Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 78, pp. 1–11.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
DELAY DIFFERENTIAL EQUATIONS WITH HOMOGENEOUS
INTEGRAL CONDITIONS
ABDUR RAHEEM, DHIRENDRA BAHUGUNA
Abstract. In this article we prove the existence and uniqueness of a strong
solution of a delay differential equation with homogenous integral conditions
using the method of semidiscretization in time. As an application, we include
an example that illustrates the main result.
1. Introduction
This article concerns the delay differential equation having homogeneous integral
conditions,
∂u ∂ 2 u
∂3u
−
− λ 2 = F (x, t, ut ) on (0, 1) × (0, T ],
2
∂t
∂x
∂x ∂t
u(x, t) = φ(x, t) on (0, 1) × [−T, 0],
(1.1)
(1.2)
with integral conditions
Z
Z
1
u(x, t) dx = 0,
t ∈ [0, T ],
(1.3)
xu(x, t) dx = 0,
t ∈ [0, T ],
(1.4)
0
1
0
where 0 < T < ∞, the unknown function u : [−T, T ] → L2 (0, 1), the history
φ : [−T, 0] → L2 (0, 1) and the nonlinear map F : (0, T ] × C0 → B(0, 1), are defined
by u(t)(x) = u(x, t), φ(t)(x) = φ(x, t) and F (t, ut )(x) = F (x, t, ut ), respectively.
Here L2 (0, 1) is the real Hilbert of all square integrable real valued functions on
(0, 1) with the standard inner product, and B(0, 1) is the completion of C0 (0, 1),
the space of all continuous functions on (0, 1) having compact support in (0, 1),
with the inner product defined by
Z 1
(u, v)B =
=x u(x)=x v(x) dx,
0
Rx
where =x u(x) = 0 u(ξ)dξ. We recall that, if k · kB denote the corresponding norm;
that is,
p
kψkB = (ψ, ψ)B ,
2000 Mathematics Subject Classification. 35D35, 34K05, 34K07, 34K10, 34K30.
Key words and phrases. Method of semidiscretization; delay differential equation;
strong solution.
c
2013
Texas State University - San Marcos.
Submitted April 6, 2012. Published March 22, 2013.
1
2
A. RAHEEM, D. BAHUGUNA
EJDE-2013/78
then it follows that
1
kψk2 .
2
Also for t ∈ (0, T ], Ct := C([−T, t]; B(0, 1)) is the Banach space of all continuous
functions from [−T, t] into B(0, 1) endowed with the supremum norm
kψk2B ≤
kψkt =
sup kψ(θ)kB .
−T ≤θ≤t
For ψ ∈ CT , we denote ψt ∈ C0 given by ψt (θ) = ψ(t + θ), θ ∈ [−T, 0].
For the consideration of integral conditions, we use the space V introduce by
Merazga and Bouziani [11],
Z 1
Z 1
V = φ ∈ L2 (0, 1) :
φ(x) dx =
xφ(x) dx = 0 .
0
0
Note that V is a Hilbert space with respect to the standard inner product.
Since 1930, various classical types of initial boundary value problems have been
investigated by many authors using the method of semidiscretization, see for instance, [6, 13, 14, 15] and references therein.
In this paper our aim is to extend the application of the method of semidiscretization in time to delay differential equations with homogeneous integral conditions
and to establish the existence and uniqueness of a strong solution for a delay differential equation with homogeneous integral conditions given by (1.1)-(1.4).
The method of semidiscretization in time is a constructive method and has a
strong numerical aspect. For the application of the method of semidiscretization
to integrodifferential equations with nonclassical boundary conditions, we refer the
readers to [2, 3, 4, 8, 9] and references therein.
Dubey [5] established the existence and uniqueness of a strong solution for the
following nonlinear differential equation in a reflexive Banach space with a nonlocal
history condition using the method of semidiscretization in time
u0 (t) + Au(t) = f (t, u(t), ut ),
h(u0 ) = φ
on
t ∈ (0, T ],
[−τ, 0],
where 0 < T < ∞, φ ∈ C0 := C([−τ, 0]; X), τ > 0, the nonlinear operator A is
single-valued and m-accretive defined from the domain D(A) ⊂ X into X, the
nonlinear map f is defined from [0, T ] × X × C0 := C([−τ, 0]; X) into X, the map
h is defined from C0 and C0 . Bahuguna, Abbas, and Dabas [1] applied the method
of semidiscretization to a semilinear functional partial differential equation with an
integral condition.
