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![Invention Journal of Research Technology in Engineering & Management (IJRTEM)
ISSN: 2455-3689
www.ijrtem.com Volume 2 Issue 7 ǁ July 2018 ǁ PP 05-15
|Volume 2| Issue 7 | www.ijrtem.com | 5 |
Existence of positive solutions for fractional q-difference
equations involving integral boundary conditions with p-
Laplacian operator
Yizhu Wang1
, Chengmin Hou1*
(Department of Mathematics, Yanbian University, Yanji, 133002, P.R. China)
ABSTRACT: The existence of positive solutions is considered for a fractional q-difference equation with p-
Laplacian operator in this article. By employing the Avery-Henderson fixed point theorem, a new result is obtained
for the boundary value problems.
KEY WORDS: positive solutions; Riemann-Liouville fractional q-derivatives; Caputo fractional q-derivatives;
p-Laplacian; Avery-Henderson fixed point theorem
I. INTRODUCTION
Fractional calculus is the extension of integer order calculus to arbitrary order calculus. With the development of
fractional calculus, fractional differential equations have wide applications in the modeling of different physical
and natural science fields, such as fluid mechanics, chemistry, control system, heat conduction, etc. There are
many papers concerning fractional differential equations with the p-Laplacian operator [1-5] and fractional
differential equations with integral boundary conditions [6-10].
In [11], Yang studied the following fractional q-difference boundary value problem with p-Laplacian operator
where 1 , 2. The existence results for the above boundary value problem were obtained by using the
upper and lower solutions method associates with the Schauder fixed point theorem.
In [12], Yuan and Yang considered a class of four-point boundary value problems of fractional q-difference
equations with p-Laplacian operator
Where is the fractional q-derivative of the Riemann-Liouville type with 1 , 2. By applying the
upper and lower solutions method associated with the Schauder fixed point theorem, the existence results of at
least one positive solution for the above fractional q-difference boundary value problem with p-Laplacian operator
are established. Motivated by the aforementioned work, this work discusses the existence of positive solutions for
this fractional q-difference equation:
( ( ( ))) ( , ( )),0 1,2 3,
(0) (1) 0,
c c
q p qD D x t f t u t t
x x
=
= = (1) (0) 0,c c
q qD x D x
= =
( ( ( ))) ( , ( )),0 1,2 3,
(0) 0, (1) ,
q P qD D x t f t u t t
x x ax
=
= = (0) 0, (1) ( ),q q qD x D x bD x
= =
,q qD D ](https://image.slidesharecdn.com/ijrtemb0270515-190123175553/75/Existence-of-positive-solutions-for-fractional-q-difference-equations-involving-integral-boundary-conditions-with-pLaplacian-operator-1-2048.jpg)
![Existence of positive solutions for fractional q-difference…
|Volume 2| Issue 7 | www.ijrtem.com | 6 |
(1.1)
where 2 3, 2 3, and 5 6. +
2
( ) ,
p
p u u u
−
= 1.p is the Caputo fractional q-
derivative, is the Riemann-Liouville fractional q-derivative.
We will always suppose the following conditions are satisfied:
1( )H ( ):[0,1] [0, )g t → + with 1
( ) [0,1],g t L and
2( )H , (0, ),a b + and ;b a
3( )H ( , ) :[0,1] (0, ) (0, )f t u + → + is continuous.
II. BACKGROUND AND DEFINITIONS
To show the main result of this work, we give in the following some basic definitions and theorems, which can be
found in [13,14].
Definition 2.1 [13] Let 0 and f be a function defined on [0,1] . The fractional q-integral of Riemann-
Liouville type is and
Lemma 2.2 [13] Let 0, then the following equality holds:
Lemma 2.3 [14]Let 0, If then
where 1.n = +
Theorem 2.4 (Avery-Henderson fixed point theorem [15]) Let ( , )E be a Banach space, and be a
cone. Let and be increasing non-negative, continuous functionals on ,P and be a non-negative
continuous functionals on P with such that, for some 3 0r and 0,M ( ) ( ) ( ),u u u
( 1)
,0 0
( )
( )( ) ( ) ,
( )
t
qq
q
t qs
I f t f s d s
+
−
−
=
0, [0,1].t
1
,0 ,0 ,0
0
( ) ( ) ( )(0 ).
( 1)
k
c k
q q q
k q
t
I D f t f t D f
k
+ + +
−
+
=
= −
+
0
,0
( )( ) ( )q
I f t f t+ =
,0 ,0
'
,0 ,0 ,0
2
,0 ,0
1
,0 0
[ ( ( ))] ( , ( )) 0, (0,1),
( (0)) [ ( (0))] ( (1)) 0,
(0) (1) 0,
(0) (0) ( ) ( ) ,
c
pq q
c c c
p p pq q q
q q
qq
D D u t f t u t t
D u D u D u
D u D u
au bD u g t u t d t
+ +
+ + +
+ +
+
+ =
= = =
= =
+ =
,0
c
q
D
+
,0q
D
+
1
0
( ) 0,qg t d t
1
0
( ) 0;qtg t d t
1
0
( ) qa g t d t
,0
[0,1],q
D u C
+
( )
,0 ,0
1
( ) ( )
n
i
iq q
i
I D f t f t k t
+ +
−
=
= +
P E
(0) 0, =](https://image.slidesharecdn.com/ijrtemb0270515-190123175553/75/Existence-of-positive-solutions-for-fractional-q-difference-equations-involving-integral-boundary-conditions-with-pLaplacian-operator-2-2048.jpg)
![Existence of positive solutions for fractional q-difference…
|Volume 2| Issue 7 | www.ijrtem.com | 7 |
and ( ),u M u for all 3( , ),u p r where 3 3( , ) { : ( ) }.p r u P u r = Suppose that there exist
positive numbers 1 2 3,r r r such that
( ) ( )lu l u for 0 1,l and 2( , ).u P r
If 3: ( , )T p r P → is a completely continuous operator satisfying:
1( )C 3( )Tu r for all 3( , );u P r
2( )C 2( )Tu r for all 2( , );u P r
3( )C 1( , ) ,P r and 1( )Tu r for all 1( , ),u P r
then T has at least two fixed points 1u and 2u such that 1 1( )r u with 1 2( )u r and 2 2( )r u
with 2 3( ) .u r
III. PRELIMINARY LEMMAS
Lemma 3.1 The boundary value problem (1.1) is equivalent to the following equation:
(3.1)
where
(3.2)
(3.3)
(3.4)
(3.5)
is the inverse function of a.e.,
Proof From we get
( 1)
0 1 0
1
( ) ( ) ( ) ,
( )
t
q
q
u t d d t t qs v s d s
−
= + + −
1
( 1)
0 1 0 0
0
1
( ) ( ) ( )
[ ( ) ] ( )
t
q q
q q
d g t t qs v s d sd t
a g t d t
−
= −
−
1
1
( 2)0
1 0
0
[ ( ) ]
(1 ) ( ) ,
[ ( ) ] ( 1)
q
q
q q
b tg t d t
qs v s d s
a g t d t
−
−
+ −
− −
1
( 2)
1 0
1
(1 ) ( ) ,
( 1)
q
q
d qs v s d s
−
= − −
−
( )1
0
( ) ( , ) ( , ( )) ,q qv s H s q f u d =
( 1) ( 1)
( 1)
( (1 )) ( ) ,1
( , )
( ) ( (1 )) ,q
s s
H s
s
− −
−
− − −
=
−
0 1,s
0 1.s
( )q s
2
( ) ,
q
q s s s
−
=
1 1
1.
p q
+ =
,0 ,0
[ ( ( ))] ( , ( )) 0,c
pq q
D D u t f t u t
+ + + =
( 1) 1 2 3
1 2 3,0 0
1
( ( )) ( ) ( , ( )) .
