On a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions

Chaos, Solitons and Fractals 83 (2016) 234–241

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

On a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions Bashir Ahmad a,∗, Sotiris K. Ntouyas b,a, Ahmed Alsaedi a a Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia b Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece

a r t i c l e

i n f o

Article history: Received 22 October 2015 Accepted 14 December 2015

MSC: 34A08 34A12 34B15 Keywords: Fractional differential equations Coupled system Nonlocal conditions Integral boundary conditions Fixed point

a b s t r a c t We investigate a coupled system of fractional differential equations with nonlinearities depending on the unknown functions as well as their lower order fractional derivatives supplemented with coupled nonlocal and integral boundary conditions. We emphasize that the problem considered in the present setting is new and provides further insight into the study of nonlocal nonlinear coupled boundary value problems. We present two results in this paper: the first one dealing with the uniqueness of solutions for the given problem is established by applying contraction mapping principle, while the second one concerning the existence of solutions is obtained via Leray–Schauder’s alternative. The main results are well illustrated with the aid of examples. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction In this paper we study a coupled system of nonlinear fractional differential equations:

⎧ α D x(t ) = f (t, x(t ), y(t ), Dγ y(t )), t ∈ [0, T ], ⎪ ⎪ ⎪ ⎨ 1 < α ≤ 2, 0 < γ < 1,   β ⎪ D y(t ) = g t, x(t ), Dδ x(t ), y(t ) , t ∈ [0, T ], ⎪ ⎪ ⎩ 1 < β ≤ 2, 0 < δ < 1,

(1.1)

supplemented with coupled nonlocal and integral boundary conditions of the form:



x ( 0 ) = h ( y ),

y ( 0 ) = φ ( x ),

T 0

y(s )ds = μ1 x(η ),

0

x(s )ds = μ2 y(ξ ),

T

∗

η, ξ ∈ (0, T ),

Corresponding author. Tel.: +966 2640000064515. E-mail addresses: [email protected], [email protected] (B. Ahmad), [email protected] (S.K. Ntouyas). http://dx.doi.org/10.1016/j.chaos.2015.12.014 0960-0779/© 2016 Elsevier Ltd. All rights reserved.

(1.2)

where c Di , i = α , β , γ , δ denote the Caputo fractional derivatives of order i, i = α , β , γ , δ respectively, f, g : [0, T ] × R × R × R → R, h, φ : C ([0, T ], R ) → R are given continuous functions, and μ1 , μ2 are real constants. The study of boundary value problems for linear and nonlinear differential equations constitutes an important and popular field of research in view of occurrence of such problems in a variety of disciplines of pure and applied sciences. In recent years, fractional-order boundary value problems have been extensively investigated and a great deal of work ranging from theoretical development to applications can be found in the literature on the topic, for instance, see [1–14] and the references cited therein. The importance of fractional calculus is now quite perceptible as the mathematical modeling of several real world phenomena via the tools of this branch of mathematics has led to exploration of new hereditary and memory characteristics of the processes and materials involved

B. Ahmad et al. / Chaos, Solitons and Fractals 83 (2016) 234–241

in the phenomena. In consequence, integer-order models in many physical and engineering disciplines such as viscoelasticity, biophysics, blood flow phenomena, chemical processes, control theory, wave propagation, signal and image processing, etc., have been transformed to their fractional-order counterparts. For further details, we refer the reader to the texts [15–18], while the facts about the recent history of fractional calculus can be found in [19]. The study of coupled systems of fractional-order differential equations is found to be of great value and interest in view of the occurrence of such systems in a variety of problems of applied nature. Examples include quantum evolution of complex systems [20], distributedorder dynamical systems [21], Chua circuit [22], Duffing system [23], Lorenz system [24], anomalous diffusion [25,26], systems of nonlocal thermoelasticity [27,28], secure communication and control processing [29], synchronization of coupled fractional-order chaotic systems [30– 33], etc. Fractional differential systems are more suitable for describing the physical phenomena possessing memory and genetic characteristics. Nonlocal conditions play a key role in describing some peculiarities of physical, chemical or other processes happening at various positions inside the domain, which is obviously not possible with the end-point (initial/boundary) conditions. For the historical background of these conditions, we refer the reader to the works [34–36]. Integral boundary conditions are found to be important and significant in the study of Computational fluid dynamics (CFD) studies related to blood flow problems. In the analysis of such problems, cross-section of blood vessels is assumed to be circular, which is not always justifiable. In order to cope this problem, integral boundary conditions provide an effective and applicable approach. More details can be found in [37]. Also, integral boundary conditions have useful applications in regularizing ill-posed parabolic backward problems in time partial differential equations, see for example, mathematical models for bacterial selfregularization [38]. Some recent investigations on coupled systems of fractional order differential equations, including nonlocal and integral boundary conditions, can be found in [39–46] and the references cited therein. The paper is organized as follows. In Section 2, we recall some definitions from fractional calculus and present an auxiliary lemma. The main results for the coupled system of nonlinear fractional differential equations with coupled nonlocal and integral boundary conditions are obtained via contraction mapping principle and Leray– Schauder alternative, and are presented in Section 3.1. Since the methods of proofs employed in this paper are the standard ones in the contexts of fractional differential equations with boundary conditions, for instance, see [43], we omit some details in the proofs of our results. The paper concludes with illustrative examples.

