The Euler-Lagrange equations of nabla derivatives for variational approach to optimization problems on time scales

Abstract

This paper investigates the variational approach using nabla (denoted as ∇) within the framework of time scales. By employing two different methods, we derive the Euler-Lagrange equations for first-order variational approach to optimization problems involving exponential functions, as well as for those with both exponential functions and their ∇-derivatives. To establish the high-order variational approach to optimization problem, we present the Leibniz Formula for ∇-derivatives along with its proof. Additionally, we determine the high-order variational approach to optimization problem incorporating ∇-derivatives of exponential functions. Through these analyses, we aim to contribute to the understanding and application of the variational calculus on time scales, offering insights into the behavior of dynamic systems governed by exponential functions and their derivatives.

1 Introduction

The Euler–Lagrange equations represent an apex of mathematical physics, epitomizing profound insights into the intricate dynamics of physical systems and establishing themselves as a cornerstone of variational calculus. Through their elegant formulation, these equations unveil the fundamental principles governing the evolution of physical phenomena from a rigorous mathematical standpoint, thereby showcasing the profound symbiosis between theoretical abstraction and empirical observation. As a testament to the symbiotic relationship between mathematics and the natural sciences, the Euler–Lagrange equations illuminate the underlying symmetries and conservation laws that underpin the fabric of the universe, offering a powerful lens through which to decipher its mysteries and propel the frontiers of scientific inquiry ever further.

In the realm of mathematics, the Euler–Lagrange equations occupy a paramount position, serving as the cornerstone of the variational methods, a discipline dedicated to the optimization of functionals. Their significance extends far beyond mere mathematical abstraction, as they constitute the fundamental framework for addressing a diverse spectrum of optimization problems encountered in both theoretical and applied contexts. Through a systematic elucidation of extremal paths in variational formulations, these equations unveil profound insights into the governing principles dictating the behavior of dynamic systems from a rigorous mathematical standpoint. Moreover, the Euler–Lagrange equations play a pivotal role in the advancement of mathematical theory and analysis, providing researchers with indispensable tools to dissect extremal paths, derive critical equations, and construct mathematical models essential for comprehending the complexities of dynamic systems. Through their application, mathematicians gain invaluable insights into the optimization strategies of complex systems, the characterization of optimal trajectories, and the elucidation of fundamental laws dictating the evolution of physical phenomena, thereby enriching our understanding of the intricate interplay between mathematical abstraction and empirical reality.

Time scale theory offers a versatile framework for studying dynamic systems exhibiting heterogeneous temporal behavior, encompassing both continuous and discrete time domains. This approach facilitates a unified treatment of diverse dynamical phenomena across various disciplines, ranging from physics and engineering to biology and economics. Through the lens of time scale theory, researchers can investigate the interplay between continuous and discrete dynamics, uncovering underlying principles governing system behavior and fostering advancements in modeling, analysis, and control [1–20].

Atici et al. [3] investigated the variational calculus on time scales, utilizing the nabla notation. They explored the derivation of Euler-Lagrange equations for first-order problems. The research aimed to contribute to the understanding and application of variational methods on time scales, offering insights into the behavior of dynamic systems. Bai et al. [12] employed the nabla notation to investigate the variational approach on time scales. It delved into the derivation of Euler-Lagrange equations for first-order problems, which involved exponential functions and their derivatives. The aim of the research was to contribute to the comprehension and application of variational approach on time scales, providing insights into the behavior of dynamic systems governed by exponential functions and their derivatives. Martins et al. [10] investigated the application of variational methods on time scales, incorporating nabla derivatives. It explored the derivation of Euler–Lagrange equations for problems about high-order functions. Through this analysis, the study aimed to enhance understanding of the variational calculus within the context of time scales, leveraging nabla derivatives to offer new perspectives on dynamic systems governed by such functions.

Motivated by the insights gleaned from the aforementioned studies, our primary endeavor seeks to advance upon these findings by introducing an enhanced framework. In this pursuit, we augment the scope of investigation by incorporating both exponential functions and their ∇-derivatives into the first-order variational problem as well as the high-order variational problem. Specifically, our focus lies in refining the understanding of these systems through a meticulous examination of the Euler-Lagrange equations. By integrating these additional components, we aim to provide a more comprehensive and nuanced analysis, thereby enriching the existing body of knowledge and paving the way for further advancements in the field.

