/compile/output.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 7, No. 2, 2014, 115-128

ISSN 1307-5543 – www.ejpam.com

On Hybrid Caputo Fractional Differential Equations with

Variable Moments of Impulse

J. Vasundhara Devi, N.Giribabu∗

Department of Mathematics, GVP-Prof.V.Lakshmikantham Institute for Advanced Studies, GVP Col-

lege of Engineering, Visakhapatnam, AP, India.

Abstract. In this paper existence and continuation results for hybrid Caputo fractional differential

equations of order q ∈ (0,1) with variable moments of impulse are established under the weakened

hypothesis of Cq continuity.

2010 Mathematics Subject Classifications: 34A12,34A08,34A37, 34K07

Key Words and Phrases: Hybrid Caputo fractional differential equations, existence, continuation

1. Introduction

The concept of a fractional derivative, as is well known, has its inception in a question

posed during a communication between Leibnitz and L’Hospital. The five century old question

has become a major area of research both in the realm of applications and in the theoretical set

up. The potential it offers in both these branches has attracted the attention of both theoretical

and applied scientists as well as engineers and other technologists. The major contributions

in this field are given in [6, 8, 9, 11–14] and the references therein. In [6] Diethelm gave a

simple example which naturally introduces the fractional derivative. We briefly introduce it

here, so as to connect it later, to the problem considered in this paper.

Consider the stress σ(t) and strain ε(t) of a viscous liquid. It is known that the Newton’s

law

σ(t) = ηD1ε(t), (1)

describes the relation between stress and strain for a viscous liquid, where η is the viscosity of

the material.

The Hooke’s law states the stress-strain relationship for elastic solid and is given by

σ(t) = E D0ε(t), (2)

∗Corresponding author.

Email addresses: jvdevi@gmail.com (J. Devi), giri.hcu.gvpcoe@gmail.com (N. Giribabu)

http://www.ejpam.com 115 c© 2014 EJPAM All rights reserved.



J. Devi, N. Giribabu / Eur. J. Pure Appl. Math, 7 (2014), 115-128 116

where E is the modulus of elasticity of the material. Now it can be naturally concluded that

the behaviour of viscoelastic material must have a behaviour that is modeled by an equation

having derivative of order k ∈ (0,1) that lies in betweeen the equations of (1) and (2) and is

given by

σ(t) = νDkε(t), (3)

where ν is the constant of the material and k ∈ (0,1). The relation (3) in a slightly different

set up is called as Nutting’s law. It has been observed in [9] that viscoelastic materials like

polymers, some biological tissue etc. may follow the relation (3). The operator Dk is called as

the fractional derivative and is described in Section 2.

It has been observed that the theory of ordinary differential equations is being systemati-

cally extended to the set up of fractional differential equations. Further an effort is being put

to obtain better results by using fractional derivatives in place of ordinary derivatives, where

there is nonlocality or memory involved.

It is known that many evolutionary processes experience a change of state abruptly. These

undergo short term perturbations, where the time span is negligible with respect to the dura-

tion of the process. Thus, it is natural to assume that the perturbations act instantaneously and

hence can be modelled as impulses. These perturbations or impulses can be considered as of

two types. The moments of impulse can be decided in advance or can depend on the solution

of the model described by the physical phenomenon. Some examples that can be modelled in

this set up are biological phenomena involving thresholds, bursting rhythmic models in biol-

ogy and medicine, optimal control in economics, to name a few.

In some of the afore mentioned models, the occurrence of a change of state or perturbation

depends on the solution. Also, the constraints of any physical model can be considered as a

barrier or a surface. If a solution of the model encounters the surface, it must be given an

impulse or a perturbation to avoid it or to get out of it. Thus a mathematical model involving

differential equation with variable moments of impulse is a system worth studying. This model

exhibits many interesting phenomena which are discussed in [9].

