DOI 10.1007/s10958-017-3528-8 Journal of Mathematical Sciences, Vol. 226, No. 3, October, 2017

A COUPLED SYSTEM OF NONLOCAL FRACTIONAL DIFFERENTIAL EQUATIONS WITH COUPLED AND UNCOUPLED SLIT-STRIPSTYPE INTEGRAL BOUNDARY CONDITIONS B. Ahmad1 and S. K. Ntouyas2

UDC 517.9

The paper deals with with the existence and uniqueness of solutions for a coupled system of fractional differential equations with coupled and uncoupled slit-strips-type integral boundary conditions. The existence and uniqueness of solutions is established by using the Banach contraction principle, while the existence of solutions is obtained by using the Leray–Schauder alternative. The results are explained with the help of examples.

1. Introduction The study of boundary-value problems for linear and nonlinear differential equations is a popular field of research that finds extensive applications in a variety of disciplines of pure and applied sciences. The investigation of boundary-value problems of fractional order has recently picked up a great momentum. Thus, a variety of results of diverse interest, ranging from theoretical to application aspects, are available in the literature on the topic. In particular, the tools of fractional calculus have revolutionized the field of mathematical modeling, and the integer-order models in many physical and engineering phenomena have been transformed to their fractionalorder counterparts. One of the salient features accounting for this trend is probably the nonlocal characteristic of fractional-order operators, which can describe the hereditary properties of numerous important materials and processes. For examples and applications in physics, chemistry, biology, biophysics, blood-flow phenomena, control theory, wave propagation, signal and image processing, viscoelasticity, percolation, identification, fitting of experimental data, economics etc., we refer the reader to the books [1–3]. For some recent work on the topic, see [4–24] and the references therein. In a recent paper [25], the authors discussed some new fractional boundary-value problems with slit-strips-type conditions. The investigation of coupled systems of fractional-order differential equations is also very significant because systems of this kind appear in various problems of applied nature, especially in biosciences. For details and examples, the reader is referred to the papers [26–32] and the references cited therein. In this paper, motivated by [25], we study a coupled system of nonlinear fractional differential equations: c

c

D q x.t / D f .t; x.t /; y.t //; p

D y.t / D g.t; x.t /; y.t //;

t 2 Œ0; 1ç;

1 < q  2;

t 2 Œ0; 1ç;

1 < p  2;

(1.1)

1 King

Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia; e-mail: bashirahmad [email protected]. of Ioannina, 451 10 Ioannina, Greece and King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia; e-mail: [email protected].

2 University

Published in Neliniini Kolyvannya, Vol. 19, No. 3, pp. 291–310, July–September, 2016. Original article submitted January 14, 2015. 1072-3374/17/2263–0175

c 2017 �

Springer Science+Business Media New York

175

B. A HMAD AND S. K. N TOUYAS

176

supplemented with coupled and uncoupled slit-strips-type integral boundary conditions, respectively, given by

x.0/ D 0;

x.⇣/ D a

Z⌘ 0

Z1

Z⌘

x.s/ds;

0 < ⌘ < ⇣ < ⇠ < 1;

0

Z1

Z⌘

x.s/ds;

0 < ⌘ < ⇣ < ⇠ < 1;

0

Z1

Z⌘

Z1

y.s/ds C b

y.s/ds;

0 < ⌘ < ⇣ < ⇠ < 1;

⇠

(1.2) y.0/ D 0;

y.⇣/ D a

x.s/ds C b

⇠

and

x.0/ D 0;

x.⇣/ D a

x.s/ds C b

⇠

(1.3) y.0/ D 0;

y.⇣/ D a

y.s/ds C b

0

y.s/ds;

0 < ⌘ < ⇣ < ⇠ < 1;

⇠

where c D q and c D p denote the Caputo fractional derivatives of orders q and p respectively, f; gW Œ0; 1ç ⇥ R ⇥ R ! R are given continuous functions, and a; b are real constants. Here, we note that the differential equations with integral boundary conditions form an important class of boundary-value problems. The concept of coupled and uncoupled integral boundary conditions introduced in the present paper is new. We can interpret these conditions from the physical point of view as the contributions of finite strips of arbitrary lengths on a given interval related to the value of the unknown function at any (nonlocal) position in the region outside these strips. The applications of slit-strip-type boundary conditions can be found, e.g., in [33–36]. The paper is organized as follows. In Section 2, we present the main results for a coupled system of nonlinear fractional differential equations with coupled slit-strips-type integral boundary conditions. The results obtained for uncoupled integral boundary conditions are discussed in Section 3. Our results are based on the standard tools of the fixed-point theory. They are well illustrated with the help of examples. 2. Coupled Slit-Strips-Type Integral Boundary Conditions First, we recall the definitions of fractional integral and fractional derivative [1, 2]. Definition 2.1. The Riemann–Liouville fractional integral of order q for a continuous function g is defined as 1 I g.t / D Ä.q/ q

Zt 0

provided that the integral exists.

g.s/ ds; .t � s/1q

q > 0;

A C OUPLED S YSTEM OF N ONLOCAL F RACTIONAL D IFFERENTIAL E QUATIONS

177

Definition 2.2. For a function gW Œ0; 1/ ! R continuously differentiable at least n-times, the Caputo derivative of fractional order q is defined as

c

1 D g.t / D Ä.n � q/ q

Zt

.t � s/nq1 g .n/ .s/ds;

n � 1 < q < n;

n D Œqç C 1;

0

where Œqç denotes the integer part of the real number q: We now prove an auxiliary result pivotal in finding the solution of problem (1.1), (1.2). Lemma 2.1 (auxiliary Lemma). Given �;

x.0/ D 0;

y.0/ D 0;

2 C.Œ0; 1ç; R/; the system

c

D q x.t / D �.t /;

t 2 Œ0; 1ç;

1 < q  2;

c

D p y.t / D

.t /;

t 2 Œ0; 1ç;

1 < p  2;

x.⇣/ D a

y.⇣/ D a

Z⌘

(2.1)

y.s/ds;

0 < ⌘ < ⇣ < ⇠ < 1;

0

Z1

Z⌘

Z1

x.s/ds;

0 < ⌘ < ⇣ < ⇠ < 1;

y.s/ds C b

⇠

x.s/ds C b

0

⇠

can be represented via the equivalent integral equations as follows: 2

º Z⌘ Zs .s � ⌧ /p1 t 6 x.t/ D 2 ⇣ a 4 ⇣ � Å2 Ä.p/

.⌧ / d ⌧ ds C b

0 0

�

Z⇣ 0

Cb

C

Zt 0

⇠

.s � ⌧ /p1 Ä.p/

.⌧ / d ⌧ ds

0

Ω º Z⌘ Zs .⇣ � s/q1 .s � ⌧ /q1 �.s/ ds C Å a �.⌧ / d ⌧ds Ä.q/ Ä.q/ 0 0

Z1 Zs ⇠

Z1 Zs

⌧/q1

.s � Ä.q/

�.⌧ / d ⌧ ds �

0

.t � s/q1 �.s/ds; Ä.q/

Z⇣ 0

s/p1

.⇣ � Ä.p/

Ω

3

7 .s/ ds 5 (2.2)

