The sequential Henstock-Kurzweil delta integral on time scales

Definition 2.1

(Definition 1.1, [17]) We say T is a time scale, provided T ⊂ R ( T ≠ ∅ ) and T is closed.

Definition 2.2

(Definition 1.6, 1.7, 1.8, [17]) We define the forward and backward operator on T by σ ( s ) = inf { q ∈ T : q > s } and ρ ( s ) = sup { q ∈ T : q < s } , where inf ∅ = sup T , sup ∅ = inf T .

Definition 2.3

(Definition 1.1, [5]) For any s ∈ T , s is called right-scattered (left-scattered), provided σ ( s ) > s ( ρ ( s ) < s ) . s ∈ T is called right-dense if s < sup T and σ ( s ) = s , while if s > inf T and ρ ( s ) = s , we say s ∈ T is left-dense.

Definition 2.4

(Definition 1.25, [17]) For any s ∈ T , we define the right-graininess and left-graininess function on T by μ ( s ) = σ ( s ) − s , ν ( s ) = s − ρ ( s ) .

Definition 2.5

(Definition 1.1.9, [6]) If sup T = ∞ , let T K ≔ T , while if sup T < ∞ , let T K ≔ T \ ( ρ ( sup T ) , sup T ] .

Definition 2.6

(Definition 1.3, [30]) Let h : T → R and s ∈ T K , for any ε > 0 , there exists h Δ ( s ) ∈ R and there is a neighborhood N ( s , ε ) of s , we have

for all x ∈ N ( s , ε ) . We say h Δ ( s ) is the delta derivative of h at s and we call h is delta differentiable on T if h Δ ( s ) exists for every s ∈ T K .

Theorem 2.1

(Theorem 1.16, [5]) Assume h : T → R , for any q ∈ T K , we have the following properties:

1. If h is delta differentiable at q, then h is continuous at q.

2. If q is right-scattered and h is continuous at q, then h is delta differentiable at q with

   h Δ ( q ) = h ( σ ( q ) ) − h ( q ) σ ( q ) − q .

3. If q is right-dense, then h is delta differentiable at q iff the limit

   lim u → q h ( u ) − h ( q ) u − q

   is finite. Further

   h Δ ( q ) = lim u → q h ( u ) − h ( q ) u − q .

Definition 2.7

(Definition 1.15, [7]) Assume h : T → R , we say that h is right-dense (rd)-continuous if h is continuous at all right-dense points in T and all its left-dense points’ left side limits are finite, we denote all rd-continuous functions on T as C r d ( T ) . If h is delta differentiable on T and h Δ ( s ) is rd-continuous, we denote h ∈ C r d 1 ( T ) .

Definition 2.8

(Definition 1.5, [30]) A partition P for [ a , b ] T is defined by

with s i > s i − 1 , we call s i end points, ξ i tag points, and s i , ξ i ∈ [ a , b ] T , sometimes we write P = { ( [ s i − 1 , s i ] , ξ i ) } i = 1 n .

Definition 2.9

(Definition 1.4, 1.6, [30]) For any s ∈ [ a , b ) T , we say δ = ( δ L , δ R ) is a δ -gauge for [ a , b ] T if δ R ( s ) > 0 on [ a , b ) T , δ R ( b ) ⩾ 0 , δ R ( s ) ⩾ μ ( s ) , and δ L ( s ) > 0 on ( a , b ] T , δ L ( a ) ⩾ 0 . For any δ -gauge for [ a , b ] T , we say a partition P is a δ -fine partition, provided

Theorem 2.2

(Lemma 1.9, [30]) If δ is a δ -gauge for [ a , b ] T , there exists a δ -fine partition P for [ a , b ] T .

Definition 2.10

(Definition 1.2.2, [21]) Assume h : [ a , b ] → R , we say that h is Henstock-Kurzweil integrable on [ a , b ] if there exists a number V ∈ R and for any ε > 0 , there exists a δ ( x ) > 0 for every δ -fine partition P , we have ∣ S ( h , P ) − V ∣ < ε , and we write ( H ) ∫ a b h d x = V .

Definition 2.11

(Definition 9, [35]) Let h : [ a , b ] → R , we say that h is sequential Henstock-Kurzweil integrable on [ a , b ] if there exists a number V ∈ R and for any ε > 0 , there exists a sequence of positive functions { δ k } k = 1 ∞ such that for every δ k -fine partition P k , its end points are { { a = s 0 k < s 1 k < … < s m k k = b } } k = 1 ∞ , its tag points are { { ξ 1 k , ξ 2 k , … , ξ m k k } } k = 1 ∞ , we have ∣ S ( h , P k ) − V ∣ < ε ( k → ∞ ) , and we write ( S H ) ∫ a b h d x = V .

Remark 2.1

We say that h is not sequential Henstock-Kurzweil integrable on [ a , b ] if for any V ∈ R and there exists a ε 0 > 0 , we cannot find a sequence of positive functions { δ k } k = 1 ∞ satisfying ∣ S ( h , P k ) − V ∣ < ε 0 ( k → ∞ ) , where P k is a δ k -fine partition for [ a , b ] .

Example 2.1

Let h : [ 0 , 1 ] → R , then we define

We claim that h is sequential Henstock-Kurzweil integrable with

Let ε > 0 be given, assume { δ k } k = 1 ∞ is a decreasing sequence on [ 0 , 1 ] , let K be large enough such that 1 K < ε , we define

When k > K , for every δ k -fine partition P k , the tag points ξ have two cases :

If ξ = m n , h ( ξ ) = 1 n , if ξ ∈ ( 0 , 1 ) \ Q ∪ { 0 , 1 } , h ( ξ ) = 0 , thus

Therefore, ( S H ) ∫ 0 1 h ( x ) d x = 0 .

Theorem 2.3

(Theorem 1, [35]) Let h : [ a , b ] → R , h is Henstock-Kurzweil integrable on [ a , b ] if and only if h is sequential Henstock-Kurzweil integrable on [ a , b ] .

Remark 2.2

When h is Henstock-Kurzweil integrable on [ a , b ] , by Definition 2.10, there exists a function δ ( x ) > 0 on [ a , b ] . Then, we let δ k = δ . It is easy to show that h is sequential Henstock-Kurzweil integrable on [ a , b ] . Conversely, if h is sequential Henstock-Kurzweil integrable on [ a , b ] , by Definition 2.11, there exists a sequence of positive functions { δ k } k = 1 ∞ and a positive integer K , let δ ( x ) = δ K + 1 ( x ) . Obviously, h is Henstock-Kurzweil integrable on [ a , b ] .

Definition 2.12

(Definition 1.7, [30]) Let h : [ a , b ] T → R , we say that h is Henstock-Kurzweil delta integrable on [ a , b ] T if there exists a number V ∈ R and for any ε > 0 , there exists a δ -gauge δ = ( δ L , δ R ) , for each δ -fine partition P , we have ∣ S ( h , P ) − V ∣ < ε , and we write ( H D ) ∫ a b h Δ s = V .

Definition 3.1

Assume h : [ a , b ] T → R , we say that h is sequential Henstock-Kurzweil delta integrable on [ a , b ] T if there exists a number V ∈ R and for any ε > 0 , there exists a sequence of δ -gauges { δ k = ( δ L k , δ R k ) } k = 1 ∞ such that for every δ k -fine partition P k , its end points are

its tag points are { { ξ 1 k , ξ 2 k , … , ξ m k k } } k = 1 ∞ , we have ∣ S ( h , P k ) − V ∣ < ε ( k → ∞ ) , we write ( S H D ) ∫ a b h Δ s = V and h ∈ S H D ( [ a , b ] T ) .

Remark 3.1

We say that h is not sequential Henstock-Kurzweil delta integrable on [ a , b ] T if for any V ∈ R and there exists a ε 0 > 0 , we cannot find a sequence of δ -gauges { δ k } k = 1 ∞ satisfy ∣ S ( h , P k ) − V ∣ < ε 0 ( k → ∞ ) , where P k is a δ k -fine partition for [ a , b ] T .

Example 3.1

Let h : [ a , b ] T → R and S = ⋃ i = 1 ∞ q i , q i is a rational number in [ a , b ] T . We define h by

For any ε > 0 , let K be large enough such that 1 K < ε , then we define δ L K ( q i ) = δ R K ( q i ) = ε 2 i + 1 , i ⩾ 1 , δ L K ( s ) = 1 , δ R K ( s ) = max { 1 , μ ( s ) } for s ∈ [ a , b ] T \ S , then we assume { δ k } k = 1 ∞ is a decreasing sequence, so when k > K , for every δ k -fine partition P k , we have

Therefore, ( S H D ) ∫ a b h Δ s = 0 .

Theorem 3.1

Let h : [ a , b ] T → R , h is Henstock-Kurzweil delta integrable on [ a , b ] T if and only if h ∈ S H D ( [ a , b ] T ) .

Proof

( ⇒ ) Since h is Henstock-Kurzweil delta integrable on [ a , b ] T , for any ε > 0 , there exists a δ = ( δ L , δ R ) , for every δ -fine partition P , we have

Let ε = 1 k ( k = 1 , 2 , … ) , for one ε , we have one δ k , so we obtain a sequence of δ -gauges { δ k } k = 1 ∞ , for every δ k -fine partition P k , we have

Therefore, there exists a δ K ∈ { δ k } k = 1 ∞ , when k > K , for every δ k -fine partition P k , we have

So, ( S H D ) ∫ a b h Δ s = ( H D ) ∫ a b h Δ s .

( ⇐ ) For any ε > 0 , since h ∈ S H D ( [ a , b ] T ) , we have a δ N ∈ { δ n = ( δ L n , δ R n ) } n = 1 ∞ , when n > N , for every δ n -fine partition P n we have

We choose δ = δ N + 1 , denote P = P N + 1 , so every δ N + 1 -fine partition P N + 1 is a δ -fine partition. So

Therefore, ( H D ) ∫ a b h Δ s = ( S H D ) ∫ a b h Δ s , and the proof is complete.□

Theorem 3.2

If h ∈ S H D ( [ a , b ] T ) , the value of this integral is unique.

Proof

Suppose the value of this integral is not unique and ( S H D ) ∫ a b h Δ s = V 1 ∈ R . Let V 2 ∈ R , V 2 ≠ V 1 such that ( S H D ) ∫ a b h Δ s = V 2 .

For any ε > 0 , since h ∈ S H D ( [ a , b ] T ) , we have a δ N ∈ { δ n = ( δ L n , δ R n ) } n = 1 ∞ , when n > N , for every δ n -fine partition P n , we have

Similarly, we have another δ M ∈ { δ m = ( δ L m , δ R m ) } m = 1 ∞ , when m > M , for every δ m -fine partition P m , we have

For every δ n ∈ { δ n = ( δ L n , δ R n ) } n = 1 ∞ and δ m ∈ { δ m = ( δ L m , δ R m ) } m = 1 ∞ , we choose

Therefore, { δ k = ( δ L k , δ R k ) } k = 1 ∞ is a sequence of δ -gauges on [ a , b ] T , and for each δ k -fine partition P k , it is also δ n and δ m -fine partition for [ a , b ] T .

Then, for the above ε , there exists a δ K ∈ { δ k = ( δ L k , δ R k ) } k = 1 ∞ , when k > max { M , N } and P k is a δ k -fine partition, we have

Therefore, the value of this integral is unique.□

Theorem 3.3

Let h ∈ S H D ( [ a , b ] T ) , if h ( s ) ⩾ 0 for any s ∈ [ a , b ] T , then

Proof

Since h ∈ S H D ( [ a , b ] T ) , for any ε > 0 , we have a δ K ∈ { δ k = ( δ L k , δ R k ) } k = 1 ∞ , when k > K , for every δ k -fine partition P k , we have

Therefore,

and we know S ( h , P k ) = ∑ i = 1 m k ∈ N h ( ξ i k ) ( s i k − s ( i − 1 ) k ) , h ( ξ i k ) ⩾ 0 , s i k − s ( i − 1 ) k ⩾ 0 , so S ( h , P k ) ⩾ 0 , thus we have

Theorem 3.4

Let h 1 , h 2 ∈ S H D ( [ a , b ] T ) , for any α , β ∈ R , we have α h 1 + β h 2 ∈ S H D ( [ a , b ] T ) and

Proof

The case where α = 0 , β = 0 is trivial, so we assume α ≠ 0 , β ≠ 0 .

For any ε > 0 , since h 1 ∈ S H D ( [ a , b ] T ) , we have a δ N ∈ { δ n = ( δ L n , δ R n ) } n = 1 ∞ , when n > N , for every δ n -fine partition P n we have

Similarly, we have another δ M ∈ { δ m = ( δ L m , δ R m ) } m = 1 ∞ , when m > M , for every δ m -fine partition P m , we have

As in the proof of Theorem 3.2, we choose δ L k = min { δ L n , δ L m } , δ R k = min { δ R n , δ R m } , so { δ k = ( δ L k , δ R k ) } k = 1 ∞ is a sequence of δ -gauges on [ a , b ] T , and for every δ k -fine partition P k , it is also δ n and δ m -fine partition for [ a , b ] T .

For the above ε , we let K = max { M , N } , when k > K and P k is a δ k -fine partition, we have

Therefore, α h 1 + β h 2 ∈ S H D ( [ a , b ] T ) and

Theorem 3.5

Let h 1 , h 2 ∈ S H D ( [ a , b ] T ) , and h 1 ( s ) ⩽ h 2 ( s ) for any s ∈ [ a , b ] T , then ( S H D ) ∫ a b h 1 Δ s ⩽ ( S H D ) ∫ a b h 2 Δ s .

It is easy to show by Theorems 3.3 and 3.4.

Proof

Since h 1 , h 2 ∈ S H D ( [ a , b ] T ) , by Theorem 3.4 we know that h 2 − h 1 ∈ S H D ( a , b T ) . In addition, h 2 ( s ) − h 1 ( s ) ⩾ 0 for any s ∈ [ a , b ] T . By Theorem 3.3 we have

the proof is complete.□

Theorem 3.6

Let h , ∣ h ∣ ∈ S H D ( [ a , b ] T ) , then

Theorem 3.7

For any c ∈ ( a , b ) T , if h ∈ S H D ( [ a , c ] T ) , h ∈ S H D ( [ c , b ] T ) , then h ∈ S H D ( [ a , b ] T ) , and

Proof

Since h ∈ S H D ( [ a , c ] T ) , h ∈ S H D ( [ c , b ] T ) , for any ε > 0 , we have a δ N ∈ { δ n = ( δ L n , δ R n ) } n = 1 ∞ on [ a , c ] T , when n ⩾ N , for every δ n -fine partition P n , we have

In the same way, we have another δ M ∈ { δ m = ( δ L m , δ R m ) } m = 1 ∞ on [ c , b ] T , when m ⩾ M , for every δ m -fine partition P m , we have

Assume { δ k = ( δ L k , δ R k ) } k = 1 ∞ is a decreasing sequence, let K be large enough such that 1 K < ε , then define a δ -gauge δ K ∈ { δ k = ( δ L k , δ R k ) } k = 1 ∞ on [ a , b ] T , let δ L K ( s ) = δ L N ( s ) , s ∈ [ a , c ) T , and

δ L K ( s ) = min δ L M ( s ) , s − c 3 , s ∈ ( c , b ] T , δ R K ( s ) = min δ R N ( s ) , max μ ( s ) , c − s 3 , s ∈ [ a , c ) T , δ R K ( s ) = δ R N ( s ) , s ∈ [ c , b ] T .

Then, let P k be a δ k -fine partition for [ a , b ] T , when k > K , by the way we define δ k , c has two cases: c is an end point or a tag point.

( 1 ) If c is an end point, we denote that