Here, we recall some elementary results which will be used throughout the article.
Suppose
n to be a positive integer. For each
\(i\in\lbrace1, 2,\ldots, n\rbrace\), let
\(\mathbb{T}_{i}\) denote a time scale, that is, a non-empty closed subset of
\(\mathbb{R}\). Set
$$\mathbb{T}^{n}= \mathbb{T}_{1} \times\mathbb{T}_{2} \times\cdots\times \mathbb{T}_{n} = \bigl\lbrace t=(t_{1}, t_{2},\ldots, t_{n}) : t_{i} \in \mathbb{T}_{i}, i=1,2,\ldots,n\bigr\rbrace , $$
we call
\(\mathbb{T}^{n}\) an
n-dimensional time scale. The set
\(\mathbb {T}^{n}\) is a complete metric space with the metric
d defined by
$$d(t,s)= \Biggl(\sum_{i=1}^{n} | t_{i} - s_{i}|^{2} \Biggr)^{1/2} \quad \mbox{for }t,s \in\mathbb{T}^{n}. $$
Denote by
\(\sigma_{i}\) and
\(\rho_{i}\) the forward and backward jump operators defined on
\(\mathbb{T}_{i}\). Specially, for
\(t_{i}\in \mathbb{T}_{i}\), the forward jump operator
\(\sigma_{i}:\mathbb{T}_{i}\longrightarrow\mathbb{T}_{i}\) is defined by
$$\sigma_{i}(t_{i})= \inf \lbrace s_{i} \in \mathbb{T}_{i}: s_{i} > t_{i} \rbrace; $$
and the back jump operator
\(\rho_{i}:\mathbb{T}_{i} \longrightarrow \mathbb{T}_{i}\) is defined by
$$\rho_{i}(t_{i})= \sup \lbrace t_{i} \in \mathbb{T}_{i}: s_{i} < t_{i} \rbrace. $$
In this definition we put
\(\sigma_{i}( \max \mathbb{T}_{i})= \max \mathbb{T}_{i}\) if
\(\mathbb{T}_{i}\) has a finite maximum and
\(\rho _{i}( \min \mathbb{T}_{i})= \min \mathbb{T}_{i}\) whenever
\(\mathbb {T}_{i}\) has finite minimum. Also, we call each
\(t_{i}\in\mathbb {T}_{i}\) the right-scattered element in
\(\mathbb{T}_{i}\) if
\(\sigma _{i}(t_{i}) > t_{i}\), right-dense element in
\(\mathbb{T}_{i}\) if
\(\sigma_{i}(t_{i}) = t_{i}\), where
\(t_{i} < \max \mathbb{T}_{i}\), left-scattered in
\(\mathbb{T}_{i}\) if
\(\rho_{i}(t_{i}) < t_{i} \), and left-dense in
\(\mathbb{T}_{i}\) if
\(\rho_{i}(t_{i}) = t_{i}\), where
\(t_{i} > \min \mathbb{T}_{i}\). If
\(\mathbb{T}_{i}\) has a left-scattered maximum
M, then we define
\(\mathbb{T}_{i}^{k}= \mathbb{T}_{i}\setminus\lbrace M \rbrace\), otherwise
\(\mathbb{T}_{i}^{k}= \mathbb{T}_{i}\). When
\(\mathbb {T}_{i}\) has a right-scattered minimum
m, then
\((\mathbb {T}_{i})_{k}= \mathbb{T}_{i}\setminus\lbrace m \rbrace\), otherwise
\((\mathbb{T}_{i})_{k}= \mathbb{T}_{i}\).
Assume a function
\(f:\mathbb{T}^{n} \longrightarrow\mathbb{R}\). The partial delta derivative of
f with respect to
\(t_{i} \in(\mathbb {T}^{n})^{k}\) is defined as
$$\lim_{ s_{i} \longrightarrow t_{i} , s_{i} \neq\sigma_{i}(t_{i})} \frac{f( t_{1}, \ldots , t_{i-1} , \sigma_{i}(t_{i}) , t_{i+1} , \ldots , t_{n}) - f(t_{1}, \ldots , t_{i-1} , s_{i} , t_{i+1} , \ldots , t_{n})}{\sigma_{i}(t_{i}) - s_{i} } $$
whenever the limit exists, and denoted by
\(\frac{\partial f(t)}{\Delta _{i} t_{i}}\). Furthermore, the second order partial delta derivative of
f is denoted as
\(\frac{\partial^{2} f(t)}{\Delta_{i} t_{i}^{2}}\) or
\(\frac{\partial^{2} f(t)}{\Delta_{i} t_{i} \Delta_{j} t_{j}}\). In the same fashion, one can define a higher order delta derivative.
In addition to the above, the partial nebla derivative of
f with respect to the independent variable
\(t_{i} \in(\mathbb{T}^{n})_{k}\) is defined as
$$\lim_{ s_{i} \longrightarrow t_{i} , s_{i} \neq\rho_{i}(t_{i})} \frac{f( t_{1}, \ldots , t_{i-1} , \rho_{i}(t_{i}) , t_{i+1} , \ldots , t_{n}) - f(t_{1}, \ldots , t_{i-1} , s_{i} , t_{i+1} , \ldots , t_{n})}{\rho _{i}(t_{i}) - s_{i} } $$
provided the limit exists and is denoted by
\(\frac{\partial f(t)}{\rho _{i} t_{i}}\). The second order partial nebla derivative of
f is denoted as
\(\frac{\partial^{2} f(t)}{\rho_{i} t_{i}^{2}}\) or
\(\frac {\partial^{2} f(t)}{\rho_{i} t_{i} \rho_{j} t_{j}}\). Higher order partial nebla derivatives are similarly defined. Combining both delta and nebla derivatives, we can define the mixed derivatives. For instance, a second order mixed derivative is denoted by
\(\frac {\partial^{2} f(t)}{\Delta_{i} t_{i} \rho_{j} t_{j}}\) or
\(\frac {\partial^{2} f(t)}{\rho_{i} t_{i} \Delta_{j} t_{j}}\). Hence, for any multi index
\(\alpha= (\alpha_{1},\ldots,\alpha_{n})\) of order
\(|\alpha| = \alpha_{1} + \cdots + \alpha_{n}\), we define
$$D^{\alpha}_{\Delta}f = \frac{\partial^{|\alpha|}f}{\Delta_{1} t_{1}^{\alpha_{1}} \cdots \Delta_{n} t_{n}^{\alpha_{n}}}. $$
If
k is a non-negative integer, then
$$D^{k}_{\Delta}f = \bigl\lbrace D^{\alpha}_{\Delta}f : |\alpha|= k \bigr\rbrace , $$
i.e., the set of all delta partial derivatives of order
k with
$$\bigl\vert D^{k}_{\Delta}f \bigr\vert = \biggl(\sum _{ \vert \alpha \vert =k} \bigl\vert D^{\alpha }_{\Delta}f \bigr\vert ^{2} \biggr)^{1/2}. $$
For
\(k=1\), the elements of
\(D_{\Delta}f\) can be seen in the form of a vector:
$$D_{\Delta}f = \biggl( \frac{\partial f}{\Delta t_{1}} ,\ldots, \frac {\partial f}{\Delta t_{n}} \biggr). $$
For a Δ-measurable set
\(\mathbb{E}_{\mathbb{T}} \subset \mathbb{T}^{n}\) and a Δ-measurable function
\(f:\mathbb {E}_{\mathbb{T}} \longrightarrow\mathbb{R}\), the corresponding Lebesgue Δ-integral of
f over
\(\mathbb{E}_{\mathbb{T}}\) will be denoted by
$$\int_{\mathbb{E}_{\mathbb{T}}} f( t_{1} , t_{2},\ldots, t_{n} ) \Delta t_{1} \Delta t_{2} , \ldots , \Delta t_{n} \quad \mbox{or}\quad \int_{\mathbb{E}_{\mathbb {T}}} f(t) \Delta t \quad \mbox{or}\quad \int_{\mathbb{E}_{\mathbb{T}}} f(t) \mu _{\Delta}, $$
where
\(\mu_{\Delta}\) represents the Lebesgue measure [
10].
2.1 Some function spaces and results on time scales
Let
\(\Omega_{\mathbb{T}^{n}}\) be an open subset of
\(\mathbb{T}^{n} \),
\(n\geq1\).
$$\begin{aligned}& C^{k}( \Omega_{\mathbb{T}^{n}} ) = \lbrace u : \Omega_{\mathbb{T}^{n}} \longrightarrow\mathbb{R} : u \mbox{ is $k$-times continuously differentiable on } \Omega_{\mathbb{T}^{n}} \rbrace, \\& C^{\infty}( \Omega_{\mathbb{T}^{n}} ) = \lbrace u : \Omega _{\mathbb{T}^{n}} \longrightarrow\mathbb{R} : u \mbox{ is infinitely differentiable on } \Omega_{\mathbb{T}^{n}} \rbrace= \bigcap_{k=0}^{\infty} C^{k}( \Omega_{\mathbb{T}^{n}}). \end{aligned}$$
Moreover, we define some other auxiliary spaces
$$\begin{aligned}& C^{k}_{c}( \Omega_{\mathbb{T}^{n}} ) = \mbox{Functions in }C^{k}( \Omega_{\mathbb{T}^{n}} )\mbox{ with compact support}, \\& C^{\infty}_{c}( \Omega_{\mathbb{T}^{n}} ) = \mbox{Functions in }C^{\infty }( \Omega_{\mathbb{T}^{n}} )\mbox{ with compact support}, \\& \operatorname{BC}( \Omega_{\mathbb{T}^{n}} ) = \mbox{Space of all continuous and bounded functions on }\Omega_{\mathbb{T}^{n}}. \end{aligned}$$
For
\(u \in \operatorname{BC}( \Omega_{\mathbb{T}^{n}} ) \) and
\(0< \gamma\leq1\), let
$$\| u \|_{C( \Omega_{\mathbb{T}^{n}} )} := \sup_{t \in\Omega _{\mathbb{T}^{n}}} \bigl\vert u(t) \bigr\vert \quad \mbox{and}\quad [u]_{\gamma}:= \sup_{t , s \in\Omega _{\mathbb{T}^{n}},t \neq s} \biggl\lbrace \frac{| u(t)-u(s) |}{| t-s |^{\gamma}} \biggr\rbrace . $$
If
\([u]_{\gamma}<\infty\), then
u is Hölder continuous with Hölder exponent
γ. The collection of
γ-Hölder continuous functions on
\(\Omega_{\mathbb{T}^{n}}\) will be denoted by
$$C^{0,\gamma}(\Omega_{\mathbb{T}^{n}}):= \bigl\lbrace u \in \operatorname{BC}( \Omega _{\mathbb{T}^{n}}) : [u]_{\gamma}< \infty\bigr\rbrace , $$
and for
\(u \in C^{0,\gamma}(\Omega_{\mathbb{T}^{n}}) \) we can define the norm by
$$ \| u \|_{C^{0,\gamma}(\Omega_{\mathbb{T}^{n}})} := \| u \|_{C( \Omega_{\mathbb{T}^{n}} )} + [u]_{\gamma}. $$
(2.1)
For
\(\gamma=1\), the function
u is said to be a Lipschitz continuous function.
The Hölder space
\(C^{k , \gamma}(\Omega_{\mathbb{T}^{n}})\) consists of those functions
u that are
k times continuously differentiable and whose
kth-partial derivatives are Hölder continuous with exponent
γ. The norm linear spaces,
\(C^{k , \gamma}(\Omega_{\mathbb{T}^{n}})\) are Banach spaces with the norm defined by
$$\| u \|_{C^{k , \gamma}(\Omega_{\mathbb{T}^{n}})} = \sup_{t \in \Omega_{\mathbb{T}^{n}}} \bigl|D^{\alpha}_{\Delta}u\bigr| + \sup_{t , s \in \Omega_{\mathbb{T}^{n}},t \neq s} \biggl\lbrace \frac{| u(t)-u(s) |}{| t-s |^{\gamma}} \biggr\rbrace . $$