Chaos, Solitons and Fractals 77 (2015) 240–246
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations Kamal Shah∗, Hammad Khalil, Rahmat Ali Khan Department of Mathematics, University of Malakand, Chakadara Dir(L), Khyber Pakhtunkhwa, Pakistan
a r t i c l e
i n f o
Article history: Received 15 March 2015 Accepted 6 June 2015 Available online 24 June 2015
a b s t r a c t In this article, we study a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations. We obtain sufficient conditions for existence and uniqueness of positive solutions. We use the classical fixed point theorems such as Banach fixed point theorem and Krasnoselskii’s fixed point theorem for uniqueness and existence results. As in application, we provide an example to illustrate our main results. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Fractional order differential equations have attracted the attention of many researchers because of their applications in many scientific and engineering disciplines such as physics, chemistry, biology, viscoelasticity, control theory, signal and image processing phenomenons, economics, bioengineering etc, we refer to [1–5,7,11–15] and the references therein. It is investigated that fractional order differential equation model real world problems more accurately than differential equations of integral order. Recently, the study of existence and uniqueness of solutions to initial and boundary value problems for fractional order differential equations has attracted considerable attention, we refer to [6–9,32] and the references therein. The study of coupled systems of fractional order differential equations has also attracted some attention. Because mathematical models of various phenomenona in the field of physics, biology and psychology etc., are in the form of coupled systems of differential equations. For the study of coupled systems of fractional order differential equations,
∗
Corresponding author. Tel.: +923463025197. E-mail addresses:
[email protected] (K. Shah), hammadk310@ gmail.com (H. Khalil),
[email protected] (R.A. Khan). http://dx.doi.org/10.1016/j.chaos.2015.06.008 0960-0779/© 2015 Elsevier Ltd. All rights reserved.
we refer to [16–20]. Another important class of differential equations is known as impulsive differential equations. This class plays the role of an effective mathematical tools for those evolution processes that are subject to abrupt changes in their states. There are many physical systems that exhibit impulsive behavior such as the action of a pendulum clock, mechanical systems subject to impacts, the maintenance of a species through periodic stocking or harvesting, the thrust impulse maneuver of a spacecraft, and the function of the heart, we refer to [25] for an introduction to the theory of impulsive differential equations. It is well known that in the evolution processes the impulsive phenomena can be found in many situations. For example, disturbances in cellular neural networks [28], operation of a damper subjected to the percussive effects [26], change of the valve shutter speed in its transition from open to closed state [27], fluctuations of pendulum systems in the case of external impulsive effects [29], percussive systems with vibrations [30], relaxational oscillations of the electromechanical systems [31], dynamic of system with automatic regulation [32], control of the satellite orbit, using the radial acceleration [32] and so on. The theory of impulsive differential equations is well studied and the large number of research articles are available in the literature on impulsive differential equations, we refer to [5,10–15] and the references therein for some of the recent development in the theory.
