Vol. 19 (2018), No. 2, pp. 847–863 DOI: 10.18514/MMN.2018.2368
ON THE SOLUTIONS OF A MULTI-POINT BVP FOR FRACTIONAL DIFFERENTIAL EQUATIONS WITH IMPULSES
FATMA TOKMAK FEN Received 24 June, 2017
Abstract. The existence and uniqueness of solutions for a multi-point boundary value problem (BVP) of impulsive fractional differential equations are investigated by means of Schaefer’s fixed point theorem and Banach contraction mapping principle. Examples that support the theoretical results are presented.
2010Mathematics Subject Classification: 26A33; 34B37
Keywords: Caputo fractional derivative, impulsive differential equations, fractional differential equations, boundary value problems, Banach contraction mapping principle, Schaefer’s fixed point theorem
1. INTRODUCTION
Even though it is a 300-year-old topic, fractional calculus has come into promin- ence in the course of time, and nowadays it has an important role in the modeling of real world phenomena based on fractional derivatives [15]. Models arising from fractional order derivatives are excellent instruments for the description of memory and hereditary properties of various materials and processes [15]. Impulsive dif- ferential equations, on the other hand, are capable of describing the dynamics of processes in which abrupt changes occur [2,17]. For instance, according to the switching phenomenon, frequency changes or other sudden noises, the states of the electronic devices are often subject to instantaneous perturbations and experience abrupt changes at certain instants [2,11], i.e., they exhibit impulsive effects. There- fore, mathematical models with impulsive effects are more accurate to describe the evolutionary processes of the systems. Fractional order and impulsive differential equations have many applications in various fields such as mechanics, electronics, biology, economics, chaos theory, and neuroscience [1,3–5,8,12–14,19–21].
Various types of impulsive fractional differential equations were considered in the papers [22–27]. Banach, Krasnoselskii’s, and Leray-Schauder’s fixed point theorems were utilized to show the existence and uniqueness of solutions for a class of frac- tional order differential switched systems with coupled nonlocal initial and impulsive
c 2018 Miskolc University Press
conditions in [24]. The papers [6,27] were concerned with significant results on for- mulas for solutions of impulsive fractional differential equations. In [27], a boundary value problem (BVP) of order between1and2is studied. Fixed point theorems were also used in [6] and [27] to show the existence and uniqueness of solutions. On the other hand, the theory of impulsive fractional differential inclusions was improved in [25] by taking into account the existence of PC-mild solutions of such systems with nonlocal conditions when the linear part is a fractional sectorial operator defined on a separable Banach space. Moreover, existence results based on the Banach and Schauder’s fixed point theorems were revealed by Wang and Zhang [26] for frac- tional impulsive systems with Hadamard derivatives. In the present study, we make use of the Banach and Schaefer’s fixed point theorems to deal with the existence and uniqueness of solutions of a BVP of fractional differential equations with impulses.
In contrast with [27], in this paper multi-points in the boundary conditions are used and the order of the system is between2and3.
In [10], Liu et al. considered the following nonlinear fractional impulsive differ- ential equation
8 ˆˆ
<
ˆˆ :
D0ˇCp
D0˛Cu.t /
Df .t; u.t //; t2J0; u.tk/DIk.u.tk//; p
D0˛Cu.tk/
Dbk; kD1; 2; :::; m;
u.0/Du0; D0˛Cu.0/Du1;
where˛; ˇ2.0; 1 with˛Cˇ > 1:The existence and uniqueness of solutions were obtained by means of the Schauder’s fixed point theorem and Banach contraction mapping principle. Additionally, in [6], the authors considered the following frac- tional impulsive differential equation
8
<
:
CDqtu.t /Df .t; u.t //; t2J0WDJn ft1; : : : ; tmg; J DŒ0; 1 ; u.tkC/Du.tk/Cyk; kD1; 2; : : : ; m;
u.0/Du0; yk2R;
whereCDtq is the Caputo fractional derivative of orderq2.0; 1/,u02R, andf W J R!Ris a continuous function. Fixed point methods were utilized to obtain some sufficient conditions for the existence of solutions. Moreover, Rehman and Eloe [16] presented a general method for converting an impulsive fractional differential equation to an equivalent integral equation. Using this method, they investigated the existence of solutions for the following four-point impulsive fractional BVP
8
<
:
CD˛y.x/Df .x; y/; x2J;
y.xk/DIk.u.tk//; y0.xk/D NIk.y.xk//; kD1; 2; : : : ; m;
y.0/Dy./; y.1/Dˇy./;
where1 < ˛ < 2,ˇ,; ; 2R,; 2.0; 1/,J DŒ0; 1, andf WJR!Ris a continuous function.
