Open Access

Existence of positive solutions for a discrete fractional boundary value problem

Advances in Difference Equations20142014:253

https://doi.org/10.1186/1687-1847-2014-253

Received: 18 April 2014

Accepted: 9 September 2014

Published: 25 September 2014

Abstract

This paper is concerned with the existence of positive solutions to a discrete fractional boundary value problem. By using the Krasnosel’skii and Schaefer fixed point theorems, the existence results are established. Additionally, examples are provided to illustrate the effectiveness of the main results.

MSC:26A33, 39A10, 47H07.

Keywords

existencepositive solutiondiscretefractional boundary value problem

1 Introduction

Fractional differential equations have received increasing attention within the last ten years or so. The theory of fractional differential equations has been a new important mathematical branch due to its wide applications in different research areas and engineering, such as physics, chemistry, economics, control of dynamical etc. For more details, see [19] and the references therein. On the other hand, accompanied with the development of the theory for fractional calculus, fractional difference equations have attracted increasing attention slowly but steadily in the past three years or so. Some research papers have appeared, see [1019]. For example, Atici and Eloe [10] analyzed the conjugate discrete fractional boundary value problem (FBVP) with delta derivative:
{ Δ v y ( t ) = f ( t + v 1 , y ( t + v 1 ) ) , t [ 0 , b ] N 0 , y ( v 2 ) = y ( v + b + 1 ) = 0 , 1 < v 2 .
Goodrich [11] studied the discrete fractional boundary value problems:
{ Δ v y ( t ) = λ f ( t + v 1 , y ( t + v 1 ) ) , t [ 0 , T ] Z , y ( v 1 ) = y ( v + T ) + i = 1 N F ( t i , y ( t i ) ) , 0 < v < 1 .
In [12], Lv discussed the existence of solutions for discrete fractional boundary value problems with a p-Laplacian operator:
{ Δ c β [ ϕ p ( Δ c α u ) ] ( t ) = f ( t + α + β 1 , u ( t + α + β 1 ) ) , t [ 0 , b ] N 0 , Δ c α u ( t ) | t = β 1 + Δ c α u ( t ) | t = β + b = 0 , u ( α + β 2 ) + u ( α + β + b ) = 0 , 0 < α , β 1 , 1 < α + β 2 .
They obtained a series of excellent results of discrete fractional boundary value problems. Motivated by the aforementioned works, in this paper we consider a discrete fractional boundary value problem (FBVP):
{ Δ v y ( t ) = f ( t + v 1 , y ( t + v 1 ) ) , t [ 0 , b ] N 0 , y ( v 2 ) = 0 , Δ y ( v 2 ) = Δ y ( v + b 1 ) ,
(1.1)

where 1 < v 2 , Δ v denotes the Riemann-Liouville fractional difference operator, N a = { a , a + 1 , a + 2 , } and I N a = I N a for any number a R and each interval I of R, b N 1 . We appeal to the convention that s = k k 1 y ( s ) = 0 for any k N a , where y is a function defined on N a . By using the Krasnosel’skii and Schaefer fixed point theorems, the existence results are established and two examples are also provided to illustrate the effectiveness of the main results.

The rest of the paper is organized as follows. In Section 2, we introduce some lemmas and definitions which will be used later. In Section 3, the existence of positive solutions for the boundary value problem (1.1) is investigated. In Section 4, two examples are provided to illustrate the effectiveness of the main results.

2 Basic definitions and preliminaries

Firstly we present here some necessary definitions and lemmas which are used throughout this paper.

Definition 2.1 [13, 14]

Define t v ̲ : = Γ ( t + 1 ) Γ ( t + 1 v ) for any t and v for which the right-hand side is defined. If t + 1 v is a pole of the gamma function and t + 1 is not a pole, then t v ̲ = 0 .

Definition 2.2 [15]

The v th fractional sum of a function f, for v > 0 , is defined to be
Δ v f ( t ) = Δ v f ( t ; a ) : = 1 Γ ( v ) s = a t v ( t s 1 ) v 1 ̲ f ( s )
(2.1)

for t { a + v , a + v + 1 , } : = N a + v . Define the v th fractional difference for v > 0 by Δ v f ( t ) : = Δ N Δ v N f ( t ) , t N a + v and N N satisfies 0 N 1 < v N .

