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
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 c
Dq ( p ( c Dq x(t ))) f (t , u (t )), 0 t 1, 2 3,
x(0) x(1) 0,
c
Dq x(1) c Dq x(0) 0,
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
Dq (P ( Dq x(t ))) f (t , u (t )), 0 t 1, 2 3, x(0) 0, x(1) ax ,
Dq x(0) 0, Dq x(1) bDq x( ),
Where Dq , Dq are 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:
|Volume 2| Issue 7 |
www.ijrtem.com
|5|
Existence of positive solutions for fractional q-difference…
Dq,0 [ p ( c Dq,0 u (t ))] f (t , u (t )) 0, t (0,1), p ( c Dq,0 u (0)) [ p ( c Dq,0 u (0))]' p ( c Dq,0 u (1)) 0, 2 Dq ,0 u (0) Dq ,0 u (1) 0, 1 au (0) bDq ,0 u (0) 0 g (t )u (t )d qt , where 2 3, 2 3, and 5 6. p (u ) u
p 2
(1.1)
u, p 1.
c
Dq,0is the Caputo fractional
q-derivative, Dq ,0 is the Riemann-Liouville fractional q-derivative.
We will always suppose the following conditions are satisfied:
( H1 ) g (t ) :[0,1] [0, ) with g (t ) L1[0,1],
1
0
g (t )d qt 0, and
1
tg (t )d t 0; 0
q
1
( H 2 ) a, b (0, ), a 0 g (t )d qt and b a; ( H 3 ) f (t , u) :[0,1] (0, ) (0, ) 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].
0 and f
Definition 2.1 [13] Let
be a function defined on [0,1] . The fractional q-integral of
Riemann-Liouville type is ( I q0,0 f )(t ) f (t ) and
(t qs)( 1) f ( s ) d q s, 0 q ( )
( I q,0 f )(t )
t
0,
Lemma 2.2 [13] Let
then the following equality holds: 1
I q ,0 Dq ,0 f (t ) f (t )
c
k 0
0,
Lemma 2.3 [14]Let
0, t [0,1].
tk ( Dqk,0 f )(0 ). q (k 1)
If Dq,0 u C[0,1], then n
I q,0 Dq,0 f (t ) f (t ) kit ( i ) i 1
where
n 1.
Theorem 2.4 (Avery-Henderson fixed point theorem [15]) Let ( E , ) be a Banach space, and P E be a cone. Let
and
|Volume 2| Issue 7 |
be increasing non-negative, continuous functionals on P, and
www.ijrtem.com
be a non-negative
|6|
Existence of positive solutions for fractional q-difference… continuous functionals on P with (0) 0, such that, for some r3 0 and M 0, (u) (u) (u), and u M (u ), for all u p( , r3 ), where p( , r3 ) {u P : (u) r3}. Suppose that there exist positive numbers r1 r2 r3 , such that
(lu) l(u) If
for
0 l 1, and u P(, r2 ).
T : p( , r3 ) P is a completely continuous operator satisfying:
(C1 ) (Tu ) r3 for all u P(, r3 ); (C2 ) (Tu ) r2 for all u P(, r2 ); (C3 ) P( , r1 ) , and (Tu ) r1 for all u P( , r1 ), then
T
has at least two fixed points u1 and u2 such that r1 (u1 ) with
(u1 ) r2
and
r2 (u2 ) with (u2 ) r3 . III. PRELIMINARY LEMMAS Lemma 3.1
The boundary value problem (1.1) is equivalent to the following equation:
u (t ) d0 d1t
t 1 (t qs)( 1) v(s)d q s, 0 q ( )
(3.1)
where
d0
1 1
[a g (t )d qt ] q 0
( )
1
0
t
g (t ) (t qs)( 1) v( s)d q sd qt 0
1
[b tg (t )d qt ] 0
1
[a g (t )d q t ] q 0
d1
1
1
(1 qs) ( 1)
( 2)
0
( 2)
(3.3)
H (s, q ) f ( , u( ))d ,
(3.4)
H ( s, )
1
0
q
( 1) ( s )( 1) , 1 ( s(1 )) ( 1) q ( ) , ( s(1 ))
q ( s) is the inverse function of p ( s), a.e., q (s) s
|Volume 2| Issue 7 |
(3.2)
v( s ) d q s,
v ( s ) d q s,
0
q
v( s) q
1
(1 qs) ( 1)
q 2
0 s 1, 0 s 1.
