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APPROXIMATIONS OF SOLUTIONS FOR A NONLOCAL FRACTIONAL INTEGRO-DIFFERENTIAL EQUATION WITH DEVIATED ARGUMENT
APPROXIMATIONS OF SOLUTIONS FOR A NONLOCAL FRACTIONAL INTEGRO-DIFFERENTIAL EQUATION WITH DEVIATED ARGUMENT
Journal of Applied Mathematics & Informatics. 2015. Sep, 33(5_6): 699-721
Copyright © 2015, Korean Society of Computational and Applied Mathematics
  • Received : January 28, 2015
  • Accepted : February 23, 2015
  • Published : September 30, 2015
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ALKA CHADHA
DWIJENDRA N. PANDEY

Abstract
This paper investigates the existence of mild solution for a fractional integro-differential 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 Faedo-Galerkin 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.
Keywords
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 finite-dimensional approximations known as Faedo-Galerkin approximations. The Faedo-Galerkin 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 Faedo-Galerkin 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 Faedo-Galerkin approximations of the solutions to a class of functional integro-differential equation extended the results of [20] . In [8] , the Faedo-Galerkin approximations of the mild solution to non-local history-valued 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 Faedo-Galerkin approximations and analytic semigroup theory. In [28] , authors have considered an fractional differential equation and studied the Faedo-Galerkin 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 Faedo-Galerkin approximations have been obtained by authors. The Faedo-Galerkin approximations of solutions for fractional integro-differential equation have been considered by authors in [30] . For the Faedo-Galerkin 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 integro-differential equation with a deviating argument in a separable Hilbert space ( H , ∥ · ∥ H , (·, ·) H )
PPT Slide
Lager Image
PPT Slide
Lager Image
where
PPT Slide
Lager Image
is the generalized fractional derivative of order q in Caputo sense with lower limit 0. For x C ([0, T 0 ]; H ), xt : [− r , 0] → H is defined as xt ( 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 Faedo-Galerkin 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 Lp ((0, T 0 ), H ) stands for Banach space of all Bochnermeasurable functions from (0, T 0 ) to H with the norm
PPT Slide
Lager Image
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 )∥ ≤ Ceωt , for t ≥ 0. In addition, we note that
PPT Slide
Lager Image
where Mj 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 Aα as closed linear operator with domain D ( Aα ) ⊆ H for α ∈ (0, 1]. Moreover, D ( Aα ) is dense in H with the norm
PPT Slide
Lager Image
Hence, we signify the space D ( Aα ) by Hα endowed with the α -norm (∥ · ∥ α ). It is easy to show that Hα is a Banach space with norm ∥ · ∥ α [35] . Also, we have that Hκ Hα for 0 < α < κ and therefore, the embedding is continuous. Then, we define H = ( Hα )∗, for each α > 0. The space H stands for the dual space of Hα , is a Banach space with the norm ∥ z −α = ∥ A−αz ∥. 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 semi-group { 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
PPT Slide
Lager Image
PPT Slide
Lager Image
Now, we state some basic definitions and properties of fractional calculus.
Definition 2.2 ( [3] ). The Riemann-Liouville fractional integral operator J of order q > 0 with lower point 0, is defined by
PPT Slide
Lager Image
where F L 1 ((0, T 0 ), H ).
Definition 2.3 ( [3] ). The Riemann-Liouville fractional derivative is given by
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Here the notation Wδ ,1 ((0, T 0 ), H ) stands for the Sobolev space defined as
PPT Slide
Lager Image
Note that z ( t ) = yδ ( t ), dj = yj (0).
Definition 2.4 ( [3] ). The Caputo fractional derivative is given by
PPT Slide
Lager Image
for δ − 1 < q < δ, δ ∈ ℕ, where F L 1 ((0, T 0 ), H ) ∩ Cδ −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, Ct := C ([− r , t ], H ) be the Banach space of all H-valued continuous functions on [− r , t ] endowed with the supremum norm ∥ y t := sup s ∈[− r,t ] y ( s )∥ for each y Ct and t ∈ (0, T 0 ] and the space of all continuous functions from [− r, t ] into Hα denoted by
PPT Slide
Lager Image
is a Banach space with the supremum norm ∥ y t,α := sup s ∈[− r,t ] Aαy ( s )∥, for each
PPT Slide
Lager Image
For 0 ≤ α < 1, we define
PPT Slide
Lager Image
where L > 0 is a appropriate constant to be defined later.
Now, we turn to the following fractional differential equations with nonlocal conditions as
PPT Slide
Lager Image
We give few examples of function h as
  • (1) Letw∈L1(0,r) be such thatLet
PPT Slide
Lager Image
  • (ii) Let −r≤t1
PPT Slide
Lager Image
  • whereci≥ 0.
