Open Access

Existence of solutions for fractional q-integro-difference inclusions with fractional q-integral boundary conditions

Advances in Difference Equations20142014:257

https://doi.org/10.1186/1687-1847-2014-257

Received: 15 July 2014

Accepted: 30 September 2014

Published: 13 October 2014

Abstract

In this paper, we investigate the existence of solutions for a nonlocal boundary value problem of fractional q-integro-difference inclusions of two fractional orders with Riemann-Liouville fractional q-integral boundary conditions. A new existence result is obtained by making use of a nonlinear alternative for contractive maps and is well illustrated with the aid of an example.

MSC:34A60, 34A08.

Keywords

fractional differential inclusionsnonlocal boundary conditionsfixed point theorems

1 Introduction

Nonlocal nonlinear boundary value problems of fractional order have been extensively investigated in recent years. Several results of interest ranging from theoretical analysis to asymptotic behavior and numerical methods for fractional differential equations are available in the literature on the topic. The introduction of fractional derivative in the mathematical modeling of many real world phenomena has played a key role in improving the integer-order mathematical models. One of the important factors accounting for the popularity of the subject is that differential operators of fractional-order help to understand the hereditary phenomena in many materials and processes in a better way than the corresponding integer-order differential operators. For examples and details in physics, chemistry, biology, biophysics, blood flow phenomena, control theory, wave propagation, signal and image processing, viscoelasticity, identification, fitting of experimental data, economics, etc., we refer the reader to the texts [14]. Some recent work on fractional differential equations can be found in a series of papers [517] and the references cited therein.

Fractional q-difference equations, known as fractional analogue of q-difference equations, have recently been discussed by several researchers. For some recent work on the topic, see [1832]. In a recent paper [33], the authors obtained some existence results for the Langevin type q-difference (integral) equation with two fractional orders and four-point nonlocal integral boundary conditions.

Initial and boundary value problems involving multivalued maps have been studied by many researchers. In fact, the multivalued (inclusions) problems appear in the mathematical modeling of a variety of problems in economics, optimal control, etc. and are widely studied by many authors, see [3436] and the references therein. Recent works on fractional-order multivalued problems [3743] clearly indicate the interest in the subject. In [44] the authors studied the existence of solutions for a problem of nonlinear fractional q-difference inclusions with nonlocal Robin (separated) conditions given by
D q α c x ( t ) F ( t , x ( t ) ) , 0 t 1 , 1 < q 2 , α 1 x ( 0 ) β 1 D q x ( 0 ) = γ 1 x ( η 1 ) , α 2 x ( 1 ) + β 2 D q x ( 1 ) = γ 2 x ( η 2 ) ,

where D q α c is the fractional q-derivative of the Caputo type, F : [ 0 , 1 ] × R P ( R ) is a multivalued map, P ( R ) is the family of all subsets of and α i , β i , γ i , η i R ( i = 1 , 2 ). However, the study of multivalued problems in the setting of fractional q-difference equations is still at an initial stage and needs to be explored further.

In this paper, motivated by [44], we consider the multivalued analogue of the problem addressed in [33]. Precisely we discuss the existence of solutions for a boundary value problem of fractional q-integro-difference inclusions with fractional q-integral boundary conditions given by
D q β c ( c D q γ + λ ) x ( t ) A F ( t , x ( t ) ) + B I q ξ G ( t , x ( t ) ) , t [ 0 , 1 ] , x ( 0 ) = a I q α 1 x ( η ) , x ( 1 ) = b I q α 1 x ( σ ) , α > 2 , 0 < η , σ < 1 ,
(1.1)
where D q β c denotes the Caputo fractional q-difference operator of order β, 0 < q < 1 , 0 < β 1 , 0 < γ 1 , 0 < ξ < 1 , F , G : [ 0 , 1 ] × R P ( R ) are multivalued maps, P ( R ) is the family of all nonempty subsets of , a, b, A, B, α 1 , β 1 , σ 1 , σ 2 are real constants and
I q α x ( ϱ ) = 0 ϱ ( ϱ q s ) ( α 1 ) Γ q ( α ) x ( s ) d q s ( ϱ = η , σ ) .

Here we emphasize that the multivalued (inclusion) problem at hand is new in the sense that it involves fractional q-integro-difference inclusions of two fractional orders with four-point nonlocal Riemann-Liouville fractional q-integral boundary conditions (in contrast to the problem considered in [44]). Also our method of proof is different from the one employed in [44]. The paper is organized as follows. An existence result for problem (1.1), based on a nonlinear alternative for contractive maps, is established in Section 3. The background material for the problem at hand can be found in the related literature. However, for quick reference, we outline it in Section 2.

2 Preliminaries on fractional q-calculus

Here we recall some definitions and fundamental results on fractional q-calculus [4547].

Let a q-real number denoted by [ a ] q be defined by
[ a ] q = 1 q a 1 q , a R , q R + { 1 } .
The q-analogue of the Pochhammer symbol (q-shifted factorial) is defined as
( a ; q ) 0 = 1 , ( a ; q ) k = i = 0 k 1 ( 1 a q i ) , k N { } .
The q-analogue of the exponent ( x y ) k is
( x y ) ( 0 ) = 1 , ( x y ) ( k ) = j = 0 k 1 ( x y q j ) , k N , x , y R .
The q-gamma function Γ q ( y ) is defined as
Γ q ( y ) = ( 1 q ) ( y 1 ) ( 1 q ) y 1 ,
where y R { 0 , 1 , 2 , } . Observe that Γ q ( y + 1 ) = [ y ] q Γ q ( y ) . For any x , y > 0 , the q-beta function B q ( x , y ) is given by
B q ( x , y ) = 0 1 t ( x 1 ) ( 1 q t ) ( y 1 ) d q t ,
which, in terms of q-gamma function, can be expressed as
B q ( x , y ) = Γ q ( x ) Γ q ( y ) Γ q ( x + y ) .
(2.1)

Definition 2.1 ([45])

Let f be a function defined on [ 0 , 1 ] . The fractional q-integral of the Riemann-Liouville type of order β 0 is ( I q 0 f ) ( t ) = f ( t ) and
I q β f ( t ) : = 0 t ( t q s ) ( β 1 ) Γ q ( β ) f ( s ) d q s = t β ( 1 q ) β k = 0 q k ( q β ; q ) k ( q ; q ) k f ( t q k ) , β > 0 , t [ 0 , 1 ] .
Observe that β = 1 in Definition 2.1 yields q-integral
I q f ( t ) : = 0 t f ( s ) d q s = t ( 1 q ) k = 0 q k f ( t q k ) .

For more details on q-integral and fractional q-integral, see Section 1.3 and Section 4.2 respectively in [47].

