Vol4 Iss2 102 114 EXISTENCE OF GLOBAL SOLUTIONS FOR I
J. Nonlinear Sci. Appl. 4 (2011), no. 2, 102–114
The Journal of Nonlinear Science and Applications
http://www.tjnsa.com
EXISTENCE OF GLOBAL SOLUTIONS FOR IMPULSIVE
FUNCTIONAL DIFFERENTIAL EQUATIONS WITH
NONLOCAL CONDITIONS
S. SIVASANKARAN1 , M. MALLIKA ARJUNAN2 AND V. VIJAYAKUMAR3,∗
Dedicated to Professor Themistocles M. Rassias on the occasion of his sixtieth birthday
Communicated by Choonkil Park
Abstract. In this paper, we study the existence of global solutions for a class
of impulsive abstract functional differential equation with nonlocal conditions.
The results are obtained by using the Leray-Schauder alternative fixed point
theorem. An example is provided to illustrate the theory.
1. Introduction
In this work, we discuss the existence of global solutions for an impulsive abstract functional differential equation with nonlocal conditions in the form
u′ (t) = Au(t) + f (t, ut , u(ρ(t))), t ∈ I = [0, ∞),
u0 = g(u, ϕ) ∈ B,
(1.1)
(1.2)
∆u(ti ) = Ii (uti ),
(1.3)
where A is the infinitesimal generator of a C0 -semigroup of bounded linear operators (T (t))t≥0 defined on a Banach space (X, k · k); I is an interval of the
form [0, ∞); 0 < t1 < t2 < · · · < ti < · · · are pre-fixed numbers; the history
ut : (−∞, 0] → X, ut (θ) = u(t + θ), belongs to some abstract phase space B
defined axiomatically; ρ : [0, ∞) → [0, ∞), f : I × B × X → X, Ii : B → X are
Date: Received: September 19, 2010; Revised: November 24, 2010.
∗
Corresponding author
c 2011 N.A.G.
°
2000 Mathematics Subject Classification. Primary 34A37, 34K30, 35R10, 35R12, 47D06,
49K25.
Key words and phrases. Impulsive functional differential equations; mild solutions; global
solutions; semigroup theory.
102
IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS
103
appropriate functions and the symbol ∆ξ(t) represents the jump of the function
ξ at t, which is defined by ∆ξ(t) = ξ(t+ ) − ξ(t− ).
Differential equations arise in many real world problems such as physics, population dynamics, ecology, biological systems, biotechnology, optimal control, and
so forth. Much has been done the assumption that the state variables and systems parameters change continuously. However, one may easily visualize that
abrupt changes such as shock, harvesting, and disasters may occur in nature.
These phenomena are short-time perturbations whose duration is negligible in
comparison with the duration of the whole evolution process. Consequently, it
is natural to assume, in modeling these problems, that these perturbations act
instantaneously, that is in the form of impulses. The theory of impulsive differential equations has become an active area of investigation due to their applications
in the field such as mechanics, electrical engineering, medicine biology and so on;
see for instance [1, 3, 4, 7, 8, 9, 23, 26, 30, 35, 37, 38]. For more details on this
theory and on its applications we refer to the monographs of Benchohra [10],
Lakshmikantham [29], Bainov and Simeonov [5] and Samoilenko and Perestyuk
[39], and the papers of [2, 15, 16, 17, 18, 25, 31], where numerous properties of
their solutions are studied and detailed bibliographies are given.
The literature concerning differential equations with nonlocal conditions also
include ordinary differential equations, partial differential equations, abstract partial functional differential equations. Related to this matter, we cite among others
works, [6, 11, 12, 13, 14, 20, 22, 28, 33, 34]. Our main results can been seen as
a generalization of the work in [21] and the above mentioned impulsive partial
functional differential equations.
2. Preliminaries
In this paper, A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators (T (t))t≥0 defined on a Banach space (X, k · k)
and M, δ are positive constants such that kT (t)k ≤ M e−δt for every t ≥ 0. For
the theory of strongly continuous semigroup, refer the reader to Pazy [36].
To consider the impulsive condition (1.3), it is convenient to introduce some
additional concepts and notations. We say that a function u : [σ, τ ] → X is a
normalized piecewise continuous function on [σ, τ ] if u is piecewise continuous
and left continuous on (σ, τ ]. We denote by PC([σ, τ ]; X) the space formed by
all normalized piecewise continuous functions from [σ, τ ] into X. In particular,
we introduce the space PC formed by all functions u : [0, a] → X such that u
is continuous at t 6= ti , i = 1, ..., n. It is clear that, PC([σ, τ ]; X) endowed with
the norm of uniform convergence, is a Banach space. On the other hand, the
notation PC([0, ∞); X) stands for the space formed by all bounded normalized
piecewise continuous functions u : [0, ∞) → X such that u|[0,a] ∈ PC([0, a]; X)
for all a > 0, endowed with the uniform convergence topology.
In what follows, for the case I = [0, a], we set t0 = 0, tn+1 = a, and for
u ∈ PC([0, a]; X), we denote by u
ei ∈ C([ti , ti+1 ]; X), i = 0, 1, ..., n, the function
104
given by
S. SIVASANKARAN, M. MALLIKA ARJUNAN AND V. VIJAYAKUMAR
(
u(t),
u
ei (t) =
u(t+
i ),
for t ∈ (t, ti+1 ],
for t = ti .
ei , i = 0, 1, ..., n, for the sets
Moreover, for B ⊂ PC, we employ the notation B
ei = {e
B
ui : u ∈ B}.
Lemma 2.1. A set B ⊂ PC is relatively compact in PC([0, a]; X) if, and only if,
ei , is relatively compact in the space C([ti , ti+1 ]; X) for every i = 0, 1, ..., n.
the set B
In this paper, the phase space B is a linear space of functions mapping (−∞, 0]
into X endowed with a seminorm k · kB and verifying the following axioms.
(A) If x : (−∞, a) → X, a > 0, is continuous on [0, a) and x0 ∈ B, then for
every t ∈ [0, a) the following conditions hold:
(i) xt is in B.
(ii) kx(t)k ≤ Hkxt kB .
(iii) kxt kB ≤ K(t) sup{kx(s)k : 0 ≤ s ≤ t} + M (t)kx0 kB , where H > 0 is
a constant; K, M : [0, ∞) → [1, ∞), K is continuous, M is locally
bounded and H, K, M are independent of x(·).
(B) The space B is complete.
Remark 2.2. In this paper, K is a positive constant such that supt≥0 {K(t), M (t)}
≤ K. We know from that the functions M (·), K(·) are bounded if, for instance,
B is a fading memory space, see [27, pp. 190] for additional details.
