Next Article in Journal
Inertial Subgradient Extragradient Methods for Solving Variational Inequality Problems and Fixed Point Problems
Next Article in Special Issue
Generalized Nabla Differentiability and Integrability for Fuzzy Functions on Time Scales
Previous Article in Journal
Discrete and Fuzzy Models of Time Series in the Tasks of Forecasting and Diagnostics
Previous Article in Special Issue
Lyapunov Type Theorems for Exponential Stability of Linear Skew-Product Three-Parameter Semiflows with Discrete Time
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Nonlocal Fractional Boundary Value Problems Involving Mixed Right and Left Fractional Derivatives and Integrals

1
Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2
Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Submission received: 30 March 2020 / Revised: 21 April 2020 / Accepted: 29 April 2020 / Published: 1 May 2020

Abstract

:
In this paper, we study the existence of solutions for nonlocal single and multi-valued boundary value problems involving right-Caputo and left-Riemann–Liouville fractional derivatives of different orders and right-left Riemann–Liouville fractional integrals. The existence of solutions for the single-valued case relies on Sadovskii’s fixed point theorem. The first existence results for the multi-valued case are proved by applying Bohnenblust-Karlin’s fixed point theorem, while the second one is based on Martelli’s fixed point theorem. We also demonstrate the applications of the obtained results.

1. Introduction

Fractional calculus has emerged as an interesting and fruitful subject in view of wide applications of its tools in modeling complex dynamical systems. Mathematical models based on fractional-order operators provide insight into the past history of the underlying phenomena. Examples include constitutive equations (fractional law) in the viscoelastic materials [1], Caputo power law in transport processes [2], dynamic memory describing the economic processes, see [3,4].
Widespread applications of fractional differential equations motivated many researchers to develop the theoretical aspects of the topic. During the last few decades, one can witness the remarkable development on initial and boundary value problems of fractional differential equations and inclusions. Much of the literature on such problems include Caputo, Riemann–Liouville, Hadamard type fractional derivatives, and different kinds of classical and non-classical boundary conditions. For some recent works on fractional order boundary value problems, for example, see the articles [5,6,7,8,9,10,11,12] and the references cited therein. Fractional differential equations involving left and right fractional derivatives also received considerable attention, for instance, see [13,14,15,16]. These derivatives appear in the study of Euler-Lagrange equations [17], steady heat-transfer in fractal media [18], electromagnetic waves phenomena in a variety of dielectric media with susceptibility [19], etc.
Multivalued (inclusions) problems are found to be of great utility in studying dynamical systems and stochastic processes, for example, see [20,21]. In the text [22], one can find the details on stochastic processes, queueing networks, optimization and their application in finance, control, climate control, etc. Monotone differential inclusions were applied to study the nonlinear dynamics of wheeled vehicles in [23]. In [24], a fractional differential inclusion with oscillatory potential was studied. In [25], the authors investigated the mild solutions to the time fractional Navier-Stokes delay differential inclusions. Other applications include polynomial control systems [20], synchronization of piecewise continuous systems of fractional order [21], oscillation and nonoscillation of impulsive fractional differential inclusions [26], etc. For some recent existence and controllability results on fractional differential inclusions, we refer the reader to articles [27,28,29,30,31,32,33] and the references cited therein.
Recently, in [34], the authors studied existence and uniqueness of solutions for a new kind of boundary value problem involving right-Caputo and left-Riemann–Liouville fractional derivatives of different orders and right-left Riemann–Liouville fractional integrals, subject to nonlocal boundary conditions of the form
D 1 α C D 0 + β R L y ( t ) + λ I 1 p I 0 + q h ( t , y ( t ) ) = f ( t , y ( t ) ) , t J : = [ 0 , 1 ] , y ( 0 ) = y ( ξ ) = 0 , y ( 1 ) = δ y ( μ ) , 0 < ξ < μ < 1 ,
where D 1 α C and D 0 + β R L denote the right Caputo fractional derivative of order α ( 1 , 2 ] and the left Riemann–Liouville fractional derivative of order β ( 0 , 1 ] , I 1 p and I 0 + q denote the right and left Riemann–Liouville fractional integrals of orders p , q > 0 respectively, f , h : [ 0 , 1 ] × R R are given continuous functions and δ , λ R .
Here we emphasize that the importance of nonlocal conditions can be understood in the sense that such conditions are used to model the peculiarities occurring inside the domain of physical and chemical processes as the classical initial and boundary conditions fail to cater this situation. The present problem is motivated by useful applications of nonlocal boundary data in petroleum exploitation, thermodynamics, elasticity, and wave propagation, etc., for instance, see [35,36] and the details therein.
The existence results for the problem (1) were derived by applying a fixed point theorem due to Krasnoselski and Leray–Schauder nonlinear alternative, while the uniqueness result was established via Banach contraction mapping principle.
The objective of the present work is to enrich the results on this new class of problems. We firstly prove another existence result for the problem (1) with the aid of Sadovskii’s fixed point theorem. Afterwards, we initiate the study of the multi-valued analogue of the problem (1) by considering the following inclusions problem:
D 1 α C D 0 + β R L y ( t ) F ( t , y ( t ) ) λ I 1 p I 0 + q H ( t , y ( t ) ) , t [ 0 , 1 ] , y ( 0 ) = y ( ξ ) = 0 , y ( 1 ) = δ y ( μ ) , 0 < ξ < μ < 1 ,
where F , H : [ 0 , 1 ] × R P ( R ) are compact multivalued maps, P ( R ) is the family of all nonempty subsets of R , and the other quantities are the same as defined in problem (1). Existence results for the problem (2) are established via fixed point theorems due to Bohnenblust-Karlin [37] and Martelli [38].
The rest of the paper is arranged as follows. In Section 2 we recall some preliminary concepts and a known lemma [34]. In Section 3 we prove an existence result for the problem (1) by applying Sadovskii’s fixed point theorem. Section 4 presents the existence results for the problem (2). Applications and examples are discussed in Section 5.

2. Preliminaries

Let us collect some important definitions on fractional calculus.
Definition 1.
[39] The left and right Riemann–Liouville fractional integrals of order δ > 0 for g L 1 [ a , b ] , existing almost everywhere on [ a , b ] , are respectively defined by
I a + δ g ( t ) = a t ( t s ) δ 1 Γ ( δ ) g ( s ) d s a n d I b δ g ( t ) = t b ( s t ) δ 1 Γ ( δ ) g ( s ) d s .
In addition, according to the classical theorem of Vallee-Poussin and the Young convolution theorem, I a + δ g , I b δ g L 1 [ a , b ] , δ > 0 .
Definition 2.
[39] For g A C n [ a , b ] , the left Riemann–Liouville and the right Caputo fractional derivatives of order δ ( n 1 , n ] , n N , existing almost everywhere on [ a , b ] , are respectively defined by
D a + δ R L g ( t ) = d n d t n a t ( t s ) n δ 1 Γ ( n δ ) g ( s ) d s a n d C D b δ g ( t ) = ( 1 ) n t b ( s t ) n δ 1 Γ ( n δ ) g ( n ) ( s ) d s .
The following known lemma [34] plays a key role in proving the main results.
Lemma 1.
Let H , F C [ 0 , 1 ] L ( 0 , 1 ) and y C ( [ 0 , 1 ] , R ) . Then the linear problem
D 1 α C D 0 + β R L y ( t ) + λ I 1 p I 0 + q H ( t ) = F ( t ) , t J : = [ 0 , 1 ] , y ( 0 ) = y ( ξ ) = 0 , y ( 1 ) = δ y ( μ ) ,
is equivalent to the fractional integral equation:
y ( t ) = 0 t ( t s ) β 1 Γ ( β ) I 1 α F ( s ) λ I 1 α + p I 0 + q H ( s ) d s + a 1 ( t ) { δ 0 μ ( μ s ) β 1 Γ ( β ) I 1 α F ( s ) λ I 1 α + p I 0 + q H ( s ) d s 0 1 ( 1 s ) β 1 Γ ( β ) I 1 α F ( s ) λ I 1 α + p I 0 + q H ( s ) d s } + a 2 ( t ) 0 ξ ( ξ s ) β 1 Γ ( β ) I 1 α F ( s ) λ I 1 α + p I 0 + q H ( s ) d s ,
where
a 1 ( t ) = 1 Λ ξ β + 1 t β ξ β t β + 1 , a 2 ( t ) = 1 Λ t β ( 1 δ μ β + 1 ) t β + 1 ( 1 δ μ β ) ,
and it is assumed that
Λ = ξ β + 1 ( 1 δ μ β ) ξ β ( 1 δ μ β + 1 ) 0 .