For the basic knowledge of time scales, the readers can refer to [4, 21]. In what follows, let $\mathbb{R}$, $\mathbb{N}$ denote the set of real numbers and the set of positive integers, respectively. $\mathbb{N}_{0}=0\cup \mathbb{N}$, and $t\in \mathbb{N}$. A time scale $\mathbb{T}$ is defined as an arbitrary nonempty closed subset of $\mathbb{R}$.

If a is right-dense, $[a,b]_{\mathcal{K}}$ stands for $[a,b]$; otherwise, $[a,b]_{\mathcal{K}}$ stands for $[\sigma (a),b]$. $z\in C^{t}([a,b],\mathbb{R})$ represents the set $\{z:[a,b]\cap \mathbb{T}\rightarrow \mathbb{R}|z^{\nabla ^{t}} \text{ keeps continuity on } [a,b]_{\mathcal{K}^{t}}\}$, and there are at least $t+1$ points in $[a,b]\cap \mathbb{T}$. $z_{*}\in C^{t}([a,b],\mathbb{R})$ is called a weak local extremum point, if there is $\delta >0$ such that $M[z_{*}]\leq M[z]$ (or $M[z_{*}]\geq M[z]$), for $z\in C^{t}([a,b],\mathbb{R})$, and $\|z-z_{*}\|_{t,\infty}<\delta$, where $\|z\|_{t,\infty}\doteq \Sigma _{i=0}^{t}\|z^{\rho ^{t-i}\nabla ^{i}} \|_{\infty}$ and $\|z\|_{\infty}\doteq \sup_{s\in [a,b]_{\mathcal{K}^{t}}}|z(s)|$. In this paper, the notation “→extr” represents solving the extreme value of $M[z]$. Assume the function $F(s,v_{0},v_{1},\ldots ,v_{t})$ has continuous partial derivatives of $v_{0},v_{1},\ldots ,v_{t}$, i.e., $F_{v_{i}}(s,v_{0},v_{1},\ldots ,v_{t})\in C([a,b],\mathbb{R})$, for $i=0,1,\ldots ,t$. At the same time, the ∇-derivative of $F_{v_{i}}(s,v_{0},v_{1},\ldots ,v_{t})$ with respect to s exists, i.e., $F^{\nabla ^{i}}_{v_{i}}(s,v_{0},v_{1},\ldots ,v_{t})$ exists, for $i=0,1,\ldots ,t$.

In the paper, we consider the general variational problem with nabla derivatives

$$ \begin{aligned} &M\bigl[z(\cdot )\bigr]= \int ^{b}_{\sigma ^{t-1}(a)}F\bigl(s,(e_{q_{0}}z)^{\rho ^{t}}(s),(e_{q_{1}}z)^{ \rho ^{t-1}\nabla}(s), \ldots ,(e_{q_{t-1}}z)^{\rho \nabla ^{t-1}}(s),(e_{q_{t}}z)^{ \nabla ^{t}}(s) \bigr)\nabla s \\ &\hphantom{M[z(\cdot )]}\rightarrow \mathrm{extr}, \\ &z\bigl(\sigma ^{t-1}(a)\bigr)=\alpha _{0},\qquad z(b)=\beta _{0}, \\ &\vdots \\ &z^{\nabla ^{t-1}}\bigl(\sigma ^{t-1}(a)\bigr)=\alpha _{t-1},\qquad z^{\nabla ^{t-1}}(b)= \beta _{t-1}, \end{aligned} $$

(1.1)

where $t\in \mathbb{N}$, $q_{j} \in \mathbb{R}$, and $j=0, 1, \ldots , t$.

The organization of this paper is as follows. In Sect. 2, we study the calculus for first-order variational problem on time scales. The Euler–Lagrange equations of the variational approach with exponential function and with both exponential function and ∇-derivatives of exponential function (Theorems 2.1, 2.2) will be proved in two different ways. In Sect. 3, to verify the Euler–Lagrange equation of high-order variational problem (Theorem 3.2), we show the proof of Leibniz Formula (Theorem 3.1) of ∇-derivatives by induction first, and several lemmas and proofs second, as preparations. Finally, we will provide the proof process of Theorem 3.2.