Lemma 2.3 [15]

Let t and v be any numbers for which t v ̲ and t v 1 ̲ are defined. Then Δ t v ̲ = v t v 1 ̲ .

Lemma 2.4 [15]

Assume that 0 N 1 < v N . Then
Δ v Δ v y ( t ) = y ( t ) + C 1 t v 1 ̲ + C 2 t v 2 ̲ + + C N t v N ̲
(2.2)

for some C i R , with 1 i N .

Lemma 2.5 (The nonlinear alternative of Leray and Schauder [20])

Let E be a Banach space with C E closed and convex. Let U be a relatively open subset of C with 0 U and T : U ¯ C be a continuous and compact mapping. Then either
  1. (a)

    the mapping T has a fixed point in U ¯ ; or

     
  2. (b)

    there exist u U and λ ( 0 , 1 ) with u = λ T u .

     

Lemma 2.6 [13]

Let B be a Banach space and let K B be a cone. Assume that Ω 1 and Ω 2 are bounded open sets contained in B such that 0 Ω 1 and Ω ¯ 1 Ω 2 . Assume further that T : K ( Ω ¯ 2 Ω 1 ) K is a completely continuous operator. If either
  1. (i)

    T y y for y K Ω 1 and T y y for y K Ω 2 ; or

     
  2. (ii)

    T y y for y K Ω 1 and T y y for y K Ω 2 ;

     

then T has at least one fixed point in K ( Ω ¯ 2 Ω 1 ) .

We state next the structural assumptions that we impose on (1.1).

(H1) Assume that the nonlinearity function f : [ v 1 , v + b 1 ] N v 1 × R [ 0 , + ) is continuous.

(H2) Assume that there exist nonnegative continuous functions a 1 ( t ) , a 2 ( t ) , t [ v 1 , v + b 1 ] N v 1 such that | f ( t , y ) | a 1 ( t ) + a 2 ( t ) | y | , t [ v 1 , v + b 1 ] N v 1 , y R .

(H3) Assume that lim y 0 + f ( t , y ) y = 0 uniformly for t [ v 1 , v + b 1 ] N v 1 .

(H4) Assume that lim y + f ( t , y ) y = + uniformly for t [ v 1 , v + b 1 ] N v 1 .

3 Existence results

In this section, we will establish the existence of at least one positive solution for problem (1.1). At first, we state and prove some preliminary lemmas.

Lemma 3.1 Let h : [ v 1 , v + b 1 ] N v 1 R be given. Then the unique solution of the discrete fractional boundary value problem
{ Δ v y ( t ) = h ( t + v 1 ) , t [ 0 , b ] N 0 , y ( v 2 ) = 0 , Δ y ( v 2 ) = Δ y ( v + b 1 ) ,
(3.1)
is
y ( t ) = 1 Γ ( v ) s = 0 b G ( t , s ) h ( s + v 1 ) .
(3.2)
Here, for ( t , s ) [ v 2 , v + b ] N v 2 × [ 0 , b ] N 0 , G ( t , s ) is defined by
G ( t , s ) = { ( b + v s 2 ) v 2 ̲ t v 1 ̲ Γ ( v 1 ) ( b + v 1 ) v 2 ̲ + ( t s 1 ) v 1 ̲ , 0 s t v b , ( b + v s 2 ) v 2 ̲ t v 1 ̲ Γ ( v 1 ) ( b + v 1 ) v 2 ̲ , 0 t v < s b .
(3.3)
Proof Suppose that y ( t ) defined on [ v 2 , v + b ] N v 2 is a solution of (3.1). Using Lemma 2.4, for some constants C 1 , C 2 R , we have
y ( t ) = 1 Γ ( v ) s = 0 t v ( t s 1 ) v 1 ̲ h ( s + v 1 ) + C 1 t v 1 ̲ + C 2 t v 2 ̲ , t [ v 2 , v + b ] N v 2 .
(3.4)

By y ( v 2 ) = 0 and Definition 2.1, we obtain C 2 = 0 .

