Abstract
This paper deals with the derivation of some new dynamic Hilbert-type inequalities in time scale nabla calculus. In proving the results, the basic idea is to use some algebraic inequalities, Hölder’s inequality, and Jensen’s time scale inequality. This generalization allows us not only to unify all the related results that exist in the literature on an arbitrary time scale, but also to obtain new outcomes that are analytical to the results of the delta time scale calculation.
Similar content being viewed by others
1 Introduction
In recent years, Hilbert’s dual-series inequality and its integral form [1, pp. 253–254] have been granted significant attention by many scholars (for example, see [2–10]). In particular, B. G. Pachpatte [11] established a new inequality close to that of Hilbert as follows. Let \(k, r\geq 1\), \(A_{s}=\sum_{m=1}^{s}a_{m}\geq 0\) and \(B_{\vartheta }=\sum_{n=1}^{\vartheta }b_{n}\geq 0\). Then
where
In the same article [11], Pachpatte demonstrated the integral version of (1) as follows. Let \(k,r\geq 1\), \(\Pi (s)=\int _{0}^{s}\omega _{1}(\xi )\,d\xi \geq 0\) and \(\Omega (\vartheta )=\int _{0}^{\vartheta }\omega _{2}(\nu )\,d\nu \geq 0\), for \(s,\xi \in (0, x)\) and \(\vartheta ,\nu \in (0,y)\). Then
where
In [12], Young-Ho Kim gave some generalizations of (1) and (2) by introducing a parameter \(\gamma >0\) as follows. Let \(k, r\geq 1\), \(A_{s}=\sum_{m=1}^{s}a_{m}\geq 0\) and \(B_{\vartheta }=\sum_{n=1}^{\vartheta }b_{n}\geq 0\). Then
where
The integral version of (3) is established in the next consequence. Let \(k, r\geq 1\), \(\gamma >0\), \(\Pi (s)=\int _{0}^{s}\omega _{1}(\xi )\,d\xi \geq 0\), and \(\Omega (\vartheta )=\int _{0}^{\vartheta }\omega _{2}(\nu )\,d\nu \geq 0\), for \(s, \xi \in (0, x)\) and ϑ, \(\nu \in (0, y)\). Then
where
Another refinement of inequalities (1) and (2) has been made by Yang [13] as follows. Let \(k, r\geq 1\) and \(\lambda , \mu >1\) be constants such that \(1/\lambda +1/\mu =1\), \(A_{s}=\sum_{m=1}^{s}a_{m}\geq 0\), and \(B_{\vartheta }=\sum_{n=1}^{\vartheta }b_{n}\geq 0\). Then
where
The integral version of (5) is established in the next consequence. Let \(k, r\geq 1\) and \(\lambda , \mu >1\) be constants such that \(1/\lambda +1/\mu =1\), \(\Pi (s)=\int _{0}^{s}\omega _{1}(\xi )\,d\xi \geq 0\), and \(\Omega (\vartheta )=\int _{0}^{\vartheta }\omega _{2}(\nu )\,d\nu \geq 0\), for \(s, \xi \in (0, x)\) and \(\vartheta , \nu \in (0, y)\). Then
where
After construction of time scale calculus, dynamic inequalities have become the focus of interest, and classical inequalities have been established for any time scale \(\mathbb{T}\). We can refer two surveys [14, 15] and a monograph [16] for exhibition of these results.
