Impulsive boundary value problems containing Caputo fractional derivative of a function with respect to another function

(1)

R E S E A R C H

Open Access

Impulsive boundary value problems

containing Caputo fractional derivative

of a function with respect to another function

Chanon Promsakon

1

_{, Eakachai Suntonsinsoungvon}

2

_{, Sotiris K. Ntouyas}

3,4

_{and Jessada Tariboon}

1*

*_{Correspondence:}

[email protected]

1_{Intelligent and Nonlinear Dynamic}

Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, Thailand Full list of author information is available at the end of the article

Abstract

In this paper, we study the existence and uniqueness for a new class of impulsive fractional boundary value problems with separated boundary conditions containing the Caputo fractional derivative of a function with respect to another function. The existence of solutions is established by using the Leray–Schauder nonlinear alternative, and the uniqueness result is proved via Banach’s contraction mapping principle. Some examples are also constructed to demonstrate the application of main results.

MSC: 34A08; 34A37

Keywords: Impulsive fractional diﬀerential equations; Separated boundary

conditions; Fixed point theorems

1 Introduction and preliminaries

Impulsive boundary value problems corresponding to integer-order differential equations with impulsive conditions have been considered extensively in the literature; see [1–3] and references therein. Also, in the case of noninteger-order differential equations, there are many results on impulsive boundary value problems; see, for example, [4–8], to name a few, but some problems still need further investigation under impulsive conditions. To the best of our knowledge, there are no papers on impulsive fractional differential equations containing the fractional derivative of a function with respect to another function. This gap is covered by the present paper.

More precisely, the purpose of this investigation is to establish the existence and unique-ness of solutions for a new class of impulsive fractional diﬀerential equations with bound-ary conditions of the form

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

tkD αk

gkx(t) =f(t,x(t)), t=tk,k= 0, 1, 2, . . . ,m,

x(t+_k) –x(t–

k) =φk(x(tk)), k= 1, 2, . . . ,m,

Dgkx(t +

k) –Dgk–1x(t

–

k) =φk∗(x(tk)), k= 1, 2, . . . ,m,

x(0) +β1Dg0x(0) =λ1, x(T) +β2Dgmx(T) =λ2,

(1.1)

(2)

where tkD αk

gk is the Caputo-type fractional diﬀerential operator with respect to

an-other increasing diﬀerentiable function gk(t),t∈Jk, of order 1 <αk ≤2, Jk = (tk,tk+1],

k= 0, 1, 2, . . . ,m,t0= 0 <t1<t2<· · ·<tm<tm+1=Tare impulsive points, the integer order

of the diﬀerential operatorDgkis deﬁned by

Dgk=

1 g_k(t)

d

dt, (1.2)

the function f :J×R→R,J = [0,T] ={0} ∪(m₀ Jk), tm+1=T, the functions φk,φk∗ :

R→R, k= 1, . . . ,m, are continuous, βi, λi, i= 1, 2, are given constants, and x(tk+) =

lim_ε→0+x(tk+ε),x(t–_k) =x(tk). For example, ifα0=α1= 2,m= 1, andgk(t) =e(k+1)t,k= 0, 1,

then (1.1) is reduced to

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

e–2t₍d2x(t)

dt2 –dx_dt(t)) =f(t,x(t)), t∈(t0,t1], 1

4e –4t₍d2x(t)

dt2 – 2

dx(t)

dt ) =f(t,x(t)), t∈(t1,T], x(t+₁) –x(t–

1) =φ1(x(t1)), (₂1e–2t dx_dt)t=t1+– (e

–t dx dt)t=t–1 =φ

∗

k(x(t1)),

x(0) +β1(e–t dx_dt)t=0=λ1, x(T) +β2(1₂e–2t dx_dt)t=T=λ2,

(1.3)

withDgk=

e–(k+1)t (k+1)

d

dtfork= 0, 1.