( )
t
c
p qq
q
D u t t q f u d c t c t c t
+
− − − −
= − − + + +
( ),p s](https://image.slidesharecdn.com/ijrtemb0270515-190123175553/75/Existence-of-positive-solutions-for-fractional-q-difference-equations-involving-integral-boundary-conditions-with-pLaplacian-operator-3-2048.jpg)
![Existence of positive solutions for fractional q-difference…
|Volume 2| Issue 7 | www.ijrtem.com | 8 |
In view of we get 2 3 0,c c= = i.e.,
(3.6)
Conditions imply that
(3.7)
By use of (3.6) and (3.7), we get
(3.8)
In view of (3.8), we obtain
(3.9)
Let
by use of (3.9), we get
Conditions imply that 2 0,d = i.e.,
then we have
Conditions imply that
From we get
Therefore, we can obtain
'
,0 ,0
( (0)) [ ( (0))] 0,c c
p pq q
D u D u
+ += =
( 1) 1
1,0 0
1
( ( )) ( ) ( , ( )) .
( )
t
c
p qq
q
D u t t q f u d c t
+
− −
= − − +
,0
( (1)) 0c
p q
D u
+ =
1
( 1)
1 0
1
(1 ) ( , ( )) .
( )
q
q
c q f u d
−
= − −
1
,0 0
( ( )) ( , ) ( , ( )) .c
p qq
D u t H t q f u d
+ =
( )1
,0 0
( ) ( , ) ( , ( )) .c
q qq
D u t H t q f u d
+ =
( )1
0
( ) ( , ) ( , ( )) ,q qv t H t q f u d =
( 1) 2
0 1 20
1
( ) ( ) ( ) .
( )
t
q
q
u t t qs v s d s d d t d t
−
= − + + +
2
,0
(0) 0q
D u+ =
( 1)
0 10
1
( ) ( ) ( ) ,
( )
t
q
q
u t t qs v s d s d d t
−
= − + +
' ( 2)
10
1
( ) ( ) ( ) .
( 1)
t
q
q
u t t qs v s d s d
−
= − +
−
,0
(1) 0q
D u+ =
1
( 2)
1 0
1
(1 ) ( ) .
( 1)
q
q
d qs v s d s
−
= − −
−
1
,0 0
(0) (0) ( ) ( ) ,qq
au bD u g t u t d t++ =
1
( 1)
0 1 0 0
0
1
( ) ( ) ( )
[ ( ) ] ( )
t
q q
q q
d g t t qs v s d sd t
a g t d t
−
= −
−
1
1
( 2)0
1 0
0
[ ( ) ]
(1 ) ( ) .
[ ( ) ] ( 1)
q
q
q q
b tg t d t
qs v s d s
a g t d t
−
−
+ −
− −
( 1)
0 1 0
1
( ) ( ) ( )
( )
t
q
q
u t d d t t qs v s d s
−
= + + −
](https://image.slidesharecdn.com/ijrtemb0270515-190123175553/75/Existence-of-positive-solutions-for-fractional-q-difference-equations-involving-integral-boundary-conditions-with-pLaplacian-operator-4-2048.jpg)
![Existence of positive solutions for fractional q-difference…
|Volume 2| Issue 7 | www.ijrtem.com | 9 |
The proof is complete. □
Lemma 3.2 ([16]) The function ( , )H s defined by (3.5) is continuous on [0,1] [0,1] and satisfy
for , [0,1].s
Let E be the real Banach space [0,1]C with the maximum norm, define the operator :T E E→ by
Lemma 3.3 For with is non-increasing and non-negative.
Proof Since
so we get
So ( )( )Tu t is non-increasing, then we have We have
1
( 1)
1 0 0
0
1
( ) ( ) ( )
[ ( ) ] ( )
t
q q
q q
g t t qs v s d sd t
a g t d t
−
= −
−
1
1
( 2)0
1 0
0
[ ( ) ]
(1 ) ( )
[ ( ) ] ( 1)
q
q
q q
b tg t d t
qs v s d s
a g t d t
−
−
+ −
− −
1
( 2) ( 1)
0 0
1
(1 ) ( ) ( ) ( ) .
( 1) ( )
t
q q
q q
t
qs v s d s t qs v s d s
− −
− − + −
−
1 ( 1) ( 1)
(1 ) (1 ) (1 )
( , )
( ) ( 1)q q
s s
H s
− − −
− − −
−
( 1)
0 1 0
1
( ) ( ) ( )
( )
t
q
q
Tu t d d t t qs v s d s
−
= + + −
1
( 1)
1 0 0
0
1
( ) ( ) ( )
[ ( ) ] ( )
t
q q
q q
g t t qs v s d sd t
a g t d t
−
= −
−
1
1
( 2)0
1 0
0
[ ( ) ]
(1 ) ( )
[ ( ) ] ( 1)
q
q
q q
b tg t d t
qs v s d s
a g t d t
−
−
+ −
− −
1
( 2) ( 1)
0 0
1
(1 ) ( ) ( ) ( ) .
( 1) ( )
t
q q
q q
t
qs v s d s t qs v s d s
− −
− − + −
−
[0,1]u C ( )( )Tu t( ) 0,u t
( 1)
0 1 0
1
( ) ( ) ( ) ,
( )
t
q
q
Tu t d d t t qs v s d s
−
= + + −
( )
' ( 2)
1 0
1
( ) ( ) ( )
( 1)
t
q
q
Tu t d t qs v s d s
−
= + −
−
1
( 2) ( 2)
0 0
1 1
(1 ) ( ) ( ) ( )
( 1) ( 1)
t
q q
q q
qs v s d s t qs v s d s
− −
= − − + −
− −
0.
