Applied Mathematics and Computation 218 (2011) 4169–4176

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

A study of delayed cooperation diffusion system with Dirichletboundary conditions

Abdur Raheem ⇑, Dhirendra BahugunaDepartment of Mathematics and Statistics, Indian Institute of Technology, Kanpur 208016, India

a r t i c l e i n f o

Keywords:Delayed cooperation diffusion systemStrong solutionSemigroup of bounded linear operatorsMethod of semidiscretization

0096-3003/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.amc.2011.09.048

⇑ Corresponding author.E-mail addresses: araheem@iitk.ac.in (A. Raheem

a b s t r a c t

In this paper we study the existence and uniqueness of strong solution of a delayed coop-eration diffusion system with Dirichlet boundary conditions using the method of semidis-cretization in time.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

In mathematical biology, many models of population dynamics can be described by the delayed reaction diffusion equa-tions. Consider the following delayed cooperation diffusion system with Dirichlet boundary conditions:

@w@tðt; xÞ ¼ @

2w@x2 ðt; xÞ þ kwðt; xÞ½1�wðt � s; xÞ� þ f ðt; xÞ; t 2 ½t0; T�; x 2 ½0;p�; ð1Þ

wðt;0Þ ¼ wðt;pÞ ¼ 0; t 2 ½t0 � s; T�;wðt; xÞ ¼ /ðt; xÞ; ðt; xÞ 2 ½t0 � s; t0� � ½0;p�;

where w(t,x) denotes the density of species at time t and space location x, and k, s are all positive constants, and the maps f, /are defined from [t0,T] � [0,p] and [t0 � s, t0] � [0,p] into L2[0,p] respectively.

For earlier work on existence and uniqueness of strong solution of nonlinear retarded differential equation in a real Hil-bert space H, we refer to Agarwal and Bahuguna [1,2]. In [3], Bahuguna and Shukla have applied the method of semidiscret-ization in time to a class of quasilinear integrodifferential equation in a reflexive Banach space. Kartsatos and Zigler [4], usingthe method of semidiscretization, have proved the existence of a weak solution of the following equation in a reflexiveBanach space X whose dual is uniformly convex

dudtðtÞ þ AuðtÞ ¼ f ðt; uðtÞÞ; 0 < t 6 T; 0 < T <1; uð0Þ ¼ u0;

where A : D(A) # X ? X, is maximal monotone operator and f : [0,T] � X ? X, satisfies the Lipchitz-like condition

kf ðt; xÞ � f ðs; xÞkX 6 kUðtÞ �UðsÞkX þ Lkx� ykX ;

for all t, s in [0,T] and all x, y 2 X, where U : [0,T] ? X, is a continuous function of bounded variation and L is a positive constant.For applications of the method of semidiscretization to more general quasilinear evolution equations and nonlinear evo-

lution equations involving nonlocal conditions, we refer the reader to [5–10].

. All rights reserved.

), dhiren@iitk.ac.in (D. Bahuguna).

4170 A. Raheem, D. Bahuguna / Applied Mathematics and Computation 218 (2011) 4169–4176

Our aim is to study the strong solution for the delayed cooperation diffusion system with Dirichlet boundary conditionsusing the method of semidiscretization in time. In this method we replace the time derivative by the corresponding differ-ence quotients giving rise to a system of time independent operator equations. With the help of the theory of semigroups,these systems are guaranteed to have unique solutions. An approximate solution to the equation is defined in terms of thesolutions of these time independent systems. After proving a priori estimates for the approximate solution, the convergenceof the approximate solution to a unique strong solution is established.

In present work, first we consider the Eq. (1) on the interval [t0 � s, t0 + s] and prove the local Lipschitz condition for F.Then by the method of semidiscretization we prove the local existence and uniqueness of the strong solution of the Eq.(1) on [t0 � s, t0 + s]. In next steps we consider the previous solutions as history functions. Continuing this process we getthe unique strong solution either on the whole interval or on the maximal interval of existence.

