COUPLED SYSTEMS OF FRACTIONAL DIFFERENTIAL INCLUSIONS … · Di erential inclusions are found to be...

Electronic Journal of Differential Equations, Vol. 2019 (2019), No. 69, pp. 1–21.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

COUPLED SYSTEMS OF FRACTIONAL DIFFERENTIAL

INCLUSIONS WITH COUPLED BOUNDARY CONDITIONS

BASHIR AHMAD, SOTIRIS K. NTOUYAS, AHMED ALSAEDI

Abstract. We investigate the existence of solutions for a boundary-value

problem of coupled fractional differential inclusions supplemented with coupledboundary conditions. By applying standard fixed point theorems for multi-

valued maps, we derive some existence results for the given problem when the

multi-valued maps involved have convex and non-convex values. Several re-sults follow from the ones obtained in this article by specializing the parameters

involved in the problem at hand.

1. Introduction

Fractional differential equations arise naturally in the mathematical modeling ofseveral real-world phenomena and have recently gained great importance in view oftheir varied applications in scientific and engineering problems. Fractional deriva-tives help to take care of the hereditary properties of processes under investigationand give rise to more realistic models than the ones based on integer-order deriva-tives. For further details and explanations see, for instance, [2, 24, 31].

Fractional-order boundary value problems (BVPs) have been extensively studiedby many researchers. For the recent development of the topic, we refer the reader toa series of articles [4, 17, 21, 28, 37, 39, 40, 42] and the references cited therein. Inparticular, coupled systems of fractional-order differential equations have attractedspecial attention in view of their occurrence in the mathematical modeling of phys-ical phenomena like chaos synchronization [18, 20, 41], anomalous diffusion [35],ecological effects [23], disease models [11, 16, 30], etc. For some recent theoreticalresults on coupled systems of fractional-order differential equations, for example,see [3, 5, 6, 7, 8, 34, 36, 38].

Differential inclusions are found to be of great utility in studying dynamical sys-tems and stochastic processes. Examples include sweeping processes [1, 29, 32],granular systems [33], nonlinear dynamics of wheeled vehicles [9], control problems[26], etc. The details of pressing issues in stochastic processes, control, differen-tial games, optimization and their application in finance, manufacturing, queueingnetworks, and climate control can be found in the text [25]. For application offractional differential inclusions in synchronization processes, see [14].

2010 Mathematics Subject Classification. 34A08, 34B15.Key words and phrases. Fractional differential inclusions; system; existence;

coupled boundary conditions; fixed point.c©2019 Texas State University.

Submitted March 19, 2019. Published May 15, 2019.

1

2 B. AHMAD, S. K. NTOUYAS, A. ALSAEDI EJDE-2019/69

In this article, motivated by [8], we consider a new boundary value problem ofcoupled Caputo (Liouville-Caputo) type fractional differential inclusions:

cDαx(t) ∈ F (t, x(t), y(t)), t ∈ [0, T ], 1 < α ≤ 2,

cDβy(t) ∈ G(t, x(t), y(t)), t ∈ [0, T ], 1 < β ≤ 2,(1.1)

subject to the coupled boundary conditions:

x(0) = ν1y(T ), x′(0) = ν2y′(T ),

y(0) = µ1x(T ), y′(0) = µ2x′(T ),

(1.2)

where cDα,cDβ denote the Caputo fractional derivatives of order α and β respec-tively, F,G : [0, T ]×R×R→ P(R) are given multivalued maps, P(R) is the familyof all nonempty subsets of R, and νi, µi, i = 1, 2 are real constants with νiµi 6= 1,i = 1, 2.

The objective of the present work is to establish existence criteria for solutionsof the problem (1.1)-(1.2) for convex and nonconvex valued multivalued maps Fand G by applying the standard fixed-point theorems for multivalued maps. Therest of the paper is organized as follows. We present background material aboutmultivalued analysis and fractional calculus in Section 2, while the main resultsare derived in Section 3. We emphasize that the tools of the fixed point theoryemployed in our analysis are standard, however their application to systems offractional differential inclusions is new.

2. Preliminaries

Let us begin this section with some basic concepts of multivalued maps [15, 22].Let (X , ‖ · ‖) be a normed space and define Pcl(X ) = {Y ∈ P(X ) : Y is closed},

Pcp,c(X ) = {Y ∈ P(X ) : Y is compact and convex}.A multi-valued map G : X → P(X ) is

(a) convex (closed) valued if G(x) is convex (closed) for all x ∈ X ;(b) upper semi-continuous (u.s.c.) on X if the set G(x0) is a nonempty closed

subset of X for each x0 ∈ X , and there exists an open neighborhood N0 ofx0 such that G(N0) ⊆ N for each open set N of X containing G(x0);

(c) lower semi-continuous (l.s.c.) if the set {y ∈ X : G(y) ∩B 6= ∅} is open forany open set B in E.

(d) completely continuous if G(B) is relatively compact for every B ∈ Pb(X ) ={Y ∈ P(X ) : Y is bounded}.

Remark 2.1. If the multi-valued map G is completely continuous with nonemptycompact values, then G is u.s.c. if and only if G has a closed graph, i.e., xn → x∗,yn → y∗, yn ∈ G(xn) imply y∗ ∈ G(x∗); the set Gr(G) = {(x, y) ∈ X×Y, y ∈ G(x)}defines the graph of G.

A multivalued map G : [a, b] → Pcl(R) is said to be measurable if the functiont 7→ d(y,G(t)) = inf{|y − z| : z ∈ G(t)} is measurable for every y ∈ R.

A multivalued map G : [a, b] × R2 → P(R) is said to be Caratheodory if (i)t 7→ G(t, x, y) is measurable for all x, y ∈ R and (ii) (x, y) 7→ F (t, x, y) is uppersemicontinuous for almost all t ∈ [a, b].

Further a Caratheodory function G is called L1-Caratheodory if (i) for each ρ > 0,there exists ϕρ ∈ L1([a, b],R+) such that ‖G(t, x, y)‖ = sup{|v| : v ∈ G(t, x, y)} ≤ϕρ(t) for all x, y ∈ R with ‖x‖, ‖y‖ ≤ ρ and for a.e. t ∈ [a, b].

EJDE-2019/69 FRACTIONAL-ORDER COUPLED SYSTEMS 3

Next, we recall some basic definitions of fractional calculus.

Definition 2.2. The fractional integral of order r with the lower limit zero for afunction f is defined as

Irf(t) =1

Γ(r)

∫ t

0

f(s)

(t− s)1−rds, t > 0, r > 0,

provided the right hand-side is point-wise defined on [0,∞), where Γ(·) is the gammafunction, which is defined by Γ(r) =

∫∞0tr−1e−tdt.

Definition 2.3. The Riemann-Liouville fractional derivative of order r > 0, n−1 <r < n, n ∈ N, is defined as

Dr0+f(t) =

1

Γ(n− r)( ddt

)n ∫ t

0

(t− s)n−r−1f(s)ds,

where the function f(t) has absolutely continuous derivative up to order (n− 1).