Example 2.3. The phase space PC r × Lp (ρ, X)
Let r ≥ 0, 1 ≤ p < ∞, and ρ : (−∞, −r] → R be a non-negative measurable
function which satisfies the conditions (g-5)-(g-6) of [14]. Briefly, this means that
ρ is locally integrable and there exists a non-negative locally bounded function γ
on (−∞, 0] such that ρ(ξ + θ) ≤ γ(ξ)ρ(θ), for all ξ ≤ 0 and θ ∈ (−∞, −r) \ Nξ ,
where Nξ ⊆ (−∞, −r) is a set whose Lebesgue measure zero.
The space B = PC r ×Lp (ρ, X) consists of all classes of functions ψ : (−∞, 0] →
X such that ψ is continuous on [−r, 0], Lebesgue-measurable, and ρkψkp is
Lebesgue integrable on (−∞, −r). The seminorm in PC r × Lp (ρ, X) is defined as
follows:
³ Z −r
´ p1
p
kψkB = sup{kψ(θ)k : −r ≤ θ ≤ 0} +
ρ(θ)kψ(θ)k dθ .
−∞
The space B = PC r × Lp (ρ, X) satisfies axioms (A) and (B). Moreover, when
1
r = 0 and p = 2, one can take H = 1, M (t) = γ(−t) 2 and K(t) = 1 +
¢ 21
¡R0
for t ≥ 0 (see [27, Theorem 1.3.8]). We also note that if the
ρ(θ)dθ
−t
conditions (g-5)-(g-7) of [27] hold, then B is a uniform fading memory.
For additional details concerning phase space, we cite [27]
In this paper, h : [0, ∞) → R be a positive, continuous and non-decreasing
function such that h(0) = 1 and limt→∞ h(t) = ∞. In the sequel, PC([0, ∞); X)
IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS
105
and (PC h )0 (X) are the spaces defined by
x|[0,a] ∈ PC([0, a]; X),
∀a ∈ (0, +∞) \ {ti : i ∈ N},
PC([0, ∞); X) =
x : [0, ∞) → X : x0 = 0,
kxk∞ = supt≥0 kx(t)k < +∞,
n
o
kx(t)k
(PC)0h (X) = x ∈ PC([0, ∞); X) : lim
=0 ,
t→∞ h(t)
endowed with the norms kxk∞ = supt≥0 kx(t)k and kxkPC h = supt≥0
spectively.
Additionally, we define the space
n
o
BPC 0h (X) = u ∈ PC(R, X) : u0 ∈ B, u|[0,∞) ∈ PC 0h (X)
kx(t)k
,
h(t)
re-
endowed with the norm kukBPC 0h = ku0 kB + ku|[0,∞) kh .
We recall here the following compactness criterion.
Lemma 2.4. A bounded set B ⊂ (PC)0h (X) is relatively compact in (PC)0h (X) if,
and only if:
(a) The set Ba = {u|[0,a] : u ∈ B} is relatively compact in PC([0, a]; X), for
all a ∈ (0, ∞) \ {ti ; i ∈ N}.
= 0, uniformly for x ∈ B.
(b) limt→∞ kx(t)k
h(t)
Theorem 2.5. ([19, Theorem 6.5.4] Leray-Schauder Alternative Theorem). Let
D be a closed convex subset of a Banach space (Z, k · kZ ) and assume that 0 ∈ D.
Let F : D → D be a completely continuous map, then the set {x ∈ D : x = λF (x),
0 < λ < 1} is unbounded or the map F has a fixed point in D.
Theorem 2.6. ([32, Theorem 4.3.2]). Let D be a convex, bounded and closed
subset of a Banach space (Z, k · kZ ) and F : D → D be a condensing map. Then
F has a fixed point in D.
3. Existence of mild solutions
In this section, we study the existence of mild solutions for the nonlocal abstract
Cauchy problem (1.1)-(1.3). To treat this system, we introduce the following
conditions.
H1 The function ρ : [0, ∞) → [0, ∞) is continuous and ρ(t) ≤ t for every
t ≥ 0.
H2 The function f : I ×B×X → X is continuous and there exist an integrable
function m : [0, ∞) → [0, ∞) and a nondecreasing continuous function
W : [0, ∞) → (0, ∞) such that kf (t, ψ, x)k ≤ m(t)W (kψkB + kxk), for
every (t, ψ, x) ∈ [0, ∞) × B × X.
H3 The function g : BPC 0h (X) × B → B is continuous and there exists Lg ≥ 0
such that
kg(u, ϕ) − g(v, ϕ)kB ≤ Lg ku − vkBPC 0h ,
u, v ∈ BPC 0h (X).
106
S. SIVASANKARAN, M. MALLIKA ARJUNAN AND V. VIJAYAKUMAR
H4 The function g(·, ϕ) : BPC 0h (X) → B is completely continuous and uniformly bounded. In the sequel, N is a positive constant such that
kg(ψ, ϕ)kB ≤ N for every ψ ∈ BPC 0h (X).
H5 The maps Ii : B → X are completely continuous and uniformly bounded,
i ∈ N. In what follows, we use the notation Ni = sup{kIi (φ)k : φ ∈ B}.
H6 There are positive constants Li , such that
kIi (ψ1 ) − Ii (ψ2 )k ≤ Li kψ1 − ψ2 kB , ψ1 , ψ2 ∈ B, i ∈ N.
In this work, we adopt the following concept of mild solutions for (1.1)-(1.3).
Definition 3.1. A function u ∈ PC(R, X) is called a mild solution of system
(1.1)-(1.3) if u0 = g(u, ϕ) and
Z t
T (t − s)f (s, us , u(ρ(s)))ds
u(t) = T (t)g(u, ϕ)(0) +
0
X
T (t − ti )Ii (uti ), t ∈ I = [0, ∞).
+
0 0, limt→∞ h(t)
0
R ∞ ds
R∞
If M K(1 + H) 0 m(s)ds < c W (s) , where c = KN (1 + H)(1 + M H)
P
+ MK ∞
i=1 Ni , then there exists a mild solution for the system (1.1)-(1.3).
Proof. Let Γ : BPC 0h (I : X) → BPC 0h (X) be the operator defined by (Γu)0 =
g(u, ϕ) and
Z t
T (t − s)f (s, us , u(ρ(s)))ds
Γu(t) = T (t)g(u, ϕ)(0) +
0
X
+
T (t − ti )Ii (uti ), t ≥ 0.
