Surveys in Mathematics and its Applications
ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 7 (2012), 49 – 68
GLOBAL EXISTENCE FOR
VOLTERRA–FREDHOLM TYPE NEUTRAL
IMPULSIVE FUNCTIONAL
INTEGRODIFFERENTIAL EQUATIONS
V. Vijayakumar, S. Sivasankaran and M. Mallika Arjunan
Abstract. In this paper, we study the global existence of solutions for the initial value problems
for Volterra-Fredholm type neutral impulsive functional integrodifferential equations. Using the
Leray-Schauder’s Alternative theorem, we derive conditions under which a solution exists globally.
An application is provided to illustrate the theory.
1
Introduction
In recent years, a great deal of work has been done in the study of the existence of
solutions for impulsive differential equations, by which a number of chemotherapy,
population dynamics, optimal control, industrial robotics and physics phenomena
are described. For the general aspects of impulsive differential equations, we refer
the reader to the classical monographs Bainov et al. [1, 2], Lakshmikantham et
al. [20], Samoilenko and Perestyuk [26] and Hernández et al. [12, 11] for the case
of ordinary and partial differential functional differential equations with impulses.
For some general and recent works on the theory of impulsive differential and
integrodifferential equations, we refer the reader to [3, 4, 5, 6, 8, 7, 9, 19, 31].
Recently, in [21], Ntouyas studied the global existence for first order neutral
impulsive functional integrodifferential equations of the form
d
[x(t) − g(t, xt )] = f t, xt ,
dt
x0 = φ
Z
t
k(t, s)h(s, xs )ds , t ∈ I = [0, T ],
0
and also studied the global existence for second order neutral impulsive functional
2010 Mathematics Subject Classification: 34A60, 34K05, 34K10, 45N05, 45J05.
Keywords: A priori bounds; Neutral impulsive functional integrodifferential equations; Global
existence.
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50
V. Vijayakumar, S. Sivasankaran and M. Mallika Arjunan
integrodifferential equations of the form
Z t
d 0
[x (t) − g(t, xt )] = F t, xt ,
k(t, s)H(s, xs , x0 (s))ds, x0 (t) , t ∈ I = [0, T ],
dt
0
x0 = φ, x0 (0) = η
by using Leray-Schauder’s Alternative theorem. Recent results on global existence
and global solutions for ordinary and partial neutral impulsive integrodifferential
equations can be foun in [13, 14, 15, 16, 17, 18, 22, 24, 23, 25, 27, 28, 29, 30].
In this paper, we study the global existence of solutions for the initial value
problems for the first and second order Volterra-Fredholm type neutral impulsive
functional integrodifferential equations. In Section 3, we study the following initial
value problem
d
[x(t) − g(t, xt )] = f t, xt ,
dt
Z
t
Z
a(t, s)w(s, xs )ds,
T
b(t, s)h(s, xs )ds ,
0
0
t ∈ I = [0, T ] \ {t1 , ..., tm },
∆x|t=tk =
Ik (x(t−
k )),
k = 1, ..., m,
x0 = φ
(1.1)
(1.2)
(1.3)
where f : I × D × Rn × Rn → Rn , g : I × D → Rn , D = {ψ : [−r, 0] → Rn ; ψ
is continuous everywhere except for a finite number of points e
t at which ψ(e
t− ) and
ψ(e
t+ ) are exist with ψ(e
t− ) = ψ(e
t)}, w : I × D → Rn , h : I × D → Rn , a : I × I → R,
b : I × I → R are continuous functions, φ ∈ D, 0 < r < ∞, 0 = t0 < t1 < · · · < tm <
+
tm+1 = T , Ik ∈ C(Rn , Rn ), (k = 1, ..., m). x(t−
k ) and x(tk ) represent the left and
−
right limits of x(t) at t = tk , respectively, ∆x|t=tk = x(t+
k ) − x(tk ).
In Section 4, we study the global existence of solutions for the initial value
problems for the second order Volterra-Fredholm type neutral impulsive functional
integrodifferential equations of the form
Z t
d 0
0
[x (t) − g(t, xt )] = F t, xt , x (t),
a(t, s)W (s, xs , x0 (s))ds,
dt
0
Z T
0
b(t, s)H(s, xs , x (s))ds , t ∈ I = [0, T ] \ {t1 , ..., tm }, (1.4)
0
∆x|t=tk = Ik (x(t−
k )),
0
∆x |t=tk =
Jk (x(t−
k )),
0
k = 1, ..., m,
(1.5)
k = 1, ..., m, and
(1.6)
x0 = φ, x (0) = η,
(1.7)
where F : I × D × Rn × Rn × Rn → Rn , g : I × D → Rn , D = {ψ : [−r, 0] → Rn ; ψ
is continuous everywhere except for a finite number of points e
t at which ψ(e
t− ) and
ψ(e
t+ ) are exist with ψ(e
t− ) = ψ(e
t)}, W : I × D × Rn → Rn , H : I × D × Rn → Rn ,