Forγ > 0, the Riemann–Liouville fractional integral of an integrable functionh: [a,b]→ Rwith respect to an increasing functiong∈Cn_([_a_,_b_{]) such that}_g₍_t₎_{= 0 for all}_t_∈_[_a_,_b_]

is deﬁned by [9–11]

aIgγh(t) =

1

Γ(γ)

t

a

g(s)h(s)

[g(t) –g(s)]1–γ ds, (1.4)

whereΓ is the gamma function. The Riemann–Liouville fractional derivative of a function hwith respect to another functiongon [a,b] is deﬁned as

aDγgh(t) =Dng aIng–γh(t) =

1

Γ(n–γ)D

n g

t

a

g(s)h(s)

[g(t) –g(s)]1+γ–nds, (1.5)

whereas the Caputo derivative is deﬁned by

aDγgh(t) =aIgn–γDngh(t) =

1

Γ(n–γ)

t

a

g(s)Dn gh(s)

[g(t) –g(s)]1+γ–nds, (1.6)

whereDn

g =Dg· · ·Dg

ntimes

,Dg is deﬁned in (1.2), andnis a positive integer such thatn– 1 <

γ <n. There are relations of fractional integral and derivatives of the Riemann–Liouville and Caputo types, which will be used in our investigation [9]:

aIgγ

aDγgh

(t) =h(t) –

n

j=1

(g(t) –g(a))γ–j

Γ(γ –j+ 1) D

n–j g

aIgn–γh

(a) (1.7)

and

aIgγ

aDγgh

(t) =h(t) –

n–1

j=0

(g(t) –g(a))j

j! D

j

(3)

In addition, forγ,δ> 0, the relation

aIgγ

g(t) –g(a)δ= Γ(δ+ 1)

Γ(γ +δ+ 1)

g(t) –g(a)γ+δ, (1.9)

is applied in the main results [9]. For some recent results, we refer the interested the reader to [12–17].

Note that (1.4) is reduced to the Riemann–Liouville and Hadamard fractional inte-grals when g(t) =t andg(t) =logt, respectively, where log(·) =log_e(·). The fractional derivatives of Hadamard and Hadamard–Caputo types can be obtained by substituting g(t) =logtinto (1.5) and (1.6), respectively. Also, the Riemann–Liouville and Caputo frac-tional derivatives are presented by replacingg(t) =tin (1.5) and (1.6), respectively. In the particular case forg(t) =et _{of fractional diﬀerential equations and boundary value}

prob-lems, the reader can ﬁnd some results in [18,19]. Therefore problem (1.1) generates many types and also mixed types of impulsive fractional diﬀerential equations with boundary conditions.

Using standard ﬁxed point theorems, we study the existence and uniqueness of solutions for the impulsive fractional boundary value problem (1.1). An auxiliary result useful to transform problem (1.1) into an equivalent integral equation is proved in Sect.2. The main results are presented in Sect.3, where an existence and uniqueness result is proved via Banach’s ﬁxed point theorem and an existence result with the help of the Leray–Schauder nonlinear alternative. Some illustrative examples are constructed in Sect.4.

2 An auxiliary lemma

In this section, we prove an auxiliary lemma, which is the basic tool to express the solution of problem (1.1) in an equivalent integral equation. Note that two boundary conditions in (1.1) are separated: one condition is stated at the initial pointt= 0, and the other at the end pointt=T. We use the notation

tj+1= ⎧ ⎨ ⎩

t, tj+1>tk,

tj+1, tj+1≤tk.

For example, ift∈(t3,t4], thentk=t3, and 3

j=0 tj+1= t1+ t2+ t3+ t4=t1+t2+t3+t.

In addition, we deﬁneGj=gj(tj+1) –gj(tj) and also the nonzero constant

Ω= (β2–β1) +

m

j=0

Gj.

The next lemma concerns a linear variant of problem (1.1).

