Directory UMM :Journals:Journal_of_mathematics:EJQTDE:
Electronic Journal of Qualitative Theory of Differential Equations
2010, No. 4, 1-12; http://www.math.u-szeged.hu/ejqtde/
EXISTENCE OF SOLUTIONS OF ABSTRACT
FRACTIONAL IMPULSIVE SEMILINEAR EVOLUTION
EQUATIONS
K. Balachandran∗ and S. Kiruthika∗
Abstract
In this paper we prove the existence of solutions of fractional impulsive semilinear evolution equations in Banach spaces. A nonlocal Cauchy problem is
discussed for the evolution equations. The results are obtained using fractional
calculus and fixed point theorems. An example is provided to illustrate the
theory.
Keywords : Existence of solution, evolution equation, nonlocal condition, fractional calculus, fixed point theorems.
2000 AMS Subject Classification. 34G10, 34G20.
1
Introduction
Fractional differential equations are increasingly used for many mathematical models
in science and engineering. In fact fractional differential equations are considered as
an alternative model to nonlinear differential equations [8]. The theory of fractional
differential equations has been extensively studied by several authors [11, 16-19].
In [12, 14] the authors have proved the existence of solutions of abstract differential
equations by using semigroup theory and fixed point theorem. Many partial fractional
differential equations can be expressed as fractional differential equations in some
Banach spaces [13].
The nonlocal Cauchy problem for abstract evolution differential equation was
first studied by Byszewski [9]. Subsequently several authors have investigated the
problem for different types of nonlinear differential equations and integrodifferential
equations including functional differential equations in Banach spaces [2-4, 10, 20].
Mophou and N’Gu´er´ekata [22, 23, 24] and Balachandran and Park [5] discussed the
existence of solutions of abstract fractional differential equations with nonlocal initial
conditions.
Impulsive differential equations have become important in recent years as mathematical models of phenomena in both physical and social sciences. There has been a
Department of Mathematics, Bharathiar University, Coimbatore-641 046, India.
[email protected]
∗
e-mail:
EJQTDE, 2010 No. 4, p. 1
significant development in impulsive theory especially in the area of impulsive differential equations with fixed moments. Recently Benchohra and Slimani [6,7] discussed
the existence and uniqueness of solutions of impulsive fractional differential equations
and Ahmad and Sivasundaram [1] discussed the existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations.
Motivated by this work we study in this paper the existence of solutions of fractional
impulsive semilinear evolution equations in Banach spaces by using fractional calculus
and fixed point theorems.
2
Preliminaries
We need some basic definitions and properties of fractional calculus which are used
in this paper. By C(J, X) we denote the Banach space of continuous functions x(t)
with x(t) ∈ X for t ∈ J, a compact interval in R and kxkC(J,X) = max kx(t)k.
t∈J
Definition2.1. A real function f (t) is said to be in the space Cα , α ∈ R if there
exists a real number p > α, such that f (t) = tp g(t), where g ∈ C[0, ∞) and it is said
to be in the space Cαm iff f (m) ∈ Cα , m ∈ N.
Definition2.2. The Riemann-Liouville fractional integral operator of order β > 0 of
function f ∈ Cα , α ≥ −1 is defined as
Z t
1
β
(t − s)β−1 f (s)ds
I f (t) =
Γ(β) 0
where Γ(.) is the Euler gamma function.
m
Definition2.3. If the function f ∈ C−1
and m is a positive integer then we define
the fractional derivative of f (t) in the Caputo sense as
Z t
dα f (t)
1
=
(t − s)m−α−1 f m (s)ds, m − 1 < α < m.
α
dt
Γ(m − α) 0
If 0 < α < 1, then
1
dα f (t)
=
α
dt
Γ(1 − α)
Z
0
t
f ′ (s))
ds,
(t − s)α
df (s)
and f is an abstract function with values in X. For basic
where f ′ (s) =
ds
facts about fractional derivatives and fractional calculus one can refer to the books
[15,21,25,26].
Consider the Banach space
P C(J, X) = {u : J → X : u ∈ C((tk , tk+1 ], X), k = 0, . . . , m and there exist
+
−
u(t−
k ) and u(tk ), k = 1, . . . , m with u(tk ) = u(tk )}.
EJQTDE, 2010 No. 4, p. 2
with the norm kukP C = supt∈J ku(t)k. Set J ′ := [0, T ]\{t1 , . . . , tm }.