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Surveys in Mathematics and its Applications 7 (2012), 49 – 68
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Global Existence For Volterra–Fredholm Type Functional Differential Equations
51
a : I × I → R, b : I × I → R are continuous functions, φ ∈ D and η ∈ Rn ,
0 < r < ∞, 0 = t0 < t1 < · · · < tm < tm+1 = T , Ik , Jk ∈ C(Rn , Rn ), (k = 1, ..., m).
+
x(t−
k ) and x(tk ) represent the left and right limits of x(t) at t = tk , respectively,
−
0 +
0 −
0
∆x|t=tk = x(t+
k ) − x(tk ), and ∆x |t=tk = x (tk ) − x (tk ).
For any continuous function x defined on [−r, T ] \ {t1 , ..., tm } and any t ∈ [0, T ]
we denote by xt the element of D defined by xt (θ) = x(t + θ), θ ∈ [−r, 0], where
xt (·) represents the history of the state from time t − r, up to the present time t.
2
Preliminaries
In this section, we introduce some basic definitions, notations and preliminary
facts which are used throughout this paper.
Let C([−r, 0], Rn ) be the Banach space of continuous functions from [−r, 0] into
Rn endowed with the norm
kφk = sup |φ(0)| : −r ≤ θ ≤ 0
and C([0, T ], Rn ) denote the Banach space of all continuous functions from [0, T ]
into Rn normed by
kxkr = sup{|x(t)| : −r ≤ t ≤ T },
kxk0 = sup{|x(t) : t ∈ I},
kxk1 = sup{|x0 (t) : t ∈ I},
kxk∗ = max{kxkr , kxk1 },
kxkT = max{kxk0 , kxk1 }.
In order to define the solutions of (1.1)-(1.3) and (1.4)-(1.7), we introduce the
following space:
n
P C = x : [0, T ] → Rn :xk ∈ C([tk , tk+1 ], Rn ), k = 0, ..., m and there exist
o
+
−
+
x(t−
)
and
x(t
)
with
x(t
)
=
x(t
),
k
=
1,
...,
m
k
k
k
k
which is a Banach space with the norm
kxkP C = max{kxk k(tk ,tk+1 ] , k = 0, ..., m},
where xk is the restriction of x to (tk , tk+1 ], k = 0, ..., m.
Set Ω = D ∪ P C. Then Ω is a Banach space normed by
kxkΩ = max{kxkD , kxkP C }, for each x ∈ Ω.
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52
V. Vijayakumar, S. Sivasankaran and M. Mallika Arjunan
and
n
P C 1 = x : [0, T ] → Rn :xk ∈ C 1 ([tk , tk+1 ], Rn ), k = 0, ..., m and there exist
0 +
0 −
0 +
x0 (t−
k ) and x (tk ) with x (tk ) = x (tk ), k = 1, ..., m
o
which is a Banach space with the norm
kxkP C 1 = max{kxk k(tk ,tk+1 ] , k = 0, ..., m},
where xk is the restriction of x to (tk , tk+1 ], k = 0, ..., m.
Set Ω1 = D ∪ P C 1 . Then Ω1 is a Banach space normed by
kxkΩ1 = max{kxkD , kxkP C 1 }, for each x ∈ Ω1 .
The considerations of this paper are based on the following fixed point result
[10].
Lemma 1 (Leray-Schauder’s Alternative Theorem). Let S be a closed convex subset
of a normed linear space E and assume that 0 ∈ S. If F : S → S be a completely
continuous operator, i.e. it is continuous and the image of any bounded set is
included in a compact set and let
Φ(F ) = {x ∈ S : x = λF x, for some 0 < λ < 1}.
Then either Φ(F ) is unbounded or F has a fixed point.
3
Global Existence for First order IVP
In this section, we present the global existence results for the IVP (Initial value
problem) (1.1)-(1.3).
Definition 2. A function x ∈ Ω is called solution of the initial value problem
(1.1)-(1.3) if x satisfies the following integral equation
Z
t
Z
s
x(t) = φ(0) − g(0, φ) + g(t, xt ) +
f s, xs ,
a(s, σ)w(σ, xσ )dσ,
0
0
Z T
X
b(s, σ)h(σ, xσ )dσ ds +
Ik (x(t−
k )), t ∈ I.
0
0<tk <t
Theorem 3. Let f : I × D × Rn × Rn → Rn , g : I × D → Rn , w : I × D → Rn ,
h : I × D → Rn , a : I × I → R, b : I × I → R are continuous functions.
Assume that
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Global Existence For Volterra–Fredholm Type Functional Differential Equations
53
Hg1 There exists constants c1 and c2 such that
|g(t, φ)| ≤ c1 kφk + c2 , t ∈ I, φ ∈ D.
Hg2 The function g is completely continuous and such that the operator
G : D → P C([0, T ], Rn )
defined by (Gφ)(t) = g(t, φ) is compact.
Hw There exists a continuous function m1 : I → [0, ∞) such that
|w(t, φ)| ≤ m1 (t)Ω1 (kφk), t ∈ I, φ ∈ D,
where Ω1 : [0, ∞) → (0, ∞) is a continuous nondecreasing function.
Hh There exists a continuous function m2 : I → [0, ∞) such that
|h(t, φ)| ≤ m2 (t)Ω2 (kφk), t ∈ I, φ ∈ D,