2. Preliminaries and main result

First we will prove the existence and uniqueness of a strong solution of Eq. (1) on the interval [t0 � s, t0 + s], using themethod of semidiscretization in time.

It is clear that on the interval [t0 � s, t0 + s], (1) will become

@w@tðt; xÞ ¼ @

2w@x2 ðt; xÞ þ kwðt; xÞ½1� /ðt � s; xÞ� þ f ðt; xÞ; t 2 ½t0; t0 þ s�; x 2 ½0;p�; ð2Þ

wðt;0Þ ¼ wðt;pÞ ¼ 0; t 2 ½t0 � s; t0 þ s�;wðt; xÞ ¼ /ðt; xÞ; ðt; xÞ 2 ½t0 � s; t0� � ½0;p�:

Consider that H :¼ L2[0,p], the real Hilbert space of all real-valued square-integrable functions on the interval [0,p], let thelinear operator A be defined by

DðAÞ :¼ u 2 H : u00 2 H;uð0Þ ¼ uðpÞ ¼ 0f g; Au ¼ �u00:

Then we know that �A is the infinitesimal generator of a C0-semigroup S(t), t P 0 of contractions in H.Define the map F : [t0, t0 + s] � H ? H by

Fðt;vÞðxÞ ¼ kðð1�Uðt � sÞÞvÞðxÞ þ f ðt; xÞ: ð3Þ

If we define u : [t0 � s, t0 + s] ? H by u(t)(x) = w(t,x), and U : [t0 � s, t0] ? H by U(t)(x) = /(t,x), then (2) can be rewritten as

u0ðtÞ þ AuðtÞ ¼ Fðt; uðtÞÞ; t 2 ½t0; t0 þ s�; ð4ÞuðtÞ ¼ UðtÞ; t 2 ½t0 � s; t0�:

Lemma 2.1. If U : [t0 � s, t0] ? H is continuous on [t0 � s, t0], and U(t) is also continuous on [0,p] for each t 2 [t0 � s, t0], then Fdefined by (3) is uniformly Lipchitz continuous on H.

Proof. Now for any v1, v2 2 H, and t 2 [0,T], we have

kFðt;v1Þ � Fðt;v2Þk2 ¼Z p

0jFðt;v1ÞðxÞ � Fðt;v2ÞðxÞj

2ds� �1

2

¼ kZ p

0jð1�Uðt � sÞv1ÞðxÞ � ð1�Uðt � sÞv2ÞðxÞj

2dx� �1

2

6 kZ p

0jð1�Uðt � sÞÞj2jv1ðxÞ � v2ðxÞj

2dx� �1

2

6 k supt2½t0�s;t0 �

supx2½0;p�

jð1�Uðt � sÞÞðxÞjZ p

0jv1ðxÞ � v2ðxÞj

2dx ¼ KZ p

0jv1ðxÞ � v2ðxÞj

2dx� �1

2

;

where K ¼ k supt2½t0�s;t0 �

supx2½0;p�

jð1�Uðt � sÞÞðxÞj:

This complete the proof of the lemma. h

Lemma 2.2. If f, U are uniformly Lipchitz continuous on [t0, t0 + s] and [t0 � s, t0] respectively i.e. there exists L1, L2 > 0, such that

kf ðtÞ � f ðsÞk2 6 L1jt � sj;kUðtÞ �UðsÞk2 6 L2jt � sj;

then F, defined by (3) is locally Lipchitz continuous on [t0, t0 + s].

