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Existence Theory of Differential Equations of Arbitrary Order

Written By

Kamal Shah and Yongjin Li

Submitted: 01 November 2017 Reviewed: 16 February 2018 Published: 23 May 2018

DOI: 10.5772/intechopen.75523

From the Edited Volume

Differential Equations - Theory and Current Research

Edited by Terry E. Moschandreou

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The aims of this chapter are devoted to investigate a system of fractional-order differential equations (FDEs) with multipoint boundary conditions. Necessary and sufficient conditions are investigated for at most one solution to the proposed problem. Also, results for the existence of at least one or two positive solutions are developed by using a fixed-point theorem of concave-type operator for the considered problem. Further, we extend the conditions for more than two solutions and established some adequate conditions for multiplicity results to the proposed problem. Also, a result devoted to Hyers-Ulam stability is discussed. Suitable examples are provided to verify the established results.


  • fractional differential equations
  • coupled system
  • boundary condition
  • concave operator
  • Mathematics subject classification: 26A33
  • 34A08
  • 35B40

1. Introduction

Arbitrary-order differential equations are the excellent tools in the description of many phenomena and process in different fields of science, technology, and engineering (see [1, 2]). Therefore, considerable attention has been paid to the subject of differential equations of arbitrary order (see [3, 4, 5] and the references therein). The area devoted to the existence of positive solutions to fractional differential equations and their system especially coupled systems was greatly studied by many authors (for details see [6, 7, 8, 9]). In all these articles, the concerned results were obtained by using classical fixed point theorems like Banach contraction principle, Leray-Schauder fixed point theorem, and fixed point theorems of cone type. The aforesaid area has been very well explored for both ordinary- and arbitrary-order differential equations. Existence and uniqueness results for nonlinear and linear, classical, as well as arbitrary-order differential equations have been investigated in many papers (see few of them as [10, 11, 12, 13]).

Another warm area of research in the theory of fractional-order differential equations (FDEs) is devoted to the multiplicity of solutions. Plenty of research articles are available on this topic in literature. In [14], the author studied the given boundary value problem (BVP) for existence of multiple solutions:


where D is the Riemann-Liouville derivative of non-integer order and I=01. In same line, Kaufmann and Mboumi [15] studied the given boundary value problem of fractional differential equations for multiplicity of positive solutions:


where D is the Riemann-Liouville derivative and ϕCIR,HCI×RR.

In the last few decades, the theory devoted to the multiplicity of solutions is very well extended to coupled systems of nonlinear FDEs, and we refer to few papers in [16, 17, 18]. Wang et al. [19] established some conditions under which the given system of three point BVP


has a solution, where θ1,θ212 and μ,νI,ξ01,Hi:01×RR for i=1,2 are nonlinear functions.

In the last few decades, another important aspect devoted to stability analysis of FDEs with initial/boundary conditions has been given much attention. This is because stability is very important from the numerical and optimization point of view. Various forms of stabilities were studied for the aforesaid FDEs including exponential, Mittag-Leffler, and Lyapunov stability. Recently, Hyers-Ulam stability has given more attention. This concept was initially introduced by Ulam and then by Hyers (for details see [20, 21, 22]). Now, many articles have been written on this concept (see [23, 24, 25, 26, 27]). So far, the aforementioned stability has not yet well studied for multipoint BVPs of FDEs. Motivated by the aforesaid discussion, we propose the following coupled system of four-point BVP provided as


where j=0,1,2,m2,m3,I=01,η,ξ01, H1,H2:01×0R+×0R+0R+ are continuous functions, and Dθ1,Dθ2 stand for Riemann-Liouville fractional derivative of order θ1,θ2 in sequel. We obtain necessary and sufficient conditions for the existence of solution to system (1) by using another type of fixed point result based on a concave-type operator with increasing or decreasing property. The idea then extends to form some conditions which ensure multiplicity of solutions to the considered problem. Also, we discuss some results about the Hyers-Ulam stability for the considered problem. Further by providing examples, we illustrate the established results.


2. Preliminaries

In the current section, we review few fundamental lemmas and results found in [2, 4, 6, 28, 29].

Arbitrary-order integral of function ψ:0R is recalled as

where θ1>0 is a real number and also the integral is pointwise defined on R+

Arbitrary-order derivative in Riemann-Liouville sense for a function ψ0R is given by

Dθ1ψt=ddtm0ttsmθ11Γmθ1ψsds,θ1>0,where m=θ1+1.