where Ω2 : [0, ∞) → (0, ∞) is a continuous nondecreasing function.
Hf There exists an integrable function p : I → [0, ∞) such that
|f (t, u, v, w)| ≤ p(t)Ω3 (kuk + |v| + |w|), t ∈ I, u ∈ D, v, w ∈ Rn ,
where Ω3 : [0, ∞) → (0, ∞) is a continuous nondecreasing function.
Ha There exists a constant L1 such that
|a(t, s)| ≤ L1 for t ≥ s ≥ 0.
Hb There exists a constant L2 such that
|b(t, s)| ≤ L2 for t ≥ s ≥ 0.
HI There exist constants ck such that |Ik (x)| ≤ ck |x|, k = 1, ..., m for each x ∈ Rn .
Then if
Z
0
T
Z
m(s)ds
b
<
c
+∞
ds
Ω3 (s) + Ω1 (s) + Ω2 (s)
where
m(t)
b
= max
c=
1
(1 − c1 )
o
1
p(t), L1 m1 (t), L2 m2 (t) and
(1 − c1 )
n
n
o
X
(1 + c1 )kφk + 2c2 +
ck |x(t−
)|
,
k
n
k=1
the initial value problem (1.1)-(1.3) has at least one solution on [−r, T ].
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54
V. Vijayakumar, S. Sivasankaran and M. Mallika Arjunan
Proof. To prove the existence of a solution of the initial value problem (1.1)-(1.3)
we apply Lemma 1. First we obtain the a priori bounds for the solutions of the
initial value problem (1.1)λ − (1.3), λ ∈ (0, 1), where (1.1)λ stands for the equation
Z T
Z t
d
b(t, s)h(s, xs )ds , t ∈ I.
a(t, s)w(s, xs )ds,
[x(t) − g(t, xt )] = λf t, xt ,
dt
0
0
Let x be a solution of the initial value problem (1.1)λ − (1.3). From
Z t Z s
x(t) = λφ(0) − λg(0, φ) + λg(t, xt ) + λ
f s, xs ,
a(s, σ)w(σ, xσ )dσ,
0
0
Z T
X
b(s, σ)h(σ, xσ )dσ ds + λ
Ik (x(t−
k )), t ∈ I.
0
0<tk <t
we have, for every t ∈ I,
t
Z
Z
0
0
T
Z
+ L2
0
s
m1 (σ)Ω1 (kxσ k)dσ
p(s)Ω3 kxs k + L1
|x(t)| ≤ kφk + c1 kφk + 2c2 + c1 kxt k +
n
X
ck |x(t−
m2 (σ)Ω2 (kxσ k)dσ ds +
k )|.
k=1
We consider the function µ given by
µ(t) = sup{|x(s)| : −r ≤ s ≤ t}, t ∈ I.
Let t∗ ∈ [−r, t] be such that µ(t) = |x(t∗ )|. If t∗ ∈ [0, t], by the previous inequality
we have, for every t ∈ I,
Z t
Z s
µ(t) ≤ (1 + c1 )kφk + 2c2 + c1 µ(t) +
p(s)Ω3 µ(s) + L1
m1 (σ)Ω1 (µ(σ))dσ
0
Z
+ L2
T
0
m2 (σ)Ω2 (µ(σ))dσ ds +
0
n
X
ck |x(t−
k )|
k=1
or
Z t
Z s
n
1
µ(t) ≤
(1 + c1 )kφk + 2c2 +
p(s)Ω3 µ(s) + L1
m1 (σ)Ω1 (µ(σ))dσ
(1 − c1 )
0
0
Z T
n
o
X
+ L2
m2 (σ)Ω2 (µ(σ))dσ ds +
ck |x(t−
)|
k
0
If
k=1
t∗
∈ [−r, 0] then µ(t) = kφk and the previous inequality obvious holds.
Denoting by u(t) the right-hand side of the above inequality we have,
u(0) =
n
n
o
X
1
(1 + c1 )kφk + 2c2 +
ck |x(t−
)|
≡ c, µ(t) ≤ u(t), t ∈ I,
k
(1 − c1 )
k=1
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Global Existence For Volterra–Fredholm Type Functional Differential Equations
55
and for every t ∈ I
Z t
1
m1 (σ)Ω1 (µ(σ))dσ
u (t) =
p(t)Ω3 µ(t) + L1
(1 − c1 )
0
Z T
m2 (σ)Ω2 (µ(σ))dσ
+ L2
0
Z t
1
≤
m1 (σ)Ω1 (u(σ))dσ
p(t)Ω3 u(t) + L1
(1 − c1 )
0
Z T
m2 (σ)Ω2 (u(σ))dσ .
+ L2
0
0
Set
t
Z
Z
T
m2 (s)Ω2 (u(s))ds,
m1 (s)Ω1 (u(s))ds + L2
v(t) = u(t) + L1
t ∈ I.
0
0
Then
v(0) = u(0) ≡ c,
u(t) ≤ v(t), t ∈ I,
u0 (t) ≤
1
p(t)Ω3 (v(t)), t ∈ I
(1 − c1 )
and
v 0 (t) = u0 (t) + L1 m1 (t)Ω1 (u(t)) + L2 m2 (t)Ω2 (u(t))
1
≤
p(t)Ω3 (v(t)) + L1 m1 (t)Ω1 (v(t)) + L2 m2 (t)Ω2 (v(t))
(1 − c1 )
≤ m(t)
b
Ω3 (v(t)) + Ω1 (v(t)) + Ω2 (v(t)) , t ∈ I.
or
v 0 (t)
≤ m(t),
e
t ∈ I.
Ω3 (v(t)) + Ω1 (v(t)) + Ω2 (v(t))
Integrating this inequality from 0 to t and using the change of variables formula we
get
Z
v(t)
v(0)
ds
≤
Ω3 (s) + Ω1 (s) + Ω2 (s)
Z
0
T
Z
m(s)ds
b
<
c
+∞
ds
, t ∈ I.
Ω3 (s) + Ω1 (s) + Ω2 (s)
This inequality implies that there is a constant K such that u(t) ≤ K, t ∈ I and
hence µ(t) ≤ K, t ∈ I. Since for every t ∈ I, kxt k ≤ µ(t) we have
kxkr = sup{|x(t)| : −r ≤ t ≤ T } ≤ K,
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56
V. Vijayakumar, S. Sivasankaran and M. Mallika Arjunan
where K depends only on the functions m1 , m2 , p, Ω1 , Ω2 and Ω3 . We will rewrite
the initial value problem (1.1)-(1.3) as follows. For φ ∈ D define φe ∈ B, B = Ω by
(
φ(t), −r ≤ t ≤ 0,
e
φ=
φ(0), 0 ≤ t ≤ T.
e
If x(t) = y(t) + φ(t),
t ∈ [−r, T ] it is easy to see that y satisfies
y0 = 0
Z t Z s
e
e
y(t) = −g(0, φ) + g(t, yt + φt ) +
f s, ys + φs ,
a(s, τ )w(τ, yτ + φeτ )dτ,
0
0
Z T
X
b(s, τ )h(τ, yτ + φeτ )dτ ds +
Ik (x(t−
k )), t ∈ I.
0
0<tk <t
Define B0 = {y ∈ B : y0 = 0} and S : B0 → B0 , by
Now define an operator Q : E0 → E by