Motivated by the above mentioned studies, in this paper, we consider the following p-point impulsive fractional BVP
8 ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆ<
ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆ:
CD0˛Cu.t /Df .t; u.t //; t2J0;
u.tk/DIk.u.tk//; u0.tk/D NIk.u.tk//;
u00.tk/D QIk.u.tk//; kD1; 2; : : : ; m;
au.0/Cbu0.0/D
p 2
X
iD1
ciu.i/;
cu.1/Cdu0.1/D
p 2
X
iD1
diu.i/; u00.0/D0;
(1.1)
whereCD˛0Cis the Caputo fractional derivative of order˛2.2; 3/,a; b; c, andd are real constants, the functions f WŒ0; 1R!R, Ik WR!R, INk WR!R;andIQk W R!Rare continuous in all their arguments,0 < 1< 2< < p 2< 1,ci; di 2 R, iD1; : : : ; p 2; withıWDacCad bc
p 2
X
iD1
dii
a
p 2
X
iD1
ci
C
p 2
X
iD1
di
b
p 2
X
iD1
cii
Cc
p 2
X
iD1
cii¤0,J DŒ0; 1 ; 0Dt0< t1< < tk< < tm< tmC1D1;
J0DJn ft1; t2; : : : ; tmg; u.tk/Du.tkC/ u.tk/;andu.tkC/D lim
t!tkC
u.t /for each kD1; 2; : : : ; m:The impulse actions u0.tk/ andu00.tk/ are defined in a similar way tou.tk/:
The rest of this paper is organized as follows. In Section 2, we provide some necessary definitions and preliminary lemmas, which are key tools for our main res- ults. In Section3, we make use of the Schaefer’s fixed point theorem and Banach contraction mapping principle to prove the existence and uniqueness of solutions of (1.1). Examples are provided in Section4to demonstrate the applications of the main results.
2. PRELIMINARIES
The following definitions are needed in the remaining parts of the paper.
Definition 1([9]). The Riemann-Liouville fractional integral of order˛for a func- tiongWŒ0;1/!Ris defined as
I0˛Cg.t /D 1 .˛/
Z t 0
.t s/˛ 1g.s/ds; t > 0; ˛ > 0;
provided the right side is point-wise defined onŒ0;1/, where is the gamma func- tion.
Definition 2 ([9]). The Riemann-Liouville derivative of order ˛ for a function gWŒ0;1/!Rcan be written as
LD0˛Cg.t /D 1 .n ˛/
dn dtn
Z t 0
g.s/
.t s/˛C1 nds; t > 0;
wherenDŒ˛C1andŒ˛is the integer part of˛:
Definition 3([9]). The Caputo fractional derivative of order˛ > 0for a continuous functiongWŒ0;1/!Rcan be written as
CD0˛Cg.t /D LD0˛C
"
g.t /
n 1
X
iD0
g.i /.0/
i Š ti
#
; t > 0;
wherenDŒ˛C1andŒ˛is the integer part of˛:
One can confirm that ifgWŒ0;1/!Ris an ntimes continuously differentiable function, then
CD0˛Cg.t /D 1 .n ˛/
Z t 0
g.n/.s/
.t s/˛C1 ndsDI0n ˛C g.n/.t /;
fort > 0andn 1 < ˛ < n:Moreover, the Caputo derivative of a constant is equal to zero.
