This paper investigates the existence of mild solution for a fractional integrodifferential equations with a deviating argument and nonlocal initial condition in an arbitrary separable Hilbert space
H
via technique of approximations. We obtain an associated integral equation and then consider a sequence of approximate integral equations obtained by the projection of considered associated nonlocal fractional integral equation onto finite dimensional space. The existence and uniqueness of solutions to each approximate integral equation is obtained by virtue of the analytic semigroup theory via Banach fixed point theorem. Next we demonstrate the convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. We consider the FaedoGalerkin approximation of the solution and demonstrate some convergenceresults. An example is also given to illustrate the abstract theory.
AMS Mathematics Subject Classification : 34K37, 34K30, 35R11, 47N20.
1. Introduction
In recent few decades, researchers have developed great interest in fractional calculus due to its broad applicability in science and engineering. The tool of fractional calculus has been available and applicable to deal with many physical and real world problems such as anomalous diffusion process, traffic flow, nonlinear oscillation of earthquake, real system characterized by power laws, critical phenomena, scale free process, describe viscoelastic materials and many others. The details on the theory and its applications can be found in
[1]

[4]
. The existence of the solution for the differential equations with nonlocal conditions has been investigated widely by many authors as nonlocal conditions are more realistic than the classical initial conditions such as in dealing with many physical problems. Concerning the developments in the study of nonlocal problems, we refer to
[6]

[16]
and references given therein.
To the solvability of evolution problems in the time domain, we have various approaches, namely, the evolution family approach and an approach using finitedimensional approximations known as FaedoGalerkin approximations. The FaedoGalerkin approach may be used for the study of more regular solutions, imposing higher regularity on the data. In
[20]
, author has extended the results of the
[19]
and considered the FaedoGalerkin approximations of the solutions for functional Cauchy problem in a separable Hilbert space with the help of analytic semigroup theory and Banach fixed point theorem. In
[21]
, authors have studied the FaedoGalerkin approximations of the solutions to a class of functional integrodifferential equation extended the results of
[20]
. In
[8]
, the FaedoGalerkin approximations of the mild solution to nonlocal historyvalued retarded differential equations have been obtained by authors. In
[9]
, authors have established the existence of the mild solution and approximations of mild solutions via technique of FaedoGalerkin approximations and analytic semigroup theory. In
[28]
, authors have considered an fractional differential equation and studied the FaedoGalerkin approximations of the solutions for fractional differential equation. In
[26]
, the existence and approximations of the mild solution to fractional differential equation with deviated argument via technique of FaedoGalerkin approximations have been obtained by authors. The FaedoGalerkin approximations of solutions for fractional integrodifferential equation have been considered by authors in
[30]
. For the FaedoGalerkin approximation of solutions, we refer to papers
[8]

[9]
,
[21]

[31]
.
The purpose of this work is to establish the approximation of the solution for following nonlocal integrodifferential equation with a deviating argument in a separable Hilbert space (
H
, ∥ · ∥
_{H}
, (·, ·)
_{H}
)
where
is the generalized fractional derivative of order
q
in Caputo sense with lower limit 0. For
x
∈
C
([0,
T
_{0}
];
H
),
x_{t}
: [−
r
, 0] →
H
is defined as
x_{t}
(
θ
) =
x
(
t
+
θ
) for
θ
∈ [−
r
, 0]. In (1),
A
:
D
(
A
) ⊂
H
→
H
is a closed, positive definite and self adjoint linear operator with dense domain
D
(
A
). We assume that −
A
generates an analytic semigroup of bounded linear operators on
H
. The functions
f
: [0,
T
_{0}
]×
C
([−
r
, 0],
H
)×
H
→
H, a
:
H
×[0,
T
_{0}
] → [0,
T
_{0}
],
b
: [0,
T
_{0}
]×[0,
T
_{0}
] → ℝ,
g
: [0,
T
_{0}
]×
H
×
C
([−
r
, 0],
H
) →
H
and
h
:
C
([−
r
, 0],
H
) →
C
([−
r
, 0],
H
) are appropriate functions to be mentioned later. For more details on differential equation with deviated argument, we refer to papers
[17]