Consider the linear fractional impulsive evolution equation
dq u(t)
= A(t)u(t), t ∈ J = [0, T ], t 6= tk ,
dtq
∆u |t=tk = Ik (u(t−
k )),
u(0) = u0 ,
(2.1)
where 0 < q ≤ 1 and A(t) is a bounded linear operator on a Banach space X,
Ik : X → X, k = 1, 2, · · · , m and u0 ∈ X, 0 = t0 < t1 < t2 < · · · < tm < tm+1 = T ,
−
+
−
∆u|t=tk = u(t+
k ) − u(tk ), u(tk ) = lim+ u(tk + h) and u(tk ) = lim− u(tk + h) represent
h→0
h→0
the right and left limits of u(t) at t = tk .
Our Eq.(2.1) is equivalent to the integral equation
Z t
1
(t − s)q−1 A(s)u(s)ds, if t ∈ [0, t1 ],
u0 +
Γ(q)
0
k Z
1 X ti
u0 +
(ti − s)q−1 A(s)u(s)ds
(2.2)
u(t) =
Γ(q)
i=1 ti−1
Z t
k
X
1
q−1
(t
−
s)
A(s)u(s)ds
+
Ii (u(t−
if t ∈ (tk , tk+1].
+
i )),
Γ(q)
tk
i=1
Definition 2.4. By a solution of the abstract Cauchy problem (2.1), we mean an
abstract function u such that the following conditions are satisfied:
(i) u ∈ P C(J, X) and u ∈ D(A(t)) for all t ∈ J ′ ;
dq u
(ii) q exists on J ′ where 0 < q < 1;
dt
(iii) u satisfies Eq.(2.1) on J ′ , and satisfy the conditions
∆u|t=tk = Ik (u(t−
k )), k = 1, . . . , m,
u(0) = u0 .
Now, we assume the following conditions to prove the existence of a solution of the
evolution Eq.(2.1).
(HA) A(t) is a bounded linear operator on X for each t ∈ J and the function t → A(t)
is continuous in the uniform operator topology.
(HI) The functions Ik : X → X are continuous and there exists a constant L1 > 0
such that
kIk (u) − Ik (v)k ≤ L1 ku − vk, for each u, v ∈ X and k = 1, 2 · · · , m.
Tq
= γ.
Γ(q + 1)
Theorem 2.1 If the hypotheses (HA) and (HI) are satisfied, then Eq.(2.1) has a
unique solution on J.
For brevity let us take
EJQTDE, 2010 No. 4, p. 3
Proof: The proof is based on the application of Picard’s iteration method. Let
M = max kA(t)k and define a mapping F : P C([0, T ] : X) → P C([0, T ] : X) by
0≤t≤T
Z tk
1 X
(tk − s)q−1 A(s)u(s)ds
F u(t) = u0 +
Γ(q) 0
2010, No. 4, 1-12; http://www.math.u-szeged.hu/ejqtde/
EXISTENCE OF SOLUTIONS OF ABSTRACT
FRACTIONAL IMPULSIVE SEMILINEAR EVOLUTION
EQUATIONS
K. Balachandran∗ and S. Kiruthika∗
Abstract
In this paper we prove the existence of solutions of fractional impulsive semilinear evolution equations in Banach spaces. A nonlocal Cauchy problem is
discussed for the evolution equations. The results are obtained using fractional
calculus and fixed point theorems. An example is provided to illustrate the
theory.
Keywords : Existence of solution, evolution equation, nonlocal condition, fractional calculus, fixed point theorems.
2000 AMS Subject Classification. 34G10, 34G20.
1
Introduction
Fractional differential equations are increasingly used for many mathematical models
in science and engineering. In fact fractional differential equations are considered as
an alternative model to nonlinear differential equations [8]. The theory of fractional
differential equations has been extensively studied by several authors [11, 16-19].
In [12, 14] the authors have proved the existence of solutions of abstract differential
equations by using semigroup theory and fixed point theorem. Many partial fractional
differential equations can be expressed as fractional differential equations in some
Banach spaces [13].
The nonlocal Cauchy problem for abstract evolution differential equation was
first studied by Byszewski [9]. Subsequently several authors have investigated the
problem for different types of nonlinear differential equations and integrodifferential
equations including functional differential equations in Banach spaces [2-4, 10, 20].
Mophou and N’Gu´er´ekata [22, 23, 24] and Balachandran and Park [5] discussed the
existence of solutions of abstract fractional differential equations with nonlocal initial
conditions.
Impulsive differential equations have become important in recent years as mathematical models of phenomena in both physical and social sciences. There has been a
Department of Mathematics, Bharathiar University, Coimbatore-641 046, India.
[email protected]
∗
e-mail:
EJQTDE, 2010 No. 4, p. 1
significant development in impulsive theory especially in the area of impulsive differential equations with fixed moments. Recently Benchohra and Slimani [6,7] discussed
the existence and uniqueness of solutions of impulsive fractional differential equations
and Ahmad and Sivasundaram [1] discussed the existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations.
Motivated by this work we study in this paper the existence of solutions of fractional
impulsive semilinear evolution equations in Banach spaces by using fractional calculus
and fixed point theorems.