We say that a real valued functionu.t /defined onJbelongs to the spaceP C.J;R/
if it is left-continuous and continuous except, possibly, at the pointstk,kD1; 2; : : : ; m, where it has discontinuities of the first kind. Besides, for a positive integer, we will denote byP C.J;R/the set of real valued functions whose derivative of order i belongs to the set P C.J;R/ for each i D0; 1; : : : ; . It is worth noting that the space P C.J;R/ is a Banach space endowed with the norm kuk Dsupt2Jju.t /j. In the sequel, we will denote J0 DŒ0; t1; J1 D.t1; t2; : : : ; Jm 1 D.tm 1; tm;
andJmD.tm; 1. Moreover, let AC.J;R/ be the set of functions u.t / such that u. 1/2P C.J;R/and the restriction ofu. 1/onJk,kD0; 1; : : : ; m, is absolutely continuous.
Lemma 1([16, Corollary 2.5]). Assumef WJR!Ris continuous andn 1 <
˛ < n. Thenu2ACn.J;R/satisfies
CD0˛Cu.t /Df .t; u.t //; t2J;
if and only ifu2P Cn 1.J;R/and u.t /D
n 1
X
iD0
u.i /.0/
i Š tiC
n 1
X
iD0
X
0<tk<t
1
i Šu.i /.tk/.t tk/i
C Z t
0
.t s/˛ 1
.˛/ f .s; u.s//ds fort2J.
Theorem 1([18, Schaefer’s Fixed Point Theorem]). LetBbe a Banach space and T WB!B be a completely continuous operator. If the set
D.B/D fu2BWuDT u for some2Œ0; 1g is bounded, thenT has at least one fixed point.
Lemma 2([7,P C-type Ascoli-Arzela Theorem]). Let˝P C.J;R/. Suppose the following conditions are satisfied:
(i) ˝is uniformly bounded;
(ii) ˝is equicontinuous inJk; kD0; 1; : : : ; m.
Then˝is a relatively compact subset ofP C.J;R/.
3. MAIN RESULTS
This section is concerned with the existence as well as uniqueness of solutions of the problem (1.1).
In the following lemma, we provide an equivalent integral equation to problem (1.1).
Lemma 3. A functionu.t /is a solution of fractional impulsive BVP(1.1)if and only if it is a solution of the following equation
u.t /D 8 ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ
<
ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ :
Z t 0
.t s/˛ 1
.˛/ f .s; u.s//dsC1 ı
p 2
X
iD1
ciFu.i/
cCd
p 2
X
iD1
dii
c
p 2
X
iD1
di
t
C1 ı
p 2 X
iD1
diFu.i/ Au
bC
p 2
X
iD1
ciiC
a
p 2
X
iD1
ci
t
; t 2J0;
Fu.t /C1 ı
p 2
X
iD1
ciFu.i/
cCd
p 2
X
iD1
dii
c
p 2
X
iD1
di
t
C1 ı
p 2 X
iD1
diFu.i/ Au
bC
p 2
X
iD1
ciiC
a
p 2
X
iD1
ci
t
