Basic theory of differential equations with mixed perturbations of the second type on time scales

(1)

R E S E A R C H

Open Access

Basic theory of differential equations with

mixed perturbations of the second type on

time scales

Yige Zhao

1*

_{, Yibing Sun}

1

_{, Zhi Liu}

2

_{and Zhanbing Bai}

3

*_{Correspondence:}

[email protected]

1_{School of Mathematical Sciences,}

University of Jinan, Jinan, P.R. China Full list of author information is available at the end of the article

Abstract

In this paper, we develop the theory of diﬀerential equations with mixed

perturbations of the second type on time scales. We give an existence theorem for differential equations with mixed perturbations of the second type on time scales under Lipschitz condition. We also present some fundamental differential inequalities on time scales, which are utilized to investigate the existence of extremal solutions. We establish the comparison principle for differential equations with mixed

perturbations of the second type on time scales. Our results in this paper extend and improve some well-known results.

Keywords: Mixed perturbations; Existence; Diﬀerential inequalities; Comparison principle; Time scales

1 Introduction

In this paper, we discuss the following diﬀerential equations with mixed perturbations of the second type on time scales (DETS):

⎧ ⎨ ⎩

[u(t)–k(t,u(t))_f_(t,u(t)) ]₌_g₍_t_,_u₍_t_)), _t_∈_J_,

u(t0) =u0,

(1)

wheref ∈Crd(J×R,R\ {0}) andk,g∈Crd(J×R,R).

LetTbe a time scale, and letJ= [t0,t0+a]T= [t0,t0+a]∩Tbe a bounded interval inT

for somet0∈Randa> 0. We denote byCrd(J×R,R) the class of rd-continuous functions

g:J×R→R. For basic deﬁnitions and useful lemmas from the theory of calculus on time scales, we refer to [1].

By a solution of the DETS (1) we mean a-diﬀerentiable functionusuch that

(i) the functiont→u–k(t,u)_f_(t,u) is-diﬀerentiable for eachu∈R, and (ii) usatisﬁes equations (1).

The theory of time scales has been drawn a lot of attention since 1988 (see [1–9]). In recent years the theory of nonlinear diﬀerential equations with perturbations has been a hot research topic; see [10–15]. Dhage [13] discussed the following ﬁrst-order hybrid

(2)

diﬀerential equation with mixed perturbations of the second type:

⎧ ⎨ ⎩

d dt[

x(t)–k(t,x(t))

f(t,x(t)) ] =g(t,x(t)), t∈[t0,t0+a],

x(t0) =x0∈R,

where [t0,t0+a] is a bounded interval inRfor somet0∈Randa> 0,f ∈C([t0,t0+a]× R,R\ {0}), andk,g∈C([t0,t0+a]×R,R). They developed the theory of hybrid

diﬀer-ential equations with mixed perturbations of the second type and gave some original and interesting results.

As far as we know, there are no results for the DETS (1). From the works mentioned we consider the theory of DETS (1). We give an existence theorem for the DETS (1) under Lipschitz conditions. We also present some fundamental diﬀerential inequalities on time scales (DITS), which are utilized to investigate the existence of extremal solutions. We establish the comparison principle for the DETS (1). Our results in this paper extend and improve some well-known results.

The paper is organized as follows. In Sect.2, we give an existence theorem for the DETS (1) under Lipschitz conditions by the ﬁxed point theorem in Banach algebra due to Dhage. In Sect.3, we establish some fundamental DITS to strict inequalities for the DETS (1). In Sect.4, we present existence results of maximal and minimal solutions for HDTS. In Sect.5, we prove the comparison principle for the DETS (1), which is followed by the conclusion in Sect.6.

2 Existence result

In this section, we discuss the existence results for the DETS (1). We place the DETS (1) in the spaceCrd(J,R) of rd-continuous functions deﬁned onJwith the supremum norm · deﬁned as

u=sup

t∈J

u(t)

and the multiplication “·” inCrd(J,R) deﬁned as

(u·v)(t) = (uv)(t) =u(t)v(t)

foru,v∈Crd(J,R). Clearly,Crd(J,R) is a Banach algebra with respect to these norm and

multiplication. ByL1₍_J_,_R_{) we denote the space of Lebesgue}_{-integrable functions on}_J

equipped with the norm · L1 deﬁned as

uL1= t0+a

t0

u(s)s.

