Applied Mathematics E-Notes, 13(2013), 109-119 c
Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/
ISSN 1607-2510
On Mild Solutions Of Nonlocal Semilinear Impulsive
Functional Integro-Di¤erential Equations
Rupali Shikharchand Jainy, Machindra Baburao Dhaknez
Received 4 June 2013
Abstract
In the present paper, we investigate the existence, uniqueness and continuous dependence on initial data of mild solutions of …rst order nonlocal semilinear
functional impulsive integro-di¤erential equations of more general type with …nite
delay in Banach spaces. Our analysis is based on semigroup theory and Banach
contraction theorem.
1
Introduction
Impulsive equations arise in many di¤erent real processes and phenomena which appears in physics, population dynamics, medicine, economics etc. The study of impulsive
functional di¤erential equations is linked to their utility in stimulating processes and
phenomena subject to short time perturbations during their evolution. The perturbations are performed discretely and their duration is negligible in comparison with
the total duration of the processes and phenomena. That is the reason for the perturbations to be considered as taking place instantaneously in the form of impulses.
Impulsive di¤erential equations in recent years have been the object of investigation
with increasing interest. For more information see the monographs, Lakshmikantham
[14], Samoilenko and Perestyuk [14], the research papers [5, 16] and the references cited
therein.
Also, the problems of existence, uniqueness and other qualitative properties of solutions for semilinear di¤erential equations in Banach spaces has been studied extensively
in the literature for last many years, see [1, 3, 4, 6, 7, 9–13, 15, 17]. On the other hand,
di¤erential equations with nonlocal condition have been studied by many researchers as
they are more precise to describe natural phenomena than di¤erential equations with
classical initial condition, for example see [1, 4, 6, 9, 10, 12] and the references cited
therein.
Mathematics Subject Classi…cations: 45J05, 45N05, 47H10 ,47B38.
of Mathematical Sciences, Swami Ramanand Teerth Marathwada University, Nanded431606, Maharashtra, India
z Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad431004, India
y School
109
110
On Mild Solutions of Impulsive Functional Equations
Recently, Li et. al [17] studied fractional integro-di¤erential equations of the type:
q
D x(t) = Ax(t) + f (t; x(t) +
Z
t
k(t; s)h(t; s; x(s)ds);
0
x(0) = g(x) + x0 ;
using measure of noncompactness and …xed point theorem of condensing maps. Also,
Radhakrishnan and Bhalchandran [2] investigated controllability results for the impulsive integro-di¤erential evolution system with nonlocal condition of the form:
Z t
0
x (t) = A(t)x(t) + Bu(t) + f t; x(t);
a(t; s; x(s)ds ; t 2 J, t 6= ti ;
0
x(0) = g(x) + x0 ;
xjt=ti = Ii x(ti ); i = 1; 2; ::; m:
Ahmed et. al [8] obtained existence results of mild solutions of integro-di¤erential equations:
Z t
0
u (t) = Au(t)) + f t; u(t);
H(t; s)u(s)ds ; t 2 [0; K];
0
u(0) = g(u) + u0 ;
u(ti ) = Ii u(ti ); i = 1; 2; ::; m;
using measure of noncompactness and new …xed point theorem. In the present paper
we consider semilinear functional impulsive integro-di¤erential equation of …rst order
of the type:
Z t
x0 (t) = Ax(t) + f t; xt ;
k(t; s)h(s; xs )ds ; t 2 (0; T ]; t 6= k ; k = 1; 2; ::; m; (1)
0
x(t) + (g(xt1 ; :::; xtp ))(t) = (t);
x(
k)
= Ik x(
k );
r
k = 1; 2; :::; m;
t
0;
(2)
(3)
where 0 < t1 < t2 < ::: < tp
T , p 2 N, A is the in…nitesimal generator of strongly
continuous semigroup of bounded linear operators fT (t)gt 0 and Ik (k = 1; 2; :::; m) are
the linear operators acting in a Banach space X. The functions f; h; g; k and are
given functions satisfying some assumptions. The impulsive moments k are such that
0
T , m 2 N and x( k ) = x( k + 0) x( k 0)
0 < 1 < 2 < ::: < m < m+1
where x( k + 0) and x( k 0) are, respectively, the right and the left limits of x at k .