K. Shah et al. / Chaos, Solitons and Fractals 77 (2015) 240–246
Recently, Feng et al. [21] studied existence of positive solutions to the following impulsive boundary value problem (IBVP)
⎧ −(φ p (u (t ))) = f (t, u(t )), t ∈ [0, 1], t = tk , ⎪ ⎪ ⎪ k = 1, 2 . . . n, ⎨ −u(tk ) = Ik (y(tk )), k = 1, 2, . . . n, ⎪ 1 ⎪ ⎪ ⎩u (0 ) = 0, u(1 ) = g(t )u(t )dt. 0
Wang et al. [22] developed a sufficient condition for existence of positive solutions to the following impulsive boundary value problem (IBVP) via topological degree theory
⎧c q ⎨ D u(t ) = f (t, u(t )), t ∈ [0, T ] \ D := {t1 , t2 , . . . , tm }, u(0 ) = 0, ⎩ u(ti ) = Ii (u(ti )), i = 1, 2, . . . , m,
where 0 < q < 1 and Ii : R → R are nonlinear functions describing the jump size (Ii (u(ti )) = u(ti+ ) − u(ti− )) at ti , 0 = t0 < t1 < · · · < tm < tm+1 = T . Significant development has been made in last few years in the theory of impulsive fractional order differential equations with fixed moments. It gives natural description of observed evolution process and is an important tool for understanding several real world phenomenona. Y.Tian, et al. [23] developed some interesting results for the existence of positive solutions to the following impulsive fractional order differential equations (IFBVP)
⎧c q ⎨ D x(t ) = f (t, x(t )), t ∈ [0, 1], t = tk , k = 1, 2 . . . p, x(tk ) = Ik (x(tk )), x (tk ) = I¯k (x(tk )), k = 1, 2, . . . , p, ⎩ x ( 0 ) = g ( x ), x ( 1 ) = h ( x ), where 1 < q ≤ 2, f : [0, 1] × R → R is continuous, Ik , I¯k : R → R are continuous, and (x(tk )) = x(tk+ ) − x(tk− ), (x (tk )) = x (tk+ ) − x (tk− ) with x(tk+ ), x (tk+ ), x(tk− ), x (tk− ), are the respective left and right limits of x(tk ) at t = tk . Zhang et al. [24] extended the work to coupled system of 2m-point BVP for impulsive fractional differential equations at resonance as:
⎧ ⎪ Dα0+ u(t ) = f (t, v(t ), D p v(t )), ⎪ ⎪ ⎪ ⎪ Dq u(t )) t ∈ (0, 1 ), ⎪ ⎪ ⎪ ⎪ ⎪ u(ti ) = Ai (v(ti ), D p v(ti )), ⎪ ⎪ ⎪ ⎪ Dq v(ti )), i = 1, 2, . . . k; ⎪ ⎪ ⎪ ⎨v(ti ) = Ci (u(ti ), Dq u(ti )), Dq u(ti )), i = 1, 2, . . . k; ⎪ ⎪ ⎪ m ⎪ ⎪ ⎪Dα−1 u(0 ) = a Dα−1 u(ξ ), ⎪ i i ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎪ m ⎪ ⎪ β −1 v(0 ) = ⎪ D ci Dβ −1 v(ςi ), ⎪ ⎩ i=1
β
D0+ v(t ) = g(t, u(t ),
u(ti ) = Bi (v(ti ), v(ti ) = Di (u(ti ), u (1 ) =
m
bi ηi2−α u(ηi );
i=1
v (1 ) =
m
2−β
di θ i
v ( θi ),
i=1
where 1 < α , β < 2, α − q ≥ 1, β − p ≥ 1. Motivated by the importance of the study mentioned above of impulsive fractional differential equations, we develop sufficient conditions for existence and uniqueness of solutions to the following complex dynamical network in the form of a coupled system of m + 2-point boundary
241
conditions for impulsive fractional differential equations
⎧c α D u(t ) = (t, u(t ), v(t )), t ∈ [0, 1], t = t j , ⎪ ⎪ ⎪ ⎪ j = 1, 2, . . . , m, ⎪ ⎪ ⎪ ⎪c Dβ v(t ) = (t, u(t ), v(t )), t ∈ [0, 1], ⎪ ⎪ ⎪ ⎪ ⎪ t = ti , i = 1, 2, . . . , n, ⎪ ⎪ ⎪ ⎨u(0 ) = h(u ), u(1 ) = g(u ) and v(0 ) = κ (v ), v ( 1 ) = f ( v ), ⎪ ⎪ ⎪ ⎪ ⎪ u (t j ) = I j (u(t j )), u (t j ) = I¯j (u(t j )), ⎪ ⎪ ⎪ ⎪ j = 1, 2, . . . , m, ⎪ ⎪ ⎪ ⎪ ⎪ v ⎪ (ti ) = Ii (v(ti )), v (ti ) = I¯i (v(ti )), ⎪ ⎩
(1)
i = 1, 2, . . . , n,
where 1 < α , β ≤ 2, , : [0, 1] × R2 → R are continuous functions and g, h : X → R, f, κ : Y → R are continuous functionals define by
g( u ) =
p
p λ j u(ξ j ), h(u ) = j=1 λ j u ( η j ),
j=1
f (v ) =
q
δi v(ξi ), κ (v ) =
i=1
q
δi v(ηi ),
i=1
ξ i , ηi , ξ j , ηj ∈ (0, 1) for i = 1, 2, . . . , p, j = 1, 2, . . . , q, and
u(t j ) = u(t +j ) − u(t −j ), u (t j ) = u (t +j ) − u (t −j ) v(ti ) = v(ti+ ) − v(ti− ), v (ti ) = v (ti+ ) − v (ti− ). As given above, the notations u(t + ), v(ti+ ) are right and j u(t − ), v(t − ) are left limits, respectively. Moreover, Ir , I¯r (r = j
