Open access peer-reviewed chapter

A Study of Nonlinear Boundary Value Problem

Written By

Noureddine Bouteraa and Habib Djourdem

Reviewed: 16 September 2021 Published: 03 November 2021

DOI: 10.5772/intechopen.100491

From the Edited Volume

Simulation Modeling

Edited by Constantin Volosencu and Cheon Seoung Ryoo

Chapter metrics overview

270 Chapter Downloads

View Full Metrics


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.


  • 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)


We also cover the multi-valued case of problem


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


where D0+α,D0+β are the stantard Riemann-Liouville fractional derivative of order αn1n,β1n2forn3, the function fC01×RR, the multifunction F:01×R2R are allowed to be singular at t=0 and/or t=1 and 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:


where 1<α2,0<η<1,0<a,β<1,fC01×R2R and D0+α,D0+β are 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 2020 Li et al. [41] consider the existence of a positive solution for the following BVP of nonlinear fractional differential equation with integral boundary conditions:


where 2<q3, 0<σ1, α,γ,δ0, and β>0 satisfying 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


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


subject to three-point boundary conditions


where 2<α3,1<p2,0<η<1,β,γR+,F:01×R×R2R is a compact valued multifunction and cDα denotes 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:


subject to the coupled boundary conditions:


where CDα, CDβ denote the Caputo fractional derivatives of order α and β respectively, F,G:0T×R×R are given multivalued maps, PR is the family of all nonempty subsets of R, and νi,μi,i=1,2 are 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.


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 ACi01R denote the space of itimes differentiable functions u:01R whose ith derivative ui is absolutely continuous and α donotes the integer part of number α.

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

The Caputo derivative of fractional order α for the function u:0+R is defined by


The Riemann-Liouville fractional derivative order α for the function u:0+R is defined by


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


is called Euler’s gamma function.

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


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


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


is given by





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


where c1,c2,cn are arbitrary real constants.

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


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


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


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


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


  2. gtstα1Γαfor allt,s01.

  3. gtstα1g1sfor allt,s01, where


    From the above properties, we deduce the following properties:

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

  5. maxt01Gts=G1s, for alls01, where


Proof. It is easy to chek that i,v,vi holds. So we prove that ii is true. Note that (6) and 01sαβ11. It follows that gtstα1Γα for 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


Case2. When 0ts1, we have


Hence gtstα1g1s for all t,s01.

2.2 Existence results

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

A1ftu1ftu2 for any 0<t<1,0u1u2.

A2 For any r01, there exists a constant q01 such that



We shall consider the Banach space E=C01 equipped with the norm u=max0t1ut and let




In view of Lemma 2.3, we define an operator T as


where Gts is given by (5).

By A1 it is easy to see that the operator T:DC+01 is increasing. Observe that the BVP (1) has a solution if and only if the operator T has a fixed point.

Obviously, from A1 we obtain


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


Then, from A1, ftu non-decreasing respect to u and A2, we can imply that for s01,q01


From (11) and Lemma 2.4, we obtain




Eqs. (12) and (13) and assumption A3 imply that T:DD.

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

Theorem 1.1 Suppose A1A3 hold. Then problem (1) has a unique, nondecreasing solution uD, moreover, constructing successively the sequence of functions


for any initial function h0tD, then hnt must converge to ut uniformly on 01 and the rate of convergence is


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

Proof. For any h0D, we let




Since the operator T is increasing, A1,A2 and (16)(20) imply that there exist iterative sequences un,vn satisfying


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




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


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


where θ=mM.

From (21) and (25) we know that


and since 1θqnMh0t0,asn,this yields that there exists uD such that


Moreover, from (26) and


we have






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


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

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


It follows from (26)(29) that




Hence, (15) holds. Since h0t is arbitrary in D we know that ut is 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


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


thus,ftusatisfiesA2and is singular att=0.

On the other hand,


so, assumptionA3is satisfied.