Remark 2.2 The q-fractional integration possesses the semigroup property (Proposition 4.3 [47])
I q γ I q β f ( t ) = I q β + γ f ( t ) ; γ , β R + .
(2.2)
Further, it has been shown in Lemma 6 of [46] that
I q β ( x ) ( σ ) = Γ q ( σ + 1 ) Γ q ( β + σ + 1 ) ( x ) ( β + σ ) , 0 < x < a , β R + , σ ( 1 , ) .

Before giving the definition of fractional q-derivative, we recall the concept of q-derivative.

We know that the q-derivative of a function f ( t ) is defined as
( D q f ) ( t ) = f ( t ) f ( q t ) t q t , t 0 , ( D q f ) ( 0 ) = lim t 0 ( D q f ) ( t ) .
Furthermore,
D q 0 f = f , D q n f = D q ( D q n 1 f ) , n = 1 , 2 , 3 , .
(2.3)

Definition 2.3 ([47])

The Caputo fractional q-derivative of order β > 0 is defined by
D q β c f ( t ) = I q β β D q β f ( t ) ,

where β is the smallest integer greater than or equal to β.

Next we recall some properties involving Riemann-Liouville q-fractional integral and Caputo fractional q-derivative (Theorem 5.2 [47]).
I q β c D q β f ( t ) = f ( t ) k = 0 β 1 t k Γ q ( k + 1 ) ( D q k f ) ( 0 + ) , t ( 0 , a ] , β > 0 ;
(2.4)
D q β c I q β f ( t ) = f ( t ) , t ( 0 , a ] , β > 0 .
(2.5)

To define the solution for problem (1.1), we need the following lemma.

Lemma 2.4 ([33])

For a given h C ( [ 0 , 1 ] , R ) , the integral solution of the boundary value problem
D q β c ( c D q γ + λ ) x ( t ) = h ( t ) , t [ 0 , 1 ] , x ( 0 ) = a I q α 1 x ( η ) , x ( 1 ) = b I q α 1 x ( σ ) , α > 2 , 0 < η , σ < 1 ,
(2.6)
is given by
x ( t ) = 0 t ( t q u ) ( γ 1 ) Γ q ( γ ) ( I q β h ( u ) λ x ( u ) ) d q u [ δ 3 t γ δ 4 ] Δ { a 0 η ( η q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) ( I q β h ( u ) λ x ( u ) ) d q u ) d q s } + [ δ 1 t γ δ 2 ] Δ { b 0 σ ( σ q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) ( I q β h ( u ) λ x ( u ) ) d q u ) d q s } [ δ 1 t γ δ 2 ] Δ { 0 1 ( 1 q u ) ( γ 1 ) Γ q ( γ ) ( I q β h ( u ) λ x ( u ) ) d q u } ,
(2.7)
where
δ 1 = a η ( α 1 ) Γ q ( α ) 1 , δ 2 = a η ( α + γ 1 ) Γ q ( γ + 1 ) Γ q ( α + γ ) , δ 3 = b σ ( α 1 ) Γ q ( α ) 1 , δ 4 = b σ ( α + γ 1 ) Γ q ( γ + 1 ) Γ q ( α + γ ) 1
and
Δ = δ 3 δ 2 δ 4 δ 1 0 .

Let C = C ( [ 0 , 1 ] , R ) denote the Banach space of all continuous functions from [ 0 , 1 ] R endowed with the norm defined by x = sup { | x ( t ) | , t [ 0 , 1 ] } . Also by L 1 ( [ 0 , 1 ] , R ) we denote the Banach space of measurable functions x : [ 0 , 1 ] R which are Lebesgue integrable and normed by x L 1 = 0 1 | x ( t ) | d q t .

In order to prove our main existence result, we make use of the following form of the nonlinear alternative for contractive maps [[48], Corollary 3.8].

Theorem 2.5 Let X be a Banach space, and D be a bounded neighborhood of 0 X . Let H 1 : X P c p , c ( X ) (here P c p , c ( X ) denotes the family of all nonempty, compact and convex subsets of X) and H 2 : D ¯ P c p , c ( X ) be two multi-valued operators such that
  1. (a)

    H 1 is a contraction, and

     
  2. (b)

    H 2 is upper semi-continuous (u.s.c.) and compact.

     
Then, if H = H 1 + H 2 , either
  1. (i)

    has a fixed point in D ¯ , or

     
  2. (ii)

    there is a point u D and θ ( 0 , 1 ) with u θ H ( u ) .

     
Definition 2.6 A multivalued map F : [ 0 , 1 ] × R P c p , c ( R ) is said to be L 1 -Carathéodory if
  1. (i)

    t F ( t , x ) is measurable for each x R ,

     
  2. (ii)

    x F ( t , x ) is upper semi-continuous for almost all t [ 0 , 1 ] , and

     
  3. (iii)
    for each real number ρ > 0 , there exists a function h ρ L 1 ( [ 0 , 1 ] , R + ) such that
    F ( t , u ) : = sup { | v | : v F ( t , u ) } h ρ ( t ) , a.e.  t [ 0 , 1 ]
     

for all u R with u ρ .

Denote
S F , x = { v L 1 ( [ 0 , 1 ] , R ) : v ( t ) F ( t , x ( t ) )  a.e.  t [ 0 , 1 ] } .

Lemma 2.7 (Lasota and Opial [49])

Let X be a Banach space. Let F : [ 0 , 1 ] × R P c p , c ( R ) be an L 1 -Carathéodory multivalued map, and let Θ be a linear continuous mapping from L 1 ( [ 0 , 1 ] , R ) to C ( [ 0 , 1 ] , X ) . Then the operator
Θ S F : C ( [ 0 , 1 ] , R ) P c p , c ( C ( [ 0 , 1 ] , R ) ) , x ( Θ S F ) ( x ) = Θ ( S F , x )

is a closed graph operator in C ( [ 0 , 1 ] , R ) × C ( [ 0 , 1 ] , R ) .

3 Existence result

Before presenting the main results, we define the solutions of the boundary value problem (1.1).