3. Existence Result for the Single-Valued Problem (1) via Sadovskii’s Fixed Point Theorem

Our existence result for the problem (1) is based on Sadovskii’s fixed point theorem. Before proceeding further, let us recall some related auxiliary material. In the sequel, we use the norm . = sup t [ 0 , 1 ] | . | .
Definition 3.
Let M be a bounded set in metric space ( X , d ) . The Kuratowski measure of noncompactness, α ( M ) , is defined as
inf { ϵ : M covered by a finitely many sets such that the diameter of each set ϵ } .
Definition 4.
[40] Let Φ : D ( Φ ) X X be a bounded and continuous operator on Banach space X . Then Φ is called a condensing map if α ( Φ ( B ) ) < α ( B ) for all bounded sets B D ( Φ ) , where α denotes the Kuratowski measure of noncompactness.
Lemma 2.
[41], Example 11.7 The map K + C is a k-set contraction with 0 k < 1 , and thus also condensing, if
(i) 
K , C : D X X are operators on the Banach space X ;
(ii) 
K is k-contractive, that is, for all x , y D and a fixed k [ 0 , 1 ) ,
K x K y k x y ;
(iii) 
C is compact.
Lemma 3.
[42] Let B be a convex, bounded and closed subset of a Banach space X and Φ : B B be a condensing map. Then Φ has a fixed point.
In the sequel, we set
Λ 1 = Δ 1 Γ ( α + 1 ) , Λ 2 = | λ | Δ 1 Γ ( α + p + 1 ) Γ ( q + 1 ) , Λ 3 = Δ 2 Γ ( α ) , Λ 4 = | λ | Δ 2 Γ ( α + p ) Γ ( q ) ,
where
Δ 1 = 1 Γ ( β + 1 ) 1 + a ¯ 1 ( | δ | μ β + 1 ) + a ¯ 2 ξ β , Δ 2 = 1 Γ ( β + 1 ) 1 + a ¯ 1 ( | δ | + 1 ) + a ¯ 2 , a ¯ 1 = max t [ 0 , 1 ] | a 1 ( t ) | , a ¯ 2 = max t [ 0 , 1 ] | a 2 ( t ) | .
Theorem 1.
Assume that:
( B 1 )
There exist L > 0 such that | f ( t , x ) f ( t , y ) | L | x y | , t [ 0 , 1 ] , x , y R ;
( B 2 )
| f ( t , y ) | σ ( t ) and | h ( t , y ) | ρ ( t ) , where σ , ρ C ( [ 0 , 1 ] , R + ) .
Then the problem (1) has at least one solution on [ 0 , 1 ] if
Q : = L Λ 1 < 1 .
where Λ 1 is given by (7).
Proof. 
Let B r = { x C ( [ 0 , 1 ] , R ) : x r } be a closed bounded and convex subset of C ( [ 0 , 1 ] , R ) , where r is a fixed constant. In view of Lemma 1, we introduce an operator G : C ( [ 0 , 1 ] , R ) C ( [ 0 , 1 ] , R ) associated with the problem (1) as follows:
G y ( t ) = 0 t ( t s ) β 1 Γ ( β ) I 1 α f ( s , y ( s ) ) λ I 1 α + p I 0 + q h ( s , y ( s ) ) d s + a 1 ( t ) [ δ 0 μ ( μ s ) β 1 Γ ( β ) I 1 α f ( s , y ( s ) ) λ I 1 α + p I 0 + q h ( s , y ( s ) ) d s 0 1 ( 1 s ) β 1 Γ ( β ) I 1 α f ( s , y ( s ) ) λ I 1 α + p I 0 + q h ( s , y ( s ) ) d s ] + a 2 ( t ) 0 ξ ( ξ s ) β 1 Γ ( β ) I 1 α f ( s , y ( s ) ) λ I 1 α + p I 0 + q h ( s , y ( s ) ) d s .
Let us split the operator G : C ( [ 0 , 1 ] , R ) C ( [ 0 , 1 ] , R ) on B r as G = G 1 + G 2 , where
G 1 y ( t ) = 0 t ( t s ) β 1 Γ ( β ) I 1 α f ( s , y ( s ) ) d s + a 1 ( t ) [ δ 0 μ ( μ s ) β 1 Γ ( β ) I 1 α f ( s , y ( s ) ) d s 0 1 ( 1 s ) β 1 Γ ( β ) I 1 α f ( s , y ( s ) ) d s ] + a 2 ( t ) 0 ξ ( ξ s ) β 1 Γ ( β ) I 1 α f ( s , y ( s ) ) d s , G 2 y ( t ) = λ 0 t ( t s ) β 1 Γ ( β ) I 1 α + p I 0 + q h ( s , y ( s ) ) d s λ a 1 ( t ) [ δ 0 μ ( μ s ) β 1 Γ ( β ) I 1 α + p I 0 + q h ( s , y ( s ) ) d s 0 1 ( 1 s ) β 1 Γ ( β ) I 1 α + p I 0 + q h ( s , y ( s ) ) d s ] λ a 2 ( t ) 0 ξ ( ξ s ) β 1 Γ ( β ) I 1 α + p I 0 + q h ( s , y ( s ) ) d s .
We shall show that the operators G 1 and G 2 satisfy all the conditions of Lemma 3. The proof will be given in several steps.
Step 1. G B r B r .
Let us select r σ Λ 1 + ρ Λ 2 , where Λ 1 , Λ 2 are given by (7). For any y B r , we have
G y sup t [ 0 , 1 ] { 0 t ( t s ) β 1 Γ ( β ) I 1 α | f ( s , y ( s ) ) | + | λ | I 1 α + p I 0 + q | h ( s , y ( s ) ) | d s + | a 1 ( t ) | { | δ | 0 μ ( μ s ) β 1 Γ ( β ) I 1 α | f ( s , y ( s ) ) | + | λ | I 1 α + p I 0 + q | h ( s , y ( s ) ) | d s + 0 1 ( 1 s ) β 1 Γ ( β ) I 1 α | f ( s , y ( s ) ) | + | λ | I 1 α + p I 0 + q | h ( s , y ( s ) ) | d s } + | a 2 ( t ) | 0 ξ ( ξ s ) β 1 Γ ( β ) I 1 α | f ( s , y ( s ) ) | + | λ | I 1 α + p I 0 + q | h ( s , y ( s ) ) | d s } σ sup t [ 0 , 1 ] { 0 t ( t s ) β 1 Γ ( β ) I 1 α ( 1 ) d s + | a 1 ( t ) | [ | δ | 0 μ ( μ s ) β 1 Γ ( β ) I 1 α ( 1 ) d s + 0 1 ( 1 s ) β 1 Γ ( β ) I 1 α ( 1 ) d s ] + | a 2 ( t ) | 0 ξ ( ξ s ) β 1 Γ ( β ) I 1 α ( 1 ) d s } + ρ | λ | sup t [ 0 , 1 ] { 0 t ( t s ) β 1 Γ ( β ) I 1 α + p I 0 + q ( 1 ) d s + | a 1 ( t ) | [ | δ | 0 μ ( μ s ) β 1 Γ ( β ) I 1 α + p I 0 + q ( 1 ) d s + 0 1 ( 1 s ) β 1 Γ ( β ) I 1 α + p I 0 + q ( 1 ) d s ] + | a 2 ( t ) | 0 ξ ( ξ s ) β 1 Γ ( β ) I 1 α + p I 0 + q ( 1 ) d s } σ Γ ( α + 1 ) + ρ | λ | Γ ( α + p + 1 ) Γ ( q + 1 ) Δ 1 = σ Λ 1 + ρ Λ 2 < r ,
which implies that G B r B r .
Step 2. G 2 is compact.
Observe that the operator G 2 is uniformly bounded in view of Step 1. Let t 1 , t 2 J with t 1 < t 2 and y B r . Then we have
| G 2 y ( t 2 ) G 2 y ( t 1 ) | | λ | | 0 t 1 ( t 2 s ) β 1 ( t 1 s ) β 1 Γ ( β ) I 1 α + p I 0 + q | h ( s , y ( s ) ) | d s | + | λ | | t 1 t 2 ( t 2 s ) β 1 Γ ( β ) I 1 α + p I 0 + q | h ( s , y ( s ) ) | d s | + | λ | | a 1 ( t 2 ) a 1 ( t 1 ) | { | δ | 0 μ ( μ s ) β 1 Γ ( β ) I 1 α + p I 0 + q | h ( s , y ( s ) ) | d s + 0 1 ( 1 s ) β 1 Γ ( β ) I 1 α + p I 0 + q | h ( s , y ( s ) ) | d s } + | λ | | a 2 ( t 2 ) a 2 ( t 1 ) | 0 ξ ( ξ s ) β 1 Γ ( β ) I 1 α + p I 0 + q | h ( s , y ( s ) ) | d s | λ | ρ Γ ( β + 1 ) Γ ( α + p + 1 ) Γ ( q + 1 ) { 2 ( t 2 t 1 ) β + | t 2 β t 1 β | + ( | δ | μ β + 1 ) | Λ | ξ β + 1 | t 2 β t 1 β | + ξ β | t 2 β + 1 t 1 β + 1 | + ξ β | Λ | | 1 δ μ β + 1 | | t 2 β t 1 β | + | 1 δ μ β | | t 2 β + 1 t 1 β + 1 | } ,
which tends to zero independent of y as t 2 t 1 . This shows that G 2 is equicontinuous. It is clear from the foregoing arguments that the operator G 2 is relatively compact on B r . Hence, by the Arzelá-Ascoli theorem, G 2 is compact on B r .
Step 3. G 1 is Q-contractive.
Using ( B 1 ) and ( B 2 ), it is easy to show that
G 1 y G 1 x sup t [ 0 , 1 ] { 0 t ( t s ) β 1 Γ ( β ) I 1 α | f ( s , y ( s ) ) f ( s , x ( s ) ) | d s + | a 1 ( t ) | [ δ 0 μ ( μ s ) β 1 Γ ( β ) I 1 α | f ( s , y ( s ) ) f ( s , x ( s ) ) | d s + 0 1 ( 1 s ) β 1 Γ ( β ) I 1 α | f ( s , y ( s ) ) f ( s , x ( s ) ) | d s ] + | a 2 ( t ) | 0 ξ ( ξ s ) β 1 Γ ( β ) I 1 α | f ( s , y ( s ) ) f ( s , x ( s ) ) | d s } L Γ ( β + 1 ) Γ ( α + 1 ) 1 + a ¯ 1 ( | δ | μ β + 1 ) + a ¯ 2 ξ β y x = L Λ 1 y x ,
which is Q-contractive, since Q : = L Λ 1 < 1 .
Step 4. G is condensing. Since G 1 is continuous, Q-contraction and G 2 is compact, therefore, by Lemma 2, G : B r B r with G = G 1 + G 2 is a condensing map on B r .
From the above four steps, we conclude by Lemma 3 that the map G has a fixed point which, in turn, implies that the problem (1) has a solution on [ 0 , 1 ] .