Then, for all t [ v 2 , v + b 1 ] N v 2 , we obtain [21]
Δ y ( t ) = 1 Γ ( v 1 ) s = 0 t ( v 1 ) ( t s 1 ) v 2 ̲ h ( s + v 1 ) + C 1 ( v 1 ) t v 2 ̲ .
(3.5)
In view of Δ y ( v 2 ) = Δ y ( v + b 1 ) , we have
C 1 = 1 Γ ( v 1 ) [ Γ ( v ) ( v 1 ) ( b + v 1 ) v 2 ̲ ] s = 0 b ( b + v s 2 ) v 2 ̲ h ( s + v 1 ) .
Substituting the values of C 1 and C 2 in (3.4), we have
y ( t ) = 1 Γ ( v ) s = 0 t v ( t s 1 ) v 1 ̲ h ( s + v 1 ) + s = 0 b ( b + v s 2 ) v 2 ̲ t v 1 ̲ Γ ( v 1 ) [ Γ ( v ) ( v 1 ) ( b + v 1 ) v 2 ̲ ] h ( s + v 1 ) = 1 Γ ( v ) s = 0 b G ( t , s ) h ( s + v 1 ) , t [ v 2 , v + b 1 ] N v 2 .

 □

Lemma 3.2 The function G ( t , s ) given in (3.3) satisfies the following:
  1. (1)

    0 G ( t , s ) D ( v + b ) v 1 ̲ ( s + v 1 ) v 1 ̲ G ( s + v 1 , s ) , ( t , s ) [ v 2 , v + b ] N v 2 × [ 0 , b ] N 0 ;

     
  2. (2)

    min t [ v 1 , v + b ] N v 1 G ( t , s ) Γ ( v ) ( s + v 1 ) v 1 ̲ G ( s + v 1 , s ) > 0 .

     
Here,
D = max s [ 0 , b ] { 1 + Γ ( v 1 ) ( v + b 1 ) v 2 ̲ ( v + b s 2 ) v 2 ̲ } = 1 + Γ ( v 1 ) ( v + b 1 ) v 2 ̲ ( v + b 2 ) v 2 ̲ .
(3.6)

Proof First of all, (3.3) implies that G ( s + v 1 , s ) = ( v + b s 2 ) v 2 ̲ ( s + v 1 ) v 1 ̲ Γ ( v 1 ) ( v + b 1 ) v 2 ̲ . Note that Γ ( v 1 ) ( v + b 1 ) v 2 ̲ > 0 , we know G ( s + v 1 , s ) > 0 .

Second of all, by (3.3) and the definition of D in (3.6), we obtain
0 G ( t , s ) ( b + v s 2 ) v 2 ̲ ( v + b ) v 1 ̲ Γ ( v 1 ) ( b + v 1 ) v 2 ̲ + ( v + b ) v 1 ̲ D ( b + v ) v 1 ̲ ( s + v 1 ) v 1 ̲ G ( s + v 1 , s ) .
(3.7)
On the other hand,
min t [ v 1 , v + b ] N v 1 G ( t , s ) = min t [ v 1 , v + b ] N v 1 ( b + v s 2 ) v 2 ̲ t v 1 ̲ Γ ( v 1 ) ( b + v 1 ) v 2 ̲ ( v 1 ) v 1 ̲ ( s + v 1 ) v 1 ̲ ( b + v s 2 ) v 2 ̲ ( s + v 1 ) v 1 ̲ Γ ( v 1 ) ( v + b 1 ) v 2 ̲ = Γ ( v ) ( s + v 1 ) v 1 ̲ G ( s + v 1 , s ) > 0 .
(3.8)

The proof of Lemma 3.2 is completed. □

Let B be the collection of all functions y : [ v 2 , v + b ] N v 2 R with the norm y = max { | y ( t ) | : t [ v 2 , v + b ] N v 2 } .

Define the operator T : B B by
T y ( t ) = 1 Γ ( v ) s = 0 b G ( t , s ) f ( s + v 1 , y ( s + v 1 ) ) .
(3.9)

In view of the continuity of f, it is easy to know that T is continuous. Furthermore, it is not difficult to verify that T maps bounded sets into bounded sets and equi-continuous sets. Therefore, in the light of the well-known Arzelá-Ascoli theorem, we know that T is a compact operator (see [11, 12]).

Let E = { y B | y ( t ) 0 , t [ v 2 , v + b ] N v 2 } and set
A = s = 0 b D ( v + b ) v 1 ̲ G ( s + v 1 , s ) Γ ( v ) ( s + v 1 ) v 1 ̲ a 2 , B = s = 0 b D ( v + b ) v 1 ̲ G ( s + v 1 , s ) Γ ( v ) ( s + v 1 ) v 1 ̲ a 1 .