0,
t ∈ [−r, 0]



Rs
Rt e
f s, ys + φes , 0 a(s, τ )w(τ, yτ + φeτ )dτ,
−g(0,
φ)
+
g(t,
y
+
φ
)
+
t
t
(Sy)(t) =
0


 R T b(s, τ )h(τ, y + φe )dτ ds + P
t ∈ I.
Ik (x(t− )),
τ
τ
0<tk <t
0
k
S is clearly continuous. We shall prove that S is completely continuous.
By Hg2 it suffices to prove that the operator S1 : B0 → B0 defined by

0,
t ∈ [−r, 0]


 Rt R
e s
yτ + φeτ )dτ,
(S1 y)(t) =
0 f τ, ys + φs , 0 a(s, τ )w(τ,

R

T

b(s, τ )h(τ, y + φe )dτ ds,
t ∈ I.
0
τ
τ
is completely continuous.
Let {uv } be a bounded sequence in B0 , i.e.
kuv kr ≤ b,
for all
v,
where b is a positive constant. We obviously have kuvt k ≤ b, t ∈ I, for all v. Hence,
if M ∗ = sup{m
e : t ∈ I}, p0 = sup{p(t) : t ∈ I}, we obtain
h
i
M ∗T 2
M ∗T 2
kS1 uv kr ≤ p0 Ω3 (b + kφk)T +
Ω1 (b + kφk) +
Ω2 (b + kφk) .
2
2
This means that {S1 uv } is uniformly bounded.
Moreover, the sequence {S1 uv } is equicontinuous, since for t1 , t2 ∈ [−r, T ] we
have:
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Global Existence For Volterra–Fredholm Type Functional Differential Equations
57
(i) if 0 ≤ t1 ≤ t2 then
|S1 uv (t1 ) − S1 uv (t2 )|
≤ p0 Ω3 (b + kφk) + M ∗ T (Ω1 (b + kφk) + Ω2 (b + kφk) |t1 − t2 |.
(ii) if t1 ≤ 0 ≤ t2 then
|S1 uv (t1 ) − S1 uv (t2 )|
≤ p0 Ω3 (b + kφk) + M ∗ T (Ω1 (b + kφk) + Ω2 (b + kφk) |0 − t2 |.
(iii) if t1 ≤ t2 ≤ 0 then
|S1 uv (t1 ) − S1 uv (t2 )| = 0.
Thus, by Arzela-Ascoli theorem the operator S1 is completely continuous.
Consequently the operator S is completely continuous.
Finally, the set Φ(S) = {y ∈ B0 : y = λSy, λ ∈ (0, 1)} is bounded, since for every
solution y ∈ B0 the function x = y + φ is a solution of the initial value problem
(1.1)-(1.3), for which we have proved that kxkr ≤ K and hence kykr ≤ K + kφk.
Consequently, by Lemma 1, the operator S has a fixed point y ∗ in B0 . Then
∗
x = y ∗ + φe is a solution of the initial value problem (1.1)-(1.3).
Hence the proof of the theorem is complete.
4
Global Existence for Second order IVP
In this section we study the global existence for the IVP (Initial value problem)
(1.4)-(1.7).
Definition 4. A function x ∈ Ω ∩ Ω1 is called solution of the initial value problem
(1.4)-(1.7) if x satisfies the following integral equation
Z t
Z tZ s x(t) = φ(0) + [η − g(0, φ)]t +
g(s, xs )ds +
F τ, xτ , x0 (τ ),
0
0
0
Z τ
Z T
a(τ, σ)W (σ, xσ , x0 (σ))dσ,
b(τ, σ)H(σ, xσ , x0 (σ))dσ dτ ds
0
0
X
−
+
[Ik (x(tk )) + (T − tk )Jk (x(t−
k ))], t ∈ I.
0<tk <t
Theorem 5. Let F : I ×D ×Rn ×Rn ×Rn → Rn , g : I ×D → Rn , W : I ×D ×Rn →
Rn , H : I × D × Rn → Rn are continuous functions and a : I × I → R, b : I × I → R
are measurable for t ≥ s ≥ 0 function.
Assume that HI is satisfied and g satisfies Hg1 , Hg2 , a, b satisfies Ha and Hb .
Moreover we assume that:
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58
V. Vijayakumar, S. Sivasankaran and M. Mallika Arjunan
