Existence and approximation of solution to neutral ... - Semantic Scholar
Computers and Mathematics with Applications 69 (2015) 893–908
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Existence and approximation of solution to neutral fractional differential equation with nonlocal conditions Alka Chadha, D.N. Pandey ∗ Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, India
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Article history: Received 6 August 2014 Received in revised form 16 January 2015 Accepted 3 February 2015 Available online 18 March 2015 Keywords: Analytic semigroup Banach fixed point theorem Caputo derivative Faedo–Galerkin approximations Nonlocal conditions Neutral fractional differential equation
abstract This paper is concerned with the approximation of the solution for neutral fractional differential equation with nonlocal conditions in an arbitrary separable Hilbert space H. We study an associated integral equation and then, consider a sequence of approximate integral equations obtained by the projection of considered associated nonlocal neutral fractional integral equation onto finite dimensional space. The sufficient condition for the existence and uniqueness of solutions to every approximate integral equation is derived by using analytic semigroup and Banach fixed point theorem. We demonstrate convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. Moreover, we consider the Faedo–Galerkin approximations of the solution and demonstrate some convergence results. An example is also provided to illustrate the discussed abstract theory. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction In the recent years, the investigation of fractional differential equation has been picking up much attention of researchers. This is due to the fact that fractional differential equations have various applications in engineering and scientific disciplines, for example, fluid dynamics, fractal theory, diffusion in porous media, fractional biological neurons, traffic flow, polymer rheology, neural network modeling, viscoelastic panel in super sonic gas flow, real system characterized by power laws, electrodynamics of complex medium, sandwich system identification, nonlinear oscillation of earthquake, models of population growth, mathematical modeling of the diffusion of discrete particles in a turbulent fluid, nuclear reactors and theory of population dynamics. The fractional differential equation is an important tool to describe the memory and hereditary properties of various materials and phenomena. The details on the theory and its applications may be found in books [1–4] and references therein. On the other hand, the nonlocal problem for abstract evolution equations has been studied by many authors. The existence of a solution for abstract Cauchy differential equation with nonlocal conditions in a Banach space has been considered first by Byszewski [5]. In physical science, the nonlocal condition may be connected with better effect in applications than the classical initial condition since nonlocal conditions are normally more exact for physical estimations than the classical initial condition. For the study of nonlocal problems, we refer to [6–11] and references given therein. The Faedo–Galerkin approach may be used for the study of more regular solutions, imposing higher regularity on the data. Also, the Faedo–Galerkin method may be used within a variational formulation in order to provide solutions of the equations under possibly weaker assumptions on the data, and may also prove a very useful tool for numerical approximation of
∗
Corresponding author. E-mail addresses: [email protected] (A. Chadha), [email protected] (D.N. Pandey).
http://dx.doi.org/10.1016/j.camwa.2015.02.003 0898-1221/© 2015 Elsevier Ltd. All rights reserved.
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the equations. In [12], the Faedo–Galerkin approximations of the solutions for functional Cauchy problem in a separable Hilbert space have been studied by Miletta with the help of analytic semigroup theory and Banach fixed point theorem. In [13], authors have extended the results of [12] and considered the Faedo–Galerkin approximations of the solutions to a class of functional integro-differential equation. The Faedo–Galerkin approximations of the solutions for nonlinear Sobolev type evolution equation have been studied by authors of [14]. In [8], authors have considered the Faedo–Galerkin approximations of the solutions to neutral functional differential equations with nonlocal conditions. The existence and approximations of solutions for fractional differential equation have been investigated by the author in [15]. The existence and Faedo–Galerkin approximations of the solutions for fractional integral equations has been studied by authors in [16]. In [17], authors have discussed the Faedo–Galerkin approximations of solutions to fractional differential equations with a deviating argument with the help of analytic semigroup. The present work extends the previous studies by examining the neutral fractional differential equation with nonlocal conditions and determining the mild solution for the nonlocal neutral fractional differential equation. For a nice introduction to the existence of an approximate solution and associated study of different problems, we refer to the Refs. [7,18–24,31] and references given therein. Motivated by of above mentioned work, the main objective of this work is to investigate the Faedo–Galerkin approximations of the solution to the following nonlocal neutral fractional differential equation in a separable Hilbert space (H , ∥ · ∥, (·, ·)) c
q
Dt [y(t ) + G(t , y(t ), y(t − h1 ))] = −By(t ) + F (t , y(t ), y(t − h2 )),
g (y) = ψ,
on [−h, 0], h > 0,
t ∈ [0, T ],
(1.1) (1.2)
c q Dt
where 0 < q < 1, 0 < T < ∞, is the fractional derivative in Caputo sense and h = max{h1 , h2 }, h1 , h2 > 0. In (1.1), B : H ⊃ D(B) → H is assumed to be a closed, self adjoint and positive definite linear operator with dense domain D(B) such that −B is the infinitesimal generator of an analytic semigroup of bounded linear operator on H. The functions G, F : [0, T ] × H × H → H , g : C ([−h, 0]; H ) → C ([−h, 0]; H ) are nonlinear continuous functions satisfying certain conditions to be mentioned later. Neutral differential equation arises in many areas of applied mathematics, science and engineering such as the theory of aeroelasticity [25] and lossless transmission lines [26]. The theory of heat conduction in materials and the lumped control systems can be described by neutral differential equations. The system of rigid heat conduction with finite wave spaces can be modeled in the form of the integro-differential equation of neutral type with delay. For the initial study of the neutral functional differential equations with finite delay, we refer to books [27,28] and references given therein. The article is organized as follows: Section 2 provides some basic definitions, lemmas and theorems as preliminaries as these are useful for proving our results. Section 3 derives the sufficient condition for the existence and uniqueness of the approximate solutions by using analytic semigroup and Banach fixed point theorem. Section 4 proves the convergence of the solution to each of the approximate integral equations with the limiting function which satisfies the associated integral equation and Section 5 focuses on the convergence of the approximate Faedo–Galerkin solutions. Section 6 presents an example. 2. Preliminaries and assumptions In this section, some basic definitions, preliminaries, theorems and lemmas and assumptions which will be used to prove existence result, are provided. Throughout the work, we assume that (H , ∥ · ∥, ⟨·, ·⟩) is a separable Hilbert space. The symbol C ([0, T ]; H ) stands for the Banach space of all the continuous functions from [0, T ] into H equipped with the norm ∥z ∥C = supt ∈[0,T ] ∥ z (t )∥ and Lp ((0, T ); H ) denotes the Banach space of all Bochner-measurable functions from (0, T ) to H with the norm
∥z ∥Lp =
(0,T )
∥z (s)∥p ds
1/p
,
z ∈ Lp ((0, T ); H ).
In this work, −B is assumed to be the infinitesimal generator of an analytic semigroup of bounded linear operators {T (t ) : t ≥ 0} on H. Therefore, there exist constants C ≥ 1 and δ ≥ 0 such that Ceδt ≥ ∥T (t )∥ for t ≥ 0. In addition, we note that
j d T (t ) ≤ Mj , dt j
t > t0 , t0 > 0, j = 1, 2, . . . ,
(2.1)
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(−B) i.e., −B is invertible. This permits us to define the positive fractional power Bη as closed linear operator with domain D(Bη ) ⊆ H for η ∈ (0, 1]. Moreover, D(Bη ) is dense in H with the norm
∥y∥η = ∥Bη y∥,
∀ y ∈ D(Bη ). η
(2.2)
Hence, we signify the space D(B ) by Hη endowed with the η-norm (∥ · ∥η ). Also, we have that Hκ ↩→ Hη for 0 < η < κ and therefore, the embedding is continuous. Then, we define H−η = (Hη )∗ , for each η > 0 and the dual space of Hη , is a Banach space with the norm ∥z ∥−η = ∥B−η z ∥ for z ∈ H−η . For more study on the fractional powers of closed linear operators, we refer to book by Pazy [29].
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Lemma 2.1 ([29]). Let −B be the infinitesimal generator of an analytic semigroup {T (t ) : t ≥ 0} such that ∥ T (t )∥ ≤ M, for t ≥ 0 and 0 ∈ ρ(−B). Then, (i) For 0 < η ≤ 1, Hη is a Hilbert space. (ii) The operator Bη T (t ) is bounded for every t > 0 and
∥BT (t )∥ ≤ Mt −1 ,
(2.3)
∥Bη T (t )∥ ≤ Mη t −η .
(2.4)
Now, we state some basic definitions and properties of fractional calculus. q
Definition 2.1. The Riemann–Liouville fractional integral operator Jt is given by t
1
q
Jt F (t ) =
Γ (q)
(t − s)q−1 F (s)ds,
(2.5)
0
where F ∈ L1 ((0, T ); H ) and q > 0 is the order of the fractional integration. Definition 2.2. The Riemann–Liouville fractional derivative is given by RL
δ−q
Dt F (t ) = Dδt Jt q
F (t ),
δ − 1 < q < δ, δ ∈ N,
δ
δ−q
where Dδt = dtd δ , F ∈ L1 ((0, T ); H ), Jt space defined by
F ∈ W δ,1 ((0, T ); H ). Here, the notation W δ,1 ((0, T ); H ) stands for the Sobolev
W δ,1 ((0, T ); H ) =
(2.6)
y ∈ H : ∃ z ∈ L1 ((0, T ); H ) : y(t ) =
δ−1
dk
k=0
tk k!
+
t δ−1
(δ − 1)!
∗ z (t ), t ∈ (0, T ) .
Note that z (t ) = yδ (t ), dk = yk (0). Definition 2.3. The Caputo fractional derivative is given by c
t
1
q
Dt F ( t ) =
Γ (δ − q)
(t − s)δ−q−1 F δ (s)ds,
δ − 1 < q < δ,
(2.7)
0
where F ∈ C δ−1 ((0, T ); H ) ∩ L1 ((0, T ); H ) and the following holds q
q
Jt (c Dt F (t )) = F (t ) −
δ−1 k t
k! k=0
F k (0).
(2.8)
We denote by C0 := C ([−h, 0]; H ) the Banach space of all H-valued continuous functions on [−h, 0] endowed with the supremum norm ∥z ∥0 := supt ∈[−h,0] ∥z (t )∥, for z ∈ C0 . In addition, let Ct = C ([−h, t ]; H ) be the Banach space of all H-valued continuous functions on [−h, t ] endowed with the supremum norm ∥z ∥t = sups∈[−h,t ] ∥z (s)∥, for z ∈ Ct and η t ∈ (0, T ] and the space of all continuous functions from [0, t ] into Hη denoted by Ct , is a Banach space endowed with the η norm ∥z ∥t ,η = sups∈[−h,t ] ∥z (s)∥η , for z ∈ Ct . Now, we assume the following assumptions: (A1) B is a closed, densely defined, positive definite and self-adjoint linear operator from D(B) ⊂ H into H. We assume that operator B has the pure point spectrum 0 < λ0 ≤ λ1 ≤ λ2 ≤ · · · ≤ λm ≤ · · · ,
(2.9)
with λm → ∞ as m → ∞ and a corresponding complete orthonormal system of eigenfunctions {φj }, i.e., Bφj = λj φj ,
and ⟨φl , φj ⟩ = δlj ,
(2.10)
where
δlj = η
1, 0, η
j = l, otherwise. η
(A2) g : C0 → C0 and there exists a Lipschitz continuous function χ ∈ C0 with Lipschitz constant Lχ > 0 such that η g (χ ) = ψ on [−h, 0] and χ (t ) belongs to D(Bθ0 ) for 0 < η < θ0 < 1 and ψ ∈ C0 .
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(A3) The nonlinear function F : [0, T ] × Hη × Hη → H is continuous function for η ∈ (0, 1] and there exists an increasing function fR : [0, ∞) → (0, ∞) which depends on R > 0 such that
∥F (t , w1 , z1 ) − F (t , w2 , z2 )∥ ≤ fR (t )[∥w1 − w2 ∥η + ∥z1 − z2 ∥η ],
(2.11)
∥F (t , w, z )∥ ≤ fR (t ),
(2.12)
for each (t , w, z ), (t , w1 , z1 ), (t , w2 , z2 ) ∈ [0, T ] × BR (Hη × Hη : (χ (0), χ (0))). Here, BR (Y : (y0 , y0 )) = {(y1 , y2 ) ∈ Y 2 : ∥y1 − y0 ∥Y + ∥y2 − y0 ∥Y ≤ R} for any Banach space Y . (A4) For 0 < η < β < 1, the map Bβ G : [0, T ] × Hη × Hη → H is continuous and there exists a constant LG > 0 such that 2
∥Bβ G(t , w1 , z1 ) − Bβ G(s, w2 , z2 )∥ ≤ LG [|t − s|µ + ∥w1 − w2 ∥η + ∥z1 − z2 ∥η ],
(2.13) η−β
for each (t , w1 , z1 ), (s, w2 , z2 ) ∈ [0, T ] × BR (Hη × Hη : (χ (0), χ (0))), µ ∈ (0, 1] and 2LG ∥B
∥ < 1.
In order to establish the mild solution for the problem (1.1)–(1.2), we consider the following differential equation q
Dt [y(t ) + G(t , y(t ), y(t − h1 ))] = −By(t ) + F (t , y(t ), y(t − h2 )), y(t ) = χ (t ),
t ∈ [0, T ],
(2.14)
on [−h, 0], h > 0.
(2.15)
We establish the existence, uniqueness and approximation of the mild solution of the problem (2.14)–(2.15) which is equivalent to establish the existence of the solution for the system (1.1)–(1.2). We present the definition of mild solution for the problem (2.14)–(2.15) as follows Definition 2.4. A continuous function y : [−h, T ] → Hη is said to be a mild solution for the problem (2.14)–(2.15) if y(·) satisfies the following integral equation
χ (t ), t ∈ [−h, 0], Sq (t )[χ (0) + G(0, χ (0), χ (−h1 ))] − G(t , y(t ), y(t − h1 )) t + (t − s)q−1 Tq (t − s)BG(s, y(s), y(s − h1 ))ds y(t ) = 0 t + (t − s)q−1 Tq (t − s)F (s, y(s), y(s − h2 ))ds, t ∈ [0, T ].
(2.16)
0
The operators Sq (t ) and Tq (t ) are defined as follows:
Sq (t )x =
∞
ζq (ξ )T (t q ξ )xdξ ,
(2.17)
0
Tq (t )x = q
∞
ξ ζq (ξ )T (t q ξ )xdξ ,
x ∈ H,
(2.18)
0
where ζq (ξ ) =
ψq (ξ ) =
1 q
1
ξ 1−1/q × ψq (ξ − q ) is a probability density function defined on (0, ∞) i.e., ζq (ξ ) ≥ 0,
1
∞
π
n =1
(−1)n−1 ξ −nq−1
Γ (nq + 1) sin(nπ q), n!
∞ 0
ζq (ξ )dξ = 1 and
0 < ξ < ∞.
For more details on probability density function, we refer to papers [32,34–36]. Lemma 2.2 ([30]). The operators Sq (t ), t ≥ 0 and Tq (t ), t ≥ 0 are bounded linear operators such that qMη Γ (2−η)t −qη
(i) ∥Sq (t )z ∥ ≤ M ∥z ∥, ∥ Tq (t )z ∥ ≤ Γ (1+q) ∥z ∥ and ∥Bη Tq (t )z ∥ ≤ Γ (1+q(1−η)) ∥z ∥, for any z ∈ H. (ii) The families {Sq (t ) : t ≥ 0} and {Tq (t ) : t ≥ 0} are strongly continuous. (iii) If T (t ) is compact, then Sq (t ) and Tq (t ) are compact operators for any t > 0. qM
3. Approximate solutions and convergence Let Hn ⊂ H spanned by {φ0 , φ1 , . . . , φn } be the finite dimensional subspace and P n : H → Hn be the corresponding η η projection operator for n = 0, 1, 2, . . . . We define Gn : [0, T ] × CT × C0 → H by
Gn (t , y(t ), y(t − h1 )) = G(t , P n y(t ), P n y(t − h1 )). η
(3.1)
η
Similarly, we define Fn : [0, T ] × CT × C0 → H as follows
Fn (t , y(t ), y(t − h2 )) = F (t , P n y(t ), P n y(t − h2 )), for each n = 1, 2, 3, . . . .
