Periodic boundary value problems for impulsive conformable fractional integro-differential equations

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R E S E A R C H

Open Access

Periodic boundary value problems for

impulsive conformable fractional

integro-differential equations

Suphawat Asawasamrit

1

_{, Sotiris K Ntouyas}

2,3

_{, Phollakrit Thiramanus}

1

_{and Jessada Tariboon}

1*

*_{Correspondence:} [email protected] 1_{Nonlinear Dynamic Analysis} Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, 10800, Thailand Full list of author information is available at the end of the article

Abstract

This paper is concerned with the existence of solutions for periodic boundary value problems for impulsive fractional integro-diﬀerential equations using a recent novel concept ofconformable fractional derivative. We give a new deﬁnition of exponential notations and impulsive integrals for constructing the Green function and a

comparison result of the linear problems with impulses. By applying the method of lower and upper solutions in reversed order coupled with the monotone iterative technique, some new suﬃcient conditions for the existence of solutions are established. The obtained results are well illustrated by an example.

MSC: 26A33; 34A08; 34A37

Keywords: periodic boundary value problem; conformable fractional derivative; impulsive diﬀerential equation; monotone iterative technique; lower and upper solutions

1 Introduction

Fractional order differential equations plays an important role in describing many phe-nomena and processes in various fields of science and engineering such as mechanics, chemistry, control systems, dynamical processes, viscoelasticity, and so forth. For a de-tailed account of applications and recent results on initial and boundary value problems of fractional differential equations, we refer the reader to a series of books and papers [–] and references cited therein.

The definition of the fractional order derivative used is either the Caputo or the Riemann-Liouville fractional order derivative involving an integral expression and the gamma function. Recently, Khalilet al.[] introduced a new well-behaved definition of a local fractional derivative, calledthe conformable fractional derivative, depending just on the basic limit definition of the derivative. This new theory is improved by Abdeljawad []. For recent results on conformable fractional derivatives we refer the reader to [–].

Impulsive diﬀerential equations serve as basic models to study the dynamics of processes that are subject to sudden changes in their states. Recent development in this ﬁeld has been motivated by many applied problems arising in control theory, population dynamics and

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medicines. For some recent work on the theory of impulsive diﬀerential equations, we refer the reader to [–].

Due to fact that fractional differential equations and impulsive differential equations serve effective tools in order to improve the mathematical modeling of several concepts arising in engineering and various areas of science, many researchers have paid a consid-erable attention to the subject of impulsive fractional differential equations in the recent literature [–].

In this paper, we consider the following periodic boundary value problem for impulsive conformable fractional integro-diﬀerential equations:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

tkD

α_{x(t) =}_f_(t,_{x(t), (Fx)(t), (Sx)(t)),} _t_∈_J_{:= [,}_T_],_t₌_tk,

x(tk) =Ik(x(tk)), k= , , . . . ,m, x() =x(T),

(.)

whereaDα_{denotes the conformable fractional derivative of order  <}_α_≤_{ starting from} a∈ {t, . . . ,tm},t=  <t<· · ·<tm<tm+=T,f∈C(J×R,R),

(Fx)(t) =

t



l(t,s)x(s)ds, (Sx)(t) =

T



h(t,s)x(s)ds,

l∈C(D,R+_),_D₌_{_(t,_s)_∈_J_:_t_≥_s_}_,_h_∈_C(J_,_R+_),_Ik_∈_C(_R_,_R_),_{x(tk) =}_x(t+

k) –x(t–k). The monotone iterative technique coupled with the method of lower and upper solu-tions provides an effective way to obtain two sequences which approximate the extremal solutions between lower and upper solutions of nonlinear differential and impulsive dif-ferential (fractional or non-fractional ordered) equations; see, for instance, [–]. To the best of the authors’ knowledge, this is the first paper establishing the impulsive frac-tional differential equations via the conformable fracfrac-tional calculus developed by []. By means of a new maximal principle and new definitions of lower and upper solutions, the monotone iterative technique will be used in our investigation of the problem (.).

The rest of the paper is organized as follows: In Section  we recall some deﬁnitions and results from conformable fractional calculus and prove some basic results used in the sequel. In Section  we deﬁne the lower and upper solutions, obtain the Green functions and prove a comparison result. The existence results are contained in Section , while an example illustrating the main result is presented in Section .

2 Conformable fractional calculus

In this section, we recall some deﬁnitions, notations and results which will be used in our main results.

Deﬁnition . [] The conformable fractional derivative starting from a pointaof a functionf : [a,∞)→Rof order  <α≤ is deﬁned by

aDαf(t) =lim ε→

f(t+ε(t–a)–α_{) –}_f_(t)

ε , (.)