i
β
j, i ) : R → R are continuous functions and Dα , D0+ stand 0+ for Caputo’s fractional derivative of order α , β , respectively. p q We assume that r=1 λ j ηrα −1 < 1, r=1 δr ξrα −1 < 1, r = i, j. Here, we remark that the system (1) includes as a special case the following well-studied Hopfield neural networks and the design model in the presence of impulses [28]
⎧ m dui ⎪ ⎪ = −a u ( t ) + bi j f j (u j (t )) + ci , t = tk , ⎪ i i ⎨ dt j=1
⎪ i = 1, 2, . . . , m, k = 1, 2, . . . , ⎪ ⎪ ⎩ ui (tk ) = Ii (ui (tk )). We obtain necessary and sufficient conditions for the existence and uniqueness of positive solution via Krasnoselskii’s fixed point theorem and Banach contraction principle. We give an example to illustrate our main results. 2. Background materials In this section, we recall some basic definitions and results from fractional calculus, fixed point theory and functional analysis [1–5,33,34]. Definition 2.1. The Rieman–Liouvilli fractional integral of order α ∈ R+ of a function ω ∈ C ((0, ∞ ), R ) is defined as α ω (t ) = I0+
1
(α )
0
t
(t − s )α−1 ω (s ) ds,
where α > 0 and is the gamma function, provided that the right side is point wise defined on (0, ∞).
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K. Shah et al. / Chaos, Solitons and Fractals 77 (2015) 240–246
Definition 2.2. The Riemann–liouville fractional order derivative of a function ω : (0, ∞ ) → R is defined by
Dα0+ ω (t ) =
1 (n − α )
d dt
n
t 0
(t − s )n−α−1 ω (s ) ds,
where n = [α ] + 1 and [α ] represents the integer part of α and provided that the right side is point wise defined on (0, ∞). Lemma 2.1. The fractional differential equation of order α > 0
Dα ω (t ) = 0, n − 1 ≤ α < n, has a unique solution of the form ω (t ) = ci ∈ R, i = 1, 2, 3, . . . , n, n = [α ] + 1.
n
n−1 , where i=1 ci t
Lemma 2.2. The following result holds for fractional differential equations
Iα Dα ω (t ) = ω (t ) +
n
ci t n−1 ,
i=1
1 =
define 2Lψ (β +1 )
2Lφ
(α +1 ) + m(A1 + A2 ) + Kg + Kh
Lemma 2.3. (Krasnoselskii’s Fixed point Theorem) [33–35], Let D be a convex closed and non-empty subset of Banach space X × Y. Let F, G be the operators such that (i ) F x + Gy ∈ D whenever x, y ∈ D (ii) F is compact and continuous and G is contraction mapping. Then there exist z ∈ D such that z = F z + Gz, where z = (u, v ) ∈ X × Y. 3. Main results Theorem 3.1. For w ∈ C ([0, 1], R ), the following impulsive boundary value problem c
Dα u(t ) = w(t ), t ∈ [0, 1],
u(t j ) = I j (u(t j )), u (t j ) = I¯j (u(t j )), j = 1, 2, . . . , m, (2) u ( 0 ) = h ( u ), u ( 1 ) = g ( u ) has a solution of the form
u(t ) =
X = {u : [0, 1] → R : u ∈ C (I ) and left u(t + ) j
) − u(t j , ), 1 ≤ j ≤ m}. u(t − j
v(ti− ) exist and v(ti− ) − v(ti , ), 1 ≤ i ≤ n}
v( ) ti+
and right limit
is a Banach space under the norm v = maxt∈[0,1] |v(t )|. Consequently, the product X × Y is a Banach space under the norms (u, v ) = u + v and (u, v ) = max{ u , v }. Assume that the following hold (H1 ) for any u, v ∈ C ([0, 1], R ), ∃ Kg , Kh , K f , Kκ such that
g(u ) − g(v ) ≤ Kg u − v ,
f (u ) − f (v ) ≤ K f u − v ,
h(u ) − h(v ) ≤ Kh u − v ,
κ (u ) − κ (v ) ≤ Kκ u − v ; (H2 ) for all u, u¯ ∈ R ∀ t ∈ [0, 1] ∃ L , such that
¯ ]; ¯ v¯ )| ≤ L [|u − u¯ | + |v − v| | (t, u, v ) − (t, u, (H3 ) for all v, v¯ ∈ R ∀ t ∈ [0, 1] ∃ L , such that
¯ ]. ¯ v¯ )| ≤ L [|u − u¯ | + |v − v| | (t, u, v ) − (t, u, (H4 ) There exist constants Al , Bl > 0 (l = 1, 2 ) such that for ¯ v, v¯ ∈ R, u, u,
|I j (u ) − I j (u¯ )| ≤ A1 |u− u¯ |, |I¯j (u ) − I¯j (u¯ | ≤ A2 |u− u¯ |, j = 1, 2, . . . , m ¯ ≤ B2 |v − v| ¯ , |I¯i (v ) − I¯i (v| ¯ , |Ii (v ) − Ii (v¯ )| ≤ B1 |v − v| i = 1, 2, . . . , n.
