Weighted Fractional Sobolev Spaces on Timescales with Applications to Weighted Fractional p-Laplacian Systems

Abstract

The primary objective of this work is to develop a comprehensive theory of weighted fractional Sobolev spaces within the framework of timescales. To this end, we first introduce a novel class of weighted fractional operators and rigorously define associated weighted integrable spaces on timescales, generalising classical notions to this non-uniform temporal domain. Building upon these foundations, we systematically investigate the fundamental functional-analytic properties of the resulting Sobolev spaces. Specifically, we establish their completeness under appropriate norms, prove reflexivity under appropriate duality pairings, and demonstrate separability under mild conditions on the weight functions. As a pivotal application of our theoretical framework, we derive two robust existence theorems for solutions to the proposed model. These results not only extend classical partial differential equation theory to timescales but also provide a versatile tool for analysing dynamic systems with heterogeneous temporal domains.

1. Introduction

In conventional approaches to the resolution of practical problems, researchers have historically categorised mathematical models into two distinct types based on their mathematical characteristics: continuous systems, which are analysed through differential equations, and discrete systems, which are analysed through difference equations. Nevertheless, owing to the continuous advancement of scientific research in recent years, scholars have come to the realisation that a considerable number of practical problems inherently combine both continuous and discrete situations. It is evident that this hybrid property renders the traditional methods, which consider the continuous case and the discrete case, respectively, no longer suitable for addressing these problems. In order to surmount this inherent limitation, Hilger [1,2] advanced the seminal timescale theory, a pioneering concept that seamlessly integrates continuous and discrete analyses. The timescale theory has been demonstrated to enhance the capacity to model the complex dynamic systems encountered in engineering, physics, and biological sciences. To date, the timescale theory has been extensively applied in numerous fields, including economics, ecology, and astronomy [3,4,5,6,7].

The notion of extending the concept of calculus from integer order to fractional order is both natural and ingenious from a mathematical perspective. However, fractional calculus has historically been confined to theoretical discussions within the field of mathematics. This is primarily due to the fact that fractional calculus does not possess as readily apparent a practical significance as integer calculus. In the 1970s, researchers discovered that fractional calculus could be applied to fractal geometry and fractal dynamics, due to its non-local character. This finding led to a rapid development of the subject. Over the course of centuries, and particularly in the recent past, the field of fractional calculus has evolved significantly. Indeed, it has become a crucial branch of mathematics and a powerful analytical tool in a wide range of fields. The following fields have been identified as relevant to the study: forecasting of the transmission of COVID-19 [8,9,10,11,12], processing of images [13,14,15,16], fractional order control [17,18,19,20,21], seismology [22,23,24], and other fields [25,26,27,28,29]. In the extensive corpus of the literature pertaining to the definition of fractional calculus, the most prevalent definitions are those of Riemann–Liouville and Caputo. In 1892, Hadamard proposed the concept of the Hadamard fractional calculus (see ref. [30]). In their seminal work, the authors in paper [31] provided a comprehensive definition of $\zeta$-Caputo fractional derivatives. This definition encompasses three key components: the Caputo derivative ($\zeta \left(t\right)=t$), the Caputo–Katugampola derivative ($\zeta \left(t\right)=t^{\rho -1}/\rho$), and the Hadamard derivative ($\zeta \left(t\right)=lnt$). In order to obtain a more general fractional derivative, the authors of article [32] incorporated a weighted function $\varkappa \left(t\right)$ and proposed the concept of weighted $\zeta$-Caputo fractional derivatives. In the circumstance that $\varkappa \left(t\right)=1$, the aforementioned derivatives are equivalent to their conventional counterparts, namely, the $\zeta$-Caputo fractional derivatives. It is evident that the weighted $\zeta$-Caputo fractional derivative possesses the capacity to derive a multitude of fractional operators through the utilisation of diverse choices of $\varkappa$ and $\zeta$. This observation signifies that the weighted $\zeta$-Caputo fractional derivative exhibits remarkable applicability and uniformity.

As a cornerstone of modern functional analysis, the theory of Sobolev spaces provides a unifying framework for studying differential equations across continuous, discrete, and hybrid domains [33,34,35,36,37,38]. These spaces not only facilitate the analysis of solution regularity and well-posedness but also enable the development of numerical methods for complex dynamical systems. Recent advances have extended their applicability to nonlocal operators, variable-order derivatives, and heterogeneous timescales, significantly broadening their scope in both theoretical and applied mathematics [36,37,38]. In the context of timescale calculus, Zhou and Li [33] laid the groundwork by constructing integer-order Sobolev spaces and establishing their fundamental properties, including compact embeddings and density theorems. Their variational approach to solving boundary value problems for dynamic equations on timescales demonstrated the adaptability of Sobolev techniques to non-continuous domains. This work was later complemented by Wei and Bai [35], who incorporated nonlocal boundary conditions and superlinear damping effects into the analysis, further enriching the study of dynamic systems on timescales. The fractional extension of this theory was pioneered by Wang et al. [36], who introduced conformable fractional Sobolev spaces on timescales and applied them to fractional dynamic equations. Their work was refined by Hu and Li [37], who developed sharper embedding results and applied them to fractional boundary value problems, highlighting the interplay between fractional calculus and timescale topology. Meanwhile, Rahimkhani and Sedaghat [38] expanded the analytical toolkit by incorporating wavelet methods for solving fractal–fractional systems with variable-order kernels, demonstrating the versatility of Sobolev frameworks in handling nonlocal and memory-dependent phenomena.

This study is driven by the works referenced in the preceding analysis, and its aim is to establish weighted nonlocal timescaled Sobolev spaces. Further, a systematic investigation is conducted into the fundamental properties of these spaces, including completeness, reflexivity, and separability. This establishes them as well-defined Banach spaces suitable for variational analysis. Subsequently, the established foundation is utilised in conjunction with critical point theory and variational methods to establish two distinct existence conclusions for solutions to the weighted fractional p-Laplacian problem on timescales. The results obtained in this study serve to extend classical Sobolev space theory to a more general fractional and weighted setting. Furthermore, they provide new analytical tools for the study of nonlinear problems with nonlocal and anisotropic features in the context of timescale calculus. The findings contribute to the broader understanding of partial differential equations on timescales while opening avenues for further research in both theoretical and applied directions.

The remainder of this paper is structured as follows. In Section 2, the primary objective is to undertake a comprehensive review of calculus on timescales. In Section 3, the establishment of weighted nonlocal Sobolev spaces is conducted, and the completion, reflexivity, and separability of these spaces as Banach spaces is demonstrated. In Section 4, our main application demonstrates that solutions to the weighted fractional p-Laplacian system exist, as verified through variational analysis.

2. Preliminaries

To begin with, the relevant definitions of Hilger calculus on timescales are introduced. In the mathematical context of timescale theory, a timescale $\mathbb{T}$ is an arbitrary non-empty closed subset of the real numbers $\mathbb{R}$, which has topology inherited from the real numbers with the standard topology. Throughout this paper, we define $D_{\mathbb{T}}=D\cap \mathbb{T}$, where $D\subseteq \mathbb{R}$. Let $I_{\mathbb{T}}={[a,b)}_{\mathbb{T}}$ and ${\overline{I}}_{\mathbb{T}}={[a,b]}_{\mathbb{T}}$.

Definition 1

([1]). The forward jump operator $\sigma :\mathbb{T}\to \mathbb{T}$ is expressed by

$$\sigma \left(t\right)=inf\{s\in \mathbb{T}:s>t\},$$

and the backward jump operator $\rho :\mathbb{T}\to \mathbb{T}$ is of the form

$$\rho \left(t\right)=sup\{s\in \mathbb{T}:s<t\},$$

where $\forall t\in \mathbb{T}.$ When $\sigma \left(t\right)>t$ (or $\rho \left(t\right)<t$), the point t is classified as right-scattered (or left-scattered, respectively). For a point t satisfying $t<max\mathbb{T}$ (or $t>min\mathbb{T}$) with $\sigma \left(t\right)=t$ (or $\rho \left(t\right)=t$), it is termed right-dense (or left-dense). A point t that is simultaneously right-scattered and left-scattered is referred to as an isolated point. If t is both right-dense and left-dense, it is called a dense point. In the case where $\mathbb{T}$ has a left-scattered maximum M, the set ${\mathbb{T}}^k$ is defined as $\mathbb{T}\setminus \left\{M\right\}$; otherwise, ${\mathbb{T}}^k$ remains $\mathbb{T}$. The graininess function $\mu :\mathbb{T}\to [0,\infty )$ is given by $\mu \left(t\right)=\sigma \left(t\right)-t$.

Definition 2

([1]). Suppose that $g:\mathbb{T}\to \mathbf{R}$, $t\in {\mathbb{T}}^k$. We define $g^{\Delta }\left(t\right)$ to be the number $\left(\mathit{provided}\mathit{it}\mathit{exists}\right)$ with the property that for $\forall ϵ>0$ there exists a neighbourhood $U_t\left(h\right)$ of t, such that

$$\left|g\left(\sigma \left(t\right)\right)-g\left(r\right)-g^{\Delta }\left(t\right)\left(\sigma \left(t\right)-r\right)\right|\le ϵ|\sigma \left(t\right)-r|,\forall r\in U_t\left(h\right)\subset \mathbb{T}.$$

The Δ-derivative of f at t, denoted $f^{\Delta }\left(t\right)$, generalises the classical derivative for time-dependent domains.

Definition 3

([2]). If $\underset{s\to t^{-}}{lim}f\left(s\right)$ exists for $\rho \left(t\right)=t\in \mathbb{T}$ and it is also continuous for $\sigma \left(t\right)=t\in \mathbb{T}$ then we say that $f:\mathbb{T}\to \mathbf{R}$ is rd-continuous.

For ease of narration, in the remaining part of this paper, let $C_{rd}(\mathbb{T},\mathbf{R})$ be the set of functions $f:\mathbb{T}\to \mathbf{R}$ that are rd-continuous, and let $C_{rd}^1(\mathbb{T},\mathbf{R})$ denote the set of functions $f:\mathbb{T}\to \mathbf{R}$ that are rd-continuous and differentiable.

Theorem 1

([2]). Let $a,b\in \mathbb{T}$ and $p,q\in C_{rd}(\mathbb{T},\mathbf{R})$. We have

$${\mathit{\int }}_a^b{p}^{\sigma }\left(s\right)q^{\Delta }\left(s\right)\Delta s=\left(pq\right)\left(b\right)-\left(pq\right)\left(a\right)-{\mathit{\int }}_a^b{p}^{\Delta }\left(s\right)q\left(s\right)\Delta s.$$

Theorem 2

([2]). Assuming that f is Δ-integrable on $\mathbb{T}$, one has $|f|$ being Δ-integrable on $\mathbb{T}$. Moreover,

$$\left|{\mathit{\int }}_a^{b}f\left(s\right)\Delta s\right|\le {\mathit{\int }}_a^b|f\left(s\right)|\Delta s,\forall a,b\in \mathbb{T}.$$

Definition 4

([2]). A continuous function $G:\mathbb{T}\to \mathbf{R}$ is said to be a Δ-anti-derivative of $g:\mathbb{T}\to \mathbf{R}$ if G is Δ-differentiable on $\mathbb{T}$ and satisfies $G^{\Delta }\left(t\right)=g\left(t\right)$ for every $t\in \mathbb{T}$. Given such a Δ-anti-derivative G, the Cauchy Δ-integral of g over the interval $[c,d]\subseteq \mathbb{T}$ is defined as

$${\mathit{\int }}_c^{d}g\left(s\right)\Delta s=G\left(d\right)-G\left(c\right).$$

Lemma 1

([39]). Let $X\subseteq C({\overline{I}}_{\mathbb{T}},\mathbf{R})$ be bounded. X is relatively compact if for every given $ϵ>0$ there exists $\delta >0$ ensuring that for all $g\in X$ and $k_1,k_2\in {\overline{I}}_{\mathbb{T}}$ satisfying $|k_1-k_2|<\delta$ the inequality $|g\left(k_1\right)-g\left(k_2\right)|<ϵ$ holds.

Definition 5.

Given $\gamma \in (0,1]$ and a Δ-integrable mapping f on ${[a,b]}_{\mathbb{T}}$, the left and right generalised integrals with weight γ are defined as follows:

$${}_{a}I_t^{\gamma ,\zeta ,\varkappa }f\left(t\right):={\varkappa }^{-1}\left(t\right){\mathit{\int }}_a^t{\displaystyle \frac{{\left(\zeta \left(t\right)-\zeta \left(\sigma \left(s\right)\right)\right)}^{\gamma -1}}{\Gamma \left(\gamma \right)}}\varkappa \left(s\right){\zeta }^{\Delta }\left(s\right)f\left(s\right)\Delta s$$

and

$${}_{t}I_b^{\gamma ,\zeta ,\varkappa }f\left(t\right):={\varkappa }^{-1}\left(t\right){\mathit{\int }}_t^b{\displaystyle \frac{{\left(\zeta \left(s\right)-\zeta \left(\sigma \left(t\right)\right)\right)}^{\gamma -1}}{\Gamma \left(\gamma \right)}}\varkappa \left(s\right){\zeta }^{\Delta }\left(s\right)f\left(s\right)\Delta s,$$

respectively, where Γ is the gamma function, $\zeta :{\overline{I}}_{\mathbb{T}}\to \mathbf{R}$ is strictly increasing Δ-differentiable function, the weighted function ϰ is a mapping from ${\overline{I}}_{\mathbb{T}}$ to $\mathbf{R}$, and for $t\in {\overline{I}}_{\mathbb{T}}$ it is not zero.

Definition 6.

For $\gamma \in (0,1]$, $f:\mathbb{T}\to \mathbf{R}$, the weighted generalised left and right Riemann–Liouville fractional derivatives of f with order γ are

$${}_{a}D_t^{\gamma ,\zeta ,\varkappa }f\left(t\right):=D_{\varkappa }^1{}_{a}I_t^{1-\gamma ,\zeta ,\varkappa }f\left(t\right)={\displaystyle \frac{{\varkappa }^{-1}\left(t\right)}{\Gamma (1-\gamma ){\zeta }^{\Delta }\left(t\right)}}{\left({\mathit{\int }}_a^t{(\zeta \left(t\right)-\zeta \left(\sigma \left(s\right)\right))}^{-\gamma }\varkappa \left(s\right){\zeta }^{\Delta }\left(s\right)f\left(s\right)\Delta s\right)}^{\Delta }$$

and

$${}_{t}D_b^{\zeta ,\varkappa }f\left(t\right):=-D_{\varkappa }^1{}_{t}I_b^{1-\gamma ,\zeta ,\varkappa }f\left(t\right)={\displaystyle \frac{-{\varkappa }^{-1}\left(t\right)}{\Gamma (1-\gamma ){\zeta }^{\Delta }\left(t\right)}}{\left({\mathit{\int }}_t^b{(\zeta \left(s\right)-\zeta \left(\sigma \left(t\right)\right))}^{-\gamma }\varkappa \left(s\right){\zeta }^{\Delta }\left(s\right)f\left(s\right)\Delta s\right)}^{\Delta },$$

respectively, where $D_{\varkappa }^{1}f\left(t\right)={\displaystyle \frac{{\varkappa }^{-1}\left(t\right)}{{\zeta }^{\Delta }\left(t\right)}}{\left(\varkappa \left(t\right)f\left(t\right)\right)}^{\Delta }$, $t\in \mathbb{T}$.

