Open access peer-reviewed chapter - ONLINE FIRST

A Study of Nonlinear Boundary Value Problem

By Noureddine Bouteraa and Habib Djourdem

Reviewed: September 16th 2021Published: November 3rd 2021

DOI: 10.5772/intechopen.100491

Downloaded: 28

Abstract

In this chapter, firstly we apply the iterative method to establish the existence of the positive solution for a type of nonlinear singular higher-order fractional differential equation with fractional multi-point boundary conditions. Explicit iterative sequences are given to approximate the solutions and the error estimations are also given. Secondly, we cover the multi-valued case of our problem. We investigate it for nonconvex compact valued multifunctions via a fixed point theorem for multivalued maps due to Covitz and Nadler. Two illustrative examples are presented at the end to illustrate the validity of our results.

Keywords

  • Positive solution
  • Uniqueness
  • Iterative sequence
  • Green’s function
  • Fractional differential equation and inclusion
  • Existence
  • Nonlocal boundary value problem
  • Fixed point theorem

1. Introduction

In this chapter, we are interested in the existence of solutions for the nonlinear fractional boundary value problem (BVP)

D0+αut+ftut=0,t01,ui0=0,i0,1,2n2,D0+βu1=pj=1ajD0+βuηj.E1

We also cover the multi-valued case of problem

D0+αutFtut,t01,ui0=0,i0,1,2n2,D0+βu1=pj=1ajD0+βuηj,E2

where D0+α,D0+βare the standard Riemann-Liouville fractional derivative of order

αn1n,β1n2fornNandn3,

where D0+α,D0+βare the stantard Riemann-Liouville fractional derivative of order αn1n,β1n2forn3, the function fC01×RR, the multifunction F:01×R2Rare allowed to be singular at t=0and/or t=1and ajR+,j=1,2,,p,0<η1<η2<<ηp<1,forpN.

The first definition of fractional derivative was introduced at the end of the nineteenth century by Liouville and Riemann, but the concept of non-integer derivative and integral, as a generalization of the traditional integer order differential and integral calculus, was mentioned already in 1695 by Leibniz [1] and L’Hospital [2]. In fact, fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. The mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, Bode’s analysis of feedback amplifiers, capacitor theory, electrical circuits, electro-analytical chemistry, biology, control theory, fitting of experimental data, involves derivatives of fractional order. In consequence, the subject of fractional differential equations is gaining much importance and attention. For more details we refer the reader to [1, 2, 3, 4, 5, 6] and the references cited therein.

Boundary value problems for nonlinear differential equations arise in a variety of areas of applied mathematics, physics and variational problems of control theory. A point of central importance in the study of nonlinear boundary value problems is to understand how the properties of nonlinearity in a problem influence the nature of the solutions to the boundary value problems. The multi-point boundary conditions are important in various physical problems of applied science when the controllers at the end points of the interval (under consideration) dissipate or add energy according to the sensors located, at intermediate points, see [7, 8] and the references therein. We quote also that realistic problems arising from economics, optimal control, stochastic analysis can be modeled as differential inclusion. The study of fractional differential inclusions was initiated by El-Sayed and Ibrahim [9]. Also, recently, several qualitative results for fractional differential inclusion were obtained in [10, 11, 12, 13] and the references therein.

The techniques of nonlinear analysis, as the main method to deal with the problems of nonlinear differential equations (DEs), nonlinear fractional differential equations (FDEs), nonlinear partial differential equations (PDEs), nonlinear fractional partial differential equations (FPDEs), nonlinear stochastic fractional partial differential equations (SFPDEs), plays an essential role in the research of this field, such as establishing the existence, uniqueness and multiplicity of solutions (or positive solutions) and mild solutions for nonlinear of different kinds of FPDEs, FPDEs, SFPDEs, inclusion differential equations and inclusion fractional differential equations with various boundary conditions, by using different techniques (approaches). For more details, see [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37] and the references therein. For example, iterative method is an important tool for solving linear and nonlinear Boundary Value Problems. It has been used in the research areas of mathematics and several branches of science and other fields. However, Many authors showed the existence of positive solutions for a class of boundary value problem at resonance case. Some recent devolopment for resonant case can be found in [38, 39]. Let us cited few papers. In [40], the authors studied the boundary value problems of the fractional order differential equation:

D0+αut=ftut=0,t01,u0=0,D0+βu1=aD0+βuη,

where 1<α2,0<η<1,0<a,β<1,fC01×R2Rand D0+α,D0+βare the stantard Riemann-Liouville fractional derivative of order α. They obtained the multiple positive solutions by the Leray-Schauder nonlinear alternative and the fixed point theorem on cones.