The momentum that the research on fractional differential equation is gaining had prompted

us to take up the study of fractional differential equations with variable moments of impulse,

in this paper. We proceeded along the lines of the theory established in [9] and have con-

structed examples in the present set up. The fact that there are many physical phenomena

that can be modeled using fractional derivatives had encouraged us to obtain conditions using

fractional derivatives. The remaining part of the paper is organized as follows. In Section 2,

we deal with the preliminaries of fractional differential equations. Adapting the description of

solution of an evolutionary process in [9]we described the solution of an impulsive or a hybrid

fractional differential equation in Section 3. An example illustrating the proposed system is

given in Section 4. In Section 5 we deal with existence and continuation of solutions. Section

6 concludes the work done in the paper.



J. Devi, N. Giribabu / Eur. J. Pure Appl. Math, 7 (2014), 115-128 117

2. Preliminaries

In this section, we introduce notations, definitions, results and preliminary facts from [10],

[4] that are required in the remainder of this paper.

Definition 1. The Riemann-Liouville fractional integral of order q, where q is a positive real

number, of a function x given on the interval [t0, T], t0 ≥ 0 is defined as

D−q x(t) =
1

Γ(q)

t
∫

t0

(t − s)q−1 x(s)ds, t0 ≤ t ≤ T,

where Γ is the Gamma function.

Definition 2. The Riemann-Liouville fractional derivative of order q, where q is a positive real

number, of a function x given on the interval [t0, T], t0 ≥ 0 is defined as

Dq x(t) =
1

Γ(p)

dm

d tm









t
∫

t0

(t − s)p−1 x(s)ds









, t0 ≤ t ≤ T,

where m− p = q and m is the least positive integer greater than q so that 0< p ≤ 1.

Definition 3. The Caputo’s fractional derivative of order q, where q is a positive real number, of

a function x given on the interval [t0, T], t0 ≥ 0 is defined as

c Dq x(t) =
1

Γ(n− q)

t
∫

t0

(t − s)n−q−1 x (n)(s)ds, t0 ≤ t ≤ T,

where n is a positive integer such that n− 1< q < n.

In particular, the Caputo’s fractional derivative of order q, where 0< q < 1 is defined as

c Dq x(t) =
1

Γ(1− q)

t
∫

t0

(t − s)−q x ′(s)ds, t0 ≤ t ≤ T.

Definition 4. A function u is said to be Cp continuous i.e., u ∈ Cp

�

[t0, t0 + a],R
�

if and only if

u ∈ C((t0, t0 + a],R) and (t − t0)
pu(t) ∈ C([t0, t0 + a],R) with p+ q = 1, 0< q < 1.

Definition 5. A function u is said to be Cq continuous i.e., u ∈ Cq
�

[t0, T],R
�

if and only if the

Caputo derivative c Dqu(t) exists and satisfies

c Dqu(t) =
1

Γ(1− q)

t
∫

t0

(t − s)−qu′(s)ds, t0 ≤ t ≤ T.



J. Devi, N. Giribabu / Eur. J. Pure Appl. Math, 7 (2014), 115-128 118

We observe that u ∈ Cq
�

[t0, t0 + a],R
�

, implies that u is continuous and differentiable.

Next we state the following result from [10], which is used in Section 5.

Lemma 1. x(t) ∈ Cq([t0, t0 + a],R) is solution of the initial value problem

c Dq x = f (t, x), x(t0) = x0, 0< q < 1

if and only if it satisfies corresponding Volterra fractional integral equation

x(t) = x0 +
1

Γ(q)

t
∫

t0

(t − s)q−1 f (s, x(s))ds, t0 ≤ t ≤ t0 + a.

Finally, we state the following Lemma [1], which is used in Section 5 and Section 6 to

prove our main results.

Lemma 2. Let 0< q < 1. Consider the Caputo fractional differential equation

c D
q
t0

u(t) = g(t,u(t)), t ≥ t0,

where g(t,u) ≥ 0 and t0 ∈ R. If the solutions exist and u(t0) ≥ 0, then they are nonnegative.

Furthermore, if g(t,u) = λu for λ ≥ 0, then the solutions are nondecreasing in t.