HW There exists a continuous function m3 : I → [0, ∞) such that
|W (t, φ, ψ)| ≤ m3 (t)Ω4 (kφk + |ψ|), t ∈ I, φ ∈ D, ψ ∈ Rn ,
where Ω4 : [0, ∞) → (0, ∞) is a continuous nondecreasing function.
HH There exists a continuous function m4 : I → [0, ∞) such that
|H(t, φ, ψ)| ≤ m4 (t)Ω5 (kφk + |ψ|), t ∈ I, φ ∈ D, ψ ∈ Rn ,
where Ω5 : [0, ∞) → (0, ∞) is a continuous nondecreasing function.
HF There exists an integrable function P : I → [0, ∞) such that
|F (t, u, v, w, y)| ≤ P (t)Ω6 (kuk + |v| + |w| + |y|), t ∈ I, u ∈ D, v, w, y ∈ Rn ,
where Ω6 : [0, ∞) → (0, ∞) is a continuous nondecreasing function.
HJ There exist constants dk such that |Jk (x)| ≤ dk |x|, k = 1, ..., m for each x ∈ Rn .
Then if
Z
T
Z
∞
c(s)ds <
M
0
c0
ds
s + 2Ω6 (s) + Ω4 (s) + Ω5 (s)
where
Z t
n
o
c(t) = max (1 + c1 )c1 , (1 + c1 )
M
P (τ )dτ, P (t), L1 m3 (t), L2 m4 (t) , and
0
c0 = kφk + |η| + c1 kφk + 2c2 ) (1 + T ) + c1 u(0)
n
X
−
ck |x(t−
+
k )| + (T − tk )dk |x(tk )|
k=1
the initial value problem (1.4)-(1.7) has at least one solution on [−r, T ].
Proof. To prove the existence of a solution of the initial value problem (1.4)-(1.7)
we apply Lemma 1. First we obtain the a priori bounds for the solutions of the
initial value problem (1.4)λ − (1.7), λ ∈ (0, 1), where (1.4)λ stands for the equation
Z t
d 0
[x (t) − g(t, xt )] =λF t, xt , x0 (t),
a(t, s)g(s, xs , x0 (s))ds,
dt
0
Z T
b(t, s)h(s, xs , x0 (s))ds , t ∈ I.
0
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Global Existence For Volterra–Fredholm Type Functional Differential Equations
59
Let x be a solution of the initial value problem (1.4)λ − (1.7). From
Z
Z tZ
t
s
F τ, xτ , x0 (τ ),
g(s, xs )ds + λ
x(t) = λφ(0) + λ[η − g(0, φ)]t + λ
0
0
0
Z T
Z τ
b(τ, σ)h(σ, xσ , x0 (σ))dσ dτ ds
a(τ, σ)w(σ, xσ , x0 (σ))dσ,
0
0
X
−
+λ
[Ik (x(tk )) + (T − tk )Jk (x(t−
k ))], t ∈ I
0<tk <t
we have, for every t ∈ I,
Z tZ s
Z t
P (τ )Ω6 kxτ k + |x0 (τ )|
kxs kds +
|x(t)| ≤ kφk + [|η| + c1 kφk + 2c2 ]T + c1
0
0
0
Z T
Z τ
m4 (σ)Ω5 (kxσ k)dσ dτ ds
m3 (σ)Ω4 (kxσ k)dσ + L2
+ L1
0
0
+
n
X
−
[ck |x(t−
k )| + (T − tk )dk |x(tk )|].
k=1
We consider the function µ given in the proof of Theorem 3. Then by the previous
inequality we have,
µ(t) = sup{|x(s)| : −r ≤ s ≤ t}, t ∈ I.
Let t∗ ∈ [−r, t] be such that µ(t) = |x(t∗ )|. If t∗ ∈ [0, t], by the previous inequality
we have, for every t ∈ I,
|µ(t)|
Z t
Z tZ s
≤ kφk + [|η| + c1 kφk + 2c2 ]T + c1
µ(s)ds +
P (τ )Ω6 µ(τ ) + |x0 (τ )|
0
0
0
Z T
Z τ
m4 (σ)Ω5 (µ(σ) + |x0 (σ)|)dσ dτ ds
+ L1
m3 (σ)Ω4 (µ(σ) + |x0 (σ)|)dσ + L2
0
+
n
X
0
−
[ck |x(t−
k )| + (T − tk )dk |x(tk )|].
k=1
If t∗ ∈ [−r, 0] then µ(t) = kφk and the previous inequality obvious holds.
Denoting by u(t) the right hand side of the above inequality we have,
n
X
−
u(0) = kφk + [|η| + c1 kφk + 2c2 ]T +
[ck |x(t−
k )|+(T − tk )dk |x(tk )|],
k=1
µ(t) ≤ u(t), t ∈ I.