In 2020Li et al. [41] consider the existence of a positive solution for the following BVP of nonlinear fractional differential equation with integral boundary conditions:

CD0+qut+ftut=0,t01,u0=0,αu0βu0=01h1susds,γu1+δCD0+σu1=01h2sds,

where 2<q3, 0<σ1, α,γ,δ0, and β>0satisfying 0<ρα+βγ+αδΓ2σ<βγ+δΓqΓqσ, f:01×0+0+and hii=12:010+are continuous. To obtain the existence results, the authors used the well-known GuoKrasnoselskiis fixed point theorem.

In 2017, Rezapour et al. [42] investigated a Caputo fractional inclusion with integral boundary condition for the following problem

cDαutFtut,cDβutut,u0+u0+cDβu0=0ηusds,u1+u1+cDβu1=0νusds,

where 1<α2,η,ν,β01,F:01×R×R×R2Ris a compact valued multifunction and cDαdenotes the Caputo fractional derivative of order α.

In 2018, Bouteraa and Benaicha [10] studied the existence of solutions for the Caputo fractional differential inclusion

cDαutFtutut,tJ=01,

subject to three-point boundary conditions

βu0+γu1=uη,u0=0ηusds,βcDpu0+γcDpu1=cDpuη,

where 2<α3,1<p2,0<η<1,β,γR+,F:01×R×R2Ris a compact valued multifunction and cDαdenotes the Caputo fractional derivative of order α.

In 2019, Ahmad et al. [43] investigated the existence of solutions for the boundary value problem of coupled Caputo (Liouville-Caputo) type fractional differential inclusions:

CDαxtFtxtyt,t0T,1<α2,CDβytFtxtyt,t0T,1<β2,

subject to the coupled boundary conditions:

x0=ν1yT,x0=ν2yT,y0=μ1xT,y0=μ2xT,

where CDα, CDβdenote the Caputo fractional derivatives of order αand βrespectively, F,G:0T×R×Rare given multivalued maps, PRis the family of all nonempty subsets of R, and νi,μi,i=1,2are real constants with νiμi1,i=1,2.

Inspired and motivated by the works mentioned above, we focus on the uniqueness of positive solutions for the nonlocal boundary value problem (1) with the iterative method and properties of ftu, explicit iterative sequences are given to approximate the solutions and the error estimations are also given. We also cover the multi-valued case of problem (2) when the right-hand side is nonconvex compact valued multifunctions via a fixed point theorem for multivalued maps due to Covitz and Nadler.

The chapter is organized as follows. In Section 2, we present some notations and lemmas that will be used to prove our main results of problem (1) and we discuss the uniqueness of problem (1). Finally, we give an example to illustrate our result. In Section 3, we introduce some definitions and preliminary results about essential properties of multifunction that will be used in the remainder of the chapter and we present existence results for the problem (2) when the right-hand side is a non-convex compact multifunction. We shall use the fixed point theorem for contraction multivalued maps due to Covitz and Nadler [44] to prove the uniqueness of solution of problem (2). Finally, we give an example to ascertain the main result.

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2. Existence and uniqueness results for problem (2)

2.1 Preliminaries

In this section, we recall some definitions and facts which will be used in the later analysis. These details can be found in the recent literature; see [2, 4, 6, 45, 46, 47] and the references therein.

Let ACi01Rdenote the space of itimesdifferentiable functions u:01Rwhose ithderivative uiis absolutely continuous and αdonotes the integer part of number α.

