Impulsive functional differential equations of fractional order with variable moments
We establish some existence results for the solutions of initial-value problems for fractional-order impulsive functional differential equations with neutral-delay at variable moments.
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| author | Ergören, H. Ергорен, Г. |
| author_facet | Ergören, H. Ергорен, Г. |
| author_sort | Ergören, H. |
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| description | We establish some existence results for the solutions of initial-value problems for fractional-order impulsive functional
differential equations with neutral-delay at variable moments. |
| first_indexed | 2026-03-24T02:15:06Z |
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UDC 517.9
H. Ergören (Yuzuncu Yil Univ., Turkey)
IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS
OF FRACTIONAL ORDER WITH VARIABLE MOMENTS
IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS
OF FRACTIONAL ORDER WITH VARIABLE MOMENTS
We establish some existence results for the solutions of initial-value problems for fractional-order impulsive functional
differential equations with neutral-delay at variable moments.
Встановлено деякi результати про iснування розв’язкiв початкової задачї для iмпульсивних фунцкцiонально-диференцiальних
рiвнянь з нейтральним запiзненням в змiннi моменти часу.
1. Introduction. We deal with the existence of solutions to the following initial value problem (IVP)
for the neutral impulsive fractional differential equations with variable times:
D\alpha
\bigl[
x(t) - g(t, xt)
\bigr]
= f(t, xt), t \in J, t \not = \tau k
\bigl(
x(t)
\bigr)
, 0 < \alpha \leq 1, (1.1)
x(t+) = Ik
\bigl(
x(t)
\bigr)
, t = \tau k
\bigl(
x(t)
\bigr)
, (1.2)
x(t) = \phi (t), t \in [ - \rho , 0], (1.3)
where D\alpha is Caputo fractional derivative, J = [0, T ], 0 < \rho < \infty , \scrU =\{ \psi : [ - \rho , 0] \rightarrow Rn is
continuous everywhere except for a finite number of points s at which \psi (s - ) and \psi (s+) exist and
\psi (s - ) = \psi (s)\} , and \phi \in \scrU , f, g : J \times \scrU \rightarrow Rn, Ik : Rn \rightarrow Rn, \tau k : Rn \rightarrow R, k = 1, 2, . . . , p,
are given functions satisfying some hypotheses to be specified later. For any function x defined on
[ - \rho , T ] and any t \in J we denote by xt the element of \scrU defined by xt = x(t+ \theta ), \theta \in [ - \rho , 0].
As well as fractional calculus [1 – 8], impulsive differential equations [9 – 15] play an important
role in mathematical modeling of many practical phenomena arising in engineering and various areas
of science. That is why, many scientists and researchers have devoted a great deal of attention to the
topic of impulsive fractional differential equations during the past decades [16 – 24].
Incidentally, we should note that impulsive effects for differential equations are classified as fixed
moments (t = tk ) and variable moments (t = \tau k(x(t))) in the mentioned literature above. What
is more, as far as we know, whereas some authors have addressed the functional(delay or neutral)
impulsive differential equations of integer orders with both fixed and variable moments [25 – 29] and
those of fractional orders with fixed moments [30, 31], only one author has considered impulsive
retarded functional differential equations of fractional order with variable moments up to now [32].
Hence we are in the position to continue on this way, that is, we will take into account a
class of fractional order neutral functional impulsive differential equations with variable moments in
(1.1) – (1.3) by generalizing the integer order functional impulsive differential equations with variable
moments
d
dt
[y(t) - g(t, yt)] = f(t, yt), t \in J = [0, T ], t \not = \tau k(y(t)), (1.4)
y(t+) = Ik(y(t)), t = \tau k(y(t)), (1.5)
c\bigcirc H. ERGÖREN, 2016
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 9 1169
1170 H. ERGÖREN
y(t) = \phi (t), t \in [ - \rho , 0], 0 < \rho <\infty , (1.6)
in [25] to the fractional order ones.
Throughout this paper, in Section 2 we firstly introduce some notations, definitions and basic
facts to be used this work. Then we will establish sufficient conditions for existence of solution
to the initial value problem (1.1) – (1.3) by extending the appreciable results in [25] consisting of
(1.4) – (1.6). At the end, we will present an effective example illustrating the main result.
2. Basic results and preliminaries. By C(J,Rn), C
\bigl(
[ - \rho , 0], Rn
\bigr)
and C
\bigl(
[ - \rho , T ], Rn
\bigr)
we
denote the Banach space of all continuous functions from J into Rn with the norm
\| x\| C := \mathrm{s}\mathrm{u}\mathrm{p}
\bigl\{
| x(t)| : t \in J
\bigr\}
,
the Banach space of all continuous functions from [ - \rho , 0] into Rn with the norm
\| \phi \| \scrU := \mathrm{s}\mathrm{u}\mathrm{p}
\bigl\{
\| \phi (\theta )\| : \theta \in l - \rho , 0]
\bigr\}
and the Banach space of all continuous functions from [ - \rho , T ] into Rn with the norm
\| x\| := \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
\| x\| C , \| \phi \| \scrU
\bigr\}
,
respectively.