We have the following theorem.

Theorem 3.3 Assume that (H1) and (H2) hold. Then system (1.1) has at least one positive solution provided that
s = 0 b D ( v + b ) v 1 ̲ G ( s + v 1 , s ) Γ ( v ) ( s + v 1 ) v 1 ̲ a 2 < 1 .
(3.10)
Proof Let Ω = { y E | y < r } with r = B 1 A > 0 . If y Ω ¯ , that is, y r . From (H1), (H2) and (3.9), we have
T y ( t ) = max t [ v 2 , v + b ] N v 2 | 1 Γ ( v ) s = 0 b G ( t , s ) f ( s + v 1 , y ( s + v 1 ) ) | 1 Γ ( v ) s = 0 b D ( v + b ) v 1 ̲ ( s + v 1 ) v 1 ̲ G ( s + v 1 , s ) ( | a 1 ( t ) | + | a 2 ( t ) | | y ( t ) | ) s = 0 b ( v + b ) v 1 ̲ Γ ( v ) D G ( s + v 1 , s ) ( s + v 1 ) v 1 ̲ a 1 + s = 0 b ( v + b ) v 1 ̲ Γ ( v ) D G ( s + v 1 , s ) ( s + v 1 ) v 1 ̲ a 2 y = B + A y r ,

which shows that T y Ω ¯ .

Consider the eigenvalue problem
y = λ T y , λ ( 0 , 1 ) .
(3.11)
Assume that y is a solution of (3.11), we obtain
y = λ T y < T y r .
(3.12)

It shows that y Ω . By Lemma 2.5, T has a fixed point in Ω ¯ . The proof is completed. □

We define the cone K B by
K = { y E : min t [ v 1 , v + b ] N v 1 y ( t ) Γ ( v ) D ( v + b ) v 1 ̲ y } .
(3.13)

Lemma 3.4 Let T be the operator defined in (3.9) and K be the cone defined in (3.13). Then T : K K .

Proof Note that for each t [ v 2 , v + b ] N v 2 , we have
T y ( t ) = max t [ v 2 , v + b ] N v 2 | 1 Γ ( v ) s = 0 b G ( t , s ) f ( s + v 1 , y ( s + v 1 ) ) | 1 Γ ( v ) s = 0 b D ( v + b ) v 1 ̲ ( s + v 1 ) v 1 ̲ G ( s + v 1 , s ) | f ( s + v 1 , y ( s + v 1 ) ) | = s = 0 b D ( v + b ) v 1 ̲ G ( s + v 1 , s ) Γ ( v ) ( s + v 1 ) v 1 ̲ | f ( s + v 1 , y ( s + v 1 ) ) | .
Therefore, it holds that
min t [ v 1 , v + b ] N v 1 ( T y ) ( t ) = min t [ v 1 , v + b ] N v 1 1 Γ ( v ) s = 0 b G ( t , s ) f ( s + v 1 , y ( s + v 1 ) ) 1 Γ ( v ) s = 0 b Γ ( v ) ( s + v 1 ) v 1 ̲ G ( s + v 1 , s ) f ( s + v 1 , y ( s + v 1 ) ) = Γ ( v ) D ( v + b ) v 1 ̲ s = 0 b D ( v + b ) v 1 ̲ G ( s + v 1 , s ) Γ ( v ) ( s + v 1 ) v 1 ̲ f ( s + v 1 , y ( s + v 1 ) ) Γ ( v ) D ( v + b ) v 1 ̲ T y .

The conclusion of Lemma 3.4 holds. □

Theorem 3.5 Suppose that conditions (H1), (H3) and (H4) hold. Then problem (1.1) has at least one positive solution.

Proof We have already shown T ( K ) K in Lemma 3.4. By condition (H3), we can select η 1 > 0 sufficiently small so that both | f ( t , y ) | η 1 y and η 1 s = 0 b ( v + b ) v 1 ̲ Γ ( v ) D G ( s + v 1 , s ) ( s + v 1 ) v 1 ̲ < 1 hold for all t [ v 1 , v + b 1 ] N v 1 and 0 < y < r 1 , where r 1 : = r 1 ( η 1 ) .