2 Variational approach to the first-order optimization problems

In this section, we present two different methods to prove two special first-order variational problems.

Theorem 2.1

Denote $v_{0}=e^{\rho}_{q_{0}}(s,0)z^{\rho}(s)$, $v_{1}=z^{\nabla}(s)$, and consider the extreme problem as follows:

$$ M[z]= \int ^{\rho (b)}_{\rho ^{2}(a)}F(s,v_{0},v_{1}) \nabla s \rightarrow {\mathrm{extr}}, \qquad z\bigl(\rho ^{2}(a)\bigr)=E,\qquad z \bigl(\rho (b)\bigr)=G. $$

If $z_{*}(s)\in C^{1}([\rho ^{2}(a),b],\mathbb{R})$ is the solution, then the Euler–Lagrange equation

$$ F_{v_{0}}(s,v_{0},v_{1})e^{\rho}_{q_{0}}(s,0) -F_{v_{1}}^{\nabla}(s,v_{0},v_{1})=0, $$

establishes, for $s \in [\rho ^{2}(a),b]_{\mathcal{K}}$.

Proof

Let $\eta : [\rho ^{2}(a),b]\rightarrow \mathbb{R}$ and $\eta \in C^{1}([\rho ^{2}(a),b],\mathbb{R})$ with $\eta (\rho ^{2}(a))=\eta (\rho (b))=0$. Define $\Psi (\xi )=M[z_{*}(s)+\xi \eta (s)]$, with $\xi \in \mathbb{R}$. Due to $\Psi (0)=M(z_{*})$ is a local extremum, it indicates that the first derivative of Ψ at point 0 equals to 0.

Next, we discuss

$$\begin{aligned} \Psi (\xi )= \int ^{\rho (b)}_{\rho ^{2}(a)}F(s,w_{0},w_{1}) \nabla s, \end{aligned}$$

where $w_{0}=e_{q_{0}}^{\rho}(s,0)(z^{\rho}+\xi \eta ^{\rho})(s)$ and $w_{1}=z^{\nabla}(s)+\xi \eta ^{\nabla}(s)$.

Differentiating $\Psi (\xi )$ with respect to ξ, we get

$$\begin{aligned} \Psi '(\xi )= {}& \int ^{\rho (b)}_{\rho ^{2}(a)} \frac{{\mathrm{d}}}{{\mathrm{d}}\xi}F \bigl(s,e_{q_{0}}^{\rho}(s,0) \bigl(z^{\rho}+\xi \eta ^{\rho}\bigr) (s),z^{\nabla}(s)+\xi \eta ^{\nabla}(s)\bigr) \nabla s \\ = {}& \int ^{\rho (b)}_{\rho ^{2}(a)}\bigl\{ F_{w_{0}}(s,w_{0},w_{1})e_{q_{0}}^{ \rho}(s,0) \eta ^{\rho}(s) + F_{w_{1}}(s,w_{0},w_{1}) \eta ^{\nabla}(s) \bigr\} \nabla s. \end{aligned}$$

Obviously, we have

$$\begin{aligned} \Psi '(0)= \int ^{\rho (b)}_{\rho ^{2}(a)}\bigl\{ F_{v_{0}}(s,v_{0},v_{1})e_{q_{0}}^{ \rho}(s,0) \eta ^{\rho}(s)+ F_{v_{1}}(s,v_{0},v_{1})\eta ^{\nabla}(s) \bigr\} \nabla s=0. \end{aligned}$$