(α )
t 0
t
(t − s )α−1 w(s )ds
(α )
1 0
(1 − s )α−1 w(s )ds
+ tg(u ) + (1 − t )h(u ) +
Then, clearly (X, , ) is the Banach space under the norm
u = maxt∈[0,1] . Similarly for ti ∈ (0, 1) such that t1 < t2 < · · · < tn , and I = I − {t1 , t2 , . . . , tn }, the space
Y = {v : [0, 1] → R : v ∈ C (I ) and left
1
−
) exist and
t = t j , j = 1, 2, . . . , m,
1 < α ≤ 2,
For tj ∈ (0, 1) such that t1 < t2 < · · · < tm , and I = I − {t1 , t2 , . . . , tm } define the space
and right limit u(
2 =
+ n(B1 + B2 ) + K f + Kk .
for arbitrary ci ∈ R, i = 0, 1, 2, . . . , n, n = [α ] + 1.
t− j
and
m
G j (t, u j ),
j=1
where
G j (t, u j ) =
−t[I j (u(t j )) + (1 − t j )I¯j u(t j )], (1 − t )[I j (u(t j )) − t j I¯j u(t j )],
0 ≤ t ≤ t j; t j < t ≤ 1,
Proof. Using Lemma (2.2) and the differential equation in (2), there exist real constants c0 , c1 such that
u(t ) = Iα w(t ) − c0 − c1 t t 1 = (t − s )α−1 w(s )ds − c0 − c1 t, t ∈ [0, t1 ], (3)
α
0
from which it follows that
u (t ) = Iα −1 w(t ) − c1 t 1 = (t − s )α−2 w(s )ds − c1 , t ∈ [0, t1 ]. (4) (α − 1 ) 0 Similarly, for t ∈ (t1 , t2 ], there exist real numbers d0 , d1 such that
u(t ) = u (t ) =
1
α
t
t1
(t − s )α−1 w(s )ds − d0 − d1 (t − t1 ),
1 (α − 1 )
t
t1
(t − s )α−2 w(s )ds − d1 .
(5)
Hence, it follows that
u(t1− ) =
1
α
t1 0
(t1 − s )α w(s )ds − c0 − c1 t1 ,
u(t1+ ) = −d0 , 1 (α − 1 ) u (t1+ ) = −d1 .
u (t1− ) =
0
t1
(t1 − s )α−2 w(s )ds − c1 , (6)
K. Shah et al. / Chaos, Solitons and Fractals 77 (2015) 240–246
Using the impulsive conditions u(t1 ) = u(t1+ ) − u(t1− ) = I1 (u(t1 )) and u (t1 ) = u (t1+ ) − u (t1− ) = I¯1 (u(t1 )), we obtain
−d0 = −d1 =
1
α
t1
0
1 (α − 1 )
t1 0
α
t
t1
1
α
t1 0
(t1 − s )α w(s )ds
t1 t − t1 (t − s )α−2 w(s )ds (α − 1 ) 0 1 + (t −t1 )I¯1 (u(t1 )) +I1 (u(t1 )) −c0 − c1 t, t ∈ (t1 , t2 ].
