inequalities and \((\alpha,\rho,\eta)\)invex multiobjective programming problem are discussed. SectionÂ 4 gives the existence theorems of solution of Stampacchia and Minty vector variationallike inequality under the hypothesis of \((\alpha,\rho,\eta)\)monotonicity.
2 Notations and preliminaries
Throughout the paper, \(\mathbb{R}^{n}\), \(\mathbb{R}\), \(\mathbb{R}_{+}\) and \(\mathbb{R}_{++}\) represent the ndimensional Euclidean space, the set of real numbers, the set of nonnegative real numbers and the set of positive real numbers, respectively. For any \(x, y\in\mathbb{R}^{n}\), the inner product of x and y is denoted by \(x^{T}y\). We use the following conventions for vectors in \(\mathbb{R}^{n}\):
We begin with recalling some known definitions, which will be applicable in the sequel of the paper. Suppose that \(X\subseteq\mathbb{R}^{n}\) is a nonempty open subset and \(\varphi:X\rightarrow\mathbb{R}\) is a realvalued function. In the rest of paper, we always assume that \(\alpha,\rho: X\times X\rightarrow \mathbb{R}\) are realvalued functions and \(\eta: X\times X\rightarrow \mathbb{R}^{n}\) is a vectorvalued function.
Definition 2.1
see [9]
The subset X is said to be invex with respect to \(\eta: X\times X\rightarrow \mathbb{R}^{n}\) if for every \(x, y\in X\), \(\lambda\in[0, 1]\) it satisfies
In this case, X is called an Î·invex set.
Definition 2.2
see [9]
Let \(\varphi: X\rightarrow \mathbb{R}\) be a function. The directional derivative of Ï† at \(\bar{x}\in X\) in the direction of a vector \(v\in X\), denoted by \(\varphi'(\bar{x},v)\), is defined as
Ï† is called directionally differentiable at \(\bar{x}\in X\) if the directional derivative of Ï† at xÌ„ in any direction exists. Ï† is called directionally differentiable on X if it is directionally differentiable at every point of X.
Definition 2.3
see [9]
Let \(X\subseteq\mathbb{R}^{n}\) be an Î·invex set and \(\varphi: X\rightarrow \mathbb{R}\) be directionally differentiable on X. It is said that Ï† is Î·invex at xÌ„ if
Ï† is said to be Î·invex on X if it is Î·invex at every point of X.
Motivated by the above definition of Î·invex function, we make two extensions of invexity to \((\alpha,\rho,\eta)\)invexity and pseudo\((\alpha, \rho, \eta)\)invexity as follows.
Definition 2.4
Let \(X\subseteq\mathbb{R}^{n}\) be an Î·invex set and \(\varphi: X\rightarrow \mathbb{R}\) be directionally differentiable on X. It is said that Ï† is \((\alpha,\rho,\eta)\)invex at \(\bar{x}\in X\) if there exist realvalued functions \(\alpha, \rho: X\times X\rightarrow \mathbb{R}\) such that
If Ï† is \((\alpha,\rho,\eta)\)invex for each \(x\in X\), then Ï† is called \((\alpha,\rho,\eta)\)invex on X; Ï† is called strictly \((\alpha,\rho,\eta)\)invex at \(\bar{x}\in X\) if equation (2.1) takes strict inequality, that is,
If Ï† is strictly \((\alpha,\rho,\eta)\)invex for each \(x\in X\), then Ï† is called strictly \((\alpha,\rho,\eta)\)invex on X.
Remark 2.1
Setting \(\alpha=1\) and \(\rho=0\) in Definition 2.4, we arrive at the notion of Î·invexity, defined by Farajzadeh and Lee (see [9]).