B. A HMAD AND S. K. N TOUYAS

178

2

º Z⌘ Zs t .s � ⌧ /p1 6 y.t/ D 2 Å a 4 ⇣ � Å2 Ä.p/

.⌧ / d ⌧ ds C b

0 0

�

Z⇣ 0

Cb

C

Zt

0

s

.s � ⌧ /p1 Ä.p/

.⌧ / d ⌧ ds

⇠

Ω º Z⌘ Zs .⇣ � s/q1 .s � ⌧ /q1 �.s/ ds C ⇣ a �.⌧ / d ⌧ds Ä.q/ Ä.q/ 0 0

Z1 Zs ⇠

Z1 Z

⌧/q1

.s � Ä.q/

�.⌧ / d ⌧ ds �

0

Z⇣

s/p1

.⇣ � Ä.p/

0

.t � s/p1 Ä.p/

Ω

3

7 .s/ds 5 (2.3)

.s/ ds;

0

where �⇤ ⇥ � Å D a⌘2 C b 1 � ⇠ 2 =2 ¤ 0:

(2.4)

Proof. It is well known that the general solution of the fractional differential equations in (2.1) has the form x.t / D c0 C c1 t C

Zt

.t � s/q1 �.s/ ds; Ä.q/

(2.5)

Zt

.t � s/p1 Ä.p/

(2.6)

0

y.t / D c2 C c3 t C

.s/ ds;

0

where c0 ; c1 2 R are arbitrary constants. Applying the conditions x.0/ D 0 and y.0/ D 0; we conclude that c0 D 0 and c2 D 0: In view of the nonlocal conditions x.⇣/ D a

Z⌘

y.s/ds C b

0

Z1

y.s/ds;

y.⇣/ D a

Z⌘

x.s/ds C b

0

⇠

Z1

x.s/ds;

⇠

we obtain a system of equations

⇣c1 � Åc3 D a

Z⌘ Zs

.s � ⌧/p1 Ä.p/

.⌧ / d ⌧ ds C b

0 0

�

Z⇣ 0

Z1 Zs ⇠

.⇣ � s/q1 �.s/ds; Ä.q/

0

.s � ⌧ /p1 Ä.p/

.⌧ / d ⌧ ds

A C OUPLED S YSTEM OF N ONLOCAL F RACTIONAL D IFFERENTIAL E QUATIONS

�Åc1 C ⇣c3 D a

Z⌘ Zs

.s � ⌧/q1 �.⌧ / d ⌧ ds C b Ä.q/

0 0

�

Z⇣

Z1 Zs

.s � ⌧ /q1 �.⌧ / d ⌧ds Ä.q/

0

⇠

.⇣ � s/p1 Ä.p/

179

.s/ ds;

0

where ⇥ �⇤ � Å D a⌘2 C b 1 � ⇠ 2 =2:

As a result of the solution of system (2.7), (2.8), we find 2

º Z⌘ Zs .s � ⌧ /p1 1 6 c1 D 2 ⇣ a 4 ⇣ � Å2 Ä.p/

.⌧ / d ⌧ ds C b

0 0

�

Z⇣ 0

Cb

⇠

.s � ⌧/q1 �.⌧ / d ⌧ ds � Ä.q/

0

Z⇣

.⇣ � s/p1 Ä.p/

0

2

º Z⌘ Zs 1 .s � ⌧ /p1 6 c3 D 2 Å a 4 ⇣ � Å2 Ä.p/

.⌧ / d ⌧ ds C b

0 0

0

Cb

.⌧ / d ⌧ ds

0

0 0

⇠

�

.s � ⌧ /p1 Ä.p/

Ω º Z⌘ Zs .⇣ � s/q1 .s � ⌧ /q1 �.s/ds C Å a �.⌧ / d ⌧ds Ä.q/ Ä.q/

Z1 Zs

Z⇣

Z1 Zs

Ω

7 .s/ ds 5 ;

Z1 Zs ⇠

3

.s � ⌧ /p1 Ä.p/

.⌧ / d ⌧ ds

0

Ω º Z⌘ Zs .⇣ � s/q1 .s � ⌧ /q1 �.s/ds C ⇣ a �.⌧ / d ⌧ ds Ä.q/ Ä.q/ 0 0

Z1 Zs ⇠

0

⌧/q1

.s � Ä.q/

�.⌧ /d ⌧ ds �

Z⇣

s/p1

.⇣ � Ä.p/

0

Ω

3

7 .s/ ds 5 :

Substituting the values of c0 ; c1 ; c2 ; and c3 in (2.5) and (2.6), we get (2.2) and (2.3). The converse assertion is obtained by direct computations. Lemma 2.1 is proved. 2.1. Existence Results.

We now introduce a space X D fx.t /jx.t / 2 C.Œ0; 1ç/g endowed with the norm kxk D maxfjx.t /j; t 2 Œ0; 1çg:

B. A HMAD AND S. K. N TOUYAS

180

Obviously, .X; k � k/ is a Banach space. Also let Y D fy.t /jy.t / 2 C.Œ0; 1ç/g be endowed with the norm kyk D maxfjy.t /j; t 2 Œ0; 1çg: Clearly, the product space .X ⇥ Y; k.x; y/k/ is a Banach space with the norm k.x; y/k D kxk C kyk: In view of Lemma 2.1, we define an operator T W X ⇥ Y ! X ⇥ Y as 0

T .x; y/.t / D @

T1 .x; y/.t / T2 .x; y/.t /

1

A;

where 2 º ⌘ s Z Z t .s � ⌧ /p1 4 g.⌧; x.⌧ /; y.⌧// d ⌧ ds ⇣ a T1 .x; y/.t/ D 2 ⇣ � Å2 Ä.p/ 0 0

Cb

Z1 Zs ⇠

.s � ⌧/p1 g.⌧; x.⌧/; y.⌧// d ⌧ ds � Ä.p/

0

Z⇣

.⇣ � s/q1 f .s; x.s/; y.s// ds Ä.q/

0

Ω

º Z⌘ Zs .s � ⌧/q1 CÅ a f .⌧; x.⌧ /; y.⌧// d ⌧ds Ä.q/ 0 0

Cb

Z1 Zs ⇠

�

Z⇣

.s � ⌧/q1 f .⌧; x.⌧ /; y.⌧// d ⌧ds Ä.q/

0

s/p1

.⇣ � Ä.p/

0

Ω

3

7 g.s; x.s/; y.s// ds 5 C

Zt

.t � s/q1 f .s; x.s/; y.s// ds; Ä.q/

0

2 º ⌘ s Z Z t .s � ⌧ /p1 4 g.⌧; x.⌧ /; y.⌧// d ⌧ds Å a T2 .x; y/.t/ D 2 ⇣ � Å2 Ä.p/ 0 0