[18]
,
[26]
and references cited therein.
The organization of the article is as follows: Section 2 provides some basic definitions, lemmas and theorems as preliminaries as these are useful for proving our results. We firstly obtain an integral equation associated with (1). A mild solution of equation (1) is defined as a solution of associated integral equation. We consider a sequence of approximate integral equations. Section 3 proves the existence and uniqueness of the approximate solutions by using analytic semigroup and fixed point theorem. In section 4, we show the convergence of the solution to each of the approximate integral equations with the limiting function which satisfies the associated integral equation and the convergence of the approximate FaedoGalerkin solutions will be shown in section 5. Section 5 gives an example.
2. Preliminaries and Assumptions
Some basic definitions, theorems, lemmas and assumptions which will be used to prove existence result, are stated in this section.
Throughout the work, we assume that (
H
, ∥ · ∥
_{H}
, (·, ·)
_{H}
) is a separable Hilbert space. The symbol
C
([0,
T
_{0}
],
H
) stands for the Banach space of all the continuous functions from [0,
T
_{0}
] into
H
equipped with the norm ∥
z
(
t
)∥
C
:= sup
_{t}
∈[0,
T
_{0}
] ∥
z
(
t
)∥
_{H}
and
L^{p}
((0,
T
_{0}
),
H
) stands for Banach space of all Bochnermeasurable functions from (0,
T
_{0}
) to
H
with the norm
Since −
A
is the infinitesimal generator of an analytic semigroup of bounded linear operators {
T
(
t
);
t
≥ 0}. Therefore, there exist constants
C
> 0 and
ω
≥ 0 such that ∥
T
(
t
)∥ ≤
Ce^{ωt}
, for
t
≥ 0. In addition, we note that
where
M_{j}
are some positive constants. Henceforth, without loss of generality, we may assume that
T
(
t
) is uniformly bounded by
M
i.e., ∥
T
(
t
)∥ ≤
M
and 0 ∈
ρ
(−
A
) which means that −
A
is invertible. This permits us to define the positive fractional power
A^{α}
as closed linear operator with domain
D
(
A^{α}
) ⊆
H
for
α
∈ (0, 1]. Moreover,
D
(
A^{α}
) is dense in
H
with the norm
Hence, we signify the space
D
(
A^{α}
) by
H_{α}
endowed with the
α
norm (∥ · ∥
_{α}
). It is easy to show that
H_{α}
is a Banach space with norm ∥ · ∥
_{α}
[35]
. Also, we have that
H_{κ}
→
H_{α}
for 0 <
α
<
κ
and therefore, the embedding is continuous. Then, we define
H_{α}
= (
H_{α}
)∗, for each
α
> 0. The space
H_{α}
stands for the dual space of
H_{α}
, is a Banach space with the norm ∥
z
∥
_{−α}
= ∥
A^{−α}z
∥. For additional parts on the fractional powers of closed linear operators, we refer to book by Pazy
[35]
.
Lemma 2.1
(
[35]
).
Let
−
A be the infinitesimal generator of an analytic semigroup
{
T
(
t
)}
_{t}
_{≥0}
such that
∥
T
(
t
)∥ ≤
M, for t
≥ 0
and
0 ∈
ρ
(−
A
).
Then
,

(i)For0 <α≤ 1,Hαis a Hilbert space.

(ii)The operator AαT(t)is bounded for every t> 0and
Now, we state some basic definitions and properties of fractional calculus.
Definition 2.2
(
[3]
). The RiemannLiouville fractional integral operator
J
of order
q
> 0 with lower point 0, is defined by
where
F
∈
L
^{1}
((0,
T
_{0}
),
H
).
Definition 2.3
(
[3]
). The RiemannLiouville fractional derivative is given by
where
Here the notation
W^{δ}
^{,1}
((0,
T
_{0}
),
H
) stands for the Sobolev space defined as
Note that
z
(
t
) =
y^{δ}
(
t
),
d_{j}
=
y^{j}
(0).
Definition 2.4
(
[3]
). The Caputo fractional derivative is given by
for
δ
− 1 <
q
<
δ, δ
∈ ℕ, where
F
∈
L
^{1}
((0,
T
_{0}
),
H
) ∩
C^{δ}
^{−1}
((0,
T
_{0}
),
H
).
Let
C
_{0}
:=
C
([−
r
, 0],
H
) be the collection of continuous mappings from [−
r
, 0] into
H
equipped with the supremum norm ∥
y
∥
_{0}
:= sup
_{t}
_{∈[−}
_{r}
,
_{0]}
∥
y
(
t
)∥ for
y
∈
C
_{0}
. In addition,
C_{t}
:=
C
([−
r
,
t
],
H
) be the Banach space of all Hvalued continuous functions on [−
r
,
t
] endowed with the supremum norm ∥
y
∥
_{t}
:= sup
_{s}
_{∈[−}
_{r,t}
_{]}
∥
y
(
s
)∥ for each
y
∈
C_{t}
and
t
∈ (0,
T
_{0}
] and the space of all continuous functions from [−
r, t
] into
H_{α}
denoted by
is a Banach space with the supremum norm ∥
y
∥
_{t,α}
:= sup
_{s}
_{∈[−}
_{r,t}
_{]}
∥
A^{α}y
(
s
)∥, for each
For 0 ≤
α
< 1, we define
where
L
> 0 is a appropriate constant to be defined later.
Now, we turn to the following fractional differential equations with nonlocal conditions as
We give few examples of function
h
as