By virtue of the property of ∇-integral, we obtain

$$\begin{aligned} \Psi '(0)= {}& \int ^{\rho (a)}_{\rho ^{2}(a)}\bigl\{ F_{v_{0}}(s,v_{0},v_{1})e_{q_{0}}^{ \rho}(s,0) \eta ^{\rho}(s)+ F_{v_{1}}(s,v_{0},v_{1})\eta ^{\nabla}(s) \bigr\} \nabla s \\ &{}+ \int ^{\rho (b)}_{\rho (a)}\bigl\{ F_{v_{0}}(s,v_{0},v_{1})e_{q_{0}}^{ \rho}(s,0) \eta ^{\rho}(s)+ F_{v_{1}}(s,v_{0},v_{1})\eta ^{\nabla}(s) \bigr\} \nabla s \\ = {}& \bigl(\rho (a)-\rho ^{2}(a)\bigr)\bigl\{ F_{v_{0}}\bigl( \rho (a),v_{0},v_{1}\bigr) (e_{q_{0}} \eta )^{\rho ^{2}}(a)+ F_{v_{1}}\bigl(\rho (a),v_{0},v_{1} \bigr)\eta ^{\nabla \rho}(a)\bigr\} \\ &{}+ \int ^{\rho (b)}_{\rho (a)}\bigl\{ F_{v_{0}}(s,v_{0},v_{1})e_{q_{0}}^{ \rho}(s,0) \eta ^{\rho}(s)+ F_{v_{1}}(s,v_{0},v_{1})\eta ^{\nabla}(s) \bigr\} \nabla s \\ = {}& \bigl(\rho (a)-\rho ^{2}(a)\bigr)F_{v_{1}}\bigl(\rho (a),v_{0},v_{1}\bigr)\eta ^{ \nabla}\bigl(\rho (a) \bigr) \quad \bigl(\text{by }\eta \bigl(\rho ^{2}(a)\bigr)=0\bigr) \\ &{}+ \int ^{\rho (b)}_{\rho (a)}\bigl\{ F_{v_{0}}(s,v_{0},v_{1})e_{q_{0}}^{ \rho}(s,0) \eta ^{\rho}(s)+ F_{v_{1}}(s,v_{0},v_{1})\eta ^{\nabla}(s) \bigr\} \nabla s \\ ={} & F_{v_{1}}\bigl(\rho (a),v_{0},v_{1}\bigr)\eta ^{\rho}(a) \\ &{}+ \int ^{\rho (b)}_{\rho (a)}\bigl\{ F_{v_{0}}(s,v_{0},v_{1})e_{q_{0}}^{ \rho}(s,0) \eta ^{\rho}(s)+ F_{v_{1}}(s,v_{0},v_{1})\eta ^{\nabla}(s) \bigr\} \nabla s. \end{aligned}$$

With the help of the formula (see [21], Theorem 8.47), we acquire

$$\begin{aligned} \Psi '(0)= {}&F_{v_{1}}\bigl(\rho (a),v_{0},v_{1} \bigr)\eta \bigl(\rho (a)\bigr)+ \int ^{ \rho (b)}_{\rho (a)}F_{v_{0}}(s,v_{0},v_{1})e_{q_{0}}^{\rho}(s,0) \eta ^{\rho}(s)\nabla s \\ &{}+ F_{v_{1}}\bigl(\rho (b),v_{0},v_{1}\bigr)\eta \bigl(\rho (b)\bigr))- F_{v_{1}}\bigl(\rho (a),v_{0},v_{1} \bigr) \eta \bigl(\rho (a)\bigr) \\ &{}- \int ^{\rho (b)}_{\rho (a)}F^{\nabla}_{v_{1}}(s,v_{0},v_{1}) \eta ^{ \rho}(s)\nabla s \\ ={} & \int ^{\rho (b)}_{\rho (a)}\bigl\{ F_{v_{0}}(s,v_{0},v_{1})e_{q_{0}}^{ \rho}(s,0) \eta ^{\rho}(s)- F^{\nabla}_{v_{1}}(s,v_{0},v_{1}) \eta ^{ \rho}(s)\bigr\} \nabla s \quad \bigl(\text{by }\eta ^{\rho}(b)=0 \bigr). \end{aligned}$$

It is obtained that

$$\begin{aligned} & \int ^{b}_{\rho (b)}\bigl\{ F_{v_{0}}(s,v_{0},v_{1})e_{q_{0}}^{\rho}(s,0)- F^{ \nabla}_{v_{1}}(s,v_{0},v_{1})\bigr\} \eta ^{\rho}(s)\nabla s \\ &\quad = \bigl(b-\rho (b)\bigr)\bigl\{ F_{v_{0}}(b,v_{0},v_{1})e_{q_{0}}^{\rho}(b,0)- F^{ \nabla}_{v_{1}}(b,v_{0},v_{1})\bigr\} \eta ^{\rho}(b) \\ &\quad =0 \quad \bigl(\text{by }\eta ^{\rho}(b)=0\bigr). \end{aligned}$$