Cb

Z1 Zs ⇠

0

.s � ⌧/p1 g.⌧; x.⌧/; y.⌧// d ⌧ ds � Ä.p/

Z⇣ 0

.⇣ � s/q1 f .s; x.s/; y.s//ds Ä.q/

Ω

A C OUPLED S YSTEM OF N ONLOCAL F RACTIONAL D IFFERENTIAL E QUATIONS

181

º Z⌘ Zs .s � ⌧/q1 C⇣ a f .⌧; x.⌧ /; y.⌧// d ⌧ds Ä.q/ 0 0

Cb

Z1 Zs ⇠

�

Z⇣ 0

.s � ⌧/q1 f .⌧; x.⌧ /; y.⌧// d ⌧ds Ä.q/

0

Ω

3

.⇣ � s/p1 7 g.s; x.s/; y.s//ds 5 C Ä.p/

Zt

.t � s/p1 g.s; x.s/; y.s// ds: Ä.p/

0

For the sake of convenience, we set  qC1 � ⇣ 1 ⌘qC1 1 � ⇠ qC1 1 M1 D ; C C jÅjjaj C jÅjjbj Ä.p C 1/ j⇣ 2 � Å2 j Ä.q C 1/ Ä.q C 2/ Ä.q C 2/

(2.7)

 � ⌘pC1 1 � ⇠ pC1 ⇣p 1 M2 D 2 ⇣jaj ; C ⇣jbj C jÅj j⇣ � Å2 j Ä.p C 2/ Ä.p C 2/ Ä.p C 1/

(2.8)

 � 1 ⌘qC1 1 � ⇠ qC1 ⇣q M3 D 2 ⇣jaj ; C ⇣jbj C jÅj j⇣ � Å2 j Ä.q C 2/ Ä.q C 2/ Ä.q C 1/

(2.9)

 pC1 � 1 ⇣ 1 ⌘pC1 1 � ⇠ pC1 M4 D ; C 2 C jÅjjaj C jÅjjbj Ä.p C 1/ j⇣ � Å2 j Ä.p C 1/ Ä.p C 2/ Ä.p C 2/

(2.10)

and »

º

M0 D min 1 � .M1 C M3 /k1 � .M2 C M4 /�1 ; 1 � .M1 C M3 /k2 � .M2 C M4 /�2 ; ki ; �i � 0;

(2.11)

i D 1; 2:

The first result establishes the existence and uniqueness of solutions for problem (1.1), (1.2) and is based on Banach’s contraction-mapping principle. Theorem 2.1. Assume that f; gW Œ0; 1ç ⇥ R2 ! R are continuous functions and there exist constants mi ; ni ; i D 1; 2; such that for all t 2 Œ0; 1ç and ui ; vi 2 R; i D 1; 2; jf .t; u1 ; u2 / � f .t; v1 ; v2 /j  m1 ju1 � v1 j C m2 ju2 � v2 j and jg.t; u1 ; u2 / � g.t; v1 ; v2 /j  n1 ju1 � v1 j C n2 ju2 � v2 j:

B. A HMAD AND S. K. N TOUYAS

182

In addition, assume that .M1 C M3 /.m1 C m2 / C .M2 C M4 /.n1 C n2 / < 1; where Mi ; i D 1; 2; 3; 4; are given by (2.7)–(2.10). Then the boundary-value problem (1.1), (1.2) possesses a unique solution. Proof. We define and

sup f .t; 0; 0/ D N1 < 1

t 2Œ0;1ç

sup g.t; 0; 0/ D N2 < 1

t 2Œ0;1ç

such that r�

.M1 C M3 /N1 C .M2 C M4 /N2 : 1 � Œ.M1 C M3 /.m1 C m2 / C .M2 C M4 /.n1 C n2 /ç

Further, we show that TBr ⇢ Br ; where Br D f.x; y/ 2 X ⇥ Y W k.x; y/k  rg: For .x; y/ 2 Br ; we get 2 8 < Z⌘ Zs .s � ⌧ /p1 t 4⇣ jaj jT1 .x; y/.t/j D max 4 2 .jg.⌧; x.⌧ /; y.⌧// � g.⌧; 0; 0/j : Ä.p/ t 2Œ0;1ç j⇣ � Å2 j 2

0 0

C jg.⌧; 0; 0/j/d ⌧ds C jbj

Z1 Zs ⇠

Cjg.⌧; 0; 0/j/d ⌧ds C

Z⇣

.s � ⌧ /p1 .jg.⌧; x.⌧ /; y.⌧// � g.⌧; 0; 0/j Ä.p/

0

s/q1

.⇣ � Ä.q/

.jf .s; x.s/; y.s// � f .s; 0; 0/j C jf .s; 0; 0/j/ds

0

8 <

Z⌘ Zs

Z1 Zs

.s � ⌧/q1 t .jf .⌧; x.⌧ /; y.⌧// � f .⌧; 0; 0/j C jf .⌧; 0; 0/j/ d ⌧ ds Ä.q/

C jÅj jaj : C jbj

⇠

C

Z⇣ 0

.s � ⌧/q1 .jf .⌧; x.⌧ /; y.⌧// � f .⌧; 0; 0/j C jf .⌧; 0; 0/j/d ⌧ ds Ä.q/

0 0

0

.⇣ � s/p1 Ä.p/

93 > = 7 .jg.s; x.s/; y.s// � g.s; 0; 0/j C jg.s; 0; 0/j/ ds 5 > ;

9 > = > ;

A C OUPLED S YSTEM OF N ONLOCAL F RACTIONAL D IFFERENTIAL E QUATIONS

C

Zt 0

3 .t � s/q1 .jf .s; x.s/; y.s// � f .s; 0; 0/j C jf .s; 0; 0/j/ds 5 Ä.q/

2 8 < Z⌘ Zs .s � ⌧ /p1 1 4⇣ jaj .n1 kxk C n2 kyk C N2 / d ⌧ ds  2 : j⇣ � Å2 j Ä.p/ 0 0

C jbj

Z1 Zs ⇠

C

Z⇣

.s � ⌧/p1 .n1 kxk C n2 kyk C N2 /d ⌧ ds Ä.p/

0

s/q1

.⇣ � Ä.q/

.m1 kxk C m2 kyk C N1 /ds

8 <

Z⌘ Zs

.s � ⌧/q1 .m1 kxk C m2 kyk C N1 / d ⌧ ds Ä.q/

Z1 Zs

.s � ⌧/q1 .m1 kxk C m2 kyk C N1 / d ⌧ ds Ä.q/

0

C jÅj jaj : C jbj

⇠

C

> ;

0 0

0

93 > = 7 .n1 kxk C n2 kyk C N2 /ds 5 > ;