(1) Letw∈L1(0,r) be such thatLet
If we take
defined as
ψ
(
θ
) ≡
ϕ
for all
θ
∈ [−
r
, 0] and
given by
G
(
y
)(
θ
) ≡
h
(
y
) for all
θ
∈ [−
r
, 0] and
Then the condition
h
(
y
) =
ϕ
is equivalent to the condition
G
(
y
) =
ψ
. Thus, the functional differential equation with a more general nonlocal history condition may be considered which is illustrated as follows,
which includes (12). For example,
is a particular case of (15). Thus, the problem (12) and (15) are equivalent. Next, we make the following assumptions:

(A1) A is a closed, densely defined, positive definite and selfadjoint linear operator fromD(A) ⊂HintoH. We assume that operatorAhas the pure point spectrum

withλm→ ∞ asm→ ∞ and corresponding complete orthonormal system of eigenfunctions {χj}, i.e.,

(A2) (i) There exists a functionsuch thath(k) =ϕ, for allt∈ [−r, 0]. We assume thathis Lipschitz continuous function onC([−r, 0],D(Aα)).

(ii) The functionk(t) ∈D(Aα) for eacht∈ [−r, 0] andkis locally Hölder continuous with exponent 1 on [−r, 0] i.e. there exists a constantLk> 0 such that

(A3) The nonlinear functionis Lipschitz continuous and there exist constantsLf> 0 andμ1∈ (0, 1] such that

for allHere,andZis a Banach space andR> 0 is a constant to be defined later. The functionis defined by

(A4) The functiona:Hα× [0,T0] → [0,T0] is continuous function and there exist constantsLα> 0 andμ2∈ (0, 1] such that

for allanda(·, 0) = 0.

