Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 531894, 11 pages
http://dx.doi.org/10.1155/2013/531894
Research Article
Approximate Controllability of Fractional Neutral Evolution
Equations in Banach Spaces
N. I. Mahmudov
Eastern Mediterranean University, Gazimagusa, TRNC, 10 Mersin, Turkey
Correspondence should be addressed to N. I. Mahmudov; nazim.mahmudov@emu.edu.tr
Received 24 December 2012; Revised 13 March 2013; Accepted 13 March 2013
Academic Editor: Juan J. Nieto
Copyright © 2013 N. I. Mahmudov. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We discuss the approximate controllability of semilinear fractional neutral differential systems with infinite delay under the
assumptions that the corresponding linear system is approximately controllable. Using Krasnoselkii’s fixed-point theorem,
fractional calculus, and methods of controllability theory, a new set of sufficient conditions for approximate controllability of
fractional neutral differential equations with infinite delay are formulated and proved. The results of the paper are generalization
and continuation of the recent results on this issue.
1. Introduction and Preliminaries
Many social, physical, biological and engineering problems
can be described by fractional partial differential equations.
In fact, fractional differential equations are considered as an
alternative model to nonlinear differential equations. In the
last two decades, fractional differential equations (see, e.g.,
Samko et al. [1] and references therein) have attracted many
scientists, and notable contributions have been made to both
theory and applications of fractional differential equations.
Nowadays, controllability theory for linear systems has
already been well established, for finite and infinite dimensional systems; see, for instance, [2]. Several authors have
extended these concepts to infinite-dimensional systems
represented by nonlinear evolution equations in infinitedimensional spaces, see [3–30]. On the other hand, approximate controllability problems for fractional evolution equations in Hilbert spaces are not yet sufficiently investigated,
and there are only few works on it [13–21, 28–30]. So far,
the overwhelming majority of the approximate controllability
results have only been available for semilinear evolution
differential systems in Hilbert spaces, with the exception of
the case of [11]. Motivated by the fact that many partial
fractional differential equations can be converted into fractional PDE in some Banach spaces, we consider that there
is a realistic need to discuss the approximate controllability
problem of fractional-order differential systems in Banach
spaces. Note that our results are new even for the approximate
controllability of fractional neutral differential equations with
infinite delay in Hilbert spaces.
Consider the following fractional neutral evolution differential system with infinite delay:
𝑑𝛼
[𝑥 (𝑡) − ℎ (𝑡, 𝑥𝑡 )] = −𝐴𝑥 (𝑡) + 𝐵𝑢 (𝑡) + 𝑓 (𝑡, 𝑥𝑡 ) ,
𝑑𝑡𝛼
𝑡 ∈ [0, 𝑇] ,
(1)
𝑥 (𝑡) = 𝜙 (𝑡) ∈ 𝐵𝑙 ,
where the state 𝑥 takes values in a Banach space 𝑋 and
the control function takes values in a Hilbert space 𝑈. The
functions ℎ, 𝑓 will be specified in the sequel. Let 𝑥𝑡 (⋅) denote
𝑥𝑡 (𝜃) = 𝑥(𝑡 + 𝜃), 𝜃 ∈ (−∞, 0]. Assume that 𝑙 : (−∞, 0] →
0
(−∞, 0) is a continuous function satisfying 𝑙 = ∫−∞ 𝑙(𝑡)𝑑𝑡 <
∞. The Banach space (𝐵𝑙 , ‖ ⋅ ‖𝑙 ) induced by function ℎ is
defined as follows:
{
{
{
{
{
𝐵𝑙 := {
{
{
{
{
{
𝜑 : (−∞, 0] 󳨀→ 𝑋 : for any 𝑐 > 0, }
}
𝜑 (𝜃) is a bounded and measurable }
}
}
function on [−𝑐, 0] ,
}
0
}
}
󵄩
󵄩
}
and ∫ 𝑙 (𝑡) sup 󵄩󵄩󵄩𝜑 (𝜃)󵄩󵄩󵄩 𝑑𝑡 < ∞ }
−∞
𝑡≤𝜃≤0
}
(2)
2
Abstract and Applied Analysis
0
endowed with the norm ‖𝜑‖𝐵𝑙 := ∫−∞ ℎ(𝑡)sup𝑡≤𝜃≤0 ‖𝜑(𝜃)‖𝑑𝑡.
It should be mentioned that (approximate) controllability
results for first- and second-order partial neutral functional
differential equations with infinite delay were considered by
Sakthivel et al. [18], Chalishajar [8], and Chalishajar and
Acharya [9].
Throughout this paper, unless otherwise specified, the
following notations will be used. Let (𝑋, ‖ ⋅ ‖) be a separable
reflexive Banach space, and let (𝑋∗ , ‖ ⋅ ‖∗ ) stand for its dual
space with respect to the continuous pairing ⟨⋅, ⋅⟩. We may
assume, without loss of generality, that 𝑋 and 𝑋∗ are smooth
and strictly convex, by virtue of renorming theorem (see, e.g.,
[10]). In particular, this implies that the duality mapping 𝐽 of
𝑋 into 𝑋∗ given by the following relations:
⟨𝐽 (𝑧) , 𝑧⟩ = ‖𝑧‖2 ,
‖𝐽 (𝑧)‖∗ = ‖𝑧‖ ,
∀𝑧 ∈ 𝑋
(3)
is bijective, homogeneous, and demicontinuous, that is,
continuous from 𝑋 with a strong topology into 𝑋∗ with weak
topology and strictly monotonic. Moreover, 𝐽−1 : 𝑋∗ → 𝑋
is also duality mapping.
In this paper, we also assume that −𝐴 : 𝐷(𝐴) ⊂ 𝑋 → 𝑋
is the infinitesimal generator of a compact analytic semigroup
𝑆(𝑡), 𝑡 > 0, of uniformly bounded linear operator in 𝑋, that
is, there exists 𝑀 > 1 such that ‖𝑆(𝑡)‖𝐿(𝑋) ≤ 𝑀 for all 𝑡 ≥ 0.
Without loss of generality, let 0 ∈ 𝜌(𝐴), where 𝜌(𝐴) is the
resolvent set of 𝐴. Then, for any 𝛽 > 0, we can define 𝐴−𝛽
by
𝐴−𝛽 :=
From Lemma 1(iv), since 𝐴−𝛽 is a bounded linear operator for 0 ≤ 𝛽 ≤ 1, there exists a constant 𝐶𝛽 such that
‖𝐴−𝛽 ‖ ≤ 𝐶𝛽 for 0 ≤ 𝛽 ≤ 1.
Let us recall the following known definitions in fractional
calculus. For more details, see [1].
Definition 2. The fractional integral of order 𝛼 > 0 with the
lower limit 0 for a function 𝑓 is defined as
𝐼𝛼 𝑓 (𝑡) =
𝑡 > 0, 𝛼 > 0,
(6)
provided that the right-hand side is pointwise defined on
[0, ∞), where Γ is the gamma function.
Definition 3. Riemann-Liouville derivative of order 𝛼 with
the lower limit 0 for a function 𝑓 : [0, ∞) → 𝑅 can be written
as
𝐿
𝐷𝛼 𝑓 (𝑡) =
𝑓 (𝑠)
𝑑𝑛 𝑡
1
𝑑𝑠,
∫
Γ (𝑛 − 𝛼) 𝑑𝑡𝑛 0 (𝑡 − 𝑠)𝛼+1−𝑛
(7)
𝑡 > 0, 𝑛 − 1 < 𝛼 < 𝑛,
Definition 4. The Caputo derivative of order 𝛼 for a function
𝑓 : [0, ∞) → 𝑅 can be written as
𝑐
∞
1
∫ 𝑡𝛽−1 𝑆 (𝑡) 𝑑𝑡.
Γ (𝛽) 0
𝑡
𝑓 (𝑠)
1
𝑑𝑠,
∫
Γ (𝛼) 0 (𝑡 − 𝑠)1−𝛼
𝑛−1 𝑘
𝑡 (𝑘)
𝑓 (0)) ,
𝑘!
𝑘=0
𝐷𝛼 𝑓 (𝑡) = 𝐿 𝐷𝛼 (𝑓 (𝑡) − ∑
(4)
(8)
𝑡 > 0, 𝑛 − 1 < 𝛼 < 𝑛.
−𝛽
It follows that each 𝐴 is an injective continuous endomor−1
phism of 𝑋. Hence, we can define 𝐴𝛽 := (𝐴−𝛽 ) , which is a
closed bijective linear operator in 𝑋. It can be shown that each
𝐴𝛽 has dense domain and that 𝐷(𝐴𝛾 ) ⊂ 𝐷(𝐴𝛽 ) for 0 ≤ 𝛽 ≤ 𝛾.
Moreover, 𝐴𝛽+𝛾 𝑥 = 𝐴𝛽 𝐴𝛾 𝑥 = 𝐴𝛾 𝐴𝛽 𝑥 for every 𝛽, 𝛾 ∈ 𝑅 and
𝑥 ∈ 𝐷(𝐴𝜇 ) with 𝜇 := max(𝛽, 𝛾, 𝛽 + 𝛾), where 𝐴0 = 𝐼 and 𝐼 is
the identity in 𝑋.
We denote by 𝑋𝛽 the Banach space of 𝐷(𝐴𝛽 ) equipped
with norm ‖𝑥‖𝛽 := ‖𝐴𝛽 𝑥‖ for 𝑥 ∈ 𝐷(𝐴𝛽 ), which is equivalent
to the graph norm of 𝐴𝛽 . Then, we have 𝑋𝛾 󳨅→ 𝑋𝛽 , for 0 ≤
𝛽 ≤ 𝛾 (with 𝑋0 = 𝑋 ), and the embedding is continuous.
Moreover, 𝐴𝛽 has the following basic properties.
Lemma 1 (see [31]). 𝐴𝛽 has the following properties:
𝛽
∞
S𝛼 (𝑡) = ∫ Ψ𝛼 (𝜃) 𝑆 (𝑡𝛼 𝜃) 𝑑𝜃,
0
𝛼
A𝛼 (𝑡) = 𝛼 ∫ 𝜃Ψ𝛼 (𝜃) 𝑆 (𝑡 𝜃) 𝑑𝜃,
0
where
Ψ𝛼 (𝜃) =
Γ (𝑛𝛼 + 1)
1 ∞
∑ (−1)𝑛−1
sin (𝑛𝜋𝛼) ,
𝜋𝛼 𝑛=1
𝑛!
𝜃 ∈ (0, ∞) ,
is the function of Wright type defined on (0, ∞) which
satisfies
𝛽
(ii) 𝐴 𝑆(𝑡)𝑥 = 𝑆(𝑡)𝐴 𝑥 for each 𝑥 ∈ 𝐷(𝐴 ) and 𝑡 ≥ 0.
(iii) For every 𝑡 > 0, 𝐴𝛽 𝑆(𝑡) is bounded in 𝑋, and there
exists 𝑀𝛽 > 0 such that
󵄩󵄩 𝛽
󵄩
󵄩󵄩𝐴 𝑆 (𝑡)󵄩󵄩󵄩 ≤ 𝑀𝛽 𝑡−𝛽 .
󵄩
󵄩
(9)
∞
(10)
(i) 𝑆(𝑡) : 𝑋 → 𝑋𝛽 for each 𝑡 > 0 and 𝛽 ≥ 0.
𝛽
For 𝑥 ∈ 𝑋, we define two families {S𝛼 (𝑡) : 𝑡 ≥ 0} and
{A𝛼 (𝑡) : 𝑡 ≥ 0} of operators by
Ψ𝛼 (𝜃) ≥ 0,
∞
(5)
(iv) 𝐴−𝛽 is a bounded linear operator for 0 ≤ 𝛽 ≤ 1 in 𝑋.
