Existence and Ulam stability for fractional differential equations of mixed Caputo-Riemann derivatives

1.   Introduction

In 1889, Hölder ^[1] proved that

where $\left\{ x_{k}\right\} _{k = 1}^{n}\ $and $\left\{ y_{k}\right\} _{k = 1}^{n}$ are positive sequences, $p$ and $q$ are two positive numbers, such that

The inequality reverses if $p \; q < 0$. The integral form of (1.1) is

where $\eta, \; \epsilon \in \mathbb{R}$, $\gamma > 1, \ 1/\gamma +1/\nu = 1$, and $\lambda, \; \omega \in C\left([a, b], \mathbb{R}\right)$. If $0 < \gamma < 1,$ then (1.2) is reversed.

The subsequent inequality, widely recognized as Minkowski's inequality, asserts that, for $p\geq 1,$ if $\varphi, \; \varpi$ are nonnegative continuous functions on $[\eta, \epsilon ]$ such that

then

Sulaiman ^[2] introduced the following result pertaining to the reverse Minkowski's inequality: for any $\varphi, \varpi > 0,$ if $p\geq 1$ and

then

Sroysang ^[3] proved that if $p\geq 1$ and $\varphi, \varpi > 0$ with

then

In 2023, Kirmaci ^[4] proved that for nonnegative sequences $\left\{ x_{k}\right\} _{k = 1}^{n}\ $and $\left\{ y_{k}\right\} _{k = 1}^{n}$, if $p > 1, \; s\geq 1,$

then

and if $s\geq 1, \; sp < 0$ and

then (1.4) is reversed. Also, the author ^[4] generalized Minkowski's inequality and showed that for nonnegative sequences $\left\{ x_{k}\right\} _{k = 1}^{n}\ $and $\left\{ y_{k}\right\} _{k = 1}^{n}$, if $m\in \mathbb{N}$, $p > 1,$

then

Hilger was a trailblazer in integrating continuous and discrete analysis, introducing calculus on time scales in his doctoral thesis, later published in his influential work ^[5]. The exploration of dynamic inequalities on time scales has garnered significant interest in academic literature. Agarwal et al.'s book ^[6] focuses on essential dynamic inequalities on time scales, including Young's inequality, Jensen's inequality, Hölder's inequality, Minkowski's inequality, and more. For further insights into dynamic inequalities on time scales, refer to the relevant research papers ^{[7,8,9,10,11,12,13,14,15,16,17,18,19,20]}. For the solutions of some differential equations, see, for instance, ^{[21,22,23,24,25,26,27]} and the references therein.

Expanding upon this trend, the present paper aims to broaden the scope of the inequalities (1.4) and (1.5) on diamond alpha time scales. We will establish the generalized form of Hölder's inequality and Minkowski's inequality for the forward jump operator and the backward jump operator in the discrete calculus. In addition, we prove these inequalities in diamond $-\alpha$ calculus, where for $\alpha = 1,$ we can get the inequalities for the forward operator, and when $\alpha = 0,$ we get the inequalities for the backward jump operator. Also, we can get the special cases of our results in the discrete and continuous calculus.

The paper is organized as follows: In Section 2, we present some definitions, theorems, and lemmas on time scales, which are needed to get our main results. In Section 3, we state and prove new dynamic inequalities and present their special cases in different (continuous, discrete) calculi.

2.   Preliminaries and basic lemmas

In 2001, Bohner and Peterson ^[28] defined the time scale $\mathbb{ T}$ as an arbitrary, nonempty, closed subset of the real numbers $\mathbb{R}$. The forward jump operator and the backward jump operator are defined by

and

respectively. The graininess function $\mu$ for a time scale $\mathbb{T}$ is defined by

and for any function $\varphi$: $\mathbb{T} \rightarrow \mathbb{R}$, the notation $\varphi ^{\sigma }(\xi)$ and $\varphi ^{\rho }(\xi)$ denote $\varphi (\sigma (\xi))$ and $\varphi (\rho (\xi)),$ respectively. We define the time scale interval $[\eta, \epsilon ]_{\mathbb{T}}$ by

Definition 2.1. (The Delta derivative^[28]) Assume that $\varphi$: $\mathbb{T}\rightarrow \mathbb{R}$ is a function and let $\xi \in \mathbb{T}.$ We define $\varphi ^{\Delta }(\xi)$ to be the number, provided it exists, as follows: for any $\epsilon > 0$, there is a neighborhood $U$ of $\xi,$

for some $\delta > 0$, such that

In this case, we say $\varphi ^{\Delta }(\xi)$ is the delta or Hilger derivative of $\varphi$ at $\xi$.