(A5)is continuous function and there exists a positive constantLgsuch that
Now, we provide the definition of mild solution for the nonlocal system (1)(2).
Definition 2.5.
A continuous function
x
: [0,
T
_{0}
] →
H
is said to be a mild solution for the system (1)(2) if
and the following integral equation
is verified.
The operator
S_{q}
(
t
) and
T_{q}
(
t
) are defined as follows:
where
is a a probability density function defined on (0,∞) i.e.,
and
Lemma 2.6
(
[11]
).
If −A is the infinitesimal generator of analytic semigroup of uniformly continuous bounded operators. Then,
(1)
The operator
S_{q}
(
t
),
t
≥ 0
and
T_{q}
(
t
),
t
≥ 0
are bounded linear operators.
(2) ║
S_{q}
(
t
)
y
║ ≤
M
║
y
║,
,
for any y
∈
H
.
(3)
The families
{
S_{q}
(
t
) :
t
≥ 0}
and
{
T_{q}
(
t
) :
t
≥ 0}
are strongly continuous
.
(4)
If
T
(
t
)
is compact, then
S_{q}
(
t
)
and
T_{q}
(
t
)
are compact operators for any t
> 0.
3. Approximate Solutions and Convergence
In this section, we study the existence of approximate solutions for the system (1)(2).
Let
H_{n}
be the finite dimensional subspace of
H
spanned by {
χ
_{0}
,
χ
_{1}
, · · · ,
χ_{n}
} and
P^{n}
:
H
→
H_{n}
be the corresponding projection operator for
n
= 0, 1, 2, · · · , . We define
by
and
by
We choose
T
, 0 <
T
≤
T
_{0}
sufficiently small such that
Now, we consider
By the assumptions (
A
3)−(
A
4), we have that
f
is continuous on [0,
T
]. Therefore, there exist a positive constant
N_{f}
such that
with
Similarly with the help of the assumption (
A
5), we can show that
Therefore, we can indicate effectively that
G_{g}
=
b_{T}N_{g}
, where
Let us consider the operator
defined by (
A^{α}y
)(
t
) =
A^{α}
(
y
(
t
)) and (
P^{n}x_{t}
)(
s
) =
P^{n}
(
x
(
t
+
s
)), for all
s
∈ [−
r
, 0] and
t
∈ [0,
T
]. We consider the operator
Q_{n}
:
B_{R}
→
B_{R}
defined by
for each
x
∈
B_{R}
, where
Theorem 3.1.
Suppose
(
A
1)(
A
5)
holds and k
(
t
) ∈
D
(
A
)
for all t
∈ [−
r
, 0].
Then, there exists a unique x_{n}
∈
B_{R} such that Q_{n}x_{n}
=
x_{n} for each n
= 0, 1, 2, · · · ,
and x_{n} satisfies the following approximate integral equation
Proof
. To demonstrate the theorem, we first need to show that
It is easy to show that
by using the fact that
f
and
g
are continuous function. Now, it remains to show that
For
t, τ
∈ [−
r
, 0] with
t
>
τ
, we have
by using fact that
k
is Hölder continuous with exponent 1 i. e., Lipschitz continuous on [−
r
, 0].
For
0 <
τ
<
t
<
T
, then we have
From the first term of above inequality, we have
Also, we have that for each
x
∈
H
Therefore, we estimate the first term as
where
The second integrals is estimated as
where
K
_{2}
= ∥
A^{α}
^{−2}
∥
M
_{2}
N_{f}T
. The third integrals is estimated as
where
K
_{3}
=
M
_{1}
∥
A^{α}
^{−2}
∥
N_{f}
. Similarly, we estimate forth integral as
where
K
_{4}
= ∥
A^{α}
^{−2}
∥
M
_{2}
TG_{g}
and
where
K
_{5}
=
M
_{1}
∥
A^{α}
^{−2}
∥
G_{g}
and
Thus, from the inequality (38) to (42), we obtain that
for a positive suitable constant
Therefore, we conclude that
Hence, we deduce that the operator
is well defined map.
Next, we prove that
Q_{n}
:
B_{R}
→
B_{R}
. For 0 ≤
t
≤
T
and
x
∈
B_{R}
, we get that
From the inequalities (28) and (44), we conclude that
Q_{n}
(
B_{R}
) ⊂
B_{R}
. Finally, we will show that
Q_{n}
is a contraction map. For
x, y
∈
B_{R}
and 0 ≤
t
≤
T
, we have
We have the following inequalities:
and
Using (46)(47) in (45), we get,
From the inequality (30), we get
with Θ < 1. Therefore, it implies that the map
Q_{n}
is a contraction map i.e.
Q_{n}
has a unique fixed point
x_{n}
∈
B_{R}
i.e.,
Q_{n}x_{n}
=
x_{n}
and
x_{n}
satisfies the approximate integral equation
Hence, the proof of the theorem is completed. □
Lemma 3.2.
Assume that hypotheses
(
A
1)(
A
5)
are satisfied. If k
(
t
) ∈
D
(
A
)