+
α +
+
+
t
tj
k 1
α
j=1
tj
t j−1
u(t ) =
1 (α − 1 ) k
(t − t j )
j=1
(t − t j )I¯j (u(t j )) +
j=1
tj
t j−1
(α ) −
t
(α )
t 0
0
1
Gi (t, vi ) =
u(t ) =
0
t
(t − s )α−1 (s, u(s ), v(s ))ds
(α )
1 0
(1 − s )α−1 (s, u(s ), v(s ))ds
(β )
m
G j (t, u j ),
j=1
1
t 0
t
(t − s )β −1 (s, u(s ), v(s ))ds
(β )
1
0
m
1
(α ) −
v(t ) =
G j (t, u j ),
(10)
(1 − s )β −1 (s, u(s ), v(s ))ds
t
0
n
Gi (t, vi ).
t
(t − s )α−1 (s, u, v )ds
(α )
1 0
(1 − s )α−1 (s, u, v )ds
1
(β ) −
t 0
Dβ v(t ) = w(t ), t ∈ [0, 1], t = ti , i = 1, 2, . . . , n, 1 < β ≤ 2, v(ti ) = Ii (v(ti )), v (ti ) = I¯i (v(ti )), i = 1, 2, . . . , n, c
(11)
G j (t, u(t j ))
t
(β )
(t − s )β −1 (s, u, v )ds
0
1
(1 − s )β −1 (s, u, v )ds n
Gi (t, v(ti )).
(13)
i=1
0 ≤ t ≤ ti ; ti < t ≤ 1.
m j=1
+ t f ( v ) + ( 1 − t )κ ( v ) +
Similarly, the impulsive boundary value problem
has a solution of the form
t
+ tg(u ) + (1 − t )h(u ) +
−t[Ii (v(ti )) + (1 − ti )I¯i (v(ti ))], (1 − t )[Ii (v(ti )) − ti I¯i v(ti )],
v ( 0 ) = κ ( v ), v ( 1 ) = f ( v )
(α )
(9)
(1 − s )α−1 w(s )ds
0 ≤ t ≤ ti , ti < t ≤ 1.
Theorem 3.2. Let , are continuous, then (u, v) ∈ X × Y is a solution of IBVP(1) if and only if (u, v) is the solution of the integral equations
I j (u(t j )) − c0 − c1 t,
j=1
where
Gi (t, vi ),
(12)
k = 1, 2, . . . , m.
+ tg(u ) + (1 − t )h(u ) +
n
i=1
(t − s )α−1 w(s )ds
+ t f ( u ) + ( 1 − t )κ ( v ) +
Now, using the boundary conditions u(0 ) = g(u ), u(1 ) = h(u ) on (9), calculating the values of c0 , c1 , Eq. (3) can be rewritten as follows
1
(1 − s )β −1 w(s )ds
+ tg(u ) + (1 − t )h(u ) +
(t j − s )α−2 w(s )ds
k
0
−t[Ii (v(ti )) + (1 − ti )I¯i v(ti )], (1 − t )[Ii (v(ti )) − ti I¯i v(ti )],
1
−
j=1
t ∈ (t j , t j+1 ],
u(t ) =
−
(t j − s )α−1 w(s )ds
k
1
In view of Theorem (3.1), (2) and (11), if (u, v) is a solution of the coupled system (1), then we have
v(t ) =
(t − s )α−1 w(s )ds
(β )
where
Repeating in the same fashion, we obtain expression for the solution u(t) for t ∈ (ti , ti+1 ] as follows
1
(t − s )β −1 w(s )ds
i=1
(8)
u(t ) =
0
t
Gi (t, vi ) =
(t − s )α−1 w(s )ds +
t
+ t f ( v ) + ( 1 − t )κ ( v ) +
(t1 − s )α−2 w(s )ds − c1 + I¯1 (u(t1 )).
