Open access peer-reviewed chapter

Existence Results for Boundary Value Problem of Nonlinear Fractional Differential Equation

Written By

Noureddine Bouteraa and Habib Djourdem

Submitted: 23 June 2022 Reviewed: 08 July 2022 Published: 16 November 2022

DOI: 10.5772/intechopen.106412

From the Edited Volume

Nonlinear Systems - Recent Developments and Advances

Edited by Bo Yang and Dušan Stipanović

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Abstract

In this chapter, we investigate the existence and uniqueness of solutions for class of nonlinear fractional differential equations with nonlocal boundary conditions. The existence results are obtained by using Leray-Schauder nonlinear alternative and Banach contraction principle. An illustrative example is presented at the end to illustrated the validity of our results.

Keywords

  • fractional differential equations
  • existence
  • nonlocal boundary
  • fixed-point theorem

1. Introduction

In this chapter, we are interested in the existence of solutions for nonlinear fractional difference equations

cD0+αutAcD0+βut=ftut,cD0+βut,cD0+αut,tJ=0T,E1

subject to the three-point boundary conditions

λu0μuTγuη=d,λu0μuTγuη=l,E2

where T>0,0ηT,λμ+γ,d,l,λ,μ,γR,β+1<α, A is an Rn×n matrix and cD0+α,cD0+β are the Caputo fractional derivatives of order 1<α2,0<β1, respectively.

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 and L’Hospital. 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] and the references cited therein.

Fractional differential equation theory have recieved increasing attention. This theory has been developed very quickly and attracted a considerable interest from researches in this field, which revealed the flexibility of fractional calculus theory in designing various mathematical models. The main methods conducted in this articles are by terms of fixed point techniques [6]. 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-Sayad 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] 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 [37, 38]. Let us cited few papers. In [39], 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×R2R and D0+α,D0+β are the standard 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 2017, Resapour et al. [40] 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×R2R is a compact valued multifunction and cDα denotes the Caputo fractional derivative of order α.

In 2017, Sheng and Jiang [41] studied the existence and uniqueness of the solutions for fractional damped dynamical systems

cD0+αutAcD0+βut=ftut,t0T,u0=u0,u0=u0',

where 0<β1<α2,0<T<,uRn,A is an Rn×n matrix, f:01×RnRn jointly continuous function and cD0+α,cD0+β are the Caputo derivatives of order α,β, respectively.

In 2018, Abbes et al. [42] studied the existence and uniqueness of the solutions for fractional damped dynamical systems

cD0+αutAcD0+βut=ftut,cD0+βut,cD0+αut,t0T,u0=uT,u0=uT,

where 0<β1<α2,0<T<,uRn,A is an Rn×n matrix and f:01×RnRn jointly continuous.

In 2019, Tao Zhu [43] studied the existence and uniqueness of positive solutions of the following fractional differential equations

cD0+αutAcD0+βut=ftut,t0T,0<β<α<1,u0=u0.

Inspired and motivated by the works mentioned above, we establish the existence results for the nonlocal boundary value problem (1.1)–(1.2) by using Leray-Schauder nonlinear alternative and the Banach fixed point theorem. Note that our work generalized the three works cited above [41, 42, 43]. The chapter is organized as follows. In Section 2, we recall some preliminary facts that we need in the sequel. In Section 3, deals with main results and we give an example to illustrate our results.

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2. Existence and uniqueness results for our problem

2.1 Preliminaries

Let as introduce notations, definitions and preliminary facts that will be need in the sequel. For more details, see for example [44, 45, 46].

Definition 2.1. The Caputo fractional derivative of order α for the function uCn0R is defined by

cD0+αut=1Γnα0ttsnα1unsds.

where Γ is the Eleur gamma function and α>0,n=α+1,α denotes the integer part of the real number α.

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

I0+αut=1Γα0ttsα1usds,t>0.

where Γ is the Eleur gamma function, provided that the right side is pointwise defined on 0.

Lemma 2.1. Let uACn0T,nN and uC0T. Then, we have

cD0+βI0+αut=I0+αβut,
I0+αcD0+αut=utk=0n1tkk!uk0,t>0,n1<α<n,

Especially, when 1<α<2, then we have

I0+αcD0+αut=utu0tu0.