Definition 2.5
Let \(X\subseteq\mathbb{R}^{n}\) be an Î·invex set and \(\varphi: X\rightarrow \mathbb{R}\) be directionally differentiable on X. It is said that Ï† is \((\alpha, \rho, \eta)\)pseudoinvex at \(\bar{x}\in X\) if there exist realvalued functions \(\alpha, \rho: X\times X\rightarrow \mathbb{R}\) such that
or equivalently,
If function Ï† is \((\alpha, \rho, \eta)\)pseudoinvex at every \(x\in X\), then Ï† is called \((\alpha, \rho, \eta)\)pseudoinvex on X.
The following two examples illustrate the existences of \((\alpha,\rho,\eta)\)invex functions and \((\alpha, \rho, \eta )\)pseudoinvex functions, respectively.
Example 2.1
Let \(X=\{x=(x_{1},x_{2})\in\mathbb{R}^{2}: x_{2}>x_{1}>0\}\cup\{(0,0)\}\). For arbitrary vectors \(x, y\in X\), \(x=(x_{1},x_{2})\) and \(y=(y_{1},y_{2})\), consider the functions \(\varphi: X\rightarrow \mathbb{R}\), \(\alpha: X\times X\rightarrow \mathbb{R}\), \(\rho: X\times X\rightarrow\mathbb{R}\), \(\eta: X\times X\rightarrow \mathbb{R}^{n}\), defined by
By a direct calculation, we get that
Therefore,
If \(\varphi(y)\varphi(x)=x_{2}y_{2}\geq 0\), then
Thus, it holds that
If \(\varphi(y)\varphi(x)=x_{2}y_{2}< 0\), then
Thus, one has
So, we have verified that Ï† is an \((\alpha,\rho,\eta)\)invex function on X.
Example 2.2
Let \(X=\{x=(x_{1},x_{2})\in\mathbb{R}^{2}: x_{2}>x_{1}>0\}\). For arbitrary vectors \(x, y\in X\), \(x=(x_{1},x_{2})\) and \(y=(y_{1},y_{2})\), consider the functions \(\varphi: X\rightarrow \mathbb{R}\), \(\alpha: X\times X\rightarrow \mathbb{R}\), \(\rho: X\times X\rightarrow\mathbb{R}\), \(\eta: X\times X\rightarrow \mathbb{R}^{n}\), defined by
By a direct calculation, we derive
Therefore, it yields
If
then
This shows that Ï† is \((\alpha, \rho, \eta )\)pseudoinvex on X.
In view of the concepts of monotonicities given by AlHomidan et al. [20], we introduce \((\alpha, \rho, \eta )\)monotonicity for a realvalued function, which will be helpful in proving our results.
Definition 2.6
Let \(X\subseteq\mathbb{R}^{n}\), Î±, \(\rho: X\times X\rightarrow \mathbb{R}\), \(\eta:X\times X\rightarrow X\). The function \(\varphi:X\rightarrow\mathbb{R}\) is said to be \((\alpha,\rho,\eta)\)monotone on X if
From now on, unless otherwise specified, we always assume that \(X\subseteq\mathbb{R}^{n}\) is an Î·invex set, \(f(x)= (f_{1}(x),f_{2}(x),\ldots,f_{p}(x) )\) , \(f_{i}:X \rightarrow \mathbb{R}\) and \(i\in P=\{1,2,\ldots,p\}\) .
Consider the following multiobjective programming problem (MP):
In multiobjective programming problems, objectives often conflict with each other. In this regard, the concepts of efficient and weak efficient solutions are widely used.
Definition 2.7
A point \(\bar{x}\in X\) is said to be an efficient solution of (MP) if there exists no \(x\in X\) such that
with strict inequality holding for at least one i.
Definition 2.8
A point \(\bar{x}\in X\) is said to be a weak efficient solution of (MP) if there exists no \(x\in X\) such that
Next, we are about to introduce the following Stampacchia and Minty vector variationallike inequalities, respectively, with also their weak formulations, which will be used to ensure the efficient solutions of the problem (MP). Let \(\rho_{i}:X\times X \rightarrow \mathbb{R}\), \(i\in P\).