Hence we have
1
(β )
(t1 − s )α w(s )ds − c0 − c1 t1 + I1 (u(t1 )),
1
−
(7)
u(t ) =
v(t ) =
243
Proof. If (u, v) is a solution of IBVP(1), then it is a solution of the system of integral Eq. (13). Conversely let (u, v) is solution of the system of integral Eq. (13), and then c c
Dα u(t ) = (t, u(t ), v(t ))
Dβ v(t ) = (t, u(t ), v(t ))
and u(0 ) = h(u ), u(1 ) = g(u ) and v(0 ) = κ (v ), v(1 ) = f (v ) where t ∈ [0, 1], t = t j , j = 1, 2, . . . , m and t = ti , i = 1, 2, . . . , n
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K. Shah et al. / Chaos, Solitons and Fractals 77 (2015) 240–246
¯ v¯ )| ≤ 2 u − Similarly, we can show that |T2 (u, v ) − T2 (u,
It is also clear that
¯ . Hence, it follows that u¯ + v − v
G j (0, u(t j )) = G j (1, u(t j )) = 0, j = 1, 2, . . . , m
¯ ), ¯ v¯ )| ≤ max(1 , 2 )( u − u¯ + v − v
|T (u, v ) − T (u,
Gi (0, v(ti )) = Gi (1, v(ti )) = 0, i = 1, 2, . . . , n, and
u(t j ) = I j (u(t j )),
u (t j ) = I¯j (u(t j )), j = 1, 2, . . . , m
v(ti ) = Ii (v(ti )),
v (ti ) = I¯i (v(ti )), i = 1, 2, . . . , n.
Thus (u, v) is the solution of the IBVP(1).
1
(α ) −
t
0
t
(α )
1 0
(1 − s )α−1 (s, u(s ), v(s ))ds
+ tg(u ) + (1 − t )h(u ) + T2 (u, v ) =
(β ) −
m
(H6 ) there exists constants ρl , μl (l = 1, 2 ) such that
|g(u )| ≤ ρ1 , |h(u )| ≤ ρ2 ∀u ∈ X, | f (v )| ≤ μ1 , |κ (v )| ≤ μ2 ∀v ∈ Y.
G j (t, u j ),
j=1
1
|I j (u j )| ≤ C1 , |I¯j (u j )| ≤ C2 , |Ii (ui )| ≤ D1 , |I¯i (vi )| ≤ D2 .
(t − s )α−1 (s, u(s ), v(s ))ds
t 0
t
(β )
If m(A1 + A2 ) + Kg + Kh < 1 and n(B1 + B2 ) + K f + Kκ < 1, then system(1) has at least one positive solution.
(t − s )β −1 (s, u(s ), v(s ))ds
1
0
Proof. Choose ρ (t ) = (t, 0, 0 ), σ (t ) = (t, 0, 0 ), ρ L1 = 1 1 and max 0 | (s, 0, 0 )|ds, σ L1 = max 0 | (s, 0, 0 )|ds
(1 − s )β −1 (s, u(s ), v(s ))ds
+ t f ( v ) + ( 1 − t )κ ( v ) +
n
Theorem 3.4. In addition to the assumption (H1 )−(H4 ), assume that the following hold (H5 ) there exist constants Cl , Dl > 0 for l = 1, 2 such that j=1,2, … ,m, i=1,2, … ,n,
Define T1 , T2 : X × Y → X × Y by
T1 (u, v ) =
which implies that T is contractions hence it has a unique fixed point.
2 ρ 1
2 σ 1
R ≥ max{ (αL) +m(C1 +C2 ) + ρ1 + ρ2 , (β L) +n(D1 + D2 ) + μ1 + μ2 }. Construct a closed ball B⊂X × Y such that
Gi (t, vi ),
i=1
and T (u, v ) = (T1 (u, v ), T2 (u, v )). Then, solutions of the coupled system of integral Eq. (13) are fixed points of T. Theorem 3.3. If max (1 , 2 ) < 1, then under the assumptions (H1 )−(H4 ), the impulsive coupled system (1) has a unique positive solution.