Lemma 2.1 The separated boundary value problem of impulsive fractional diﬀerential equations with respect to other functions

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

tkD αk

gkx(t) =y(t), t=tk,k= 0, 1, 2, . . . ,m,

x(t+_k) –x(t–_k) =φk(x(tk)), k= 1, 2, . . . ,m,

Dgkx(t +

k) –Dgk–1x(t

–

k) =φ∗k(x(tk)), k= 1, 2, . . . ,m,

x(0) +β1Dg0x(0) =λ1, x(T) +β2Dgmx(T) =λ2,

(4)

is equivalent to the integral equation

(5)

By substituting impulsive conditions we obtain

From the ﬁrst boundary condition in (1.1) we have

c0+β1c1=λ1. (2.5)

by substituting in the second boundary condition of (1.1) we obtain

(6)

–

Substituting the values ofc0 andc1 into (2.4), we obtain (2.2). The converse follows by

direct computation. The proof is completed.

Remark2.2 Ifk= 0, then problem (2.1) is reduced to a nonimpulsively fractional diﬀer-ential equation with boundary conditions of the form

⎧

Consequently, the result in Theorem2.1can be presented by

x(t) = λ1

3 Main results

(7)

k = 1, 2, . . . ,m}. Note that PC(J,R) is a Banach space equipped with the norm x=

sup{|x(t)|:t∈J}.

In the following, for the convenience of the reader, we express the fractional integral deﬁned in (1.4) of a nonlinear two-variable functionf(t,x(t)) by a subscript notation by

In view of Lemma2.1, to establish existence theorems, we consider the operator equa-tionx=Ax, whereA:PC(J,R)→PC(J,R) is deﬁned by

and also apply ﬁxed point theory.

Next, we use the Lipschitz condition of a nonlinear functionf to prove the existence of a unique solution using the Banach ﬁxed point theorem.

For convenience, we denote

(8)

and

Q5= (Q2+ 1)m.

Theorem 3.1 Let gk∈C2([0,T])with gk(t) > 0for t∈[0,T],k= 0, 1, . . . ,m.Assume that

functions f :J×R→R,φk:R→R,andφk∗:R→R,k= 1, 2, . . . ,m,satisfy

(H1) |f(t,x) –f(t,y)| ≤L1|x–y|,L1> 0,∀t∈J,x,y∈R,

(H2) |φk(x) –φk(y)| ≤L2|x–y|,|φ_k∗(x) –φ∗_k(y)| ≤L3|x–y|,L2,L3> 0,∀x,y∈R.

If

L1Q3+L2Q5+L3Q4< 1, (3.1)

then the separated boundary value problem of impulsive fractional diﬀerential equations with respect to other functions(1.1)has a unique solution on J.

Proof The Banach ﬁxed point theorem guarantees a unique solution in a set Bρ = {x∈PC(J,R) :x ≤ρ}, whereρis chosen such that

ρ>Q1+|λ2|Q2+ρ(L1Q3+L2Q5+L3Q4) +M1Q3+M2Q5+M3Q4 1 – (L1Q3+L2Q5+L3Q4)

.

We set sup_t_∈_J|f0|=M1, maxk|φk(0)|=M2, and maxk|φ_k∗(0)|=M3, wheref0=f(t, 0). Using |fx| ≤ |fx–f0|+|f0|,|φk(x)| ≤ |φk(x) –φk(0)|+|φk(0)|, and|φk∗(x)| ≤ |φk∗(x) –φk∗(0)|+|φk∗(0)|, k= 1, 2, 3, . . . ,m, for anyx∈Br, we have

_Ax(t)

≤|λ1| |Ω|

|β2|+ _m

j=0 |Gj|

+

_k

j=0 |Gj|

+ 1

|Ω|

|β1|+ _k

j=0 |Gj|

|λ2|+

m–1

j=0

tjI αj

gj|fx|

(tj+1) +φj+1

x(tj+1)

+tmI αm

gm|fx|

(T) +

m–1

j=0 |Gj+1|

_j

r=0

trI αr–1

gr |fx|

(tr+1) +φr∗+1

x(tr+1)

+|β2|

tmIgαm–1m |fx|

(T) +|β2|

m–1

j=0

tjI αj–1

gj |fx|

(tj+1) +φj∗+1

x(tj+1)