In [17] the researchers concluded some generalizations of inequalities (1) and (2) for time scale delta calculus. Specifically, they proved that if \(s, \vartheta , \vartheta _{0}\in \mathbb{T}\), \(\omega _{1}(s)\in \mathrm{C}_{rd}([\vartheta _{0}, x)_{\mathbb{T}}, \mathbb{R}^{+})\), \(\omega _{2}(\vartheta )\in \mathrm{C}_{rd}([\vartheta _{0}, y)_{\mathbb{T}}, \mathbb{R}^{+})\), \(k, r\geq 1\) and \(\lambda , \mu >1\) are constants such that \(1/\lambda +1/\mu =1\), then for \(s\in {}[ \vartheta _{0}, x)_{\mathbb{T}}\) and \(\vartheta \in {}[ \vartheta _{0}, y)_{\mathbb{T}}\), one has
where \(\Pi (s)=\int _{\vartheta _{0}}^{s}\omega _{1}(\xi )\Delta \xi \), \(\Omega (\vartheta )=\int _{\vartheta _{0}}^{\vartheta }\omega _{2}( \xi )\Delta \xi \), and
Another refinement of (7) for time scale delta calculus has been made by Rezk et al. [18] as follows. Let \(s, \vartheta , \vartheta _{0}\in \mathbb{T}\), \(\omega _{1}(s)\in \mathrm{C}_{rd}([\vartheta _{0}, x)_{\mathbb{T}}, \mathbb{R}^{+})\), \(\omega _{2}(\vartheta )\in \mathrm{C}_{rd}([\vartheta _{0}, y)_{\mathbb{T}}, \mathbb{R}^{+})\), \(k, r\geq 1\) and \(\lambda , \mu >1\) be constants such that \(1/\lambda +1/\mu =1\), then for \(s\in {}[ \vartheta _{0}, \rho )_{\mathbb{T}}\) and \(\vartheta \in {}[ \vartheta _{0}, \tau )_{\mathbb{T}}\), one has
where \(\Pi (s)=\int _{\vartheta _{0}}^{s}\omega _{1}(\xi )\Delta \xi \), \(\Omega (\vartheta )=\int _{\vartheta _{0}}^{\vartheta }\omega _{2}( \xi )\Delta \xi \), and
For developing of Hilbert’s inequalities for time scale delta calculus, we refer the reader to the articles [19–29]. Although there are many results for time scale calculus in the sense of delta derivative, there is not much done for the nabla derivative. Therefore the major contribution of this article is to extend Hilbert-type inequalities for the nabla time scale calculus and to unify them for an arbitrary time scale. The main theorems are inspired from the paper [18] which presents the corresponding results for time scale delta calculus. By obtaining their nabla versions, we can show the generalizations of these inequalities for different types of time scales \(\mathbb{T}\), such as real numbers and integers.
The structure of this paper can be listed as follows. Section 2 presents the fundamental concepts of the time scale calculus in terms of delta and nabla derivatives. Section 3 is devoted to main results, which are to generalize inequalities (5) and (6) for the nabla time scale calculus and so, to obtain nabla calculus versions of (9) and several inequalities of Hilbert’s type in [18].
2 Preliminaries
In this section, the fundamental theories of the time scale delta and nabla calculi will be presented. Time scale calculus whose detailed information can be found in [30, 31] has been invented in order to unify continuous and discrete analysis.
A nonempty closed subset of \(\mathbb{R}\) is named a time scale and is denoted by \(\mathbb{T}\). For \(\vartheta \in \mathbb{T}\), if \(\inf \emptyset =\sup \mathbb{T}\) and \(\sup \emptyset =\inf \mathbb{T}\), then the forward jump operator \(\sigma :\mathbb{T}\rightarrow \mathbb{T}\) and the backward jump operator \(\rho :\mathbb{T}\rightarrow \mathbb{T}\) are defined as \(\sigma (\vartheta )=\inf (\vartheta , \infty )_{\mathbb{T}}\) and \(\rho (\vartheta )=\sup (-\infty , \vartheta )_{\mathbb{T}}\), respectively. From the above two concepts, it can be mentioned that a point \(\vartheta \in \mathbb{T}\) with \(\inf \mathbb{T}<\vartheta <\sup \mathbb{T}\) is named right-scattered if \(\sigma (\vartheta )>\vartheta \), right-dense if \(\sigma (\vartheta )=\vartheta \), left-scattered if \(\rho (\vartheta )<\vartheta \) and left-dense if \(\rho (\vartheta )=\vartheta \).