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


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 C01R denotes the Banach space of all continuous functions from 01 into R with the norm u=suput:forallt01,L101R, the Banach space of measurable functions u:01R which are Lebesgue integrable, normed by uL1=01utdt.

Let Xd be 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 PX is the family of all subsets of X.

Definition 3.1. A multivalued mapG:XPX.

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

  2. is bounded on bounded sets if GB=uBGu is bounded in X for all BPbX i.e., supuBsupvvGu<,

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

Definition 3.2. A multivalued map G:01PclR is said to be measurable if for every yR the function


is measurable.

Definition 3.3. Let Y be a nonempty closed subset of a Banach space E and G:YPclE be a multivalued operator with nonempty closed values.

  1. G is said to be lower semi-continuous (l.s.c) if the set xX:GxUϕ is open for any open set U in E.

  2. G has a fixed point if there is xY such that xGx.

For each uC01R, define the set of selection of F by


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

Hd:2X×2X0 given by


where daB=infbBdab and dbA=infaAdab. Then Pb,clXHd is a metric space and PclXHd is a generalized metric space see [8].

Definition 3.4. Let A be a subset of 01×R. A is LB measurable if A belongs to the σalgebra generated by all sets of the J×D, where J is Lebesgue measurable in 01 and D is Borel measurable in R.

Definition 3.5. A subset A of L101R is decomposable if all u,vA and measurable J01=j, the function uχJ+vχj\JA, where χJ stands for the caracteristic function of J.

Definition 3.6. Let Y be a separable metric space and N:YPL101R be a multivalued operator. We say N has property (BC) if N is lower semi-continuous (l.s.c) and has nonempty closed and decomposable values.

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


by letting


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

Definition 3.8. A multivalued operator N:XPclX is called.

  1. ρLipschitz if and only if there exists ρ>0 such that HdNuNvρduv for 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:01PX is said to be integrably bounded if there exists a function gL101X such that, for all vFt,vgt for a.e. t01.

Let us introduce the following hypotheses.

A4F:01×RPcpR be a multivalued map verifying.

  1. tuFtu is LB measurable.

  2. uFtu is lower semi-continuous for a.e.t01.

A5F is integrably bounded, that is, there exists a function mL101R+ such that Ftu=supv:vFtumt for 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 uAC201R is called a solution to the boundary value problem (2) if u satisfies the differential inclusion in (2)a.e. on 01 and 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 3.4 and 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×RPcpR is an integrable bounded multifunction such that the map tFtu is measurable,

H2HdFtu1Ftu2mtu1u2 for almost all tJ and u1,u2R with mL1JR and d0Ft0mt for almost all tJ. Then the problem (2) has a solution provided that


Proof. We transform problem (2) into a fixed point problem. Consider the operator N:C01PC01R defined by


where Gts defined by (5). It is clear that fixed points of N are solution of (2).

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

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

First, we show that Nu is a closed subset of X for all uAC10R. Let uX and unn1 be a sequence in Nu with unu,asn in uC0. For each n, choose ynSF,u such that


Since F has compact values, we may pass onto a subsequence (if necessary) to obtain that yn converges to yL101R in L101R. In particular, ySF,u and for any t01, we have


i.e., uNu and Nu is closed.

Next, we show that N is a contractive multifunction with constant l<1. Let u,vC01R and h1Nu. Then there exist y1SF,u such that


By H2, we have


for almost all tJ.

So, there exists wSF,v such that


for almost all tJ.

Define the multifunction U:JPR by


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

Thus, there exists a function y2t which is measurable selection for V. So, y2SF,v and for each tJ, we have


Now, consider h2Nu which is defined by


and one can obtain




Analogously, interchanging the roles of u and v, we obtain


Since N is a contraction, it follows by Lemma 3.1 (by using the result of Covitz and Nadler) that N has 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


subject to the three-point boundary conditions


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





we have


which shows thatH2holds

So, ifmt=t+12for allt01, then


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



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


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.



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.


