Skip to main content

General convergence analysis of projection methods for a system of variational inequalities in q-uniformly smooth Banach spaces

Abstract

In this paper, we introduce and consider a system of variational inequalities involving two different operators in q-uniformly smooth Banach spaces. We suggest and analyze a new explicit projection method for solving the system under some more general conditions. Our results extend and unify the results of Verma (Appl. Math. Lett. 18:1286-1292, 2005) and Yao, Liou and Kang (J. Glob. Optim., 2011, doi:10.1007/s10898-011-9804-0) and some other previously known results.

MSC:47H05, 47H10, 47J25.

1 Introduction

Let E and E be a real Banach space and the dual space of E, respectively. Let C be a subset of E and q>1 be a real number. The generalized duality mapping J q :E E is defined by

J q (x)= { x E : x , x = x q , x = x q 1 }

for all x in E. In particular, J= J 2 is called the normalized duality mapping and J q (x)= x q 2 J 2 (x) for x0. If E is a Hilbert space, then J=I, the identity mapping. It is well known that if E is smooth, then J q is single-valued, which is denoted by j q .

Recall the variational inequality problem of finding x C such that

T x , x x 0,xC,
(1.1)

where T:CH is a nonlinear mapping. Variational inequalities theory, which was introduced by Stampacchia [1], emerged as an interesting and fascinating branch of mathematical and engineering sciences. The ideas and techniques of variational inequalities have been applied in structural analysis, economics, optimization, operations research fields. It has been shown that variational inequalities provide the most natural, direct, simple and efficient framework for a general treatment of some unrelated problems arising in various fields of pure and applied sciences. In recent years, there have been considerable activities in the development of numerical techniques including projection methods, Wiener-Hopf equations, auxiliary principle and descent framework for solving variational inequalities; see [114] and the references therein. These activities have motivated us to generalize and extend the variational inequalities and related optimization problems in several directions using novel techniques.

Recently, Verma [11] proved the strong convergence of two-step projection method for solving the following system of variational inequality problems in a Hilbert space: Find ( x , y )C×C such that

{ ρ T ( y ) + x y , x x 0 , x C , ρ > 0 , η T ( x ) + y x , x y 0 , x C , η > 0 .
(1.2)

In order to solve problem (1.2), Verma [11] introduced the following projection method:

{ x n + 1 = ( 1 α n ) x n + α n P C [ y n ρ T ( y n ) ] , y n = ( 1 β n ) x n + β n P C [ x n η T ( x n ) ] , n 0 ,
(1.3)

where P C is the projection of a Hilbert space H onto C. This method contains several previously known projection schemes as special cases, while some have been applied to problems arising, especially, from complementarity problems, convex quadratic programming and other variational problems; see [24, 11] and the references therein.

Very recently, Yao et al. [12] considered the following system of variational inequality problems in 2-uniformly smooth Banach spaces: Find ( x , y )C×C such that

{ ρ T 1 ( y ) + x y , j ( x x ) 0 , x C , ρ > 0 , η T 2 ( x ) + y x , j ( x y ) 0 , x C , η > 0 ,
(1.4)

where T 1 , T 2 are two different nonlinear operators. Moreover, they modified projection method to system (1.4) in Banach spaces and introduced the following iterative method:

{ x n + 1 = ( 1 α n ) x n + α n Π C [ y n ρ T 1 ( y n ) ] , y n = Π C [ x n η T 2 ( x n ) ] , n 0 ,
(1.5)

where Π C is a sunny nonexpansive retraction from E onto C.

One question arises naturally: Do Yao et al.’s new projection methods work for two bivariate nonlinear operators in 2-uniformly smooth Banach spaces, or more generally, in q-uniformly smooth Banach spaces with q>1, under more general control conditions?

In order to give some affirmative answers to the question raised above, we introduce the following system of bivariate variational inequality problems in q-uniformly smooth Banach spaces: Find ( x , y )C×C such that

{ ρ T 1 ( y , x ) + x y , j q ( x x ) 0 , x C , ρ > 0 , η T 2 ( x , y ) + y x , j q ( x y ) 0 , x C , η > 0 .
(1.6)

The purpose of this paper is not only to show that the projection technique can be extended to the system of bivariate variational inequality problems in q-uniformly smooth Banach spaces, but also to suggest and analyze a new explicit iterative method, which includes the previously known projection methods as special cases, and whose convergence analysis is proved under some more general conditions. Our results extend and unify the corresponding results of [5, 7, 11, 12] and many others.