s,
(3.5)
1 1 1. p q
www.ijrtem.com
|7|
Existence of positive solutions for fractional q-difference… Proof
From D [ p ( c D u(t ))] f (t , u(t )) 0, we get q ,0 q ,0 t 1 (t q )( 1) f ( , u ( ))d q c1t 1 c2t 2 c3t 3 . q ( ) 0
p ( c Dq,0 u (t ))
In view of p ( c Dq,0 u(0)) [ p ( c Dq,0 u(0))]' 0, we get c2 c3 0, i.e., t 1 (t q )( 1) f ( , u ( ))d q c1t 1. 0 q ( )
p ( c Dq,0 u (t ))
(3.6)
Conditions p ( c D u (1)) 0 imply that q ,0
c1
1 1 (1 q )( 1) f ( , u ( ))d q . q ( ) 0
(3.7)
By use of (3.6) and (3.7), we get 1
p ( c Dq,0 u(t )) H (t , q ) f ( , u ( ))d q .
(3.8)
0
In view of (3.8), we obtain c
Dq,0 u (t ) q
Let
v(t ) q
H (t, q ) f ( , u( ))d . 1
q
0
(3.9)
H (t, q ) f ( , u( ))d , 1
q
0
by use of (3.9), we get t 1 (t qs)( 1) v( s)d q s d 0 d1t d 2t 2 . q ( ) 0
u (t )
Conditions Dq2,0 u (0) 0 imply that d 2 0, i.e., t 1 (t qs)( 1) v( s)d q s d 0 d1t , q ( ) 0
u (t ) then we have
u ' (t )
1
q
t
(t qs) ( 1)
( 2)
0
v( s)d q s d1.
Conditions D u (1) 0 imply that q ,0
d1
1
1
(1 qs) ( 1)
( 2)
0
q
v( s)d q s.
1
From au (0) bDq ,0 u (0) g (t )u (t )d qt , we get 0
d0
1 1
[a g (t )d qt ] q 0
( )
1
0
t
g (t ) (t qs)( 1) v( s)d q sd qt 0
1
[b tg (t )d qt ] 0
1
[a g (t )d q t ] q 0
1
(1 qs) ( 1) 0
( 2)
v( s)d q s.
Therefore, we can obtain
|Volume 2| Issue 7 |
www.ijrtem.com
|8|
Existence of positive solutions for fractional q-difference… t 1 (t qs)( 1) v(s)d q s q ( ) 0
u (t ) d0 d1t
1
[a
1
0
g (t )d t ] ( )
1
0
q
t
g (t ) (t qs)( 1) v( s)d q sd qt 0
q
1
[b tg (t )d qt ]
1
(1 qs) g (t )d t ] ( 1) 0
[a
1
0
( 2)
0
q
t
q
1
(1 qs) ( 1)
( 2)
0
q
v( s )d q s
v( s ) d q s
t 1 (t qs)( 1) v( s)d q s. q ( ) 0
□
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
s 1 (1 s) (1 )( 1) (1 )( 1) H ( s, ) q ( ) q ( 1)
for
s, [0,1].
Let E be the real Banach space C[0,1] with the maximum norm, define the operator T : E E by t 1 (t qs)( 1) v( s)d q s q ( ) 0 1 t 1 g (t ) (t qs)( 1) v( s)d q sd qt 1 0 0 [a g (t )d qt ] q ( )
Tu (t ) d0 d1t
0
1
[b tg (t )d qt ] [a
0
1
(1 qs) g (t )d t ] ( 1) 0
1
( 2)
0
q
t
1
(1 qs) ( 1)
( 2)
0
q
v( s )d q s
q
v( s ) d q s
t 1 (t qs)( 1) v( s)d q s. q ( ) 0
Lemma 3.3 For u C[0,1] with u (t ) 0, (Tu)(t ) is non-increasing and non-negative. Proof
Since
Tu (t ) d0 d1t
t 1 (t qs)( 1) v( s)d q s, 0 q ( )
so we get
Tu (t ) d1
1
'
q
t
(t qs) ( 1)
1
1
(1 qs) ( 1) q
( 2)
0
0
( 2)
v ( s )d q s
v( s ) d q s
1
t
(t qs) ( 1) q
0
( 2)
v( s )d q s
0. So (Tu)(t ) is non-increasing, then we have min Tu(t ) Tu(1). We have t[0,1]
1 1 Tu (1) d0 d1 (1 qs)( 1) v( s)d q s 0 q ( )
|Volume 2| Issue 7 |
www.ijrtem.com
|9|
Existence of positive solutions for fractional q-difference…
1
[a
1
g (t )d t ] ( )
0
q
0
1
t
g (t ) (t qs)( 1) v( s)d q sd qt 0
q
1
[b tg (t )d qt ] [a
1
0
q
1
1
( 2)
0
1
[a
g (t )d t ] ( )
1
0
q
0
v( s ) d q s
1 1 (1 qs)( 1) v( s)d q s q ( ) 0
t
g (t ) (t qs)( 1) v( s)d q sd qt 0
q
1
v( s )d q s
q
(1 qs) ( 1) 1
( 2)
0
q
1
(1 qs) g (t )d t ] ( 1) 0
1
[b tg (t )d qt ] [a g (t )d qt ] 0
0
1
[a g (t )d qt ] q ( 1)
1
(1 qs)
( 2)
0
v( s )d q s
0
1 1 (1 qs)( 1) v( s)d q s 0 q ( )
0. □
The proof is complete.