Conflict of interest

The authors declare no conflict of interest.


  1. 1. R. P. Agarwal, D. Baleanu, V. Hedayati and S. H. Rezapour, Two fractional derivative inclusion problems via integral boundary condition, Appl. Math. Comput. vol.257 (2015), 205-212
  2. 2. V. Kac and P. Cheung, Quantum Calculus, Springer, New-York, 2002
  3. 3. V. Lakshmikantham and A. S. Vatsala, General uniqueness and monotone iterative technique for fractional differential equations. Appl. Math. Lett. 21 (8)(2008), 828-834
  4. 4. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, John Wiley and Sons, Inc. New-York, (1993)
  5. 5. W. Rudin, Functional Analysis, 2nd edn. International Series in Pure and Applied Mathematics. Mc Graw-Hill, New York (1991)
  6. 6. S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Yverdon (1993)
  7. 7. F. Jarad, T. Abdeljaw and D. Baleanu, On the generalized fractional derivatiives and their Caputo modification. J. Nonl. Sci. Appl. 10 (5) (2017), 2607-2619
  8. 8. Y. Tian, Positive solutions to m-point boundary value problem of fractional differential equation. Acta Math. Appl. Sinica (Engl. Ser.) 29 (2013), 661-672
  9. 9. A.M.A. El-Sayed and A.G. Ibrahim, Multivalued Fractional differential equations of arbitrary orders. Springer-Verlag, Appl. Math. Comput. 68 (1995), 15-25
  10. 10. N. Bouteraa and S. Benaicha, Existence of solutions for nonlocal boundary value problem for Caputo nonlinear fractional differential inclusion, Journal of Mathematical Sciences and Modelling, 1 (1) (2018), 45-55
  11. 11. N. Bouteraa and S. Benaicha, Existence results for fractional differential inclusion with nonlocal boundary conditions, Ri. Mat. Parma, Vol.11(2020),181-206
  12. 12. A. Cernia, Existence of solutions for a certain boundary value problem associated to a fourth order differential inclusion, Inter. Jour. Anal. Appl. vol.14 (2017), 27-33
  13. 13. S. K. Ntouyas, S. Etemad, J. Tariboon and W. Sutsutad, Boundary value problems for Riemann-Liouville nonlinear fractional diffrential inclusions with nonlocal Hadamard fractional integral conditions, Medittter. J. Math. 2015 (2015), 16 pages
  14. 14. N. Bouteraa and S. Benaicha, Triple positive solutions of higher-order nonlinear boundary value problems, Journal of Computer Science and Computational Mathematics, Volume 7, Issue 2, June 2017, 25-31
  15. 15. N. Bouteraa and S. Benaicha, Existence of solutions for three-point boundary value problem for nonlinear fractional equations, Analele Universitatii Oradia Fasc. Mathematica, Tom XXIV (2017), Issue No. 2, 109-119
  16. 16. S. Benaicha and N. Bouteraa, Existence of solutions for three-point boundary value problem for nonlinear fractional differential equations, Bulltin of the Transilvania University of Brasov, Serie III: Mathematics, Informtics, Physics. Volume 10 (59), No. 2-2017
  17. 17. N. Bouteraa and S. Benaicha, Existence of solutions for third-order three-point boundary value problem, Mathematica. 60 (83), N 0 1, 2018, pp. 21-31
  18. 18. N. Bouteraa and S. Benaicha, The uniqueness of positive solution for higher-order nonlinear fractional differential equation with nonlocal boundary conditions, Advances in the Theory of Nonlinear and it Application, 2(2018) No 2, 74-84
  19. 19. N. Bouteraa, S. Benaicha and H. Djourdem, Positive solutions for nonlinear fractional differential equation with nonlocal boundary conditions, Universal Journal of Mathematics and Applications, 1 (1) (2018), 39-45
