Existence of Positive Solutions for Boundary Value Problem of Nonlinear Fractional q Difference Equation

(1)

Existence of Positive Solutions for Boundary Value

Problem of Nonlinear Fractional q-Difference Equation

*

Liu Yang

Department of Mathematics and Computing Sciences, Hengyang Normal University, Hengyang, China Email: [email protected]

Received April 14, 2013; revised May 14, 2013; accetped May 21, 2013

Copyright © 2013 Liu Yang. This is an open access article distributed under the Creative Commons Attribution License, which per-mits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

In this paper, we investigate the existence of positive solutions for a class of nonlinear q-fractional boundary value problem. By using some fixed point theorems on cone, some existence results of positive solutions are obtained.

Keywords: Fractional q-Difference Equation; Positive Solution; Fixed Point Theorems on Cone

1. Introduction

Considering the following boundary value problem of nonlinear fractional q-difference equation:

 



 



 

2

 

3

 

, , 0 1,3

0 0 0 0, 1

q

q q q

D y x f x y x x

y D y D y D y

 _

 _{ } _{ } _{ }

 

  



4, 0,

 (P)

where f is a nonnegative continuous function and D_q

fractional q-derivative of the Riemann-Liouville type.

is the

Fractional differential calculus is a discipline to which many researchers are dedicating their time, perhaps be-cause of its demonstrated applications in various fields of science and engineering [1]. Recently, there are many papers dealing with the boundary value problem of frac-tional differential equations, see [2-5] and references therein.

The q-difference calculus or quantum calculus is an old subject that was initially developed by Jackson [6,7], and basic definitions and properties of q-difference calculus can be found in [8]. The fractional q-difference calculus had its origin in the works by Al-Salam [9] and Agarwal [10]. More recently, maybe due to the explosion in research within the fractional differential calculus set-ting, new developments in this theory of fractional q- difference calculus were made, see [11,12].

The question of the existence of positive solutions for

fractional q-difference boundary value problems is in its infancy, see [13-16]. No contributions exist, as far as we know, concerning the existence of positive solutions for problem (P).

This paper is organized as follows. In Section 2, some preliminaries are presented. In Section 3, we discuss the existence of positive solutions for problem (P).

2. Preliminaries

Let q

 

0,1 and define

 

1

, 1

a

q q a

q



.

a

 

  (2.1)

The q-analogue of the power function



a b



n with

0

n is









1





0

1, , , , .

n

n k

k

a b a b a bq n a b

 

   



  

(2.2) More generally, if a, then





 

0

.

n

n n

a bq

a b a

a bq

 





 



 





(2.3)

Note that, if b0 then a  a. The q-gamma function is defined by

  



 









1 1

1

, 0, 1, 2

1

x

q x

q

x x

q

 



    

  , (2.4)

*_{This work was supported by the Natural Science Foundation of Hunan}

Province (12JJ9001), Hunan Provincial Science and Technology De-partment of Science and Technology Project (2012SK3117) and Con-struct program of the key discipline in Hunan Province.

(2)





 

_

_

 





 





 

0

, 0 lim

1

q q

x

f x f qx

D f x D f D f x

q x 



 

 q

1 _, _ _. (2.5) and q-derivative of higher order by



0

 

 

_,



n

 



n

 

q q q q

D f x  f x D f x D D  f x n 

(2.6) The q-integral of a function f defined in the interval

 

0,b is given by

 

₀

 





 

 

0

1 ,

x _n _n

q q

n

I f x f t d t x q f xq q x b



_

 



 0, .



(2.7) If a

 

0,b and f defined in the interval

 

0,b , its integral from a to is defined by b

 

₀

 

₀

 

.

b b a

q q

af t d t f t d t f t d tq



(2.8)

Remark 2.1. (see [17]) If abq nn,  and

 

f t g t on

 

a b, , then

 

_q

 

_q .

a a

b b

f t d t g t d t



Similarly as done for derivatives, an operator n q I can be defined, namely,

 

0

 

_,

  

n



n 1

 

q q q q

I f x  f x I f x I I  f x n, .