235

Definition 2.1. The Riemann–Liouville fractional integral of order q for a continuous function g is defined as

1 (q )

Iq g(t ) =

0

t

g( s ) ds, (t − s )1−q

q > 0,

provided the integral exists. Definition 2.2. For at least n-times continuously differentiable function g : [0, ∞ ) → R, the Caputo derivative of fractional order q is defined as c

t 1 (t − s )n−q−1 g(n) (s )ds, (n − q ) 0 < n, n = [q] + 1,

Dq g(t ) =

where [q] denotes the integer part of the real number q. Now we prove an auxiliary result which is pivotal to define the solution for the problem (1.1) and (1.2). Lemma 2.3 (Auxiliary Lemma). Let ω, z ∈ L[0, 1] and x, y ∈ AC2 [0, 1]. Then the unique solution of the problem

⎧c α D x(t ) = ω (t ), t ∈ [0, T ], 1 < α ≤ 2, ⎪ ⎪ ⎨c Dβ y(t ) = z(t ), t ∈ [0, T ], 1 < β ≤ 2, T x ( 0 ) = h ( y ), 0 y (s )ds = μ1 x (η ), ⎪ ⎪ T ⎩ y ( 0 ) = φ ( x ), 0 x (s )ds = μ2 y (ξ ), is

x(t ) =

and

y(t ) =



t



t

(t − s )β −1 z(s )ds + (σ3 t + 1 )φ (x ) + t σ4 h(y ) (β ) 0 ξ  t (ξ − s )β −1 + z(s )ds μ1 η μ 2 (β ) 0

T ( T − s )α − ω (s )ds ( α + 1) 0 η T2 (η − s )α−1 + ω (s )ds μ1 2 (α ) 0



T ( T − s )β z(s )ds , (2.3) − 0 (β + 1 )

First of all, we recall definitions of fractional integral and derivative [15,16].

T 4 − 4μ1 μ2 ηξ , 4 T μ2 ( T − 2 ξ ) = , 2

=

σ2

(2.1)

(t − s )α−1 ω (s )ds + (σ1t + 1 )h(y ) + σ2t φ (x ) (α ) 0 η  t (η − s )α−1 + ω (s )ds μ2 ξ μ1 (α ) 0

T ( T − s )β − z(s )ds 0 (β + 1 ) ξ T2 (ξ − s )β −1 μ2 + z(s )ds 2 (β ) 0



T ( T − s )α − ω (s )ds , (2.2) 0 (α + 1 )

where 2. Preliminaries

n−1
2 μ1 μ2 ξ − T 3 , 2 2 μ1 μ2 η − T 3 = , 2

σ1 = σ3

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B. Ahmad et al. / Chaos, Solitons and Fractals 83 (2016) 234–241

σ4 =

T μ1 ( T − 2 η ) . 2

Proof. It is well known that the general solutions of the fractional differential equations in (2.1) can be written as

x(t ) = c1 t + c2 +



y(t ) = d1 t + d2 +

t 0



t 0

(t − s )α−1 ω (s )ds, (α )

(2.4)

(t − s )β −1 z(s )ds, (β )

(2.5)

η T2 (η − s )α−1 ω (s )ds + μ1 h(y ) μ1 2 (α ) 0

T ( T − s )β − z(s )ds − T φ (x ) 0 (β + 1 ) ξ (ξ − s )β −1 + μ1 η μ2 z(s )ds + μ2 φ (x ) 1



0

− and

d1 =

T

0

(β )

( T − s )α ω (s )ds − T h(y ) (α + 1 )





0

−

T 0

(α )

( T − s )β z(s )ds − T φ (x ) (β + 1 )

 .