Definition 2.1. Let α>0,n1<α<n,n=α+1and uACn0R.

The Caputo derivative of fractional order αfor the function u:0+Ris defined by

cDαut=1Γnα0ttsnα1unsds.

The Riemann-Liouville fractional derivative order αfor the function u:0+Ris defined by

D0+αut=1Γnαdndtn0ttsnα1usds,t>0,

provided that the right hand side is pointwise defined in 0and the function Γ:0R, defined by

Γu=0tu1etdt,

is called Euler’s gamma function.

Definition 2.2. The Riemann-Liouville fractional integral of order α>0of a function u:0Ris given by

Iαut=1Γα0ttsα1usds,t>0,

provided that the right hand side is pointwise defined in 0.

We recall in the following lemma some properties involving Riemann-Liouville fractional integral and Riemann-Liouville fractional derivative or Caputo fractional derivative which are need in Lemma 2.4.

Lemma 2.1. (([45], Prop.4.3), [46]) Letα,β0anduL101. Then the next formulas hold.

  1. DβIαut=Iαβut,

  2. DαIαut=ut,

  3. I0+αI0+βut=I0+α+βut.

  4. Ifβ>α>0, thenDαtβ1=Γβtβα1Γβα. whereDαandDβrepresents Riemann-Liouville’s or Caputo’s fractional derivative of orderαandβrespectively.

Lemma 2.2 [47]. Letα>0andyL101. Then, the general solution of the fractional differential equationD0+αut+yt=0,0<t<1is given by

ut=1Γα0ttsα1ysds+c1tα1+c2tα2++cntαn,0<t<1,

wherec0,c1,,cn1are real constants andn=α+1.

Based on the previous Lemma 2.2, we will define the integral solution of our problem 1.

Lemma 2.3. Letpj=1ajηjαβ101,αn1n,β1n2,n3andyC01. Then the solution of the fractional boundary value problem

D0+αut+yt=0,ui0=0,i0,1,2n2,D0+βu1=pj=1ajD0+βuηj,E3

is given by

ut=01Gtsysds,E4

where

Gts=gts+tα1dpj=1ajhηjs,E5
gts=1Γαtα11sαβ1tsα1,0st1,tα11sαβ1,0ts1,E6
hts=1Γαtαβ11sαβ1tsαβ1,0st1,tαβ11sαβ1,0ts1,E7

whered=1j=1pajηjαβ1.

Proof. By using Lemma 2.2, the solution of the equation D0+αut+yt=0is

ut=1Γα0ttsα1ysds+c1tα1+c2tα2++cntαn,

where c1,c2,cnare arbitrary real constants.

From the boundary condition in (1), one can c2=c3=cn2=cn1=cn=0. Hence

ut=1Γα0ttsα1ysds+c1tα1.

By the last above equation and Lemma 2.1i, we get

D0+βut=1Γαβc1Γαtαβ10ttsαβ1ysds,

this and by D0+βu1=j=1pajD0+βuηj, we have

c1=1dΓα011sαβ1ysdspj=1aj0ηjηjsαβ1ysds.

Then, the unique solution of the problem (1) is given by

ut=tα1dΓα011sαβ1ysdspj=1aj0ηjηjsαβ1ysds1Γα0ttsα1ysds,=1Γα0ttα11sαβ1tsα1ysds+t1tα11sαβ1ysds+1dd01tα11sαβ1ysdstα1dpj=1aj0ηjηjsαβ1ysds=01gtsysds+tα1dpj=1ajηj1ηjαβ11sαβ1ysds+0ηjηjαβ11sαβ1ηjsαβ1ysds=01gtsysds+tα1dpj=1aj01hηjsysds=01Gtsysds.

The proof is completed.

Lemma 2.4. Letj=1pajηjαβ101,αn1n,β1n2,n3. Then, the functionsgtsandhtsdefined by(6)and(7)have the following properties:

  1. The functionsgtsandhtsare continuous on01×01and for allt,s01

    gts>0,hts>0.

  • gtstα1Γαfor allt,s01.

  • gtstα1g1sfor allt,s01, where

    g1s=1Γα1sαβ11sα1.