The Δ-derivative of \(\psi :\mathbb{T}\rightarrow \mathbb{R}\) at \(\vartheta \in \mathbb{T}^{k}=\mathbb{T}/(\rho (\sup \mathbb{T}), \sup \mathbb{T]}\) denoted by \(\psi ^{\Delta }(\vartheta )\) is the number enjoying the property that for all \(\varepsilon >0\) there is a neighborhood U of \(\vartheta \in \mathbb{T}^{k}\) such that
The ∇-derivative of \(\psi :\mathbb{T}\rightarrow \mathbb{R}\) at \(\vartheta \in \mathbb{T}_{k}=\mathbb{T}/[\inf \mathbb{T}, \sigma (\inf \mathbb{T}))\) denoted by \(\psi ^{\nabla }(\xi )\) is the number enjoying the property that for all \(\varepsilon >0\) there is a neighborhood V of \(\vartheta \in \mathbb{T}_{k}\) such that
A function \(\psi :\mathbb{T}\rightarrow \mathbb{R}\) is rd-continuous if it is continuous at each right-dense point in \(\mathbb{T}\) and \(\underset{s\rightarrow \vartheta ^{-}}{\lim }\psi (s)\) exists as a finite number for all left-dense points in \(\mathbb{T}\). The set \(\mathrm{C}_{rd}(\mathbb{T}, {\mathbb{R)}}\) represents the class of real, rd-continuous functions defined on \(\mathbb{T}\). If \(\psi \in \mathrm{C}_{rd}(\mathbb{T}, {\mathbb{R)}}\), then there exists a function \(\Psi (\vartheta )\) such that \(\Psi ^{\Delta }(\vartheta )=\psi (\vartheta )\) and the delta integral of ψ is defined by
A function \(\psi :\mathbb{T}\rightarrow \mathbb{R}\) is ld-continuous if it is continuous at each left-dense point in \(\mathbb{T}\) and \(\underset{s\rightarrow \vartheta ^{+}}{\lim }\psi (s)\) exists as a finite number for all right-dense points in \(\mathbb{T}\). The set \(\mathrm{C}_{ld}(\mathbb{T}, {\mathbb{R)}}\) represents the class of real, ld-continuous functions defined on \(\mathbb{T}\). If ψ∈ \(\mathrm{C}_{ld}(\mathbb{T}, {\mathbb{R)}}\), then there exists a function \(\Psi (\vartheta )\) such that \(\Psi ^{\nabla }(\vartheta )=\psi (\vartheta )\) and the nabla integral of ψ is defined by
In the following, we display some basic lemmas and algebraic inequalities that play a key role in proving the major findings of this paper.
Lemma 2.1
(Nabla Hölder’s Inequality [32])
Let \(x_{0}, x\in \mathbb{T}\). For \(\xi , \psi \in \mathrm{C}_{ld}([x_{0}, x]_{\mathbb{T}}, \mathbb{R})\), we have
where \(\lambda , \mu >1\) with \(1/\lambda +1/\mu =1\).
Lemma 2.2
(Nabla Jensen’s Inequality [33, Theorem 3.4])
Let \(x_{0}\), \(x\in \mathbb{T}\) and \(m, n\in \mathbb{R}\). Assume that \(\xi \in \mathrm{C}_{ld}([x_{0}, x]_{ \mathbb{T}}, (m, n))\) and \(\psi \in \mathrm{C}_{ld}([x_{0}, x]_{\mathbb{T}} , \mathbb{R})\) are nonnegative with \(\int _{x_{0}}^{x} \xi (\eta )\Delta \eta >0\). If \(\Theta \in \mathrm{C}((m, n), \mathbb{R})\) is a convex function, then
Lemma 2.3
(The power rule for nabla derivative [33, Lemma 3.1])
Let \(x_{0}\), \(x\in \mathbb{T,}\) \(\psi \in \mathrm{C}_{ld}([x_{0}, x]_{\mathbb{T}}, \mathbb{R})\) be a nonnegative function, and \(\gamma \geq 1\) a real constant. Then
Lemma 2.4
(Young’s inequality [34])
Let \(\delta >0\), \(\Lambda _{q}>0\) and \(\sum_{q=1}^{n}\Lambda _{q}=\Upsilon _{n}\). Then
Lemma 2.5
([33, Lemma 3.2])
Let s, ϑ, \(\vartheta _{0}\in \mathbb{T}\) with \(s, \vartheta \geq \vartheta _{0}\) and \(\psi \in \mathrm{C}_{ld}([a, b]_{\mathbb{T}}, \mathbb{R})\). Then
3 Key results
In this section, we focus on obtaining the corresponding outcomes for the nabla time scale calculation in [18]. We must assume that all functions found in the theorem statements are nonnegative, ld-continuous, ∇-differentiable, and locally nabla integrable.