0 0 such that
Z t
∞
2M X
M
2M N H
ε
m(s)W (2Kηh(s))ds +
+
Ni ≤ ,
h(t)
h(t) 0
h(t) i=1
2
107
t≥L
(3.1)
From the properties of the functions f , g, we can choose Nε ∈ N such that
ε
kg(xn , ϕ) − g(x, ϕ)kB ≤
,
(3.2)
6(M H + 1)
Z L
ε
,
(3.3)
kf (s, xns , xn (ρ(s))) − f (s, xs , x(ρ(s)))kds ≤
6M
0
and
X
kIi (xnti ) − Ii (xti )k ≤
0≤ti ≤L
ε
,
6M
n ≥ Nε
for every n ≥ Nε . Under these conditions, we see that
o ε
n kΓxn (t) − Γx(t)k
sup
: t ∈ [0, L], n ≥ Nε ≤ .
h(t)
2
(3.4)
(3.5)
Moreover, for t ≥ L and n ≥ Nε , we find that
∞
kg(xn , ϕ)(0) − g(x, ϕ)(0)k 2M X
kΓxn (t) − Γx(t)k
Ni
≤M
+
h(t)
h(t)
h(t) i=1
Z t
M
+
kf (s, xns , xn (ρ(s))) − f (s, xs , x(ρ(s)))kds,
h(t) 0
Z t
∞
2M X
2M N H
M
m(s)W (2Kηh(s))ds +
≤
+
Ni ,
h(t)
h(t) 0
h(t) i=1
and hence,
sup
n kΓxn (t) − Γx(t)k
h(t)
o
ε
: t ≥ L, n ≥ Nε ≤ .
2
(3.6)
From the inequalities (3.2)-(3.6), it follows that kΓxn − ΓxkBPC 0h ≤ ε for every
n ≥ Nε , which proves that Γ is continuous.
To prove that Γ is completely continuous, we introduce the decomposition
Γ = Γ1 + Γ2 + Γ3 , where (Γ1 u)0 = g(u, ϕ), (Γ2 u)0 = 0, (Γ3 u)0 = 0 and
Γ1 u(t) =T (t)g(u, ϕ)(0), t ≥ 0
Z t
T (t − s)f (s, us , u(ρ(s)))ds, t ≥ 0
Γ2 u(t) =
0
X
Γ3 u(t) =
T (t − ti )Ii (uti ), t ≥ 0
(3.7)
(3.8)
(3.9)
0 0. Next, by using Lemma 2.1, we
prove that Γ3 is also completely continuous. We observe that the continuity of
Γ3 is obvious. On the other hand, for r > 0, t ∈ [ti , ti+1 ], i ≥ 1, and u ∈ Br =
Br (0, PC([0, a]; X)), we have that there exists re > 0 such that
Pi
t ∈ (ti , ti+1 ),
Pj=1 T (t − tj )Ij (Bre(0; B)),
i
g
[Γ3 u]i (t) ∈
t = ti+1 ,
j=1 T (ti+1 − tj )Ij (Bre(0; B)),
Pi−1 T (t − t )I (B (0; B)) + I (B (0; B)), t = t ,
i
j j
re
i
re
i
j=1
where Bre(0; B) is an open ball of radius re. From condition H5 it follows that
[Γ^
3 (Br )]i (t) is relatively compact in X, for all t ∈ [ti , ti+1 ], i ≥ 1. Moreover,
using the compactness of the operators {Ii }i and the continuity of (T (t))t≥0 , we
conclude that for ε > 0, there exists δ > 0 such that
kT (t)z − T (s)zk ≤ ε,
z∈
n
[
i=1
Ii (Bre(0, B)),
(3.10)
for all ti , i = 1, · · · , n, t, s ∈ (ti , ti+1 ] with |t − s| < δ. Under these conditions,
for u ∈ Br , t ∈ [ti , ti+1 ], i ≥ 0, and 0 < |h| < δ with t + h ∈ [ti , ti+1 ], we have that
g
g
k[Γ
3 u]i (t + h) − [Γ3 u]i (t)k ≤ M nε,
t ∈ I,
(3.11)
which proves that the set of functions [Γ^
3 (Br )]i , i ≥ 0, is uniformly equicontinuous. Now, from Lemma 2.1, we conclude that Γ3 is completely continuous. Thus,
Γ is completely continuous.
To finish the proof, we obtain a priori estimates for the solutions of the integral
equation {u = λΓu, λ ∈ (0, 1)}. Let λ ∈ (0, 1) and uλ ∈ BPC 0h (X) be a solution
of λΓz = z. By using the notation αλ (t) = K(sups∈[0,t] kuλ (s)k+kuλ0 kB ), for t ∈ I
we see that
λ
ku (t)k ≤ M N H + M
Z
t
m(s)W (kuλs kB + kuλ (ρ(s))k)ds + M
0
≤ MNH + M
Z
0
∞
X
i=1
t
m(s)W ((1 + H)αλ (s))ds + M
∞
X
i=1
Ni ,
Ni ,
IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS
109
and hence,
αλ (t)
≤ Kkg(u, ϕ)kB + KM N H + KM
Z
t
λ
m(s)(W (1 + H)α (s))ds + KM
0
≤ KN (1 + M H) + KM
Z
∞
X
Ni ,
i=1
t
λ
m(s)W ((1 + H)α (s))ds + KM
0
∞
X
Ni .
i=1
Denoting by βλ the right hand side of last inequality, we see that
βλ′ (t) ≤ KM m(t)W ((1 + H)βλ (t)),
which implies that
Z
(1+H)βλ (t)
(1+H)βλ (0)=c
ds
≤ M K(1 + H)
W (s)
Z
∞
0
m(s)ds <
Z
c
∞
ds
.
W (s)
This inequality permits us to conclude that the set of functions {βλ : λ ∈ (0, 1)}
is bounded in C(R) and as consequence that the set {uλ ; λ ∈ (0, 1)} is bounded
in BPC 0h (X).
Now the assertion is a consequence of Theorem 2.5. The proof is complete. ¤
Theorem 3.3. Assume assumptions H1 , H2 , H3 , H5 and H6 be hold. If the
conditions (a) and (b) of Theorem 3.2 are valid and
Lg (1 + M H) + M lim inf
r→∞
Z
0
+∞
∞
X
m(s)W ((1 + 2K)rh(s))
ds + M
Li < 1,
rh(s)
i=1
(3.10)
then there exists a mild solution of (1.1)-(1.3).