B = {(u, v ) ∈ X × Y : (u, v ) ≤ R}. Split the operator T into two parts as T = F + G with F = (F1 , F2 ) and G = (G1 , G2 ) where
F1 (u, v ) =
Proof. Let u, u¯ ∈ X, v, v¯ ∈ Y and each t ∈ J, and consider
(α ) −
¯ v¯ )| |T1 (u, v ) − T1 (u, t
1 ¯ v¯ ) ds = (t − s )α−1 (s, u, v ) − (s, u, (α ) 0 1
t ¯ v¯ ) ds (1 − s )α−1 (s, u, v ) − (s, u, + (α ) 0
+ t g(u ) − g(u¯ ) + (1 − t ) h(u ) − h(u¯ ) m + G j (t, u(t j )) − G j (t, u¯ (t j ))
F2 (u, v ) =
1
0
t
(t − s )α−1 (s, u(s ), v(s ))ds
(α )
(β )
t
0
t
1 0
1
−
t
(1 − s )α−1 (s, u(s ), v(s ))ds
(t − s )β −1 (s, u, v )ds
(β )
1 0
(1 − s )β −1 (s, u, v )ds
G1 u(t ) = tg(u ) + (1 − t )h(u ) +
2Lφ ¯ ]+Kg u − u¯ + Kh u − u¯
[ u− u¯ + v − v
(α +1 ) m m + |I j (u(t j )) −I j (u¯ (t j ))| + |I¯j (u(t j )) − I¯j (u(t j ))|
G2 v(t ) = t f (v ) + (1 − t )κ (v ) +
≤
j=1
2Lφ ¯ ] + Kg u − u¯
≤ [ u − u¯ + v − v
(α + 1 ) + Kh u − u¯ + m(A1 + A2 ) u − u¯ , which implies that
¯ v¯ )| ≤ |T1 (u, v ) − T1 (u,
2Lφ K + Kh + m(A1 + A2 ) (α + 1 ) g
¯ ) ×( u − u¯ + v − v
¯ ). = 1 ( u − u¯ + v − v
G j (t, u(t j ))
j=1
j=1
j=1
m
Gi (t, v(ti )).
i=1
Clearly T1 = F1 + G1 , T2 = F2 + G2 . Now we show that T (u, v ) = F (u, v ) + G(u, v ) ∈ B, ∀(u, v ) ∈ B. For any (u, v) ∈ B, we have
|T1 (u, v )| ≤
1
(α ) +
n
0
1
(α )
t
(t − s )α−1 | (s, u, v )|ds
1 0
(1 − s )α−1 | (s, u, v )|ds
+|tg(u¯ )|(1 − t )h(u¯ )| +
m j=1
+
m j=1
|I¯j (u(t j ))|
|I j (u(t j ))|
K. Shah et al. / Chaos, Solitons and Fractals 77 (2015) 240–246
2 ρ L1
≤
+ ρ1 + ρ2 + m(C1 + C2 ) ≤ R.
(α )
2 σ
Similarly, we have |T2 (u, v )| ≤ (β L)1 + μ1 + μ2 + n(D1 + D2 ) ≤ R. Hence, we have |T(u, v) ≤ R which implies that T(B)⊆B. Now, we show that G is contraction. Using (H1 )–(H3 ) for ¯ v¯ ) ∈ B, we have any (u, v ), (u,
|G1 (u ) − G1 (u¯ )| ≤ |g(u ) − g(u¯ )| + |h(u ) − h(u¯ )| +
m
|G j (t, u(t j )) − G j (t, u(t j ))|
j=1
≤ Kg u − u¯ + Kh u − u¯ + m(A1 + A2 ) u − u¯
= (Kg + Kh + m(A1 + A2 )) u − u¯ . ¯ ≤ (K f + Kκ + n(B1 + B2 )) v − v
¯ . The Similarly, |G2 v − G2 v| contraction of G follows from the assumption that (Kg + Kh + m(A1 + A2 )) < 1 and (K f + Kκ + n(B1 + B2 )) < 1. Next F = (F1 + F2 ) is compact. The continuity of F follows from the continuity of , . For (u, v) ∈ B, we have
1
|F1 (u, v )| =
(α ) +
≤
t
0
1
(α )
2
(α ) 0 2 ρ L1 . ≤ (α ) Similarly |F2 v(t )| ≤ Hence, F (u, v )
(t − s )α−1 | (s, u(s ), v(s ))|ds 1 0
1
(1 − s )α−1 | (s, u(s ), v(s ))|ds
(1 − s )α−1 | (s, u(s ), v(s ))|ds
2 σ L1
(β )
ρ 1
σ 1 ≤ 2( (αL) + (βL)
) which implies that F is
τ
(τ − s )α−1 (s, u(s ), v(s ))ds (α ) t (t − τ ) 1 (1 − s )α−1 (s, u(s ), v(s ))ds + (α ) 0
≤
1
(α +1 )
(2(τ −t )α +t α − τ α + t − τ ) → 0 as t → τ .