Equations of the form (1)-(3) or their special forms arise in some physical applications as a natural generalization of the classical initial value problems. Our aim of
the present paper is to generalize and extend results reported in [5, 16]. The models
in this paper are studied for impulse e¤ect. We study the existence, uniqueness and
continuous dependence of mild solution of nonlocal IVP problem for an impulsive functional integro-di¤erential equation. The main tool used in our analysis is based on an
application of the Banach contraction theorem and semigroup theory.
R. S. Jain and M. B. Dhakne
111
As usual, in the theory of impulsive di¤erential equations at the points of discontinuity i of the solution t ! x(t), we assume that x( i ) x( i 0). It is clear that, in
general the derivatives x0 ( i ) do not exist. On the other hand, from (1), there exist the
limits x0 ( i 0). According to the above convention, we assume x0 ( i ) = x0 ( i 0).
This paper is organized as follows. Section 2 presents the preliminaries and hypotheses. In Section 3, we prove existence and uniqueness of mild solution. In section
4, we prove continuous dependence of solutions on initial data. Finally, in section 5,
we give application based on our result .
2
Preliminaries and Hypotheses
Let X be a Banach space with the norm k k. Let C = C([ r; 0]; X); 0 < r < 1; be
the Banach space of all continuous functions : [ r; 0] ! X endowed with supremum
norm k kC = supfk (t)k : r t 0g and B denote the set fx : [ r; T ] ! X : x(t)
is continuous at t 6= k , left continuous at t = k , and the right limit x( k + 0) exists
for k = 1; 2; :::; mg. Clearly, B is a Banach space with the supremum norm kxkB =
supfkx(t)k : t 2 [ r; T ]nf 1 ; 2 ; :::; m gg. For any x 2 B and t 2 [0; T ]nf 1 ; 2 ; :::; m g,
we denote xt the element of C given by xt ( ) = x(t + ) for 2 [ r; 0] and is a given
element of C.
In this paper, we assume that, there exist positive constant K
1 such that
kT (t)k
K for every t 2 [0; T ]. Also assume k : [0; T ] [0; T ] ! R is continuous
function and as the set [0; T ] [0; T ] is compact, there exists a constant L > 0 such
that jk(t; s)j L for 0 s t T .
DEFINITION 2.1. A function x 2 B satisfying the equations
x(t)
=
T (t) (0) T (t)(g(xt1 ; :::; xtp ))(0)
Z t
Z s
+
T (t s)f s; xs ;
k(s; )h( ; x )d
0
0
X
+
T (t
k )Ik x( k )
0<
ds
k <t
for t 2 (0; T ], and
x(t) + (g(xt1 ; :::; xtp ))(t) = (t) for
r
t
0;
is said to be the mild solution of the initial value problem (1)-(3).
REMARK. A mild solution of equations (1)-(3) satis…es (2) and (3). However, a
mild solution may not be di¤erentiable at zero.
The following inequality will be useful while proving our result.
112
On Mild Solutions of Impulsive Functional Equations
LEMMA 2.2 ([15, p.12]). Let a nonnegative piecewise continuous function u(t)
satis…es the inequality
u(t)
Z
D+
t
v(s)u(s)ds +
t0
X
t0 <
i u( i )
i <t
for t t0 where D 0, i 0, v(t) > 0 and i are the …rst kind discontinuity points
of the function u(t). Then the following estimate holds for the function u(t),
u(t)
D
Y
t0 <
(1 +
i ) exp
Z
t
v(s)ds :
to
i <t
We list the following hypotheses for our convenience.