Proof. Let Γ, Γ1 , Γ2 , Γ3 be the operators introduced in the proof of Theorem
3.2, we claim that there exists r > 0 such that Γ(Br ) ⊂ Br , where Br =
Br (0, BPC 0h (X)). In fact, if this property is false, then for every r > 0, there
110
S. SIVASANKARAN, M. MALLIKA ARJUNAN AND V. VIJAYAKUMAR
r
r
exist xr ∈ Br and tr ≥ 0 such that r < k(Γxr )0 kB + k Γxh(t(tr ) ) k. Consequently
kg(xr , ϕ)(0)k
r ≤ kg(xr , ϕ)kB + M
h(tr )
Z tr
M
m(s)W (Kkxr0 kB + K sup kxr (θ)k + kxr (ρ(s))k)ds
+
h(tr ) 0
θ∈[0,s]
∞
∞
X Li kxt k
M X
i
+
kI
(0)k
+
M
i
r
r
h(t ) i=1
h(t )
i=1
M
MH
Lg kxr kBPC 0h +
kg(0, ϕ)kB
≤ Lg kxr kBPC oh + kg(0, ϕ)kB +
r
h(t )
h(tr )
Z tr
M
m(s)W (Kr + Krh(s) + rh(s))ds
+
h(tr ) 0
∞
∞
X
M X
+ Mr
kIi (0)k
Li +
h(tr ) i=1
i=1
≤ Lg (1 + M H)r + (1 + M )kg(0, ϕ)kB + M
Z
0
and then,
1 ≤ Lg (1 + M H) + M lim inf
r→∞
Z
+∞
0
tr
m(s)((2K + 1)rh(s))
ds
h(s)
∞
X
m(s)((2K + 1)rh(s))
ds + M
Li ,
rh(s)
i=1
which is contrary to (3.10).
Let r > 0 such that Γ(Br ) ⊂ Br . From the proof of the Theorem 3.2, we know
that Γ2 is a completely continuous on Br . Moreover, from the estimate,
kΓ1 u − Γ1 vkBPC 0h ≤ Lg ku − vkBPC 0h + Hkg(u, ϕ) − g(v, ϕ)kB
≤ Lg (1 + H)ku − vkBPC 0h ,
we infer that Γ1 is a contraction on Br . It is easy to see that
∞
X
Li ku − vkBPC 0h , u, v ∈ BPC 0h ,
kΓ3 u − Γ3 vkBPC 0h ≤ M
i=1
which implies that Γ3 is a contraction in BPC 0h (X) from which we conclude that
Γ is a condensing operator on Br .
The assertion is now a consequence of Theorem 2.6.
¤
4. An application
To complete this work, we study the existence of global solutions for a concrete
partial differential equation with nonlocal conditions. In the sequel, X = L2 [0, π]
and A : D(A) ⊂ X → X is the operator Ax = x′′ with domain D(A) = {x ∈
X : x′′ ∈ X, x(0) = x(π) = 0}. It is well known that A is the infinitesimal
generator of an analytic semigroup (T (t))t≥0 on X. Furthermore, A has discrete
spectrum with eigenvalues of the form −n2 , n ∈ N, and corresponding normalized
IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS
111
1
eigenfunctions given by zn (x) = ( π2 ) 2 sin(nx); the set of functions {zn : n ∈ N}
P
−n2 t
hx, zn izn for every x ∈ X
is an orthonormal basis for X and T (t)x = ∞
n=1 e
and all t ≥ 0. It follows from this last expression that (T (t))t≥0 is a compact
semigroup on X and that kT (t)k ≤ e−t for every t ≥ 0.
As phase space, we choose the space B = PC 0 × L2 (ρ, X), see Example 2.3,
and we assume that the conditions (g5) and (g7) in [27] are verified. Under these
conditions, B is a fading memory space, and as consequence, there exists K > 0
such that supt≥0 {M (t), K(t)} ≤ K.
Consider the delayed impulsive partial differential equation with nonlocal conditions
Z t
∂2
∂
a(t, s − t)w(s, ξ)ds + b(t, w(ρ(t), ξ)),
(4.1)
w(t, ξ) = 2 w(t, ξ) +
∂t
∂ξ
−∞
w(t, 0) =w(t, π) = 0, t ≥ 0,
(4.2)
∞
X
w(s, ξ) =
Li w(ti + s, ξ) + ϕ(s, ξ), s ≤ 0, ξ ∈ [0, π]
(4.3)
i=1
∆w(ti , ·)
=w(t+
i , ·)
−
w(·, t−
i )
=
Z
π
pi (ξ, w(ti , s))ds,
(4.4)
0
for t ≥ 0 and ξ ∈ [0, π], and where (Li )i∈N , (ti )i∈N are sequences of real numbers
and ϕ ∈ B.
To treat this system in the abstract form (1.1)–(1.3), we assume the following
conditions:
P
2
(a) Assume ∞
i=1 |Li |h(ti ) < ∞, the functions ρ : [0, ∞) → [0, ∞), a : R → R
1
³R
´
2
0
a2 (t,s)
ds < ∞, for every
are continuous and ρ(t) ≤ t and L1 (t) =
−∞ ρ(s)
t ≥ 0.
(b) Assume b : [0, ∞)×R → R is continuous and that there exists a continuous
function L2 : [0, ∞) → [0, ∞) such that |b(t, x)| ≤ L2 (t)|x| for every t ≥ 0
and x ∈ R.
(c) The P
functions g : BPC 0h (X) → B, f : [0, ∞) × B × X → X by g(u, ϕ) =
ϕ+ ∞
i=1 Li uti and
Z 0
a(t, s)ψ(s, ξ)ds + b(t, x(ρ(t), ξ)).
f (t, ψ, x)(ξ) =
−∞
(d) The functions pi : [0, π] × R → R, i ∈ N, are continuous and there are
positive constants Li such that
|pi (ξ, s) − pi (ξ, s)| ≤ Li |s − s|, ξ ∈ [0, π], s, s ∈ R.
Rπ
By defining the operator Ii : X → X by Ii (x)(ξ) = 0 pi (ξ, x(0, s))ds, i ∈ N,
ξ ∈ [0, π], we can transform system (4.1)-(4.4) into the abstract system (1.1)-(1.3).
Moreover, it is easy to see that f, g are continuous,
kf (t, ψ, x)k ≤ L1 (t)kψkB + L2 (t)kxk, for all (t, ψ, x) ∈ [0, ∞)
P
and g verifies conditions H3 with Lg : K ∞
i=1 Li h(ti ).
The next result is a direct consequence from Theorem 3.3. We omit the proof.
112
S. SIVASANKARAN, M. MALLIKA ARJUNAN AND V. VIJAYAKUMAR
R∞
P
Proposition
4.1. If K ∞
i=1 Li h(ti ) + (1 + 2K) 0 (L1 (s) + L2 (s))ds
P∞
+ K i=1 Li < 1, then there exists a mild solution of (4.1)-(4.4).