∞
∫ Ψ𝛼 (𝜃) 𝑑𝜃 = 1,
0
Γ (1 + 𝜁)
,
∫ 𝜃 Ψ𝛼 (𝜃) 𝑑𝜃 =
Γ (1 + 𝛼𝜁)
0
𝜁
(11)
𝜁 ∈ (−1, ∞) .
The following lemma follows from the results in [32–34].
Abstract and Applied Analysis
3
Lemma 5. The operators S𝛼 and A𝛼 have the following
properties:
(i) for any fixed 𝑡 ≥ 0 and any 𝑥 ∈ 𝑋𝛽 , one has the
operators S𝛼 (𝑡) and A𝛼 (𝑡) which are linear and bounded operators; that is, for any 𝑥 ∈ 𝑋𝛽 ,
󵄩󵄩
󵄩
󵄩󵄩S𝛼 (𝑡) 𝑥󵄩󵄩󵄩𝛽 ≤ 𝑀‖𝑥‖𝛽 ,
(H2 ) The function 𝑔 : [0, 𝑇] × 𝐵𝑙 → 𝑋 is continuous, and
there exists some constant 𝑀𝑔 > 0, 0 < 𝛽 < 1, such
that 𝑔 is 𝑋𝛽 -valued and
󵄩󵄩 𝛽
󵄩
󵄩󵄩𝐴 𝑔 (𝑡, 𝑥) − 𝐴𝛽 𝑔 (𝑡, 𝑦)󵄩󵄩󵄩 ≤ 𝑀𝑔 󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩𝐵 ,
󵄩
󵄩
𝑙
𝑥, 𝑦 ∈ 𝐵𝑙 , 𝑡 ∈ [0, 𝑇] ,
𝑀
󵄩󵄩
󵄩
‖𝑥‖𝛽 ; (12)
󵄩󵄩A𝛼 (𝑡) 𝑥󵄩󵄩󵄩𝛽 ≤
Γ (𝛼)
(ii) the operators S𝛼 (𝑡) and A𝛼 (𝑡) are strongly continuous
for all 𝑡 ≥ 0;
(iii) S𝛼 (𝑡) and A𝛼 (𝑡) are norm continuous in 𝑋 for 𝑡 > 0;
󵄩
󵄩󵄩 𝛽
󵄩󵄩𝐴 𝑔 (𝑡, 𝑥)󵄩󵄩󵄩 ≤ 𝑀𝑔 (1 + ‖𝑥‖𝐵𝑙 ) .
󵄩
󵄩
(H3 ) The function 𝑓 : [0, 𝑇] × 𝐵𝑙 → 𝑋 satisfies the following:
(a) 𝑓(𝑡, ⋅) : 𝐵𝑙 → 𝑋 is continuous for each 𝑡 ∈
[0, 𝑇] and for each 𝑥 ∈ 𝐵𝑙 , 𝑓(⋅, 𝑥) : [0, 𝑇] → 𝑋
is strongly measurable;
(b) there is a positive integrable function 𝑛 ∈
𝐿∞ ([0, 𝑇], [0, +∞)) and a continuous nondecreasing function Λ 𝑓 : [0, ∞) → (0, ∞) such
that for every (𝑡, 𝑥) ∈ [0, 𝑇] × 𝐵𝑙 , we have
(iv) S𝛼 (𝑡) and A𝛼 (𝑡) are compact operators in 𝑋 for 𝑡 > 0;
(v) for every 𝑡 > 0, the restriction of S𝛼 (𝑡) to 𝑋𝛽 and the
restriction of A𝛼 (𝑡) to 𝑋𝛽 are norm continuous;
(vi) for every 𝑡 > 0, the restriction of S𝛼 (𝑡) to 𝑋𝛽 and the
restriction of A𝛼 (𝑡) to 𝑋𝛽 are compact operators in 𝑋𝛽 ;
(vii) for all 𝑥 ∈ 𝑋 and 𝑡 ∈ [0, 𝑇],
󵄩󵄩 𝛽
󵄩
󵄩󵄩𝐴 A𝛼 (𝑡) 𝑥󵄩󵄩󵄩 ≤ 𝐶𝛽 𝑡−𝛼𝛽 ‖𝑥‖ ,
󵄩
󵄩
𝐶𝛽 :=
𝑀𝛽 𝛼Γ (2 − 𝛽)
.
Γ (1 + 𝛼 (1 − 𝛽))
(13)
In this paper, we adopt the following definition of mild
solution of (1).
Definition 6. A solution 𝑥(⋅; 𝑢) ∈ 𝐶([0, 𝑇], 𝑋) is said to be a
mild solution of (1) if for any 𝑢 ∈ 𝐿 2 ([0, 𝑇], 𝑈) the integral
equation
𝑥 (𝑡) = S𝛼 (𝑡) (𝜙 (0) + 𝑔 (0, 𝜙)) − 𝑔 (𝑡, 𝑥𝑡 )
󵄩
󵄩󵄩
󵄩󵄩𝑓 (𝑡, 𝑥)󵄩󵄩󵄩 ≤ 𝑛 (𝑡) Λ 𝑓 (‖𝑥‖𝐵𝑙 ) ,
0
𝑡
0
is satisfied.
Let 𝑥(𝑇; 𝑢) be the state value of (14) at terminal time 𝑇
corresponding to the control 𝑢. Introduce the set R(𝑇) =
{𝑥(𝑇; 𝑢) : 𝑢 ∈ 𝐿 2 ([0, 𝑇], 𝑈)}, which is called the reachable
set of the system (14) at terminal time 𝑇, and its closure in 𝑋
is denoted by R(𝑇).
Definition 7. The system (1) is said to be approximately
controllable on [0, 𝑇] if R(𝑇) = 𝑋; that is, given an arbitrary
𝜀 > 0, it is possible to steer from the point 𝑥0 to within a
distance 𝜀 from all points in the state space 𝑋 at time 𝑇.
To investigate the approximate controllability of the
system (14), we assume the following conditions.
(H1 ) 𝐴 is the infinitesimal generator of an analytic semigroup of bounded linear operators 𝑆(𝑡) in 𝑋, 0 ∈ 𝜌(𝐴),
𝑆(𝑡) is compact for 𝑡 > 0, and there exists a positive
constant 𝑀 such that ‖𝑆(𝑡)‖ ≤ 𝑀.
Λ 𝑓 (𝑟)
𝑟
𝑟→∞
= 𝜎𝑓 < ∞.
(16)
2𝛼−1
1
2 𝑇
(1 + 𝑀𝐵2 𝑀A
)
𝜀
2𝛼 − 1
𝑇𝛼𝛽
𝑀 𝑇𝛼
󵄩
󵄩
×(𝑀𝑔 󵄩󵄩󵄩󵄩𝐴−𝛽 󵄩󵄩󵄩󵄩 𝑙+𝐶1−𝛽 𝑀𝑔
𝑙+
𝜎 sup 𝑛 (𝑠)) < 1.
𝛼𝛽 Γ (𝛼) 𝛼 𝑓 𝑠∈𝐽
(17)
Here, 𝑀𝐵 := ‖𝐵‖, 𝑀A := ‖A𝛼 ‖, and 𝐶1−𝛽 = 𝛼𝑀1−𝛽 Γ
(1 + 𝛽)/Γ(1 + 𝛼𝛽).
(14)
+ ∫ (𝑡 − 𝑠)𝛼−1 A𝛼 (𝑡 − 𝑠) [𝐵𝑢 (𝑠) + 𝑓 (𝑠, 𝑥𝑠 )] 𝑑𝑠
lim inf
(H4 ) The following relationship holds:
𝑡
− ∫ (𝑡 − 𝑠)𝛼−1 𝐴A𝛼 (𝑡 − 𝑠) 𝑔 (𝑠, 𝑥𝑠 ) 𝑑𝑠
(15)
−1
(Hac ) For every ℎ ∈ 𝑋𝑧𝛼 (ℎ) = 𝜀(𝜀𝐼 + Γ0𝑇 𝐽) (ℎ) converges to
zero as 𝜀 → 0+ in strong topology, where
𝑇
Γ0𝑇 := ∫ (𝑇 − 𝑠)2(𝛼−1) A𝛼 (𝑇 − 𝑠) 𝐵𝐵∗ A∗𝛼 (𝑇 − 𝑠) 𝑑𝑠
0
(18)
and 𝑧𝜀 (ℎ) is a solution of the equation
𝜀𝑧𝜀 + Γ0𝑇 𝐽 (𝑧𝜀 ) = 𝜀ℎ.
(19)
Let
𝐶𝑇 = {𝑥 : 𝑥 ∈ 𝐶 ((−∞, 𝑇] , 𝑋) , 𝑥0 = 𝜙 ∈ 𝐵𝑙 } .
(20)
set ‖ ⋅ ‖𝑇 be a seminorm defined by
󵄩 󵄩
‖𝑥‖𝑇 = 󵄩󵄩󵄩𝑥0 󵄩󵄩󵄩𝐵𝑙 + sup ‖𝑥 (𝑠)‖ ,
0≤𝑠≤𝑇
𝑥 ∈ 𝐶𝑇 .
(21)
Lemma 8 (see [8]). Assume that 𝑥 ∈ 𝐶𝑇 , then for all 𝑡 ∈
[0, 𝑇], 𝑥𝑡 ∈ 𝐵𝑙 and
󵄩 󵄩
󵄩 󵄩
𝑙 ‖𝑥 (𝑡)‖ ≤ 󵄩󵄩󵄩𝑥𝑡 󵄩󵄩󵄩𝐵𝑙 ≤ 𝑙 sup ‖𝑥 (𝑠)‖ + 󵄩󵄩󵄩𝑥0 󵄩󵄩󵄩𝐵𝑙 .
0≤𝑠≤𝑡
(22)
4
Abstract and Applied Analysis
2. Existence Theorem
In order to formulate the controllability problem in the
form suitable for application of fixed-point theorem, it is
assumed that the corresponding linear system is approximately controllable. Then, it will be shown that the system
(1) is approximately controllable if for all 𝜀 > 0 there exists a
continuous function 𝑥 ∈ 𝐶([0, 𝑇], 𝑋) such that
−1
𝑢𝜀 (𝑡, 𝑥) = (𝑇 − 𝑡)𝛼−1 𝐵∗ A∗𝛼 (𝑇 − 𝑡) 𝐽 ((𝜀𝐼 + Γ0𝑇 𝐽) 𝑝 (𝑥)) ,
It will be shown that for all 𝜀 > 0, the operator Φ𝜀 : 𝐶𝑇 → 𝐶𝑇
has a fixed point.
̃
̃ is
Suppose that 𝑥(𝑡) = 𝜙(𝑡)+𝑧(𝑡),
𝑡 ∈ (−∞, 𝑇], where 𝜙(𝑡)
taken as 𝜙(𝑡) for 𝑡 ∈ (−∞, 0], while for 𝑡 ∈ [0, 𝑇], it is defined
as S𝛼 (𝑡)𝜙(0). Set
𝐶𝑇0 = {𝑧 ∈ 𝐶𝑇 : 𝑧0 = 0 ∈ 𝐵𝑙 } .
For any 𝑧 ∈ 𝐶𝑇0 , we have
󵄩 󵄩
‖𝑧‖𝑇 = 󵄩󵄩󵄩𝑧0 󵄩󵄩󵄩𝐵𝑙 + sup ‖𝑧 (𝑠)‖ = sup ‖𝑧 (𝑠)‖ .
