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

R E S E A R C H

Open Access

Basic theory of differential equations

with linear perturbations of second type

on time scales

Yige Zhao

_{, Yibing Sun}

_{, Zhi Liu}

_{and Zhanbing Bai}

*_{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 linear perturbations of second type on time scales. An existence theorem for differential equations with linear perturbations of second type on time scales is given underD-Lipschitz conditions. Some fundamental differential inequalities on time scales, which are utilized to investigate the existence of extremal solutions, are also presented. The comparison principle on differential equations with linear perturbations of second type on time scales is established. Our results in this paper extend and improve some well-known results.

MSC: 34N05; 34A12

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

1 Introduction

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

[u(t) –f(t,u(t))]_{=g(t,u(t)), t∈J,}

u(t0) =u0,

(1)

wheref,g∈Crd(J×R,R).

LetTbe a time scale andJ= [t0,t0+a]T= [t0,t0+a]∩Tbe a bounded interval inTfor somet0,a∈Rwitha> 0. LetCrd(J×R,R) denote the class of rd-continuous functions

f:J×R→R. For the basic definitions and useful lemmas from the theory of calculus on time scales, one can be referred to [1].

By a solution of DETS (1), we mean a-differentiable functionusuch that (i) the functiont→u(t) –f(t,u(t))is-differentiable for eachu∈R, and (ii) usatisfies the equations in (1).

The theory of time scales has drawn a lot of attention since 1988 (see [1–9]). In recent years, the theory of nonlinear differential equations with linear perturbations has been a hot research topic, see [9–15]. Dhage and Jadhav [12] discussed the following first order

hybrid differential equation with linear perturbations of second type:

d

dt[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 in Rfor some t0,a∈Rwith a> 0, andf,g ∈

C([t0,t0+a]×R,R). They developed the theory of hybrid differential equations with linear perturbations of second type and gave some original and interesting results.

As far as we know, there are no results for DETS (1). From the above works, we consider the theory of DETS (1). An existence theorem for DETS (1) is given underD-Lipschitz conditions. Some fundamental differential inequalities on time scales (in short DITS), which are utilized to investigate the existence of extremal solutions, are also presented. The comparison principle on DETS (1) is established. Our results in this paper extend and improve some well-known results.

The paper is organized as follows: Sect.2gives an existence theorem for DETS (1) un-derD-Lipschitz conditions by the fixed point theorem in Banach algebra due to Dhage. Section3establishes some fundamental DITS to strict inequalities for DETS (1). Section4

presents existence results of maximal and minimal solutions for HDTS. Section5proves the comparison principle for DETS (1), which is followed by the conclusion in Sect.6.

2 Existence result

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

u=sup t∈J

u(t).

ClearlyCrd(J,R) is a Banach algebra with respect to the above norm.L1(J,R) denotes the space of Lebesgue-integrable functions onJequipped with the norm · _L1defined by

uL1=

t0+a t0

_u(s)s.

Some definitions and lemmas that will be used in our main results are given in what follows.

Definition 2.1 ([12]) A mapping ϕ :R+ _→R+ _{is called a dominating function or, in} short,D-function if it is an upper semi-continuous and nondecreasing function satisfying

ϕ(0) = 0. A mappingT:P→Pis calledD-Lipschitz if there is aD-functionϕ:R+→R+ satisfying

Tu–Tv ≤ϕu–v

for allu,v∈P. The functionϕis called aD-function ofQonP. Ifϕ(t) =lt,l> 0, thenT

is called Lipschitz with the Lipschitz constantl. In particular, ifl< 1, thenT is called a contraction onXwith the contraction constantl. Furthermore, ifϕ(t) <tfort> 0, thenT

The details of different types of contractions appear in the monographs of Dhage [16] and Granas and Dugundji [17]. There do existD-functions, and the commonly usedD -functions areϕ(t) =ltandϕ(t) = ₁₊t_t. TheseD-functions have been widely used in the theory of nonlinear differential and integral equations for proving the existence results via fixed point methods.

Another notion that we need in the sequel is the following definition.

Definition 2.2([12]) An operatorT on a Banach spaceP mapping into itself is called compact ifT(P) is a relatively compact subset ofP.T is called totally bounded if, for any bounded subsetQofP,T(Q) is a relatively compact subset ofP. IfT is continuous and totally bounded, then it is called completely continuous onP.

The following fixed point theorem in 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 and bounded subset of the Banach space P,

and let A:P→P and B:Q→P be two operators such that

(a) Ais a nonlinearD-contraction, (b) Bis compact and continuous,and

(c) u=Au+Bvfor allv∈Q⇒u∈Q.

Then the operator equation Au+Bu=u has a solution in Q.

We present the following hypotheses:

(A0) The functionu→u–f(t,u)is increasing inRfor allt∈J. (A1) There exists a constantL> 0such that

f(t,u) –f(t,v)≤ L|u–v|

M+|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) –ft,u(t)=v(t), t∈J, (2)

and

u(t0) =u0∈R, (3)

if and only if u satisfies the integral equation

u(t) =u0–f(t0,u0) +f

t,u(t)+

t t0

Proof Letube a solution of problem (2) and (3). Applying-integral to (2) fromt0tot, we obtain

u(t) –ft,u(t)–u0–f(t0,u0)

=

t t0

v(s)s,

i.e.,

u(t) =u0–f(t0,u0) +f

t,u(t)+

t t0

v(s)s, t∈J.

Thus, (4) holds.

Conversely, suppose thatusatisfies equation (4). By direct differentiation and applying

-derivative on both sides of (4), then (2) is satisfied. Thus, substitutingt=t0in (4) implies

u(t0) –f

t0,u(t0)

=u0–f(t0,u0).

Since the mapu→u–f(t,u) is increasing inRfort∈J, the mapu→u–f(t0,u) is injective inRandu(t0) =u0. Hence, (3) also holds.

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

Theorem 2.1 Suppose that(A0)–(A2)hold.Then DETS(1)has a solution defined on J.

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

S= u∈U|u ≤N,

whereN=|u0–f(t0,u0)|+L+F0+hL1, andF₀=sup_t∈J|f(t, 0)|.

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

u(t) =u0–f(t0,u0) +f

t,u(t)+

t t0

gs,u(s)s, t∈J. (5)

Define two operatorsA:U→UandB:S→Uby

Au(t) =ft,u(t), t∈J, (6)

and

Bu(t) =u0–f(t0,u0) +

t t0

gs,u(s)s, t∈J. (7)

Then equation (5) is transformed into the operator equation as follows:

Au(t) +Bu(t) =u(t), t∈J.

Firstly, we prove thatAis a nonlinearD-contraction onUwith aD-functionϕ. Let

u,v∈U. Then, by (A1),

Au(t) –Av(t)=ft,u(t)–ft,v(t)≤ L|u(t) –v(t)|

M+|u(t) –v(t)|≤

Lu–v

M+u–v

for allt∈J. Taking supremum overt, we have

Au–Av ≤ Lu–v

M+u–v

for allu,v∈U. This shows thatAis a nonlinearD-contraction onUwith aD-function

ϕdefined byϕ(t) =_MLt_+t.

Next, we prove thatBis a compact and continuous operator onSintoU. Firstly, we prove thatBis continuous onS. Let{un}be a sequence inSconverging to a pointu∈S.

Then, by Lebesgue dominated convergence theorem adapted to time scales, we have

lim

n→∞Bun(t) =nlim→∞

u0–f(t0,u0) +

t t0

gs,un(s)

s

=u0–f(t0,u0) + lim

n→∞

t t0

gs,un(s)

s

=u0–f(t0,u0) +

t t0

lim n→∞g

s,un(s)

s

=u0–f(t0,u0) +

t t0

gs,u(s)s

=Bu(t)

for allt∈J. This shows thatBis a continuous operator onS.

Next we prove that Bis a compact operator onS. It is enough to show thatB(S) is a uniformly bounded and equicontinuous set inU. On the one hand, letu∈Sbe arbitrary. Then, by (A2),

Bu(t)=u0–f(t0,u0) +

t t0

gs,u(s)s

≤u0–f(t0,u0)+

t t0

gs,u(s)s

≤u0–f(t0,u0)+

t t0

h(s)s

≤u0–f(t0,u0)+hL1

for allt∈J. Taking supremum overt,

Bu ≤u0–f(t0,u0)+hL1

On the other hand, lett1,t2∈J. Then, for anyu∈S, we get

Bu(t1) –Bu(t2)=

_t0t1gs,u(s)s–

t2 t0

gs,u(s)s

≤

t1 t2

gs,u(s)s

≤

t1 t2

h(s)s

=p(t1) –p(t2),

wherep(t) =_t0t h(s)s. Since the functionpis continuous on compactJ, it is uniformly 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 a uniformly bounded and equicontinuous set inU, so it is compact by the Arzela– Ascoli theorem. As a result,Bis a complete continuous operator onS.

Next, we show that (c) of Lemma2.1is satisfied. Letu∈Uandv∈Sbe arbitrary such thatu=Au+Bv. Then, by assumption (A1), we have

u(t)≤Au(t)+Bv(t)

=ft,u(t)+u0–f(t0,u0)+

t t0

gs,v(s)s

≤ft,u(t)–f(t, 0)+f(t, 0)+

t t0

gs,v(s)s

≤u0–f(t0,u0)+L+F0+

t t0

h(s)s

≤u0–f(t0,u0)+L+F0+hL1.

Taking supremum overt, we get

u ≤u0–f(t0,u0)+L+F0+hL1=N.

This shows that (c) of Lemma2.1is satisfied.

Thus, all the conditions of Lemma2.1are satisfied, and hence the operator equation

Au+Bu=uhas a solution inS. Therefore, DETS (1) has a solution defined onJ.

3 Differential inequalities on time scales

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

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

v,w such that

and

w(t) –ft,w(t)≥gt,w(t), t∈J, (9)

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

v(t) <w(t) (10)

for all t∈J.

Proof Assume that inequality (9) is strict. Suppose that the claim is false. Then there exists

t1∈J,t1>t0such thatv(t1) =w(t1) andv(t) <w(t) fort0≤t<t1. Define

V(t) =v(t) –ft,v(t) and W(t) =w(t) –ft,w(t)

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

V(t1+h) –V(t1)

h >

W(t1+h) –W(t1)

h

for sufficiently smallh< 0. The above inequality implies that

V(t1)≥W(t1)

because of (A0). Then we obtain

gt1,v(t1)

≥V(t1)≥W(t1) >g

t1,w(t1)

.

This is a contradiction withv(t1) =w(t1). Hence conclusion (10) is valid.

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

Theorem 3.2 Suppose that the conditions of Theorem3.1hold with inequalities(8)and

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

g(t,u1) –g(t,u2)≤K sup

t0≤s≤t

u1(s) –f

s,u1(s)

–u2(s) –f

s,u2(s)

, t∈J (11)

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

Proof Letε> 0 and a real numberK> 0 be given. Define

wε(t) –ft,wε(t)=w(t) –ft,w(t)+εe2L(t–t0), so that we get

LetWε(t) =wε(t) –f(t,wε(t)) so thatW(t) =w(t) –f(t,w(t)) fort∈J. By (9), we obtain

Wε(t) =W

(t) + 2Kεe2L(t–t0)≥gt,w(t)+ 2Lεe2L(t–t0). From (11), we have

gt,wε(t)–gt,w(t)≤K sup t0≤s≤t

Wε(s) –W(s)=Kεe2L(t–t0)

for allt∈J, then we obtain

Wε(t)≥g

t,wε(t)–Kεe2L(t–t0)+ 2Kεe2L(t–t0)>gt,wε(t), i.e.,

wε(t) –ft,wε(t)>gt,wε(t)

for allt∈J. Also, we getwε(t0) >w(t0) >v(t0). Hence, an application of Theorem3.1with

w=wεimplies thatv(t) <wε(t) for allt∈J. By the arbitrariness ofε> 0, taking the limits asε→0, we havev(t)≤w(t) for allt∈J.

4 Existence of maximal and minimal solutions

In this section, we give the existence of maximal and minimal solutions for DETS (1) on

J= [t0,t0+a]T.

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

to DETS (1), one hasu(t)≤r(t) for allt∈J. Similarly, a solutionρof DETS (1) is said to be minimal ifρ(t)≤u(t) for allt∈J, whereuis any solution of 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 arbitrarily small real numberε> 0, discuss the following initial value problem of DETS:

[u(t) –f(t,u(t))]_{=g(t,u(t)) +ε, t∈J,}

u(t0) =u0+ε,

(12)

wheref,g∈Crd(J×R,R).

An existence theorem for DETS (12) can be stated as follows.

Theorem 4.1 Suppose that(A0)–(A2)hold.Then,for every small numberε> 0,DETS(12)

has a solution defined on J.

Proof The proof is similar to that of Theorem2.1, and we omit the proofs.

Our main existence theorem for maximal solution for DETS (1) is as follows.

Theorem 4.2 Suppose that(A0)–(A2)hold.Then DETS(1)has a maximal solution

Proof Let {εn}∞0 be a decreasing sequence of positive real numbers such that

limn→∞εn= 0. Then, for any solutionxof DETS (1), by Theorem4.1, we get

x(t) <r(t,εn)

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

[u(t) –f(t,u(t))]_{=g(t,u(t)) +ε}

n, t∈J, u(t0) =u0+εn.

(13)

By Theorem3.2,{r(t,εn)}is a decreasing sequence of positive real numbers, the limit

r(t) = lim

n→∞r(t,εn) (14)

exists. We prove that the convergence in (14) is uniform on J. Next, we show that the sequence {r(t,εn)} is equicontinuous inCrd(J,R). Let t1,t2∈J witht1<t2 be arbitrary. Then we have

r(t1,εn) –r(t2,εn)

=u0+εn–f(t0,u0+εn) +f

t1,r(t1,εn)

+

t1 t0

gs,r(s,εn)s

+

t1 t0

εns–

u0+εn–f(t0,u0+εn) +f

t2,r(t2,εn)

+

t2 t0

gs,r(s,εn)

s+

t2 t0

εns

≤ft1,r(t1,εn)

–ft2,r(t2,εn)+

t1

t0

gs,r(s,εn)

s

–

t2 t0

gs,r(s,εn)

s+

t1 t0

εns–

t2 t0

εns

=ft1,r(t1,εn)

–ft2,r(t2,εn)+

t2

t1

gs,r(s,εn)

s+

t2 t1

εns

≤ft1,r(t1,εn)

–ft2,r(t2,εn)+

_t1t2h(s)s+|t1–t2|εn

≤ft1,r(t1,εn)

–ft2,r(t2,εn)+p(t1) –p(t2)+|t1–t2|εn,

wherep(t) =_t0t h(s)s.

Sincef is continuous on a compact setJ×[–N,N], it is uniformly continuous there. Hence,

_f

t1,r(t1,εn)

–ft2,r(t2,εn)→0 ast1→t2

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

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 DETS (1) defined onJ. Sincer(t,εn)

is a solution of DETS (13), we get

r(t,εn) =u0+εn–f(t0,u0+εn) +f

t,r(t,εn)

+

t t0

gs,r(s,εn)

s+

t t0

εns (15)

for allt∈J. Taking the limit asn→ ∞in equation (15) implies

r(t) =u0–f(t0,u0) +f

t,r(t)+

t t0

gs,r(s)s

for allt∈J. Thus, the functionris a solution of DETS (1) onJ. Finally, from inequality (13), it follows thatx(t)≤r(t) for allt∈J. Hence, DETS (1) has a maximal solution onJ.

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 DETS (1). In this section, we present the maximal and minimal solutions serve as bounds for the solutions of the related DITS to DETS (1) onJ= [t0,t0+a]T.

Theorem 5.1 Suppose that(A0)–(A2)hold.Assume that there exists a-differentiable

function u such that

[x(t) –f(t,x(t))]_{≤g(t,x(t)), t∈J,}

x(t0)≤u0.

(16)

Then

x(t)≤r(t) (17)

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

Proof Letε> 0 be arbitrarily small. From Theorem4.2,r(t,ε) is a maximal solution of DETS (12) and the limit

r(t) =lim

ε→0r(t,ε) (18)

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

[r(t,ε) –f(t,r(t,ε))]_{=g(t,r(t,ε)) +ε, t∈J,}

The above inequality implies that

[r(t,ε) –f(t,r(t,ε))]_{>g(t,r(t,ε)),t∈J,}

r(t0,ε) >u0.

(19)

Now, we apply Theorem3.2to inequalities (16) and (19) and conclude thatx(t) <r(t,ε) for allt∈J. Thus, (18) implies that inequality (17) holds onJ.

Theorem 5.2 Suppose that(A0)–(A2)hold.Assume that there exists a-differentiable

function u such that

[y(t) –f(t,y(t))]_{≥g(t,y(t)), t∈J,}

y(t0)≥u0.

Then

ρ(t)≤y(t)

for all t∈J,whereρis a minimal solution of DETS(1)on J.

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

Theorem 5.3 Suppose that (A0)–(A2) hold. Assume that there exists a function G :

J×R+_→R+_{such that}

g(t,u1) –g(t,u2)≤G

t,u1–f(t,u1)

–u2–f(t,u2)

, t∈J

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

differential equation

m(t) =Gt,m(t), t∈J, m(t0) = 0, (20)

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

Proof From Theorem2.1, DETS (1) has a solution defined onJ. Suppose that there are

two solutionsx1andx2of DETS (1) existing onJwithx1>x2. Definem:J→R+by

m(t) =x1(t) –f

t,x1(t)

–x2(t) –f

t,x2(t)

.

By (A0), we conclude thatm(t) > 0. Then we obtain

m(t) =x1(t) –f

t,x1(t)

–x2(t) –f

t,x2(t)

=g(t,x1) –g(t,x2)

≤G

t, x1

f(t,x1) – x2

f(t,x2)

=Gt,m(t)

Now, we apply Theorem5.1withf(t,u)≡0 to get thatm(t)≤0 for allt∈J, where an identically zero function is the only solution of DETS (20).m(t)≤0 is a contradiction with

m(t) > 0. Then we havex1=x2.

Remark5.1 Whenf ≡0 andT=Rin our results of this paper, we obtain the differential

inequalities and other related results given in Lakshmikantham and Leela [18] for the IVP of ordinary nonlinear differential equation

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

Remark5.2 The main results in this paper extend and improve some well-known results

in [12].

6 Conclusion

In this paper, we have developed the theory of DETS (1). By the fixed point theorem in Banach algebra due to Dhage, we have presented an existence theorem for DETS (1) under

D-Lipschitz conditions. We have also established some DITS for DETS (1) which are used to investigate the existence of extremal solutions. The comparison principle on DETS (1) has been given. 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), 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.

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Received: 5 April 2019 Accepted: 3 October 2019

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