4. Existence Results for the Multi-Vaued Problem (2)

For a normed space ( X , · ) , we have P c l ( X ) = { Y P ( X ) : Y is closed } , P b ( X ) = { Y P ( X ) : Y is bounded } , P c p ( X ) = { Y P ( X ) : Y is compact } , P c p , c ( X ) = { Y P ( X ) : Y is compact and convex}, P b , c l , c ( R ) = { Y P ( X ) : Y is bounded, closed and convex}. We also define the sets of selections of the multi-valued maps F and H as
S F , y : = { f L 1 ( [ 0 , 1 ] , R ) : f ( t ) F ( t , y ) } , S ^ H , y : = { h L 1 ( [ 0 , 1 ] , R ) : h ( t ) H ( t , y ) } .
By Lemma 1, we define a solution of the boundary value problem (2) as follows (see also [43,44]).
Definition 5.
A function y C ( [ 0 , 1 ] , R ) is a solution of the boundary value problem (2) if y ( 0 ) = y ( ξ ) = 0 , y ( 1 ) = δ y ( μ ) , and there exist functions f S F , y , h S ^ H , y a.e. on [ 0 , 1 ] and
y ( t ) = 0 t ( t s ) β 1 Γ ( β ) I 1 α f ( s ) λ I 1 α + p I 0 + q h ( s ) d s + a 1 ( t ) { δ 0 μ ( μ s ) β 1 Γ ( β ) I 1 α f ( s ) λ I 1 α + p I 0 + q h ( s ) d s 0 1 ( 1 s ) β 1 Γ ( β ) I 1 α f ( s ) λ I 1 α + p I 0 + q h ( s ) d s } + a 2 ( t ) 0 ξ ( ξ s ) β 1 Γ ( β ) I 1 α f ( s ) λ I 1 α + p I 0 + q h ( s ) d s .
Now we provide the lemmas which will be used in the main existence results in this section.
Lemma 4.
(Bohnenblust-Karlin) ([37]) Let D be a nonempty, bounded, closed, and convex subset of X. Let Φ : D P ( R ) be upper semi-continuous with closed, convex values such that Φ ( D ) D and Φ ( D ) ¯ is compact. Then Φ has a fixed point.
Lemma 5.
([45]) Let X be a separable Banach space. Let F : J × X P c p , c ( X ) be measurable with respect to t for each y X and upper semi-continuous with respect to y for almost all t J and S F , y , for any y C ( J , X ) , and let Θ be a linear continuous mapping from L 1 ( J , X ) to C ( J , X ) . Then the operator
Θ S F : C ( J , X ) P c p , c ( C ( J , X ) ) , y ( Θ S F ) ( y ) = Θ ( S F , y )
is a closed graph operator in C ( J , X ) × C ( J , X ) .
In the first result, we study the existence of the solution for the multi-valued problem (2) by applying Bohnenblust–Karlin fixed point theorem.
Theorem 2.
Suppose that:
( M 1 )
F , H : [ 0 , 1 ] × R P b , c , c p ( R ) ; ( t , y ) f ( t , y ) and ( t , y ) h ( t , y ) be measurable with respect to t for each y R , upper semi-continuous with respect to y for almost everywhere t [ 0 , 1 ] , and for each fixed y R , the sets S F , y and S ^ H , y are nonempty for almost everywhere t [ 0 , 1 ] .
( M 2 )
For each ρ > 0 , there exist functions ϕ ρ , ψ ρ L 1 ( [ 0 , 1 ] , R + ) such that
F ( t , y ) = sup { | f | : f ( t ) F ( t , y ) } ϕ ρ ( t ) ,
H ( t , y ) = sup { | h | : h ( t ) H ( t , y ) } ψ ρ ( t ) ,
for each ( t , y ) [ 0 , 1 ] × R with y ρ , and
lim inf ρ + 1 ρ 0 1 ϕ ρ ( t ) d t = ζ 1 < , lim inf ρ + 1 ρ 0 1 ψ ρ ( t ) d t = ζ 2 < .
Then the boundary value problem (2) has at least one solution on [0,1] provided that
ζ 1 Λ 3 + ζ 2 Λ 4 < 1 ,
where ζ 1 , ζ 2 are defined by (8), and Λ 3 , Λ 4 are given by (7).
Proof. 
To transform the problem (2) into a fixed point problem, we define a multi-valued map U : C ( [ 0 , 1 ] , R ) P ( C ( [ 0 , 1 ] , R ) ) as
U ( y ) = { g C ( [ 0 , 1 ] , R ) : g ( t ) = 0 t ( t s ) β 1 Γ ( β ) I 1 α f ( s ) λ I 1 α + p I 0 + q h ( s ) d s + a 1 ( t ) [ δ 0 μ ( μ s ) β 1 Γ ( β ) I 1 α f ( s ) λ I 1 α + p I 0 + q h ( s ) d s 0 1 ( 1 s ) β 1 Γ ( β ) I 1 α f ( s ) λ I 1 α + p I 0 + q h ( s ) d s ] + a 2 ( t ) 0 ξ ( ξ s ) β 1 Γ ( β ) I 1 α f ( s ) λ I 1 α + p I 0 + q h ( s ) d s } ,
for f S F , y , h S ^ H , y .
Now we prove that the operator U satisfies the hypothesis of Lemma 4 and thus it will have a fixed point which corresponds to a solution of problem (2). Here we show that U is a compact and upper semi-continuous multi-valued map with convex closed values. This will be established in a sequence of steps.
Step 1: U ( y ) is convex for each y C ( [ 0 , 1 ] , R ) . For that, let g 1 , g 2 U ( y ) . Then there exist f 1 , f 2 S F , y , h 1 , h 2 S ^ H , y such that, for each t [ 0 , 1 ] , we get
g i ( t ) = 0 t ( t s ) β 1 Γ ( β ) I 1 α f i ( s ) λ I 1 α + p I 0 + q h i ( s ) d s + a 1 ( t ) { δ 0 μ ( μ s ) β 1 Γ ( β ) I 1 α f i ( s ) λ I 1 α + p I 0 + q h i ( s ) d s 0 1 ( 1 s ) β 1 Γ ( β ) I 1 α f i ( s ) λ I 1 α + p I 0 + q h i ( s ) d s } + a 2 ( t ) 0 ξ ( ξ s ) β 1 Γ ( β ) I 1 α f i ( s ) λ I 1 α + p I 0 + q h i ( s ) d s , i = 1 , 2 .
For each t [ 0 , 1 ] and 0 ν 1 , we can find that
ν g 1 + ( 1 ν ) g 2 ( t ) = 0 t ( t s ) β 1 Γ ( β ) I 1 α ν f 1 ( s ) + ( 1 ν ) f 2 ( s ) λ I 1 α + p I 0 + q ν h 1 ( s ) + ( 1 ν ) h 2 ( s ) d s + a 1 ( t ) { δ 0 μ ( μ s ) β 1 Γ ( β ) I 1 α ν f 1 ( s ) + ( 1 ν ) f 2 ( s ) λ I 1 α + p I 0 + q ν h 1 ( s ) + ( 1 ν ) h 2 ( s ) d s 0 1 ( 1 s ) β 1 Γ ( β ) I 1 α ν f 1 ( s ) + ( 1 ν ) f 2 ( s ) λ I 1 α + p I 0 + q ν h 1 ( s ) + ( 1 ν ) h 2 ( s ) d s } + a 2 ( t ) 0 ξ ( ξ s ) β 1 Γ ( β ) I 1 α ν f 1 ( s ) + ( 1 ν ) f 2 ( s ) λ I 1 α + p I 0 + q ν h 1 ( s ) + ( 1 ν ) h 2 ( s ) d s .