3. Hybrid Caputo Fractional Differential Equations

As observed in the introduction it is natural to expect viscoelastic materials to be modeled

after fractional differential equations. Let the natural constraints that arise in the model be

construed as a barrier or a threshold. When the solution of the model, come into contact

with the barrier, some additional input or impulse must be given for the solution to get out

of the barrier. This general behaviour can be understood through an evolutionary process

of a physical phenomenon given below. We adapt the process given in [9] to the set up of

fractional differential equations. Consider an evolutionary process or a physical phenomenon

that exhibits behaviour which can be described by

(i) a Caputo fractional differential equation of order q ∈ (0,1)

c Dq x = f (t, x), (4)

where f : R+×Ω→ R,Ω ⊂ R is an open set, and R is the space of real numbers and R+
is the nonnegative real line;

(ii) the sets M(t), N(t) ⊂ Ω for each t ∈ R+; and

(iii) the operator A(t) : M(t)→ N(t) for each t ∈ R+.



J. Devi, N. Giribabu / Eur. J. Pure Appl. Math, 7 (2014), 115-128 119

Let x(t) = x(t, t0, x0) be any solution of (4) starting at (t0, x0). The behaviour of the

evolutionary process or the physical phenomenon can be described by the point Pt = (t, x(t))

starting at Pt0
= (t0, x0), it moves along the curve

{(t, x) : t ≥ t0, x = x(t)} until the point Pt meets the set M(t) at t = t1. Then at the

point t = t1, the operator A(t) transfers the point Pt1
= (t1, x(t1)) ∈ M(t1) to the position

Pt1
+ = (t1, x+

1
) ∈ N(t1) where x+

1
= A(t1)x(t1). Now the motion of the point Pt continues

forward from Pt+
1

along the the curve {(t, x) : t ≥ t1, x = x(t, t1, x+
1
)} as the solution of (4)

with starting point (t1, x+
1
) until it again meets the set M(t) at t = t2. This again yields,

by the effect of operator A(t), that the point Pt2
= (t2, x(t2)) is transferred to the position

Pt2
+ = (t2, x+

2
) where x+

2
= A(t2)x(t2) ∈ N(t2). From then on, the motion of the evolutionary

process moves the point Pt forward following the equation c Dq x = f (t, x) till it touches M(t)

at t = t3. This process can be continued forward as long as the solution of (4) exists.

The set up given by (i), (ii), (iii) characterizes the considered evolutionary process as a

hybrid Caputo fractional differential system. The motion of the point Pt is a curve represents

the solution curve of the hybrid Caputo fractional differential equation (HCFDE for short).

Thus a solution of a hybrid Caputo fractional differential equation may exhibit a variety of

behaviour as given below.

(i) The solution may be a continuous function, if the integral curve does not intersect M(t)

or hits it at the fixed points of the operator A(t).

(ii) The solution can be a piecewise continuous function having a finite number of disconti-

nuities of first kind which are are not fixed points of the operator A(t).

(iii) The solution may also have a countable number of discontinuities of first kind. Then

the solution is a piecewise continuous function with a countable number of discontinu-

ities. Clearly all these points are not fixed points of the operator A(t). The question

of uncountable number of discontinuities does not arise as we have assumed x(t) is a

solution of (4).

The moments of time at which the integral curve hits the set M(t) are called as the moments

of the impulse and are denoted by t = tk. Following the standard notation in impulsive

differential equations [9], we assume that the solution of the HCFDE is left continuous at

t = tk, k = 1,2,3, . . . that is x(t−
k
) = limh→0+ x(tk − h) = x(tk).

The generality in the above set up, gives rise to two types of HCFDE known as HCFDE with

fixed moments of impulse and the other by HCFDE with variable moments of impulse. The

former is quite popular and runs parallel to ordinary differential equations with fixed moments

of impulse. There are many papers on fractional differential equations with fixed moments of

impulse, of which some are [2, 3, 5, 7].

The latter gives rise to very interesting phenomenon and exhibits complex behaviour. We

concentrate on the latter types of equations in this paper.