[16] Let θ1>0, then for arbitrary CjR,j=1,2,,m,m=θ1+1, and the solution of

is provided by

[17, 28] Consider a Banach space E with a closed set CE. Then, C is said to be partially ordered if pq such that qpC. Further, C is said to be a cone if it holds the given conditions:

  1. pC and for a real constant κ0 the relation κpC holds.

  2. p and pC yield that 0C, where 0 is zero element of Banach space E

[17, 28] A closed and convex set C of E is said to be a normal cone if it obeys the given properties:

  1. For 0pqE, there exists β>0, such that pEβqE;

  2. pq, for all p,qE yields that there exist constants a,b>0 such that apqbq.

As is an equivalence relation, therefore defines a set Cf=pE:pf for fC. Obviously, one can derive that CfC for f0.

The operator S:CC is said to be λ concave for every θ,λ01,pC, if and only if SλpθλSp.

The operator S:CC is said to be to be increasing if p,qC,pq gives that SpSq.

[17, 28] Assume that S:CC is increasing λconcave operator for a normal cone C produced by Banach space E, such that there exists p0 with SfCf. Then, S has a unique fixed point pCf

[30] Let E be a Banach space with CB, which is closed and convex. Let E be a relatively open subset of C with 0E and S:E¯C be a continuous and compact operator. Then.

  1. The operator S has a fixed point in E¯,

  2. There exist w∂ℰ and λ01 with w=λSw.

[30] For a Banach space E together with a cone C, there exist two relatively open subsets A1 and A2 of E such that 0A1A¯1A2. Moreover, for a completely continuous operator S:CA¯2\A1C, one of the given conditions holds:

  1. Spp for all pCA1;Spp, for all pCA2;

  2. Spp for all pCA1;Spp, for all pCA2

Then, S has at least one fixed point in CA¯2\A1.


3. Main results

Let φC01R,η01 and λ1=1ηθ11<1, and then the unique solution to BVP of linear FDE

is given by
where Gts is the Green’s function defined by

Proof. In view of Lemma 2.3, we may write Eq. (2) as


In view of conditions pjtt=0=0,j=0,1,m2,m3,, Eq. (5) suffers from singularity; therefore, we have C2=C3==Cn=0. Hence, Eq. (5) becomes


Applying boundary condition ptt=1=ptt=η and d=1η1θ in Eq. (6), one has


where Gts is Green’s function given in Eq. (4).

In view of Theorem 3.1 and using λ1=1ηθ11,λ2=1ξθ21, the corresponding coupled system of integral equations to the proposed system (1) is given as


where G1ts,G2ts are Green’s functions, which can be similarly computed like in Theorem 3.1. Further, they are continuous on I×I and satisfy the following properties:

  1. maxtIG1tsλ1+11sθ11λ1=G11s,forallsI,maxtIG1tsλ2+11sθ21λ2=G21s,forallsI;

  2. mintθ1θG1tsγ1s2G1s for every θs01;

    mintθ1θG2tsγ2s2G1s for every θs01;

    Further, taking that γ=infγ1=θθ11γ2=θθ21.

Let us define a Banach space by E=ptpCI endowed with a norm pE=maxtIpt. Further, in the norm for the product space, we define it as pqE×E=pE+qE. Clearly, E×EE×E is a Banach space. Onward, we define the cone CE×E by


Consider an operator S:E×EE×E defined by


It is to be noted that the fixed points of the operator S correspond with the solution of the system (1) under consideration.

Under the continuity of H1,H2:I×R+0×R+0R+0, the operator S satisfies that SCCandS:CC is completely continuous.

Proof. To derive SCC, let pqC, and then we have


Also, we get


Thus, from Eqs. (10) and (11), we have

S1ptqtγS1pqE,for everytI.

Similarly, we can obtain

S2ptqtγS2pqE,for everytI.

Hence, we have SpqCSCC.

Let us consider


Then, we consider t1<t2I, such that


By the same fashion, we obtain for S2 as


The right hand sides of Eqs. (12) and (13) are approaching to zero at t1t2. Thus, the operator S is equi-continuous. Therefore, thanks to the Arzelá-Ascoli theorem, we receive that S=S1S2:CC is completely continuous.

Due to continuity of H1 and H2 on I×R+0×R+0R+, there exist φj,ψj,σjj=12:01R+0 for t01,p,q0 such that

along with the following conditions:
  1. Δ1=01G11sφ1sds<,Λ1=01G11sψ1s+σ1sds<1;

  2. Δ2=01G21sφ2sds<,Λ2=01G21sψ2s+σ2sds<1

are satisfied. Then, the system (1) has at least one solution pq which lies in

Proof. Let E=pqC:pqE×E<r with min2Δ112Λ12Δ212Λ2<r.