[0,1]
min ( ) (1).
t
Tu t Tu
=
1
( 1)
0 1 0
1
(1) (1 ) ( )
( )
q
q
Tu d d qs v s d s
−
= + + −
](https://image.slidesharecdn.com/ijrtemb0270515-190123175553/75/Existence-of-positive-solutions-for-fractional-q-difference-equations-involving-integral-boundary-conditions-with-pLaplacian-operator-5-2048.jpg)
![Existence of positive solutions for fractional q-difference…
|Volume 2| Issue 7 | www.ijrtem.com | 10 |
The proof is complete. □
IV. MAIN RESULTS
Theorem 4.1 Suppose that there exist numbers 1 2 30 r r r such that f satisfies the following conditions:
1( )H 3( , ) ,f t u M for [0,1],t
2( )H 2( , ) ,f t u M for [0,1],t 2[0, ];u r
3( )H 1( , ) ,f t u M for [0,1],t 1[0, ],u r
where
Then the problem (1.1) has at least two positive solutions 1u and 2u such that 1 1( )r u with 1 2( )u r
1
( 1)
1 0 0
0
1
( ) ( ) ( )
[ ( ) ] ( )
t
q q
q q
g t t qs v s d sd t
a g t d t
−
= −
−
1
1
( 2)0
1 0
0
[ ( ) ]
(1 ) ( )
[ ( ) ] ( 1)
q
q
q q
b tg t d t
qs v s d s
a g t d t
−
−
+ −
− −
1 1
( 2) ( 1)
0 0
1 1
(1 ) ( ) (1 ) ( )
( 1) ( )
q q
q q
qs v s d s qs v s d s
− −
− − + −
−
1
( 1)
1 0 0
0
1
( ) ( ) ( )
[ ( ) ] ( )
t
q q
q q
g t t qs v s d sd t
a g t d t
−
−
−
1 1
1
( 2)0 0
1 0
0
[ ( ) ] [ ( ) ]
(1 ) ( )
[ ( ) ] ( 1)
q q
q
q q
b tg t d t a g t d t
qs v s d s
a g t d t
−
− − −
+ −
− −
1
( 1)
0
1
(1 ) ( )
( )
q
q
qs v s d s
−
+ −
0.
3
3[ , ];
r
u r
k
1
3
3
3
( )
,
(2, )
p
q
q
r
M
qB L
−
=
1
2
2
2
( 1)
,
(2, )
p
q
q
r
M
qB L
−
−
=
1
1
1
1
( )
,
(2, )
p
q
q
r
M
qB L
−
=
( )
1 11
3 1 0
0
1
(1 ) ,
( 1)[ ( ) ] ( )
q
q
q
q q
b a
L s s d s
a g t d t
−−
−
= + −
+−
1 1
0 0
2 1 1
0 0
( ) ( )
,
[ ( ) ] ( 1) [ ( ) ] ( )
q q
q q q q
g t d t b tg t d t
L
a g t d t a g t d t
−
= +
− + −
( )
1
1 110
1 1 0
0
( )
(1 ) .
[ ( ) ] ( )
qq
q
q q
b tg t d t
L s s d s
a g t d t
−−
−
= −
−
](https://image.slidesharecdn.com/ijrtemb0270515-190123175553/75/Existence-of-positive-solutions-for-fractional-q-difference-equations-involving-integral-boundary-conditions-with-pLaplacian-operator-6-2048.jpg)
![Existence of positive solutions for fractional q-difference…
|Volume 2| Issue 7 | www.ijrtem.com | 11 |
and 2 2( )r u with 2 3( ) .u r
Proof Define the cone P E by
where
For any ,u P in view of Lemma 3.3, we get
Therefore, :T P P→ . In view of the Arzela-Ascoli theorem, we have :T P P→ is completely continuous.
We define the functions on the cone P:
Obviously, we have (0) 0, = ( ) ( ) ( ).u u u
For any we get that is, therefore we obtain
[0,1]
,min ( ) , [0,1] ,
t
P u u E u t k u t
=
1 1
0 0
1
0
[ ( ) ] [ ( ) ]
,
( )
q q
q
b tg t d t a g t d t
k
b tg t d t
− − −
=
−
0 1.k
1
( 1)
0 1 0[0,1]
1
min ( ) (1) (1 ) ( )
( )
q
t
q
Tu t Tu d d qs v s d s
−
= = + + −
1
( 1)
1 0 0
0
1
( ) ( ) ( )
[ ( ) ] ( )
t
q q
q q
g t t qs v s d sd t
a g t d t
−
= −
−
1
1
( 2)0
1 0
0
[ ( ) ]
(1 ) ( )
[ ( ) ] ( 1)
q
q
q q
b tg t d t
qs v s d s
a g t d t
−
−
+ −
− −
1 1
( 2) ( 1)
0 0
1 1
(1 ) ( ) (1 ) ( )
( 1) ( )
q q
q q
qs v s d s qs v s d s
− −
− − + −
−
1
( 1)
1 0 0
0
1
( ) ( ) ( )
[ ( ) ] ( )
t
q q
q q
g t t qs v s d sd t
a g t d t
−
−
−
1 1
1
( 2)0 0
1 0
0
[ ( ) ] [ ( ) ]
(1 ) ( )
[ ( ) ] ( 1)
q q
q
q q
b tg t d t a g t d t
qs v s d s
a g t d t
−
− − −
+ −
− −
1
( 1)
1 0 0
0
1
( ) ( ) ( )
[ ( ) ] ( )
t
q q
q q
k g t t qs v s d sd t
a g t d t
−
−
−
1
1
( 2)0
1 0
0
[ ( ) ]
(1 ) ( )
[ ( ) ] ( 1)
q
q
q q
b tg t d t
qs v s d s
a g t d t
−
−
+ −
− −
(0) .kTu k Tu= =
[0,1]
( ) min ( ) (1),
t
u u t u
= =
[0,1]
( ) max ( ) (0),
t
u u t u
= =
[0,1]
( ) max ( ) (0).
t
u u t u
= =
[0,1]
min ( ) ,
t
u t k u
1
( ).u u
k
( ) ,u k u 3( , ),u p r](https://image.slidesharecdn.com/ijrtemb0270515-190123175553/75/Existence-of-positive-solutions-for-fractional-q-difference-equations-involving-integral-boundary-conditions-with-pLaplacian-operator-7-2048.jpg)
![Existence of positive solutions for fractional q-difference…
|Volume 2| Issue 7 | www.ijrtem.com | 12 |
For any 2( , ),u P r we get ( ) ( )lu l u = for 0 1.l
In the following, we prove that the conditions of Theorem 2.1 hold.
