Skip to main content

Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces

Abstract

By using fixed point results on cones, we study the existence and uniqueness of positive solutions for some nonlinear fractional differential equations via given boundary value problems. Examples are presented in order to illustrate the obtained results.

1 Introduction

The field of fractional differential equations has been subjected to an intensive development of the theory and the applications (see, for example, [16] and the references therein). It should be noted that most of papers and books on fractional calculus are devoted to the solvability of linear initial fractional differential equations on terms of special functions. There are some papers dealing with the existence of solutions of nonlinear initial value problems of fractional differential equations by using the techniques of nonlinear analysis such as fixed point results, the Leray-Schauder theorem, stability, etc. (see, for example, [719] and the references therein). In fact, fractional differential equations arise in many engineering and scientific disciplines such as physics, chemistry, biology, economics, control theory, signal and image processing, biophysics, blood flow phenomena and aerodynamics (see, for example, [2023] and the references therein). The main advantage of using the fractional nonlinear differential equations is related to the fact that we can describe the dynamics of complex non-local systems with memory. In this line of taught, the equations involving various fractional orders are important from both theoretical and applied view points. We need the following notions.

Definition 1.1 ([1, 4])

For a continuous function f:[0,)R, the Caputo derivative of fractional order α is defined by

D α c f(t)= 1 Γ ( n α ) 0 t ( t s ) n α 1 f ( n ) (s)ds,

where n1<α<n, n=[α]+1 and [α] denotes the integer part of α.

Definition 1.2 ([1, 4])

The Riemann-Liouville fractional derivative of order α for a continuous function f is defined by

D α f(t)= 1 Γ ( n α ) ( d d t ) n 0 t f ( s ) ( t s ) α n 1 ds ( n = [ α ] + 1 ) ,

where the right-hand side is pointwise defined on (0,).

Definition 1.3 ([1, 4])

Let [a,b] be an interval in and α>0. The Riemann-Liouville fractional order integral of a function f L 1 ([a,b],R) is defined by

I a α f(t)= 1 γ ( α ) a t f ( s ) ( t s ) 1 α ds

whenever the integral exists.

Suppose that E is a Banach space which is partially ordered by a cone PE, that is, xy if and only if yxP. We denote the zero element of E by θ. A cone P is called normal if there exists a constant N>0 such that θxy implies xNy (see [24]). Also, we define the order interval [ x 1 , x 2 ]={xE| x 1 x x 2 } for all x 1 , x 2 E [24]. We say that an operator A:EE is increasing whenever xy implies AxAy. Also, xy means that there exist λ>0 and μ>0 such that λxyμx (see [24]). Finally, put P h ={xE|xh} for all h>θ. It is easy to see that P h P is convex and λ P h = P h for all λ>0. We recall the following in our results. Let E be a real Banach space and let P be a cone in E. Let (a,b) be an interval and let τ and φ be two positive-valued functions such that φ(t)τ(t) for all t(a,b) and τ:(a,b)(0,1) is a surjection. We say that an operator A:PP is τ-φ-concave whenever A(τ(t)x)φ(t)Ax for all t(a,b) and xP [13]. We say that A is φ-concave whenever τ(t)=t for all t [13]. We recall the following result.

Theorem 1.1 ([13])

Let E be a Banach space, let P be a normal cone in E, and let A:PP be an increasing and τ-φ-concave operator. Suppose that there exists θhP such that Ah P h . Then there are u 0 , v 0 P h and r(0,1) such that r v 0 u 0 v 0 and u 0 A u 0 A v 0 v 0 , the operator A has a unique fixed point x [ u 0 , v 0 ], and for x 0 P h and the sequence { x n } with x n =A x n 1 , we have x n x 0.

2 Main results

We study the existence and uniqueness of a solution for the fractional differential equation

D α u(t)+f ( t , u ( t ) ) =0

on partially ordered Banach spaces with two types of boundary conditions and two types of fractional derivatives, Riemann-Liouville and Caputo.