If we take
PPT Slide
Lager Image
defined as ψ ( θ ) ≡ ϕ for all θ ∈ [− r , 0] and
PPT Slide
Lager Image
given by G ( y )( θ ) ≡ h ( y ) for all θ ∈ [− r , 0] and
PPT Slide
Lager Image
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,
PPT Slide
Lager Image
which includes (12). For example,
PPT Slide
Lager Image
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 self-adjoint linear operator fromD(A) ⊂HintoH. We assume that operatorAhas the pure point spectrum
PPT Slide
Lager Image
  • withλm→ ∞ asm→ ∞ and corresponding complete orthonormal system of eigenfunctions {χj}, i.e.,
PPT Slide
Lager Image
  • where
PPT Slide
Lager Image
  • (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
PPT Slide
Lager Image
  • (A3) The nonlinear functionis Lipschitz continuous and there exist constantsLf> 0 andμ1∈ (0, 1] such that
PPT Slide
Lager Image
  • for allHere,andZis a Banach space andR> 0 is a constant to be defined later. The functionis defined by
PPT Slide
Lager Image
  • (A4) The functiona:Hα× [0,T0] → [0,T0] is continuous function and there exist constantsLα> 0 andμ2∈ (0, 1] such that
PPT Slide
Lager Image
  • for allanda(·, 0) = 0.
  • (A5)is continuous function and there exists a positive constantLgsuch that
PPT Slide
Lager Image
  • for allandt∈ [0,T0].
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
PPT Slide
Lager Image
and the following integral equation
PPT Slide
Lager Image
is verified.
The operator Sq ( t ) and Tq ( t ) are defined as follows:
PPT Slide
Lager Image
where
PPT Slide
Lager Image
is a a probability density function defined on (0,∞) i.e.,
PPT Slide
Lager Image
and
PPT Slide
Lager Image
Lemma 2.6 ( [11] ). If −A is the infinitesimal generator of analytic semigroup of uniformly continuous bounded operators. Then,
(1) The operator Sq ( t ), t ≥ 0 and Tq ( t ), t ≥ 0 are bounded linear operators.
(2) ║ Sq ( t ) y ║ ≤ M y ║,
PPT Slide
Lager Image
, for any y H .
(3) The families { Sq ( t ) : t ≥ 0} and { Tq ( t ) : t ≥ 0} are strongly continuous .
(4) If T ( t ) is compact, then Sq ( t ) and Tq ( 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 Hn be the finite dimensional subspace of H spanned by { χ 0 , χ 1 , · · · , χn } and Pn : H Hn be the corresponding projection operator for n = 0, 1, 2, · · · , . We define
PPT Slide
Lager Image
by
PPT Slide
Lager Image
and
PPT Slide
Lager Image
by
PPT Slide
Lager Image
We choose T , 0 < T T 0 sufficiently small such that
PPT Slide
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
Now, we consider
PPT Slide
Lager Image
By the assumptions ( A 3)−( A 4), we have that f is continuous on [0, T ]. Therefore, there exist a positive constant Nf such that
PPT Slide
Lager Image
with
PPT Slide
Lager Image
Similarly with the help of the assumption ( A 5), we can show that
PPT Slide
Lager Image
Therefore, we can indicate effectively that Gg = bTNg , where
PPT Slide
Lager Image
Let us consider the operator
PPT Slide
Lager Image
defined by ( Aαy )( t ) = Aα ( y ( t )) and ( Pnxt )( s ) = Pn ( x ( t + s )), for all s ∈ [− r , 0] and t ∈ [0, T ]. We consider the operator Qn : BR BR defined by
PPT Slide
Lager Image
for each x BR , where
PPT Slide
Lager Image
Theorem 3.1. Suppose ( A 1)-( A 5) holds and k ( t ) ∈ D ( A ) for all t ∈ [− r , 0]. Then, there exists a unique xn BR such that Qnxn = xn for each n = 0, 1, 2, · · · , and xn satisfies the following approximate integral equation
PPT Slide
Lager Image
Proof . To demonstrate the theorem, we first need to show that
PPT Slide
Lager Image
It is easy to show that
PPT Slide
Lager Image
by using the fact that f and g are continuous function. Now, it remains to show that
PPT Slide
Lager Image
For t, τ ∈ [− r , 0] with t > τ , we have
PPT Slide
Lager Image
by using fact that k is Hölder continuous with exponent 1 i. e., Lipschitz continuous on [− r , 0].