Firstly, let 3( , ),u P r that is, for [0,1].t By means of 1( ),H we have
where So we get
3
3[ , ]
r
u r
k
( )1
0
( ) ( , ) ( , ( ))q qv s H s q f u d =
1 ( 1)
1
3 0
(1 ) (1 )
( )
q q
q
s s q q
M d
− − − −
1
1
3 (1 ) (2, )
,
( )
q
q
q
M s s qB
−
−
−
=
1
( 1)
0
(2, ) (1 ) .q qB q d
−
= −
1
( 1)
0 1 0[0,1]
1
( ) min ( ) (1) (1 ) ( )
( )
q
t
q
Tu Tu t Tu d d qs v s d s
−
= = = + + −
1
( 1)
1 0 0
0
1
( ) ( ) ( )
[ ( ) ] ( )
t
q q
q q
g t t qs v s d sd t
a g t d t
−
= −
−
1
1
( 2)0
1 0
0
[ ( ) ]
(1 ) ( )
[ ( ) ] ( 1)
q
q
q q
b tg t d t
qs v s d s
a g t d t
−
−
+ −
− −
1 1
( 2) ( 1)
0 0
1 1
(1 ) ( ) (1 ) ( )
( 1) ( )
q q
q q
qs v s d s qs v s d s
− −
− − + −
−
1 1
1
( 2)0 0
1 0
0
[ ( ) ] [ ( ) ]
(1 ) ( )
[ ( ) ] ( 1)
q q
q
q q
b tg t d t a g t d t
qs v s d s
a g t d t
−
− − −
−
− −
1
( 1)
0
1
(1 ) ( )
( )
q
q
qs v s d s
−
+ −
1
1
1 3( 2)
1 0
0
(1 ) (2, )
(1 )
( )[ ( ) ] ( 1)
q
q
q
q
q q
M s s qBb a
qs d s
a g t d t
−
−
−
−−
− − −
1
1
1 3( 1)
0
(1 ) (2, )1
(1 )
( ) ( )
q
q
q
q q
M s s qB
qs d s
−
−
−
−
+ −
( )
1
1 13 1
0
(2, )1
(1 )
( 1) ( )
q
qq
q
q q
M qB
s s d s
−
−−
+ − +
( )
1
1 13 1
1 0
0
(2, )
(1 )
( )[ ( ) ] ( )
q
qq
q
q
q q
M qBb a
s s d s
a g t d t
−
−−
−
= − −
1
3
3 3
(2, )
.
( )
q
q
q
M qB
L r
−
= = ](https://image.slidesharecdn.com/ijrtemb0270515-190123175553/75/Existence-of-positive-solutions-for-fractional-q-difference-equations-involving-integral-boundary-conditions-with-pLaplacian-operator-8-2048.jpg)
![Existence of positive solutions for fractional q-difference…
|Volume 2| Issue 7 | www.ijrtem.com | 13 |
Secondly, let 2( , ),u P r that is, 2[0, ]u r for [0,1].t By means of 2( ),H we get
So, we have
Finally, let 1( , ),u P r that is, 1[0, ]u r for [0,1].t By means of 3( ),H we get
So we get
( )1
0
( ) ( , ) ( , ( ))q qv s H s q f u d =
1
( 1)
1 2
2 0
(2, )(1 )
.
( 1) ( 1)
q
q
q q
q q
M qBq q
M d
−
− −
= − −
0
[0,1]
( ) max ( ) (0)
t
Tu Tu t Tu d
= = =
1
( 1)
1 0 0
0
1
( ) ( ) ( )
[ ( ) ] ( )
t
q q
q q
g t t qs v s d sd t
a g t d t
−
= −
−
1
1
( 2)0
1 0
0
[ ( ) ]
(1 ) ( )
[ ( ) ] ( 1)
q
q
q q
b tg t d t
qs v s d s
a g t d t
−
−
+ −
− −
1
1 2( 1)
1 0 0
0
(2, )1
( ) ( )
( 1)[ ( ) ] ( )
q
t q
q q
q
q q
M qB
g t t qs d sd t
a g t d t
−
−
− −−
1 1
1 2( 2)0
1 0
0
[ ( ) ] (2, )
(1 )
( 1)[ ( ) ] ( 1)
q
q q
q
q
q q
b tg t d t M qB
qs d s
a g t d t
−
−
−
+ − −− −
1 1
20
1
0
( ) (2, )
( 1)[ ( ) ] ( 1)
q
q q
q
q q
g t d t M qB
a g t d t
−
−− +
1 1
20
1
0
[ ( ) ] (2, )
( 1)[ ( ) ] ( )
q
q q
q
q q
b tg t d t M qB
a g t d t
−
−
+ −−
1
2
2 2
(2, )
.
( 1)
q
q
q
M qB
L r
−
= = −
( )1
0
( ) ( , ) ( , ( ))q qv s H s q f u d =
1
11 ( 1)
1 1
1 0
(1 ) (2, )(1 ) (1 )
.
( ) ( )
q
q
q q
q q
M s s qBs s q q
M d
−
−− − −− −
=
0
[0,1]
( ) max ( ) (0)
t
Tu Tu t Tu d
= = =
1
( 1)
1 0 0
0
1
( ) ( ) ( )
[ ( ) ] ( )
t
q q
q q
g t t qs v s d sd t
a g t d t
−
= −
−
](https://image.slidesharecdn.com/ijrtemb0270515-190123175553/75/Existence-of-positive-solutions-for-fractional-q-difference-equations-involving-integral-boundary-conditions-with-pLaplacian-operator-9-2048.jpg)
![Existence of positive solutions for fractional q-difference…
|Volume 2| Issue 7 | www.ijrtem.com | 14 |
Therefore, in view of Theorem 2.1, we see that the problem (1.1) has at least two positive solutions 1u and 2u
such that 1 1( )r u with 1 2( )u r and 2 2( )r u with 2 3( ) .u r □
VI. EXAMPLE
In this section, we give a simple example to explain the main result.
Example 5.1 For the problem (1.1), let 2.8, = 2.3, = 4,a = 10,b = 2,p = ( ) ,g t t= then we get 2,q =
Let
From a direct calculation, we get
for
for
for
In view of Theorem 4.1, we see that the aforementioned problem has at least two positive solutions 1u and 2u
such that 10.5 ( )u with 1( ) 9u and 29 ( )u with 2( ) 10.u
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[1] Chen, T, Liu, W: An anti-periodic boundary value problem for the fractional differential equation with a
p-Laplacian operator. Appl. Math. Lett. 25, 1671-1675 (2012)
[2] Chai, G: Positive solutions for boundary value problem of fractional differential equation with p-
Laplacian operator. Bound. Value Probl. 2012, 18 (2012)
[3] Feng, X, Feng, H, Tan, H: Existence and iteration of positive solutions for third-order Sturm-Liouville
1
1
( 2)0
1 0
0
[ ( ) ]
(1 ) ( )
[ ( ) ] ( 1)
q
q
q q
b tg t d t
qs v s d s
a g t d t
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−
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1 1
1
1 1( 2)0
1 0
0
[ ( ) ] (1 ) (2, )
(1 )
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q
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b tg t d t M s s qB
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−
−
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− − −
( )
1 1
1 11 10
1 0
0
[ ( ) ] (2, )
(1 )
( )[ ( ) ] ( )
q
qq q
q
q
q q
b tg t d t M qB
s s d s
a g t d t
−
−−
−
= − −
1
1
1 1
(2, )
.
