Study of a class of arbitrary order differential equations by a coincidence degree method

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R E S E A R C H

Open Access

Study of a class of arbitrary order

differential equations by a coincidence

degree method

Nigar Ali

1

_{, Kamal Shah}

1

_{, Dumitru Baleanu}

2

_{, Muhammad Arif}

3*

_{and Rahmat Ali Khan}

1

*_{Correspondence:}

[email protected]

3_{Department of Mathematics,}

Abdulwali Khan University Mardan, Khyber Pakhtunkhwa, Pakistan Full list of author information is available at the end of the article

Abstract

In this manuscript, we investigate some appropriate conditions which ensure the existence of at least one solution to a class of fractional order diﬀerential equations (FDEs) provided by

–C_Dq_z₍_t_{) =}

_θ

₍_t_,_z₍_t_)); _t_∈_J_{= [0, 1],}_q_∈_{(1, 2],} z(t)|t=0=

φ(

z), z(1) =

δ

CDpz(η), p,η∈(0, 1).

The nonlinear function

θ

:J×R→Ris continuous. Further,

δ

∈(0, 1) and

φ

∈C(J,R) is a non-local function. We establish some adequate conditions for the existence of at least one solution to the considered problem by using Grönwall’s inequality and a priori estimate tools called the topological degree method. We provide two examples to verify the obtained results.

MSC: 34A08; 35R11

Keywords: fractional order diﬀerential equations; Caputo derivative; condensing operator; Grönwall’s inequality; topological degree method

1 Introduction

The applications of non-integer order differential equations are increasing day by day in various areas of research. These applications can be traced out in many disciplines of sci-ence including technology and engineering. To see the recent applications of the men-tioned area in various fields like physics, mechanics, chemical science, biological dynam-ics, material engineering, theory of control, signal and image propagation, communication and transform, economical problems and optimization theory, etc.; the reader is referred to [–]. Non-integer order differential equations provide a strong tool for the description of memory and hereditary properties of various materials and processes; see []. One of the key reasons of taking interest in fractional differential equations by the researchers is the presence of greater degree of freedom of fractional differential operator. In fact the fractional derivative is a global operator instead of classical derivative which is local in nature; see []. In the last few decades, various aspects of fractional differential equa-tions have been investigated like existence theory, stability and numerical analysis, etc. One of the attractive areas of research is the existence theory of solutions for FDEs. In

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the last few decades, the aforesaid area has been extensively investigated by using various techniques of classical analysis, for further explanations, we refer the reader to [–]. It is well known that the standard Riemann-Liouville fractional derivative fails to provide the required physical interpretation for boundary value problems (BVPs) and initial value problems (IVPs) in most of the cases. However, these requirements of interpreting the ini-tial and boundary conditions can better be fulfilled by the use of Caputo non-integer order derivatives. Existence theory of solutions or positive solutions to multi-points boundary value problems using different types of fixed point theorems like as Banach theorem of contraction type, Schaefer and Leray-Schauder theorem of fixed point is studied in detail [–]. Moreover, the existence of solutions to FDEs using coincidence degree theory for a contraction operator is studied in [–]. Recently the existence of a center stable manifold for planar fractional damped equations has been investigated. For the required solution, the authors in [] constructed a suitable Lyapunov-Perron operator by giving the asymptotic behavior of the Mittag-Leffler function. Then they obtained an interesting center stable manifold result to prove center stable manifold theorem for planar fractional damped equations involving two Caputo fractional derivatives. In a similar manner, the authors of [], studied finite time stability and existence theory of delay type differen-tial equations of fractional order by using classical analysis. Wanget al.[] investigated the existence theory and proved some conditions for uniqueness and derived some data dependency results of solutions using topological degree technique by considering some classes of non-local Cauchy problems including BVPs and impulsive Cauchy problems (ICPs) to FDEs. Chenet al.[], obtained the existence results by coincidence degree the-ory to the following BVP involving a p-Laplacian operator:

⎧ ⎨ ⎩

C_Dq

+φp _C

Dp_+z(t)=θt,z(t),C_Dp

+z(t)