(3.2)
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We choose T0 , 0 < T0 ≤ T sufficiently small such that A1 = max ∥Bβ G(t , χ (0), χ (0))∥
(3.3)
t ∈[0,T ]
and R
∥(Sq (t ) − I )Bη (χ (0) + Gn (0, χ (0), χ (−h1 )))∥ ≤ (1 − ν) ,
(3.4)
3
µ
∥Bη−β ∥LG (T0 + Lχ T0 ) + +
Mη Γ (2 − η)
Γ (1 + q(1 − η))
M1+η−β Γ (1 − (η − β))
Γ (1 + q(β − η))
f R (T )
T0 q(1−η)
(1 − η)
T0 q(β−η)
q(β−η)
T0 R + A1 ) (LG (β − η)
R
< (1 − ν) ,
(3.5)
6
Γ (1 + q(β − η))
LG
Mη Γ (2 − η)
T0 q(1−η)
(1 − ν) < , (β − η) Γ (1 + q(1 − η)) (1 − η) 2 where ν = ∥Bη−β ∥LG < 1/2, R = 2(R2 + 3∥χ∥20,η ) and Mη , M1+η−β are constants. Let Bη : C ([−h, t ]; D(Bη )) → C ([−h, t ]; H ) be defined as (Bη ψ)(s) = Bη (ψ(s)) for all s ∈ [−h, t ], t ∈ [0, T0 ]. M1+η−β Γ (1 − (η − β))
+
f R (T )
(3.6)
Now, we consider η
η
BR = BR (CT0 , χ (0)) = {y ∈ CT0 : ∥y − χ (0)∥η ≤ R},
(3.7)
and define the operator Qn on BR as follows
χ (t ), t ∈ [−h, 0], Sq (t )[χ (0) + Gn (0, χ (0), χ (−h1 ))] − Gn (t , y(t ), χ (t − h1 )) t + (t − s)q−1 Tq (t − s)BGn (s, y(s), χ (s − h1 ))ds Qn y(t ) = 0 t + (t − s)q−1 Tq (t − s)Fn (s, y(s), χ (s − h2 ))ds, t ∈ [0, T0 ],
(3.8)
0
for each y ∈ BR . Theorem 3.1. Suppose that conditions (A1)–(A4) are fulfilled and χ (0) ∈ D(Bη ). Then, there exists a unique fixed point yn ∈ BR of the map Qn i.e., Qn yn = yn for each n = 0, 1, 2, . . . , and yn satisfies the following approximate integral equation
χ (t ), t ∈ [−h, 0], Sq (t )[χ (0) + Gn (0, χ (0), χ (−h1 ))] − Gn (t , yn (t ), χ (t − h1 )) t + (t − s)q−1 Tq (t − s)BGn (s, yn (s), χ (s − h1 ))ds yn (t ) = 0 t + (t − s)q−1 Tq (t − s)Fn (s, yn (s), χ (s − h2 ))ds, t ∈ [0, T0 ].
(3.9)
0
η
η
Proof. To demonstrate the theorem, we first need to show that Qn : BR (CT0 , χ (0)) → BR (CT0 , χ (0)). To prove this, we show that the map t → (Qn y)(t ) is continuous from [−h, T0 ] into D(Bη ) with the η-norm. For t1 , t2 ∈ [−h, 0] and η y ∈ BR (CT0 , χ (0)), we have
∥(Qn y)(t2 ) − (Qn y)(t1 )∥η = ∥χ (t2 ) − χ (t1 )∥η . η
For t1 , t2 ∈ [0, T0 ] with t1 < t2 and y ∈ BR (CT0 , χ (0)), we get Bη [(Qn y)(t2 ) − (Qn y)(t1 )]
= [Sq (t2 ) − Sq (t1 )]Bη (χ (0) + Gn (0, χ (0), χ (−h1 ))) − Bη−β [Bβ Gn (t2 , y(t2 ), χ (t2 − h1 )) − Bβ Gn (t1 , y(t1 ), χ (t1 − h1 ))] t1 + [(t2 − s)q−1 − (t1 − s)q−1 ]B1+η−β Tq (t2 − s)Bβ Gn (s, y(s), χ (s − h1 ))ds 0 t1 + (t1 − s)q−1 B1+η−β [Tq (t2 − s) − Tq (t1 − s)]Bβ Gn (s, y(s), χ (s − h1 ))ds 0
(3.10)
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A. Chadha, D.N. Pandey / Computers and Mathematics with Applications 69 (2015) 893–908 t2
+
(t2 − s)q−1 B1+η−β Tq (t2 − s)Bβ Gn (s, y(s), χ (s − h1 ))ds
t1 t1
+
[(t2 − s)q−1 − (t1 − s)q−1 ]Bη Tq (t2 − s)Fn (s, y(s), χ (s − h2 ))ds
0 t1
+
(t1 − s)q−1 Bη [Tq (t2 − s) − Tq (t1 − s)]Fn (s, y(s), χ (s − h2 ))ds
0 t2
+
(t2 − s)q−1 Bη Tq (t2 − s)Fn (s, y(s), χ (s − h2 ))ds
t1
=
8
Ji .
(3.11)
i =1
For z ∈ H, we have
[T (t2 ξ ) − T (t1 ξ )]z = q
q
t2
d dt
t1
T (t ξ )zdt = q
t2
qξ t q−1 BT (t q ξ )zdt .
t1
Then, the first term of inequality (3.11) is estimated as ∞
ζq (ξ )∥[T (t2 q ξ ) − T (t1 q ξ )]Bη (χ (0) + Gn (0, χ (0), χ (−h1 )))∥dξ , ∞ t2 ≤ ζq (ξ ) qξ t q−1 ∥Bη T (t q ξ )∥dt ∥B(χ (0) + Gn (0, χ (0), χ (−h1 )))∥dξ ,
∥J 1 ∥ ≤
0
0
t1
∞
ξ 1−η ζq (ξ )
≤ qMη 0
t2
t q(1−η)−1 ∥B(χ (0) + Gn (0, χ (0), χ (−h1 )))∥dtdξ ,
t1
Γ (1 − q) q(1−η) q(1−η) Mη ∥B(χ (0) + Gn (0, χ (0), χ (−h1 )))∥(t2 − t1 ), Γ (1 + q(1 − η)) qMη Γ (2 − η) ≤ ∥B(χ (0) + Gn (0, χ (0), χ (−h1 )))∥[t1 + Φ (t2 − t1 )]q(1−η)−1 (t2 − t1 ), Γ (1 + q(1 − η))
≤
≤
qMη ∥B(χ (0) + Gn (0, χ (0), χ (−h1 )))∥Γ (2 − η)
Γ (1 + q(1 − η))
Φ q(1−η)−1 (t2 − t1 )q(1−η) ,
(3.12)
where 0 < Φ < 1 (see [16,33]). The second term of the inequality (3.11) can be estimated as
∥J2 ∥ = ∥Bη−β ∥ · ∥Bβ Gn (t2 , y(t2 ), χ (t2 − h1 )) − Bβ Gn (t1 , y(t1 ), χ (t1 − h1 ))∥ ≤ LG ∥Bη−β ∥[|t2 − t1 |µ + ∥P n y(t2 ) − P n y(t1 )∥η + ∥P n χ (t2 − h1 ) − P n χ (t1 − h1 )∥η ], ≤ LG ∥Bη−β ∥[|t2 − t1 |µ + ∥y(t2 ) − y(t1 )∥η + Lχ |t2 − t1 |]
(3.13)
and
∥Bβ Gn (t , y(t ), χ (t − h1 ))∥ ≤ ∥Bβ Gn (t , y(t ), χ (t − h1 )) − Bβ Gn (t , χ (0), χ (0))∥ + ∥Bβ Gn (t , χ (0), χ (0))∥, ≤ LG [∥P n y(s) − χ (0)∥η + ∥P n χ (t − h1 ) − χ (0)∥η ] + A1 ≤ LG R + A1 .
(3.14)
Also,
∥J 3 ∥ ≤ ≤
qM1+η−β Γ (1 − (η − β))
Γ (1 + q(β − η)) qM1+η−β Γ (1 − (η − β))
Γ (1 + q(β − η))
(LG R + A1 )
t1
(t1 − s)m1 −1 [(t2 − s)−m1 k1 − (t1 − s)−m1 k1 ]ds,
0
(LG R + A1 )k1 Ψ k1 −1 (1 − d)−m1 (1−k1 )−1 (t2 − t1 )m1 (1−k1 ) , (1−q)
where m1 = 1 − q(1 + η − β), k1 = 1−q(1+η−β) , d = (1 − ( m1 )1/k1 m1 ) and 0 < Ψ ≤ 1 [16,33]. 1
∥J 4 ∥ ≤
qM1+η−β Γ (1 − (η − β))
Γ (1 + q(β − η))
(LG R + A1 )
k
t1
0
(t1 − s)q−1
(3.15)
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× [(t1 − s)−q(1+η−β) − (t2 − s)−q(1+η−β) ]ds, R + A1 ) η−β qM1+η−β Γ (1 − (η − β)) (LG ≤ d (1 − d3 )−q(β−η)−1 (t2 − t1 )q(β−η) , Γ (1 + q(β − η)) 1+η−β 2 where 0 < d2 ≤ 1 and d3 = (1 − (1 + η − β/q)1/q(1+η−β) ).
∥J 5 ∥ ≤ ≤
qM1+η−β Γ (1 − (η − β))
Γ (1 + q(β − η))
(LG R + A1 )
t2
(t2 − s)q(β−η)−1 ds,
t1
M1+η−β Γ (1 − (η − β))
Γ (1 + q(β − η))
q(β−η)
(t2 − t1 ) (LG R + A1 ) (β − η)
.
(3.16)
Similarly
∥J 6 ∥ ≤ ≤
qMη Γ (2 − η)
Γ (1 + q(1 − η)) qMη Γ (2 − η)
f R (T )
t1
(t1 − s)m2 −1 [(t2 − s)−m1 k2 − (t1 − s)−m2 k2 ]ds,
0 k −1
Γ (1 + q(1 − η))
2 f R (T )k2 Ψ1
(1 − d4 )−m2 (1−k2 )−1 (t2 − t1 )m2 (1−k2 ) ,
(3.17)
where m2 = 1 − qη, k2 = 1−qη , 0 < Φ1 ≤ 1 and d4 = (1 − ( m2 )1/k2 m2 ). 2 k
1−q
∥J 7 ∥ ≤
qMη Γ (2 − η)
Γ (1 + q(1 − η))
η−1
f R (T )Φ2
(1 − d5 )−q(1−η)−1 (t2 − t1 )q(1−η) ,
(3.18)
where d5 = (1 − (η/q)1/(qη) ) and 0 < Φ2 ≤ 1.
∥J 8 ∥ ≤
Mη Γ (2 − η)
Γ (1 + q(1 − η))
f R (T )
(t2 − t1 )q(1−η) . (1 − η)
(3.19)
From the inequality (3.10)–(3.19), we deduce that the map Qn is Hölder continuous from [−h, T0 ] into D(Bη ) with the η η η η-norm. Now, we claim that Qn (BR (CT0 , χ (0))) ⊆ BR (CT0 , χ (0)). Thus, for t ∈ [−h, 0] and y ∈ BR (CT0 , χ (0)), we have
∥(Qn y)(t ) − χ (t )∥η = 0.
(3.20)
η
For t ∈ (0, T0 ] and y ∈ BR (CT0 , χ (0)),
∥(Qn y)(t ) − χ (0)∥η ≤ ∥(Sq (t ) − I )Bη (χ (0) + Gn (0, χ (0), χ (−h1 )))∥ + ∥Bη−β ∥ ∥Bβ Gn (t , y(t ), χ (t − h1 )) − Bβ Gn (0, χ (0), χ (−h1 ))∥ t + (t − s)q−1 ∥B1+η−β Tq (t − s)∥ ∥Bβ Gn (s, y(s), χ (s − h1 ))∥ds 0 t + (t − s)q−1 ∥Bη Tq (t − s)∥ ∥Fn (s, y(s), χ (s − h2 ))∥ds 0
R
µ
≤ (1 − ν) + ∥Bη−β ∥LG [T0 + ∥y(t ) − χ (0)∥η + ∥χ (t − h1 ) − χ (−h1 )∥η ] 3
q(β−η)
M1+η−β Γ (1 − (η − β))
q(1−η)
T0 Mη Γ (2 − η) T0 (LG R + A1 ) + f(T ) , Γ (1 + q(β − η)) (β − η) Γ (1 + q(1 − η)) R (1 − η) R M1+η−β Γ (1 − (η − β)) T0 q(β−η) µ ≤ (1 − ν) + ∥Bη−β ∥LG [T0 + R + Lχ T0 ] + (LG R + A1 ) 3 Γ (1 + q(β − η)) (β − η) q(1−η) Mη Γ (2 − η) T0 + f(T ) , Γ (1 + q(1 − η)) R (1 − η)
+
R
R
R
3
6
2
≤ (1 − ν) + (1 − ν) + ν
≤ R.
(3.21) η
η
Thus, we conclude that ∥Qn y − χ (0)∥T0 ,η ≤ R. Therefore, we deduce that Qn : BR (CT0 , χ (0)) → BR (CT0 , χ (0)). Next, we
will show that Qn is a strict contraction map. For y , y ∗
∥Qn y∗ (t ) − Qn y∗∗ (t )∥η = 0,
∗∗
η
∈ BR (CT0 , χ (0)) and t ∈ [−h, 0], we get that
(3.22)
900
A. Chadha, D.N. Pandey / Computers and Mathematics with Applications 69 (2015) 893–908
and for t ∈ (0, T0 ],
∥(Qn y∗ )(t ) − (Qn y∗∗ )(t )∥η ≤ ∥Bη−β ∥ ∥Bβ Gn (t , y∗ (t ), χ (t − h1 )) − Bβ Gn (t , y∗∗ (t ), χ (t − h1 ))∥ t (t − s)q−1 ∥B1+η−β Tq (t − s)∥ ∥Bβ Gn (s, y∗ (s), χ (s − h1 )) + 0
− Bβ Gn (s, y∗∗ (s), χ (s − h1 ))∥ds +
t
(t − s)q−1 ∥Bη Tq (t − s)∥
0
× ∥Fn (s, y∗ (s), χ (s − h2 )) − Fn (s, y∗∗ (s), χ (s − h2 ))∥ds, M1+η−β Γ (1 − (η − β)) T0 q(β−η) Mη Γ (2 − η) T0 q(1−η) η−β ≤ ∥B ∥LG + LG + f(T ) Γ (1 + q(β − η)) (β − η) Γ (1 + q(1 − η)) R (1 − η) × ∥y∗ − y∗∗ ∥T0 ,η .