    From the above properties, we deduce the following properties:

  • The functionGts0is continuous on01×01andGts>0for allt,s01.

  • maxt01Gts=G1s, for alls01, where

  • G1s=g1s+1dpj=1ajhηjs1sαβ1dΓα.

    Proof. It is easy to chek that i,v,viholds. So we prove that iiis true. Note that (6) and 01sαβ11. It follows that gtstα1Γαfor all t,s01. It remains to prove iii. We divide the proof into two cases and by (1), we have.

    Case1. When 0st1, we have

    gts=1Γαtα11sαβ11stα1tα1g1s.

    Case2. When 0ts1, we have

    gts=1Γαtα11sαβ1tα1g1s.

    Hence gtstα1g1sfor all t,s01.

    2.2 Existence results

    First, for the uniqueness results of problem (1), we need the following assumptions.

    A1ftu1ftu2for any 0<t<1,0u1u2.

    A2For any r01, there exists a constant q01such that

    ftrurqftu,tu01×0.E8

    A30<01fssα1ds<.

    We shall consider the Banach space E=C01equipped with the norm u=max0t1utand let

    D=uC+01:Mumu0\mutα1utMutα1fort01,E9

    where

    C+01=uE:ut0t01.

    In view of Lemma 2.3, we define an operator Tas

    Tut=01Gtsysds,E10

    where Gtsis given by (5).

    By A1it is easy to see that the operator T:DC+01is increasing. Observe that the BVP (1) has a solution if and only if the operator Thas a fixed point.

    Obviously, from A1we obtain

    ftrurqftu,r>1,q01,tu01×0.

    In what follows, we first prove T:DD. In fact, for any uD, there exist a positive constants 0<mu<1<Musuch that

    musα1usMusα1,s01.

    Then, from A1, ftunon-decreasing respect to uand A2, we can imply that for s01,q01

    muqfssα1fsusMuqfssα1,s01.E11

    From (11) and Lemma 2.4, we obtain

    Tut=01gtsfsusds+tα1dpj=1aj01hηjsfsusds,tα11Γα01fsusds+1dpj=1aj01hηjsfsusds,tα1MuqΓα01fssα1ds+Muqdpj=1aj01hηjsfssα1ds,t01,E12

    and

    Tut=01gtsfsusds+tα1dpj=1aj01hηjsfsusds,tα11Γα01g1sfsusds+1dpj=1aj01hηjsfsusds,tα1muqΓα01g1sfssα1ds+muqdpj=1aj01hηjsfssα1ds,t01.E13

    Eqs. (12) and (13) and assumption A3imply that T:DD.

    Now, we are in the position to give the first main result of this chapter.

    Theorem 1.1 Suppose A1A3hold. Then problem (1) has a unique, nondecreasing solution uD, moreover, constructing successively the sequence of functions

    hnt=01Gtsfshn1sds,t01,n=1,2,,E14

    for any initial function h0tD, then hntmust converge to utuniformly on 01and the rate of convergence is

    maxt01hntut=O1θqn,E15

    where 0<θ<1, which depends on the initial function h0t.

    Proof. For any h0D, we let

    lh0=supl>0:lh0tTh0tt01,E16
    Lh0=infL>0:Lh0tTh0tt01,E17
    m=min1lh011q,M=max1Lh011q,E18

    and

    u0t=mh0t,v0t=Mh0t,E19
    unt=Tun1t,vnt=Tvn1t,n=0,1,,.E20

    Since the operator Tis increasing, A1,A2and (16)(20) imply that there exist iterative sequences un,vnsatisfying

    u0tu1tuntvntv1tv0t,t01.E21

    In fact, from (19) and (20), we have

    u0tv0t,E22
    u1t=Tu0t=01G1tsfsmh0sds+tα1dni=1aj01G2ηjsfsmh0sds,mq01G1tsfsh0sds+tα1dni=1aj01G2ηjsfsh0sds,mqTh0tmh0t=u0t,E23

    and

    v1t=Tv0t=01G1tsfsMh0sds+tα1dni=1aj01G2ηjsfsMh0sds,Mq01G1tsfsh0sds+tα1dni=1aj01G2ηjsfsh0sdsMqTh0tMh0t=v0t.E24

    Then, by (22)(24) and induction, the iterative sequences un,vnsatisfy

    u0tu1tuntvntv1tv0t,t01.