Define the operator S:E¯C as in Eq. (9).

Let pqE that is pqE×E<r. Then, we have


Thus, from Eq. (14), we have


Similarly, one can derive that


Thus, from Eqs. (15) and (16), we get


Therefore, SpqE¯. Hence, by Theorem 3.2 the operator S:E¯E is completely continuous.

Consider the eigenvalue problem:


Under the assumption that pq is a solution of Eq. (18) for ρ01, we have

ptρmaxtI01G1tsH1(spsqs)dsρ01G11sφ1sds+01G1(1s)(ψ1sps+σ1sqs)dsρΔ1+rΛ1which implies thatpE<r2.

Similarly, we can obtain that qE<r2, so pqE×E<r, which implies that pq does not belong to E for all ρ01. Therefore, due to Theorem 2.10, S has a fixed point in E¯

Assume that the given hypothesis holds:

(H1) The nonlinear functions H1 and H2 are continuous on I×R+0×R+0R+0

(H2) For all tI, we have




(H3) For all tI such that


(H4) For p,q0, there exist real numbers 0<λ,μ<1, such that for each tI,τ01, we have


Under the assumptions H1H4, the BVP (1) has a unique solution in Cf where ft=tθ11tθ21.

Proof. Let maxλμ=κ and pqC. For each tI, using H4, we have


Analogously, we also get


In view of partial order on E×E induced by the cone C, we get SτpτqτκSptqt,τ01,pqC. Which yields that S is τ concave and nondecreasing operator with respect to the partial order by using hypothesis H4. Hence, taking fC for each tI defined by


Suppose that




Also, from Green’s functions, we can obtain that


Due to nondecreasing property of H1,H2 in view of H3, we get μ>0,ν>0. Therefore, applying (19) together with H4, one has


Similarly, we can get


Then, we obtain


Like the aforesaid process, applying Eq. (19) together with H4, for each tI, one has


With same fashion, we can obtain


Thus, we have


From Eqs. (20) and (21), we produce


which implies that SfCf. So, thanks to Lemma 2.9, we see that the operator S is concave; hence, it has at most one fixed point pqCf which is the corresponding solution of BVPs (1).

Now, we define the following:

(C1) Hjj=12:I×R+0×R+0R+0 is uniformly bounded and continuous on I with respect to t.

(C2) Green’s functions G11s,G21s satisfy


(C3) Let these limits hold:


Assume that the conditions C1C3 together with given assumptions are satisfied:

(H5) H1,0γ12θ1θG11sds>1,H1,γ12θ1θG11sds>1 and


Moreover, H1,0=H2,0=H1,=H2,= also hold:

(H6) There exists constant α>0 such that




Then, the system (1) of BVPs has at least two positive solutions pq,p¯q¯ which obeying


Proof. Assume that H5 holds, and consider ϵ,α,λ such that 0<ϵ<α<λ. Further we define a set by


Now, if


Then, obviously, we can obtain that


Now, if H1,γ12θ1θG11sds>1andH2,γ22θ1θG21sds>1.

Then, like the proof of Eq. (23), we have


Also, from H5 and pqCΩα, we get


From which we have


Similarly, we have S1pqE×E<α2 as pqCΩα. Hence, we have


Now, applying Lemma 2.11 to Eqs. (23) and (25) yields that S has a fixed point pqCΩ¯α\Cε and a fixed point in p¯q¯CΩ¯λ\Ωα. Hence, we conclude that the system of BVPs (1) has at least two positive solutions pq,p¯q¯ such that pqE×Eα and p¯q¯E×Eα. Thus, relation (22) holds.

Consider that C1C3 together with the following hypothesis are satisfied:

(H7) δ1H1,0<1, δ1H1,<1;δ2H1,0<1, and δ2H2,<1;

(H8) There exist ρ>0 such that


such that


Then, the proposed coupled system of BVPs (1) has at least two positive solutions.

Proof. Proof is like the proof of Theorem 3.4.

Analogously, we deduce from Theorem 3.5 and 3.6 the following results for multiplicity of solutions to the system (1) of BVPs.

Under the conditions C1C3, there exist 2k positive numbers aj,âj,j=1,2k with a1<γ1â1<â1<a2<γ1â2<â2ak<γ1âk<âk and a1<γ2â1<â1<a2<γ2â2<â2ak<γ2âk<âk such that.