𝑥 (𝑡) = S𝛼 (𝑡) (𝜙 (0) + 𝑔 (0, 𝜙)) − 𝑔 (𝑡, 𝑥𝑡 )
0≤𝑠≤𝑇
𝑡
− ∫ (𝑡 − 𝑠)
0
𝛼−1
𝐴A𝛼 (𝑡 − 𝑠) 𝑔 (𝑠, 𝑥𝑠 ) 𝑑𝑠
𝑡
+ ∫ (𝑡 − 𝑠)𝛼−1 A𝛼 (𝑡 − 𝑠) [𝐵𝑢𝜀 (𝑠, 𝑥) + 𝑓 (𝑠, 𝑥𝑠 )] 𝑑𝑠,
0
where
𝑇
+ ∫ (𝑇 − 𝑠)𝛼−1 𝐴A𝛼 (𝑇 − 𝑠) 𝑔 (𝑠, 𝑥𝑠 ) 𝑑𝑠
0
0
B𝑟 := {𝑧 ∈ 𝐶𝑇0 : ‖𝑧‖𝑇 ≤ 𝑟} .
ℎ ∈ 𝑋.
Having noticed this fact, our goal in this section is to find
conditions for solvability of (23). Note that it will be shown
that the control in (23) drives the system (1) from 𝜙(0) to
−1
(25)
provided that the system (23) has a solution.
Theorem 9. Assume that assumptions (H1 )–(H4 ) hold and
1/2 < 𝛼 < 1. Then, there exists a solution to (23).
0≤𝑠≤𝑡
󵄩 󵄩
󵄩
󵄩
≤ 𝑙 (𝑀 󵄩󵄩󵄩𝜙 (0)󵄩󵄩󵄩 + 𝑟) + 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝐵𝑙 := 𝑅 (𝑟) .
Consider the maps Π𝜀 , Θ𝜀 : 𝐶𝑇0 → 𝐶𝑇0 defined by
(Π𝜀 𝑧) (𝑡)
𝑡 ∈ (−∞, 0] ,
{0
{
{S (𝑡) 𝑔 (0, 𝜙) − 𝑔 (𝑡, 𝜙̃ + 𝑧 )
𝑡
𝑡
:= { 𝛼 𝑡
{
{
𝛼−1
− ∫ (𝑡−𝑠) 𝐴A𝛼 (𝑡−𝑠) 𝑔 (𝑠, 𝜙̃𝑠 +𝑧𝑠 ) 𝑑𝑠 𝑡 ∈ [0, 𝑇] ,
{
0
(Θ𝜀 𝑧) (𝑡)
For all 𝜀 > 0, consider the operator Φ𝜀 : 𝐶𝑇 → 𝐶𝑇
defined as follows:
{0 𝑡
{
{
:= {∫ (𝑡 − 𝑠)𝛼−1 A𝛼 (𝑡 − 𝑠)
{
{ 0
̃
(𝑠, 𝜙̃𝑠 +𝑧𝑠 )] 𝑑𝑠,
{ × [𝐵𝑢𝜀 (𝑠, 𝜙+𝑧)+𝑓
(Φ𝜀 𝑥) (𝑡)
(30)
󵄩 󵄩
+ 𝑙 sup ‖𝑧 (𝑠)‖ + 󵄩󵄩󵄩𝑧0 󵄩󵄩󵄩𝐵𝑙
Proof. The proof of Theorem 9 follows from Lemmas 10–14
and infinite dimensional analogue of Arzela-Ascoli theorem.
𝜙 (𝑡)
{
{
{
{
{
S𝛼 (𝑡) (𝜙 (0)+𝑔 (0, 𝜙)) − 𝑔 (𝑡, 𝑥𝑡 )
{
{
𝑡
{
{
{ − ∫ (𝑡−𝑠)𝛼−1 𝐴A (𝑡−𝑠) 𝑔 (𝑠, 𝑥 ) 𝑑𝑠
𝛼
𝑠
:= {
0
{
𝑡
{
{
𝛼−1
{
{
{ + ∫ (𝑡 − 𝑠) A𝛼 (𝑡 − 𝑠)
{
{
0
{
× [𝐵𝑢𝜀 (𝑠, 𝑥)+𝑓 (𝑠, 𝑥𝑠 )] 𝑑𝑠,
{
(29)
󵄩󵄩 ̃
󵄩
󵄩 󵄩
󵄩󵄩𝜙𝑡 + 𝑧𝑡 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝜙̃𝑡 󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑧𝑡 󵄩󵄩󵄩󵄩𝐵
󵄩
󵄩𝐵𝑙 󵄩 󵄩𝐵𝑙
𝑙
󵄩󵄩 ̃ 󵄩󵄩 󵄩󵄩 ̃ 󵄩󵄩
≤ 𝑙 sup 󵄩󵄩󵄩𝜙 (𝑠)󵄩󵄩󵄩 + 󵄩󵄩󵄩𝜙0 󵄩󵄩󵄩𝐵
𝑙
0≤𝑠≤𝑡
(24)
ℎ − 𝜀𝐽 ((𝜀𝐼 + Γ0𝑇 𝐽) 𝑝 (𝑥))
(28)
It is clear that 𝐵𝑟 is bounded closed convex set in 𝐶𝑇0 . For any
𝑧 ∈ 𝐵𝑟 , we see that
𝑝 (𝑥) = ℎ − S𝛼 (𝑇) (𝜙 (0) + 𝑔 (0, 𝜙)) + 𝑔 (𝑇, 𝑥𝑇 )
− ∫ (𝑇 − 𝑠)𝛼−1 A𝛼 (𝑇 − 𝑠) 𝑓 (𝑠, 𝑥𝑠 ) 𝑑𝑠,
0≤𝑠≤𝑇
Thus, (𝐶𝑇0 , ‖ ⋅ ‖𝑇 ) is a Banach space. For each positive number
𝑟 > 0, set
(23)
𝑇
(27)
𝑡 ∈ (−∞, 0] ,
𝑡 ∈ [0, 𝑇] .
(31)
𝑡 ∈ (−∞, 0]
𝑡 ∈ [0, 𝑇] .
(26)
Obviously, the operator Φ𝜀 has a fixed point if and only if
operator Π𝜀 + Θ𝜀 has a fixed point. In order to prove that
Π𝜀 + Θ𝜀 has a fixed point we will employ the Krasnoselskii’s
fixed-point theorem.
Lemma 10. Under assumptions (H1 )–(H4 ), for any 𝜀 > 0,
there exists a positive number 𝑟 := 𝑟(𝜀) such that Π𝜀 𝑧𝑟 +Θ𝜀 𝑦𝑟 ⊂
B𝑟 for all 𝑧, 𝑦 ∈ B𝑟𝑟 .
Abstract and Applied Analysis
5
Proof. Let 𝜀 > 0 be fixed. If it is not true, then for each 𝑟 > 0,
there are functions 𝑧𝑟 , 𝑦𝑟 ∈ 𝐵𝑟 , but Π𝜀 (𝑧𝑟 ) + Θ𝜀 (𝑦𝑟 ) ∉ B𝑟 . So
for some 𝑡 = 𝑡(𝑟) ∈ [0, 𝑇], one can show that
󵄩
󵄩
𝑟 ≤ 󵄩󵄩󵄩Π𝜀 (𝑧𝑟 ) (𝑡) + Θ𝜀 (𝑦𝑟 ) (𝑡)󵄩󵄩󵄩
Combining the estimates (32)–(36) yields
󵄩
󵄩
󵄩 󵄩
𝐼1 + 𝐼2 + 𝐼3 + 𝐼4 < 𝑀𝑀𝑔 󵄩󵄩󵄩󵄩𝐴−𝛽 󵄩󵄩󵄩󵄩 (1 + 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝐵𝑙 )
󵄩
󵄩
+ 𝑀𝑔 󵄩󵄩󵄩󵄩𝐴−𝛽 󵄩󵄩󵄩󵄩 (1 + 𝑅 (𝑟))
󵄩
󵄩
󵄩 󵄩
≤ 󵄩󵄩󵄩S𝛼 (𝑡) 𝑔 (0, 𝜙)󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑔 (𝑡, 𝜙̃𝑡 + 𝑧𝑡 )󵄩󵄩󵄩󵄩
󵄩󵄩 𝑡
󵄩󵄩
󵄩
󵄩
+ 󵄩󵄩󵄩∫ (𝑡 − 𝑠)𝛼−1 𝐴A𝛼 (𝑡 − 𝑠) 𝑔 (𝑠, 𝜙̃𝑠 + 𝑧𝑠 ) 𝑑𝑠󵄩󵄩󵄩
󵄩󵄩 0
󵄩󵄩
󵄩󵄩 𝑡
󵄩󵄩
󵄩
󵄩
+ 󵄩󵄩󵄩∫ (𝑡 − 𝑠)𝛼−1 A𝛼 (𝑡 − 𝑠) 𝑓 (𝑠, 𝜙̃𝑠 + 𝑦𝑠 ) 𝑑𝑠󵄩󵄩󵄩
󵄩󵄩 0
󵄩󵄩
󵄩󵄩 𝑡
󵄩󵄩
󵄩
󵄩
+ 󵄩󵄩󵄩∫ (𝑡 − 𝑠)𝛼−1 A𝛼 (𝑡 − 𝑠) 𝐵𝑢𝜀 (𝑠, 𝜙 + 𝑦) 𝑑𝑠󵄩󵄩󵄩
󵄩󵄩 0
󵄩󵄩
+ 𝐶1−𝛽 𝑀𝑔
+
(32)
󵄩󵄩
󵄩
󵄩
󵄩
󵄩
󵄩 󵄩
𝐼1 ≤ 𝑀 󵄩󵄩󵄩󵄩𝐴−𝛽 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩𝐴𝛽 𝑔 (0, 𝜙)󵄩󵄩󵄩󵄩 ≤ 𝑀𝑀𝑔 󵄩󵄩󵄩󵄩𝐴−𝛽 󵄩󵄩󵄩󵄩 (1 + 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝐵𝑙 ) ,
󵄩
󵄩󵄩
󵄩
󵄩
󵄩
󵄩
󵄩
𝐼2 ≤ 󵄩󵄩󵄩󵄩𝐴−𝛽 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩𝐴𝛽 𝑔 (𝑡, 𝜙̃𝑡 + 𝑧𝑡 )󵄩󵄩󵄩󵄩 ≤ 𝑀𝑔 󵄩󵄩󵄩󵄩𝐴−𝛽 󵄩󵄩󵄩󵄩 (1 + 󵄩󵄩󵄩󵄩𝜙̃𝑡 + 𝑧𝑡 󵄩󵄩󵄩󵄩𝐵 )
𝑙
(33)
(34)
Using Lemma 5 and the Hölder inequality, one can deduce
that
𝑡
󵄩
󵄩
𝐼3 ≤ ∫ 󵄩󵄩󵄩󵄩(𝑡 − 𝑠)𝛼−1 𝐴1−𝛽 A𝛼 (𝑡 − 𝑠) 𝐴𝛽 𝑔 (𝑠, 𝜙̃𝑠 + 𝑧𝑠 )󵄩󵄩󵄩󵄩 𝑑𝑠
0
𝑀1−𝛽 𝛼Γ (1 + 𝛽)
Γ (1 + 𝛼𝛽)
𝑡
󵄩
󵄩
∫ (𝑡 − 𝑠)𝛼𝛽−1 󵄩󵄩󵄩󵄩𝐴𝛽 𝑔 (𝑠, 𝜙̃𝑠 + 𝑧𝑠 )󵄩󵄩󵄩󵄩 𝑑𝑠
0
𝑡
󵄩
󵄩
≤ ∫ 󵄩󵄩󵄩󵄩(𝑡 − 𝑠)𝛼−1 (𝑇 − 𝑠)𝛼−1 A𝛼 (𝑡 − 𝑠) 𝐵𝐵∗ A∗𝛼 (𝑇 − 𝑡)󵄩󵄩󵄩󵄩 𝑑𝑠
0
󵄩󵄩
󵄩󵄩
−1
× 󵄩󵄩󵄩𝐽 ((𝜀𝐼 + Γ0𝑇 𝐽) 𝑝 (𝜙 + 𝑦))󵄩󵄩󵄩
󵄩
󵄩
2
≤ 𝑀𝐵2 𝑀A
󵄩󵄩
𝑇2𝛼−1 󵄩󵄩󵄩
𝑇 −1
󵄩󵄩𝐽 ((𝜀𝐼 + Γ0 𝐽) 𝑝 (𝜙 + 𝑦))󵄩󵄩󵄩
󵄩
2𝛼 − 1 󵄩
2
= 𝑀𝐵2 𝑀A
󵄩󵄩
𝑇2𝛼−1 󵄩󵄩󵄩
𝑇 −1
󵄩󵄩(𝜀𝐼 + Γ0 𝐽) 𝑝 (𝜙 + 𝑦)󵄩󵄩󵄩
󵄩
󵄩
2𝛼 − 1
2𝛼−1
1
󵄩
󵄩󵄩
2 𝑇
≤ 𝑀𝐵2 𝑀A
󵄩𝑝 (𝜙 + 𝑦)󵄩󵄩󵄩
𝜀
2𝛼 − 1 󵄩
2𝛼−1
1
2 𝑇
≤ 𝑀𝐵2 𝑀A
Δ.