Definition 2.4. The Caputo derivative of order r for a function f : [0,∞) → Rcan be written as

cDr0+f(t) = Dr

0+

(f(t)−

n−1∑k=0

tk

k!f (k)(0)

), t > 0, n− 1 < r < n.

In the rest of this article, we will use cDr instead of cDr0+ for the sake of conve-

nience.

Remark 2.5. If f(t) ∈ Cn[0,∞), then

cDrf(t) =1

Γ(n− r)

∫ t

0

f (n)(s)

(t− s)r+1−n ds = In−rf (n)(t), t > 0, n− 1 < q < n.

Now we present an auxiliary lemma which plays a key role in the forthcominganalysis; see [8] for its proof.

Lemma 2.6. Let φ, h ∈ C([0, T ],R) and νiµi 6= 1, i = 1, 2. Then the solution ofthe linear fractional differential system

cDαx(t) = φ(t), t ∈ [0, T ], 1 < α ≤ 2,

cDβy(t) = h(t), t ∈ [0, T ], 1 < β ≤ 2,

x(0) = ν1y(T ), x′(0) = ν2y′(T ),

y(0) = µ1x(T ), y′(0) = µ2x′(T ),

(2.1)

is equivalent to the system of integral equations

x(t)

=µ2

1− ν2µ2

(ν1T (µ1ν2 + 1)

1− ν1µ1+ ν2t

)B2 +

ν21− ν2µ2

(T (µ1 + µ2)ν11− ν1µ1

+ t)A2

+ν1

1− ν1µ1(A1 + µ1B1) +

∫ t

0

(t− s)α−1

Γ(α)φ(s)ds

(2.2)


and

y(t)

=µ2

1− ν2µ2

(Tµ1(ν1 + ν2)

1− ν1µ1+ t)B2 +

ν21− ν2µ2

(Tµ1(ν1µ2 + 1)

1− ν1µ1+ µ2t

)A2

+µ1

1− ν1µ1(ν1A1 +B1) +

∫ t

0

(t− s)β−1

Γ(β)h(s)ds,

(2.3)

where

A1 =

∫ T

0

(T − s)β−1

Γ(β)h(s)ds, B1 =

∫ T

0

(T − s)α−1

Γ(α)φ(s)ds, (2.4)

A2 =

∫ T

0

(T − s)β−2

Γ(β − 1)h(s)ds, B2 =

∫ T

0

(T − s)α−2

Γ(α− 1)φ(s)ds. (2.5)

Definition 2.7. A function (x, y) ∈ C2([0, T ],R) × C2([0, T ],R) is a solution ofthe coupled system (1.1)-(1.2) if it satisfies the coupled boundary conditions (1.2)and there exist functions f, g ∈ L1([0, T ],R) such that f(t) ∈ F (t, x(t), y(t)), g(t) ∈G(t, x(t), y(t)) a.e. on [0, T ] and

x(t) =µ2

1− ν2µ2

(ν1T (µ1ν2 + 1)

1− ν1µ1+ ν2t

)∫ T

0

(T − s)α−2

Γ(α− 1)f(s)ds

+ν2

1− ν2µ2

(T (µ1 + µ2)ν11− ν1µ1

+ t)∫ T

0

(T − s)β−2

Γ(β − 1)g(s)ds

+ν1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)g(s)ds+

ν1µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)f(s)ds

+

∫ t

0

(t− s)α−1

Γ(α)f(s)ds

(2.6)

and

y(t) =µ2

1− ν2µ2

(Tµ1(ν1 + ν2)

1− ν1µ1+ t)∫ T

0

(T − s)α−2

Γ(α− 1)f(s)ds

+ν2

1− ν2µ2

(Tµ1(ν1µ2 + 1)

1− ν1µ1+ µ2t

)∫ T

0

(T − s)β−2

Γ(β − 1)g(s)ds

+ν1µ1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)g(s)ds

+µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)f(s)ds+

∫ t

0

(t− s)β−1

Γ(β)g(s)ds.

(2.7)

3. Main results

Let us introduce the space X = {x(t)|x(t) ∈ C([0, T ],R)} endowed with thenorm ‖x‖ = sup{|x(t)|, t ∈ [0, T ]}. Obviously (X, ‖ · ‖) is a Banach space. Also theproduct space (X ×X, ‖(x, y)‖) is a Banach space equipped with norm ‖(x, y)‖ =‖x‖+ ‖y‖.

For each (x, y) ∈ X ×X, define the sets of selections of F,G by

SF,(x,y) = {f ∈ L1([0, T ],R) : f(t) ∈ F (t, x(t), y(t)) for a.e. t ∈ [0, T ]}


and

SG,(x,y) = {g ∈ L1([0, T ],R) : g(t) ∈ G(t, x(t), y(t)) for a.e. t ∈ [0, T ]}.

In view of Lemma 2.6, we define the operators K1,K2 : X ×X → P(X ×X) by

K1(x, y) ={h1 ∈ X ×X : there exist f ∈ SF,(x,y), g ∈ SG,(x,y) such that

h1(x, y)(t) = Q1(t, x, y),∀t ∈ [0, T ]} (3.1)

and

K2(x, y) ={h2 ∈ X ×X : there exists f ∈ SF,(x,y), g ∈ SG,(x,y) such that

h2(x, y)(t) = Q2(t, x, y),∀t ∈ [0, T ]},

(3.2)

where

Q1(x, y)(t) =µ2

1− ν2µ2

(ν1T (µ1ν2 + 1)

1− ν1µ1+ ν2t

)∫ T

0

(T − s)α−2

Γ(α− 1)f(s)ds

+ν2

1− ν2µ2

(T (µ1 + µ2)ν11− ν1µ1

+ t)∫ T

0

(T − s)β−2

Γ(β − 1)g(s)ds

+ν1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)g(s)ds

+ν1µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)f(s)ds+

∫ t

0

(t− s)α−1

Γ(α)f(s)ds

and

Q2(x, y)(t) =µ2

1− ν2µ2

(Tµ1(ν1 + ν2)

1− ν1µ1+ t)∫ T

0

(T − s)α−2

Γ(α− 1)f(s)ds

+ν2

1− ν2µ2

(Tµ1(ν1µ2 + 1)

1− ν1µ1+ µ2t

)∫ T

0

(T − s)β−2

Γ(β − 1)g(s)ds

+ν1µ1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)g(s)ds

+µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)f(s)ds+

∫ t

0

(t− s)β−1

Γ(β)g(s)ds.

Then we define an operator K : X ×X → P(X ×X) by

K(x, y)(t) =

(K1(x, y)(t)K2(x, y)(t)

),

where K1 and K2 are respectively defined by (3.1) and (3.2).For the sake of computational convenience, we set

M1 =|µ2|

|1− ν2µ2|

( |ν1|(|µ1||ν2|+ 1)

|1− ν1µ1|+ |ν2|

) Tα

Γ(α)

+( |ν1||µ1||1− ν1µ1|

+ 1) Tα

Γ(α+ 1),

(3.3)

M2 =|ν2|

|1− ν2µ2|

( (|µ1|+ |µ2|)|ν1||1− ν1µ1|

+ 1) T β

Γ(β)+

|ν1||1− ν1µ1|

T β

Γ(β + 1), (3.4)

M3 =|µ2|

|1− ν2µ2|

( |µ1|(|ν1|+ |ν2|)|1− ν1µ1|

+ 1) Tα

Γ(α)+

|µ1||1− ν1µ1|

Tα

Γ(α+ 1), (3.5)


M4 =|ν2|

|1− ν2µ2|

( |µ1|(|ν1||µ2|+ 1)

|1− ν1µ1|+ |µ2|

) T β

Γ(β)

+( |ν1||µ1||1− ν1µ1|

+ 1) T β

Γ(β + 1).