for each t
∈ [−
r
, 0],
then x_{n}
(
t
) ∈
D
(
A^{υ}
)
for all t
∈ [−
r
,
T
]
with
0 ≤
υ
< 1.
Proof
. If
t
∈ [−
r
, 0], then results are obvious. Thus, it remains to show results for
t
∈ [0,
T
]. From Theorem (3.1), we have that there exists a unique
such that
x_{n}
satisfy the integral equation (35). Theorem 2.6.13 in Pazy
[35]
implies that
T
(
t
) :
H
→
D
(
A^{υ}
) for
t
> 0 and 0 ≤
υ
< 1 and for 0 ≤
υ
≤
η
< 1,
D
(
A^{η}
) ⊆
D
(
A^{υ}
). It is easy to see that Hölder continuity of
x_{n}
can be established using the similar arguments from equation (38)(42). Also from Theorem 1.2.4 in Pazy
[35]
, we have that
T
(
t
)
x
∈
D
(
A
) if
x
∈
D
(
A
). The result follows from these facts and
D
(
A
) ⊆
D
(
A^{υ}
) for 0 ≤
υ
≤ 1. This completes the proof of Lemma. □
Corollary 3.3.
Suppose that
(
A
1)(
A
5)
are satisfied. If k
(
t
) ∈
D
(
A
), ∀
t
∈ [−
r
, 0],
then for any t
∈ [−
r, T
],
there exists a constant U
_{0}
independent of n such that
with
0 <
α
<
υ
< 1.
Proof
. Let
k
(
t
) ∈
D
(
A
) for every
t
∈ [−r, 0]. For
t
∈ [−
r
, 0], applying
A^{υ}
on the both the sides of (35) and obtaining,
For
t
∈ (0,
T
], we apply
A^{υ}
on the both the sides of (35) and get
This finishes the proof of lemma. □
4. Convergence of Solutions
The convergence of the solution
x_{n}
∈
H_{α}
of the approximate integral equations (35) to a unique solution
x
(·) of the equation (23) on [0,
T
] is discussed in this section.
Theorem 4.1.
Let us assume that the conditions
(
A
1)(
A
5)
are satisfied. If k
(0) ∈
D
(
A
),
for each t
∈ [−
r
, 0],
then
Proof
. For 0 <
α
<
υ
< 1,
n
≥
p
. Let
t
∈ [−
r
, 0], we conclude
For
t
∈ (0,
T
], we obtain,
We also have the following estimation:
Thus, we obtain
And
Therefore, we estimate
We choose
t′
_{0}
such that 0 <
t′
_{0}
<
t
<
T
, we have
We estimate the first integral as
By using Corollary 3.3, the second integral is estimated as
Third and forth term are estimated as
and
Thus, we have
where
We now put
t
=
t
+
θ
in the above inequality, where
θ
∈ [
t′
_{0}
−
t
, 0] and get
Taking
s
−
θ
=
ν
in above inequality and obtaining,
Thus, we have
Since, for
t
+
θ
≤ 0, we have
x_{n}
(
t
+
θ
) =
k
(
t
+
θ
) for all
n
≥
n
_{0}
. Thus, we obtain sup
_{−}
_{r}
_{−}
_{t}
_{≤}
_{θ}
_{≤0}
∥
x_{n}
(
t
+
θ
) −
x_{p}
(
t
+
θ
)∥
_{α}
Thus, for each
t
∈ (0,
t′
_{0}
], we have
where
D
_{5}
and
D
_{6}
are arbitrary positive constants. Using (64) and (66) in (65) and thus getting
Thus, we get
By Lemma 5.6.7 in
[35]
, we have that there exists a constant
K
such that
Since
t′
_{0}
is arbitrary and letting
p
→ ∞, therefore the right hand side may be made as small as desired by taking
t′
_{0}
sufficiently small. This complete the proof of the Theorem. □
By the Theorem 4.1, we conclude that {
x_{n}
} is a Cauchy sequence in
B_{R}
. Now, we show the convergence of the solution for the approximate integral equation
x_{n}
(·) to the solution of associated integral equation
x
(·).
Theorem 4.2.
Suppose that conditions
(
A
1)(
A
5)
are satisfied and k
(
t
) ∈
D
(
A
)
for each t
∈ [−
r
, 0].
Then, there exists a unique x_{n}
∈
B_{R}, satisfying
and x
∈
B_{R}, satisfying
such that x_{n} converges to x in BR i.e., x_{n}
→
x as n
→ ∞.
Proof
. Let
k
(
t
) ∈
D
(
A
) for all
t
∈ [−
r
, 0]. For 0 <
t
≤
T
, it follows that there exists
x_{n}
∈
B_{R}
such that
A^{α}x_{n}
(
t
) →
A^{α}x
(
t
) ∈
B_{R}
as
n
→ ∞ and
x
(
t
) =
x_{n}
(
t
) =
k
(
t
), for each
t
∈ [−
r
, 0] and for all
n
. Also, for
t
∈ [−
r
,
T
], we have
A^{α}x_{n}
(
t
) →
A^{α}x
(
t
) as
n
→ ∞ in
H
. Since
B_{R}
is a closed subspace of
and
x_{n}
∈
B_{R}
, therefore it follows that
x
∈
B_{R}
and
Also, we have
as
n
→ ∞ and
as
n
→ ∞. For 0 <
t
_{0}
<
t
, we rewrite (35) as
We may estimate the first and third integral as
Thus, we deduce that
Letting
n