For simplicity, we define

$$I_{a^{+}}^{\gamma ,\zeta ,\varkappa }{=}_a{D}_t^{\gamma ,\zeta ,\varkappa },I_{b^{-}}^{\gamma ,\zeta ,\varkappa }{=}_t{D}_b^{\gamma ,\zeta ,\varkappa },D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{=}_a{D}_t^{\gamma ,\zeta ,\varkappa },D_{b^{-}}^{\gamma ,\zeta ,\varkappa }{=}_t{D}_b^{\gamma ,\zeta ,\varkappa }.$$

For a function g defined on the interval $I_{\mathbb{T}}$ with values in $\mathbf{R}$, we introduce the following function spaces: $C_{rd}^{\gamma }:=C_{rd}^{\gamma }\left(I_{\mathbb{T}},\mathbf{R}\right)$ denotes the set of all functions in which $D_{c^{+}}^{\gamma ,\zeta ,\varkappa }g$ exists and belongs to $C_{rd}\left(I_{\mathbb{T}},\mathbf{R}\right)$. $C_{0,rd}^{\gamma }\left(I_{\mathbb{T}},\mathbf{R}\right)$ represents the subset of $C_{rd}^{\gamma }$ and meeting $g\left(a\right)=g\left(b\right)=0$. $C_{a,b,rd}^{\gamma }\left(I_{\mathbb{T}},\mathbf{R}\right)$ denotes the subset of $C_{rd}^{\gamma }$ and meeting $g\left(a\right)=g\left(b\right)$.

Definition 7.

Let $p\ge 1$, function $f:E\to \mathbf{R}$ be $\Delta -$measurable, where $E\subset \mathbb{T}$ is $\Delta -$measurable. Let

$${\mathcal{H}}_{\Delta }^p(E,\mathbf{R})=\left\{f:{\mathit{\int }}_E{|f\left(s\right)|}^p\Delta s<+\infty \right\},$$

$$X_{\Delta }^{\varkappa ,p}(E,\mathbf{R})=\left\{f:{\mathit{\int }}_E{|\varkappa \left(s\right)f\left(s\right)|}^p{\zeta }^{\Delta }\left(s\right)\Delta s<+\infty \right\}$$

for $1\le p<+\infty$; and C is a positive constant,

$${\mathcal{H}}_{\Delta }^p(E,\mathbf{R})=\left\{f:|f|\le C\Delta -a.e.\mathrm{on}E\right\},$$

$$X_{\Delta }^{\varkappa ,p}(E,\mathbf{R})=\left\{f:|\varkappa f|\le C\Delta -a.e.\mathrm{on}E\right\}$$

for $p=+\infty$.

Remark 1.

We note that $f\in X_{\Delta }^{\varkappa ,p}$ if $\varkappa f{\left({\zeta }^{\Delta }\right)}^{{\displaystyle \frac{1}{p}}}\in {\mathcal{H}}_{\Delta }^p$ and that $f\in X_{\Delta }^{\varkappa ,+\infty }$ if $\varkappa f\in {\mathcal{H}}_{\Delta }^{+\infty }$.

We now present some fundamental analytical properties of the weighted Riemann–Liouville fractional calculus.

Lemma 2.

For $n\in \mathbb{N}$, we have $D_{\varkappa }^1{I}_{a^{+}}^{1,\zeta ,\varkappa }f=f.$

Proposition 1.

 (i) For $\gamma ,\beta >0$, we have

$${}_{a}I_s^{\gamma ,\zeta ,\varkappa }\left({\varkappa }^{-1}\left(s\right){\left(\zeta \left(s\right)-\zeta \left(a\right)\right)}^{\beta -1}\right)={\displaystyle \frac{\Gamma \left(\beta \right)}{\Gamma (\beta +\gamma )}}{\varkappa }^{-1}\left(s\right){\left(\zeta \left(s\right)-\zeta \left(a\right)\right)}^{\beta +\gamma -1}.$$

(ii) 

For $\gamma <n$ and $\beta >0$, we have

$${}_{a}D_s^{\gamma ,\zeta ,\varkappa }\left({\varkappa }^{-1}\left(s\right){\left(\zeta \left(s\right)-\zeta \left(a\right)\right)}^{\beta -1}\right)={\displaystyle \frac{\Gamma \left(\beta \right)}{\Gamma (\beta -\gamma )}}{\varkappa }^{-1}\left(t\right){\left(\zeta \left(s\right)-\zeta \left(a\right)\right)}^{\beta -\gamma -1}.$$

Proposition 2.

Let $\gamma ,\beta >0$, $p\ge 1$, $f\in X_{\Delta }^{\varkappa ,p}$. Then, $\left(I_{a^{+}}^{\gamma ,\zeta ,\varkappa }{I}_{a^{+}}^{\beta ,\zeta ,\varkappa }\right)f=I_{a^{+}}^{\gamma +\beta ,\zeta ,\varkappa }f.$

Proposition 3.

Let $\gamma >\beta >0$; then, $\left(D_{a^{+}}^{\beta ,\zeta ,\varkappa }{I}_{a^{+}}^{\gamma ,\zeta ,\varkappa }\right)f=I_{a^{+}}^{\gamma -\beta ,\zeta ,\varkappa }f.$ Especially, we have $D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{I}_{a^{+}}^{\gamma ,\zeta ,\varkappa }f=f.$

Lemma 3

([36]). Assume that $g\in L_{\Delta }^1$, and for each $q\in C_0(I_{\mathbb{T}},\mathbf{R})$ we have

$${\mathit{\int }}_a^{b}g\left(s\right)q\left(s\right)\Delta s$$

which is equal to 0; then, $g(·)=0,\Delta -a.e.$

Theorem 3.

Assume that $g\in X_{\Delta }^{\varkappa ,1}$, and for each $q\in C_0(I_{\mathbb{T}},\mathbf{R}),$ we have

$${\mathit{\int }}_a^b\varkappa \left(s\right)g\left(s\right){\zeta }^{\Delta }\left(s\right)q\left(s\right)\Delta s,$$

which is equal to 0; then, $g(·)=0,\Delta -a.e.$

Proof. 

From Remark 1, we obtain $\varkappa g{\zeta }^{\Delta }\in {\mathcal{H}}_{\Delta }^1$. In light of Lemma 3, we have $\left(\varkappa g{\zeta }^{\Delta }\right)\left(t\right)=0,$$\Delta -a.e.t\in I_{\mathbb{T}}$, which means that $g\left(t\right)=0,\Delta -a.e.t\in I_{\mathbb{T}}.$ □

Lemma 4

([36]). Let $g\in L_{\Delta }^1$ and $q\in C_{0,rd}^1(I_{\mathbb{T}},\mathbf{R})$. For $\Delta -a.e.t\in I_{\mathbb{T}}$, g is a constant if

$${\mathit{\int }}_a^{b}g\left(s\right)q^{\Delta }\left(s\right)\Delta s=0$$

holds.

Theorem 4.

Let $g\in X_{\Delta }^{\varkappa ,1}$. For $\Delta -a.e.t\in I_{\mathbb{T}}$, g is a constant if for each $I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }q\in C_{0,rd}^1(I_{\mathbb{T}},\mathbf{R})$,

$${\mathit{\int }}_a^b\varkappa \left(s\right)g\left(s\right){\zeta }^{\Delta }\left(s\right)D_{a^{+}}^{\gamma ,\zeta ,\varkappa }q\left(s\right)\Delta s=0$$

holds.

Proof. 

From Remark 1, we obtain $g\in {\mathcal{H}}_{\Delta }^1$. Furthermore, according to Definitions 5 and 6, it holds that ${\mathit{\int }}_a^b\varkappa \left(s\right)g\left(s\right){\zeta }^{\Delta }\left(s\right)D_{a^{+}}^{\gamma ,\zeta ,\varkappa }q\left(s\right)\Delta s={\mathit{\int }}_a^{b}g\left(s\right){\left(I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }q\left(s\right)\right)}^{\Delta }\Delta s$. Applying Lemma 4, we complete the proof. □

Lemma 5

([40]). Space ${\mathcal{H}}_{\Delta }^p:={\mathcal{H}}_{\Delta }^p\left(I_{\mathbb{T}},\mathbf{R}\right)$ becomes a Banach space with

$${\parallel g\parallel }_{{\mathcal{H}}_{\Delta }^p}:=\left\{\begin{array}{cc}{\left[{\displaystyle {\mathit{\int }}_a^b{|g|}^p\left(s\right)\Delta s}\right]}^{{\displaystyle \frac{1}{p}}},\hfill & ifp\in [1,+\infty ),\hfill \\ inf\{C\in \mathbf{R}:|g|\le C,\Delta -a.e.t\in I_{\mathbb{T}}\},\hfill & ifp=+\infty ,\hfill \end{array}\right.$$

where $g\in {\mathcal{H}}_{\Delta }^p$. Moreover, ${\mathcal{H}}_{\Delta }^2$ is a Hilbert space with the inner product

$${(f,g)}_{{\mathcal{H}}_{\Delta }^2}:={\mathit{\int }}_{f^0}f\left(s\right)·g\left(s\right)\Delta s,$$

where $(f,g)\in$${\mathcal{H}}_{\Delta }^2\times {\mathcal{H}}_{\Delta }^2$.

Theorem 5.

Space $X_{\Delta }^{\varkappa ,p}:=X_{\Delta }^{\varkappa ,p}\left(I_{\mathbb{T}},\mathbf{R}\right)$ becomes a Banach space with

$${\parallel g\parallel }_{X_{\Delta }^{\varkappa ,p}}:=\left\{\begin{array}{cc}{\left[{\mathit{\int }}_a^b{|\varkappa \left(s\right)g\left(s\right)|}^p{\zeta }^{\Delta }\left(s\right)\Delta s\right]}^{{\displaystyle \frac{1}{p}}},\hfill & ifp\in [1,+\infty ),\hfill \\ inf\{C\in \mathbf{R}:|\varkappa \left(s\right)g\left(s\right)|\le C,\Delta -a.e.t\in I_{\mathbb{T}}\},\hfill & ifp=+\infty ,\hfill \end{array}\right.$$

where $g\in X_{\Delta }^{\varkappa ,p}{I}_{\mathbb{T}}$. Moreover, we define

$${(g,h)}_{X_{\Delta }^{\varkappa ,2}}:={\mathit{\int }}_a^b{\varkappa }^2\left(s\right){\zeta }^{\Delta }\left(s\right)g\left(s\right)h\left(s\right)\Delta s$$

for $(g,h)\in X_{\Delta }^{\varkappa ,2}\times X_{\Delta }^{\varkappa ,2}$. Then, $X_{\Delta }^{\varkappa ,2}$ becomes a Hilbert space.

Lemma 6

([40]). Let $p\ge 1, q\in \mathbf{R}$, such that ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$ and $f\in {\mathcal{H}}_{\Delta }^p, g\in {\mathcal{H}}_{\Delta }^q$; then, $fg\in {\mathcal{H}}_{\Delta }^1$ and

$${\parallel fg\parallel }_{{\mathcal{H}}_{\Delta }^1}\le {\parallel f\parallel }_{{\mathcal{H}}_{\Delta }^p}{\parallel g\parallel }_{{\mathcal{H}}_{\Delta }^q}.$$

Theorem 6.

Let $p\ge 1, q\in \mathbf{R}$, such that ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$ and $u\in X_{\Delta }^{\varkappa ,p}, v\in X_{\Delta }^{\varkappa ,q}$; then, $uv\in X_{\Delta }^{\varkappa ,1}$ and

$${\parallel uv\parallel }_{X_{\Delta }^{\varkappa ,1}}\le C_{\varkappa }{\parallel u\parallel }_{X_{\Delta }^{\varkappa ,p}}{\parallel v\parallel }_{X_{\Delta }^{\varkappa ,q}},$$

where $C_{\varkappa }={max}_{t\in I_{\mathbb{T}}}|{\varkappa }^{-1}\left(t\right)|$.

Proof. 

From Lemma 6, we infer

$$\begin{array}{ccc}\hfill {\parallel uv\parallel }_{X_{\Delta }^{\varkappa ,1}}& =& {\mathit{\int }}_a^b|\varkappa \left(s\right)u\left(s\right)v\left(s\right)|{\zeta }^{\Delta }\left(s\right)\Delta s\hfill \\ & =& {\mathit{\int }}_a^b|\varkappa \left(s\right){\varkappa }^{-1}\left(s\right)\varkappa \left(s\right)u\left(s\right)v\left(s\right)|{\left({\zeta }^{\Delta }\left(s\right)\right)}^{{\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}}\Delta s\hfill \\ & \le & C_{\varkappa }{\mathit{\int }}_a^b|\varkappa \left(s\right)u\left(s\right)|{\left({\zeta }^{\Delta }\left(s\right)\right)}^{{\displaystyle \frac{1}{p}}}|\varkappa \left(s\right)v\left(s\right)|{\left({\zeta }^{\Delta }\left(s\right)\right)}^{{\displaystyle \frac{1}{q}}}\Delta s\hfill \\ & \le & C_{\varkappa }{\left({\mathit{\int }}_a^b{|\varkappa \left(s\right)u\left(s\right)|}^p{\zeta }^{\Delta }\left(s\right)\Delta s\right)}^{{\displaystyle \frac{1}{p}}}{\left({\mathit{\int }}_a^b{|\varkappa \left(s\right)v\left(s\right)|}^q{\zeta }^{\Delta }\left(s\right)\Delta s\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & =& C_{\varkappa }{\parallel u\parallel }_{X_{\Delta }^{\varkappa ,p}}{\parallel v\parallel }_{X_{\Delta }^{\varkappa ,q}}.\hfill \end{array}$$

The proof is finished. □

Let $I_{a^{+}}^{\gamma ,\zeta ,\varkappa }({\overline{I}}_{\mathbb{T}},\mathbf{R})$ represent the function space comprising all weighted left Riemann–Liouville $\Delta -$integrals with order $\gamma$ for any continuous mappings.

Theorem 7.

Let $\gamma >0$, $f\in C({\overline{I}}_{\mathbb{T}},\mathbf{R})$. Then, $f\in I_{a^{+}}^{\gamma ,\zeta ,\varkappa }({\overline{I}}_{\mathbb{T}},\mathbf{R})$ if $I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }f\left(a\right)=0$ and $I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }f\in C^1({\overline{I}}_{\mathbb{T}},\mathbf{R})$.

Proof. 

Let $f\in I_{a^{+}}^{\gamma ,\zeta ,\varkappa }({\overline{I}}_{\mathbb{T}},\mathbf{R})$; then, there is a continuous function g ensuring that $f\left(t\right)=I_{a^{+}}^{\gamma ,\zeta ,\varkappa }g\left(t\right)$. According to Proposition 2, we obtain

$$I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }f\left(t\right)=I_{a^{+}}^{1,\zeta ,\varkappa }g\left(t\right)={\varkappa }^{-1}\left(t\right){\mathit{\int }}_a^t\varkappa \left(s\right){\zeta }^{\Delta }\left(s\right)g\left(s\right)\Delta s\in C^1({\overline{I}}_{\mathbb{T}},\mathbf{R})$$

and $I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }f\left(a\right)=0$.

Inversely, if $I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }f\left(a\right)=0$ and $I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }f\in C^1({\overline{I}}_{\mathbb{T}},\mathbf{R})$ then

$$I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }f\left(t\right)={\mathit{\int }}_a^t{\left(I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }f\left(s\right)\right)}^{\Delta }\Delta s,$$

where $t\in {\overline{I}}_{\mathbb{T}}$. Let $\phi \left(t\right)={\displaystyle \frac{1}{{\zeta }^{\Delta }\left(t\right)\varkappa \left(s\right)}}{\left(I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }f\left(t\right)\right)}^{\Delta }$; then, $\phi \in C({\overline{I}}_{\mathbb{T}},\mathbf{R})$ and, again by Proposition 2, one can see that

$$I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }f\left(t\right)=I_{a^{+}}^{1,\zeta ,\varkappa }\varkappa \left(t\right){\zeta }^{\Delta }\left(t\right)\phi \left(t\right)=I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }{I}_{a^{+}}^{\gamma ,\zeta ,\varkappa }\varkappa \left(t\right){\zeta }^{\Delta }\left(t\right)\phi \left(t\right).$$

Thus, $I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }\left(f\left(t\right)-I_{a^{+}}^{\gamma ,\zeta ,\varkappa }\varkappa \left(t\right){\zeta }^{\Delta }\left(t\right)\phi \left(t\right)\right)=0$, namely, $f\left(t\right)=I_{a^{+}}^{\gamma ,\zeta ,\varkappa }\varkappa \left(t\right){\zeta }^{\Delta }\left(t\right)\phi \left(t\right)$ and $f\in I_{a^{+}}^{\gamma ,\zeta ,\varkappa }({\overline{I}}_{\mathbb{T}},\mathbf{R})$. This completes the proof. □

Theorem 8.