    Note that u0t=mMv0t, from A1, (10), (19) and (20), it can obtained by induction that

    untθqnvnt,t01,n=0,1,2,,E25

    where θ=mM.

    From (21) and (25) we know that

    0un+ptuntvntunt1θqnMh0t,n,pN,E26

    and since 1θqnMh0t0,asn,this yields that there exists uDsuch that

    untut,uniformlyon01.

    Moreover, from (26) and

    0vntut=vntunt+untut,1θqnMh0t0,asn,

    we have

    vntut,uniformlyon01,

    so,

    untut,vntut,uniformlyon01.E27

    Therefore

    untutvnt,t01,n=0,1,2,,,E28

    From A1,(19) and (20), we have

    un+1t=TuntTutTvnt=vn+1t,n=0,1,2,,.

    This together with (27) and uniqueness of limit imply that usatisfy u=Tu,that is uDis a solution of BVP (1) and (2).

    From (19)(21) and A1, we obtain

    unthntvnt,n=0,1,2,,.E29

    It follows from (26)(29) that

    hntuthntunt+untut,hntunt+utunt,2vntunt,2M1θqnh0t.

    Therefore

    maxt01hntut2M1θqnmaxt01h0t.

    Hence, (15) holds. Since h0tis arbitrary in Dwe know that utis the unique solution of the boundary value problem (1) in D.

    We construct an example to illustrate the applicability of the result presented.

    Example 2.1. Consider the following boundary value problem

    D0+52ut+u2316costt=0,t01,u0=u0=0,u1=22u12,E30

    whereα=52,β=1,a1=22,η1=12andftu=u2316costtis increasing function with respect toufor allt01, so, assumptionA1satisfied.

    By simple calculation we haved=12212=12.

    For anyr01, there existsq=1201such that

    ftru=ru2316costtr12u2316costt=r12ftu,

    thus,ftusatisfiesA2and is singular att=0.

    On the other hand,

    01ftt2,51dt01t14dt=45<,

    so, assumptionA3is satisfied.

    Hence, all the assumptions of Theorem1.1are satisfied. Which implies that the boundary value30has an unique, nondecreasing solutionuD.

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    3. Existence result for inclusion problem (2)

    We provide another result about the existence of solutions for the problem (2) by using the assumption of nonconvex compact values for multifunction. Our strategy to deal with this problem is based on the Covitz-Nadler theorem for the contraction multivalued maps [44] for lower semi-continuous maps with decomposable values.

    First, we will present notations, definitions and preliminary facts from multivalued analysis which are used throughout this chapter. For more details on the multivalued maps, see the book of Aubin and Cellina [48], Demling [49], Gorniewicz [50] and Hu and Papageorgiou [51], see also [44, 48, 49, 52, 53, 54].

    Here C01Rdenotes the Banach space of all continuous functions from 01into Rwith the norm u=suput:forallt01,L101R, the Banach space of measurable functions u:01Rwhich are Lebesgue integrable, normed by uL1=01utdt.

    Let Xdbe a metric space induced from the normed space X. We denote

    P0X=APX:Aϕ,PbX=AP0X:Ais bounded,PclX=AP0X:Ais closed,PcpX=AP0X:Ais compact,Pb,clX=AP0X:Ais closedandbounded,

    where PXis the family of all subsets of X.

    Definition 3.1. A multivalued mapG:XPX.

    1. Guis convex (closed) valued if Guis convex (closed) for all uX,

    2. is bounded on bounded sets if GB=uBGuis bounded in Xfor all BPbXi.e., supuBsupvvGu<,

    3. has a fixed point if there is uXsuch that uGu. The fixed point set of the multivalued operator Gwill be denote by Fix G.