2.1 Existence results for the fractional differential equation with the Riemann-Liouville fractional derivative

First, we study the existence and uniqueness of a positive solution for the fractional differential equation

(2.1)
(2.2)

where D α is the Riemann-Liouville fractional derivative of order α. Let E=C[ε,T]. Consider the Banach space of continuous functions on [ε,T] with the sup norm and set P={yC[ε,T]: min t [ ε , T ] y(t)0}. Then P is a normal cone.

Lemma 2.1 Let 0<ε<T, T1, t[ε,T], η(ε,t) and 0<α<1. Then the problem D α u(t)+f(t,u(t))=0 with the boundary value condition u(η)=u(T) has a solution u 0 if and only if u 0 is a solution of the fractional integral equation

u(t)= ε T G(t,s)f ( s , u ( s ) ) ds,

where

G(t,s)= { t α 1 ( η s ) α 1 t α 1 ( T s ) α 1 ( η α 1 T α 1 ) Γ ( α ) ( t s ) α 1 Γ ( α ) , ε s η t T , t α 1 ( T s ) α 1 ( η α 1 T α 1 ) Γ ( α ) ( t s ) α 1 Γ ( α ) , ε η s t T , t α 1 ( T s ) α 1 ( η α 1 T α 1 ) Γ ( α ) , ε η t s T .

Proof From D α u(t)+f(t,u(t))=0 and the boundary condition, it is easy to see that u(t) c 1 t α 1 = I ε α f(t,u(t)). By the definition of a fractional integral, we get

u(t)= c 1 t α 1 ε t ( t s ) α 1 Γ ( α ) f ( s , u ( s ) ) ds.

Thus, u(η)= c 1 η α 1 ε η ( η s ) α 1 Γ ( α ) f(s,u(s))ds and

u(T)= c 1 T α 1 ε T ( T s ) α 1 Γ ( α ) f ( s , u ( s ) ) ds.

Since u(η)=u(T), we obtain

c 1 = 1 η α 1 T α 1 ε η ( η s ) α 1 Γ ( α ) f ( s , u ( s ) ) ds 1 η α 1 T α 1 ε T ( T s ) α 1 Γ ( α ) f ( s , u ( s ) ) ds.

Hence,

u ( t ) = t α 1 η α 1 T α 1 ε η ( η s ) α 1 Γ ( α ) f ( s , u ( s ) ) d s t α 1 η α 1 T α 1 ε T ( T s ) α 1 Γ ( α ) f ( s , u ( s ) ) d s ε t ( t s ) α 1 Γ ( α ) f ( s , u ( s ) ) d s = ε T G ( t , s ) f ( s , u ( s ) ) d s .

This completes the proof. □

Now, we are ready to state and prove our first main result.

Theorem 2.2 Let 0<ε<T be given and let τ and φ be two functions on (ε,T) such that φ(t)τ(t) for all t(ε,T). Suppose that τ:(ε,T)(0,1) is a surjection and f(t,u(t))C([ε,T]×[0,]) is increasing in u for each fixed t, f(t,u(t))0 and f(t,τ(λ)u(t))φ(λ)f(t,u(t)) for all t,λ(ε,T) and uP. Assume that there exist M 1 >0, M 2 >0 and θhP such that

M 1 h(t) ε T G(t,s)f ( s , h ( s ) ) ds M 2 h(t)

for all t[ε,T], where G(t,s) is the green function defined in Lemma 2.1. Then the problem (2.1) with the boundary value condition (2.2) has a unique positive solution u P h . Moreover, for the sequence u n + 1 = ε T G(t,s)f(s, u n (s))ds, we have u n u 0 for all u 0 P h .

Proof By using Lemma 2.1, the problem is equivalent to the integral equation

u(t)= ε T G(t,s)f ( s , u ( s ) ) ds,

where

G(t,s)= { t α 1 ( η s ) α 1 t α 1 ( T s ) α 1 ( η α 1 T α 1 ) Γ ( α ) ( t s ) α 1 Γ ( α ) , ε s η t T , t α 1 ( T s ) α 1 ( η α 1 T α 1 ) Γ ( α ) ( t s ) α 1 Γ ( α ) , ε η s t T , t α 1 ( T s ) α 1 ( η α 1 T α 1 ) Γ ( α ) , ε η t s T .