(SVVI) For given realvalued functions Î± and \(\rho_{i}\), \(i\in P\), find \(\bar{x}\in X\) such that there is no \(x\in X\)fulfilling
(MVVI) For given realvalued functions Î± and \(\rho_{i}\), \(i\in P\), find \(\bar{x}\in X\) such that there is no \(x\in X\) fulfilling
(WSVVI) For given realvalued functions Î± and \(\rho_{i}\), \(i\in P\), find \(\bar{x}\in X\) such that there is no \(x\in X\) fulfilling
(WMVVI) For given realvalued functions Î± and \(\rho_{i}\), \(i\in P\), find \(\bar{x}\in X\) such that there is no \(x\in X\) fulfilling
Remark 2.2
Let \(\alpha \in \mathbb{R}_{++}\) and \(\rho_{i}=0\) in above (SVVI) (respectively (WSVVI)), then this problem reduces to the vector variational inequality (respectively weak) introduced by Farajzadeh and Lee [9].
Example 2.3
Let \(X=\{x=(x_{1},x_{2})\in\mathbb{R}^{2}: x_{2}>x_{1}>0\}\) and \(\bar{x}=(1,0)\). Consider the functions \(f: X\rightarrow \mathbb{R}^{2}\), \(\alpha: X\times X\rightarrow \mathbb{R}\) and \(\rho_{i}: X\times X\rightarrow \mathbb{R}\), \(i=\{1, 2\}\), defined by
respectively, where
Further, define \(\eta(x, \bar{x})=x=(x_{1}, x_{2})\), \(\forall x\in X\), then we have
and
Therefore,
Thus, for every \(x\in X\), we arrive at
which shows that xÌ„ is a solution to (SVVI).
3 The relationships between vector variationallike inequalities and multiobjective programming problems
In this section, we shall examine the relationships between Stampacchia and Minty vector variationallike inequalities and multiobjective programming problems in terms of \((\alpha,\rho,\eta)\)invex functions, which are formulated in SectionÂ 2.
Theorem 3.1
Let Î±, \(\rho_{i}: X\times X\rightarrow \mathbb{R}\), \(\eta: X\times X\rightarrow \mathbb{R}^{n}\) and for each \(i\in P\), \(f_{i}\) be \((\alpha,\rho_{i},\eta)\)invex function at \(\bar{x}\in X\). If xÌ„ solves (SVVI), then xÌ„ is an efficient solution of (MP).
Proof
Suppose that xÌ„ is not an efficient solution of (MP), then there exists \(\hat{x}\in X\) such that
and there exists at least one \(i_{0}\in P\) such that
Since \(f_{i}\) is an \((\alpha,\rho_{i},\eta)\)invex function at \(\bar{x}\in X\), it follows from Definition 2.4 that
Combining inequalities (3.1), (3.2) and (3.3), we obtain that there exists \(\hat{x}\in X\) such that
and there is at least one \(i_{0}\in P\) such that
This indicates that there exists \(\hat{x}\in X\) such that
which is a contradiction to the fact that xÌ„ is a solution to (SVVI).â€ƒâ–¡
Theorem 3.2
Let Î±, \(\rho_{i}: X\times X\rightarrow \mathbb{R}\), \(\eta: X\times X\rightarrow \mathbb{R}^{n}\) and for each \(i\in P\), \(f_{i}\) be an \((\alpha,\rho_{i},\eta)\)pseudoinvex function at \(\bar{x}\in X\). If xÌ„ solves (WSVVI), then xÌ„ is a weak efficient solution of (MP).