(3.23)
Taking supremum on t over [−h, T0 ], we obtain
M1+η−β Γ (1 − (η − β)) T0 q(β−η) LG ∥(Qn y∗ ) − (Qn y∗∗ )∥T0 ,η ≤ ∥Bη−β ∥LG + Γ (1 + q(β − η)) (β − η) Mη Γ (2 − η) T0 q(1−η) + f × ∥y∗ − y∗∗ ∥T0 ,η . R (T ) Γ (1 + q(1 − η)) (1 − η)
(3.24)
Therefore, by the inequality (3.6), we conclude that the map Qn is a strict contraction map and has a unique fixed point η yn ∈ BR (CT0 , χ (0)) i.e., Qn yn = yn and yn satisfies the approximate integral equation (3.9) on [−h, T0 ]. Lemma 3.2. Assume that hypotheses (A1)–(A4) are fulfilled. If χ (0) ∈ D(Bη ) for all 0 < η < β < 1, then yn (t ) ∈ D(Bυ ) for all t ∈ [−h, T0 ] with 0 ≤ υ ≤ β < 1. η
Proof. If t ∈ [−h, 0], then it is obvious. For t ∈ (0, T0 ], by Theorem 3.1, we obtain that there is a unique yn ∈ BR (CT0 , χ (0)) such that yn satisfies Eq. (3.9). Theorem 2.6.13 in Pazy [29] gives that T (t ) : H → D(Bυ ) for t > 0 and 0 ≤ υ < 1 and for 0 ≤ υ ≤ ϑ < 1, D(Bϑ ) ⊆ D(Bυ ). By the hypotheses (A4), we have that the mapping t → Bβ Gn (t , (yn )(t ), χ (t − h1 )) is Hölder continuous on [0, T0 ] with the exponent ϱ1 = min{υ, µ}. Thus, it is not difficult to see that Hölder continuity of yn might be made using the similar arguments from Eqs. (3.10)–(3.17). By the Theorem 4.3.2 in [29], we have t
(t − s)η−1 Tq (t − s)Bη Gn (s, (yn )(s), χ (s − h1 ))ds ∈ D(B).
0
Additionally, from Theorem 1.2.4 in Pazy [29], we have that T (t )y ∈ D(B) if y ∈ D(B). The result follows from these facts and the fact that D(B) ⊆ D(Bυ ) for 0 ≤ υ ≤ 1. This finishes the proof of lemma. Corollary 3.1. Suppose that the hypotheses (A1)–(A4) are satisfied. If χ (0) ∈ D(Bη ) for 0 < η < β < 1, then there exists a constant Ut0 such that
∥Bυ yn (t )∥ ≤ Ut0 ,
n = 1, 2, 3, . . . ,
for all t0 ≤ t ≤ T0 independent of n, where 0 ≤ υ < β < 1 and for any t0 ∈ [−h, 0] there exists a constant U independent of n and t0 such that
∥Bυ yn (t )∥ ≤ U ,
t0 ≤ t ≤ 0, 0 ≤ υ < β < 1.
Proof. Let χ (0) ∈ D(Bη ). Applying Bυ on the both the sides of (3.9) and for t0 ≤ t ≤ 0, −h ≤ t0 < 0, we get
∥Bυ yn (t )∥ = ∥χ (t )∥υ ≤ ∥χ∥0,υ = U . For t0 ∈ (0, T0 ]
∥Bυ yn (t )∥ ≤ ∥Bυ Sq (t )(χ (0) + Gn (0, χ (0), χ (−h1 )))∥ + ∥Bυ−β ∥ ∥Bβ Gn (t , (yn )(t ), χ (t − h1 ))∥ t + (t − s)q−1 ∥B1+υ−β Tq (t − s)∥ ∥Bβ Gn (s, (yn )(s), χ (s − h1 ))∥ds 0 t + (t − s)q−1 ∥Bυ Tq (t − s)∥ ∥Fn (s, (yn )(s), χ (s − h2 ))∥ds, 0
(3.25)
A. Chadha, D.N. Pandey / Computers and Mathematics with Applications 69 (2015) 893–908
901
≤ M ∥χ (0) + Gn (0, χ (0), χ (−h1 ))∥υ + ∥Bυ−β ∥(LG R + A1 ) +
M1+υ−β Γ (1 − (υ − β))
Γ (1 + q(β − υ))
q(β−υ)
q(1−υ)
T Mυ Γ (2 − υ) T R + A1 ) 0 (LG + f(T ) 0 ≤ U t0 . (β − υ) Γ (1 + q(1 − υ)) R (1 − υ)
Thus, we obtain the required result for t ∈ [−h, T0 ]. This finish the proof of the theorem.
(3.26)
4. Convergence of solutions η
η
The convergence of the solution yn ∈ CT0 of the approximate integral equations (3.9) to a unique solution y(·) ∈ CT0 of Eq. (2.16) on [−h, T0 ] is discussed in this section. Theorem 4.1. Suppose that (A1)–(A4) are satisfied. If χ (0) ∈ D(Bη ), then lim
sup
p→∞ {n≥p,t ≤t ≤T } 0 0
∥yn (t ) − yp (t )∥η = 0,
(4.1)
for all t0 ∈ (0, T0 ]. Proof. Let n ≥ p ≥ n0 , where n0 is large enough and n, p, n0 ∈ N. For −h ≤ t0 ≤ 0, we have
∥Bη [yn (t ) − yp (t )]∥ = ∥χ (t ) − χ (t )∥η = 0.
(4.2)
For t0 ∈ (0, T0 ] and 0 < η < υ < β < 1,
∥Fn (t , (yn )(t ), χ (t − h2 )) − Fp (t , (yp )(t ), χ (t − h2 ))∥ ≤ ∥Fn (t , (yn )(t ), χ (t − h2 )) − Fn (t , (yp )(t ), χ (t − h2 ))∥ + ∥Fn (t , (yp )(t ), χ (t − h2 )) − Fp (t , (yp )(t ), χ (t − h2 ))∥, ≤ fR (T )[∥(yn )(t ) − (yp )(t )∥η + ∥(P n − P p )(yp )(t )∥η + ∥(P n − P p )χ (t − h2 )∥η ].
(4.3)
Also,
∥(P n − P p )(ym )(t )∥η ≤ ∥Bη−υ (P n − P p )Bυ (yp )(t )∥ ≤
1
υ−η ∥(yp )(t )∥υ ,
(4.4)
λp
and
∥(P n − P p )χ (t − h2 )∥η ≤
1
υ−η ∥χ (t
λp
− h2 )∥υ .
(4.5)
Thus
∥Fn (t , (yn )(t ), χ (t − h2 )) − Fp (t , (yp )(t ), χ (t − h2 ))∥ ≤ fR (T ) ∥(yn )(t ) − (yp )(t )∥η +
1
υ−η ∥(yp )(t )∥υ
λp
+
1
υ−η ∥χ (t
λp
− h2 )∥υ .
(4.6)
Similarly,
∥Bβ Gn (t , yn (t ), χ (t − h1 )) − Bβ Gp (t , yp (t ), χ (t − h1 ))∥ ≤ LG ∥(yn )(t ) − (yp )(t )∥η +
1
υ−η ∥(yp )(t )∥υ
λp
+
1
υ−η ∥χ (t
λp
− h1 )∥υ .
We choose t0′ such that 0 < t0′ < t0 < T , we have
∥yn (t ) − yp (t )∥η ≤ ∥Sq (t )Bη (Gn (0, χ (0), χ (−h1 )) − Gp (0, χ (0), χ (−h1 )))∥ + ∥Bη−β ∥ ∥Bβ Gn (t , yn (t ), χ (t − h1 )) − Bβ Gp (t , yp (t ), χ (t − h1 ))∥ ′ t0 t + + (t − s)q−1 0
t0′
× ∥B1+η−β Tq (t − s)∥[Bβ Gn (s, yn (s), χ (s − h1 )) − Bβ Gp (s, yp (s), χ (s − h1 ))]ds
(4.7)
902
A. Chadha, D.N. Pandey / Computers and Mathematics with Applications 69 (2015) 893–908
t
t0′
+
+
t0′
0
(t − s)q−1 ∥Bη Tq (t − s)∥[∥Fn (s, (yn )(s), χ (s − h2 ))
− Fp (s, (yp )(s), χ (s − h2 ))∥]ds.
(4.8)
The first term of above inequality can be evaluated as
∥Sq (t )Bη (Gn (0, χ (0), χ (−h1 )) − Gp (0, χ (0), χ (−h1 )))∥ ≤ M ∥Bη−β ∥ ∥Bβ Gn (0, χ (0), χ (−h1 )) − Bβ Gp (0, χ (0), χ (−h1 ))∥, ≤ M ∥Bη−β ∥LG [∥(P n − P p )χ (0)∥η + ∥(P n − P p )χ (−h1 )∥η ].
(4.9)
First and third integral of the inequality (4.8) can be evaluated as t0′
(t − s)q−1 ∥B1+η−β Tq (t − s)[Bβ Gn (s, yn (s), χ (s − h1 )) − Bβ Gp (s, yp (s), χ (s − h1 ))]∥ds
0
≤
2qM1+η−β Γ (1 − (η − β))
Γ (1 + q(β − η)) t0′
×
(LG R + A1 )(T0 − t0′ )q(β−η)−1 t0′ ,
(t − s)q−1 ∥Bη Tq (t − s)∥[∥Fn (s, (yn )(s), χ (s − h2 )) − Fp (s, (yp )(s), χ (s − h2 ))∥]ds
0
≤
2qMη Γ (2 − η)
Γ (1 + q(1 − η))
′ q(1−η)−1 ′ f t0 . R (T )(T0 − t0 )
(4.10)
We estimate the second and fourth integral of the inequality (4.8) as
t t0′
(t − s)q−1 ∥B1+η−β Tq (t − s)∥[∥Bβ Gn (s, yn (s), χ(s − h1 )) − Bβ Gp (s, yp (s), χ (s − h1 ))∥]ds
≤
qM1+η−β Γ (1 − (η − β))
Γ (1 + q(β − η))
+
≤
1
υ−η ∥χ (s
λp
t t0′
Γ (1 + q(1 − η))
≤
1
υ−η ∥χ (s
λp
∥yn (s) − yp (s)∥η +
1
υ−η ∥yp (s)∥υ
λp
LG
q(β−η)
(Ut0′ + U )T0
+
υ−η
q(β − η)λp
t
t0′
( t − s)
q(β−η)−1
∥yn − yp ∥s,η ds
(t − s)q−1 ∥Bη Tq (t − s)∥[∥Fn (s, yn (s), χ (s − h2 )) − Fp (s, yp (s), χ (s − h2 ))∥]ds
qMη Γ (2 − η)
+
( t − s)
− h1 )∥υ ds,
Γ (1 + q(β − η))
t′
q(β−η)−1
0
qM1+η−β Γ (1 − (η − β))
×
≤
LG
t
f R (T )
t
t0′
(t − s)
q(1−η)−1
∥(yn )(s) − (yp )(s)∥η +
1
υ−η ∥(yp )(s)∥υ
λp
− h2 )∥υ ds,
qMη Γ (2 − η)
Γ (1 + q(1 − η))
f R (T )
q(1−η)
(Ut0′ + U )T0
υ−η
q(1 − η)λp
t
+ t0′
(t − s)
q(1−η)−1
∥yn − yp ∥s,η ds .
(4.11)
Thus,
∥yn (t ) − yp (t )∥η ≤ M ∥B
η−β
η−β
∥LG ∥(P − P ) χ ∥0,η + ∥B n
p
∥LG ∥yn − yp ∥t ,η +
(Ut0′ + U ) υ−η
λp
qM1+η−β Γ (1 − (η − β)) (LG R + A1 ) (T0 − t0′ )−q(β−η)+1 Γ (1 + q(β − η)) (Ut0′ + U ) qMη Γ (2 − η) ′ f + R (T ) t0 + Kη,β υ−η ′ −q(1−η)+1 (T0 − t0 ) Γ (1 + q(1 − η)) λp
+2
A. Chadha, D.N. Pandey / Computers and Mathematics with Applications 69 (2015) 893–908
t +
qM1+η−β Γ (1 − (η − β))
Γ (1 + q(β − η))
t0′
+
qMη Γ (2 − η)
903
LG (t − s)q(β−η)−1
Γ (1 + q(1 − η))
q(1−η)−1 f ∥yn − yp ∥s,η ds, R (T )(t − s)
(4.12)
where Kη,β =
qM1+η−β Γ (1 − (η − β))
Γ (1 + q(β − η))
q(β−η)
LG
T0
q(β − η)
+
qMη Γ (2 − η)
Γ (1 + q(1 − η))
q(1−η)
f R (T )
T0
q(1 − η)
.
Since we have LG ∥Bη−β ∥ < 1. Moreover, we have
∥yn (t ) − yp (t )∥η ≤
1
(1 − LG ∥Bη−β ∥)
M (∥(P n − P p )Bη χ (0)∥ + ∥(P n − P p )χ (−h1 )∥)
(Ut0′ + U )
qM1+η−β Γ (1 − (η − β)) − t0′ )−q(β−η)+1 Γ (1 + q(β − η)) ( T λp 0 qMη Γ (2 − η) f(T ) t0′ × (LG R + A1 ) + (T0 − t0′ )−q(1−η)+1 Γ (1 + q(1 − η)) R t qM1+η−β Γ (1 − (η − β)) + LG (t − s)q(β−η)−1 ′ Γ (1 + q(β − η)) t0 qMη Γ (2 − η) q(1−η)−1 + f ( T )( t − s ) ∥ y − y ∥ ds . n p s,η Γ (1 + q(1 − η)) R η−β
+ (∥B
∥LG + Kη,β )
υ−η
+2
(4.13)
We apply the Gronwall’s inequality and obtain
∥yn (t ) − yp (t )∥η ≤
1
(1 − LG
∥Bη−β ∥)
M (∥(P n − P p )Bη χ (0)∥ + ∥(P n − P p )χ (−h1 )∥)
+ (∥Bη−β ∥LG + Kη,β )
(Ut0′ + U ) υ−η
+2
(Ut0′ + U ) υ−η
λp
qM1+η−β Γ (1 − (η − β))
(T0 − t0′ )−q(β−η)+1 Γ (1 + q(β − η)) qMη Γ (2 − η) × (LG R + A1 ) + f ( T ) t0′ (T0 − t0′ )−q(1−η)+1 Γ (1 + q(1 − η)) R Kη,β . × exp (1 − LG ∥Bη−β ∥) λp
(4.14)
We take the supremum over [t0 , T0 ] and p → ∞, we get lim
sup
p→∞ {t ∈[t ,T ]} 0 0
≤
∥yn (t ) − yp (t )∥η
qM1+η−β Γ (1 − (η − β)) (LG R + A1 ) (1 − LG ∥Bη−β ∥) (T0 − t0′ )−q(β−η)+1 Γ (1 + q(β − η)) qMη Γ (2 − η) Kη,β + f ( T ) × exp t0′ . R (T0 − t0′ )−q(1−η)+1 Γ (1 + q(1 − η)) (1 − LG ∥Bη−β ∥)
2
(4.15)
As t0′ is arbitrary, in this manner the right hand side may be made as small as desired by taking t0′ sufficiently small. This finishes the proof of the theorem. Theorem 4.2. Suppose that hypothesis (A1)–(A4) are satisfied and χ (0) ∈ D(Bη ). Then, there exists a unique yn ∈ C ([−h, T0 ]; Hη ), satisfying
χ (t ), t ∈ [−h, 0], Sq (t )[χ (0) + Gn (0, χ (0), χ (−h1 ))] − Gn (t , yn (t ), χ (t − h1 )) t + (t − s)q−1 BTq (t − s)Gn (s, yn (s), χ (s − h1 ))ds yn (t ) = 0 t + (t − s)q−1 Tq (t − s)Fn (s, yn (s), χ (s − h2 ))ds, t ∈ [0, T0 ], 0
(4.16)
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A. Chadha, D.N. Pandey / Computers and Mathematics with Applications 69 (2015) 893–908
and y ∈ C ([−h, T0 ]; Hη ), satisfying
χ (t ), t ∈ [−h, 0], Sq (t )[χ (0) + G(0, χ (0), χ (−h1 ))] − G(t , y(t ), χ (t − h1 )) t (t − s)q−1 BTq (t − s)G(s, y(s), χ (s − h1 ))ds + y(t ) = 0 t (t − s)q−1 Tq (t − s)F (s, y(s), χ (s − h2 ))ds, t ∈ [0, T0 ], +
(4.17)
0
such that yn converges to y in C ([−h, T0 ]; Hη ) i.e., yn → y as n → ∞. Proof. Let χ (0) ∈ D(Bη ). Since yn (t ) = y(t ) = χ (t ) for all t ∈ [−h, 0] and Bη yn (t ) converges to Bη y(t ) as n → ∞ for all η t ∈ (0, T0 ]. Thus, Bη yn (t ) converges to Bη y(t ) in H as n → ∞. Since yn ∈ BR (CT0 , χ (0)), therefore we have that y ∈ BR η
(CT0 , χ (0)) for any t0 ∈ (0, T0 ], lim
sup
n→∞ {t ∈[t ,T ]} 0 0
∥yn (t ) − y(t )∥η = 0.