Z⇣

.⇣ � s/p1 Ä.p/

Zt

.t � s/q1 .m1 kxk C m2 kyk C N1 / ds Ä.q/

0

C

9 > =

0

 � ⌘pC1 1 � ⇠ pC1 ⇣p 1 ⇣jaj .n1 kxk C n2 kyk C N2 / C ⇣jbj C jÅj  2 j⇣ � Å2 j Ä.p C 2/ Ä.p C 2/ Ä.p C 1/ º

1 C 2 j⇣ � Å2 j



� » ⇣ qC1 1 ⌘qC1 1 � ⇠ qC1 C C jÅjjaj C jÅjjbj Ä.q C 1/ Ä.q C 2/ Ä.q C 2/ Ä.q C 1/

⇥ .m1 kxk C m2 kyk C N1 / D M2 .n1 kxk C n2 kyk C N2 / C M1 .m1 kxk C m2 kyk C N1 / D .M2 n1 C M1 m1 /kxk C .M2 n2 C M1 m2 /kyk C M2 N2 C M1 N1

183

B. A HMAD AND S. K. N TOUYAS

184

 .M2 n1 C M1 m1 C M2 n2 C M1 m2 /r C M2 N2 C M1 N1 : In the same way, we conclude that  � » ⌘pC1 1 1 � ⇠ pC1 ⇣ pC1 1 jT2 .x; y/.t /j  jÅjjaj C C jÅjjbj C⇣ j⇣ 2 � Å2 j Ä.p C 2/ Ä.p C 2/ Ä.p C 1/ Ä.q C 1/ º

⇥ .n1 kxk C n2 kyk C N2 / C

j⇣ 2

1 � Å2 j



� ⇣q ⌘qC1 1 � ⇠ qC1 ⇥ jÅj .m1 kxk C m2 kyk C N1 / C ⇣jaj C ⇣jbj Ä.q C 1/ Ä.q C 2/ Ä.q C 2/ D M4 .n1 kxk C n2 kyk C N2 / C M3 .m1 kxk C m2 kyk C N1 / D .M4 n1 C M3 m1 /kxk C .M4 n2 C M3 m2 /kyk C M4 N2 C M3 N1  .M4 n1 C M3 m1 C M4 n2 C M3 m2 /r C M4 N2 C M3 N1 : Consequently, kT .x; y/.t /k  r: Now, for .x2 ; y2 /; .x1 ; y1 / 2 X ⇥ Y and any t 2 Œ0; 1ç; we get jT1 .x2 ; y2 /.t / � T1 .x1 ; y1 /.t /j 2 8 < Z⌘ Zs .s � ⌧/p1 t 4⇣ jaj jg.⌧; x2 .⌧ /; y2 .⌧ // � g.⌧; x1 .⌧ /; y1 .⌧ //j d ⌧ ds  2 : j⇣ � Å2 j Ä.p/ 0 0

C jbj

Z1 Zs ⇠

C

Z⇣ 0

0

9 > =

.⇣ � s/q1 jf .s; x2 .s/; y2 .s// � f .s; x1 .s/; y1 .s//jds > Ä.q/ ; 8 <

Z⌘ Zs

Z1 Zs

.s � ⌧/q1 jf .⌧; x2 .⌧ /; y2 .⌧ // � f .⌧; x1 .⌧ /; y1 .⌧ //j d ⌧ ds Ä.q/

C jÅj jaj : C jbj

.s � ⌧/p1 jg.⌧; x2 .⌧ /; y2 .⌧ // � g.⌧; x1 .⌧ /; y1 .⌧ //jd ⌧ ds Ä.p/

⇠

0

.s � ⌧/q1 jf .⌧; x2 .⌧ /; y2 .⌧ // � f .⌧; x1 .⌧ /; y1 .⌧ //jd ⌧ ds Ä.q/

0 0

A C OUPLED S YSTEM OF N ONLOCAL F RACTIONAL D IFFERENTIAL E QUATIONS

C

C

93 > = 7 jg.s; x2 .s/; y2 .s// � g.s; x1 .s/; y1 .s//jds 5 > ;

Z⇣

.⇣ � s/p1 Ä.p/

Zt

1 .t � s/q1 jg.s; x2 .s/; y2 .s// � g.s; x1 .s/; y1 .s//j ds  2 Ä.q/ j⇣ � Å2 j

0

0

185



� ⌘pC1 1 � ⇠ pC1 ⇣p ⇥ ⇣jaj .n1 kx2 � x1 k C n2 ky2 � y1 k/ C ⇣jbj C jÅj Ä.p C 2/ Ä.p C 2/ Ä.p C 1/ º

 qC1 � » 1 ⇣ 1 ⌘qC1 1 � ⇠ qC1 C C C jÅjjaj C jÅjjbj j⇣ 2 � Å2 j Ä.q C 1/ Ä.q C 2/ Ä.q C 2/ Ä.q C 1/ ⇥ .m1 kx2 � x1 k C m2 ky2 � y1 k/  M2 .n1 kx2 � x1 k C n2 ky2 � y1 k/ C M1 .m1 kx2 � x1 k C m2 ky2 � y1 k/ D .M2 n1 C M1 m1 /kx2 � x1 k C .M2 n2 C M1 m2 /ky2 � y1 k; and, therefore, we obtain kT1 .x2 ; y2 /.t / � T1 .x1 ; y1 /k  .M2 n1 CM1 m1 CM2 n2 CM1 m2 /Œkx2 � x1 k C ky2 � y1 kç:

(2.12)

Similarly, kT2 .x2 ; y2 /.t /�T2 .x1 ; y1 /k  .M4 n1 CM3 m1 CM4 n2 CM3 m2 /Œkx2 � x1 k C ky2 � y1 kç:

(2.13)

It follows from (2.12) and (2.13) that kT .x2 ; y2 /.t / � T .x1 ; y1 /.t /k  Œ.M1 C M3 /.m1 C m2 / C .M2 C M4 /.n1 C n2 /ç.ku2 � u1 k C kv2 � v1 k/: Since .M1 C M3 /.m1 C m2 / C .M2 C M4 /.n1 C n2 / < 1; we conclude that T is a contraction operator. Hence, by the Banach fixed-point theorem, the operator T possesses a unique fixed point, which is the unique solution of problem (1.1), (1.2). Theorem 2.1 is proved. In the next result, we prove the existence of solutions for problem (1.1), (1.2) by applying the Leray–Schauder alternative.