In order to define the solutions of problem (1.1) – (1.3) we will consider the piecewise continuous
spaces:
\Omega = \{ x : [ - \rho , T ] \rightarrow Rn : there exists 0 = t0 < t1 < t2 < . . . < tp < tp+1 = T such that
tk = \tau k(x(tk)) and xk+1 \in C((tk, tk+1], R
n), k = 0, 1, 2, . . . , p\} . Also, there exist x(t+k ) and x(t - k )
with x(t - k ) = x(tk) for k = 1, 2, . . . , p, and x(t) = \phi (t), t \leq t0, where xk+1 is the restriction of x
over (tk, tk+1] and denoted by xk+1 := x| (tk,tk+1], k = 0, 1, 2, . . . , p.
The space \Omega forms a Banach space with the norm
\| x\| \Omega := \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
\| xk+1\| , k = 0, 1, . . . , p
\bigr\}
+ \| \phi \| \scrU .
Definition 2.1 [1, 2]. The fractional (arbitrary) order integral of the function h \in L1(J,R) of
order \alpha \in R+ is defined by
I\alpha 0+h(t) =
t\int
0
(t - s)\alpha - 1
\Gamma (\alpha )
h(s)ds,
where \Gamma (.) is the Euler gamma function.
Definition 2.2 [1, 2]. For a function h given on the interval J, Caputo fractional derivative of
order \alpha > 0 is defined by
D\alpha
0+h(t) =
t\int
0
(t - s)n - \alpha - 1
\Gamma (n - \alpha )
h(n)(s)ds, n = [\alpha ] + 1,
where the function h(t) has absolutely continuous derivatives up to order (n - 1).
Theorem 2.1 [33]. If U is closed, bounded, convex subset of a Banach space X and the mapping
A : U \rightarrow U is completely continuous, then A has a fixed point in U.
Theorem 2.2 [34]. If x(t) \in C1[0, T ], then for \alpha 1, \alpha 2 \in R+ and \alpha 1 + \alpha 2 \leq 1 we have
D\alpha 1D\alpha 2x(t) = D\alpha 2D\alpha 1x(t) = D\alpha 1+\alpha 2x(t).
As a matter of convenience, we shall use: J1 = [t1, T ], J2 = [t2, T ], . . . , Jk = [tk, T ], 1 \leq k \leq p.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 9
IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER WITH VARIABLE . . . 1171
3. Main results.
Definition 3.1. A function x \in \Omega is said to be a solution of problem (1.1) – (1.3) if x satisfies
the equation (1.1) and the conditions (1.2) and (1.3) are satisfied for x.
Now, let us state the following assumptions in order to establish some existence results for the
solutions of the IVP (1.1) – (1.3):
(A1) The function g : J \times \scrU \rightarrow Rn is completely continuous with the set \{ t\rightarrow g(t, u) : u \in S\}
equicontinuous for any bounded set S in C([ - \rho , T ], Rn) such that | g(t, u)| \leq q(t) for all t \in J,
u \in \scrU , where q(t), t \in J, is a function with q0 = \mathrm{s}\mathrm{u}\mathrm{p}
\bigl\{
| q(t)| : t \in J
\bigr\}
.
(A2) The function f : J \times \scrU \rightarrow Rn and \scrI k : Rn \rightarrow Rn, k = 1, 2, . . . , p, are continuous and
there exist a function \kappa (t) \geq 0, t \in J, with \kappa 0 = \mathrm{s}\mathrm{u}\mathrm{p}
\bigl\{
| \kappa (t)| : t \in J
\bigr\}
such that
\bigm| \bigm| f(t, u)\bigm| \bigm| \leq \kappa (t) for
all t \in J, u \in \scrU .
(A3) There exist the functions \tau k \in C1(Rn, R) for k = 1, 2, . . . , p such that 0 < \tau 1(x) <
< \tau 2(x) < . . . < \tau k(x) < T for all x \in Rn.
Lemma 3.1 [35]. The function x(t) \in C
\bigl(
[ - \rho , T ], Rn
\bigr)
is a solution of the problem
D\alpha
\bigl[
x(t) - g(t, xt)
\bigr]
= f(t, xt), t \in J, 0 < \alpha \leq 1,
x(t) = \phi (t), t \in [ - \rho , 0],
(3.1)
if and only if x(t) satisfies the following integral equation:
x(t) =
\left\{
\phi (t), t \in [ - \rho , 0],
\phi (0) - g(0, \phi ) + g(t, xt) +
\int t
0
(t - s)\alpha - 1
\Gamma (\alpha )
f(s, xs) ds, t \in J.
(3.2)
Theorem 3.1. In addition to the assumptions (A1) – (A3) let the following ones be satisfied:
(A4) Either g is a nonnegative function and \tau k is a nonincreasing function, or g is a nonpositive
function and \tau k is a non-decreasing function.