( )
q
q
q
M qB
L r
−
= =
1
0
1
( ) ,
2
qg t d t =
1
0
1
( ) ,
3
qtg t d t =
1 1
0 0
1
0
[ ( ) ] [ ( ) ] 37
0.637931.
58( )
q q
q
b tg t d t a g t d t
k
b tg t d t
− − −
= =
−
23,
( , ) 23 600( 9),
623,
f t u u
= + −
[0,1], [0,9],t u
580
[0,1], [10, ],
37
t u 3( , ) 583.266938f t u M
2( , ) 36.538326f t u M
1( , ) 21.322041f t u M
[0,1], [9,10],t u
[0,1], [10, ).t u +
[0,1], [0,9],t u
[0,1], [0,0.5].t u ](https://image.slidesharecdn.com/ijrtemb0270515-190123175553/75/Existence-of-positive-solutions-for-fractional-q-difference-equations-involving-integral-boundary-conditions-with-pLaplacian-operator-10-2048.jpg)
![Existence of positive solutions for fractional q-difference…
|Volume 2| Issue 7 | www.ijrtem.com | 15 |
boundary value problems with p-Laplacian. Appl. Math. Comput. 266, 634-641 (2015)
[4] Li, Y, Qi, A: Positive solutions for multi-point boundary value problems of fractional differential
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[5] Han, Z, Lu, H, Zhang, C: Positive solutions for eigenvalue problems of fractional differential equation
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[6] Jankowski, T: Positive solutions to fractional differential equations involving Stieltjes integral conditions.
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integral boundary conditions. Fract. Calc. Appl. Anal. 19, 989-1009 (2016)
[10] Cabada, A, Dimitrijevic, S, Tomovic, T, Aleksic, S: The existence of a positive solution for nonlinear
fractional differential equations with integral boundary value conditions. Math. Methods Appl. Sci. (2016)
[11] Yang, W: Positive solution for fractional q-difference boundary value problems with -Laplacian
operator. Bull.Malays.Math.Soc. 36(4)1195-1203(2013)
[12] Yang, Q,Yang, W: Positive solution for q-fractional four-point boundary value problems with p-
Laplacian operator. J.Inequal.Appl. 481(2014)
[13] Bashir, A, Sotiris, K,N, Loannis,K,P: Existence results for nonlocal boundary value problems of
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Chengmin Hou. “Existence of Positive Solutions for Fractional q-Difference Equations
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05–15., www.ijrtem.com.](https://image.slidesharecdn.com/ijrtemb0270515-190123175553/75/Existence-of-positive-solutions-for-fractional-q-difference-equations-involving-integral-boundary-conditions-with-pLaplacian-operator-11-2048.jpg)
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Existence of positive solutions for fractional q-difference equations involving integral boundary conditions with pLaplacian operator
- 1. Invention Journal ofResearch Technology in Engineering & Management (IJRTEM) ISSN: 2455-3689 www.ijrtem.com Volume 2 Issue 7 ǁ July 2018 ǁ PP 05-15 |Volume 2| Issue 7 | www.ijrtem.com | 5 | Existence of positive solutions for fractional q-difference equations involving integral boundary conditions with p- Laplacian operator Yizhu Wang1 , Chengmin Hou1* (Department of Mathematics, Yanbian University, Yanji, 133002, P.R. China) ABSTRACT: The existence of positive solutions is considered for a fractional q-difference equation with p- Laplacian operator in this article. By employing the Avery-Henderson fixed point theorem, a new result is obtained for the boundary value problems. KEY WORDS: positive solutions; Riemann-Liouville fractional q-derivatives; Caputo fractional q-derivatives; p-Laplacian; Avery-Henderson fixed point theorem I. INTRODUCTION Fractional calculus is the extension of integer order calculus to arbitrary order calculus. With the development of fractional calculus, fractional differential equations have wide applications in the modeling of different physical and natural science fields, such as fluid mechanics, chemistry, control system, heat conduction, etc. There are many papers concerning fractional differential equations with the p-Laplacian operator [1-5] and fractional differential equations with integral boundary conditions [6-10]. In [11], Yang studied the following fractional q-difference boundary value problem with p-Laplacian operator where 1 , 2. The existence results for the above boundary value problem were obtained by using the upper and lower solutions method associates with the Schauder fixed point theorem. In [12], Yuan and Yang considered a class of four-point boundary value problems of fractional q-difference equations with p-Laplacian operator Where is the fractional q-derivative of the Riemann-Liouville type with 1 , 2. By applying the upper and lower solutions method associated with the Schauder fixed point theorem, the existence results of at least one positive solution for the above fractional q-difference boundary value problem with p-Laplacian operator are established. Motivated by the aforementioned work, this work discusses the existence of positive solutions for this fractional q-difference equation: ( ( ( ))) ( , ( )),0 1,2 3, (0) (1) 0, c c q p qD D x t f t u t t x x = = = (1) (0) 0,c c q qD x D x = = ( ( ( ))) ( , ( )),0 1,2 3, (0) 0, (1) , q P qD D x t f t u t t x x ax = = = (0) 0, (1) ( ),q q qD x D x bD x = = ,q qD D