For
PPT Slide
Lager Image
0 < τ < t < T , then we have
PPT Slide
Lager Image
From the first term of above inequality, we have
PPT Slide
Lager Image
Also, we have that for each x H
PPT Slide
Lager Image
Therefore, we estimate the first term as
PPT Slide
Lager Image
where
PPT Slide
Lager Image
The second integrals is estimated as
PPT Slide
Lager Image
where K 2 = ∥ Aα −2 M 2 NfT . The third integrals is estimated as
PPT Slide
Lager Image
where K 3 = M 1 Aα −2 Nf . Similarly, we estimate forth integral as
PPT Slide
Lager Image
where K 4 = ∥ Aα −2 M 2 TGg and
PPT Slide
Lager Image
where K 5 = M 1 Aα −2 Gg and
Thus, from the inequality (38) to (42), we obtain that
PPT Slide
Lager Image
for a positive suitable constant
PPT Slide
Lager Image
Therefore, we conclude that
PPT Slide
Lager Image
Hence, we deduce that the operator
PPT Slide
Lager Image
is well defined map.
Next, we prove that Qn : BR BR . For 0 ≤ t T and x BR , we get that
PPT Slide
Lager Image
PPT Slide
Lager Image
From the inequalities (28) and (44), we conclude that Qn ( BR ) ⊂ BR . Finally, we will show that Qn is a contraction map. For x, y BR and 0 ≤ t T , we have
PPT Slide
Lager Image
We have the following inequalities:
PPT Slide
Lager Image
and
PPT Slide
Lager Image
Using (46)-(47) in (45), we get,
PPT Slide
Lager Image
From the inequality (30), we get
PPT Slide
Lager Image
with Θ < 1. Therefore, it implies that the map Qn is a contraction map i.e. Qn has a unique fixed point xn BR i.e., Qnxn = xn and xn satisfies the approximate integral equation
PPT Slide
Lager Image
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 xn ( t ) ∈ D ( Aυ ) 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
PPT Slide
Lager Image
such that xn satisfy the integral equation (35). Theorem 2.6.13 in Pazy [35] implies that T ( t ) : H D ( Aυ ) for t > 0 and 0 ≤ υ < 1 and for 0 ≤ υ η < 1, D ( Aη ) ⊆ D ( Aυ ). It is easy to see that Hölder continuity of xn 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 ( Aυ ) 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
PPT Slide
Lager Image
with 0 < α < υ < 1.
Proof . Let k ( t ) ∈ D ( A ) for every t ∈ [−r, 0]. For t ∈ [− r , 0], applying Aυ on the both the sides of (35) and obtaining,
PPT Slide
Lager Image
For t ∈ (0, T ], we apply Aυ on the both the sides of (35) and get
PPT Slide
Lager Image
This finishes the proof of lemma. □
4. Convergence of Solutions
The convergence of the solution xn Hα 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
PPT Slide
Lager Image
Proof . For 0 < α < υ < 1, n p . Let t ∈ [− r , 0], we conclude
PPT Slide
Lager Image
For t ∈ (0, T ], we obtain,
PPT Slide
Lager Image
We also have the following estimation:
PPT Slide
Lager Image
Thus, we obtain
PPT Slide
Lager Image
And
PPT Slide
Lager Image
Therefore, we estimate
PPT Slide
Lager Image
We choose t′ 0 such that 0 < t′ 0 < t < T , we have
PPT Slide
Lager Image
We estimate the first integral as
PPT Slide
Lager Image
By using Corollary 3.3, the second integral is estimated as
PPT Slide
Lager Image
Third and forth term are estimated as
PPT Slide
Lager Image
and
PPT Slide
Lager Image
Thus, we have
PPT Slide
Lager Image
where
PPT Slide
Lager Image
We now put t = t + θ in the above inequality, where θ ∈ [ t′ 0 t , 0] and get
PPT Slide
Lager Image
Taking s θ = ν in above inequality and obtaining,
PPT Slide
Lager Image
Thus, we have
PPT Slide
Lager Image
Since, for t + θ ≤ 0, we have xn ( t + θ ) = k ( t + θ ) for all n n 0 . Thus, we obtain sup r t θ ≤0 xn ( t + θ ) − xp ( t + θ )∥ α
PPT Slide
Lager Image
Thus, for each t ∈ (0, t′ 0 ], we have
PPT Slide
Lager Image
where D 5 and D 6 are arbitrary positive constants. Using (64) and (66) in (65) and thus getting
PPT Slide
Lager Image
Thus, we get
PPT Slide
Lager Image
By Lemma 5.6.7 in [35] , we have that there exists a constant K such that
PPT Slide
Lager Image
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 { xn } is a Cauchy sequence in BR . Now, we show the convergence of the solution for the approximate integral equation xn (·) 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 xn BR, satisfying
PPT Slide
Lager Image
and x BR, satisfying
PPT Slide
Lager Image
such that xn converges to x in BR i.e., xn x as n → ∞.