Lemma 2.2. 10 Let 0<β<1<α<2, then we have

I0+αcD0+βut=I0+αβutu0tαβΓαβ+1.

2.2 Existence results

Let CJRn be the Banach space for all continuous function from J into Rn equipped with the norm

u=suput:tJ,

where denotes a suitable complete norm on Rn. Denote L1JRn the Banach space of the measurable functions u:JRn that are Lebesgue integrable with norm

uL1=0Tutdt.

Let ACJRn be the Banach space of absolutely continuous valued functions on J and set

ACnJ=u:JRn:uuuun1CJRnandun1ACJRn.

By

C1J=u:JRnwhereuCJRn,

we denote the Banach space equipped with the norm

u1=maxuu.

For the sake of brevity, we set

δ=λμγΓαβ+1+AμTαβ+γηαβ,Δ=Γαβ+1δσ=AαβμTαβ1+γηαβ1,Λ=λμγμT+γησδ,R1=1+ATαβΓαβ+1M1+TM2+ATαβΓαβ+1R2=M2+αβATαβ1Γαβ+1M1+ATαβ1ΓαβM1=ΔΛ1σδμT+γη+1Φ+Λ1μT+γηΘ,

and

M2=Λ1σδΦ+Θ,

with

Φ=AΓαβ+1μTαβ+γηαβ+μTα+γηαL1Γ2β+T1βL3A+L2Γα+1Γ2β1L3,
Θ=AΓαβμTαβ1+γηαβ1+μTα1+γηα1L1Γ2β+T1βL3A+L2ΓαΓ2β1L3.

Lemma 2.3. Let yCJRn. The function uC1JRn is a solution of the fractional differential problem

cD0+αutAcD0+βut=yt,tJ=0T,λu0μuTγuη=d,λu0μuTγuη=l,E3

if and only if, u is a solution of the fractional integral equation

ut=1AtαβΓαβ+1u0+tu0+AΓαβ0ttsαβ1usds+1Γα0ttsα1ysds,E4

with

u0=ΔΛ1σδμT+γη+1AI0+αβμuT+γuη+I0+αμyT+γyη+d +Λ1μT+γηAI0+αβ1μuT+γuη+I0+α1μyT+γyη+l,E5

and

u0=Λ1σδAI0+αβμuT+γuη+I0+αμyT+γyη+d +AI0+αβ1μuT+γuη+I0+α1μyT+γyη+l,E6

where

I0+αuT=1Γα0TTsα1usds,I0+αuη=1Γα0ηηsα1usds,I0+αβuT=1Γαβ0TTsαβ1usds,I0+αβuη=1Γαβ0ηηsαβ1usds.

Proof. From Lemmas 2.1 and 2.2, we have

ut=u0+tu0AtαβΓαβ+1u0+AΓαβ0ttsαβ1usds+1Γα0ttsα1ysds

Applying conditions (2), we obtain (5) and (6).

Conversely, assume that u satisfies the fractional integral (4), and using the facts that cD0+α is the left inverse of I0+α and the fact that cD0+αC=0, where C is a constant, we get

cD0+αutAcD0+βut=ftutut,tJ=0T.

Also, we can easily show that

λu0μuTγuη=d,λu0μuTγuη=l.

The proof is complete.

To simplify the proofs in the forthcoming theorem, we etablish the bounds for the integrals and the bounds for the term arising in the sequel.

Lemma 2.4. For yCJRn, we have

I0+αyη=0ηητα1ΓαyτηαΓα+1y.

Proof. Obviously,

0ηητα1Γαyτ=ητααΓα0η=ηααΓα=sαΓα+1.

Hence

0ηsτα1ΓαyτηαΓα+1y.

Lemma 2.5. For uC1JRn and 0<β1, we have

cD0+βutT1βΓ2βu,

and, so

cD0+βutT1βΓ2βu1.

Proof.

Clearly, when β=1, the conclusion are true. So, consider the case 0<β<1. By Definition 2.1, for each uC1JRn and tJ, we have

D0+βut=1Γ1β0ttsβusdsu1Γ1β0ttsβds=ut1βΓ1β T1βΓ1βuT1βΓ1βu1.