2 Preliminaries

Let E and E be a real Banach space and the dual space of E, respectively. Let C be a nonempty closed convex subset of E, and let S(E)={xE:x=1}. Then the norm of E is said to be Gâteaux differentiable if the following limit

lim t 0 x + t y x t

exists for each x,yS(E). In this case, E is called smooth. The norm of E is said to be uniformly Gâteaux differentiable if for each yS(E), the limit above is attained uniformly for xS(E). The norm of E is called Fréchet differentiable if for each xS(E), the limit above is attained uniformly for yS(E). The norm of E is called uniformly Fréchet differentiable if the limit above is attained uniformly for x,yS(E). It is well known that (uniform) Fréchet differentiability of the norm of E implies (uniform) Gâteaux differentiability of the norm of E.

Recall that ψ E :[0,)[0,), the modulus of smoothness of E, is defined by

ψ E (t)=sup { 1 2 ( x + y + x y ) 1 : x S ( E ) , y t } .

A Banach space E is said to be uniformly smooth if ψ E ( t ) t as t0. A Banach space E is said to be q-uniformly smooth, if there exists a fixed constant c>0 such that ψ E (t)c t q (q>1). It is well known that E is uniformly smooth if and only if the norm of E is uniformly Fréchet differentiable. If E is q-uniformly smooth, then q2 and E is uniformly smooth, and hence the norm of E is uniformly Fréchet differentiable, in particular, the norm of E is Fréchet differentiable. It is well known that Hilbert and Lebesgue L p (p>1) spaces are uniformly smooth. More precisely, L p is min{p,2}-uniformly smooth for every p>1.

In order to prove our main results, we also need the following concepts and lemmas.

Let C be a nonempty closed and convex subset of a real Banach space E, and let K be a nonempty subset of C. Let Π C :CK be a mapping, and Π C is said to be:

  1. (a)

    sunny if for each xC and t[0,1], we have Π C [tx+(1t) Π C ]= Π C x;

  2. (b)

    a retraction of C onto K if Π C x=x, xK;

  3. (c)

    a sunny nonexpansive retraction if Π C is sunny, nonexpansive and retraction onto K.

Definition 2.1 An operator T:CE is said to be μ-Lipschitz continuous in the first variable if there exists a constant μ>0 such that

T ( x , ) T ( y , ) μxy,x,yC.

Definition 2.2 Let C be a nonempty closed convex subset of a smooth Banach space E, and J q :E E is a generalized duality mapping. A bivariate operator T:C×CE is said to be:

  1. (i)

    r-strongly accretive in the first variable if there exists a constant r>0 such that

    T ( x , ) T ( y , ) , j q ( x y ) r x y q ,x,yC.
  2. (ii)

    γ-cocoercive in the first variable if there exists a constant γ>0 such that

    T ( x , ) T ( y , ) , j q ( x y ) γ T ( x , ) T ( y , ) q ,x,yC.
  3. (iii)

    relaxed (γ,r)-cocoercive in the first variable if there exist constants γ,r>0 such that

    T ( x , ) T ( y , ) , j q ( x y ) γ T ( x , ) T ( y , ) q +r x y q ,x,yC.

Remark 2.1 In Definition 2.2, the r-strongly accretive operator includes the r-strongly monotone and the r-strongly accretive ones defined in [11, 12] as special cases.

Remark 2.2 Obviously, an r-strongly accretive operator must be a relaxed (γ,r)-cocoercive whenever γ=0, but the converse is not true. Therefore the relaxed (γ,r)-cocoercive operator is more general than r-strongly accretive one.

Lemma 2.1 [15]

Let E be a real q-uniformly smooth Banach space, then there exists a constant C q >0 such that

x + y q x q +q y , j q ( x ) + C q y q ,x,yE.

In particular, if E is a real 2-uniformly smooth Banach space, then there exists the best smooth constant K>0 such that

x + y 2 x 2 +2 y , j ( x ) +2 K y 2 ,x,yE.