(A5) For all x \in Rn, \tau k(x) < \tau k+1(Ik(x)), k = 1, 2, . . . , p.
(A6) Let x \in \Omega , then for any t \in J we have\bigl\langle
\tau \prime k(x(t) - g(t, xt)), D
1 - \alpha f(t, xt)
\bigr\rangle
\not = 1
for k = 1, 2, . . . , p, where \langle ., .\rangle denotes the scalar product in Rn.
Then the IVP (1.1) – (1.3) has at least one solutions on J.
Proof. The proof will be carried out in several steps:
Step 1: Consider the following problem:
D\alpha [x(t) - g(t, xt)] = f(t, xt), t \in J, 0 < \alpha \leq 1, (3.3)
x(t) = \phi (t), t \in [ - \rho , 0]. (3.4)
Let us transform the problem (3.3), (3.4) into a fixed point problem. In view of Lemma 3.1,
consider the operator F : C
\bigl(
[ - \rho , T ], Rn
\bigr)
\rightarrow C
\bigl(
[ - \rho , T ], Rn
\bigr)
defined by
F (x)(t) =
\left\{
\phi (t), t \in [ - \rho , 0],
\phi (0) - g(0, \phi ) + g(t, xt) +
\int t
0
(t - s)\alpha - 1
\Gamma (\alpha )
f(s, xs) ds, t \in J.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 9
1172 H. ERGÖREN
We will use Schauder’s fixed point theorem in order to show that the operator F has fixed points giving
the solution to problem (3.3), (3.4). First of all, we define the set Cr =
\bigl\{
x(t) \in C
\bigl(
[ - \rho , T ], Rn
\bigr)
:
\| x\| \leq r for r > 0
\bigr\}
which is obviously closed, bounded and convex. Then, we will prove the
completely continuity of F in order to satisfy the rest of conditions of the Schauder’s fixed point
theorem. To do this, it is enough to show that the operator
\widetilde F (x)(t) =
\left\{ \phi (t), t \in [ - \rho , 0],
\phi (0) +
\int t
0
(t - s)\alpha - 1
\Gamma (\alpha )
f(s, xs) ds, t \in J,
is completely continuous.
To begin with, for each t \in J, the continuity of the functions \phi and f implies that \widetilde F is continuous.
For the compactness of \widetilde F :
(i) There exists a constant L > 0 such that we have
\bigm\| \bigm\| \bigm\| \widetilde Fx\bigm\| \bigm\| \bigm\| \leq L for each x \in Cr. In view of
(A1) and (A2) we have, for each t \in J,
\bigm| \bigm| \bigm| \widetilde F (x)(t)\bigm| \bigm| \bigm| \leq | \phi (0)| +
t\int
0
(t - s)\alpha - 1
\Gamma (\alpha )
| \kappa (s)| ds \leq
\bigm\| \bigm\| \phi (0)\bigm\| \bigm\| + \kappa 0
T\alpha
\Gamma (\alpha + 1)
:= L,
\bigm\| \bigm\| \bigm\| \widetilde F (x)(t)\bigm\| \bigm\| \bigm\| \leq L
which implies that the operator \widetilde F is uniformly bounded.
(ii) Let l1, l2 \in J, l1 < l2 and x \in Cr. Then, for each t \in J, we obtain
\bigm| \bigm| \bigm| \widetilde F (x)(l2) - \widetilde F (x)(l1)\bigm| \bigm| \bigm| \leq l1\int
0
\bigl[
(l2 - s)\alpha - 1 - (l1 - s)\alpha - 1
\bigr]
\Gamma (\alpha )
\bigm| \bigm| \kappa (s)\bigm| \bigm| ds+ l2\int
l1
(l2 - s)\alpha - 1
\Gamma (\alpha )
\bigm| \bigm| \kappa (s)\bigm| \bigm| ds,
\bigm\| \bigm\| \bigm\| \widetilde F (x)(l2) - \widetilde F (x)(l1)\bigm\| \bigm\| \bigm\| \leq \kappa 0
\Gamma (\alpha + 1)
\bigm| \bigm| 2(l2 - l1)
\alpha + l\alpha 1 - l\alpha 2
\bigm| \bigm| := K,
implying that \widetilde F is equicontinuous on J since the right-hand side of the inequality converges to zero
as l1 \rightarrow l2.
Consequently, as a result of Arzela – Ascoli theorem, the operator \widetilde F is compact and continuous,
that is, it is completely continuous.
Therefore, thanks to Schauder’s fixed point theorem, we deduce that F has a fixed point which
is a solution of problem (3.3), (3.4). We note this solution by x1.
Now we will discuss possible discontinuity moment the solution x(t) may beat. Let us define
the following function so that our discussion will become easier:
\sigma k,1(t) = \tau k(x1(t)) - t, t \geq 0.
From (A3) we get
\sigma k,1(0) = \tau k(x1(0)) \not = 0, k = 1, 2, . . . , p.
If \sigma k,1(t) \not = 0, that is, \tau k(x1(t)) \not = t on J for k = 1, 2, . . . , p, then x1(t) is a solution of both (3.3),
(3.4) and (1.1) – (1.3).