(2.9) The fundamental theorem of calculus applies to these operators Iq and Dq, i.e.,



D I fq q



 

x  f x

 

(2.10)

and if f is continuous at x0, then



I D f_q _q



 

x  f x

 

 f

 

0 . (2.11) Basic properties of the two operators can be found in [14]. We now point out three formulas that will be used later ( denotes the derivative with respect to variable

)

iDq i









 

,

a ts  a ts 

 

 





 

_{ }

_

_

 1

,

tD tq s q t s

 _ 

  

 



_xD_q



₀xf x t d t, _q



 

x 



₀x_xD f x t d t_q

 

, _q  f qx x



,



.

Remark 2.2. (see [14]) We note that if  0 and , then

a b t





 

_

_

 

.

ta   t b  (2.12)

Definition 2.3. (see [10]) Let  0 and f be a function defined on

 

0,1 . The fractional q-integral of the Riemann-Liouville type is

 

I fq0

 

x  f x





and

 

_{  }



 

_{ }

 

1 0

1

d , 0, 0,1 .

x

q q

q

I f x x qt f t t

x

 



 



 



(2.13)

Definition 2.4. (see [14-16]) The fractional q-derivative of the Riemann-Liouville type of order  0 is defined by



D fq0



 

x  f x

 

and



D f_q



 

x 



D I_qm _qmf



 

x ,  0, (2.14) where m is the smallest integer greater than or equal to

 .

Next, we list some properties that are already known in the literature.

Lemma 2.5. (see [14-16]) Let  , 0 and f be a function defined on

 

0,1 , Then, the next formulas hold:

1)



I I f_q _q



 

x 



I _q f



 

x ,

2)



D I fq q



 

x f x

 

.

  _

Lemma 2.6. (see [14-16]) Let  0 and be a positive integer. Then, the following equality holds:

p





 





 

1

_

_





 

0

0 . 1

p q q

p k p

p k

q q q

k q

I D f x

x

D I f x D f

k p



 



  



 

   



(2.15)

Let p4, in view of Lemma 2.5 and Lemma 2.6, we see that

 



 







 



 



 

  



 



 



4 4

1 2 3

1 4

4 ₀

,

1

, d

q

q q q q

x

q q

D y x f x y x

I D I y x I f x y x

y x c x c x c x

c x x qt f t y t t



  

 





  

 

 

  

   

  





for some constants 1 2 3 4 Using the boundary

condition

, , , .

c c c c 

 

0

y 0 we have 4 Differentiating

both side of the above equality, one gets

0.

c 













 









 



 



2 3

1 2

2 0

1 2 3

1

1 ,

q q q q

x

q q

q

D y c x c x c x 4

3

d .

x qt f t y t t

 



  

 



  



     

  





(2.16) Using the boundary condition

 

D yq

 

0 0, we have

3 0.

c  similarly, we have c20. From

 



 



 



 







 

_

_{ }

_

3 4

1

0

4

1 2 3

1

1 2 3

, d

q q q q

x

q q

q

D y c x

x qt f t y t t



  





   

  



 

(3)

and boundary value problem , one can

obtain

 

3

1 0

q

D y 

  



 



 



1 4

1 ₀

1

1 , q

q

c qt  f t y





 





t d .t (2.18)

Putting all things together we finally have

  



 



 



  



 



 



 





 





 





 







 

_

_{ }

_

1 4 1

0

1 0

4 1 1

0

1 4 1

1 1 , 1 , 1 1

1 , d .

q q x q q x q q q x

yx qt f t y t d t x

x qt f t y t d t

qt x x qt f t y t d t

qt x f t y t t

                     _  _          _    _     



, (2.19) If we define a function G by

 

_{ }





 

_

_

 





 

4 1 1

4 1 1 , 0 1, 1 , 1 , 0 1. q

t x x t

t x G x t

t x x t                       _  _{ }       (2.20)

Hence, in order to solve the problem (P), it is suffi-cient to find positive solutions of the following integral equation

 

1







 



0 , , dq

y t 



G x qt f t y t t.

,

1.



1 (2.21) Some properties of the function needed in the sequel are now stated and proved.