Proof
We proceed by contradiction. Assume that xÌ„ is not a weak efficient solution of (MP), then there exists \(\hat{x}\in X\) such that
Because \(f_{i}\) is \((\alpha,\rho_{i},\eta)\)pseudoinvex at \(\bar{x}\in X\), we derive
That is,
which contradicts the hypothesis that xÌ„ solves (WSVVI).â€ƒâ–¡
Theorem 3.3
Let Î±, \(\rho_{i}: X\times X\rightarrow \mathbb{R}\), \(\eta: X\times X\rightarrow \mathbb{R}^{n}\) and for each \(i\in P\), \(f_{i}\) be an \((\alpha,\rho_{i},\eta)\)invex function on X. If \(\bar{x}\in X\) is an efficient solution of (MP), then xÌ„ solves (MVVI).
Proof
Suppose that xÌ„ does not solve (MVVI), then there exists \(\hat{x}\in X\) satisfying
that is,
and there is at least one \(i_{0}\in P\) such that
Noticing that \(f_{i}\) is \((\alpha,\rho_{i},\eta)\)invex on X, we have
Together with inequalities (3.4), (3.5) and (3.6), it follows that there exists \(\hat{x}\in X\) satisfying
which leads to a contradiction, that xÌ„ is an efficient solution of (MP).â€ƒâ–¡
Theorem 3.4
Let Î±, \(\rho_{i}: X\times X\rightarrow \mathbb{R}\), \(\eta: X\times X\rightarrow \mathbb{R}^{n}\) and for each \(i\in P\), \(f_{i}\) be an \((\alpha,\rho_{i},\eta)\)invex function on X. If \(\bar{x}\in X\) solves (WSVVI), then xÌ„ solves (WMVVI).
Proof
Suppose that xÌ„ solves (WSVVI), then there exists no \(x\in X\) such that
i.e., there exists no \(x\in X\) such that
Since \(f_{i}\) is \((\alpha,\rho_{i},\eta)\)invex on X, one has
Similarly, we can derive
Adding inequalities (3.8) and (3.9), we get
Combining inequalities (3.7) and (3.10), we derive that there exists no \(x\in X\) such that
i.e., there exists no \(x\in X\) satisfying
which implies that xÌ„ solves (WMVVI).â€ƒâ–¡
Theorem 3.5
Let Î±, \(\rho_{i}: X\times X\rightarrow \mathbb{R}\), \(\eta: X\times X\rightarrow \mathbb{R}^{n}\) and for each \(i\in P\), \(f_{i}\) be a strictly \((\alpha,\rho_{i},\eta)\)invex function on X. If \(\bar{x}\in X\) is a weak efficient solution of (MP), then xÌ„ solves (MVVI).
Proof
Suppose that xÌ„ is a weak efficient solution of (MP) but does not solve (MVVI), then there exists \(\hat{x}\in X\) \((\hat{x}\neq\bar{x})\) satisfying
or equivalently,
and there is at least one \(i_{0}\in P\) such that
Noticing that \(f_{i}\) is strictly \((\alpha,\rho_{i},\eta)\)invex on X, we obtain
By inequalities (3.11), (3.12) and (3.13), we get that there exists \(\hat{x}\in X\) satisfying
which contradicts the fact that xÌ„ is an efficient solution of (MP).â€ƒâ–¡
Theorem 3.6
Let Î±, \(\rho_{i}: X\times X\rightarrow \mathbb{R}\), \(\eta: X\times X\rightarrow \mathbb{R}^{n}\) and for each \(i\in P\), \(f_{i}\) be an \((\alpha,\rho_{i},\eta)\)invex function on X. If \(\bar{x}\in X\) is a weak efficient solution of (MP), then xÌ„ solves (WMVVI).