,

C_Dp

+z(t)|t==CDp+z() = ,

whereC_Dq

+andCD

p

+represent non-integer order derivatives in the Caputo sense,p,q∈ (, ],p+q∈(, ]. Tanget al.[] applied the aforesaid degree theory and established results for the following two point BVP of non-integer order p-Laplace DEs:

⎧ ⎨ ⎩

C_Dq

+φp _C

Dp_+z(t)=θt,z(t),C_Dp

+z(t)

,

z(t)_t_== , C_Dp

+z(t)t==

C_Dp

+z(),

whereC_Dq

+andCD

p

+are non-integer order derivatives of Caputo type,p,q∈(, ],p+q∈ (, ]. The mentioned theory of degree type has been studied recently in many papers; see [–].

The current manuscript is inspired from the aforesaid work. Our aim is to investigate the existence and uniqueness of at least one solution by applying the coincidence degree theory for a condensing mapping to the three-points BVP supplied as

⎧ ⎨ ⎩

–C_Dq_z₍_t_{) =}_θ_t_,_z₍_t₎_; _t_∈_J_,_q_∈_{(, ],} z(t)|t==φ(z), z() =δCDpz(η).

()

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Section  is concerned with some background material and lemmas required for the main results. In Section , the problem under consideration of FDEs is transformed to its equivalent Fredholm integral equation. Then the required theory devoted to the aims of this paper is developed via using coincidence degree of condensing maps and using the standard singular Grönwall inequality. At the end, an example is provided for justiﬁcation of the established results.

2 Background material

This section contains basics materials and preliminaries related to of non-integer order calculus and degree theory of topological type. For further details, we refer to [–, – ].

The space consisting of all continuous functionsJ→Ris a Banach space endowed with a normzZ=sup{|z(t)|:t∈J}. For simplicity, we denote the deﬁned space by Z =C(J, R).

Deﬁnition . Letz∈C(R+_{, R) be a function. Then the non-integer order integral of} or-derq∈R+of the functionz(t) is deﬁned as

Iq_a₊z(t) = 

(q)

t

a z(τ) (t–τ)q–dτ,

provided that integral on the right is pointwise deﬁned on (,∞).

Deﬁnition . The Caputo type non-integer order derivative of a functionz: R+_→_R_is deﬁned by

C_Dq a+z(t) =



(m–q)

t

a

(t–τ)m–q–z(m)(τ)dτ,

wherem= [q] +  and [q] represents the integer part ofq.

For further details on fractional derivatives and integrals; see [–].

Lemma .([]) The unique solution of FDE of order q> 

C_Dq_z₍_t_{) = ,} _q_∈₍_m_{– ,}_m_],

is given as

z(t) =e+et+et+· · ·+em–tm–, where ei∈R,i= , , , . . . ,m– .

Theorem .([]) The given FDE

C_Dq_z₍_t_{) =}_δ₍_t_), _q_∈₍_m_{– ,}_m_],

has a solution given by

z(t) = Iqδ(t) +e+et+et+· · ·+em–tm–,

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Next, we present some important deﬁnitions, propositions and theorems from []. For the Banach space Z, with C∈P(Z) represents the collection of all bounded sets.

Deﬁnition . The measure with non-compactness of Kuratowski typeβ : C→R+as

given by

β(B) =min{d> },

where B∈Cinserts a ﬁnite cover with a sets of diameter≤d.

Proposition . The measure of Kuratowski type denoted byβsatisﬁes the following prop-erties:

(i) the setBis relatively compact if and only ifB∈Chas Kuratowski measure zero; (ii) βis a seminorm,because it satisﬁesβ(λB) =|λ|β(B),λ∈Rand

β(B+ B)≤β(B) +β(B);

(iii) β(B)≤β(B)forB⊂Bandβ(B∪B) =sup{β(B),β(B)};

(iv) β(convB) =β(B); (v) β(B¯) =β(B).

Deﬁnition . Assume that the functionF:→Zis a continuous and bounded

map-ping for⊂Z. ThenFisβ-Lipschitz if there existsK≥ such that

βF(B)≤Kβ(B), for all B⊂bounded.

Also ifK< , thenFis said to be a strictβ-contraction.