Lemma 2.2 [16, 17]

Let C be a nonempty subset of a smooth Banach space E, and let Π C :EC be a retraction. Then Π C is sunny and nonexpansive if and only if

u Π C [ u ] , J ( y Π C [ u ] ) 0,uE,yC.

Lemma 2.3 [18]

Assume that { a n } is a sequence of nonnegative real numbers such that

a n + 1 (1 γ n ) a n + σ n ,n0,

where { γ n } is a sequence in (0,1) and { σ n } is a sequence in R such that

  1. (i)

    n = 0 γ n =;

  2. (ii)

    lim sup n σ n / γ n 0 or n = 0 | σ n |<.

Then lim n a n =0.

3 Main results

By Lemma 2.2, we establish the equivalence between the system of variational inequalities (1.6) and the fixed point problem with projection technique, that is, ( x , y )C×C is a solution of the system of variational inequalities (1.6) if and only if

x = Π C [ y ρ T 1 ( y , x ) ] ,
(3.1a)
y = Π C [ x η T 2 ( x , y ) ] .
(3.1b)

This alternative formula enables us to suggest and analyze a two-step explicit projection method for solving system (1.6), and this is the main motivation of our next result.

Theorem 3.1 Let C be a nonempty closed convex subset of a q-uniformly smooth Banach space E. Let T i :C×CE be relaxed ( γ i , r i )-cocoercive and μ i -Lipschitz continuous in the first variable, i=1,2. For arbitrarily chosen initial points ( x 0 , y 0 )C×C, define sequences { x n } and { y n } in the following manner:

{ x n + 1 = ( 1 α n ) x n + α n Π C [ y n ρ T 1 ( y n , x n ) ] , y n + 1 = ( 1 β n ) x n + 1 + β n Π C [ x n + 1 η T 2 ( x n + 1 , y n ) ] ,
(3.2)

where Π C is a sunny nonexpansive retraction from E onto C, the following conditions are satisfied:

  1. (i)

    α n , β n [0,1], n = 0 α n = and n = 0 α n (1 β n 1 )<;

  2. (ii)

    0<ρ< ( q ( r 1 γ 1 μ 1 q ) C q μ 1 q ) 1 q 1 and 0<η< ( q ( r 2 γ 2 μ 2 q ) C q μ 2 q ) 1 q 1 .

Then the sequences { x n } and { y n } converge to x and y , respectively, where ( x , y ) is a solution of the system of variational inequalities (1.6).

Proof Let ( x , y )C×C be a solution of (1.6). From (3.1a) and (3.2), we have

x n + 1 x = ( 1 α n ) x n + α n Π C [ y n ρ T 1 ( y n , x n ) ] x ( 1 α n ) x n x + α n Π C [ y n ρ T 1 ( y n , x n ) ] Π C [ y ρ T 1 ( y , x ) ] ( 1 α n ) x n x + α n y n y ρ ( T 1 ( y n , x n ) T 1 ( y , x ) ) .
(3.3)

Since the operator T 1 is relaxed ( γ 1 , r 1 )-cocoercive and μ 1 -Lipschitz continuous definition in the first variable, it follows from Lemma 2.1 that

y n y ρ ( T 1 ( y n , x n ) T 1 ( y , x ) ) q = y n y q ρ q T 1 ( y n , x n ) T 1 ( y , x ) , j q ( y n y ) + C q ρ q T 1 ( y n , x n ) T 1 ( y , x ) q y n y q ρ q ( γ 1 T 1 ( y n , x n ) T 1 ( y , x ) q + r 1 y n y q ) + C q ρ q T 1 ( y n , x n ) T 1 ( y , x ) q ( 1 ρ q r 1 + ρ q γ 1 μ 1 q + C q ρ q μ 1 q ) y n y q ,

which implies that

y n y ρ ( T 1 ( y n , x n ) T 1 ( y , x ) ) θ 1 y n y ,
(3.4)

where θ 1 = 1 ρ q r 1 + ρ q γ 1 μ 1 q + C q ρ q μ 1 q q . We can obtain θ 1 (0,1) by condition (ii). Substituting (3.4) into (3.3), we have

x n + 1 x (1 α n ) x n x + α n θ 1 y n y .
(3.5)