B. A HMAD AND S. K. N TOUYAS

186

Lemma 2.2 (Leray–Schauder alternative, [37, p. 4]). Let F W E ! E be a completely continuous operator (i.e., a map restricted to any bounded set in E is compact). Let E.F / D fx 2 EW x D �F .x/ for some

0 < � < 1g:

Then either the set E.F / is unbounded or F has at least one fixed point. Theorem 2.2. Assume that there exist real constants ki ; �i � 0; i D 1; 2; and k0 > 0; �0 > 0 such that, for any xi 2 R; i D 1; 2; the following inequalities are true: jf .t; x1 ; x2 /j  k0 C k1 jx1 j C k2 jx2 j; jg.t; x1 ; x2 /j  �0 C �1 jx1 j C �2 jx2 j: In addition, it is assumed that .M1 C M3 /k1 C .M2 C M4 /�1 < 1

and

.M1 C M3 /k2 C .M2 C M4 /�2 ç < 1;

where Mi ; i D 1; 2; 3; 4; are given by (2.7)–(2.10). Then there exists at least one solution of the boundary-value problem (1.1), (1.2). Proof. First, we show that the operator T W X ⇥Y ! X ⇥Y is completely continuous. In view of the continuity of the functions f and g; the operator T is continuous. Let  ⇢ X ⇥ Y be bounded. Then there exist positive constants L1 and L2 such that jf .t; x.t /; y.t //j  L1 ;

jg.t; x.t /; y.t //j  L2

8.x; y/ 2 :

Thus, for any .x; y/ 2 ; we find 2 º Z⌘ Zs .s � ⌧ /p1 t 4⇣ jaj jg.⌧; x.⌧ /; y.⌧//jd ⌧ds jT1 .x; y/.t /j  2 ⇣ � Å2 Ä.p/ 0 0

C jbj

Z1 Zs ⇠

C

Z⇣

.s � ⌧ /p1 jg.⌧; x.⌧ /; y.⌧//jd ⌧ ds Ä.p/

0

.⇣ � s/q1 jf .s; x.s/; y.s//jds Ä.q/

0

º

C jÅj jaj

Z⌘ Zs 0 0

Ω

.s � ⌧ /q1 jf .⌧; x.⌧ /; y.⌧//j d ⌧ ds Ä.q/

A C OUPLED S YSTEM OF N ONLOCAL F RACTIONAL D IFFERENTIAL E QUATIONS

C jbj

Z1 Zs ⇠

Z⇣ 0

C

.s � ⌧ /q1 jf .⌧; x.⌧ /; y.⌧//jd ⌧ dsC Ä.q/

0

Ω

3

.⇣ � s/p1 7 jg.s; x.s/; y.s//jds 5 Ä.p/ Zt

.t � s/q1 jf .s; x.s/; y.s//jds Ä.q/

0

 � 1 ⌘pC1 1 � ⇠ pC1 ⇣p  2 ⇣jaj L2 C ⇣jbj C jÅj j⇣ � Å2 j Ä.p C 2/ Ä.p C 2/ Ä.p C 1/  qC1 ⇣ ⌘qC1 1 C jÅjjaj C j⇣ 2 � Å2 j Ä.q C 1/ Ä.q C 2/ º

� » 1 � ⇠ qC1 1 CjÅjjbj C L1 : Ä.q C 2/ Ä.q C 1/ This yields  � ⌘pC1 1 � ⇠ pC1 ⇣p 1 ⇣jaj L2 C ⇣jbj C jÅj kT1 .x; y/k  2 j⇣ � Å2 j Ä.p C 2/ Ä.p C 2/ Ä.p C 1/ C

º

1 j⇣ 2 � Å2 j



� » ⇣ qC1 1 ⌘qC1 1 � ⇠ qC1 C L1 C jÅjjaj C jÅjjbj Ä.q C 1/ Ä.q C 2/ Ä.q C 2/ Ä.q C 1/

D M2 L2 C M1 L1 : Similarly, we get  � » ⌘pC1 1 1 � ⇠ pC1 ⇣ pC1 1 kT2 .x; y/k  jÅjjaj C L2 C jÅjjbj C⇣ j⇣ 2 � Å2 j Ä.p C 2/ Ä.p C 2/ Ä.p C 1/ Ä.q C 1/ º

 � 1 ⇣q ⌘qC1 1 � ⇠ qC1 C 2 jÅj L1 C ⇣jaj C ⇣jbj j⇣ � Å2 j Ä.q C 1/ Ä.q C 2/ Ä.q C 2/ D M 4 L2 C M 3 L1 : Thus, it follows from the inequalities presented above that the operator T is uniformly bounded. Further, we show that T is equicontinuous. Let t1 ; t2 2 Œ0; 1ç with t1 < t2 : This yields jT1 .x.t2 /; y.t2 // � T1 .x.t1 /; y.t1 //j

187

B. A HMAD AND S. K. N TOUYAS

188

1  Ä.q/

Zt1 Œ.t2 � s/q1 � .t1 � s/q1 çjf .s; x.s/; y.s//jds 0

1 C Ä.q/

Zt2 .t2 � s/q1 jf .s; x.s/; y.s//jds

t1

2 º Z⌘ Zs t2 � t1 4 .s � ⌧ /p1 C 2 jg.⌧; x.⌧ /; y.⌧//jd ⌧ ds ⇣ jaj ⇣ � Å2 Ä.p/ 0 0

C jbj

Z1 Zs 0

⇠

º

C jÅj jaj

Cjbj

Zt1 0

Z⇣

.⇣ � s/q1 jf .s; x.s/; y.s//jds Ä.q/

0

Z⌘ Zs

Ω

.s � ⌧/q1 jf .⌧; x.⌧ /; y.⌧//jd ⌧ ds Ä.q/

0 0

Z1 Zs

⌧/q1

.s � Ä.q/

jf .⌧; x.⌧/; y.⌧//jd ⌧ds C

0

⇠

L1  Ä.q/

.s � ⌧/p1 jg.⌧; x.⌧ /; y.⌧//j d ⌧ ds C Ä.p/

⇥

s/p1

.⇣ � Ä.p/

0

.t2 � s/

t2 � t1 C 2 j⇣ � Å2 j

Z⇣

q1

º

� .t1 � s/

q1

⇤

L1 ds C Ä.q/

Ω

3

7 jg.s; x.s/; y.s//jds 5

Zt2 .t2 � s/q1 ds

t1

� 1 � ⇠ pC1 ⇣p ⌘pC1 L2 C ⇣jbj C jÅj ⇣jaj Ä.p C 2/ Ä.p C 2/ Ä.p C 1/

� Ω ⌘qC1 1 � ⇠ qC1 ⇣ qC1 L1 : C jÅjjaj C jÅjjbj C Ä.q C 1/ Ä.q C 2/ Ä.q C 2/ 

Similarly, we obtain jT2 .x.t2 /; y.t2 // � T2 .x.t1 /; y.t1 //j L2  Ä.p/

Zt1 0

⇥

.t2 � s/p1 � .t1 � s/

t2 � t1 C 2 j⇣ � Å2 j

º

⇤ p1

L2 ds C Ä.p/

Zt2 .t2 � s/p1 ds

t1

� 1 � ⇠ pC1 ⇣ pC1 ⌘pC1 L2 C jÅjjbj C⇣ jÅjjaj Ä.p C 2/ Ä.p C 2/ Ä.p C 1/

A C OUPLED S YSTEM OF N ONLOCAL F RACTIONAL D IFFERENTIAL E QUATIONS

189

� Ω ⌘qC1 1 � ⇠ qC1 ⇣q L1 : C ⇣jaj C ⇣jbj C jÅj Ä.q C 1/ Ä.q C 2/ Ä.q C 2/ 

Therefore, the operator T .x; y/ is equicontinuous and, hence, the operator T .x; y/ is completely continuous. Finally, we show that the set E D f.x; y/ 2 X ⇥ Y j.x; y/ D �T .x; y/; 0  �  1g is bounded. Let .x; y/ 2 E: Then .x; y/ D �T .x; y/: For any t 2 Œ0; 1ç; we get x.t / D �T1 .x; y/.t /;

y.t / D �T2 .x; y/.t /:

Then 2 º Z⌘ Zs 1 .s � ⌧/p1 4⇣ jaj jx.t/j  2 jg.⌧; x.⌧ /; y.⌧//jd ⌧ ds ⇣ � Å2 Ä.p/ 0 0

C jbj

Z1 Zs ⇠

0

º

C

C

.s � ⌧/q1 jf .⌧; x.⌧ /; y.⌧//jd ⌧ ds Ä.q/

.s � ⌧/q1 jf .⌧; x.⌧ /; y.⌧//jd ⌧ ds Ä.q/

0

Ω

3

Z⇣

.⇣ � Ä.p/

Zt

.t � s/q1 1 jf .s; x.s/; y.s//jds  2 Ä.q/ j⇣ � Å2 j

0

0



.⇣ � s/q1 jf .s; x.s/; y.s//jds Ä.q/

0 0

Z1 Zs ⇠

Z⇣ 0

Z⌘ Zs

C jÅj jaj

C jbj

.s � ⌧/p1 jg.⌧; x.⌧ /; y.⌧//jd ⌧ds C Ä.p/

s/p1

7 jg.s; x.s/; y.s//jds 5

� ⌘pC1 1 � ⇠ pC1 ⇣p ⇥ ⇣jaj .�0 C �1 kxk C �2 kyk/ C ⇣jbj C jÅj Ä.p C 2/ Ä.p C 2/ Ä.p C 1/

Ω

B. A HMAD AND S. K. N TOUYAS

190

C

º

Ω  qC1 � ⇣ 1 ⌘qC1 1 � ⇠ qC1 1 C C jÅjjaj C jÅjjbj j⇣ 2 � Å2 j Ä.q C 1/ Ä.q C 2/ Ä.q C 2/ Ä.q C 1/

⇥ .k0 C k1 kxk C k2 kyk/ and

jy.t/j 

º

Ω  � 1 ⌘pC1 1 1 � ⇠ pC1 ⇣ pC1 jÅjjaj C C jÅjjbj C⇣ j⇣ 2 � Å2 j Ä.p C 2/ Ä.p C 2/ Ä.p C 1/ Ä.q C 1/  � 1 ⇣q ⌘qC1 1 � ⇠ qC1 ⇥ .�0 C �1 kxk C �2 kyk/ C 2 jÅj C ⇣jaj C ⇣jbj j⇣ � Å2 j Ä.q C 1/ Ä.q C 2/ Ä.q C 2/ ⇥ .k0 C k1 kxk C k2 kyk/:

Hence we find kxk  M1 .k0 C k1 kxk C k2 kyk/ C M2 .�0 C �1 kxk C �2 kyk/ and kyk  M3 .k0 C k1 kxk C k2 kyk/ C M4 .�0 C �1 kxk C �2 kyk/: This means that kxk C kyk  .M1 C M3 /k0 C .M2 C M4 /�0 C Œ.M1 C M3 /k1 C .M2 C M4 /�1 çkxk C Œ.M1 C M3 /k2 C .M2 C M4 /�2 çkyk: Consequently, k.x; y/k 

.M1 C M3 /k0 C .M2 C M4 /�0 M0

for any t 2 Œ0; 1ç; where M0 is defined by (2.11), which proves that E is bounded. Thus, by Lemma 2.2, the operator T has at least one fixed point. Hence, the boundary-value problem (1.1), (1.2) has at least one solution. Theorem 2.2 is proved. 2.2. Examples. Example 2.1. Consider the following system of coupled fractional differential equations with slit-strips-type integral boundary conditions:

A C OUPLED S YSTEM OF N ONLOCAL F RACTIONAL D IFFERENTIAL E QUATIONS

c

c

D 5=4 x.t / D

D 3=2 y.t / D

jy.t /j 3 3 2 x.t / C C ; 55 61 .1 C jy.t /j/ 2

191

t 2 Œ0; 1ç;

1 j cos x.t /j 2 C sin y.t / C 3; 27 .1 C j cos x.t /j/ 41

x.0/ D 0;

Z1=3 Z1 y.s/ds C y.s/ds; x.1=2/ D 0

y.0/ D 0;

t 2 Œ0; 1ç; (2.14)

2=3

Z1=3 Z1 y.1=2/ D x.s/ds C x.s/ds: 0

2=3

Here, q D 5=4; p D 3=2; a D 1; b D 1; ⇣ D 1=2; ⌘ D 1=3; ⇠ D 2=3: For the indicated values, it was obtained that Å D 1=3;

m1 D 2=55;

m2 D 3=61

M2 ' 1:397944;

n1 D 1=27;

M3 ' 1:854888;

n2 D 2=41;

M1 ' 2:731029;

M4 ' 2:216142;

and .M1 C M3 /.m1 C m2 / C .M2 C M4 /.n1 C n2 / ' 0:702454 < 1: Thus all conditions of Theorem 2.1 are satisfied. Therefore, by the conclusion of Theorem 2.1, problem (2.14) possesses a unique solution on Œ0; 1ç: Example 2.2. We now consider problem (2.14) with the following values: f .t; x.t /; y.t // D

1 2 2 C sin x.t / C y.t / tan1 x.t /; 2 41 43⇡

g.t; x.t /; y.t // D

2 1 1 C x.t / C sin y.t /: 3 11 17

Clearly, jf .t; x; y/j  k0 C k1 jxj C k2 jyj

and

jg.t; x; y/j D �0 C �1 jxj C �2 jyj;

where k0 D 1=2; k1 D 2=41; k2 D 1=43; �0 D 2=3; �1 D 1=11; �2 D 1=17: Furthermore, .M1 C M3 /k1 C .M2 C M4 /�1 ' 0:552257 < 1 and .M1 C M3 /k2 C .M2 C M4 /�2 ' 0:319243 < 1:

B. A HMAD AND S. K. N TOUYAS

192

Thus, all conditions of Theorem 2.2 are true and, therefore, the conclusion of Theorem 2.2 can be applied to problem (2.14) with the indicated values of f .t; x; y/ and g.t; x; y/: 3. Uncoupled Slit-Strips-Type Integral Boundary Conditions In connection with problem (1.1)–(1.3), we consider the following lemma: Lemma 3.1 (auxiliary lemma). For � 2 C.Œ0; 1ç; R/; the unique solution of the problem c

x.0/ D 0;

D q x.t / D �.t /;

x.⇣/ D a

Z⌘

1 < q  2;

x.s/ds C b

0

Z1

t 2 Œ0; 1ç; (3.1)

x.s/ds;