The following ﬁxed point theorem in a Banach algebra due to Dhage [16] is useful in the proofs of our main results.

Lemma 2.1([16]) Let Q be a closed convex bounded subset of a Banach space P,and let

(3)

(a) AandCare Lipschitz with Lipschitz constantsαandβ,respectively, (b) Bis compact and continuous,

(c) u=AuBv+Cufor allv∈Q⇒u∈Q,and

(d) αM+β< 1,whereM=B(Q)=sup{B(u):u∈Q}.

Then the operator equation AuBu+Cu=u has a solution in Q. We present the following hypotheses.

(A0) The functionu→u–k(t,u)_f_(t,u) is increasing inRfor allt∈J.

(A1) There exist constantsL1> 0andL2> 0such that

f(t,u) –f(t,v)≤L1|u–v|

and

k(t,u) –k(t,v)≤L2|u–v|

for allt∈Jandu,v∈R. Moreover,L≤M.

(A2) There exists a functionh∈L1(J,R)such that

g(t,u)≤h(t), t∈J,

for allu∈R.

Lemma 2.2 Suppose that(A0)holds.Then for any v∈L1(J,R),the-diﬀerentiable

func-tion u is a solufunc-tion of the DETS

u(t) –k(t,u(t))

f(t,u(t))

=v(t), t∈J, (2)

and

u(t0) =u0∈R, (3)

if and only if u satisﬁes the integral equation

u(t) =k t,u(t)+f t,u(t)u0–k(t0,u0)

f(t0,u0)

+

t

t0

v(s)s

, t∈J. (4)

Proof Letube a solution of problem (2)–(3). Applying the-integral to (2) fromt0tot,

we obtain

u(t) –k(t,u(t))

f(t,u(t))

–

u0–k(t0,u0)

f(t0,u0)

=

t

t0

v(s)s,

that is,

f(t0,u0)

+

t

t0

v(s)s

, t∈J.

(4)

Conversely, suppose thatusatisfies equation (4). By direct differentiation, applying the -derivative to both sides of (4), we get that (2) is satisfied. Thus, substitutet=t0in (4)

implies

u(t0) –k(t0,u(t0))

f(t0,u(t0))

=u0–k(t0,u0)

f(t0,u0)

.

Since the mapu→u–k(t,u)_f(t,u) is increasing inRfort∈J, the mapu→u–k(t0,u)

f(t0,u) is injective in

R, andu(t0) =u0. Hence (3) also holds.

Now we will give the following existence theorem for the DETS (1).

Theorem 2.1 Suppose that(A0)–(A2)hold.If

L1

u0–k(t0,u0)

f(t0,u0)

+hL1

+L2< 1, (5)

then the DETS(1)has a solution deﬁned on J.

Proof SetU=Crd(J,R) and deﬁne the subsetSofUby

S=u∈U|u ≤N,

where

N= F0(|

u0–k(t0,u0)

f(t0,u0) |+hL1) +K0 1 –L1(|u0_f–k(t_(t₀_,u0₀,u₎0)|+hL1) –L₂

,

F0=supt∈J|f(t, 0)|, andK0=supt∈J|k(t, 0)|.

Clearly,Sis a closed, convex, and bounded subset of the Banach spaceU. By Lemma2.2 the DETS (1) is equivalent to the nonlinear integral equation

f(t0,u0)

+

t

t0

g s,u(s)s

, t∈J. (6)

Deﬁne three operatorsA,C:U→UandB:S→Uby

Au(t) =f t,u(t), t∈J, (7)

Bu(t) = u0–k(t0,u0)

f(t0,u0)

+

t

t0

g s,u(s)s, t∈J, (8)

and

Cu(t) =k t,u(t), t∈J. (9) Then equation (6) is transformed into the operator equation as

(5)

Next, we prove that the operatorsA,B, andCsatisfy all the conditions of Lemma2.1. First, we prove that Ais a Lipschitz operator on U with Lipschitz constant L1. Let

u,v∈U. Then by (A1)

Au(t) –Av(t)=f t,u(t)–f t,v(t)≤L1u(t) –v(t)≤L1u–v

for allt∈J. Taking the supremum overt, then we have

Au–Av ≤L1u–v

for allu,v∈U. This shows thatAis a Lipschitz operator onUwith Lipschitz constantL1.

Similarly, we can get thatCis also a Lipschitz operator onUwith Lipschitz constantL2.