Definition 3.1 A function x A C 2 ( [ 0 , 1 ] , R ) is said to be a solution of problem (1.1) if x ( 0 ) = a I q α 1 x ( η ) , x ( 1 ) = b I q α 1 x ( σ ) , and there exist functions f S F , x , g S G , x such that
x ( t ) = 0 t ( t q u ) ( γ 1 ) Γ q ( γ ) ( A 0 u ( u q m ) ( β 1 ) Γ q ( β ) f ( m ) d q m + B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) g ( m ) d q m λ x ( u ) ) d q u [ δ 3 t γ δ 4 ] Δ { a 0 η ( η q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) ( A 0 u ( u q m ) ( β 1 ) Γ q ( β ) × f ( m ) d q m + B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) g ( m ) d q m λ x ( u ) ) d q u ) d q s } + [ δ 1 t γ δ 2 ] Δ { b 0 σ ( σ q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) ( A 0 u ( u q m ) ( β 1 ) Γ q ( β ) × f ( m ) d q m + B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) g ( m ) d q m λ x ( u ) ) d q u ) d q s } [ δ 1 t γ δ 2 ] Δ { 0 1 ( 1 q u ) ( γ 1 ) Γ q ( γ ) ( A 0 u ( u q m ) ( β 1 ) Γ q ( β ) f ( m ) d q m + B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) g ( m ) d q m λ x ( u ) ) d q u } .
In the sequel, we set
χ 1 = | A | { 1 + α 2 Γ q ( β + γ + 1 ) + 1 Γ q ( β + γ + α ) ( α 1 | a | η ( β + γ + α 1 ) + α 2 | b | σ ( β + γ + α 1 ) ) } ,
(3.1)
χ 2 : = 1 Γ q ( γ + 1 ) ( 1 + α 2 ) + 1 Γ q ( γ + α ) ( α 1 | a | η ( γ + α 1 ) + α 2 | b | σ ( γ + α 1 ) ) ,
(3.2)
ω i : = | B | { ( 1 + α 2 ) ( I q ( β + γ ) b i ) ( 1 ) + α 1 | a | ( I q ( β + γ + α 1 ) b i ) ( η ) + α 2 | b | ( I q ( β + γ + α 1 ) b i ) ( σ ) } , i = 1 , 2 ,
(3.3)
and
α 1 = | δ 3 δ 4 | | Δ | , α 2 = | δ 1 δ 2 | | Δ | .

Theorem 3.2 Assume that

(H1) F : [ 0 , 1 ] × R P c p , c ( R ) is an L 1 -Carathéodory multivalued map;

(H2) there exists a function k C ( [ 0 , 1 ] , R + ) such that
H ( F ( t , x ) , F ( t , y ) ) k ( t ) x y a.e.  t [ 0 , 1 ]

for all x , y C ( [ 0 , 1 ] , R ) and χ 1 k < 1 , where χ 1 is given by (3.1) and H denotes the Hausdorff metric;

(H3) G : [ 0 , 1 ] × R P c p , c ( R ) is an L 1 -Carathéodory multivalued map;

(H4) there exist functions b 1 , b 2 L 1 ( [ 0 , 1 ] , R ) and a nondecreasing function ψ : R + ( 0 , ) such that
G ( t , x ) : = sup { | v | : v G ( t , x ) } b 1 ( t ) ψ ( x ) + b 2 ( t ) a.e.  t [ 0 , 1 ]

for all x R ;

(H5) there exists a number M > 0 such that
( 1 χ 1 k | λ | χ 2 ) M χ 1 F 0 + ψ ( M ) ω 1 + ω 2 > 1 ,
(3.4)

where χ 1 , χ 2 , ω i , i = 1 , 2 are given by (3.1), (3.2) and (3.3) respectively, F 0 = 0 1 F ( t , 0 ) d t and χ 1 k + | λ | χ 2 < 1 .

Then problem (1.1) has a solution on [ 0 , 1 ] .

Proof To transform problem (1.1) to a fixed point problem, let us introduce an operator L : C ( [ 0 , 1 ] , R ) P ( C ( [ 0 , 1 ] , R ) ) as
L = L 1 + L 2 ,
with
L 1 ( x ) = { h C ( [ 0 , 1 ] , R ) : h ( t ) = ( F 1 x ) ( t ) } ,
where
( F 1 x ) ( t ) = 0 t ( t q u ) ( γ 1 ) Γ q ( γ ) ( A 0 u ( u q m ) ( β 1 ) Γ q ( β ) f ( m ) d q m λ x ( u ) ) d q u [ δ 3 t γ δ 4 ] Δ { a 0 η ( η q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) ( A 0 u ( u q m ) ( β 1 ) Γ q ( β ) × f ( m ) d q m λ x ( u ) ) d q u ) d q s } + [ δ 1 t γ δ 2 ] Δ { b 0 σ ( σ q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) ( A 0 u ( u q m ) ( β 1 ) Γ q ( β ) × f ( m ) d q m λ x ( u ) ) d q u ) d q s } [ δ 1 t γ δ 2 ] Δ { 0 1 ( 1 q u ) ( γ 1 ) Γ q ( γ ) ( A 0 u ( u q m ) ( β 1 ) Γ q ( β ) f ( m ) d q m λ x ( u ) ) d q u }
for f S F , x and
L 2 ( x ) = { h C ( [ 0 , 1 ] , R ) : h ( t ) = ( F 2 x ) ( t ) } ,
where
( F 2 x ) ( t ) = 0 t ( t q u ) ( γ 1 ) Γ q ( γ ) ( B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) g ( m ) d q m ) d q u [ δ 3 t γ δ 4 ] Δ { a 0 η ( η q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) ( B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) × g ( m ) d q m ) d q u ) d q s } + [ δ 1 t γ δ 2 ] Δ { b 0 σ ( σ q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) ( B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) × g ( m ) d q m ) d q u ) d q s } [ δ 1 t γ δ 2 ] Δ { 0 1 ( 1 q u ) ( γ 1 ) Γ q ( γ ) ( B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) g ( m ) d q m ) d q u }

for g S G , x .

We shall show that the operators L 1 and L 2 satisfy all the conditions of Theorem 2.5 on [ 0 , 1 ] . For the sake of clarity, we split the proof into a sequence of steps and claims.

Step 1. We show that L 1 is a multivalued contraction on C ( [ 0 , 1 ] , R ) .

