Existence theory to a class of boundary value problems of hybrid fractional sequential integro-differential equations

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R E S E A R C H

Open Access

Existence theory to a class of boundary value

problems of hybrid fractional sequential

integro-differential equations

M. Jamil

1*

_{, R.A. Khan}

1

_{and K. Shah}

1

*_{Correspondence:}

[email protected] 1_{Department of Mathematics,}

University of Malakand, Chakdara, Dir(Lower), khyber Pakhtunkhwa, Pakistan

Abstract

In this article, we study the existence result for a boundary value problem (BVP) of hybrid fractional sequential integro-diﬀerential equations. A ﬁxed point theorem provided by Dhage in (Nonlinear Anal. 4:414–424,2010) is used for the solution existence of our boundary value problem. Also we illustrated our result through an example.

MSC: Primary 26A33; 34A08; secondary 34A38

Keywords: Boundary value problem; Hybrid fractional sequential integro-diﬀerential equation; Fixed point theorem

1 Introduction

Fractional calculus is a branch of mathematics which investigates the properties of inte-grals and derivatives of non-integer order. These inteinte-grals and derivatives of non-integer order are called fractional integrals and derivatives. The potential of fractional calculus has drawn the attention of applied scientists, engineers, and other researchers. Also the-oretical scientists or researchers are interested in the field of fractional derivatives and fractional integration. Chronologically, in [2] we can find different approaches of frac-tional calculus proposed by Liouville, Euler, Fourier, Riemann, Abel, etc. We can find some relevant work on fractional calculus in a series of papers [3–28].

In the area of fractional calculus, several researchers have studied hybrid fractional dif-ferential equations, see [1,29–32] and the references therein. Recently in [33], the authors developed an existence result for a boundary value problem of hybrid fractional diﬀeren-tial equations of the form

⎧ ⎨ ⎩

Dα₍φ(t)–ψ(t,φ(t))

θ(t,φ(t)) ) =ϕ(t,φ(t)), t∈[0, 1],α∈(1, 2],

[φ(t_θ)–₍_tψ_,_φ(₍t_t,φ₎₎(t))]|t=0= 0, [φ(tθ)–(tψ,φ((tt,))φ(t))]|t=1= 0,

whereθ∈C(J×R,R\ {0}),ψ,ϕ∈C(J×R,R).

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In [34], Sitho derived an existence result for an initial value problem of hybrid fractional sequential integro-diﬀerential equations of the form

⎧ ⎨ ⎩

Dα_[Dωu(t)–mi=1Iβifi(t,u(t))

g(t,u(t)) ] =h(t,u(t),I

γ_u₍_t_)), _t_∈_[0,_T_],

u(0) = 0, Dω_u_{(0) = 0,} (1)

whereDα_,_Dω_{denote the Riemann–Liouville fractional derivatives of order}_α_,_ω

respec-tively and 0 <α,ω≤1.

In this article, we study the existence result for the boundary value problem of hybrid fractional sequential integro-diﬀerential equations involving Caputo derivatives given by

⎧ ⎨ ⎩

c_Dα_[cDωu(t)–im=1Iβifi(t,u(t))

g(t,u(t)) ] =h(t,u(t),I

γ_u₍_t_)), _t_∈_J_{= [0, 1],}

u(0) = 0, Dω_u_{(0) = 0,} _u_{(1) =}_δ_u₍_η_), _{0 <}_δ_{< 1, 0 <}_η_{< 1,} (2)

where c_Dα _{is the Caputo fractional derivative of order}_α_, c_Dω _{is the Caputo fractional}

derivative of orderω, 0 <α≤1, 1 <ω≤2,Iγ _{denotes the Riemann–Liouville fractional}

integral of orderγ > 0,Iβi_{denotes the Riemann–Liouville fractional sequential integrals}

of orderβi> 0,g∈C(J×R,R\ {0}),h∈C(J×R2_,_R_{) and}_f

i∈C(J×R,R) withfi(0, 0) = 0,

i= 1, 2, . . . ,m.

In comparison to problem (1), our considered BVP (2) is more general than the problem studied in [34], as we consider a problem with three point boundary conditions, while the authors in [34] investigated an initial value problem. Moreover, in our problem the fractional orders of derivatives are 0 <α≤1 and 1 <ω≤2, whereas in problem (1) the fractional orders are 0 <α,ω≤1.

We obtain an existence result for the boundary value problem (2) in Sect.3by using generalized Krasnoselskii’s ﬁxed point theorem provided by Dhage in [1].