Next, we prove thatBis a compact continuous operator fromSintoU. First, we prove thatBis continuous onS. Let{un}be a sequence inSconverging to a pointu∈S. Then by the Lebesgue dominated convergence theorem adapted to time scale we have

(6)

On the other hand, lett1,t2∈J. Then for anyu∈S, we get continuous. Hence, forε> 0, there existsδ> 0 such that

|t1–t2|<δ ⇒ Bu(t1) –Bu(t2)<ε

for allt1,t2∈Jandu∈S. This shows thatB(S) is an equicontinuous set inU. Now the set

B(S) is uniformly bounded and equicontinuous set inU, so it is compact by Arzelà–Ascoli theorem. ThusBis a compact operator onS.

Next, we show that (c) of Lemma2.1is satisﬁed. Letu∈Uandv∈Sbe such thatu=

Taking the supremum overt, we have

(7)

Finally, we obtain

M=B(S)=supB(u):u∈S≤u0–k(t0,u0) f(t0,u0)

+hL1,

and so

L1M+L2≤L1

u0–k(t0,u0)

f(t0,u0)

+hL1

+L2< 1.

Thus all the conditions of Lemma 2.1 are satisﬁed, and hence the operator equa-tionAuBu+Cu=uhas a solution inS. Therefore the DETS (1) has a solution deﬁned

onJ.

3 Differential inequalities on time scales

In this section, we establish DITS for the DETS (1).

Theorem 3.1 Suppose that(A0)holds.Assume that there exist-diﬀerentiable functions

v,w such that

v(t) –k(t,v(t))

f(t,v(t))

≤g t,v(t), t∈J, (10)

and

w(t) –k(t,w(t))

f(t,w(t))

≥g t,w(t), t∈J, (11)

one of the inequalities being strict.Then v(t0) <w(t0)implies

v(t) <w(t) (12)

for all t∈J.

Proof Assume that inequality (11) is strict. Suppose that the claim is false. Then there existst1∈J,t1>t0, such thatv(t1) =w(t1) andv(t) <w(t) fort0≤t<t1.

Deﬁne

V(t) =v(t) –k(t,v(t))

f(t,v(t)) and W(t) =

w(t) –k(t,w(t))

f(t,w(t))

for allt∈J. Then we obtainV(t1) =W(t1), and by (A0) we haveV(t) <W(t) for allt<t1.

SinceV(t1) =W(t1), we get

V(t1+h) –V(t1)

h >

W(t1+h) –W(t1)

h

for suﬃciently smallh< 0. This inequality implies that

(8)

because of (A0). Then we obtain

g t1,v(t1)

≥V(t1)≥W(t1) >g t1,w(t1)

.

This is a contradiction withv(t1) =w(t1). Hence inequality (12) is valid.

The next result is concerned with nonstrict DITS, which needs a Lipshitz condition.

Theorem 3.2 Suppose that the conditions of Theorem3.1hold with inequalities(10)and

(11).Suppose that there exists a real number K> 0such that

g(t,u1) –g(t,u2)≤K sup t0≤s≤t

u1(t) –k(t,u1(t))

f(t,u1(t))

–u2(t) –k(t,u2(t))

f(t,u2(t))

, t∈J (13)

for all u1,u2∈Rwith u1≥u2.Then v(t0)≤w(t0)implies v(t)≤w(t)for all t∈J.

Proof Letε> 0 andK> 0 be given. Deﬁne

wε(t) –k(t,wε(t))

f(t,wε(t))

=w(t) –k(t,w(t))

f(t,w(t)) +εe

2L(t–t0)_,

so that we get

wε(t) –k(t,wε(t))

f(t,wε(t))

>w(t) –k(t,w(t))

f(t,w(t)) ⇒ wε(t) >w(t). LetWε(t) =wε(t)–k(t,w_f_(t,w ε(t))

ε(t)) , so thatW(t) =

w(t)–k(t,w(t))

f(t,w(t)) fort∈J. Then by (11) we obtain

Wε(t) =W

(t) + 2Kεe2L(t–t0)≥_g _t_,_w₍_t₎_{+ 2}_L_ε_e2L(t–t0)_.