Now, we are in position to consider the case when
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 9
IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER WITH VARIABLE . . . 1173
\sigma 1,1(t) = 0, i.e., t = \tau 1(x1(t)) for some t \in J.
Since \sigma 1,1 is continuous and \sigma 1,1(0) \not = 0 by (A3), there exists t1 > 0 such that
\sigma 1,1(t1) = 0 and \sigma 1,1(t) \not = 0 for all t \in [0, t1).
Hence by (A3) we obtain
\sigma k,1(t) \not = 0 for all t \in [0, t1) and k = 1, 2, . . . , p.
Thus, we have formed the discontinuity point t1 where the solution x(t) beats.
Step 2: Consider the following problem:
D\alpha [x(t) - g(t, xt)] = f(t, xt), t \in J1, 0 < \alpha \leq 1, (3.5)
x(t+1 ) = I1(x1(t1)), (3.6)
x(t) = x1(t), t \in [t1 - \rho , t1]. (3.7)
Let us transform the problem (3.5) – (3.7) into a fixed point problem by considering the operator
F1 : C
\bigl(
[t1 - \rho , T ], Rn
\bigr)
\rightarrow C
\bigl(
[t1 - \rho , T ], Rn
\bigr)
defined by
F1(x)(t) =
\left\{
x1(t), t \in [t1 - \rho , t1],
I1(x1(t1)) - g(t1, xt1) + g(t, xt) +
\int t
t1
(t - s)\alpha - 1
\Gamma (\alpha )
f(s, xs) ds, t \in J1.
Pursuing the process in the 1st Step, as a consequence of Schauder’s fixed point theorem, one can
conclude that F1 has a fixed point which is a solution of the problem (3.5) – (3.7) on J1 by proving
the completely continuity of the operator
\widetilde F1(x)(t) =
\left\{
x1(t), t \in [t1 - \rho , t1],
I1(x1(t1)) +
\int t
t1
(t - s)\alpha - 1
\Gamma (\alpha )
f(s, xs) ds, t \in J1.
Let us indicate this solution as x2.
Then we will investigate a possible discontinuity moment coming after t1 that the solution x(t)
meets. Let us state the function
\sigma k,2(t) = \tau k(x2(t)) - t, t \geq t1.
If \sigma k,2(t) \not = 0, that is, \tau k(x2(t)) \not = t on (t1, T ] for k = 1, 2, . . . , p, then x2(t) is a solution of
problem (3.5) – (3.7). That is,
x(t) =
\left\{ x1(t), t \in [t0, t1],
x2(t), t \in (t1, T ],
is a solution of problem (1.1) – (1.3).
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 9
1174 H. ERGÖREN
Now, let us consider the following case:
\sigma 2,2(t) = 0, i.e., t = \tau 2(x2(t)) for some t \in (t1, T ].
Then from (A5) we have
\sigma 2,2(t
+
1 ) = \tau 2(x2(t
+
1 )) - t1 = \tau 2
\bigl(
I1(x1(t1))
\bigr)
- t1 > \tau 1
\bigl(
x1(t1)
\bigr)
- t1 = \sigma 1,1(t1) = 0.
Since \sigma 2,2 is continuous, there exists t2 > t1 such that
\sigma 2,2(t2) = 0 and \sigma 2,2(t) \not = 0 for all t \in (t1, t2).
Hence by (A3) we get
\sigma k,2(t) \not = 0 for all t \in (t1, t2) and k = 2, 3, . . . , p.
Also, let us show that there does not exist any \xi \in (t1, t2) such that \sigma 1,2(\xi ) = 0. Now, assume
that there exists \xi \in (t1, t2) such that \sigma 1,2(\xi ) = 0. Considering the function \gamma 1(t) = \tau 1
\bigl(
x2(t) -
- g(t, x2t)
\bigr)
- t, by (A4) it follows that
\gamma 1(\xi ) = \tau 1
\bigl(
x2(\xi ) - g(\xi , x2\xi )
\bigr)
- \xi \geq \tau 1
\bigl(
(x2(\xi ))
\bigr)
- \xi = \sigma 1,2(\xi ) = 0.
Thus the function \gamma 1 gains a nonnegative maximum at some point \eta \in (t1, t2]. Moreover, from
Theorem 2.2 and in view of the Eq. (3.5) and the function x2(t), since
d
dt
\bigl[
x(t) - g(t, x2t)
\bigr]
= D1 - \alpha f(t, x2t),
we obtain that, for some point \eta \in (t1, t2],
\gamma \prime 1(\eta ) = \tau
\prime
1
\bigl(
x2(\eta ) - g(\eta , x2\eta )
\bigr) d
dt
\bigl[
x2(\eta ) - g(\eta , x2\eta )
\bigr]
=
= \tau
\prime
1(x2(\eta ) - g(\eta , x2\eta ))
CD1 - \alpha f(t, x2\eta ) - 1 = 0,
that is, \Bigl\langle
\tau
\prime
1(x2(\eta ) - g(\eta , x2\eta )), D
1 - \alpha f(t, x2\eta )
\Bigr\rangle
= 1,
which contradicts (A6).