Let $\gamma >0$, $f\in C({\overline{I}}_{\mathbb{T}},\mathbf{R})$ satisfy the condition in Theorem 7. Then,

$$\left(I_{a^{+}}^{\gamma ,\zeta ,\varkappa }\circ D_{a^{+}}^{\gamma ,\zeta ,\varkappa }\right)f=f.$$

Proof. 

This is the direct corollary of Theorem 7 and Proposition 3. □

Theorem 9.

Consider the parameters $p,q\ge 1$ and $\gamma >0$ satisfying the condition ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}\le 1+\gamma$. In the special case where ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1+\gamma$, it follows that neither p nor q can equal 1. Let $I_{\gamma +}^{\gamma ,\zeta ,\varkappa }\left(X_{\Delta }^{\varkappa ,p}\right)$ denote the function space consisting of weighted left Riemann–Liouville Δ-integrals of order γ for functions $g\in X_{\Delta }^{\varkappa ,p}$. Similarly, let $I_{b-}^{\gamma ,\zeta ,\varkappa }\left(X_{\Delta }^{\varkappa ,p}\right)$ represent the corresponding space for right Riemann–Liouville Δ-integrals. Under these conditions, the following integration-by-parts formulas hold valid.

(i) 

Let $f\in X_{\Delta }^{\varkappa ,p}\left({\overline{I}}_{\mathbb{T}}\right)$, ${\varkappa }^2\left(\varsigma \right)g\in X_{\Delta }^{\varkappa ,q}\left({\overline{I}}_{\mathbb{T}}\right)$; then,

$${\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\zeta }^{\Delta }\left(\varsigma \right)f\left(\varsigma \right)\left(I_{a^{+}}^{\gamma ,\zeta ,\varkappa }g\right)\left(\varsigma \right)\Delta \varsigma ={\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\zeta }^{\Delta }\left(\varsigma \right)g\left(\varsigma \right)\left(I_{b^{-}}^{\gamma ,\zeta ,\varkappa }f\right)\left(\varsigma \right)\Delta \varsigma .$$

(ii) 

If $f\in I_{a^{+}}^{\gamma ,\zeta ,\varkappa }\left(X_{\Delta }^{\varkappa ,q}\right)$ and $g\in I_{b^{-}}^{\gamma ,\zeta ,\varkappa }\left(X_{\Delta }^{\varkappa ,p}\right)$ then

$${\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\left({\zeta }^{\Delta }\left(\varsigma \right)\right)}^{2}g\left(\varsigma \right)\left(D_{a^{+}}^{\gamma ,\zeta ,\varkappa }f\right)\left(\varsigma \right)\Delta \varsigma ={\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\left({\zeta }^{\Delta }\left(\varsigma \right)\right)}^{2}f\left(\varsigma \right)\left(D_{b^{-}}^{\gamma ,\zeta ,\varkappa }g\right)\left(\varsigma \right)\Delta \varsigma .$$

Proof. 

Combining Definition 5 and Fubini’s Theorem in the field of timescales, we have

$$\begin{array}{ccc}& & {\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\zeta }^{\Delta }\left(\varsigma \right)f\left(\varsigma \right)\left(I_{a^{+}}^{\gamma ,\zeta ,\varkappa }g\right)\left(\varsigma \right)\Delta \varsigma \hfill \\ & =& {\mathit{\int }}_a^b\varkappa \left(\varsigma \right){\zeta }^{\Delta }\left(\varsigma \right)f\left(\varsigma \right){\mathit{\int }}_a^t{\displaystyle \frac{{\left(\zeta \left(\varsigma \right)-\zeta \left(\sigma \right(s\left)\right)\right)}^{\gamma -1}}{\Gamma \left(\gamma \right)}}\varkappa \left(s\right){\zeta }^{\Delta }\left(s\right)g\left(s\right)\Delta s\Delta \varsigma \hfill \\ & =& {\mathit{\int }}_a^b\varkappa \left(s\right){\zeta }^{\Delta }\left(s\right)g\left(s\right){\mathit{\int }}_s^b{\displaystyle \frac{{\left(\zeta \left(\varsigma \right)-\zeta \left(\sigma \right(s\left)\right)\right)}^{\gamma -1}}{\Gamma \left(\gamma \right)}}\varkappa \left(\varsigma \right){\zeta }^{\Delta }\left(\varsigma \right)g\left(\varsigma \right)\Delta \varsigma \Delta s\hfill \\ & =& {\mathit{\int }}_a^b\varkappa \left(\varsigma \right){\zeta }^{\Delta }\left(\varsigma \right)g\left(\varsigma \right){\mathit{\int }}_t^b{\displaystyle \frac{{\left(\zeta \left(s\right)-\zeta \left(\sigma \right(\varsigma \left)\right)\right)}^{\gamma -1}}{\Gamma \left(\gamma \right)}}\varkappa \left(s\right){\zeta }^{\Delta }\left(s\right)g\left(s\right)\Delta s\Delta \varsigma \hfill \\ & =& {\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\zeta }^{\Delta }\left(\varsigma \right)g\left(\varsigma \right)\left(I_{b^{-}}^{\gamma ,\zeta ,\varkappa }f\right)\left(\varsigma \right)\Delta \varsigma .\hfill \end{array}$$

Hence, item (i) is obtained. In addition, from Definition 6, we obtain

$$\begin{array}{ccc}& & {\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\left({\zeta }^{\Delta }\left(\varsigma \right)\right)}^{2}g\left(\varsigma \right)\left(D_{a^{+}}^{\gamma ,\zeta ,\varkappa }f\right)\left(\varsigma \right)\Delta \varsigma \hfill \\ & =& {\mathit{\int }}_a^b\varkappa \left(\varsigma \right){\zeta }^{\Delta }\left(\varsigma \right)g\left(\varsigma \right){\left({\mathit{\int }}_a^t{\displaystyle \frac{{\left(\zeta \left(\varsigma \right)-\zeta \left(\sigma \right(s\left)\right)\right)}^{-\gamma }}{\Gamma (1-\gamma )}}\varkappa \left(s\right){\zeta }^{\Delta }\left(s\right)f\left(s\right)\Delta s\right)}^{\Delta }\Delta \varsigma \hfill \\ & =& {\mathit{\int }}_a^b\varkappa \left(s\right){\zeta }^{\Delta }\left(s\right)g\left(s\right){\left({\mathit{\int }}_a^t{\displaystyle \frac{{\left(\zeta \left(\varsigma \right)-\zeta \left(\sigma \right(s\left)\right)\right)}^{-\gamma }}{\Gamma (1-\gamma )}}\varkappa \left(\varsigma \right){\zeta }^{\Delta }\left(\varsigma \right)f\left(\varsigma \right)\Delta \varsigma \right)}^{\Delta }\Delta s\hfill \\ & =& {\mathit{\int }}_a^b\varkappa \left(\varsigma \right){\zeta }^{\Delta }\left(\varsigma \right)g\left(\varsigma \right){\left({\mathit{\int }}_a^t{\displaystyle \frac{{\left(\zeta \left(\varsigma \right)-\zeta \left(\sigma \right(s\left)\right)\right)}^{-\gamma }}{\Gamma (1-\gamma )}}\varkappa \left(s\right){\zeta }^{\Delta }\left(s\right)f\left(s\right)\Delta s\right)}^{\Delta }\Delta \varsigma \hfill \\ & =& {\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\left({\zeta }^{\Delta }\left(\varsigma \right)\right)}^{2}f\left(\varsigma \right)\left(D_{b^{-}}^{\gamma ,\zeta ,\varkappa }g\right)\left(\varsigma \right)\Delta \varsigma .\hfill \end{array}$$

Evidently, item (ii) is also fulfilled. The proof is finished. □

3. Weighted Nonlocal Sobolev Space

Lemma 7

([40]). A real-valued function g defined on a timescale $\mathbb{T}$ is absolutely continuous if

• g is Δ-differentiable almost everywhere in the interval ${[a,b]}_{\mathbb{T}}$;
• The derivative $g^{\Delta }$ belongs to the space $L^1\left(\mathbb{T}\right)$;
• For every $s\in \mathbb{T}$, the function can be expressed as

  $$g\left(s\right)=g\left(a\right)+{\mathit{\int }}_a^s{g}^{\Delta }\left(r\right)\Delta r.$$

Lemma 8

([41]). Let g be a function in $L^1$ and let γ be a real number in the interval $(0,1)$; ${}_{a}D_t^{\gamma }g$ exists if $\xi \in {\mathbf{R}}^m$ and $\phi \in L^1$, such that for almost every s in $[a,b]$ the function g can be written as

$$g\left(s\right)={\displaystyle \frac{\xi }{\Gamma \left(\gamma \right){(s-a)}^{1-\gamma }}}+\left({}_{a}I_s^{\gamma }\phi \right)\left(s\right).$$

Moreover, under these conditions, we have

$${}_{a}I_s^{1-\gamma }g\left(a\right)=\xi \mathit{and}\left({}_{a}D_s^{\gamma }g\right)\left(s\right)=\phi \left(s\right)\mathrm{a}.\mathrm{e}.\mathrm{on}[a,b].$$

Theorem 10.

Let γ be a real number in $(0,1]$, and let f belong to the weighted function space $X_{\Delta }^{\varkappa ,p}({\overline{I}}_{\mathbb{T}},{\mathbf{R}}^N)$; ${}_{a}D_t^{\gamma ,\zeta ,\varkappa }f$ of order γ exists if $c\in {\mathbf{R}}^N$ and $\phi \in X_{\Delta }^{\varkappa ,p}({\overline{I}}_{\mathbb{T}},{\mathbf{R}}^N)$, ensuring that for almost every t in ${\overline{I}}_{\mathbb{T}}$ the following representation holds:

$$f\left(t\right)={\displaystyle \frac{c}{\Gamma \left(\gamma \right){(\zeta \left(t\right)-\zeta \left(a\right))}^{1-\gamma }}}+\left({}_{a}I_t^{\gamma ,\zeta ,\varkappa }\phi \right)\left(t\right).$$

(1)

Furthermore, in this case we have

$${}_{a}I_t^{1-\gamma ,\zeta ,\varkappa }f\left(a\right)=c\mathit{and}\left({}_{a}D_t^{\gamma ,\zeta ,\varkappa }f\right)\left(t\right)=\phi \left(t\right)\mathrm{a}.\mathrm{e}.\mathrm{on}{\overline{I}}_{\mathbb{T}}.$$

Definition 8.

For $\gamma \in (0,1]$, the space $A{C}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}({\overline{I}}_{\mathbb{T}},{\mathbf{R}}^N)$, abbreviated as $A{C}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$, consists of all functions $f:{\overline{I}}_{\mathbb{T}}\to {\mathbf{R}}^N$ that can be expressed in the following form:

$$f\left(t\right)={\displaystyle \frac{c}{\Gamma \left(\gamma \right){(\zeta \left(t\right)-\zeta \left(a\right))}^{1-\gamma }}}+\left({}_{a}I_t^{\gamma ,\zeta ,\varkappa }\phi \right)\left(t\right),t\in {\overline{I}}_{\mathbb{T}}\mathrm{a}.\mathrm{e}.,$$

where $c\in {\mathbf{R}}^N$ and $\phi \in X_{\Delta }^{\varkappa ,p}$.

Definition 9.

Let $p\in [1,+\infty )$, $\gamma \in (0,1]$, and $g:{\overline{I}}_{\mathbb{T}}\to {\mathbf{R}}^N$; $g\in {\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}({\overline{I}}_{\mathbb{T}},{\mathbf{R}}^N)$, abbreviated as ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$, if

• $g\in X_{\Delta }^{\varkappa ,p}({\overline{I}}_{\mathbb{T}},{\mathbf{R}}^N)$;
• For every test function $\phi \in C_{c,r,d}^{\infty }({\overline{I}}_{\mathbb{T}},{\mathbf{R}}^N)$ there exists a function $h\in X_{\Delta }^{\varkappa ,p}({\overline{I}}_{\mathbb{T}},{\mathbf{R}}^N)$, such that the following integral equality holds:

  $${\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\left({\zeta }^{\Delta }\left(\varsigma \right)\right)}^{2}g\left(\varsigma \right)\left(D_b^{\gamma ,\zeta ,\varkappa }\phi \right)\left(\varsigma \right)\Delta \varsigma ={\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\left({\zeta }^{\Delta }\left(\varsigma \right)\right)}^{2}h\left(\varsigma \right)\phi \left(\varsigma \right)\Delta \varsigma .$$

Theorem 11.

If $p\in [1,+\infty )$, $\gamma \in (0,1]$ then one has ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}=A{C}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}\cap X_{\Delta }^{\varkappa ,p}.$

Proof. 

Let $u\in A{C}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}\cap X_{\Delta }^{\varkappa ,p}$; combining Theorem 10 and Definition 8, we have $D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\in X_{\Delta }^{\varkappa ,p}$. From Theorem 9, we have

$${\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\left({\zeta }^{\Delta }\left(\varsigma \right)\right)}^{2}u\left(\varsigma \right)\left(D_{b^{-}}^{\gamma ,\zeta ,\varkappa }\phi \right)\left(\varsigma \right)\Delta \varsigma ={\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\left({\zeta }^{\Delta }\left(\varsigma \right)\right)}^2\phi \left(\varsigma \right)\left(D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\right)\left(\varsigma \right)\Delta \varsigma ,$$

where $\phi \in C_{c,rd}^{\infty }$. Hence, $u\in {\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$ with $g=D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\in X_{\Delta }^{\varkappa ,p}.$

Inversely, let $u\in {\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$; then, $u\in X_{\Delta }^{\varkappa ,p}(I_{\mathbb{T}},{\mathbf{R}}^N)$; also, we have $g\in X_{\Delta }^{\varkappa ,p}(I_{\mathbb{T}},{\mathbf{R}}^N)$ meeting

$${\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\left({\zeta }^{\Delta }\left(\varsigma \right)\right)}^{2}u\left(\varsigma \right)\left(D_{b^{-}}^{\gamma ,\zeta ,\varkappa }\phi \right)\left(\varsigma \right)\Delta \varsigma ={\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\left({\zeta }^{\Delta }\left(\varsigma \right)\right)}^{2}g\left(\varsigma \right)\phi \left(\varsigma \right)\Delta \varsigma$$

for every $\phi \in C_{c,rd}^{\infty }\left({\overline{I}}_{\mathbb{T}},{\mathbf{R}}^N\right)$.