Substituting the values of c1 , c2 , d1 , d2 in (2.4) and (2.5), we get (2.2) and (2.3). This completes the proof.  3. Existence results We define the space

X = {x|x ∈ C ([0, T ], R )

and Dδ x ∈ C ([0, T ], R )},



t



t

(t − s )α−1 f (s, x(s ), y(s ), Dγ y(s ))ds (α ) 0 + ( σ 1 t + 1 )h ( y ) + σ2 t φ ( x )  η t (η − s )α−1 μ2 ξ μ1 + f (s, x(s ), y(s ), Dγ y(s ))ds (α ) 0

T ( T − s )β δ g(s, x(s ), D x(s ), y(s ))ds − 0 (β + 1 ) ξ T2 (ξ − s )β −1 + g(s, x(s ), Dδ x(s ), y(s ))ds μ2 2 (β ) 0



T ( T − s )α − f (s, x(s ), y(s ), Dγ y(s ))ds , 0 (α + 1 )

F1 (x, y )(t ) =

and

(t − s )β −1 g(s, x(s ), Dδ x(s ), y(s ))ds (β ) 0 + ( σ 3 t + 1 )φ ( x ) + t σ 4 h ( y ) ξ  t (ξ − s )β −1 + g(s, x(s ), Dδ x(s ), y(s ))ds μ η μ2 1 (β ) 0

T ( T − s )α − f (s, x(s ), y(s ), Dγ y(s ))ds 0 (α + 1 ) η T2 (η − s )α−1 μ1 + f (s, x(s ), y(s ), Dγ y(s ))ds 2 (α ) 0



T ( T − s )β δ − g(s, x(s ), D x(s ), y(s ))ds . (3.2) 0 (β + 1 )

F2 (x, y )(t ) =

ξ T2 (ξ − s )β −1 z(s )ds + μ2 φ (x ) μ2 2 (β ) 0

T ( T − s )α − ω (s )ds − T h(y ) 0 (α + 1 ) η (η − s )α−1 + μ2 ξ μ1 ω (s )ds + μ1 h(y ) 1

F (x, y )(t ) := (F1 (x, y )(t ), F2 (x, y )(t )), where

where ci , di ∈ R, i = 1, 2 are arbitrary constants. Applying the conditions x(0 ) = h(y ), y(0 ) = φ (x ), it is found that c2 = h(y ), d2 = φ (x ). In view of the conditions T μ1 0 y(s )ds = x(η ) and μ2 0T x(s )ds = y(ξ ), we get

c1 =

Obviously the product space (X × Y,  · X × Y ) is a Banach space with norm (x, y )X×Y = xX + yY for (x, y) ∈ X × Y. In view of Lemma 2.3, we define an operator F : X × Y → X × Y by

In the sequel, we need the following assumptions: (H1 ) f, g : [0, T ] × R3 → R are continuous functions and there exist constants > 0, 1 > 0 such that for all t ∈ [0, T] and ui , vi ∈ R, i = 1, 2, 3

| f (t, u1 , u2 , u3 ) − f (t, v1 , v2 , v3 )| ≤ (|u1 − v1 | + |u2 − v2 | + |u3 − v3 | ) and

and the norm

xX = x + Dδ x = sup |x(t )| + sup |Dδ x(t )|. t∈[0,T ]

t∈[0,T ]

Then it is well known that (X,  · X ) is a Banach space ([47]). Also we define the space

Y = {y|y ∈ C ([0, T ], R ) and Dγ y ∈ C ([0, T ], R )}, and the norm

|g(t, u1 , u2 , u3 ) − g(t, v1 , v2 , v3 )| ≤ 1 (|u1 − v1 | + |u2 − v2 | + |u3 − v3 | ). (H2 ) h, φ : C ([0, T ], R ) → R are continuous functions with h(0 ) = φ (0 ) = 0 and there exist constants L > 0, L1 > 0 such that

|h(x1 ) − h(x2 )| ≤ Lx1 − x2 , |φ (x1 ) − φ (x2 )| ≤ L1 x1 − x2 , ∀x1 , x2 ∈ C ([0, T ], R ).

yY = y + Dγ y = sup |y(t )| + sup |Dγ y(t )|.

For computation convenience, we introduce the notations:

Again (Y,  · Y ) is a Banach space.

 = ϑ1 + ϑ2 1 + (|σ1 |T + 1 )L + |σ2 |T L1 ,

t∈[0,T ]

t∈[0,T ]

B. Ahmad et al. / Chaos, Solitons and Fractals 83 (2016) 234–241

M = ϑ1 N1 + ϑ2 N2 ,

1 = ϑ¯ 1 +

ϑ2 1

M1 = ϑ¯ 1 N1 +

T

(3.3)

T

≤ 1 r + N2 ,

,

(3.4)

ϑ4 T

(3.5)

+ |σ3 |L1 + |σ4 |L, M1 = ϑ¯ 3 N2 +

ϑ4 N1 T

,

(3.6) where



 ηα T T 2 T α +1 |μ1 μ2 |ξ ϑ1 = + + , (α + 1 ) | | (α + 1 ) 2 (α + 2 )

T |μ2 | T2 T β +1 ξβ ξ ϑ2 = + , | | (β + 2 ) 2 (β + 1 )  α−1

 ηα 1 T 2 T α +1 T |μ1 μ2 |ξ ϑ¯ 1 = + + , (α ) | | (α + 1 ) 2 (α + 2 ) 