Our problem is motivated by the work of Lakoud and Belakroum [10] and Dubey
[5]. Lakoud and Balakroum [10] established the existence and uniqueness of a weak
solution for the integro-differential evolution with a memory term,
Z t
∂2v
∂3v
∂v
−
−
λ
=
g(x,
t)
+
a(t − s)k(s, v(x, s)) ds on (0, 1) × (0, T ],
∂t
∂x2
∂x2 ∂t
0
subject to the initial conditions
v(x, 0) = V0 (x),
and integral conditions
Z
1
v(x, t) dx = E(t),
0
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DELAY DIFFERENTIAL EQUATIONS
3
1
Z
xv(x, t) dx = G(t),
0
where f, V0 , G, E are given functions and T, λ are positive constants.
The plan of the paper is as follows. In section 2, we state all the assumptions
and preliminaries. In section 3, we state the main result. In section 4, we state and
prove all the lemmas that are required to prove the main result and at the end of
this section, we prove the main result. In the last section, we give an application
of the main result.
Throughout the paper we denote a generic constant by C. This constant may
have different values in the same discussion.
2. Preliminaries
We will use the following assumptions:
(H1) The nonlinear map F : (0, T ] × C0 → B(0, 1) satisfies a local Lipschitz
condition
kF (t1 , ψ1 ) − F (t2 , ψ2 )kB ≤ LF (r)[|t1 − t2 | + kψ1 − ψ2 k0 ],
for all t1 , t2 ∈ (0, T ] and ψ1 , ψ2 ∈ C0 with kψi − φ(0)k0 ≤ r, i = 1, 2 and
LF (r) is a nondecreasing function of r > 0.
(H2) The history function φ : [−T, 0] → L2 (0, 1) is uniformly Lipschitz continuous with Lipschitz constant K > 0; i.e., kφ(t) − φ(s)k ≤ K|t − s|.
R1
R1
(H3) 0 φ(x, 0) dx = 0, 0 xφ(x, 0) dx = 0.
Lemma 2.1. If −A is the infinitesimal generator of a C0 -semigroup of contractions
in a Banach space X then A is m-accretive; i.e.,
(Au, J(u)) ≥ 0,
for u ∈ D(A),
where J is the duality mapping and R(I + λA) = X for λ > 0, I is the identity
operator on X and R(·) is the range of an operator.
The proof of the above lemma follows from Lumer Phillips theorem [12, Thm.
1.4.3].
3. Main result
Theorem 3.1. Suppose that assumptions (H1)–(H3) are satisfied. Then problem
(1.1)-(1.4) has a unique strong solution on the interval [−T, T ].
4. Discretization and a priori estimates
To apply the method of semidiscretization we divide the interval [0, T ] into the
subintervals of length hn = Tn . We set un0 = φ(0) for all n ∈ N and define {unj }
successively as the unique solution of the problem
δunj −
∂ 2 unj
∂ 2 δunj
−
λ
= F (tnj , ũnj−1 ),
∂x2
∂x2
Z 1
unj dx = 0,
Z
(4.1)
(4.2)
0
1
0
xunj dx = 0,
(4.3)
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A. RAHEEM, D. BAHUGUNA
EJDE-2013/78
where ũn0 = φ(t) for t ∈ [−T, 0] and 2 ≤ j ≤ n,
(
φ(tnj−1 + t),
if t ∈ [−T, −tnj−1 ]
ũnj−1 (t) =
uni−1 + (tnj−1 + t − tni−1 )δuni , if t ∈ [−tnj−i , −tnj−i−1 ], 1 ≤ i ≤ j − 1,
and
unj − unj−1
.
hn
Let wjn = unj + λδunj . This implies that δunj = hn1+λ wjn −
will reduce to
∂ 2 wjn
1
+
−
wn = fjn ,
∂x2
hn + λ j
Z 1
wjn dx = 0,
δunj =
Z
1
n
hn +λ uj−1 .
So (4.1)-(4.3)
(4.4)
(4.5)
0
1
xwjn dx = 0,
(4.6)
0
where
1
un + F (tnj , ũnj−1 ).
hn + λ j−1
Now we show the existence and uniqueness of functions wjn satisfying (4.4)-(4.6).