Theorem 3.1
Let \(s, \vartheta , \vartheta _{0}\in \mathbb{T}\) and \(\omega _{1}\in \mathrm{C}_{ld}([\vartheta _{0}, x]_{\mathbb{T}}, \mathbb{R}^{+})\), \(\omega _{2}\in \mathrm{C}_{ld}([\vartheta _{0}, y]_{\mathbb{T}}, \mathbb{R}^{+})\). Define
Then for \(s\in {}[ \vartheta _{0}, x]_{\mathbb{T}}\) and \(\vartheta \in {}[ \vartheta _{0}, y]_{\mathbb{T}}\), we have
where
Proof
By using (13), we obtain
and
Then, we have
Applying (11) to \(\int _{\vartheta _{0}}^{s}\Pi ^{k-1}(\xi )\omega _{1}(\xi )\nabla \xi \) with indices λ and \(\lambda /(\lambda -1)\), we find that
while doing the same to the integral \(\int _{\vartheta _{0}}^{\vartheta }\Omega ^{r-1}(\xi )\omega _{2}( \xi )\nabla \xi \) with indices μ and \(\mu /(\mu -1)\), we find that
From (20), (21), and (22), we get
Using inequality (14), we note
Now, by setting \(s_{1}=(s-\vartheta _{0})^{\lambda -1}\), \(s_{2}=(\vartheta -\vartheta _{0})^{\mu -1}\), \(\Lambda _{1}=1/\lambda \), \(\Lambda _{1}=1/\mu \), and \(\delta =\Lambda _{1}+\Lambda _{2}\) in (24), we get
Substituting (25) into (23) yields
Dividing both sides of (26) by \(\mu (s-\vartheta _{0})^{ [ (\lambda -1)(\lambda +\mu ) ] / \lambda \mu }+\lambda (\vartheta -\vartheta _{0})^{{ [ (\mu -1)( \lambda +\mu ) ] /\lambda \mu }}\), we obtain
Integrating both sides of (27) and using (11) again, we get
Applying Lemma 2.5 on (28), we conclude that
that is, (17) is true. □
Remark 3.2
By setting \(1/\lambda +1/\mu =1\) in (24), we obtain
Hence, by applying (29) on the right-hand side of (17) in Theorem 3.1, we get
Corollary 3.1
If we take \(1/\lambda +1/\mu =1\) in (17), then
where
Remark 3.3
As a particular case of Corollary 3.1, if \(\lambda =\mu =2\), then we have
which is [33, Theorem 3.3].
Remark 3.4
Clearly, for \(\mathbb{T}=\mathbb{Z}\) or \(\mathbb{T}=\mathbb{R,}\) and \(\vartheta _{0}=0\), together with \(\rho (u)=u-1\) or \(\rho (u)=u\), (17) reduces to (5) or (6), respectively.
Remark 3.5
In Theorem 3.1, if we take \(k=r=1\), then we have
where
For \(\lambda =\mu =2\), this is Anderson’s result [33, Remark 4].
In what follows, we give a further generalization of (32) obtained in Remark 3.5. Before giving our results, we presume that there are Φ and Ψ which are real-valued, nonnegative, convex and submultiplicative functions defined on \([ 0, \infty ) \). A function ψ is submultiplicative if \(\psi (s\vartheta )\leq \psi (s)\psi (\vartheta )\) for \(s, \vartheta \geq 0\).
Theorem 3.6
Let \(s, \vartheta , \vartheta _{0}\in \mathbb{T}\) and \(\Pi (s)\), \(\Omega ( \vartheta )\) be as in Theorem 3.1and let \(k(\xi )\), \(l(\xi )\) be two positive functions defined for \(\xi \in {}[ \vartheta _{0}, x]_{\mathbb{T}}\) and \(\xi \in {}[ \vartheta _{0}, y]_{\mathbb{T}}\). Suppose that
Then for \(s\in {}[ \vartheta _{0}, x]_{\mathbb{T}}\) and \(\vartheta \in {}[ \vartheta _{0}, y]_{\mathbb{T}}\), we have
where
Proof
Using Jensen’s inequality (12) and the properties of Φ, we obtain
Further, by (11), we find that
Analogously,
By multiplying (36) and (37), we get
Applying (24) on the term \((s-\vartheta _{0})^{(\lambda -1)/\lambda }\times (\vartheta - \vartheta _{0})^{(\mu -1)/\mu }\) gives
From (39), we observe that
Integrating both sides of (40) and using (11) again with indices λ, \(\lambda /(\lambda -1)\) and μ, \(\mu /(\mu -1)\), we find that
Applying Lemma 2.5 to (41), we get
which is (34). □
Corollary 3.2
If we take \(1/\lambda +1/\mu =1\) in (34), then we get
where
Remark 3.7
As a particular case of Corollary 3.2, if \(\lambda =\mu =2\), then we get
where
which is [33, Theorem 3.5].
Remark 3.8
As a particular case of Theorem 3.6 if \(\mathbb{T}=\mathbb{Z}\), \(\vartheta _{0}=0\), then \(\rho (u)=u-1\) and (34) reduces to
where
which is [13, Theorem 2.2].
Remark 3.9
As a particular case of Theorem 3.6 if \(\mathbb{T}=\mathbb{R}\), \(t_{0}=0\), then \(\rho (u)=u\) and (34) reduces to
where
which is [13, Theorem 3.2].
Our next outcome deals with a further generalization of the inequality in (34).
Theorem 3.10
Let \(s, \vartheta , \vartheta _{0}\in \mathbb{T}\), and \(\omega _{1}\), \(\omega _{2}\) be as in Theorem 3.1. Define
Then for \(s\in {}[ \vartheta _{0}, x]_{\mathbb{T}}\) and \(\vartheta \in {}[ \vartheta _{0}, y]_{\mathbb{T}}\), we have
where
Proof
Based on the assumptions and the inequality of Jensen (12), we can see that
By applying (11) to (48) with indices λ, \(\lambda /(\lambda -1)\), we have
This implies that
Analogously,
Applying (24) to the term \((s-\vartheta _{0})^{(\lambda -1)/\lambda }\times (\vartheta - \vartheta _{0})^{(\mu -1)/\mu }\) gives
From (53), we have
Integrating both sides of (54) and using (11) again with indices λ, \(\lambda /(\lambda -1)\) and μ, \(\mu /(\mu -1)\), we get
Applying Lemma 2.5 to (55), we find that
which is (47). □
Corollary 3.3
If we take \(1/\lambda +1/\mu =1\) in (47), then we get
where
Remark 3.11
As a particular case of Corollary 3.3, if \(\lambda =\mu =2\), then we get
which is [33, Theorem 3.6].
Remark 3.12
As a particular case of Theorem 3.10, if \(\mathbb{T}=\mathbb{Z}\), \(\vartheta _{0}=0\), then \(\rho (u)=u-1\) and (47) reduces to
where
which is [13, Theorem 2.3].
Remark 3.13
As a particular state of Theorem 3.10, if \(\mathbb{T}=\mathbb{R}\), \(t_{0}=0\), then \(\rho (u)=u\) and (47) reduces to
where
which is [13, Theorem 3.3].
Theorem 3.14
Let \(s, \vartheta , \vartheta _{0}\in \mathbb{T}\) and \(\omega _{1}\), \(\omega _{2}\), k, l, H, L be as in Theorem 3.6. Define
Then for \(s\in {}[ \vartheta _{0}, y]_{\mathbb{T}}\) and \(\vartheta \in {}[ \vartheta _{0}, x]_{\mathbb{T}}\), we get
where
Proof
Based on the assumptions and the inequality of Jensen (12), we find that
By applying (11) to (62) with indices λ, \(\lambda /(\lambda -1)\), we have
From (63), we get
Similarly, we also obtain
From (64) and (65), we find that
Applying (24) to the term \((s-\vartheta _{0})^{(\lambda -1)/\lambda }\times (\vartheta - \vartheta _{0})^{(\mu -1)/\mu }\) gives
This implies that
Integrating both sides of (68) and using (11) again with indices λ, \(\lambda /(\lambda -1)\) and μ, \(\mu /(\mu -1)\), we get
Applying Lemma 2.5 to (69), we find that
which is (61). □
Corollary 3.4
If we take \(1/\lambda +1/\mu =1\) in (61), then
where
Remark 3.15
As a particular case of Corollary 3.4, if \(\lambda =\mu =2\), then we get
which is [33, Theorem 3.7].
Remark 3.16
As a particular case of Theorem 3.14, if \(\mathbb{T}=\mathbb{Z}\), \(\vartheta _{0}=0\), then \(\rho (u)=u-1\) and (61) reduces to
where
which is [13, Theorem 2.4].
Remark 3.17
As a particular case of Theorem 3.14, if \(\mathbb{T}=\mathbb{R}\), \(t_{0}=0\), then \(\rho (u)=u\) and (61) reduces to
where
which is [13, Theorem 3.4].
Remark 3.18
Clearly, Theorems 3.1, 3.6, 3.10, and 3.14 present the corresponding results of Theorems 6, 9, 12, and 15 in [18], respectively, for time scale delta calculus. Likewise, Corollaries 3.1, 3.2, 3.3, and 3.4 display the corresponding results of Theorems 3.1, 3.2, 3.3, and 3.4 in [17], respectively, for delta time scale calculus.
Availability of data and materials
Not applicable.
References
Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1934)
Gao, M., Yang, B.: On the extended Hilbert’s inequality. Proc. Am. Math. Soc. 126, 751–759 (1998)
Jichang, K.: On new extensions of Hilbert’s integral inequality. J. Math. Anal. Appl. 235, 608–614 (1999)
Handley, G.D., Koliha, J.J., Pečarić, J.E.: New Hilbert–Pachpatte type integral inequalities. J. Math. Anal. Appl. 257, 238–250 (2001)
Yang, B.: On new generalizations of Hilbert’s inequality. J. Math. Anal. Appl. 248, 29–40 (2000)
AlNemer, G., Zakarya, M., Abd El-Hamid, H.A., Kenawy, M.R., Rezk, H.M.: Dynamic Hardy-type inequalities with non-conjugate parameters. Alex. Eng. J., 1–10 (2020)
Abdel-Aty, A., Khater, M.M.A., Attia, R.A.M., Abdel-Aty, M., Eleuch, H.: On the new explicit solutions of the fractional nonlinear space–time nuclear model. Fractals 28, 2040035 (2020)
Ereu, J., Gimenez, J., Perez, L.: On solutions of nonlinear integral equations in the space of functions of shiba-bounded variation. Appl. Math. Inf. Sci. 14, 393–404 (2020)
Nchama, G.A.M., Mecıas, A.L., Richard, M.R.: The Caputo–Fabrizio fractional integral to generate some new inequalities. Inf. Sci. Lett. 8, 73–80 (2019)
Abu-Donia, H.M., Atia, H.A., Khater, O.M.A.: Some fixed-point theorems in fuzzy 2-metric spaces under ψ-contractive mappings. Inf. Sci. Lett. 9, 21–25 (2020)
Pachpatte, B.G.: On some new inequalities similar to Hilbert’s inequality. J. Math. Anal. Appl. 226, 166–179 (1998)
Kim, Y.-H.: An improvement of some inequalities similar to Hilbert’s inequality. Int. J. Math. Math. Sci. 28(4), 211–221 (2001)
Yang, W.: Some new Hilbert–Pachpatte’s inequalities. J. Inequal. Pure Appl. Math. 10, 1–14 (2009)
Agarwal, R.P., Bohner, M., Peterson, A.: Inequalities on time scales: a survey. Math. Inequal. Appl. 4, 535–557 (2001)
Saker, S.: Dynamic inequalities on time scales: a survey. J. Fract. Calc. Appl. 3 (S) (2), 1–36 (2012)
Agarwal, R.P., O’Regan, D., Saker, S.: Dynamic Inequalities on Time Scales. Springer, Berlin (2014)
Saker, S., Ahmed, A.M., Rezk, H.M., O’Regan, D., Agarwal, R.P.: New Hilbert’s dynamic inequalities on time scales. J. Inequal. Pure Appl. Math. 20(40), 1017–1039 (2017)
Ahmed, A.M., AlNemer, G., Zakarya, M., Rezk, H.M.: Some dynamic inequalities of Hilbert’s type. J. Funct. Spaces 2020, 1–13 (2020)
O’Regan, D., Rezk, H.M., Saker, S.: Some dynamic inequalities involving Hilbert and Hardy–Hilbert operators with kernels. Results Math. 73(146), 1–22 (2018)
Saker, S., Rezk, H.M., O’Regan, D., Agarwal, R.P.: A variety of inverse Hilbert type inequality on time scales. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 24, 347–373 (2017)
Saker, S., Rezk, H.M., Krnić, M.: More accurate dynamic Hardy-type inequalities obtained via superquadraticity. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113, 2691–2713 (2019)
Saker, S., Rezk, H.M., Abohela, I., Baleanu, D.: Refinement multidimensional dynamic inequalities with general kernels and measures. J. Inequal. Appl. 2019, 306 (2019)
Saker, S., Kenawy, M., AlNemer, G., Zakarya, M.: Some fractional dynamic inequalities of Hardy’s type via conformable calculus. Mathematica 8, 434 (2020)
Li Nian, W.: Bounds for certain new integral inequalities on time scales. Adv. Differ. Equ. 2009, 484185 (2009)
Li Nian, W.: Nonlinear integral inequalities in two independent variables on time scales. Adv. Differ. Equ. 2011, 283926 (2011)
Sarfaraz, S., Ahmad, N., Rahman, G.: Study of nonlinear Pachpatte’s inequalities on time scales. Adv. Differ. Equ. 2019, 402 (2019)
Agarwal, R.P., Hyder, A., Zakarya, M.: Well-posedness of stochastic modified Kawahara equation. Adv. Differ. Equ. 2020, 18 (2020)
Abd El-Hamid, H.A., Rezk, H.M., Ahmed, A.M., AlNemer, G., Zakarya, M., El Saify, H.A.: Dynamic inequalities in quotients with general kernels and measures. J. Funct. Spaces 2020, 1–12 (2020)
AlNemer, G., Zakarya, M., Abd El-Hamid, H.A., Agarwal, P., Rezk, H.M.: Some dynamic Hilbert-type inequalities on time scales. Symmetry 12(9), 1410 (2020)
Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston (2001)
Bohner, M., Peterson, A.: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston (2003)
Özkan, U.M., Sarikaya, M.Z., Yildirim, H.: Extensions of certain integral inequalities on time scales. Appl. Math. Lett. 21, 993–1000 (2008)
Anderson, D.R.: Dynamic double integral inequalities in two independent variables on time scales. J. Math. Inequal. 2, 163–184 (2008)
Mitrinovic, D.S., Pecaric, J.E., Fink, A.M.: Classical and New Inequalities in Analysis. Kluwer Academic, Dordrecht (1993)
Acknowledgements
This research was funded by the Deanship of Scientific Research at Princess Nourah Bint Abdulrahman University through the Fast-track Research Funding Program.
Funding
This research was funded by the Deanship of Scientific Research at Princess Nourah Bint Abdulrahman University through the Fast-track Research Funding Program.
Author information
Authors and Affiliations
Contributions
All authors conceived the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Rezk, H.M., AlNemer, G., Abd El-Hamid, H.A. et al. Hilbert-type inequalities for time scale nabla calculus. Adv Differ Equ 2020, 619 (2020). https://doi.org/10.1186/s13662-020-03079-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-020-03079-w