𝜀
2𝛼 − 1
󵄩
󵄩
𝑟 ≤ 󵄩󵄩󵄩Π𝜀 (𝑧𝑟 ) (𝑡) + Θ𝜀 (𝑦𝑟 ) (𝑡)󵄩󵄩󵄩
𝑙
𝑇𝛼𝛽
(1 + 𝑅 (𝑟)) .
𝛼𝛽
Using the assumption (H3 ), one has
󵄩
󵄩
𝐼4 ≤ ∫ 󵄩󵄩󵄩󵄩(𝑡 − 𝑠)𝛼−1 A𝛼 (𝑡 − 𝑠) 𝑓 (𝑠, 𝜙̃𝑠 + 𝑦𝑠 )󵄩󵄩󵄩󵄩 𝑑𝑠
0
≤
𝑀
󵄩
󵄩
∫ (𝑡 − 𝑠)𝛼−1 𝑛 (𝑠) Λ 𝑓 (󵄩󵄩󵄩󵄩𝜙̃𝑠 + 𝑦𝑠 󵄩󵄩󵄩󵄩) 𝑑𝑠
Γ (𝛼) 0
≤
𝑀 𝑇𝛼
Λ (𝑅 (𝑟)) sup 𝑛 (𝑠) .
Γ (𝛼) 𝛼 𝑓
𝑠∈𝐽
(39)
2𝛼−1
1
2 𝑇
= (1 + 𝑀𝐵2 𝑀A
) Δ.
𝜀
2𝛼 − 1
2𝛼−1
1
2 𝑇
)
(1 + 𝑀𝐵2 𝑀A
𝜀
2𝛼 − 1
𝑡
𝑡
2𝛼−1
1
2 𝑇
≤ Δ + 𝑀𝐵2 𝑀A
Δ
𝜀
2𝛼 − 1
Dividing both sides of (39) by 𝑟 and taking 𝑟 → ∞, we obtain
that
𝑡
𝑀
󵄩
󵄩
∫ (𝑡 − 𝑠)𝛼−1 󵄩󵄩󵄩󵄩𝑓 (𝑠, 𝜙̃𝑠 + 𝑦𝑠 )󵄩󵄩󵄩󵄩 𝑑𝑠
Γ (𝛼) 0
(38)
Thus,
(35)
≤
𝑡󵄩
󵄩
= ∫ 󵄩󵄩󵄩(𝑡 − 𝑠)𝛼−1 (𝑇 − 𝑠)𝛼−1 A𝛼 (𝑡 − 𝑠) 𝐵𝐵∗ A∗𝛼
0 󵄩
󵄩󵄩
−1
× (𝑇 − 𝑡) 𝐽 ((𝜀𝐼 + Γ0𝑇 𝐽) 𝑝 (𝜙 + 𝑦))󵄩󵄩󵄩 𝑑𝑠
󵄩
0
𝑡
󵄩
󵄩
≤ 𝐾 (𝛼, 𝛽) ∫ (𝑡 − 𝑠)𝛼𝛽−1 𝑀𝑔 (1 + 󵄩󵄩󵄩󵄩𝜙̃𝑠 + 𝑧𝑠 󵄩󵄩󵄩󵄩𝐵 ) 𝑑𝑠
≤ 𝐾 (𝛼, 𝛽) 𝑀𝑔
𝑀 𝑇𝛼
Λ (𝑅 (𝑟)) sup 𝑛 (𝑠) := Δ.
Γ (𝛼) 𝛼 𝑓
𝑠∈𝐽
𝑡
󵄩
󵄩
𝐼5 ≤ ∫ 󵄩󵄩󵄩󵄩(𝑡 − 𝑠)𝛼−1 A𝛼 (𝑡 − 𝑠) 𝐵𝑢𝜀 (𝑠, 𝜙 + 𝑦)󵄩󵄩󵄩󵄩 𝑑𝑠
0
Let us estimate 𝐼𝑖 , 𝑖 = 1, . . . , 4. By the assumption (H2 ), we
have
≤
(37)
On the other hand,
=: 𝐼1 + 𝐼2 + 𝐼3 + 𝐼4 + 𝐼5 .
󵄩
󵄩
≤ 𝑀𝑔 󵄩󵄩󵄩󵄩𝐴−𝛽 󵄩󵄩󵄩󵄩 (1 + 𝑅 (𝑟)) .
𝑇𝛼𝛽
(1 + 𝑅 (𝑟))
𝛼𝛽
(36)
𝑇𝛼𝛽
𝑀 𝑇𝛼
󵄩
󵄩
× (𝑀𝑔 󵄩󵄩󵄩󵄩𝐴−𝛽 󵄩󵄩󵄩󵄩 𝑙 + 𝑀𝑔 𝐶1−𝛽
𝑙+
𝜎 sup 𝑛 (𝑠)) ≥ 1,
𝛼𝛽
Γ (𝛼) 𝛼 𝑓 𝑠∈𝐽
(40)
which is a contradiction by assumption (H4 ). Thus, Π𝜀 𝑧𝑟 +
Θ𝜀 𝑦𝑟 ⊂ B𝑟 for some 𝑟 > 0.
6
Abstract and Applied Analysis
Lemma 11. Let assumptions (H1 )–(H4 ) hold. Then, Θ1 is
contractive.
Proof. Let 𝑥, 𝑦 ∈ B𝑟 . Then,
󵄩
󵄩󵄩
󵄩󵄩(Π𝜀 𝑥) (𝑡) − (Π𝜀 𝑦) (𝑡)󵄩󵄩󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩󵄩𝑔 (𝑡, 𝜙̃𝑡 + 𝑥𝑡 ) − 𝑔 (𝑡, 𝜙̃𝑡 + 𝑦𝑡 )󵄩󵄩󵄩󵄩
󵄩󵄩 𝑡
󵄩
+ 󵄩󵄩󵄩∫ (𝑡 − 𝑠)𝛼−1 𝐴A𝛼 (𝑡 − 𝑠)
󵄩󵄩 0
󵄩
󵄩󵄩
󵄩󵄩(Θ𝜀 𝑧) (𝑡 + ℎ) − (Θ𝜀 𝑧) (𝑡)󵄩󵄩󵄩
󵄩󵄩 𝑡
󵄩
≤ 󵄩󵄩󵄩∫ ((𝑡 + ℎ − 𝑠)𝛼−1 − (𝑡 − 𝑠)𝛼−1 ) A𝛼 (𝑡 + ℎ − 𝑠)
󵄩󵄩 0
󵄩󵄩
󵄩
× [(𝑇 − 𝑠)𝛼−1 𝐵V𝜀 (𝑠, 𝜙̃ + 𝑧) + 𝑓 (𝑠, 𝜙̃𝑠 + 𝑧𝑠 )] 𝑑𝑠󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩
󵄩
× (𝑔 (𝑠, 𝜙̃𝑠 + 𝑥𝑠 ) − 𝑔 (𝑠, 𝜙̃𝑠 + 𝑦𝑠 )) 𝑑𝑠󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩 −𝛽 󵄩󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩𝐴 󵄩󵄩󵄩 𝑀𝑔 󵄩󵄩󵄩𝑥𝑡 − 𝑦𝑡 󵄩󵄩󵄩𝐵𝑙 + 𝐶1−𝛽
󵄩󵄩 𝑡+ℎ
󵄩
+ 󵄩󵄩󵄩󵄩∫ (𝑡 + ℎ − 𝑠)𝛼−1 A𝛼 (𝑡 + ℎ − 𝑠)
󵄩󵄩 𝑡
𝑡
󵄩
󵄩
× ∫ (𝑡 − 𝑠)𝛼𝛽−1 󵄩󵄩󵄩󵄩𝐴𝛽 (𝑔 (𝑠, 𝜙̃𝑠 +𝑥𝑠 )−𝑔 (𝑠, 𝜙̃𝑠 +𝑦𝑠 ))󵄩󵄩󵄩󵄩 𝑑𝑠
0
󵄩
󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩󵄩𝐴−𝛽 󵄩󵄩󵄩󵄩 𝑀𝑔 󵄩󵄩󵄩𝑥𝑡 − 𝑦𝑡 󵄩󵄩󵄩𝐵𝑙 + 𝐶1−𝛽 𝑀𝑔
𝑡
× ∫ (𝑡 − 𝑠)
0
for every 𝑠1 , 𝑠2 ∈ [0, 𝑇] with |𝑠1 − 𝑠2 | < 𝛿. For 𝑧 ∈ B𝑟 , 0 < |ℎ| <
𝛿, 𝑡 + ℎ ∈ [0, 𝑇], we have
󵄩󵄩
󵄩
× [(𝑇 − 𝑠)𝛼−1 𝐵V𝜀 (𝑠, 𝜙̃ + 𝑧) + 𝑓 (𝑠, 𝜙̃𝑠 + 𝑧𝑠 )] 𝑑𝑠󵄩󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩 𝑡
󵄩
+ 󵄩󵄩󵄩∫ (𝑡 − 𝑠)𝛼−1 (A𝛼 (𝑡 + ℎ − 𝑠) − A𝛼 (𝑡 − 𝑠))
󵄩󵄩 0
󵄩󵄩
󵄩
× [(𝑇 − 𝑠)𝛼−1 𝐵V𝜀 (𝑠, 𝜙̃ + 𝑧) + 𝑓 (𝑠, 𝜙̃𝑠 + 𝑧𝑠 )] 𝑑𝑠󵄩󵄩󵄩 .
󵄩󵄩
(46)
𝛼𝛽−1 󵄩
󵄩
󵄩
󵄩󵄩𝑥𝑠 − 𝑦𝑠 󵄩󵄩󵄩𝐵𝑙 𝑑𝑠.