Since S F , y , S ^ H , y are convex valued ( F , H have convex values), it follows that ν g 1 + ( 1 ν ) g 2 U ( y ) . Step 2: U ( y ) maps bounded sets (balls) into bounded sets in C ( [ 0 , 1 ] , R ) . Let us define B ρ = { y C ( [ 0 , 1 ] , R ) : y ρ } as a bounded closed convex set in C ( [ 0 , 1 ] , R ) for each positive constant ρ . We shall prove that there exists a positive number ρ ¯ such that U ( B ρ ¯ ) B ρ ¯ . If it is not true, then we can find a function y ρ B ρ , g ρ U ( y ρ ) with U ( y ρ ) > ρ , such that
g ρ ( t ) = 0 t ( t s ) β 1 Γ ( β ) I 1 α f ρ ( s ) λ I 1 α + p I 0 + q h ρ ( s ) d s + a 1 ( t ) { δ 0 μ ( μ s ) β 1 Γ ( β ) I 1 α f ρ ( s ) λ I 1 α + p I 0 + q h ρ ( s ) d s 0 1 ( 1 s ) β 1 Γ ( β ) I 1 α f ρ ( s ) λ I 1 α + p I 0 + q h ρ ( s ) d s } + a 2 ( t ) 0 ξ ( ξ s ) β 1 Γ ( β ) I 1 α f ρ ( s ) λ I 1 α + p I 0 + q h ρ ( s ) d s ,
for some f ρ S F , y ρ , h ρ S ^ H , y ρ .
According to condition ( M 2 ) , we obtain
ρ < U ( y ρ ) 0 t | t s | β 1 Γ ( β ) I 1 α ϕ ρ ( s ) + | λ | I 1 α + p I 0 + q ψ ρ ( s ) d s + | a 1 ( t ) | { | δ | 0 μ | μ s | β 1 Γ ( β ) I 1 α ϕ ρ ( s ) + | λ | I 1 α + p I 0 + q ψ ρ ( s ) d s + 0 1 | 1 s | β 1 Γ ( β ) I 1 α ϕ ρ ( s ) + | λ | I 1 α + p I 0 + q ψ ρ ( s ) d s } + | a 2 ( t ) | 0 ξ | ξ s | β 1 Γ ( β ) I 1 α ϕ ρ ( s ) + | λ | I 1 α + p I 0 + q ψ ρ ( s ) d s 1 + a ¯ 1 ( | δ | + 1 ) + a ¯ 2 Γ ( β + 1 ) Γ ( α ) 0 1 ϕ ρ ( t ) d t + | λ | ( 1 + a ¯ 1 ( | δ | + 1 ) + a ¯ 2 ) Γ ( β + 1 ) Γ ( α + p ) Γ ( q ) 0 1 ψ ρ ( t ) d t Λ 3 0 1 ϕ ρ ( t ) d t + Λ 4 0 1 ψ ρ ( t ) d t ,
where Λ 3 , Λ 4 are given by (7). In (10), we have used the following estimates ( α ( 1 , 2 ] , β ( 0 , 1 ] , p > 0 , q > 1 ):
0 t ( t s ) β 1 Γ ( β ) I 1 α ϕ ρ ( s ) d s = 0 t ( t s ) β 1 Γ ( β ) s 1 ( u s ) α 1 Γ ( α ) ϕ ρ ( u ) d u d s 0 t ( t s ) β 1 Γ ( β ) ( 1 s ) α 1 Γ ( α ) d s 0 1 ϕ ρ ( u ) d u 0 t ( t s ) β 1 Γ ( β ) 1 Γ ( α ) d s 0 1 ϕ ρ ( u ) d u 1 Γ ( β + 1 ) Γ ( α ) 0 1 ϕ ρ ( u ) d u 0 t ( t s ) β 1 Γ ( β ) I 1 α + p I 0 + q ψ ρ ( s ) d s = 0 t ( t s ) β 1 Γ ( β ) s 1 ( u s ) α + p 1 Γ ( α + p ) 0 u ( u r ) q 1 Γ ( q ) ψ ρ ( r ) d r d u d s 0 t ( t s ) β 1 Γ ( β ) s 1 ( u s ) α + p 1 Γ ( α + p ) u q 1 Γ ( q ) d u d s 0 1 ψ ρ ( r ) d r 0 t ( t s ) β 1 Γ ( β ) s 1 1 Γ ( α + p ) 1 Γ ( q ) d u d s 0 1 ψ ρ ( r ) d r 0 t ( t s ) β 1 Γ ( β ) 1 s Γ ( α + p ) Γ ( q ) d s 0 1 ψ ρ ( r ) d r 1 Γ ( β + 1 ) Γ ( α + p ) Γ ( q ) 0 1 ψ ρ ( t ) d t .
Dividing both sides of (10) by ρ and then taking the lower limit as ρ , we find by (8) that ζ 1 Λ 3 + ζ 2 Λ 4 > 1 , which is a contradiction to the assumption (9). Hence there exists a positive number ρ ¯ such that U ( B ρ ¯ ) B ρ ¯ .
Step 3: U ( y ) maps bounded sets into equicontinuous sets of C ( [ 0 , 1 ] , R ) . For that, let 0 t 1 t 2 1 , y B ρ ¯ , and g U ( y ) . Then there exist f S F , y , h S ^ H , y such that, for each t [ 0 , 1 ] , we find that
g ( t ) = 0 t ( t s ) β 1 Γ ( β ) I 1 α f ( s ) λ I 1 α + p I 0 + q h ( s ) d s + a 1 ( t ) { δ 0 μ ( μ s ) β 1 Γ ( β ) I 1 α f ( s ) λ I 1 α + p I 0 + q h ( s ) d s 0 1 ( 1 s ) β 1 Γ ( β ) I 1 α f ( s ) λ I 1 α + p I 0 + q h ( s ) d s } + a 2 ( t ) 0 ξ ( ξ s ) β 1 Γ ( β ) I 1 α f ( s ) λ I 1 α + p I 0 + q h ( s ) d s ,
and that
| g ( t 2 ) g ( t 1 ) | = 0 t 1 | ( t 2 s ) β 1 ( t 1 s ) β 1 | Γ ( β ) I 1 α | f ( s ) | + | λ | I 1 α + p I 0 + q | h ( s ) | d s + t 1 t 2 | t 2 s | β 1 Γ ( β ) I 1 α | f ( s ) | + | λ | I 1 α + p I 0 + q | h ( s ) | d s + | a 1 ( t 2 ) a 1 ( t 1 ) | { | δ | 0 μ | μ s | β 1 Γ ( β ) I 1 α | f ( s ) | + | λ | I 1 α + p I 0 + q | h ( s ) | d s + 0 1 | 1 s | β 1 Γ ( β ) I 1 α | f ( s ) | + | λ | I 1 α + p I 0 + q | h ( s ) | d s } + | a 2 ( t 2 ) a 2 ( t 1 ) | 0 ξ | ξ s | β 1 Γ ( β ) I 1 α | f ( s ) | + | λ | I 1 α + p I 0 + q | h ( s ) | d s [ | t 2 β t 1 β | + 2 ( t 2 t 1 ) β + ( | δ | + 1 ) | Λ | ξ β + 1 | t 2 β t 1 β | + ξ β | t 1 β + 1 t 2 β + 1 | + 1 | Λ | | 1 δ μ β + 1 | | t 2 β t 1 β | + | 1 δ μ β | | t 1 β + 1 t 2 β + 1 | ] × 1 Γ ( β + 1 ) Γ ( α ) 0 1 ϕ ρ ( s ) d s + | λ | Γ ( β + 1 ) Γ ( α + p ) Γ ( q ) 0 1 ψ ρ ( s ) d s .
Clearly, the right-hand side of the above inequality tends to zero as t 2 t 1 independently of y B ρ ¯ . Hence U is equi-continuous. As U satisfies the above three steps, it follows by the Ascoli-Arzelá theorem that U is a compact multi-valued map.
Step 4: U has a closed graph. Let y n y * , g n U ( y n ) and g n g * . Then we need to show that g * U ( y * ) . Associated with g n U ( y n ) , we can find f n S F , y n , h n S ^ H , y n such that, for each t [ 0 , 1 ] , we have
g n ( t ) = 0 t ( t s ) β 1 Γ ( β ) I 1 α f n ( s ) λ I 1 α + p I 0 + q h n ( s ) d s + a 1 ( t ) { δ 0 μ ( μ s ) β 1 Γ ( β ) I 1 α f n ( s ) λ I 1 α + p I 0 + q h n ( s ) d s 0 1 ( 1 s ) β 1 Γ ( β ) I 1 α f n ( s ) λ I 1 α + p I 0 + q h n ( s ) d s } + a 2 ( t ) 0 ξ ( ξ s ) β 1 Γ ( β ) I 1 α f n ( s ) λ I 1 α + p I 0 + q h n ( s ) d s .