For this consider H = L2 (0, 1), the Hilbert space of all real valued square integrable
functions on the interval (0, 1). Let the linear operator A be defined by
Z 1
Z 1
D(A) := {u ∈ H : u00 ∈ H,
u(x) dx =
xu(x) dx = 0}, Au = −u00 .
fjn =
0
0
Then we know that −A is the infinitesimal generator of a C0 -semigroup S(t), t ≥ 0
of contractions in H.
The existence of unique wjn satisfying equations (4.4)-(4.6) is a consequence of
Lemma 2.1. As
hn
λ
unj =
wn +
un ,
λ + hn j
λ + hn j−1
there exist unique unj ∈ D(A) satisfying (4.1)–(4.3). Now we define
(
φ(t),
if t ∈ [−T, 0]
n
n
U (t) =
(4.7)
un
j −uj−1
n
n
uj−1 + (t − tj−1 ) hn , if t ∈ (tnj−1 , tnj ].
Lemma 4.1. For n ∈ N and j = 1, 2, · · · n,
kunj − φ(0)k ≤ C,
where C is a generic constant independent of n, j, hn .
Proof. Now for any ψ ∈ V , from (4.1) we have
∂ 2 un ∂ 2 δun j
j
(δunj , ψ)B −
,
ψ
,ψ
−
λ
= (Fjn , ψ)B ,
(4.8)
∂x2
∂x2
B
B
where Fjn = F (tnj , ũnj−1 ). By the definition of the inner product (, )B , we have
Z 1
∂ 2 un j
,ψ
=−
unj ψ dx = −(unj , ψ).
(4.9)
∂x2
B
0
EJDE-2013/78
DELAY DIFFERENTIAL EQUATIONS
5
(δunj , ψ)B + (unj , ψ) + λ(δunj , ψ) = (Fjn , ψ)B .
(4.10)
So (4.8) reduces to
Taking j = 1, ψ =
un1
−
un0
in (4.10),
(un1 − un0 , un1 − un0 )B + hn (un1 , un1 − un0 ) + λ(un1 − un0 , un1 − un0 )
= hn (F1n , un1 − un0 )B .
Now using (4.9), we obtain
(un1 − un0 , un1 − un0 )B + hn (un1 − un0 , un1 − un0 ) + λ(un1 − un0 , un1 − un0 )
d2 un0 n
n
= hn F1n +
,
u
−
u
.
0
dx2 1
B
Now, we obtain
d2 un kun1 − un0 k2B + hn kun1 − un0 k2 + λkun1 − un0 k2 ≤ hn kF1n kB + k 20 kB kun1 − un0 kB .
dx
By ignoring first two terms on the left hand side, we obtain
d2 un λkun1 − un0 k2 ≤ hn kF1n kB + hn k 20 kB kun1 − un0 kB .
dx
As kun1 − un0 kB ≤
√1 kun
1
2
− un0 k, we have
d2 un hn kun1 − un0 k ≤ √ kF1n kB + k 20 kB .
dx
λ 2
By using assumption (H1), and the inequality hn ≤ T , we obtain
T d2 un kun1 − un0 k ≤ √ LF (r)(T + r) + kF (0, φ(0))kB + k 20 kB ≤ C.
dx
λ 2
By putting ψ = unj − un0 in (4.10), we obtain
(unj − unj−1 , unj − un0 )B + hn (unj , unj − un0 ) + λ(unj − unj−1 , unj − un0 )
= hn (Fjn , unj − un0 )B .
Using (4.9), we obtain
unj − un0 , unj − un0
+ hn unj − un0 , unj − un0 + λ unj − un0 , unj − un0
d2 un0 n
, uj − un0 B
= hn Fjn , unj − un0 B + unj−1 − un0 , unj − un0 B + hn
2
dx
+ λ unj−1 − un0 , unj − un0 .
B
By ignoring the first two terms on the left hand side, we obtain
λkunj − un0 k2 ≤ hn kFjn kB kunj − un0 kB + kunj−1 − un0 kB kunj − un0 kB
+ hn k
As kunj − un0 kB ≤
√1 kun
j
2
kunj − un0 k ≤
d2 un0
kB kunj − un0 kB + λkunj−1 − un0 kkunj − un0 k.
dx2
− un0 k, we have
1 hn
d2 un
1
√ (kFjn kB + k 20 kB ) + ( + λ)kunj−1 − un0 k .