Define the operator A:PE by Au(t)= ε T G(t,s)f(s,u(s))ds. Then u is a solution for the problem if and only if u=Au. It is easy to check that the operator A is increasing on P. On the other hand,

A ( τ ( λ ) u ) ( t ) = ε T G ( t , s ) f ( s , τ ( λ ) u ( s ) ) d s φ ( λ ) ε T G ( t , s ) f ( s , u ( s ) ) d s = φ ( λ ) A u ( t )

for all λ[ε,T] and uP. Thus, the operator A is τ-φ-concave. Since

M 1 h(t)Ah(t)= ε T G(t,s)f ( s , h ( s ) ) ds M 2 h(t)

for all t[ε,T], we get Ah P h . Now, by using Theorem 1.1, the operator A has a unique positive solution u P h . This completes the proof. □

Here, we give the following example to illustrate Theorem 2.2.

Example 2.1 Let 0<ε<1 be given. Consider the periodic boundary value problem

D 1 3 u ( t ) + { g ( t ) + [ u ( t ) ] α } = 0 ( t [ ε , 1 ] ) , u ( η ) = u ( 1 ) ,

where η(ε,t), g is continuous on [ε,1] and min t [ ε , 1 ] g(t)>0. Put

G(t,s)= { t 2 / 3 ( η s ) 2 / 3 t 2 / 3 ( 1 s ) 2 / 3 ( η 2 / 3 1 2 / 3 ) Γ ( 1 / 3 ) ( t s ) 2 / 3 Γ ( 1 / 3 ) , ε s η t 1 , t 2 / 3 ( 1 s ) 2 / 3 ( η 2 / 3 1 2 / 3 ) Γ ( 1 / 3 ) ( t s ) 2 / 3 Γ ( 1 / 3 ) , ε η s t 1 , t 2 / 3 ( 1 s ) 2 / 3 ( η 2 / 3 1 2 / 3 ) Γ ( 1 / 3 ) , ε η t s 1 .

Then ε 1 G(t,s)ds= t 2 / 3 ( η ε ) 1 / 3 t 2 / 3 ( 1 ε ) 1 / 3 ( t ε ) 1 / 3 ( η 2 / 3 1 ) Γ ( 4 / 3 ) ( η 2 / 3 1 ) . Now, define τ(t)=t, φ(t)= t 1 / 3 , γ 1 = min t [ ε , 1 ] g(t), γ 2 = max t [ ε , 1 ] g(t) and also f(t,u)=g(t)+ u 1 / 3 for all t. Then τ:(0,1)(0,1) is a surjection and φ(t)>τ(t) for all t(ε,1). For each u0, we have

f ( t , τ ( λ ) u ( t ) ) = f ( t , λ u ( t ) ) = g ( t ) + λ 1 / 3 [ u ( t ) ] 1 / 3 λ 1 / 3 ( g ( t ) + [ u ( t ) ] 1 / 3 ) = φ ( λ ) f ( t , u ( t ) ) .

Now, put h1, M 1 =( γ 1 +1) min t [ ε , 1 ] , η [ ε , 1 ] t 2 / 3 ( 1 ε ) 1 / 3 ( t ε ) 1 / 3 ( η 2 / 3 1 ) Γ ( 4 / 3 ) ( η 2 / 3 1 ) and M 2 =( γ 2 +1) max η [ ε , 1 ] ε 2 / 3 η 1 / 3 Γ ( 4 / 3 ) ( η 2 / 3 1 ) . Then we get

ε 1 G ( t , s ) { g ( s ) + [ h ( s ) ] 1 / 3 } d s ε 1 G ( t , s ) ( γ 2 + 1 ) d s ( γ 2 + 1 ) max t [ ε , 1 ] ε 1 G ( t , s ) d s ( γ 2 + 1 ) ( max η [ ε , 1 ] ε 2 / 3 η 1 / 3 Γ ( 4 / 3 ) ( η 2 / 3 1 ) ) = M 2 h

and

ε 1 G ( t , s ) { g ( s ) + [ h ( s ) ] 1 / 3 } d s ( γ 1 + 1 ) min t [ ε , 1 ] ε 1 G ( t , s ) d s ( γ 1 + 1 ) min t [ ε , 1 ] , η [ ε , 1 ] t 2 / 3 ( 1 ε ) 1 / 3 ( t ε ) 1 / 3 ( η 2 / 3 1 ) Γ ( 4 / 3 ) ( η 2 / 3 1 ) = M 1 h .