(4.18)
We also have sup ∥Fn (t , yn , χ (t − h2 )) − F (t , y(t ), χ (t − h2 ))∥
t ∈[t0 ,T0 ]
≤ fR (T )[∥yn − y∥T0 ,η + ∥(P n − I )y∥T0 + ∥(P n − I )χ (t − h2 )∥T0 ] → 0,
(4.19)
and sup ∥Bβ Gn (t , yn , χ (t − h1 )) − Bβ G(t , y(t ), χ (t − h1 ))∥
t ∈[t0 ,T0 ]
≤ LG [∥yn − y∥T0 ,η + ∥(P n − I )y∥T0 + ∥(P n − I )χ (t − h1 )∥T0 ] → 0,
(4.20)
as n → ∞. Thus, for t0 ∈ (0, T0 ), the integral equation (3.9) can be written as yn (t ) = Sq (t )[χ (0) + Gn (0, χ (0), χ (−h1 ))] − Gn (t , yn (t ), χ (t − h1 )) t0
+
t +
0 t0
(t − s)q−1 Tq (t − s)BGn (s, yn (s), χ (s − h1 ))ds
t0
+
t +
0
(t − s)q−1 Tq (t − s)Fn (s, yn (s), χ (s − h2 ))ds.
(4.21)
t0
We can evolute the first and third integral as
t0
BTq (t − s)Gn (s, yn , χ (s − h1 ))ds
( t − s) t0 ≤ (t − s)q−1 ∥B1−β Tq (t − s)Bβ Gn (s, yn , χ (s − h1 ))∥ds q−1
0
0
qM1−β Γ (1 + β)
≤
Γ (1 + qβ)
t0 0
qβ−1 (LG R + A1 )T0 t0 ,
M q−1 (t − s)q−1 Tq (t − s)Fn (s, (yn ), χ (s − h2 ))ds ≤ Γ (q) fR (T )T0 t0 .
(4.22)
Therefore, we have that
t yn (t ) − Sq (t )[χ (0) + Gn (0, χ (0), χ (−h1 ))] + Gn (t , yn (t ), χ (t − h1 )) − (t − s)q−1 BTq (t − s) t0 t q −1 × Gn (s, yn (s), χ (s − h1 ))ds − (t − s) Tq (t − s)Fn (s, yn (s), χ (s − h2 ))ds t0
≤
qM1−β Γ (1 + β)
Γ (1 + qβ)
M qβ−1 q −1 (LG R + A1 )T0 + f ( T ) T t0 . 0 Γ (q) R
(4.23)
A. Chadha, D.N. Pandey / Computers and Mathematics with Applications 69 (2015) 893–908
905
Letting n → ∞ in the inequality (4.23), we obtain
t y(t ) − Sq (t )[χ (0) + G(0, χ (0), χ (−h1 ))] + G(t , y(t ), χ (t − h1 )) − (t − s)q−1 BTq (t − s) t0 t (t − s)q−1 Tq (t − s)F (s, y(s), χ (s − h2 ))ds × G(s, y(s), χ (s − h1 ))ds − t0
≤
qM1−β Γ (1 + β)
Γ (1 + qβ)
M qβ−1 q −1 (LG R + A1 )T0 + f(T )T0 t0 . Γ (q) R
(4.24)
Since t0 ∈ (0, T0 ] is arbitrary, therefore, we conclude that y is the solution of the integral equation (2.16). η Next, we show the uniqueness of the solutions to (2.16). Let y1 , y2 ∈ BR (CT0 , χ (0)) be two solutions to (2.16) on the interval [−h, T0 ]. For t ∈ [−h, 0], it is obvious. Let t ∈ (0, T0 ]. Thus, we have
∥y1 (t ) − y2 (t )∥η ≤ ∥Bη−β ∥ ∥Bβ G(t , y1 (t ), χ (t − h1 )) − Bβ G(t , y2 (t ), χ (t − h1 ))∥ t + (t − s)q−1 ∥B1+η−β Tq (t − s)[Bβ G(s, y1 (s), χ (s − h1 )) − Bβ G(s, y2 (s), χ (s − h1 ))]∥ds 0 t + (t − s)q−1 ∥Bη Tq (t − s)[F (s, y1 (s), χ (s − h2 )) − F (s, y2 (s), χ (s − h2 ))]∥ds, 0 qM1+η−β Γ (1 − (η − β)) t (t − s)q(β−η)−1 ≤ ∥Bη−β ∥LG ∥y1 (t ) − y2 (t )∥η + Γ (1 + q(β − η)) 0 t qMη Γ (2 − η) × ∥y1 (s) − y2 (s)∥η ds + (t − s)q(1−η)−1 ∥y1 (s) − y2 (s)∥η ds. Γ (1 + q(1 − η)) 0
(4.25)
We use the Gronwall inequality and get
∥y1 (t ) − y2 (t )∥α = 0,
for all t ∈ [0, T0 ].
(4.26)
By the fact 1
∥y1 (t ) − y2 (t )∥ ≤
η ∥y1 (t )
λ0
− y2 (t )∥η ,
hence y1 = y2 on [0, T0 ]. This finish the proof of the theorem.
5. Faedo–Galerkin approximations In this section, we study the Faedo–Galerkin approximation of a solution and show the convergence results for such an approximation. η We know that for any 0 < T0 < T , we have a unique y ∈ CT0 satisfying the following integral equation
χ (t ), t ∈ [−h, 0], Sq (t )[χ (0) + G(0, χ (0), χ (−h1 ))] − G(t , y(t ), y(t − h1 )) t + (t − s)q−1 Tq (t − s)BG(s, y(s), y(s − h1 ))ds y(t ) = 0 t + (t − s)q−1 Tq (t − s)F (s, y(s), y(s − h2 ))ds, t ∈ [0, T0 ].
(5.1)
0
η
Also, we have a unique solution yn ∈ CT0 of the approximate integral equation
χ (t ), t ∈ [−h, 0], Sq (t )[χ (0) + Gn (0, χ (0), χ (−h1 ))] − Gn (t , yn (t ), χ (t − h1 )) t + (t − s)q−1 Tq (t − s)BGn (s, yn (s), χ (s − h1 ))ds yn (t ) = 0 t + (t − s)q−1 Tq (t − s)Fn (s, yn (s), χ (s − h2 ))ds, t ∈ [0, T0 ]. 0
(5.2)
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Applying the projection on above equation, then Faedo–Galerkin approximation is given by vn (t ) = P n yn (t ) satisfying
n P χ (t ), t ∈ [−h, 0], Sq (t )[P n χ (0) + P n G(0, P n χ (0), P n χ (−h1 ))] − P n G(t , P n (yn ), P n χ (t − h1 )) t n (t − s)q−1 BTq (t − s)P n G(s, P n (yn ), P n χ (s − h1 ))ds + P yn (t ) = vn (t ) = 0 t (t − s)q−1 Tq (t − s)P n F (s, P n (yn ), P n χ (s − h2 ))ds, t ∈ [0, T0 ]. +
(5.3)
0
Let solution y(·) of (5.1) and vn (·) of (5.3), have the following representation y(t ) =
∞
ηi (t )φi ,
ηi (t ) = (y(t ), φi ), i = 0, 1, 2 · · · ,
(5.4)
i=0
vn ( t ) =
n
ηin (t )φi ,
ηin (t ) = (vn (t ), φi ), i = 0, 1, 2, . . . ,
(5.5)
i=0
respectively. For the convergence of ηin to ηi , we have the following convergence theorem. Theorem 5.1. Let us assume that (A1)–(A4) are satisfied and χ (0) ∈ D(Bη ). Then there exist a unique function vn ∈ C ([−h, T0 ]; H ) given as
n P χ (t ), t ∈ [−h, 0], Sq (t )[P n χ (0) + P n G(0, P n χ (0), P n χ (−h1 ))] − P n G(t , vn , P n χ (t − h1 )) t + (t − s)q−1 BTq (t − s)P n G(s, vn , P n χ (s − h1 ))ds vn ( t ) = 0 t + (t − s)q−1 Tq (t − s)P n F (s, vn , P n χ (s − h2 ))ds, t ∈ [0, T0 ],
(5.6)
0
and y ∈ C ([−h, T0 ]; H ) satisfying
χ (t ), t ∈ [−h, 0], Sq (t )[χ (0) + G(0, χ (0), y(−h1 ))] − G(t , y(t ), y(t − h1 )) t + (t − s)q−1 Tq (t − s)BG(s, y(s), y(s − h1 ))ds y(t ) = 0 t + (t − s)q−1 Tq (t − s)F (s, y(s), y(s − h2 ))ds, t ∈ [0, T0 ],
(5.7)
0
such that vn → y as n → ∞ in C ([−h, T0 ]; H ) and y satisfies Eq. (2.16) on [0, T0 ]. To determine the ηin ’s, it can easily be investigated that
η
η
B [y(t ) − vn (t )] = B
n ∞ n (ηi (t ) − ηi (t ))φi + Bη ηi (t )φi , i=0
=
n
i=n+1
∞
η
λi (ηi (t ) − ηin )φi +
i =0
η
λi ηi (t )φi .
(5.8)
i=n+1
Thus, we conclude that
∥Bη [y(t ) − vn (t )]∥2 ≥
n
2η
λi ∥(ηi (t ) − ηin (t ))∥2 .
(5.9)
i =0
Theorem 5.2. Let us assume that (A1)–(A4) are satisfied and χ (0) ∈ D(Bη ). Then, we have
lim
sup
n→∞ t ∈[t ,T ] 0 0
n
2η i
λ (ηi (t ) − η (t )) n i
2
= 0.
i=0
The statement of this hypothesis takes after from the facts specified above and the following results.
(5.10)
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907
Corollary 5.1. Assume that (A1)–(A4) are satisfied. If χ (0) ∈ D(Bη ), then sup
n≥p, t ∈[t0 ,T0 ]
∥vn (t ) − vp (t )∥η → 0,
as p → ∞.
(5.11)
Proof. For n ≥ p and 0 ≤ η < υ , we get
∥vn (t ) − vp (t )∥η = ∥P n yn (t ) − P p yn (t )∥η , ≤ ∥P n [yn (t ) − yp (t )]∥η + ∥(P n − P p )yp (t )∥η , ≤ ∥yn (t ) − yp (t )∥η +
1 υ υ−η ∥B yp (t )∥.
(5.12)
λp
Then, the result follows from Theorem 4.1. This gives the complete proof the theorem.
6. Application To generalization of theory, we present an example in this section. Let Ω be a bounded domain in the RN with the sufficiently smooth boundary ∂ Ω and ∆ be N-dimensional Laplacian. Consider the fractional differential equation with nonlocal conditions of the form
∂q [b(t )w(x, t ) − 1w(x, t ) + aw(x, t − h1 )] ∂tq = −∆2 w(x, t ) + F (x, t , w(x, t ), w(x, t − h2 )), w(x, t ) = 0, x ∈ ∂ Ω , t ≥ −h, g (w(x, t )) = ψ(x), x ∈ Ω , t ∈ [−h, 0],
t > 0, x ∈ Ω ,
(6.1) (6.2) (6.3)
where h = min{h1 , h2 }, h1 , h2 > 0, 0 < q < 1, b is a Lipschitz continuous function on [0, ∞), a is a positive constant and function F is sufficiently smooth in all its arguments. We consider H = L2 (Ω ) and the operator E : D(E ) ⊂ H → H defined by Ey = −1y with domain D(E ) = H01 (Ω ) ∩ H 2 (Ω ).
(6.4)
For nonlocal function g, we have following examples: η
0
0
k(s)ds ̸= 0. k(s)ϕ(s)(x)ds for x ∈ Ω and ϕ ∈ C0 , where k ∈ L1 (−h, 0) such that ζ := −h (1) g (ϕ)(x) = −h n n η (2) g (ϕ)(x) = w ϕ(θ )( x ) for some x ∈ Ω and ϕ ∈ C , where − h ≤ θ < θ < · · · < θ ≤ 0 and W = i i 1 2 n 0 i=1 wi ̸= 0. ζi in=1 η (3) g (ϕ)(x) = (w /ε ) ϕ( s )( x ) ds for x ∈ Ω and ϕ ∈ C , where ζ and w are same as in (2) and ε > 0 for each i i i i i 0 i=0 ζi −εi i = 1, 2, . . . , n. Thus, Eq. (6.1) can be reformulated as dq dt q
[b(t )y(t ) + Ey(t ) + ay(t − h1 )] = −EB2 y(t ) + F (t , y(t ), y(t − h2 )),
g (y) = ψ,
t ∈ [−h, 0].
(6.5)
Let us consider that 0 < d < 1. Then, (E + dI ) is invertible such that ∥(E + dI )−1 ∥ ≤ D . Let (E + D I ) = B. Thus, it is clear that B fulfills the hypotheses (A1). Now, we define the function G and F such that
G(t , y(t ), y(t − h1 )) = (b(t ) − d)B−1 y(t ) + aB−1 y(t − h1 ), F (t , y(t ), y(t − h2 )) = dy(t ) + d(B − dI )B−1 y(t ) + F (t , B−1 y(t ), B−1 y(t − h2 )). Now, we consider g (ϕ)(t ) = g (ϕ) for t ∈ [−h, 0]. Take ψ(t ) ≡ y0 on [−h, 0], y0 ∈ D(Bθ0 ), 0 < η < θ0 < 1. For the cases (1) for nonlocal function, it may be taken χ (t ) ≡ (1/ζ )y0 . For (2) and (3), we have χ (t ) ≡ (1/W )y0 on [−h, 0]. Thus, the nonlocal function g satisfies the hypotheses (A2). Thus, the results of the earlier sections to guarantee the existence of Faedo–Galerkin approximations and their convergence to the unique solution of (6.1) may be applied with appropriate functions G, F satisfying suitable conditions. Acknowledgments The authors would like to thank the referee for valuable comments and suggestions. The work of the first author is supported by the University Grants Commission (UGC), Government of India, New Delhi (6405-11-61) and Indian Institute of Technology, Roorkee.