G

Lemma 2.7. Function defined above satisfies the following conditions:

G



,



0,



,





1,



for all 0 , 1,

G x qt G x qt G qt x t

 

  (2.22)





1





, 1, , for all 0 ,

G x qt x G qt x t (2.23)

Proof. Let

  



 4 1

_

_

 1

1 , 1 , 0

g x t  t  x  x t    t x (2.24)

and

  



 4 1

2 , 1 , 0

g x t  t  x   x t 1.

0

(2.25) It is clear that g2



x qt,



 . Now, g2



0,qt



0. For

, in view of (2.3) and Remark 2.2, we have 0

x



 



   





 

_

_

 

1 4 1 1

1 4 1 1 , 1 1 1 qt

g x qt qt x x x

x

x qt qt

                   _  _     _    _

Therefore, G x t

 

, 0. Moreover,









 

_

_

_

_

 









 

_

_

 

1

2

4 2 2

4 2

2

,qt

1 1 1

1 1 1 0.

x q

q q

q

D g x

qt

qt x x x

x

x qt qt

                        _  _       _    _ (2.27) i.e., g1



x qt,



is an increasing function of x. Obviously,



,



2

g x qt is increasing in x, therefore, is an increasing function of x for fixed



, G x qt



 

0,1 .

t This con-cludes the proof of (2.22).

Suppose now that xqt, then













 

_

_

 





 

_

_

 





   





 

_

_

 





 

_

_

 





 

_

_

 

4 ₁ 1

4 1 1 4 1 4 1 4 1 1 1 4 1 , 1

1, ₁ ₁

1 1

.

1 1

G x qt qt x x qt

G qt _qt _qt

qt

x qt

x

qt qt

x qt qt

x qt qt                                         _ _      _ _             _ _{ }         (2.28) If xqt, then









1

, 1, G x qt

x G qt



 (2.29)

and this finishes the proof of (2.23).

Let  C

 

0,1 be the Banach space endowed with norm u supt_{ }_0,1 u

 

t . Define the cone C by

 



1



: .

C u  u x x  u

It follows from the non-negativeness and continuity of and

G f that the operator T C:   defined by

  

1







 



0 , , dq

Tu x 



G x qt f t u t t (2.30) is completely continuous [18]. Moreover, for uC, in view of (2.22) and (2.23), we have

  

Tu x 0 on

 

0,1 and

  







 









 



1 0 1 1 0 1 , , 1, , , q q

Tu x G x qt f t u t d t

x G qt f t u t d t

x Tu       



(2.31)

that is T C

 

C.

0.



(2.26)

Lemma 2.8. (see [19]) Let be a Banach space, a cone, and 1 2

E

PE  , two bounded open balls of centered at the origin with

E   ₁ ₂. Suppose that



2 1



:

(4)

op-erator such that either

1) Tu  u u, P1and Tu  u u, P2,

or

2) Tu  u u, P1and Tu  u u, P2

holds. Then T has a fixed point in P



 ₂ ₁



.

3. Main Results

Let  i



yC

 

0,1 : y r1



,i1, 2, 3, 4, where

will be defined later.

0

i r 

Theorem 3.1. Suppose that f t u

 

, is a nonnegative continuous function on

  

0,1 0,



. In addition, sup-pose that one of the following two conditions holds:

(H1)

 

 

 

 

0 0,1

0,1

,

lim min ,

,

lim min 0;

y x

f x y y f x y

y

 

 

 



(H2)

 

 

 

 

0 0,1 0,1

,

lim min 0,

,

lim min .

y x

f x y y f x y

y

 

 



 

Then problem (P) has at least one positive solution.

Proof. Note that the operator T C



ompletely continuous. Now, assume that condition (H1) holds. Since

 



C is c

 

0 0,1

,

limy minx f x y , there exists an such y

   

that

1 0

r 

 

, 1 , for

 

0,1 , 0 1,

f x y  y x  yr (3.1) where the constant  1 0 such that

(3.2) Thus,



1 0



G 1,qt t



d tq 1.

1 _ ₁

  _

 







,y x 1y, for yC1,x 0,1 .