A. Raheem, D. Bahuguna / Applied Mathematics and Computation 218 (2011) 4169–4176 4171

Proof. Take t, s 2 [t0, t0 + s]. Then

kFðt;vÞ � Fðs;vÞk2 ¼Z p

0kð1�Uðt � sÞvÞðxÞ þ f ðt; xÞ � kð1�Uðs� sÞvÞðxÞ þ f ðs; xÞj j2dx

� �12

6 kZ p

0jððUðt � sÞ �Uðs� sÞÞvÞðxÞj2dx

� �12

þZ p

0jf ðt; xÞ � f ðs; xÞj2dx

� �12

6 kZ p

0jðUðt � sÞ �Uðs� sÞÞðxÞ

� �j2jvðxÞj2dxÞ

12 þ kf ðt; xÞ � f ðs; xÞk2

6 kZ p

0jðUðt � sÞ �Uðs� sÞÞðxÞj2dx

� �12Z p

0jvðxÞj2dx

� �12

þ kf ðt; xÞ � f ðs; xÞk2

6 kkU1ðt � sÞ �U2ðs� sÞk2kvk2 þ kf ðt; xÞ � f ðs; xÞk2 6 ðkL2kvk2 þ L1Þjt � sj 6 L2ðRÞjt � sj;

where L2(R) = (kL2 R + L1) and kvk 6 R. h

Remark 2.3. From Lemmas 2.1 and 2.2, it is clear that for everyv1,v2 2 H and t, s 2 [t0, t0 + s], there exists a constant L(R) > 0, s.t.

kFðt;v1Þ � Fðs;v2Þk2 6 LðRÞ½jt � sj þ kv1 � v2k2�; ð5Þ

where R = max{kv1k,kv2k}.

Lemma 2.4. If �A is the infinitesimal generator of a C0-semigroup of contractions then A is m-accretive i.e.

ðAu; JðuÞÞP 0; for u 2 DðAÞ;

where J is the duality mapping and R(I + kA) = X for k > 0, I is the identity operator on X and R(.) is the range of an operator.

Proof. It follows from Lumer Phillips theorem (cf. Theorem 1.4.3 [11]). h

Lemma 2.5. If �A is the infinitesimal generator of a C0-semigroup of contractions. If Xn 2 D(A), n = 1,2,3, . . . ,Xn ? u 2 H and ifkAXnk are bounded, then u 2 D(A) and AXn N Au.

Proof. (cf. Lemma 2.5(a) [12]). h

Suppose that there exist v 2 C([t0 � s, t0 + s],H) such that

vðtÞ ¼ UðtÞ on ½t0 � s; t0�:

A function u 2 C([t0 � s, t0 + s],H) such that

uðtÞ ¼vðtÞ; if t 2 ½t0 � s; t0�;SðtÞvðt0Þ þ

R tt0

Sðt � sÞFðs;uðsÞÞds; if t 2 ½t0; t0 þ s�;

(

is called the mild solution of (4).By a strong solution u of (4) on [t0 � s, t0 + s], we mean a function, u 2 C([t0 � s, t0 + s]) such that u(t) 2 D(A) for a.e.

t 2 [t0, t0 + s], u is differentiable a.e. on [t0, t0 + s] and

u0ðtÞ þ AuðtÞ ¼ Fðt;uðtÞÞ; a:e: t 2 ½t0; t0 þ s�:

Theorem 2.6. Under the conditions of Lemmas 2.1 and 2.2 Eq. (4) has a unique strong solution either on [t0 � s, t0 + s], or on themaximal interval of existence [t0 � s, tmax[, 0 < tmax 6 t0 + s. If 0 < tmax < t0 + s, then

limt"tmaxkuðtÞk ¼ 1:

3. Approximation

Let t0 < t1 < t0 + s. For each n, let hn ¼ t1n ; tn

j ¼ jhn.

Define unj

n os:t: un

0 ¼ vð0Þ as the unique solution of the equation

u� unj�1

hnþ Au ¼ F tn

j ;unj�1

� �: ð6Þ

4172 A. Raheem, D. Bahuguna / Applied Mathematics and Computation 218 (2011) 4169–4176

Existence and uniqueness of unj 2 DðAÞ, satisfying Eq. (6) is a consequence of Lemma 2.4.