Let Ω 1 = { y B : y < r 1 } . Then, for y Ω 1 K , we have
T y ( t ) = max t [ v 2 , v + b ] N v 2 | 1 Γ ( v ) s = 0 b G ( t , s ) f ( s + v 1 , y ( s + v 1 ) ) | 1 Γ ( v ) s = 0 b D ( v + b ) v 1 ̲ ( s + v 1 ) v 1 ̲ G ( s + v 1 , s ) | f ( s + v 1 , y ( s + v 1 ) ) | s = 0 b D ( v + b ) v 1 ̲ G ( s + v 1 , s ) Γ ( v ) ( s + v 1 ) v 1 ̲ η 1 y < y .

It implies that T is a cone contraction on y Ω 1 K .

On the other hand, from condition (H4), we may select a number η 2 > 0 such that both | f ( t , y ) | > η 2 y and η 2 s = 0 b G ( s + v 1 , s ) ( s + v 1 ) v 1 ̲ > 1 hold for all t [ v 1 , v + b 1 ] N v 1 and 0 < y < r 2 , where r 2 : = r 2 ( η 2 ) and r 2 > r 1 > 0 . Define Ω 2 = { y B : y < r 2 } , we obtain
T y ( t ) = max t [ v 2 , v + b ] N v 2 1 Γ ( v ) s = 0 b G ( t , s ) | f ( s + v 1 , y ( s + v 1 ) ) | 1 Γ ( v ) s = 0 b min t [ v 1 , v + b ] N v 1 G ( t , s ) | f ( s + v 1 , y ( s + v 1 ) ) | > s = 0 b G ( s + v 1 , s ) ( s + v 1 ) v 1 ̲ η 2 y > y ,

whenever y Ω 2 K , so that T is a cone expansion on Ω 2 K .

In summary, we may invoke Lemma 2.6 to deduce the existence of a function y 0 K ( Ω ¯ 2 Ω 1 ) such that T y 0 = y 0 , where y 0 is a positive solution to problem (1.1). The proof is completed. □

4 Example

Example 4.1 Consider the fractional difference boundary value problem
{ Δ 3 2 y ( t ) = f ( t + 1 2 , y ( t + 1 2 ) ) , t [ 0 , 3 ] N 0 , y ( 1 2 ) = 0 , Δ y ( 1 2 ) = Δ y ( 7 2 ) .
(4.1)
Set a 1 ( t ) = 1 , a 2 ( t ) = t 30 , f ( t , y ) = t | y | 30 + | sin t | , t [ 1 2 , 7 2 ] N 1 2 . We have
| f ( t , y ) | 1 + t 30 | y | .

By a simple computation, we can obtain D 1.7750 , s = 0 3 ( 9 2 ) 1 2 ̲ G ( s + 1 2 , s ) Γ ( 3 2 ) ( s + 1 2 ) 1 2 ̲ 4.7688 , a 2 = 7 60 . Therefore, A 0.9875 < 1 . The conditions of Theorem 3.3 hold, the boundary value problem (4.1) has at least one positive solution.

Example 4.2 Consider the fractional difference boundary value problem
{ Δ 3 2 y ( t ) = f ( t + 1 2 , y ( t + 1 2 ) ) , t [ 0 , 11 ] N 0 , y ( 1 2 ) = 0 , Δ y ( 1 2 ) = Δ y ( 23 2 ) .
(4.2)
Set f ( t , y ) = t y 2 , t [ 1 2 , 23 2 ] N 1 2 . We have
( 1 ) lim y 0 + f ( t , y ) y = lim y 0 + t y 2 y = 0 , ( 2 ) lim y + f ( t , y ) y = lim y + t y 2 y = + .

The conditions of Theorem 3.5 hold, the boundary value problem (4.2) has at least one positive solution.

Misc

Equal contributors

Declarations

Acknowledgements

This work was jointly supported by the Natural Science Foundation of China under Grants 11471278, the Natural Science Foundation of Hunan Province under Grants 13JJ3120 and 14JJ2133, and the Construct Program of the Key Discipline in Hunan Province. We would like to show our great thanks to the anonymous referee for his/her valuable suggestions and comments, which improve the former version of this paper and make us rewrite the paper in a more clear way.

