APPROXIMATIONS OF SOLUTIONS FOR A NONLOCAL FRACTIONAL INTEGRO-DIFFERENTIAL EQUATION WITH DEVIATED ARGUMENT

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.

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)

where is the generalized fractional derivative of order q in Caputo sense with lower limit 0. For x ∈ C([0, T0];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, T0]×C([−r, 0],H)×H → H, a : H×[0, T0] → [0, T0], b : [0, T0]×[0, T0] → ℝ, g : [0, T0]×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, T0],H) stands for the Banach space of all the continuous functions from [0, T0] into H equipped with the norm ∥ z(t)∥C := supt∈[0,T0] ∥ z(t)∥H and Lp((0, T0),H) stands for Banach space of all Bochnermeasurable functions from (0, T0) 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)∥ ≤ Ceωt, for t ≥ 0. In addition, we note that

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

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-α = (Hα)∗, for each α > 0. The space H-α 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,

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

where F ∈ L1((0, T0),H).

Definition 2.3 ([3]). The Riemann-Liouville fractional derivative is given by

where Here the notation Wδ,1((0, T0),H) stands for the Sobolev space defined as

Note that z(t) = yδ(t), dj = yj(0).

Definition 2.4 ([3]). The Caputo fractional derivative is given by

for δ − 1 < q < δ, δ ∈ ℕ, where F ∈ L1((0, T0),H) ∩ Cδ−1((0, T0),H).

Let C0 := C([−r, 0],H) be the collection of continuous mappings from [−r, 0] into H equipped with the supremum norm ∥y∥0 := supt∈[−r,0] ∥y(t)∥ for y ∈ C0. 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 := sups∈[−r,t]∥y(s)∥ for each y ∈ Ct and t ∈ (0, T0] and the space of all continuous functions from [−r, t] into Hα denoted by is a Banach space with the supremum norm ∥y∥t,α := sups∈[−r,t] ∥Aαy(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

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:

Now, we provide the definition of mild solution for the nonlocal system (1)-(2).

Definition 2.5. A continuous function x : [0, T0] → H is said to be a mild solution for the system (1)-(2) if and the following integral equation

is verified.

The operator Sq(t) and Tq(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 Sq(t), t ≥ 0 and Tq(t), t ≥ 0 are bounded linear operators.

(2) ║Sq(t)y║ ≤ M║y║, , 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 {χ0, χ1, · · · , χn} and Pn : H → Hn be the corresponding projection operator for n = 0, 1, 2, · · · , . We define

by

and

by

We choose T, 0 < T ≤ T0 sufficiently small such that

Now, we consider

By the assumptions (A3)−(A4), we have that f is continuous on [0, T]. Therefore, there exist a positive constant Nf such that

with

Similarly with the help of the assumption (A5), we can show that Therefore, we can indicate effectively that Gg = bTNg, where

Let us consider the operator 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

for each x ∈ BR, where

Theorem 3.1. Suppose (A1)-(A5) 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

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 K2 = ∥ Aα−2∥M2NfT. The third integrals is estimated as

where K3 = M1∥Aα−2∥Nf. Similarly, we estimate forth integral as

where K4 = ∥Aα−2∥M2TGg and

where K5 = M1∥Aα−2∥Gg 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 Qn : BR → BR. For 0 ≤ t ≤ T and x ∈ BR, we get that

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

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 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

Hence, the proof of the theorem is completed. □

Lemma 3.2. Assume that hypotheses (A1)-(A5) 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 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 (A1)-(A5) are satisfied. If k(t) ∈ D(A), ∀ t ∈ [−r, 0], then for any t ∈ [−r, T], there exists a constant U0 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 Aυ on the both the sides of (35) and obtaining,

For t ∈ (0, T], we apply Aυ on the both the sides of (35) and get

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 (A1)-(A5) 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 xn(t+θ) = k(t+θ) for all n ≥ n0. Thus, we obtain sup−r−t≤θ≤0 ∥xn(t + θ) − xp(t + θ)∥α

Thus, for each t ∈ (0, t′0], we have

where D5 and D6 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 {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 (A1)-(A5) are satisfied and k(t) ∈ D(A) for each t ∈ [−r, 0]. Then, there exists a unique xn ∈ BR, satisfying

and x ∈ BR, satisfying

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 and xn ∈ BR, therefore it follows that x ∈ BR and

Also, we have

as n → ∞ and

as n → ∞. For 0 < t0 < 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 t0 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 < T0, there exists a unique satisfying the following integral equation

with 0 < T < T0.

Also, we have a unique solution of the approximate integral equation

Applying the projection on above equation, then Faedo-Galerkin approximation is given by vn(t) = Pnxn(t) satisfying

or

Let solution x(·) of (77) and vn(·) of (79), have the following representation

Using (82) in (79), we obtain a system of fractional order integro-differential equation of the form

where

For the convergence of to αi, we have the following convergence theorem.

Corollary 5.1. Assume that (A1)-(A5) are satisfied. If k(t) ∈ D(A) for each t ∈ [−r, 0], then

Proof. For n ≥ p and 0 ≤ α < υ, we get

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 (A1)-(A5) are satisfied and k(t) ∈ D(A) for all t ∈ [−r, 0]. Then there exist a unique function vn ∈ BR given as

for all t ∈ [0, T] and x ∈ BR satisfying

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

Thus, we conclude that

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 Thus, we have following theorem.

Theorem 5.3. Let us assume that (A1)-(A5) 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 vn → x as n → ∞. Thus, we conclude that 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

where t ∈ [0, 1], x ∈ [0, 1], q ∈ (0, 1), p ∈ ℕ, r > 0, b is real valued, γ1 : [−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 ∈ C1([0, 1] × [0, 1],ℝ).

Let H = L2((0, 1),ℝ). Now, we define operator by Aw = −d2w/dx2 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 λn = n2π2 for n ∈ ℕ. Thus, 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 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 (A3).

Similarly, for x ∈ (0, 1), we define by

Then, it can be seen that g fulfills hypotheses (A5). Thus, we can apply the results of previous sections to study the existence and convergence of the mild solution to system (98)-(100).