According to Theorem 10 and Definition 8, the condition $u\in A{C}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}\cap X_{\Delta }^{\varkappa ,p}$ can be established by confirming two key requirements: First, the integral operator $I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }$ must be absolutely continuous. Second, its weighted fractional derivative of order $\gamma$ needs to be an element of the space $X_{\Delta }^{\varkappa ,p}$. Indeed, for $\phi \in C_{c,rd}^{\infty }$, we have $\phi \in D_{b^{-}}^{\gamma ,\zeta ,\varkappa }\left(C_{rd}\right)$, $D_{b^{-}}^{\gamma ,\zeta ,\varkappa }\phi =-{\displaystyle \frac{{\varkappa }^{-1}\left(r\right)}{{\zeta }^{\Delta }\left(r\right)}}{\left(I_{b^{-}}^{1-\gamma ,\zeta ,\varkappa }\phi \right)}^{\Delta }$. From Theorem 9, we obtain

$$\begin{array}{ccc}& & {\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\left({\zeta }^{\Delta }\left(\varsigma \right)\right)}^{2}u\left(\varsigma \right)\left(D_{b^{-}}^{\gamma ,\zeta ,\varkappa }\phi \right)\left(\varsigma \right)\Delta \varsigma \hfill \\ & =& {\mathit{\int }}_a^b\varkappa \left(\varsigma \right){\zeta }^{\Delta }\left(\varsigma \right)u\left(\varsigma \right){(-I_{b^{-}}^{1-\gamma ,\zeta ,\varkappa }\phi )}^{\Delta }\Delta \varsigma \hfill \\ & =& {\mathit{\int }}_a^b\varkappa \left(\varsigma \right){\zeta }^{\Delta }\left(\varsigma \right)\left(D_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }{I}_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }u\right)\left(\varsigma \right){(-I_{b^{-}}^{1-\gamma ,\zeta ,\varkappa }\phi )}^{\Delta }\Delta \varsigma \hfill \\ & =& -{\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\left({\zeta }^{\Delta }\left(\varsigma \right)\right)}^2\left(I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }u\right)\left(\varsigma \right){\phi }^{\Delta }\left(t\right)\Delta \varsigma .\hfill \end{array}$$

As a result, we infer

$${\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\left({\zeta }^{\Delta }\left(\varsigma \right)\right)}^2\left(I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }u\right)\left(\varsigma \right){\phi }^{\Delta }\left(t\right)\Delta =-{\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\left({\zeta }^{\Delta }\left(\varsigma \right)\right)}^{2}g\left(\varsigma \right)\phi \left(\varsigma \right)\Delta \varsigma ,\forall \phi \in C_{c,rd}^{\infty };$$

that is, $I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }u\in {\mathcal{W}}_{\Delta ,a^{+}}^{1,\zeta ,\varkappa ,p}$. The proof is finished. □

Definition 10.

Define the norm ${\parallel ·\parallel }_{{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}^p$ of set ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$ as

$${\parallel u\parallel }_{{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}^p={\parallel u\parallel }_{X_{\Delta }^{\varkappa ,p}}^p+{\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\parallel }_{X_{\Delta }^{\varkappa ,p}}^p,\forall u\in {\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}.$$

From hereon, $0<\gamma \le 1$, $p\ge 1$.

Theorem 12.

Suppose that $\gamma >0$, $p\ge 1$, $f\in X_{\Delta }^{\varkappa ,p}$. Then, $I_{a^{+}}^{\gamma ,\zeta ,\varkappa }f$ is bounded in $X_{\Delta }^{\varkappa ,p}$ and

$$\parallel I_{a^{+}}^{\gamma ,\zeta ,\varkappa }{f\parallel }_{X_{\Delta }^{\varkappa ,p}}\le {\displaystyle \frac{{\left(\zeta \left(b\right)-\zeta \left(a\right)\right)}^{\gamma }}{\Gamma (\gamma +1)}}{\parallel f\parallel }_{X_{\Delta }^{\varkappa ,p}}.$$

Proof. 

If $1\le p<\infty$, using Hölder’s inequality and generalised Minkowski inequality, we have

$$\begin{array}{ccc}\hfill \parallel I_{a^{+}}^{\gamma ,\zeta ,\varkappa }{f\parallel }_{X_{\Delta }^{\varkappa ,p}}& =& {\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\left({\mathit{\int }}_{I_{\mathbb{T}}}{\left|{\mathit{\int }}_a^t{\left(\zeta \left(t\right)-\zeta \left(\sigma \right(s\left)\right)\right)}^{\gamma -1}\varkappa \left(s\right){\zeta }^{\Delta }\left(s\right)f\left(s\right)\Delta s\right|}^p{\zeta }^{\Delta }\left(t\right)\Delta t\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & =& {\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\left({\mathit{\int }}_{\zeta \left(a\right)}^{\zeta \left(b\right)}{\left|{\mathit{\int }}_{\zeta \left(a\right)}^v{\left(v-u\right)}^{\gamma -1}\varkappa \left({\zeta }^{-1}\left(u\right)\right)f\left({\zeta }^{-1}\left(u\right)\right)\Delta u\right|}^p\Delta v\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le & {\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\mathit{\int }}_{\zeta \left(a\right)}^{\zeta \left(b\right)}{\left({\left|\varkappa \left({\zeta }^{-1}\left(u\right)\right)f\left({\zeta }^{-1}\left(u\right)\right)\right|}^p{\mathit{\int }}_u^{\zeta \left(b\right)}{(v-u)}^{(\gamma -1)p}\Delta v\right)}^{{\displaystyle \frac{1}{p}}}\Delta u\hfill \\ & =& {\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\mathit{\int }}_{\zeta \left(a\right)}^{\zeta \left(b\right)}\left|\varkappa \left({\zeta }^{-1}\left(u\right)\right)f\left({\zeta }^{-1}\left(u\right)\right)\right|{\left({\displaystyle \frac{{\left(\zeta \left(b\right)-u\right)}^{(\gamma -1)p+1}}{(\gamma -1)p+1}}\right)}^{{\displaystyle \frac{1}{p}}}\Delta u\hfill \\ & \le & {\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\left({\mathit{\int }}_{\zeta \left(a\right)}^{\zeta \left(b\right)}{\left|\varkappa \left({\zeta }^{-1}\left(u\right)\right)f\left({\zeta }^{-1}\left(u\right)\right)\right|}^p\Delta u\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & & \times {\left({\mathit{\int }}_{\zeta \left(a\right)}^{\zeta \left(b\right)}{\left({\displaystyle \frac{{\left(\zeta \left(b\right)-u\right)}^{(\gamma -1)p+1}}{(\gamma -1)p+1}}\right)}^{{\displaystyle \frac{q}{p}}}\Delta u\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & \le & {\displaystyle \frac{{\left(\zeta \left(b\right)-\zeta \left(a\right)\right)}^{\gamma }}{\Gamma (\gamma +1)}}{\parallel f\parallel }_{X_{\Delta }^{\varkappa ,p}},\hfill \end{array}$$

where ${\zeta }^{-1}$ is the inverse function of $\zeta$.

If $p=\infty$, from Theorem 5 we have

$$\begin{array}{ccc}\hfill |\varkappa I_{a^{+}}^{\gamma ,\zeta ,\varkappa }f|& =& \left|{\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\mathit{\int }}_a^t{\left(\zeta \left(t\right)-\zeta \left(\sigma \right(s\left)\right)\right)}^{\gamma -1}\varkappa \left(t\right){\zeta }^{\Delta }\left(s\right)f\left(s\right)\Delta s\right|\hfill \\ & \le & {\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\mathit{\int }}_a^t{\left(\zeta \left(t\right)-\zeta \left(\sigma \right(s\left)\right)\right)}^{\gamma -1}{\zeta }^{\Delta }\left(s\right)|\varkappa \left(t\right)f\left(s\right)|\Delta s\hfill \\ & \le & {\displaystyle \frac{{\parallel f\parallel }_{X_{\Delta }^{\infty }}}{\Gamma \left(\gamma \right)}}{\mathit{\int }}_a^t{\left(\zeta \left(t\right)-\zeta \left(\sigma \right(s\left)\right)\right)}^{\gamma -1}{\zeta }^{\Delta }\left(s\right)\Delta s\hfill \\ & =& {\displaystyle \frac{{\parallel f\parallel }_{X_{\Delta }^{\infty }}}{\Gamma \left(\gamma \right)}}{\displaystyle \frac{{\left(\zeta \left(t\right)-\zeta \left(a\right)\right)}^{\gamma }}{\gamma }}\hfill \\ & \le & {\displaystyle \frac{{\parallel f\parallel }_{X_{\Delta }^{\infty }}{\left(\zeta \left(b\right)-\zeta \left(a\right)\right)}^{\gamma }}{\Gamma (\gamma +1)}}.\hfill \end{array}$$

The proof is finished. □

Definition 11.

Define ${\parallel ·\parallel }_{a,{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}^p$ of set ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$ as

$${\parallel u\parallel }_{a,{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}^p=|I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }{u\left(a\right)|}^p+{\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\parallel }_{X_{\Delta }^{\varkappa ,p}}^p,\forall u\in {\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}.$$

Hereby, $0<\gamma \le 1$.

Based on Theorem 12, we now prove that norm ${\parallel ·\parallel }_{{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}^p$ equals norm ${\parallel ·\parallel }_{a,{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}^p$.

Theorem 13.

Let $0<\gamma \le 1$, $g\in {\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$. Then, norms ${\parallel ·\parallel }_{{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}^p$ and ${\parallel ·\parallel }_{a,{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}^p$ are equivalent.

Proof. 

On the one hand, with the help of the continuity of $\varkappa ,\zeta$, similar to the argument of Theorem 24 in [37], we obtain

$$|I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }{g\left(a\right)|}^p\le M\left({\parallel g\parallel }_{X_{\Delta }^{\varkappa ,p}}^p+{\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }g\parallel }_{X_{\Delta }^{\varkappa ,p}}^p\right).$$

Consequently,

$$\begin{array}{ccc}& & {\parallel g\parallel }_{a,{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}^p\hfill \\ & =& |I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }{g|}^p+{\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }g\parallel }_{X_{\Delta }^{\varkappa ,p}}^p\hfill \\ & \le & (M+1)\left({\parallel g\parallel }_{X_{\Delta }^{\varkappa ,p}}^p+{\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }g\parallel }_{X_{\Delta }^{\varkappa ,p}}^p\right)\hfill \\ & =& {(M+1)\parallel g\parallel }_{{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}^p.\hfill \end{array}$$

If $g\in {\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$ then, by Theorems 10 and 11, g can be expressed by

$$g\left(s\right)={\displaystyle \frac{\xi }{\Gamma \left(\gamma \right){(\zeta \left(s\right)-\zeta \left(a\right))}^{1-\gamma }}}+{(}_a{I}_s^{\gamma ,\zeta ,\varkappa }\phi )\left(s\right),s\in {\overline{I}}_{\mathbb{T}}a.e.,$$

where $\xi \in {\mathbf{R}}^N,\phi \in X_{\Delta }^{\varkappa ,p}$ and $\xi =I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }g\left(a\right),\phi =D_{a^{+}}^{\gamma ,\zeta ,\varkappa }g$. Based on Theorem 12, we have

$$\begin{array}{ccc}\hfill {\parallel g\parallel }_{X_{\Delta }^{\varkappa ,p}}^p& =& {\mathit{\int }}_a^b{|\varkappa \left(s\right)g\left(s\right)|}^p{\zeta }^{\Delta }\left(s\right)\Delta s\hfill \\ & =& {\mathit{\int }}_a^b{|{\displaystyle \frac{\varkappa \left(s\right)\xi }{\Gamma \left(\gamma \right)}}{(\zeta \left(s\right)-\zeta \left(a\right))}^{\gamma -1}+\left(\varkappa \left(s\right)I_{a^{+}}^{\gamma ,\zeta ,\varkappa }\phi \right)\left(s\right)|}^p{\zeta }^{\Delta }\left(s\right)\Delta s\hfill \\ & \le & 2^p\left({\displaystyle \frac{C_{{\varkappa }_1}{|\xi |}^p}{{\Gamma }^p\left(\gamma \right)}}{\mathit{\int }}_a^b{(\zeta \left(s\right)-\zeta \left(a\right))}^{(\gamma -1)p}{\zeta }^{\Delta }\left(s\right)\Delta s+{\parallel I_{a^{+}}^{\gamma ,\zeta ,\varkappa }\phi \parallel }_{X_{\Delta }^{\varkappa ,p}}^p\right)\hfill \\ & \le & 2^p\left({\displaystyle \frac{C_{{\varkappa }_1}{|\xi |}^p\left((\gamma -1)p+1\right)}{{\Gamma }^p\left(\gamma \right)}}{(\zeta \left(b\right)-\zeta \left(a\right))}^{(\gamma -1)p+1}+{\displaystyle \frac{{\left(\zeta \left(b\right)-\zeta \left(a\right)\right)}^{\gamma }}{\Gamma (\gamma +1)}}{\parallel \phi \parallel }_{X_{\Delta }}^{\varkappa ,p}\right)\hfill \\ & \le & C\left(|I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }{g\left(a\right)|}^p+{\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }g\parallel }_{X_{\Delta }}^{\varkappa ,p}\right),\hfill \end{array}$$

where $C_{{\varkappa }_1}={max}_{t\in I_{\mathbb{T}}}|\varkappa \left(t\right)|$ and

$$C=2^p\left({\displaystyle \frac{C_{{\varkappa }_1}\left((\gamma -1)p+1\right)}{{\Gamma }^p\left(\gamma \right)}}{(\zeta \left(b\right)-\zeta \left(a\right))}^{(\gamma -1)p+1}+{\displaystyle \frac{{\left(\zeta \left(b\right)-\zeta \left(a\right)\right)}^{\gamma }}{\Gamma (\gamma +1)}}\right);$$

hence,

$$\begin{array}{ccc}\hfill {\parallel g\parallel }_{{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}^p& =& {\parallel g\parallel }_{X_{\Delta }^{\varkappa ,p}}^p+{\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }g\parallel }_{X_{\Delta }^{\varkappa ,p}}^p\hfill \\ & \le & (C+1)\left(|I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }{g\left(a\right)|}^p+{\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }g\parallel }_{X_{\Delta }}^{\varkappa ,p}\right)\hfill \\ & =& {(C+1)\parallel g\parallel }_{a,{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}^p.\hfill \end{array}$$

This completes the proof. □

With our subsequent analysis, we are poised to demonstrate several fundamental properties of ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$.

Theorem 14.

Let $0<\gamma \le 1,1\le p<\infty$; then, ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$ is completely endowed with norm ${\parallel ·\parallel }_{{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}$.

Proof. 

In light of Theorem 13, the norm ${\parallel ·\parallel }_{{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}^p$ equals ${\parallel ·\parallel }_{a,{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}^p$. Therefore, it is sufficient to demonstrate that ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$ is complete when equipped with the norm ${\parallel ·\parallel }_{a,{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}$.

Let $\left\{u_k\right\}$ be a Cauchy sequence (CS) in ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$ with respect to ${\parallel ·\parallel }_{a,{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}$. Then, $\left\{I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }{u}_k\left(a\right)\right\}$ and $\left\{D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u}_k\right\}$ are, separately, CS in ${\mathbf{R}}^N$ and $X_{\Delta }^{\varkappa ,p}$, which means that there exist $c\in {\mathbf{R}}^N$ and $\phi \in X_{\Delta }^{\varkappa ,p}$, such that

$$\underset{k\to \infty }{lim}{I}_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }{u}_k\left(a\right)=c,\underset{k\to \infty }{lim}{D}_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u}_k=\phi ,$$

which implies that

$$u\left(t\right)=\left({\displaystyle \frac{c}{\Gamma \left(\gamma \right)}}{\left(\zeta \left(t\right)-\zeta \left(a\right)\right)}^{\gamma -1}+I_{a^{+}}^{\gamma ,\zeta ,\varkappa }\phi \left(t\right)\right)\in {\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p},$$

with $I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }u\left(a\right)=c$ and $D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u=\phi$. Therefore, we obtain

$$\begin{array}{ccc}\hfill \parallel u_k{-u\parallel }_{a,{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}^p& =& |I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }\left(u_k-u\right){\left(a\right)|}^p+{\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }\left(u_k-u\right)\parallel }_{X_{\Delta }^{\varkappa ,p}}^p\hfill \\ & =& |I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }{u}_k\left(a\right)-I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }{u\left(a\right)|}^p+{\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u}_k-D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\parallel }_{X_{\Delta }^{\varkappa ,p}}^p\hfill \\ & =& |I_{a^{+}}^{1-\gamma ,\zeta ,\varkappa }{u}_k{\left(a\right)-c|}^p+{\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u}_k-\phi \parallel }_{X_{\Delta }^{\varkappa ,p}}^p\hfill \\ & \to & 0,\hfill \end{array}$$

as $k\to \infty$, which completes the proof. □

Theorem 15.

${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$ is reflexive for $0<\gamma \le 1,1\le p<\infty$.

Proof. 