Authors’ Affiliations

(1)
Department of Mathematics, Xiangnan University
(2)
The Editorial Department of Journal of Xiangnan University

References

  1. Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives. Theory and Applications. Gordon & Breach, Yverdon; 1993.Google Scholar
  2. Podlubny I Mathematics in Science and Engineering 198. In Fractional Differential Equations. Academic Press, New York; 1999.Google Scholar
  3. Bai ZB, Lü HS: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 2005, 311: 495-505. 10.1016/j.jmaa.2005.02.052MathSciNetView ArticleGoogle Scholar
  4. Zhang SQ: Positive solutions for boundary value problems of nonlinear fractional differential equations. Electron. J. Differ. Equ. 2006., 2006: Article ID 36Google Scholar
  5. Jafari H, Gejji VD: Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method. Appl. Math. Comput. 2006, 180: 700-706. 10.1016/j.amc.2006.01.007MathSciNetView ArticleGoogle Scholar
  6. Dehghani R, Ghanbari K: Triple positive solutions for boundary value problem of a nonlinear fractional differential equation. Bull. Iran. Math. Soc. 2007, 33(2):1-14.MathSciNetGoogle Scholar
  7. Wang JH, Xiang HJ, Liu ZG: Positive solutions for three-point boundary value problems of nonlinear fractional differential equations with P -Laplacian. Far East J. Appl. Math. 2009, 37(1):33-47.MathSciNetGoogle Scholar
  8. Wang JH, Xiang HJ, Liu ZG: Upper and lower solutions method for a class of singular fractional boundary value problems with P -Laplacian. Abstr. Appl. Anal. 2010., 2010: Article ID 971824Google Scholar
  9. Cai G: Positive solutions for boundary value problems of fractional differential equations with P -Laplacian operator. Bound. Value Probl. 2012., 2012: Article ID 18Google Scholar
  10. Atici FM, Eloe PW: Two-point boundary value problems for finite fractional difference equations. J. Differ. Equ. Appl. 2011, 17: 445-456. 10.1080/10236190903029241MathSciNetView ArticleGoogle Scholar
  11. Goodrich CS: On a first-order semipositone discrete fractional boundary value problem. Arch. Math. 2012, 99: 509-518. 10.1007/s00013-012-0463-2MathSciNetView ArticleGoogle Scholar
  12. Lv WD: Existence of solutions for discrete fractional boundary value problems with a P -Laplacian operator. Adv. Differ. Equ. 2012., 2012: Article ID 163Google Scholar
  13. Goodrich CS: Existence of a positive solution to a system of discrete fractional boundary value problems. Appl. Math. Comput. 2011, 217: 4740-4753. 10.1016/j.amc.2010.11.029MathSciNetView ArticleGoogle Scholar
  14. Atici FM, Eloe PW: A transform method in discrete fractional calculus. Int. J. Differ. Equ. 2007, 2(2):165-176.MathSciNetGoogle Scholar
  15. Goodrich CS: On discrete sequential fractional boundary value problems. J. Math. Anal. Appl. 2012, 385: 111-124. 10.1016/j.jmaa.2011.06.022MathSciNetView ArticleGoogle Scholar
  16. Goodrich CS: Positive solutions to boundary value problems with nonlinear boundary conditions. Nonlinear Anal. 2012, 75: 417-432. 10.1016/j.na.2011.08.044MathSciNetView ArticleGoogle Scholar
  17. Goodrich CS: On discrete fractional three-point boundary value problems. J. Differ. Equ. Appl. 2012, 18: 397-415. 10.1080/10236198.2010.503240MathSciNetView ArticleGoogle Scholar
  18. Goodrich CS: Nonlocal systems of BVPs with asymptotically superlinear boundary conditions. Comment. Math. Univ. Carol. 2012, 53: 79-97.MathSciNetGoogle Scholar
  19. Atici FM, Sengul S: Modeling with fractional difference equations. J. Math. Anal. Appl. 2010, 369: 1-9. 10.1016/j.jmaa.2010.02.009MathSciNetView ArticleGoogle Scholar
  20. Zeidler E: Nonlinear Functional Analysis and Applications, I: Fixed Point Theorems. Springer, New York; 1986.View ArticleGoogle Scholar
  21. Atici FM, Eloe PW: Initial value problems in discrete fractional calculus. Proc. Am. Math. Soc. 2009, 137: 981-989.MathSciNetView ArticleGoogle Scholar

Copyright

© Wang et al.; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.