We define the sequence Unj

n oas:

UnðtÞ ¼vðtÞ; if t 2 ½t0 � s; t0�;un

j�1 þ 1hn

t � tnj�1

� �un

j � unj�1

� �; if t 2 ½t0; t0 þ s�:

(

Now we will prove that sequence {Un(t)} converges to a unique strong u solution of Eq. (6).

Lemma 3.1. For each n 2 N, j = 1,2,3, . . . ,n,

unj � vð0Þ

��� ��� 6 C;

where C, is a generic constant independent of n, j, and hn.

Proof. We will prove this result by induction. For j = 1,

un1 � un

0

hnþ Aun

1 ¼ F tn1;u

n0

� �:

Subtracting Au0 from both sides, we get

un1 � un

0

hnþ Aun

1 � Au0 ¼ F tn1; u

n0

� �� Au0:

Applying J un1 � un

0

� �on both the sides,we get

un1 � un

0

hn; J un

1 � un0

� � þ A un

1 � un0

� �; J un

1 � un0

� �� �¼ F tn

1;un0

� �; J un

1 � un0

� �� �� Aun

0; J un1 � un

0

� �� �:

By the definition of duality mapping and Lemma 2.4, we get

1hn

un1 � un

0

�� ��26 F tn

1;un0

� ��� �� un1 � un

0

�� ��þ Aun0

�� �� un1 � un

0

�� ��1hnkun

1 � un0k 6 Fðtn

1;un0Þ � Fð0;vð0ÞÞ

�� ��þ kFð0;vð0ÞÞk þ Aun0

�� ��1hn

un1 � un

0

�� �� 6 LðRÞ tn1

þ kFð0;vð0ÞÞk þ kAvð0Þkj

un1 � un

0

�� �� 6 hn½LðRÞðt0 þ sÞ þ kFð0;vð0ÞÞk þ kAvð0Þk� 6 C:

Suppose that

uni � vð0Þ

�� �� 6 C; for i ¼ 1;2; . . . ; j� 1:

Now we see for i = j,

unj � un

j�1

hnþ Aun

j ¼ F tnj ;u

nj�1

� �:

Subtracting Aun0 from both sides, we get

unj � un

j�1

hnþ Aun

j � Aun0 ¼ F tn

j ; unj�1

� �� Aun

0:

Applying J unj � un

0

� �on both sides, we get

unj � un

j�1

hn; J un

j � un0

� � þ A un

j � un0

� �; J un

j � un0

� �D E¼ F tn

j ;unj�1

� �� Aun

0; J unj � un

0

� �D E:

By using the definition of duality mapping and Lemma 2.4, we get

unj � vð0Þ

��� ��� 6 unj�1 � vð0Þ

��� ���þ hn kF tnj ;u

nj�1

� �� Fð0;vð0ÞÞk þ kFð0;vð0ÞÞk þ kAvð0Þk

h i6 kun

j�1 � vð0Þk þ hn LðRÞ tnj

þ unj�1 � vð0Þ

��� ���� �þ kFð0;vð0ÞÞk þ kAvð0Þk

h i6 un

j�1 � vð0Þ��� ���þ hn LðRÞðt1 þ RÞ þ kFð0;vð0ÞÞk þ kAvð0Þk½ �:

By the induction hypothesis, we get

A. Raheem, D. Bahuguna / Applied Mathematics and Computation 218 (2011) 4169–4176 4173

unj � vð0Þ

��� ��� 6 C:

This complete the proof of the lemma. h

Lemma 3.2. For each n 2 N, and j = 1,2,3, . . . ,n,

unj � un

j�1

hn

�������� 6 C;

where C, is generic constant independent of n, j, and hn.