IV. MAIN RESULTS Theorem 4.1
Suppose that there exist numbers 0 r1 r2 r3 such that f satisfies the following
conditions:
r ( H1 ) f (t , u) M 3 , for t [0,1], u [r3 , 3 ]; k ( H 2 ) f (t , u) M 2 , for t [0,1], u [0, r2 ]; ( H 3 ) f (t , u) M1 , for t [0,1], u [0, r1 ], where
q ( ) r3 M3 qBq (2, ) L3
p 1
M2
,
q ( 1) r2 qBq (2, ) L2
p 1
,
q ( ) r1 M1 qBq (2, ) L1
p 1
,
q 1 ba 1 1 1 L3 0 s (1 s) d q s, 1 [a g (t )d q t ] q ( ) q ( 1) 0 L2
1
0
1
g (t )d qt
1
b tg (t )d qt 0
1
[a g (t )d qt ] q ( 1) [a g (t )d qt ] q ( ) 0
,
0
1
L1
b tg (t )d q t 0
1
[a g (t )d q t ] q 0
|Volume 2| Issue 7 |
s ( ) 1
0
1
(1 s) d q s. q 1
www.ijrtem.com
| 10 |
Existence of positive solutions for fractional q-difference… Then the problem (1.1) has at least two positive solutions u1 and u2 such that r1 (u1 ) with
(u1 ) r2
(u2 ) r3 .
and r2 (u2 ) with
Define the cone P E by
Proof
P u u E , min u (t ) k u , t [0,1] , t[0,1]
where
k
1
1
0
0
[b tg (t )d qt ] [a g (t )d qt ] 1
b tg (t )d qt
0 k 1.
,
0
For any
u P, in view of Lemma 3.3, we get 1 1 (1 qs)( 1) v( s)d q s 0 q ( )
min Tu (t ) Tu (1) d0 d1
t[0,1]
1
[a
1
0
g (t )d t ] ( )
1
0
q
t
g (t ) (t qs)( 1) v( s)d q sd qt 0
q
1
[b tg (t )d qt ] [a
1
1
1
( 2)
0
1
[a
1
0
g (t )d t ] ( )
1
0
q
v( s )d q s
q
(1 qs) ( 1) q
( 2)
0
q
0
1
(1 qs) g (t )d t ] ( 1) 0
v( s ) d q s
1 1 (1 qs)( 1) v( s)d q s 0 q ( )
t
g (t ) (t qs)( 1) v( s)d q sd qt 0
q
1
1
0
0
[b tg (t )d qt ] [a g (t )d qt ] 1
[a g (t )d qt ] q ( 1)
1
(1 qs)
( 2)
0
v( s )d q s
0
1 t 1 k g ( t ) (t qs)( 1) v( s)d q sd qt 1 0 0 [a g (t )d q t ] q ( ) 0 1
[b tg (t )d qt ] 0
1
[a g (t )d q t ] q ( 1)
1
(1 qs) 0
0
( 2)
v( s )d q s
kTu(0) k Tu . 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:
(u) min u(t ) u(1), t[0,1]
(u) max u(t ) u(0), t[0,1]
(u) max u(t ) u(0). t[0,1]
Obviously, we have (0) 0, (u) (u) (u).
|Volume 2| Issue 7 |
www.ijrtem.com
| 11 |
Existence of positive solutions for fractional q-difference… For any u p( , r3 ), we get min u (t ) k u , that is, t[0,1]
1 k
(u) k u , therefore we obtain u (u ).