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Iff is diﬀerentiable thenaDα_f_{(t) = (t}_–_a)–α_f_{(t). In addition, if the conformable} frac-tional derivative off of orderαexists on [a,∞), then we say thatf isα-diﬀerentiable on [a,∞).

It is easy to prove the following results.

Lemma . Letα ∈(, ],k,k,p,λ∈R, and the functions f,g be α-diﬀerentiable on

[a,∞).Then:

(i) aDα_(k

f +kg) =kaDα(f) +kaDα(g);

(ii) aDα_(t_–_a)p₌_p(t_–_a)p–α_;

(iii) aDα_λ_{= }_{for all constant functions}_f_{(t) =}_λ_;

(iv) aDα_{(fg) =}_f

aDαg+gaDαf;

(v) aDα₍f g) =

gaDαf–faDαg

g for all functionsg(t)= .

Deﬁnition .[] Let α∈(, ]. The conformable fractional integral starting from a pointaof a functionf : [a,∞)→Rof orderαis deﬁned as

aIαf(t) =

t

a

(s–a)α–_f_(s)_ds. _(.)

Remark . Ifa= , the deﬁnitions of the conformable fractional derivative and integral above will be reduced to the results in [].

Theorem .(Rolle’s theorem) Let an interval[c,d]⊂[a,∞)and f : [a,∞)→Rbe a given function that satisﬁes:

(i) f is continuous on[c,d],

(ii) f isα-diﬀerentiable for someα∈(, )on[c,d], (iii) f(c) =f(d).

Then there exists a constant e∈(c,d),such thataDα_f_{(e) = .}

Proof From (i)f is continuous on [c,d], and since by (iii)f(c) =f(d), there exists a point e∈(c,d) such thatf(e) is a maximum or minimum value off on [c,d]. Without loss of generality, we assume thateis a point of local minimum. Then we have

aDαf(e) = lim ε→+

f(e+ε(e–a)–α_{) –}_f_(e)

ε =εlim→–

f(e+ε(e–a)–α_{) –}_f_(e)

ε .

Observe that the ﬁrst limit is nonnegative and also the second limit is non-positive.

There-fore, we obtainaDαf(e) = .

Theorem .(Mean value theorem) Let an interval[c,d]⊂[a,∞)and let f : [a,∞)→R be a given function satisfying:

(i) f is continuous on[c,d],

(ii) f isα-diﬀerentiable for someα∈(, ).

Then there exists a constant e∈(c,d),such thataDα_f_{(e) =} f(d)–f(c) 

α(d–a)α–α(c–a)α

.

Proof Setting the function

g(t) =f(t) –f(c) – _ f(d) –f(c) α(d–a)

α_– 

α(c–a) α



α(t–a)

α – 

α(c–a)

α

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we see that g satisfy all the conditions of the Rolle’s theorem. Therefore, there exists a constante∈(c,d), such thataDα_{g(e) = . Using the fact that}_aDα₍

α(t–a)

α_{) = , we get the}

desired result. This completes the proof.

Remark . Ifa= , then Theorems . and . are reduced to Theorems . and . in [], respectively.

3 Impulsive results

LetJ–=J\ {t,t, . . . ,tm},J= [t,t],Jk= (tk,tk+] fork= , , . . . ,mbe sub-intervals ofJ

and the set PC(J,R) = {x:J→R:x(t) is continuous everywhere except for sometk at whichx(t–

k) andx(tk+) exist andx(t–k) =x(tk),k= , , . . . ,m}. LetE=PC(J,R), thenEis the Banach space with the normx=sup_t_∈_J|x(t)|. A functionx∈Eis called a solution of the impulsive periodic boundary value problem (.) if it satisﬁes (.).

Deﬁnition . A functionμ∈Eis called a lower solution of the periodic boundary value

problem (.) if it satisﬁes

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

tkD

α_μ

(t)≤f(t,μ(t), (Fμ)(t), (Sμ)(t)), t∈J–,

μ(tk)≤Ik(μ(tk)), k= , , . . . ,m,

μ()≤μ(T).

(.)

Analogously, a functionν∈Eis called an upper solution of the periodic boundary value

problem (.) if the inequalities

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

tkD

α_ν

(t)≥f(t,ν(t), (Fν)(t), (Sν)(t)), t∈J–,

ν(tk)≥Ik(ν(tk)), k= , , . . . ,m,

ν()≥ν(T),

(.)

hold.