Similarly, it follows from (3.1b) and (3.2) that

y n + 1 y = ( 1 β n ) x n + 1 + β n Π C [ x n + 1 η T 2 ( x n + 1 , y n ) ] y ( 1 β n ) x n + 1 y + β n Π C [ x n + 1 η T 2 ( x n + 1 , y n ) ] Π C [ x η T 2 ( x , y ) ] ( 1 β n ) x n + 1 y + β n x n + 1 x η ( T 2 ( x n + 1 , y n ) T 2 ( x , y ) ) .
(3.6)

Since the operator T 2 is relaxed ( γ 2 , r 2 )-cocoercive and μ 2 -Lipschitz continuous definition in the first variable, it follows that

x n + 1 x η ( T 2 ( x n + 1 , y n ) T 2 ( x , y ) ) q = x n + 1 x q η q T 2 ( x n + 1 , y n ) T 2 ( x , y ) , j q ( x n + 1 x ) + C q η q T 2 ( x n + 1 , y n ) T 2 ( x , y ) q x n + 1 x q η q ( γ 2 T 2 ( x n + 1 , y n ) T 2 ( x , y ) q + r 2 x n + 1 x q ) + C q η q T 2 ( x n + 1 , y n ) T 2 ( x , y ) q ( 1 η q r 2 + η q γ 2 μ 2 q + C q η q μ 2 q ) x n + 1 x q ,

which implies that

x n + 1 x η ( T 2 ( x n + 1 , y n ) T 2 ( x , y ) ) θ 2 x n + 1 x ,
(3.7)

where θ 2 = 1 η q r 2 + η q γ 2 μ 2 q + C q η q μ 2 q q . We can obtain θ 2 (0,1) from condition (ii). Substituting (3.7) into (3.6), we have

y n + 1 y ( 1 β n ) x n + 1 y + β n θ 2 x n + 1 x ( 1 β n ) x n + 1 x + ( 1 β n ) x y + β n θ 2 x n + 1 x = [ 1 ( 1 θ 2 ) β n ] x n + 1 x + ( 1 β n ) x y < x n + 1 x + ( 1 β n ) x y ,
(3.8)

that is,

y n y x n x +(1 β n 1 ) x y .
(3.9)

It follows from (3.5) and (3.9) that

x n + 1 x ( 1 α n ) x n x + α n θ 1 ( x n x + ( 1 β n 1 ) x y ) [ 1 ( 1 θ 1 ) α n ] x n x + α n ( 1 β n 1 ) x y .
(3.10)

Since 1 θ 1 >0, n = 0 α n = and n = 0 α n (1 β n 1 )<, we apply Lemma 2.3 to get

lim n x n x =0.
(3.11)

Combining condition (ii), (3.9) and (3.11), we have

lim n y n y =0.
(3.12)

It shows that lim n x n = x , lim n y n = y , respectively, satisfying the system of variational inequalities (1.6). This completes the proof. □

Theorem 3.2 Let C be a nonempty closed convex subset of a q-uniformly smooth Banach space E. Let T i :C×CE be r i -strongly accretive and μ i -Lipschitz continuous in the first variable, i=1,2. For arbitrarily chosen initial points ( x 0 , y 0 )C×C, define sequences { x n } and { y n } in the following manner:

{ x n + 1 = ( 1 α n ) x n + α n Π C [ y n ρ T 1 ( y n , x n ) ] , y n + 1 = Π C [ x n + 1 η T 2 ( x n + 1 , y n ) ] ,
(3.13)

where Π C is a sunny nonexpansive retraction from E onto C, the following conditions are satisfied:

  1. (i)

    α n (0,1), n = 0 α n =;

  2. (ii)

    0<ρ< ( q r 1 C q μ 1 q ) 1 q 1 and 0<η< ( q r 2 C q μ 2 q ) 1 q 1 .

Then the sequences { x n } and { y n } converge to x and y , respectively, where ( x , y ) is a solution of the system of variational inequalities (1.6).