Similarly, we have
F2 (u(t ), v(t )) − F2 (u(τ ), v(τ ))
≤
1
(β + 1 ) →0
Consider the following impulsive coupled system of fractional differential equations
⎧ 1 + |u(t )| + |v(t )| sin t 1 c 32 ⎪ D u(t ) = , t = , ⎪ ⎪ 3 ( 2 + t )2 (2 + |u| + |v| ) ⎪ ⎪ ⎪ ⎪ 1 (2|u(t )| + 2|v(t )) −t ⎪ c 32 ⎪ D v(t ) = e , t = , ⎪ ⎪ 4 (4 + t 2 ) ⎪ ⎪ ⎪ 60 60 ⎪ u (ξ j ) u (η j ) ⎪ ⎪ ⎪ , ( 0 ) = g ( u ) = ( 1 ) = h ( u ) = u u ⎪ 2 + 100 ⎪ t + 50 t ⎪ j=1 j=1 ⎪ ⎪ ⎪ 100 100 ⎪ ⎪ v ( ξi ) v ( ηi ) ⎪ ⎨v(0 ) = f (v ) = , v ( 1 ) = κ ( v ) = , + 4 i=1
t + 100
i=1
⎪ 1 1 ⎪ ⎪ 1 ⎪ ⎪ u = Iu = , ⎪ ⎪ 3 3 6 + |u| ⎪ ⎪ 1 1 1 ⎪ ⎪ ¯ u = Iu = ⎪ ⎪ 3 3 6 + |u| ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ 1 ⎪v ⎪ = Iv = , ⎪ ⎪ 4 4 50 + |v| ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎩v 1 = I¯v 1 = 4 4 100 + |v|
2t 60
(14) where
1 1 for j = 1, 2, 3, . . . , 60, and ti = 3 4 for i = 1, 2, 3, . . . , 100.
t j =
we obtain
.
F1 (u(t ), v(t )) − F1 (u(τ ), v(τ ))
t 1 = ((t −s )α−1 − (τ − s )α−1 ) (s, u(s ), v(s ))ds (α ) 0 1
4. Example
¯ v, v¯ ∈ R, t ∈ [0, 1], For any u, u,
uniformly bounded on B. Take a bounded subset C of B and (u, v) ∈ C. Then, for t, τ ∈ J with 0 ≤ t ≤ τ ≤ 1, using (H3 ) and (H5 ), we obtain
+
245
( 2 ( τ − t )β + t β − τ β + t − τ )
as t → τ .
Hence, F is equicontinuous and by the Arzela Ascoli theorem, F is compact. By Lemma (2.3), system (1) has at least one positive solution.
1 8 1 ¯ ], ¯ v¯ )| ≤ [ u − u¯ + v − v
| (t, u, v ) − (t, u, 2 1 1 ¯ (t j ) − I¯u¯ (t j )| ≤ |Iu(t j ) − Iu¯ (t j )| ≤ |u − u¯ |, |Iu |u − u¯ |, 8 16 1 1 ¯ , |I¯v(ti ) − I¯v(ti )| ≤ ¯ , |Iv(ti ) − Iv¯ (ti )| ≤ |v − v| |v − v| 50 100 |g(u ) − g(u¯ )| ≤ Kg u − u¯ , |h(u ) − h(u¯ )| ≤ Kh u − u¯ , ¯ ], ¯ v¯ )| ≤ [ u − u¯ + v − v
| (t, u, v ) − (t, u,
¯ , Kk (v ) − K f (v¯ )| ≤ Kk v − v
¯ , | f (v ) − f (v¯ )| ≤ K f v − v
1 1 1 1 where Kg = 100 , Kh = 50 , K f = 100 , Kκ = 60 , A1 = 16 , A2 = 1 1 1 , B = , B = and m = 1, n = 1. Furthermore, we 1 2 6 50 100 obtain L = 18 , L = 12 ,
1 = =
2L
(α + 1 ) 2×
1 8
( 32 + 1 )
+ m(A1 + A2 ) + Kg + Kh +1
1 6
+
1 6
+
1 1 + 100 50
1 3 1