Definition 2.2. (The nabla derivative ^[29]) A function $\lambda$: $\mathbb{T\rightarrow R}$ is said to be $\nabla$- differentiable at $\xi \in \mathbb{T}$, if $\psi$ is defined in a neighborhood $U$ of $\xi$ and there exists a unique real number $\psi ^{\nabla }(\xi),$ called the nabla derivative of $\psi$ at $\xi$, such that for each $\epsilon > 0$, there exists a neighborhood $N$ of $\xi$ with $N\subseteq U$ and

for all $s\in N.$

Definition 2.3. (^[6]) Let $\mathbb{T}$ be a time scale and $\Xi (\xi)$ be differentiable on $\mathbb{T}$ in the $\Delta$ and $\nabla$ sense. For $\xi \in \mathbb{T}$, we define the diamond $-\alpha$ derivative $\Xi ^{\lozenge _{\alpha }}(\xi)$ by

Thus, $\Xi$ is diamond $-\alpha$ differentiable if, and only if, $\Xi$ is $\Delta$ and $\nabla$ differentiable.

For $\alpha = 1,$ we get that

and for $\alpha = 0,$ we have that

Theorem 2.1. (^[6]) Let $\Xi$, $\Omega$: $\mathbb{T\rightarrow R}$ be diamond-$\alpha$ differentiable at $\xi \in \mathbb{T}$, then

(1) $\Xi +\Omega$: $\mathbb{T\rightarrow R}$ is diamond-$\alpha$ differentiable at $\xi \in \mathbb{T}$, with

(2) $\Xi.\Omega$: $\mathbb{T\rightarrow R}$ is diamond-$\alpha$ differentiable at $\xi \in \mathbb{T}$, with

(3) For $\Omega (\xi)\Omega ^{\sigma }(\xi)\Omega ^{\rho }(\xi)\neq 0$, $\Xi /\Omega$: $\mathbb{T\rightarrow R}$ is diamond-$\alpha$ differentiable at $\xi \in \mathbb{T}$, with

Theorem 2.2. (^[6]) Let $\Xi$, $\Omega$: $\mathbb{T\rightarrow R}$ be diamond-$\alpha$ differentiable at $\xi \in \mathbb{T}$, then the following holds

(1) $(\Xi)^{\lozenge _{\alpha }\Delta }(\xi) = \alpha \Xi ^{\Delta \Delta }(\xi)+(1-\alpha)\Xi ^{\nabla \Delta }(\xi),$

(2) $(\Xi)^{\lozenge _{\alpha }\nabla }(\xi) = \alpha \Xi ^{\Delta \nabla }(\xi)+(1-\alpha)\Xi ^{\nabla \nabla }(\xi),$

(3) $(\Xi)^{\Delta \lozenge _{\alpha }}(\xi) = \alpha \Xi ^{\Delta \Delta }(\xi)+(1-\alpha)\Xi ^{\Delta \nabla }(\xi)\neq (\Xi)^{\lozenge _{\alpha }\Delta }(\xi),$

(4) $(\Xi)^{\nabla \lozenge _{\alpha }}(\xi) = \alpha \Xi ^{\nabla \Delta }(\xi)+(1-\alpha)\Xi ^{\nabla \nabla }(\xi)\neq (\Xi)^{\lozenge _{\alpha }\nabla }(\xi),$

(5) $(\Xi)^{\lozenge _{\alpha }\lozenge _{\alpha }}(\xi) = \alpha ^{2}\Xi ^{\Delta \Delta }(\xi)+\alpha (1-\alpha)[\Xi ^{\Delta \nabla }(\xi)+\Xi ^{\nabla \Delta }(\xi)].$

Theorem 2.3. (^[6]) Let $a, \; \xi \; \in \mathbb{T}$ and $h$: $\mathbb{T\rightarrow R}$, then the diamond$-\alpha$ integral from $a$ to $\xi$ of $h$ is defined by

provided that there exist delta and nabla integrals of $h$ on $\mathbb{T}$.

It is known that

and

but in general

Example 2.1. ^[30] Let $\mathbb{T} = \left\{ 0, \text{ }1, \text{ }2\right\}, \; a = 0$, and $h(\xi) = \xi ^{2}$ for $\xi \in \mathbb{T}$. This gives

so that the equality above holds only when $\lozenge _{\alpha } = \Delta$ or $\lozenge _{\alpha } = \nabla.$

Theorem 2.4. (^[6]) Let $a, b, \xi \in \mathbb{T}$, $\beta, \; \gamma, \; c\in \mathbb{R}$, and $\Xi$ and $\Omega$ be continuous functions on $[a, \; b]\cup \mathbb{T}$, then the following properties hold.

(1) $\int\nolimits_{a}^{b}\left[ \gamma \Xi (\xi)+\beta \Omega (\xi) \right] \lozenge _{\alpha }\xi = \gamma \int\nolimits_{a}^{b}\Xi (\xi)\lozenge _{\alpha }\xi +\beta \int\nolimits_{a}^{b}\Omega (\xi)\lozenge _{\alpha }\xi$,

(2) $\int\nolimits_{a}^{b}c\Xi (\xi)\lozenge _{\alpha }\xi = c\int\nolimits_{a}^{b}\Xi (\xi)\lozenge _{\alpha }\xi$,

(3) $\int\nolimits_{a}^{b}\Xi (\xi)\lozenge _{\alpha }\xi = -\int\nolimits_{b}^{a}\Xi (\xi)\lozenge _{\alpha }\xi$,

(4) $\int\nolimits_{a}^{c}\Xi (\xi)\lozenge _{\alpha }\xi = \int\nolimits_{a}^{b}\Xi (\xi)\lozenge _{\alpha }\xi +\int\nolimits_{b}^{c}\Xi (\xi)\lozenge _{\alpha }\xi$.

Theorem 2.5. (^[6]) Let $\mathbb{T}$ be a time scale $a, \; b\in \mathbb{T}$ with $a < b.$ Assume that $\Xi$ and $\Omega$ are continuous functions on $[a, \; b]_{\mathbb{T}}$,

$(1)$ If $\Xi (\xi)\geq 0$ for all $\xi \in \lbrack a, \; b]_{\mathbb{T}}$, then $\int_{a}^{b}\Xi (\xi)\lozenge _{\alpha }\xi \geq 0;$

$(2)$ If $\Xi (\xi)\leq \Omega (\xi)$ for all $\xi \in \lbrack a, \; b]_{ \mathbb{T}}$, then $\int_{a}^{b}\Xi (\xi)\lozenge _{\alpha }\xi \leq \int_{a}^{b}\Omega (\xi)\lozenge _{\alpha }\xi;$

$(3)$ If $\Xi (\xi)\geq 0$ for all $\xi \in \lbrack a, \; b]_{\mathbb{T}}$, then $\Xi (\xi) = 0$ if, and only if, $\int_{a}^{b}\Xi (\xi)\lozenge _{\alpha }\xi = 0.$

Example 2.2. (^[6]) If $\mathbb{T = R}$, then

and if $\mathbb{T = N}$ and $m < n,$ then we obtain

Lemma 2.1. (Hölder's inequality ^[6]) If $\eta, \; \epsilon \in \mathbb{T}$, $0\leq \alpha \leq 1$, and $\lambda, \; \omega \in \mathrm{C}(\mathbb{[}\eta, \epsilon \mathbb{]}_{\mathbb{T}}, \; \mathbb{R}^{+}),$ then

where $\gamma > 1$ and $1/\gamma +1/\nu = 1.$ The inequality (2.2) is reversed for $0 < \gamma < 1$ or $\gamma < 0.$

Lemma 2.2. (Minkowski's inequality^[6]) Let $\eta, \epsilon \in \mathbb{T}$, $\epsilon > \eta, \; 0\leq \alpha \leq 1, \; p > 1$, and $\Xi, \; \Omega \in \mathrm{C}(\left[ \eta, \epsilon \right] _{\mathbb{T}}, \mathbb{R}^{+})$, then

Lemma 2.3. (Generalized Young inequality ^[4]) If $a, \; b\geq 0, \; sp, \; sq > 1$ with

then

Lemma 2.4. (See ^[31]) If $x, \; y > 0$ and $s > 1,$ then

and if $0 < s < 1,$ then

3.   Main results

In the manuscript, we will operate under the assumption that the considered integrals are presumed to exist. Also, we denote $\mathbf{\lozenge }_{\alpha }\mathbf{\tau } = \lozenge _{\alpha }\tau _{1}, \cdots, \lozenge _{\alpha }\tau _{N}\ $ and $\varpi \left(\mathbf{\tau }\right) = \varpi (\tau _{1}, \cdots, \tau _{N}).$

Theorem 3.1. Assume that $\eta _{i}, \ \epsilon _{i}\in \mathbb{T}, \; \epsilon _{i} > \eta _{i}, \; i = 1, 2, \cdots, N, \; 0\leq \alpha \leq 1$, $p, \; s > 0, \; ps > 1$ such that

and $\varpi, \; \Theta$: $\mathbb{T}^{N}\rightarrow \mathbb{R}^{+}$ are continuous functions, then

Proof. Applying (2.4) with

and

we get

Integrating (3.2) over $\xi _{i}$ from $\eta _{i}$ to $\epsilon _{i}, \; i = 1, 2, \cdots, N,$ we observe that

and then (note that $\frac{1}{sp}+\frac{1}{sq} = 1$)

which is (3.1). □

Remark 3.1. If $s = 1$ and $N = 1,$ we get the dynamic diamond alpha Hölder's inequality (2.2).

Remark 3.2. If $\mathbb{T = R}, \ \eta _{i}, \ \epsilon _{i}\in \mathbb{R}, \; \epsilon _{i} > \eta _{i}, \; i = 1, 2, \cdots, N, \; p, \; s > 0, \; ps > 1$ such that

and $\varpi, \; \Theta$: $\mathbb{R}^{N}\rightarrow \mathbb{R }^{+}$ are continuous functions$,$ then

where $\mathbf{d\tau = }d\tau _{1}, \cdots, d\tau _{N}.$

Remark 3.3. If $\mathbb{T = N}$, $N = 1, \; \eta, \ \epsilon \in \mathbb{N}, \; \epsilon > \eta, \; 0\leq \alpha \leq 1$, $p, \; s > 0, \; ps > 1$ such that

and $\varpi, \; \Theta \ $are positive sequences$,$ then

Example 3.1. If $\mathbb{T = R}, \; \epsilon \in \mathbb{R}$, $N = 1, \ \eta = 0, \; sp = 3, \; sq = , \ \varpi \left(\tau \right) = \tau ^{s}$, and $\Theta \left(\tau \right) = \tau ^{s},$ then

Proof. Note that

Using the above assumptions, we observe that

then

Also, we can get

and so

From (3.4) and (3.5) (note $\frac{1}{sp}+\frac{1}{sq} = 1$), we have that

Since $sp = 3$ and $sq = 3/2,$

then

From (3.3), (3.6) and (3.7), we see that

The proof is complete. □

Corollary 3.1. If $\eta _{i}, \ \epsilon _{i}\in \mathbb{T}, \; \alpha = 1, \; \epsilon _{i} > \eta _{i}, \; i = 1, 2, \cdots, N, \; p, \; s > 0, \; ps > 1$ such that

and $\varpi, \; \Theta$: $\mathbb{T}^{N}\rightarrow \mathbb{R }^{+}$, then

Corollary 3.2. If $\eta _{i}, \ \epsilon _{i}\in \mathbb{T}, \; \alpha = 0, \; \epsilon _{i} > \eta _{i}, \; i = 1, 2, \cdots, N, \; p, \; s > 0, \; ps > 1$ such that

and $\varpi, \; \Theta$: $\mathbb{T}^{N}\rightarrow \mathbb{R }^{+}$, then

Theorem 3.2. Assume that $\eta _{i}, \ \epsilon _{i}\in \mathbb{T}, \; \epsilon _{i} > \eta _{i}, \; i = 1, 2, \cdots, N, \; 0\leq \alpha \leq 1, \; p, \; s > 0, \; ps > 1$ such that

and $\Xi, \; \Omega$: $\mathbb{T} ^{N}\rightarrow \mathbb{R}^{+}\ $are continuous functions$.$ If $0 < m\leq \Xi /\Omega \leq M < \infty,$ then

Proof. Applying (3.1) with $\varpi \left(\mathbf{\tau }\right) = \Xi ^{\frac{1}{ p}}\left(\mathbf{\tau }\right)$ and $\Theta \left(\mathbf{\tau }\right) = \Omega ^{\frac{1}{q}}\left(\mathbf{\tau }\right)$, we see that

Since

then (3.11) becomes

Since

and

we have from (3.12) that

then we have for

that

which is (3.10). □

Remark 3.4. Take $\mathbb{T = R}, \; \eta _{i}, \ \epsilon _{i}\in \mathbb{R}, \; \epsilon _{i} > \eta _{i}, \; i = 1, 2, \cdots, N, \; p, \; s > 0, \; ps > 1$ such that

and $\Xi, \; \Omega$: $\mathbb{R}^{N}\rightarrow \mathbb{R} ^{+}\ $are continuous functions$.$ If $0 < m\leq \Xi /\Omega \leq M < \infty,$ then

Remark 3.5. Take $\mathbb{T = N}, \; N = 1, \; \eta, \ \epsilon \in \mathbb{N}, \; \epsilon > \eta, \; 0\leq \alpha \leq 1$, $p, \; s > 0, \; ps > 1$ such that

and $\Xi, \; \Omega \ $are positive sequences$.$ If $0 < m\leq \Xi /\Omega \leq M < \infty,$ then

Theorem 3.3. Assume that $\eta _{i}, \ \epsilon _{i}\in \mathbb{T}, \; \epsilon _{i} > \eta _{i}, \; i = 1, 2, \cdots, N, \; 0\leq \alpha \leq 1$, $sp < 0$ such that

and $\phi, \; \lambda$: $\mathbb{T}^{N}\rightarrow \mathbb{R} ^{+}$ are continuous functions, then

Proof. Since $sp < 0,$ by applying (3.1) with indices

(note that $\frac{1}{sP}+\frac{1}{sQ} = 1$), $\varpi \left(\mathbf{\tau }\right) = \phi ^{-sq}\left(\mathbf{\tau }\right)$ and $\Theta \left(\mathbf{\tau }\right) = \phi ^{sq}\left(\mathbf{\tau } \right) \lambda ^{sq}\left(\mathbf{\tau }\right),$ then

and substituting with values $\varpi \left(\mathbf{\tau }\right)$ and $\Theta \left(\mathbf{\tau }\right),$ the last inequality gives us

Thus

then we have for $sp < 0\ $and $sq = sp/\left(sp-1\right) > 0$ that

which is (3.13). □

Remark 3.6. If $\mathbb{T = R}, \ \eta _{i}, \ \epsilon _{i}\in \mathbb{R}, \; \epsilon _{i} > \eta _{i}, \; i = 1, 2, \cdots, N, \; 0\leq \alpha \leq 1$, $sp < 0$ such that

and $\phi, \; \lambda$: $\mathbb{R} ^{N}\rightarrow \mathbb{R}^{+}$are continuous functions, then

Remark 3.7. If $\mathbb{T = N}$, $N = 1, \eta, \ \epsilon \in \mathbb{N}, \ \epsilon > \eta, \; 0\leq \alpha \leq 1$, $sp < 0$, such that

and $\phi, \; \lambda \ $are positive sequences, then

Theorem 3.4. Assume that $\eta _{i}, \ \epsilon _{i}\in \mathbb{T}, \; \epsilon _{i} > \eta _{i}, \; i = 1, 2, \cdots, N, \; 0\leq \alpha \leq 1$, $r > u > t > 0$, and $\Xi, \; \Omega$: $\mathbb{T}^{N}\rightarrow \mathbb{R}^{+}$are continuous functions, then

Proof. Applying (3.1) with

$($note that $s = 1)$,

we see that

and then

which is (3.14). □

Remark 3.8. If $\mathbb{T = R}, \ \eta _{i}, \ \epsilon _{i}\in \mathbb{T}, \ \epsilon _{i} > \eta _{i}, i = 1, 2, \cdots, N, \; r > u > t > 0$ and $\Xi, \; \Omega$: $\mathbb{R} ^{N}\rightarrow \mathbb{R}^{+}\ $are continuous functions, then

Remark 3.9. If $\mathbb{T = N}$, $N = 1, \eta, \ \epsilon \in \mathbb{N}, \ \epsilon > \eta, \; 0\leq \alpha \leq 1$, $r > u > t > 0$ and $\Xi, \; \Omega \ $are positive sequences, then

Theorem 3.5. Assume that $\eta _{i}, \ \epsilon _{i}\in \mathbb{T}, \; \epsilon _{i} > \eta _{i}, \; i = 1, 2, \cdots, N, \; 0\leq \alpha \leq 1$, $s\geq 1, \; sp > 1$ such that

and $\Xi, \; \Omega$: $\mathbb{T}^{N}\rightarrow \mathbb{R}^{+}$ are continuous functions, then