λ 2
dx
2
(4.11)
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A. RAHEEM, D. BAHUGUNA
EJDE-2013/78
By assumption (H1), we have
kFjn kB = kF (tnj , ũnj−1 )kB
= kF (tnj , ũnj−1 ) − F (0, φ(0))kB + kF (0, φ(0))kB
≤ LF (r)[|tnj | + kũnj−1 − φ(0)k0 ] + kF (0, φ(0))kB
(4.12)
≤ LF (r)[T + r] + kF (0, φ(0))kB .
Using (4.12) in (4.11), we obtain
d2 un hn kunj − un0 k ≤ √ LF (r)(T + r) + kF (0, φ(0))kB + k 20 kB
dx
λ 2
1 n
kuj−1 − un0 k
+ 1+
2λ
1 n
kuj−1 − un0 k,
kunj − un0 k ≤ hn K + 1 +
2λ
d2 un
1
where K = λ√
LF (r)(T + r) + kF (0, φ(0))kB + k dx20 kB . Repeating the above
2
procedure, we obtain
kunj − un0 k ≤ C.
This completes the proof.
Lemma 4.2. For j = 1, 2, · · · , n,
unj − unj−1
k
k ≤ C.
hn
Proof. By putting j = 1, ψ = unj − un0 in (4.10) and using (4.9), we obtain
un − un
un − un
1
0
0
, un1 − un0
, un1 − un0
+ (un1 − un0 , un1 − un0 ) + λ 1
hn
hn
B
d2 un
0
, un − un0 .
= (F1n , un1 − un0 )B +
dx2 1
B
By ignoring the first two terms on the left hand sides, we obtain
λ n
d2 un0 n
ku1 − un0 k2 ≤ F1n +
, u1 − un0 .
(4.13)
2
hn
dx
B
As kunj − un0 kB ≤
√1 kun
j
2
− un0 k, we have
1 d2 un un1 − un0
k ≤ √ kF1n kB + k 20 kB .
hn
dx
λ 2
Using assumption (H1), we obtain
1 d2 un un − un0
k ≤ √ LF (r)(T + r) + kF (0, φ(0))kB + k 20 kB ≤ C.
k 1
hn
dx
λ 2
Subtracting (4.10) written for j, from the same identity for j − 1 and then putting
ψ = unj − unj−1 , we obtain
k
(δunj , unj − unj−1 )B + (unj − unj−1 , unj − unj−1 ) + λ(δunj , unj − unj−1 )
n
= (Fjn − Fj−1
, unj − unj−1 )B + (δunj−1 , unj − unj−1 )B + λ δunj−1 , unj − unj−1 .
Ignoring the first two terms on the left hand side,
unj−1 − unj−2
λ n
n
kuj − unj−1 k2 ≤ kFjn − Fj−1
kB kunj − unj−1 kB + k
kB kunj − unj−1 kB
hn
hn
EJDE-2013/78
DELAY DIFFERENTIAL EQUATIONS
+ λk
7
unj−1 − unj−2
kkunj − unj−1 k.
hn
√
As kψkB ≤ kψk/ 2, we have
unj − unj−1
1
1 unj−1 − unj−2
n
k ≤ √ kFjn − Fj−1
kB + (1 +
)k
k.
k
hn
2λ
hn
λ 2
(4.14)
Now using assumption (H1),
n
kFjn − Fj−1
kB = kF (tnj , ũnj−1 ) − F (tnj−1 , ũnj−2 )kB
≤ LF (r)[|tnj − tnj−1 | + kũnj−1 − ũnj−2 k0 ]
≤ LF (r)[T + 2r].
Using the above inequality in (4.14), we obtain
unj − unj−1
1 unj−1 − unj−2
1
)k
k ≤ √ LF (r)[T + 2r] + (1 +
k.
k
hn
2λ
hn
λ 2
Repeating the above procedure, we finally obtain
unj − unj−1
k ≤ C.
k
hn
This completes the proof.
Now we introduce a sequence of step functions {X n (t)} defined by
(
φ(0), if t = 0
n
X (t) =
unj ,
if t ∈ (tnj−i , tnj ].
(4.15)
(4.16)
(4.17)
Remark 4.3. From Lemma 4.2 it follows that the functions U n are uniformly
Lipschitz continuous on [−T, T ] and U n (t) − X n (t) → 0, as n → ∞ on [0, T ].