; t2Jk; kD1; 2; : : : ; m;
(3.1)
where
Fu.t /D Z t
0
.t s/˛ 1
.˛/ f .s; u.s//dsC X
0<tk<t
Ik.u.tk//
C X
0<tk<t
INk.u.tk//.t tk/C X
0<tk<t
IQk.u.tk//.t tk/2
2 ;
(3.2)
and
AuD Z 1
0
c.1 s/˛ 1
.˛/ Cd.1 s/˛ 2 .˛ 1/
f .s; u.s//ds Cc
m
X
kD1
Ik.u.tk//C
m
X
kD1
INk.u.tk// .c.1 tk/Cd /
C1 2
m
X
kD1
IQk.u.tk// c.1 tk/2C2d.1 tk/ :
(3.3)
Proof. Suppose thatu.t /is solution of (1.1). One can confirm by means of Lemma 1that
u.t /De1Ce2tCe3t2C X
0<tk<t
Ik.u.tk//C X
0<tk<t
INk.u.tk//.t tk/
C X
0<tk<t
Q
Ik.u.tk//.t tk/2
2 C
Z t 0
.t s/˛ 1
.˛/ f .s; u.s//ds;
(3.4)
u0.t /De2C2e3tC X
0<tk<t
INk.u.tk//C X
0<tk<t
IQk.u.tk//.t tk/
C Z t
0
.t s/˛ 2
.˛ 1/ f .s; u.s//ds;
(3.5)
and
u00.t /D2e3C X
0<tk<t
IQk.u.tk//C Z t
0
.t s/˛ 3
.˛ 2/ f .s; u.s//ds: (3.6) Applying the boundary condition u00.0/D0 in (3.6), we obtain e3 D0: Now, using the boundary conditionsau.0/Cbu0.0/D
p 2
X
iD1
ciu.i/andcu.1/Cdu0.1/D
p 2
X
iD1
diu.i/;it can be verified that
a
p 2
X
iD1
ci
e1C
b
p 2
X
iD1
cii
e2D
p 2
X
iD1
ciFu.i/; (3.7)
and
c
p 2
X
iD1
di
e1C
cCd
p 2
X
iD1
dii
e2D
p 2
X
iD1
diFu.i/ Au: (3.8) Solving the linear system of equations (3.7) and (3.8) fore1ande2, we get
e1D1 ı
cCd
p 2
X
iD1
dii
p 2 X
iD1
ciFu.i/C
bC
p 2
X
iD1
cii
p 2 X
iD1
diFu.i/ Au
;
e2D1 ı
a
p 2
X
iD1
ci
p 2 X
iD1
diFu.i/ Au
c
p 2
X
iD1
di
p 2 X
iD1
ciFu.i/
:
Substituting the values ofe1 ande2in (3.4), we obtain (3.1). Conversely, ifu.t / satisfies (3.1), then it is easy to show thatusatisfies (1.1). This completes the proof.
Let us define an operatorT WP C.J;R/!P C.J;R/as
T u.t /DFu.t /C1 ı
p 2
X
iD1
ciFu.i/
cCd
p 2
X
iD1
dii
c
p 2
X
iD1
di
t
C1 ı
p 2 X
iD1
diFu.i/ Au
bC
p 2
X
iD1
ciiC
a
p 2
X
iD1
ci
t
; t2J;
(3.9)
whereFu.t /andAuare defined as in (3.2) and (3.3), respectively. Clearly, the fixed points of the operatorT are the solutions of the problem (1.1).
For the sake of convenience, we define
1D sup
t2Œ0;1
ˇ ˇ ˇ ˇ
cCd
p 2
X
iD1
dii
c
p 2
X
iD1
di
t
ˇ ˇ ˇ ˇ
jıj ;
2D sup
t2Œ0;1
ˇ ˇ ˇ ˇ
bC
p 2
X
iD1
ciiC
a
p 2
X
iD1
ci
t
ˇ ˇ ˇ ˇ
jıj ;
1D ˇ ˇ ˇ ˇ c
p 2
X
iD1
di
ˇ ˇ ˇ ˇ
jıj ; 2D ˇ ˇ ˇ ˇ a
p 2
X
iD1
ci
ˇ ˇ ˇ ˇ jıj :
A property of the operatorT defined by (3.9) is given in the next lemma.
Lemma 4. The operatorT WP C.J;R/!P C.J;R/is completely continuous.