(41)
Hence,
󵄩
󵄩󵄩
󵄩󵄩(Π𝜀 𝑥) (𝑡) − (Π𝜀 𝑦) (𝑡)󵄩󵄩󵄩
Applying (38) and the Hölder inequality, we obtain
𝛼𝛽
𝑇
󵄩
󵄩
󵄩
󵄩
≤ 𝑀𝑔 𝑙 (󵄩󵄩󵄩󵄩𝐴−𝛽 󵄩󵄩󵄩󵄩 + 𝐶1−𝛽
) sup 󵄩󵄩𝑥 (𝑠) − 𝑦 (𝑠)󵄩󵄩󵄩 ,
𝛼𝛽 0≤𝑠≤𝑡 󵄩
(42)
≤
where we have used the fact that 𝑥0 = 𝑦0 = 0. Thus,
󵄩
󵄩
sup 󵄩󵄩󵄩(Π𝜀 𝑥) (𝑡) − (Π𝜀 𝑦) (𝑡)󵄩󵄩󵄩
𝛼𝛽
(43)
𝑡
0
Lemma 12. Let assumptions (H1 )–(H4 ) hold. Then, Θ𝜀 maps
bounded sets to bounded sets in 𝐵𝑟 .
Proof. By the similar argument as Lemma 10, we obtain
𝑀 𝑇𝛼
Λ (𝑅 (𝑟)) sup 𝑛 (𝑠) := 𝑟1 (𝜀)
×
Γ (𝛼) 𝛼 𝑓
𝑠∈𝐽
𝑀 1
𝑀 𝑀 Δ
Γ (𝛼) 𝜀 𝐵 A
× ∫ ((𝑡 + ℎ − 𝑠)𝛼−1 − (𝑡 − 𝑠)𝛼−1 ) (𝑇 − 𝑠)𝛼−1 𝑑𝑠
so Θ1 is a contraction by assumption (H4 ).
1 2 2 𝑇2𝛼−1
󵄩
󵄩󵄩
)
󵄩󵄩(Θ𝜀 𝑧) (𝑡)󵄩󵄩󵄩 < (1 + 𝑀𝐵 𝑀A
𝜀
2𝛼 − 1
𝑡
𝑀
Λ 𝑓 (𝑅 (𝑟)) ∫ ((𝑡 + ℎ − 𝑠)𝛼−1 − (𝑡 − 𝑠)𝛼−1 ) 𝑛 (𝑠) 𝑑𝑠
Γ (𝛼)
0
+
0≤𝑡≤𝑇
𝑇
󵄩
󵄩
󵄩
󵄩
≤ 𝑀𝑔 𝑙 (󵄩󵄩󵄩󵄩𝐴−𝛽 󵄩󵄩󵄩󵄩 + 𝐶1−𝛽
) sup 󵄩󵄩𝑥 (𝑠) − 𝑦 (𝑠)󵄩󵄩󵄩 ,
𝛼𝛽 0≤𝑠≤𝑇 󵄩
󵄩󵄩
󵄩
󵄩󵄩(Θ𝜀 𝑧) (𝑡 + ℎ) − (Θ𝜀 𝑧) (𝑡)󵄩󵄩󵄩
+
𝑡+ℎ
𝑀
Λ 𝑓 (𝑅 (𝑟)) ∫ (𝑡 + ℎ − 𝑠)𝛼−1 𝑛 (𝑠) 𝑑𝑠
Γ (𝛼)
𝑡
+
𝑡+ℎ
𝑀 1
𝑀𝐵 𝑀A Δ ∫ (𝑡 + ℎ − 𝑠)𝛼−1 (𝑇 − 𝑠)𝛼−1 𝑑𝑠
Γ (𝛼) 𝜀
𝑡
+
𝑡
𝜂𝑇𝛼
Λ 𝑓 (𝑅 (𝑟)) ∫ (𝑡 − 𝑠)𝛼−1 𝑛 (𝑠) 𝑑𝑠
𝛼
0
+
𝑡
𝜂𝑇𝛼 1
𝑀𝐵 𝑀A Δ ∫ (𝑡 − 𝑠)𝛼−1 (𝑇 − 𝑠)𝛼−1 𝑑𝑠.
𝛼 𝜀
0
(44)
(47)
which implies that Θ𝜀 𝑧 ∈ B𝑟1 (𝜀) .
Lemma 13. Let assumptions (H1 )–(H4 ) hold. Then, the set
{Θ𝜀 𝑧 : 𝑧 ∈ B𝑟 } is an equicontinuous family of functions on
[0, 𝑇].
Proof. Let 0 < 𝜂 < 𝑡 < 𝑇 and 𝛿 > 0 such that
󵄩
󵄩󵄩
󵄩󵄩A𝛼 (𝑠1 ) − A𝛼 (𝑠2 )󵄩󵄩󵄩 < 𝜂
(45)
Therefore, for 𝜀 sufficiently small, the right-hand side of (47)
tends to zero as ℎ → 0. On the other hand, the compactness
of A𝛼 (𝑡), 𝑡 > 0 implies the continuity in the uniform operator
topology. Thus, the set {Θ𝜀 𝑧 : 𝑧 ∈ B𝑟 } is equicontinuous.
Lemma 14. Let assumptions (H1 )–(H4 ) hold. Then, Θ𝜀 maps
B𝑟 onto a precompact set in B𝑟 .
Abstract and Applied Analysis
7
Proof. Let 0 < 𝑡 ≤ 𝑇 be fixed and 𝜀 be a real number satisfying
0 < 𝜆 < 𝑡. For 𝛿 > 0, define an operator Θ𝜆,𝛿
𝜀 on 𝐵𝑟 by
where we have used the equality
∞
∫ 𝜃𝛽 𝜂𝛼 (𝜃) 𝑑𝜃 =
(Θ𝜆,𝛿
𝜀 𝑧) (𝑡)
= 𝛼∫
𝑡−𝜆
0
0
∞
𝛿
󵄩󵄩
󵄩󵄩
󵄩󵄩(Θ𝜀 𝑧) (𝑡) − (Θ𝜆,𝛿
󵄩 󳨀→ 0
𝜀 𝑧) (𝑡)󵄩
󵄩
󵄩
× [𝐵𝑢𝜀 (𝑠, 𝜙̃ + 𝑧) + 𝑓 (𝑠, 𝜙̃𝑠 + 𝑧𝑠 )] 𝑑𝑠
𝑡−𝜆
0
𝛿
Since 𝑆(𝑡), 𝑡 > 0 is a compact operator, the set {(Θ𝜆,𝛿
𝜀 𝑧)(𝑡) : 𝑧 ∈
B𝑟 } is precompact in 𝐻 for every 0 < 𝜆 < 𝑡, 𝛿 > 0. Moreover,
for each 𝑧 ∈ B𝑟 , we have
󵄩󵄩󵄩(Θ 𝑧) (𝑡) − (Θ𝜆,𝛿 𝑧) (𝑡)󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩 𝜀
𝜀
󵄩󵄩 𝑡 𝛿
󵄩
≤ 𝛼𝐸 󵄩󵄩󵄩󵄩∫ ∫ 𝜃(𝑡 − 𝑠)𝛼−1 𝜂𝛼 (𝜃) 𝑆 ((𝑡 − 𝑠)𝛼 𝜃)
󵄩󵄩 0 0
󵄩󵄩
󵄩
× [𝐵𝑢𝜀 (𝑠, 𝜙̃ + 𝑧) + 𝑓 (𝑠, 𝜙̃𝑠 + 𝑧𝑠 )] 𝑑𝜃 𝑑𝑠󵄩󵄩󵄩󵄩
󵄩󵄩
(49)
󵄩󵄩 𝑡
∞
󵄩
+ 𝛼𝐸 󵄩󵄩󵄩∫ ∫ 𝜃(𝑡 − 𝑠)𝛼−1 𝜂𝛼 (𝜃) 𝑆 ((𝑡 − 𝑠)𝛼 𝜃)
󵄩󵄩 𝑡−𝜆 𝛿
󵄩󵄩
󵄩
× [𝐵𝑢𝜀 (𝑠, 𝜙̃ + 𝑧) + 𝑓 (𝑠, 𝜙̃𝑠 + 𝑧𝑠 )] 𝑑𝜃 𝑑𝑠󵄩󵄩󵄩
󵄩󵄩
A similar argument as before
𝐽1 ≤ 𝛼𝑀 ∫ (𝑡 − 𝑠)
𝛼−1
󵄩
󵄩 󵄩
󵄩
(󵄩󵄩󵄩󵄩𝐵𝑢𝜀 (𝑠, 𝜙̃ + 𝑧)󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑓 (𝑠, 𝜙̃𝑠 + 𝑧𝑠 )󵄩󵄩󵄩󵄩) 𝑑𝑠
𝛿
× (∫ 𝜃𝜂𝛼 (𝜃) 𝑑𝜃)
𝑡
1
≤ 𝛼𝑀 ( 𝑀𝐵 𝑀A Δ ∫ (𝑡 − 𝑠)𝛼−1 (𝑇 − 𝑠)𝛼−1 𝑑𝑠
𝜀
0
𝑡−𝜆
𝑡
𝛿
0
0
󵄩
󵄩 󵄩
󵄩
(𝑡 − 𝑠)𝛼−1 (󵄩󵄩󵄩󵄩𝐵𝑢𝜀 (𝑠, 𝜙̃ + 𝑧)󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑓 (𝑠, 𝜙̃𝑠 + 𝑧𝑠 )󵄩󵄩󵄩󵄩) 𝑑𝑠
∞
× (∫ 𝜃𝜂𝛼 (𝜃) 𝑑𝜃)
𝐷𝑡𝛼 𝑥 (𝑡) = 𝐴𝑥 (𝑡) + 𝐵𝑢 (𝑡) ,
𝑡 ∈ [0, 𝑇] ,
𝑥 (0) = 𝜙 (0) .
(53)
(54)
The approximate controllability for the linear fractional system (53) is a natural generalization of approximate controllability of linear first-order control system. It is convenient at
this point to introduce the controllability operator associated
with (53) as
𝑇
𝐿𝑇0 = ∫ (𝑇 − 𝑠)𝛼−1 A𝛼 (𝑇 − 𝑠) 𝐵𝑢 (𝑠) 𝑑𝑠,
0
∗
𝑇
Γ0𝑇 = 𝐿𝑇0 (𝐿𝑇0 ) = ∫ (𝑇−𝑠)2(𝛼−1) A𝛼 (𝑇−𝑠) 𝐵𝐵∗ A∗𝛼 (𝑇 − 𝑠) 𝑑𝑠,
(55)
respectively, where 𝐵∗ denotes the adjoint of 𝐵 and A∗𝛼 (𝑡) is
the adjoint of A𝛼 (𝑡). It is straightforward that the operator 𝐿𝑇0
is a linear-bounded operator for 1/2 < 𝛼 ≤ 1.
(ii) For all ℎ ∈ 𝑋, 𝐽(𝑧𝜀 (ℎ)) converges to the zero as 𝜀 → 0+
−1
in the weak topology, where 𝑧𝜀 (ℎ) = 𝜀(𝜀𝐼 + Γ0𝑇 𝐽) (ℎ)
is a solution of the equation
𝜀𝑧𝜀 + Γ0𝑇 𝐽 (𝑧𝜀 ) = 𝛼ℎ.
(56)
−1
(iii) For all ℎ ∈ 𝑋, 𝑧𝜀 (ℎ) = 𝜀(𝜀𝐼 + Γ0𝑇 𝐽) (ℎ) converges to
the zero as 𝜀 → 0+ in the strong topology.
𝛿
≤
Consider the following linear fractional differential system:
(i) Γ0𝑇 is positive, that is, ⟨𝑧∗ , Γ0𝑇 𝑧∗ ⟩ > 0 for all nonzero
𝑧∗ ∈ 𝑋∗ .