Consequently, we have built a second discontinuity point t2 > t1 where the solution x(t) meets
in such a way that \sigma 2,2(t2) = 0 and \sigma k,2(t) \not = 0 for all t \in (t1, t2) and k = 1, 2, 3, . . . , p.
Step 3: Let us continue the procedure as in the previous steps by taking into consideration that
xp := x| (tp - 1,T ] is a solution of the following problem:
D\alpha
\bigl[
x(t) - g(t, xt)
\bigr]
= f(t, xt), t \in Jp - 1, 0 < \alpha \leq 1, (3.8)
x(t+p - 1) = Ip - 1(xp - 1(tp - 1)), (3.9)
x(t) = xp - 1(t), t \in
\bigl[
tp - 1 - \rho , tp - 1
\bigr]
. (3.10)
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IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER WITH VARIABLE . . . 1175
We transform the problem (3.8) – (3.10) into a fixed point problem by considering the operator
Fp - 1 : C
\bigl(
[tp - 1 - \rho , T ], Rn
\bigr)
\rightarrow C([tp - 1 - \rho , T ], Rn) defined by
Fp - 1(x)(t) =
\left\{
xp - 1(t), t \in [tp - 1 - \rho , tp - 1],
Ip - 1(xp - 1(tp - 1)) - g(tp - 1, xtp - 1)+
+g(t, xt) +
\int t
tp - 1
(t - s)\alpha - 1
\Gamma (\alpha )
f(s, xs) ds, t \in Jp - 1.
As in Step1, as a result of Schauder’s fixed point theorem we can conclude that Fp - 1 has a fixed
point which is a solution of problem (3.8) – (3.10) on Jp - 1. Denote now this solution by xp.
Then we will explore a possible discontinuity moment after the point tp - 1 the solution x(t)
encounters by making use of the function
\sigma k,p(t) = \tau k(xp(t)) - t, t \geq tp - 1.
If \sigma k,p(t) \not = 0, that is, \tau k(xp(t)) \not = t on (tp - 1, T ] for k = 1, 2, . . . , p, then xp(t) is a solution of
problem (3.8) – (3.10). That is,
x(t) =
\left\{
x1(t), t \in [t0, t1],
x2(t), t \in (t1, t2],
. . . . . . . . . . . . . . . . . .
xp(t), t \in (tp - 1, T ],
is a solution of problem (1.1) – (1.3).
Now we are in the position to focus on the circumstance when
\sigma p,p(t) = 0, i.e., t = \tau p(xp(t)) for some t \in (tp - 1, T ].
From (A5) we have
\sigma p,p(t
+
p - 1) = \tau p(xp
\bigl(
t+p - 1)
\bigr)
- tp - 1 =
= \tau p
\bigl(
Ip - 1(xp - 1(tp - 1))
\bigr)
- tp - 1 > \tau p - 1(xp - 1
\bigl(
tp - 1)
\bigr)
- tp - 1 = \sigma p - 1,p - 1(tp - 1) = 0.
Since \sigma p,p is continuous, there exists tp > tp - 1 such that
\sigma p,p(tp) = 0 and \sigma p,p(t) \not = 0 for all t \in (tp - 1, tp).
Thus by (A3) we get
\sigma k,p(t) \not = 0 for all t \in (tp - 1, tp) and k = 3, 4, . . . , p.
Also, we need to show that there does not exist any \xi \in (tp - 1, tp) such that \sigma p - 1,p(\xi ) = 0.
Suppose now that there exists \xi \in (tp - 1, tp) such that \sigma p - 1,p(\xi ) = 0. Considering the function
\gamma p - 1(t) = \tau p - 1(xp(t) - g(t, xpt)) - t, by (A4) it follows that
\gamma p - 1(\xi ) = \tau p - 1(xp(\xi ) - g(\xi , xp\xi )) - \xi \geq \tau p - 1
\bigl(
(xp(\xi ))
\bigr)
- \xi = \sigma p - 1,p(\xi ) = 0.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 9
1176 H. ERGÖREN
Therefore, the function \gamma p - 1 attains a nonnegative greatest value at some point \eta \in (tp - 1, tp].
Furthermore, from Theorem 2.2 and in view of the Eq. (3.8) and the function xp(t), since
d
dt
\bigl[
x(t) - g(t, xpt)
\bigr]
= D1 - \alpha f(t, xpt),
we find that, for some point \eta \in (tp - 1, tp],
\gamma \prime p - 1(\eta ) = \tau
\prime
p - 1(xp(\eta ) - g(\eta , xp\eta ))
d
dt
\bigl[
xp(\eta ) - g(\eta , xp\eta )
\bigr]
- 1 =
= \tau
\prime
p - 1(xp(\eta ) - g(\eta , xp\eta ))D
1 - \alpha f(t, xp\eta ) - 1 = 0,
that is, \Bigl\langle
\tau
\prime
p - 1(xp(\eta ) - g(\eta , xp\eta )), D
1 - \alpha f(t, xp\eta )
\Bigr\rangle
= 1,
which implies a contradiction with (A6).