Proof
Assuming that xÌ„ does not solve (WMVVI), therefore, there exists \(\hat{x}\in X\) such that
that is,
Noticing that each \(f_{i}\) is \((\alpha,\rho_{i},\eta)\)invex on X, we derive
We conclude from (3.14) and (3.15) that there exists \(\hat{x}\in X\) satisfying
which leads to a contradiction that xÌ„ is the weak efficient solution of (MP).â€ƒâ–¡
4 The solution existence of vector variationallike inequalities
This section is devoted to a discussion of the existence of solutions for Stampacchia and Minty vector variationallike inequalities under the assumption of \((\alpha,\rho,\eta)\)monotonicity. Now, we assume that Y is a topological vector space, \(X\subseteq\mathbb{R}^{n}\) is a convex set, and \(f_{i}:X \rightarrow \mathbb{R}\), \(i\in P\).
Definition 4.1
see [21]
Let E be a nonempty subset of a topological vector space Y. AÂ multifunction \(\psi: E\rightarrow2^{Y}\) is a KKM mapping if for any finite subset \(\{x_{1}, x_{2}, \ldots , x_{n}\}\) of E, it satisfies
where \(co\{x_{1}, x_{2}, \ldots , x_{n}\}\) denotes the convex hull of \(\{x_{1}, x_{2}, \ldots , x_{n}\}\).
Lemma 4.1
see [21] (KKMFan theorem)
Let E be a nonempty convex subset of a topological vector space Y, and let \(\psi: E\mapsto 2^{Y}\) be a KKM mapping with closed values. If there is a point \(x_{0}\in E\) such that \(\psi(x_{0})\) is compact, then
Theorem 4.1
Let Î±, \(\rho_{i}: X\times X\rightarrow \mathbb{R}\), \(i\in P\), \(\eta: X\times X\rightarrow \mathbb{R}^{n}\). Assume that

(i)
for each \(i\in P\), \(f_{i}\) is \((\alpha,\rho_{i},\eta)\)monotone on X;

(ii)
\(\alpha(x,y) \in \mathbb{R}_{++}\), \(x\neq y\), \(x,y\in X\). \(\rho_{i}(x,y)\in \mathbb{R}_{+}\), \(x,y\in X\);

(iii)
Î± and \(\rho_{i}\) are affine functions with respect to their second argument such that \(\alpha(x,x)=0\) and \(\rho_{i}(x,x)=0\), \(\forall x\in X\);

(iv)
the setvalued map \(\Gamma: X\rightarrow2^{X}\) defined by
$$\begin{aligned} \Gamma(x) =&\bigl\{ y\in X:\bigl(\alpha(x,y)f_{1}^{\prime}\bigl(y,\eta(x,y) \bigr)+\rho_{1}(x,y),\ldots, \\ & \alpha(x,y)f_{p}^{\prime} \bigl(y,\eta(x,y)\bigr)+\rho_{p}(x,y)\bigr)\nleqslant 0, \forall x\in X\bigr\} , \end{aligned}$$is closed valued;

(v)
there exist a nonempty compact set \(M\subset X\) and a nonempty compact convex set \(N\subset X\) such that for each \(y\in X\backslash M\), there exists \(x\in N\) such that \(y \notin \Gamma(x)\).
Then (SVVI) is solvable on X.