Proof. The operator T is continuous in view of the continuity of the functions f; Ik; INk and IQk: Let ˝ be a bounded subset of P C.J;R/: There exist positive constantsMi; i D1; 2; 3; 4;such thatjf .t; u/j M1;jIk.u/j M2;ˇ
ˇINk.u/ˇ ˇM3
andˇ ˇIQk.u/ˇ
ˇM4for allu2˝:
Thus, we have foru2˝that jFu.i/j
Z i 0
.i s/˛ 1
.˛/ jf .s; u.s//jdsC X
0<tk<i
jIk.u.tk//j
C X
0<tk<i
ˇ
ˇINk.u.tk//ˇ
ˇ.i tk/C X
0<tk<i
ˇ
ˇIQk.u.tk//ˇ ˇ
.i tk/2 2 i˛
.˛C1/M1Cm M2CiM3Ci2 2 M4
!
;
and
jAuj Z 1
0
jcj.1 s/˛ 1
.˛/ Cjdj.1 s/˛ 2 .˛ 1/
jf .s; u.s//jds C jcj
m
X
kD1
jIk.u.tk//j C
m
X
kD1
ˇ
ˇINk.u.tk//ˇ
ˇjc.1 tk/Cdj C1
2
m
X
kD1
ˇ
ˇIQk.u.tk//ˇ ˇ
ˇˇc.1 tk/2C2d.1 tk/ˇ ˇ
jcj
.˛C1/C jdj .˛/
M1Cm .jcjM2C.jcj C jdj/ M3/ Cm
2 .jcj C2jdj/ :
Therefore, one can confirm fort2J that jT u.t /j
Z t 0
.t s/˛ 1
.˛/ jf .s; u.s//jdsC X
0<tk<t
jIk.u.tk//j
C X
0<tk<t
ˇ
ˇINk.u.tk//ˇ
ˇ.t tk/C X
0<tk<t
ˇ
ˇIQk.u.tk//ˇ ˇ
.t tk/2 2 C 1
jıj
p 2
X
iD1
jcij jFu.i/j ˇ ˇ ˇ ˇ ˇ
cCd
p 2
X
iD1
dii
! c
p 2
X
iD1
di
! t
ˇ ˇ ˇ ˇ ˇ C 1
jıj
p 2
X
iD1
jdij jFu.i/j C jAuj
! ˇ ˇ ˇ ˇ ˇ
bC
p 2
X
iD1
ciiC a
p 2
X
iD1
ci
! t
ˇ ˇ ˇ ˇ ˇ
M1
.˛C1/Cm
M2CM3CM4
2
C
p 2
X
iD1
.1jcij C2jdij/
"
i˛
.˛C1/M1Cm M2CiM3Ci2 2 M4
!#
C2M1
jcj
.˛C1/C jdj .˛/
C2m
jcjM2C.jcj C jdj/ M3Cjcj C2jdj
2 M4
:
Hence, the operator T is uniformly bounded. On the other hand, for any t 2Jk; 0km, one can confirm that
j.T u/0.t /j Z t
0
.t s/˛ 2
.˛ 1/ jf .s; u.s//jdsC X
0<tk<t
ˇ
ˇINk.u.tk//ˇ ˇ
C X
0<tk<t
ˇ
ˇIQk.u.tk//ˇ
ˇ.t tk/C 1 jıj
p 2
X
iD1
jcij jFu.i/j ˇ ˇ ˇ ˇ ˇ c
p 2
X
iD1
di
ˇ ˇ ˇ ˇ ˇ C 1
jıj
p 2
X
iD1
jdij jFu.i/j C jAuj
! ˇ ˇ ˇ ˇ ˇ a
p 2
X
iD1
ci
ˇ ˇ ˇ ˇ ˇ : Therefore, we have
j.T u/0.t /j M1
.˛/Cm .M3CM4/ C
p 2
X
iD1
.1jcij C2jdij/
"
i˛
.˛C1/M1Cm M2CiM3Ci2 2 M4
!#
C2M1
jcj
.˛C1/C jdj .˛/
C2m
jcjM2C.jcj C jdj/ M3Cjcj C2jdj
2 M4
:
The last inequality implies the equicontinuity of T .˝/ on Jk; kD0; 1; : : : ; m:
In accordance with the PC-type Arzela-Ascoli theoremT .˝/is relatively compact inP C.J;R/:Consequently, the operator T WP C.J;R/!P C.J;R/is completely
continuous.