We need to give the following hypothesis:

H1 there existe a constants L1,L2>0 and 0<L3<1 such that

ftuvwftu¯v¯w¯L1uu¯+L2vv¯+L3ww¯,

for any u,v,u¯,v¯,w¯Rn and tJ.

Now we are in a position to present the first main result of this paper. The existence results is based on Banach contraction principle.

Theorem 1.1. ([47] Banach’s fixed point theorem). Let C be a non-empty closed subset of a Banach space E, then any contraction mapping T of C into itself has a unique fixed point.

Theorem 1.2. Assume that H1 holds. If

maxR1R2<1,E7

then the boundary value problem (1.1)–(1.2) has a unique solution on J.

Proof. We transform the problem (1.1)–(1.2) into fixed point problem. Let N:C1JRnC1JRn the operator defined by

Nut=1AtαβΓαβ+1B+tD+AΓαβ0ttsαβ1usds+1Γα0ttsα1gsds,E8

with

B=ΔΛ1σδμT+γη+1AI0+αβμuT+γuη+I0+αμyT+γyη+d+Λ1μT+γηAI0+αβ1μuT+γuη+I0+α1μyT+γyη+l,

and

D=Λ1σδAI0+αβμuT+γuη+I0+αμgT+γgη+d+AI0+αβ1μuT+γuη+I0+α1μgT+γgη+l,

where gCJRn be such that

gt=ftut,cD0+βutgt+AcD0+βut

For every uC1JRn and any tJ, we have

Nut=DαβAtαβ1Γαβ+1B+AΓαβ10ttsαβ2usds+1Γα10ttsα2gsds.E9

Clearly, the fixed points of operator N are solutions of problem (1.1)–(1.2).

It is clear that NuCJRn, consequently, N is well defined.

Let u,vCJRn. Then for tJ, we have

NutNvt1+ATαβΓαβ+1BB1+TDD1+AΓαβ0TTsαβ1usvsds+1Γα0TTsα1gshsds,

with

B1=ΔΛ1σδμT+γη+1AI0+αβμvT+γvη+I0+αμhT+γhη+d +Λ1μT+γηAI0+αβ1μvT+γvη+I0+α1μhT+γhη+l,

and

D1=Λ1σδAI0+αβμvT+γvη+I0+αμhT+γhη+d+AI0+αβ1μvT+γvη+I0+α1μhT+γhη+l,

From H, for any tJ, we have

gtht=L1utvt+L2cD0+βutcD0+βvt+L3gt+AcD0+βuthtAcD0+βvtL1utvt+L2cD0+βutcD0+βvt+L3gtht+L3AcD0+βutcD0+βvtL1utvt+L3gtht+L3A+L2cD0+βutvt.

Thus

gthtL11L3utvt+L3A+L21L3cD0+βutvtL11L3uv+L3A+L21L3cD0+βuv.

Then, according to the Lemma 3.2, we get

gthtL11L3uv1+T1βL3A+L2Γ2β1L3uv1=L1Γ2β+T1βL3A+L2Γ2β1L3uv1.E10

By employing (10) and Lemma 3.1, we get

B1B2ΔΛ1σδμT+γη+1Φ+Λ1μT+γηΘuv1=M1uv1.

and

D1D2Λ1σδΦ+Θuv1=M2uv1,

where Φ and Θ defined above.

Thus, for tJ, we have

NutNvt1+ATαβΓαβ+1M1+TM2+ATαβΓαβ+1+TαL1Γ2β+T1β+αL3A+L2Γα+1Γ2β1L2uv1=R1uv1.

Also

NutNvtD2D1+αβATαβ1Γαβ+1B1B2+AΓαβ10TTsαβ2usvsds+1Γα10TTsα2gshsds.

By employing (10) and Lemma 3.2, we get

NutNvtM2+αβATαβ1Γαβ+1M1+ATαβ1Γαβ+Tα1L1Γ2β+TαβL3A+L2ΓαΓ2β1L2uv1=R2uv1.

Therefore

NutNvtmaxR1R2uv1.