(3.6)

3.1. The Caratheodory case. To prove our first result dealing with the convexvalues of F and G, we need the following known results.

Lemma 3.1 ([15, Proposition 1.2]). If G : X → Pcl(Y ) is u.s.c., then Gr(G) is aclosed subset of X × Y ; i.e., for every sequence {xn}n∈N ⊂ X and {yn}n∈N ⊂ Y , ifxn → x∗, yn → y∗ when n→∞ and yn ∈ G(xn), then y∗ ∈ G(x∗). Conversely, if Gis completely continuous and has a closed graph, then it is upper semi-continuous.

Lemma 3.2 ([27]). Let X be a separable Banach space. Let G : [0, T ] × R2 →Pcp,c(R) be an L1− Caratheodory multivalued map and let χ be a linear continuousmapping from L1([0, T ],R) to C([0, T ],R). Then the operator

χ ◦ SG,x : C([0, T ],R)→ Pcp,c(C([0, T ],R)), x 7→ (χ ◦ SG,x)(x) = χ(SG,x)

is a closed graph operator in C([0, T ],R)× C([0, T ],R).

Lemma 3.3 (Nonlinear alternative for Kakutani maps [19]). Let E1 be a closedconvex subset of a Banach space E, and U be an open subset of E1 with 0 ∈ U .Suppose that F : U → Pcp,c(E1) is an upper semicontinuous compact map. Theneither

(i) F has a fixed point in U , or(ii) there is a u ∈ ∂U and λ ∈ (0, 1) with u ∈ λF (u).

For the next Theorem we use the following assumptions:

(H1) the maps F,G : [0, T ]×R2 → P(R) are L1−Caratheodory and have convexvalues;

(H2) there exist continuous nondecreasing functions ψ1, ψ2, φ1, φ2 : [0,∞) →(0,∞) and functions p1, p2 ∈ C([0, T ],R+) such that

‖F (t, x, y)‖P := sup{|f | : f ∈ F (t, x, y)} ≤ p1(t)[ψ1(‖x‖) + φ1(‖y‖)],and

‖G(t, x, y)‖P := sup{|g| : g ∈ G(t, x, y)} ≤ p2(t)[ψ2(‖x‖) + φ2(‖y‖)],for each (t, x, y) ∈ [0, T ]× R2;

(H3) there exists a positive number N such that

N

(M1 +M3)‖p1‖(ψ1(N) + φ1(N)) + (M2 +M4)‖p2‖(ψ2(N) + φ2(N))> 1,

where Mi (i = 1, 2, 3, 4) are given by (3.3)-(3.6).

Theorem 3.4. Under assumptions (H1)–(H3), the coupled system (1.1)-(1.2) hasat least one solution on [0, T ].

Proof. Consider the operators K1,K2 : X ×X → P(X ×X) defined by (3.1) and(3.2). From (H1), it follows that the sets SF,(x,y) and SG,(x,y) are nonempty foreach (x, y) ∈ X × X. Then, for f ∈ SF,(x,y), g ∈ SG,(x,y) for (x, y) ∈ X × X, wehave

h1(x, y)(t) =µ2

1− ν2µ2

(ν1T (µ1ν2 + 1)

1− ν1µ1+ ν2t

)∫ T

0

(T − s)α−2

Γ(α− 1)f(s)ds


+ν2

1− ν2µ2

(T (µ1 + µ2)ν11− ν1µ1

+ t)∫ T

0

(T − s)β−2

Γ(β − 1)g(s)ds

+ν1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)g(s)ds

+ν1µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)f(s)ds+

∫ t

0

(t− s)α−1

Γ(α)f(s)ds

and

h2(x, y)(t) =µ2

1− ν2µ2

(Tµ1(ν1 + ν2)

1− ν1µ1+ t)∫ T

0

(T − s)α−2

Γ(α− 1)f(s)ds

+ν2

1− ν2µ2

(Tµ1(ν1µ2 + 1)

1− ν1µ1+ µ2t

)∫ T

0

(T − s)β−2

Γ(β − 1)g(s)ds

+ν1µ1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)g(s)ds

+µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)f(s)ds+

∫ t

0

(t− s)β−1

Γ(β)g(s)ds,

where h1 ∈ K1(x, y), h2 ∈ K2(x, y) and so (h1, h2) ∈ K(x, y).Now we verify that the operator K satisfies the assumptions of the nonlinear

alternative of Leray-Schauder type. It will be done in several steps. In the firststep, we show that K(x, y) is convex valued. Let (hi, hi) ∈ (K1,K2), i = 1, 2. Thenthere exist fi ∈ SF,(x,y), gi ∈ SG,(x,y), i = 1, 2 such that, for each t ∈ [0, T ], we have

hi(t) =µ2

1− ν2µ2

(ν1T (µ1ν2 + 1)

1− ν1µ1+ ν2t

)∫ T

0

(T − s)α−2

Γ(α− 1)fi(s)ds

+ν2

1− ν2µ2

(T (µ1 + µ2)ν11− ν1µ1

+ t)∫ T

0

(T − s)β−2

Γ(β − 1)gi(s)ds

+ν1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)gi(s)ds+

ν1µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)fi(s)ds

+

∫ t

0

(t− s)α−1

Γ(α)fi(s)ds

and

hi(t) =µ2

1− ν2µ2

(Tµ1(ν1 + ν2)

1− ν1µ1+ t)∫ T

0

(T − s)α−2

Γ(α− 1)fi(s)ds

+ν2

1− ν2µ2

(Tµ1(ν1µ2 + 1)

1− ν1µ1+ µ2t

)∫ T

0

(T − s)β−2

Γ(β − 1)gi(s)ds

+ν1µ1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)gi(s)ds+

µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)fi(s)ds

+

∫ t

0

(t− s)β−1

Γ(β)gi(s)ds.