The following conditions are required for the next theorem, which is concerned with the existence of solutions of (1.1).
(C1) There exist nonnegative constantshandlsuch that jf .t; u/j hCljuj; t 2J; u2RI
(C2) There exist nonnegative constantshk; lk;hNk;lNk;hQk;lQk such that foru2R;
kD1; 2; : : : ; m,
jIk.u/j hkClkjuj;ˇ ˇINk.u/ˇ
ˇ NhkC Nlkjuj;ˇ ˇIQk.u/ˇ
ˇ QhkC Qlkjuj: Let us define
KD 1C2jcj
.˛C1/C2jdj .˛/ C
p 2
X
iD1
.1jcij C2jdij/ i˛ .˛C1/
! l
C
p 2
X
iD1
.1jcij C2jdij/ X
0<tk<i
lkCilNkCi2 2 lQk
!
C
m
X
kD1
.1C jcj2/ lkC.1C2.jcj C jdj//lNkC1C2.jcj C2jdj/
2 lQk
:
(3.10)
The numberKdefined by (3.10) is used in the following theorem.
Theorem 2. Assume that the conditions.C1/and.C 2/are valid. IfK< 1;then the problem(1.1)has at least one solution.
Proof. First of all, we know by Lemma 4 that the operator T W P C.J;R/ ! P C.J;R/defined by equation (3.9) is completely continuous. For01;con- sider the equation
uDT u: (3.11)
Ifuis a solution of (3.11), then we have fort2J that jFu.i/j
Z i 0
.i s/˛ 1
.˛/ .hClju.s/j/ dsC X
0<tk<i
hkClkju.tk/j
C X
0<tk<i
hNkC Nlkju.tk/j
.i tk/C X
0<tk<i
hQkC Qlkju.tk/j.i tk/2 2 i˛h
.˛C1/C X
0<tk<i
hkCihNkCi2 2 hQk
!
C i˛l .˛C1/kuk
C X
0<tk<i
lkCilNkCi2 2 lQk
! kuk
and
jAuj Z 1
0
jcj.1 s/˛ 1
.˛/ Cjdj.1 s/˛ 2 .˛ 1/
jf .s; u.s//jds C jcj
m
X
kD1
jIk.u.tk//j C
m
X
kD1
ˇ
ˇINk.u.tk//ˇ
ˇjc.1 tk/Cdj C1
2
m
X
kD1
ˇ
ˇIQk.u.tk//ˇ ˇ
ˇˇc.1 tk/2C2d.1 tk/ˇ ˇ
jcj
.˛C1/C jdj .˛/
.hClkuk/ C
m
X
kD1
jcjhkC.jcj C jdj/hNkC1
2.jcj C2jdj/hQk
C
m
X
kD1
jcjlkC.jcj C jdj/lNkC1
2.jcj C2jdj/lQk
kuk: Therefore, we have
ju.t /j DjT u.t /j
Z t 0
.t s/˛ 1
.˛/ .hClju.s/j/ dsC X
0<tk<t
.hkClkju.tk/j/
C X
0<tk<t
hNkC Nlkju.tk/j
.t tk/C X
0<tk<t
hQkC Qlkju.tk/j.t tk/2 2 C1
p 2
X
iD1
jcij jFu.i/j C2 p 2
X
iD1
jdij jFu.i/j C2jAuj
and thus,
ju.t /j 1C2jcj
.˛C1/C2jdj .˛/ C
p 2
X
iD1
.1jcij C2jdij/ i˛ .˛C1/
! h
C
p 2
X
iD1
.1jcij C2jdij/ X
0<tk<i
hkCihNkCi2 2 hQk
!