(H1 ) Let f : [0; T ] C X ! X such that for every w 2 B, x 2 X and t 2 [0; T ],
f ( ; wt ; x) 2 B and there exists a constant F > 0 such that
kf (t; ; x)
f (t; ; y)k
F (k
kC + kx
yk);
;
2 C; x; y 2 X:
(H2 ) Let h : [0; T ] C ! X such that for every w 2 B and t 2 [0; T ], h( ; wt ) 2 B and
there exists a constant H > 0 such that
kh(t; )
h(t; )k
Hk
kC ;
(H3 ) Let g : C p ! C such that there exists a constant G
k(g(xt1 ; xt2 ; :::; xtp ))(t)
(g(yt1 ; yt2 ; :::; ytp ))(t)k
;
2 C:
0 satisfying
Gkx
ykB ; t 2 [ r; 0]:
(H4 ) Let Ik : X ! X are functions such that there exists constants Lk satisfying
kIk (v)k
3
Lk kvk; v 2 X; k = 1; 2; :::; m:
Existence and Uniqueness
We have the following
THEOREM 3.1. Suppose that the hypotheses (H1 )–(H4 ) are satis…ed and
where
X
= KG + KF [1 + LHT ]T + K
Lk :
0<
k <t
Then the initial-value problem (1)-(3) has a unique mild solution x on [ r; T ].
PROOF. We introduce an operator F on a Banach space B as follows
<1
R. S. Jain and M. B. Dhakne
113
8
>
< (t) (g(xt1 ; :::; xtp ))(t) if r R t 0;
Rs
t
(Fx)(t) = T (t)[ (0) g(xt1 ; :::; xtp )(0)] + 0 T (t s)f s; xs ; 0 k(s; )h( ; s )d
>
: P
+ 0< k <t T (t
k )Ik x( k ) if t 2 (0; T ]:
ds
It is easy to see that F : B ! B. Now we will show that F is a contraction on B. Let
x; y 2 B. Then
k(Fx)(t)
(Fy)(t)k = kg(xt1 ; :::; xtp )(t)
g(yt1 ; :::; ytp )(t)k
Gkx
ykB
(4)
for t 2 [ r; 0] and
k(Fx)(t)
(Fy)(t)k
T (t) g(xt1 ; :::; xtp )(0) g(yt1 ; :::; ytp )(0)
Z s
t
+
T (t s)f s; xs ;
k(s; )h( ; x )d
0
0
Z s
f s; ys ;
k(s; )h( ; y )d
ds
0
X
+
T (t
Ik y( k )]k
k )[Ik x( k )
Z
0<
k <t
kT (t)kkg(xt1 ; :::; xtp )(0) g(yt1 ; :::; ytp )(0)k
Z t
Z s
+
kT (t s)k f s; xs ;
k(s; )h( ; x )d
0
0
Z s
ds
f s; ys ;
k(s; )h( ; y )d
0
X
+
kT (t
Ik y( k )]k
k )kk[Ik x( k )
0<
k <t
KGkx
for t 2 (0; T ] where
Z t
J1 =
kT (t
0
f
K
s; ys ;
Z
s)kk f
Z
s
Z
[kxs
[kx
Z
Z
k(s; )h( ; x )d
ds
s
0
ys kC + L
t
0
KF
ys kC +
t
0
KF
s
k(s; )h( ; y )d
F [kxs
Z
(5)
0
0
KF
Z
s; xs ;
0
t
ykB + J1 + J2
Z
jk(s; )jkh( ; x )
h( ; y )kd ]ds
s
Hkx
0
ykB + LHT kx
y kC d ]ds
ykB ]ds
t
[1 + LHT ]kx
ykB ds
0
KF [1 + LHT ]T kx
ykB
(6)
114
On Mild Solutions of Impulsive Functional Equations
and
J2 =
X
0<
k <t
0<
k <t
X
kT (t
k )kkIk x( k )
KkIk x(
k)
X
K
0<
k <t
0<
k <t
K
X
Lk kx(
Lk kx
k)
Ik y(
y(
Ik y(
k )k
k )k
k )k
ykB :
(7)
Using (6)-(7), inequality (5) becomes
k(Fx)(t) (Fy)(t)k
KGkx ykB + KF [1 + LHT ]T + K
X
0<
Lk
k <t
!
kx ykB (8)
for t 2 [0; T ]: As K 1, in view of inequality (4) and (8), we can say that inequality
(8) holds good for t 2 [ r; T ]. Therefore, for t 2 [ r; T ],
!