+ Λ 𝑓 (𝑅 (𝑟)) ∫ (𝑡−𝑠)𝛼−1 𝑛 (𝑠) 𝑑𝑠) (∫ 𝜃𝜂𝛼 (𝜃) 𝑑𝜃) ,
𝑡
3. Main Results
Theorem 15 (see [11]). The following three conditions are
equivalent:
0
𝐽2 ≤ 𝛼𝑀 ∫
Therefore, there are relatively compact sets arbitrary close to
the set {(Θ𝜀 𝑧)(𝑡) : 𝑧 ∈ B𝑟 }. Hence, the set {(Θ𝜀 𝑧)(𝑡) : 𝑧 ∈ B𝑟 }
is also precompact in B𝑟 .
0
:= 𝐽1 + 𝐽2 .
0
as 𝜆 󳨀→ 0+ , 𝛿 󳨀→ 0+ .
(52)
∞
∫ 𝜃(𝑡 − 𝑠)𝛼−1 𝜂𝛼 (𝜃) 𝑆 ((𝑡 − 𝑠)𝛼 𝜃 − 𝜆𝛼 𝛿)
× [𝐵𝑢𝜀 (𝑠, 𝜙̃ + 𝑧) + 𝑓 (𝑠, 𝜙̃𝑠 + 𝑧𝑠 )] 𝑑𝑠.
(48)
𝑡
(51)
Form (49) to (50), one can see that for each 𝑧 ∈ B𝑟 ,
∫ 𝜃(𝑡 − 𝑠)𝛼−1 𝜂𝛼 (𝜃) 𝑆 ((𝑡 − 𝑠)𝛼 𝜃)
= 𝛼𝑆 (𝜆𝛼 𝛿) ∫
Γ (1 + 𝛽)
.
Γ (1 + 𝛼𝛽)
𝑡
𝛼𝑀
1
( 𝑀𝐵 𝑀A Δ ∫ (𝑡−𝑠)𝛼−1 (𝑇−𝑠)𝛼−1 𝑑𝑠
Γ (1 + 𝛼) 𝜀
𝑡−𝜆
+ Λ 𝑓 (𝑅 (𝑟)) ∫
𝑡
𝑡−𝜆
(𝑡 − 𝑠)𝛼−1 𝑛 (𝑠) 𝑑𝑠) ,
(50)
Remark 16. It is known that Theorem 15(i) holds if and only if
Im 𝐿𝑇0 = 𝑋. In other words, Theorem 15(i) holds if and only if
the corresponding linear system is approximately controllable
on [0, 𝑇].
8
Abstract and Applied Analysis
Theorem 17 (see [11]). Let 𝑝 : 𝑋 → 𝑋 be a nonlinear
operator. Assume that 𝑧𝜀 is a solution of the following equation:
𝜀𝑧𝜀 + Γ0𝑇 𝐽 (𝑧𝜀 ) = 𝛼𝑝 (𝑧𝜀 ) ,
(57)
󵄩󵄩
󵄩
+
󵄩󵄩𝑝 (𝑧𝜀 ) − 𝑝󵄩󵄩󵄩 󳨀→ 0 as 𝜀 󳨀→ 0 , 𝑝 ∈ 𝑋.
(58)
Then, there exists a subsequence of the sequence {𝑧𝜀 } strongly
converging to zero as 𝜀 → 0+ .
In other words, 𝑧𝜀 = ℎ − 𝑥𝜀 (𝑇) is a solution of the equation
(𝜀𝐼 + Γ0𝑇 𝐽) (𝑧𝜀 ) = 𝜀𝑝 (𝑥𝜀 ) .
It follows that
𝜀 ⟨𝐽 (𝑧𝜀 ) , 𝑧𝜀 ⟩ + ⟨𝐽 (𝑧𝜀 ) , Γ0𝑇 𝐽 (𝑧𝜀 )⟩ = 𝜀 ⟨𝐽 (𝑧𝜀 ) , 𝑝 (𝑥𝜀 )⟩ ,
󵄩 󵄩2
𝜀󵄩󵄩󵄩𝑧𝜀 󵄩󵄩󵄩 + ⟨𝐽 (𝑧𝜀 ) , Γ0𝑇 𝐽 (𝑧𝜀 )⟩ = 𝜀 ⟨𝐽 (𝑧𝜀 ) , 𝑝 (𝑥𝜀 )⟩ ,
We are now in a position to state and prove the main result
of the paper.
󵄩 󵄩󵄩
󵄩
󵄩 󵄩2
𝜀󵄩󵄩󵄩𝑧𝜀 󵄩󵄩󵄩 ≤ 𝜀 ⟨𝐽 (𝑧𝜀 ) , 𝑝 (𝑥𝜀 )⟩ ≤ 𝜀 󵄩󵄩󵄩𝑧𝜀 󵄩󵄩󵄩 󵄩󵄩󵄩𝑝 (𝑥𝜀 )󵄩󵄩󵄩 ,
Theorem 18. Let 1/2 < 𝛼 < 1. Suppose that conditions (H1 )–
(H4 ) and (H𝑎𝑐 ) are satisfied. Besides, assume additionally that
(Hgc ) 𝑔 : [0, 𝑇] × 𝑋 → 𝑋 and 𝐴𝛽 𝑔(𝑇, ⋅) is continuous from
the weak topology of 𝑋 to the strong topology of 𝑋.
(Hub ) There exists 𝑁 ∈ 𝐿∞ ([0, 𝑇], [0, +∞)) such that
󵄩
󵄩
󵄩
󵄩
sup 󵄩󵄩󵄩𝑓 (𝑡, 𝑥)󵄩󵄩󵄩 +sup 󵄩󵄩󵄩󵄩𝐴𝛽 𝑔 (𝑡, 𝑦)󵄩󵄩󵄩󵄩 ≤ 𝑁 (𝑡) , for a.e. 𝑡 ∈ [0, 𝑇] .
𝑥∈𝐵
𝑦∈𝑋
󵄩󵄩 󵄩󵄩 󵄩󵄩
󵄩 󵄩
𝜀 󵄩
󵄩󵄩𝑧𝜀 󵄩󵄩 = 󵄩󵄩𝐽 (𝑧𝜀 )󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑝 (𝑥 )󵄩󵄩󵄩 .
󵄩
󵄩
󵄩󵄩
𝜀 󵄩
󵄩󵄩𝑝 (𝑥 )󵄩󵄩󵄩 ≤ ‖ℎ‖ + 𝑀 󵄩󵄩󵄩𝜙 (0)󵄩󵄩󵄩 + 𝑁 (𝑇)
(59)
+
Then, the system (1) is approximately controllable on [0, 𝑇].
+
Proof. Let 𝑥𝜀 be a fixed point of Φ𝜀 in 𝐵𝑟(𝜀) . Then, 𝑥𝜀 is a mild
solution of (1) on [0, 𝑇] under the control
−1
𝑝 (𝑥𝜀 ) = ℎ − S𝛼 (𝑇) (𝜙 (0) + 𝑔 (0, 𝜙 (0))) + 𝑔 (𝑇, 𝑥𝜀 (𝑇))
𝑇
Γ (1 + 𝛼𝛽)
𝑇
− ∫ (𝑇 − 𝑠)𝛼−1 A𝛼 (𝑇 − 𝑠) 𝑓 (𝑠, 𝑥𝑠𝜀 ) 𝑑𝑠
(60)
and satisfies the following equality:
𝑥𝜀 (𝑇) = S𝛼 (𝑇) (𝜙 (0) + 𝑔 (0, 𝜙 (0))) − 𝑔 (𝑇, 𝑥𝜀 (𝑇))
𝑇
− ∫ (𝑇 − 𝑠)𝛼−1 𝐴A𝛼 (𝑇 − 𝑠) 𝑔 (𝑠, 𝑥𝑠𝜀 ) 𝑑𝑠
0
𝑇
+ ∫ (𝑇 − 𝑠)𝛼−1 A𝛼 (𝑡 − 𝑠) [𝐵𝑢𝜀 (𝑠, 𝑥) + 𝑓 (𝑠, 𝑥𝑠𝜀 )] 𝑑𝑠
0
= S𝛼 (𝑇) (𝜙 (0) + 𝑔 (0, 𝜙 (0))) − 𝑔 (𝑇, 𝑥𝜀 (𝑇))
𝑇
− ∫ (𝑇 − 𝑠)𝛼−1 𝐴A𝛼 (𝑇 − 𝑠) ℎ (𝑠, 𝑥𝑠𝜀 ) 𝑑𝑠
0
−1
+ (−𝜀𝐼 + 𝜀𝐼 + Γ0𝑇 𝐽) ((𝜀𝐼 + Γ0𝑇 𝐽) 𝑝 (𝑥𝜀 ))
𝑇
+ ∫ (𝑇 − 𝑠)𝛼−1 A𝛼 (𝑡 − 𝑠) 𝑓 (𝑠, 𝑥𝑠𝜀 ) 𝑑𝑠
0
−1
= ℎ − 𝜀(𝜀𝐼 + Γ0𝑇 𝐽) 𝑝 (𝑥𝜀 ) .
(61)
𝑇
∫ (𝑇 − 𝑠)𝛼𝛽−1 𝑁 (𝑠) 𝑑𝑠
0
(64)
𝑀 𝑇
∫ (𝑇 − 𝑠)𝛼−1 𝑁 (𝑠) 𝑑𝑠.
Γ (𝛼) 0
From (63) and (64), it follows that 𝑥𝜀 (𝑇) ⇀ 𝑥̃ weakly as 𝜀 →
̃
0+ and by the assumption (Hgc ) 𝐴𝛽 𝑔(𝑇, 𝑥𝜀 (𝑇)) → 𝐴𝛽 𝑔(𝑇, 𝑥)
+
strongly as 𝜀 → 0 . Moreover, because of assumption (Hub ),
𝑇
0
0
𝑀1−𝛽 𝛼Γ (1 + 𝛽)
𝑇
󵄩
󵄩2
∫ 󵄩󵄩󵄩𝑓 (𝑠, 𝑥𝑠𝜀 )󵄩󵄩󵄩 𝑑𝑠 + ∫
0
0
+ ∫ (𝑇 − 𝑠)𝛼−1 𝐴A𝛼 (𝑇 − 𝑠) 𝑔 (𝑠, 𝑥𝑠𝜀 ) 𝑑𝑠
(63)
On the other hand, by (Hub ),
𝑙
𝑢𝜀 (𝑡, 𝑥𝜀 ) = (𝑇 − 𝑡)𝛼−1 𝐵∗ 𝑆∗ (𝑇 − 𝑡) 𝐽 ((𝜀𝐼 + Γ0𝑇 𝐽) 𝑝 (𝑥𝜀 )) ,
(62)
𝑇
󵄩󵄩 𝛽
󵄩2
󵄩󵄩𝐴 𝑔 (𝑠, 𝑥𝑠𝜀 )󵄩󵄩󵄩 𝑑𝑠 ≤ ∫ 𝑁 (𝑠) 𝑑𝑠.