To prove the reflexivity of ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$, we will state that space ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$ is uniformly convex. In other words, we only need to prove that if $\forall ϵ>0$ then $\delta >0$, such that for $\forall u,v\in {\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$ with ${\parallel u\parallel }_{{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}={\parallel v\parallel }_{{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}=1$ and ${\parallel u-v\parallel }_{{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}\ge ϵ$; then, $\parallel {\displaystyle \frac{1}{2}}{(u+v)\parallel }_{{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}<\delta .$

Actually, when $p\in [2,\infty )$, according to Clarkson’s inequality and Remark 1, we obtain

$$\begin{array}{ccc}& & \parallel {\displaystyle \frac{1}{2}}{(u+v)\parallel }_{{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}^p\hfill \\ & =& \parallel {\displaystyle \frac{1}{2}}{(u+v)\parallel }_{X_{\Delta }^{\varkappa ,p}}^p+{\parallel {\displaystyle \frac{1}{2}}{D}_{a^{+}}^{\gamma ,\zeta ,\varkappa }(u+v)\parallel }_{X_{\Delta }^{\varkappa ,p}}^p\hfill \\ & =& {\displaystyle \frac{1}{2}}\parallel \varkappa {\left({\zeta }^{\Delta }\right)}^{{\displaystyle \frac{1}{p}}}u+\varkappa {\left({\zeta }^{\Delta }\right)}^{{\displaystyle \frac{1}{p}}}{v\parallel }_{{\mathcal{H}}_{\Delta }^p}^p+{\displaystyle \frac{1}{2}}{\parallel \varkappa {\left({\zeta }^{\Delta }\right)}^{{\displaystyle \frac{1}{p}}}{D}_{a^{+}}^{\gamma ,\zeta ,\varkappa }u+\varkappa {\left({\zeta }^{\Delta }\right)}^{{\displaystyle \frac{1}{p}}}{D}_{a^{+}}^{\gamma ,\zeta ,\varkappa }v\parallel }_{{\mathcal{H}}_{\Delta }^p}^p\hfill \\ & \le & 2^{p-2}\left(\parallel \varkappa {\left({\zeta }^{\Delta }\right)}^{{\displaystyle \frac{1}{p}}}{u\parallel }_{{\mathcal{H}}_{\Delta }^p}^p+{\parallel \varkappa {\left({\zeta }^{\Delta }\right)}^{{\displaystyle \frac{1}{p}}}v\parallel }_{{\mathcal{H}}_{\Delta }^p}^p\right)-{\displaystyle \frac{1}{2}}{\parallel \varkappa {\left({\zeta }^{\Delta }\right)}^{{\displaystyle \frac{1}{p}}}(u-v)\parallel }_{{\mathcal{H}}_{\Delta }^p}^p\hfill \\ & & +2^{p-2}\left(\parallel \varkappa {\left({\zeta }^{\Delta }\right)}^{{\displaystyle \frac{1}{p}}}{D}_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u\parallel }_{{\mathcal{H}}_{\Delta }^p}^p+{\parallel \varkappa {\left({\zeta }^{\Delta }\right)}^{{\displaystyle \frac{1}{p}}}{D}_{a^{+}}^{\gamma ,\zeta ,\varkappa }v\parallel }_{{\mathcal{H}}_{\Delta }^p}^p\right)-{\displaystyle \frac{1}{2}}{\parallel \varkappa {\left({\zeta }^{\Delta }\right)}^{{\displaystyle \frac{1}{p}}}{D}_{a^{+}}^{\gamma ,\zeta ,\varkappa }(u-v)\parallel }_{{\mathcal{H}}_{\Delta }^p}^p\hfill \\ & =& 2^{p-2}\left({\parallel u\parallel }_{X_{\Delta }^{\varkappa ,p}}^p+{\parallel v\parallel }_{X_{\Delta }^{\varkappa ,p}}^p\right)-{\displaystyle \frac{1}{2}}{\parallel (u-v)\parallel }_{X_{\Delta }^{\varkappa ,p}}^p+2^{p-2}\left(\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u\parallel }_{X_{\Delta }^{\varkappa ,p}}^p+{\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }v\parallel }_{X_{\Delta }^{\varkappa ,p}}^p\right)\hfill \\ & & -{\displaystyle \frac{1}{2}}{\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }(u-v)\parallel }_{{\mathcal{H}}_{\Delta }^{\varkappa ,p}}^p\hfill \\ & =& 2^{p-2}{\parallel u\parallel }_{{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}^p+2^{p-2}{\parallel v\parallel }_{{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}^p-{\displaystyle \frac{1}{2}}{\parallel (u-v)\parallel }_{{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}^p\hfill \\ & \le & 2^{p-2}-{\left({\displaystyle \frac{ϵ}{2}}\right)}^p.\hfill \end{array}$$

(2)

When $p\in (1,2]$, we have $q\in [2,\infty )$. By applying inequality (2), we obtain

$$\begin{array}{ccc}& & \parallel {\displaystyle \frac{1}{2}}{(u+v)\parallel }_{{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}^q\hfill \\ & =& {\left(\parallel {\displaystyle \frac{1}{2}}{(u+v)\parallel }_{{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}^p\right)}^{{\displaystyle \frac{q}{p}}}\hfill \\ & =& {\left(\parallel {\displaystyle \frac{1}{2}}{(u+v)\parallel }_{{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}^p\right)}^{q-1}\hfill \\ & =& {\left(\parallel {\displaystyle \frac{1}{2}}{(u+v)\parallel }_{X_{\Delta }^{\varkappa ,p}}^p+{\parallel {\displaystyle \frac{1}{2}}{D}_{a^{+}}^{\gamma ,\zeta ,\varkappa }(u+v)\parallel }_{X_{\Delta }^{\varkappa ,p}}^p\right)}^{q-1}\hfill \\ & \le & 2^{q-1}\left(\parallel {\displaystyle \frac{1}{2}}{(u+v)\parallel }_{X_{\Delta }^{\varkappa ,p}}^q+{\parallel {\displaystyle \frac{1}{2}}{D}_{a^{+}}^{\gamma ,\zeta ,\varkappa }(u+v)\parallel }_{X_{\Delta }^{\varkappa ,p}}^q\right)\hfill \\ & \le & 2^{q-1}\left(2^{q-2}-{\left({\displaystyle \frac{ϵ}{2}}\right)}^q\right).\hfill \end{array}$$

(3)

From Theorem 14 and inequalities (2), (3) we know that ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$ is a uniformly convex Banach space. The proof is finished. □

Theorem 16.

Assuming that $\gamma \in (0,1]$, $p\in [1,+\infty )$, we have ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$ being separable with respect to ${\parallel ·\parallel }_{{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}$.

Proof. 

The operator A is to be set in the form of

$$A:u\mapsto \left(u,D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\right)\in X_{\Delta }^{\varkappa ,p}\times X_{\Delta }^{\varkappa ,p},\forall u\in {\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}.$$

Then, we have

$${∥u∥}_{{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}={∥Au∥}_{X_{\Delta }^{\varkappa ,p}\times X_{\Delta }^{\varkappa ,p}},$$

where

$${∥Au∥}_{X_{\Delta }^{\varkappa ,p}\times X_{\Delta }^{\varkappa ,p}}={\left(\sum _{i=1}^2{∥{\left(Au\right)}_i∥}_{X_{\Delta }^{\varkappa ,p}\times X_{\Delta }^{\varkappa ,p}}^p\right)}^{{\displaystyle \frac{1}{p}}}.$$

This result demonstrates that the operator A acts as an isometric isomorphism. As a result, the space ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$ is isometrically isomorphic to the subspace $\Omega$ defined by

$$\Omega =\left\{\left(u,D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\right)\mid u\in {\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}\right\}.$$

Furthermore, $\Omega$ is a closed subspace of the product space $X_{\Delta }^{\varkappa ,p}\times X_{\Delta }^{\varkappa ,p}$, provided that ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$ itself is closed. In light of the separability of $X_{\Delta }^{\varkappa ,p}$, it can be concluded that the Cartesian product space $X_{\Delta }^{\varkappa ,p}\times X_{\Delta }^{\varkappa ,p}$ is also separable in relation to the norm ${\parallel ·\parallel }_{X_{\Delta }^{\varkappa ,p}\times X_{\Delta }^{\varkappa ,p}}$. Therefore, it can be demonstrated that ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$ is separable regarding the norm ${\parallel ·\parallel }_{{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}$. The proof is finished. □

Proposition 4.

Suppose that $\gamma \in (0,1]$, $p\in (1,+\infty )$. Then,

$${\parallel u\parallel }_{X_{\Delta }^{\varkappa ,p}}\le {\displaystyle \frac{{\zeta }^{\gamma }\left(b\right)}{\Gamma (\gamma +1)}}{\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\parallel }_{X_{\Delta }^{\varkappa ,p}},\forall u\in {\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}.$$

Proof. 

By Theorems 8 and 12, we have

$${\parallel u\parallel }_{X_{\Delta }^{\varkappa ,p}}=\parallel I_{a^{+}}^{\gamma ,\zeta ,\varkappa }{D}_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u\parallel }_{X_{\Delta }^{\varkappa ,p}}\le {\displaystyle \frac{{\left(\zeta \left(b\right)-\zeta \left(a\right)\right)}^{\gamma }}{\Gamma (\gamma +1)}}\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{\parallel }_{X_{\Delta }^{\varkappa ,p}}\le {\displaystyle \frac{{\zeta }^{\gamma }\left(b\right)}{\Gamma (\gamma +1)}}{\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\parallel }_{X_{\Delta }^{\varkappa ,p}}.$$

The proof is finished. □

Remark 2.

Based on Proposition 4, it is evident that both ${\parallel u\parallel }_{{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}$ and $\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u\parallel }_{X_{\Delta }^{\varkappa ,p}}$ are equal.

Theorem 17.

We have

$${\parallel u\parallel }_{\infty }\le {\displaystyle \frac{{\zeta }^{\gamma -\frac{1}{p}}\left(b\right)}{\Gamma \left(\gamma \right){((\gamma -1)q+1)}^{\frac{1}{q}}}}{\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\parallel }_{X_{\Delta }^{\varkappa ,p}},$$

where $\gamma >{\displaystyle \frac{1}{p}}$, $1<p<\infty$.

Proof. 

In view of Theorem 8 and Hölder’s inequality, we have

$$\begin{array}{ccc}\hfill |I_{a^{+}}^{\gamma ,\zeta ,\varkappa }{D}_{a^{+}}^{\gamma ,\zeta ,\varkappa }u|& =& {\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}\left|{\mathit{\int }}_a^t{(\zeta \left(t\right)-\zeta \left(\sigma \left(s\right)\right))}^{\gamma -1}\varkappa \left(s\right){\zeta }^{\Delta }\left(s\right)D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(s\right)\Delta s\right|\hfill \\ & =& {\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}\left|{\mathit{\int }}_a^t{(\zeta \left(t\right)-\zeta \left(\sigma \left(s\right)\right))}^{\gamma -1}{\left({\zeta }^{\Delta }\left(s\right)\right)}^{{\displaystyle \frac{1}{q}}}\varkappa \left(s\right)D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(s\right){\left({\zeta }^{\Delta }\left(s\right)\right)}^{{\displaystyle \frac{1}{p}}}\Delta s\right|\hfill \\ & \le & {\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\left({\mathit{\int }}_a^t{(\zeta \left(t\right)-\zeta \left(\sigma \left(s\right)\right))}^{(\gamma -1)q}{\zeta }^{\Delta }\left(s\right)\Delta s\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & & \times {\left({\mathit{\int }}_a^t{|\varkappa \left(s\right)D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(s\right)|}^p{\zeta }^{\Delta }\left(s\right)\Delta s\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le & {\displaystyle \frac{{\zeta }^{\gamma -1+\frac{1}{q}}\left(b\right)}{\Gamma \left(\gamma \right){\left((\gamma -1)q+1\right)}^{\frac{1}{q}}}}{\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\parallel }_{X_{\Delta }^{\varkappa ,p}}\hfill \\ & =& {\displaystyle \frac{{\zeta }^{\gamma -\frac{1}{p}}\left(b\right)}{\Gamma \left(\gamma \right){\left((\gamma -1)q+1\right)}^{\frac{1}{q}}}}{\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\parallel }_{X_{\Delta }^{\varkappa ,p}}.\hfill \end{array}$$

The proof is finished. □

Remark 3.

Theorem 17 implies that ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$ is continuously embedded in $C(I_{\mathbb{T}},{\mathbf{R}}^N)$.

Theorem 18.

For $\gamma \in (0,1]$ and $p\in (1,+\infty )$, let ζ be a Lipschitz continuous function. If a sequence $\left\{u_k\right\}\subset {\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$ converges weakly to u in ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$ then $\left\{u_k\right\}$ also converges uniformly to u in the Banach space $C(I_{\mathbb{T}},{\mathbf{R}}^N)$.

Proof. 

By Theorem 15, $u_k$ admits the property of boundedness in space ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$; therefore, by using Remark 3, we claim that $u_k$ is bounded in $C(I_{\mathbb{T}},{\mathbf{R}}^N)$ and $u_k\rightharpoonup u$ in $C(I_{\mathbb{T}},{\mathbf{R}}^N)$.

Let $t_1,t_2\in {\overline{I}}_{\mathbb{T}}$ with $t_1\le t_2$; from Theorem 8 and the boundedness of $u_k$ in ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$, we obtain

$$\begin{array}{ccc}& & |u_k\left(t_1\right)-u_k\left(t_2\right)|\hfill \\ & =& \left|{}_{a}I_{t_1}^{\gamma ,\zeta ,\varkappa }\left({}_{a}D_{t_1}^{\gamma ,\zeta ,\varkappa }\right){-}_a{I}_{t_2}^{\gamma ,\zeta ,\varkappa }\left({}_{a}D_{t_2}^{\gamma ,\zeta ,\varkappa }\right)\right|\hfill \\ & =& {\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}|{\mathit{\int }}_a^{t_1}{(\zeta \left(t_1\right)-\zeta \left(\sigma \left(s\right)\right))}^{\gamma -1}\varkappa \left(s\right){\zeta }^{\Delta }\left(s\right)D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u}_k\left(s\right)\Delta s\hfill \\ & & -{\mathit{\int }}_a^{t_2}{(\zeta \left(t_2\right)-\zeta \left(\sigma \left(s\right)\right))}^{\gamma -1}\varkappa \left(s\right){\zeta }^{\Delta }\left(s\right)D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u}_k\left(s\right)\Delta s|\hfill \\ & \le & {\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}|{\mathit{\int }}_a^{t_1}\left({(\zeta \left(t_1\right)-\zeta \left(\sigma \left(s\right)\right))}^{\gamma -1}-{(\zeta \left(t_2\right)-\zeta \left(\sigma \left(s\right)\right))}^{\gamma -1}\right)\varkappa \left(s\right){\zeta }^{\Delta }\left(s\right)D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u}_k\left(s\right)\Delta s|\hfill \\ & & +{\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}|{\mathit{\int }}_{t_1}^{t_2}{(\zeta \left(t_2\right)-\zeta \left(\sigma \left(s\right)\right))}^{\gamma -1}\varkappa \left(s\right){\zeta }^{\Delta }\left(s\right)D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u}_k\left(s\right)\Delta s|\hfill \\ & \le & {\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\mathit{\int }}_a^{t_1}\left({(\zeta \left(t_1\right)-\zeta \left(\sigma \left(s\right)\right))}^{\gamma -1}-{(\zeta \left(t_2\right)-\zeta \left(\sigma \left(s\right)\right))}^{\gamma -1}\right)|\varkappa \left(s\right){\zeta }^{\Delta }\left(s\right)D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u}_k\left(s\right)|\Delta s\hfill \\ & & +{\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\mathit{\int }}_{t_1}^{t_2}{(\zeta \left(t_2\right)-\zeta \left(\sigma \left(s\right)\right))}^{\gamma -1}|\varkappa \left(s\right){\zeta }^{\Delta }\left(s\right)D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u}_k\left(s\right)|\Delta s\hfill \\ & \le & {\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\left({\mathit{\int }}_a^{t_1}{\left({(\zeta \left(t_1\right)-\zeta \left(\sigma \left(s\right)\right))}^{\gamma -1}-{(\zeta \left(t_2\right)-\zeta \left(\sigma \left(s\right)\right))}^{\gamma -1}\right)}^q{\zeta }^{\Delta }\left(s\right)\Delta s\right)}^{{\displaystyle \frac{1}{q}}}{\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u}_k\parallel }_{X_{\Delta }^{\varkappa ,p}}\hfill \\ & & +{\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\left({\mathit{\int }}_{t_1}^{t_2}{(\zeta \left(t_2\right)-\zeta \left(\sigma \left(s\right)\right))}^{(\gamma -1)q}{\zeta }^{\Delta }\left(s\right)\Delta s\right)}^{{\displaystyle \frac{1}{q}}}{\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u}_k\parallel }_{X_{\Delta }^{\varkappa ,p}}\hfill \\ & \le & {\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\left({\mathit{\int }}_a^{t_1}\left({(\zeta \left(t_1\right)-\zeta \left(\sigma \left(s\right)\right))}^{(\gamma -1)q}-{(\zeta \left(t_2\right)-\zeta \left(\sigma \left(s\right)\right))}^{(\gamma -1)q}\right){\zeta }^{\Delta }\left(s\right)\Delta s\right)}^{{\displaystyle \frac{1}{q}}}\hfill \end{array}$$