 ξβ Tβ T T 2 T β +1 |μ1 μ2 |η ϑ3 = + + , (β + 1 ) | | (β + 1 ) 2 (β + 2 )  β −1

 ξβ T 1 T 2 T β +1 |μ1 μ2 |η ϑ¯ 3 = + + , (β ) | | (β + 1 ) 2 (β + 2 )

T |μ1 | T2 T α +1 ηα ϑ4 = + , η | | (α + 2 ) 2 (α + 1 ) 

Tα

N1 = sup f (t, 0, 0, 0 ) < ∞,

N2 = sup g(t, 0, 0, 0 ) < ∞.

t∈[0,T ]

t∈[0,T ]

The first result is concerned with the existence and uniqueness of solutions for the problem (1.1) and (1.2) and is based on Banach’s contraction mapping principle. Theorem 3.1. Assume that (H1 ) and (H2 ) hold and that

+

T 1−γ

(2 − γ )

1 2

1 < ,

 +

T 1−δ

(2 − δ )

1 2

 1 < ,

where , 1 ,  ,  1 are given by (3.3)–(3.6). Then the boundary value problem (1.1) and (1.2) has a unique solution. Proof. Let us fix supt∈[0,T ] f (t, 0, 0, 0 ) = N1 < ∞ supt∈[0,T ] g(t, 0, 0, 0 ) = N2 < ∞ and define

r ≥ max

⎧ ⎪ ⎪ ⎨

M+

T 1−γ (2−γ ) M1

⎪ ⎪ 1 ⎩2 −  +

T 1−γ (2−γ )

and

⎫ ⎪ ⎪ ⎬ M + (T2−δ ) M1

, ⎪ 1−δ ⎭ −  + (T2−δ )  1 ⎪ 1−δ

,

1

1 2

where , 1 ,  ,  1 and M, M1 , M and M1 are given by (3.3)–(3.6). We first show that FBr ⊂ Br , where Br = {(x, y ) ∈ X × Y : (x, y )X×Y ≤ r}. For (x, y) ∈ Br , we find the following estimates based on the assumptions (H1 ) and (H2 ):

| f (t, x(t ), y(t ), Dγ y(t ))| ≤ | f (t, x(t ), y(t ), Dγ y(t )) − f (t, 0, 0, 0 )| + | f (t, 0, 0, 0 )| ≤ (|x(t )| + |y(t )| + |Dγ y(t )| ) + N1 ≤ (xX + yY ) + N1 ≤ r + N1 ,

|h(y )| ≤ Ly ≤ LyY ≤ Lr, |φ (x )| ≤ L1 xX ≤ L1 r. Using these estimates, we get

 = ϑ3 1 + ϑ4 + (|σ3 |T + 1 )L1 + |σ4 |T L, M = ϑ3 N2 + ϑ4 N1 ,  1 = ϑ¯ 3 1 +

Similarly, we have

|g(t, x(t ), Dδ x(t ), y(t ))| ≤ 1 (xX + yY ) + N2

+ |σ1 |L + |σ2 |L1 ,

ϑ2 N2

237



(t − s )α−1 | f (s, x(s ), y(s ), Dγ y(s ))|ds (α ) 0 + (|σ1 |T + 1 )|h(y )| + T |σ2 ||φ (x )| 

η T (η − s )α−1 + | f (s, x(s ), y(s ), |μ2 |ξ |μ1 | | | (α ) 0

|F1 (x, y )(t )| ≤

× Dγ y(s ))|ds + + +

2



T 2

T 0

|μ2 |

t



T 0

( T − s )β |g(s, x(s ), Dδ x(s ), y(s ))|ds (β + 1 )

ξ (ξ − s )β −1

(β )

0

|g(s, x(s ), Dδ x(s ), y(s )|ds

 ( T − s )α γ | f (s, x(s ), y(s ), D y(s ))|ds (α + 1 )

≤ r + M, where  and M are given by (3.3). Using the notations (3.4), we get



(t − s )α−2 | f (s, x(s ), y(s ), Dγ y(s ))|ds 0 (α − 1 ) + |σ1 ||h(y )| + |σ2 ||φ (x )| 

η 1 (η − s )α−1 + | f (s, x(s ), y(s ), |μ2 |ξ |μ1 | | | (α ) 0

|F1 (x, y ) (t )| ≤

t

× Dγ y(s ))ds

T ( T − s )β δ |g(s, x(s ), D x(s ), y(s ))ds|ds + 0 (β + 1 )

ξ (ξ − s )β −1 T2 + |g(s, x(s ), Dδ x(s ), y(s ))ds|ds |μ2 | 2 (β ) 0



T ( T − s )α | f (s, x(s ), y(s ), Dγ y(s ))|ds + 0 (α + 1 ) ≤ 1 r + M1 , which implies that

|Dγ F1 (x, y )(t )| ≤ ≤

0

t

(t − s )−γ |F (x, y ) (s )|ds (1 − γ ) 1

T 1−γ ( r + M1 ). (2 − γ ) 1

Thus

F1 (x, y )X = F1 (x, y ) + Dγ F1 (x, y )