2
Preliminaries
We need some basic definitions and properties of fractional calculus which are used
in this paper. By C(J, X) we denote the Banach space of continuous functions x(t)
with x(t) ∈ X for t ∈ J, a compact interval in R and kxkC(J,X) = max kx(t)k.
t∈J
Definition2.1. A real function f (t) is said to be in the space Cα , α ∈ R if there
exists a real number p > α, such that f (t) = tp g(t), where g ∈ C[0, ∞) and it is said
to be in the space Cαm iff f (m) ∈ Cα , m ∈ N.
Definition2.2. The Riemann-Liouville fractional integral operator of order β > 0 of
function f ∈ Cα , α ≥ −1 is defined as
Z t
1
β
(t − s)β−1 f (s)ds
I f (t) =
Γ(β) 0
where Γ(.) is the Euler gamma function.
m
Definition2.3. If the function f ∈ C−1
and m is a positive integer then we define
the fractional derivative of f (t) in the Caputo sense as
Z t
dα f (t)
1
=
(t − s)m−α−1 f m (s)ds, m − 1 < α < m.
α
dt
Γ(m − α) 0
If 0 < α < 1, then
1
dα f (t)
=
α
dt
Γ(1 − α)
Z
0
t
f ′ (s))
ds,
(t − s)α
df (s)
and f is an abstract function with values in X. For basic
where f ′ (s) =
ds
facts about fractional derivatives and fractional calculus one can refer to the books
[15,21,25,26].
Consider the Banach space
P C(J, X) = {u : J → X : u ∈ C((tk , tk+1 ], X), k = 0, . . . , m and there exist
+
−
u(t−
k ) and u(tk ), k = 1, . . . , m with u(tk ) = u(tk )}.
EJQTDE, 2010 No. 4, p. 2
with the norm kukP C = supt∈J ku(t)k. Set J ′ := [0, T ]\{t1 , . . . , tm }.
Consider the linear fractional impulsive evolution equation
dq u(t)
= A(t)u(t), t ∈ J = [0, T ], t 6= tk ,
dtq
∆u |t=tk = Ik (u(t−
k )),
u(0) = u0 ,
(2.1)
where 0 < q ≤ 1 and A(t) is a bounded linear operator on a Banach space X,
Ik : X → X, k = 1, 2, · · · , m and u0 ∈ X, 0 = t0 < t1 < t2 < · · · < tm < tm+1 = T ,
−
+
−
∆u|t=tk = u(t+
k ) − u(tk ), u(tk ) = lim+ u(tk + h) and u(tk ) = lim− u(tk + h) represent
h→0
h→0
the right and left limits of u(t) at t = tk .
Our Eq.(2.1) is equivalent to the integral equation
Z t
1
(t − s)q−1 A(s)u(s)ds, if t ∈ [0, t1 ],
u0 +
Γ(q)
0
k Z
1 X ti
u0 +
(ti − s)q−1 A(s)u(s)ds
(2.2)
u(t) =
Γ(q)
i=1 ti−1
Z t
k
X
1
q−1
(t
−
s)
A(s)u(s)ds
+
Ii (u(t−
if t ∈ (tk , tk+1].
+
i )),
Γ(q)
tk
i=1
Definition 2.4. By a solution of the abstract Cauchy problem (2.1), we mean an
abstract function u such that the following conditions are satisfied:
(i) u ∈ P C(J, X) and u ∈ D(A(t)) for all t ∈ J ′ ;
dq u
(ii) q exists on J ′ where 0 < q < 1;
dt
(iii) u satisfies Eq.(2.1) on J ′ , and satisfy the conditions
∆u|t=tk = Ik (u(t−
k )), k = 1, . . . , m,
u(0) = u0 .
Now, we assume the following conditions to prove the existence of a solution of the
evolution Eq.(2.1).
(HA) A(t) is a bounded linear operator on X for each t ∈ J and the function t → A(t)
is continuous in the uniform operator topology.
(HI) The functions Ik : X → X are continuous and there exists a constant L1 > 0
such that
kIk (u) − Ik (v)k ≤ L1 ku − vk, for each u, v ∈ X and k = 1, 2 · · · , m.
Tq
= γ.
Γ(q + 1)
Theorem 2.1 If the hypotheses (HA) and (HI) are satisfied, then Eq.(2.1) has a
unique solution on J.
For brevity let us take
EJQTDE, 2010 No. 4, p. 3
Proof: The proof is based on the application of Picard’s iteration method. Let
M = max kA(t)k and define a mapping F : P C([0, T ] : X) → P C([0, T ] : X) by
0≤t≤T
Z tk
1 X
(tk − s)q−1 A(s)u(s)ds
F u(t) = u0 +
Γ(q) 0