$$\begin{array}{ccc}& & \times \parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u}_k{\parallel }_{X_{\Delta }^{\varkappa ,p}}+{\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\left({\mathit{\int }}_{t_1}^{t_2}{(\zeta \left(t_2\right)-\zeta \left(\sigma \left(s\right)\right))}^{(\gamma -1)q}{\zeta }^{\Delta }\left(s\right)\Delta s\right)}^{{\displaystyle \frac{1}{q}}}{\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u}_k\parallel }_{X_{\Delta }^{\varkappa ,p}}\hfill \\ & =& {\displaystyle \frac{\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u}_k{\parallel }_{X_{\Delta }^{\varkappa ,p}}}{\Gamma \left(\gamma \right){\gamma }_q^{\frac{1}{q}}}}{\left({\left(\zeta \left(t_1\right)-\zeta \left(a\right)\right)}^{{\gamma }_q}-{\left(\zeta \left(t_2\right)-\zeta \left(a\right)\right)}^{{\gamma }_q}+{\left(\zeta \left(t_2\right)-\zeta \left(t_1\right)\right)}^{{\gamma }_q}\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & & +{\displaystyle \frac{\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u}_k{\parallel }_{X_{\Delta }^{\varkappa ,p}}}{\Gamma \left(\gamma \right){\gamma }_q^{\frac{1}{q}}}}{\left({\left(\zeta \left(t_2\right)-\zeta \left(t_1\right)\right)}^{{\gamma }_q}\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & \le & {\displaystyle \frac{2\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u}_k{\parallel }_{X_{\Delta }^{\varkappa ,p}}}{\Gamma \left(\gamma \right){\gamma }_q^{\frac{1}{q}}}}{\left({\left(\zeta \left(t_2\right)-\zeta \left(t_1\right)\right)}^{{\gamma }_q}\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & =& {\displaystyle \frac{2\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u}_k{\parallel }_{X_{\Delta }^{\varkappa ,p}}}{\Gamma \left(\gamma \right){\gamma }_q^{\frac{1}{q}}}}{\left(\zeta \left(t_2\right)-\zeta \left(t_1\right)\right)}^{\gamma -{\displaystyle \frac{1}{p}}}\hfill \\ & \le & C{\left(t_2-t_1\right)}^{\gamma -{\displaystyle \frac{1}{p}}},\hfill \end{array}$$

where ${\gamma }_q=(\gamma -1)q+1$, ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1,$ C is a constant. Hence, applying Lemma 1, we know that $\left\{u_k\right\}$ is relatively compact in $C(I_{\mathbb{T}},{\mathbf{R}}^N)$. Furthermore, $\left\{u_k\right\}$ converges uniformly to u in $C(I_{\mathbb{T}},{\mathbf{R}}^N)$. The proof is finished. □

Remark 4.

Based on the result of Theorem 18, it follows that the space ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$ is compactly embedded into the space $C(I_{\mathbb{T}},{\mathbf{R}}^N)$.

Theorem 19.

Assume that ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$, where $1<p<\infty$. Consider a function $\mathcal{H}$ defined on ${\overline{I}}_{\mathbb{T}}\times {\mathbf{R}}^N\times {\mathbf{R}}^N\to \mathbf{R}$, which meets the following conditions:

(i) 

$\mathcal{H}(s,·,·)$ is Δ-measurable with respect to s;

(ii) 

$\mathcal{H}(·,r,v)$ is continuously differentiable with respect to $(r,v)$.

• Additionally, assume the following functions exist:

• $\alpha \in C({\mathbf{R}}^{+},{\mathbf{R}}^{+})$;
• $\theta \in X_{\Delta }^{\varkappa ,1}({\overline{I}}_{\mathbb{T}},{\mathbf{R}}^{+})$;
• $\vartheta \in X_{\Delta }^{\varkappa ,q}({\overline{I}}_{\mathbb{T}},{\mathbf{R}}^{+})$,

ensuring that the following inequalities hold:

$$\begin{array}{cc}\hfill |\mathcal{H}(s,r,v)|& \le \alpha (|r|)\left(\widehat{b}\left(s\right)+{|v|}^p\right);\hfill \\ \hfill |{\mathcal{H}}_x(s,r,v)|& \le \theta (|r|)\left(\widehat{b}\left(s\right)+{|v|}^p\right);\hfill \\ \hfill |{\mathcal{H}}_y(s,r,v)|& \le \vartheta (|r|)\left(\widehat{c}\left(s\right)+{|v|}^{p-1}\right).\hfill \end{array}$$

(4)

• Under these conditions, $\Phi :{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}\to \mathbf{R}$, defined as

$$\Phi \left(u\right)={\mathit{\int }}_a^b\mathcal{H}\left(s,u\left(s\right),D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(s\right)\right)\Delta s,$$

is continuously differentiable in the space ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$. Moreover, its derivative is given by

$$\begin{array}{ccc}\hfill \langle {\Phi }^{\prime }\left(u\right),v\rangle & =& {\mathit{\int }}_a^b[\left({\mathcal{H}}_x(s,u\left(s\right),D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(s\right)),v\left(s\right)\right)\hfill \\ & & +\left({\mathcal{H}}_y(s,u\left(s\right),D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(s\right)),D_{a^{+}}^{\gamma ,\zeta ,\varkappa }v\left(s\right)\right)]\Delta s\hfill \end{array}$$

(5)

for any $v\in {\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$.

Proof. 

From the above condition, we know that $\Phi$ exists and is limited. Fixing $u,v\in {\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$ for $\varrho \in [-1,1]$, $s\in I_{\mathbb{T}},$ we define

$$G(\varrho ,s)=\mathcal{H}\left(s,u\left(s\right)+\varrho v\left(s\right),D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(s\right)+\varrho D_{a^{+}}^{\gamma ,\zeta ,\varkappa }v\left(s\right)\right)$$

and

$$\Psi \left(\varrho \right)={\mathit{\int }}_a^{b}G(\varrho ,s)\Delta s=\Phi (u+\varrho v).$$

Then,

$$\begin{array}{ccc}\hfill |G_{\varrho }(\varrho ,s)|& \le & \left|\left({\mathcal{H}}_x(s,u\left(s\right)+\varrho v\left(s\right),D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(s\right)+\varrho D_{a^{+}}^{\gamma ,\zeta ,\varkappa }v\left(s\right)),v\left(s\right)\right)\right|\hfill \\ & & +\left|\left({\mathcal{H}}_y(s,u\left(s\right)+\varrho v\left(s\right),D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(s\right)+\varrho D_{a^{+}}^{\gamma ,\zeta ,\varkappa }v\left(s\right)),D_{a^{+}}^{\gamma ,\zeta ,\varkappa }v\left(s\right)v\left(s\right)\right)\right|\hfill \\ & \le & \alpha \left(\left|u\left(s\right)+\varrho v\left(s\right)\right|\right)\left(\theta +{\left|D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(s\right)+\varrho D_{a^{+}}^{\gamma ,\zeta ,\varkappa }v\left(s\right)\right|}^p\right)|v\left(s\right)|\hfill \\ & & +\alpha \left(\left|u\left(s\right)+\varrho v\left(s\right)\right|\right)\left(\vartheta +{\left|D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(s\right)+\varrho D_{a^{+}}^{\gamma ,\zeta ,\varkappa }v\left(s\right)\right|}^{p-1}\right)|D_{a^{+}}^{\gamma ,\zeta ,\varkappa }v\left(s\right)|\hfill \\ & \le & \overline{\alpha }\left(\theta +{\left(\left|D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(s\right)\right|+\left|D_{a^{+}}^{\gamma ,\zeta ,\varkappa }v\left(s\right)\right|\right)}^p\right)|v\left(s\right)|\hfill \\ & & +\overline{\alpha }\left(\vartheta +{\left(\left|D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(s\right)\right|+\left|D_{a^{+}}^{\gamma ,\zeta ,\varkappa }v\left(s\right)\right|\right)}^{p-1}\right)|D_{a^{+}}^{\gamma ,\zeta ,\varkappa }v\left(s\right)|\hfill \\ & :=& d\left(s\right)\hfill \end{array}$$

(6)

where $\overline{\alpha }={max}_{(\varrho ,s)\in [-1,1]\times {\overline{I}}_{\mathbb{T}}}\alpha \left(\left|u\left(s\right)+\varrho v\left(s\right)\right|\right)$ and

$$\begin{array}{ccc}\hfill {\Psi }^{\prime }\left(0\right)& =& {\mathit{\int }}_{[a,b)}{G}_{\varrho }(0,s)\Delta s\hfill \\ & =& {\mathit{\int }}_a^b\left({\mathcal{H}}_x(s,u\left(s\right),D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(s\right)),v\left(s\right)\right)+\left({\mathcal{H}}_y(s,u\left(s\right),D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(s\right)),D_{a^{+}}^{\gamma ,\zeta ,\varkappa }v\left(s\right)\right)\Delta s.\hfill \end{array}$$

(7)

Moreover,

$$\left|{\mathcal{H}}_x(s,u\left(s\right),D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(s\right))\right|\le \alpha |u\left(s\right)|\left(\theta +{\left|D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(s\right)\right|}^p\right),$$

(8)

$$\left|{\mathcal{H}}_y(s,u\left(s\right),D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(s\right))\right|\le \alpha |u\left(s\right)|\left(\vartheta +{\left|D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(s\right)\right|}^{p-1}\right);$$

(9)

hence, from Theorem 17 and (6)–(9) it can be concluded that there are constants $M_1,M_2,M_3$ satisfying

$$\begin{array}{ccc}\hfill \langle {\Psi }^{\prime }\left(u\right),v\rangle & =& {\mathit{\int }}_a^b\left({\mathcal{H}}_x(s,u\left(s\right),D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(s\right)),v\left(s\right)\right)+\left({\mathcal{H}}_y(s,u\left(s\right),D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(s\right)),D_{a^{+}}^{\gamma ,\zeta ,\varkappa }v\left(s\right)\right)\Delta s.\hfill \\ & \le & M_1{\parallel v\parallel }_{\infty }+M_2{\parallel D_{a^{+}}^{\gamma ,\zeta ,\varkappa }v\parallel }_{X_{\Delta }^{\varkappa ,p}}\hfill \\ & \le & M_3{\parallel v\parallel }_{{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}.\hfill \end{array}$$

Therefore, ${\Psi }^{\prime }\left(u\right)$ is the directional derivative of $\Psi$ at u, and ${\Psi }^{\prime }\left(u\right)\in {\left({\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}\right)}^{*}$ given by (5).

Finally, via (4),

$$u\to \left({\mathcal{H}}_x\left(·,u,D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\right),{\mathcal{H}}_y\left(·,u,D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\right)\right)$$

is continuous. Therefore, ${\Psi }^{\prime }:{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}\to {\left({\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}\right)}^{*}$ is continuous. □

4. Application

Leveraging the conceptual framework and analytical findings established in Section 3, we now proceed to demonstrate their practical applicability through a concrete application. More precisely, employing variational methods grounded in critical point theory, an investigation in this section is conducted into a weighted fractional-order dynamic system, subject to the following boundary value conditions:

$$\left\{\begin{array}{c}{D}_{b^{-}}^{\gamma ,\zeta ,\varkappa }{\varphi }_p\left(D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(t\right)\right)=\nabla F(\sigma \left(t\right),u^{\sigma }\left(t\right)),\Delta -a.e.t\in I_{\mathbb{T}},\hfill \\ u\left(a\right)=u\left(b\right)=0,\hfill \end{array}\right.$$

(10)

where $p>1$, $\gamma \in (0,1]$, $D_{a^{+}}^{\gamma ,\zeta ,\varkappa },D_{b^{-}}^{\gamma ,\zeta ,\varkappa }$ denote, separately, the weighted nonlocal operators, ${\varphi }_p={|y|}^{p-2}y$ if $y\ne 0$; ${\varphi }_p=0$ if $y=0$. Additionally,

(F₁)

The function $F:{\overline{I}}_{\mathbb{T}}\times {\mathbf{R}}^N\to \mathbf{R}$ meets the following conditions:

(a)

For every $y\in {\mathbf{R}}^M$, the function $F(·,y)$ is $\Delta$-measurable with respect to the first argument.

(b)

For $\Delta$-almost every $s\in {\overline{I}}_{\mathbb{T}}$, the function $F(s,·)$ is continuously differentiable in its second argument.

(c)

There exist a continuous function r and $q\in X_{\Delta }^{\varkappa ,1}(K,{\mathbf{R}}_{+})$ ensuring that for all $y\in {\mathbf{R}}^M$ and $\Delta$-almost every $s\in {[c,d]}_{\mathbb{T}}$ the following inequalities hold:

$$|F(s,y)|\le r(\parallel y\parallel )q\left(s\right),\parallel \nabla F(s,y)\parallel \le r(\parallel y\parallel )q\left(s\right),$$

where $\nabla F(s,y)$ represents the gradient of F with respect to y.

The variational framework for problem (10) is constructed in the space ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$. This transformation of the solvability analysis of problem (10) into the critical points for the associated energy functional is, therefore, undertaken. It should be noted that this study unifies the examination of weighted fractional boundary value problems (FBVPs) on $\mathbf{R}$ and those that occur on a timescale. In addition to this, it unifies the study of several other types of fractional order boundary value problem, including those of the Riemann–Liouville, Hadamard, Erdelyi–Kober types, and more.

Specifically, when $\mathbb{T}=\mathbf{R}$, problem (10) is turned into a classical weighted fractional boundary value problem (FBVP) as studied in [1]. Furthermore, if we set $\zeta \left(t\right)=t$, $\varkappa \left(t\right)=1$, problem (10) simplifies to the standard Riemann–Liouville fractional p-Laplacian differential system, which has been investigated in [3]. Similarly, taking $\zeta \left(t\right)=lnt$, $\varkappa \left(t\right)=t^{\mu }$, problem (10) becomes the standard Hadamard fractional p-Laplacian differential system, also examined in [3]. Additionally, when $p=2$, problem (10) further reduces to a weighted FBVP, as explored in [3].

Substantial progress has been made in the analysis of the existence of solutions for FBVPs through second-order Hamiltonian systems on timescales, yet the result of problem (10) remains conspicuously absent. Consequently, the present study constitutes a pioneering and applicable result.

The present study employs critical point theory as a methodological framework for the analysis of problem (10). In order to facilitate this analysis, the following procedure is first required. The functional $\phi :{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}\to \mathbf{R}$ is hereby defined as follows:

$$\phi \left(u\right)={\displaystyle \frac{1}{p}}{\mathit{\int }}_a^b{\varkappa }^2\left(r\right){\left({\zeta }^{\Delta }\left(r\right)\right)}^2{\left|D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(r\right)\right|}^p\Delta r-{\mathit{\int }}_a^b{\varkappa }^2\left(r\right){\left({\zeta }^{\Delta }\left(r\right)\right)}^{2}F(t,u\left(r\right))\Delta r.$$

Theorem 20.