Let us introduce the new notations which will be used in this paper. For nonnegative real numbersa≤banda,b∈J, we deﬁne

eMα_(a,_{b) =} ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

eMα(b–a)α, a,b∈(t_i,t_i_+],i= , , . . . ,m,

eMα(b–ti)α·_eMα(ti–a)α_, _a∈_(ti

–,ti],b∈(ti,ti+],

i= , , . . . ,m,

eMα(b–tq)α·

a<ti–<ti<be M

α(ti–ti–)α·_eMα(tp–a)α_, _a_<_ti

–<ti<b,

i= , , . . . ,m,

(.)

where tq =max{tk;a< tk <b}, tp =min{tk;a<tk <b} for some k ∈ {, , . . . ,m} and

(·) = .

Letf :J→Rbe a function given by

f(t) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

f(t), t∈J,

f(t), t∈J,

. . . ,

fm(t), t∈Jm.

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The impulsive integral notation is deﬁned as

b

a

f(s)dsˆ =

tp

a

fp–(s)ds+ tp+

tp

fp(s)ds+· · ·+

b

tq

fq(s)ds, a,b∈J, (.)

wherea≤tp<· · ·<tq≤b.

Example . ForJ= [, ],tk=k,k= , , , , , we have

eMα_{(, ) =}_eMα(–)α·_eMα(–)α·_eMα(–)α_,

e–Mα _{(., .) =}_e–Mα (.–)α·_e–Mα (–.)α_, .

.

f(s)dsˆ =



.

f(s)ds+ 



f(s)ds+ 



f(s)ds+ .



f(s)ds.

It is easy to prove the following result.

Property . Leta≤c≤b≤dbe nonnegative real numbers. The following relations hold:

(i) eMα_(a,_c)eMα_(c,_{b) =}_eMα_(a,_b)_,

(ii) eMα(a,b)eMα(c,d) =eMα(a,d)eMα(c,b).

Now, we consider the following boundary value problem of a linear impulsive con-formable fractional integro-diﬀerential equation subject to periodic boundary condition of the form

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

tkD

α_{x(t) –}_{Mx(t) =}_{H(Fx)(t) +}_{K(Sx)(t) +}_v(t), _{ <}_α_≤_,_t_∈_J–_,

x(tk) =Lkx(tk) +Ik(σ(tk)) –Lkσ(tk), k= , , . . . ,m, x() =x(T),

(.)

whereM> ,H,K≥,Lk≥,k= , , . . . ,m, are given constants and functionsv,σ∈E.

Lemma . The solution x∈E of the problem(.)can be written as the following

impul-sive integral equation:

x(t) =

T



G(t,s)P(s)dsˆ

+ m

k=

G(t,tk)

Lkx(tk) +Ikσ(tk)–Lkσ(tk), t∈J, (.)

where P(t) =H(Fx)(t) +K(Sx)(t) +v(t),

G(t,s) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

(s–th)α–e M

α(th,t)e– M α(th,s) –eMα(t,T)

, ≤s<t≤T,

(s–th)α–e M

α(t,t)e M α(th,T)e–

M α(th,s) –eMα(t,T)

, ≤t≤s≤T,

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and

G(t,s) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

eMα(s,t) –eMα(t,T)

, ≤s<t≤T,

eMα(t,t)e M

α(s,T) –eMα(t,T)

, ≤t≤s≤T,

(.)

with th=max{tk;k= , , . . . ,m and tk≤s}.

Proof Letx(t) be a solution of the problem (.). Fort∈J, multiplying bye–M (t–t_)α

α both

sides of the ﬁrst equation of the problem (.) and using Lemma .(v),

e–M(t–t)

α α _t

D

α

x(t) –e–M(t–t)

α

α _{Mx(t) =}_t D

α e–M(t–t)

α α _x(t)_,

we obtain

tD

α e–M(t–t)

α

α _x(t)₌_e–M (t–t_)α

α _{H(Fx)(t) +}_K_{(Sx)(t) +}_v(t)_. _(.)

Applying the conformable fractional integral of orderαto both sides of (.) fort∈J,

we have

x(t) =x()eMt

α α ₊

t



sα–eMt

α α_e–Ms

α α_P(s)_ds

=x()eMα_(t ,t) +

t



sα–_eM_α_(t ,t)e–

M α_(t

,s)P(s)ds.

Fort∈J, multiplying both sides of the ﬁrst equation of the problem (.) bye–M (t–t_)α

α and

using the product rule, we get

tD

α_e–M(t–t)α

α _x(t)₌_e–M (t–t)α

α _P(t). _(.)

The conformable fractional integration of orderαfromttotof (.) implies

x(t) =xt+_eM(t–t)

α α ₊

t

t

(s–t)α–eM (t–t)α

α _e–M (s–t)α

α _P(s)_ds

=xt+_eMα_(t__,_{t) +} t

t

(s–t)α–e M

α_(t__,_t)e–Mα_(t__,_s)P(s)_ds.