+

k–1

j=0

tjI αj

gj|fx|

(tj+1) +φj+1

x(tj+1)+

tkI αk

gk|fx|

(t)

+

k–1

j=0 |Gj+1|

_j

r=0

trI αr–1

gr |fx|

(tr+1) +φ∗r+1

x(tr+1)

≤Q1+|λ2|Q2+ρ(L1Q3+L2Q5+L3Q4) +M1Q3+M2Q5+M3Q4

(9)

This means theABρ⊂Bρ. To prove the contractivity of the operatorA, for anyx,y∈Bρ,

we consider

_A_x₍_t_{) –}_A_y₍_t₎

≤_|1 Ω|

|β1|+

_k

j=0 |Gj|

_m_–1

j=0

tjI αj

gj|fx–fy|

(tj+1) +φj+1

x(tj+1)

–φj+1

y(tj+1)

+tmI αm

gm|fx–fy|

(T) +|β2|

tmI αm–1

gm |fx–fy|

(T)

+

m–1

j=0 |Gj+1|

_j

r=0

trI αr–1

gr |fx–fy|

(tr+1) +φr∗+1

x(tr+1)

–φ_r∗₊₁y(tr+1)

+|β2|

m–1

j=0

tjI αj–1

gj |fx–fy|

(tj+1) +φj∗+1

x(tj+1)

–φ_j∗₊₁y(tj+1)

+

k–1

j=0

tjI αj

gj|fx–fy|

(tj+1) +φj+1

x(tj+1)

–φj+1

y(tj+1)

+

k–1

j=0 |Gj+1|

_j

r=0

trI αr–1

gr |fx–fy|

(tr+1) +φr∗+1

x(tr+1)

–φ_r∗₊₁y(tr+1)

+tkI αk

gk|fx–fy|

(t)

≤(L1Q3+L2Q5+L3Q4)x–y,

which impliesAx–Ay ≤(L1Q3+L2Q5+L3Q4)x–y. Since inequality (3.1) holds, we

get that the operatorAis a contraction. The beneﬁt of the Banach contraction principle leads to the existence of a unique ﬁxed point of operatorA, which is a unique solution of

problem (1.1) onJ. The proof is ﬁnished.

Ifm= 0, thenQ4andQ5in the previous theorem are reduced to zero, and the constants

Q1,Q2,Q3are reduced to

Q∗₁=|λ1|

|Ω|

|β2|+ 2g0(T) –g0(0),

Q∗₂= 1

|Ω|

|β1|+g0(T) –g0(0),

Q∗₃=Q∗₂+ 1(g0(T) –g0(0))

α0

Γ(α0+ 1)

+Q∗₂|β2|

(g0(T) –g0(0))α0–1 Γ(α0)

,

respectively. Consequently, we get a result for nonimpulse eﬀects.

Corollary 3.2 Assume that the function f satisﬁes(H1).If Q∗3L1< 1,then there exists a

unique solution on[0,T]of the problem

⎧ ⎨ ⎩

t0D

α0

g0x(t) =f(t,x(t)), t∈(0,T),

x(0) +β1Dg0x(0) =λ1, x(T) +β2Dg0x(T) =λ2.

(3.2)

(10)

Lemma 3.3(Nonlinear alternative for single-valued maps [20]) Let E be a Banach space, let C be a closed convex subset of E,and let U be an open subset of C such that0∈U. Suppose that F:U→C is a continuous and compact map,that is, F(U)is a relatively compact subset of C.Then either

(i) Fhas a ﬁxed point inU,or

(ii) there isu∈∂U(the boundary ofUinC)andλ∈(0, 1)withu=λF(u).

Theorem 3.4 Suppose that:

(H3) There exist a continuous nondecreasing functionψ1: [0,∞)→(0,∞)and a

contin-uous functionp:J→R+such that

_f(t,x)≤p(t)ψ1

|x|, (t,x)∈J×R.