Proof . Let k ( t ) ∈ D ( A ) for all t ∈ [− r , 0]. For 0 < t T , it follows that there exists xn BR such that Aαxn ( t ) → Aαx ( t ) ∈ BR as n → ∞ and x ( t ) = xn ( t ) = k ( t ), for each t ∈ [− r , 0] and for all n . Also, for t ∈ [− r , T ], we have Aαxn ( t ) → Aαx ( t ) as n → ∞ in H . Since BR is a closed subspace of
PPT Slide
Lager Image
and xn BR , therefore it follows that x BR and
PPT Slide
Lager Image
Also, we have
PPT Slide
Lager Image
as n → ∞ and
PPT Slide
Lager Image
as n → ∞. For 0 < t 0 < t , we rewrite (35) as
PPT Slide
Lager Image
We may estimate the first and third integral as
PPT Slide
Lager Image
Thus, we deduce that
PPT Slide
Lager Image
Letting n → ∞ in the above inequality, we obtain
PPT Slide
Lager Image
Since t 0 is arbitrary and hence, we conclude that x (·) satisfies the integral equation (23). □
5. Faedo-Galerkin Approximations
In this section, we consider the Faedo-Galerkin 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
PPT Slide
Lager Image
satisfying the following integral equation
PPT Slide
Lager Image
with 0 < T < T 0 .
Also, we have a unique solution
PPT Slide
Lager Image
of the approximate integral equation
PPT Slide
Lager Image
Applying the projection on above equation, then Faedo-Galerkin approximation is given by vn ( t ) = Pnxn ( t ) satisfying
PPT Slide
Lager Image
or
PPT Slide
Lager Image
Let solution x (·) of (77) and vn (·) of (79), have the following representation
PPT Slide
Lager Image
PPT Slide
Lager Image
Using (82) in (79), we obtain a system of fractional order integro-differential equation of the form
PPT Slide
Lager Image
PPT Slide
Lager Image
where
PPT Slide
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
For the convergence of
PPT Slide
Lager Image
to αi , 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
PPT Slide
Lager Image
Proof . For n p and 0 ≤ α < υ , we get
PPT Slide
Lager Image
Since xn xp and λp → ∞ 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 vn BR given as
PPT Slide
Lager Image
for all t ∈ [0, T ] and x BR satisfying
PPT Slide
Lager Image
for t ∈ [0, T ], such that vn x as n → ∞ in BR and x satisfies the equation (23) on [0, T ].
Proof . By the Theorem 4.2, we have that
PPT Slide
Lager Image
Thus, we conclude that
PPT Slide
Lager Image
Since xn 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
PPT Slide
Lager Image
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
PPT Slide
Lager Image
Proof . It can easily be determined that
PPT Slide
Lager Image
Thus, we conclude that
PPT Slide
Lager Image
From the Theorem 4.2, we have vn x as n → ∞. Thus, we conclude that
PPT Slide
Lager Image
as n → ∞. This gives the proof of the theorem. □
6. Example
Let us consider the following integro-differential equation with deviated argument of the form
PPT Slide
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
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
PPT Slide
Lager Image
H is given by
PPT Slide
Lager Image
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
PPT Slide
Lager Image
We also have
PPT Slide
Lager Image
and
PPT Slide
Lager Image
For each w D ( A ) and λ ∈ ℝ with − Aw = λw , we get
PPT Slide
Lager Image
The
PPT Slide
Lager Image
is the solution of the problem Aw = − λw . By utilizing the boundary conditions, we get D = 0 and λn = n 2 π 2 for n ∈ ℕ. Thus,
PPT Slide
Lager Image
is the eigenvector corresponding to eigenvalue λn . We also have < wn,wm >= 0 for n m and < wn,wm >= 1. Thus, we have that for w D ( A ), there exists a sequence βn of real numbers such that
PPT Slide
Lager Image
The, we have following representation of the semigroup
PPT Slide
Lager Image
Now, for x ∈ (0, 1), we define
PPT Slide
Lager Image
by
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Thus, it can be verified that f satisfies the hypotheses ( A 3).
Similarly, for x ∈ (0, 1), we define
PPT Slide
Lager Image
by
PPT Slide
Lager Image
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, Roorkee-247667, India.
e-mail: alkachadda23@gmail.com
Dwijendra N. Pandey
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, India.
e-mail: dwij.iitk@gmail.com
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