Sincex(t_+) =x(t) +Lx(t) +I(σ(t)) –Lσ(t) and

x(t) =x()e M

α_(t ,t) +

t



sα–_eM_α_(t ,t)e–

M α_(t

,s)P(s)ds,

by using Property .(i), we get

x(t) =x()eMα_(t__,_{t) +} t

t

(s–t)α–e M

α_(t__,_t)e–Mα_(t__,_s)P(s)_ds

+eMα_(t ,t)

Lx(t) +I

σ(t)

–Lσ(t)

+

t

t

(s–t)α–e M α_(t

,t)e– M α_(t

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Repeating the above process, fort∈Jk, we have

x(t) =x()eMα_(t__,_{t) +}

t<tk<t

eMα_(tk,_t)_{Lkx(tk) +}_Ik_σ_(tk)_–_Lk_σ_(tk)

+

t<tk<t tk

tk–

(s–tk–)α–e M

α_(tk_–_,_t)e–Mα_(tk_–_,_s)P(s)ds

+

t

tl

(s–tl)α–e

M α_(t

l,t)e–

M α_(t

l,s)P(s)ds, (.)

wheretl=max{tk;k= , , . . . ,mandtk<t}. Puttingt=Tin (.), we obtain

x(T) =x()eMα_(t ,T) +

m

k=

eMα_(t

k,T)

Lkx(tk) +Ik

σ(tk)

–Lkσ(tk)

+ m

k= tk

tk–

α_(tk_–_,_T_)e–Mα_(tk_–_,_s)P(s)ds

+

T

tm

(s–tm)α–eMα_(tm,_T_)e–Mα_(tm,_s)P(s)_ds.

From periodic boundary value conditionx() =x(T), we deduce that

x() = 

 –eMα_(t__,_T₎ _m

k=

eMα_(tk,_T₎_{Lkx(tk) +}_Ik_σ_(tk)_–_Lk_σ_(tk)

+ m

k= tk

tk–

α_(tk_–_,_T_)e–Mα_(tk_–_,_s)P(s)_ds

+

T

tm

(s–tm)α–e

M α_(t

m,T)e–

M α_(t

m,s)P(s)ds

. (.)

Substituting (.) into (.), it follows that

x(t) = e

M α(t_,t)

 –eMα_(t__,_T) _m

k=

e(tk,T)Lkx(tk) +Ikσ(tk)–Lkσ(tk)

+ m

k= tk

tk–

α_(t

k–,T)e– M

α_(t

k–,s)P(s)ds

+

T

tm

(s–tm)α–eMα_(tm,_T)e–Mα_(tm,_s)P(s)_ds

+

t<tk<t

+

t<tk<t tk

tk–

α_(t

k–,t)e– M α_(t

k–,s)P(s)ds

+

t

tl

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=   –eMα_(t__,_T)

_m

k=

eMα_(t__,_t)eMα_(tk,_T)_{Lkx(tk) +}_Ik_σ_(tk)_–_Lk_σ_(tk)

+ m

k= tk

tk–

α_(t__,_t)eMα_(tk_–_,_T_)e–Mα_(tk_–_,_s)P(s)_ds

+

T

tm

(s–tm)α–e

M α_(t

,t)e M α_(t

m,T)e–

M α_(t

m,s)P(s)ds

–

t<tk<t

eMα_(t__,_T_)eMα_(tk,_t)_{Lkx(tk) +}_Ik_σ_(tk)_–_Lk_σ_(tk)

–

t<tk<t tk

tk–

α_(t__,_T_)eMα_(tk_–_,_t)e–Mα_(tk_–_,_s)P(s)_ds

–

t

tl

(s–tl)α–eMα_(t__,_T_)eMα_(tl,_t)e–Mα_(tl,_s)P(s)ds

+

t<tk<t

eMα_(t

k,t)

Lkx(tk) +Ik

σ(tk)

–Lkσ(tk)

+

t<tk<t tk

tk–

α_(tk_–_,_t)e–Mα_(tk_–_,_s)P(s)ds

+

t

tl

(s–tl)α–eMα_(tl,_t)e–Mα_(tl,_s)P(s)_ds

.

Using Property .(ii), we have

x(t) = 

 –eMα_(t__,_T)

t<tk<T

eMα_(t ,t)e

M

α_(tk,_T₎_{Lkx(tk) +}_Ik_σ_(tk)_–_Lk_σ_(tk)

+

tl+

t

(s–tl)α–eMα_(t__,_t)eMα_(tl,_T_)e–Mα_(tl,_s)P(s)_ds

+

tl+<tk<T tk

tk–

α_(t__,_t)eMα_(tk_–_,_T_)e–Mα_(tk_–_,_s)P(s)ds

+

T

tm

(s–tm)α–e

M α_(t

,t)e M α_(t

m,T)e–

M α_(t

m,s)P(s)ds

+

t<tk<t

+

t<tk<t tk

tk–

α_(tk_–_,_t)e–Mα_(tk_–_,_s)P(s)ds

+

t

tl

(s–tl)α–eMα_(tl,_t)e–Mα_(tl,_s)P(s)_ds

= 

 –eMα_(t__,_T)

t<tk<T

eMα_(t__,_t)eMα_(tk,_T₎_{Lkx(tk) +}_Ik_σ_(tk)_–_Lk_σ_(tk)

+

T

t

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+ t<tk<t

+

t

t

(s–th)α–e

M α_(t

h,t)e–

M α_(t

h,s)P(s)dsˆ

.