(H9) H1tptqtγ101G11sdsaj, for tpqI×γ1ajaj×γ2ajaj, and


(H10) H2tptqtγ201G21sdsaj, for tpqI×γ1ajaj×γ2ajaj, and


Then, system (1) of BVPs has at least k solutions pjqj, satisfying


Further, if assumptions C1C3 hold such that there exist 2k positive numbers bj,b̂j,j=1,2k, with


together with following hypothesis hold:

(H11) H1tpq and H2tpq are nondecreasing on 0b̂k for all tI;


Then, system (1) of BVPs has at least k solutions pjqj, satisfying



4. Hyers-Ulam stability

([31, Definition 2]) Consider a Banach space E×E such that S1,S2:E×EE×E be the two operators. Then, the operator system provided by

is called Hyers-Ulam stability if we can find Cii=1,2,3,4>0, such that for each ρii=12>0 and for each solution pqE×E of the inequalities given by
there exist a solution p¯q¯E×E of system (26) which satisfy

If λi, for i=1,2,,n be the (real or complex) eigenvalues of a matrix MCn×n, then the spectral radius ρM is defined by


Further, the matrix will converge to zero if ρM<1..

([31, Theorem 4]) Consider a Banach space E×E with S1,S2:E×EE×E be the two operators such that

and if the matrix
converges to zero ([31, Theorem 1]), then the fixed points corresponding to operatorial system (26) are Hyers-Ulam stable.

For the stability results, the following should be hold:

(H13) Under the continuity of Hi,i=1,2, there exist ai,bic01,i=1,2 and pq,p¯q¯ such that


In this section, we study Hyers-Ulam stability for the solutions of our proposed system. Thanks to Definition 4.1 and Theorem 4.3, the respective results are received.

Suppose that the assumptions H13 along with condition that matrix

is converging to zero. Then, the solutions of (1) are Hyers-Ulam stable.

Proof. In view of Theorem 4.3, we have


From which we get


Hence, we get


where M=01G11sa1sds01G11sb1sds01G21sa2sds01G21sb2sds. Hence, we received the required results.


5. Illustrative examples

Consider the given system of BVPs


Clearly, H1tpq0,H2tpq0, at pq=00, and H1tpq0,H2tpq0, at pq=11. Simple computation yields that H1,H2 are nondecreasing for every t01. Also, for τ,t01, and p,q0, one has max1413=13,


Thus, all the conditions of Theorem 3.4 are fulfilled, so the system (32) of BVPs has unique positive solution in Bf where ft=t52t92.

Consider the following system of BVPs:


It is obvious that H1tpq0,H2tpq0, at pq=00, and H1tpq0,H2tpq0, at pq=11. Also, an easy computation yields that H1,H2 are nondecreasing for each t01. Moreover, for τ,t01, and p,q0, we see that max32=3,


Thus, all the assumption of Theorem 3.4 is fulfilled, so the coupled system (33) has a unique positive solution in Bf where ft=t34t43.

Consider the following system of BVPs:


From system (33), we see that

where φ1t=t40, φ2t=t250, ψ1t=t20, ψ2t=t260, σ1t=t220, σ2t=t60. Also, η=ξ=12,λ1=λ2=0.17677. Thus, by computation, we have

Upon computation, we get


Similarly, we can also compute.

Λ1=01G11sψ1s+σ1sds=0.03092853<1,Λ2=01G21sψ2s+σ2sds=0.00289<1.Further, we see that max0.0076260.00185=0.007626. So, all the conditions of Theorem 3.3 are satisfied. So, the BVP (34) has at least one solution and the solution lies in


Taking the following system of BVPs


It is simple to check that H1,0=H2,0=H1,=H2,=. Also, for any tpqI×I×I, we see that


Thus, all the assumptions of Theorem 3.5 are satisfied with taking α=1, so the coupled system (35) has two solutions satisfying 0<pqE×E<1<pqE×E.

Consider the following coupled systems of boundary value problems:


Here, a1t=Γ52t16,b1t=Γ52t232,a2t=Γ529t216π,b2t=Γ529t32π. Moreover


Here, ρM=4.61×102<1. Therefore, matrix M converges to zero, and hence the solutions of (36) are Hyers-Ulam stable by using Theorem 4.4.


6. Conclusion

We have developed a comprehensive theory on existence of solutions and its Hyers-Ulam stability for system of multipoint BVP of FDEs. The concerned theory has been enriched by providing suitable examples.



We are very thankful to the reviewers for his/her careful reading and suggestion which improved this chapter very well.


Conflict of interest

We declare the there is no conflict of interest regarding this chapter.


Research funder

This work has been supported by the National Natural Science Foundation of China (11571378).


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Written By

Kamal Shah and Yongjin Li

Submitted: 01 November 2017 Reviewed: 16 February 2018 Published: 23 May 2018