Deﬁnition . The functionFisβ-condensing if

βF(B)<β(B), for every B⊂bounded withβ(B) > .

In other words,β(F(B))≥β(B) impliesβ(B) = .

Here, we represent the family of all strict β-contraction mappings F : →Z by

Cβ(). Further, denoting the family of allβ-condensing mappingsF:→Zby Cβ().

Remark  EachF∈Cβ() isβ-Lipschitz with constantK= , whereCβ()⊂Cβ().

Moreover, if there existsK> , thenF:→Zis said to be Lipschitz if and only if

_F(z) –F(z¯)≤K|z–¯z|, for everyz,¯z∈.

AlsoFis strict contraction if and only ifK< .

The provided propositions are necessarily required for our analysis throughout this pa-per.

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Proposition . The operatorFisβ-Lipschitz with constantK= .Then the same oper-atorF:→Zis compact.

Proposition . If an operatorF:→Zis Lipschitz with constantK.Then the same operatorFwill also beβ-Lipschitz with the same constantK.

Theorem . We recall some basic properties of proposed degree theory from Isaia[].

Let for the family of admissible triplets given by

=(I–F,,z) :⊂Zbe an open and bounded set,F∈Cβ(¯),

z∈Z\(I–F)(∂),

there exists a functiondeg:→Zof one degree which has the following properties.

(D) Normalization:deg(I,,z) = at eachz∈;

(D) additivity on domain:For each pair of disjoint open sets,⊂and each

z∈/(I–F)((¯)\(∪)),we have

deg(I–F,,z) =deg(I–F,,z) +deg(I–F,,z);

(D) invariance property under homoptopy:deg(I–H(t,z),,z)is independent oft∈J for each continuous and bounded mappingH:J× ¯→Zwhich satisﬁes

βH(J×B)<β(B), for allB⊂ ¯withβ(B) > 

and every continuous functionz:J→Zwhich satisﬁes

z=z–H(t,z), for allt∈J,for everyz∈∂;

(D) existence:deg(I–F,,z)= yields

z∈(I–F)();

(D) excision:deg(I–F,,z) =deg(I–F,,z)for each open set⊂and for all

z∈/(I–F)(¯ \).

Theorem . Assume thatF: Z→Zis aβ-condensing operator and

={z∈Z:there existsλ∈Jwith z=λFz} ⊂Z

is a bounded set and there exists a real number r> with⊂Br().Then

degI–λF, Br(), 

= , for allλ∈J.

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Theorem .([]) Let z∈Zsatisﬁes the following inequality:

z(t)≤ ˆa+bˆ t



(t–τ)q–z(s)λdτ+ˆc t



(T–τ)q–z(τ)λdτ, q∈(, ]. ()

Here <λ<  –_r for some r∈(,_–_q)andaˆ,bˆ,ˆc∈(,∞)are constants.Then we have z(t)≤(aˆ+ )eMT.

3 Existence of at least one solution to BVP (1)

The purpose of this section is concerning to establish the required theory for existence of at least one solutions to the BVP ().

Lemma .

Forω∈L(J, R),the solution of the linear BVP of FDEs

C_Dq_z₍_t_{) +}_ω₍_t_{) = ;} _t_∈_J_,_q_∈_{(, ],}

u(t)|t==φ(z), z() =δCDpz(η),

()

is given as

z(t) =φ(z)( –td) + 



H(t,τ)ω(τ)dτ,

whereH(t,s)is the Green’s function given by

H(t,τ) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

td(–τ)q–

(q) – (t–τ)q–

(q) –

δtd(η–τ)q–p–

(q–p) ; ≤τ≤t≤η≤,

td(–τ)q–

(q) –

δtd(η–τ)q–p–

(q–p) ; ≤t≤τ≤η≤,

td(–τ)q–

(q) – (t–τ)q–

(q) ; ≤η≤τ≤t≤,

td(–τ)q–

(q) ; ≤η≤t≤τ≤.