Thus, by using Theorem 2.2, the problem has a unique solution in P h = P 1 .

2.2 Existence results for the fractional differential equation with the Caputo fractional derivative

Here, we study the existence and uniqueness of a positive solution for the fractional differential equation

D α c u(t)+f ( t , u ( t ) ) =0 ( t [ 0 , T ] , T 1 , 1 < α < 2 ) ,
(2.3)
u(0)= β 1 u(η),u(T)= β 2 u(η) ( η ( 0 , t ) , 0 < β 1 < β 2 < 1 ) ,
(2.4)

where D α c is the Caputo fractional derivative of order α. Let E=C[0,T] be the Banach space of continuous functions on [0,T] with the sup norm and

P= { y C [ 0 , T ] : min t [ 0 , T ] y ( t ) 0 } .

It is known that P is a normal cone. Similar to the proof of Lemma 2.1, we can prove the following result.

Lemma 2.3 Let 1<α<2, T1, t[0,T], η(0,t) and 0< β 1 < β 2 <1. Then the problem D α c u(t)+f(t,u(t))=0 with the boundary value conditions u(0)= β 1 u(η) and u(T)= β 2 u(η) has a solution u 0 if and only if u 0 is a solution of the fractional integral equation u(t)= 0 T G(t,s)f(s,u(s))ds, where

G(t,s)= { [ β 1 T + t ( β 2 β 1 ) ] ( η s ) α 1 + t ( T s ) α 1 T ( t s ) α 1 T Γ ( α ) , 0 s η t T , t ( T s ) α 1 T ( t s ) α 1 T Γ ( α ) , 0 η s t T , t ( T s ) α 1 T Γ ( α ) , 0 η t s T .

Theorem 2.4 Let T1 be given and let τ and φ be two positive-valued functions on (0,T) such that φ(t)τ(t) for all t(0,T). Suppose that τ:(0,T)(0,1) is a surjection and f(t,u(t))C([ε,T]×[0,]) is increasing in u for each fixed t, f(t,u(t))=0 whenever 0<η<s<t<T and f(t,u(t))0 otherwise, and also f(t,τ(λ)u(t))φ(λ)f(t,u(t)) for all t,λ(0,T) and uP. Assume that there exist M 1 >0, M 2 >0 and θhP such that

M 1 h(t) 0 T G(t,s)f ( s , h ( s ) ) ds M 2 h(t)

for all t[0,T], where G(t,s) is the green function defined in Lemma 2.3. Then the problem (2.3) with the boundary value conditions (2.4) has a unique positive solution u P h . Moreover, for the sequence u n + 1 = ε T G(t,s)f(s, u n (s))ds, we have u n u 0 for all u 0 P h .

Proof It is sufficient to define the operator A:PE by

Au(t)= 0 T G(t,s)f ( s , u ( s ) ) ds.

Now, by using a similar proof of Theorem 2.2, one can show that Au(t)0 for all uP and t[0,T], and also the operator A is τ-φ-concave. By using Theorem 1.1, the operator A has a unique positive solution u P h . This completes the proof by using Lemma 2.3. □

Below we present an example to illustrate Theorem 2.4.