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Existence and approximation of solution to neutral fractional differential equation with nonlocal conditions Alka Chadha, D.N. Pandey ∗ Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, India
article
info
Article history: Received 6 August 2014 Received in revised form 16 January 2015 Accepted 3 February 2015 Available online 18 March 2015 Keywords: Analytic semigroup Banach fixed point theorem Caputo derivative Faedo–Galerkin approximations Nonlocal conditions Neutral fractional differential equation
abstract This paper is concerned with the approximation of the solution for neutral fractional differential equation with nonlocal conditions in an arbitrary separable Hilbert space H. We study an associated integral equation and then, consider a sequence of approximate integral equations obtained by the projection of considered associated nonlocal neutral fractional integral equation onto finite dimensional space. The sufficient condition for the existence and uniqueness of solutions to every approximate integral equation is derived by using analytic semigroup and Banach fixed point theorem. We demonstrate convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. Moreover, we consider the Faedo–Galerkin approximations of the solution and demonstrate some convergence results. An example is also provided to illustrate the discussed abstract theory. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction In the recent years, the investigation of fractional differential equation has been picking up much attention of researchers. This is due to the fact that fractional differential equations have various applications in engineering and scientific disciplines, for example, fluid dynamics, fractal theory, diffusion in porous media, fractional biological neurons, traffic flow, polymer rheology, neural network modeling, viscoelastic panel in super sonic gas flow, real system characterized by power laws, electrodynamics of complex medium, sandwich system identification, nonlinear oscillation of earthquake, models of population growth, mathematical modeling of the diffusion of discrete particles in a turbulent fluid, nuclear reactors and theory of population dynamics. The fractional differential equation is an important tool to describe the memory and hereditary properties of various materials and phenomena. The details on the theory and its applications may be found in books [1–4] and references therein. On the other hand, the nonlocal problem for abstract evolution equations has been studied by many authors. The existence of a solution for abstract Cauchy differential equation with nonlocal conditions in a Banach space has been considered first by Byszewski [5]. In physical science, the nonlocal condition may be connected with better effect in applications than the classical initial condition since nonlocal conditions are normally more exact for physical estimations than the classical initial condition. For the study of nonlocal problems, we refer to [6–11] and references given therein. The Faedo–Galerkin approach may be used for the study of more regular solutions, imposing higher regularity on the data. Also, the Faedo–Galerkin method may be used within a variational formulation in order to provide solutions of the equations under possibly weaker assumptions on the data, and may also prove a very useful tool for numerical approximation of
∗
Corresponding author. E-mail addresses: [email protected] (A. Chadha), [email protected] (D.N. Pandey).
http://dx.doi.org/10.1016/j.camwa.2015.02.003 0898-1221/© 2015 Elsevier Ltd. All rights reserved.
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A. Chadha, D.N. Pandey / Computers and Mathematics with Applications 69 (2015) 893–908
the equations. In [12], the Faedo–Galerkin approximations of the solutions for functional Cauchy problem in a separable Hilbert space have been studied by Miletta with the help of analytic semigroup theory and Banach fixed point theorem. In [13], authors have extended the results of [12] and considered the Faedo–Galerkin approximations of the solutions to a class of functional integro-differential equation. The Faedo–Galerkin approximations of the solutions for nonlinear Sobolev type evolution equation have been studied by authors of [14]. In [8], authors have considered the Faedo–Galerkin approximations of the solutions to neutral functional differential equations with nonlocal conditions. The existence and approximations of solutions for fractional differential equation have been investigated by the author in [15]. The existence and Faedo–Galerkin approximations of the solutions for fractional integral equations has been studied by authors in [16]. In [17], authors have discussed the Faedo–Galerkin approximations of solutions to fractional differential equations with a deviating argument with the help of analytic semigroup. The present work extends the previous studies by examining the neutral fractional differential equation with nonlocal conditions and determining the mild solution for the nonlocal neutral fractional differential equation. For a nice introduction to the existence of an approximate solution and associated study of different problems, we refer to the Refs. [7,18–24,31] and references given therein. Motivated by of above mentioned work, the main objective of this work is to investigate the Faedo–Galerkin approximations of the solution to the following nonlocal neutral fractional differential equation in a separable Hilbert space (H , ∥ · ∥, (·, ·)) c
q
Dt [y(t ) + G(t , y(t ), y(t − h1 ))] = −By(t ) + F (t , y(t ), y(t − h2 )),
g (y) = ψ,
on [−h, 0], h > 0,
t ∈ [0, T ],
(1.1) (1.2)
c q Dt
where 0 < q < 1, 0 < T < ∞, is the fractional derivative in Caputo sense and h = max{h1 , h2 }, h1 , h2 > 0. In (1.1), B : H ⊃ D(B) → H is assumed to be a closed, self adjoint and positive definite linear operator with dense domain D(B) such that −B is the infinitesimal generator of an analytic semigroup of bounded linear operator on H. The functions G, F : [0, T ] × H × H → H , g : C ([−h, 0]; H ) → C ([−h, 0]; H ) are nonlinear continuous functions satisfying certain conditions to be mentioned later. Neutral differential equation arises in many areas of applied mathematics, science and engineering such as the theory of aeroelasticity [25] and lossless transmission lines [26]. The theory of heat conduction in materials and the lumped control systems can be described by neutral differential equations. The system of rigid heat conduction with finite wave spaces can be modeled in the form of the integro-differential equation of neutral type with delay. For the initial study of the neutral functional differential equations with finite delay, we refer to books [27,28] and references given therein. The article is organized as follows: Section 2 provides some basic definitions, lemmas and theorems as preliminaries as these are useful for proving our results. Section 3 derives the sufficient condition for the existence and uniqueness of the approximate solutions by using analytic semigroup and Banach fixed point theorem. Section 4 proves the convergence of the solution to each of the approximate integral equations with the limiting function which satisfies the associated integral equation and Section 5 focuses on the convergence of the approximate Faedo–Galerkin solutions. Section 6 presents an example. 2. Preliminaries and assumptions In this section, some basic definitions, preliminaries, theorems and lemmas and assumptions which will be used to prove existence result, are provided. Throughout the work, we assume that (H , ∥ · ∥, ⟨·, ·⟩) is a separable Hilbert space. The symbol C ([0, T ]; H ) stands for the Banach space of all the continuous functions from [0, T ] into H equipped with the norm ∥z ∥C = supt ∈[0,T ] ∥ z (t )∥ and Lp ((0, T ); H ) denotes the Banach space of all Bochner-measurable functions from (0, T ) to H with the norm
∥z ∥Lp =
(0,T )
∥z (s)∥p ds
1/p
,
z ∈ Lp ((0, T ); H ).
In this work, −B is assumed to be the infinitesimal generator of an analytic semigroup of bounded linear operators {T (t ) : t ≥ 0} on H. Therefore, there exist constants C ≥ 1 and δ ≥ 0 such that Ceδt ≥ ∥T (t )∥ for t ≥ 0. In addition, we note that
j d T (t ) ≤ Mj , dt j
t > t0 , t0 > 0, j = 1, 2, . . . ,
(2.1)
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(−B) i.e., −B is invertible. This permits us to define the positive fractional power Bη as closed linear operator with domain D(Bη ) ⊆ H for η ∈ (0, 1]. Moreover, D(Bη ) is dense in H with the norm
∥y∥η = ∥Bη y∥,
∀ y ∈ D(Bη ). η
(2.2)
Hence, we signify the space D(B ) by Hη endowed with the η-norm (∥ · ∥η ). Also, we have that Hκ ↩→ Hη for 0 < η < κ and therefore, the embedding is continuous. Then, we define H−η = (Hη )∗ , for each η > 0 and the dual space of Hη , is a Banach space with the norm ∥z ∥−η = ∥B−η z ∥ for z ∈ H−η . For more study on the fractional powers of closed linear operators, we refer to book by Pazy [29].
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Lemma 2.1 ([29]). Let −B be the infinitesimal generator of an analytic semigroup {T (t ) : t ≥ 0} such that ∥ T (t )∥ ≤ M, for t ≥ 0 and 0 ∈ ρ(−B). Then, (i) For 0 < η ≤ 1, Hη is a Hilbert space. (ii) The operator Bη T (t ) is bounded for every t > 0 and
∥BT (t )∥ ≤ Mt −1 ,
(2.3)
∥Bη T (t )∥ ≤ Mη t −η .
(2.4)
Now, we state some basic definitions and properties of fractional calculus. q
Definition 2.1. The Riemann–Liouville fractional integral operator Jt is given by t
1
q
Jt F (t ) =
Γ (q)
(t − s)q−1 F (s)ds,
(2.5)
0
where F ∈ L1 ((0, T ); H ) and q > 0 is the order of the fractional integration. Definition 2.2. The Riemann–Liouville fractional derivative is given by RL
δ−q
Dt F (t ) = Dδt Jt q
F (t ),
δ − 1 < q < δ, δ ∈ N,
δ
δ−q
where Dδt = dtd δ , F ∈ L1 ((0, T ); H ), Jt space defined by
F ∈ W δ,1 ((0, T ); H ). Here, the notation W δ,1 ((0, T ); H ) stands for the Sobolev
W δ,1 ((0, T ); H ) =
(2.6)
y ∈ H : ∃ z ∈ L1 ((0, T ); H ) : y(t ) =
δ−1
dk
k=0
tk k!
+
t δ−1
(δ − 1)!
∗ z (t ), t ∈ (0, T ) .
Note that z (t ) = yδ (t ), dk = yk (0). Definition 2.3. The Caputo fractional derivative is given by c
t
1
q
Dt F ( t ) =
Γ (δ − q)
(t − s)δ−q−1 F δ (s)ds,
δ − 1 < q < δ,
(2.7)
0
where F ∈ C δ−1 ((0, T ); H ) ∩ L1 ((0, T ); H ) and the following holds q
q
Jt (c Dt F (t )) = F (t ) −
δ−1 k t
k! k=0
F k (0).
(2.8)
We denote by C0 := C ([−h, 0]; H ) the Banach space of all H-valued continuous functions on [−h, 0] endowed with the supremum norm ∥z ∥0 := supt ∈[−h,0] ∥z (t )∥, for z ∈ C0 . In addition, let Ct = C ([−h, t ]; H ) be the Banach space of all H-valued continuous functions on [−h, t ] endowed with the supremum norm ∥z ∥t = sups∈[−h,t ] ∥z (s)∥, for z ∈ Ct and η t ∈ (0, T ] and the space of all continuous functions from [0, t ] into Hη denoted by Ct , is a Banach space endowed with the η norm ∥z ∥t ,η = sups∈[−h,t ] ∥z (s)∥η , for z ∈ Ct . Now, we assume the following assumptions: (A1) B is a closed, densely defined, positive definite and self-adjoint linear operator from D(B) ⊂ H into H. We assume that operator B has the pure point spectrum 0 < λ0 ≤ λ1 ≤ λ2 ≤ · · · ≤ λm ≤ · · · ,
(2.9)
with λm → ∞ as m → ∞ and a corresponding complete orthonormal system of eigenfunctions {φj }, i.e., Bφj = λj φj ,
and ⟨φl , φj ⟩ = δlj ,
(2.10)
where
δlj = η
1, 0, η
j = l, otherwise. η
(A2) g : C0 → C0 and there exists a Lipschitz continuous function χ ∈ C0 with Lipschitz constant Lχ > 0 such that η g (χ ) = ψ on [−h, 0] and χ (t ) belongs to D(Bθ0 ) for 0 < η < θ0 < 1 and ψ ∈ C0 .
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(A3) The nonlinear function F : [0, T ] × Hη × Hη → H is continuous function for η ∈ (0, 1] and there exists an increasing function fR : [0, ∞) → (0, ∞) which depends on R > 0 such that
∥F (t , w1 , z1 ) − F (t , w2 , z2 )∥ ≤ fR (t )[∥w1 − w2 ∥η + ∥z1 − z2 ∥η ],
(2.11)
∥F (t , w, z )∥ ≤ fR (t ),
(2.12)
for each (t , w, z ), (t , w1 , z1 ), (t , w2 , z2 ) ∈ [0, T ] × BR (Hη × Hη : (χ (0), χ (0))). Here, BR (Y : (y0 , y0 )) = {(y1 , y2 ) ∈ Y 2 : ∥y1 − y0 ∥Y + ∥y2 − y0 ∥Y ≤ R} for any Banach space Y . (A4) For 0 < η < β < 1, the map Bβ G : [0, T ] × Hη × Hη → H is continuous and there exists a constant LG > 0 such that 2
∥Bβ G(t , w1 , z1 ) − Bβ G(s, w2 , z2 )∥ ≤ LG [|t − s|µ + ∥w1 − w2 ∥η + ∥z1 − z2 ∥η ],
(2.13) η−β
for each (t , w1 , z1 ), (s, w2 , z2 ) ∈ [0, T ] × BR (Hη × Hη : (χ (0), χ (0))), µ ∈ (0, 1] and 2LG ∥B
∥ < 1.
In order to establish the mild solution for the problem (1.1)–(1.2), we consider the following differential equation q
Dt [y(t ) + G(t , y(t ), y(t − h1 ))] = −By(t ) + F (t , y(t ), y(t − h2 )), y(t ) = χ (t ),
t ∈ [0, T ],
(2.14)
on [−h, 0], h > 0.
(2.15)
We establish the existence, uniqueness and approximation of the mild solution of the problem (2.14)–(2.15) which is equivalent to establish the existence of the solution for the system (1.1)–(1.2). We present the definition of mild solution for the problem (2.14)–(2.15) as follows Definition 2.4. A continuous function y : [−h, T ] → Hη is said to be a mild solution for the problem (2.14)–(2.15) if y(·) satisfies the following integral equation
χ (t ), t ∈ [−h, 0], Sq (t )[χ (0) + G(0, χ (0), χ (−h1 ))] − G(t , y(t ), y(t − h1 )) t + (t − s)q−1 Tq (t − s)BG(s, y(s), y(s − h1 ))ds y(t ) = 0 t + (t − s)q−1 Tq (t − s)F (s, y(s), y(s − h2 ))ds, t ∈ [0, T ].
(2.16)
0
The operators Sq (t ) and Tq (t ) are defined as follows:
Sq (t )x =
∞
ζq (ξ )T (t q ξ )xdξ ,
(2.17)
0
Tq (t )x = q
∞
ξ ζq (ξ )T (t q ξ )xdξ ,
x ∈ H,
(2.18)
0
where ζq (ξ ) =
ψq (ξ ) =
1 q
1
ξ 1−1/q × ψq (ξ − q ) is a probability density function defined on (0, ∞) i.e., ζq (ξ ) ≥ 0,
1
∞
π
n =1
(−1)n−1 ξ −nq−1
Γ (nq + 1) sin(nπ q), n!
∞ 0
ζq (ξ )dξ = 1 and
0 < ξ < ∞.
For more details on probability density function, we refer to papers [32,34–36]. Lemma 2.2 ([30]). The operators Sq (t ), t ≥ 0 and Tq (t ), t ≥ 0 are bounded linear operators such that qMη Γ (2−η)t −qη
(i) ∥Sq (t )z ∥ ≤ M ∥z ∥, ∥ Tq (t )z ∥ ≤ Γ (1+q) ∥z ∥ and ∥Bη Tq (t )z ∥ ≤ Γ (1+q(1−η)) ∥z ∥, for any z ∈ H. (ii) The families {Sq (t ) : t ≥ 0} and {Tq (t ) : t ≥ 0} are strongly continuous. (iii) If T (t ) is compact, then Sq (t ) and Tq (t ) are compact operators for any t > 0. qM
3. Approximate solutions and convergence Let Hn ⊂ H spanned by {φ0 , φ1 , . . . , φn } be the finite dimensional subspace and P n : H → Hn be the corresponding η η projection operator for n = 0, 1, 2, . . . . We define Gn : [0, T ] × CT × C0 → H by
Gn (t , y(t ), y(t − h1 )) = G(t , P n y(t ), P n y(t − h1 )). η
(3.1)
η
Similarly, we define Fn : [0, T ] × CT × C0 → H as follows
Fn (t , y(t ), y(t − h2 )) = F (t , P n y(t ), P n y(t − h2 )), for each n = 1, 2, 3, . . . .