Proof As γ=0 and β n =1, from Remark 2.2, we know that a relaxed (0,r)-cocoercive operator reduces to r-strongly accretive, and iterative algorithm (3.2) reduces to (3.13), respectively. Then the conclusion follows immediately from Theorem 3.1. This completes the proof. □

If T 1 , T 2 :CE are univariate operators, applying Theorem 3.1 to a 2-uniformly smooth Banach space with constant C q =2 K 2 , we obtain the following result.

Theorem 3.3 Let C be a nonempty closed convex subset of a 2-uniformly smooth Banach space E. Let T i :CE be relaxed ( γ i , r i )-cocoercive and μ i -Lipschitz continuous, i=1,2. For arbitrarily chosen initial points x 0 , y 0 C, define sequences { x n } and { y n } in the following manner:

{ x n + 1 = ( 1 α n ) x n + α n Π C [ y n ρ T 1 ( y n ) ] , y n + 1 = Π C [ x n + 1 η T 2 ( x n + 1 ) ] ,
(3.14)

where Π C is a sunny nonexpansive retraction from E onto C, the following conditions are satisfied:

  1. (i)

    α n [0,1], n = 0 α n =;

  2. (ii)

    0<ρ< r 1 γ 1 μ 1 2 K 2 μ 1 2 and 0<η< r 2 γ 2 μ 2 2 K 2 μ 2 2 .

Then the sequences { x n } and { y n } converge to x and y , respectively, where ( x , y ) is a solution of the system of variational inequalities (1.4).

Since the Hilbert space H is a 2-uniformly smooth Banach space with the best smooth constant K= 2 2 , from Theorem 3.1 we obtain the following result.

Theorem 3.4 Let C be a nonempty closed convex subset of a real Hilbert space H. Let T i :CH be relaxed ( γ i , r i )-cocoercive and μ i -Lipschitz continuous, i=1,2. For arbitrarily chosen initial points x 0 , y 0 C, define sequences { x n } and { y n } in the following manner:

{ x n + 1 = ( 1 α n ) x n + α n P C [ y n ρ T 1 ( y n ) ] , y n + 1 = ( 1 β n ) x n + 1 + β n P C [ x n + 1 η T 2 ( x n + 1 ) ] ,
(3.15)

where P C is the projection from H onto C, the following conditions are satisfied:

  1. (i)

    α n , β n [0,1], n = 0 α n = and n = 0 α n (1 β n 1 )<;

  2. (ii)

    0<ρ< 2 ( r 1 γ 1 μ 1 2 ) μ 1 2 and 0<η< 2 ( r 2 γ 2 μ 2 2 ) μ 2 2 .

Then the sequences { x n } and { y n } converge to x and y , respectively, where ( x , y ) is a solution of the system of variational inequalities (1.2).

Remark 3.1 Theorems 3.1 and 3.3 extend Theorem 3.1 of [12] from a 2-uniformly smooth Banach space to a q-uniformly smooth Banach space. Moreover, the underlying operator T is extended to a bivariate operator, and the property defined on T is more general than [12] in convergence analysis.

Remark 3.2 Theorems 3.1 and 3.4 extend Theorem 3.1 of [5, 7, 11] from a real Hilbert space to a q-uniformly smooth Banach space. Moreover, the property defined on the underlying operator T is extended from r-strongly monotone to relaxed (γ,r)-cocoercive, respectively.

Remark 3.3 Algorithm (3.2) includes the projection methods in [2, 5, 7, 11, 12] as special cases and unifies the previously known one-step and two-step projection-type methods in a q-uniformly smooth Banach space. Furthermore, the computation workload of the present explicit projection method is much less than the implicit algorithm in [5] at each iteration step.

References

  1. Stampacchia G: Formes bilinearies coercivities sur les ensembles convexes. C. R. Acad. Sci. Paris 1964, 258: 4413–4416.

    MathSciNet  Google Scholar 

  2. Verma RU: Projection methods, algorithms and a new system of nonlinear variational inequalities. Comput. Math. Appl. 2001, 41: 1025–1031. 10.1016/S0898-1221(00)00336-9

    Article  MathSciNet  Google Scholar 

  3. Nie H, Liu Z, Kim KH, Kang SM: A system of nonlinear variational inequalities involving strongly monotone and pseudocontractive mappings. Adv. Nonlinear Var. Inequal. 2003, 6: 91–99.