Thus, by (10) the operator N is a contraction. Hence it follows by Banach’s contraction principle that the boundary value problem (1)(12) has a unique solution on J.

Now we are in a position to present the second main result of this paper. The existence results is based on Leray-Schauder nonlinear alternative.

Theorem 1.3. ([6] Nonlinear alternative for single valued maps). Let E be a Banach space, C a closed, convex subset of E and U an open subset of C with 0U. Suppose that F:U¯C is a continuous and compact (that is FU¯ is relatively compact subset of C) map. Then either

  1. F has a fixed point in U¯, or

  2. there is a uU (the boundary of U in C) and λ01 with u=λFu.

Set

l1=M3+TM4+ATαβΓαβ+1M3+TM4+ArTαβΓαβ+1+TαΓα+1M,

and

l2=M4+αβATαβ1Γαβ+1M3+ArTαβ1Γαβ+TαMΓα.

Theorem 1.4. Assume that H1 holds and there exists a positive constant M>0 such that maxl1l2=l<M. Then the boundary value problem (1.1)–(1.2) has at least one solution on J.

Proof. Let N be the operator defined in (8).

N is continuous. Let un be a sequence such that unu in CJRn. Then for tJ, we have

NutNunt1+ATαβΓαβ+1B1Bn2+TD1Dn2+AΓαβ0TTsαβ1usunsds+1Γα0TTsα1gsgnsds,

where Bn2,Dn2Rn, with

Bn2=ΔΛ1σδμT+γη+1AI0+αβμunT+γunη+I0+αμgnT+γgnη+d+Λ1μT+γηAI0+αβ1μunT+γunη+I0+α1μgnT+γgnη+l,Dn2=Λ1σδAI0+αβμunT+γunη+I0+αμgnT+γgnη+d+AI0+αβ1μunT+γunη+I0+α1μgnT+γgnη+l,

and

gnt=ftunt,cD0+βuntgnt+AcD0+βunt.

From H, for any tJ, we have

gtgnt=L1utunt+L3gt+AcD0+βutgntAcD0+βunt+L2cD0+βutcD0+βuntL1utunt+L2cD0+βutcD0+βunt+L3gtgnt+L3AcD0+βutcD0+βuntL1utunt+L3gtgnt+L3A+L2cD0+βutunt.

Thus

gtgntL11L3utunt+L3A+L21L3cD0+βutuntL11L3uun+L3A+L21L3cD0+βuun.

Then, according to the Lemma 3.2, we get

gtgntL11L3uun1+T1βL3A+L21L3Γ2βuun1=L1Γ2β+T1βL3A+L21L3Γ2βuun1.

By employing (10) and Lemma 3.1, we get

B1Bn2ΔΛ1σδμT+γη+1Φ+Λ1μT+γηΘuun1,=M1uun1.

and

D1Dn2Λ1σδΦ+Θuun1=M2uun1,

Thus, for tJ, we have

NutNunt1+ATαβΓαβ+1M1+TM2+ATαβΓαβ+1+TαL1Γ2β+T1β+αL3A+L2Γα+1Γ2β1L2uun1=R1uun1.

Also

NutNuntDn2D1+αβATαβ1Γαβ+1B1Bn2+AΓαβ10TTsαβ2usunsds+1Γα10TTsα2gsgnsds.

By employing (10), we get

NutNuntM2+αβATαβ1Γαβ+1M1+ATαβ1Γαβ+Tα1L1Γ2βL3A+L2Tαβ1Γαβ+1M1+ATαβ1Γαβuun1.=R2uun1.

Thus NuNun10 as n, which implies that the operator N is continuous.

Now, we show N maps bounded sets into bounded sets in CJRn. For a positive number r, let Br=uC1JRn:u1r be a bounded set in CJRn. Then we have

gtftutgt+AcD0+βutD0+βutft,0,0,0+ft,0,0,0L1ut+L3gt+AcD0+βut+L2D0+βut+ft,0,0,0L1u+L3gt+L3A+L2D0+βu+f,

where suptJft,0,0,0=f<. Thus

gtL11L3u+L3A+L21L3D0+βu+f1L3.