X
k(Fx)(t) (Fy)(t)k KGkx ykB + KF [1 + LHT ]T + K
Lk kx ykB
0<
X
KG + KF [1 + LHT ]T + K
0<
k <t
Lk
!
k <t
kx
ykB ;
which implies
kFx
where
FykB
kx
ykB
= KG + KF [1 + LHT ]T + K
X
0<
Lk :
k <t
Since < 1, the operator F satis…es all the assumptions of Banach contraction theorem
and therefore F has unique …xed point in the space B and clearly it is the mild solution
of nonlocal IVP problem (1)-(3) with impulse e¤ect. This completes the proof.
4
Continuous Dependence on Initial Data
We have the following
THEOREM 4.1. Suppose that hypotheses (H1 )-(H4 ) are satis…ed and < 1. Then
for each 1 ; 2 2 C and for the corresponding mild solutions x1 and x2 of the problems
Z t
x0 (t) = Ax(t) + f t; xt ;
k(t; s)h(s; xs )ds ; t 2 (0; T ];
(9)
0
x(
k ) = Ik x(
x(t) + g(xt1 ; :::; xtp )(t) =
i (t);
k ); k = 1; 2; :::; m;
i = 1; 2; t 2 [ r; 0];
(10)
(11)
R. S. Jain and M. B. Dhakne
115
the following inequality holds
Q
K
kx1
0<
x2 kB
(1 + KLk ) exp(KF T )
k <t
Q
1
0<
where
k
(1 + KLk ) exp(KF T )
2 kC
1
(12)
k <t
= GK + KF LHT 2 :
PROOF. Let 1 ; 2 2 B be arbitrary functions and let x1 and x2 be the mild
solutions of the problem(9)-(11). Then we have
h
i
x1 (t) x2 (t) = T (t)[ 1 (0)
(0)]
T
(t)
g(x
;
:::;
x
)(0)
g(x
;
:::;
x
)(0)
1t1
1tp
2t1
2tp
2
Z t
Z s
+
T (t s) f s; x1s ;
k(s; )h(s; x1s )d
0
0
Z s
f s; x2s ;
k(s; )h(s; x2s )d
ds
0
X
+
T (t
Ik x2 ( k ))
(13)
k )(Ik x1 ( k )
0<
k <t
for t 2 (0; T ] and
x1 (t)
x2 (t) =
1 (t)
2 (t)
[g(x1t1 ; :::; x1tp )(t)
g(x2t1 ; :::; x2tp )(t)]
(14)
for t 2 [ r; 0]. From (13) and using hypothesis (H1 )-(H4 ), we get
kx1 (t)
x2 (t)k
Kk
+ HL
2 kC + GKkx1
1
Z
kx1
x2 k C d
1
X
0<
0
0<
k <t
F kx1s
Lk kx1 (
k)
x2s kC
x2 (
k )k
Z t
x2 k B + K
F kx1s x2s kC ds
0
X
x2 k B + K
Lk kx1 ( k ) x2 ( k )k
+ F KLHT 2 kx1
+K
ds + K
t
2 kC + GKkx1
1
Kk
Z
X
s
0
Kk
x2 k B + K
k <t
2 kC
0<
+ kx1
Lk kx1 (
k)
k <t
x2 kB + KF
x2 (
Z
0
t
kx1s
x2s kC ds
k )k:
(15)
Simultaneously, by (14) and hypothesis (H3 ),we get
kx1 (t)
x2 (t)k
k
1
2 kC
+ Gkx1
x2 kB ; t 2 [ r; 0]:
(16)
116
On Mild Solutions of Impulsive Functional Equations
Since K
1, the inequalities (15) and(16) imply, for t 2 [ r; T ]
Z t
kx1 (t) x2 (t)k Kk 1
k
+
kx
x
k
+
KF
kx1s
1
2 B
2 C
0
X
+K
Lk kx1 ( k ) x2 ( k )k; 0 < k t;
0<
x2s kC ds
t 2 [0; T ]:
k <t
(17)
De…ne the function z : [ r; T ] ! R by z(t) = supfkx1 (s) x2 (s)k : r s tg; t 2
[0; T ]. Let t 2 [ r; t] be such that z(t) = kx1 (t ) x2 (t )k. If t 2 [0; t], then from
inequality (17), we have
z(t) = kx1 (t )
Kk
+K
k
z(t)
2 kC
1
X
0<
for 0 <
x2 (t )k
k <t
+ kx1
Lk kx1 (
k)
x2 kB + KF
x2 (
Z
t
0
kx1s