The functional φ is continuously differentiable on ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$ and

$$\begin{array}{ccc}\hfill \langle {\phi }^{\prime }\left(u\right),v\rangle & =& {\mathit{\int }}_a^b{\varkappa }^2\left(r\right){\left({\zeta }^{\Delta }\left(r\right)\right)}^2{\left|D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(r\right)\right|}^{p-2}{D}_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(r\right)D_{a^{+}}^{\gamma ,\zeta ,\varkappa }v\left(r\right)\Delta r\hfill \\ & & -{\mathit{\int }}_a^b{\varkappa }^2\left(r\right){\left({\zeta }^{\Delta }\left(r\right)\right)}^2\nabla F(t,u\left(r\right))v\left(r\right)\Delta r,\forall v\in {\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}.\hfill \end{array}$$

(11)

Proof. 

Define

$$\mathcal{H}(s,r,v)={\displaystyle \frac{{\varkappa }^2\left(s\right){\left({\zeta }^{\Delta }\left(s\right)\right)}^2}{p}}{|v|}^p-{\varkappa }^2\left(s\right){\left({\zeta }^{\Delta }\left(s\right)\right)}^{2}F(s,r).$$

By condition $\left(F_1\right)$, it is clear that $\mathcal{H}$ fulfils the conditions in Theorem 19. Furthermore, (11) is true for every $v\in {\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$. □

Definition 12.

Let $u\in {\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$, ${\varkappa }^2(·){\left({\zeta }^{\Delta }(·)\right)}^2\nabla F(·,u(·))\in X_{\Delta }^{\varkappa ,1}$ satisty

$$\begin{array}{ccc}& & {\mathit{\int }}_a^b{\varkappa }^2\left(r\right){\left({\zeta }^{\Delta }\left(r\right)\right)}^2{D}_{b^{-}}^{\gamma ,\zeta ,\varkappa }\varphi \left(D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(r\right)\right)v\left(r\right)\Delta r\hfill \\ & & -{\mathit{\int }}_a^b{\varkappa }^2\left(r\right){\left({\zeta }^{\Delta }\left(r\right)\right)}^2\nabla F(t,u\left(r\right))v\left(r\right)\Delta r,\hfill \\ & =& {\mathit{\int }}_a^b{\varkappa }^2\left(r\right){\left({\zeta }^{\Delta }\left(r\right)\right)}^2{\left|D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(r\right)\right|}^{p-2}{D}_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(r\right)D_{a^{+}}^{\gamma ,\zeta ,\varkappa }v\left(r\right)\Delta r\hfill \\ & & -{\mathit{\int }}_a^b{\varkappa }^2\left(r\right){\left({\zeta }^{\Delta }\left(r\right)\right)}^2\nabla F(t,u\left(r\right))v\left(r\right)\Delta r\hfill \\ & =& 0,\forall v\in C_{0,rd}^{\infty }(I_{\mathbb{T}},{\mathbf{R}}^N).\hfill \end{array}$$

Then, u is called the weak solution of (10).

Theorem 21.

If $u\in {\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$ is a critical point of φ then u is a weak solution of problem (10).

Proof. 

Based on Theorem 9 and (20), we have

$$\begin{array}{ccc}\hfill 0& =& {\mathit{\int }}_a^b{\varkappa }^2\left(r\right){\left({\zeta }^{\Delta }\left(r\right)\right)}^2{\varphi }_p\left(D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(r\right)\right)D_{a^{+}}^{\gamma ,\zeta ,\varkappa }v\left(r\right)\Delta r\hfill \\ & & -{\mathit{\int }}_a^b{\varkappa }^2\left(r\right){\left({\zeta }^{\Delta }\left(r\right)\right)}^2\nabla F(t,u\left(r\right))v\left(r\right)\Delta r\hfill \\ & =& {\mathit{\int }}_a^b{\varkappa }^2\left(r\right){\left({\zeta }^{\Delta }\left(r\right)\right)}^2{D}_{b^{-}}^{\gamma ,\zeta ,\varkappa }{\varphi }_p\left(D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(r\right)\right)v\left(r\right)\Delta r\hfill \\ & & -{\mathit{\int }}_a^b{\varkappa }^2\left(r\right){\left({\zeta }^{\Delta }\left(r\right)\right)}^2\nabla F(t,u\left(r\right))v\left(r\right)\Delta r,\forall v\in {\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p},\hfill \end{array}$$

so u is a weak solution of problem (10). □

In order to illustrate the primary result, it is first necessary to introduce several critical definitions, such as the Palais–Smale condition ($\left(PS\right)$ condition; see ref. [1]), and, for the sake of convenience, a number of conclusions will be revisited.

Lemma 9.

Let $\phi \in C^1(E,\mathbf{R})$ satisfy the $\left(PS\right)$ condition, and let E be a real Banach space. Assume, also, that $c={inf}_E\phi$; then, c is a critical value of φ.

Lemma 10.

Let Y be a real Banach space, and let $\zeta \in C^1(Y,{\mathbf{R}}^M)$ be a functional satisfying the (PS) condition, $\zeta \left(0\right)$ is equal to 0. Additionally,

• For all $y\in Y$ with $\parallel y\parallel =\gamma$, the inequality $\zeta \left(y\right)\ge \beta$ is satisfied and $\gamma ,\beta >0$ are constants.
• There exists an element $y_1\in Y$ with $\parallel y_1\parallel \ge \gamma$, such that $\zeta \left(y_1\right)<\beta$.

Then, φ has a critical value $c\ge \sigma$ given by

$$c=\underset{g\in \Gamma }{inf}\underset{s\in [0,1]}{max}\phi \left(g\left(s\right)\right)$$

with $\Gamma =\{g\in C([0,1],Y):g(0)=0, g(1)=e\}.$

Theorem 22.

Let $\gamma \in \left({\displaystyle \frac{1}{p}},1\right]$, F meet $\left(F_1\right)$ and the following conditions:

$\left(F_2\right)$ There exist two constants $\mu >p,M>0$, such that for all $t\in I_{\mathbb{T}}$ and $|x|\ge M$,

$$0<\mu F(t,x)\le x\nabla F(t,x).$$

$\left(F_3\right)$ For $x\to 0$, $\nabla F(t,x)=o(|x{|}^{p-1})$ with respect to t uniformly.

Problem (10) possesses at least one non-trivial weak solution.

By Lemma 10, $\phi \left(0\right)=0$, and $\phi \in C^1({\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p},\mathbf{R})$. We now proceed to show that $\phi$ meets the $\left(PS\right)$ condition in ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}.$ From now on, we denote ${\parallel ·\parallel }_{{\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}$ by $\parallel ·\parallel$.

Theorem 23.

Let F satisfy $\left(F_1\right)$–$\left(F_3\right)$. Then, φ fulfils the $\left(PS\right)$ condition in ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$.

Proof. 

Let $\left\{u_k\right\}\subset {\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$, $|\phi (u_k)|\le K$, where $K>0$ is a constant, and $\phi \left(u_k\right)\to 0$ as $k\to \infty$. Applying $\left(F_2\right)$, we can observe that there is a positive constant c satisfying

$$F(s,r)\le {\displaystyle \frac{r}{\mu }}\nabla F(s,r)+c,\forall (s,r)\in I_{\mathbb{T}}\times \mathbf{R}.$$

(12)

It follows from (11) and (12) that

$$\begin{array}{ccc}\hfill K& \ge & \phi \left(u_k\right)\hfill \\ & =& {\displaystyle \frac{1}{p}}{\mathit{\int }}_a^b{\varkappa }^2\left(t\right){\left({\zeta }^{\Delta }\left(t\right)\right)}^2{\left|D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u}_k\left(t\right)\right|}^p\Delta t-{\mathit{\int }}_a^b{\varkappa }^2\left(t\right){\left({\zeta }^{\Delta }\left(t\right)\right)}^{2}F(t,u_k\left(t\right))\Delta t\hfill \\ & \ge & {\displaystyle \frac{1}{p}}{\mathit{\int }}_a^b{\varkappa }^2\left(t\right){\left({\zeta }^{\Delta }\left(t\right)\right)}^2{\left|D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u}_k\left(t\right)\right|}^p\Delta t\hfill \\ & & -{\mathit{\int }}_a^b{\varkappa }^2\left(t\right){\left({\zeta }^{\Delta }\left(t\right)\right)}^2\left({\displaystyle \frac{u_k}{\mu }}\nabla F(t,u_k\left(t\right))+c\right)\Delta t\hfill \\ & =& \left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{\mu }}\right){\mathit{\int }}_a^b{\varkappa }^2\left(t\right){\left({\zeta }^{\Delta }\left(t\right)\right)}^2{\left|D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u}_k\left(t\right)\right|}^p\Delta t+{\displaystyle \frac{1}{\mu }}\langle {\phi }^{\prime }\left(u_k\right),u_k\rangle \hfill \\ & & -{\mathit{\int }}_a^{b}c{\varkappa }^2\left(t\right){\left({\zeta }^{\Delta }\left(t\right)\right)}^2\Delta t\hfill \\ & \ge & \left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{\mu }}\right){\mathit{\int }}_a^b{|\varkappa |}^{2-p+p}\left(t\right){\left({\zeta }^{\Delta }\left(t\right)\right)}^2{\left|D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u}_k\left(t\right)\right|}^p\Delta t-{\displaystyle \frac{1}{\mu }}\parallel {\phi }^{\prime }\left(u_k\right){\parallel }_{\left({\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}\right)*}\parallel u_k\parallel \hfill \\ & & -C_2(b-a)c\hfill \\ & \ge & C_1\left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{\mu }}\right)\parallel u_k{\parallel }^p-{\displaystyle \frac{1}{\mu }}\parallel {\phi }^{\prime }\left(u_k\right){\parallel }_{\left({\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}\right)*}\parallel u_k\parallel -C_{2}bc,\hfill \end{array}$$

where

$$C_1=\underset{t\in I_{\mathbb{T}}}{min}\left\{{|\varkappa \left(t\right)|}^{2-p}{\zeta }^{\Delta }\left(t\right)\right\},C_2=\underset{t\in I_{\mathbb{T}}}{max}\left\{{\varkappa }^2\left(t\right){\left({\zeta }^{\Delta }\left(t\right)\right)}^2\right\}.$$

Since ${\phi }^{\prime }\left(u_k\right)\to 0$ as $k\to \infty$, there exists $N_0\in \mathbb{N}$ meeting

$$K\ge C_1\left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{\mu }}\right)\parallel u_k{\parallel }^p-\parallel u_k\parallel -C_{2}bc,k>N_0,$$

i.e., the boundedness of $\left\{u_k\right\}$ is obtained. In view of the fact that ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$ is a reflexive Banach space, it may be assumed that, if necessary, a subsequence still denotes $\left\{u_k\right\}$ and $u_k$ weakly converges to u in ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$. Hence,

$$\begin{array}{c}\hfill {\tau }_k=\langle {\phi }^{\prime }\left(u_k\right)-{\phi }^{\prime }\left(u\right),u_k-u\rangle \mathrm{tends}\mathrm{to}0,\mathrm{as}k\to \infty .\end{array}$$

(13)

Furthermore, combining $\left(F_1\right)$ with Theorem 17, we have $\left\{u_k\right\}$ meets boundedness and ${lim}_{k\to \infty }{u}_k=u$. Thus,

$${\mathit{\int }}_a^b{\varkappa }^2\left(r\right){\left({\zeta }^{\Delta }\left(r\right)\right)}^2\nabla F(r,u_k\left(r\right))\Delta r\to {\mathit{\int }}_a^b{\varkappa }^2\left(r\right){\left({\zeta }^{\Delta }\left(r\right)\right)}^2\nabla F(r,u\left(r\right))\Delta r.$$

(14)

Furthermore,

$$\begin{array}{ccc}\hfill {\tau }_k& =& {\mathit{\int }}_a^b{\varkappa }^2\left(r\right){\left({\zeta }^{\Delta }\left(r\right)\right)}^2{D}_{a^{+}}^{\gamma ,\zeta ,\varkappa }(u_k-u)\left({\varphi }_p\left(D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u}_k\right)-{\varphi }_p\left(D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\right)\right)\Delta r\hfill \\ & & +{\mathit{\int }}_a^b{\varkappa }^2\left(r\right){\left({\zeta }^{\Delta }\left(r\right)\right)}^2(u_k-u)\left(\nabla F(r,u_u\left(r\right))-\nabla F(r,u_k\left(r\right))\right)\Delta r.\hfill \end{array}$$

(15)

Combining (13)–(15), we infer

$${\mathit{\int }}_a^b{\varkappa }^2\left(r\right){\left({\zeta }^{\Delta }\left(r\right)\right)}^2\left(D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u}_k-D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\right)\left({\varphi }_p\left(D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u}_k\right)-{\varphi }_p\left(D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\right)\right)\Delta r\to 0.$$

(16)

We review that $u_k\to u$ in $C(I_{\mathbb{T}},\mathbf{R})$; hence,

$${\mathit{\int }}_a^b{\varkappa }^2\left(r\right){\left({\zeta }^{\Delta }\left(r\right)\right)}^2{|u_k|}^{p-2}\left(u_k,u_k-u\right)\Delta r\to 0,k\to \infty .$$

(17)

We define

$$\xi \left(u\right)={\mathit{\int }}_a^b{\varkappa }^2\left(r\right){\left({\zeta }^{\Delta }\left(r\right)\right)}^2|D_{a^{+}}^{\gamma ,\zeta ,\varkappa }{u}_k{|}^p\Delta r+{\mathit{\int }}_a^b{\varkappa }^2\left(r\right){\left({\zeta }^{\Delta }\left(r\right)\right)}^2{|u_k|}^p\Delta r.$$

(18)

By Hölder’s inequality and (16)–(18), we know that

$$0\le \left(\parallel u_k\parallel -\parallel u\parallel \right)\left(\parallel u_k{\parallel }^{p-1}-{\parallel u\parallel }^{p-1}\right)\le \langle {\xi }^{\prime }\left(u_k\right)-{\xi }^{\prime }\left(u\right),u_k-u\rangle \to 0,k\to \infty ,$$

which means that $u_k\to u$ in ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$. □

Next, we will state that $\phi$ meets condition ($1$) in Lemma 10.

Theorem 24.

Suppose that F satisfies $\left(F_1\right)$–$\left(F_3\right)$. Then, φ satisfies $\left(1\right)$ in Lemma 10.

Proof. 

From $\left(F_3\right)$, we have $\epsilon \in (0,1)$, $\delta \in (0,+\infty )$ satisfying

$$F(t,x)\le {\displaystyle \frac{C_1(1-\epsilon )}{bp{C}_2{C}_3^p}}{|x|}^p,\forall t\in I_{\mathbb{T}},|x|\le \delta .$$

(19)

where $C_1,C_2$ are defined as in Theorem 23 and $C_3={\displaystyle \frac{{\zeta }^{\gamma -\frac{1}{p}}\left(b\right)}{\Gamma \left(\gamma \right){\left((\gamma -1)q+1\right)}^{\frac{1}{q}}}}$. From Theorem 17, we have

$${|u|}_{\infty }\le C_3\parallel u\parallel ,$$

(20)

where $u\in {\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$, $\parallel u\parallel =\rho .$ By virtue of (19) and (20), we obtain

$$\begin{array}{ccc}\hfill \phi \left(u\right)& =& {\displaystyle \frac{1}{p}}{\mathit{\int }}_a^b{\varkappa }^2\left(r\right){\left({\zeta }^{\Delta }\left(r\right)\right)}^2{\left|D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(r\right)\right|}^p\Delta r-{\mathit{\int }}_a^b{\varkappa }^2\left(r\right){\left({\zeta }^{\Delta }\left(r\right)\right)}^{2}F(t,u\left(r\right))\Delta r\hfill \\ & \ge & {\displaystyle \frac{C_1}{p}}{\parallel u\parallel }^p-{\displaystyle \frac{C_1(1-\epsilon )}{bp{C}_2{C}_3^p}}{\mathit{\int }}_a^b{\varkappa }^2\left(r\right){\left({\zeta }^{\Delta }\left(r\right)\right)}^2{|u|}^p\Delta r\hfill \\ & \ge & {\displaystyle \frac{C_1}{p}}{\parallel u\parallel }^p-{\displaystyle \frac{C_1(1-\epsilon )}{bp{C}_3^p}}{\mathit{\int }}_a^b{|u|}^p\Delta r\hfill \\ & \ge & {\displaystyle \frac{C_1}{p}}{\parallel u\parallel }^p-{\displaystyle \frac{C_1(1-\epsilon )}{bp}}(b-a){\parallel u\parallel }^p\hfill \\ & \ge & {\displaystyle \frac{C_1}{p}}{\parallel u\parallel }^p-{\displaystyle \frac{C_1(1-\epsilon )}{p}}{\parallel u\parallel }^p\hfill \\ & =& {\displaystyle \frac{C_1\epsilon }{p}}{\parallel u\parallel }^p:=\sigma ,\hfill \end{array}$$

where $u\in {\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$, $\parallel u\parallel =\rho$. □

The subsequent step is to demonstrate that $\phi$ fulfils condition ($2)$ in Lemma 10.

Theorem 25.

Let $\left(F_1\right)$–$\left(F_3\right)$ hold; then, φ fulfils ($2)$ in Lemma 10.

Proof. 

By $\left(F_2\right)$, there are positive constants $p_0,p_1,p_2>0$ meeting

$$F(l,\varsigma )\ge p_1{|\varsigma |}^{\mu }-p_2>p_0{\zeta }^{\Delta }{\left(l\right)|\varkappa \left(l\right)|}^{\mu }{|\varsigma |}^{\mu }-p_2,\varsigma \in {\mathbf{R}}^N,l\in I_{\mathbb{T}}.$$

(21)

Let $s>0$; then, $\forall u\in {\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$ with $u\ne 0$; noting that $\mu >p$ and (21), we have

$$\begin{array}{ccc}\hfill \phi \left(su\right)& =& {\displaystyle \frac{1}{p}}{\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\left({\zeta }^{\Delta }\left(\varsigma \right)\right)}^2{\left|D_{a^{+}}^{\gamma ,\zeta ,\varkappa }su\left(\varsigma \right)\right|}^p\Delta \varsigma -{\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\left({\zeta }^{\Delta }\left(\varsigma \right)\right)}^{2}F(t,su\left(\varsigma \right))\Delta \varsigma \hfill \\ & \le & {\displaystyle \frac{C_4}{p}}{\parallel su\parallel }^p-{\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\left({\zeta }^{\Delta }\left(\varsigma \right)\right)}^2\left(p_0{\zeta }^{\Delta }{\left(\varsigma \right)|\varkappa \left(\varsigma \right)|}^{\mu }{|su|}^{\mu }-p_2\right)\Delta \varsigma \hfill \\ & =& {\displaystyle \frac{C_4}{p}}{\parallel su\parallel }^p-{\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\left({\zeta }^{\Delta }\left(\varsigma \right)\right)}^2{p}_0{\zeta }^{\Delta }{\left(\varsigma \right)|\varkappa \left(\varsigma \right)|}^{\mu }{|su|}^{\mu }\Delta \varsigma +{\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\left({\zeta }^{\Delta }\left(\varsigma \right)\right)}^2{p}_2\Delta \varsigma \hfill \\ & \le & {\displaystyle \frac{C_4}{p}}{\parallel su\parallel }^p-p_0{|s|}^{\mu }{C}_5{\parallel u\parallel }_{X_{\Delta }^{\varkappa ,\mu }}^{\mu }+p_2{C}_2(b-a)\hfill \\ & \to & -\infty ,ass\to \infty ,\hfill \end{array}$$

(22)

where

$$C_4=\underset{t\in I_{\mathbb{T}}}{max}\left\{{|\varkappa \left(\varsigma \right)|}^{2-p}{\zeta }^{\Delta }\left(\varsigma \right)\right\},C_5=\underset{t\in I_{\mathbb{T}}}{min}\left\{{\varkappa }^2\left(\varsigma \right){\left({\zeta }^{\Delta }\left(\varsigma \right)\right)}^2\right\}.$$

Hence, there exists a sufficiently large $s_0$, such that $\phi \left(s_{0}u\right)\le 0$. □

From Theorems 23–25, we demonstrate that $\phi$ fulfils all the conditions required by Lemma 10; thereby, problem (10) possesses at least one non-trivial solution.

Theorem 26.

Let $\gamma \in \left({\displaystyle \frac{1}{p}},1\right]$, $\left(F_1\right)$ and the conditions below hold.

$\left(F_4\right)$ There exist ${\mu }_1\in (1,p)$ and $d\in X_{\Delta }^{\varkappa ,1}(I_{\mathbb{T}},{\mathbf{R}}^{+})$ so that for all $(s,\varsigma )\in I_{\mathbb{T}}\times \mathbf{R}$,

$$|F(s,\varsigma )|\le {d\left(s\right)|\varsigma |}^{{\mu }_1}.$$

$\left(F_5\right)$ There are constants $\eta ,\delta >0,{\mu }_2\in (1,p)$ and open interval $\mathbb{I}\subset I_{\mathbb{T}}$ ensuring that $(s,\varsigma )\in \mathbb{I}\times [-\delta ,\delta ]$,

$$F(s,\varsigma )\ge {\eta |\varsigma |}^{{\mu }_2}.$$

Then, problem (10) has one non-trivial weak solution.

Proof. 

From $\left(F_4\right)$, we obtain

$$\begin{array}{ccc}\hfill \phi \left(u\right)& =& {\displaystyle \frac{1}{p}}{\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\left({\zeta }^{\Delta }\left(\varsigma \right)\right)}^2{\left|D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(\varsigma \right)\right|}^p\Delta \varsigma -{\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\left({\zeta }^{\Delta }\left(\varsigma \right)\right)}^{2}F(\varsigma ,u\left(\varsigma \right))\Delta \varsigma \hfill \\ & \ge & {\displaystyle \frac{1}{p}}{\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\left({\zeta }^{\Delta }\left(\varsigma \right)\right)}^2{\left|D_{a^{+}}^{\gamma ,\zeta ,\varkappa }u\left(\varsigma \right)\right|}^p\Delta \varsigma -{\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\left({\zeta }^{\Delta }\left(\varsigma \right)\right)}^{2}d\left(\varsigma \right){|u\left(\varsigma \right)|}^{{\mu }_1}\Delta \varsigma \hfill \\ & \ge & {\displaystyle \frac{C_1}{p}}{\parallel u\parallel }^p-{|u|}_{\infty }^{{\mu }_1}{\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\left({\zeta }^{\Delta }\left(\varsigma \right)\right)}^{2}d\left(\varsigma \right)\Delta \varsigma \hfill \\ & \ge & {\displaystyle \frac{C_1}{p}}{\parallel u\parallel }^p-C_6{|u|}_{\infty }^{{\mu }_1}{\parallel d\parallel }_{X_{\Delta }^{\varkappa ,1}}\hfill \\ & \ge & {\displaystyle \frac{C_1}{p}}{\parallel u\parallel }^p-C_3^{{\mu }_1}{C}_6{\parallel d\parallel }_{X_{\Delta }^{\varkappa ,1}}{\parallel u\parallel }^{{\mu }_1},\hfill \end{array}$$

(23)

where $C_6={max}_{t\in I_{\mathbb{T}}}\left\{|\varkappa \left(\varsigma \right){\zeta }^{\Delta }\left(\varsigma \right)|\right\}$; thus, $\phi$ has a lower bound in ${\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$. By means of (23), it is evident that $\phi$ satisfies the $\left(PS\right)$ condition. Hence, by Lemma 9 we know that $c={inf}_{u\in {\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}}\phi \left(u\right)$ is a critical value of $\phi$, which means that $\phi$ has critical point $u^{*}$, such that $\phi \left(u^{*}\right)=c$. Finally, we will verify that $u^{*}\ne 0$. Actually, if $u^{*}=0$ then $c=0$. But for any $u_0\in {\mathcal{W}}_{\Delta ,a^{+}}^{\gamma ,\zeta ,\varkappa ,p}$ with $u_0\ne 0$, from $\left(F_5\right)$ we derive that there exists $r^{\prime }>0$ meeting $F(s,\varsigma )\ge {\eta |\varsigma |}^{{\mu }_2}>r^{\prime }{\zeta }^{\Delta }\left(r^{\prime }\right)|\varkappa \left(r^{\prime }\right){|}^{{\mu }_2}{|\varsigma |}^{{\mu }_2}$; hence, we have

$$\begin{array}{ccc}\hfill \phi \left(s{u}_0\right)& =& {\displaystyle \frac{1}{p}}{\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\left({\zeta }^{\Delta }\left(\varsigma \right)\right)}^2{\left|D_{a^{+}}^{\gamma ,\zeta ,\varkappa }s{u}_0\left(\varsigma \right)\right|}^p\Delta \varsigma -{\mathit{\int }}_a^b{\varkappa }^2\left(\varsigma \right){\left({\zeta }^{\Delta }\left(\varsigma \right)\right)}^{2}F(\varsigma ,s{u}_0\left(\varsigma \right))\Delta \varsigma \hfill \\ & \le & {\displaystyle \frac{C_1}{p}}{\parallel s{u}_0\parallel }^p-{\mathit{\int }}_{\mathbb{I}}{\varkappa }^2\left(\varsigma \right){\left({\zeta }^{\Delta }\left(\varsigma \right)\right)}^{2}F(\varsigma ,s{u}_0\left(\varsigma \right))\Delta \varsigma \hfill \\ & \le & {\displaystyle \frac{C_1}{p}}\parallel s{u}_0{\parallel }^p-r^{\prime }{s}^{{\mu }_2}{C}_5{\parallel u_0\parallel }_{X_{\Delta }^{\varkappa ,{\mu }_2}}^{{\mu }_2}.\hfill \end{array}$$

Note that ${\mu }_2<p$; therefore, $\phi \left(s{u}_0\right)<0$ holding for $s>0$ is small enough, which is contradicted by the definition of c. Therefore, $u^{*}\ne 0$. In other words, $u^{*}$ is the non-trivial weak solution of problem (10). □

5. Examples

In this section, we give some examples to illustrate the usefulness of our main results.

Example 1.

Let $\mathbb{T}={\bigcup }_{k=0}^{1012}[2k,2k+1]$. We consider the following 2-Laplacian fractional boundary value problem:

$$\left\{\begin{array}{c}{D}_{{2025}^{-}}^{0.6,t,1}\left(D_{0^{+}}^{0.6,t,1}u\left(t\right)\right)=\nabla F(\sigma \left(t\right),u^{\sigma }\left(t\right)),\Delta -a.e.t\in {[0,2025]}_{\mathbb{T}},\hfill \\ u\left(0\right)=u\left(2025\right)=0,\hfill \end{array}\right.$$

(24)

where $F(t,x)={\displaystyle \frac{1}{3}}{|x|}^3$. Let $r\left(s\right)=s^3+s^2,q\left(t\right)=1$; then,

$$|F(t,x)|\le r(\parallel x\parallel )q\left(t\right),|\nabla F(t,x)|\le r(\parallel x\parallel )q\left(t\right);$$

hence, $\left(F_1\right)$ is satisfied. Take $\mu ={\displaystyle \frac{5}{2}},M=2025$; then, we can easily show that

$$0<\mu F(t,x)={\displaystyle \frac{5}{6}}{|x|}^3<{|x|}^3=x\nabla F(t,x)$$

holds for $t\in \mathbb{T}$ and $|x|\ge 2025$, which means that condition $\left(F_2\right)$ holds. Through a simple calculation, we have

$$\underset{x\to 0}{lim}{\displaystyle \frac{\nabla F(t,x)}{|x|}}=\underset{x\to 0}{lim}x=0,$$

which implies that the condition $\left(F_3\right)$ holds. Therefore, all these conditions in Theorem 22 are satisfied, and problem $\left(24\right)$ has at least one non-trivial weak solution.

Example 2.

Let $\mathbb{T}=[1,e]$. We consider the following 3-Laplacian fractional boundary value problem:

$$\left\{\begin{array}{c}{D}_{e^{-}}^{0.4,lnt,t^{0.5}}{\varphi }_3\left(D_{0^{+}}^{0.6,lnt,t^{0.5}}u\left(t\right)\right)=\nabla F(\sigma \left(t\right),u^{\sigma }\left(t\right)),\Delta -a.e.t\in \mathbb{T},\hfill \\ u\left(1\right)=u\left(e\right)=0,\hfill \end{array}\right.$$

(25)

where $F(t,x)=e^{-t}\left({|x|}^7+{|x|}^5\right)$. By the fact that

$$|F(t,x)|=e^{-t}\left({|x|}^7+{|x|}^5\right)\le r(|x|)q\left(t\right),|\nabla F(t,x)|=e^{-t}\left({7|x|}^6+5{|x|}^4\right)\le r(|x|)q\left(t\right),$$

where $r\left(s\right)=s^7+7s^6+s^5+5s^4,q\left(t\right)=e^{-t}$, we know that $\left(F_1\right)$ holds. Choosing $\mu =5,M=1$, we obtain

$$\mu F(t,x)=5e^{-t}\left({|x|}^7+{|x|}^5\right)\le e^{-t}\left({7|x|}^7+5{|x|}^5\right)=x\nabla F(t,x).$$

Then, condition $\left(F_2\right)$ is satisfied. To verify that condition $\left(F_3\right)$ holds, we need only check that $\nabla F(t,x)=o(|x{|}^2)$; actually,

$$\underset{x\to 0}{lim}{\displaystyle \frac{\nabla F(t,x)}{{|x|}^2}}=\underset{x\to 0}{lim}{e}^{-t}\left({7|x|}^{3}x+5|x|x\right)=0$$

implies that condition $\left(F_3\right)$ holds. Therefore, all these conditions in Theorem 22 are satisfied, and problem $\left(25\right)$ has at least one non-trivial weak solution.

Example 3.

Let $\mathbb{T}=\{\sqrt{n}:n\in {\mathbb{N}}_0\}$. We consider the following 5-Laplacian fractional boundary value problem:

$$\left\{\begin{array}{c}{D}_{e^{-}}^{0.5,t,1}{\varphi }_5\left(D_{0^{+}}^{0.5,t,1}u\left(t\right)\right)=\nabla F(\sigma \left(t\right),u^{\sigma }\left(t\right)),\Delta -a.e.t\in {[0,2025]}_{\mathbb{T}},\hfill \\ u\left(0\right)=u\left(2025\right)=0,\hfill \end{array}\right.$$

(26)

where $F(t,x)=x^2$. Let $r\left(s\right)=s^2+1,q\left(t\right)=1,{\mu }_1=2,d\left(t\right)=1,\eta =1,\delta =2025,{\mu }_2=2,\mathbb{I}=\{2k:k=1,2,\dots ,1000\};$ then, all the conditions in Theorem 26 are satisfied, and problem $\left(26\right)$ has at least one non-trivial weak solution.

6. Conclusions

This work introduces a novel framework of weighted fractional Sobolev spaces defined on timescales, followed by a systematic investigation of their fundamental properties. A study of weighted fractional order differential dynamic models was conducted, and it demonstrated the existence of non-trivial weak solutions using variational methods (Mountain Pass Theorem). Notably, the methodology developed in this work possesses broad applicability and can be adapted to study various weighted FBVPs on timescales. Furthermore, our results exhibit significant generality, as the proposed weighted fractional operators encompass and unify several well-known fractional operators on timescales. Specifically:

• When $\varkappa \left(t\right)=1$, $\zeta \left(t\right)=t$, the weighted fractional operators are turned to the classical Riemann–Liouville fractional operators.
• For $\varkappa \left(t\right)=t^{\mu },\zeta \left(t\right)=lnt$, they specialise to the Hadamard fractional operators.
• By selecting different weight functions $\varkappa$ and $\zeta$, our framework can represent various other fractional operators.

This unified approach not only extends the existing theories but also provides a versatile tool for studying fractional problems on timescales.