1. Introduction
In this section, we give several foundational definitions and notations of basic calculus of time scales. Stefan Hilger in his PhD thesis [
1] discovered a new calculus named after that time-scale calculus to unify the discrete and continuous analysis (see [
2]). Since then, this theory has received a lot of attention. The book by Bohner and Peterson [
3], on the subject of time scales, briefs and organizes much of time scale calculus.
We begin with the definition of time scale.
Definition 1. A time scale is an arbitrary nonempty closed subset of the set of all real numbers .
Now, we define two operators playing a central role in the analysis on time scales.
Definition 2. If is a time scale, then we define the forward jump operator and the backward jump operator by In the above definitions, we put (i.e., if t is the maximum of , then ) and (i.e., if t is the minimum of , then ), where ∅ is the empty set.
If , then . We note that and in when because is a closed nonempty subset of .
Next, we define the graininess functions as follows:
Definition 3. The forward graininess function is defined by The backward graininess function is defined by
With the operators defined above, we can begin to classify the points of any time scale depending on the proximity of their neighboring points in the following manner.
Definition 4. Let be a time scale. A point is said to be:
Right-scattered if ;
Left-scattered if ;
Isolated if ;
Right-dense if ;
Left-dense if ;
Dense if .
The closed interval on time scales is defined by
Open intervals and half-open intervals are defined similarly.
Two sets we need to consider are
and
which are defined as follows:
if
has
M as a left-scattered maximum and
otherwise. Similarly,
if
has
m as a right-scattered minimum and
otherwise. In fact, we can write
and
Definition 5. Let be a function defined on a time scale . Then we define the function byand the function by We introduce the nabla derivative of a function at a point as follows:
Definition 6. Let be a function and let . We define as the real number (provided it exists) with the property that for any , there exists a neighborhood N of t (i.e., for some ) such thatWe say that is the nabla derivative of f at t. Theorem 1. Let be a function, and . Then:
f being nabla differentiable at t implies f is continuous at t.
f being continuous at left-scattered t implies f is nabla differentiable at t with If t is left-dense, then f is nabla differentiable at t if and only if the limit exists as a finite number. In such a case, whenever f is nabla differentiable at t.
Example 1. Let . Thenwhere ∇
is the backward difference operator.
Theorem 2. Let f, be functions that are nabla differentiable at . Then:
The sum is nabla differentiable at t with If is a constant, then the function is nabla differentiable at t with The product is nabla differentiable at t, and we obtain the product rule The function is nabla differentiable at t with The quotient is nabla differentiable at t, and we obtain the quotient rule
Definition 7. We say that a function is a nabla antiderivative of if for all . In this case, the nabla-integral of f is defined by Now, we introduce the set of all ld-continuous functions to find a class of functions that have nabla antiderivatives.
Definition 8 (Ld-Continuous Function ()). We say that the function is ld-continuous if it is continuous at all left-dense points of and its right-sided limits exist (finite) at all right-dense points of .
Theorem 3 (Existence of Nabla Antiderivatives). Every ld-continuous function possess a nabla antiderivative.
Theorem 4. Let be a ld-continuous function, and let . Then Theorem 5. If (respectively, ), then f is nondecreasing (respectively, nonincreasing).
Theorem 6. If a, b, , , and f, , then
;
;
- )
;
;
;
if on , then ;
if on , then .
Theorem 7. Let be a ld-continuous function, and .
In the case that , we havewhere the integral on the right-hand side is the Riemann integral from calculus. In the case that consists of only isolated points, we have In the case that , where , we have In the case that , we have
The formula for nabla integration by parts is as follows:
The following theorem gives a relationship between the delta and nabla derivative.
Theorem 8. Let be delta differentiable on . Then f is nabla differentiable at t and for any that satisfies . If, in addition, is continuous on , then f is nabla differentiable at t, and for each .
Let be nabla differentiable on . Then f is delta differentiable at t and for any that satisfies . If, in addition, is continuous on , then f is delta differentiable at t, and for each .
We will use the following relations between calculus on time scales and either continuous calculus on or discrete calculus on . Please note that:
If
, then
where ∇ is the forward difference operators, respectively.
Recently, depending just on the basic limit definition of the derivative, Khalil et al. [
4] proposed the conformable derivative
of a function
for all
,
, this definition found wide resonance in the scientific community interested in fractional calculus, see [
5,
6,
7]. Iyiola and Nwaeze in [
5] proposed an extended mean value theorem and Racetrack type principle for a class of
-differentiable functions. Therefore, calculating the derivative by this definition is easy compared to the definitions that are based on integration. The researchers in [
4] also suggested a definition for the
-conformable integral of a function
as follows:
After that, Abdeljawad [
8] studied extensive research of the newly introduced conformable calculus. In his work, he introduced a generalization of the conformable derivative
definition. For
as
Benkhettou et al. [
9] introduced a conformable calculus on an arbitrary time scale, which is a natural extension of the conformable calculus.
However, in the last few decades, many authors pointed out that derivatives and integrals of non-integer order are very suitable for the description of properties of various real materials, e.g., polymers. Fractional derivatives provides an excellent instrument for the description of memory and hereditary properties of various materials and processes. This is the main advantages of fractional derivatives in comparison with classical integer-order models.
In [
10], the authors studied a version of the nabla conformable fractional derivative on arbitrary time scales. Specifically, for a function
, the nabla conformable fractional derivative,
of order
at
and
was defined as: Given any
there is a
- neighborhood
of
such that
for all
The nabla conformable fractional integral is defined by
Rahmat et al. [
11] presented a new type of conformable nabla derivative and integral which involve the time-scale power function
for
s,
.
Definition 9. Suppose and The generalized time-scale power function for is defined byand its inverse function is then given byNotice that: Corollary 1. For , we have ThenandwhereFor we have Then we write Remark 1. Regarding the generalization of the power function, to real values of (instead of integers), we havewhere Definition 10 (Conformable nabla derivative)
. Given a function and f is -nabla differentiable at if it is nabla differentiable at t, and its -nabla derivative is defined bywhere the function as defined in (4). If exists in some interval then we defineif the exists. Moreover, we call f is -nabla differentiable on provided exists for all The function is then called -nabla derivative of f on Next, we present the -nabla derivatives of products, sums, and quotients as follows.
Theorem 9. Assume are -nabla differentiable at Then:
- (i)
The sum is -nabla differentiable at t with - (ii)
For all then is -nabla differentiable at t with - (iii)
The product is -nabla differentiable at t with - (iv)
If then is -nabla differentiable at t with
Lemma 1 (Integration by parts)
. Suppose that where If ξ are conformable -nabla fractional differentiable and then: Lemma 2 (Chain rule)
. Let and assume that is continuously differentiable function. Then is -nabla differentiable and satisfies Lemma 3. Let Assume is continuous and -nabla differentiable of order γ at where and is continuously differentiable. Then there is c in the real interval such that Definition 11 (
-nabla-integral from
a)
. Assume that and then we the function f is called -nabla integrable on . if:exists and is finite. Theorem 10. Suppose and with and then We need the relations between different types of calculus on time scales
and continuous calculus, discrete calculus and quantum calculus as follows. Please note that: For the case
we obtain
If
we obtain
Theorem 11. Let and Then, for any ld-continuous function there exists a function such thatThe function F is called an -nabla antiderivative of f, and we have Theorem 12. Let If , and then
- (i)
- (ii)
- (iii)
- (iv)
- (v)
- (vi)
If on , then
Lemma 4. Let Assume is continuous and -nabla differentiable of order γ at where and is continuously differentiable. Then there is c in the real interval such that Next, we introduce the Fenchel–Legendre transform [
12,
13,
14].
Definition 12. Suppose is a function: i.e., . Then the Fenchel–Legendre transform is defined as:The scalar product is denoted by on , and is said to be the conjugate operation. Lemma 5 ( [
12])
. Suppose a function ψ and suppose Fenchel–Legendre transform of ψ, we obtainfor all , and Definition 13. We said Ω
is submultiplicative if Lemma 6 ([
15])
. Let is left-dense continuous function. Then the equality that allows interchanging the order of nabla integration given byholds for all s, w, . Lemma 7 ([
16])
. Let w and be such that and , then Over several decades Hilbert-type inequalities have been attracted many researchers and several refinements and extensions have been done to the previous results, we refer the reader to the works [
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26].
The celebrated Hardy–Hilbert’s integral inequality [
27] is
where
,
and the constant
is best possible. As special case, if
, the inequality (
29) is reduced to the classical Hilbert integral inequality
where the coefficient
t is best possible.
In [
28], Pachappte established a discrete Hilbert-type inequality and its integral version as in the following two theorems:
Theorem 13. Let be two nonnegative sequences of real numbers defined for and with and let be two positive sequences of real numbers defined for and where r are natural numbers. Define and Let Φ
and Ψ
be two real-valued nonnegative, convex, and submultiplicative functions defined on Thenwhereand Theorem 14. Let with and let be two positive functions defined for and Let and for and where y are positive real numbers. Let and Ψ
be as in Theorem 13. Thenwhere In [
29], Handley et al. gave general versions of inequalities (
30) and (
31) in the following two theorems:
Theorem 15. Let be n sequences of nonnegative real numbers defined for with and let be n sequences of positive real numbers defined for where are natural numbers. Set Let be n real-valued nonnegative convex and submultiplicative functions defined on Let and set , and Thenwhere Theorem 16. Let with let be n positive functions defined for Set for where are positive real numbers. Let and be as in Theorem 15. Thenwhere Hamiaz et al. [
22] discussed the inequalities:
Theorem 17. Let and be sequences of real numbers. Define Thenandunless or is null, where In this important article, by implying (
24), we study some new dynamic inequalities of Hardy–Hilbert type using nabla-integral on time scales. We further show some relevant inequalities as special cases: discrete inequalities and integral inequalities. These inequalities maybe be used to obtain more generalized results of several obtained inequalities before by replace
,
by specific substitution.
2. Main Results
In the following, we will let , and
We start with a foundational results before introducing the main inequalities.
Lemma 8. Suppose the time scales with t, such that Let be left-dense continuous function with and Then Proof. We fix the point
,
. Assume
t is left-dense, by making a modification of the chain rule, we obtain
Letting
t be a left-scattered point. Define
as
and
where we used (
17). Using the differentiability of the real-valued function
, where
and applying the mean value theorem, we obtain
From (
1), (
11), (
34), (
35) and (
36), we obtain
where we used (
6) in the last step. Thus, (
37) holds in either case. From (
37), taking
integral for both sides, we obtain the required inequality (
32). This completes the proof. □
Lemma 9 (Generalized
-nabla Hölder fractional inequality on timescales)
. Let where . If and thenwhere and . This inequality is reversed if and if or Proof. Setting
and applying the Young inequality
where
B are nonnegative,
and
we see that
which is the desired inequality (
38). On the other hand, without loss of generality, we assume that
Set
and
Then
From (
38) we have that
Letting
and
in the last inequality, we obtain the inverse inequality of (
38). The proof is complete. □
Lemma 10 (Generalized
-nabla Jensen’s fractional inequality on timescales)
. Let δ, and c, . Assume that and are nonnegative with . If is a convex function, thenIt is easy to see that inequality (39) are turned around if Φ
is concave. Proof. Because the convexity of
. For
there exists
such that
From (
40) and item (vi) in Theorem 12, we obtain
This gives the required inequality (
39). The reversed inequality obtained directly if we put
and apply inequality (
39), since
is convex. □
Theorem 18. Suppose the time scales with and z Assume and are r-d continuous and respectively and define then for and we have that andwhere Proof. Using the inequality (
32), we obtain
We use Lemma 9. Then from (
44), we obtain
Applying Lemma 9. Thus, from (
45), we obtain
From (
46) and (
47), we obtain
From inequality (
48), we have
Using Lemma 5 in (
48) and (
49) gives
Using Lemma 7 in (
50) and (
51) gives
Dividing both sides of (
52) and (
53) by
and
respectively, we obtain that
From (
54) using Lemma 9 we obtain
This completes the proof. □
Remark 2. In (43), as special case, if we take , we have see [13], so we getConsequently, for inequality (56) producesPutting , we obtain Remark 3. In Remark 2, if we take , we obtain [15] [Theorem 3.3]. Theorem 19. Suppose , and are defined as in Theorem 18, thusand Proof. In (
42) and (
43) taking
. This gives our claim. □
In Theorem 18, if we chose , then the next results:
Corollary 2. If , . Define and , thenwhere In Theorem 18, if we chose , and the next result:
Corollary 3. Let , be sequences of nonnegative real numbers where N, . DefineThenwhere Corollary 4. With the hypotheses of Theorem 18, we have: Proof. Using the Fenchel-Young inequality (
24) in (
42) and (
43). This proves the claim. □
Theorem 20. Assuming the time scale with and are defined as in Theorem 18. Suppose and are right-dense continuous functions on and respectively. Suppose that and are convex, and submultiplicative on Furthermore, assume thatthen for and we have thatwhere Proof. From the properties of
and using (10), we obtain
Using (9) in (
59), we see that
Additionally, from the convexity and submultiplicative property of
, we obtain using (10) and (9):
From (
60) and (
61), we have
Using (
24) on
gives:
Applying Lemma 7 on the right-hand side of (
63), we see that
From (
66), using (9), we have
From (
67), using Lemma 6, we obtain
where
This completes the proof. □
Remark 4. In Theorem 20, as special case, if we take , , and by following the same procedure employed in Remark 2, then we obtain [15] [Theorem 3.5]. In Theorem 20, taking and we have the result:
Corollary 5. Assume that and , we defineThenwhere In Theorem 20, taking , and gives the result:
Corollary 6. Let , , , be sequences of nonnegative real numbers where N, . DefineThenwhere Remark 5. In Corollary 6, if we obtain the result due to Hamiaz and Abuelela [22] [Theorem 5]. Corollary 7. Under the hypotheses of Theorem 20 the following inequality hold: Proof. Using (
24) in (
58). This proves our claim. □
Lemma 11. With hypotheses of Theorem 20, we obtain:where Proof. From (
60) and (
61) and using Fenchel-Young inequality with
we have
From (
69), using (9) with
, we obtain
Applying Lemma 6 on (
70), we obtain
where
This proves our claim. □
Theorem 21. Let δ, ξ, Ψ
and Φ
be as in Theorem 20. Furthermore, assume that for w, then for and we have thatwhere Proof. Applying Lemma 9 on (
73), we obtain
From (
75) and (
76), we observe that
Applying Lemma 5 on the term
gives:
From 7 and (
78), we obtain
Dividing both sides of (
79) by
we obtain
Taking double nabla-integral for (
80), yields:
Using Lemma 9 in (
81), yield:
From Lemma 6 and (
82), we obtain:
This completes the proof. □
Remark 6. In Theorem 21, as special case, if we take , , and by following the same procedure employed in Remark 2, then we obtain [15] [Theorem 3.7]. Taking and in Theorem 21, we have:
Corollary 8. Assume , , . DefineThenwhere Taking and in Theorem 21, gives:
Corollary 9. Let , , , be sequences of nonnegative real numbers where N, . DefineThenwhere Remark 7. In Corollary 9, if we obtain the result due to Hamiaz and Abuelela [22] [Theorem 7]. Corollary 10. With the hypotheses of Theorem 21, we obtain: Proof. We apply the Fenchel-Young inequality (
24) in (
72). This completes the proof. □