Proof. We will prove this result by induction.For j = 1, same as Lemma 3.1, we can show that

un1 � un

0

hn

�������� 6 C:

Now, suppose that

uni � un

i�1

hn

�������� 6 C; for i ¼ 1;2; . . . ; j� 1:

Now we see for i = j,

unj � un

j�1

hnþ Aun

j ¼ F tnj ;u

nj�1

� �:

This implies that

unj � un

j�1

hn�

unj�1 � un

j�2

hnþ Aun

j � Aunj�1 ¼ F tn

j ;unj�1

� �� F tn

j�1;unj�2

� �:

Applying J unj � un

j�1

� �on both sides, we get

unj � un

j�1

hn�

unj�1 � un

j�2

hn; J un

j � unj�1

� � þ Aun

j � Aunj�1; J un

j � unj�1

� �D E¼ F tn

j ;unj�1

� �� F tn

j�1;unj�2

� �; J un

j � unj�1

� �D E:

By using the definition of duality mapping and Lemma 2.4, we get

unj � un

j�1

hn

�������� 6 un

j�1 � unj�2

hn

��������þ F tn

j ;unj�1

� �� F tn

j�1; unj�2

� ���� ��� 6 unj�1 � un

j�2

hn

��������þ LðRÞ tn

j � tnj�1

þ unj�1 � un

j�2

��� ���2

h i

6

unj�1 � un

j�2

hn

��������þ LðRÞ t1 þ un

j�1 � unj�2

��� ���2

h i:

Using the induction hypothesis and Lemma 3.1, we get

unj � un

j�1

hn

�������� 6 C:

This complete the proof of the lemma. h

Now we define a sequence of step functions

XnðtÞ ¼vðt0Þ; if t ¼ t0;

unj ; if t 2 ðtn

j�1; tnj �:

(

Remark 3.3. From Lemma 3.2, it is clear that Un(t) is uniformly Lipchitz continuous and Un(t) � Xn(t) ? 0, n ?1.

If we suppose that

f nðtÞ ¼ F tnj ;u

nj�1

� �;

then Eq. (6) can be written as:

d�

dtUnðtÞ þ AXnðtÞ ¼ f nðtÞ; t 2 ðt0; t1�; ð7Þ

where d�

dt denote the left derivative in (t0, t1].

4174 A. Raheem, D. Bahuguna / Applied Mathematics and Computation 218 (2011) 4169–4176

Also, for t 2 (t0, t1], we have

Z t

t0

AXnðsÞds ¼ vðt0Þ � UnðtÞ þZ t

t0

f nðsÞds: ð8Þ

Next we prove the convergence of Un to u in C([t0 � s, t1],H).

Lemma 3.4. There exist u 2 C([t0 � s, t1],H), such that Un ? u in C([t0 � s, t1],H) as n ?1. Moreover, u is Lipchitz continuous on[t0, t1].

Proof. Now from Eq. (7), we see that

d�

dtUnðtÞ � d�

dtUkðtÞ; JðXnðtÞ � XkðtÞÞ

þ AXnðtÞ � AXkðtÞ; JðXnðtÞ � XkðtÞÞD E

¼ f nðtÞ � f kðtÞ; J XnðtÞ � XkðtÞ� �D E

:

Using Lemma 2.4, we get

d�

dtUnðtÞ � d�

dtUkðtÞ; JðXnðtÞ � XkðtÞÞ

6 f nðtÞ � f kðtÞ; JðXnðtÞ � XkðtÞÞD E

:

Now using above equation, we can write

d�

dtkUnðtÞ � UkðtÞk2 ¼ d�

dtUnðtÞ � d�

dtUkðtÞ; JðUnðtÞ � UkðtÞÞ

6d�

dtUnðtÞ � d�

dtUkðtÞ þ f nðtÞ � f kðtÞ � f nðtÞ þ f kðtÞ; JðUnðtÞ � UkðtÞÞ � JðXnðtÞ � XkðtÞÞ

þ hf nðtÞ � f kðtÞ; JðXnðtÞ � XkðtÞÞi

¼ d�

dtUnðtÞ � d�

dtUkðtÞ � f nðtÞ þ f kðtÞ; JðUnðtÞ � UkðtÞÞ � JðXnðtÞ � XkðtÞÞ

þ hf nðtÞ � f kðtÞ; JðUnðtÞ � UkðtÞÞi:

Now using (5), we have

kf nðtÞ � f kðtÞk2 ¼ F tnj ;u

nj�1

� �� F tk

l ; ukl�1

� ���� ���26 LðRÞ tn

j � tkl

þ unj�1 � uk

l�1

��� ���2

h i6 �nkðtÞ þ LðRÞkUnðtÞ � UkðtÞk2;

where

�nkðtÞ ¼ hnK þ knK þ tnj � tk

l

þ XnðtÞ � UnðtÞ�� ��

2 þ kXkðtÞ � UkðtÞk2

h i:

It is clear that �nk(t) ? 0 as n, k ?1.This implies that

d�

dtkUnðtÞ � UkðtÞk2

2 6d�

dtðUnðtÞ � UkðtÞÞ � f nðtÞ þ f kðtÞ

��������

2� kUnðtÞ � UkðtÞ � XnðtÞ þ XkðtÞk2

þ kf nðtÞ � f kðtÞk2kUnðtÞ � UkðtÞk2

d�

dtkUnðtÞ � UkðtÞk2

2 6 LðRÞ �nkðtÞ þ kUnðtÞ � UkðtÞk22

h i:

Hence, we have the estimates

kUnðtÞ � UkðtÞk22 6 LðRÞ �nkt1 þ

Z t

t0

kUnðtÞ � UkðtÞk22ds

� �:

Applying Grownwall’s inequality, we get that Un ? u in C([t0 � s, t1],H). As each Un is uniformly Lipchitz continuous, u is Lip-chitz continuous. h

Now we show that this u is the strong solution of Eq. (4).

Remark 3.5. Clearly Xn(t) 2 D(A), for each n. As Un(t) � Xn(t) ? 0 as n ?1, Xn(t) ? u(t) 2 H. Also kAXnk are boundedtherefore by Lemma 2.5, it is clear that AXn N Au.

So for every x⁄ 2 X⁄ and t 2 (t0, t1], we have

Z t

t0

ðAXnðsÞ; x�Þds ¼ ðvðt0Þ; x�Þ � ðUnðtÞ; x�Þ þZ t

t0

ðf nðsÞ; x�Þds:

A. Raheem, D. Bahuguna / Applied Mathematics and Computation 218 (2011) 4169–4176 4175

Using Lemma 3.4, Remark 3.5 and the bounded convergence theorem, we obtain as n ?1,

Z t

t0

ðAuðsÞ; x�Þds ¼ ðvðt0Þ; x�Þ � ðuðtÞ; x�Þ þZ t

t0

ðFðs;uðsÞÞ; x�Þds: ð9Þ

As Au(t) is Bochner integrable on [t0, t1], from Eq. (9) we have

ddt

uðtÞ þ AuðtÞ ¼ Fðt;uðtÞÞ; a:e: t 2 ðt0; t1�: ð10Þ

Clearly u 2 C([t0 � s, t1];H) and differentiable a.e. on (t0, t1] with u(t) 2 D(A) a.e. on (t0, t1] and u = U on [t0 � s, t0] satisfying(10). Hence it will be a strong solution of Eq. (4) on [t0 � s, t1].

Now we will prove the uniqueness. For this suppose that u1, u2 are two strong solutions of Eq. (4).

Let R ¼maxfku1k2; ku2k2g:

Then for u = u1 � u2, we have

duðtÞdt

; JðuðtÞÞ

þ Aðu1ðtÞ � u2ðtÞÞ; Jðu1ðtÞ � u2ðtÞÞh i ¼ Fðt;u1ðtÞÞ � Fðt;u2ðtÞÞ; JðuðtÞÞh i:

By Lemma 2.4 and by the definition of duality mapping, we get

ddtkuðtÞk2

2 6 kFðt;u1ðtÞÞ � Fðt;u2ðtÞÞk2kuðtÞk2:

Using Lemma 2.1, we get

ddtkuðtÞk2

2 6 KkuðtÞk22:

This implies that

kuðtÞk22 6 K

Z t

t0

kuðtÞk22ds:

Applying Grownwall’s inequality, we get u � 0 on [t0 � s, t1].Hence we get a unique strong solution on the interval [t0 � s, t1].In the next step, we prove the existence and uniqueness of a strong solution on the interval [t0 � s, t1 + s]. By the first step,

we know the solution of (1) on the interval [t0 � s, t1] say u. In the second step we consider this solution as history solutionand obtain the solution on the interval [t0 � s, t2], where t1 < t2 < t1 + s.