T 1−γ 1 r ≤ + (2 − γ )

1−γ + M+

T

(2 − γ )

M1

≤

r . 2

(3.7)

In a similar manner, using (3.5) and (3.6), we obtain

|F2 (x, y )(t )| ≤  r + M , |(F2 (x, y )) (t )| ≤  1 r + M1 ,

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B. Ahmad et al. / Chaos, Solitons and Fractals 83 (2016) 234–241

and

|Dδ F2 (x, y )(t )| ≤



t 0

≤

(t − s )−δ |F (x, y ) (s )|ds (1 − δ ) 2

F2 (x1 , y1 ) − F2 (x2 , y2 )Y = F2 (x1 , y1 ) − F2 (x2 , y2 ) + Dδ F2 (x1 , y1 ) − Dδ (F2 (x2 , y2 )   T 1−δ  1 (x1 − x2 X + y1 − y2 Y ). ≤  + (2 − δ )

In consequence, we get

F2 (x, y )Y = F2 (x, y ) + Dδ F2 (x, y ))

T 1−δ  1 r ≤  + (2 − δ )

1−δ +

T M (2 − δ ) 1

r ≤ . 2

Consequently, we obtain

(3.8)

Thus, it follows from (3.7) and (3.8) that F (x, y )(t )X×Y ≤ r. For xi , yi ∈ Br , i = 1, 2 and for each t ∈ [0, T], we have

|F1 (x1 , y1 )(t ) − F1 (x2 , y2 )(t )|

t (t − s )α−1 ≤ | f (s, x1 (s ), y1 (s ), Dγ y1 (s )) (α ) 0 − f (s, x2 (s ), y2 (s ), Dγ y2 (s ))|ds + (|σ1 |T + 1 )|h(y1 ) − h(y2 )| + |σ2 |T |φ (x1 ) − φ (x2 )|

T (T − s )α−1 ≤

(x1 − x2  + y1 − y2  + Dγ y1 (α ) 0 − Dγ y2  )ds + (|σ1 |T + 1 )Ly1 − y2  + |σ2 |T L1 x1 − x2  

η (η − s )α−1 T +

( x 1 − x 2  + y 1 |μ2 |ξ |μ1 | | | (α ) 0

( T − s )β

1 (x1 0 (β + 1 )

− x2  + y1 − y2  + Dδ x1 − Dδ x2  )ds − y2  + Dγ y1 − Dγ y2  )ds +



T2 + |μ2 | 2

ξ 0

T

(ξ − s )β −1

1 (x1 − x2  + y1 − y2  (β )

( T − s )α

( x 1 − x 2  0 (α + 1 )  + y1 − y2  + Dγ y1 − Dγ y2  )ds ) + Dδ x1 − Dδ x2  )ds +



T 1−γ  (x1 − x2 X + y1 − y2 Y ). (2 − γ ) 1

In a similar manner, we can find that

T 1−δ ( r + M1 ). ≤ (2 − δ ) 1

M +

 +

T

≤ (x1 − x2 X + y1 − y2 Y ).

F (x, y )(t )X×Y ≤  +

+  +

T 1−γ  (2 − γ ) 1

T 1−δ  (x1 − x2 X + y1 − y2 Y ), (2 − δ ) 1

which shows that F is a contraction in view of the as1−γ 1−δ sumptions:  + (T 2−γ ) 1 +  + (T 2−δ )  1 < 1. Hence, by

Banach’s fixed point theorem, the operator F has a unique fixed point which corresponds to the unique solution of problem (1.1) and (1.2). This completes the proof. 

In the next result, we show the existence of solutions for the problem (1.1) and (1.2) by applying Leray–Schauder alternative. Lemma 3.2 (Leray–Schauder alternative). ([48] p. 4.) Let F: E → E be a completely continuous operator (i.e., a map that restricted to any bounded set in E is compact). Let

E (F ) = {x ∈ E : x = λF (x )for some0 < λ < 1}. Then either the set E (F ) is unbounded, or F has at least one fixed point. For the forthcoming result, we assume that (A1 ) f, g : [0, T ] × R3 → R are continuous and there exist real constants ki , λi ≥ 0(i = 1, 2, 3 ) and k0 > 0, λ0 > 0 such that ∀xi ∈ R, (i = 1, 2, 3 ) we have

| f (t, x1 , x2 , x3 )| ≤ k0 + k1 |x1 | + k2 |x2 | + k3 |x3 |, |g(t, x1 , x2 , x3 )| ≤ λ0 + λ1 |x1 | + λ2 |x2 | + λ3 |x3 |. h, φ : C ([0, T ], R ) → R are continuous h(0 ) = φ (0 ) = 0 and there exist constants K, K1 > 0, such

(A2 )

Also we have

that

|F1 (x1 , y1 ) (t ) − F1 (x2 , y2 ) (t )| ≤ 1 (x1 − x2 X + y1 − y2 Y ).

|h(y )| ≤ K y, |φ (x )| ≤ K1 x, ∀y, x ∈ C ([0, T ], R ).