For any u P(, r2 ), we get (lu) l (u) for 0 l 1. In the following, we prove that the conditions of Theorem 2.1 hold. Firstly, let u P(, r3 ), that is, u [r3 ,
v( s) q
H (s, q ) f ( , u( ))d
r3 ] for t [0,1]. By means of ( H1 ), we have k
1
q
0
1 1s (1 s)q (1 q )( 1) q M 3 d q 0 q ( )
M 3 s 1 (1 s)qBq (2, ) q ( )
q 1
,
1
where Bq (2, ) (1 q )( 1) d q . So we get 0
1 1 (1 qs)( 1) v( s)d q s 0 q ( )
(Tu ) min Tu (t ) Tu (1) d0 d1 t[0,1]
1
[a
1
0
g (t )d t ] ( )
1
0
q
t
g (t ) (t qs)( 1) v( s)d q sd qt 0
q
1
[b tg (t )d qt ] [a
0
( 2)
0
q
1
v( s )d q s
q
1
(1 qs) ( 1)
( 2)
0
q
1
(1 qs) g (t )d t ] ( 1) 0
1
1
1
0
0
v( s ) d q s
[b tg (t )d qt ] [a g (t )d qt ] 1
[a g (t )d qt ] q ( 1)
1 1 (1 qs)( 1) v( s)d q s 0 q ( )
1
(1 qs)
( 2)
0
v( s )d q s
0
1 1 (1 qs)( 1) v( s)d q s q ( ) 0
ba
1
1
[a g (t )d qt ] q ( 1)
(1 qs)
( 2)
0
0
M 3 s 1 (1 s)qBq (2, ) q ( )
M 3 s 1 (1 s)qBq (2, ) 1 1 ( 1) (1 qs ) q ( ) 0 q ( ) M 3qBq (2, ) 1 [a g (t )d qt ] q ( ) q ( ) 0 ba
M 3qBq (2, ) q ( 1) q ( ) 1
M 3qBq (2, ) ( ) q
|Volume 2| Issue 7 |
q 1
s 1
0
q 1
1
s 1
1
0
(1 s)
dq s
q 1
(1 s)
q 1
q 1
dq s q 1
dq s
dq s
q 1
L3 r3 .
www.ijrtem.com
| 12 |
Existence of positive solutions for fractional q-difference… Secondly, let u P(, r2 ), that is, u [0, r2 ] for
v( s) q
t [0,1]. By means of ( H 2 ), we get
H (s, q ) f ( , u( ))d 1
q
0
( 1) M 2 qBq (2, ) 1 q (1 q ) q M 2 d q ( 1) 0 q ( 1) q
q 1
.
So we have
(Tu) max Tu(t ) Tu(0) d0 t[0,1] 1 t 1 g ( t ) (t qs)( 1) v( s)d q sd qt 1 0 0 [a g (t )d qt ] q ( ) 0
1
[b tg (t )d qt ]
(1 qs) g (t )d t ] ( 1)
[a
1
( 2)
0
q
0
q
1
1
0
1
[a g (t )d qt ] q ( )
1
0
t
g (t ) (t qs)
( 1)
0
0
v( s )d q s M 2 qBq (2, ) q ( 1)
1
[b tg (t )d qt ]
1
(1 qs)
0
1
[a g (t )d qt ] q ( 1)
( 2)
0
0
1
M 2 qBq (2, ) 1 [a g (t )d qt ] q ( 1) q ( 1) 0 0
g (t )d q t
1
[b tg (t )d qt ]
q 1
M 2 qBq (2, ) q ( 1)
d q sd qt q 1
dq s
q 1
M 2 qBq (2, ) 1 [a g (t )d qt ] q ( ) q ( 1) 0
q 1
0
q 1
M 2 qBq (2, ) L2 r2 . ( 1) q Finally, let u P( , r1 ), that is, u [0, r1 ] for t [0,1]. By means of ( H 3 ), we get
H (s, q ) f ( , u( ))d 1
v( s) q
q
0
1 M1s 1 (1 s)qBq (2, ) 1s (1 s)q (1 q )( 1) q M1 d q 0 q ( ) q ( )
q 1
.