Therefore, we obtain the integral equation (.) as required.

Conversely, it can easily be shown by direct computation that the integral equation (.) satisﬁes the impulsive periodic boundary value problem (.). This completes the proof.

Denotea=maxk{tk+–tk},k= , , . . . ,m. Now we prove the comparison result.

Lemma . Let <α≤.Assume that x∈E satisﬁes

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

tkD

α_x(t)_≥_{Mx(t) +}_{H(Fx)(t) +}_K(Sx)(t), _t_∈_J–_,

x(tk)≥Lkx(tk), k= , , . . . ,m, x()≥x(T),

(.)

where M> ,H,K≥,Lk≥,k= , , . . . ,m are given constants.Suppose in addition that

m

k=

Lk+a α

α (m+ )(M+H¯lT+KhT)¯ ≤, (.)

where¯l=sup{l(t,s) : (t,s)∈J_}_and_h¯₌_sup_{_h(t,_{s) : (t,}_s)_∈_J_}_._{Then x(t)}_≤__{for all t}_∈_J.

Proof Suppose, to the contrary, thatx(t) >  for somet∈J. Then there are two cases:

(i) There exists a pointt∗∈J, such thatx(t∗) > andx(t)≥for allt∈J. (ii) There exist two pointst∗,t∗∈J, such thatx(t∗) > andx(t∗) < .

Case (i): Settingu(t) =e–Mα(t–tk)α_{x(t) for}_t∈_{J, then we have}

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

tkD

α_u(t)_≥_Ht

l(t,s)e

–M_α[(t–tk)α–(s–tk)α]_u(s)ds

+K_Th(t,s)e–M_α[(t–tk)α–(s–tk)α]_u(s)_ds, _t∈_J–_,

u(tk)≥Lku(tk), k= , , . . . ,m, u()≥eMα(T–tm)α_u(T_).

Obviously, the functionsu(t) andx(t) have the same sign. According to the above, it fol-lows thattkD

α_u(t)_≥_{ for}_t_∈_J–_and_u(tk)_≥_{ for}_k_{= , , . . . ,m. This implies that}_{u(t) is}

nondecreasing inJ. Therefore, we haveu(T)≥u(t∗) >  andu(T)≥u()≥eMα(T–tm)α_u(T_),

which is a contradiction.

Case (ii): Letinf{x(t) :t∈J}= –b. Then we can assume thatb>  and also there exists a pointt∗∈(ti,ti+],i∈ {, , . . . ,m}, such thatx(t∗) = –borx(ti+) = –b. Now, we only consider the casex(t∗) = –b. For the casex(t+_i) = –b, the proof is similar. It is easy to see that

tkD

α_x(t)_≥_–b

M+H

t



l(t,s)ds+K

T



h(t,s)ds

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Suppose thatt∗∈(tj,tj+) for somej∈ {, , . . . ,m}. We assumet∗<t∗, which impliesi≤j.

For the caset∗>t∗, the proof is similar and thus we omit it. By Theorem ., we get

x(T) –x(tm)≥x(T) –xt+_m+Lmx(tm) = 

α(T–tm)

α tmD

α

x(rm) +Lmx(tm)

≥–b

aα

α (M+H¯lT+KhT) +¯ Lm

, rm∈(tm,T),

x(tm) –x(tm–)≥x(tm) –x

t+_m_–+Lm–x(tm–)

=

α(tm–tm–)

α tm–D

α

x(rm–) +Lm–x(tm–)

≥–b

aα

α(M+H¯lT+KhT) +¯ Lm–

, rm–∈(tm–,tm),

. . . ,

x(tj+) –x

t∗=



α(tj+–tj)

α_–

α

t∗–tj

α tjD

α_x_r∗

≥–b

aα

α (M+H¯lT+KhT¯ )

, r∗∈t∗,tj+

.

Summing up the above inequalities, we get

x(T) –xt∗≥ m

k=j+

–b

aα

α (M+H¯lT+KhT¯ ) +Lk

–b

aα

. (.)