- 2. Existence of positivesolutions for fractional q-difference… |Volume 2| Issue 7 | www.ijrtem.com | 6 | (1.1) where 2 3, 2 3, and 5 6. + 2 ( ) , p p u u u − = 1.p is the Caputo fractional q- derivative, is the Riemann-Liouville fractional q-derivative. We will always suppose the following conditions are satisfied: 1( )H ( ):[0,1] [0, )g t → + with 1 ( ) [0,1],g t L and 2( )H , (0, ),a b + and ;b a 3( )H ( , ) :[0,1] (0, ) (0, )f t u + → + is continuous. II. BACKGROUND AND DEFINITIONS To show the main result of this work, we give in the following some basic definitions and theorems, which can be found in [13,14]. Definition 2.1 [13] Let 0 and f be a function defined on [0,1] . The fractional q-integral of Riemann- Liouville type is and Lemma 2.2 [13] Let 0, then the following equality holds: Lemma 2.3 [14]Let 0, If then where 1.n = + Theorem 2.4 (Avery-Henderson fixed point theorem [15]) Let ( , )E be a Banach space, and be a cone. Let and be increasing non-negative, continuous functionals on ,P and be a non-negative continuous functionals on P with such that, for some 3 0r and 0,M ( ) ( ) ( ),u u u ( 1) ,0 0 ( ) ( )( ) ( ) , ( ) t qq q t qs I f t f s d s + − − = 0, [0,1].t 1 ,0 ,0 ,0 0 ( ) ( ) ( )(0 ). ( 1) k c k q q q k q t I D f t f t D f k + + + − + = = − + 0 ,0 ( )( ) ( )q I f t f t+ = ,0 ,0 ' ,0 ,0 ,0 2 ,0 ,0 1 ,0 0 [ ( ( ))] ( , ( )) 0, (0,1), ( (0)) [ ( (0))] ( (1)) 0, (0) (1) 0, (0) (0) ( ) ( ) , c pq q c c c p p pq q q q q qq D D u t f t u t t D u D u D u D u D u au bD u g t u t d t + + + + + + + + + = = = = = = + = ,0 c q D + ,0q D + 1 0 ( ) 0,qg t d t 1 0 ( ) 0;qtg t d t 1 0 ( ) qa g t d t ,0 [0,1],q D u C + ( ) ,0 ,0 1 ( ) ( ) n i iq q i I D f t f t k t + + − = = + P E (0) 0, =
- 3. Existence of positivesolutions for fractional q-difference… |Volume 2| Issue 7 | www.ijrtem.com | 7 | and ( ),u M u for all 3( , ),u p r where 3 3( , ) { : ( ) }.p r u P u r = Suppose that there exist positive numbers 1 2 3,r r r such that ( ) ( )lu l u for 0 1,l and 2( , ).u P r If 3: ( , )T p r P → is a completely continuous operator satisfying: 1( )C 3( )Tu r for all 3( , );u P r 2( )C 2( )Tu r for all 2( , );u P r 3( )C 1( , ) ,P r and 1( )Tu r for all 1( , ),u P r then T has at least two fixed points 1u and 2u such that 1 1( )r u with 1 2( )u r and 2 2( )r u with 2 3( ) .u r III. PRELIMINARY LEMMAS Lemma 3.1 The boundary value problem (1.1) is equivalent to the following equation: (3.1) where (3.2) (3.3) (3.4) (3.5) is the inverse function of a.e., Proof From we get ( 1) 0 1 0 1 ( ) ( ) ( ) , ( ) t q q u t d d t t qs v s d s − = + + − 1 ( 1) 0 1 0 0 0 1 ( ) ( ) ( ) [ ( ) ] ( ) t q q q q d g t t qs v s d sd t a g t d t − = − − 1 1 ( 2)0 1 0 0 [ ( ) ] (1 ) ( ) , [ ( ) ] ( 1) q q q q b tg t d t qs v s d s a g t d t − − + − − − 1 ( 2) 1 0 1 (1 ) ( ) , ( 1) q q d qs v s d s − = − − − ( )1 0 ( ) ( , ) ( , ( )) ,q qv s H s q f u d = ( 1) ( 1) ( 1) ( (1 )) ( ) ,1 ( , ) ( ) ( (1 )) ,q s s H s s − − − − − − = − 0 1,s 0 1.s ( )q s 2 ( ) , q q s s s − = 1 1 1. p q + = ,0 ,0 [ ( ( ))] ( , ( )) 0,c pq q D D u t f t u t + + + = ( 1) 1 2 3 1 2 3,0 0 1 ( ( )) ( ) ( , ( )) . ( ) t c p qq q D u t t q f u d c t c t c t + − − − − = − − + + + ( ),p s
- 4. Existence of positivesolutions for fractional q-difference… |Volume 2| Issue 7 | www.ijrtem.com | 8 | In view of we get 2 3 0,c c= = i.e., (3.6) Conditions imply that (3.7) By use of (3.6) and (3.7), we get (3.8) In view of (3.8), we obtain (3.9) Let by use of (3.9), we get Conditions imply that 2 0,d = i.e., then we have Conditions imply that From we get Therefore, we can obtain ' ,0 ,0 ( (0)) [ ( (0))] 0,c c p pq q D u D u + += = ( 1) 1 1,0 0 1 ( ( )) ( ) ( , ( )) . ( ) t c p qq q D u t t q f u d c t + − − = − − + ,0 ( (1)) 0c p q D u + = 1 ( 1) 1 0 1 (1 ) ( , ( )) . ( ) q q c q f u d − = − − 1 ,0 0 ( ( )) ( , ) ( , ( )) .c p qq D u t H t q f u d + = ( )1 ,0 0 ( ) ( , ) ( , ( )) .c q qq D u t H t q f u d + = ( )1 0 ( ) ( , ) ( , ( )) ,q qv t H t q f u d = ( 1) 2 0 1 20 1 ( ) ( ) ( ) . ( ) t q q u t t qs v s d s d d t d t − = − + + + 2 ,0 (0) 0q D u+ = ( 1) 0 10 1 ( ) ( ) ( ) , ( ) t q q u t t qs v s d s d d t − = − + + ' ( 2) 10 1 ( ) ( ) ( ) . ( 1) t q q u t t qs v s d s d − = − + − ,0 (1) 0q D u+ = 1 ( 2) 1 0 1 (1 ) ( ) . ( 1) q q d qs v s d s − = − − − 1 ,0 0 (0) (0) ( ) ( ) ,qq au bD u g t u t d t++ = 1 ( 1) 0 1 0 0 0 1 ( ) ( ) ( ) [ ( ) ] ( ) t q q q q d g t t qs v s d sd t a g t d t − = − − 1 1 ( 2)0 1 0 0 [ ( ) ] (1 ) ( ) . [ ( ) ] ( 1) q q q q b tg t d t qs v s d s a g t d t − − + − − − ( 1) 0 1 0 1 ( ) ( ) ( ) ( ) t q q u t d d t t qs v s d s − = + + −