So we get
(Tu) max Tu(t ) Tu(0) d0 t[0,1]
1
[a
1
0
|Volume 2| Issue 7 |
g (t )d t ] ( )
1
0
q
t
g (t ) (t qs)( 1) v( s)d q sd qt 0
q
www.ijrtem.com
| 13 |
Existence of positive solutions for fractional q-difference… 1
[b tg (t )d qt ]
1
(1 qs) g (t )d t ] ( 1) 0
[a
1
( 2)
0
q
0
v( s )d q s
q
1
[b tg (t )d qt ]
1
(1 qs) g (t )d t ] ( 1) 0
[a
1
0
( 2)
0
q
q
1
v( s )d q s
[b tg (t )d qt ]
1
0
1
[a g (t )d qt ] q ( 1)
(1 qs)
( 2)
0
0
1
[b tg (t )d qt ]
M1qBq (2, ) 1 [a g (t )d qt ] q ( ) q ( ) 0
M1s 1 (1 s)qBq (2, ) q ( )
q 1
0
M1qBq (2, ) ( ) q
s 1
1
0
(1 s)
q 1
q 1
dq s
dq s
q 1
L1 r1.
Therefore, in view of Theorem 2.1, we see that the problem (1.1) has at least two positive solutions u1 and u2 such that r1 (u1 ) with
(u1 ) r2 and r2 (u2 ) with (u2 ) r3 .
□
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, a 4, b 10, p 2, g (t ) t , then we get q 2,
1
0
1 g (t )d qt , 2
1
1
tg (t )d t 3 , 0
q
1
k
1
[b tg (t )d qt ] [a g (t )d qt ] 0
0
1
b tg (t )d qt
37 0.637931. 58
0
Let
23, f (t , u ) 23 600(u 9), 623,
From a direct calculation, we get
f (t , u) M 3 583.266938 for f (t , u) M 2 36.538326 for f (t , u) M1 21.322041 for
t [0,1], u [0,9], t [0,1], u [9,10], t [0,1], u [10, ).
580 ], 37 t [0,1], u [0,9], t [0,1], u [0,0.5].
t [0,1], u [10,
In view of Theorem 4.1, we see that the aforementioned problem has at least two positive solutions u1 and u2 such that
0.5 (u1 ) with (u1 ) 9 and 9 (u2 ) with (u2 ) 10.
|Volume 2| Issue 7 |
www.ijrtem.com
| 14 |
Existence of positive solutions for fractional q-difference…
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 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 equations with p-Laplacian. Math. Methods Appl. Sci. 39, 1425-1434 (2015)
[5]
Han, Z, Lu, H, Zhang, C: Positive solutions for eigenvalue problems of fractional differential equation with generalized p-Laplacian. Appl. Math. Comput. 257,526-536 (2015)
[6]
Jankowski, T: Positive solutions to fractional differential equations involving Stieltjes integral conditions. Appl. Math. Comput. 241, 200-213 (2014)
[7]
Ntouyas, S, Etemad, S: On the existence of solutions for fractional differential inclusions with sum and integral boundary conditions. Appl. Math. Comput. 266, 235-243 (2015)
[8]
Cabada, A, Hamdi, Z: Nonlinear fractional differential equations with integral boundary value conditions. Appl. Math. Comput. 228, 251-257 (2014)
[9]
Gunendi, M, Yaslan, I: Positive solutions of higher-order nonlinear multi-point fractional equations with 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.
[11]
(2016) 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 nonlinear fractional q-difference equations. Adv.Differ.Equ.140(2012)
[14]
Rajkovic, PM, Marinkovic, SD, Stankovic, MS: Fractional integrals and derivatives in q-calulus. Appl. Anal. DiscreteMath. 1,311-323 (2007)
[15]
Avery, Rl, Chyan, CJ, Henderson, J: Twin solutions of boundary value problems for ordinary differential equations and finite difference equations. Comput. Math. Appl. 42,695-704 (2001)
[16]
Yuan, C: Multiple Positive solutions for (n-1,1)-type semipositone conjugate boundary value problems of nonlinear fractional differential equations. Electron. J. Qual. Theory Differ. Equ. 2010, 36 (2010)
Chengmin Hou. “Existence of Positive Solutions for Fractional q-Difference Equations Involving Integral Boundary Conditions with p-Laplacian Operator.” Invention Journal of Research Technology in Engineering & Management (IJRTEM), vol. 2, no. 7, 5 July 2018, pp. 05–15., www.ijrtem.com.
|Volume 2| Issue 7 |
www.ijrtem.com
| 15 |