C
m
X
kD1
.1C2jcj/ hkC.1C2.jcj C jdj//hNkC1C2.jcj C2jdj/
2 hQk
C
1C2jcj
.˛C1/C2jdj .˛/C
p 2
X
iD1
.1jcij C2jdij/ i˛ .˛C1/
! l C
m
X
kD1
.1C2jcj/ lkC.1C2.jcj C jdj//lNkC1C2.jcj C2jdj/
2 lQk
C
p 2
X
iD1
.1jcij C2jdij/ X
0<tk<i
lkCilNkCi2 2 lQk
! kuk: Hence, we have fort2J that
ju.t /j KCKkuk; where
KD 1C2jcj
.˛C1/C2jdj .˛/ C
p 2
X
iD1
.1jcij C2jdij/ i˛ .˛C1/
! h
C
p 2
X
iD1
.1jcij C2jdij/ X
0<tk<i
hkCihNkCi2 2hQk
!
C
m
X
kD1
.1C2jcj/ hkC.1C2.jcj C jdj//hNkC1C2.jcj C2jdj/
2 hQk
: BecauseK< 1, the last inequality implies that
kuk K 1 K:
Therefore, any solution of (3.11) is bounded regardless of the value of:According to Theorem1, the operatorT has a fixed point. Consequently, the fractional impulsive
multi-point BVP (1.1) has at least one solution.
Now, we will use the Banach contraction mapping principle to prove the existence and uniqueness of solutions for problem (1.1).
The following conditions are required.
(C3) There exists a positive numberLsuch thatjf .t; u/ f .t; v/j Lju vjfor allt2J andu; v2RI
(C4) There exist positive numbers L1; L2;and L3 such that jIk.u/ Ik.v/j L1ju vj; ˇ
ˇINk.u/ INk.v/ˇ
ˇ L2ju vj, andˇ
ˇIQk.u/ IQk.v/ˇ
ˇL3ju vj for allu; v2R;andkD1; 2; : : : ; m:
Theorem 3. Suppose that the conditions.C 3/,.C 4/hold, and
L <1 4
1C2jcj CPp 2
iD1.1jcij C2jdij/ i˛
.˛C1/ C2jdj
.˛/
! 1
;
L1< 1 4m
p 2
X
iD1
.1jcij C2jdij/C1C2jcj
! 1
;
L2< 1 4m
p 2
X
iD1
.1jcij C2jdij/ iC1C2.jcj C jdj/
! 1
;
L3< 1 2m
p 2
X
iD1
.1jcij C2jdij/ i2C1C2.jcj C2jdj/
! 1 : Then, the problem(1.1)has a unique solution.
Proof. Letu; v2P C.J;R/:For eacht2J;we have jT u.t / T v.t /j jFu.t / Fv.t /j
C 1 jıj
p 2
X
iD1
jcij jFu.i/ Fv.i/j ˇ ˇ ˇ ˇ
cCd
p 2
X
iD1
dii
c
p 2
X
iD1
di
t
ˇ ˇ ˇ ˇ C 1
jıj
p 2
X
iD1
jdij jFu.i/ Fv.i/j C jAu Avj
!