Let 0 ≤ ω ≤ 1. Then, for each t ∈ [0, T ], we have

[ωh1 + (1− ω)h2](t)


=µ2

1− ν2µ2

(ν1T (µ1ν2 + 1)

1− ν1µ1+ ν2t

)∫ T

0

(T − s)α−2

Γ(α− 1)[ωf1(s) + (1− ω)f2(s)]ds

+ν2

1− ν2µ2

(T (µ1 + µ2)ν11− ν1µ1

+ t)∫ T

0

(T − s)β−2

Γ(β − 1)[ωg1(s) + (1− ω)g2(s)]ds

+ν1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)[ωg1(s) + (1− ω)g2(s)]ds

+ν1µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)[ωf1(s) + (1− ω)f2(s)]ds

+

∫ t

0

(t− s)α−1

Γ(α)[ωf1(s) + (1− ω)f2(s)]ds

and

[ωh1 + (1− ω)h2](t)

=µ2

1− ν2µ2

(Tµ1(ν1 + ν2)

1− ν1µ1+ t)∫ T

0

(T − s)α−2

Γ(α− 1)[ωf1(s) + (1− ω)f2(s)]ds

+ν2

1− ν2µ2

(Tµ1(ν1µ2 + 1)

1− ν1µ1+ µ2t

)∫ T

0

(T − s)β−2

Γ(β − 1)[ωg1(s) + (1− ω)g2(s)]ds

+ν1µ1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)[ωg1(s) + (1− ω)g2(s)]ds

+µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)[ωf1(s) + (1− ω)f2(s)]ds

+

∫ t

0

(t− s)β−1

Γ(β)[ωg1(s) + (1− ω)g2(s)]ds.

Since F,G are convex valued, we deduce that SF,(x,y), SG,(x,y) are convex valued.

Obviously ωh1 + (1− ω)h2 ∈ K1, ωh1 + (1− ω)h2 ∈ K2 and hence ω(h1, h1) + (1−ω)(h2, h2) ∈ K.

Now we show that K maps bounded sets into bounded sets in X × X. For apositive number r, let Br = {(x, y) ∈ X × X : ‖(x, y)‖ ≤ r} be a bounded set inX ×X. Then, there exist f ∈ SF,(x,y), g ∈ SG,(x,y) such that

h1(x, y)(t) =µ2

1− ν2µ2

(ν1T (µ1ν2 + 1)

1− ν1µ1+ ν2t

)∫ T

0

(T − s)α−2

Γ(α− 1)f(s)ds

+ν2

1− ν2µ2

(T (µ1 + µ2)ν11− ν1µ1

+ t)∫ T

0

(T − s)β−2

Γ(β − 1)g(s)ds

+ν1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)g(s)ds+

ν1µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)f(s)ds

+

∫ t

0

(t− s)α−1

Γ(α)f(s)ds

and

h2(x, y)(t) =µ2

1− ν2µ2

(Tµ1(ν1 + ν2)

1− ν1µ1+ t)∫ T

0

(T − s)α−2

Γ(α− 1)f(s)ds

+ν2

1− ν2µ2

(Tµ1(ν1µ2 + 1)

1− ν1µ1+ µ2t

)∫ T

0

(T − s)β−2

Γ(β − 1)g(s)ds


+ν1µ1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)g(s)ds+

µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)f(s)ds

+

∫ t

0

(t− s)β−1

Γ(β)g(s)ds.

Then we have

|h1(x, y)(t)| ≤ |µ2|T|1− ν2µ2|

( |ν1|(|µ1||ν2|+ 1)

|1− ν1µ1|+ |ν2|

)∫ T

0

(T − s)α−2

Γ(α− 1)|f(s)|ds

+|ν2|T

|1− ν2µ2|

( (|µ1|+ |µ2|)|ν1||1− ν1µ1|

+ 1)∫ T

0

(T − s)β−2

Γ(β − 1)|g(s)|ds

+|ν1|

|1− ν1µ1|

∫ T

0

(T − s)β−1

Γ(β)|g(s)|ds

+|ν1||µ1||1− ν1µ1|

∫ T

0

(T − s)α−1

Γ(α)|f(s)|ds+

∫ t

0

(t− s)α−1

Γ(α)|f(s)|ds

≤ |µ2||1− ν2µ2|

( |ν1|(|µ1||ν2|+ 1)

|1− ν1µ1|+ |ν2|

) Tα

Γ(α)‖p1‖(ψ1(r) + φ1(r))

+|ν2|

|1− ν2µ2|

( (|µ1|+ |µ2|)|ν1||1− ν1µ1|

+ 1) T β

Γ(β)‖p2‖(ψ2(r) + φ2(r))

+|ν1|

|1− ν1µ1|T β

Γ(β + 1)‖p2‖(ψ2(r) + φ2(r))

+|ν1||µ1||1− ν1µ1|

Tα

Γ(α+ 1)‖p1‖(ψ1(r) + φ1(r))

+Tα

Γ(α+ 1)‖p1‖(ψ1(r) + φ1(r))

= M1‖p1‖(ψ1(r) + φ1(r)) +M2‖p2‖(ψ2(r) + φ2(r))

and

|h2(x, y)(t)| ≤ |µ2|T|1− ν2µ2|

( |µ1|(|ν1|+ |ν2|)|1− ν1µ1|

+ 1) Tα

Γ(α)‖p1‖(ψ1(r) + φ1(r))

+|ν2|T

|1− ν2µ2|

( |µ1|(|ν1||µ2|+ 1)

|1− ν1µ1|+ |µ2|

) T β

Γ(β)‖p2‖(ψ2(r) + φ2(r))

+|ν1||µ1||1− ν1µ1|

T β

Γ(β + 1)‖p2‖(ψ2(r) + φ2(r))

+|µ1|

|1− ν1µ1|Tα

Γ(α+ 1)‖p1‖(ψ1(r) + φ1(r))

+T β

Γ(β + 1)‖p2‖(ψ2(r) + φ2(r))

= M3‖p1‖(ψ1(r) + φ1(r)) +M4‖p2‖(ψ2(r) + φ2(r)).

Thus,

‖h1(x, y)‖ ≤M1‖p1‖(ψ1(r) + φ1(r)) +M2‖p2‖(ψ2(r) + φ2(r)),

‖h2(x, y)‖ ≤M3‖p1‖(ψ1(r) + φ1(r)) +M4‖p2‖(ψ2(r) + φ2(r)).


Hence we obtain

‖(h1, h2)‖ = ‖h1(x, y)‖+ ‖h2(x, y)‖≤ (M1 +M3)‖p1‖(ψ1(r) + φ1(r)) + (M2 +M4)‖p2‖(ψ2(r) + φ2(r))

= ` (a constant).

Next, we show that K is equicontinuous. Let t1, t2 ∈ [0, T ] with t1 < t2. Then,there exist f ∈ SF,(x,y), g ∈ SG,(x,y) such that

h1(x, y)(t) =µ2

1− ν2µ2

(ν1T (µ1ν2 + 1)

1− ν1µ1+ ν2t

)∫ T

0

(T − s)α−2

Γ(α− 1)f(s)ds

+ν2

1− ν2µ2

(T (µ1 + µ2)ν11− ν1µ1

+ t)∫ T

0

(T − s)β−2

Γ(β − 1)g(s)ds

+ν1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)g(s)ds+

ν1µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)f(s)ds

+

∫ t

0

(t− s)α−1

Γ(α)f(s)ds

and

h2(x, y)(t) =µ2

1− ν2µ2

(Tµ1(ν1 + ν2)

1− ν1µ1+ t)∫ T

0

(T − s)α−2

Γ(α− 1)f(s)ds

+ν2

1− ν2µ2

(Tµ1(ν1µ2 + 1)

1− ν1µ1+ µ2t

)∫ T

0

(T − s)β−2

Γ(β − 1)g(s)ds

+ν1µ1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)g(s)ds+

µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)f(s)ds

+

∫ t

0

(t− s)β−1

Γ(β)g(s)ds.