J. Devi, N. Giribabu / Eur. J. Pure Appl. Math, 7 (2014), 115-128 120

4. HCFDE with Variable Moments of Impulse

Consider a sequence of surfaces {Sk} given by Sk : t = τk(x), k = 1,2,3, . . ., τk : R→ R
such that τk(x)< τk+1(x) and limk→∞ τk(x) =∞. Then the HCFDE with variable moments

of impulse is given by

c Dq x = f (t, x), t 6= τk(x),

x(t+) =x(t) + Ik(x(t)), t = τk(x),
(5)

where f : R+ ×Ω→ R,Ω ⊂ R is an open set, τk ∈ C[Ω, (0,∞)], k = 1,2,3, . . . ,

Ik(x(t)) = ∆(x(t)) = x(t+)− x(t−), and Ik ∈ C[Ω,R].

In this case, the moments of the impulsive effect for the system (5) depend on the solutions

satisfying tk = τk(x(tk)), for each k. Thus, the solutions starting at different points will have

different points of discontinuity. Also a solution may hit the same surface t = τk(x) several

times and we shall call such a behaviour as “pulse phenomena”. In addition, different solutions

may coincide after some time and behaves as a single solution there after. This phenomena is

called “confluence”. In the following example, the different solutions that arise in this context

are described and the graphs are drawn.

Consider the hybrid Caputo fractional differential equation with variable moments of im-

pulse

c Dq x =0, t 6= τk(x), t ≥ 0,

x(t+) =x2 sgn x , t = τk(x), k = 1,2,3, . . .
(6)

where τk(x) = x2 + 20(k − 1) for |x | < 6 describe the surfaces Sk : t = τk(x). Here

Ik(x) = ∆(x) = x2 sgn x − x .

If c Dq x = 0 then x(t) = x0, a constant.

Case (1) The solutions x(t) with initial condition x(0) = x0, |x0| ≥ 6 are free from impulsive

effect since they do not intersect the surfaces Sk for any k. For example, consider the so-

lution x(t) of (6) starting at the point (0,8.5). It does not hit any surface, see Figure 1a.

Case (2) The solutions starting at the points (0, x0) where 1< x0 < 6 or −6< x0 < −1, undergo

impulsive effect a finite number of times. For example, consider the solution x(t) with

initial condition x(0) =
p

2. The point pt = (t, x(t)) starts its motion from

(t0, x0) = (0,
p

2) and moves along the curve

{(t, x) : t ≥ 0, x = x(t)}= {(t, x)t ≥ 0, x =
p

2}
until the time t1 = 2 > t0 = 0 at which the point Pt meets the surface S1 : t = x2. The

point pt1
= (t1, x1) = (2,

p
2) which lies on the surface S1 is transferred to the point

Pt+
1
= (t1, x+

1
) = (2,2) where x+

1
= x2

1 sgn x1 = 2. Then the point Pt continues to move

further along the curve with x(t) = x(t; t1, x+
1
) as the solution of (6) starting at (t1, x+

1
)

until it hits the same surface S1 at the next moment t2 = 4 > t1 = 2. Then once again



J. Devi, N. Giribabu / Eur. J. Pure Appl. Math, 7 (2014), 115-128 121

the point Pt2
= (t2, x2) = (4,2) is transferred to the point Pt+

2
= (t2, x+

2
) = (4,4) where

x+
2
= x2

2 sgn x2. As before the point Pt continuous to move forward with

x(t) = x(t, t2, x+
2
) = x(t, 4, 4) as the solution of (6) starting at (t2, x+

2
) = (4,4) until it

hits the same surface S1 at the moment t3 = 16> t2 = 4. Then the point

(t3, x3) = (16,4) is transferred to the point Pt+
3
= (t3, x+

3
) = (16,16), where

x+
3
= x2

3 sgn x3 and it does not encounter any surface, beyond time t3 = 16. In this case

the solution x(t) is a piecewise continuous function having finite number of disconti-

nuities of the first kind since the integral curve meets the surfaces at a finite number of

times which are not the fixed points of the operator A(t) given by A(t)x = x2 sgn x . In

this case the solution hit the same surface S1 three times exhibiting pulse phenomenon,

see Figure 1b.