(H4) There exist two continuous nondecreasing functionsψ2,ψ3: [0,∞)→(0,∞)such

that

φk(x)≤ψ2

|x| and φ_k∗(x)≤ψ3

|x|

for allx∈R,k= 1, 2, . . . ,m.

(H5) There exists a constantM> 0such that

M

(Q1+|λ2|Q2) + (pψ1(M)Q3+ψ2(M)Q5+ψ3(M)Q4)

> 1.

Then the impulsive fractional boundary value problem(1.1)has at least one solution on J. Proof First, we prove the continuity ofA. LetBρ∗={x∈PC(J,R) :x ≤ρ∗,ρ∗> 0}be a

bounded ball inPC(J,R), and let{xn}be a sequence such thatxn→xinBρ∗. From the

continuity off onJ×Rwe obtain

ft,xn(t)

→ft,x(t) asn→ ∞.

Then, for anyt∈J, we get

_Axn(t) –Ax(t)

≤_|1 Ω|

|β1|+

_k

j=0 |Gj|

_m_–1

j=0

tjI αj

gj|fxn–fx|

(tj+1) +φj+1

xn(tj+1)

–φj+1

x(tj+1)+

tmI αm

gm|fxn–fx|

(T) +|β2|

tmI αm–1

gm |fxn–fx|

(T)

+

m–1

j=0 |Gj+1|

_j

r=0

trI αr–1

gr |fxn–fx|

(tr+1) +φr∗+1

xn(tr+1)

–φ_r∗₊₁x(tr+1)

+|β2|

m–1

j=0

tjI αj–1

gj |fxn–fx|

(tj+1) +φj∗+1

xn(tj+1)

–φ_j∗₊₁x(tj+1)

+

k–1

j=0

tjI αj

gj|fxn–fx|

(tj+1) +φj+1

xn(tj+1)

–φj+1

(11)

(12)

=Q1+|λ2|Q2+pψ1

In the next step, we will show thatAmaps bounded sets into equicontinuous sets of

PC(J,R). Letτ1,τ2∈Jk for some k∈ {0, 1, 2, . . . ,m}, τ1 <τ2. Then, for any x∈Bρ∗, we

As τ1→τ2, the right-hand side of this inequality tends to zero independently of

un-known variablex. Therefore by the Arzelá–Ascoli theorem it follows that the operator

(13)

+

k–1

j=0 |Gj+1|

_j

r=0

trI αr–1

gr |fx|

(tr+1) +φr∗+1

x(tr+1)

≤Q1+|λ2|Q2

+pψ1

xQ3+ψ2

xQ5+ψ3

xQ4

,

which can be expressed as

x

(Q1+|λ2|Q2) + (pψ1(x)Q3+ψ2(x)Q5+ψ3(x)Q4)≤

1.

In view of (H5), there existsMsuch thatx =M. Let

U=x∈PC(J,R) :x<M.

Note that the operatorA:U→PC(J,R) is continuous and completely continuous. From the choice ofUthere is nox∈∂Usuch thatx=λA(x) for someλ∈(0, 1). Consequently, by the nonlinear alternative of Leray–Schauder type (Lemma3.3) we deduce thatAhas a ﬁxed pointx∈U, which is a solution of problem (1.1). This completes the proof. In the particular casep(t) = 1 andψi(x) =Λix+Ni,i= 1, 2, 3, we have the following

corollary.

Corollary 3.5 Let f :J×R→Randφk,φ∗k:R→R,k= 1, 2, . . . ,m,be continuous func-tions.Assume that:

(A1) There exist constantsΛi,Ni> 0,i= 1, 2, 3,such that

f(t,x)≤Λ1x+N1 for allt∈J,x∈R,

φk(x)≤Λ2x+N2, φk∗≤Λ3x+N3 for allx∈R,k= 1, 2, . . . ,m.