x2s kC ds
k )k
t and
Kk
2 kC + kx1
1
x2 kB + KF
Z
t
z(s)ds + K
0
X
0<
Lk z(
k)
(18)
k <t
Now applying Lemma 2.2 to the inequality (18), we get
Y
z(t) (Kk 1
x2 kB )
(1 + KLk ) exp(KF T ):
2 kC + kx1
0<
k <t
Hence, we get
kx1
x2 kB
(Kk
1
2 kC
+ kx1
x2 kB )
Y
0<
(1 + KLk ) exp(KF T ):
k <t
The inequality given by (12) is the immediate consequence of the above inequality.
This completes the proof.
5
Application
To illustrate the application of our result proved in section 3, consider the following
semilinear partial functional di¤erential equation of the form
Z t
@
@2
w(u; t) =
w(u;
t)
+
H
t;
w(u;
t
r);
k(t; s)P (s; w(s r))ds ;
(19)
@t
@u2
0
w(0; t) = w( ; t) = 0; 0
w(u; ) +
p
X
t
T;
w(u; ti + ) = (u; t);
(20)
(21)
i=1
w(u;
k)
= Ik (w(u;
k ));
k = 1; 2; :::; m;
(22)
R. S. Jain and M. B. Dhakne
117
for 0 u
; 0 t T and r
0 where 0 < t1 t2 tp T , the functions
H : [0; T ] R R ! R, P : [0; T ] R ! R and Ik : R ! R are continuous . We assume
that the functions H, P and Ik satisfy the following conditions: For every t 2 [0; T ]
and u; v; x; y 2 R, there exists positive constants l, p, ck and d such that
jH(t; u; x)
H(t; v; y)j
jP (t; u)
P (t; v)j
jIk (x)j
p
X
l(ju
vj + jx
p(ju
vj);
ck jxj ; k = 1; 2; :::; m;
jw(u; ti + t)j
d;
Kd + Kl(1 + LpT )T + K
X
i=1
and
yj);
0<
ck < 1:
k <t
00
Let us take X = L2 [0; ]. De…ne the operator A : X ! X by Az = z with domain
0
00
D(A) = fz 2 X : z; z are absolutely continuous, z 2 X and z(0) = z( ) = 0g. Then
the operator A can be written as
Az =
1
X
n2 (z; zn )zn ; z 2 D(A)
1
X
exp( n2 t)(z; zn )zn ; z 2 X:
n=1
p
where zn (u) = ( 2= ) sin nu; n = 1; 2; ::: is the orthogonal set of eigenvectors of A and
A is the in…nitesimal generator of an analytic semigroup T (t); t 0 and is given by
T (t)z =
n=1
Now, the analytic semigroup T (t) being compact, there exists constant K such that
jT (t)j
De…ne the functions f : [0; T ]
C
K; for each t 2 [0; T ]:
X ! X, h : [0; T ]
C ! X, Ik : X ! X as follows
f (t; ; x)(u) = H(t; ( r)u; x(u));
h(t; )(u) = P (t; ( r)u);
for t 2 [0; T ]; ; 2 C, x 2 X and 0
u
. With these choices of the functions
the equations (19)-(22) can be formulated as an abstract integro-di¤erential equation
in Banach space X:
Z t
0
x (t) = Ax(t) + f t; xt ;
k(t; s)h(s; xs )ds ; t 2 [0; T ];
0
x(t) + (g(xt1 ; :::; xtp ))(t) = (t); t 2 [ r; 0]:
Since all the hypotheses of the theorem 3.1 are satis…ed, the theorem 3.1, can be applied
to guarantee the existence of mild solution w(u; t) = x(t)u, t 2 [0; T ] and u 2 [0; ], of
the semilinear partial integro-di¤erential equation (19)-(22).