0 < ⌘ < ⇣ < ⇠ < 1;

⇠

is given by the formula

x.t / D

Zt 0

º Z⌘ Zs .t � s/q1 .s � ⌧ /q1 t �.s/ds C �.⌧ / d ⌧ds a Ä.q/ A Ä.q/ 0 0

Cb

Z1 Zs ⇠

.s � ⌧/q1 �.⌧ / d ⌧ ds � Ä.q/

Z⇣ 0

0

Ω .⇣ � s/q1 �.s/ d ds ; Ä.q/

(3.2)

where AD⇣�

a⌘2 b.1 � ⇠ 2 / � ¤ 0: 2 2

(3.3)

Proof. We present only a sketch of the proof. The general solution of the fractional differential equation in (3.1) can be represented in the form

x.t / D e0 C e1 t C

Zt

.t � s/q1 y.s/ ds; Ä.q/

(3.4)

0

where e0 ; e1 2 R are arbitrary constants. By using the presented boundary conditions, we conclude that e0 D 0 and Ω º Z⌘ Zs Z1 Zs Z⇣ 1 .s � ⌧/q1 .s � ⌧ /q1 .⇣ � s/q1 y.⌧/ d ⌧ ds C b y.⌧ /d ⌧ ds � y.s/ d ds : a e1 D A Ä.q/ Ä.q/ Ä.q/ 0 0

⇠ 0

Substituting the values of e0 ; e1 in (3.4), we get (3.2). Lemma 3.1 is proved.

0

A C OUPLED S YSTEM OF N ONLOCAL F RACTIONAL D IFFERENTIAL E QUATIONS

193

3.1. Existence Results for the Uncoupled Case. In view of Lemma 3.1, we define an operator T W X ⇥ Y ! X ⇥ Y as 1 0 T1 .u; v/.t / A; T .u; v/.t / D @ T2 .u; v/.t /

where

T1 .u; v/.t/ D

Zt 0

º Z⌘ Zs .s � ⌧ /q1 .t � s/q1 t f .s; u.s/; v.s// ds C f .⌧; u.⌧ /; v.⌧ // d ⌧ds a Ä.q/ A Ä.q/ 0 0

Cb

Z1 Zs

.s � ⌧/q1 f .⌧; u.⌧ /; v.⌧ // d ⌧ds � Ä.q/

⇠ 0

Z⇣

.⇣ � s/q1 f .s; u.s/; v.s// ds Ä.q/

0

Ω

and

T2 .u; v/.t/ D

Zt 0

º Z⌘ Zs .t � s/p1 .s � ⌧ /p1 t h.s; u.s/; v.s// ds C h.⌧; u.⌧ /; v.⌧ // d ⌧ ds a Ä.p/ A Ä.p/ 0 0

Cb

Z1 Zs ⇠

0

.s � ⌧/p1 h.⌧; u.⌧ /; v.⌧ // d ⌧ds � Ä.p/

Z⇣ 0

Ω .⇣ � s/p1 f .s; u.s/; v.s//ds : Ä.p/

In what follows, we set º » 1 ⌘qC1 1 1 � ⇠ qC1 ⇣q jaj ; C C jbj C �1 D Ä.q C 1/ jAj Ä.q C 2/ Ä.q C 2/ Ä.q C 1/

(3.5)

º » ⌘pC1 1 1 � ⇠ pC1 ⇣p 1 jaj : C C jbj C Ä.p C 1/ jAj Ä.p C 2/ Ä.p C 2/ Ä.p C 1/

(3.6)

�2 D

We now present the existence and uniqueness result for problem (1.1)–(1.3). We do not present the proof of this result because it is similar to the proof of Theorem 2.1. N i and Theorem 3.1. Assume that f; gW Œ0; 1ç ⇥ R2 ! R are continuous functions and there exist constants m nN i ; i D 1; 2; such that, for all t 2 Œ0; 1ç and ui ; vi 2 R; i D 1; 2; the following inequalities are true: N 1 ju1 � v1 j C m N 2 ju2 � v2 j jf .t; u1 ; u2 / � g.t; v1 ; v2 /j  m and jg.t; u1 ; u2 / � h.t; v1 ; v2 /j  nN 1 ju1 � v1 j C nN 2 ju2 � v2 j:

B. A HMAD AND S. K. N TOUYAS

194

In addition, assume that � 1 .m N1 Cm N 2 / C �2 .nN 1 C nN 2 / < 1; where �1 and �2 are given by (3.5) and (3.6), respectively. Then the boundary-value problem (1.1)–(1.3) possesses a unique solution. Example 3.1. Consider the following system of coupled fractional differential equations with uncoupled slitstrips-type integral boundary conditions: c

D 5=4 x.t / D c

jx.t /j 1 C tan1 y C 1; 24.1 C jx.t /j/ 20

D 3=2 y.t / D

x.0/ D 0;

1 1 sin x.t / C y.t / C 4; 35 25

t 2 Œ0; 1ç;

Z1=3 Z1 x.s/ds C x.s/ds; x.1=2/ D 0

y.0/ D 0;

t 2 Œ0; 1ç;

(3.7)

2=3

Z1=3 Z1 y.1=2/ D y.s/ds C y.s/ds; 0

2=3

where q D 5=4; p D 3=2; a D 1; b D 1; ⇣ D 1=2; ⌘ D 1=3; and ⇠ D 2=3: For the indicated values, it was shown that A D 1=6; m N 1 D 1=24; m N 2 D 1=20; nN 1 D 1=35; nN 2 D 1=25; �1 ' 4:716276; and �2 ' 3:614087: Hence, we obtain N1 Cm N 2 / C �2 .nN 1 C nN 2 / ' 0:680148 < 1: �1 .m Thus, all conditions of Theorem 3.1 are satisfied. Therefore, there exists a unique solution of problem (3.7) on Œ0; 1ç: The second result dealing with the existence of solutions of problem (1.1)–(1.3) is similar to Theorem 2.2. We now present this result. Theorem 3.2. Assume that there exist real constants ⇢i ; ⌫i � 0; i D 1; 2; and ⇢0 > 0; ⌫0 > 0 such that for any xi 2 R; i D 1; 2; the following inequalities are true: jf .t; x1 ; x2 /j  ⇢0 C ⇢1 jx1 j C ⇢2 jx2 j; jg.t; x1 ; x2 /j  ⌫0 C ⌫1 jx1 j C ⌫2 jx2 j: In addition, it is assumed that �1 ⇢1 C �2 ⌫1 < 1

and

�1 ⇢2 C �2 ⌫2 < 1;