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60
V. Vijayakumar, S. Sivasankaran and M. Mallika Arjunan
and
Z τ
Z t
m3 (σ)Ω4 (µ(σ) + |x0 (σ)|)dσ
P (τ )Ω6 µ(τ ) + |x0 (τ )| + L1
u0 (t) = c1 µ(t) +
0
0
Z T
m4 (σ)Ω5 (µ(σ) + |x0 (σ)|)dσ dτ
+ L2
0
Z τ
Z t
0
m3 (σ)Ω4 (u(σ) + |x0 (σ)|)dσ
P (τ )Ω6 u(τ ) + |x (τ )| + L1
≤ c1 u(t) +
0
0
Z T
m4 (σ)Ω5 (u(σ) + |x0 (σ)|)dσ dτ, t ∈ I.
+ L2
0
Therefore if
v(t) = sup{|x0 (s)| : s ∈ I}, t ∈ I
we obtain
Z
0
t
Z
τ
P (τ )Ω6 u(τ ) + v(τ ) + L1
m3 (σ)Ω4 (u(σ) + v(σ))dσ
u (t) ≤ c1 u(t) +
0
0
Z T
+ L2
m4 (σ)Ω5 (u(σ) + v(σ))dσ dτ, t ∈ I.
(4.1)
0
On the other hand, by
t x (t) =λ[η − g(0, φ)] + λg(t, xt ) + λ
F τ, xτ , x0 (τ ),
0
Z τ
Z T
0
a(τ, s)W (s, xs , x (s))ds,
b(τ, s)H(s, xs , x0 (s))ds dτ
Z
0
0
0
for any t ∈ I and every s ∈ [0, t], we obtain
Z t
Z τ
0
0
|x (t)| ≤|η| + c1 kφk + 2c2 + c1 kxt k +
P (τ )Ω6 u(τ ) + |x (τ )| + L1
m3 (σ)
0
0
Z T
(×)Ω4 (u(σ) + |x0 (τ )|)dσ + L2
m4 (σ)Ω5 (u(σ) + |x0 (τ )|)dσ dτ,
0
or
Z
t
Z
τ
v(t) ≤|η| + c1 kφk + 2c2 + c1 u(t) +
P (τ )Ω6 u(τ ) + v(τ ) + L1
m3 (σ)
0
0
Z T
(×)Ω4 (u(σ) + v(τ ))dσ + L2
m4 (σ)Ω5 (u(σ) + v(τ ))dσ dτ, t ∈ I.
0
Denoting by z(t) the right hand side of the above inequality we have:
z(0) = |η| + c1 kφk + 2c2 + c1 u(0),
v(t) ≤ z(t), t ∈ I
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61
and
Z t
m3 (σ)Ω4 (u(σ) + v(τ ))dσ
z 0 (t) = c1 u0 (t) + P (t)Ω6 u(t) + v(t) + L1
0
Z T
m4 (σ)Ω5 (u(σ) + v(τ ))dσ ,
+ L2
0
Z t
0
m3 (σ)Ω4 (u(σ) + z(τ ))dσ
≤ c1 u (t) + P (t)Ω6 u(t) + z(t) + L1
0
Z T
m4 (σ)Ω5 (u(σ) + z(τ ))dσ , t ∈ I.
+ L2
0
From (4.1), since v(t) ≤ z(t) we have
Z τ
Z t
m3 (σ)Ω4 (u(σ) + z(σ))dσ
u0 (t) ≤ c1 u(t) +
P (τ )Ω6 u(τ ) + z(τ ) + L1
0
0
Z T
+ L2
m4 (σ)Ω5 (u(σ) + z(σ))dσ dτ, t ∈ I.
0
Let
Z
t
w(t) = u(t) + z(t)+L1
m1 (σ)Ω4 (u(σ) + z(σ))dσ
Z T
+ L2
m2 (σ)Ω5 (u(σ) + z(σ))dσ,
0
t ∈ I.
0
Then
u(t) + z(t) ≤ w(t), t ∈ I
w(0) = u(0) + z(0) = c0 ,
and
w0 (t) = u0 (t) + z 0 (t) + L1 m3 (t)Ω4 (u(t) + z(t)) + L2 m4 (t)Ω5 (u(t) + z(t))
Z t
≤ c1 u(t) +
P (τ )Ω6 (w(τ ))dτ + c1 u0 (t) + P (t)Ω6 (w(t))
0
+ L1 m3 (t)Ω4 (u(t) + z(t)) + L2 m4 (t)Ω5 (u(t) + z(t))
Z t
≤ (1 + c1 )c1 w(t) + (1 + c1 )
P (τ )Ω6 (w(τ ))dτ + P (t)Ω6 (w(t))
0
+ L1 m3 (t)Ω4 (u(t) + z(t)) + L2 m4 (t)Ω5 (u(t) + z(t))
≤ m(t)
b
w(t) + 2Ω6 (w(t)) + Ω4 (w(t)) + Ω5 (w(t)) , t ∈ I.
This implies
Z w(t)
w(0)
ds
≤
s + 2Ω6 (s) + Ω4 (s) + Ω5 (s)
Z
T
Z
∞
c(s)ds <
M
0
c0
ds
s + 2Ω6 (s) + Ω4 (s) + Ω5 (s)
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62
V. Vijayakumar, S. Sivasankaran and M. Mallika Arjunan
This inequality implies that there is a constant K such that w(t) ≤ K, t ∈ I. Then
|x(t)| ≤ µ(t) ≤ u(t), t ∈ I
|x0 (t)| ≤ v(t) ≤ z(t), t ∈ I
and hence
kx∗ k ≤ kxkr + kxk1 ≤ K.
In the second step we rewrite the initial value problem (1.4)-(1.7) as an integral
operator and will prove that this operator is completely continuous.
Define the operator Q : B → B, B = Ω by