Let x , y C ( [ 0 , 1 ] , R ) and u 1 L 1 ( x ) . Then u 1 P ( C ( [ 0 , 1 ] , R ) ) and
u 1 ( t ) = 0 t ( t q u ) ( γ 1 ) Γ q ( γ ) ( A 0 u ( u q m ) ( β 1 ) Γ q ( β ) v 1 ( m ) d q m λ x ( u ) ) d q u [ δ 3 t γ δ 4 ] Δ { a 0 η ( η q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) ( A 0 u ( u q m ) ( β 1 ) Γ q ( β ) × v 1 ( m ) d q m λ x ( u ) ) d q u ) d q s } + [ δ 1 t γ δ 2 ] Δ { b 0 σ ( σ q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) ( A 0 u ( u q m ) ( β 1 ) Γ q ( β ) × v 1 ( m ) d q m λ x ( u ) ) d q u ) d q s } [ δ 1 t γ δ 2 ] Δ { 0 1 ( 1 q u ) ( γ 1 ) Γ q ( γ ) ( A 0 u ( u q m ) ( β 1 ) Γ q ( β ) v 1 ( m ) d q m λ x ( u ) ) d q u }
for some v 1 S F , x . Since H ( F ( t , x ) , F ( t , y ) ) k ( t ) x y , there exists w F ( t , y ) such that | v 1 ( t ) w ( t ) | k ( t ) x y . Thus the multivalued operator U is defined by U ( t ) = S F , y K ( t ) , where
K ( t ) = { w R | | v 1 ( t ) w ( t ) | k ( t ) x y }

has nonempty values and is measurable. Let v 2 be a measurable selection for U (which exists by Kuratowski-Ryll-Nardzewski’s selection theorem [50, 51]). Then v 2 F ( t , y ) and | v 1 ( t ) v 2 ( t ) | k ( t ) x y a.e. on [ 0 , 1 ] .

Define
u 2 ( t ) = 0 t ( t q u ) ( γ 1 ) Γ q ( γ ) ( A 0 u ( u q m ) ( β 1 ) Γ q ( β ) v 2 ( m ) d q m λ x ( u ) ) d q u [ δ 3 t γ δ 4 ] Δ { a 0 η ( η q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) ( A 0 u ( u q m ) ( β 1 ) Γ q ( β ) × v 2 ( m ) d q m λ x ( u ) ) d q u ) d q s } + [ δ 1 t γ δ 2 ] Δ { b 0 σ ( σ q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) ( A 0 u ( u q m ) ( β 1 ) Γ q ( β ) × v 2 ( m ) d q m λ x ( u ) ) d q u ) d q s } [ δ 1 t γ δ 2 ] Δ { 0 1 ( 1 q u ) ( γ 1 ) Γ q ( γ ) ( A 0 u ( u q m ) ( β 1 ) Γ q ( β ) v 2 ( m ) d q m λ x ( u ) ) d q u } .
It follows that u 2 L 1 ( y ) and
| u 1 ( t ) u 2 ( t ) | = 0 t ( t q u ) ( γ 1 ) Γ q ( γ ) ( A 0 u ( u q m ) ( β 1 ) Γ q ( β ) | v 1 ( m ) v 2 ( m ) | d q m ) d q u [ δ 3 t γ δ 4 ] Δ { a 0 η ( η q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) ( A 0 u ( u q m ) ( β 1 ) Γ q ( β ) × | v 1 ( m ) v 2 ( m ) | d q m ) d q u ) d q s } + [ δ 1 t γ δ 2 ] Δ { b 0 σ ( σ q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) ( A 0 u ( u q m ) ( β 1 ) Γ q ( β ) × | v 1 ( m ) v 2 ( m ) | d q m ) d q u ) d q s } [ δ 1 t γ δ 2 ] Δ { 0 1 ( 1 q u ) ( γ 1 ) Γ q ( γ ) ( A 0 u ( u q m ) ( β 1 ) Γ q ( β ) | v 1 ( m ) v 2 ( m ) | d q m ) d q u } | A | { 1 + α 2 Γ q ( β + γ + 1 ) + 1 Γ q ( β + γ + α ) ( α 1 | a | η ( β + γ + α 1 ) + α 2 | b | σ ( β + γ + α 1 ) ) } k x y .
Taking the supremum over the interval [ 0 , 1 ] , we obtain
u 1 u 2 | A | { 1 + α 2 Γ q ( β + γ + 1 ) + 1 Γ q ( β + γ + α ) ( α 1 | a | η ( β + γ + α 1 ) + α 2 | b | σ ( β + γ + α 1 ) ) } k x y .
Combining the previous inequality with the corresponding one obtained by interchanging the roles of x and y, we find that
H ( L 1 ( x ) , L 1 ( y ) ) | A | { 1 + α 2 Γ q ( β + γ + 1 ) + 1 Γ q ( β + γ + α ) ( α 1 | a | η ( β + γ + α 1 ) + α 2 | b | σ ( β + γ + α 1 ) ) } k x y
for all x , y C ( [ 0 , 1 ] , R ) . This shows that L 1 is a multivalued contraction as
χ 1 k = | A | { 1 + α 2 Γ q ( β + γ + 1 ) + 1 Γ q ( β + γ + α ) ( α 1 | a | η ( β + γ + α 1 ) + α 2 | b | σ ( β + γ + α 1 ) ) } k < 1 .

Step 2. We shall show that the operator L 2 is u.s.c. and compact. It is well known [[52], Proposition 1.2] that a completely continuous operator having a closed graph is u.s.c. Therefore we will prove that L 2 is completely continuous and has a closed graph. This step involves several claims.

Claim I: L 2 maps bounded sets into bounded sets in C ( [ 0 , 1 ] , R ) .

Let B r = { x C ( [ 0 , 1 ] , R ) : x r } be a bounded set in C ( [ 0 , 1 ] , R ) and u L 2 ( x ) for some x B r . Then we have
| F 2 ( x ) ( t ) | 0 t ( t q u ) ( γ 1 ) Γ q ( γ ) ( | B | 0 u ( u q m ) ( β 1 ) Γ q ( β ) | g ( m ) | d q m + | λ | | x ( u ) | ) d q u + α 1 | a | 0 η ( η q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) × ( | B | 0 u ( u q m ) ( β 1 ) Γ q ( β ) | g ( m ) | d q m + | λ | | x ( u ) | ) d q u ) d q s + α 2 | b | 0 σ ( σ q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) × ( | B | 0 u ( u q m ) ( β 1 ) Γ q ( β ) | g ( m ) | d q m + | λ | | x ( u ) | ) d q u ) d q s + α 2 0 1 ( 1 q u ) ( γ 1 ) Γ q ( γ ) ( | B | 0 u ( u q m ) ( β 1 ) Γ q ( β ) | g ( m ) | d q m + | λ | | x ( u ) | ) d q u 0 t ( t q u ) ( γ 1 ) Γ q ( γ ) × ( | B | 0 u ( u q m ) ( β 1 ) Γ q ( β ) [ b 1 ( m ) ψ ( x ) + b 2 ( m ) ] d q m + | λ | | x ( u ) | ) d q u + α 1 | a | 0 η ( η q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) × ( | B | 0 u ( u q m ) ( β 1 ) Γ q ( β ) [ b 1 ( m ) ψ ( x ) + b 2 ( m ) ] d q m + | λ | | x ( u ) | ) d q u ) d q s + α 2 | b | 0 σ ( σ q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) × ( | B | 0 u ( u q m ) ( β 1 ) Γ q ( β ) [ b 1 ( m ) ψ ( x ) + b 2 ( m ) ] d q m + | λ | | x ( u ) | ) d q u ) d q s + α 2 0 1 ( 1 q u ) ( γ 1 ) Γ q ( γ ) × ( | B | 0 u ( u q m ) ( β 1 ) Γ q ( β ) [ b 1 ( m ) ψ ( x ) + b 2 ( m ) ] d q m + | λ | | x ( u ) | ) d q u ψ ( r ) { 0 1 ( 1 q u ) ( γ 1 ) Γ q ( γ ) ( | B | 0 u ( u q m ) ( β 1 ) Γ q ( β ) b 1 ( m ) d q m ) d q u + α 1 | a | 0 η ( η q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) × ( | B | 0 u ( u q m ) ( β 1 ) Γ q ( β ) b 1 ( m ) d q m ) d q u ) d q s + α 2 | b | 0 σ ( σ q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) × ( | B | 0 u ( u q m ) ( β 1 ) Γ q ( β ) b 1 ( m ) d q m ) d q u ) d q s + α 2 0 1 ( 1 q u ) ( γ 1 ) Γ q ( γ ) ( | B | 0 u ( u q m ) ( β 1 ) Γ q ( β ) b 1 ( m ) d q m ) d q u } + { 0 1 ( 1 q u ) ( γ 1 ) Γ q ( γ ) ( | B | 0 u ( u q m ) ( β 1 ) Γ q ( β ) b 2 ( m ) d q m ) d q u + α 1 | a | 0 η ( η q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) × ( | B | 0 u ( u q m ) ( β 1 ) Γ q ( β ) b 2 ( m ) d q m ) d q u ) d q s + α 2 | b | 0 σ ( σ q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) × ( | B | 0 u ( u q m ) ( β 1 ) Γ q ( β ) b 2 ( m ) d q m ) d q u ) d q s + α 2 0 1 ( 1 q u ) ( γ 1 ) Γ q ( γ ) ( | B | 0 u ( u q m ) ( β 1 ) Γ q ( β ) b 2 ( m ) d q m ) d q u } + r | λ | { 0 1 ( 1 q u ) ( γ 1 ) Γ q ( γ ) d q u + α 1 | a | 0 η ( η q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) d q u ) d q s + α 2 | b | 0 σ ( σ q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) d q u ) d q s + α 2 0 1 ( 1 q u ) ( γ 1 ) Γ q ( γ ) d q u } ψ ( r ) ω 1 + ω 2 + r | λ | χ 2 .
Consequently,
F 2 x ψ ( r ) ω 1 + ω 2 + r | λ | χ 2 ,

and hence L 2 is bounded.

Claim II: L 2 maps bounded sets into equicontinuous sets.

As in the proof of Claim I, let B r be a bounded set and u L 2 ( x ) for some x B r . Let t 1 , t 2 [ 0 , 1 ] with t 1 < t 2 . Then we have
| ( F 2 x ) ( t 2 ) ( F 2 x ) ( t 1 ) | | 0 t 1 | ( t 2 q u ) γ 1 ( t 1 q u ) γ 1 | Γ q ( γ ) × ( | B | 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) [ b 1 ( m ) ψ ( r ) + b 2 ( m ) ] d q m + | λ | r ) d q u + t 1 t 2 ( t 2 q u ) ( γ 1 ) Γ q ( γ ) × ( | B | 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) [ b 1 ( m ) ψ ( r ) + b 2 ( m ) ] d q m + | λ | r ) d q u | + 1 | Δ | { | a | | δ 3 ( t 2 γ t 1 γ ) | 0 η ( η q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) × ( | B | 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) [ b 1 ( m ) ψ ( r ) + b 2 ( m ) ] d q m + | λ | r ) d q u ) d q s + | b | | δ 1 ( t 2 γ t 1 γ ) | 0 σ ( σ q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) × ( | B | 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) [ b 1 ( m ) ψ ( r ) + b 2 ( m ) ] d q m + | λ | r ) d q u ) d q s + | δ 1 ( t 2 γ t 1 γ ) | 0 1 ( 1 q u ) ( γ 1 ) Γ q ( γ ) × ( | B | 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) [ b 1 ( m ) ψ ( r ) + b 2 ( m ) ] d q m + | λ | r ) d q u } .

Obviously the right-hand side of the above inequality tends to zero independently of x B r as t 1 t 2 0 . Therefore it follows by the Arzelá-Ascoli theorem that L 2 : C ( [ 0 , 1 ] , R ) P ( C ( [ 0 , 1 ] , R ) ) is completely continuous.

Claim III: Next we prove that L 2 has a closed graph.

Let x n x , h n L 2 ( x n ) and h n h . Then we need to show that h L 2 ( x ) . Associated with h n L 2 ( x n ) , there exists v n S G , x n such that for each t [ 0 , 1 ] ,
h n ( t ) = 0 t ( t q u ) ( γ 1 ) Γ q ( γ ) ( B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) v n ( m ) d q m ) d q u [ δ 3 t γ δ 4 ] Δ { a 0 η ( η q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) ( B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) × v n ( m ) d q m ) d q u ) d q s } + [ δ 1 t γ δ 2 ] Δ { b 0 σ ( σ q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) ( B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) × v n ( m ) d q m ) d q u ) d q s } [ δ 1 t γ δ 2 ] Δ { 0 1 ( 1 q u ) ( γ 1 ) Γ q ( γ ) ( B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) v n ( m ) d q m ) d q u } .
Thus it suffices to show that there exists v S G , x such that for each t [ 0 , 1 ] ,
h ( t ) = 0 t ( t q u ) ( γ 1 ) Γ q ( γ ) ( B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) v ( m ) d q m ) d q u [ δ 3 t γ δ 4 ] Δ { a 0 η ( η q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) ( B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) × v ( m ) d q m ) d q u ) d q s } + [ δ 1 t γ δ 2 ] Δ { b 0 σ ( σ q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) ( B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) × v ( m ) d q m ) d q u ) d q s } [ δ 1 t γ δ 2 ] Δ { 0 1 ( 1 q u ) ( γ 1 ) Γ q ( γ ) ( B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) v ( m ) d q m ) d q u } .
Let us consider the linear operator Θ : L 1 ( [ 0 , 1 ] , R ) C ( [ 0 , 1 ] , R ) given by
v Θ ( v ) ( t ) = 0 t ( t q u ) ( γ 1 ) Γ q ( γ ) ( B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) v ( m ) d q m ) d q u [ δ 3 t γ δ 4 ] Δ { a 0 η ( η q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) ( B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) × v ( m ) d q m ) d q u ) d q s } + [ δ 1 t γ δ 2 ] Δ { b 0 σ ( σ q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) ( B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) × v ( m ) d q m ) d q u ) d q s } [ δ 1 t γ δ 2 ] Δ { 0 1 ( 1 q u ) ( γ 1 ) Γ q ( γ ) ( B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) v ( m ) d q m ) d q u } .
Observe that
h n ( t ) h ( t ) = 0 t ( t q u ) ( γ 1 ) Γ q ( γ ) ( B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) ( v n ( m ) v ( m ) ) d q m ) d q u [ δ 3 t γ δ 4 ] Δ { a 0 η ( η q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) ( B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) × ( v n ( m ) v ( m ) ) d q m ) d q u ) d q s } + [ δ 1 t γ δ 2 ] Δ { b 0 σ ( σ q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) ( B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) × ( v n ( m ) v ( m ) ) d q m ) d q u ) d q s } [ δ 1 t γ δ 2 ] Δ { 0 1 ( 1 q u ) ( γ 1 ) Γ q ( γ ) × ( B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) ( v n ( m ) v ( m ) ) d q m ) d q u } 0 ,
as n . Thus, it follows by Lemma 2.7 that Θ S G is a closed graph operator. Further, we have h n ( t ) Θ ( S G , x n ) . Since x n x , therefore, we have
h ( t ) = 0 t ( t q u ) ( γ 1 ) Γ q ( γ ) ( B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) v ( m ) d q m ) d q u [ δ 3 t γ δ 4 ] Δ { a 0 η ( η q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) ( B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) × v ( m ) d q m ) d q u ) d q s } + [ δ 1 t γ δ 2 ] Δ { b 0 σ ( σ q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) ( B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) × v ( m ) d q m ) d q u ) d q s } [ δ 1 t γ δ 2 ] Δ { 0 1 ( 1 q u ) ( γ 1 ) Γ q ( γ ) ( B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) v ( m ) d q m ) d q u }

for some v S G , x .

Hence L 2 has a closed graph (and therefore has closed values). In consequence, L 2 is compact valued.

Therefore the operators L 1 and L 2 satisfy all the conditions of Theorem 2.5. So the conclusion of Theorem 2.5 applies and either condition (i) or condition (ii) holds. We show that conclusion (ii) is not possible. If x θ L 1 ( x ) + θ L 2 ( x ) for θ ( 0 , 1 ) , then there exist f S F , x and g S G , x such that
x ( t ) = θ 0 t ( t q u ) ( γ 1 ) Γ q ( γ ) ( A 0 u ( u q m ) ( β 1 ) Γ q ( β ) f ( m ) d q m λ x ( u ) ) d q u [ δ 3 t γ δ 4 ] Δ { a 0 η ( η q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) ( A 0 u ( u q m ) ( β 1 ) Γ q ( β ) × f ( m ) d q m λ x ( u ) ) d q u ) d q s } + θ [ δ 1 t γ δ 2 ] Δ { b 0 σ ( σ q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) ( A 0 u ( u q m ) ( β 1 ) Γ q ( β ) × f ( m ) d q m λ x ( u ) ) d q u ) d q s } θ [ δ 1 t γ δ 2 ] Δ { 0 1 ( 1 q u ) ( γ 1 ) Γ q ( γ ) ( A 0 u ( u q m ) ( β 1 ) Γ q ( β ) f ( m ) d q m λ x ( u ) ) d q u } + θ 0 t ( t q u ) ( γ 1 ) Γ q ( γ ) ( B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) g ( m ) d q m ) d q u λ [ δ 3 t γ δ 4 ] Δ { a 0 η ( η q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) ( B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) × g ( m ) d q m ) d q u ) d q s } + θ [ δ 1 t γ δ 2 ] Δ { b 0 σ ( σ q s ) ( α 2 ) Γ q ( α 1 ) ( 0 s ( s q u ) ( γ 1 ) Γ q ( γ ) ( B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) × g ( m ) d q m ) d q u ) d q s } θ [ δ 1 t γ δ 2 ] Δ { 0 1 ( 1 q u ) ( γ 1 ) Γ q ( γ ) ( B 0 u ( u q m ) ( β + ξ 1 ) Γ q ( β + ξ ) g ( m ) d q m ) d q u } .
By hypothesis (H2), for all t [ 0 , 1 ] , we have
F ( t , x ) = H ( F ( t , x ) , 0 ) H ( F ( t , x ) , F ( t , 0 ) ) + H ( F ( t , 0 ) , 0 ) H ( F ( t , x ) , F ( t , 0 ) ) + F ( t , 0 ) .
Hence, for any a F ( t , x ) ,
| a | F ( t , x ) H ( F ( t , x ) , F ( t , 0 ) ) + F ( t , 0 ) k ( t ) x + F ( t , 0 )
for all t [ 0 , 1 ] . Then, by using the computations from Step 1 and Step 2, Claim I, we have
| x ( t ) | χ 1 ( k x + F 0 ) + ψ ( x ) ω 1 + ω 2 + | λ | χ 2 x ,
or
x χ 1 ( k x + F 0 ) + ψ ( x ) ω 1 + ω 2 + | λ | χ 2 x .
(3.5)
Now, if condition (ii) of Theorem 2.5 holds, then there exist θ ( 0 , 1 ) and x B M such that x = θ L ( x ) . This implies that x is a solution with x = M and consequently, inequality (3.5) yields
( 1 χ 1 k | λ | χ 2 ) M χ 1 F 0 + ψ ( M ) ω 1 + ω 2 1 ,

which contradicts (3.4). Hence, has a fixed point in [ 0 , 1 ] by Theorem 2.5, which in fact is a solution of problem (1.1). This completes the proof. □

Example 3.3 Consider a nonlocal integral boundary value problem of fractional integro-differential equations given by
{ D q 1 / 2 c ( c D q 1 / 2 + 1 8 ) x ( t ) F ( t , x ( t ) ) + I 3 / 4 G ( t , x ( t ) ) , t [ 0 , 1 ] , x ( 0 ) = I q 2 x ( 1 / 3 ) , x ( 1 ) = 1 2 I q 2 x ( 2 / 3 ) ,
(3.6)

where A = 1 , B = 1 , β = γ = q = b = 1 / 2 , a = 1 , α = 3 , η = 1 / 3 , σ = 2 / 3 , λ = 1 / 8 .

We found δ 1 = 0.925926 , δ 2 = 0.030136 , δ 3 = 0.851852 , δ 4 = 0.914762 , Δ = 0.872674 , α 1 = 0.072089 , α 2 = 1.095555 , χ 1 = 2.158402 , χ 2 = 2.37938 .

Let
F ( t , x ) = [ 1 2 cos x ( 5 + t ) 2 , 1 6 ] , G ( t , x ) = [ 1 4 cos t 2 sin ( | x | 2 ) + e x 2 ( t 2 + 1 ) 1 + ( t 2 + 1 ) + 1 3 , 1 15 sin ( | x | ) + ( t 3 + 1 ) 1 + ( t 3 + 1 ) ] .
Then we have
sup { | u | : u F ( t , x ) } 1 2 + 1 ( 5 + t ) 2 , H ( F ( t , x ) , F ( t , x ¯ ) ) k ( t ) | x x ¯ |
with k ( t ) = 1 ( 5 + t ) 2 . Clearly, k = 1 / 25 , b 1 ( t ) = 1 8 , b 2 ( t ) = 1 , ψ ( M ) = M , F 0 = 0.55 , w 1 = 0.2698 , w 2 = 2.1584 and χ 1 k + χ 2 | λ | < 1 . By the condition
( 1 χ 1 k | λ | χ 2 ) M χ 1 F 0 + ψ ( M ) ω 1 + ω 2 > 1 ,

it is found that M > M 1 , where M 1 9.65681 . Thus, all the assumptions of Theorem 3.2 are satisfied. Hence, the conclusion of Theorem 3.2 applies to problem (3.6).

Declarations

Acknowledgements

This paper was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The authors, therefore, acknowledge technical and financial support of KAU.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, King Abdulaziz University, Jeddah, Saudi Arabia
(2)
Department of Mathematics, University of Ioannina, Ioannina, Greece

References

  1. Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives, Theory and Applications. Gordon & Breach, Yverdon; 1993.MATHGoogle Scholar
  2. Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.MATHGoogle Scholar
  3. Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematics Studies 204. In Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.Google Scholar
  4. Baleanu D, Diethelm K, Scalas E, Trujillo JJ Series on Complexity, Nonlinearity and Chaos. In Fractional Calculus Models and Numerical Methods. World Scientific, Boston; 2012.Google Scholar
  5. Agarwal RP, Zhou Y, He Y: Existence of fractional neutral functional differential equations. Comput. Math. Appl. 2010, 59: 1095-1100. 10.1016/j.camwa.2009.05.010MathSciNetView ArticleMATHGoogle Scholar
  6. Baleanu D, Mustafa OG, Agarwal RP:On L p -solutions for a class of sequential fractional differential equations. Appl. Math. Comput. 2011, 218: 2074-2081. 10.1016/j.amc.2011.07.024MathSciNetView ArticleMATHGoogle Scholar
  7. Ahmad B, Nieto JJ: Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions. Bound. Value Probl. 2011., 2011: Article ID 36Google Scholar
  8. Ahmad B, Ntouyas SK, Alsaedi A: New existence results for nonlinear fractional differential equations with three-point integral boundary conditions. Adv. Differ. Equ. 2011., 2011: Article ID 107384Google Scholar
  9. Sudsutad W, Tariboon J: Existence results of fractional integro-differential equations with m -point multi-term fractional order integral boundary conditions. Bound. Value Probl. 2012., 2012: Article ID 94Google Scholar
  10. O’Regan D, Stanek S: Fractional boundary value problems with singularities in space variables. Nonlinear Dyn. 2013, 71: 641-652. 10.1007/s11071-012-0443-xMathSciNetView ArticleMATHGoogle Scholar
  11. Ahmad B, Ntouyas SK, Alsaedi A: A study of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multistrip boundary conditions. Math. Probl. Eng. 2013., 2013: Article ID 320415Google Scholar
  12. Ahmad B, Nieto JJ: Boundary value problems for a class of sequential integrodifferential equations of fractional order. J. Funct. Spaces Appl. 2013., 2013: Article ID 149659Google Scholar
  13. Zhang L, Ahmad B, Wang G, Agarwal RP: Nonlinear fractional integro-differential equations on unbounded domains in a Banach space. J. Comput. Appl. Math. 2013, 249: 51-56.MathSciNetView ArticleMATHGoogle Scholar
  14. Liu X, Jia M, Ge W: Multiple solutions of a p -Laplacian model involving a fractional derivative. Adv. Differ. Equ. 2013., 2013: Article ID 126Google Scholar
  15. Ahmad B, Ntouyas SK: Existence results for higher order fractional differential inclusions with multi-strip fractional integral boundary conditions. Electron. J. Qual. Theory Differ. Equ. 2013., 2013: Article ID 20Google Scholar
  16. Zhai C, Xu L: Properties of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations with a parameter. Commun. Nonlinear Sci. Numer. Simul. 2014, 19: 2820-2827. 10.1016/j.cnsns.2014.01.003MathSciNetView ArticleGoogle Scholar
  17. Punzo F, Terrone G: On the Cauchy problem for a general fractional porous medium equation with variable density. Nonlinear Anal. 2014, 98: 27-47.MathSciNetView ArticleMATHGoogle Scholar
  18. Ferreira R: Nontrivial solutions for fractional q -difference boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2010., 2010: Article ID 70Google Scholar
  19. Goodrich CS: Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions. Comput. Math. Appl. 2011, 61: 191-202. 10.1016/j.camwa.2010.10.041MathSciNetView ArticleMATHGoogle Scholar
  20. Ma J, Yang J: Existence of solutions for multi-point boundary value problem of fractional q -difference equation. Electron. J. Qual. Theory Differ. Equ. 2011., 2011: Article ID 92Google Scholar
  21. Abdeljawad T, Baleanu D: Caputo q -fractional initial value problems and a q -analogue Mittag-Leffler function. Commun. Nonlinear Sci. Numer. Simul. 2011, 16: 4682-4688. 10.1016/j.cnsns.2011.01.026MathSciNetView ArticleMATHGoogle Scholar
  22. Jarad F, Abdeljawad T, Gundogdu E, Baleanu D: On the Mittag-Leffler stability of q -fractional nonlinear dynamical systems. Proc. Rom. Acad., Ser. A: Math. Phys. Tech. Sci. Inf. Sci. 2011, 12: 309-314.MathSciNetGoogle Scholar
  23. Graef JR, Kong L: Positive solutions for a class of higher order boundary value problems with fractional q -derivatives. Appl. Math. Comput. 2012, 218: 9682-9689. 10.1016/j.amc.2012.03.006MathSciNetView ArticleMATHGoogle Scholar
  24. Li X, Han Z, Sun S: Existence of positive solutions of nonlinear fractional q -difference equation with parameter. Adv. Differ. Equ. 2013., 2013: Article ID 260Google Scholar
  25. Alsaedi A, Ahmad B, Al-Hutami H: A study of nonlinear fractional q -difference equations with nonlocal integral boundary conditions. Abstr. Appl. Anal. 2013., 2013: Article ID 410505Google Scholar
  26. Jarad F, Abdeljawad T, Baleanu D: Stability of q -fractional non-autonomous systems. Nonlinear Anal., Real World Appl. 2013, 14: 780-784. 10.1016/j.nonrwa.2012.08.001MathSciNetView ArticleMATHGoogle Scholar
  27. Abdeljawad T, Baleanu D, Jarad F, Agarwal RP: Fractional sums and differences with binomial coefficients. Discrete Dyn. Nat. Soc. 2013., 2013: Article ID 104173Google Scholar
  28. Ferreira R: Positive solutions for a class of boundary value problems with fractional q -differences. Comput. Math. Appl. 2011, 61: 367-373. 10.1016/j.camwa.2010.11.012MathSciNetView ArticleMATHGoogle Scholar
  29. Ahmad B, Nieto JJ, Alsaedi A, Al-Hutami H: Existence of solutions for nonlinear fractional q -difference integral equations with two fractional orders and nonlocal four-point boundary conditions. J. Franklin Inst. 2014, 351: 2890-2909. 10.1016/j.jfranklin.2014.01.020MathSciNetView ArticleGoogle Scholar
  30. Agarwal RP, Ahmad B, Alsaedi A, Al-Hutami H: Existence theory for q -antiperiodic boundary value problems of sequential q -fractional integrodifferential equations. Abstr. Appl. Anal. 2014., 2014: Article ID 207547Google Scholar
  31. Zhang L, Baleanu D, Wang G:Nonlocal boundary value problem for nonlinear impulsive q k -integrodifference equation. Abstr. Appl. Anal. 2014., 2014: Article ID 478185Google Scholar
  32. Ahmad B, Ntouyas SK, Alsaedi A, Al-Hutami H: Nonlinear q -fractional differential equations with nonlocal and sub-strip type boundary conditions. Electron. J. Qual. Theory Differ. Equ. 2014., 2014: Article ID 26Google Scholar
  33. Ahmad, B, Nieto, JJ, Alsaedi, A, Al-Hutami, H: Boundary value problems of nonlinear fractional q-difference (integral) equations with two fractional orders and four-point nonlocal integral boundary conditions. Filomat (in press)Google Scholar
  34. Abbasbandy S, Nieto JJ, Alavi M: Tuning of reachable set in one dimensional fuzzy differential inclusions. Chaos Solitons Fractals 2005, 26: 1337-1341. 10.1016/j.chaos.2005.03.018MathSciNetView ArticleMATHGoogle Scholar
  35. Frigon M: Systems of first order differential inclusions with maximal monotone terms. Nonlinear Anal. 2007, 66: 2064-2077. 10.1016/j.na.2006.03.002MathSciNetView ArticleMATHGoogle Scholar
  36. Nieto JJ, Rodriguez-Lopez R: Euler polygonal method for metric dynamical systems. Inf. Sci. 2007, 177: 4256-4270. 10.1016/j.ins.2007.05.002MathSciNetView ArticleMATHGoogle Scholar
  37. Henderson J, Ouahab A: Fractional functional differential inclusions with finite delay. Nonlinear Anal. 2009, 70: 2091-2105. 10.1016/j.na.2008.02.111MathSciNetView ArticleMATHGoogle Scholar
  38. Chang Y-K, Nieto JJ: Some new existence results for fractional differential inclusions with boundary conditions. Math. Comput. Model. 2009, 49: 605-609. 10.1016/j.mcm.2008.03.014MathSciNetView ArticleMATHGoogle Scholar
  39. Cernea A: On the existence of solutions for nonconvex fractional hyperbolic differential inclusions. Commun. Math. Anal. 2010, 9(1):109-120.MathSciNetMATHGoogle Scholar
  40. Agarwal RP, Benchohra M, Hamani S: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 2010, 109: 973-1033. 10.1007/s10440-008-9356-6MathSciNetView ArticleMATHGoogle Scholar
  41. Ahmad B, Ntouyas SK: Some existence results for boundary value problems of fractional differential inclusions with non-separated boundary conditions. Electron. J. Qual. Theory Differ. Equ. 2010., 2010: Article ID 71Google Scholar
  42. Ntouyas SK: Boundary value problems for nonlinear fractional differential equations and inclusions with nonlocal and fractional integral boundary conditions. Opusc. Math. 2013, 33: 117-138. 10.7494/OpMath.2013.33.1.117MathSciNetView ArticleMATHGoogle Scholar
  43. Ahmad B, Ntouyas SK: An existence theorem for fractional hybrid differential inclusions of Hadamard type with Dirichlet boundary conditions. Abstr. Appl. Anal. 2014., 2014: Article ID 705809Google Scholar
  44. Ahmad B, Ntouyas SK: Existence of solutions for nonlinear fractional q -difference inclusions with nonlocal Robin (separated) conditions. Mediterr. J. Math. 2013, 10: 1333-1351. 10.1007/s00009-013-0258-0MathSciNetView ArticleMATHGoogle Scholar
  45. Agarwal RP: Certain fractional q -integrals and q -derivatives. Proc. Camb. Philos. Soc. 1969, 66: 365-370. 10.1017/S0305004100045060View ArticleMathSciNetMATHGoogle Scholar
  46. Rajkovic PM, Marinkovic SD, Stankovic MS: On q -analogues of Caputo derivative and Mittag-Leffler function. Fract. Calc. Appl. Anal. 2007, 10: 359-373.MathSciNetMATHGoogle Scholar
  47. Annaby MH, Mansour ZS Lecture Notes in Mathematics 2056. In q-Fractional Calculus and Equations. Springer, Berlin; 2012.View ArticleGoogle Scholar
  48. Petryshyn WV, Fitzpatric PM: A degree theory, fixed point theorems, and mapping theorems for multivalued noncompact maps. Trans. Am. Math. Soc. 1974, 194: 1-25.View ArticleGoogle Scholar
  49. Lasota A, Opial Z: An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 1965, 13: 781-786.MathSciNetMATHGoogle Scholar
  50. Kuratowski K, Ryll-Nardzewski C: A general theorem on selectors. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 1965, 13: 397-403.MathSciNetMATHGoogle Scholar
  51. Gorniewicz L: Topological Fixed Point Theory of Multivalued Mappings. Springer, Dordrecht; 2006.MATHGoogle Scholar
  52. Deimling K: Multivalued Differential Equations. de Gruyter, Berlin; 1992.View ArticleMATHGoogle Scholar

Copyright

© Ahmad et al.; licensee Springer 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.