A C OUPLED S YSTEM OF N ONLOCAL F RACTIONAL D IFFERENTIAL E QUATIONS

195

where �1 and �2 are given by relations (3.5) and (3.6), respectively. Then the boundary-value problem (1.1)–(1.3) has at least one solution. Proof. Setting �0 D minf1 � .�1 ⇢1 C �2 ⌫1 /; 1 � .�1 ⇢2 C �2 ⌫2 /g; ⇢i ; ⌫i � 0; i D 1; 2; with �1 and �2 given by (3.5) and (3.6), respectively, we conclude that the proof is similar to the proof of Theorem 2.2. Thus, we omit the proof. REFERENCES 1. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego (1999). 2. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, 204, North-Holland Math. Stud., Elsevier Sci. B. V., Amsterdam (2006). 3. J. Sabatier, O. P. Agrawal, and J. A. T. Machado (editors), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht (2007). 4. D. Baleanu and O. G. Mustafa, “On the global existence of solutions to a class of fractional differential equations,” Comput. Math. Appl., 59, 1835–1841 (2010). 5. B. Ahmad and J. J. Nieto, “Riemann–Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions,” Boundary Value Problems, 2011, 36, 9 (2011). 6. A. Alsaedi, S. K. Ntouyas, and B. Ahmad, “Existence results for Langevin fractional differential inclusions involving two fractional orders with four-point multiterm fractional integral boundary conditions,” in: Abstr. Appl. Anal., Art. ID 869837 (2013). 7. D. Baleanu, O. G. Mustafa, and R. P. Agarwal, “On Lp -solutions for a class of sequential fractional differential equations,” Appl. Math. Comput., 218, 2074–2081 (2011). 8. J. R. Graef, L. Kong, and Q. Kong, “Application of the mixed monotone operator method to fractional boundary value problems,” Fract. Calc. Different. Calc., 2, 554–567 (2011). 9. F. T. Akyildiz, H. Bellout, K. Vajravelu, and R. A. Van Gorder, “Existence results for third order nonlinear boundary value problems arising in nano boundary layer fluid flows over stretching surfaces,” Nonlin. Anal. Real World Appl., 12, 2919–2930 (2011). 10. Z. B. Bai and W. Sun, “Existence and multiplicity of positive solutions for singular fractional boundary value problems,” Comput. Math. Appl., 63, 1369–1381 (2012). 11. R. Sakthivel, N. I. Mahmudov, and J. J. Nieto, “Controllability for a class of fractional-order neutral evolution control systems,” Appl. Math. Comput., 218, 10334–10340 (2012). 12. R. P. Agarwal, D. O’Regan, and S. Stanek, “Positive solutions for mixed problems of singular fractional differential equations,” Math. Nachr., 285, 27–41 (2012). 13. J. R. Wang, Y. Zhou, and M. Medved, “Qualitative analysis for nonlinear fractional differential equations via topological degree method,” Topol. Methods Nonlin. Anal., 40, 245–271 (2012). 14. A. Cabada and G. Wang, “Positive solutions of nonlinear fractional differential equations with integral boundary value conditions,” J. Math. Anal. Appl., 389, 403–411 (2012). 15. G. Wang, B. Ahmad, L. Zhang, and R. P. Agarwal, “Nonlinear fractional integro-differential equations on unbounded domains in a Banach space,” J. Comput. Appl. Math., 249, 51–56 (2013). 16. B. Ahmad, S. K. Ntouyas, and A. Alsaedi, “A study of nonlinear fractional differential equations of arbitrary order with Riemann– Liouville type multistrip boundary conditions,” in: Math. Probl. Eng., Art. ID 320415 (2013). 17. D. O’Regan and S. Stanek, “Fractional boundary value problems with singularities in space variables,” Nonlin. Dynam., 71, 641–652 (2013). 18. Z. Liu, L. Lu, and I. Sz´ant´o, “Existence of solutions for fractional impulsive differential equations with p-Laplacian operator,” Acta Math. Hung., 141, 203–219 (2013). 19. J. R. Wang, Y. Zhou, and M. Feckan, “On the nonlocal Cauchy problem for semilinear fractional order evolution equations,” Cent. Eur. J. Math., 12, 911–922 (2014). 20. J. R. Graef, L. Kong, and M. Wang, “Existence and uniqueness of solutions for a fractional boundary value problem on a graph,” Fract. Calc. Appl. Anal., 17, 499–510 (2014). 21. G. Wang, S. Liu, and L. Zhang, “Eigenvalue problem for nonlinear fractional differential equations with integral boundary conditions,” in: Abstr. Appl. Anal., Art. ID 916260 (2014). 22. F. Punzo and G. Terrone, “On the Cauchy problem for a general fractional porous medium equation with variable density,” Nonlin.Anal., 98, 27–47 (2014).

196

B. A HMAD AND S. K. N TOUYAS

23. X. Liu, Z. Liu, and X. Fu, “Relaxation in nonconvex optimal control problems described by fractional differential equations,” J. Math. Anal. Appl., 409, 446–458 (2014). 24. C. Zhai and L. Xu, “Properties of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations with a parameter,” Comm. Nonlin. Sci. Numer. Simul., 19, 2820–2827 (2014). 25. B. Ahmad and R. P. Agarwal, “Some new versions of fractional boundary value problems with slit-strips conditions,” Boundary Value Problem, 2014, 175, 12 (2014). 26. B. Ahmad and J. J. Nieto, “Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions,” Comput. Math. Appl., 58, 1838–1843 (2009). 27. X. Su, “Boundary value problem for a coupled system of nonlinear fractional differential equations,” Appl. Math. Lett., 22, 64–69 (2009). 28. J. Wang, H. Xiang, and Z. Liu, “Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations,” in: Int. J. Different. Equat., 2010, Article ID 186928 (2010). 29. J. Sun, Y. Liu, and G. Liu, Existence of Solutions for Fractional Differential Systems with Antiperiodic Boundary Conditions, Comput. Math. Appl. (2012), /doi: 10.1016/j.camwa.2011.12.083 30. S. K. Ntouyas and M. Obaid, “A coupled system of fractional differential equations with nonlocal integral boundary conditions,” Adv. Different. Equat., 130 (2012). 31. M. Faieghi, S. Kuntanapreeda, H. Delavari, and D. Baleanu, “LMI-based stabilization of a class of fractional-order chaotic systems,” Nonlin. Dynam., 72, 301–309 (2013). 32. B. Senol and C. Yeroglu, “Frequency boundary of fractional order systems with nonlinear uncertainties,” J. Franklin Inst., 350, 1908– 1925 (2013). 33. T. Otsuki, “Diffraction by multiple slits,” JOSA A, 7, 646–652 (1990). 34. S. Asghar, T. Hayat, and B. Ahmad, “Acoustic diffraction from a slit in an infinite absorbing sheet,” Jap. J. Indust. Appl. Math., 13, 519–532 (1996). 35. T. Mellow and L. Karkkainen, “On the sound fields of infinitely long strips,” J. Acoust. Soc. Amer., 130, 153–167 (2011). 36. D. J. Lee, S. J. Lee, W. S. Lee, and J. W. Yu, “Diffraction by dielectric-loaded multiple slits in a conducting plane: TM case,” Progr. Electromagnet. Res., 131, 409–424 (2012). 37. A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York (2003).