Then from (13) we have

g t,wε(t)

–g t,w(t)≤K sup

t0≤s≤t

Wε(s) –W(s)

=Kεe2L(t–t0)

for allt∈J, and thus

Wε(t)≥g t,wε(t)

–Kεe2L(t–t0)_{+ 2}_K_ε_e2L(t–t0)_>_g _t_,_w

ε(t)

,

that is,

wε(t) –f t,wε(t)

>g t,wε(t)

for allt∈J. Also, we getwε(t0) >w(t0) >v(t0). Hence Theorem3.1withw=wεimplies that

v(t) <wε(t) for allt∈J. By the arbitrariness ofε> 0, taking the limits asε→0, we have

(9)

4 Existence of maximal and minimal solutions

In this section, we prove the existence of maximal and minimal solutions for the DETS (1) onJ= [t0,t0+a]T.

Deﬁnition 4.1 A solutionrof the DETS (1) is said to be maximal if for any other solution

uto the DETS (1), we haveu(t)≤r(t) for allt∈J. Similarly, a solutionρof the DETS (1) is said to be minimal ifρ(t)≤u(t) for allt∈J, whereuis any solution of the DETS (1) onJ. We discuss the case of maximal solution only. Similarly, the case of minimal solution can be obtained with the same arguments with appropriate modiﬁcations. Given an arbitrary small numberε> 0, we discuss the following initial value problem of DETS:

⎧ ⎨ ⎩

[u(t)–k(t,u(t))_f_(t,u(t)) ]₌_g₍_t_,_u₍_t_{)) +}_ε_, _t_∈_J_,

u(t0) =u0+ε,

(14)

wheref ∈Crd(J×R,R\ {0}) andk,g∈Crd(J×R,R).

An existence theorem for the DETS (14) can be stated as follows.

Theorem 4.1 Suppose that(A0)–(A2)and inequality(5)hold.Then for every small

num-berε> 0,the DETS(14)has a solution deﬁned on J.

Proof By hypothesis, since

L1

u0–k(t0,u0)

f(t0,u0)

+hL1

+L2< 1,

there existsε0> 0 such that

L1

u0+ε–k(t0,u0+ε)

f(t0,u0+ε)

+hL1+εa

+L2< 1

for all 0 <ε≤ε0. The rest of the proof is similar to that of Theorem2.1, and we omit

it.

Our main existence theorem for maximal solution for the DETS (1) is the following:

Theorem 4.2 Suppose that(A0)–(A2)and inequality(5)hold.Then the DETS(1)has a

maximal solution deﬁned on J.

Proof Let{εn}∞₀ be a decreasing sequence of positive numbers such thatlimn→∞εn= 0,

whereε0is a positive number satisfying the inequality

L1

u0+ε0–k(t0,u0+ε0)

f(t0,u0+ε0)

+hL1+ε₀a

+L2< 1.

Such a numberε0exists in view of inequality (5). Then for any solutionxof the DETS (1),

by Theorem4.1we get

(10)

for allt∈Jandn∈N∪ {0}, wherer(t,εn) deﬁned onJis a solution of the DETS

(11)

·u0+εn–k(t0,u0+εn)

f(t0,u0+εn)

+hL1+εna

+Fp(t1) –p(t2)+|t1–t2|εn

,

whereF=sup_(t,u)_∈_J_×_[–N,N]|f(t,u)|andp(t) =_tt 0h(s)s.

Sincef andkare continuous on the compact setJ×[–N,N], they are uniformly con-tinuous. Hence

f t1,r(t1,εn)

–f t2,r(t2,εn)→0 ast1→t2

and

k t1,r(t1,εn)

–k t2,r(t2,εn)→0 ast1→t2

uniformly for alln∈N. Similarly, since the functionpis continuous on the compact setJ, it is uniformly continuous, and hence

p(t1) –p(t2)→0 ast1→t2.

Therefore we obtain

r(t1,εn) –r(t2,εn)→0 ast1→t2

uniformly for alln∈N. Therefore

r(t,εn)→r(t) asn→ ∞

for allt∈J.

Next, we prove that the functionr(t) is a solution of the DETS (1) deﬁned onJ. Since

r(t,εn) is a solution of the DETS (15), we get

r(t,εn) =

f t,r(t,εn)

u0+εn–k(t0,u0+εn)

f(t0,u0+εn)

+

t

t0

g s,r(s,εn)

s

+

t

t0 εns

+k t,r(t,εn)

(17)

for allt∈J. Taking the limit asn→ ∞in (17) implies

r(t) =f(t,r(t)u0–k(t0,u0)

f(t0,u0)

+

t

t0

g s,r(s)s

+k t,r(t)

for allt∈J. Thus the functionris a solution of the DETS (1) onJ. Finally, from inequality (15) it follows thatx(t)≤r(t) for allt∈J. Hence the DETS (1) has a maximal solution

(12)

5 Comparison theorems on time scales

The main problem of the DITS is to estimate a bound for the solution set for the DITS related to the DETS (1). In this section, we present the maximal and minimal solutions serving as bounds for the solutions of the related DITS to the DETS (1) onJ= [t0,t0+a]T.