Example 2.2 Let α= 3 2 . Consider the periodic boundary value problem

D α c u ( t ) + g ( t ) + [ u ( t ) ] α = 0 ( t [ 0 , 1 ] ) , u ( 0 ) = 1 3 u ( 1 2 ) u ( 1 ) = 1 2 u ( 1 2 ) ,

where g is a continuous function on [0,1] with min t [ 0 , 1 ] g(t)>0. Put β 2 =η=1/2, β 1 =1/3 and

G(t,s)= { [ 1 3 + 1 6 t ] ( 1 2 s ) 1 / 2 + t ( 1 s ) 1 / 2 ( t s ) 1 / 2 Γ ( 3 / 2 ) , 0 s η t 1 , t ( 1 s ) 1 / 2 ( t s ) 1 / 2 Γ ( 3 / 2 ) , 0 η s t 1 , t ( 1 s ) 1 / 2 Γ ( 3 / 2 ) , 0 η t s 1 .

Then 0 1 G(t,s)ds= [ 1 3 + 1 6 t ] ( 1 2 ) 3 / 2 + t t 3 / 2 Γ ( 5 / 2 ) . Now, define τ(t)=t, φ(t)= t α , γ 1 = min t [ 0 , 1 ] g(t), γ 2 = max t [ 0 , 1 ] g(t) and f(t,u)=g(t)+ u α . Then it is easy to see that τ:(0,1)(0,1) is a surjection map and φ(t)>τ(t) for t(0,1). Also, we have

f ( t , τ ( λ ) u ( t ) ) = f ( t , λ u ( t ) ) = g ( t ) + λ α [ u ( t ) ] α λ α ( g ( t ) + [ u ( t ) ] α ) = φ ( λ ) f ( t , u ( t ) )

for all u0. Now, put h1, M 1 =( γ 1 +1) min t [ 0 , 1 ] 1 3 t ( 1 2 ) 3 / 2 t 3 / 2 Γ ( 5 / 2 ) and also M 2 =( γ 2 +1) 5 6 ( 1 2 ) 3 / 2 + 1 Γ ( 5 / 2 ) . Then we have

0 1 G ( t , s ) { g ( s ) + [ h ( s ) ] 3 / 2 } d s 0 1 G ( t , s ) ( γ 2 + 1 ) d s ( γ 2 + 1 ) max t [ 0 , 1 ] 0 1 G ( t , s ) d s ( γ 2 + 1 ) 5 6 ( 1 2 ) 3 / 2 + 1 Γ ( 5 / 2 ) = M 2 h

and

0 1 G ( t , s ) { g ( s ) + [ h ( s ) ] 3 / 2 } d s ( γ 1 + 1 ) min t [ 0 , 1 ] 0 1 G ( t , s ) d s ( γ 1 + 1 ) min t [ 0 , 1 ] 1 3 t ( 1 2 ) 3 / 2 t 3 / 2 Γ ( 5 / 2 ) = M 1 h .

Thus, by using Theorem 2.4, the problem has a unique solution in P h = P 1 .

References

  1. Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematics Studies 204. In Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.

    Google Scholar 

  2. Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equation. Wiley, New York; 1993.

    Google Scholar 

  3. Oldham KB, Spainer J: The Fractional Calculus. Academic Press, New York; 1974.

    Google Scholar 

  4. Podlubny I: Fractional Differential Equations. Academic Press, New York; 1999.

    Google Scholar 

  5. Samko SG, Kilbas AA, Marichev OI: Fractional Integral and Derivative: Theory and Applications. Gordon & Breach, Switzerland; 1993.

    Google Scholar 

  6. Weitzner H, Zaslavsky GM: Some applications of fractional equations. Commun. Nonlinear Sci. Numer. Simul. 2010, 15: 939-945. 10.1016/j.cnsns.2009.05.004

    Article  MathSciNet  Google Scholar 

  7. Ahmad B, Nieto JJ: Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations. Abstr. Appl. Anal. 2009., 2009: Article ID 494720

    Google Scholar 

  8. Al-Mdallal M, Syam MI, Anwar MN: A collocation-shooting method for solving fractional boundary value problems. Commun. Nonlinear Sci. Numer. Simul. 2010, 15: 3814-3822. 10.1016/j.cnsns.2010.01.020

    Article  MathSciNet  Google Scholar 

  9. Belmekki M, Nieto JJ, Rodriguez-Lopez R: Existence of periodic solution for a nonlinear fractional differential equation. Bound. Value Probl. 2009., 2009: Article ID 324561