As a result, we have constituted a p-th discontinuity point tp > tp - 1 > . . . > t2 > t1, where
the solution x(t) beats in such a way that \sigma p,p(tp) = 0 and \sigma k,p(t) \not = 0 for all t \in (tp - 1, tp) and
k = 1, 2, 3, . . . , p.
Finally, the solution x of problem (1.1) – (1.3) is defined by
x(t) =
\left\{
x1(t), t \in [t0, t1],
x2(t), t \in (t1, t2],
. . . . . . . . . . . . . . . . . . . .
xp(t), t \in (tp - 1, tp],
xp+1(t), t \in (tp, T ].
Theorem 3.1 is proved.
In the sequel, we shall give some sufficient conditions for the uniqueness of the solutions of
IVP (1.1) – (1.3).
Theorem 3.2. In addition to the assumptions (A3) – (A6), suppose that
(A7) There exists constant c > 0 such that | g(t, u) - g(t, v)| \leq c| u - v| for each t \in J and
u, v \in Rn.
(A8) There exists constant d > 0 such that | f(t, u) - f(t, v)| \leq d| u - v| for each t \in J and
u, v \in Rn.
(A9) There exist constants dk > 0, k = 1, 2, 3, . . . , p, such that | Ik(u) - Ik(v)| \leq dk| u - v| for
each u, v \in Rn.
Further, if the condition
\Lambda := dk + 2c+
dT\alpha
\Gamma (\alpha + 1)
< 1
is fulfilled, then the IVP (1.1) – (1.3) has a unique solution on J.
Proof. Taking the steps in Theorem 3.1 into consideration, we consider the following problem
CD\alpha
\bigl[
x(t) - g(t, xt)
\bigr]
= f(t, xt), t \in [tk, tk+1], 0 < \alpha \leq 1, (3.11)
x(t+k ) = Ik(xk(tk)), (3.12)
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 9
IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER WITH VARIABLE . . . 1177
x(t) = xk(t), t \in [tk - \rho , tk], (3.13)
whose solution is xk+1 := x| (tk,tk+1]. We transform problem (3.11) – (3.13) into a fixed point problem
in view of the operator \scrF k : C
\bigl(
[tk - \rho , tk+1], R
n
\bigr)
\rightarrow C
\bigl(
[tk - \rho , tk+1], R
n
\bigr)
defined by
\scrF k(x)(t) =
\left\{
xk(t), t \in [tk - \rho , tk],
Ik(xk(tk)) - g(tk, xtk) + g(t, xt) +
\int t
tk
(t - s)\alpha - 1
\Gamma (\alpha )
f(s, xs) ds, t \in [tk, tk+1].
Here, it suffices to show that the operator Fk is a contracting mapping in order to prove that x(t) is
a unique solution of the IVP (1.1) – (1.3) on [tk, tk+1]. Now, let x, y \in C
\bigl(
[tk - \rho , tk+1], R
n
\bigr)
. Then,
for each t \in [tk, tk+1], it is obvious that \scrF k is a contraction since
\| \scrF k(x) - \scrF k(y)\| \leq \Lambda \| x - y\| .
As a consequence of Banach’s fixed point theorem, \scrF k has a fixed point. Therefore, it leads that the
IVP (1.1) – (1.3) has a unique solution.
Theorem 3.2 is proved.
Example 3.1. Consider the following IVP for impulsive neutral fractional differential equation
at variable moments:
D1/2
\left[ x(t) +
\mathrm{s}\mathrm{i}\mathrm{n}x
\biggl(
t - 1
5
\biggr)
\biggl(
t+
1
4
\biggr) 2
\right] =
=
e - t
\bigm| \bigm| \bigm| \bigm| x\biggl( t - 1
5
\biggr) \bigm| \bigm| \bigm| \bigm|
(et + 2)3
\biggl(
1 +
\bigm| \bigm| \bigm| \bigm| x\biggl( t - 1
5
\biggr) \bigm| \bigm| \bigm| \bigm| \biggr) , t \in J, t \not = \tau k(x(t)), (3.14)
x(t+) = Ik(x(t)), t = \tau k(x(t)), k = 1, 2, . . . , p, (3.15)
x(s) = \phi (s), s \in
\biggl[
- 1
2
, 0
\biggr]
, (3.16)
where J = [0, 1] and
\tau k(x) = 1 - 1
3k(1 + x2)
, Ik(x) = ckx, ck \in
\biggl(
1\surd
3
, 1
\biggr]
, ck > 0, k = 1, 2, . . . , p.