Proof
Define a setvalued map \(\hat{\Gamma}(x): X\rightarrow2^{X}\) as
It is obvious that \(x\in \Gamma(x)\cap\hat{\Gamma}(x)\). Therefore, \(\Gamma(x)\) and \(\hat{\Gamma}(x)\) are nonempty. Firstly, we will demonstrate that Î“Ì‚ is a KKM map on X. Indeed, suppose to the contrary that \(\hat{\Gamma}(x)\) is not a KKM map, then there exists \(\{x_{1}, x_{2}, \ldots, x_{n}\}\subset X\), \(t_{j}\geq0\), \(j=1, 2, \ldots, n\), with \(\sum_{j=1}^{n}t_{j}=1\) such that
Hence, for any \(j=1, 2, \ldots, n\),
that is,
and there is at least one \(i_{0}\in P\) such that
Therefore, for all \(i\in P\), \(j=1,2,\ldots,n\), we have
This is a contradiction. Therefore, Î“Ì‚ is a KKM map on X. Afterwards, it is necessary to verify that \(\hat{\Gamma}(x)\subset\Gamma(x)\), \(\forall x\in X\). If \(\bar{x}\notin \Gamma(x)\), then there exists \(x\in X\) such that
with strict inequality holding for at least one i. Because \(f_{i}\) is \((\alpha,\rho_{i},\eta)\)monotone on X, for all \(i\in P\) and \(x \in X\), we have
This leads to
Thus,
Therefore, we arrive at
Combining the above inequality with (4.2) yields
with strict inequality holding for at least one i. So, we get that there exists \(x\in X\) satisfying
Therefore, \(\bar{x}\notin \hat{\Gamma}(x)\). Up to now, we have shown that \(\hat{\Gamma}(x)\subset\Gamma(x) \) for any \(x\in X\). Hence, Î“ is also a KKM map. From hypotheses (iv) and (v), we know that \(\Gamma(x)\) is a closed subset of the compact set. Thus, we derive that \(\hat{\Gamma}(x)\) is also a compact set. Now, it follows from LemmaÂ 4.1 that
which means there exists no \(x\in X\) satisfying
Therefore, xÌ„ is a solution of (SVVI).â€ƒâ–¡
We end this paper by presenting the following existence theorem for Minty vector variationallike inequality (MVVI). We omit its proof because it can be proven along similar lines of Theorem 4.1.
Theorem 4.2
Let Î±, \(\rho_{i}: X\times X\rightarrow \mathbb{R}\), \(i\in P\), \(\eta: X\times X\rightarrow \mathbb{R}^{n}\). Assume that

(i)
for each \(i\in P\), \(f_{i}\) is \((\alpha,\rho_{i},\eta)\)monotone on X;

(ii)
Î± and \(\rho_{i}\) are affine functions with respect to their first argument such that \(\alpha(x,x)=0\) and \(\rho_{i}(x,x)=0\), \(\forall x\in X\);

(iii)
the setvalued map \(\Gamma: X\rightarrow2^{X}\) defined by
$$\begin{aligned} \Gamma(x) =&\bigl\{ y\in X: \bigl(\alpha(y, x)f_{1}^{\prime}\bigl(x, \eta(y, x)\bigr)+\rho_{1}(y, x), \ldots, \\ &\alpha(y, x)f_{p}^{\prime} \bigl(x, \eta(y, x)\bigr)+\rho_{p}(y, x)\bigr)\nleqslant 0, \forall x\in X \bigr\} , \end{aligned}$$is closed valued;

(iv)
there exist a nonempty compact set \(M\subset X\) and a nonempty compact convex set \(N\subset X\) such that for each \(y\in X\backslash M\), there exists \(x\in N\) such that \(y \notin \Gamma(x)\).
Then (MVVI) is solvable in X.
5 Conclusion and remarks
In this paper, we have introduced a kind of generalized invex functions, termed \((\alpha,\rho,\eta)\)invex functions, and Stampacchia and Minty vector variationallike inequalities associated with the introduced extended convexities. Moreover, we have also demonstrated the relationships between the discussed vector variationallike inequalities and nonsmooth multiobjective programming problems involving \((\alpha,\rho,\eta)\)invex functions. Finally, the existence results for Stampacchia and Minty vector variationallike inequalities are proposed by utilizing the KKMFan theorem. The results of this note extend some earlier results of Farajzadeh and Lee [9] to a more general class of functions.
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Acknowledgements
This research was supported by the Natural Science Foundation of China under Grant No. 11361001; Natural Science Foundation of Ningxia under Grant No. NZ14101.
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Li, R., Yu, G. A class of generalized invex functions and vector variationallike inequalities. J Inequal Appl 2017, 70 (2017). https://doi.org/10.1186/s1366001713458
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DOI: https://doi.org/10.1186/s1366001713458