→ ∞ in the above inequality, we obtain
Since
t
_{0}
is arbitrary and hence, we conclude that
x
(·) satisfies the integral equation (23). □
5. FaedoGalerkin Approximations
In this section, we consider the FaedoGalerkin Approximation of a solution and show the convergence results for such an approximation.
We know that for any 0 <
T
<
T
_{0}
, there exists a unique
satisfying the following integral equation
with 0 <
T
<
T
_{0}
.
Also, we have a unique solution
of the approximate integral equation
Applying the projection on above equation, then FaedoGalerkin approximation is given by
v_{n}
(
t
) =
P^{n}x_{n}
(
t
) satisfying
or
Let solution
x
(·) of (77) and
v_{n}
(·) of (79), have the following representation
Using (82) in (79), we obtain a system of fractional order integrodifferential equation of the form
where
For the convergence of
to
α_{i}
, we have the following convergence theorem.
Corollary 5.1.
Assume that
(
A
1)(
A
5)
are satisfied. If k
(
t
) ∈
D
(
A
)
for each t
∈ [−
r
, 0],
then
Proof
. For
n
≥
p
and 0 ≤
α
<
υ
, we get
Since
x_{n}
→
x_{p}
and
λ_{p}
→ ∞ as
p
→ ∞, thus, for
t
∈ [−
r
, 0] and
k
(
t
) ∈
D
(
A
), the result follows from Theorem 4.1. □
Theorem 5.2.
Let us assume that
(
A
1)(
A
5)
are satisfied and k
(
t
) ∈
D
(
A
)
for all t
∈ [−
r
, 0].
Then there exist a unique function v_{n}
∈
B_{R} given as
for all t
∈ [0,
T
]
and x
∈
B_{R} satisfying
for
t
∈ [0,
T
],
such that v_{n}
→
x as n
→ ∞
in B_{R} and x satisfies the equation (23) on
[0,
T
].
Proof
. By the Theorem 4.2, we have that
Thus, we conclude that
Since
x_{n}
→
x
as
n
→ ∞, then, for
t
∈ [−
r
, 0] and
k
(
t
) ∈
D
(
A
), the result follows from Theorem 4.2. □
The system (83)(84) determines the
Thus, we have following theorem.
Theorem 5.3.
Let us assume that
(
A
1)(
A
5)
are satisfied. If k
(
t
) ∈
D
(
A
)
for each t
∈ [−
r
, 0],
then
Proof
. It can easily be determined that
Thus, we conclude that
From the Theorem 4.2, we have
v_{n}
→
x
as
n
→ ∞. Thus, we conclude that
as
n
→ ∞. This gives the proof of the theorem. □
6. Example
Let us consider the following integrodifferential equation with deviated argument of the form
where
t
∈ [0, 1],
x
∈ [0, 1],
q
∈ (0, 1),
p
∈ ℕ,
r
> 0,
b
is real valued,
γ
_{1}
: [−
r
, 0] → ℝ are continuous functions with
H is given by
and the function G : ℝ
^{+}
× [0, 1] ×
C
([−
r
, 0],ℝ) → ℝ is measurable in
x
, locally Lipschitz continuous in
w
, uniformly in
x
and locally Hölder continuous in
t
. Here, we assume that
g
: ℝ
^{+}
→ ℝ
^{+}
is locally Hölder continuous in
t
such that
g
(0) = 0 and
K
∈
C
^{1}
([0, 1] × [0, 1],ℝ).
Let
H
=
L
^{2}
((0, 1),ℝ). Now, we define operator by
Aw
= −
d
^{2}
w
/
dx
^{2}
with domain
We also have
and
For each
w
∈
D
(
A
) and
λ
∈ ℝ with −
Aw
=
λw
, we get
The
is the solution of the problem
Aw
= −
λw
. By utilizing the boundary conditions, we get
D
= 0 and
λ_{n}
=
n
^{2}
π
^{2}
for
n
∈ ℕ. Thus,
is the eigenvector corresponding to eigenvalue
λ_{n}
. We also have <
w_{n},w_{m}
>= 0 for
n
≠
m
and <
w_{n},w_{m}
>= 1. Thus, we have that for
w
∈
D
(
A
), there exists a sequence
β_{n}
of real numbers such that
The, we have following representation of the semigroup
Now, for
x
∈ (0, 1), we define
by
where
Thus, it can be verified that
f
satisfies the hypotheses (
A
3).
Similarly, for x ∈ (0, 1), we define
by
Then, it can be seen that
g
fulfills hypotheses (
A
5). Thus, we can apply the results of previous sections to study the existence and convergence of the mild solution to system (98)(100).
BIO
Alka Chadha
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee247667, India.
email: alkachadda23@gmail.com
Dwijendra N. Pandey
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee247667, India.
email: dwij.iitk@gmail.com
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