Proof. Note that

Using the inequality (2.5) with replacing $x, \ y$ by $\Xi ^{\frac{1}{s }}\left(\mathbf{\tau }\right)$ and $\Omega ^{\frac{1}{s}}\left(\mathbf{ \tau }\right),$ respectively, we have

therefore, (3.16) becomes

then

Applying (3.1) on the two terms of the righthand side of (3.17) with indices $sp > 1$ and $\left(sp\right) ^{\ast } = sp/\left(sp-1\right)$ (note $\frac{1}{sp}+\frac{1}{\left(sp\right) ^{\ast }} = 1$), we obtain

and

Adding (3.18) and (3.19) and substituting into (3.17), we see that

thus,

which is (3.15). □

Remark 3.10. If $s = 1$ and $N = 1,$ then we get the dynamic Minkowski inequality (2.3).

In the following, we establish the reversed form of inequality (3.15).

Theorem 3.6. Assume that $\eta _{i}, \ \epsilon _{i}\in \mathbb{T}, \; \epsilon _{i} > \eta _{i}, \; i = 1, 2, \cdots, N, \; 0\leq \alpha \leq 1$, $p < 0, \; 0 < s < 1$ and $\Xi, \; \Omega$: $\mathbb{T}^{N}\rightarrow \mathbb{R}^{+}$ are continuous functions, then

Proof. Applying (2.6) with replacing $x, y$ by $\Xi ^{\frac{1}{s}}\left(\mathbf{\tau }\right)$ and $\Omega ^{\frac{1}{s}}\left(\mathbf{\tau } \right)$, respectively, we have that

Since

by using (3.21), we get

then

Applying (3.13) on $\int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Xi ^{\frac{1}{s}}\left(\mathbf{\tau }\right) \left(\Xi \left(\mathbf{\tau }\right) +\Omega \left(\mathbf{\tau }\right) \right) ^{\frac{1}{s}\left(sp-1\right) }\mathbf{\lozenge }_{\alpha }\mathbf{ \tau },$ with

and the indices $sp < 0, \ \left(sp\right) ^{\ast } = sp/\left(sp-1\right) ($note $\frac{1}{sp}+\frac{1}{\left(sp\right) ^{\ast }} = 1),$ we see that

Again by applying (3.13) on $\int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Omega ^{\frac{1}{s}}\left(\mathbf{ \tau }\right) \left(\Xi \left(\mathbf{\tau }\right) +\Omega \left(\mathbf{ \tau }\right) \right) ^{\frac{1}{s}\left(sp-1\right) }\mathbf{\lozenge } _{\alpha }\mathbf{\tau }$ with

and the indices $sp < 0, \ \left(sp\right) ^{\ast } = sp/\left(sp-1\right) ($note $\frac{1}{sp}+\frac{1}{\left(sp\right) ^{\ast }} = 1),$ we have that

Substituting (3.23) and (3.24) into (3.22), we see that

then

which is (3.20). □

Remark 3.11. If $\mathbb{T = R}$, $\eta _{i}, \ \epsilon _{i}\in \mathbb{T}, \; \epsilon _{i} > \eta _{i}, \; i = 1, 2, \cdots, N, \; p < 0, \; 0 < s < 1$ and $\Xi, \; \Omega$: $\mathbb{ R}^{N}\rightarrow \mathbb{R}^{+}\ $are continuous functions, then

Remark 3.12. If $\mathbb{T = N}$, $N = 1, \eta, \ \epsilon \in \mathbb{N}, \ \epsilon > \eta, \; 0\leq \alpha \leq 1$, $p < 0, \; 0 < s < 1$, and $\Xi, \; \Omega$ are positive sequences, then

4.   Conclusions and future work

In this paper, we present novel generalizations of Hölder's and Minkowski's dynamic inequalities on diamond alpha time scales. These inequalities give us the inequalities on delta calculus when $\alpha = 1$ and the inequalities on nabla calculus when $\alpha = 0$. Also, we introduced some of the continuous and discrete inequalities as special cases of our results. In addition, we added an example in our results to indicate the work.

In the future, we will establish some new generalizations of Hölder's and Minkowski's dynamic inequalities on conformable delta fractional time scales. Also, we will prove some new reversed versions of Hölder's and Minkowski's dynamic inequalities on time scales.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors extend their appreciation to the Deputyship for Research & Innovation, Ministry of Education in Saudi Arabia for funding this research work through the project number: ISP23-86.

Conflict of interest

The authors declare that they have no conflicts of interest.