Case (3) The solutions x(t) starting at the points (0, x0), 0 < x0 < 1, meets the surfaces Sk at an

infinite number of times tk and we have tk→∞ as k→∞ as well as limk→∞ x(tk) =

0. For example, let us take x0 = 0.9. The solution x(t) begins its motion at (0,0.9)

and continuous to move along the curve x = 0.9 until it hits the surface S1 : t = x2 at

(0.81,0.9). This point (0.81,0.9) is transferred to (0.81,0.81). Then the solution starts

at (0.81,0.81) and continuous to move along the line x = 0.81 until it hits the surface

S2 : t = x2 + 20 at (20.6561,0.81). It is then transferred to (20.6561,0.6561). The

solution then starts at (20.6561,0.6561) and continuous to move until it hits the surface

S3 : t = x2+40 at (40.43047,0.6561) and so on. As k→∞we have limk→∞ x(tk) = 0.

In this case the solution undergo an impulsive effect an infinite number of times, see

Figure 1c.

Case (4) The solutions starting at (0,0), (0,1) and (0,−1) hit the surface Sk at times tk,

(k = 0,1,2,3, . . . )which are the fixed points of the operator A(t) x = x2 sgn x , and there

is no impulsive effect, see Figure 1d.

Case (5) The solutions starting at
�

0,2
1
8

�

,
�

0,2
1
4

�

,
�

0,2
1
2

�

unite for t ≥ p2 and thus exhibit the

phenomenon of confluence, see Figure 1e.



J. Devi, N. Giribabu / Eur. J. Pure Appl. Math, 7 (2014), 115-128 122

0 50 100 150
−10

−8

−6

−4

−2

0

2

4

6

8

10

t−axis

x−
ax

is

S
1

S
2

S
3

S
4

S
5

x=x(t,0,8.5)

(a) Case (1)

0 50 100 150
−10

−8

−6

−4

−2

0

2

4

6

8

10

t−axis

x−
ax

is

S
1

S
2

S
3

S
4

S
5

(b) Case (2)

0 50 100 150
−10

−8

−6

−4

−2

0

2

4

6

8

10

t−axis

x−
ax

is

S
1

S
2

S
3

S
4

S
5

(c) Case (3)

0 50 100 150
−10

−8

−6

−4

−2

0

2

4

6

8

10

t−axis

x−
ax

is

S
1

S
2

S
3

S
4

S
5

(d) Case (4)

0 50 100 150
−10

−8

−6

−4

−2

0

2

4

6

8

10

t−axis

x−
ax

is

S
1

S
2

S
3

S
4

S
5

(e) Case (5)

Figure 1: Solutions to (6)



J. Devi, N. Giribabu / Eur. J. Pure Appl. Math, 7 (2014), 115-128 123

5. Existence and Continuation of Solutions

The example in Section 3, points to the rich potential that Caputo fractional differential

equation with variable moments of impulse offers. The first question that one comes across

is to discuss the meaning of a solution of this equation and obtain an existence result. In this

section we answer this question. We define the solution of Caputo fractional differential equa-

tion with variable moments of impulse and proceed to prove an existence result.

Consider an open set Ω ⊂ R and set D = R+ ×Ω. Suppose that for each

k = 1,2,3, . . . , τk ∈ C[Ω, (0,∞)], τk(x) < τk+1(x) and limk→∞ τk(x) = ∞ for x ∈ Ω.

Suppose that k varies from 1 to∞. Also assume that Sk : t = τk(x) are the surfaces.

Consider the initial value problem (IVP) for the hybrid Caputo fractional differential equation

with variable moments of impulse

c Dq x = f (t, x), t 6= τk(x)

x(t+) = x(t) + Ik(x(t)), t = τk(x)

x(t+
0
) = x0, t0 ≥ 0







(7)

where f : D→ R and Ik : Ω→ R. A function x : [t0, t0 + a)→ R, t0 ≥ 0, a > 0 is said to be a

solution of (7) if

(i) x(t+
0
) = x0 and (t, x(t)) ∈ D for t ∈ [t0, t0 + a),

(ii) x(t) ∈ Cq([t0, t0 + a),R), c Dq x(t) is continuous, and x(t) satisfies c Dq x = f (t, x) for

t ∈ [t0, t0 + a) and t 6= τk(x(t)),

(iii) if t ∈ [t0, t0 + a) and t = τk(x(t)), then x(t+) = x(t) + Ik(x(t)), and at such t ’s we

always assume that x(t) is left continuous and s 6= τ j(x(s)) for any j, t < s < t + δ, for

some δ > 0.