Immediately, since g is completely continuous and f is continuous such that
\bigm| \bigm| g(t, xt)\bigm| \bigm| =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| -
\mathrm{s}\mathrm{i}\mathrm{n}x
\biggl(
t - 1
5
\biggr)
\biggl(
t+
1
4
\biggr) 2
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
1\biggl(
t+
1
4
\biggr) 2 =: q(t)
with q0 = 16 and
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 9
1178 H. ERGÖREN
\bigm| \bigm| f(t, xt)\bigm| \bigm| =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
e - t
\bigm| \bigm| \bigm| \bigm| x\biggl( t - 1
5
\biggr) \bigm| \bigm| \bigm| \bigm|
(et + 2)3
\biggl(
1 +
\bigm| \bigm| \bigm| \bigm| x\biggl( t - 1
5
\biggr) \bigm| \bigm| \bigm| \bigm| \biggr)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
e - t
(et + 2)3
=: \kappa (t)
with \kappa 0 =
1
27
. So, (A1) and (A2) are satisfied. Since
\tau k+1(x) - \tau k(x) =
2
3k+1(1 + x2)
> 0 \forall x \in R, k = 1, 2, . . . , p,
and
\tau k+1(Ik(x)) - \tau k(x) =
2 +
\bigl(
3c2k - 1
\bigr)
x2
3k+1(1 + x2)
\bigl(
1 + c2kx
2
\bigr) > 0 \forall x \in R,
the assumptions (A3) and (A5) are fulfilled. Also, in view of g and \tau \prime k(x), one can see that the
condition (A4) holds. Finally, it is clear that (A6) is valid.
Consequently, since all assumptions of the Theorem 3.1 hold, the problem (3.14) – (3.16) has at
least one solution.
Conclusion. We have investigated existence of solution to the IVP (1.1) – (1.3) consisting of
a class of impulsive fractional neutral functional differential equations with variable moments. In
this work, we have extended the notable results of Benchohra and Ouahab [25] considering a class
of integer order neutral functional impulsive differential equations with variable times to a class of
fractional order ones.
References
1. Kilbas A. A., Srivastava H. M., Trujillo J. J. Theory and applications of fractional differential equations // North-
Holland Math. Stud. – Amsterdam: Elsevier, 2006. – 204.
2. Podlubny I. Fractional differential equations. – San Diego: Acad. Press, 1999.
3. Lakshmikantham V., Leela S., Devi J. V. Theory of fractional dynamic systems. – Cambridge: Cambridge Sci. Publ.,
2009.
4. Hilfer R. Applications of fractional calculus in physics. – Singapore: World Sci., 2000.
5. Agarwal R. P., Benchohra M., Hamani S. A survey on existence results for boundary value problems of nonlinear
fractional differential equations and inclusions // Acta Appl. Math. – 2010. – 109, № 3. – P. 973 – 1033.
6. Ahmad B., Sivasundaram S. On four-point nonlocal boundary value problems of nonlinear integro-differential
equations of fractional order // Appl. Math. and Comput. – 2010. – 217, № 2. – P. 480 – 487.
7. Ergören H. On The Lidstone Boundary Value Problems for Fractional Differential Equations // Int. J. Math. and
Comput. – 2014. – 22, № 1. – P. 66 – 74.
8. Ergören H. On the positive solutions for fractional differential equations with weakly contractive mappings // AIP
Conf. Proc. 1676 / 020080 (2015); doi: 10.1063/1.4930506.
9. Benchohra M., Henderson J., Ntouyas S. Impulsive differential equations and inclusions. – New York: Hindawi Publ.
Corporation, 2006.
10. Samoilenko A. M., Perestyuk N. A. Impulsive differential equations with actions (Russian). – Kiev: Vyscha Shkola,
1997.
11. Samoilenko A. M., Perestyuk N. A. Impulsive differential equations. – Singapore: World Sci., 1995.
12. Lakshmikantham V., Bainov D. D., Simeonov P. S. Theory of impulsive differential equations. – Singapore: World
Sci., 1989.
13. Rogovchenko Y. V. Impulsive evolution systems: main results and new trends // Dynam. Contin. Discrete Impuls.
Syst. – 1997. – 3. – P. 57 – 88.
14. Akhmet M. Principles of discontinuous dynamical systems. – New York etc.: Springer, 2010.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 9
IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER WITH VARIABLE . . . 1179
15. Lafci M., Bereketoğlu H. On a certain impulsive differential system with piecewise constant arguments // Math. Sci. –
2014. – 8, № 1.
16. Benchohra M., Slimani B. A. xistence and uniqueness of solutions to impulsive fractional differential equations //
Electron. J. Different. Equat. – 2009. – 10. – P. 1 – 11.
17. Tian Y., Bai Z. Existence results for the three-point impulsive boundary value problem involving fractional differential
equations // Comput. and Math. Appl. – 2010. – 59, № 8. – P. 2601 – 2609.
18. Ahmad B., Sivasundaram S. Existence of solutions for impulsive integral boundary value problems of fractional
order // Nonlinear Anal. – 2010. – 4, № 1. – P. 134 – 141.
19. Balachandran K., Kiruthika S., Trujillo J. J. Existence results for fractional impulsive integrodifferential equations
in Banach spaces // Commun. Nonlinear Sci. Numer. Simul. – 2011. – 16. – P. 1970 – 1977.