118
On Mild Solutions of Impulsive Functional Equations
References
[1] L. Byszewski and H. Akca, Existence of solutions of a semilinear functional evolution nonlocal problem, Nonlinear Anal., 34(1998), 65–72.
[2] B. Radhakrishnan and K. Bhalchandran, Controllability results for semilinear impulsive integro-di¤erential evolution systems with nonlocal conditions, J. Control
Theory Appl., 10(1)(2012), 28–34.
[3] T. A. Burton, Volterra Integral and Di¤erential Equations, Academic Press, New
York, 1983.
[4] L. Byszewski, On mild solution of a semilinear functional di¤erential evolution
nonlocal problem, J. Appl. Math. Stoch. Anal., 10(3)(1997), 265–271.
[5] H. Acka, A. Boucherif and V. Covachev, Impulsive functional di¤erential equations
with nonlocal conditions, Int. J. Math. Math. Sci. 29(2002), 251–256.
[6] R. S. Jain and M. B. Dhakne, On global existence of solutions for abstract nonlinear functional integro-di¤erential equations with nonlocal condition, Contempoary
Math. and stat., 1(2013), 44–53.
[7] J. Hale, Theory of Functional Di¤erential Equations, Applied Mathematical Sciences, Vol. 3. Springer-Verlag, 1977.
[8] B. Ahmed, K. Malar and K. Karthikeyan, A study of nonlocal problems of impulsive integrodi¤erential equations with measure of noncompactness, Adv. Di¤erence
Equ., 2013:205.
[9] L. Byszewski and V. Lakshamikantham, Theorems about the existence and uniqueness of a solution of a nonlocal abstract cauchy problem in a Banach space, Appl.
Anal. 40(1991), 11–19.
[10] Y. Lin and J. H. Liu, Semilinear integrodi¤erential equations with nonlocal Cauchy
problem, Nonlinear Analysis: Theory, Methods and appl., 26(1996), 1023–1033.
[11] S. K. Ntouyas, Initial and boundary value problems for functional di¤erential
equations via the topological transversality method: a survey, Bull. Greek Math.
Soc., (1998), 3–41.
[12] S. K. Ntouyas, Nonlocal initial and boundary value problems: A survey, Handbook
of di¤erential equations:Ordinary di¤erential equations, Vol.2, (2006), 461–557.
[13] A. Pazy, Semigroups of Linear Operators and Applications to Partial Di¤erential
Equations, Springer Verlag, New York, 1983.
[14] A. M. Samoilenko and N. A. Perestyuk, Impulsive Di¤erential Equations, World
Scienti…c, Singapore, 1995.
[15] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Thoery of Impulsive Differential Equations, World Scienti…c, Singapore, 1989.
R. S. Jain and M. B. Dhakne
119
[16] S. C. Ji and S. Wen, Nonlocal Cauchy problem for impulsive di¤erential equations
in Banach spaces, Int. J. Nonlinear Sci. 10 (2010), no. 1, 88–95.
[17] F. Li, J. Liang, T. T. Lu and H. Zhu, A nonlocal cauchy problem for fractional
integrodi¤erential equations, J. Appl. Math. 2012, Art. ID 901942, 18 pp.