Thus it suffices to show that there exist f * S F , y * , h * S ^ H , y * such that for each t [ 0 , 1 ] ,
g * ( t ) = 0 t ( t s ) β 1 Γ ( β ) I 1 α f * ( s ) λ I 1 α + p I 0 + q h * ( s ) d s + a 1 ( t ) { δ 0 μ ( μ s ) β 1 Γ ( β ) I 1 α f * ( s ) λ I 1 α + p I 0 + q h * ( s ) d s 0 1 ( 1 s ) β 1 Γ ( β ) I 1 α f * ( s ) λ I 1 α + p I 0 + q h * ( s ) d s } + a 2 ( t ) 0 ξ ( ξ s ) β 1 Γ ( β ) I 1 α f * ( s ) λ I 1 α + p I 0 + q h * ( s ) d s .
Let us consider the continuous linear operator Θ : L 1 ( [ 0 , 1 ] , R ) C ( [ 0 , 1 ] ) so that
Θ ( f , h ) ( t ) = 0 t ( t s ) β 1 Γ ( β ) I 1 α f ( s ) λ I 1 α + p I 0 + q h ( s ) d s + a 1 ( t ) { δ 0 μ ( μ s ) β 1 Γ ( β ) I 1 α f ( s ) λ I 1 α + p I 0 + q h ( s ) d s 0 1 ( 1 s ) β 1 Γ ( β ) I 1 α f ( s ) λ I 1 α + p I 0 + q h ( s ) d s } + a 2 ( t ) 0 ξ ( ξ s ) β 1 Γ ( β ) I 1 α f ( s ) λ I 1 α + p I 0 + q h ( s ) d s .
Observe that
g n ( t ) g * ( t ) = 0 t ( t s ) β 1 Γ ( β ) I 1 α ( f n ( s ) f * ( s ) ) λ I 1 α + p I 0 + q ( h n ( s ) h * ( s ) ) d s + a 1 ( t ) { δ 0 μ ( μ s ) β 1 Γ ( β ) I 1 α ( f n ( s ) f * ( s ) ) λ I 1 α + p I 0 + q ( h n ( s ) h * ( s ) ) d s 0 1 ( 1 s ) β 1 Γ ( β ) I 1 α ( f n ( s ) f * ( s ) ) λ I 1 α + p I 0 + q ( h n ( s ) h * ( s ) ) d s } + a 2 ( t ) 0 ξ ( ξ s ) β 1 Γ ( β ) I 1 α ( f n ( s ) f * ( s ) ) λ I 1 α + p I 0 + q ( h n ( s ) h * ( s ) ) d s 0 as n .
Thus, it follows by Lemma 5 that Θ S B is a closed graph operator where S B = S F S ^ H . Moreover, we have g n ( t ) Θ ( S B , y n ) . Since y n y * , g n g * , therefore, Lemma 5 yields
g * ( t ) = 0 t ( t s ) β 1 Γ ( β ) I 1 α f * ( s ) λ I 1 α + p I 0 + q h * ( s ) d s + a 1 ( t ) { δ 0 μ ( μ s ) β 1 Γ ( β ) I 1 α f * ( s ) λ I 1 α + p I 0 + q h * ( s ) d s 0 1 ( 1 s ) β 1 Γ ( β ) I 1 α f * ( s ) λ I 1 α + p I 0 + q h * ( s ) d s } + a 2 ( t ) 0 ξ ( ξ s ) β 1 Γ ( β ) I 1 α f * ( s ) λ I 1 α + p I 0 + q h * ( s ) d s ,
for some f * S F , y * , h * S ^ H , y * .
Hence, we conclude that U is a compact and upper semi-continuous multi-valued map with convex closed values. Thus, the hypothesis of Lemma 4 holds true, and therefore its conclusion implies that the operator U has a fixed point y, which corresponds to a solution of problem (2). This completes the proof. ☐
Next, we give an existence result based upon the following form of fixed point theorem due to Martelli [38], which is applicable to completely continuous operators.
Lemma 6.
Let X a Banach space, and T : X P b , c l , c ( X ) be a completely continuous multi-valued map. If the set E = { x X : κ x T ( x ) , κ > 1 } is bounded, then T has a fixed point.
Theorem 3.
Assume that the following hypotheses hold:
( M 3 )
F , H : [ 0 , 1 ] × R P b , c l , c ( R ) are L 1 -Carathéodory multi-valued maps; that is, (i) t F ( t , y ) , t H ( t , y ) , are measurable for each y R ; (ii) y F ( t , y ) , y H ( t , y ) are upper semicontinuous for almost all t [ 0 , 1 ] ; (iii) for each r > 0 , there exist ϕ r , ψ r L 1 ( [ 0 , 1 ] , R + ) such that F ( t , y ) = sup { | v | : v F ( t , y ) } ϕ r ( t ) , H ( t , y ) = sup { | v | : v F ( t , y ) } ψ r ( t ) , for all y R with y r and for almost every t [ 0 , 1 ] .
( M 4 )
There exist functions z , u L 1 ( [ 0 , 1 ] , R + ) such that
F ( t , y ) z ( t ) , H ( t , y ) u ( t ) , for a . e . t [ 0 , 1 ] a n d each y R .
Then the problem (2) has at least one solution on [ 0 , 1 ] .
Proof. 
Consider U defined in the proof of Theorem 2. As in Theorem 2, we can show that U is convex and completely continuous. It remains to show that the set
E = { y C ( [ 0 , 1 ] , R ) : κ y U ( y ) , κ > 1 }
is bounded. Let y E , then κ y U ( y ) for some κ > 1 and there exist functions f S F , y , h S ^ H , y such that
y ( t ) = 0 t ( t s ) β 1 Γ ( β ) I 1 α f ( s ) λ I 1 α + p I 0 + q h ( s ) d s + a 1 ( t ) { δ 0 μ ( μ s ) β 1 Γ ( β ) I 1 α f ( s ) λ I 1 α + p I 0 + q h ( s ) d s 0 1 ( 1 s ) β 1 Γ ( β ) I 1 α f ( s ) λ I 1 α + p I 0 + q h ( s ) d s } + a 2 ( t ) 0 ξ ( ξ s ) β 1 Γ ( β ) I 1 α f ( s ) λ I 1 α + p I 0 + q h ( s ) d s .
For each t [ 0 , 1 ] , we have
| y ( t ) | 0 t | t s | β 1 Γ ( β ) I 1 α z ( s ) + | λ | I 1 α + p I 0 + q u ( s ) d s + | a 1 ( t ) | { | δ | 0 μ | μ s | β 1 Γ ( β ) I 1 α z ( s ) + | λ | I 1 α + p I 0 + q u ( s ) d s + 0 1 | 1 s | β 1 Γ ( β ) I 1 α z ( s ) + | λ | I 1 α + p I 0 + q u ( s ) d s } + | a 2 ( t ) | 0 ξ | ξ s | β 1 Γ ( β ) I 1 α z ( s ) + | λ | I 1 α + p I 0 + q u ( s ) d s 1 + a ¯ 1 ( | δ | + 1 ) + a ¯ 2 Γ ( β + 1 ) Γ ( α ) z L 1 + | λ | ( 1 + a ¯ 1 ( | δ | + 1 ) + a ¯ 2 ) Γ ( β + 1 ) Γ ( α + p ) Γ ( q ) u L 1 Λ 3 z L 1 + Λ 4 u L 1 ,
Taking the supremum over t J , we get
y Λ 3 z L 1 + Λ 4 u L 1 < .
Hence the set E is bounded. As a consequence of Lemma 6 we deduce that U has at least one fixed point which implies that the problem (2) has a solution on [ 0 , 1 ] . ☐