Then we have

|h1(x, y)(t2)− h1(x, y)(t1)|

≤ Tα‖p1‖(ψ1(r) + φ1(r))|ν2||µ2||1− ν2µ2|Γ(α)

(t2 − t1) +T β‖p2‖(ψ2(r) + φ2(r))|ν2|

|1− ν2µ2|Γ(β)(t2 − t1)

+ ‖p1‖(ψ1(r) + φ1(r))∣∣ 1

Γ(α)

∫ t2

0

(t2 − s)α−1ds−1

Γ(α)

∫ t1

0

(t1 − s)α−1ds∣∣

≤[Tα‖p1‖(ψ1(r) + φ1(r))|ν2||µ2|

|1− ν2µ2|Γ(α)+T β‖p2‖(ψ2(r) + φ2(r))|ν2|

|1− ν2µ2|Γ(β)

](t2 − t1)

+‖p1‖(ψ1(r) + φ1(r))

Γ(α+ 1)[2(t2 − t1)α + |tα2 − tα1 |].

Analogously, we can obtain

|h2(x, y)(t2)− h2(x, y)(t1)|

≤[Tα‖p1‖(ψ1(r) + φ1(r))|µ2|

|1− ν2µ2|Γ(α)+T β‖p2‖(ψ2(r) + φ2(r))|ν2||µ2|

|1− ν2µ2|Γ(β)

](t2 − t1)

+‖p2‖(ψ2(r) + φ2(r))

Γ(β + 1)[2(t2 − t1)β + |tβ2 − t

β1 |].


From the foregoing arguments, it follows that the operator K(x, y) is equicontinu-ous, and so it is completely continuous by the Ascoli-Arzela theorem.

In our next step, we show that the operator K(x, y) has a closed graph. Let(xn, yn) → (x∗, y∗), (hn, hn) ∈ K(xn, yn) and (hn, hn) → (h∗, h∗), then we need toshow (h∗, h∗) ∈ K(x∗, y∗). Observe that (hn, hn) ∈ K(xn, yn) implies that thereexist fn ∈ SF,(xn,yn) and gn ∈ SG,(xn,yn) such that

hn(xn, yn)(t) =µ2

1− ν2µ2

(ν1T (µ1ν2 + 1)

1− ν1µ1+ ν2t

)∫ T

0

(T − s)α−2

Γ(α− 1)fn(s)ds

+ν2

1− ν2µ2

(T (µ1 + µ2)ν11− ν1µ1

+ t)∫ T

0

(T − s)β−2

Γ(β − 1)gn(s)ds

+ν1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)gn(s)ds

+ν1µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)fn(s)ds+

∫ t

0

(t− s)α−1

Γ(α)fn(s)ds

and

hn(xn, yn)(t) =µ2

1− ν2µ2

(Tµ1(ν1 + ν2)

1− ν1µ1+ t)∫ T

0

(T − s)α−2

Γ(α− 1)fn(s)ds

+ν2

1− ν2µ2

(Tµ1(ν1µ2 + 1)

1− ν1µ1+ µ2t

)∫ T

0

(T − s)β−2

Γ(β − 1)gn(s)ds

+ν1µ1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)gn(s)ds

+µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)fn(s)ds+

∫ t

0

(t− s)β−1

Γ(β)gn(s)ds.

Let us consider the continuous linear operators Φ1,Φ2 : L1([0, T ], X × X) →C([0, T ], X ×X) given by

Φ1(x, y)(t) =µ2

1− ν2µ2

(ν1T (µ1ν2 + 1)

1− ν1µ1+ ν2t

)∫ T

0

(T − s)α−2

Γ(α− 1)f(s)ds

+ν2

1− ν2µ2

(T (µ1 + µ2)ν11− ν1µ1

+ t)∫ T

0

(T − s)β−2

Γ(β − 1)g(s)ds

+ν1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)g(s)ds+

ν1µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)f(s)ds

+

∫ t

0

(t− s)α−1

Γ(α)f(s)ds

and

Φ2(x, y)(t) =µ2

1− ν2µ2

(Tµ1(ν1 + ν2)

1− ν1µ1+ t)∫ T

0

(T − s)α−2

Γ(α− 1)f(s)ds

+ν2

1− ν2µ2

(Tµ1(ν1µ2 + 1)

1− ν1µ1+ µ2t

)∫ T

0

(T − s)β−2

Γ(β − 1)g(s)ds

+ν1µ1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)g(s)ds+

µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)f(s)ds


+

∫ t

0

(t− s)β−1

Γ(β)g(s)ds.

From Lemma 3.2, we know that (Φ1,Φ2)◦(SF , SG) is a closed graph operator. Fur-ther, we have (hn, hn) ∈ (Φ1,Φ2)◦(SF,(xn,yn), SG,(xn,yn)) for all n. Since (xn, yn)→(x∗, y∗), (hn, hn) → (h∗, h∗), it follows that f∗ ∈ SF,(x,y) and g∗ ∈ SG,(x,y) suchthat

h∗(x∗, y∗)(t) =µ2

1− ν2µ2

(ν1T (µ1ν2 + 1)

1− ν1µ1+ ν2t

)∫ T

0

(T − s)α−2

Γ(α− 1)f∗(s)ds

+ν2

1− ν2µ2

(T (µ1 + µ2)ν11− ν1µ1

+ t)∫ T

0

(T − s)β−2

Γ(β − 1)g∗(s)ds

+ν1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)g∗(s)ds

+ν1µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)f∗(s)ds+

∫ t

0

(t− s)α−1

Γ(α)f∗(s)ds

and

h∗(x∗, y∗)(t) =µ2

1− ν2µ2

(Tµ1(ν1 + ν2)

1− ν1µ1+ t)∫ T

0

(T − s)α−2

Γ(α− 1)f∗(s)ds

+ν2

1− ν2µ2

(Tµ1(ν1µ2 + 1)

1− ν1µ1+ µ2t

)∫ T

0

(T − s)β−2

Γ(β − 1)g∗(s)ds

+ν1µ1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)g∗(s)ds

+µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)f∗(s)ds+

∫ t

0

(t− s)β−1

Γ(β)g∗(s)ds,

that is, (hn, hn) ∈ K(x∗, y∗).Finally, we discuss a priori bounds on solutions. Let (x, y) ∈ νK(x, y) for ν ∈

(0, 1). Then there exist f ∈ SF,(x,y) and g ∈ SG,(x,y) such that

x(t) = νµ2

1− ν2µ2

(ν1T (µ1ν2 + 1)

1− ν1µ1+ ν2t

)∫ T

0

(T − s)α−2

Γ(α− 1)f(s)ds

+ νν2

1− ν2µ2

(T (µ1 + µ2)ν11− ν1µ1

+ t)∫ T

0

(T − s)β−2

Γ(β − 1)g(s)ds

+ νν1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)g(s)ds+

ν1µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)f(s)ds

+ ν

∫ t

0

(t− s)α−1

Γ(α)f(s)ds

and

y(t) = νµ2

1− ν2µ2

(Tµ1(ν1 + ν2)

1− ν1µ1+ t)∫ T

0

(T − s)α−2

Γ(α− 1)f(s)ds

+ νν2

1− ν2µ2

(Tµ1(ν1µ2 + 1)

1− ν1µ1+ µ2t

)∫ T

0

(T − s)β−2

Γ(β − 1)g(s)ds

+ νν1µ1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)g(s)ds+

µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)f(s)ds


+ ν

∫ t

0

(t− s)β−1

Γ(β)g(s)ds.