Obviously, it follows that

$$ \int ^{b}_{\rho ^{2}(a)}\bigl\{ F_{v_{0}}(s,v_{0},v_{1})e_{q_{0}}^{\rho}(s,0) \eta ^{\rho}(s)- F^{\nabla}_{v_{1}}(s,v_{0},v_{1}) \eta ^{\rho}(s)\bigr\} \nabla s=0. $$

We are led to the conclusion that

$$\begin{aligned} F_{v_{0}}(s,v_{0},v_{1})e^{\rho}_{q_{0}}(s,0)-F_{v_{1}}^{\nabla}(s,v_{0},v_{1})=0, \quad (\text{by Lemma 2.1 of [3]}), \end{aligned}$$

for $s \in [\rho ^{2}(a),b]_{\mathcal{K}}$. □

Theorem 2.2

Denote $v_{1}^{1}=(e_{q_{1}}(s,0)z(s))^{\nabla}$, and consider the problem

$$ M[z]= \int ^{\rho (b)}_{\rho ^{2}(a)}F\bigl(s,v_{0},v_{1}^{1} \bigr)\nabla s \rightarrow {\mathrm{extr}}, \qquad z\bigl(\rho ^{2}(a) \bigr)=E,\qquad z\bigl(\rho (b)\bigr)=G. $$

If $z_{*}(s)\in C^{1}([\rho ^{2}(a),b],\mathbb{R})$ is the solution, then the Euler–Lagrange equation

$$\begin{aligned} F_{v_{0}}\bigl(s,v_{0},v_{1}^{1}\bigr) \cdot e^{\rho}_{q_{0}}(s,0) - F_{v_{1}^{1}}^{ \nabla} \bigl(s,v_{0},v_{1}^{1}\bigr)\cdot e^{\rho}_{q_{1}}(s,0)=0, \end{aligned}$$

holds, for $s \in [\rho ^{2}(a),b]_{\mathcal{K}}$.

Proof

We can verify this theorem by the method similar to proving Theorem 2.1. However, we want to explore a new way to prove it.

Let $\eta : [\rho ^{2}(a),b]\rightarrow \mathbb{R}$ and $\eta \in C^{1}([\rho ^{2}(a),b],\mathbb{R})$, s.t. $\eta ^{\nabla}(b)=0$, and $\eta (\rho (b))=0$.

Define $\Psi (\xi )=M[z_{*}(s)+\xi \eta (s)]$, where $\xi \in \mathbb{R}$, and $\Psi '(0)=0$ is induced. Denote $w_{1}^{1}=(e_{q_{1}}(s,0)z(s))^{\nabla}+\xi (e_{q_{1}}(s,0)\eta (s))^{ \nabla}$.

Next we obtain

$$\begin{aligned} \Psi '(\xi )= {}& \int ^{\rho (b)}_{\rho ^{2}(a)} \frac{{\mathrm{d}}}{{\mathrm{d}}\xi}F \bigl(s,w_{0},w_{1}^{1}\bigr)\nabla s \\ = {}& \int ^{\rho (b)}_{\rho ^{2}(a)}\bigl\{ F_{w_{0}} \bigl(s,w_{0},w_{1}^{1}\bigr) \cdot e_{q_{0}}^{\rho}(s,0)\eta ^{\rho}(s)+ F_{w_{1}^{1}} \bigl(s,w_{0},w_{1}^{1}\bigr) \cdot \bigl(e_{q_{1}}(s,0)\eta (s)\bigr)^{\nabla}\bigr\} \nabla s, \end{aligned}$$