2 Preliminaries

This section provides some important deﬁnitions of fractional calculus [2, 35, 36] and results of ﬁxed point theory [1,20,21], which is base for the forthcoming sections.

Deﬁnition 1 The Caputo fractional derivative of positive real orderω> 0 of a function

u(t) is given by

c_Dω

u(t) = 1

Γ(n–ω)

t

0

u(n)₍_s₎

(t–s)ω–n+1ds, ω> 0,

provided that the integral on the right-hand side exists.

Deﬁnition 2 The Riemann–Liouville fractional derivative of positive real orderω> 0 of a functionu(t) is given by

Dωu(t) = 1

Γ(n–ω)

dn dtn

t

0

u(s)

(t–s)ω–n+1ds, ω> 0,

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Deﬁnition 3 The Riemann–Liouville fractional integral of positive real orderω> 0 of a functionu(t) is given by

Iω_u₍_t_{) =} 1 Γ(ω)

t

0

(t–s)ω–1_u₍_s₎_ds_, _ω_{> 0,}

provided that the integral on the right-hand side exists.

Remark1 For Deﬁnitions1–3,n= [ω] + 1, [ω] is the greatest integer less thanωandΓ is the gamma function deﬁned byΓ(ω) =₀∞e–s_sω–1_ds_.

Property 1([36]) Letβ,ω> 0 andu(t) be a function, then the semi-group property for the Riemann–Liouville fractional integrals of ordersβandωrespectively is given by

IβIωu(t) =Iβ+ωu(t).

Next we present two important results as lemmas which we need later.

Lemma 1([37]) For u∈C(0,T)∩L(0,T),the solution of the Caputo fractional diﬀerential equation

c_Dω

u(t) =g(t), n– 1 <ω<n,

is given by

u(t) =Iωg(t) +

n–1

i=0

kiti,

or

u(t) =Iωg(t) +k0+k1t+· · ·+kn–1t,

where n= [ω] + 1, [ω]is the greatest integer less thanω.

LetE=C(J,R) be the Banach space of continuous real-valued functions deﬁned onJ= [0, 1]. We deﬁne a norm·and a multiplication inEbyu=sup_t_∈_J|u(t)|and (uv)(t) =

u(t)v(t),∀t∈J. Then clearlyEis a Banach algebra with above deﬁned supremum norm and multiplication in it.

Lemma 2([1]) Let S be a nonempty,convex,closed,and bounded set such that S⊆E,and let A:E→E and B:S→E be two operators which satisfy the following:

(c1) Ais contraction,

(c2) Bis compact,and

(c3) u=Au+Bv∀v∈S ⇒ u∈S.

(4)

3 Solution existence of hybrid fractional sequential integro-differential equations

In this section, we consider the hybrid fractional sequential integro-diﬀerential equation boundary value problem (2). First we derive its solution and then discuss the existence of solution using the above stated deﬁnitions and results of the preliminary section.

Lemma 3 Suppose that0 <α≤1, 1 <ω≤2,γ > 0,and functions g,h,fi,i= 1, 2, . . . ,m,

satisfy BVP(2).Then the unique solution of BVP(2)is given by

u(t) =

t

0

(t–s)ω–1

Γ(ω) g

s,u(s)

s

0

(s–μ)α–1

Γ(α) h

μ,u(μ),Iγu(μ)dμds

+

m

i=1

Iβi+ω_f

i

t,u(t)

+ t

δη– 1

1

0

(1 –s)ω–1

Γ(ω) g

s,u(s)

s

0

(s–μ)ω–1

Γ(α) h

+

m

i=1

Iβi+ω_f

i

1,u(1)

–δ

η

0

(η–s)ω–1

Γ(ω) g

s,u(s)

s

0

(s–μ)ω–1

Γ(α) h

+

m

i=1

Iβi+ω_f

i

η,u(η)

, (3)

where

Iβi+ω_f

i

t,u(t)=

t

0

(t–s)βi+ω–1

Γ(βi+ω)

fi

s,u(s)ds,

Iβi+ω_f

i

1,u(1)=

1

0

(1 –s)βi+ω–1

Γ(βi+ω) fi

s,u(s)ds,

and

Iβi+ω_f

i

η,u(η)=

η

0

(η–s)βi+ω–1

Γ(βi+ω)

fi

s,u(s)ds.