If

Λ1Q3+Λ2Q5+Λ3Q4< 1,

then the impulsive fractional boundary value problem(1.1)has at least one solution on J. The result for nonimpulse eﬀects is given by the following:

Corollary 3.6 Assume that condition(H3)holds.In addition,we suppose that:

(B1) There exists a constantM> 0such that

M

Q∗₁+|λ2|Q∗2+pψ1(M)Q∗3

> 1.

Then boundary value problem(3.2)has at least one solution on J.

In addition,suppose that(A1)holds with m= 0 (i.e.,φk(·) =φk∗(·)≡0)and thatΛ1Q∗3< 1.

(14)

4 Examples

In this section, we present some examples to illustrate our results.

Example4.1 Consider the following impulsive boundary value problem containing Ca-puto fractional derivative with respect to another function of the form

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

tkD (2_kk₊₂+3)

(et₊_tk₎x(t) =f(t,x(t)), t=tk=k,k= 0, 1, 2, 3,

x(t+

k) –x(t–k) =φk(x(tk)), k= 1, 2, 3,

D₍_et₊_tk₎x(t+_k) –D₍_et₊_tk–1₎x(t–_k) =φ_k∗(x(tk)), k= 1, 2, 3,

x(0) +2₃(D(et₊₁₎x)(0) =1

2, x(4) + 3

5(D(et₊_t3₎x)(4) =5

7.

(4.1)

Hereαk= (2k+ 3)/(k+ 2),gk(t) =et+tk,k= 0, 1, 2, 3,m= 3,T = 4,β1= 2/3,β2= 3/5, λ1= 1/2, andλ2= 5/7. Then we can compute thatΩ≈96.53148336,Q1≈1.003798415,

Q2≈1.007596831,Q3≈5114.088162,Q4≈514.9585869, andQ5≈6.022790493.

(i) Letf: [0, 4]×R→R,φk:R→R, andφk∗:R→Rbe the functions deﬁned by

f(t,x) = cos

2_π_t

2(t+ 100)2

x2_{+ 2}_|_x_| |x|+ 1

+3 2,

φk(x) =

1

(k+ 20)sin|x|+ 1

3, k= 1, 2, 3,

φ∗_k(x) = 1 (k+ 60)2tan

–1_|_x_|₊2

5, k= 1, 2, 3.

(4.2)

It follows that|f(t,x) –f(t,y)| ≤(1/100)2_|_x_–_y_|_,_|_φ

k(x) –φk(y)| ≤(1/21)|x–y|, and|φk∗(x) –

φ_k∗(y)| ≤(1/61)2_|_x_–_y_|_with_L

1= (1/100)2,L2= 1/21, andL3= (1/61)2. Therefore we obtain

L1Q3+L2Q5+L3Q4≈0.9366008889 < 1. Hence by Theorem3.1we get that problem (4.1)

withf(t,x),φk(x), andφk∗(x) deﬁned in (4.2) has a unique solution on [0, 4].

(ii) Now we consider the functionsf : [0, 4]×R→R,φk:R→R, andφk∗ :R→R

deﬁned by

f(t,x) = e

–t

5(t+ 40)

1 900x

2₊ 1

150

,

φk(x) =

1 800x

2₊ 1

k+ 19, k= 1, 2, 3,

φ∗_k(x) = 1 9000x

2₊ 1

(39 +k)2, k= 1, 2, 3.

(4.3)

Setting p(t) = 1/(5(t+ 40)),ψ1(x) = (1/900)x2+ (1/150),ψ2(x) = (1/800)x2+ (1/20), and ψ3(x) = (1/9000)x2_{+ (1/1600), conditions (}_H

3)–(H4) are fulﬁlled. Fromp= 1/200 by

di-rect computing we ﬁnd that there exists a constantM∈(4.029734051, 6.704750535) sat-isfying condition (H5) in Theorem3.4. By Theorem3.4problem (4.1) with all functions

(15)