5. Applications

We consider four different cases for F ( t , y ) and H ( t , y ) (in (2)) to demonstrate applications of theorem (2): ( a ) F and H have sub-linear growth in their second variable. ( b ) F and H have linear growth in their second variable. ( c ) F has sub-linear growth in its second variable and H has linear growth. ( d ) F has linear growth in its second variable and H has sub-linear growth.
Case ( a ) . For each ( t , y ) [ 0 , 1 ] × R , there exist functions σ i ( t ) , ϑ i ( t ) L 1 ( [ 0 , 1 ] , R + ) , i = 1 , 2 , γ [ 0 , 1 ) such that F ( t , y ) σ 1 ( t ) | y | γ + ϑ 1 ( t ) and H ( t , y ) σ 2 ( t ) | y | γ + ϑ 2 ( t ) which correspond in this case to ϕ ρ ( t ) = σ 1 ( t ) ρ γ + ϑ 1 ( t ) and ψ ρ ( t ) = σ 2 ( t ) ρ γ + ϑ 2 ( t ) and the condition (9) will take the form 0 · Λ 3 + 0 · Λ 4 < 1 , that is, ζ 1 = ζ 2 = 0 .
Case ( b ) . F and H will satisfy the assumptions F ( t , y ) σ 1 ( t ) | y | + ϑ 1 ( t ) and H ( t , y ) σ 2 ( t ) | y | + ϑ 2 ( t ) , which, in view of ( M 2 ) , implies that ϕ ρ ( t ) = σ 1 ( t ) ρ + ϑ 1 ( t ) and ψ ρ ( t ) = σ 2 ( t ) ρ + ϑ 2 ( t ) , and the condition (9) becomes σ 1 L 1 · Λ 3 + σ 2 L 1 · Λ 4 < 1 .
Similarly, one can verify the cases ( c ) and ( d ) . Thus, the boundary value problem (2) has at least one solution on [ 0 , 1 ] for all the cases ( a ) ( d ) .
Let us consider the following inclusions problem:
D 1 5 / 4 C D 0 + 3 / 4 R L y ( t ) F ( t , y ( t ) ) 2 I 1 3 / 2 I 0 + 5 / 2 H ( t , y ( t ) ) , t [ 0 , 1 ] , y ( 0 ) = y ( 1 / 3 ) = 0 , y ( 1 ) = 1 4 y ( 2 / 3 ) ,
where α = 5 / 4 , β = 3 / 4 , λ = 2 , p = 3 / 2 , q = 5 / 2 , ξ = 1 / 3 , μ = 2 / 3 , δ = 1 / 4 . It is easy to find that
a ¯ 1 = max t [ 0 , 1 ] | a 1 ( t ) | = | a 1 ( t ) | t = 1 1.101592729739686 , a ¯ 2 = max t [ 0 , 1 ] | a 2 ( t ) | = | a 2 ( t ) | t = t a 2 1.055901462873258 ,
where
t a 2 = β ( 1 δ μ β + 1 ) ( 1 δ μ β ) ( β + 1 ) 0.460880265746053 < 1 .
Using the above given data, we find that Λ 3 4.120918689155884 , Λ 4 3.494023466997676 , where Λ 3 , Λ 4 are given by (7).
( a ) . We consider F ( t , y ) σ 1 ( t ) | y | 1 / 3 + ϑ 1 ( t ) and H ( t , y ) σ 2 ( t ) | y | 1 / 3 + ϑ 2 ( t ) with σ i ( t ) , ϑ i ( t ) L 1 ( [ 0 , 1 ] , R + ) , i = 1 , 2 , γ [ 0 , 1 ) . In this case, F and H in (11) satisfy all the assumptions of Theorem 2 with 0 · Λ 3 + 0 · Λ 4 < 1 , which implies that the boundary value problem (11) has at least one solution on [ 0 , 1 ] .
( b ) As a second example, let F and H be such that F ( t , y ) 1 4 ( 1 + t ) 2 | y | + 2 e t and H ( t , y ) 2 ( 4 + t ) 2 | y | + e t . In this case, the condition (9) will take the form 1 8 · Λ 3 + 1 10 · Λ 4 0.864517182844253 < 1 . Thus, by the conclusion of Theorem 2, there exists at least one solution for the problem (11) on [ 0 , 1 ] .
In a similar manner, one can verify that the problem (2) has at least one solution on [ 0 , 1 ] when we choose the cases: ( c ) F ( t , y ) σ 1 ( t ) | y | 1 / 3 + ϑ 1 ( t ) , H ( t , y ) 2 ( 4 + t ) 2 | y | + e t , and ( d ) F ( t , y ) 1 4 ( 1 + t ) 2 | y | + 2 e t , H ( t , y ) σ 2 ( t ) | y | 1 / 3 + ϑ 2 ( t ) .

6. Conclusions

In this paper, we have discussed the existence of solutions for a new class of boundary value problems involving right-Caputo and left-Riemann–Liouville fractional derivatives of different orders and right-left Riemann–Liouville fractional integrals with nonlocal boundary conditions. The existence result for the single-valued case of the given problem is proven via Sadovski’s fixed point theorem, while the existence results for the multi-valued case of the problem at hand are derived by means of Bohnenblust-Karlin and Martelli fixed point theorems. Applications for the obtained results are also presented. By taking δ = 0 in the results of this paper, we obtain the ones for a problem associated with three-point nonlocal boundary conditions: y ( 0 ) = 0 , y ( ξ ) = 0 , y ( 1 ) = 0 ( 0 < ξ < 1 ) as a special case.