- 5. Existence of positivesolutions for fractional q-difference… |Volume 2| Issue 7 | www.ijrtem.com | 9 | The proof is complete. □ Lemma 3.2 ([16]) The function ( , )H s defined by (3.5) is continuous on [0,1] [0,1] and satisfy for , [0,1].s Let E be the real Banach space [0,1]C with the maximum norm, define the operator :T E E→ by Lemma 3.3 For with is non-increasing and non-negative. Proof Since so we get So ( )( )Tu t is non-increasing, then we have We have 1 ( 1) 1 0 0 0 1 ( ) ( ) ( ) [ ( ) ] ( ) t q q q q g t t qs v s d sd t a g t d t − = − − 1 1 ( 2)0 1 0 0 [ ( ) ] (1 ) ( ) [ ( ) ] ( 1) q q q q b tg t d t qs v s d s a g t d t − − + − − − 1 ( 2) ( 1) 0 0 1 (1 ) ( ) ( ) ( ) . ( 1) ( ) t q q q q t qs v s d s t qs v s d s − − − − + − − 1 ( 1) ( 1) (1 ) (1 ) (1 ) ( , ) ( ) ( 1)q q s s H s − − − − − − − ( 1) 0 1 0 1 ( ) ( ) ( ) ( ) t q q Tu t d d t t qs v s d s − = + + − 1 ( 1) 1 0 0 0 1 ( ) ( ) ( ) [ ( ) ] ( ) t q q q q g t t qs v s d sd t a g t d t − = − − 1 1 ( 2)0 1 0 0 [ ( ) ] (1 ) ( ) [ ( ) ] ( 1) q q q q b tg t d t qs v s d s a g t d t − − + − − − 1 ( 2) ( 1) 0 0 1 (1 ) ( ) ( ) ( ) . ( 1) ( ) t q q q q t qs v s d s t qs v s d s − − − − + − − [0,1]u C ( )( )Tu t( ) 0,u t ( 1) 0 1 0 1 ( ) ( ) ( ) , ( ) t q q Tu t d d t t qs v s d s − = + + − ( ) ' ( 2) 1 0 1 ( ) ( ) ( ) ( 1) t q q Tu t d t qs v s d s − = + − − 1 ( 2) ( 2) 0 0 1 1 (1 ) ( ) ( ) ( ) ( 1) ( 1) t q q q q qs v s d s t qs v s d s − − = − − + − − − 0. [0,1] min ( ) (1). t Tu t Tu = 1 ( 1) 0 1 0 1 (1) (1 ) ( ) ( ) q q Tu d d qs v s d s − = + + −
- 6. Existence of positivesolutions for fractional q-difference… |Volume 2| Issue 7 | www.ijrtem.com | 10 | The proof is complete. □ IV. MAIN RESULTS Theorem 4.1 Suppose that there exist numbers 1 2 30 r r r such that f satisfies the following conditions: 1( )H 3( , ) ,f t u M for [0,1],t 2( )H 2( , ) ,f t u M for [0,1],t 2[0, ];u r 3( )H 1( , ) ,f t u M for [0,1],t 1[0, ],u r where Then the problem (1.1) has at least two positive solutions 1u and 2u such that 1 1( )r u with 1 2( )u r 1 ( 1) 1 0 0 0 1 ( ) ( ) ( ) [ ( ) ] ( ) t q q q q g t t qs v s d sd t a g t d t − = − − 1 1 ( 2)0 1 0 0 [ ( ) ] (1 ) ( ) [ ( ) ] ( 1) q q q q b tg t d t qs v s d s a g t d t − − + − − − 1 1 ( 2) ( 1) 0 0 1 1 (1 ) ( ) (1 ) ( ) ( 1) ( ) q q q q qs v s d s qs v s d s − − − − + − − 1 ( 1) 1 0 0 0 1 ( ) ( ) ( ) [ ( ) ] ( ) t q q q q g t t qs v s d sd t a g t d t − − − 1 1 1 ( 2)0 0 1 0 0 [ ( ) ] [ ( ) ] (1 ) ( ) [ ( ) ] ( 1) q q q q q b tg t d t a g t d t qs v s d s a g t d t − − − − + − − − 1 ( 1) 0 1 (1 ) ( ) ( ) q q qs v s d s − + − 0. 3 3[ , ]; r u r k 1 3 3 3 ( ) , (2, ) p q q r M qB L − = 1 2 2 2 ( 1) , (2, ) p q q r M qB L − − = 1 1 1 1 ( ) , (2, ) p q q r M qB L − = ( ) 1 11 3 1 0 0 1 (1 ) , ( 1)[ ( ) ] ( ) q q q q q b a L s s d s a g t d t −− − = + − +− 1 1 0 0 2 1 1 0 0 ( ) ( ) , [ ( ) ] ( 1) [ ( ) ] ( ) q q q q q q g t d t b tg t d t L a g t d t a g t d t − = + − + − ( ) 1 1 110 1 1 0 0 ( ) (1 ) . [ ( ) ] ( ) qq q q q b tg t d t L s s d s a g t d t −− − = − −
- 7. Existence of positivesolutions for fractional q-difference… |Volume 2| Issue 7 | www.ijrtem.com | 11 | and 2 2( )r u with 2 3( ) .u r Proof Define the cone P E by where For any ,u P in view of Lemma 3.3, we get Therefore, :T P P→ . In view of the Arzela-Ascoli theorem, we have :T P P→ is completely continuous. We define the functions on the cone P: Obviously, we have (0) 0, = ( ) ( ) ( ).u u u For any we get that is, therefore we obtain [0,1] ,min ( ) , [0,1] , t P u u E u t k u t = 1 1 0 0 1 0 [ ( ) ] [ ( ) ] , ( ) q q q b tg t d t a g t d t k b tg t d t − − − = − 0 1.k 1 ( 1) 0 1 0[0,1] 1 min ( ) (1) (1 ) ( ) ( ) q t q Tu t Tu d d qs v s d s − = = + + − 1 ( 1) 1 0 0 0 1 ( ) ( ) ( ) [ ( ) ] ( ) t q q q q g t t qs v s d sd t a g t d t − = − − 1 1 ( 2)0 1 0 0 [ ( ) ] (1 ) ( ) [ ( ) ] ( 1) q q q q b tg t d t qs v s d s a g t d t − − + − − − 1 1 ( 2) ( 1) 0 0 1 1 (1 ) ( ) (1 ) ( ) ( 1) ( ) q q q q qs v s d s qs v s d s − − − − + − − 1 ( 1) 1 0 0 0 1 ( ) ( ) ( ) [ ( ) ] ( ) t q q q q g t t qs v s d sd t a g t d t − − − 1 1 1 ( 2)0 0 1 0 0 [ ( ) ] [ ( ) ] (1 ) ( ) [ ( ) ] ( 1) q q q q q b tg t d t a g t d t qs v s d s a g t d t − − − − + − − − 1 ( 1) 1 0 0 0 1 ( ) ( ) ( ) [ ( ) ] ( ) t q q q q k g t t qs v s d sd t a g t d t − − − 1 1 ( 2)0 1 0 0 [ ( ) ] (1 ) ( ) [ ( ) ] ( 1) q q q q b tg t d t qs v s d s a g t d t − − + − − − (0) .kTu k Tu= = [0,1] ( ) min ( ) (1), t u u t u = = [0,1] ( ) max ( ) (0), t u u t u = = [0,1] ( ) max ( ) (0). t u u t u = = [0,1] min ( ) , t u t k u 1 ( ).u u k ( ) ,u k u 3( , ),u p r