(iii) Let us consider the functionsf : [0, 4]×R→R,φk:R→R, andφk∗:R→Rdeﬁned

by

f(t,x) =

1 (130 +t)2

x8_e–x2

|x|7_{+ 1}+tan

–1_|_x_|_{+ 2,}

φk(x) =

1 19 +k

x10_cos2_|_x_| |x|9_{+ 3} +

1 2sin

4_|_x_|_{+ 1,} _k_{= 1, 2, 3,}

φ∗_k(x) =

1 (39 +k)2

|x|5_sin2_|_x_|

x4_{+ 5} +

1 4e

–|x| _k_{= 1, 2, 3.}

(4.4)

It is obvious that |f(t,x)| ≤(1/16900)|x|+ ((π/2) + 2), |φk(x)| ≤(1/20)|x|+ (3/2), and

|φ_k∗(x)| ≤(1/1600)|x|+ (1/4), so that conditionA1is satisﬁed. Further, by directly

com-putation we haveΛ1Q3+Λ2Q5+Λ3Q4≈0.9255974084 < 1. Then from Corollary3.5it

follows that problem (4.1) with nonlinear functions in (4.4) has at least one solution in [0, 4].

5 Discussion

Problem (1.1) can be extended to impulsive fractional diﬀerential equations with delays by replacing the functionf(t,x(t)) withf(t,x(t),x(θ(t))), whereθ :J→Jandθ(t)≤t. Next, we give the existence theorems without proofs for impulsive delay fractional diﬀerential equations with boundary conditions of the form

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

tkD αk

gkx(t) =f(t,x(t),x(θ(t))), t=tk,k= 0, 1, 2, . . . ,m,

x(t+_k) –x(t–_k) =φk(x(tk)), k= 1, 2, . . . ,m,

Dgkx(t +

k) –Dgk–1x(t

–

k) =φ∗k(x(tk)), k= 1, 2, . . . ,m, x(0) +β1Dg0x(0) =λ1, x(T) +β2Dgmx(T) =λ2.

(5.1)

Theorem 5.1 Let gk∈C2([0,T])with g_k(t) > 0for t∈[0,T],k= 0, 1, . . . ,m.Suppose that the function f :J×R2_→_R_satisﬁes

(H6) |f(t,x,u) –f(t,y,v)| ≤L1|x–y|+L4|u–v|,L1,L4> 0,∀t∈J,x,y,u,v∈R.

In addition,assume that the functionsφk:R→Randφk∗:R→R,k= 1, 2, . . . ,m,satisfy

(H2).

If

(L1+L4)Q3+L2Q5+L3Q4< 1, (5.2)

then the boundary value problem of impulsive delay fractional diﬀerential equations with respect to other functions(5.1)has a unique solution on J.

Theorem 5.2 Assume that the functionsφkandφ∗ksatisfy(H4)for k= 1, 2, . . . ,m and

sup-pose that

(H7) There exist two continuous nondecreasing functions ψ1,ψ4: [0,∞)→(0,∞)and

two continuous functionsp1,p2:J→R+such that

f(t,x,y)≤p1(t)ψ1

|x|+p2(t)ψ4

(16)

(H8) There exists a constantN> 0such that

N

(Q1+|λ2|Q2) + [{p1ψ1(N) +p2ψ4(N)}Q3+ψ2(N)Q5+ψ3(N)Q4]

> 1.

Then the impulsive fractional boundary value problem with delay(5.1)has at least one solution on J.

6 Conclusion

In this paper, we initiate the study of existence and uniqueness of solutions for a new class of impulsive fractional boundary value problems with separated boundary conditions con-taining the Caputo fractional derivative of a function with respect to another function. The existence of solutions is established by using the Leray–Schauder nonlinear alterna-tive, whereas the uniqueness result is proved via Banach’s contraction mapping principle. Examples are also constructed to illustrate the main results. The considered problem gen-erates many types and also mixed types of impulsive fractional diﬀerential equations with boundary conditions.

Acknowledgements Not applicable.

Funding

This research was funded by King Mongkut’s University of Technology North Bangkok. Contract no. KMUTNB-60-ART-073.