Author Contributions

Conceptualization, S.K.N. and B.A.; formal analysis, A.A., A.B., S.K.N. and B.A.; funding acquisition, A.A.; methodology, A.A., A.B., S.K.N. and B.A. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Paola, M.D.; Pinnola, F.P.; Zingales, M. Fractional differential equations and related exact mechanical models. Comput. Math. Appl. 2013, 66, 608–620. [Google Scholar] [CrossRef] [Green Version]
  2. Ahmed, N.; Vieru, D.; Fetecau, C.; Shah, N.A. Convective flows of generalized time-nonlocal nanofluids through a vertical rectangular channel. Phys. Fluids 2018, 30, 052002. [Google Scholar] [CrossRef]
  3. Tarasov, V.E. Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media; Springer: New York, NY, USA, 2010. [Google Scholar]
  4. Tarasova, V.V.; Tarasov, V.E. Logistic map with memory from economic model. Chaos Solitons Fractals 2017, 95, 84–91. [Google Scholar] [CrossRef] [Green Version]
  5. Henderson, J.; Luca, R.; Tudorache, A. On a system of fractional differential equations with coupled integral boundary conditions. Fract. Calc. Appl. Anal. 2015, 18, 361–386. [Google Scholar] [CrossRef]
  6. Peng, L.; Zhou, Y. Bifurcation from interval and positive solutions of the three-point boundary value problem for fractional differential equations. Appl. Math. Comput. 2015, 257, 458–466. [Google Scholar] [CrossRef]
  7. Ahmad, B.; Alsaedi, A.; Ntouyas, S.K.; Tariboon, J. Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities; Springer: Cham, Switzerland, 2017. [Google Scholar]
  8. Ahmad, B.; Ntouyas, S.K. Nonlocal initial value problems for Hadamard-type fractional differential equations and inclusions. Rocky Mt. J. Math. 2018, 48, 1043–1068. [Google Scholar] [CrossRef]
  9. Cui, Y.; Ma, W.; Sun, Q.; Su, X. New uniqueness results for boundary value problem of fractional differential equation. Nonlinear Anal. Model. Control 2018, 23, 31–39. [Google Scholar]
  10. Ahmad, B.; Alghamdi, N.; Alsaedi, A.; Ntouyas, S.K. A system of coupled multi-term fractional differential equations with three-point coupled boundary conditions. Fract. Calc. Appl. Anal. 2019, 22, 601–618. [Google Scholar] [CrossRef]
  11. Alsaedi, A.; Ahmad, B.; Alghanmi, M. Extremal solutions for generalized Caputo fractional differential equations with Steiltjes-type fractional integro-initial conditions. Appl. Math. Lett. 2019, 91, 113120. [Google Scholar] [CrossRef]
  12. Liang, S.; Wang, L.; Yin, G. Fractional differential equation approach for convex optimization with convergence rate analysis. Optim. Lett. 2020, 14, 145–155. [Google Scholar] [CrossRef]
  13. Zhang, L.; Ahmad, B.; Wang, G. The existence of an extremal solution to a nonlinear system with the right-handed Riemann-Liouville fractional derivative. Appl. Math. Lett. 2014, 31, 1–6. [Google Scholar] [CrossRef]
  14. Khaldi, R.; Guezane-Lakoud, A. Higher order fractional boundary value problems for mixed type derivatives. J. Nonlinear Funct. Anal. 2017. [Google Scholar] [CrossRef]
  15. Lakoud, A.G.; Khaldi, R.; Kilicman, A. Existence of solutions for a mixed fractional boundary value problem. Adv. Differ. Equ. 2017, 2017, 164. [Google Scholar] [CrossRef]
  16. Ahmad, B.; Ntouyas, S.K.; Alsaedi, A. Existence theory for nonlocal boundary value problems involving mixed fractional derivatives. Nonlinear Anal. Model. Control 2019, 24, 937–957. [Google Scholar] [CrossRef]
  17. Atanackovic, T.M.; Stankovic, B. On a differential equation with left and right fractional derivatives. Fract. Calc. Appl. Anal. 2007, 10, 139–150. [Google Scholar] [CrossRef]
  18. Yang, A.M.; Han, Y.; Zhang, Y.Z.; Wang, L.T.; Zhang, D.; Yang, X.J. On nonlocal fractional Volterra integro-differential equations in fractional steady heat transfer. Therm. Sci. 2016, 20, S789–S793. [Google Scholar] [CrossRef] [Green Version]
  19. Tarasov, V.E. Fractional integro-differential equations for electromagnetic waves in dielectric media. Theor. Math. Phys. 2009, 158, 355–359. [Google Scholar] [CrossRef] [Green Version]
  20. Korda, M.; Henrion, D.; Jones, C.N. Convex computation of the maximum controlled invariant set for polynomial control systems. SIAM J. Control Optim. 2014, 52, 2944–2969. [Google Scholar] [CrossRef]
  21. Danca, M.F. Synchronization of piecewise continuous systems of fractional order. Nonlinear Dyn. 2014, 78, 2065–2084. [Google Scholar] [CrossRef] [Green Version]
  22. Kisielewicz, M. Stochastic Differential Inclusions and Applications; Springer: New York, NY, USA, 2013. [Google Scholar]
  23. Bastien, J. Study of a driven and braked wheel using maximal monotone differential inclusions: Applications to the nonlinear dynamics of wheeled vehicles. Arch. Appl. Mech. 2014, 84, 851–880. [Google Scholar] [CrossRef]
  24. Yue, Y.; Tian, Y.; Bai, Z. Infinitely many nonnegative solutions for a fractional differential inclusion with oscillatory potential. Appl. Math. Lett. 2019, 88, 64–72. [Google Scholar] [CrossRef]