φ(t),
t ∈ [−r, 0]



R
R R

 φ(0) + [η − g(0, φ)]t + t g(s, ys )ds + t s F τ, xτ , x0 (τ ),
0
0 0
(Qx)(t) =
Rτ
RT
0
0 (σ))dτ dτ ds

a(s,
σ)W
(σ,
x
,
x
(σ))dσ,
b(s,
σ)H(τ,
x
,
x

σ
σ

0
0


 +P
−
−
t ∈ I.
0<tk <t [Ik (x(tk )) + (T − tk )Jk (x(tk ))],
Let a bounded sequence {zυ } in B, i.e kzυ k ≤ b, for all υ. Then the sequence
{zυ (t)}, t ∈ I is bounded in B. Using the fact that
Z t
Z
0
0
a(t, s)W (s, zυs , zυ (s))ds,
F t, zυt , zυ (s),
0
0
T
b(t, s)H(s, zυs , zυ0 (s))ds ≤ M1
where
f(t), t ∈ I}}
M1 = M ∗ Ω6 2b + M ∗ Ω4 (2b)T + M ∗ Ω5 (2b)T , M ∗ = max{kφk, max{M
we can easily prove that there exists a constant
n
T2
M = max kφk + [|η| + c1 kφk + 2c2 ]T + c1 bT + M1
2
n
o
X
+
(ck + (T − tk )dk )|z(t−
)|,
|η|
+
(1
+
T
)c
kφk
+
(2
+
T
)c
+
M
T
1
2
1
k
k=1
such that
kQzυ kr ≤ M
and k(Qzυ )0 k1 ≤ M
which means that {Qzυ } and {(Qzυ )0 } are uniformly bounded.
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63
Global Existence For Volterra–Fredholm Type Functional Differential Equations
Moreover rewriting the operator Qx for t ∈ I as (Qx)(t) = (Q1 x)(t) + (Q2 x)(t)
where
Z t
g(s, ys )ds
(Q1 x)(t) =φ(0) + [η − g(0, φ)]t +
0
X
−
+
[Ik (x(t−
k )) + (T − tk )Jk (x(tk ))],
Z tZ
(Q2 x)(t) =
0
s
F τ, xτ , x0 (τ ),
Z
0<tk <t
τ
0
0
a(s, σ)W (σ, xσ )dσ,
Z T
b(s, σ)H(τ, xσ )dτ dτ ds, t ∈ I,
0
it is easy to see that the sequences {Q1 zυ }, {(Q1 zυ )0 } and {Q2 zυ }, {(Q2 zυ )0 } are
equicontinuous.
Thus, by the Arzela-Ascoli theorem the operator Q is completely continuous.
Finally, the set Φ(Q) = {x ∈ B : x = λQx, λ ∈ (0, 1)} is bounded, as we proved
in the first part. Hence by Lemma 1, the operator Q has a fixed point in B. Then
it is clear that the initial value problem (1.4)-(1.7) has at least one solution.
Hence the proof of the theorem is complete.
5
Application
In this section, we apply some of the results established in this paper.
Example 6. First, we consider the partial first order differential equation of the
form
t
Z
wt (u, t) − g1 (t, w(x, t − r)) =P t, w(u, t),
0
k1 t, w(x, t − r) ds,
Z b
h1 t, w(x, t − r) ds ,
(5.1)
0
w(0, t) =w(π, t) = 0,
w(u, t) =φ(x, t),
−
w(t+
k , y)−w(tk , y)
0 ≤ t ≤ b,
(5.2)
0 ≤ u ≤ π,
=
Ik (w(t−
k , y)),
(5.3)
k = 1, 2, ..., m,
(5.4)
where P : [0, b] × D × Rn × Rn → Rn is a function and g1 , k1 , h1 : [0, b] × D → Rn are
continuous functions. We assume that the functions P, g1 , k1 and h1 in (5.1)-(5.4)
satisfy the following conditions.
(1) There exist constants c1 and c2 such that |g(t, φ)| ≤ c1 |φ| + c2 , for t ∈ [0, b],
φ ∈ D.
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64
V. Vijayakumar, S. Sivasankaran and M. Mallika Arjunan
(2) There exists a nonnegative function p1 defined on [0, b] such that
Z t
k1 (t, x)ds ≤ p1 (t)|x|
0