    Google Scholar 

  10. Baleanu D, Mohammadi H, Rezapour S: Positive solutions of a boundary value problem for nonlinear fractional differential equations. Abstr. Appl. Anal. 2012., 2012: Article ID 837437

    Google Scholar 

  11. Baleanu D, Mohammadi H, Rezapour S: Some existence results on nonlinear fractional differential equations. Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci. 2013., 371(1990): Article ID 20120144

    Google Scholar 

  12. Baleanu D, Mustafa OG, Agarwal RP: On the solution set for a class of sequential fractional differential equations. J. Phys. A, Math. Theor. 2010., 43(38): Article ID 385209

    Google Scholar 

  13. Zhai C-B, Cao X-M: Fixed point theorems for τ - φ -concave operators and applications. Comput. Math. Appl. 2010, 59: 532-538. 10.1016/j.camwa.2009.06.016

    Article  MathSciNet  Google Scholar 

  14. Delbosco D, Rodino L: Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl. 1996, 204: 609-625. 10.1006/jmaa.1996.0456

    Article  MathSciNet  Google Scholar 

  15. Hashim I, Abdulaziz O, Momani S: Homotopy analysis method for fractional IVPs. Commun. Nonlinear Sci. Numer. Simul. 2009, 14: 674-684. 10.1016/j.cnsns.2007.09.014

    Article  MathSciNet  Google Scholar 

  16. Jafari H, Daftardar-Gejji V: Positive solution of nonlinear fractional boundary value problems using Adomin decomposition method. J. Appl. Math. Comput. 2006, 180: 700-706. 10.1016/j.amc.2006.01.007

    Article  MathSciNet  Google Scholar 

  17. Zhao Y, Sun SH, Han Z: The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 2011, 16: 2086-2097. 10.1016/j.cnsns.2010.08.017

    Article  MathSciNet  Google Scholar 

  18. Zhang S: The existence of a positive solution for nonlinear fractional differential equation. J. Math. Anal. Appl. 2000, 252: 804-812. 10.1006/jmaa.2000.7123

    Article  MathSciNet  Google Scholar 

  19. Zhang S: Existence of positive solutions for some class of nonlinear fractional equation. J. Math. Anal. Appl. 2003, 278: 136-148. 10.1016/S0022-247X(02)00583-8

    Article  MathSciNet  Google Scholar 

  20. Agarwal RP, Lakshmikantam V, Nieto JJ: On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal. 2010, 72: 2859-2862. 10.1016/j.na.2009.11.029

    Article  MathSciNet  Google Scholar 

  21. Baleanu D, Diethelm K, Scalas E, Trujillo JJ Series on Complexity, Nonlinearity and Chaos. In Fractional Calculus: Models and Numerical Methods. World Scientific, Singapore; 2012.

    Google Scholar 

  22. Qiu T, Bai Z: Existence of positive solution for singular fractional equations. Electron. J. Differ. Equ. 2008, 146: 1-9.

    MathSciNet  Google Scholar 

  23. Sabatier J, Agarwal OP, Machado JAT: Advances in Fractional Calculus. Theorical Developments and Applications in Physics and Engineering. Springer, Berlin; 2007.

    Book  Google Scholar 

  24. Rezapour S, Hamlbarani Haghi R: Some notes on the paper ‘Cone metric spaces and fixed point theorems of contractive mappings’. J. Math. Anal. Appl. 2008, 345: 719-724. 10.1016/j.jmaa.2008.04.049

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work is partially supported by the Scientific and Technical Research Council of Turkey. Research of the third and forth authors was supported by Azarbaidjan Shahid Madani University. Also, the authors express their gratitude to the referees for their helpful suggestions which improved final version of this paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Dumitru Baleanu.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Authors contributed equally in writing this article. Authors read and approved the final version of the 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

Baleanu, D., Agarwal, R.P., Mohammadi, H. et al. Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces. Bound Value Probl 2013, 112 (2013). https://doi.org/10.1186/1687-2770-2013-112

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-2770-2013-112

Keywords