(3.2)
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897
We choose T0 , 0 < T0 ≤ T sufficiently small such that A1 = max ∥Bβ G(t , χ (0), χ (0))∥
(3.3)
t ∈[0,T ]
and R
∥(Sq (t ) − I )Bη (χ (0) + Gn (0, χ (0), χ (−h1 )))∥ ≤ (1 − ν) ,
(3.4)
3
µ
∥Bη−β ∥LG (T0 + Lχ T0 ) + +
Mη Γ (2 − η)
Γ (1 + q(1 − η))
M1+η−β Γ (1 − (η − β))
Γ (1 + q(β − η))
f R (T )
T0 q(1−η)
(1 − η)
T0 q(β−η)
q(β−η)
T0 R + A1 ) (LG (β − η)
R
< (1 − ν) ,
(3.5)
6
Γ (1 + q(β − η))
LG
Mη Γ (2 − η)
T0 q(1−η)
(1 − ν) < , (β − η) Γ (1 + q(1 − η)) (1 − η) 2 where ν = ∥Bη−β ∥LG < 1/2, R = 2(R2 + 3∥χ∥20,η ) and Mη , M1+η−β are constants. Let Bη : C ([−h, t ]; D(Bη )) → C ([−h, t ]; H ) be defined as (Bη ψ)(s) = Bη (ψ(s)) for all s ∈ [−h, t ], t ∈ [0, T0 ]. M1+η−β Γ (1 − (η − β))
+
f R (T )
(3.6)
Now, we consider η
η
BR = BR (CT0 , χ (0)) = {y ∈ CT0 : ∥y − χ (0)∥η ≤ R},
(3.7)
and define the operator Qn on BR as follows
χ (t ), t ∈ [−h, 0], Sq (t )[χ (0) + Gn (0, χ (0), χ (−h1 ))] − Gn (t , y(t ), χ (t − h1 )) t + (t − s)q−1 Tq (t − s)BGn (s, y(s), χ (s − h1 ))ds Qn y(t ) = 0 t + (t − s)q−1 Tq (t − s)Fn (s, y(s), χ (s − h2 ))ds, t ∈ [0, T0 ],
(3.8)
0
for each y ∈ BR . Theorem 3.1. Suppose that conditions (A1)–(A4) are fulfilled and χ (0) ∈ D(Bη ). Then, there exists a unique fixed point yn ∈ BR of the map Qn i.e., Qn yn = yn for each n = 0, 1, 2, . . . , and yn satisfies the following approximate integral equation
χ (t ), t ∈ [−h, 0], Sq (t )[χ (0) + Gn (0, χ (0), χ (−h1 ))] − Gn (t , yn (t ), χ (t − h1 )) t + (t − s)q−1 Tq (t − s)BGn (s, yn (s), χ (s − h1 ))ds yn (t ) = 0 t + (t − s)q−1 Tq (t − s)Fn (s, yn (s), χ (s − h2 ))ds, t ∈ [0, T0 ].
(3.9)
0
η
η
Proof. To demonstrate the theorem, we first need to show that Qn : BR (CT0 , χ (0)) → BR (CT0 , χ (0)). To prove this, we show that the map t → (Qn y)(t ) is continuous from [−h, T0 ] into D(Bη ) with the η-norm. For t1 , t2 ∈ [−h, 0] and η y ∈ BR (CT0 , χ (0)), we have
∥(Qn y)(t2 ) − (Qn y)(t1 )∥η = ∥χ (t2 ) − χ (t1 )∥η . η
For t1 , t2 ∈ [0, T0 ] with t1 < t2 and y ∈ BR (CT0 , χ (0)), we get Bη [(Qn y)(t2 ) − (Qn y)(t1 )]
= [Sq (t2 ) − Sq (t1 )]Bη (χ (0) + Gn (0, χ (0), χ (−h1 ))) − Bη−β [Bβ Gn (t2 , y(t2 ), χ (t2 − h1 )) − Bβ Gn (t1 , y(t1 ), χ (t1 − h1 ))] t1 + [(t2 − s)q−1 − (t1 − s)q−1 ]B1+η−β Tq (t2 − s)Bβ Gn (s, y(s), χ (s − h1 ))ds 0 t1 + (t1 − s)q−1 B1+η−β [Tq (t2 − s) − Tq (t1 − s)]Bβ Gn (s, y(s), χ (s − h1 ))ds 0
(3.10)
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A. Chadha, D.N. Pandey / Computers and Mathematics with Applications 69 (2015) 893–908 t2
+
(t2 − s)q−1 B1+η−β Tq (t2 − s)Bβ Gn (s, y(s), χ (s − h1 ))ds
t1 t1
+
[(t2 − s)q−1 − (t1 − s)q−1 ]Bη Tq (t2 − s)Fn (s, y(s), χ (s − h2 ))ds
0 t1
+
(t1 − s)q−1 Bη [Tq (t2 − s) − Tq (t1 − s)]Fn (s, y(s), χ (s − h2 ))ds
0 t2
+
(t2 − s)q−1 Bη Tq (t2 − s)Fn (s, y(s), χ (s − h2 ))ds
t1
=
8
Ji .
(3.11)
i =1
For z ∈ H, we have
[T (t2 ξ ) − T (t1 ξ )]z = q
q
t2
d dt
t1
T (t ξ )zdt = q
t2
qξ t q−1 BT (t q ξ )zdt .
t1
Then, the first term of inequality (3.11) is estimated as ∞
ζq (ξ )∥[T (t2 q ξ ) − T (t1 q ξ )]Bη (χ (0) + Gn (0, χ (0), χ (−h1 )))∥dξ , ∞ t2 ≤ ζq (ξ ) qξ t q−1 ∥Bη T (t q ξ )∥dt ∥B(χ (0) + Gn (0, χ (0), χ (−h1 )))∥dξ ,
∥J 1 ∥ ≤
0
0
t1
∞
ξ 1−η ζq (ξ )
≤ qMη 0
t2
t q(1−η)−1 ∥B(χ (0) + Gn (0, χ (0), χ (−h1 )))∥dtdξ ,
t1
Γ (1 − q) q(1−η) q(1−η) Mη ∥B(χ (0) + Gn (0, χ (0), χ (−h1 )))∥(t2 − t1 ), Γ (1 + q(1 − η)) qMη Γ (2 − η) ≤ ∥B(χ (0) + Gn (0, χ (0), χ (−h1 )))∥[t1 + Φ (t2 − t1 )]q(1−η)−1 (t2 − t1 ), Γ (1 + q(1 − η))
≤
≤
qMη ∥B(χ (0) + Gn (0, χ (0), χ (−h1 )))∥Γ (2 − η)
Γ (1 + q(1 − η))
Φ q(1−η)−1 (t2 − t1 )q(1−η) ,
(3.12)
where 0 < Φ < 1 (see [16,33]). The second term of the inequality (3.11) can be estimated as
∥J2 ∥ = ∥Bη−β ∥ · ∥Bβ Gn (t2 , y(t2 ), χ (t2 − h1 )) − Bβ Gn (t1 , y(t1 ), χ (t1 − h1 ))∥ ≤ LG ∥Bη−β ∥[|t2 − t1 |µ + ∥P n y(t2 ) − P n y(t1 )∥η + ∥P n χ (t2 − h1 ) − P n χ (t1 − h1 )∥η ], ≤ LG ∥Bη−β ∥[|t2 − t1 |µ + ∥y(t2 ) − y(t1 )∥η + Lχ |t2 − t1 |]
(3.13)
and
∥Bβ Gn (t , y(t ), χ (t − h1 ))∥ ≤ ∥Bβ Gn (t , y(t ), χ (t − h1 )) − Bβ Gn (t , χ (0), χ (0))∥ + ∥Bβ Gn (t , χ (0), χ (0))∥, ≤ LG [∥P n y(s) − χ (0)∥η + ∥P n χ (t − h1 ) − χ (0)∥η ] + A1 ≤ LG R + A1 .
(3.14)
Also,
∥J 3 ∥ ≤ ≤
qM1+η−β Γ (1 − (η − β))
Γ (1 + q(β − η)) qM1+η−β Γ (1 − (η − β))
Γ (1 + q(β − η))
(LG R + A1 )
t1
(t1 − s)m1 −1 [(t2 − s)−m1 k1 − (t1 − s)−m1 k1 ]ds,
0
(LG R + A1 )k1 Ψ k1 −1 (1 − d)−m1 (1−k1 )−1 (t2 − t1 )m1 (1−k1 ) , (1−q)
where m1 = 1 − q(1 + η − β), k1 = 1−q(1+η−β) , d = (1 − ( m1 )1/k1 m1 ) and 0 < Ψ ≤ 1 [16,33]. 1
∥J 4 ∥ ≤
qM1+η−β Γ (1 − (η − β))
Γ (1 + q(β − η))
(LG R + A1 )
k
t1
0
(t1 − s)q−1
(3.15)
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899
× [(t1 − s)−q(1+η−β) − (t2 − s)−q(1+η−β) ]ds, R + A1 ) η−β qM1+η−β Γ (1 − (η − β)) (LG ≤ d (1 − d3 )−q(β−η)−1 (t2 − t1 )q(β−η) , Γ (1 + q(β − η)) 1+η−β 2 where 0 < d2 ≤ 1 and d3 = (1 − (1 + η − β/q)1/q(1+η−β) ).
∥J 5 ∥ ≤ ≤
qM1+η−β Γ (1 − (η − β))
Γ (1 + q(β − η))
(LG R + A1 )
t2
(t2 − s)q(β−η)−1 ds,
t1
M1+η−β Γ (1 − (η − β))
Γ (1 + q(β − η))
q(β−η)
(t2 − t1 ) (LG R + A1 ) (β − η)
.
(3.16)
Similarly
∥J 6 ∥ ≤ ≤
qMη Γ (2 − η)
Γ (1 + q(1 − η)) qMη Γ (2 − η)
f R (T )
t1
(t1 − s)m2 −1 [(t2 − s)−m1 k2 − (t1 − s)−m2 k2 ]ds,
0 k −1
Γ (1 + q(1 − η))
2 f R (T )k2 Ψ1
(1 − d4 )−m2 (1−k2 )−1 (t2 − t1 )m2 (1−k2 ) ,
(3.17)
where m2 = 1 − qη, k2 = 1−qη , 0 < Φ1 ≤ 1 and d4 = (1 − ( m2 )1/k2 m2 ). 2 k
1−q
∥J 7 ∥ ≤
qMη Γ (2 − η)
Γ (1 + q(1 − η))
η−1
f R (T )Φ2
(1 − d5 )−q(1−η)−1 (t2 − t1 )q(1−η) ,
(3.18)
where d5 = (1 − (η/q)1/(qη) ) and 0 < Φ2 ≤ 1.
∥J 8 ∥ ≤
Mη Γ (2 − η)
Γ (1 + q(1 − η))
f R (T )
(t2 − t1 )q(1−η) . (1 − η)
(3.19)
From the inequality (3.10)–(3.19), we deduce that the map Qn is Hölder continuous from [−h, T0 ] into D(Bη ) with the η η η η-norm. Now, we claim that Qn (BR (CT0 , χ (0))) ⊆ BR (CT0 , χ (0)). Thus, for t ∈ [−h, 0] and y ∈ BR (CT0 , χ (0)), we have
∥(Qn y)(t ) − χ (t )∥η = 0.
(3.20)
η
For t ∈ (0, T0 ] and y ∈ BR (CT0 , χ (0)),
∥(Qn y)(t ) − χ (0)∥η ≤ ∥(Sq (t ) − I )Bη (χ (0) + Gn (0, χ (0), χ (−h1 )))∥ + ∥Bη−β ∥ ∥Bβ Gn (t , y(t ), χ (t − h1 )) − Bβ Gn (0, χ (0), χ (−h1 ))∥ t + (t − s)q−1 ∥B1+η−β Tq (t − s)∥ ∥Bβ Gn (s, y(s), χ (s − h1 ))∥ds 0 t + (t − s)q−1 ∥Bη Tq (t − s)∥ ∥Fn (s, y(s), χ (s − h2 ))∥ds 0
R
µ
≤ (1 − ν) + ∥Bη−β ∥LG [T0 + ∥y(t ) − χ (0)∥η + ∥χ (t − h1 ) − χ (−h1 )∥η ] 3
q(β−η)
M1+η−β Γ (1 − (η − β))
q(1−η)
T0 Mη Γ (2 − η) T0 (LG R + A1 ) + f(T ) , Γ (1 + q(β − η)) (β − η) Γ (1 + q(1 − η)) R (1 − η) R M1+η−β Γ (1 − (η − β)) T0 q(β−η) µ ≤ (1 − ν) + ∥Bη−β ∥LG [T0 + R + Lχ T0 ] + (LG R + A1 ) 3 Γ (1 + q(β − η)) (β − η) q(1−η) Mη Γ (2 − η) T0 + f(T ) , Γ (1 + q(1 − η)) R (1 − η)
+
R
R
R
3
6
2
≤ (1 − ν) + (1 − ν) + ν
≤ R.
(3.21) η
η
Thus, we conclude that ∥Qn y − χ (0)∥T0 ,η ≤ R. Therefore, we deduce that Qn : BR (CT0 , χ (0)) → BR (CT0 , χ (0)). Next, we
will show that Qn is a strict contraction map. For y , y ∗
∥Qn y∗ (t ) − Qn y∗∗ (t )∥η = 0,
∗∗
η
∈ BR (CT0 , χ (0)) and t ∈ [−h, 0], we get that
(3.22)
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A. Chadha, D.N. Pandey / Computers and Mathematics with Applications 69 (2015) 893–908
and for t ∈ (0, T0 ],
∥(Qn y∗ )(t ) − (Qn y∗∗ )(t )∥η ≤ ∥Bη−β ∥ ∥Bβ Gn (t , y∗ (t ), χ (t − h1 )) − Bβ Gn (t , y∗∗ (t ), χ (t − h1 ))∥ t (t − s)q−1 ∥B1+η−β Tq (t − s)∥ ∥Bβ Gn (s, y∗ (s), χ (s − h1 )) + 0
− Bβ Gn (s, y∗∗ (s), χ (s − h1 ))∥ds +
t
(t − s)q−1 ∥Bη Tq (t − s)∥
0
× ∥Fn (s, y∗ (s), χ (s − h2 )) − Fn (s, y∗∗ (s), χ (s − h2 ))∥ds, M1+η−β Γ (1 − (η − β)) T0 q(β−η) Mη Γ (2 − η) T0 q(1−η) η−β ≤ ∥B ∥LG + LG + f(T ) Γ (1 + q(β − η)) (β − η) Γ (1 + q(1 − η)) R (1 − η) × ∥y∗ − y∗∗ ∥T0 ,η .