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1
Department of Mathematics, University College, Sungkyunkwan University,
Suwon 440-746, South Korea
E-mail address: sdsiva@gmail.com
114
S. SIVASANKARAN, M. MALLIKA ARJUNAN AND V. VIJAYAKUMAR
2
Department of Mathematics, Karunya University, Karunya Nagar, Coimbatore - 641 114, Tamil Nadu, India
E-mail address: arjunphd07@yahoo.co.in
3
Department of Mathematics, Info Institute of Engineering, Kovilpalayam,
Coimbatore - 641 107, Tamil Nadu, India
E-mail address: vijaysarovel@gmail.com
The Journal of Nonlinear Science and Applications
http://www.tjnsa.com
EXISTENCE OF GLOBAL SOLUTIONS FOR IMPULSIVE
FUNCTIONAL DIFFERENTIAL EQUATIONS WITH
NONLOCAL CONDITIONS
S. SIVASANKARAN1 , M. MALLIKA ARJUNAN2 AND V. VIJAYAKUMAR3,∗
Dedicated to Professor Themistocles M. Rassias on the occasion of his sixtieth birthday
Communicated by Choonkil Park
Abstract. In this paper, we study the existence of global solutions for a class
of impulsive abstract functional differential equation with nonlocal conditions.
The results are obtained by using the Leray-Schauder alternative fixed point
theorem. An example is provided to illustrate the theory.
1. Introduction
In this work, we discuss the existence of global solutions for an impulsive abstract functional differential equation with nonlocal conditions in the form
u′ (t) = Au(t) + f (t, ut , u(ρ(t))), t ∈ I = [0, ∞),
u0 = g(u, ϕ) ∈ B,
(1.1)
(1.2)
∆u(ti ) = Ii (uti ),
(1.3)
where A is the infinitesimal generator of a C0 -semigroup of bounded linear operators (T (t))t≥0 defined on a Banach space (X, k · k); I is an interval of the
form [0, ∞); 0 < t1 < t2 < · · · < ti < · · · are pre-fixed numbers; the history
ut : (−∞, 0] → X, ut (θ) = u(t + θ), belongs to some abstract phase space B
defined axiomatically; ρ : [0, ∞) → [0, ∞), f : I × B × X → X, Ii : B → X are
Date: Received: September 19, 2010; Revised: November 24, 2010.
∗
Corresponding author
c 2011 N.A.G.
°
2000 Mathematics Subject Classification. Primary 34A37, 34K30, 35R10, 35R12, 47D06,
49K25.
Key words and phrases. Impulsive functional differential equations; mild solutions; global
solutions; semigroup theory.
102
IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS
103
appropriate functions and the symbol ∆ξ(t) represents the jump of the function
ξ at t, which is defined by ∆ξ(t) = ξ(t+ ) − ξ(t− ).
Differential equations arise in many real world problems such as physics, population dynamics, ecology, biological systems, biotechnology, optimal control, and
so forth. Much has been done the assumption that the state variables and systems parameters change continuously. However, one may easily visualize that
abrupt changes such as shock, harvesting, and disasters may occur in nature.
These phenomena are short-time perturbations whose duration is negligible in
comparison with the duration of the whole evolution process. Consequently, it
is natural to assume, in modeling these problems, that these perturbations act
instantaneously, that is in the form of impulses. The theory of impulsive differential equations has become an active area of investigation due to their applications
in the field such as mechanics, electrical engineering, medicine biology and so on;
see for instance [1, 3, 4, 7, 8, 9, 23, 26, 30, 35, 37, 38]. For more details on this
theory and on its applications we refer to the monographs of Benchohra [10],
Lakshmikantham [29], Bainov and Simeonov [5] and Samoilenko and Perestyuk
[39], and the papers of [2, 15, 16, 17, 18, 25, 31], where numerous properties of
their solutions are studied and detailed bibliographies are given.
The literature concerning differential equations with nonlocal conditions also
include ordinary differential equations, partial differential equations, abstract partial functional differential equations. Related to this matter, we cite among others
works, [6, 11, 12, 13, 14, 20, 22, 28, 33, 34]. Our main results can been seen as
a generalization of the work in [21] and the above mentioned impulsive partial
functional differential equations.
2. Preliminaries
In this paper, A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators (T (t))t≥0 defined on a Banach space (X, k · k)
and M, δ are positive constants such that kT (t)k ≤ M e−δt for every t ≥ 0. For
the theory of strongly continuous semigroup, refer the reader to Pazy [36].
To consider the impulsive condition (1.3), it is convenient to introduce some
additional concepts and notations. We say that a function u : [σ, τ ] → X is a
normalized piecewise continuous function on [σ, τ ] if u is piecewise continuous
and left continuous on (σ, τ ]. We denote by PC([σ, τ ]; X) the space formed by
all normalized piecewise continuous functions from [σ, τ ] into X. In particular,
we introduce the space PC formed by all functions u : [0, a] → X such that u
is continuous at t 6= ti , i = 1, ..., n. It is clear that, PC([σ, τ ]; X) endowed with
the norm of uniform convergence, is a Banach space. On the other hand, the
notation PC([0, ∞); X) stands for the space formed by all bounded normalized
piecewise continuous functions u : [0, ∞) → X such that u|[0,a] ∈ PC([0, a]; X)
for all a > 0, endowed with the uniform convergence topology.
In what follows, for the case I = [0, a], we set t0 = 0, tn+1 = a, and for
u ∈ PC([0, a]; X), we denote by u
ei ∈ C([ti , ti+1 ]; X), i = 0, 1, ..., n, the function
104
given by
S. SIVASANKARAN, M. MALLIKA ARJUNAN AND V. VIJAYAKUMAR
(
u(t),
u
ei (t) =
u(t+
i ),
for t ∈ (t, ti+1 ],
for t = ti .
ei , i = 0, 1, ..., n, for the sets
Moreover, for B ⊂ PC, we employ the notation B
ei = {e
B
ui : u ∈ B}.
Lemma 2.1. A set B ⊂ PC is relatively compact in PC([0, a]; X) if, and only if,
ei , is relatively compact in the space C([ti , ti+1 ]; X) for every i = 0, 1, ..., n.
the set B
In this paper, the phase space B is a linear space of functions mapping (−∞, 0]
into X endowed with a seminorm k · kB and verifying the following axioms.
(A) If x : (−∞, a) → X, a > 0, is continuous on [0, a) and x0 ∈ B, then for
every t ∈ [0, a) the following conditions hold:
(i) xt is in B.
(ii) kx(t)k ≤ Hkxt kB .
(iii) kxt kB ≤ K(t) sup{kx(s)k : 0 ≤ s ≤ t} + M (t)kx0 kB , where H > 0 is
a constant; K, M : [0, ∞) → [1, ∞), K is continuous, M is locally
bounded and H, K, M are independent of x(·).
(B) The space B is complete.
Remark 2.2. In this paper, K is a positive constant such that supt≥0 {K(t), M (t)}
≤ K. We know from that the functions M (·), K(·) are bounded if, for instance,
B is a fading memory space, see [27, pp. 190] for additional details.