    Definition 3.2. A multivalued map G:01PclRis said to be measurable if for every yRthe function

    tdyGt=infyz:zGt,

    is measurable.

    Definition 3.3. Let Ybe a nonempty closed subset of a Banach space Eand G:YPclEbe a multivalued operator with nonempty closed values.

    1. Gis said to be lower semi-continuous (l.s.c) if the set xX:GxUϕis open for any open set Uin E.

    2. Ghas a fixed point if there is xYsuch that xGx.

    For each uC01R, define the set of selection of Fby

    SF,u=vAC01R:vFtutforalmostallt01.

    For PX=2X, consider the Pompeiu-Hausdorff metric (see [55]).

    Hd:2X×2X0given by

    HdAB=maxsupaAdaBsupbBdbA,

    where daB=infbBdaband dbA=infaAdab. Then Pb,clXHdis a metric space and PclXHdis a generalized metric space see [8].

    Definition 3.4. Let Abe a subset of 01×R. Ais LBmeasurable if Abelongs to the σalgebra generated by all sets of the J×D, where Jis Lebesgue measurable in 01and Dis Borel measurable in R.

    Definition 3.5. A subset Aof L101Ris decomposable if all u,vAand measurable J01=j, the function uχJ+vχj\JA, where χJstands for the caracteristic function of J.

    Definition 3.6. Let Ybe a separable metric space and N:YPL101Rbe a multivalued operator. We say Nhas property (BC) if Nis lower semi-continuous (l.s.c) and has nonempty closed and decomposable values.

    Let F:01×RPRbe a multivalued map with nonempty compact values. Define a multivalued operator

    Φ:C01RPL101R,

    by letting

    Φu=wL101R:wtFtutfora.e.t01.

    Definition 3.7. The operator Φis called the Niemytzki operator associated with F. We say Fis of the lower semi-continuous type (l.s.c type) if its associated Niemytzki operator Φhas (BC) property.

    Definition 3.8. A multivalued operator N:XPclXis called.

    1. ρLipschitz if and only if there exists ρ>0such that HdNuNvρduvfor each u,vX,

    2. a contraction if and only if it is ρLipschitz with ρ<1.

    Lemma 3.1. ([44] Covitz-Nadler). LetXdbe a complete metric space. IfN:XPclXis a contraction, then FixNϕ, where FixNis the fixed point of the operatorN.

    Definition 3.9. A measurable multivalued function F:01PXis said to be integrably bounded if there exists a function gL101Xsuch that, for all vFt,vgtfor a.e. t01.

    Let us introduce the following hypotheses.

    A4F:01×RPcpRbe a multivalued map verifying.

    1. tuFtuis LBmeasurable.

    2. uFtuis lower semi-continuous for a.e.t01.

    A5Fis integrably bounded, that is, there exists a function mL101R+such that Ftu=supv:vFtumtfor almost all t01.

    Lemma 3.2. [56] LetF:01×RPcpRbe a multivalued map. AssumeA4andA5hold. ThenFis of thel.s.c.type.

    Definition 3.10. A function uAC201Ris called a solution to the boundary value problem (2) if usatisfies the differential inclusion in (2)a.e.on 01and the conditions in (2).

    Finally, we state and prove the second main result of this Chapter. We prove the existence of solutions for the inclusion problem (2) with a nonconvex valued right hand side by applying a fixed point theorem for multivalued maps due to Covitz and Nadler. For investigation of the problem (2) we shall provide an application of the Lemma 3.4and the following Lemma.

    Lemma 3.3. ([13]) A multifunctionF:XCXis called a contraction whenever there existsγ01such thatHdNuNvγduvfor allu,vX.

    Now, we present second main result of this section.

    Theorem 1.2 Assume that the following hypothyses hold.

    H1F:J×RPcpRis an integrable bounded multifunction such that the map tFtuis measurable,

    H2HdFtu1Ftu2mtu1u2for almost all tJand u1,u2Rwith mL1JRand d0Ft0mtfor almost all tJ. Then the problem (2) has a solution provided that

    l=01G1smsds<1.