Using the arguments employed in Step 2, for each t ∈ [0, T ], we obtain

‖x‖ ≤M1‖p1‖(ψ1(‖x‖) + φ1(‖y‖)) +M2‖p2‖(ψ2(‖x‖) + φ2(‖y‖)),and

‖y‖ ≤M3‖p1‖(ψ1(‖x‖) + φ1(‖y‖)) +M4‖p2‖(ψ2(‖x‖) + φ2(‖y‖)).In consequence, we have

‖(x, y)‖ = ‖x‖+ ‖y‖≤ (M1 +M3)‖p1‖(ψ1(‖x‖) + φ1(‖y‖))

+ (M2 +M4)‖p2‖(ψ2(‖x‖) + φ2(‖y‖)),which implies that

‖(x, y)‖(M1 +M3)‖p1‖(ψ1(‖x‖) + φ1(‖y‖)) + (M2 +M4)‖p2‖(ψ2(‖x‖) + φ2(‖y‖))

≤ 1.

In view of (H3), there exists N such that ‖(x, y)‖ 6= N . Let us set

U = {(x, y) ∈ X ×X : ‖(x, y)‖ < N}.

Note that the operator K : U → Pcp,cv(X) × Pcp,cv(X) is upper semicontinuousand completely continuous. From the choice of U , there is no (x, y) ∈ ∂U such that(x, y) ∈ νK(x, y) for some ν ∈ (0, 1). Consequently, by the nonlinear alternative ofLeray-Schauder type [19], we deduce that K has a fixed point (x, y) ∈ U which is asolution of the problem (1.1)-(1.2). This completes the proof. �

3.2. The lower semi-continuous case. Here we discuss the case when F andG are not necessarily convex valued by applying the nonlinear alternative of LeraySchauder type together with a selection theorem due to Bressan and Colombo [10]for lower semi-continuous maps with decomposable values. Before presenting ourmain result in this section, we recall some definitions.

(i) A1 ⊂ [0, T ]×R is L⊗D measurable if A1 belongs to the σ−algebra generatedby all sets of the form J × D, where J is Lebesgue measurable in [0, T ]and D is Borel measurable in R.

(ii) A2 ⊂ L1([0, T ],R) is decomposable if, for all u, v ∈ A2 and measurableJ ⊂ [0, T ] = J , the function uχJ + vχJ−J ∈ A2, where χJ stands for thecharacteristic function of J .

Theorem 3.5. Assume that (H2), (H3) and the following condition hold:

(H4) F,G : [0, T ] × R2 → P(R) are nonempty compact-valued multivalued mapssuch that(a) (t, x, y) 7→ F (t, x, y) and (t, x, y) 7→ G(t, x, y) are L⊗D⊗D measurable,(b) (x, y) 7→ F (t, x, y) and (x, y) 7→ G(t, x, y) are lower semicontinuous

for a.e. t ∈ [0, T ].

Then the system (1.1)-(1.2) has at least one solution on [0, T ].

Proof. It follows from (H2) and (H4) that the maps

F1 : X → P(L1([0, T ],R)), x→ F1(x, y) = SF,(x,y),

F2 : X → P(L1([0, T ],R)), y → F2(x, y) = SG,(x,y),


are lower semicontinuous and have nonempty closed and decomposable values.Then, by selection theorem due to Bressan and Colombo, there exist continuousfunctions f : X → L1([0, T ],R) and g : X → L1([0, T ],R) such that f ∈ F1(x, y)and g ∈ F2(x, y) for all x, y ∈ X. Thus we have f(t, x(t), y(t)) ∈ F (t, x(t), y(t))and g(t, x(t), y(t)) ∈ G(t, x(t), y(t)) for a.e. t ∈ [0, T ].

Consider the problem

cDαx(t) = f(t, x(t), y(t)), t ∈ [0, T ], 1 < α ≤ 2,

cDβy(t) = g(t, x(t), y(t)), t ∈ [0, T ], 1 < β ≤ 2,(3.7)

subject to the coupled boundary conditions (1.2).Obviously, if (x, y) ∈ X ×X is a solution of the system (3.7) with the boundary

conditions (1.2), then (x, y) is a solution to the problem (1.1)-(1.2). In order totransform the problem (3.7)-(1.2) into a fixed point problem, we define the operatorK : X ×X → X ×X by

K(x, y)(t) =

(K1(x, y)(t)K2(x, y)(t)

),

where

K1(x, y)(t) =µ2

1− ν2µ2

(ν1T (µ1ν2 + 1)

1− ν1µ1+ ν2t

)∫ T

0

(T − s)α−2

Γ(α− 1)f(t, x(t), y(t))ds

+ν2

1− ν2µ2

(T (µ1 + µ2)ν11− ν1µ1

+ t)∫ T

0

(T − s)β−2

Γ(β − 1)g(t, x(t), y(t))ds

+ν1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)g(t, x(t), y(t))ds

+ν1µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)f(t, x(t), y(t))ds

+

∫ t

0

(t− s)α−1


and

K2(x, y)(t) =µ2

1− ν2µ2

(Tµ1(ν1 + ν2)

1− ν1µ1+ t)∫ T

0

(T − s)α−2

Γ(α− 1)f(t, x(t), y(t))ds

+ν2

1− ν2µ2

(Tµ1(ν1µ2 + 1)

1− ν1µ1+ µ2t

)∫ T

0

(T − s)β−2

Γ(β − 1)g(t, x(t), y(t))ds

+ν1µ1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)g(t, x(t), y(t))ds

+µ1

1− ν1µ1

∫ T

0

(T − s)α−1


+

∫ t

0

(t− s)β−1

Γ(β)g(t, x(t), y(t))ds.

It can easily be shown that K is continuous and completely continuous. The re-maining part of the proof is similar to that of Theorem 3.4. So we omit it. Thiscompletes the proof. �


3.3. The Lipschitz case. This subsection is concerned with the case when themultivalued maps in the system (1.1) have nonconvex values. Let us first recallsome auxiliary material.

Let (X, d) be a metric space induced from the normed space (X; ‖ · ‖) and letHd : P(X)× P(X)→ R ∪ {∞} be defined by

Hd(U, V ) = max{supu∈U

d(u, V ), supv∈V

d(U, v)},

where d(U, v) = infu∈U d(u, v) and d(u, V ) = infv∈V d(u, v). Then (Pb,cl(X), Hd) isa metric space and (Pcl(X), Hd) is a generalized metric space (see [25]).

Definition 3.6. A multivalued operator G : X → Pcl(X) is called (i) γ−Lipschitz ifand only if there exists γ > 0 such that Hd(G(a),G(b)) ≤ γd(a, b) for each a, b ∈ X;and (ii) a contraction if and only if it is γ-Lipschitz with γ < 1.