Let wðt; xÞ ¼ vðt; xÞ on ½t0 � s; t1 þ s�:

So, on the interval [t0 � s, t1 + s], Eq. 1 will reduce to

@v@tðt; xÞ ¼ @

2v@x2 ðt; xÞ þ kvðt; xÞ½1� uðt � s; xÞ� þ f ðt; xÞ; x 2 ½0;p�; t 2 ½t1; t1 þ s�; ð11Þ

vðt;0Þ ¼ vðt;pÞ ¼ 0; t 2 ½t0 � s; t1 þ s�;vðt; xÞ ¼ /ðt; xÞ; ðt; xÞ 2 ½t0 � s; t1� � ½0;p�:

It is clear that this equation is similar to Eq. (2) therefore as the earlier step, we get the solution of (11) on the interval[t0 � s, t2].

Continuing this process we get a unique strong solution of Eq. (1) on the whole interval or on the maximal interval of existence.

Acknowledgements

The authors thank the referee for his valuable suggestions. The first author acknowledges the sponsorship from CSIR, In-dia, under its research grant 09/092 (0652)/2008-EMR-1. The second author acknowledges the financial help from theDepartment of Science and Technology, New Delhi, under its research project SR/S4/MS:581/09.

References

[1] S. Agarwal, D. Bahuguna, Exact and approximate solution of delay differential equations with nonlocal history conditions, J. Appl. Math. Stoch. Anal. 2(2005) 181–194.

4176 A. Raheem, D. Bahuguna / Applied Mathematics and Computation 218 (2011) 4169–4176

[2] S. Agarwal, D. Bahuguna, Method of semidiscretization in time to nonlinear retarded differential equation with nonlocal history conditions, IJMMS 37(2004) 1943–1956.

[3] D. Bahuguna, R. Shukla, Method of semidiscretization in time to quasilinear integrodifferential equations, IJMMS 9 (2004) 469478.[4] A.G. Kartsatos, W.R. Zigler, Rothe’s method to abstract parabolic equation, Czech. Math. J. 24 (1974) 496–500.[5] A.G. Kartsatos, On the construction of methods of lines for functional evolutions in general Banach spaces, Nonlinear Anal. 25 (12) (1995) 13211331.[6] A.G. Kartsatos, X. Liu, On the construction and the convergence of the method of lines for quasi-nonlinear functional evolutions in general Banach

spaces, Nonlinear Anal. TMA 29 (1997) 385–414.[7] D. Bahuguna, V. Raghavendra, Application of Rothe’s method to nonlinear integro-differential equations in Hilbert spaces, Nonlinear Anal. Theory

Method Appl. 23 (1) (1994) 75–81.[8] N. Merazga, A. Bouziani, On a time-discretization method for a semilinear heat equation with purely integral conditions in a nonclassical function

space, Nonlinear Anal. TMA 66 (2007) 604–623.[9] N. Merazga, A. Bouziani, Rothe time-discretization method for a nonlocal problem arising in thermoelasticity, J. Appl. Math. Stoch. Anal. 1 (2005) 13–

28.[10] K. Rektorys, The Method of Discretization in Time and Partial Differential Equations, Dordrecht-Boston-London, Reidel, 1982.[11] A. Pazy, Semigroup of Linear Operators and Application to Partial Differential Equations, Springer-Verlag, New York, 1983.[12] T. Kato, Nonlinerar semigroup and evolution equations, Math. Soc. Jpn 19 (1967) 508–520.