Then, By Lemma 3.2, we have

gtL11L3u+L3A+L2T1β1L3Γ2βu+f1L3L11L3u1+L3A+L2T1β1L3Γ2βu1+f1L3L1r1L3+L3A+L2rT1β1L3Γ2β+f1L3=M,E11

which implies that

BrAΔΛ1σδμT+γη+1μTαβ+γηαβΓαβ+1+Λ1μT+γημTαβ1+γηαβ1Γαβ+MΔΛ1σδμT+γη+1μTα+γηαΓα+1+Λ1μT+γημTα1+γηα1Γα+ΔΛ1μT+γηl+dσδ+1=M3,

and

DrAΛ1σδμTαβ+γηαβΓαβ+1+μTαβ1+γηαβ1Γαβ+MΛ1σδμTα+γηαΓα+1+μTα1+γηα1Γα+Λ1σδd+l=M4.

Thus (8) implies

NutM3+ATαβΓαβ+1M3+TM4+ArTαβΓαβ+1+TαΓα+1M=l1,

and

NutM4+αβATαβ1Γαβ+1M3+ArTαβ1Γαβ+TαMΓα=l2.

Therefore

Nu1maxl1l2=l.E12

Now, we show that N maps bounded sets into equicontinuous sets of C1JRn. Let t1,t201 with t1<t2 and uBr is bounded sets of C1JRn. Then

Nut2Nut1M4t2t1+1+AM3Γαβ+1t2αβt1αβ+ArΓαβt1t2t2sαβ1ds+ArΓαβ0t1t2sαβ1t1sαβ1ds+M1Γαt1t2t2sα1ds+0t1t2sα1t1sα1ds

Obviously, the right-hand side of the above inequality tends to zero as t2t1.

Similarly, we have

Nut2Nut1αβAM3Γαβ+1t2αβ1t1αβ1+ArΓαβ1t1t2t2sαβ2ds+ArΓαβ20t1t2sαβ2t1sαβ2ds+MΓα1t1t2t2sα2ds+0t1t2sα2t1sα2ds

Again, it is seen that the right-hand side of the above inequality tends to zero as t2t1. Thus, Nut2Nut10, as t2t1. This shows that the operator N is completely continuous, by the Ascoli-Arzela theorem. Thus, the operator N satisfies all the conditions of Theorem 3.4, and hence by its conclusion, either condition (i) or condition (ii) holds. We show that the condition (ii) is not possible.

Let U=uC1JRn:u<M with maxl1l2=l<M. In view of condition l<M and by (12), we have

Numaxl1l2<M.

Now, suppose there exists uU and λ01 such that u=λNu. Then for such a choice of u and the constant λ, we have

M=u=λNu<maxl1l2<M,

which is a contradiction. Consequently, by the Leray-Schauder alternative, we deduce that F has a fixed point uU¯ which is a solution of the boundary value problem (1)(12). The proof is completed.

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

Example 2.1. Consider the following fractional differential equation

cD0+αutAcD0+βut=ftut,cD0+βut,cD0+αut,tJ=01,E13

subject to the three-point boundary conditions

u0u1u12=1,u0u1u12=1,E14

where α=2,β=1,λ=μ=d=l=1,A=2102 and

fituvw=cit8arctanu+v+w,i=1,2,

such that f=f1f2 with 0<ci<1,i=1,2.

For every ui,viR2,i=1,2,3, we have

fitu1u2u3fitv1v2v3ci8u1v1+u2v2+u3v3,i=1,2,

where L1=L2=L3=ci8 for appropriate choice of constants ci,i=1,2. we check the condition of Theorem 2.2. Clearly, assumption H1 holds. A simple computations of R1,R2,l1 and l2 shows tha the second condition of Theorems 3.3 and 3.5 is satisfied. Thus the conclusion of Theorems 3.3 and 3.5 applies, and hence the problem (13)(14) has a unique solution and at least one solution on 01.

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3. 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 Leray-Schauder nonlinear alternative and the Banach fixed point theorem, we shows the existence and uniqueness of positive solutions of our problem. In addition, an example is provided to demonstrate the effectiveness of the main results. 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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Written By

Noureddine Bouteraa and Habib Djourdem

Submitted: 23 June 2022 Reviewed: 08 July 2022 Published: 16 November 2022