  25. Wang, Y.; Liang, T. Mild solutions to the time fractional Navier-Stokes delay differential inclusions. Discrete Contin. Dyn. Syst. Ser. B 2019, 24, 3713–3740. [Google Scholar] [CrossRef] [Green Version]
  26. Benchohra, M.; Hamani, S.; Zhou, Y. Oscillation and nonoscillation for Caputo-Hadamard impulsive fractional differential inclusions. Adv. Differ. Equ. 2019, 2019, 74. [Google Scholar] [CrossRef]
  27. Kamenskii, M.; Obukhovskii, V.; Petrosyan, G.; Yao, J.-C. On semilinear fractional order differential inclusions in Banach spaces. Fixed Point Theory 2017, 18, 269–291. [Google Scholar] [CrossRef] [Green Version]
  28. Cheng, Y.; Agarwal, R.P.; O’Regan, D. Existence and controllability for nonlinear fractional differential inclusions with nonlocal boundary conditions and time-varying delay. Fract. Calc. Appl. Anal. 2018, 21, 960–980. [Google Scholar] [CrossRef]
  29. Abbas, S.; Benchohra, M.; Graef, J.R. Coupled systems of Hilfer fractional differential inclusions in Banach spaces. Commun. Pure Appl. Anal. 2018, 17, 2479–2493. [Google Scholar] [CrossRef] [Green Version]
  30. Ntouyas, S.K.; Alsaedi, A.; Ahmed, B. Existence theorems for mixed Riemann-Liouville and Caputo fractional differential equations and inclusions with nonlocal fractional integro-differential boundary conditions. Fractal Fract. 2019, 3, 21. [Google Scholar] [CrossRef] [Green Version]
  31. Vijayakumar, V. Approximate controllability results for non-densely defined fractional neutral differential inclusions with Hille-Yosida operators. Internat. J. Control 2019, 92, 2210–2222. [Google Scholar] [CrossRef]
  32. Ahmad, B.; Ntouyas, S.K.; Alsaedi, A. Coupled systems of fractional differential inclusions with coupled boundary conditions. Electron. J. Differ. Equ. 2019, 2019, 1–21. [Google Scholar]
  33. Ahmad, B.; Ntouyas, S.K.; Tariboon, J. On inclusion problems involving Caputo and Hadamard fractional derivatives. Acta Math. Univ. Comenian. (N.S.) 2020, 89, 169–183. [Google Scholar]
  34. Ahmad, B.; Broom, A.; Alsaedi, A.; Ntouyas, S.K. Nonlinear integro-differential equations involving mixed right and left fractional derivatives and integrals with nonlocal boundary data. Mathematics 2020, 8, 336. [Google Scholar] [CrossRef] [Green Version]
  35. Li, T. A class of nonlocal boundary value problems for partial differential equations and its applications in numerical analysis. J. Comput. Appl. Math. 1989, 28, 49–62. [Google Scholar]
  36. Ahmad, B.; Nieto, J.J. Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations. Abstr. Appl. Anal. 2009, 2009, 494720. [Google Scholar] [CrossRef]
  37. Bohnenblust, H.F.; Karlin, S. On a theorem of Ville. In Contributions to the Theory of Games; Princeton University Press: Princeton, NJ, USA, 1950; Volume I, pp. 155–160. [Google Scholar]
  38. Martelli, M. A Rothe’s theorem for non compact acyclic-valued maps. Boll. Un. Mat. Ital. 1975, 4, 70–76. [Google Scholar]
  39. Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; North-Holland Mathematics Studies, 204; Elsevier Science B.V.: Amsterdam, The Netherlands, 2006. [Google Scholar]
  40. Granas, A.; Dugundji, J. Fixed Point Theory; Springer-Verlag: New York, NY, USA, 2005. [Google Scholar]
  41. Zeidler, E. Nonlinear Functional Analysis and Its Application: Fixed Point-Theorems; Springer-Verlag: New York, NY, USA, 1986; Volume 1. [Google Scholar]
  42. Sadovskii, B.N. On a fixed point principle. Funct. Anal. Appl. 1967, 1, 74–76. [Google Scholar] [CrossRef]
  43. Deimling, K. Multivalued Differential Equations; De Gruyter: Berlin, Germany, 1992. [Google Scholar]
  44. O’Regan, D.; Precup, R. Fixed point theorems for set-valued maps and existence principles for integral inclusions. J. Math. Anal. Appl. 2000, 245, 594–612. [Google Scholar] [CrossRef] [Green Version]
  45. Lasota, A.; Opial, Z. An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 1965, 13, 781–786. [Google Scholar]

Share and Cite

MDPI and ACS Style

Alsaedi, A.; Broom, A.; Ntouyas, S.K.; Ahmad, B. Nonlocal Fractional Boundary Value Problems Involving Mixed Right and Left Fractional Derivatives and Integrals. Axioms 2020, 9, 50. https://doi.org/10.3390/axioms9020050

AMA Style

Alsaedi A, Broom A, Ntouyas SK, Ahmad B. Nonlocal Fractional Boundary Value Problems Involving Mixed Right and Left Fractional Derivatives and Integrals. Axioms. 2020; 9(2):50. https://doi.org/10.3390/axioms9020050

Chicago/Turabian Style

Alsaedi, Ahmed, Abrar Broom, Sotiris K. Ntouyas, and Bashir Ahmad. 2020. "Nonlocal Fractional Boundary Value Problems Involving Mixed Right and Left Fractional Derivatives and Integrals" Axioms 9, no. 2: 50. https://doi.org/10.3390/axioms9020050

APA Style

Alsaedi, A., Broom, A., Ntouyas, S. K., & Ahmad, B. (2020). Nonlocal Fractional Boundary Value Problems Involving Mixed Right and Left Fractional Derivatives and Integrals. Axioms, 9(2), 50. https://doi.org/10.3390/axioms9020050

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop