Applied Mathematical Sciences, Vol. 5, 2011, no. 15, 717 - 727
Basic Riemann-Liouville and Caputo Impulsive Fractional Calculus M. De la Sen Instituto de Investigación y Desarrollo de Procesos. Universidad del País Vasco Campus of Leioa (Bizkaia) – Aptdo.644- Bilbao, Spain
[email protected]
Abstract: This paper gives formulas for Riemann- Liouville impulsive fractional integral calculus and for Riemann- Liouville and Caputo impulsive fractional derivatives. Keywords: Rimann- Liouville fractional calculus, Caputo fractional derivative, Dirac delta, Distributional derivatives, High- order distributional derivatives.
1. Introduction Fractional calculus has been used in a set of applications, mainly, to deal with modelling errors in differential equations and dynamic systems. There are also applications in Signal Processing and sampling and hold algorithms, [1-3]. Fractional integrals and derivatives can be of non-integer orders and even of complex order. This facilitates the description of some problems which are not easily descxribed by ordinary calculus due to modelling errors, [1-5]. There are several approaches for the integral fractional calculus, the most popular ones being the Riemann-Liouville fractional integral. There is also a fractional Riemann- Liouville derivative. However, the wellknown Caputo fractional derivative are less involved since the associated integral operator manipulates the derivatives of the primitive function under the integral symbol. This paper extends the basic fractional differ-integral calculus to impulsive functions described through the use of Dirac distributions and Dirac distributional derivatives, [5], of real fractional orders. In the general case, it is admitted a presence of infinitely many impulsive terms at certain isolated point of the relevant function domains.
718
M. De la Sen
2. Extended Riemann- Liouville fractional integral Let us denote the set of positive real numbers by R + = { r ∈ R : r > 0 } and left-sided and right-sided Lebesgue integrals, respectively, as: x ∫ 0 g (τ ) dτ : =
t lim ∫ 0 g (τ ) dτ (the identification x ≡ x −
−
t → x≡ x
is used for all x in order to
simplify the notation), and +
x t ∫ 0 g (τ ) dτ : = lim+ ∫ 0 g (τ ) dτ t→ x
Now, consider real functions f , f : R + → R ,such that ∫ 0x (x − t ) μ −1 f (t )dt exists, ∀ x ∈ R + , fulfilling: f ( x ) = f (x ) + ∑ x ∈ IMP K i δ (x − x i ) = f (x ) + ∑i ∈ I (∞ ) K i δ (x − x i ) i
denotes the Dirac delta distribution, K iδ (0 ) = f (x i+ ) − f (x i ) with K i ∈ R ; ∀i ∈ I (∞ ) ⊂ Z + , [5], and IMP : = U IMP (x ) = U IMP x + of indexing set I (∞ ) is the whole
δ (x )
x ∈R +
impulsive set defined via denumerable sets:
{
( )
x ∈R +
empty or non-empty) partial impulsive strictly ordered
( )
IMP (x ) : = x i ∈ R + : f x i+ − f (x i ) = K iδ (0 ), x i < x
}
(1)
( )
of indexing set I (x ): = { i ∈ Z 0+ : x i ∈ IMP (x ) }⊂ I x + ⊂ Z + , for each x ∈ R + ; and
( )
( ): = {x
IMP x ⊂ IMP x
+
i ∈R + :
( ) − f (x
f x
( ) {
+ i
i
) = K iδ (0) , x i ≤ x + }⊂ R +
(2)
( )}
of indexing set I ( x ) ⊂ I x + : = i ∈ Z 0+ : x i ∈ IMP x + ⊂ Z + , for each x ∈ R + . with the indexing set of IMP being I (∞ ) = U I (x ) = U I (x + ) . If we are interested in
( )
x ∈ IMP ( x )
x ∈ IMP x +
studying the fractional derivative of the impulsive function f : R + → R then f : R + → R is non- uniquely defined as f (x ) = f (x ) for x ∈ R + \ IMP , and f (x i ) = f (x i ) ,
( )
( ) ( )
f x i+ = f (x i ) + K i δ (0) = f (x i ) + K i δ (0 )
, for x i ∈ IMP with f x + ∈ R ( non-uniquely) defined being bounded arbitrary (for instance, being zero or f x + = f (x ) ) if x∈ IMP . Note that IMP and I (∞ ) have infinite cardinals if there are infinitely many impulsive values of the function f(t). Note ∫ 0 (x − t ) of x
that μ −1
the
existence
f (t )dt = ∫ 0 (x − t ) μ − 1 f (t )dt x
of
∫ 0 (x − t ) x
i
f (t )dt
implies
that
of
if x ∉ IMP (x ) , since ∫ 0x (x − t ) μ −1 f (t )dt exists, and that
x x μ −1 μ −1 μ −1 ∫0 (x − t ) f (t )dt = ∫0 (x − t ) f (t )dt + (x − x i ) ( f + i
μ −1
(x )− f (x )) if x i ∈ IMP (x + ) + i
i
(3)
Theorem 2.1. The extended fractional Riemann- Liouville integrals by considering impulsive functions are defined for any fixed order μ ∈ R+ and all x ∈ R+ by
Basic Riemann-Liouville and Caputo impulsive fractional calculus
(J μ f )(x) := Γ (1μ ) ∫ = =
(x − t ) μ −1 f (t ) d t
x 0
⎛ ⎜ Γ ( μ ) ⎜⎝ 1
719
∫0 ( x − t ) x
μ −1
f (t ) d t +
∑ (x − x i ) μ −1 ( f
i ∈ I ( x)
(x )− f (x )) ⎞⎟⎟ + i
i
⎠
⎛ ⎞ x i +1 ⎜ (x − t ) μ − 1 f (t ) d t + ∫ xx + (x − t ) μ − 1 f (t ) d t + ∑ (x − x i ) μ −1 f x i+ − f (xi ) ⎟⎟ ∑ + ∫ x ⎜ n (x) Γ ( μ ) ⎝ i ∈ I ( x) ∪ {0 } i i∈ I ( x) ⎠
( ( )
1
)
(4)
(J μ f )(x ):= Γ (1μ ) ∫ +
= =
⎛ ⎜ Γ ( μ ) ⎜⎝ 1
x+ 0
(x − t ) μ −1 f (t ) d t
x μ −1 μ −1 ∫0 (x − t ) f (t ) d t + ∑ (x − x i ) ( f i∈ I ( x+ )
⎞
(x )− f (x )) ⎟⎟ + i
i
⎠
⎞ ⎛ x i +1 ⎜ ( x − t ) μ − 1 f (t ) d t + ∑ (x − x i ) μ −1 f x i+ − f (xi ) ⎟ ∑ + ∫ x ⎟ Γ ( μ ) ⎜⎝ i ∈ I ( x + ) ∪ { 0 } i i∈I ( x+ ) ⎠
( ( )
1
)
(5)
(J f )(x ) ≡ (J f )( x) := f (x) 0
+
0
where Γ : R 0+ → R + is the Γ - function , [1-5] and n (x ) = card I (x ) = card IMP (x ) . □
n : IMP → Z
+
is defined by
Note that if x∈ IMP then ⎛
(J μ f )(x ) = Γ (1μ ) ⎜⎜ +
∑
x
∫x
⎝ i∈ I (x+ ) ∪ {0 } f (x ) + x − x n ( x )
i +1 + i
(x − t ) μ − 1 f (t ) d t + ∑ (x − x i ) μ −1 ( f i∈I (x+ )
( ) ( ) ( f (x )− f (x ( ) )) 1 ⎛⎜ (x − t ) μ f (t ) d t + ≠ ( J μ f ) (x ) = ∑ ∫ ⎜ Γ (μ ) = J
μ
μ −1
+ n(x )
⎝ i ∈ I ( x) ∪ {0 }
and if x ∉ IMP , since
+
I ( x )= I( x ) ,
−1
+ i
(
then J
+ i
i
⎠
n x
x i +1
x
⎞
(x ) − f (x ))⎟⎟
μ
∑ (x − x i ) μ −1 ( f
i ∈ I ( x)
)( ) = (J f )(x) .
f x
+
(x )− f (x ))⎞⎟⎟ + i
i
⎠
(6)
μ
3. Extended Riemann- Liouville fractional derivative Assume that f ∈C m−1 (R + , R ) and its m − th derivative exists everywhere in R + . Then, the Caputo fractional derivative of order μ ≥ 0 with m − 1 ≤ μ (∈ R + ) ≤ m , m∈ Z + is for any x∈ R + :
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M. De la Sen
(D f ) (x ) := ⎛⎜⎜ ddx ⎞⎟⎟ ( J m
μ
m−μ
f
)(x ) = Γ ( m1− μ ) ⎛⎜⎜ ddx ⎞⎟⎟ m ⎛⎜⎝ ∫ 0x (x − t ) m−μ −1 f (t )d t ⎞⎟⎠
⎝ ⎠ ⎠ ⎝ The following particular cases follow from this formula for μ = m − 1 :
(D f ) (x ) = ∫ f (t ) d t x
−1
(a) μ = −1; m = 0 yields
0
(7)
which is the standard integral of
the
function f . This case does not verifies the “derivative constraint ” 0 ≤ m − 1 ≤ μ (∈ R + ) < m leading to an integral result. (b) ) μ = 0 ; m = 1 yields D 0 f (x ) = f ( x ) which so that D 0 f is the identity operator
(
(
)
)
(c) μ = 1; m = 2 yields D 1 f (x ) = f (1) (x )
(
)
(d) μ = 2 ; m = 3 yields D 2 f (x ) = f (2 ) (x ) which is the standard first- derivative of the function f. Compared to the parallel cases with the Caputo fractional derivative, note that the Riemann- Liouville fractional derivative, compared to the Caputo corresponding one, does not depend on the conditions at zero of the function and its derivatives. Define the Kronecker delta δ (a ,b ) of any pair of real numbers (a ,b ) as δ (a ,b ) = 1 if a = b and δ (a ,b ) = 0 if a ≠ b and then evaluate recursively the Riemann – Liouville fractional derivative of order μ ≥ 0 from the above formula by using Leibniz´s differentiation rule by noting that , since μ ≠ m − j ; ∀ j (∈ Z + ) > 1 , only the differential part corresponding to the differentiation of the integrand is non zero for j > m − μ . This yields the following: Theorem 3.1. Assume that f ∈C m− 2 (R + , R ) and f (m −1) exists everywhere in R+ and that f (t ) is integrable on R + , then:
(D f ) (x) = Γ (m1− μ ) ⎛⎜⎜⎝ ddx ⎞⎟⎟⎠ μ
= = =
1 Γ (m − μ
1 Γ (m − μ
m
⎛⎜ x (x − t ) m − μ −1 f (t ) d t ⎞⎟ ⎝ ∫0 ⎠
⎛ d ⎞ ⎜ ⎟ ) ⎜⎝ d x ⎟⎠
m −1
⎡ x (m − μ − 1)(x − t ) m − μ − 2 f (t ) d t + f (x )δ (μ , m − 1)⎤ ⎢⎣ ∫ 0 ⎥⎦
⎡ (m −1) ⎛ ⎞ (x )δ (μ , m − 1) + ⎜⎜ d ⎟⎟ ⎢f ) ⎢⎣ ⎝ d x⎠
m −1
⎛⎜ x (m − μ − 1)(x − t ) m − μ − 2 f (t ) d t ⎞⎟⎤⎥ ⎝ ∫0 ⎠⎥⎦
⎡ ⎤ ⎛ m−2 m−1 ⎞ ⎡m−1 ⎤ x 1 ⎢ f (m−1) (x )δ (μ , m − 1)+ ⎜ ∑ ∏ [ j − μ ]⎟ f (i ) (x )δ (μ , m − i ) + ⎢∏ [ j − μ ]⎥ ⎛⎜ ∫ (x − t ) −(μ + 1) f (t )d t ⎞⎟⎥ 0 ⎜ i =1 j = i +1 ⎟ ⎠⎥ Γ (m − μ ) ⎢ ⎢⎣ j =0 ⎥⎦ ⎝ ⎝ ⎠ ⎣ ⎦
(8) If f ∈ PC ( R + , R ) being discontinuous of first class ( m −1 j ( x )) (x ) = δ (x ) with j (x ) = m − 1 − k (x ) , one uses to obtain the right value of (8) then f the perhaps high-order distributional derivatives formula: k
( )
with f (k ) (x )
( )
(− 1) k k ! f (m −1− k ) x + − f (m −1− k ) (x ) δ (0) = ∞ f (m −1) x + − f (m −1)(x ) = xk
(9)
Basic Riemann-Liouville and Caputo impulsive fractional calculus to yield
(D f )(x ) = Γ (m1− μ ) ⎡⎢⎢ μ
+
(− 1) k ( x ) k ( x )!
⎣
x
k( x)
721
( )
f (m −1− k ( x )) x + − f (m −1− k ( x )) (x ) δ (0 )δ (μ ,m − 1)
⎤ ⎛ m − 2 m −1 ⎞ ⎡m −1 ⎤ x + ⎜ ∑ ∏ [ j − μ ]⎟ f (i ) (x )δ (μ ,m − i ) + ⎢ ∏ [ j − μ ]⎥ ⎛⎜ ∫ 0 (x − t ) − (μ + 1) f (t )d t ⎞⎟⎥ ⎜ i =1 j = i +1 ⎟ ⎠⎥ ⎢⎣ j = 0 ⎥⎦ ⎝ ⎝ ⎠ ⎦
(10)
If μ = m − 1 then ⎡m −1 ⎤ x (11) = f (m −1) (x ) + ⎢ ∏ [ j − μ ]⎥ ⎛⎜ ∫ (x − t ) − m f (t ) d t ⎞⎟ 0 ⎝ ⎠ ⎢⎣ j = 0 ⎥⎦ x provided that ⎛⎜ ∫0 (x − t ) −(μ + 1) f (t ) d t ⎞⎟ exists for x∈ R + (which is guaranteed if f (t ) is ⎝ ⎠
(D
m −1
f
) (x )
Lebesgue-integrable on R + ), f ∈C m−2 (R + , R ) and f m−1 exists everywhere in R + . The correction (10) applies when the derivative does not exist. □ If μ ≠ m − 1 with m − 1 ≤ μ (∈ R + ) ≤ m then after defining the impulsive sets, its associated indexing sets and the function f : R + → R as for the extended Riemann- Liouville fractional integral, one gets:
(D f ) ( x ) μ
=
⎡ m −1 ⎤ x − (μ + 1) f (t ) d t ⎞⎟ ⎢ ∏ [ j − μ ]⎥ ⎛⎜ ∫0 (x − t ) ⎠ ) ⎢⎣ j =0 ⎥⎦ ⎝ ⎡m −1 ⎤ 1 = ⎢ ∏ [ j − μ ]⎥ Γ ( m − μ ) ⎢⎣ j =0 ⎥⎦
1 Γ (m − μ
⎛ ⎞ x i +1 x ×⎜ ( x − t ) − (μ + 1) f (t ) d t + ∫ + ( x − t ) −(μ + 1) f (t ) d t + ∑ (x − x i ) − (μ + 1) f x i+ − f (xi ) ⎟ ∑ + ∫ x x ⎜ i ∈ I ( x) ∪ {0 } i ⎟ n (x) i ∈ I ( x) ⎝ ⎠
( ( )
)
(12) ⎡ m −1
(D μ f )(x ) = Γ ( m1− μ ) ⎢∏ [ j − μ ]⎤⎥ +
⎢⎣ j =0
⎛ ×⎜ ⎜ ⎝
∫0 ( x − t ) x
− ( μ + 1)
⎥⎦
f (t ) d t +
∑ (x − x i ) −(μ + 1) ( f
i ∈ I ( x)
(x )− f (x )) ⎞⎟⎟ + i
i
(13)
⎠
4. Extended Caputo fractional derivative Assume that f ∈C m−1 (R + , R ) and its m − th derivative exists everywhere in R + . Then, the Caputo fractional derivative of order μ ≥ 0 with m − 1 ≤ μ (∈ R + ) < m , m∈ Z + is for any x∈ R + :
(D μ f ) ( x ) : = ( J *
m−μ
)
f (m ) ( x ) =
1
Γ (m − μ )
∫ 0 (x − t ) x
m − μ −1
f (m ) (t ) d t
(14) ;
x∈ R +
m −1 ≤ μ < m ,
m∈ Z + ,
722
M. De la Sen
The following particular cases occur with μ = m − 1 leading to
(D
m −1 *
)
( )
x f (x ) = ∫ 0 f (m ) (t ) dt = f (m −1) (x ) − f (m −1) 0 +
(
)
(15)
( )
(a) μ = −1; m = 0 yields D *−1 f (x ) = f (−1) (x ) − f (−1) 0 + which is an integral result f . Note that this case does not verifies the “derivative constraint” 0 ≤ μ (∈ R + ) < m leading to an integral result. (b) ) μ = 0 ; m = 1 yields D *0 f (x ) = f (0 ) (x ) − f (0 ) 0 + = f (x ) − f 0 + (c) μ = 1; m = 2 yields D *1 f (x ) = f (1) (x ) − f (1) 0 + (d) μ = 2 ; m = 3 yields D *2 f (x ) = f (2 ) (x ) − f (2 ) 0 +
(
( ) ( )
)
( ) ( ) ( )
( )
We can extend the above formula to real functions with impulsive m − th derivative as follows. Assume that f ∈C m − 2 (R + , R ) with bounded piecewise (m − 1) − th derivative existing
everywhere
(
in
R+
and
( )
)
d m f (x) f (m ) ( x ) ≡ d xm
being
impulsive
with
f (m ) ( x i ) = K i δ (0 ) = f (m −1) x i+ − f (m −1) ( x i ) δ (0 ) ; ∀ xi ∈ IMP
, equivalently , ∀ i ∈ I ( ∞ ) , at the eventual discontinuity points x i > 0 at the impulsive set IMP : = U IMP (x ) , where the x ∈R +
partial impulsive sets are re-defined as follows:
{
( ) ( )
}
( ) } ( )
IMP (x ) : = x i ∈ R + : f (m −1) x i+ − f (m −1) (x i ) = K i , x i < x ⊂ IMP x +
( ) {
IMP x
+
(16) (17)
: = x i ∈ R + : f (m −1) x i+ − f (m −1) (x i ) = K i , x i ≤ x + ⊂ IMP x +
Now, consider
f ∈C m −1 (0 , ∞
) with
d m f (x) being almost everywhere f (m ) ( x ) ≡ d xm
piecewise continuous in R + except possibly on a non-empty discrete impulsive set IMP .Define a non-impulsive real function f : R + → R defined as f (m ) (x ) = f (m ) (x ) for x ∈ R + \ IMP , and f (m )(x i ) = f (m ) (x i ) , f (m )(x i ) = f (m ) (x i ) + K i δ (0) for x i ∈ IMP with f (m ) x + = f (m ) (x ) ; x ∈ IMP ( defined being bounded arbitrary (for instance, zero) if x∈ IMP . Through a similar reasoning as that used for Riemann- Liouville fractional integral by replacing the function f : R + → R by its m-th derivative, one obtains the following result:
( )
Theorem 4.1. The Caputo fractional derivative satisfying m − 1 < μ ≤ m ; m∈ Z + and all x ∈ R+ is given below:
(D μ f )(x) := Γ (m1− μ ) ∫ *
x 0
of
order
(x − t ) m− μ −1 f (m )(t ) d t
= 1 Γ (m − μ
=
⎛
⎜ (x − t ) ) ⎜⎝ ∫ 0
1 Γ (m − μ )
x
m− μ −1
f (m )(t ) d t +
∑ (x − x i ) m−μ −1 ( f
i∈I ( x)
(m−1) x + − f (m−1) ( x ) δ (x − x ) ⎞⎟ i i i
( )
)
⎟ ⎠
μ ∈ R+
Basic Riemann-Liouville and Caputo impulsive fractional calculus
723
⎛ ⎞ x i +1 x m − μ −1 m − μ − 1 (m ) ×⎜ f (t ) d t + ∫ + (x − t ) m − μ −1 f (m )(t ) d t + ∑ (x − x i ) f (m−1) xi+ − f (m −1) ( xi ) δ (x − xi )⎟ ∫ x i+ (x − t ) x n (x) ⎜ i ∈ I ( x∑ ⎟ ) ∪ {0 } i∈ I ( x) ⎝ ⎠
(
(D μ f )(x ):= Γ (m1− μ ) ∫ +
*
= =
1 Γ (m − μ 1 Γ (m − μ
⎛ ⎜ ) ⎜⎝
( )
)
(18) x 0
+
(x − t ) m− μ −1 f (m ) (t ) d t
m − μ − 1 (m ) f (t ) d t + ∑ (x − x i ) m − μ −1 ( f ∫0 ( x − t ) x
i∈ I ( x+ )
( )
⎞
)
(m −1) x + − f (m −1) ( x ) δ (x − x ) ⎟ i i i
⎟ ⎠
⎞ ⎛ x i +1 ⎜ ( x − t ) m − μ − 1 f (m )(t ) d t + ∑ (x − x i ) m − μ −1 f (m −1) x i+ − f (m −1) ( x i ) δ (x − xi )⎟ ∑ + ∫ ⎟ ) ⎜⎝ i ∈ I ( x + ) ∪ {0 } x i i∈I (x+ ) ⎠
(
( )
)
(19) where n : IMP → Z + is a discrete function defined by n (x ) = card I (x ) = card IMP (x ) . □ Note that if x∈ IMP then
(D μ f )(x ) = Γ ( m1− μ ) +
*
⎞ ⎛ x i +1 ( x − t ) m − μ − 1 f (m )(t )d t + ∑ (x − x i ) m − μ −1 f (m −1) x i+ − f (m −1) ( x i ) δ (x − xi )⎟ ×⎜ ∑ + ∫ ⎟ ⎜ i∈ I ( x+ ) ∪ {0 } x i i∈ I (x+ ) ⎠ ⎝ = D *μ f (x ) + x − x n ( x ) m − μ −1 f (m −1) x +n ( x ) − f (m −1) x n ( x ) δ (0 )
(
(
(
≠ D *μ f
)
(
)
(
(
( )
)
)
))
(
) (x ) = Γ ( m1− μ )
⎛ ⎞ x i +1 ( ×⎜ x − t ) m − μ − 1 f (m )(t ) d t + ∑ (x − x i ) m − μ −1 f (m −1) x i+ − f (m −1) ( x i ) δ (x − xi )⎟ ∑ + ∫ ⎜ i ∈ I ( x) ∪ {0 } x i ⎟ i ∈ I ( x) ⎝ ⎠
(
and if x ∉ IMP , since
+
I ( x )= I( x ) ,
then
( )
)
(D μ f )(x ) = (D μ f )(x) . The above formalism +
*
*
f (m −1) : R + → R is piecewise continuous with isolated first- class
applies when discontinuity points, that is f ∈ PC m−1 (R + , R ) implying that f ∈C m−2 (R + , R ) . A more general situation arises when the discontinuities can point-wise arise for points of the function itself of for any successive derivative up- till order m. This would lead to a more general description than that given as follows. Define partial sets of positive integers as k := {1, 2 , .... , k } Assume that f ∈ PC j (R + , R ) and x is a discontinuity point of first class of f ( j ) (x ) for some j ∈ m − 1 ∪ { 0 } . Then , f ( j +l ) (x ) are impulsive for l∈ m − j of high order being increasing with l . Define the (j+1) – th impulsive sets of the function f on (0, x )⊂ R as:
{
( )
}
IMPj +1 (x ): = z ∈ R+ : z < x , 0 < f ( j ) z + − f ( j )(z ) < ∞ ; j ∈m − 1 ∪ { 0 } , x ∈ R +
This leads directly the definition of the following impulsive sets:
(20)
724
M. De la Sen
{
}
( )
IMPj +1 : = x ∈ R+ : 0 < f ( j ) x + − f ( j )(x ) < ∞ ≡ U x ∈ R IMPj +1 (x ) + ( ( j) + j) IMP : = x ∈ R : 0 < f x − f (x ) < ∞ , some j ∈ m − 1 ∪ { 0 } ≡
{
( )
+
}
(21)
(
U x∈ R + U j ∈m −1∪ { 0 } IMPj +1 (x )
)
(22)
( )
which can be empty . Thus , if z ∈ IMPj +1 then f ( j −1) x + = f ( j −1) (x ) exists with identical left and right limits, f ( j ) x + − f ( j ) (x ) = K = K (x ) ≠ 0 and f ( j ) (x ) = Kδ (0) with successive
( )
higher-order derivatives represented by higher- order Dirac distributional derivatives The above definitions yield directly the following simple results: Assertion 5.2. x ∈ IMP ⇒ x ∈ IMPj for a unique j = j (x ) ∈ m . Proceed
Proof:
by
contradiction.
i , j ( ≠ i )∈ m − 1 ∪ { 0 } .Then:
( )
Assume
(
that
x ∈ IMPi +1 ∩ IMPj +1
)
for
( )
0 < f (i ) x + − f (i )(x ) < ∞ ; 0 < f ( j ) x + − f ( j )(x ) < ∞
Assume with no loss of generality that j = i + k > i for some k (≤ m − i − 1)∈ Z + . Then,
( )
( )
( )
(− 1) k k ! f (i ) x + − f (i ) (x ) δ (0) = ∞ f ( j ) x + − f ( j )(x ) = f (i + k ) x + − f (i + k )(x ) = xk
( )
( )
with x ∈ R + . If f (i ) x + − f (i ) (x ) ≠ 0 which contradicts 0 < f (i ) x + − f (i )(x ) < ∞ so that i = j .□ ⎛
( )
⎞
Assertion 5.3. x ∈ IMP ⇒ ⎜⎜ x ∈ IMPj ⇔ ∃ a unique j = j (x ) = max f (i −1) x + − f (i −1) (x ) < ∞ ⎟⎟ . ⎜ ⎟ ⎝
( )
i∈m
Furthermore, such a unique j=j(x) satisfies f ( j −1) x + − f ( j −1) (x ) > 0 .
⎠
Proof: The existence is direct by contradiction. If ¬ ∃ j = j (x ) ∈ m − 1 ∪ { 0 } such that
( )
f ( j ) x + − f ( j ) (x ) < ∞
then x ∉ IMP . Now, assume there exist two nonnegative integers
( )
i = i (x ) = f (i −1) x + − f (i −1) (x ) < ∞
and
( )
j = j (x ) = i + k = f (i + k −1) x + − f (i + k −1) (x ) < ∞ ;
for
some k ∈ m − i . But for x > 0 , ∞=
(− 1) k k ! x
k
( )
( )
f (i −1) x + − f (i −1) (x ) δ (0 ) = f (i + k −1) x + − f (i + k −1) (x ) < ∞
which is a contradiction. Then, x ∈ IMPj ⇒ ∃ j = j (x ) = max
( )
f (i −1) x + − f (i −1) (x ) < ∞ which
i∈m
( )
is unique. Also, from the
definition of the impulsive sets IMP i (x ) , f ( j −1) x + − f ( j −1) (x ) < ∞ ⇒ x ∈Ui ∈ j ∪ { 0 } IMP i (x ) .
Now , assume that x ∈Ui ∈ j −1∪ { 0 } IMP i (x ) .
Basic Riemann-Liouville and Caputo impulsive fractional calculus
( )
725
( )
Thus, 0 < f ( j −1) x + − f ( j −1) (x ) < ∞ ⇒ f ( j ) x + − f ( j ) (x ) = ∞ from the definition of the impulsive j = j (x ) = max
sets.
Then, x ∈ IMP j (x ) .
The
( )
opposite
logic
implication
f (i −1) x + − f (i −1) (x ) < ∞ ⇒ x ∈ IMPj is proved. Then, it has been fully proved
i∈m
that ⎞ ⎟ f (i −1) x + − f (i −1) (x ) < ∞ ⎟ . ⎟ ⎠
⎛ ⎜ x ∈ IMP ⇒ ⎜ x ∈ IMPj ⇔ ∃ a unique j = j (x ) = max ⎜ i∈m ⎝
( )
Now, establish again a contradiction by assuming that
( )
( )
j = j (x ) = f (k −1) x + − f (k −1) (x ) = max
f (i −1) x + − f (i −1) (x ) = 0 < ∞ ; ∀ k ∈ m
what contradicts x ∈ IMP . This proves that the unique j=j(x) implying and being implied
( )
by x ∈ IMPj satisfies f ( j −1) x + − f ( j −1) (x ) > 0 . □ Using the necessary – high order distributional derivatives, one gets that
(
( ) ) f ( ) (x )− f ( ) (x ) < ∞ .
(− 1) m− j (m − j )! f ( j ) x + − f ( j ) (x ) δ (0) ; x ∈ IMP ⇒ f (m ) (x ) = x m− j
uniquely defined so that 0