Let F n (t) = F (tnj , ũnj−1 ). By assumption (H1) and remark 4.3, we see that
F (t) → F (t, ut ). Using (4.7) and (4.17), in (4.1), we obtain
n
d− n
∂2 n
∂3
U (t) −
X (t) − λ 2 X n (t) = F n (t).
2
dt
∂x
∂x ∂t
Integrating with respect to t, we obtain
Z t
Z t
∂2 n
∂3
n
n
−
X (s) + λ 2 X (s) ds = φ(0) − U (t) +
F n (s) ds.
∂x2
∂x ∂s
0
0
(4.18)
(4.19)
Lemma 4.4. There exists u ∈ C([−T, T ]; B(0, 1)) such that U n (t) → u(t) uniformly on [−T, T ]. Moreover u(t) is Lipschitz continuous on [−T, T ].
Proof. From (4.18), we have
d−
d− k
U n (t) −
U (t), U n (t) − U k (t)
+ (X n (t) − X k (t), U n (t) − U k (t))
dt
dt
B
∂
∂
+λ
X n (t) − X k (t), U n (t) − U k (t)
∂t
∂t
n
k
= (F (t) − F (t), U n (t) − U k (t))B .
Now,
1 d− n
λ ∂
kU (t) − U k (t)k2B + kX n (t) − X k (t)k2 +
kX n (t) − X k (t)k2
2 dt
2 ∂t
= (X n (t) − X k (t), X n (t) − X k (t) − U n (t) + U k (t))
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A. RAHEEM, D. BAHUGUNA
EJDE-2013/78
∂
(X n (t) − X k (t)), X n (t) − X k (t) − U n (t) + U k (t)
∂t
+ (F n (t) − F k (t), U n (t) − U k (t))B .
+λ
By ignoring the last two terms on the left hand side, we obtain
1 d− n
kU (t) − U k (t)k2B ≤ δnk (t) + kF n (t) − F k (t)kB kU n (t) − U k (t)kB .
2 dt
where
δnk (t) = kX n (t) − X k (t)k[kX n (t) − U n (t)k + kX k (t) − U k (t)k]
λ ∂
+ k (X n (t) − X k (t))k kX n (t) − U n (t)k + kX k (t) − U k (t)k .
2 ∂t
By Remark 4.3, it is clear that δnk (t) → 0 as n, k → ∞ uniformly on the interval
[0, T ]. Now by assumption (H1), we have
kF n (t) − F k (t)kB
≤
= kF (tnj , ũnj−1 ) − F (tkl , ũkl−1 )kB
(4.20)
0
δnk
(t)
(4.21)
n
k
+ LF (r)kU (t) − U (t)kB ,
where
0
δnk
(t) = LF (r)[|tnj − tkl | + kU n (t) − ũnj−1 k0 + kU k (t) − ũkl−1 k0 ].
0
Clearly δnk
(t) → 0 as n, k → ∞ uniformly on [0, T ]. This implies that for a.e.
t ∈ [0, T ],
1 d− n
0
kU (t) − U k (t)k2B ≤ δnk
(t) + LF (r)kU n (t) − U k (t)k2B ,
2 dt
Integrating the above inequality over (0, t) with 0 ≤ t ≤ T , we obtain
Z t
0
kU n (t) − U k (t)k2B ≤ 2δnk
T + 2LF (r)
kU n (s) − U k (s)k2B ds.
(4.22)
0
Applying Gronwall’s inequality, we obtain that U n → u in C([−τ, T ], B(0, 1)). As
each U n is uniformly Lipschitz continuous, and by assumption (H2), u is Lipschitz
continuous. This completes the proof.
Proof of Theorem 3.1. Taking limits as n → ∞ in (4.19), we obtain
Z t
Z t
∂2
∂3
−
u(t)
+
λ
u(t)
ds
=
φ(0)
−
u(t)
+
F (s, ut ) ds.
∂x2
∂x2 ∂s
0
0
This implies that
∂u(t) ∂ 2 u(t)
∂ 3 u(t)
−
− λ 2 = F (t, ut ), a.e. t ∈ [0, T ].
2
∂t
∂x
∂x ∂t
Clearly u(t) is differentiable with u(t) ∈ V a.e. on [0, T ] and u(t) = φ(t), t ∈ [−T, 0].
This implies that u(t)(x) = u(x, t) is a strong solution of (1.1)–(1.4). Now we show
the uniqueness of the strong solution. To do this, suppose that u1 , u2 are two strong
solutions of (1.1)-(1.4).