Lemma 3.7 (Covitz and Nadler [13]). Let (X, d) be a complete metric space. IfG : X → Pcl(X) is a contraction, then the fixed point set of G is not empty.

Theorem 3.8. Assume that the following conditions hold:

(H5) F,G : [0, T ] × R2 → Pcp(R) are such that F (·, x, y) : [0, T ] → Pcp(R) andG(·, x, y) : [0, T ]→ Pcp(R) are measurable for all x, y ∈ R;

(H6)

Hd(F (t, x, y), F (t, x, y) ≤ m1(t)(|x− x|+ |y − y|)and

Hd(G(t, x, y), G(t, x, y) ≤ m2(t)(|x− x|+ |y − y|)for almost all t ∈ [0, T ] and x, y, x, y ∈ R with m1,m2 ∈ C([0, T ],R+) andd(0, F (t, 0, 0)) ≤ m1(t), d(0, G(t, 0, 0)) ≤ m2(t) for almost all t ∈ [0, T ].

Then the problem (1.1)-(1.2) has at least one solution on [0, T ] if

(M1 +M3)‖m1‖+ (M2 +M4)‖m2‖ < 1, (3.8)

where Mi (i = 1, 2, 3, 4) are given by (3.3)-(3.6).

Proof. Observe that the sets SF,(x,y) and SG,(x,y) are nonempty for each (x, y) ∈X × Y by the assumption (H5), so F and G have measurable selections (see [12,Theorem III.6]). Now we show that the operator K satisfies the hypothesis ofLemma 3.7.

First we show that K(x, y) ∈ Pcl(X) × Pcl(X) for each (x, y) ∈ X × X. Let(hn, hn) ∈ K(xn, yn) such that (hn, hn) → (h, h) in X ×X. Then (h, h) ∈ X ×Xand there exist fn ∈ SF,(xn,yn) and gn ∈ SG,(xn,yn) such that

hn(xn, yn)(t) =µ2

1− ν2µ2

(ν1T (µ1ν2 + 1)

1− ν1µ1+ ν2t

)∫ T

0

(T − s)α−2

Γ(α− 1)fn(s)ds

+ν2

1− ν2µ2

(T (µ1 + µ2)ν11− ν1µ1

+ t)∫ T

0

(T − s)β−2

Γ(β − 1)gn(s)ds

+ν1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)gn(s)ds

+ν1µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)fn(s)ds+

∫ t

0

(t− s)α−1

Γ(α)fn(s)ds


and

hn(xn, yn)(t) =µ2

1− ν2µ2

(Tµ1(ν1 + ν2)

1− ν1µ1+ t)∫ T

0

(T − s)α−2

Γ(α− 1)fn(s)ds

+ν2

1− ν2µ2

(Tµ1(ν1µ2 + 1)

1− ν1µ1+ µ2t

)∫ T

0

(T − s)β−2

Γ(β − 1)gn(s)ds

+ν1µ1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)gn(s)ds

+µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)fn(s)ds+

∫ t

0

(t− s)β−1

Γ(β)gn(s)ds.

Since F and G have compact values, we pass onto a subsequences (denoted assequences) to get that fn and gn converge to f and g in L1([0, T ],R) respectively.Thus f ∈ SF,(x,y) and g ∈ SG,(x,y) for each t ∈ [0, T ] and

hn(xn, yn)(t)→ h(x, y)(t)

=µ2

1− ν2µ2

(ν1T (µ1ν2 + 1)

1− ν1µ1+ ν2t

)∫ T

0

(T − s)α−2

Γ(α− 1)f(s)ds

+ν2

1− ν2µ2

(T (µ1 + µ2)ν11− ν1µ1

+ t)∫ T

0

(T − s)β−2

Γ(β − 1)g(s)ds

+ν1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)gn(s)ds+

ν1µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)f(s)ds

+

∫ t

0

(t− s)α−1

Γ(α)f(s)ds

and

hn(xn, yn)(t)→ h(x, y)(t)

=µ2

1− ν2µ2

(Tµ1(ν1 + ν2)

1− ν1µ1+ t)∫ T

0

(T − s)α−2

Γ(α− 1)f(s)ds

+ν2

1− ν2µ2

(Tµ1(ν1µ2 + 1)

1− ν1µ1+ µ2t

)∫ T

0

(T − s)β−2

Γ(β − 1)g(s)ds

+ν1µ1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)gn(s)ds+

µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)f(s)ds

+

∫ t

0

(t− s)β−1

Γ(β)g(s)ds.

Hence (h, h) ∈ K, which implies that K is closed.Next we show that there exists γ < 1 such that

Hd(K(x, y),K(x, y)) ≤ γ(‖x− x‖+ ‖y − y‖) for all x, x, y, y ∈ X.Let (x, x), (y, y) ∈ X × X and (h1, h1) ∈ K(x, y). Then there exists f1 ∈ SF,(x,y)and g1 ∈ SG,(x,y) such that, for each t ∈ [0, T ], we have

h1(x, y)(t) =µ2

1− ν2µ2

(ν1T (µ1ν2 + 1)

1− ν1µ1+ ν2t

)∫ T

0

(T − s)α−2

Γ(α− 1)f1(s)ds

+ν2

1− ν2µ2

(T (µ1 + µ2)ν11− ν1µ1

+ t)∫ T

0

(T − s)β−2

Γ(β − 1)g1(s)ds


+ν1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)g1(s)ds+

ν1µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)f1(s)ds

+

∫ t

0

(t− s)α−1

Γ(α)f1(s)ds

and

h1(x, y)(t) =µ2

1− ν2µ2

(Tµ1(ν1 + ν2)

1− ν1µ1+ t)∫ T

0

(T − s)α−2

Γ(α− 1)f1(s)ds

+ν2

1− ν2µ2

(Tµ1(ν1µ2 + 1)

1− ν1µ1+ µ2t

)∫ T

0

(T − s)β−2

Γ(β − 1)g1(s)ds

+ν1µ1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)g1(s)ds+

µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)f1(s)ds

+

∫ t

0

(t− s)β−1

Γ(β)g1(s)ds.

By (H6), we have

Hd(F (t, x, y), F (t, x, y) ≤ m1(t)(|x(t)− x(t)|+ |y(t)− y(t)|),

and

Hd(G(t, x, y), G(t, x, y) ≤ m2(t)(|x(t)− x(t)|+ |y(t)− y(t)|).So, there exists f ∈ F (t, x(t), y(t)) and g ∈ G(t, x(t), y(t)) such that

|f1(t)− f | ≤ m1(t)(|x(t)− x(t)|+ |y(t)− y(t)|),|g1(t)− g| ≤ m2(t)(|x(t)− x(t)|+ |y(t)− y(t)|).

Define V1, V2 : [0, T ]→ P(R) by

V1(t) = {f ∈ L1([0, T ],R) : |f1(t)− f | ≤ m1(t)(|x(t)− x(t)|+ |y(t)− y(t)|)},V2(t) = {g ∈ L1([0, T ],R) : |g1(t)− g| ≤ m2(t)(|x(t)− x(t)|+ |y(t)− y(t)|)}.