    Proof. We transform problem (2) into a fixed point problem. Consider the operator N:C01PC01Rdefined by

    Nu=hXySF,u\ht=01GtsysdstJ,E31

    where Gtsdefined by (5). It is clear that fixed points of Nare solution of (2).

    We shall prove that Nfulfills the assumptions of Covitz-Nadler contraction principle.

    Note that, the multivalued map tFtutis measurable and closed for all uAC10(e.g., [52] Theorem III.6). Hence, it has a measurable selection and so the set SF,uis nonempty, so, Nuis nonempty for any uC0.

    First, we show that Nuis a closed subset of Xfor all uAC10R. Let uXand unn1be a sequence in Nuwith unu,asnin uC0. For each n, choose ynSF,usuch that

    unt=01Gtsynsds.

    Since Fhas compact values, we may pass onto a subsequence (if necessary) to obtain that ynconverges to yL101Rin L101R. In particular, ySF,uand for any t01, we have

    untut=01Gtsysds,

    i.e., uNuand Nuis closed.

    Next, we show that Nis a contractive multifunction with constant l<1. Let u,vC01Rand h1Nu. Then there exist y1SF,usuch that

    h1t=01Gtsy1sds,tJ.

    By H2, we have

    HdFtutFtvtmtutvt,

    for almost all tJ.

    So, there exists wSF,vsuch that

    y1twmtutvt,

    for almost all tJ.

    Define the multifunction U:JPRby

    Ut=wR:y1twmtutvtforalmostalltJ.

    It is easy to chek that the multifunction V=UFvis measurable (e.g., [52] Theorem III.4).

    Thus, there exists a function y2twhich is measurable selection for V. So, y2SF,vand for each tJ, we have

    y1ty2tmtutvt.

    Now, consider h2Nuwhich is defined by

    h2t=01Gtsy2sds,tJ,

    and one can obtain

    h1th2t01Gtsy1sy2sds1G1smsusvsds.

    Hence

    h1th2tpuv01G1smsds.

    Analogously, interchanging the roles of uand v, we obtain

    HdNuNvuv01G1smsds.

    Since Nis a contraction, it follows by Lemma 3.1(by using the result of Covitz and Nadler) that Nhas a fixed point which is a solution to problem (2).

    We construct an example to illustrate the applicability of the result presented.

    Example 3.1. Consider the problem

    DαutFtut,t01,E32

    subject to the three-point boundary conditions

    ui0=0,i01,D0+βu1=2j=1ajD0+βuηj,E33

    whereα=52,β=1a1=12,a2=32,η1=116,η2=516. andFtut:01×R2Rmultivalued map given by

    uFtu=0tu21+u,uR,

    verifyingH1.

    Obviously,

    supf:fFtut+12,

    we have

    HdFtuFtvt+12uv,u,vR,t01,

    which shows thatH2holds

    So, ifmt=t+12for allt01, then

    HdFtuFtvmtuv.

    It can be easily found thatd=112116523251652=0,9176244637.

    Finally,

    l=01G1smsds=0,4636273746<1.

    Hence, all assumptions and conditions of Theorem1.2are satisfied. So, Theorem1.2implies that the inclusion problem(32)and(33)has at least one solution.

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    4. Conclusions

    This chapter concerns the boundary value problem of a class of fractional differential equations involving the Riemann-Liouville fractional derivative with nonlocal boundary conditions. By using the properties of the Green’s function and the monotone iteration technique, one shows the existence of positive solutions and constructs two successively iterative sequences to approximate the solutions. In the multi-valued case, an existence result is proved by using fixed point theorem for contraction multivalued maps due to Covitz and Nadler. The results of the present chapter are significantly contribute to the existing literature on the topic.

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    Acknowledgments

    The authors want to thank the anonymous referee for the thorough reading of the manuscript and several suggestions that help us improve the presentation of the chapter.

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    Conflict of interest

    The authors declare no conflict of interest.

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    Noureddine Bouteraa and Habib Djourdem (November 3rd 2021). A Study of Nonlinear Boundary Value Problem [Online First], IntechOpen, DOI: 10.5772/intechopen.100491. Available from:

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