()

Proof Consider equation () with the associated given boundary conditions; applying Iq

on –C_Dq_z₍_t_{) =}_ω₍_t_{) and thanks to Theorem ., we have}

z(t) = –Iq(t) +e+et ()

for somee,e∈R. From the non-local conditionz(t)t==φ(z) implies thate=φ(z) and

z() =δCDp_u₍_η_{) yields}_e

=d[Iqω() –δIq–pω(η) –φ(z)], whered=_(–(–_p_)–p_δη)–p> . It implies

that

z(t) = –Iqω(t) +tdIqω() –δtdIq–pω(η) + ( –td)φ(z). ()

Thus we get a solutionzin the form

z(t) =φ(z)( –td) + 



H(t,τ)ω(τ)dτ,

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Lemma . A function z∈Zwill be the solution of the fractional integral equation()if and only if z is a solution of ().

Proof The proof is obvious.

To derive formally the required results as regards the data dependence and existence of at least one of solutions to the proposed BVP (), we state the following hypotheses:

(A) For arbitraryu,v∈Zand if there exists a constantKφ∈[, ), then one has

φ(z) –φ(¯z)≤Kφz–¯zZ;

(A) for constantsCφ,q∈[, )andMφ> , we have the following growth condition:

φ(z)≤CφzZq+Mφ, for eachz∈Z;

(A) in the same fashion, for constantsCθ,q∈[, )andMθ> , we have the following growth condition:

θ(t,z(t))≤CθzqZ+Mθ.

To show that equation () has at least one solutionz∈Zbased on assumptions (A)-(A), we deﬁne the operators by

F: Z→Z

as follows:

(Fz)(t) = ( –td)φ(z), d> 

andG: Z→Zis deﬁned by

(Gz)(t) = td

(q) 



( –s)q–θτ,z(τ)dτ– 

(q)

t



(t–τ)q–θτ,z(τ)dτ

– 

(q–p)

η



(η–τ)q–p–θτ,z(τ)dτ,

T: Z→Z, Tz=Fz+Gz.

Obviously, the operator T is well deﬁned because the functionθis continuous. So, we can write () as an operator equation given by

z= Tz=Fz+Gz. ()

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Lemma . The operatorF: Z→Zis Lipschitz and consequentlyβ-Lipschitz with con-stantK_F< .Moreover,the operatorFsatisﬁes the following growth condition:

Fz ≤Qzq

Z +Mφ, for every z∈Z. ()

Proof ForFto be Lipschitz, we consider|Fz(t) –Fz¯(t)|, and apply assumptions (A) and (A), we have

_Fz(t) –Fz¯(t)=( –td)φz(t)–φz¯(t)

=( –td)φz(t)–φz¯(t)

≤( –td)Kφz–z¯Z

≤KFz–z¯Z, whereK_F=( –td)Kφ< .

Hence, we get

Fz–Fz¯Z≤KFz–z¯Z, for everyz∈Z.

Thanks to proposition .,Fis alsoβ-Lipschitz with the same coeﬃcientK_F.

To derive the growth condition, we consider (Fz)(t) = ( –td)φ(z), and applying assump-tion (A), we get

Fz ≤Qzq

Z +Mφ, for everyz∈Z,

whereQ=|d|Cφ.

Lemma . The operatorG: Z→Zis continuous.Moreover,it also satisﬁes the growth condition as

GzZ≤

d+ 

(q–p+ )

CθzqZ+Mθ

, ()

for every z∈Z.

Proof Let{zm}be a sequence in the bounded setB¯={zZ≤κ:z∈Z}such thatzm→z

asm→ ∞inB¯. We need to show thatGzm–GzZ→ asm→ ∞. Sinceθis continuous

andzm→z, therefore,θ(τ,zm(τ))→θ(τ,z(τ)) asm→. Now consider

(Gzm)(t) – (Gz)(t)

≤ td

(q) 



( –τ)q–θτ,zm(τ)–θτ,z(τ)dτ

+ tδd

(q–p)

η



(η–τ)q–p–θτ,zm(τ)–θτ,z(τ)dτ

+ 

(q)

t



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In view of assumption (A) and thanks to the Lebesgue dominated convergence theorem, one has

(Gzm)(t) – (Gz)(t)_Z→ asm→ ∞,

which shows thatGis continuous. To obtain the growth condition for the nonlinear op-eratorG, consider

(Gz)(t)= td

(q) 



( –τ)q–θτ,z(τ)dτ

– tδd

(q–p)

η



(η–τ)q–p–θτ,z(τ)dτ

– 

(q)

t



(t–τ)q–θτ,z(τ)dτ

≤ d+ 

(q–p+ )

zq

Z +Mθ

.

Applying assumption (A), we obtain the condition ().

Lemma . The operatorG: Z→Zis compact.Consequently,Gisβ-Lipschitz with zero constant.

Proof To prove the required result, we take a bounded set D⊂ ¯B⊆Z. Let{zm}be a se-quence on D⊂ ¯B, then from () for everyzm∈D, we have

GzmZ≤

d+ 

(q–p+ )

CθzmqZ+Mθ

,

which implies thatG(D) is bounded in Z. Next, we will show that{Gzm}is equi-continuous. For this purpose, lett<t∈(, ), and using these relationsδηq–p< ,(q+)<



(q–p+), we have

(Gzm)(t) – (Gz)(t)≤

(t–t)d

(q) 



( –τ)q–θτ,z(τ)dτ

+(t–t)δd

(q–p)

η



(η–τ)q–p–θτ,z(τ)dτ

+ 

(q)

t



(t–τ)q–– (t–τ)q–θ

τ,z(τ)dτ

–

t

t

(t–τ)q–θ

τ,z(τ)dτ,

which on simpliﬁcation takes the form

(Gzm)(t) – (Gz)(t)≤

(Cθκq+Mθ)

(q–p+ )

(t–t)d+

t_q–t_q– (t–t)q

. ()

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Furthermore, in view of Proposition ., the nonlinear operatorGisβ-Lipschitz with

con-stant zero.

From now on, we will prove our main results.

Theorem . Under the hypotheses(A)-(A)equation()has at least one solution u∈Z.

Also,the set of solutions for()is bounded inZ.

Proof Thank to Proposition ., the operator T is a strictβ-contraction with constantKφ.

Consider the set

S=

z∈Z: there existsλ∈[, ] such thatz=λTz.

We need to show that S⊂Zis bounded. For this purpose, consider

|z|=|λTz|=λ|Tz| ≤λFzZ+GzZ

,

using () and (), we have

|z| ≤λ

Qzq

Z +Mφ+

d+ 

(q–p+ )

zq

Z +Mθ

,

which implies usingλ<  that

zZ≤

Qzq

Z +Mφ+

d+ 

(q–p+ )

zq

Z +Mθ

. ()

Hence, () andq,q ∈(, ) shows that Sis bounded in Z. If it is not bounded then assume thatzZ=ρand consider thatρ→ ∞. Then from (), we have

≤[Qz

q

Z +Mφ+₍_qd_–+_p_+)(zqZ+Mθ)]

ρ , ()

where due to assumption

ρ→ ∞ yields ≤.

This is impossible, so we take Sto be bounded.

Therefore, we conclude that the operator T has at least one ﬁxed point and the set of

ﬁxed points is bounded in Z.

We make the following assumption for discussion of data dependence of solutions:

(A) There exist constantsLθ> ,λ∈[,  –_r]for somer∈(,  –_–_q)such that

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Theorem . Assuming that(A)-(A)hold,let z(t)z¯(t)be the solutions of(FDE) ()with

associated boundary conditions.Then there exists a constantM∗> such that

z(t) –z¯(t)≤M∗

 (p( –q) + )

 p

.

Proof Consider|z(t) –z¯(t)|, and thanks to (A), (A), and (A), we get

z(t) –z¯(t)≤Kφz(t) –z¯(t)+ L θ

(q)

t



(t–τ)q–z(τ) –z¯(τ)λdτ

+dLθ

(q) 



( –τ)q–z(τ) –z¯(τ)λdτ

+ δLθd

(q–p)

η



(η–τ)q–p–z(τ) –¯z(τ)λdτ.

Upon further simpliﬁcation and using (), we get

z(t) –z¯(t)≤   –Kφ

_L θ

(q)

t



+ 

 –Kφ

dLθ

(q) 



( –τ)q–z(τ) –¯z(τ)λdτ

+ δLθd

(q–p)

η



(η–τ)q–p–z(τ) –¯z(τ)λdτ

.

Hence using () and Theorem ., we obtain

|z–z¯| ≤M∗,

whereM∗=eMand

M=   –Kφ

_L θ

(q)

t



+ 

 –Kφ

dLθ

(q) 



( –τ)q–z(τ) –¯z(τ)λdτ

+ δLθd

(q–p)

η



(η–τ)q–p–z(τ) –¯z(τ)λdτ

.

Now, re-write assumption (A) as follows.

(A) For aLθ> , the following relation holds:

θ(t,z) –θ(t,z¯)≤Lθ|z–z¯|, for eacht∈J,and for eachz,z¯∈R.

Theorem . Assume that the hypotheses(A)-(A)hold,then FDE()has a unique

solu-tion z∈Zif _–M_Kφ∗ < .

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assump-tions (A), (A) and (A), we obtain

Tz(t) – Tz¯(t)≤   –Kφ

Kφz–z¯Z+

Lθ

(q)

t



+ 

 –Kφ _L

θtd

q





( –τ)q–z(τ) –z(τ)

λ dτ

+ δdLθ

(q–p)

η



(η–s)q–p–z(τ) –z¯(τ)λdτ

,

using the inequality in Theorem ., we get

Tz– T¯zZ≤ M

∗

 –Kφ

z–z¯Z, t∈J,

which produces the uniqueness ofz.

4 Illustrative example

Example  Take the following FDE subject to the multi-points boundary conditions:

C_Dq_z₍_t_{) = –} cos(t)

 +t_|_z₍_t₎_|, t∈J,

z(t)|t==φ(z) = 

j= 

z(ηj), z() =  

C_D__z   . ()

Here, we takeq=_ andδ=η=_,Kφ=_ ,ηj=_j,j= , , , .r= ∈(, ),λ=_∈[, ],

Lθ=Cθ=_,Cφ=_,p=_,Mφ=Mθ= .

By simple computation,d= .,Q=dCφ= .× _ = .. For the

consid-ered problem () all the data dependence results (A)-(A) are satisﬁed. It is also obvious that the solutionz

zZ≤.,

is bounded. Thus due to Theorem . there exists at least one solution for () which is bounded. Along the same line, one can derive the assumptions of Theorem . and ..

Example  Consider the following boundary value problem of FDEs:

C_D__z₍_t_{) = –} sin|z(t)|

 +exp(t₎, t∈J,

z(t)|t==φ(z) = 

cos|z|, z() =  

C_D__z   . ()

From the given problem (), we see thatq=

and we takeδ=η=  ,Kφ=



.r= ∈(, ),

λ=_∈[, ],Lθ=Cθ=_,Cφ=_,p=_,Mφ=Mθ= .

Upon computation, we getd= .,Q=dCφ= .. Thus for the given

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Further, it is easy to show by using Theorem . that there exists at least one solution for () which is bounded. Also, one can easily derive the assumptions of Theorems . and ..

5 Concluding remarks

In this paper, we have successfully applied ana prioriestimate method known as topolog-ical degree method rather than Schauder and Brouwer degree theory. Highly interesting results for the existence of at least one solution have been derived. In the future, we can extend the concerned theory to highly applicable nonlinear problems of applied analysis to investigate them for solutions.

Acknowledgements

We are thankful to the reviewers for their useful corrections and suggestions which improved the quality of this paper. This research work has been supported ﬁnancially by Abdul Wali Khan University Mardan, Pakistan and Cankaya University, Turkey.

Competing interests

We declare that we have no competing interest corresponding to this paper.

Authors’ contributions

All authors equally contributed to this paper and approved the last version.

Author details

1_{Department of Mathematics, University of Malakand, Chakadara Dir(L), Khyber Pakhtunkhwa, Pakistan.}2_{Department of}

Mathematics, Cankaya University, Ankara, Turkey.3_{Department of Mathematics, Abdulwali Khan University Mardan,}

Khyber Pakhtunkhwa, Pakistan.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 8 May 2017 Accepted: 18 July 2017 References

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