Whenever t0 6= τk(x0) for any k, we mean the initial condition x(t+
0
) = x0 in the usual

sense, that is, x(t0) = x0. If t0 = τk(x0) for some k then x(t+
0
) = x0, which, in general,

is natural for the system (7), since (t0, x0) may be such that t0 = τk(x0) Unlike ordinary

fractional differential equations, the system (7) may not possess any solution at all, even if,

f is continuous (or continuously differentiable) since the only solution x(t) of the problem
c Dq x = f (t, x), x(t0) = x0, may totally lie on a surface and hence by the definition, we

conclude that the Caputo fractional differential equation with variable moments of impulse

does not have any solution for all t ∈ [t0, T].

Example 1. Consider the following IVP of Caputo fractional differential equation with q = 1
2

c D
1
2 x =1, t 6= τk(x),

∆x =Ik(x) =
π

4
(x − 1)2 + 1− x , t = τk(x)

x(1+) =1 where Sk : τk(x) =
π

4
(x − 1)2 + k, k = 1,2, . . .

(8)



J. Devi, N. Giribabu / Eur. J. Pure Appl. Math, 7 (2014), 115-128 124

There is no solution to the above system 8 passing though (1,1), since

c D
1
2 x = 1⇔ x(t) = x0 +

1
Γ(q)

t
∫

t0

f (s,x(s))

(t−s)1−q ds. Now

x(t) =x0 +
1

Γ(q)

t
∫

t0

f (s, x(s))

(t − s)1−q
ds

=1+
1

Γ
�

1
2

�

t
∫

1

1

(t − s)1−
�

1
2

� ds

=1+
1p
π

t
∫

1

(t − s)−
�

1
2

�

ds

=1+
1p
π

�

(−1)
(t − s)

1
2

�
1
2

�

�s=t

s=1

=1+
1p
π

�

0+
(t − 1)

1
2

�
1
2

�

�

=1+
2p
π

p
t − 1

⇒ x(t) =1+
2p
π

p
t − 1,

which lies entirely on the surface S1.

The above example clearly shows that we need to obtain some conditions that will guar-

antee that the solution will exist after it hits a surface or a barrier. As the solution satisfies the

Caputo fractional differential equation, it is natural that the conditions must be in terms of the

fractional derivatives. This is done in the following theorem, and the criteria obtained are not

only new but are very interesting. Hence we need some extra conditions on τk and f , τk or

f besides continuity in order to establish any general existence theory for the system (7). We

now proceed to state and prove a result on existence of a solution for the considered IVP. The

proof of the theorem is analogous to the proof of the corresponding Theorem 1.2.1 in [9].

Theorem 1. Assume that

(i) f : D→ R is continuous at t 6= τk(x), k = 1,2, . . . ,.

(ii) for each (t, x) ∈ D there exists a function ℓ ∈ L1
loc

such that | f (s, y) |≤ ℓ(s) in a neigh-

bourhood of (t,x).

(iii) t1 = τk(x1) for any k ≥ 1 implies that there exists δ > 0 such that t 6= τk(x) for any (t, x)

with 0< t − t1 < δ and | x − x1 | < δ.



J. Devi, N. Giribabu / Eur. J. Pure Appl. Math, 7 (2014), 115-128 125

Then for each (t0, x0) ∈ D, there exists a solution x : [t0, t0+α)→ R of the initial value problem

(7) for some α > 0.

Proof. If t0 6= τk(x0) for all k ≥ 1 then there exists δ1 > 0 such that s 6= τi(x(s)) for all

i ≥ 1, t0 < s < t0 + δ1. The continuity of f imply the existence of a local solution x(t) of
c Dq x = f (t, x) and x(t0) = x0. Hence x(t) is a local solution of the system (7).