- 8. Existence of positivesolutions for fractional q-difference… |Volume 2| Issue 7 | www.ijrtem.com | 12 | For any 2( , ),u P r we get ( ) ( )lu l u = for 0 1.l In the following, we prove that the conditions of Theorem 2.1 hold. Firstly, let 3( , ),u P r that is, for [0,1].t By means of 1( ),H we have where So we get 3 3[ , ] r u r k ( )1 0 ( ) ( , ) ( , ( ))q qv s H s q f u d = 1 ( 1) 1 3 0 (1 ) (1 ) ( ) q q q s s q q M d − − − − 1 1 3 (1 ) (2, ) , ( ) q q q M s s qB − − − = 1 ( 1) 0 (2, ) (1 ) .q qB q d − = − 1 ( 1) 0 1 0[0,1] 1 ( ) min ( ) (1) (1 ) ( ) ( ) q t q Tu Tu t Tu d d qs v s d s − = = = + + − 1 ( 1) 1 0 0 0 1 ( ) ( ) ( ) [ ( ) ] ( ) t q q q q g t t qs v s d sd t a g t d t − = − − 1 1 ( 2)0 1 0 0 [ ( ) ] (1 ) ( ) [ ( ) ] ( 1) q q q q b tg t d t qs v s d s a g t d t − − + − − − 1 1 ( 2) ( 1) 0 0 1 1 (1 ) ( ) (1 ) ( ) ( 1) ( ) q q q q qs v s d s qs v s d s − − − − + − − 1 1 1 ( 2)0 0 1 0 0 [ ( ) ] [ ( ) ] (1 ) ( ) [ ( ) ] ( 1) q q q q q b tg t d t a g t d t qs v s d s a g t d t − − − − − − − 1 ( 1) 0 1 (1 ) ( ) ( ) q q qs v s d s − + − 1 1 1 3( 2) 1 0 0 (1 ) (2, ) (1 ) ( )[ ( ) ] ( 1) q q q q q q M s s qBb a qs d s a g t d t − − − −− − − − 1 1 1 3( 1) 0 (1 ) (2, )1 (1 ) ( ) ( ) q q q q q M s s qB qs d s − − − − + − ( ) 1 1 13 1 0 (2, )1 (1 ) ( 1) ( ) q qq q q q M qB s s d s − −− + − + ( ) 1 1 13 1 1 0 0 (2, ) (1 ) ( )[ ( ) ] ( ) q qq q q q q M qBb a s s d s a g t d t − −− − = − − 1 3 3 3 (2, ) . ( ) q q q M qB L r − = =
- 9. Existence of positivesolutions for fractional q-difference… |Volume 2| Issue 7 | www.ijrtem.com | 13 | Secondly, let 2( , ),u P r that is, 2[0, ]u r for [0,1].t By means of 2( ),H we get So, we have Finally, let 1( , ),u P r that is, 1[0, ]u r for [0,1].t By means of 3( ),H we get So we get ( )1 0 ( ) ( , ) ( , ( ))q qv s H s q f u d = 1 ( 1) 1 2 2 0 (2, )(1 ) . ( 1) ( 1) q q q q q q M qBq q M d − − − = − − 0 [0,1] ( ) max ( ) (0) t Tu Tu t Tu d = = = 1 ( 1) 1 0 0 0 1 ( ) ( ) ( ) [ ( ) ] ( ) t q q q q g t t qs v s d sd t a g t d t − = − − 1 1 ( 2)0 1 0 0 [ ( ) ] (1 ) ( ) [ ( ) ] ( 1) q q q q b tg t d t qs v s d s a g t d t − − + − − − 1 1 2( 1) 1 0 0 0 (2, )1 ( ) ( ) ( 1)[ ( ) ] ( ) q t q q q q q q M qB g t t qs d sd t a g t d t − − − −− 1 1 1 2( 2)0 1 0 0 [ ( ) ] (2, ) (1 ) ( 1)[ ( ) ] ( 1) q q q q q q q b tg t d t M qB qs d s a g t d t − − − + − −− − 1 1 20 1 0 ( ) (2, ) ( 1)[ ( ) ] ( 1) q q q q q q g t d t M qB a g t d t − −− + 1 1 20 1 0 [ ( ) ] (2, ) ( 1)[ ( ) ] ( ) q q q q q q b tg t d t M qB a g t d t − − + −− 1 2 2 2 (2, ) . ( 1) q q q M qB L r − = = − ( )1 0 ( ) ( , ) ( , ( ))q qv s H s q f u d = 1 11 ( 1) 1 1 1 0 (1 ) (2, )(1 ) (1 ) . ( ) ( ) q q q q q q M s s qBs s q q M d − −− − −− − = 0 [0,1] ( ) max ( ) (0) t Tu Tu t Tu d = = = 1 ( 1) 1 0 0 0 1 ( ) ( ) ( ) [ ( ) ] ( ) t q q q q g t t qs v s d sd t a g t d t − = − −
- 10. Existence of positivesolutions for fractional q-difference… |Volume 2| Issue 7 | www.ijrtem.com | 14 | Therefore, in view of Theorem 2.1, we see that the problem (1.1) has at least two positive solutions 1u and 2u such that 1 1( )r u with 1 2( )u r and 2 2( )r u with 2 3( ) .u r □ VI. EXAMPLE In this section, we give a simple example to explain the main result. Example 5.1 For the problem (1.1), let 2.8, = 2.3, = 4,a = 10,b = 2,p = ( ) ,g t t= then we get 2,q = Let From a direct calculation, we get for for for In view of Theorem 4.1, we see that the aforementioned problem has at least two positive solutions 1u and 2u such that 10.5 ( )u with 1( ) 9u and 29 ( )u with 2( ) 10.u REFERENCES [1] Chen, T, Liu, W: An anti-periodic boundary value problem for the fractional differential equation with a p-Laplacian operator. Appl. Math. Lett. 25, 1671-1675 (2012) [2] Chai, G: Positive solutions for boundary value problem of fractional differential equation with p- Laplacian operator. Bound. Value Probl. 2012, 18 (2012) [3] Feng, X, Feng, H, Tan, H: Existence and iteration of positive solutions for third-order Sturm-Liouville 1 1 ( 2)0 1 0 0 [ ( ) ] (1 ) ( ) [ ( ) ] ( 1) q q q q b tg t d t qs v s d s a g t d t − − − − − 1 1 1 1 1( 2)0 1 0 0 [ ( ) ] (1 ) (2, ) (1 ) ( )[ ( ) ] ( 1) q q q q q q q b tg t d t M s s qB qs d s a g t d t − − − − − − − − ( ) 1 1 1 11 10 1 0 0 [ ( ) ] (2, ) (1 ) ( )[ ( ) ] ( ) q qq q q q q q b tg t d t M qB s s d s a g t d t − −− − = − − 1 1 1 1 (2, ) . ( ) q q q M qB L r − = = 1 0 1 ( ) , 2 qg t d t = 1 0 1 ( ) , 3 qtg t d t = 1 1 0 0 1 0 [ ( ) ] [ ( ) ] 37 0.637931. 58( ) q q q b tg t d t a g t d t k b tg t d t − − − = = − 23, ( , ) 23 600( 9), 623, f t u u = + − [0,1], [0,9],t u 580 [0,1], [10, ], 37 t u 3( , ) 583.266938f t u M 2( , ) 36.538326f t u M 1( , ) 21.322041f t u M [0,1], [9,10],t u [0,1], [10, ).t u + [0,1], [0,9],t u [0,1], [0,0.5].t u
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