ˇ ˇ ˇ ˇ
bCPp 2 iD1ciiC
a Pp 2 iD1ci
t
ˇ ˇ ˇ ˇ
1C2jcjCPp 2
iD1.1jcijC2jdij/i˛
.˛C1/ C2jdj .˛/
L C
p 2 X
iD1
.1jcij C2jdij/C1C2jcj
mL1
C
Pp 2
iD1.1jcij C2jdij/ iC1C2.jcj C jdj/
mL2
C p 2
X
iD1
.1jcij C2jdij/ i2C1C2.jcj C2jdj/ mL3
2
ku vk:
Since the inequality
1C2jcj CPp 2
iD1.1jcij C2jdij/ i˛
.˛C1/ C2jdj
.˛/
L C
p 2 X
iD1
.1jcij C2jdij/C1C2jcj
mL1
C
Pp 2
iD1.1jcij C2jdij/ iC1C2.jcj C jdj/
mL2
C p 2
X
iD1
.1jcij C2jdij/ i2C1C2.jcj C2jdj/ mL3
2
< 1
holds, T is a contraction mapping. According to the Banach contraction mapping principle,T has a unique fixed point inP C.J;R/;which is a solution of the problem
(1.1).
Examples that support the theoretical results are presented in the next section.
4. EXAMPLES
Example 1
Let us take into account the fractional order impulsive BVP 8
ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆ<
ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆ:
CD
9 4
0Cu.t /D1 2cos
p t
C 3
20tanh.u.t //; t2Œ0; 1n ft1; t2g; u .tk/D. 1/k
5 C 1
40u.tk/; u0.tk/D 1
10sin.u.tk//;
u00.tk/D 1
30arctan.u .tk//; kD1; 2;
3u.0/C2u0.0/D 8u 1
4
C4u 3
4
; u.1/ 2u0.1/D7u
1 4
u
3 4
; u00.0/D0;
(4.1)
wheret1D 3
10 andt2D4 5:
BVP (4.1) is in the form of (1.1) with˛D9
4; mD2; pD4; aD 3; bD2; cD1;
dD 2 c1D 8; c2D4; d1D7; d2D 1; 1D1
4; 2D3
4; f .t; u/D1
2cosp t
C 3
20tanh.u/; Ik.u/D. 1/k
5 C 1
40u;INk.u/D 1
10sin.u/;andIQk.u/D 1
30arctan.u/:It can be verified thatıD7; 1D3
7;and2D1
7 for (4.1). Moreover, conditions.C1/
and.C 2/are satisfied withhD1
2,lD 3
20,h1Dh2D1
5; l1Dl2D 1
40;hN1D Nh2D hQ1D Qh2D0,lN1D Nl2D 1
10;lQ1D Ql2D 1
30. Therefore, according to Theorem2, BVP (4.1) has at least one solution.
Example 2
Consider the fractional order impulsive BVP 8
ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆ<
ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆ:
CD
5 2
0Cu.t /DsintC 7
200arctan.u.t //; t2Œ0; 1n 3
4
; u
3 4
D eu.34/ 50
1Ceu.34/; u0 3
4
D 1 100u
3 4
; u00
3 4
D 1
20cosu 3
4
; u.0/C1
2u0.0/D u 1
2
C8u 7
8
; u.1/Cu0.1/D4u
1 2
8u
7 8
; u00.0/D0:
(4.2)
One can confirm that (4.2) is in the form of (1.1) with ˛ D 5
2; mD1; t1D 3 4; pD4; aDd D1; bD1
2; cD 1Dc1; c2D8; d1D4; d2D 8; 1D1
2; 2D7 8; f .t; u/DsintC 7
200arctan.u/; Ik.u/D eu
50.1Ceu/;INk.u/D 1
100u;andIQk.u/D 1
20cosu:One can confirm thatıD 12; 1D 5
12;and2D1
2 for (4.2). Conditions of Theorem3are valid for the problem withLD0:035; L1D0:02; L2D0:01and L3D0:05:Thus, the fractional impulsive BVP (4.2) has a unique solution.
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Author’s address
Fatma Tokmak Fen
Department of Mathematics, Gazi University, 06500 Teknikokullar, Ankara, Turkey E-mail address:fatma.tokmakk@gmail.com