󵄩
󵄩
0
(65)
Consequently, the sequences {𝑓(⋅, 𝑥⋅𝜀 )}, {𝐴𝛽 𝑔(⋅, 𝑥⋅𝜀 )} are
bounded. Then, there is a subsequence still denoted by
{𝑓(⋅, 𝑥⋅𝜀 ), 𝐴𝛽 𝑔(⋅, 𝑥⋅𝜀 )} which weakly converges to, say, (𝑓(⋅),
𝑔(⋅)) in 𝐿 2 ([0, 𝑇], 𝑋). Then,
󵄩
󵄩󵄩
𝜀
󵄩󵄩𝑝 (𝑥 ) − 𝑝󵄩󵄩󵄩
󵄩
̃ 󵄩󵄩󵄩󵄩
= 󵄩󵄩󵄩𝑔 (𝑇, 𝑥𝜀 (𝑇)) − 𝑔 (𝑇, 𝑥)
󵄩󵄩 𝑇
󵄩󵄩
󵄩
󵄩
+ 󵄩󵄩󵄩󵄩∫ (𝑇 − 𝑠)𝛼−1 𝐴1−𝛽 A𝛼 (𝑇 − 𝑠) [𝐴𝛽 𝑔 (𝑠, 𝑥𝑠𝜀 ) − 𝑔 (𝑠)] 𝑑𝑠󵄩󵄩󵄩󵄩
󵄩󵄩 0
󵄩󵄩
󵄩󵄩 𝑇
󵄩󵄩
󵄩
󵄩
+ 󵄩󵄩󵄩󵄩∫ (𝑇 − 𝑠)𝛼−1 A𝛼 (𝑇 − 𝑠) [𝑓 (𝑠, 𝑥𝑠𝜀 ) − 𝑓 (𝑠)] 𝑑𝑠󵄩󵄩󵄩󵄩
󵄩󵄩 0
󵄩󵄩
󵄩
󵄩
̃ 󵄩󵄩󵄩󵄩
≤ 󵄩󵄩󵄩󵄩𝐴−𝛽 (𝐴𝛽 𝑔 (𝑇, 𝑥𝜀 (𝑇)) − 𝐴𝛽 𝑔 (𝑇, 𝑥))
󵄩󵄩
󵄩󵄩 𝑡
󵄩
󵄩
+ sup 󵄩󵄩󵄩∫ (𝑡 − 𝑠)𝛼−1 𝐴1−𝛽 A𝛼 (𝑡−𝑠) [𝐴𝛽 𝑔 (𝑠, 𝑥𝑠𝜀 )− 𝑔 (𝑠)] 𝑑𝑠󵄩󵄩󵄩
󵄩󵄩
󵄩
0≤𝑡≤𝑇 󵄩 0
󵄩󵄩
󵄩󵄩 𝑡
󵄩
󵄩
+ sup 󵄩󵄩󵄩∫ (𝑡 − 𝑠)𝛼−1 A𝛼 (𝑡 − 𝑠) [𝑓 (𝑠, 𝑥𝑠𝜀 ) − 𝑓 (𝑠)] 𝑑𝑠󵄩󵄩󵄩 󳨀→ 0,
󵄩󵄩
󵄩 0
0≤𝑡≤𝑇 󵄩
(66)
Abstract and Applied Analysis
9
(d) The operator 𝐴1/2 is given as 𝐴1/2 𝑧 = ∑∞
𝑛=1 𝑛⟨𝑧, 𝑒𝑛 ⟩𝑒𝑛
on the space 𝐷[𝐴1/2 ] = {𝑧 ∈ 𝑋 : ∑∞
𝑛⟨𝑧,
𝑒𝑛 ⟩𝑒𝑛 ∈ 𝑋}.
𝑛=1
where
̃
𝑝 = ℎ − S𝛼 (𝑇) (𝜙 (0) + 𝑔 (0, 𝜙 (0))) + 𝑔 (𝑇, 𝑥)
For 1/2 < 𝛼 < 1, consider the neutral system
𝑇
+ ∫ (𝑇 − 𝑠)𝛼−1 𝐴1−𝛽 A𝛼 (𝑇 − 𝑠) 𝑔 (𝑠) 𝑑𝑠
0
(67)
𝑇
− ∫ (𝑇 − 𝑠)𝛼−1 A𝛼 (𝑇 − 𝑠) 𝑓 (𝑠) 𝑑𝑠
0
as 𝜀 → 0+ because of compactness of an operator 𝑓(⋅) →
⋅
∫0 (⋅ − 𝑠)𝛼−1 A𝛼 (⋅ − 𝑠)𝑓(𝑠)𝑑𝑠 : 𝐿 2 ([0, 𝑇], 𝑋) → 𝐶([0, 𝑇], 𝑋).
Then, by Theorem 17
󵄩 󵄩 󵄩
󵄩󵄩 𝜀
󵄩󵄩𝑥 (𝑇) − ℎ󵄩󵄩󵄩 = 󵄩󵄩󵄩𝑧𝜀 󵄩󵄩󵄩 󳨀→ 0
(68)
+
as 𝜀 → 0 . This gives the approximate controllability. The
theorem is proved.
Remark 19. Theorem 18 assumes that the operator 𝐴 generates a compact semigroup. If the compactness condition
holds on the bounded operator that maps the control function or the generated 𝐶0 -semigroup, then the controllability
operator 𝐿𝑇0 is also compact, and its inverse does not exist if
the state space is infinite dimensional, and, consequently, the
associated linear control system (53) is not exactly controllable. Therefore, the concept of complete controllability is too
strong in infinite dimensional spaces, and the approximate
controllability notion is more appropriate. Thus, Theorem 18
has no analogue for the concept of complete controllability.
4. Applications
In this section, we illustrate the obtained result. Let 𝑋 =
𝐿 2 [0, 𝜋], and let 𝐴 be defined as follows:
𝐴𝑧 = −𝑧󸀠󸀠
(69)
𝜋
𝜕𝛼
[𝑥
𝜉)
+
𝑏 (𝜃, 𝜉) 𝑥 (𝑡, 𝜃) d𝜃]
∫
(𝑡,
𝜕𝑡𝛼
0
=
𝜕2
𝑥 (𝑡, 𝜉) + 𝑝 (𝑡, 𝑥 (𝑡, 𝜉)) + 𝐵𝑢 (𝑡, 𝜉) ,
𝜕𝜉2
𝑥 (𝑡, 0) = 𝑥 (𝑡, 𝜋) = 0,
𝑥 (0, 𝜉) = 𝜑 (𝜉) ,
(72)
𝑡 ≥ 0,
0 ≤ 𝜉 ≤ 𝜋,
where 𝑝 : [0, 𝑇]×𝑅 → 𝑅 is continuous functions. 𝐵 is a linear
2
continuous mapping from 𝑈 = {𝑢 = ∑∞
𝑛=2 𝑢𝑛 𝑒𝑛 | ‖𝑢‖𝑈 :=
∞
2
∑𝑛=2 𝑢𝑛 < ∞} to 𝑋 as follows:
∞
𝐵𝑢 = 2𝑢2 + ∑ 𝑢𝑛 𝑒𝑛 .
(73)
𝑛=2
To write the initial-boundary value problem (72) in the
abstract form, we assume the following.
(A1) The function 𝑏 is measurable and
𝜋
𝜋
0
0
∫ ∫ 𝑏2 (𝜃, 𝜉) d𝜃 d𝜉 < ∞.
(74)
(A2) The function (𝜕/𝜕𝜉)𝑏(𝜃, 𝜉) is measurable, 𝑏(𝜃, 0) =
𝑏(𝜃, 𝜋) = 0, and let
1/2
2
𝜕
𝐿 1 = [∫ ∫ ( 𝑏 (𝜃, 𝜉)) d𝜃 d𝜉]
𝜕𝜉
0 0
𝜋
𝜋
.
(75)
Define 𝑓, 𝑔 : [0, 𝑇] × 𝑋 → 𝑋 by
𝜋
with domain
𝑔 (𝑥) (𝜉) = ∫ 𝑏 (𝜃, 𝜉) 𝑥 (𝜃) d𝜃,
𝑑𝑧
𝐷 (𝐴) = {𝑧 ∈ 𝑋 | 𝑧,
are absolutely continuous,
𝑑𝜉
𝑑2 𝑧
∈ 𝑋 and 𝑧 (0) = 𝑧 (𝜋) = 0} .
𝑑𝜉2
0
𝑓 (𝑡, 𝑥) (𝜉) = 𝑝 (𝑡, 𝑥 (𝜉)) .
(70)
Recall that 𝐴 is the infinitesimal generator of a strongly
continuous semigroup 𝑆(𝑡), 𝑡 > 0, on 𝑋 which is analytic
compact and self-adjoint, and the eigenvalues are −𝑛2 , 𝑛 ∈
𝑁, with corresponding normalized eigenvectors 𝑒𝑛 (𝜉) :=
(2/𝜋)1/2 sin(𝑛𝜉) and
2
𝑆 (𝑡) 𝑒𝑛 = 𝑒−𝑛 𝑡 𝑒𝑛 ,
𝑛 = 1, 2, . . . .
Moreover, the following hold.
(a) {𝑒𝑛 : 𝑛 ∈ 𝑁} is an orthonormal basis of 𝑋.
2
(b) If 𝑧 ∈ 𝐷(𝐴), then 𝐴(𝑧) = ∑∞
𝑛=1 𝑛 ⟨𝑧, 𝑒𝑛 ⟩𝑒𝑛 .
(c) For 𝑧 ∈ H, 𝐴−1/2 𝑧 = ∑∞
𝑛=1 (1/𝑛)⟨𝑧, 𝑒𝑛 ⟩𝑒𝑛 .
(76)
(71)
From (A1), it is clear that 𝑔 is bounded linear operator on
𝑋. Furthermore, 𝑔(𝑥) ∈ 𝐷[𝐴1/2 ], and ‖𝐴1/2 𝑔‖ ≤ 𝐿 1 . In fact,
from the definition of 𝑔 and (A2), it follows that
𝜋
𝜋
0
0
⟨𝑔 (𝑥) , 𝑒𝑛 ⟩ = ∫ [∫ 𝑏 (𝜃, 𝜉) 𝑥 (𝜃) d𝜃] 𝑒𝑛 (𝜉) d𝜉
=
𝜋
𝜕
1 2 1/2
( ) ⟨∫
𝑏 (𝜃, 𝜉) 𝑥 (𝜃) d𝜃, cos (𝑛𝜉)⟩
𝑛 𝜋
0 𝜕𝜉
=
1 2 1/2
( ) ⟨𝑔1 (𝑥) , cos (𝑛𝜉)⟩ ,
𝑛 𝜋
𝜋
(77)
where 𝑔1 (𝑥) = ∫0 (𝜕/𝜕𝜉)𝑏(𝜃, 𝜉)𝑥(𝜃)d𝜃. From (A2), we know
that 𝑔1 : 𝑋 → 𝑋 is a bounded linear operator with
‖𝑔1 ‖ ≤ 𝐿 1 . Hence, ‖𝐴1/2 𝑔(𝑥)‖ = ‖𝑔1 (𝑥)‖, which implies the
10
Abstract and Applied Analysis
assertion. Moreover, assume that 𝑓 and 𝑔 satisfy conditions
of Theorem 18. Thus, the problem (72) can be written in the
abstract form
𝑑𝛼
(𝑥 (𝑡) + 𝑔 (𝑡, 𝑥 (𝑡))) = 𝐴𝑥 (𝑡) + 𝑓 (𝑡, 𝑥 (𝑡)) + 𝐵𝑢 (𝑡) ,
𝑑𝑡𝛼
𝑥 (0) = 𝑥0 ,
𝑡 ∈ [0, 𝑇] .
(78)
Now, consider the associated linear system
𝑑𝛼
𝑥 (𝑡) = 𝐴𝑥 (𝑡) + 𝐵𝑢 (𝑡) ,
𝑑𝑡𝛼
𝑥 (0) = 𝑥0 ,
(79)
𝑡 ∈ [0, 𝑇] .