Proof Assigning the Riemann–Liouville fractional integral operator of orderαto hybrid fractional sequential integro-diﬀerential equations (2) and using Lemma1, we have

c_Dω_u₍_t_{) –}m

i=1Iβifi(t,u(t))

g(t,u(t)) =I

α

ht,u(t),Iγu(t)+k0. (4)

In view of the initial conditionsu(0) = 0,Dω_u_{(0) = 0 of problem (}₂_{) with}_f

i(0, 0) = 0, we

havek0= 0, and then equation (4) takes the form

c_Dω

u(t) =gt,u(t)

t

0

(t–s)α–1

Γ(α) h

s,u(s),Iγu(s)ds+

m

i=1

Iβi_f

i

(5)

Now assigning the Riemann–Liouville fractional integral operator of orderωto equation (5) and using Lemma1with semi-group Property1, we have

u(t) =

t

0

(t–s)ω–1

Γ(ω) g

s,u(s)

s

0

(s–μ)α–1

Γ(α) h

μ,u(μ),Iγ_u₍_μ₎_d_μ_ds

+

m

i=1

Iβi+ω_f

i

t,u(t)+k1+k2t. (6)

The initial conditionu(0) = 0 of problem (2) givesk1= 0, for which equation (6) becomes

u(t) =

t

0

(t–s)ω–1

Γ(ω) g

s,u(s)

s

0

(s–μ)α–1

Γ(α) h

+

m

i=1

Iβi+ω_f i

t,u(t)+k2t, (7)

and then the boundary conditionu(1) =δu(η) of problem (2) gives

k2=

1

δη– 1

1

0

(1 –s)ω–1

Γ(ω) g

s,u(s)

s

0

(s–μ)ω–1

Γ(α) h

+

m

i=1

Iβi+ω_f

i

1,u(1)

–δ

η

0

(η–s)ω–1

Γ(ω) g

s,u(s)

s

0

(s–μ)ω–1

Γ(α) h

+

m

i=1

Iβi+ω_f

i

η,u(η)

,

for which equation (7) takes the form

u(t) =

t

0

(t–s)ω–1

Γ(ω) g

s,u(s)

s

0

(s–μ)α–1

Γ(α) h

+

m

i=1

Iβi+ω_f

i

t,u(t)

+ t

δη– 1

1

0

(1 –s)ω–1

Γ(ω) g

s,u(s)

s

0

(s–μ)ω–1

Γ(α) h

+

m

i=1

Iβi+ω_f

i

1,u(1)

–δ

η

0

(η–s)ω–1

Γ(ω) g

s,u(s)

s

0

(s–μ)ω–1

Γ(α) h

+

m

i=1

Iβi+ω_f

i

η,u(η)

.

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For developing the existence result, we consider some assumptions which are the fol-lowing.

(A1) The functionsfi∈C(J×R,R), g∈C(J×R,R\ {0}), andh∈C(J×R2,R)are continuous, and there exist positive functionsλi(t),Θ(t), andψ(t)with boundsλi,

Θ, andψrespectively such that

fi

t,u(t)–fi(t,v(t)≤λi(t)u(t) –v(t),

gt,u(t)–g(t,v(t)≤Θ(t)u(t) –v(t),

and ht,u(t),u¯(t)–h(t,v(t),v¯(t)≤ψ(t)u(t) –v(t)+u¯(t) –v¯(t)

fort∈Jandu,v∈R.

(A3) There existsr> 0such that

1 + 1 +δ

|δη– 1|

χΩ Γ(α+ 1)Γ(ω+ 1)+

m

i=1

φi Γ(βi+ω+ 1)

≤r

and

1 + 1 +δ

|δη– 1|

ΩΘ+χψ Γ(α+ 1)Γ(ω+ 1) +

χψ

Γ(α+ 1)Γ(ω+ 1)Γ(γ+ 1)

< 1. (8)

Our main existence result is based on generalized Krasnoselikii’s ﬁxed point theorem by Dhage [1], which we have provided in Lemma2.

Theorem 2 Let assumptions(A1)–(A3)hold,then there exists at least one solution for BVP

(2)inJ= [0, 1].

Proof First we setsup_t_∈_J|φi(t)|=φi,supt∈J|λi(t)|=λi,i= 1, 2, . . . ,m,supt∈J|Θ(t)|= Θ,sup_t_∈_J|ψ(t)|=ψ,sup_t_∈_J|Ω(t)|=Ω, andsup_t_∈_J|χ(t)|=χ.

Now we considerE=C(J,R) and deﬁneS⊆Eas

S=u∈E:u ≤r.

ClearlySis a closed, convex, and bounded subset of the Banach spaceE. Let us deﬁne two operatorsC:E−→EandD:E−→Esuch that

Cu(t) =

t

0

(t–s)α–1

Γ(α) h

s,u(s),Iγu(t)ds

and

Du(t) =gt,u(t).

Then, using assumptions (A1), (A2) and proceeding with maximum overJ, we have

Cu(t) –Cv(t)≤ ψ

Γ(α+ 1)

1 + 1

Γ(γ + 1)

(7)

Du(t) –Dv(t)≤ Θu–v, (10)

Cu(t)≤ Ω

Γ(α+ 1) (11)

and Du(t)≤ χ. (12) Now we deﬁne two more operatorsA:E−→EandB:S−→Esuch that

Au(t) =

t

0

(t–s)ω–1

Γ(ω) Du(s)Cu(s)ds+

t

δη– 1

1

0

(1 –s)ω–1

Γ(ω) Du(s)Cu(s)ds – δt

δη– 1

η

0

(η–s)ω–1

Γ(ω) Du(s)Cu(s)ds, t∈J (13) and

Bu(t) =

m

i=1

Iβi+ω_f

i

t,u(t)+ t

δη– 1

m

i=1

Iβi+ω_f

i

1,u(1)

– δt

δη– 1

m

i=1

Iβi+ω_f

i

η,u(η), t∈J,

or

Bu(t) =

m

i=1

t

0

(t–s)βi+ω–1

Γ(βi+ω)

fi

s,u(s)ds+ t

δη– 1

m

i=1

1

0

(1 –s)βi+ω–1

Γ(βi+ω)

fi

s,u(s)ds

– δt

δη– 1

m

i=1

η

0

(η–s)βi+ω–1

Γ(βi+ω)

fi

s,u(s)ds, t∈J. (14)

Then the integral equation (3) can be written in the operator form as follows:

u(t) =Au(t) +Bu(t), t∈J. (15)

We show that the two operatorsAandBin (15) satisfy all the conditions of Lemma2. This can be achieved in the following steps.

Step 1. First we show thatAis a contraction mapping. Letu(t),v(t)∈S, then we have

Au(t) –Av(t)

≤ t

0

(t–s)ω–1

Γ(ω) Du(s)Cu(s) –Dv(s)Cv(s)ds + t

|δη– 1|

1

0

(1 –s)ω–1

Γ(ω) Du(s)Cu(s) –Dv(s)Cv(s)ds + δt

|δη– 1|

η

0

(η–s)ω–1

Γ(ω) Du(s)Cu(s) –Dv(s)Cv(s)ds

≤

1 + 1

|δη– 1|+

δ

|δη– 1|

1

0

(1 –s)ω–1

Γ(ω) Du(s)Cu(s) –Dv(s)Cv(s)ds

≤

1 + 1 +δ

|δη– 1|

1

0

(1 –s)ω–1

(8)

+Cu(s)Dv(s) –Dv(s)Cv(s)ds

≤

1 + 1 +δ

|δη– 1|

1

0

(1 –s)ω–1

Γ(ω) Cu(s)Du(s) –Dv(s)ds +

1

0

(1 –s)ω–1

Γ(ω) Dv(s)Cu(s) –Cv(s)ds

,

which gives

Au(t) –Av(t)≤

1 + 1 +δ

|δη– 1|

ΩΘ+χψ Γ(α+ 1)Γ(ω+ 1) + χψ

Γ(α+ 1)Γ(ω+ 1)Γ(γ+ 1)

u–v.

Hence by (8) the operatorAis a contraction mapping.

Step 2. Next we show that the operatorBsatisﬁes condition (c2) of Lemma2, that is,

the operatorBis compact onS. Therefore ﬁrst we show that the operatorBis continuous onS. Let{un(t)}be a sequence of functions inSconverging to a functionu(t)∈S. Then,

by the Lebesgue dominant convergence theorem,∀t∈J, we have

lim

n→∞Bun(t) =nlim→∞

_m

i=1

t

0

(t–s)βi+ω–1

Γ(βi+ω)

fi

s,un(s)

ds

+ t

δη– 1

m

i=1

1

0

(1 –s)βi+ω–1

Γ(βi+ω) fi

s,un(s)

ds

– δt

δη– 1

m

i=1

η

0

(η–s)βi+ω–1

Γ(βi+ω)

fi

s,un(s)

ds = m i=1 t 0

(t–s)βi+ω–1

Γ(βi+ω)

lim

n→∞fi

s,un(s)

ds

+ t

δη– 1

m

i=1

1

0

(1 –s)βi+ω–1

Γ(βi+ω)

lim

n→∞fi

s,un(s)

ds

– δt

δη– 1

m

i=1

η

0

(η–s)βi+ω–1

Γ(βi+ω)

lim

n→∞fi

s,un(s)

ds = m i=1 t 0

(t–s)βi+ω–1

Γ(βi+ω) fi

s,u(s)ds

+ t

δη– 1

m

i=1

1

0

(1 –s)βi+ω–1

Γ(βi+ω) fi

s,u(s)ds

– δt

δη– 1

m

i=1

η

0

(η–s)βi+ω–1

Γ(βi+ω)

fi

s,u(s)ds.

Hencelimn→∞Bun(t) =Bu(t). ThusBis a continuous operator onS. Further, we show that

the operatorBis uniformly bounded onS. For anyu∈S, we have

Bu(t)≤

m

i=1

t

0

(t–s)βi+ω–1

Γ(βi+ω)

fi

(9)

+ t

|δη– 1|

m

i=1

1

0

(1 –s)βi+ω–1

Γ(βi+ω) fi

s,u(s)ds

+ δt

|δη– 1|

m

i=1

η

0

(η–s)βi+ω–1

Γ(βi+ω)

fi

s,u(s)ds

≤

1 + 1 +δ

|δη– 1|

m

i=1

φi Γ(βi+ω+ 1)

:=M.

ThereforeBu(t) ≤M,∀t∈J, which shows thatBis uniformly bounded onS. Now, we show that the operatorBis equi-continuous. Lett1,t2∈Jwitht1<t2andu(t)∈S. Then

we have

Bu(t2) –Bu(t1)≤

m i=1 t2 0

(t2–s)βi+ω–1

Γ(βi+ω)

fi

s,u(s)ds

– m i=1 t1 0

(t1–s)βi+ω–1

Γ(βi+ω)

fi

s,u(s)ds

+t2–t1

δη– 1

m

i=1

1

0

(1 –s)βi+ω–1

Γ(βi+ω)

fi

s,u(s)ds

–δ(t2–t1)

δη– 1

m

i=1

η

0

(η–s)βi+ω–1

Γ(βi+ω)

fi

s,u(s)ds ≤ m i=1 φi Γ(βi+ω)

t1 0

(t2–s)βi+ω–1– (t1–s)βi+ω–1 ds

+

t2

t1

(t2–s)βi+ω–1ds+

t2–t1

δη– 1

m

i=1

1

0

(1 –s)βi+ω–1_ds

–δ(t2–t1)

δη– 1

m

i=1

η

0

(η–s)βi+ω–1_ds

≤ m i=1 φi Γ(βi+ω+ 1)

tβi+ω

2 –t

βi+ω

1 + (t2–t1)βi+ω

+t2–t1

δη– 1–

δηβi+ω₍_t

2–t1)

δη– 1

.

Now ast2–t1−→0, so the right-hand side tends to zero. ThusBis equi-continuous.

Therefore, it follows from the Arzelá–Ascoli theorem thatBis a compact operator onS.

Step 3.Condition (c3) of Lemma2holds. So, for anyv∈S, we have

_u(t)=Au(t) +Bv(t)

≤Au(t)+Bv(t)

≤ t

0

(t–s)ω–1

Γ(ω) Du(s)Cu(s)ds+

t

δη– 1

1

0

(1 –s)ω–1

Γ(ω) Du(s)Cu(s)ds – δt

δη– 1

η

0

(η–s)ω–1

Γ(ω) Du(s)Cu(s)ds

+ m i=1 t 0

(t–s)βi+ω–1

Γ(βi+ω)

fi

(10)

+ t

δη– 1

m

i=1

1

0

(1 –s)βi+ω–1

Γ(βi+ω) fi

s,v(s)ds

– δt

δη– 1

m

i=1

η

0

(η–s)βi+ω–1

Γ(βi+ω)

fi

s,v(s)ds

≤

1 + 1 +δ

|δη– 1|

χΩ Γ(α+ 1)Γ(ω+ 1)+

m

i=1

φi Γ(βi+ω+ 1)

≤r,

which impliesu ≤r, and sou∈S. Hence all the conditions of Lemma2are satisﬁed. Therefore, the operator equationu(t) =Au(t) +Bu(t) has at least one solution inS. Con-sequently, there exists a solution of BVP (2) inJ= [0, 1]. Thus the proof is completed.

4 Example

We present an example of the BVP of hybrid fractional sequential integro-diﬀerential equations to test our main result.

Example1 ⎧ ⎪ ⎨ ⎪ ⎩

c_D35_[

c_D32_u₍_t_)–3

i=1Iβifi(t,u(t))

1

4t2sec(π3t)(|u (t)|+1

|u(t)|+2)

] = ₅2cos(t₆)(_|_u|u₍_t(₎t_|)₊₁| +I52_u₍_t_)), _t∈_J_,

u(0) = 0, Dω_u_{(0) = 0,} _u_{(1) =}_δ_u₍_η_), _{0 <}_δ_{< 1, 0 <}_η_{< 1,}

(16)

where

3

i=1

Iβi_f

i

t,u(t)=I13

cos t 3 _|

u(t)| 1 +|u(t)|

+etsint

+I43

√

tsint|u(t)|

|u(t)|+ 1 + 2 3 –t2

+I35

_|

u(t)|tan(√₂t)

|u(t) + 2| +

t

1 +et

. (17)

From equations (16) and (17), we haveα=3

5,ω=

3

2,m= 3,β1=

1 3,β2=

4 3,β3=

3

5,γ =

5 2,

f1(t,u(t)) =cos(₃t)(₁₊|u_|(_ut₍)_t|₎_|) +etsint,f2(t,u(t)) = √

tsint|u(t)| |u(t)|+1 +

2

3–t2,f3(t,u(t)) = |u(t)|tan(

√

t

2 )

|u(t)+2| +

t

1+et,

g(t,u(t)) =1₄t2_sec₍πt

3)( |u(t)|+1

|u(t)|+2), andh(t,u(t),I

γ_u₍_t_{)) =}2 5cos(

t

6)( |u(t)| |u(t)|+1+I

5

2_u₍_t_{)). It is easy to}

show that

f1

t,u(t)–f1(t,v(t)≤cos

t

3

u(t) –v(t),

f2

t,u(t)–f2(t,v(t)≤ √

tsintu(t) –v(t),

f3

t,u(t)–f3(t,v(t)≤tan

√

t

2

u(t) –v(t),

gt,u(t)–g(t,v(t)≤1 4t 2_sec πt 3

u(t) –v(t),

and ht,u(t),u¯(t)–h(t,v(t),v¯(t)≤2 5cos

t

6

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Therefore, we can choose

λ1(t) =cos

t

3

, λ2(t) = √

tsint, λ3(t) =tan

√

t

2

,

Θ(t) =1 4t

2_sec

πt

3

, ψ(t) =2 5cos

t

6

.

Also the functionsg(t,u(t)),h(t,u(t),Iγ_u₍_t_{)), and}_f

i(t,u(t)),i= 1, 2, 3, are bounded by the

positive functions as follows:

f1

t,u(t)≤cos

t

3

+etsint=φ1(t),

f2

t,u(t)≤√tsint+ 2

3 –t2 =φ2(t),

f3

t,u(t)≤tan

√

t

2

+ t

1 +et =φ3(t),

gt,u(t)≤1 4t

2_sec

πt

3

=χ(t),

and ht,u(t),v(t)≤2 5cos

t

6

=Ω(t).

Choosingδ= 0.1,η= 0.9 and puttingΩ=ψ= 0.4,χ=Θ= 0.5 in (8), we have

1 + 1 +δ

|δη– 1|

ΩΘ+χψ Γ(α+ 1)Γ(ω+ 1) +

χψ

Γ(α+ 1)Γ(ω+ 1)Γ(γ+ 1)

≈0.8594 < 1.

Hence all the conditions of Theorem2hold. Thus the boundary value problem (16) has at least one solution inJ= [0, 1].

5 Conclusion

In consequence to generalized Krasnoselskii’s ﬁxed point theorem provided by Dhage [1], we developed an existence result for the aforementioned boundary value problem (2) of hybrid fractional sequential integro-diﬀerential equations. The respective result has been tested by providing an illustrative example.

Acknowledgements

We thank the referees for their useful suggestions which improved the ﬁnal version of the manuscript.

Funding

There is no funding source to support this manuscript ﬁnancially.

Availability of data and materials

Not applicable.

Competing interests

It is declared that no competing interests exist regarding this manuscript.

Authors’ contributions

All authors have equal contribution in this manuscript. All authors read and approved the ﬁnal manuscript.

Publisher’s Note

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