Example 2.3. The phase space PC r × Lp (ρ, X)
Let r ≥ 0, 1 ≤ p < ∞, and ρ : (−∞, −r] → R be a non-negative measurable
function which satisfies the conditions (g-5)-(g-6) of [14]. Briefly, this means that
ρ is locally integrable and there exists a non-negative locally bounded function γ
on (−∞, 0] such that ρ(ξ + θ) ≤ γ(ξ)ρ(θ), for all ξ ≤ 0 and θ ∈ (−∞, −r) \ Nξ ,
where Nξ ⊆ (−∞, −r) is a set whose Lebesgue measure zero.
The space B = PC r ×Lp (ρ, X) consists of all classes of functions ψ : (−∞, 0] →
X such that ψ is continuous on [−r, 0], Lebesgue-measurable, and ρkψkp is
Lebesgue integrable on (−∞, −r). The seminorm in PC r × Lp (ρ, X) is defined as
follows:
³ Z −r
´ p1
p
kψkB = sup{kψ(θ)k : −r ≤ θ ≤ 0} +
ρ(θ)kψ(θ)k dθ .
−∞
The space B = PC r × Lp (ρ, X) satisfies axioms (A) and (B). Moreover, when
1
r = 0 and p = 2, one can take H = 1, M (t) = γ(−t) 2 and K(t) = 1 +
¢ 21
¡R0
for t ≥ 0 (see [27, Theorem 1.3.8]). We also note that if the
ρ(θ)dθ
−t
conditions (g-5)-(g-7) of [27] hold, then B is a uniform fading memory.
For additional details concerning phase space, we cite [27]
In this paper, h : [0, ∞) → R be a positive, continuous and non-decreasing
function such that h(0) = 1 and limt→∞ h(t) = ∞. In the sequel, PC([0, ∞); X)
IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS
105
and (PC h )0 (X) are the spaces defined by
x|[0,a] ∈ PC([0, a]; X),
∀a ∈ (0, +∞) \ {ti : i ∈ N},
PC([0, ∞); X) =
x : [0, ∞) → X : x0 = 0,
kxk∞ = supt≥0 kx(t)k < +∞,
n
o
kx(t)k
(PC)0h (X) = x ∈ PC([0, ∞); X) : lim
=0 ,
t→∞ h(t)
endowed with the norms kxk∞ = supt≥0 kx(t)k and kxkPC h = supt≥0
spectively.
Additionally, we define the space
n
o
BPC 0h (X) = u ∈ PC(R, X) : u0 ∈ B, u|[0,∞) ∈ PC 0h (X)
kx(t)k
,
h(t)
re-
endowed with the norm kukBPC 0h = ku0 kB + ku|[0,∞) kh .
We recall here the following compactness criterion.
Lemma 2.4. A bounded set B ⊂ (PC)0h (X) is relatively compact in (PC)0h (X) if,
and only if:
(a) The set Ba = {u|[0,a] : u ∈ B} is relatively compact in PC([0, a]; X), for
all a ∈ (0, ∞) \ {ti ; i ∈ N}.
= 0, uniformly for x ∈ B.
(b) limt→∞ kx(t)k
h(t)
Theorem 2.5. ([19, Theorem 6.5.4] Leray-Schauder Alternative Theorem). Let
D be a closed convex subset of a Banach space (Z, k · kZ ) and assume that 0 ∈ D.
Let F : D → D be a completely continuous map, then the set {x ∈ D : x = λF (x),
0 < λ < 1} is unbounded or the map F has a fixed point in D.
Theorem 2.6. ([32, Theorem 4.3.2]). Let D be a convex, bounded and closed
subset of a Banach space (Z, k · kZ ) and F : D → D be a condensing map. Then
F has a fixed point in D.
3. Existence of mild solutions
In this section, we study the existence of mild solutions for the nonlocal abstract
Cauchy problem (1.1)-(1.3). To treat this system, we introduce the following
conditions.
H1 The function ρ : [0, ∞) → [0, ∞) is continuous and ρ(t) ≤ t for every
t ≥ 0.
H2 The function f : I ×B×X → X is continuous and there exist an integrable
function m : [0, ∞) → [0, ∞) and a nondecreasing continuous function
W : [0, ∞) → (0, ∞) such that kf (t, ψ, x)k ≤ m(t)W (kψkB + kxk), for
every (t, ψ, x) ∈ [0, ∞) × B × X.
H3 The function g : BPC 0h (X) × B → B is continuous and there exists Lg ≥ 0
such that
kg(u, ϕ) − g(v, ϕ)kB ≤ Lg ku − vkBPC 0h ,
u, v ∈ BPC 0h (X).
106
S. SIVASANKARAN, M. MALLIKA ARJUNAN AND V. VIJAYAKUMAR
H4 The function g(·, ϕ) : BPC 0h (X) → B is completely continuous and uniformly bounded. In the sequel, N is a positive constant such that
kg(ψ, ϕ)kB ≤ N for every ψ ∈ BPC 0h (X).
H5 The maps Ii : B → X are completely continuous and uniformly bounded,
i ∈ N. In what follows, we use the notation Ni = sup{kIi (φ)k : φ ∈ B}.
H6 There are positive constants Li , such that
kIi (ψ1 ) − Ii (ψ2 )k ≤ Li kψ1 − ψ2 kB , ψ1 , ψ2 ∈ B, i ∈ N.
In this work, we adopt the following concept of mild solutions for (1.1)-(1.3).
Definition 3.1. A function u ∈ PC(R, X) is called a mild solution of system
(1.1)-(1.3) if u0 = g(u, ϕ) and
Z t
T (t − s)f (s, us , u(ρ(s)))ds
u(t) = T (t)g(u, ϕ)(0) +
0
X
T (t − ti )Ii (uti ), t ∈ I = [0, ∞).
+
0 0, limt→∞ h(t)
0
R ∞ ds
R∞
If M K(1 + H) 0 m(s)ds < c W (s) , where c = KN (1 + H)(1 + M H)
P
+ MK ∞
i=1 Ni , then there exists a mild solution for the system (1.1)-(1.3).
Proof. Let Γ : BPC 0h (I : X) → BPC 0h (X) be the operator defined by (Γu)0 =
g(u, ϕ) and
Z t
T (t − s)f (s, us , u(ρ(s)))ds
Γu(t) = T (t)g(u, ϕ)(0) +
0
X
+
T (t − ti )Ii (uti ), t ≥ 0.