  20. 20. N. Bouteraa and S. Benaicha, The uniqueness of positive solution for nonlinear fractional differential equation with nonlocal boundary conditions, Analele universitatii Oradia. Fasc. Matematica. Tom XXV (2018), Issue No. 2, 53-65
  21. 21. N. Bouteraa, S. Benaicha, H. Djourdem and N. Benatia, Positive solutions of nonlinear fourth-order two-point boundary value problem with a parameter, Romanian Journal of Mathematics and Computer Science, 2018, Volume 8, Issue 1, p.17-30
  22. 22. N. Bouteraa and S. Benaicha, Positive periodic solutions for a class of fourth-order nonlinear differential equations, Siberian Journal of Numerical Mathematics, Volume 22, No. 1 (2019), 1-14
  23. 23. N. Bouteraa, Existence of solutions for some nonlinear boundary value problems, [thesis]. University of Oran1, Ahmed Benbella, Algeria; 2018
  24. 24. N. Bouteraa, S. Benaicha and H. Djourdem, ON the existence and multiplicity of positive radial solutions for nonlinear elliptic equation on bounded annular domains via fixed point index, Maltepe Journal of Mathematics, Volume I, Issue 1,(2019), 30-47
  25. 25. N. Bouteraa, S. Benaicha and H. Djourdem, Positive solutions for systems of fourth-order two-point boundary value problems with parameter, Journal of Mathematical Sciences and Modeling, 2(1) (2019), 30-38
  26. 26. N. Bouteraa and S. Benaicha, Existence and multiplicity of positive radial solutions to the Dirichlet problem for the nonlinear elliptic equations on annular domains, Stud. Univ. Babes-Bolyai Math, 65(2020), No. 1, 109-125
  27. 27. S. Benaicha, N. Bouteraa and H. Djourdem, Triple positive solutions for a class of boundary value problems with integral boundary conditions, Bulletin of Transilvania University of Brasov, Series III : Mathematics, Informatics, Physics, Vol. 13 (62), No. 1 (2020), 51-68
  28. 28. N. Bouteraa, H. Djourdem and S. Benaicha, Existence of solution for a system of coupled fractional boundary value problem, Proceedings of International Mathematical Sciences, Vol. II, Issue 1 (2020), 48-59
  29. 29. N. Bouteraa and S. Benaicha, Existence results for second-order nonlinear differential inclusion with nonlocal boundary conditions, Numerical Analysis and Applications, 2021, Vol. 14, No. 1, pp. 3039
  30. 30. N. Bouteraa M. In, M. A. Akinlar, B. Almohsen, Mild solutions of fractional PDE with noise, Math. Meth. Appl. Sci. 2021;115
  31. 31. N. Bouteraa, S. Benaicha, Existence Results for Second-Order Nonlinear Differential Inclusion with Nonlocal Boundary Conditions, Numerical Analysis and Applications, 2021, Vol. 14, No. 1, pp. 3039
  32. 32. N. Bouteraa, S. Benaicha, A study of existence and multiplicity of positive solutions for nonlinear fractional differential equations with nonlocal boundary conditions, Stud. Univ. Babes-Bolyai Math. 66(2021), No. 2, 361-380
  33. 33. H. Djourdem, S. Benaicha and N. Bouteraa, Existence and iteration of monotone positive solution for a fourth-order nonlinear boundary value problem, Fundamental Journal of Mathematics and Applications, 1 (2) (2018), 205-211
  34. 34. H. Djourdem, S. Benaicha, and N. Bouteraa, Two Positive Solutions for a Fourth-Order Three-Point BVP with Sign-Changing Greens Function, Communications in Advanced Mathematical Sciences. Vol. II, No. 1 (2019), 60-68
  35. 35. H. Djourdem and N. Bouteraa, Mild solution for a stochastic partial differential equation with noise, WSEAS Transactions on Systems, Vol. 19, (2020), 246-256