Availability of data and materials

Data sharing not applicable to this paper as no data sets were generated or analyzed during the current study.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to this work. All authors read and approved the ﬁnal manuscript.

Author details

1_{Intelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied}

Science, King Mongkut’s University of Technology North Bangkok, Bangkok, Thailand.2_{Department of Mathematics}

Statistic and Computer Science, Faculty of Liberal Arts and Science, Kasetsart University, Nakhonpathom, Thailand.

3_{Department of Mathematics, University of Ioannina, Ioannina, Greece.}4_{Nonlinear Analysis and Applied Mathematics}

(NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia.

Publisher’s Note

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

Received: 14 August 2019 Accepted: 13 November 2019 References

1. Benchohra, M., Henderson, J., Ntouyas, S.K.: Impulsive Diﬀerential Equations and Inclusions, vol. 2. Hindawi Publishing Corporation, New York (2006)

2. Bohner, M., Heidarkhani, S., Salari, A., Caristi, G.: Existence of three solutions for impulsive multi-point boundary value problems. Opusc. Math.37, 353–379 (2017)

3. Shen, J., Wang, W.: Impulsive boundary value problems with nonlinear boundary conditions. Nonlinear Anal. TMA69, 4055–4062 (2008)

4. Wang, J., Zhou, Y., Feckan, M.: On recent developments in the theory of boundary value problems for impulsive fractional diﬀerential equations. Comput. Math. Appl.64, 3008–3020 (2012)

5. Liu, Y.: Solvability of impulsive periodic boundary value problems for higher order fractional diﬀerential equations. Arab. J. Math.6, 195–214 (2016)

6. Thaiprayoon, Ch., Tariboon, J., Ntouyas, S.K.: Impulsive fractional boundary-value problems with fractional integral jump conditions. Bound. Value Probl.2014, Article ID 17 (2014)

7. Asma, Ali, A., Shah, K., Jarad, F.: Ulam–Hyers stability analysis to a class of nonlinear implicit impulsive fractional diﬀerential equations with three point boundary conditions. Adv. Diﬀer. Equ.2019, Article ID 7 (2019)

(17)

9. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Diﬀerential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)

10. Almeida, R.: A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul.44, 460–481 (2017)

11. Almeida, R., Malinowska, A., Monteiro, T.: Fractional diﬀerential equations with a Caputo derivative with respect to a kernel function and their applications. Math. Methods Appl. Sci.41, 336–352 (2018)

12. Ameen, R., Jarad, F., Abdeljawad, T.: Ulam stability for delay fractional diﬀerential equations with a generalized Caputo derivative. Filomat32, 5265–5274 (2018)

13. Samet, B., Aydi, H.: Lyapunov-type inequalities for an anti-periodic fractional boundary value problem involving ψ-Caputo fractional derivative. J. Inequal. Appl.2018, Article ID 286 (2018)

14. Katugampola, U.: A new approach to generalized fractional derivatives. Bull. Math. Anal. Appl.6, 1–15 (2014) 15. Wang, G.: Explicit iteration and unbounded solutions for fractional integral boundary value problem on an inﬁnite

interval. Appl. Math. Lett.47, 1–7 (2015)

16. Wang, G., Pei, K., Agarwal, R.P., Zhang, L., Ahmad, B.: Nonlocal Hadamard fractional boundary value problem with Hadamard integral and discrete boundary conditions on a half-line. J. Comput. Appl. Math.343, 230–239 (2018) 17. Wang, G., Pei, K., Chen, Y.: Stability analysis of nonlinear Hadamard fractional diﬀerential system. J. Franklin Inst.356,

6538–6546 (2019)

18. Tariboon, J., Ntouyas, S.K.: Initial and boundary value problems via exponential fractional calculus. Int. Electron. J. Pure Appl. Math.12(1), 1–19 (2018)

19. Ntouyas, S.K., Tariboon, J., Sawaddee, C.: Nonlocal initial and boundary value problems via fractional calculus with exponential singular kernel. J. Nonlinear Sci. Appl.11, 1015–1030 (2018)