    MathSciNet  Google Scholar 

  4. Noor MA: Some developments in general variational inequalities. Appl. Math. Comput. 2004, 152: 199–277. 10.1016/S0096-3003(03)00558-7

    Article  MathSciNet  Google Scholar 

  5. Verma RU: Generalized system for relaxed cocoercive variational inequalities and projection methods. J. Optim. Theory Appl. 2004, 121(1):203–210.

    Article  MathSciNet  Google Scholar 

  6. Noor MA: Projection methods for nonconvex variational inequalities. Optim. Lett. 2009, 3: 411–418. 10.1007/s11590-009-0121-1

    Article  MathSciNet  Google Scholar 

  7. Huang Z, Noor MA:An explicit projection method for a system of nonlinear variational inequalities with different (γ,r)-cocoercive mappings. Appl. Math. Comput. 2007, 190: 356–361. 10.1016/j.amc.2007.01.032

    Article  MathSciNet  Google Scholar 

  8. Noor MA, Noor KI: Projection algorithms for solving system of general variational inequalities. Nonlinear Anal. 2009, 70: 2700–2706. 10.1016/j.na.2008.03.057

    Article  MathSciNet  Google Scholar 

  9. Noor MA: Some iterative methods for nonconvex variational inequalities. Comput. Math. Model. 2010, 21(1):97–108. 10.1007/s10598-010-9057-7

    Article  MathSciNet  Google Scholar 

  10. Wen DJ: Projection methods for a generalized system of nonconvex variational inequalities with different nonlinear operators. Nonlinear Anal. 2010, 73: 2292–2297. 10.1016/j.na.2010.06.010

    Article  MathSciNet  Google Scholar 

  11. Verma RU: General convergence analysis for two-step projection methods and applications to variational problems. Appl. Math. Lett. 2005, 18: 1286–1292. 10.1016/j.aml.2005.02.026

    Article  MathSciNet  Google Scholar 

  12. Yao Y, Liou YC, Kang SM: Two-step projection methods for a system of variational inequality problems in Banach spaces. J. Glob. Optim. 2011. 10.1007/s10898-011-9804-0

    Google Scholar 

  13. Ceng LC, Teboulle M, Yao JC: Weak convergence of an iterative method for pseudomonotone variational inequalities and fixed-point problems. J. Optim. Theory Appl. 2010, 146: 19–31. 10.1007/s10957-010-9650-0

    Article  MathSciNet  Google Scholar 

  14. Wen DJ, Long XJ, Gong QF: Convergence analysis of projection methods for a new system of general nonconvex variational inequalities. Fixed Point Theory Appl. 2012. 10.1186/1687-1812-2012-59

    Google Scholar 

  15. Xu HK: Inequalities in Banach spaces with applications. Nonlinear Anal. 1991, 16: 1127–1138. 10.1016/0362-546X(91)90200-K

    Article  MathSciNet  Google Scholar 

  16. Reich S: Asymptotic behavior of contractions in Banach spaces. J. Math. Anal. Appl. 1973, 44: 57–70. 10.1016/0022-247X(73)90024-3

    Article  MathSciNet  Google Scholar 

  17. Xu HK: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 2004, 298: 279–291. 10.1016/j.jmaa.2004.04.059

    Article  MathSciNet  Google Scholar 

  18. Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002, 2: 240–256.

    Article  Google Scholar 

Download references

Acknowledgements

Supported by the National Science Foundation of China (11001287), Natural Science Foundation Project of Chongqing (CSTC, 2012jjA00039) and Science and Technology Research Project of Chongqing Municipal Education Commission (KJ130712, KJ130731).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Dao-Jun Wen.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

GQF carried out the primary studies for projection methods for a system of variational inequality, participated in its design and coordination. WDJ participated in the convergence analysis and drafted the manuscript. All authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Gong, QF., Wen, DJ. General convergence analysis of projection methods for a system of variational inequalities in q-uniformly smooth Banach spaces. J Inequal Appl 2013, 481 (2013). https://doi.org/10.1186/1029-242X-2013-481

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2013-481

Keywords