If t0 = τk(x0) for some k ≥ 1, then x(t+
0
) = x(t0)+ Ik(x(t0)). The continuity of f and the

condition (ii) imply the existence of a local solution x(t) of c Dq x = f (t, x) and

x(t+
0
) = x(t0) + Ik(x(t0)). Since τi(x)< τ j(x) for i < j and, t0 = τk(x0) we have

t 6= τ j(x(t)) for j 6= k and t sufficiently close to t0. Since t0 = τk(x0), by condition (iii) there

exists δ > 0 such that t 6= τk(x) for all (t, x) with 0< t− t0 < δ and | x− x0 | < δ. Therefore,

s 6= τi(x(s)) for all i, t0 < s < t0 + δ, for some δ > 0. Hence x(t) is a local solution of the

system (7).

Remark 1. The condition (iii) in Theorem 1 is possible for only irregular functions τk(x) since

the theory of implicit functions implies that if τk is differentiable at x0 and τ′
k
(x0) 6= 0, then the

condition (iii) can never hold.

We now proceed to state and prove a result on existence of a solution for the considered

IVP with some regularity conditions on τk(x). The condition on τk(x) is given in terms of

fractional derivative. This is natural as the solution follows the fractional differential equation

model. This result is new and has been developed for hybrid Caputo fractional differential

equation with variable moments of impulse.

Theorem 2. Assume that

(i) f : D→ R is continuous.

(ii) c Dqτk(x) exists, τk : Ω→ (0,∞) are differentiable and linear surfaces for all k ≥ 1.

(iii) if t1 = τk(x1) for some(t1, x1) ∈ D and k ≥ 1, then there existsδ > 0 such that
∂ τk(x)

∂ x . f (t, x) 6=
(t−t1)

(1−q)

Γ(2−q)
for (t, x) ∈ D with | x − x1 |< δ and 0< t − t1 < δ.

Then for each (t0, x0) ∈ D, there exists a solution x : [t0, t0+α)→ R of the system (7) for some

α > 0.

Proof. If t0 6= τk(x0) for all k ≥ 1 then there exists δ1 > 0 such that s 6= τi(x(s)) for

all i, t0 < s < t0 + δ1. The continuity of f imply the existence of a local solution x(t) of
c Dq x = f (t, x) and x(t0) = x0. Hence x(t) is a local solution of the system (7).

If t0 = τk(x0) for some k ≥ 1, then x(t+
0
) = x(t0) + Ik(x(t0)). The continuity of f imply

the existence of a local solution x(t) of C Dq x = f (t, x) and x(t+o ) = x(t0) + Ik(x(t0)). Set

σ(t) = t − τk(x(t), then σ(t0) = 0, since t0 = τk(x0). Then by using the fact that τk(x) are

linear surfaces, and the hypothesis in a small right neighborhood of t0 we obtain

c Dqσ(t) =c Dq[t −τk(x(t)]



J. Devi, N. Giribabu / Eur. J. Pure Appl. Math, 7 (2014), 115-128 126

=c Dq(t)− c Dq[τk(x(t))]

=
1

Γ(1− q)

∫ t

t0

(t − s)−qds− 1

Γ(1− q)

∫ t

t0

(t − s)−q d

ds
[τk(x(s)]ds

=
1

Γ(1− q)

�

−(t − s)1−q

1− q

�t

t0

− 1

Γ(1− q)

∫ t

t0

(t − s)−q ∂

∂ x
[τk(x)] x ′(s)ds

=
1

Γ(1− q)

(t − t0)
1−q

1− q
− ∂
∂ x
[τk(x)]

c Dq x

=
(t − t0)

1−q

Γ(2− q)
− ∂
∂ x
[τk(x)] f (t, x) 6= 0

Since c Dqσ 6= 0 in a small right neighborhood of t0 and σ(t0) = 0, we have by Lemma 2, σ(t)

is either strictly increasing or decreasing in that neighbourhood and therefore, t 6= τk(x(t))

for 0< t − t0 < δ2 for some δ2 > 0.

Since τi(x) < τ j(x) for i < j and t0 = τk(x0) we have t 6= τ j(x(t)) for j 6= k and t

sufficiently close to t0. Therefore, s 6= τi(x(s)) for any i, t0 < s < t0 + δ, for some δ > 0.

Hence x(t) is a local solution of the system (7).

Regarding the initial value problem (7) we have the following two cases:

(i) If t0 6= τk(x0), for all k ≥ 1, then a solution of (7) is understood in the classical sense;

(ii) If t0 = τk(x0), for some k ≥ 1, then a solution of (7) is understood in some extended

sense depending on the smoothness of f .

A solution x(t, t0, x0) of (7) existing on some interval [t0, t0 + a) and experiencing impulses

at the points {t i}, t0 < t i < t0 + a, t i < t j for i < j, is described as follows:

x(t, t0, x0) =



























x(t, t0, x0), t0 ≤ t ≤ t1,

x(t, t1, x+
1
), t1 < t ≤ t2,

. . . . . .

. . . . . .

x(t, t i , x+
i
), t i < t ≤ t i+1,

. . . . . .

. . . . . .

.

where x+
i
= x i + Ik(x i) and x i = x(t i). Consequently, even when t0 6= τk(x0), for any k ≥ 1,

it is possible that for some i, (t i , x+
i
) lies on a surface S j . In that case, that part of the solution

x(t, t0, x0) on the interval (t i , t i+1] consists of x(t, t i , x+
i
)which is a solution of (7) on [t i , t i+1]

in the extended sense. Given a solution x(t) of (7), defined on [t0, t0 + a) with a > 0, we

say that a solution y(t) of (7) is a proper continuation to the right of x(t) if y(t) is defined

on [t0, t0 + b) for some b > a and x(t) = y(t) for t ∈ [t0, t0 + a). The interval [t0, t0 + a)

is called the maximal interval of existence of a solution x(t) of (7), if x(t) is well defined on



REFERENCES 127

[t0, t0 + a) and it does not have any proper continuation to the right. If x(t) is a solution of

the system (7) with maximal interval of existence [t0, t0 + a) and if a <∞, then either x(t)

approaches the boundary of Ω or |x(t)| becomes unbounded as t → (t0 + a)−.

Definition 6 (Regular and Irregular Points). A point (t1, x1) is said to be a regular point if

t1 6= τk(x(t1)) for all k ≥ 1, otherwise it is said to be an irregular point.

We now proceed to state a result on continuation of solutions of the Caputo fractional

differential equations with variable moments of impulse. The proof is parallel to the Theorem

1.2.3 in [9] and hence omitted.

Theorem 3. Assume that

(i) f : D→ R is continuous.

(ii) Ik ∈ C[R,R], τk ∈ C[R,R+] for all k ≥ 1.

If x(t) is any solution of the system (7) with a finite [t0, b) as its maximal interval of existence,

with one of the following three conditions is satisfied,

a) If t1 = τk(x1) for some k ≥ 1 then there exists δ > 0 such that t 6= τk(x) for all (t, x)

with 0< t − t1 < δ and |x − x1|< δ.

b) If t1 = τk(x1) for some k ≥ 1 then t1 6= τ j(x1 + Ik(x1)) for all j ≥ 1.

c) If t1 = τk(x1) for some k ≥ 1, c Dqτk(x) exists, τi ∈ C1[R,R+] and τi(x) are linear

surfaces for all i ≥ 1 then t1 = τ j(x1 + Ik(x1)) for some j ≥ 1 and
∂ τ j(x)

∂ x . f (t, x) 6= (t−t1)
(1−q)

Γ(2−q)
at (t1, x+

1
) where x+

1
= x1 + Ik(x1).

Then limt→b− |x(t)|=∞.

6. Conclusion

In this paper we have introduced hybrid Caputo fractional differential equations of order

q ∈ (0,1) with variable moments of impulse and have shown by examples the potential it has

for further work. We have studied existence and continuation of solutions for initial value

problems in this set up.

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