Thus we obtain

|Dγ F1 (x1 , y1 )(t ) − Dγ F1 (x2 , y2 )(t )|

t (t − s )−γ ≤ |F1 (x1 , y1 ) (s ) − F1 (x2 , y2 ) (s )|ds ( 1−γ) 0 ≤

1−γ

T  (x − x2 X + y1 − y2 Y ). (2 − γ ) 1 1

From the above inequalities, we get

F 1 ( x 1 , y 1 ) − F 1 ( x 2 , y 2 )  X =  F 1 ( x 1 , y 1 ) − F 1 ( x 2 , y 2 )  + Dγ (F1 (x1 , y1 ) − Dγ (F1 (x2 , y2 )

To facilitate the proof, we introduce the following notations:

A=

ϑ1 k0 + ϑ2 λ0 +

+ ϑ3 λ0 + ϑ4 k0 + B=

T 1−γ (2 − γ ) T

1−δ

(2 − δ )



ϑ ϑ¯ 1 k0 + 2 λ0 T



ϑ4 ¯ ϑ3 λ0 + k0 , T

(3.9)

ϑ1 k1 + ϑ2 max{λ1 , λ2 } + (|σ1 |T + 1 )K   T 1−γ ϑ2 ¯ max{λ1 , λ2 } + |σ1 |K ϑk + + (2 − γ ) 1 1 T + ϑ3 max{λ1 , λ2 } + ϑ4 k1 + |σ4 |T K (3.10)

B. Ahmad et al. / Chaos, Solitons and Fractals 83 (2016) 234–241

+ D=

T 1−δ (2 − δ )

 ϑ¯ 3 max{λ1 , λ2 } +

ϑ4 T

 k1 + |σ4 |K .

ϑ1 max{k2 , k3 } + ϑ3 λ3 + |σ2 |T K1   ϑ2 T 1−γ ¯ + ϑ max{k2 , k3 } + λ3 + |σ2 |T K1 (2 − γ ) 1 T + ϑ3 λ3 + ϑ3 max{k1 , k2 } + (|σ3 |T + 1 )|K1   T 1−δ ϑ + ϑ¯ 3 λ3 + 4 max{k2 , k3 } + |σ4 |K1 . (2 − δ ) T (3.11)

Theorem 3.3. Assume that the conditions (A1 ) − (A2 ) hold. In addition it is assumed that max {B, D} < 1, where B, D are defined by (3.10) and (3.11) respectively. Then there exists at least one solution for problem (1.1) and (1.2) on [0, T]. Proof. In the first step, we show that the operator F : X × Y → X × Y is completely continuous. By continuity of the functions f, g, h and φ , it follows that the operator F is continuous. Let  ⊂ X × Y be bounded. Then there exist positive constants Mf and Mg such that |f(t, x(t), y(t), Dγ y(t)| ≤ Mf , |g(t, x(t), Dδ x(t), y(t)| ≤ Mg , ∀(x, y) ∈ , and constants Mh , Mφ > 0 such that |h(y)| ≤ Mh , |φ (x)| ≤ Mφ for all y, x ∈ C ([0, T ], R ). Then, for any (x, y) ∈ , one can find that

|F1 (x, y )(t )| ≤ ϑ1 M f + ϑ2 Mg + (|σ1 |T + 1 )Mh + |σ2 |T Mφ , |F1 (x, y ) (t )| ≤ ϑ¯ 1 M f + |Dγ (F



t

ϑ2 T

Mg + |σ1 |Mh + |σ2 |Mφ ,

(t − s )−γ |F (x, y ) (s )ds (1 − γ ) 1

(x, y )(t )| ≤ 0   T 1−γ ϑ ≤ ϑ¯ 1 M f + 2 Mg + |σ1 |Mh + |σ2 |Mφ . (2 − β ) T 1

Thus

F1 (x, y )X = F1 (x, y ) + Dγ (F1 (x, y ) ϑ2 Mg + (|σ1 |T + 1 )Mh + |σ2 |T Mφ ≤ ϑ1 M f + T   T 1−γ ϑ + ϑ¯ 1 M f + 2 Mg + |σ1 |Mh + |σ2 |Mφ . (2 − β ) T Similarly, one can obtain that