(3.23)
Taking supremum on t over [−h, T0 ], we obtain
M1+η−β Γ (1 − (η − β)) T0 q(β−η) LG ∥(Qn y∗ ) − (Qn y∗∗ )∥T0 ,η ≤ ∥Bη−β ∥LG + Γ (1 + q(β − η)) (β − η) Mη Γ (2 − η) T0 q(1−η) + f × ∥y∗ − y∗∗ ∥T0 ,η . R (T ) Γ (1 + q(1 − η)) (1 − η)
(3.24)
Therefore, by the inequality (3.6), we conclude that the map Qn is a strict contraction map and has a unique fixed point η yn ∈ BR (CT0 , χ (0)) i.e., Qn yn = yn and yn satisfies the approximate integral equation (3.9) on [−h, T0 ]. Lemma 3.2. Assume that hypotheses (A1)–(A4) are fulfilled. If χ (0) ∈ D(Bη ) for all 0 < η < β < 1, then yn (t ) ∈ D(Bυ ) for all t ∈ [−h, T0 ] with 0 ≤ υ ≤ β < 1. η
Proof. If t ∈ [−h, 0], then it is obvious. For t ∈ (0, T0 ], by Theorem 3.1, we obtain that there is a unique yn ∈ BR (CT0 , χ (0)) such that yn satisfies Eq. (3.9). Theorem 2.6.13 in Pazy [29] gives that T (t ) : H → D(Bυ ) for t > 0 and 0 ≤ υ < 1 and for 0 ≤ υ ≤ ϑ < 1, D(Bϑ ) ⊆ D(Bυ ). By the hypotheses (A4), we have that the mapping t → Bβ Gn (t , (yn )(t ), χ (t − h1 )) is Hölder continuous on [0, T0 ] with the exponent ϱ1 = min{υ, µ}. Thus, it is not difficult to see that Hölder continuity of yn might be made using the similar arguments from Eqs. (3.10)–(3.17). By the Theorem 4.3.2 in [29], we have t
(t − s)η−1 Tq (t − s)Bη Gn (s, (yn )(s), χ (s − h1 ))ds ∈ D(B).
0
Additionally, from Theorem 1.2.4 in Pazy [29], we have that T (t )y ∈ D(B) if y ∈ D(B). The result follows from these facts and the fact that D(B) ⊆ D(Bυ ) for 0 ≤ υ ≤ 1. This finishes the proof of lemma. Corollary 3.1. Suppose that the hypotheses (A1)–(A4) are satisfied. If χ (0) ∈ D(Bη ) for 0 < η < β < 1, then there exists a constant Ut0 such that
∥Bυ yn (t )∥ ≤ Ut0 ,
n = 1, 2, 3, . . . ,
for all t0 ≤ t ≤ T0 independent of n, where 0 ≤ υ < β < 1 and for any t0 ∈ [−h, 0] there exists a constant U independent of n and t0 such that
∥Bυ yn (t )∥ ≤ U ,
t0 ≤ t ≤ 0, 0 ≤ υ < β < 1.
Proof. Let χ (0) ∈ D(Bη ). Applying Bυ on the both the sides of (3.9) and for t0 ≤ t ≤ 0, −h ≤ t0 < 0, we get
∥Bυ yn (t )∥ = ∥χ (t )∥υ ≤ ∥χ∥0,υ = U . For t0 ∈ (0, T0 ]
∥Bυ yn (t )∥ ≤ ∥Bυ Sq (t )(χ (0) + Gn (0, χ (0), χ (−h1 )))∥ + ∥Bυ−β ∥ ∥Bβ Gn (t , (yn )(t ), χ (t − h1 ))∥ t + (t − s)q−1 ∥B1+υ−β Tq (t − s)∥ ∥Bβ Gn (s, (yn )(s), χ (s − h1 ))∥ds 0 t + (t − s)q−1 ∥Bυ Tq (t − s)∥ ∥Fn (s, (yn )(s), χ (s − h2 ))∥ds, 0
(3.25)
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901
≤ M ∥χ (0) + Gn (0, χ (0), χ (−h1 ))∥υ + ∥Bυ−β ∥(LG R + A1 ) +
M1+υ−β Γ (1 − (υ − β))
Γ (1 + q(β − υ))
q(β−υ)
q(1−υ)
T Mυ Γ (2 − υ) T R + A1 ) 0 (LG + f(T ) 0 ≤ U t0 . (β − υ) Γ (1 + q(1 − υ)) R (1 − υ)
Thus, we obtain the required result for t ∈ [−h, T0 ]. This finish the proof of the theorem.
(3.26)
4. Convergence of solutions η
η
The convergence of the solution yn ∈ CT0 of the approximate integral equations (3.9) to a unique solution y(·) ∈ CT0 of Eq. (2.16) on [−h, T0 ] is discussed in this section. Theorem 4.1. Suppose that (A1)–(A4) are satisfied. If χ (0) ∈ D(Bη ), then lim
sup
p→∞ {n≥p,t ≤t ≤T } 0 0
∥yn (t ) − yp (t )∥η = 0,
(4.1)
for all t0 ∈ (0, T0 ]. Proof. Let n ≥ p ≥ n0 , where n0 is large enough and n, p, n0 ∈ N. For −h ≤ t0 ≤ 0, we have
∥Bη [yn (t ) − yp (t )]∥ = ∥χ (t ) − χ (t )∥η = 0.
(4.2)
For t0 ∈ (0, T0 ] and 0 < η < υ < β < 1,
∥Fn (t , (yn )(t ), χ (t − h2 )) − Fp (t , (yp )(t ), χ (t − h2 ))∥ ≤ ∥Fn (t , (yn )(t ), χ (t − h2 )) − Fn (t , (yp )(t ), χ (t − h2 ))∥ + ∥Fn (t , (yp )(t ), χ (t − h2 )) − Fp (t , (yp )(t ), χ (t − h2 ))∥, ≤ fR (T )[∥(yn )(t ) − (yp )(t )∥η + ∥(P n − P p )(yp )(t )∥η + ∥(P n − P p )χ (t − h2 )∥η ].
(4.3)
Also,
∥(P n − P p )(ym )(t )∥η ≤ ∥Bη−υ (P n − P p )Bυ (yp )(t )∥ ≤
1
υ−η ∥(yp )(t )∥υ ,
(4.4)
λp
and
∥(P n − P p )χ (t − h2 )∥η ≤
1
υ−η ∥χ (t
λp
− h2 )∥υ .
(4.5)
Thus
∥Fn (t , (yn )(t ), χ (t − h2 )) − Fp (t , (yp )(t ), χ (t − h2 ))∥ ≤ fR (T ) ∥(yn )(t ) − (yp )(t )∥η +
1
υ−η ∥(yp )(t )∥υ
λp
+
1
υ−η ∥χ (t
λp
− h2 )∥υ .
(4.6)
Similarly,
∥Bβ Gn (t , yn (t ), χ (t − h1 )) − Bβ Gp (t , yp (t ), χ (t − h1 ))∥ ≤ LG ∥(yn )(t ) − (yp )(t )∥η +
1
υ−η ∥(yp )(t )∥υ
λp
+
1
υ−η ∥χ (t
λp
− h1 )∥υ .
We choose t0′ such that 0 < t0′ < t0 < T , we have
∥yn (t ) − yp (t )∥η ≤ ∥Sq (t )Bη (Gn (0, χ (0), χ (−h1 )) − Gp (0, χ (0), χ (−h1 )))∥ + ∥Bη−β ∥ ∥Bβ Gn (t , yn (t ), χ (t − h1 )) − Bβ Gp (t , yp (t ), χ (t − h1 ))∥ ′ t0 t + + (t − s)q−1 0
t0′
× ∥B1+η−β Tq (t − s)∥[Bβ Gn (s, yn (s), χ (s − h1 )) − Bβ Gp (s, yp (s), χ (s − h1 ))]ds
(4.7)
902
A. Chadha, D.N. Pandey / Computers and Mathematics with Applications 69 (2015) 893–908
t
t0′
+
+
t0′
0
(t − s)q−1 ∥Bη Tq (t − s)∥[∥Fn (s, (yn )(s), χ (s − h2 ))
− Fp (s, (yp )(s), χ (s − h2 ))∥]ds.
(4.8)
The first term of above inequality can be evaluated as
∥Sq (t )Bη (Gn (0, χ (0), χ (−h1 )) − Gp (0, χ (0), χ (−h1 )))∥ ≤ M ∥Bη−β ∥ ∥Bβ Gn (0, χ (0), χ (−h1 )) − Bβ Gp (0, χ (0), χ (−h1 ))∥, ≤ M ∥Bη−β ∥LG [∥(P n − P p )χ (0)∥η + ∥(P n − P p )χ (−h1 )∥η ].
(4.9)
First and third integral of the inequality (4.8) can be evaluated as t0′
(t − s)q−1 ∥B1+η−β Tq (t − s)[Bβ Gn (s, yn (s), χ (s − h1 )) − Bβ Gp (s, yp (s), χ (s − h1 ))]∥ds
0
≤
2qM1+η−β Γ (1 − (η − β))
Γ (1 + q(β − η)) t0′
×
(LG R + A1 )(T0 − t0′ )q(β−η)−1 t0′ ,
(t − s)q−1 ∥Bη Tq (t − s)∥[∥Fn (s, (yn )(s), χ (s − h2 )) − Fp (s, (yp )(s), χ (s − h2 ))∥]ds
0
≤
2qMη Γ (2 − η)
Γ (1 + q(1 − η))
′ q(1−η)−1 ′ f t0 . R (T )(T0 − t0 )
(4.10)
We estimate the second and fourth integral of the inequality (4.8) as
t t0′
(t − s)q−1 ∥B1+η−β Tq (t − s)∥[∥Bβ Gn (s, yn (s), χ(s − h1 )) − Bβ Gp (s, yp (s), χ (s − h1 ))∥]ds
≤
qM1+η−β Γ (1 − (η − β))
Γ (1 + q(β − η))
+
≤
1
υ−η ∥χ (s
λp
t t0′
Γ (1 + q(1 − η))
≤
1
υ−η ∥χ (s
λp
∥yn (s) − yp (s)∥η +
1
υ−η ∥yp (s)∥υ
λp
LG
q(β−η)
(Ut0′ + U )T0
+
υ−η
q(β − η)λp
t
t0′
( t − s)
q(β−η)−1
∥yn − yp ∥s,η ds
(t − s)q−1 ∥Bη Tq (t − s)∥[∥Fn (s, yn (s), χ (s − h2 )) − Fp (s, yp (s), χ (s − h2 ))∥]ds
qMη Γ (2 − η)
+
( t − s)
− h1 )∥υ ds,
Γ (1 + q(β − η))
t′
q(β−η)−1
0
qM1+η−β Γ (1 − (η − β))
×
≤
LG
t
f R (T )
t
t0′
(t − s)
q(1−η)−1
∥(yn )(s) − (yp )(s)∥η +
1
υ−η ∥(yp )(s)∥υ
λp
− h2 )∥υ ds,
qMη Γ (2 − η)
Γ (1 + q(1 − η))
f R (T )
q(1−η)
(Ut0′ + U )T0
υ−η
q(1 − η)λp
t
+ t0′
(t − s)
q(1−η)−1
∥yn − yp ∥s,η ds .
(4.11)
Thus,
∥yn (t ) − yp (t )∥η ≤ M ∥B
η−β
η−β
∥LG ∥(P − P ) χ ∥0,η + ∥B n
p
∥LG ∥yn − yp ∥t ,η +
(Ut0′ + U ) υ−η
λp
qM1+η−β Γ (1 − (η − β)) (LG R + A1 ) (T0 − t0′ )−q(β−η)+1 Γ (1 + q(β − η)) (Ut0′ + U ) qMη Γ (2 − η) ′ f + R (T ) t0 + Kη,β υ−η ′ −q(1−η)+1 (T0 − t0 ) Γ (1 + q(1 − η)) λp
+2
A. Chadha, D.N. Pandey / Computers and Mathematics with Applications 69 (2015) 893–908
t +
qM1+η−β Γ (1 − (η − β))
Γ (1 + q(β − η))
t0′
+
qMη Γ (2 − η)
903
LG (t − s)q(β−η)−1
Γ (1 + q(1 − η))
q(1−η)−1 f ∥yn − yp ∥s,η ds, R (T )(t − s)
(4.12)
where Kη,β =
qM1+η−β Γ (1 − (η − β))
Γ (1 + q(β − η))
q(β−η)
LG
T0
q(β − η)
+
qMη Γ (2 − η)
Γ (1 + q(1 − η))
q(1−η)
f R (T )
T0
q(1 − η)
.
Since we have LG ∥Bη−β ∥ < 1. Moreover, we have
∥yn (t ) − yp (t )∥η ≤
1
(1 − LG ∥Bη−β ∥)
M (∥(P n − P p )Bη χ (0)∥ + ∥(P n − P p )χ (−h1 )∥)
(Ut0′ + U )
qM1+η−β Γ (1 − (η − β)) − t0′ )−q(β−η)+1 Γ (1 + q(β − η)) ( T λp 0 qMη Γ (2 − η) f(T ) t0′ × (LG R + A1 ) + (T0 − t0′ )−q(1−η)+1 Γ (1 + q(1 − η)) R t qM1+η−β Γ (1 − (η − β)) + LG (t − s)q(β−η)−1 ′ Γ (1 + q(β − η)) t0 qMη Γ (2 − η) q(1−η)−1 + f ( T )( t − s ) ∥ y − y ∥ ds . n p s,η Γ (1 + q(1 − η)) R η−β
+ (∥B
∥LG + Kη,β )
υ−η
+2
(4.13)
We apply the Gronwall’s inequality and obtain
∥yn (t ) − yp (t )∥η ≤
1
(1 − LG
∥Bη−β ∥)
M (∥(P n − P p )Bη χ (0)∥ + ∥(P n − P p )χ (−h1 )∥)
+ (∥Bη−β ∥LG + Kη,β )
(Ut0′ + U ) υ−η
+2
(Ut0′ + U ) υ−η
λp
qM1+η−β Γ (1 − (η − β))
(T0 − t0′ )−q(β−η)+1 Γ (1 + q(β − η)) qMη Γ (2 − η) × (LG R + A1 ) + f ( T ) t0′ (T0 − t0′ )−q(1−η)+1 Γ (1 + q(1 − η)) R Kη,β . × exp (1 − LG ∥Bη−β ∥) λp
(4.14)
We take the supremum over [t0 , T0 ] and p → ∞, we get lim
sup
p→∞ {t ∈[t ,T ]} 0 0
≤
∥yn (t ) − yp (t )∥η
qM1+η−β Γ (1 − (η − β)) (LG R + A1 ) (1 − LG ∥Bη−β ∥) (T0 − t0′ )−q(β−η)+1 Γ (1 + q(β − η)) qMη Γ (2 − η) Kη,β + f ( T ) × exp t0′ . R (T0 − t0′ )−q(1−η)+1 Γ (1 + q(1 − η)) (1 − LG ∥Bη−β ∥)
2
(4.15)
As t0′ is arbitrary, in this manner the right hand side may be made as small as desired by taking t0′ sufficiently small. This finishes the proof of the theorem. Theorem 4.2. Suppose that hypothesis (A1)–(A4) are satisfied and χ (0) ∈ D(Bη ). Then, there exists a unique yn ∈ C ([−h, T0 ]; Hη ), satisfying
χ (t ), t ∈ [−h, 0], Sq (t )[χ (0) + Gn (0, χ (0), χ (−h1 ))] − Gn (t , yn (t ), χ (t − h1 )) t + (t − s)q−1 BTq (t − s)Gn (s, yn (s), χ (s − h1 ))ds yn (t ) = 0 t + (t − s)q−1 Tq (t − s)Fn (s, yn (s), χ (s − h2 ))ds, t ∈ [0, T0 ], 0
(4.16)
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and y ∈ C ([−h, T0 ]; Hη ), satisfying
χ (t ), t ∈ [−h, 0], Sq (t )[χ (0) + G(0, χ (0), χ (−h1 ))] − G(t , y(t ), χ (t − h1 )) t (t − s)q−1 BTq (t − s)G(s, y(s), χ (s − h1 ))ds + y(t ) = 0 t (t − s)q−1 Tq (t − s)F (s, y(s), χ (s − h2 ))ds, t ∈ [0, T0 ], +
(4.17)
0
such that yn converges to y in C ([−h, T0 ]; Hη ) i.e., yn → y as n → ∞. Proof. Let χ (0) ∈ D(Bη ). Since yn (t ) = y(t ) = χ (t ) for all t ∈ [−h, 0] and Bη yn (t ) converges to Bη y(t ) as n → ∞ for all η t ∈ (0, T0 ]. Thus, Bη yn (t ) converges to Bη y(t ) in H as n → ∞. Since yn ∈ BR (CT0 , χ (0)), therefore we have that y ∈ BR η
(CT0 , χ (0)) for any t0 ∈ (0, T0 ], lim
sup
n→∞ {t ∈[t ,T ]} 0 0
∥yn (t ) − y(t )∥η = 0.