  36. 36. R. Ghorbanian, V. Hedayati M. Postolache, and S. H. Rezapour, On a fractional differential inclusion via a new integral boundary condition, J. Inequal. Appl. (2014), 20 pages
  37. 37. M. Inc, N. Bouteraa, M. A. Akinlar, Y. M. Chu, G. W. Weber and B. Almohsen, New positive solutions of nonlinear elliptic PDEs, Applied Sciences, 2020, 10, 4863; doi :10.3390/app10144863, 17 pages
  38. 38. N. Bouteraa and S. Benaicha, Nonlinear boundary value problems for higher-order ordinary differential equation at resonance, Romanian Journal of Mathematic and Computer Science. 2018. Vol 8, Issue 2 (2018), p. 83-91
  39. 39. N. Bouteraa, S. Benaicha, A class of third-order boundary value problem with integral condition at resonance, Maltepe Journal of Mathematics, Volume II, Issue 2,(2020), 43-54
  40. 40. X. Lin, Z. Zhao and Y. Guan, Iterative Technology in a Singular Fractional Boundary Value Problem With q-Difference, Appl. Math. 7 (2016), 91-97
  41. 41. M. Li, JP. Sun and YH. Zhao, Existence of positive solution for BVP of nonlinear fractional differential equation with integral boundary conditions. Adv Differ Equ 2020, 177 (2020).
  42. 42. S. H. Rezapour and V. Hedayati, On a Caputo fractional differential inclusion with integral boundary condition for convex-compact and nonconvex-compact valued multifunctions, Kragujevac Journal of Mathematics. vol.41 (2017), 143-158
  43. 43. B. AHMAD, SK. NTOUYAS and A. ALSAEDI, Coupled systems of fractional differential inclusions with coupled boundary conditions, Electronic Journal of Differential Equations, Vol. 2019 (2019), No. 69, pp. 121
  44. 44. H. Covitz and S. B. Jr. Nadler, Multivalued contraction mappings in generalized metric spaces, Israel J. Math. vol.8 (1970), 5-11
  45. 45. M. H. Annaby, and Z. S. Mansour, q-Fractional calculus and equations, Lecture Notes in Mathematics. vol.2056, Springer-Verlag, Berlin 2012
  46. 46. O. Agrawal, Some generalized fractional calculus operators and their applications in integral equations, Fract. Cal. Appl. Anal. Vol.15 (2012), 700-711
  47. 47. A. A. Kilbas, H. M. Srivastava and J. J. Trijull, Theory and applications of fractional differential equations, Elsevier Science B. V, Amsterdam, 2006
  48. 48. J. P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, 2012
  49. 49. K. Demling, Multivalued Differential equations, Walter De Gryter, Berlin-New-York 1982
  50. 50. L. Gorniewicz, Topological Fixed Point Theory of Multivalued Map pings, Mathematics and Its Applications, Vol. 495, Kluwer Academic Publishers, Dordrecht 1999
  51. 51. S. Hu and N. Papageorgiou, Handbook of Multivalued Analysis, vol.I, Theory, Kluwer Academic. J. Diff. Equ. No.147, (2013), 1-11
  52. 52. C. Castaing and M. Valadier, Convex analysis and measurable multifunctions,Lecture Notes in Mathematics, Springer-Verlage, Berlin-Heidelberg, New-York, 580, 1977
  53. 53. A. Lasota and Z. Opial, An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Pol. Sci. Set. Sci. Math. Astronom. Phy. vol.13 (1965), 781-786
  54. 54. J. Musielak, Introduction to functional analysis, PWN, Warsaw, 1976, (in Polish)
  55. 55. V. Berinde and M. Pacurar, The role of the Pompeiu-Hausdorff metric in fixed point theory, Creat. Math. Inform. vol.22 (2013), 35-42
  56. 56. M. Frigon and A. Granas, Theoremes d’existence pour des inclusions differentielles sans convexite, C. R. Acad. Sci. Paris, SerI. vol.310 (1990), 819-822

Written By

Noureddine Bouteraa and Habib Djourdem

Reviewed: 16 September 2021 Published: 03 November 2021