for t ∈ [0, b] and x ∈ D.
(3) There exists a nonnegative function q1 defined on [0, b] such that
Z b
h1 (t, x)ds ≤ q1 (t)|x|
0
for t ∈ [0, b] and x ∈ D.
(4) There exists nonnegative real valued continuous function l1 defined on [0, b]
and a positive continuous increasing function K1 defined on R+ such that
|P (t, x, y, z)| ≤ l1 (t)K1 (|x| + |y| + |z|)
for t, x, y, z ∈ [0, b] × D × Rn × Rn .
(5) There exist constants ck such that |Ik (x)| ≤ ck |x|, k = 1, ..., m for each x ∈ Rn .
Let us take X = L2 [0, π]. Suppose that
Z
0
T
Z
m(s)ds
b
<
+∞
c
ds
Ω3 (s) + Ω1 (s) + Ω2 (s)
where
m(t)
b
= max
c=
1
(1 − c1 )
o
1
p(t), L1 m1 (t), L2 m2 (t) and
(1 − c1 )
n
n
o
X
(1 + c1 )kφk + 2c2 +
ck |x(t−
)|
k
n
k=1
is satisfied. Define the functions f : [0, b] × X × X × X → X, a, b : [0, b] × [0, b] → X,
g, w, h : [0, b] × X → X, Ik ∈ (X, X) as follows
f (t, x, y, z)(u) = P (t, x(u, t), y(u, t), z(u, t)),
k(t, x)(u) = k1 (t, x(u, t))
and
h(t, x)(u) = h1 (t, x(u, t))
for t ∈ [0, b], x, y, z ∈ X and 0 ≤ u ≤ π.
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Global Existence For Volterra–Fredholm Type Functional Differential Equations
65
With these choices of the functions, the equations (5.1)-(5.4) can be modelled
abstractly as nonlinear mixed Volterra-Fredholm integrodifferential equation with
nonlocal condition in Banach space X:
d
[x(t) − g(t, xt )] = f t, xt ,
dt
Z
t
Z
a(t, s)w(s, xs )ds,
T
b(t, s)h(s, xs )ds ,
0
0
t ∈ I = [0, T ] \ {t1 , ..., tm },
∆x|t=tk =
Ik (x(t−
k )),
x0 = φ
k = 1, ..., m,
(5.5)
(5.6)
(5.7)
Since all the hypotheses of the Theorem 3 are satisfied, the Theorem 3 can be applied
to guarantee the solution of the nonlinear mixed Volterra-Fredholm type neutral
impulsive integrodifferential equation.
Acknowledgement.
The first author would like to thank Dr. V. Palanisamy, Principal, Dr. Chitra
Manohar, Secretary, Info Institute of Engineering, Coimbatore, for their constant
encouragement and support for this research work.
The third author dedicates this paper to “Silver Jubilee Year Celebrations of
Karunya University, Coimbatore”.
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V. Vijayakumar
S. Sivasankaran
Department of Mathematics,
Department of Mathematics,
Info Institute of Engineering,
University College,
Kovilpalayam, Coimbatore-641 107,
Sungkyunkwan University,
Tamilnadu, India.
Suwon 440-746, South Korea.
e-mail: vijaysarovel@gmail.com
e-mail: sdsiva@gmail.com.
M. Mallika Arjunan
Department of Mathematics,
Karunya University,
Karunya Nagar, Coimbatore-641 114,
Tamilnadu, India.
e-mail: arjunphd07@yahoo.co.in
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Surveys in Mathematics and its Applications 7 (2012), 49 – 68
http://www.utgjiu.ro/math/sma