20. Yang L., Chen H. Nonlocal boundary value problem for impulsive differential equations of fractional order // Adv.
Difference Equat. – 2011. – 2011. – Article ID 404917, 16 p. doi:10.1155/2011/404917
21. Wang G., Ahmad B., Zhang L. Some existence results for impulsive nonlinear fractional differential equations with
mixed boundary conditions // Comput. and Math. Appl. – 2011. – 62, № 3. – P. 1369 – 1397.
22. Ergören H., Tunc C. A general boundary value problem for impulsive fractional differential equations // Palestine J.
Math. – 2016. – 5, № 1. – P. 65 – 78.
23. Ergören H., Kilicman A. Non-local boundary value problems for impulsive fractional integro-differential equations
in Banach spaces // Boundary Value Problems. – 2012. – 145.
24. Ergören H., Sakar M. G. Boundary value problems for impulsive fractional differential equations with nonlocal
conditions // Adv. App. Math. and Approxim. Theory. – 2013. – 41. – P. 283 – 297.
25. Benchohra M., Ouahab A. Impulsive neutral functional differential equations with variable times // Nonlinear Anal. –
2003. – 55. – P. 679 – 693.
26. Benchohra M., Henderson J., Ntouyas S. K. Impulsive neutral functional differential equations in Banach spaces //
Appl. Anal. – 2001. – 80, № 3. – P. 353 – 365.
27. Ballinger G., Liu X. Z. Existence and uniqueness results for impulsive delay differential equations // Dynam. Contin.
Discrete Impuls. Syst. – 1999. – 5. – P. 579 – 591.
28. Benchohra M., Henderson J., Ntouyas S. K, Ouahab A. Impulsive functional differential equations with variable
times // Comput. and Math. Appl. – 2004. – 47. – P. 1659 – 1665.
29. Benchohra M., Henderson J., Ntouyas S. K., Ouahab A. Impulsive functional differential equations with variable
times and infinite delay // Int. J. Appl. Math. Sci. – 2005. – 2, № 1. – P. 130 – 148.
30. Anguraj A., Ranjini M. C. Existence results for fractional impulsive neutral functional differential equations // JFCA
3(4). – 2012. – 4. – P. 1 – 12.
31. Dabas J., Chauhan A. Existence and uniqueness of mild solution for an impulsive neutral fractional integro-differential
equation with infinite delay // Math. Comput. Modelling. – 2013. – 57. – P. 754 – 763.
32. Ergören H. Impulsive retarded fractional differential equations with variable moments // Contemp. Anal. and Appl.
Mathf 2016. – 4, № 1. – P. 156 – 170.
33. Hale J. K., Lunel S. M. V. Introduction to functional differential equations. – New York: Springer-Verlag, 1993.
34. Li C., Deng W. Remarks on fractional derivatives // Appl. Math. Comput. – 2007. – 187. – P. 777 – 787.
35. Agarwal R. P., Zhou Y., He Y. Existence of fractional neutral functional differential equations // Comput. and Math.
Appl. – 2010. – 59. – P. 1095 – 1100.
Received 23.03.15,
after revision — 29.04.16
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|
| id | umjimathkievua-article-1912 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:15:06Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/0d/4ddcfb551f53c7936edd5ab4ed4b9f0d.pdf |
| spelling | umjimathkievua-article-19122019-12-05T09:31:35Z Impulsive functional differential equations of fractional order with variable moments Impulsive functional differential equations of fractional order with variable moments Ergören, H. Ергорен, Г. We establish some existence results for the solutions of initial-value problems for fractional-order impulsive functional differential equations with neutral-delay at variable moments. Встановлено деякi результати про iснування розв’язкiв початкової задачї для iмпульсивних фунцкцiонально-диференцiальних рiвнянь з нейтральним запiзненням в змiннi моменти часу. Institute of Mathematics, NAS of Ukraine 2016-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1912 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 9 (2016); 1169-1179 Український математичний журнал; Том 68 № 9 (2016); 1169-1179 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1912/894 Copyright (c) 2016 Ergören H. |
| spellingShingle | Ergören, H. Ергорен, Г. Impulsive functional differential equations of fractional order with variable moments |
| title | Impulsive functional differential equations of fractional order with variable moments |
| title_alt | Impulsive functional differential equations of fractional order with variable moments |
| title_full | Impulsive functional differential equations of fractional order with variable moments |
| title_fullStr | Impulsive functional differential equations of fractional order with variable moments |
| title_full_unstemmed | Impulsive functional differential equations of fractional order with variable moments |
| title_short | Impulsive functional differential equations of fractional order with variable moments |
| title_sort | impulsive functional differential equations of fractional order with variable moments |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1912 |
| work_keys_str_mv | AT ergorenh impulsivefunctionaldifferentialequationsoffractionalorderwithvariablemoments AT ergoreng impulsivefunctionaldifferentialequationsoffractionalorderwithvariablemoments |