(4.18)
We also have sup ∥Fn (t , yn , χ (t − h2 )) − F (t , y(t ), χ (t − h2 ))∥
t ∈[t0 ,T0 ]
≤ fR (T )[∥yn − y∥T0 ,η + ∥(P n − I )y∥T0 + ∥(P n − I )χ (t − h2 )∥T0 ] → 0,
(4.19)
and sup ∥Bβ Gn (t , yn , χ (t − h1 )) − Bβ G(t , y(t ), χ (t − h1 ))∥
t ∈[t0 ,T0 ]
≤ LG [∥yn − y∥T0 ,η + ∥(P n − I )y∥T0 + ∥(P n − I )χ (t − h1 )∥T0 ] → 0,
(4.20)
as n → ∞. Thus, for t0 ∈ (0, T0 ), the integral equation (3.9) can be written as yn (t ) = Sq (t )[χ (0) + Gn (0, χ (0), χ (−h1 ))] − Gn (t , yn (t ), χ (t − h1 )) t0
+
t +
0 t0
(t − s)q−1 Tq (t − s)BGn (s, yn (s), χ (s − h1 ))ds
t0
+
t +
0
(t − s)q−1 Tq (t − s)Fn (s, yn (s), χ (s − h2 ))ds.
(4.21)
t0
We can evolute the first and third integral as
t0
BTq (t − s)Gn (s, yn , χ (s − h1 ))ds
( t − s) t0 ≤ (t − s)q−1 ∥B1−β Tq (t − s)Bβ Gn (s, yn , χ (s − h1 ))∥ds q−1
0
0
qM1−β Γ (1 + β)
≤
Γ (1 + qβ)
t0 0
qβ−1 (LG R + A1 )T0 t0 ,
M q−1 (t − s)q−1 Tq (t − s)Fn (s, (yn ), χ (s − h2 ))ds ≤ Γ (q) fR (T )T0 t0 .
(4.22)
Therefore, we have that
t yn (t ) − Sq (t )[χ (0) + Gn (0, χ (0), χ (−h1 ))] + Gn (t , yn (t ), χ (t − h1 )) − (t − s)q−1 BTq (t − s) t0 t q −1 × Gn (s, yn (s), χ (s − h1 ))ds − (t − s) Tq (t − s)Fn (s, yn (s), χ (s − h2 ))ds t0
≤
qM1−β Γ (1 + β)
Γ (1 + qβ)
M qβ−1 q −1 (LG R + A1 )T0 + f ( T ) T t0 . 0 Γ (q) R
(4.23)
A. Chadha, D.N. Pandey / Computers and Mathematics with Applications 69 (2015) 893–908
905
Letting n → ∞ in the inequality (4.23), we obtain
t y(t ) − Sq (t )[χ (0) + G(0, χ (0), χ (−h1 ))] + G(t , y(t ), χ (t − h1 )) − (t − s)q−1 BTq (t − s) t0 t (t − s)q−1 Tq (t − s)F (s, y(s), χ (s − h2 ))ds × G(s, y(s), χ (s − h1 ))ds − t0
≤
qM1−β Γ (1 + β)
Γ (1 + qβ)
M qβ−1 q −1 (LG R + A1 )T0 + f(T )T0 t0 . Γ (q) R
(4.24)
Since t0 ∈ (0, T0 ] is arbitrary, therefore, we conclude that y is the solution of the integral equation (2.16). η Next, we show the uniqueness of the solutions to (2.16). Let y1 , y2 ∈ BR (CT0 , χ (0)) be two solutions to (2.16) on the interval [−h, T0 ]. For t ∈ [−h, 0], it is obvious. Let t ∈ (0, T0 ]. Thus, we have
∥y1 (t ) − y2 (t )∥η ≤ ∥Bη−β ∥ ∥Bβ G(t , y1 (t ), χ (t − h1 )) − Bβ G(t , y2 (t ), χ (t − h1 ))∥ t + (t − s)q−1 ∥B1+η−β Tq (t − s)[Bβ G(s, y1 (s), χ (s − h1 )) − Bβ G(s, y2 (s), χ (s − h1 ))]∥ds 0 t + (t − s)q−1 ∥Bη Tq (t − s)[F (s, y1 (s), χ (s − h2 )) − F (s, y2 (s), χ (s − h2 ))]∥ds, 0 qM1+η−β Γ (1 − (η − β)) t (t − s)q(β−η)−1 ≤ ∥Bη−β ∥LG ∥y1 (t ) − y2 (t )∥η + Γ (1 + q(β − η)) 0 t qMη Γ (2 − η) × ∥y1 (s) − y2 (s)∥η ds + (t − s)q(1−η)−1 ∥y1 (s) − y2 (s)∥η ds. Γ (1 + q(1 − η)) 0
(4.25)
We use the Gronwall inequality and get
∥y1 (t ) − y2 (t )∥α = 0,
for all t ∈ [0, T0 ].
(4.26)
By the fact 1
∥y1 (t ) − y2 (t )∥ ≤
η ∥y1 (t )
λ0
− y2 (t )∥η ,
hence y1 = y2 on [0, T0 ]. This finish the proof of the theorem.
5. Faedo–Galerkin approximations In this section, we study the Faedo–Galerkin approximation of a solution and show the convergence results for such an approximation. η We know that for any 0 < T0 < T , we have a unique y ∈ CT0 satisfying the following integral equation
χ (t ), t ∈ [−h, 0], Sq (t )[χ (0) + G(0, χ (0), χ (−h1 ))] − G(t , y(t ), y(t − h1 )) t + (t − s)q−1 Tq (t − s)BG(s, y(s), y(s − h1 ))ds y(t ) = 0 t + (t − s)q−1 Tq (t − s)F (s, y(s), y(s − h2 ))ds, t ∈ [0, T0 ].
(5.1)
0
η
Also, we have a unique solution yn ∈ CT0 of the approximate integral equation
χ (t ), t ∈ [−h, 0], Sq (t )[χ (0) + Gn (0, χ (0), χ (−h1 ))] − Gn (t , yn (t ), χ (t − h1 )) t + (t − s)q−1 Tq (t − s)BGn (s, yn (s), χ (s − h1 ))ds yn (t ) = 0 t + (t − s)q−1 Tq (t − s)Fn (s, yn (s), χ (s − h2 ))ds, t ∈ [0, T0 ]. 0
(5.2)
906
A. Chadha, D.N. Pandey / Computers and Mathematics with Applications 69 (2015) 893–908
Applying the projection on above equation, then Faedo–Galerkin approximation is given by vn (t ) = P n yn (t ) satisfying
n P χ (t ), t ∈ [−h, 0], Sq (t )[P n χ (0) + P n G(0, P n χ (0), P n χ (−h1 ))] − P n G(t , P n (yn ), P n χ (t − h1 )) t n (t − s)q−1 BTq (t − s)P n G(s, P n (yn ), P n χ (s − h1 ))ds + P yn (t ) = vn (t ) = 0 t (t − s)q−1 Tq (t − s)P n F (s, P n (yn ), P n χ (s − h2 ))ds, t ∈ [0, T0 ]. +
(5.3)
0
Let solution y(·) of (5.1) and vn (·) of (5.3), have the following representation y(t ) =
∞
ηi (t )φi ,
ηi (t ) = (y(t ), φi ), i = 0, 1, 2 · · · ,
(5.4)
i=0
vn ( t ) =
n
ηin (t )φi ,
ηin (t ) = (vn (t ), φi ), i = 0, 1, 2, . . . ,
(5.5)
i=0
respectively. For the convergence of ηin to ηi , we have the following convergence theorem. Theorem 5.1. Let us assume that (A1)–(A4) are satisfied and χ (0) ∈ D(Bη ). Then there exist a unique function vn ∈ C ([−h, T0 ]; H ) given as
n P χ (t ), t ∈ [−h, 0], Sq (t )[P n χ (0) + P n G(0, P n χ (0), P n χ (−h1 ))] − P n G(t , vn , P n χ (t − h1 )) t + (t − s)q−1 BTq (t − s)P n G(s, vn , P n χ (s − h1 ))ds vn ( t ) = 0 t + (t − s)q−1 Tq (t − s)P n F (s, vn , P n χ (s − h2 ))ds, t ∈ [0, T0 ],
(5.6)
0
and y ∈ C ([−h, T0 ]; H ) satisfying
χ (t ), t ∈ [−h, 0], Sq (t )[χ (0) + G(0, χ (0), y(−h1 ))] − G(t , y(t ), y(t − h1 )) t + (t − s)q−1 Tq (t − s)BG(s, y(s), y(s − h1 ))ds y(t ) = 0 t + (t − s)q−1 Tq (t − s)F (s, y(s), y(s − h2 ))ds, t ∈ [0, T0 ],
(5.7)
0
such that vn → y as n → ∞ in C ([−h, T0 ]; H ) and y satisfies Eq. (2.16) on [0, T0 ]. To determine the ηin ’s, it can easily be investigated that
η
η
B [y(t ) − vn (t )] = B
n ∞ n (ηi (t ) − ηi (t ))φi + Bη ηi (t )φi , i=0
=
n
i=n+1
∞
η
λi (ηi (t ) − ηin )φi +
i =0
η
λi ηi (t )φi .
(5.8)
i=n+1
Thus, we conclude that
∥Bη [y(t ) − vn (t )]∥2 ≥
n
2η
λi ∥(ηi (t ) − ηin (t ))∥2 .
(5.9)
i =0
Theorem 5.2. Let us assume that (A1)–(A4) are satisfied and χ (0) ∈ D(Bη ). Then, we have
lim
sup
n→∞ t ∈[t ,T ] 0 0
n
2η i
λ (ηi (t ) − η (t )) n i
2
= 0.
i=0
The statement of this hypothesis takes after from the facts specified above and the following results.
(5.10)
A. Chadha, D.N. Pandey / Computers and Mathematics with Applications 69 (2015) 893–908
907
Corollary 5.1. Assume that (A1)–(A4) are satisfied. If χ (0) ∈ D(Bη ), then sup
n≥p, t ∈[t0 ,T0 ]
∥vn (t ) − vp (t )∥η → 0,
as p → ∞.
(5.11)
Proof. For n ≥ p and 0 ≤ η < υ , we get
∥vn (t ) − vp (t )∥η = ∥P n yn (t ) − P p yn (t )∥η , ≤ ∥P n [yn (t ) − yp (t )]∥η + ∥(P n − P p )yp (t )∥η , ≤ ∥yn (t ) − yp (t )∥η +
1 υ υ−η ∥B yp (t )∥.
(5.12)
λp
Then, the result follows from Theorem 4.1. This gives the complete proof the theorem.
6. Application To generalization of theory, we present an example in this section. Let Ω be a bounded domain in the RN with the sufficiently smooth boundary ∂ Ω and ∆ be N-dimensional Laplacian. Consider the fractional differential equation with nonlocal conditions of the form
∂q [b(t )w(x, t ) − 1w(x, t ) + aw(x, t − h1 )] ∂tq = −∆2 w(x, t ) + F (x, t , w(x, t ), w(x, t − h2 )), w(x, t ) = 0, x ∈ ∂ Ω , t ≥ −h, g (w(x, t )) = ψ(x), x ∈ Ω , t ∈ [−h, 0],
t > 0, x ∈ Ω ,
(6.1) (6.2) (6.3)
where h = min{h1 , h2 }, h1 , h2 > 0, 0 < q < 1, b is a Lipschitz continuous function on [0, ∞), a is a positive constant and function F is sufficiently smooth in all its arguments. We consider H = L2 (Ω ) and the operator E : D(E ) ⊂ H → H defined by Ey = −1y with domain D(E ) = H01 (Ω ) ∩ H 2 (Ω ).
(6.4)
For nonlocal function g, we have following examples: η
0
0
k(s)ds ̸= 0. k(s)ϕ(s)(x)ds for x ∈ Ω and ϕ ∈ C0 , where k ∈ L1 (−h, 0) such that ζ := −h (1) g (ϕ)(x) = −h n n η (2) g (ϕ)(x) = w ϕ(θ )( x ) for some x ∈ Ω and ϕ ∈ C , where − h ≤ θ < θ < · · · < θ ≤ 0 and W = i i 1 2 n 0 i=1 wi ̸= 0. ζi in=1 η (3) g (ϕ)(x) = (w /ε ) ϕ( s )( x ) ds for x ∈ Ω and ϕ ∈ C , where ζ and w are same as in (2) and ε > 0 for each i i i i i 0 i=0 ζi −εi i = 1, 2, . . . , n. Thus, Eq. (6.1) can be reformulated as dq dt q
[b(t )y(t ) + Ey(t ) + ay(t − h1 )] = −EB2 y(t ) + F (t , y(t ), y(t − h2 )),
g (y) = ψ,
t ∈ [−h, 0].
(6.5)
Let us consider that 0 < d < 1. Then, (E + dI ) is invertible such that ∥(E + dI )−1 ∥ ≤ D . Let (E + D I ) = B. Thus, it is clear that B fulfills the hypotheses (A1). Now, we define the function G and F such that
G(t , y(t ), y(t − h1 )) = (b(t ) − d)B−1 y(t ) + aB−1 y(t − h1 ), F (t , y(t ), y(t − h2 )) = dy(t ) + d(B − dI )B−1 y(t ) + F (t , B−1 y(t ), B−1 y(t − h2 )). Now, we consider g (ϕ)(t ) = g (ϕ) for t ∈ [−h, 0]. Take ψ(t ) ≡ y0 on [−h, 0], y0 ∈ D(Bθ0 ), 0 < η < θ0 < 1. For the cases (1) for nonlocal function, it may be taken χ (t ) ≡ (1/ζ )y0 . For (2) and (3), we have χ (t ) ≡ (1/W )y0 on [−h, 0]. Thus, the nonlocal function g satisfies the hypotheses (A2). Thus, the results of the earlier sections to guarantee the existence of Faedo–Galerkin approximations and their convergence to the unique solution of (6.1) may be applied with appropriate functions G, F satisfying suitable conditions. Acknowledgments The authors would like to thank the referee for valuable comments and suggestions. The work of the first author is supported by the University Grants Commission (UGC), Government of India, New Delhi (6405-11-61) and Indian Institute of Technology, Roorkee.
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