Theorem 5.1 Suppose that(A0)–(A2)and inequality(5)hold.Assume that there exists a

-diﬀerentiable function u such that

⎧ ⎨ ⎩

[x(t)–k(t,x(t))_f_(t,x(t)) ]_≤_g₍_t_,_x₍_t_)), _t_∈_J_,

x(t0)≤u0.

(18)

Then

x(t)≤r(t) (19)

for all t∈J,where r is a maximal solution of the the DETS(1)on J.

Proof Letε> 0 be arbitrarily small. By Theorem4.2,r(t,ε) is a maximal solution of the DETS (14), the limit

r(t) =lim

ε→0r(t,ε) (20)

is uniform onJ, and the functionris a maximal solution of the DETS (1) onJ. Hence we have

⎧ ⎨ ⎩

[r(t,ε)–k(t,r(t,ε))_f(t,r(t,ε)) ]₌_g₍_t_,_r₍_t_,_ε_{)) +}_ε_, _t_∈_J_,

r(t0,ε) =u0+ε.

By the above inequality it implies that

⎧ ⎨ ⎩

[r(t,ε)–k(t,r(t,ε))_f(t,r(t,ε)) ]_>_g₍_t_,_r₍_t_,_ε_)), _t_∈_J_,

r(t0,ε) >u0.

(21)

Now, applying Theorem3.2to inequalities (18) and (21), we conclude thatx(t) <r(t,ε) for

allt∈J. Thus (20) implies that inequality (19) holds onJ.

-diﬀerentiable function u such that

⎧ ⎨ ⎩

[y(t)–k(t,y(t))_f_(t,y(t)) ]_≥_g₍_t_,_y₍_t_)), _t_∈_J_,

y(t0)≥u0.

Then

ρ(t)≤y(t)

(13)

Note that Theorem5.1is useful to prove the boundedness and uniqueness of the solu-tions for the DETS (1) onJ. We have a following result.

function G:J×R+_→_R+_{such that}

g(t,u1) –g(t,u2)≤G

t,max

s∈[t0,t]

u1(s) –k(s,u1(s))

f(s,u1(s))

–u2(s) –k(s,u2(s))

f(s,u2(s))

, t∈J,

for all u1,u2∈Rwith u1≥u2.If the identically zero function is the only solution of the

diﬀerential equation

m(t) =G t,m(t), t∈J, m(t0) = 0, (22)

then the DETS(1)has a unique solution on J.

Proof By Theorem2.1the DETS (1) has a solution deﬁned onJ. Suppose that there are two solutionsx1andx2of the DETS (1) existing onJwithx1>x2. Deﬁnem:J→R+by

m(t) =u1(t) –k(t,u1(t))

f(t,u1(t))

–u2(t) –k(t,u2(t))

f(t,u2(t))

.

Since (|x(t)|)_{≤ |}_x₍_t₎_|_{, we obtain that}

m(t)≤

u1(t) –k(t,u1(t))

f(t,u1(t))

–

u2(t) –k(t,u2(t))

f(t,u2(t))

=g(t,x1) –g(t,x2) ≤G

t,u1(t) –k(t,u1(t))

f(t,u1(t))

–u2(t) –k(t,u2(t))

f(t,u2(t))

=G t,m(t) fort∈Jandm(t0) = 0.

Now we apply Theorem5.1withk(t,u)≡0 andf(t,u)≡1 to get thatm(t)≤0 for all

t∈J, where the identically zero function is the only solution of the DETS (22), which is a contradiction withm(t) > 0. This implies

u1(t) –k(t,u1(t))

f(t,u1(t))

=u2(t) –k(t,u2(t))

f(t,u2(t))

for allt∈J. Then we havex1=x2.

Remark5.1 Whenk≡0,f ≡1, andT=Rin our results, we obtain the diﬀerential in-equalities and other related results of Lakshmikantam and Leela [17] for the IVP of ordi-nary nonlinear diﬀerential equation

u(t) =g t,u(t), t∈[t0,t0+a], u(t0) =u0.