This, together with the definitions of and L mma 2.

f x

C e

7, implies that for any yC1,

 







 









 





  





1 0 0,1 1 0 1 1 0

1 ₁

1 0

max , ,

1, ,

1,

1, .

q x

q

Ty G x qt f t y t d t

G qt f t y t d t

G qt y t d t

G qt t d t y

y







 





 



(3.3)

That is, for any

,

it follows that such that

1,

yC Ty  y .

On the other hand, from lim_y max_x_{ }_0,1 f x

 

, 0

there exists a L10

 

, 2 for

 

0,1 , 1,

f x y  y x yL (3.4)

where the constant 20 satisfies





1

2 0G q d tq ₂.



_

1, t 1 (3.5)

Let _{  } _



 



1 2 maxx 0,1 ,y 0,L ,

L  _ _ f x y . Then we have

 

, 2 2, for

 

0,1 ,

f x y  yL x y0. (3.6)







1



2 max 2 , 21 _{2 0} 1, q

r  r L



G qt d t and

Let

2

yC , then

 







 









 

















1 0 0,1

1 1

2 2

0 0

1 1

2 ₀ 2 ₀

max , ,

1, , 1,

1, 1, .

q x

q q

G qt f t y t d t G qt y L d t

G qt d t y L G qt d t y

 



  

  



(3.7)

Thus, the operator satisfies condition of Lemma 2.8. Consequently, the rator has at least one fixed poi



T

ope T

nt yC



 

₂1



, which is one positive

solu-tion of the problem (P).

Next, we suppose that condition (H2) holds. The proof is similar to that of the c e in which (H1) holds and will only be sketched here.

as

Let with n. Select two positive constants 3, 4

n q

 

  with _{3 0}



1G



1,qt d t



q 1 and  4 1



_1G



1,qt d t



q 1, ctively. ere exist two positive constants and

respe

such that Then, th

3

r L₃

 

, 3 , for

 

0,1 , 0 3,

f x y  y x  y r (3.8)

 

, 4 , for

 

0,1, 3.

f x y  y x yL (3.9)

It follows from that for yC3,







 





  

1

1, ,

G qt f t y t d t



_

1

3

0 q 0 1, q .

Ty

t



G q y t d t y

(3.1 In addition, let

0)



1



4 max 2 ,3 3

r  r L . If yCr₄,

then

 

1

 

4 3, for

r L x

 _,1

y x

y y

  

 

  )

and

 _

(3.11

 



,



4 , for

 

,1 , 4

f x y x  y x yC . (3.12) In view of Remark 2.1, for yC4, we have

 







 









 





  









1

4 0

1 ₁ ₁ 1

4 4

max , ,

1, , 1,

1, 1,

.

q

q q

G qt f t y t d t G qt y t d t

G qt t d t y G qt d t y

y



 

 



    



 

 





(3.13) 0

0,1

1 1

(5)

Thus, 193-203.

[8] V. Kac and P. Cheung, “Quantum Calculus,”

Ty  y for . Consequently, the operator has at least one nt

4

yC

fixed poi

T

 

₄ 3

yC  

problem (P).

Springer, New York, 2002.

http://dx.doi.org/10.1007/978-1-4613-0071-7 [9] W. A. Al

, which is itive solution of the

Example 3.2

one pos

-Salam, “Some Fractional q-Integrals and q-

No. 2, 1967, pp. 135-140.

Derivatives,” Proceedings of the Edinburgh Mathemati- cal Society, Vol. 15,

http://dx.doi.org/10.1017/S0013091500011469





 



 







3.5 0.5

2

, , 0 1,

D y x f x y x x

 _{ } _{ }

 

 







 





 

where

3

0.5 0.5 0.5

0 0 0 0, 1 0,

y D y D y D y



  



969, pp. 045060 [10] R. P. Agarwal, “Certain Fractional q-Integrals and q-

Derivatives,” Mathematical Proceedings of the Cam- bridge Philosophical Society, Vol. 66, No. 2, 1

365-370. http://dx.doi.org/10.1017/S0305004100

 





12

 

_{log 1}



 



_.

f x y x,  y x  y x Obviously,

 

 

 

 

0 0,1

0,1

,

lim_y_ max ,

,

lim max 0.

x

y x

f x y y f x y

y



 

 



, by the first part of Theorem 3.1, w above has at least one positive s

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