0 0 such that
Z t
∞
2M X
M
2M N H
ε
m(s)W (2Kηh(s))ds +
+
Ni ≤ ,
h(t)
h(t) 0
h(t) i=1
2
107
t≥L
(3.1)
From the properties of the functions f , g, we can choose Nε ∈ N such that
ε
kg(xn , ϕ) − g(x, ϕ)kB ≤
,
(3.2)
6(M H + 1)
Z L
ε
,
(3.3)
kf (s, xns , xn (ρ(s))) − f (s, xs , x(ρ(s)))kds ≤
6M
0
and
X
kIi (xnti ) − Ii (xti )k ≤
0≤ti ≤L
ε
,
6M
n ≥ Nε
for every n ≥ Nε . Under these conditions, we see that
o ε
n kΓxn (t) − Γx(t)k
sup
: t ∈ [0, L], n ≥ Nε ≤ .
h(t)
2
(3.4)
(3.5)
Moreover, for t ≥ L and n ≥ Nε , we find that
∞
kg(xn , ϕ)(0) − g(x, ϕ)(0)k 2M X
kΓxn (t) − Γx(t)k
Ni
≤M
+
h(t)
h(t)
h(t) i=1
Z t
M
+
kf (s, xns , xn (ρ(s))) − f (s, xs , x(ρ(s)))kds,
h(t) 0
Z t
∞
2M X
2M N H
M
m(s)W (2Kηh(s))ds +
≤
+
Ni ,
h(t)
h(t) 0
h(t) i=1
and hence,
sup
n kΓxn (t) − Γx(t)k
h(t)
o
ε
: t ≥ L, n ≥ Nε ≤ .
2
(3.6)
From the inequalities (3.2)-(3.6), it follows that kΓxn − ΓxkBPC 0h ≤ ε for every
n ≥ Nε , which proves that Γ is continuous.
To prove that Γ is completely continuous, we introduce the decomposition
Γ = Γ1 + Γ2 + Γ3 , where (Γ1 u)0 = g(u, ϕ), (Γ2 u)0 = 0, (Γ3 u)0 = 0 and
Γ1 u(t) =T (t)g(u, ϕ)(0), t ≥ 0
Z t
T (t − s)f (s, us , u(ρ(s)))ds, t ≥ 0
Γ2 u(t) =
0
X
Γ3 u(t) =
T (t − ti )Ii (uti ), t ≥ 0
(3.7)
(3.8)
(3.9)
0 0. Next, by using Lemma 2.1, we
prove that Γ3 is also completely continuous. We observe that the continuity of
Γ3 is obvious. On the other hand, for r > 0, t ∈ [ti , ti+1 ], i ≥ 1, and u ∈ Br =
Br (0, PC([0, a]; X)), we have that there exists re > 0 such that
Pi
t ∈ (ti , ti+1 ),
Pj=1 T (t − tj )Ij (Bre(0; B)),
i
g
[Γ3 u]i (t) ∈
t = ti+1 ,
j=1 T (ti+1 − tj )Ij (Bre(0; B)),
Pi−1 T (t − t )I (B (0; B)) + I (B (0; B)), t = t ,
i
j j
re
i
re
i
j=1
where Bre(0; B) is an open ball of radius re. From condition H5 it follows that
[Γ^
3 (Br )]i (t) is relatively compact in X, for all t ∈ [ti , ti+1 ], i ≥ 1. Moreover,
using the compactness of the operators {Ii }i and the continuity of (T (t))t≥0 , we
conclude that for ε > 0, there exists δ > 0 such that
kT (t)z − T (s)zk ≤ ε,
z∈
n
[
i=1
Ii (Bre(0, B)),
(3.10)
for all ti , i = 1, · · · , n, t, s ∈ (ti , ti+1 ] with |t − s| < δ. Under these conditions,
for u ∈ Br , t ∈ [ti , ti+1 ], i ≥ 0, and 0 < |h| < δ with t + h ∈ [ti , ti+1 ], we have that
g
g
k[Γ
3 u]i (t + h) − [Γ3 u]i (t)k ≤ M nε,
t ∈ I,
(3.11)
which proves that the set of functions [Γ^
3 (Br )]i , i ≥ 0, is uniformly equicontinuous. Now, from Lemma 2.1, we conclude that Γ3 is completely continuous. Thus,
Γ is completely continuous.
To finish the proof, we obtain a priori estimates for the solutions of the integral
equation {u = λΓu, λ ∈ (0, 1)}. Let λ ∈ (0, 1) and uλ ∈ BPC 0h (X) be a solution
of λΓz = z. By using the notation αλ (t) = K(sups∈[0,t] kuλ (s)k+kuλ0 kB ), for t ∈ I
we see that
λ
ku (t)k ≤ M N H + M
Z
t
m(s)W (kuλs kB + kuλ (ρ(s))k)ds + M
0
≤ MNH + M
Z
0
∞
X
i=1
t
m(s)W ((1 + H)αλ (s))ds + M
∞
X
i=1
Ni ,
Ni ,
IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS
109
and hence,
αλ (t)
≤ Kkg(u, ϕ)kB + KM N H + KM
Z
t
λ
m(s)(W (1 + H)α (s))ds + KM
0
≤ KN (1 + M H) + KM
Z
∞
X
Ni ,
i=1
t
λ
m(s)W ((1 + H)α (s))ds + KM
0
∞
X
Ni .
i=1
Denoting by βλ the right hand side of last inequality, we see that
βλ′ (t) ≤ KM m(t)W ((1 + H)βλ (t)),
which implies that
Z
(1+H)βλ (t)
(1+H)βλ (0)=c
ds
≤ M K(1 + H)
W (s)
Z
∞
0
m(s)ds <
Z
c
∞
ds
.
W (s)
This inequality permits us to conclude that the set of functions {βλ : λ ∈ (0, 1)}
is bounded in C(R) and as consequence that the set {uλ ; λ ∈ (0, 1)} is bounded
in BPC 0h (X).
Now the assertion is a consequence of Theorem 2.5. The proof is complete. ¤
Theorem 3.3. Assume assumptions H1 , H2 , H3 , H5 and H6 be hold. If the
conditions (a) and (b) of Theorem 3.2 are valid and
Lg (1 + M H) + M lim inf
r→∞
Z
0
+∞
∞
X
m(s)W ((1 + 2K)rh(s))
ds + M
Li < 1,
rh(s)
i=1
(3.10)
then there exists a mild solution of (1.1)-(1.3).