Show that it is approximately controllable on [0, 𝑇] for 1/2 <
𝛼 < 1. It is easy to see that if 𝑧 = ∑∞
𝑛=1 ⟨𝑧, 𝑒𝑛 ⟩𝑒𝑛 , then
∞
𝑛=3
𝐵
A∗𝛼
(𝑇 − 𝑠) 𝑧
∗
𝛼
= 𝐵 𝛼 ∫ 𝜃Ψ𝛼 (𝜃) 𝑆 ((𝑇 − 𝑠) 𝜃) 𝑧𝑑𝜃
0
∞
𝛼
𝛼
= 𝛼∫ 𝜃Ψ𝛼 (𝜃) ((2 ⟨𝑧, 𝑒1 ⟩ 𝑒−(𝑇−𝑠) 𝜃𝑒1 +⟨𝑧, 𝑒2 ⟩ 𝑒−4(𝑇−𝑠) 𝜃) 𝑒2
0
∞
𝛼
2
+ ∑ 𝑒−𝑛 (𝑇−𝑠) 𝜃 ⟨𝑧, 𝑒𝑛 ⟩ 𝑒𝑛) 𝑑𝜃
𝑛=3
∞
𝛼
= (2 ⟨𝑧, 𝑒1 ⟩ 𝛼 ∫ 𝜃Ψ𝛼 (𝜃) 𝑒−(𝑇−𝑠) 𝜃 𝑑𝜃
0
∞
𝛼
+ ⟨𝑧, 𝑒2 ⟩ 𝛼 ∫ 𝜃Ψ𝛼 (𝜃) 𝑒−4(𝑇−𝑠) 𝜃 𝑑𝜃) 𝑒2
0
∞
Acknowledgment
References
∞
∗
In this paper, abstract results concerning the approximate
controllability of fractional semilinear evolution systems with
infinite delay in a separable reflexive Banach space are
obtained. Approximate controllability result for semilinear
systems is obtained by means of the Krasnoselskii’s fixedpoint theorem under the compactness assumption. It is also
proven that the controllability of the semilinear system is
implied by the approximate controllability of the associated
linear system under some natural conditions. Upon making
some appropriate assumptions, by employing the ideas and
techniques as in this paper, one can establish the approximate
controllability results for a wide class of fractional deterministic and stochastic evolution equations.
N. I. Mahmudov is thankful to the reviewers for making
valuable suggestions leading to the better presentation of this
paper.
𝐵∗ V = (2V1 + V2 ) 𝑒2 + ∑ V𝑛 𝑒𝑛 ,
∗
5. Conclusion
∞
𝛼
2
+ 𝛼 ∑ ∫ 𝜃Ψ𝛼 (𝜃) 𝑒−𝑛 (𝑇−𝑠) 𝜃 𝑑𝜃 ⟨𝑧, 𝑒𝑛 ⟩ 𝑒𝑛
𝑛=3 0
󵄩󵄩
󵄩2
󵄩󵄩(𝑇 − 𝑠)𝛼−1 𝐵∗ A∗𝛼 (𝑇 − 𝑠) 𝑧󵄩󵄩󵄩
󵄩
󵄩
∞
𝛼
= (𝑇 − 𝑠)2(𝛼−1) (2𝛼 ∫ 𝜃Ψ𝛼 (𝜃) 𝑒−(𝑇−𝑠) 𝜃 𝑑𝜃 ⟨𝑧, 𝑒1 ⟩
0
∞
2
𝛼
+𝛼 ∫ 𝜃Ψ𝛼 (𝜃) 𝑒−4(𝑇−𝑠) 𝜃 𝑑𝜃 ⟨𝑧, 𝑒2 ⟩)
0
∞
∞
𝑛=3
𝑛=3 0
∞
2
𝛼
+ (𝑇−𝑠)2(𝛼−1) ∑ (𝛼 ∑ ∫ 𝜃Ψ𝛼 (𝜃) 𝑒−𝑛 (𝑇−𝑠) 𝜃 𝑑𝜃)
2
2
× ⟨𝑧, 𝑒𝑛 ⟩ = 0.
(80)
It follows that ⟨𝑧, 𝑒1 ⟩ = ⟨𝑧, 𝑒2 ⟩ = ⋅ ⋅ ⋅ = ⟨𝑧, 𝑒𝑛 ⟩ = ⋅ ⋅ ⋅ = 0,
and consequently, 𝑧 = 0, which means that (79) is approximately controllable on [0, 𝑇]. Therefore, from Theorem 18,
the system (72) is approximately controllable [0, 𝑇].
[1] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional
Integrals and Derivatives. Theory and Applications, Gordon and
Breach Science, Yverdon, Switzerland, 1993.
[2] R. F. Curtain and H. J. Zwart, An Introduction to InfiniteDimensional Linear Systems Theory, Springer, Berlin, Germany,
1995.
[3] A. Debbouche and D. Baleanu, “Controllability of fractional
evolution nonlocal impulsive quasilinear delay integro-differential systems,” Computers & Mathematics with Applications,
vol. 62, no. 3, pp. 1442–1450, 2011.
[4] K. Balachandran and J. P. Dauer, “Controllability of nonlinear
systems in Banach spaces: a survey,” Journal of Optimization
Theory and Applications, vol. 115, no. 1, pp. 7–28, 2002.
[5] A. E. Bashirov and N. I. Mahmudov, “On concepts of controllability for deterministic and stochastic systems,” SIAM Journal
on Control and Optimization, vol. 37, no. 6, pp. 1808–1821, 1999.
[6] M. Benchohra, L. Górniewicz, and S. K. Ntouyas, Controllability
of Some Nonlinear Systems in Banach Spaces: The Fixed Point
Theory Approach, Pawel Wlodkowicz University College, Plock,
Poland, 2003.
[7] N. Carmichael and M. D. Quinn, “Fixed point methods in
nonlinear control,” in Distributed Parameter Systems (Vorau,
1984), vol. 75 of Lecture Notes in Control and Information
Sciences, pp. 24–51, Springer, Berlin, Germany, 1985.
[8] D. N. Chalishajar, “Controllability of second order impulsive
neutral functional differential inclusions with infinite delay,”
Journal of Optimization Theory and Applications, vol. 154, no.
2, pp. 672–684, 2012.
[9] D. N. Chalishajar and F. S. Acharya, “Controllability of second
order semi-linear neutral impulsive differential inclusions on
unbounded domain with infinite delay in Banach spaces,”
Bulletin of the Korean Mathematical Society, vol. 48, no. 4, pp.
813–838, 2011.
[10] X. J. Li and J. M. Yong, Optimal Control Theory for InfiniteDimensional Systems, Systems & Control: Foundations & Applications, Birkhäuser Boston Publishers, Boston, Mass, USA,
1995.
Abstract and Applied Analysis
[11] N. I. Mahmudov, “Approximate controllability of semilinear
deterministic and stochastic evolution equations in abstract
spaces,” SIAM Journal on Control and Optimization, vol. 42, no.
5, pp. 1604–1622, 2003.
[12] J. P. Dauer and N. I. Mahmudov, “Approximate controllability
of semilinear functional equations in Hilbert spaces,” Journal of
Mathematical Analysis and Applications, vol. 273, no. 2, pp. 310–
327, 2002.
[13] J. Klamka, “Local controllability of fractional discrete-time
semilinear systems,” Acta Mechanica et Automatica, vol. 15, pp.
55–58, 2011.
[14] S. Kumar and N. Sukavanam, “Approximate controllability
of fractional order semilinear systems with bounded delay,”
Journal of Differential Equations, vol. 252, no. 11, pp. 6163–6174,
2012.
[15] N. I. Mahmudov, “Approximate controllability of evolution
systems with nonlocal conditions,” Nonlinear Analysis. Theory,
Methods & Applications, vol. 68, no. 3, pp. 536–546, 2008.
[16] V. Obukhovskiı̆ and P. Rubbioni, “On a controllability problem
for systems governed by semilinear functional differential
inclusions in Banach spaces,” Topological Methods in Nonlinear
Analysis, vol. 15, no. 1, pp. 141–151, 2000.
[17] V. Obukhovski and P. Zecca, “Controllability for systems governed by semilinear differential inclusions in a Banach space
with a noncompact semigroup,” Nonlinear Analysis. Theory,
Methods & Applications, vol. 70, no. 9, pp. 3424–3436, 2009.
[18] R. Sakthivel, J. J. Nieto, and N. I. Mahmudov, “Approximate controllability of nonlinear deterministic and stochastic systems
with unbounded delay,” Taiwanese Journal of Mathematics, vol.
14, no. 5, pp. 1777–1797, 2010.
[19] R. Sakthivel, N. I. Mahmudov, and Juan. J. Nieto, “Controllability for a class of fractional-order neutral evolution control
systems,” Applied Mathematics and Computation, vol. 218, no.
20, pp. 10334–10340, 2012.
[20] R. Sakthivel and Y. Ren, “Approximate controllability of fractional differential equations with state-dependent delay,” Results
in Mathematics, 2012.
[21] R. Sakthivel, Y. Ren, and N. I. Mahmudov, “On the approximate
controllability of semilinear fractional differential systems,”
Computers & Mathematics with Applications, vol. 62, no. 3, pp.
1451–1459, 2011.
[22] Y. Ren, L. Hu, and R. Sakthivel, “Controllability of impulsive
neutral stochastic functional differential inclusions with infinite
delay,” Journal of Computational and Applied Mathematics, vol.
235, no. 8, pp. 2603–2614, 2011.
[23] R. Sakthivel, S. Suganya, and S. M. Anthoni, “Approximate
controllability of fractional stochastic evolution equations,”
Computers & Mathematics with Applications, vol. 63, no. 3, pp.
660–668, 2012.
[24] R. Sakthivel and Y. Ren, “Complete controllability of stochastic
evolution equations with jumps,” Reports on Mathematical
Physics, vol. 68, no. 2, pp. 163–174, 2011.
[25] R. Sakthivel, N. I. Mahmudov, and J. H. Kim, “On controllability
of second order nonlinear impulsive differential systems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 1-2,
pp. 45–52, 2009.
[26] R. Sakthivel, “Controllability of nonlinear impulsive Ito type
stochastic systems,” International Journal of Applied Mathematics and Computer Science, vol. 19, no. 4, pp. 589–595, 2009.
[27] N. Sukavanam and S. Kumar, “Approximate controllability of
fractional order semilinear delay systems,” Journal of Optimization Theory and Applications, vol. 151, no. 2, pp. 373–384, 2011.
11
[28] Z. Yan, “Approximate controllability of partial neutral functional differential systems of fractional order with statedependent delay,” International Journal of Control, vol. 85, no.
8, pp. 1051–1062, 2012.
[29] J. Wang and Y. Zhou, “Complete controllability of fractional
evolution systems,” Communications in Nonlinear Science and
Numerical Simulation, vol. 17, no. 11, pp. 4346–4355, 2012.
[30] J. Wang, Z. Fan, and Y. Zhou, “Nonlocal controllability of semilinear dynamic systems with fractional derivative in Banach
spaces,” Journal of Optimization Theory and Applications, vol.
154, no. 1, pp. 292–302, 2012.
[31] A. Pazy, Semigroups of Linear Operators and Applications to
Partial Differential Equations, vol. 44 of Applied Mathematical
Sciences, Springer, New York, NY, USA, 1983.
[32] Y. Zhou and F. Jiao, “Existence of mild solutions for fractional
neutral evolution equations,” Computers & Mathematics with
Applications, vol. 59, no. 3, pp. 1063–1077, 2010.
[33] Y. Zhou and F. Jiao, “Nonlocal Cauchy problem for fractional
evolution equations,” Nonlinear Analysis. Real World Applications, vol. 11, no. 5, pp. 4465–4475, 2010.
[34] J. Wang and Y. Zhou, “A class of fractional evolution equations
and optimal controls,” Nonlinear Analysis. Real World Applications, vol. 12, no. 1, pp. 262–272, 2011.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014