F2 (x, y )Y = F2 (x, y ) + Dγ (F2 (x, y )) ≤ ϑ3 Mg + ϑ4 M f + (|σ3 |T + 1 )Mφ + |σ4 |T Mh   T 1−δ ϑ + ϑ¯ 3 Mg + 4 M f + |σ3 |Mφ + |σ4 |Mh . (2 − δ ) T

239

  γ × f (s, x(s ), y(s ), D y(s ))ds  t   2 (t2 − s )α−1  +  f (s, x(s ), y(s ), Dγ y(s ))ds (α ) t1 + |σ1 |Mh (t2 − t1 ) + |σ2 |Mφ (t2 − t1 )  t2 − t1 T β +1 ηα + M f + |μ2 |ξ M |μ2 |ξ |μ1 | | | (α + 1 ) (β + 2 ) g  T2 T 2 T α +1 ξβ + |μ2 | Mg + Mf 2 (β + 1 ) 2 (α + 2 ) Mf [(t − t )α + |t2α − t1α |] (α + 1 ) 2 1 + |σ1 |Mh (t2 − t1 ) + |σ2 |Mφ (t2 − t1 )

≤

+

t2 − t1

+



| |

|μ2 |ξ |μ1 |

T β +1 ηα M f + |μ2 |ξ M (α + 1 ) (β + 2 ) g  2 α +1

T T2 T ξβ |μ2 | M + M , 2 (β + 1 ) g 2 (α + 2 ) f

and

|F1 (x, y ) (t2 )) − F1 (x, y ) (t1 )| ≤

Mf

(α )

[(t2 − t1 )α −1 + |t2α −1 − t1α −1 |].

Thus

|Dγ F1 (x, y )(t2 ) − Dγ F1 (x, y )(t1 )|

t (t − s )−γ = |F1 (x, y ) (t2 ) − F1 (x, y ) (t1 )|ds 0 (1 − γ ) ≤

T 1−γ

Mf

(2 − γ ) (α )

[(t2 − t1 )α −1 + |t2α −1 − t1α −1 |].

Hence we have that F1 (x, y )(t2 ) − F1 (x, y )(t1 )Y → 0 independent of x and y as t2 → t1 . Analogously, one can obtain

|F2 (x, y )(t2 ) − F2 (x, y )(t1 )| M

g β β [(t − t )β + |t2 − t1 |] (β + 1 ) 2 1 + |σ3 |Mφ (t2 − t1 ) + |σ4 |Mh (t2 − t1 )  t2 − t1 T α +1 ξβ + M + |μ1 |η M |μ1 |η|μ2 | | | (β + 1 ) g (α + 2 ) f  T2 T 2 T β +1 ηα + |μ1 | Mf + Mg , 2 (α + 1 ) 2 (β + 2 )

≤

|F2 (x, y ) (t2 )) − F2 (x, y ) (t1 )| ≤

Mg

(β )

β −1

[(t2 − t1 )β −1 + |t2

β −1

− t1

|],

and

From the above inequalities, it follows that the operator T is uniformly bounded. Next, we show that F is equicontinuous. Let t1 , t2 ∈ [0, T] with t1 < t2 . Then we have

|Dδ F2 (x, y )(t2 ) − Dδ F2 (x, y )(t1 )|

|F1 (x, y )(t2 )) − F1 (x, y )(t1 )|  t  1 [(t2 − s )α−1 − (t1 − s )α−1 ] ≤  (α ) 0

which imply that F2 (x, y )(t2 ) − F2 (x, y )(t1 )X → 0 independent of x and y as t2 → t1 . Therefore, the operator F (x, y ) is equicontinuous. From the foregoing arguments,

≤

Mg T 1−δ β −1 β −1 [(t − t )β −1 + |t2 − t1 |], (2 − δ ) (β ) 2 1

240

B. Ahmad et al. / Chaos, Solitons and Fractals 83 (2016) 234–241

we infer that the operator F (x, y ) is completely continuous by Arzelá–Ascoli theorem. Finally, it will be verified that the set E = {(x, y ) ∈ X × Y |(x, y ) = λF (x, y ), 0 ≤ λ ≤ 1} is bounded. Let (x, y ) ∈ E, then (x, y ) = λF (x, y ). For any t ∈ [0, T], we have

This shows that the set E is bounded. Thus, by Lemma 3.2, the operator F has at least one fixed point. Hence problem (1.1) and (1.2) has at least one solution on [0, T]. This completes the proof. 

x(t ) = λF1 (x, y )(t ),

3.1. Examples

y(t ) = λF2 (x, y )(t ).