In the same way, we have

x(t∗) –x(ti)≥–b

aα

α(M+H¯lT+KhT) +¯ Li

,

. . . ,

x(t) –x()≥–b

aα

.

Summing up the above inequalities, it follows that

x(t∗) –x()≥

i

k=

–b

aα

α (M+H¯lT+KhT) +¯ Lk

–b

aα

. (.)

From (.), (.), and the third inequality of (.), we obtain

x(t∗) –xt∗≥–b m

k=

Lk–a α

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which leads to

 <xt∗≤–b+b m

k=

Lk+ aα

α b(m+ )(M+H¯lT+KhT).¯

Thus, we get

m

k=

Lk+a α

α (m+ )(M+H¯lT+KhT) > ,¯

which contradicts (.). The proof is completed.

4 Existence results

In view of Lemma ., we deﬁne the operatorA:E→Eby

Ax(t) =

T



G(t,s)

H(Fx)(s) +K(Sx)(s) +v(s)dsˆ

+ m

k=

G(t,tk)

Lkx(tk) +Ikσ(tk)–Lkσ(tk), (.)

where the Green functionsG(t,s) andG(t,s) are deﬁned by (.) and (.), respectively.

Next, we will prove the existence of a unique solution for the problem (.). To accomplish this, we set

:= e

M α(t_,T) | –eMα_(t__,_T)|

aα–

¯lH 

m+

k=

t_k–t_k_–+hKT¯ 

+ m

k=

Lk

.

Lemma . Assume thatα∈(, ],M> ,H,K≥,Lk≥,k= , , . . . ,m.If

< , (.)

then the periodic boundary value problem(.)has a unique solution on J.

Proof CaseI. For ≤s<t≤T, we see that

(s–th)α–_eM_α_(th_,_t)e–M_α_(th,_s)_≤_aα–_eM_α_(t ,T)

and

eMα_(s,_t)≤_eMα_(t__,_T).

CaseII. For ≤t≤s≤T, we have

eMα_(t ,t)e

M

α_(s,_T)≤_eMα_(t ,T).

Ift<th≤s, then we obtain

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Ifth≤t≤s, we haveeMα_(th,_t)e–Mα_(th,_s)≤_{, which implies that the above inequality (.)}

holds. From Cases I and II, it follows that

sup

(t,s)∈J

G(t,s)≤

aα–_eM_α_(t ,T) | –eMα(t_,T)|

and sup

(t,s)∈J

G(t,s)≤

eMα(t_,T) | –eMα(t_,T)|

.

Transform the problem (.) into a ﬁxed point problem,x=Ax, where the operatorA is deﬁned by (.). For anyx,y∈E, we have

Ax–Ay ≤ x–y

T



G(t,s)H(F)(s) +K(S)(s) _ˆ

ds

+x–y m

k=

G(t,s)Lk

≤x–y.

As< ,Ais a contraction. Therefore, by the Banach contraction mapping principle, we deduce thatAhas a ﬁxed point which is the unique solution of the problem (.). The

proof is completed.

Forν,μ∈E, we denote

[ν,μ] =

x∈E:ν(t)≤x(t)≤μ(t),t∈J

,

and we writeν≤μifν(t)≤μ(t) for allt∈J.

Theorem . Assume that the following conditions hold:

(H) the functionsμandνare lower and upper solutions of the periodic boundary value problem(.),respectively,such thatν(t)≤μ(t)onJ;

(H) the functionf∈C(J×R,R)satisﬁes

f(t,x,y,z) –f(t,x,¯ y,¯ ¯z)≤M(x–x) +¯ H(y–y) +¯ K(z–¯z),

for ν(t)≤ ¯x(t)≤x(t)≤μ(t),(Fν)(t)≤ ¯y(t)≤y(t)≤(Fμ)(t), (Sν)(t)≤ ¯z(t)≤

z(t)≤(Sμ)(t),t∈J;

(H) the functionsIk∈C(R,R)satisfy

Ikx(tk)–Iky(tk)≤Lkx(tk) –y(tk),

wheneverν(tk)≤y(tk)≤x(tk)≤μ(tk),Lk≥,k= , , . . . ,m;

(H) the inequalities(.)and(.)hold.

Then there exist monotone sequences {μn},{νn} ⊂ E such that limn→∞μn(t) = x∗(t), limn→∞νn(t) =x∗(t)uniformly on J and x∗,x∗are maximal and minimal solutions of the

problem(.),respectively,such that

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on J,where x is any solution of the periodic boundary value problem(.)such thatν(t)≤

x(t)≤μ(t)on J.

Proof For anyσ∈[ν,μ], we investigate the periodic boundary value problem (.) with

v(t) =ft,σ(t), (Fσ)(t), (Sσ)(t)–Mσ(t) –H(Fσ)(t) –K(Sσ)(t).

Applying Lemma ., the problem (.) has a unique solutionx(t) fort∈J. Let us deﬁne an operatorAbyx=Aσ. Then the operatorAis an operator from [ν,μ] toEandAhas

the following properties:

(i) ν≤Aν,Aμ≤μ;

(ii) for anyσ,σ∈[ν,μ],σ≤σimpliesAσ≤Aσ.

To prove (i), we set a functionϕ=ν–ν, whereν=Aν. Then from condition (H) and

(.), we have

tkD

α_ϕ

(t) =tkD

α_ν

(t) –tkD

α_ν

(t)

≥ft,ν(t), (Fν)(t), (Sν)(t)

–Mν(t) +H(Fν)(t) +K(Sν)(t)

+ft,ν(t), (Fν)(t), (Sν)(t)

–Mν(t) –H(Fν)(t) –K(Sν)(t)

=Mϕ(t) +H(Fϕ)(t) +K(Sϕ)(t), t∈J–, and

ϕ(tk) =ν(tk) –ν(tk) ≥Ikν(tk)

–Lkν(tk) +Ik

ν(tk)

–Lkν(tk)

=Lkϕ(tk), k= , , . . . ,m, and

ϕ() =ν() –ν() ≥ν(T) –ν(T)

=ϕ(T).

Using Lemma ., we deduce thatϕ(t)≤ for allt∈J,i.e.,ν≤Aν. Similarly, we can

prove thatAμ≤μ.

To prove (ii), we letu=Aσ,u=Aσ, whereσ≤σonJandσ,σ∈[ν,μ]. Setting

a functionϕ=u–u, then fort∈Jand by (H), we obtain

tkD

α_ϕ_{(t) =} tkD

α_u

(t) –tkD

α_u

(t)

=Mu(t) +H(Fu)(t) +K(Su)(t) +f

t,σ(t), (Fσ)(t), (Sσ)(t)

–Mσ(t) –H(Fσ)(t) –K(Sσ)(t)

–Mu(t) +H(Fu)(t) +K(Su)(t)

+ft,σ(t), (Fσ)(t), (Sσ)(t)

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≥Mu(t) –u(t)

+HF(u–u)

(t) +KS(u–u)

(t),

=Mϕ(t) +H(Fϕ)(t) +K(Sϕ)(t), t∈J–,

and by (H),

ϕ(tk) =u(tk) –u(tk)

=Lku(tk) +Ik

σ(tk)

–Lkσ(tk) –

Lku(tk) +Ik

σ(tk)

–Lkσ(tk)

≥Lku(tk) –u(tk)

=Lkϕ(tk), k= , , . . . ,m.

It is easy to see that

ϕ() =u() –u()

=u(T) –u(T)

≥ϕ(T).

Then, from Lemma ., we getϕ(t)≤, which yieldsAσ≤Aσ.

Next, we deﬁne the sequences{μn},{νn}such thatμn+=Aμnandνn+=Aνn. From (i) and (ii), we see that the sequences{μn},{νn}satisfy the following inequality:

ν≤ν≤ · · · ≤νn≤ · · · ≤μn≤ · · · ≤μ≤μ,

for alln∈N. Obviously, the functionsμn,νn(n= , , . . .) satisfy

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

tkD

α_μ

n(t) =Mμn(t) +H(Fμn)(t) +K(Sμn)(t) +f(t,μn–(t), (Fμn–)(t), (Sμn–)(t))

–Mμn–(t) –H(Fμn–)(t) –K(Sμn–)(t), t∈J–,

μn(tk) =Lkμn(tk) +Ik(μn–(tk)) –Lkμn–(tk), k= , , . . . ,m,

μn() =μn(T), and

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

tkD

α_ν_{n(t) =}_M_ν_{n(t) +}_H(F_ν_{n)(t) +}_K(S_ν_n)(t) +f(t,νn–(t), (Fνn–)(t), (Sνn–)(t))

–Mνn–(t) –H(Fνn–)(t) –K(Sνn–)(t), t∈J–,

νn(tk) =Lkνn(tk) +Ik(νn–(tk)) –Lkνn–(tk), k= , , . . . ,m,

νn() =νn(T).

Therefore, there exist functionsx∗andx∗onJ, such thatlimn→∞νn=x∗andlimn→∞μn= x∗uniformly onJ. Clearly,x∗,x∗are solutions of the periodic boundary value problem (.). Finally, we are going to show that x∗, x∗ are minimal and maximal solutions of the

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x∈[ν,μ] and also there exists a positive integernsuch thatνn(t)≤x(t)≤μn(t) onJ. Let

ϕ=νn+–x, then fort∈J, we have

tkD

α_ϕ

(t) =tkD

α_ν

n+(t) –tkD

α x(t)

=Mνn+(t) +H(Fνn+)(t) +K(Sνn+)(t)

+ft,νn(t), (Fνn)(t), (Sνn)(t) –Mνn(t) –H(Fνn)(t) –K(Sνn)(t) –f

t,x(t), (Fx)(t), (Sx)(t)

≥Mϕ(t) +H(Fϕ)(t) +K(Sϕ)(t), t∈J–, and

ϕ(tk) =νn+(tk) –x(tk)

=Lkνn+(tk) +Ik

νn(tk)

–Lkνn(tk) –Ik

x(tk)

≥Lk

νn+(tk) –x(tk)

=Lkϕ(tk), k= , , . . . ,m, and also

ϕ() =νn+() –x()

=νn+(T) –x(T) ≥ϕ(T).

Then, by applying Lemma ., we haveϕ(t)≤, which leads toνn+≤xonJ. By similar

method, we can show thatx≤μn+ onJ. Sinceν≤x≤μonJ, by mathematical

in-duction, we deduce thatνn≤x≤μnonJfor everyn. Hence, by takingn→ ∞, we have

x∗(t)≤x(t)≤x∗(t) onJ. The proof is complete.

5 An example

Example . Consider the following periodic boundary value problem for impulsive con-formable fractional integro-diﬀerential equation:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

tkD  _{x(t) =}t

 

sinx(t) +

t

 t

tsx(s)ds

+t

  



cos(st)x(s)ds, t∈[, ]\ {  },

x(_) = _ tan–_(x(

)), k= ,

x() =x().

(.)

Hereα= /,T= ,m= ,t= /. Choosingμ= ,ν=

–, t∈_{[,  ],}

–, t∈_{(  ,],} thenμandν

are lower and upper solutions of the problem (.), respectively, and alsoν≤μ. Let

f(t,u,v,w) =t

 

sinx+ t

v+ t

w,

then we have

f(t,u,v,w) –f(t,u,¯ v,¯ w)¯ ≤ 

(x–x) +¯ 

(v–v) +¯ 

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where ν(t)≤ ¯u(t)≤u(t)≤μ(t), (Fν)(t)≤ ¯v(t)≤v(t)≤(Fμ)(t), (Sν)(t)≤ ¯w(t)≤

w(t)≤(Sμ)(t),t∈[, ]. We see that

I

u(t)

–I

v(t)

≤ 



u(t) –v(t)

,

whereν(t)≤v(t)≤u(t)≤μ(t). SettingM= /,H= /,K= /,L= /,a=

/,¯l= , andh¯= , it follows that

m

k=

Lk+a α

α (m+ )(M+H¯lT+KhT) = .¯ ≤

and

= e

M α(t_,T) | –eMα_(t__,_T)|

aα–

¯ lH  m+ k=

t_k–t_k_–+hKT¯ 

+ m k= Lk

= . < ,

whereeMα(t_,T) =eMα(–)α·_eMα(–t)α_{= .. Therefore, the periodic boundary}

value problem (.) satisﬁes all conditions of Theorem .. Then by using the monotone iterative scheme, ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

tkD 

_μ_n(t) =  μn(t) +

 

t

tsμn(s)ds+_



cos(st)μn(s)ds +t

 

sinμn–(t) +

t 

t

tsμn–(s)ds+t   



cos(st)μn–(s)ds

–_μn–(t) – _ t

tsμn–(s)ds

–_ _cos_(s_t)_μ

n–(s)ds, t∈[, ]\ {_},

μn(t) =_μn(t) +_ tan–(μn–(_)) –_μn–(t), k= ,

μn() =μn(), and ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

tkD 

_ν_n(t) =  νn(t) +

 

t

tsνn(s)ds+_



cos(st)νn(s)ds +t

sinνn–(t) +

t

 t

tsνn–(s)ds+t   



cos(st)νn–(s)ds

–_νn–(t) –_ t

tsνn–(s)ds

–_ _cos_(s_t)_ν

n–(s)ds, t∈[, ]\ {_},

νn(t) =_νn(t) +_ tan–(νn–(_)) –_νn–(t), k= ,

νn() =νn(),

forn= , , , . . . , the problem (.) has the extremal solution in the segment [ν,μ].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

(17)

Author details

1_{Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s} University of Technology North Bangkok, Bangkok, 10800, Thailand.2_{Department of Mathematics, University of Ioannina,} Ioannina, 451 10, Greece.3_{Nonlinear Analysis and Applied Mathematics (NAAM) - Research Group, Department of} Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia.

Acknowledgements

This research was funded by King Mongkut’s University of Technology North Bangkok. Contract No. KMUTNB-GOV-59-39.

Received: 29 March 2016 Accepted: 23 June 2016 References

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