(14)

6 Conclusion

In this paper, we have developed the theory of the DETS (1). By the ﬁxed point theorem in Banach algebra due to Dhage we have presented an existence theorem for the DETS (1) under Lipschitz conditions. We have also established some DITS for the DETS (1), which are used to investigate the existence of extremal solutions, and the comparison principle for the DETS (1). Our results in this paper extend and improve some well-known results.

Acknowledgements

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript.

Funding

This research is supported by the National Natural Science Foundation of China (61703180, 61803176, 61877028, 61807015, 61773010), the Natural Science Foundation of Shandong Province (ZR2019MF032, ZR2017BA010, ZR2017LF012), the Project of Shandong Province Higher Educational Science and Technology Program (J18KA230, J17KA157), and the Scientiﬁc Research Foundation of University of Jinan (1008399, 160100101).

Availability of data and materials Not applicable.

Competing interests

The authors declare that they have no competing interests. Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the ﬁnal manuscript.

Author details

1_{School of Mathematical Sciences, University of Jinan, Jinan, P.R. China.}2_{School of Automation and Electrical Engineering,} Key Laboratory of Complex systems and Intelligent Computing in Universities of Shandong, Linyi University, Linyi, P.R. China.3_{College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao, P.R.} China.

Publisher’s Note

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

Received: 8 May 2019 Accepted: 20 June 2019

References

1. Hilger, S.: Analysis on measure chains – a uniﬁed approach to continuous and discrete calculus. Results Math.18, 18–56 (1990)

2. Agarwal, R.P., Bohner, M., Li, T.: Oscillatory behavior of second-order half-linear damped dynamic equations. Appl. Math. Comput.254, 408–418 (2015)

3. Bohner, M., Hassan, T.S., Li, T.: Fite–Hille–Wintner-type oscillation criteria for second-order half-linear dynamic equations with deviating arguments. Indag. Math.29, 548–560 (2018)

4. Bohner, M., Li, T.: Kamenev-type criteria for nonlinear damped dynamic equations. Sci. China Math.58, 1445–1452 (2015)

5. Sheng, K., Zhang, W., Bai, Z.: Positive solutions to fractional boundary value problems withp-Laplacian on time scales. Bound. Value Probl.2018, Article ID 70 (2018)

6. El-Deeb, A.A., Xu, H., Abdeldaim, A., et al.: Some dynamic inequalities on time scales and their applications. Adv. Diﬀer. Equ.2019, Article ID 130 (2019)

7. Agarwal, R.P., Bohner, M., Li, T., et al.: Oscillation criteria for second-order dynamic equations on time scales. Appl. Math. Lett.31, 34–40 (2014)

8. Chatzarakis, G.E., Li, T.: Oscillation criteria for delay and advanced diﬀerential equations with nonmonotone arguments. Complexity2018, Article ID 8237634 (2018)

9. Zhang, C., Agarwal, R.P., Bohner, M., et al.: Oscillation of fourth-order delay dynamic equations. Sci. China Math.58, 143–160 (2015)

10. Samei, M.E., Hedayati, V., Rezapour, S.: Existence results for a fraction hybrid diﬀerential inclusion with Caputo–Hadamard type fractional derivative. Adv. Diﬀer. Equ.2019, Article ID 163 (2019)

11. Derbazi, C., Hammouche, H., Benchohra, M., et al.: Fractional hybrid diﬀerential equations with three-point boundary hybrid conditions. Adv. Diﬀer. Equ.2019, Article ID 125 (2019)

12. Lu, H., Sun, S., Yang, D., et al.: Theory of fractional hybrid diﬀerential equations with linear perturbations of second type. Bound. Value Probl.2013, Article ID 23 (2013)

13. Dhage, B.C.: Basic results in the theory of hybrid diﬀerential equations with mixed perturbations of second type. Funct. Diﬀer. Equ.19, 87–106 (2012)

(15)

15. Sun, S., Zhao, Y., Han, Z., et al.: The existence of solutions for boundary value problem of fractional hybrid diﬀerential equations. Commun. Nonlinear Sci. Numer. Simul.17, 4961–4967 (2012)

16. Dhage, B.C.: A ﬁxed point theorem in Banach algebras with applications to functional integral equations. Kyungpook Math. J.44, 145–155 (2004)