Since the multivalued operators V1(t) ∩ F (t, x(t), y(t)) and V2(t) ∩ G(t, x(t), y(t))are measurable [12, Proposition III.4], there exists functions f2(t), g2(t) which aremeasurable selections for V1, V2 and f2(t) ∈ F (t, x(t), y(t)), g2(t) ∈ G(t, x(t), y(t))such that, for a.e. t ∈ [0, T ], we have

|f1(t)− f2(t)| ≤ m1(t)(|x(t)− x(t)|+ |y(t)− y(t)|),|g1(t)− g2(t)| ≤ mg(t)(|x(t)− x(t)|+ |y(t)− y(t)|).

Let

h2(x, y)(t) =µ2

1− ν2µ2

(ν1T (µ1ν2 + 1)

1− ν1µ1+ ν2t

)∫ T

0

(T − s)α−2

Γ(α− 1)f2(s)ds

+ν2

1− ν2µ2

(T (µ1 + µ2)ν11− ν1µ1

+ t)∫ T

0

(T − s)β−2

Γ(β − 1)g2(s)ds

+ν1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)g2(s)ds

+ν1µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)f2(s)ds+

∫ t

0

(t− s)α−1

Γ(α)f2(s)ds


and

h2(x, y)(t) =µ2

1− ν2µ2

(Tµ1(ν1 + ν2)

1− ν1µ1+ t)∫ T

0

(T − s)α−2

Γ(α− 1)f2(s)ds

+ν2

1− ν2µ2

(Tµ1(ν1µ2 + 1)

1− ν1µ1+ µ2t

)∫ T

0

(T − s)β−2

Γ(β − 1)g2(s)ds

+ν1µ1

1− ν1µ1

∫ T

0

(T − s)β−1

Γ(β)g2(s)ds

+µ1

1− ν1µ1

∫ T

0

(T − s)α−1

Γ(α)f2(s)ds+

∫ t

0

(t− s)β−1

Γ(β)g2(s)ds.

Then

|h1(x, y)(t)− h2(x, y)(t)|

≤ |µ2|T|1− ν2µ2|

( |ν1|(|µ1||ν2|+ 1)

|1− ν1µ1|+ |ν2|

)∫ T

0

(T − s)α−2

Γ(α− 1)|f1(s)− f2(s)|ds

+|ν2|T

|1− ν2µ2|

( (|µ1|+ |µ2|)|ν1||1− ν1µ1|

+ 1)∫ T

0

(T − s)β−2

Γ(β − 1)|g1(s)− g2(s)|ds

+|ν1|

|1− ν1µ1|

∫ T

0

(T − s)β−1

Γ(β)|g1(s)− g2(s)|ds

+|ν1||µ1||1− ν1µ1|

∫ T

0

(T − s)α−1

Γ(α)|f1(s)− f2(s)|ds

+

∫ t

0

(t− s)α−1

Γ(α)|f1(s)− f2(s)|ds

≤ |µ2|T|1− ν2µ2|

( |ν1|(|µ1||ν2|+ 1)

|1− ν1µ1|+ |ν2|

)×∫ T

0

(T − s)α−2

Γ(α− 1)m1(s)(|x(s)− x(s)|+ |y(s)− y(s)|)ds

+|ν2|T

|1− ν2µ2|

(T (µ1 + µ2)ν11− ν1µ1

+ t)

×∫ T

0

(T − s)β−2

Γ(β − 1)m2(s)(|x(s)− x(s)|+ |y(s)− y(s)|)ds

+|ν1|

|1− ν1µ1|

∫ T

0

(T − s)β−1

Γ(β)m2(s)(|x(s)− x(s)|+ |y(s)− y(s)|)ds

+|ν1||µ1||1− ν1µ1|

∫ T

0

(T − s)α−1

Γ(α)m1(s)(|x(s)− x(s)|+ |y(s)− y(s)|)ds

+

∫ t

0

(t− s)α−1

Γ(α)m1(s)(|x(s)− x(s)|+ |y(s)− y(s)|)ds

≤ (M1‖m1‖+M2‖m2‖)(‖x− x‖+ ‖y − y‖),

which implies

‖h1(x, y)− h2(x, y)‖ ≤ (M1‖m1‖+M2‖m2‖)(‖x− x‖+ ‖y − y‖).


In a similar manner, one can find that

‖h1(x, y)− h2(x, y)‖ ≤ (M3‖m1‖+M4‖m2‖)(‖x− x‖+ ‖y − y‖).Thus

‖(h1, h1), (h2, h2)‖ ≤ [(M1 +M3)‖m1‖+ (M2 +M4)‖m2‖](‖x− x‖+ ‖y − y‖).Analogously, interchanging the roles of (x, y) and (x, y), we can obtain

Hd(K(x, y),K(x, y)) ≤ [(M1 +M3)‖m1‖+ (M2 +M4)‖m2‖](‖x− x‖+ ‖y − y‖).Therefore K is a contraction in view of the assumption (3.8). Hence it follows byLemma 3.7 that K has a fixed point (x, y), which is a solution of problem (1.1)-(1.2).This completes the proof. �

Special Cases. Several new existence results follow as special cases from the workpresented in this paper by fixing the parameters involved in the problem. Forinstance, if we choose ν1 = 1 = ν2 and µ1 = −1 = µ2 or ν1 = −1 = ν2 and µ1 =1 = µ2, we obtain the results for coupled fractional differential inclusions equippedwith a combination of coupled periodic and anti-periodic boundary conditions ofthe form: x(0) = y(T ), x′(0) = y′(T ), y(0) = −x(T ), y′(0) = −x′(T ) or x(0) =−y(T ), x′(0) = −y′(T ), y(0) = x(T ), y′(0) = x′(T ). For ν1 = 1 = −ν2 and−µ1 = 1 = µ2, our results correspond to a coupled system of fractional differentialinclusions with the boundary conditions of the form: x(0) = y(T ), x′(0) = −y′(T ),y(0) = −x(T ), y′(0) = x′(T ), while the results for a coupled system of fractionaldifferential inclusions supplemented with the boundary conditions: x(0) = −y(T ),x′(0) = y′(T ), y(0) = x(T ), y′(0) = −x′(T ) follow by setting −ν1 = 1 = ν2 andµ1 = 1 = −µ2 in the results of this paper. Letting ν1 = 0 = µ1 in the results of thispaper, we obtain the ones for a coupled system of fractional differential inclusionsequipped with coupled flux type conditions.

Acknowledgements. This project was funded by the Deanship of Scientific Re-search (DSR) at King Abdulaziz University, Jeddah, under grant no. (RG-1-130-39). The authors, therefore, acknowledge with thanks DSR technical and financialsupport. The authors also thank the editor and the referee for their useful sugges-tions.

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Bashir AhmadNonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Sci-

ence, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Email address: bashirahmad [email protected]

Sotiris K. Ntouyas

Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece.Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of

Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah21589, Saudi Arabia

Email address: [email protected]

Ahmed AlsaediNonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Sci-

ence, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Email address: [email protected]

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