Let u = u1 − u2 , then for ψ ∈ V , we have
∂u ∂u ,ψ
+ (u, ψ) + λ
, ψ = F (t, (u1 )t ) − F (t, (u2 )t ), ψ .
∂t
∂t
B
B
Putting ψ = u and ignoring last two terms in the left hand side, we obtain
∂u ,u
≤ F (t, (u1 )t ) − F (t, (u2 )t ), u .
∂t
B
B
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DELAY DIFFERENTIAL EQUATIONS
9
By using assumption (H1), we obtain
1 ∂
ku(t)k2B ≤ LF (r)k(u1 )t − (u2 )t kB ku(t)kB
2 ∂t
≤ LF (r) sup kut (θ)kB sup ku(θ)kB .
−T ≤t+θ≤t
−T ≤θ≤t
Integrating between 0 and t, we obtain
sup ku(θ)k2B ≤ 2LF (r)
t
Z
−T ≤θ≤t
kuk2s ds
0
kuk2t ≤ 2LF (r)
Z
t
kuk2s ds.
0
Applying Gronwall’s inequality, we obtain u = 0 on [−T, T ]. Hence we obtain a
unique strong solution of problem (1.1)-(1.4) on the interval [−T, T ].
5. Application
Consider the partial differential equation
Z t
∂2v
∂3v
∂v
−
=
g(x,
t)
+
−
λ
a(t − s)k(s, v(x, s)) ds on (0, 1) × (0, T ],
∂t
∂x2
∂x2 ∂t
0
v(x, t) = φ(x, t) on (0, 1) × [−T, 0],
(5.1)
with the integral conditions
Z 1
v(x, t) dx = 0,
(5.2)
Z
0
1
xv(x, t) dx = 0.
(5.3)
0
In the above problem, we identify the unknown function v : (0, T ] → B(0, 1), by
v(t)(x) = v(x, t), g : (0, T ] → B(0, 1) by g(t)(x) = g(x, t), k : (0, T ] × R → B(0, 1)
by k(t, v(x, t)) = k(t, v(t))(x) and the history function φ : [−T, 0] → B(0, 1) by
φ(t)(x) = φ(x, t). Also we take
Z 1
Z 1
V = φ ∈ L2 (0, 1) :
φ(x) dx =
xφ(x) dx = 0 .
0
0
Putting t = s − η in the integral term, problem (5.1)-(5.3) reduces to
Z 0
∂2v
∂3v
∂v
−
− λ 2 = g(t) +
a(−η)k(t + η, v(t + η))dη on (0, T ],
∂t
∂x2
∂x ∂t
−t
v(t) = φ(t), t ∈ [−T, 0].
Now we consider the following assumptions:
(i) There exists k1 > 0, such that for all t, s ∈ (0, T ]
kg(t) − g(s)kB ≤ k1 |t − s|.
(ii) There exists k2 > 0, such that for all t, s ∈ (0, T ] and ψ1 , ψ2 ∈ C0 ,
kk(t, ψ1 (t)) − k(s, ψ2 (s))kB ≤ k2 [|t − s| + kψ1 − ψ2 k0 ].
(iii) Also there exist M > 0, such that
ka(t)kB ≤ M,
t ∈ [−T, 0].
(5.4)
10
A. RAHEEM, D. BAHUGUNA
EJDE-2013/78
Now we define G : (0, T ] × C0 → B(0, 1) by
Z 0
G(t, ψ) = g(t) +
a(−η)k(t + η, ψ)dη.
−t
Thus (5.4) reduces to
∂v
∂3v
∂2v
− λ 2 = G(t, vt ) on (0, T ],
−
2
∂t
∂x
∂x ∂t
v(t) = φ(t), t ∈ [−T, 0].
(5.5)
Now we show that G satisfies assumption (H1). For this take t, s ∈ (0, T ] and
ψ1 , ψ2 ∈ C0
kG(t, ψ1 ) − G(s, ψ2 )kB
Z
≤ kg(t) − g(s)kB + 0
Z
0
a(−η)k(t + η, ψ1 (η))dη −
−t
−s
a(−η)k(s + η, ψ2 (η))dη B .
Using the given conditions on g, a and k, we obtain
Z 0
kG(t, ψ1 ) − G(s, ψ2 )kB ≤ k1 |t − s| + M k2
{|t − s| + kψ1 − ψ2 k0 }dη
−t
Z
−t
kk(s + η, ψ2 (η))kB dη.