Proof. Let Γ, Γ1 , Γ2 , Γ3 be the operators introduced in the proof of Theorem
3.2, we claim that there exists r > 0 such that Γ(Br ) ⊂ Br , where Br =
Br (0, BPC 0h (X)). In fact, if this property is false, then for every r > 0, there
110
S. SIVASANKARAN, M. MALLIKA ARJUNAN AND V. VIJAYAKUMAR
r
r
exist xr ∈ Br and tr ≥ 0 such that r < k(Γxr )0 kB + k Γxh(t(tr ) ) k. Consequently
kg(xr , ϕ)(0)k
r ≤ kg(xr , ϕ)kB + M
h(tr )
Z tr
M
m(s)W (Kkxr0 kB + K sup kxr (θ)k + kxr (ρ(s))k)ds
+
h(tr ) 0
θ∈[0,s]
∞
∞
X Li kxt k
M X
i
+
kI
(0)k
+
M
i
r
r
h(t ) i=1
h(t )
i=1
M
MH
Lg kxr kBPC 0h +
kg(0, ϕ)kB
≤ Lg kxr kBPC oh + kg(0, ϕ)kB +
r
h(t )
h(tr )
Z tr
M
m(s)W (Kr + Krh(s) + rh(s))ds
+
h(tr ) 0
∞
∞
X
M X
+ Mr
kIi (0)k
Li +
h(tr ) i=1
i=1
≤ Lg (1 + M H)r + (1 + M )kg(0, ϕ)kB + M
Z
0
and then,
1 ≤ Lg (1 + M H) + M lim inf
r→∞
Z
+∞
0
tr
m(s)((2K + 1)rh(s))
ds
h(s)
∞
X
m(s)((2K + 1)rh(s))
ds + M
Li ,
rh(s)
i=1
which is contrary to (3.10).
Let r > 0 such that Γ(Br ) ⊂ Br . From the proof of the Theorem 3.2, we know
that Γ2 is a completely continuous on Br . Moreover, from the estimate,
kΓ1 u − Γ1 vkBPC 0h ≤ Lg ku − vkBPC 0h + Hkg(u, ϕ) − g(v, ϕ)kB
≤ Lg (1 + H)ku − vkBPC 0h ,
we infer that Γ1 is a contraction on Br . It is easy to see that
∞
X
Li ku − vkBPC 0h , u, v ∈ BPC 0h ,
kΓ3 u − Γ3 vkBPC 0h ≤ M
i=1
which implies that Γ3 is a contraction in BPC 0h (X) from which we conclude that
Γ is a condensing operator on Br .
The assertion is now a consequence of Theorem 2.6.
¤
4. An application
To complete this work, we study the existence of global solutions for a concrete
partial differential equation with nonlocal conditions. In the sequel, X = L2 [0, π]
and A : D(A) ⊂ X → X is the operator Ax = x′′ with domain D(A) = {x ∈
X : x′′ ∈ X, x(0) = x(π) = 0}. It is well known that A is the infinitesimal
generator of an analytic semigroup (T (t))t≥0 on X. Furthermore, A has discrete
spectrum with eigenvalues of the form −n2 , n ∈ N, and corresponding normalized
IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS
111
1
eigenfunctions given by zn (x) = ( π2 ) 2 sin(nx); the set of functions {zn : n ∈ N}
P
−n2 t
hx, zn izn for every x ∈ X
is an orthonormal basis for X and T (t)x = ∞
n=1 e
and all t ≥ 0. It follows from this last expression that (T (t))t≥0 is a compact
semigroup on X and that kT (t)k ≤ e−t for every t ≥ 0.
As phase space, we choose the space B = PC 0 × L2 (ρ, X), see Example 2.3,
and we assume that the conditions (g5) and (g7) in [27] are verified. Under these
conditions, B is a fading memory space, and as consequence, there exists K > 0
such that supt≥0 {M (t), K(t)} ≤ K.
Consider the delayed impulsive partial differential equation with nonlocal conditions
Z t
∂2
∂
a(t, s − t)w(s, ξ)ds + b(t, w(ρ(t), ξ)),
(4.1)
w(t, ξ) = 2 w(t, ξ) +
∂t
∂ξ
−∞
w(t, 0) =w(t, π) = 0, t ≥ 0,
(4.2)
∞
X
w(s, ξ) =
Li w(ti + s, ξ) + ϕ(s, ξ), s ≤ 0, ξ ∈ [0, π]
(4.3)
i=1
∆w(ti , ·)
=w(t+
i , ·)
−
w(·, t−
i )
=
Z
π
pi (ξ, w(ti , s))ds,
(4.4)
0
for t ≥ 0 and ξ ∈ [0, π], and where (Li )i∈N , (ti )i∈N are sequences of real numbers
and ϕ ∈ B.
To treat this system in the abstract form (1.1)–(1.3), we assume the following
conditions:
P
2
(a) Assume ∞
i=1 |Li |h(ti ) < ∞, the functions ρ : [0, ∞) → [0, ∞), a : R → R
1
³R
´
2
0
a2 (t,s)
ds < ∞, for every
are continuous and ρ(t) ≤ t and L1 (t) =
−∞ ρ(s)
t ≥ 0.
(b) Assume b : [0, ∞)×R → R is continuous and that there exists a continuous
function L2 : [0, ∞) → [0, ∞) such that |b(t, x)| ≤ L2 (t)|x| for every t ≥ 0
and x ∈ R.
(c) The P
functions g : BPC 0h (X) → B, f : [0, ∞) × B × X → X by g(u, ϕ) =
ϕ+ ∞
i=1 Li uti and
Z 0
a(t, s)ψ(s, ξ)ds + b(t, x(ρ(t), ξ)).
f (t, ψ, x)(ξ) =
−∞
(d) The functions pi : [0, π] × R → R, i ∈ N, are continuous and there are
positive constants Li such that
|pi (ξ, s) − pi (ξ, s)| ≤ Li |s − s|, ξ ∈ [0, π], s, s ∈ R.
Rπ
By defining the operator Ii : X → X by Ii (x)(ξ) = 0 pi (ξ, x(0, s))ds, i ∈ N,
ξ ∈ [0, π], we can transform system (4.1)-(4.4) into the abstract system (1.1)-(1.3).
Moreover, it is easy to see that f, g are continuous,
kf (t, ψ, x)k ≤ L1 (t)kψkB + L2 (t)kxk, for all (t, ψ, x) ∈ [0, ∞)
P
and g verifies conditions H3 with Lg : K ∞
i=1 Li h(ti ).
The next result is a direct consequence from Theorem 3.3. We omit the proof.
112
S. SIVASANKARAN, M. MALLIKA ARJUNAN AND V. VIJAYAKUMAR
R∞
P
Proposition
4.1. If K ∞
i=1 Li h(ti ) + (1 + 2K) 0 (L1 (s) + L2 (s))ds
P∞
+ K i=1 Li < 1, then there exists a mild solution of (4.1)-(4.4).
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1
Department of Mathematics, University College, Sungkyunkwan University,
Suwon 440-746, South Korea
E-mail address: sdsiva@gmail.com
114
S. SIVASANKARAN, M. MALLIKA ARJUNAN AND V. VIJAYAKUMAR
2
Department of Mathematics, Karunya University, Karunya Nagar, Coimbatore - 641 114, Tamil Nadu, India
E-mail address: arjunphd07@yahoo.co.in
3
Department of Mathematics, Info Institute of Engineering, Kovilpalayam,
Coimbatore - 641 107, Tamil Nadu, India
E-mail address: vijaysarovel@gmail.com