Then

|x(t )| ≤ ϑ1 (k0 + k1 xX + max{k2 , k3 }yY ) + ϑ2 (λ0 + max{λ1 , λ2 }xX + λ3 yY ) + (|σ1 |T + 1 )K xX + |σ2 |T K1 yY , |x (t )| ≤ ϑ¯ 1 (k0 + k1 xX + max{k2 , k3 }yY ) ϑ2 + (λ0 + max{λ1 , λ2 }xX + λ3 yY )

Example 3.4. Consider the following coupled system of fractional differential equations

 |x(t )| ⎧c 5/4 D x(t ) = √ 1 + sin(y(t )) ⎪ 1+|x(t )| 2 ⎪ 12 3600+t  e−t ⎪ ⎨ −1 1/4 + tan (D y(t )) + 1+t 4 , (3.12)   −t |D1/2 x(t )| c 3/2 ⎪ D y(t ) = √ e sin ( x ( t )) + ⎪ 1/2 x (t )| + y (t ) 1+ | D 4 ⎪ 12 1600+t ⎩ + cos t + 1,

T

+ |σ1 |K xX + |σ2 |K1 yY , and

|Dγ x(t )| ≤



T 1−γ ϑ¯ (k + k1 xX + max{k2 , k3 }yY ) (2 − γ ) 1 0

ϑ2

(λ0 + max{λ1 , λ2 }xX + λ3 yY ) T  + |σ1 |K xX + |σ2 |K1 yY . +

In view of the above estimates, we get

xX ≤ ϑ1 (k0 + k1 xX + max{k2 , k3 }yY ) + ϑ2 (λ0 + max{λ1 , λ2 }xX + λ3 yY ) + (|σ1 |T + 1 )K xX + |σ2 |T K1 yY  T 1−γ + ϑ¯ (k + k1 xX + max{k2 , k3 }yY ) (2 − γ ) 1 0 ϑ2 + (λ0 + max{λ1 , λ2 }xX + λ3 yY ) T  + |σ1 |K xX + |σ2 |K1 yY . In a similar manner, we can have

yY ≤ ϑ3 (λ0 + max{λ1 , λ2 }xX + λ3 yY ) + ϑ4 (k0 + k1 xX + max{k2 , k3 }yY ) + (|σ3 |T + 1 )K1 yY + |σ4 |T K xX  T 1−δ + ϑ¯ (λ + max{λ1 , λ2 }xX + λ3 yY ) (2 − δ ) 3 0 ϑ4 + (k0 + k1 xX + max{k2 , k3 }yY ) T  + |σ3 |K1 yY + |σ4 |K xX . Then, making use of the notations (3.9), (3.10) and (3.11), we find that

xX + yY ≤ A + max{B, D}||(x, y )X×Y , which implies that

(x, y )X×Y ≤

A . 1 − max{B, D}

t ∈ [0, 1],

supplemented with nonlocal coupled integral boundary conditions:

1 1 tan−1 (y(t )), y(s )ds = x(2/3 ), 180 0

1 1 y (0 ) = x(t ), x(s )ds = y(1/3 ). 144 0

x (0 ) =

(3.13)

Using the given data, we find that = 1/240, 1 = 1/ 160, L = 1/180, L1 = 1/144, = 1/36, −σ1 = σ2 = σ3 = −σ4 = 6, ϑ1 = 14.32371725, ϑ2 = 6.21669342, ϑ¯ 1 = 14.544 3698, ϑ3 = 9.64297953, ϑ¯ 3 = 10.01910592, ϑ4 = 18.98484 1 1 724, + (7/4 ) 1 0.36891137 < 1/2,  + (3/2 ) 1 0.46586225 < 1/2. Obviously all the conditions of Theorem 3.1 are satisfied. Thus, by the conclusion of Theorem 3.1, there exists a unique solution for the problem (3.12) and (3.13). Example 3.5. Let us consider the following system

⎧c 5/4 D x(t ) = √ 1 2 cos t ⎪ ⎪ 16+t ⎪ ⎨ + 1 e−3t sin x(t ) + 1 y(t )|x(t )| + 1 D1/4 y(t ), 240 300 1+|x(t )| 2(190+t ) x(t ) e−t 1 1/2 ⎪c D3/2 y(t ) = 30+t x(t )) 2 + 120 (1+y2 (t )) + 2 (120+t ) sin (D ⎪ ⎪ ⎩ 1 −t + 2√6400+t e sin(y(t )), t ∈ [0, 1], (3.14) subject to the conditions:

x (0 ) =

1 0

1 y(t ), 180

0

1

y(s )ds = x(2/3 ), y(0 ) =

x(s )ds = y(1/3 ).

(3.15)

Clearly

1 1 x  + 4 240

| f (t, x(t ), y(t ), D1/4 y(t ))| ≤ +

1 1 y  + D1/4 y, 300 380

|g(t, x(t ), D1/2 x(t ), y(t ))| ≤

1 1 + x  30 120

1 1 D1/2 x + y  , 240 160 1 1 |h(y )| ≤ y, |φ (x )| ≤ x  . 180 140 +

1 x(t ), 140

B. Ahmad et al. / Chaos, Solitons and Fractals 83 (2016) 234–241

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