+M
−s
After some simplifications, we obtain
kG(t, ψ1 ) − G(s, ψ2 )kB ≤ (k1 + M k2 T + M K)|t − s| + M k2 T kψ1 − ψ2 k0
≤ L[|t − s| + kψ1 − ψ2 k0 ],
where
L = max{(k1 + M k2 T + M K), M k2 T },
kk(s + η, ψ2 (η))kB ≤ K.
As G satisfies a Lipschitz like condition, we apply the result of Theorem 3.1 to
ensure the existence and uniqueness of a strong solution of (5.1)-(5.3) .
Acknowledgements. The authors want to thank the anonymous referees for their
valuable suggestions. The first author acknowledges the sponsorship from CSIR,
India, under research grant 09/092 (0652) /2008-EMR-1. The second author acknowledges the financial help from the Department of Science and Technology, New
Delhi, under research project SR/S4/MS:796/12.
References
[1] D. Bahuguna, S. Abbas, J. Dabas; Partial functional differential equation with an integral
condition and applications to population dynamics, Nonlinear Analysis 69, (2008), 2623–2635.
[2] D. Bahuguna, V. Raghavendra; Rothe’s method to parabolic integrodifferential equation via
abstract integrodifferential equations, Applicable Analysis, vol. 33, no., 3–4, 153–167, 1989.
[3] D. Bahuguna, A. K. Pani, V. Raghavendra, Rothe’s method to semilinear hyperbolic integrodifferential equations, Journal of Applied Mathematics and Stochastic Analysis, vol. 3, no.
4, pp. 245–252, 1990.
[4] A. Bouziani, R. Mechri; The Rothe’s Method to a parabolic Integrodifferential Equation with
a Nonclassical Boundary conditions, Int. J. Stoch. Anal. 2010, Art. ID 519684, 1–16.
[5] Shruti A. Dubey; The method of lines applied to nonlnear nonlocal functional differential
equations, J. Math. Anal. Appl. 376(2011) 275–281.
[6] J. Kacur; Method of Rothe in Evolution Equations, Teubner-Text zur Mathematik, Vol. 80,
1985.
EJDE-2013/78
DELAY DIFFERENTIAL EQUATIONS
11
[7] N. Kikuchi, J. Kacur; Convergence of Rothe’s method in Hl̈der spaces, Appl. Math. 48 (2003),
no. 5, 353-365.
[8] A. G. Lakoud, A. Chaoni; Rothe’s Method Applied to semilinear Hyperbolic Integrodifferetial
equation with integral condition, Int. J. open Problem Compt. Math. Vol. 4, No. 1, March
2011.
[9] A. G. Lakoud, M. S. Jasmati, A. Chaoni; Rothe’s method for an integrodifferential equation
with integral conditions, Nonlinear Analysis 72 (2010) 1552–1530.
[10] A. G. Lakoud, D. Belakroum; Time-descretization schema for an integrodifferential Sobolev
type equation with integral conditions, Appied Math. Comp. Vol. 218, 9 (2012), 4695–4702.
[11] N. Merazga, A. Bouziani; Rothe time-discretization method for a nonlocal problem arising
in thermoelasticity, Journal of Applied Mathematics and Stochastic Analysis 2005:1,(2005)
13–28.
[12] A. Pazy; Semigroup of linear operators and application to partial differential equations,
Springer-Verlag, New York, 1983.
[13] K. Rektorys; The Method of Discretization in time and Partial Differential Equations, D.
Reidel Publishing Company, 1982.
[14] K. Rektorys; On application of direct variational metheods to the solution of parabolic boundary value problems of arbitrary order in the space variables, Czech. Math. J., 21(96), 1971,
318–339.
[15] K. Rektorys; Numerical and theoretical treating of evolution problems by the method of discretization in time, Equadiff 6 (Brno, 1985), 71–84, Lecture Notes in Math., 1192, Springer,
Berlin, 1986.
Abdur Raheem
Department of Mathematics and Statistics, Indian Institute of Technology Kanpur,
Kanpur -208016, India
E-mail address: araheem@iitk.ac.in
Dhirendra Bahuguna
Department of Mathematics and Statistics, Indian Institute of Technology Kanpur,
Kanpur -208016, India
E-mail address: dhiren@iitk.ac.in