and

$$\begin{aligned} \Psi '(0)= {}& \int ^{\rho (b)}_{\rho ^{2}(a)}\bigl\{ F_{v_{0}} \bigl(s,v_{0},v_{1}^{1}\bigr) \cdot e_{q_{0}}^{\rho}(s,0)\eta ^{\rho}(s)+ F_{v_{1}^{1}} \bigl(s,v_{0},v_{1}^{1}\bigr) \cdot \bigl(e_{q_{1}}(s,0)\eta (s)\bigr)^{\nabla}\bigr\} \nabla s \\ ={} & \int ^{\rho (b)}_{\rho ^{2}(a)}\bigl\{ \bigl[F_{v_{0}} \bigl(s,v_{0},v_{1}^{1}\bigr) \cdot e_{q_{0}}^{\rho}(s,0)+F_{v_{1}^{1}}\bigl(s,v_{0},v_{1}^{1} \bigr)\cdot e_{q_{1}}^{ \nabla}(s,0)\bigr]\cdot \eta ^{\rho}(s) \\ &{}+ \bigl[F_{v_{1}^{1}}\bigl(s,v_{0},v_{1}^{1} \bigr)\cdot e_{q_{1}}(s,0)\bigr]\cdot \eta ^{ \nabla}(s)\bigr\} \nabla s \\ &\quad {}(\text{by the property of nabla differentiation} ) \\ = {}&0. \end{aligned}$$

(2.1)

Thanks to the property of ∇-integral, we get

$$\begin{aligned} & \int ^{b}_{\rho (b)}\bigl\{ \bigl[F_{v_{0}} \bigl(s,v_{0},v_{1}^{1}\bigr)\cdot e_{q_{0}}^{ \rho}(s,0)+F_{v_{1}^{1}}\bigl(s,v_{0},v_{1}^{1} \bigr)\cdot e_{q_{1}}^{\nabla}(s,0)\bigr] \cdot \eta ^{\rho}(s) \\ &\qquad {}+ \bigl[F_{v_{1}^{1}}\bigl(s,v_{0},v_{1}^{1} \bigr)\cdot e_{q_{1}}(s,0)\bigr]\cdot \eta ^{ \nabla}(s)\bigr\} \nabla s \\ &\quad = \bigl\{ \bigl[F_{v_{0}}\bigl(b,v_{0}(b),v_{1}^{1}(b) \bigr)\cdot e_{q_{0}}^{\rho}(b,0)+F_{v_{1}^{1}} \bigl(b,v_{0}(b),v_{1}^{1}(b)\bigr) \cdot e_{q_{1}}^{\nabla}(b,0)\bigr]\cdot \eta ^{\rho}(b) \\ &\qquad {}+ \bigl[F_{v_{1}^{1}}\bigl(b,v_{0}(b),v_{1}^{1}(b) \bigr)\cdot e_{q_{1}}(b,0)\bigr]\cdot \eta ^{\nabla}(b)\bigr\} \cdot \bigl(b-\rho (b)\bigr) \\ &\quad = 0, \quad \bigl(\text{by }\eta ^{\rho}(b)=0\text{ and }\eta ^{\nabla}(b)=0\bigr). \end{aligned}$$

(2.2)

Through the additivity of integral intervals, it follows that

$$\begin{aligned} & \int ^{b}_{\rho ^{2}(a)}\bigl\{ \bigl[F_{v_{0}} \bigl(s,v_{0},v_{1}^{1}\bigr)\cdot e_{q_{0}}^{ \rho}(s,0)+F_{v_{1}^{1}}\bigl(s,v_{0},v_{1}^{1} \bigr)\cdot e_{q_{1}}^{\nabla}(s,0)\bigr] \cdot \eta ^{\rho}(s) \\ &\qquad {}+ \bigl[F_{v_{1}^{1}}\bigl(s,v_{0},v_{1}^{1} \bigr)\cdot e_{q_{1}}(s,0)\bigr] \cdot \eta ^{\nabla}(s)\bigr\} \nabla s \\ &\quad = 0. \end{aligned}$$

Thanks to the property of nabla differentiation, we have

$$ \bigl[F_{v_{1}^{1}}\bigl(s,v_{0},v_{1}^{1} \bigr)\cdot e_{q_{1}}(s,0)\bigr]^{\nabla}=F^{ \nabla}_{v_{1}^{1}} \bigl(s,v_{0},v_{1}^{1}\bigr)\cdot e^{\rho}_{q_{1}}(s,0)+ F_{v_{1}^{1}}\bigl(s,v_{0},v_{1}^{1} \bigr) \cdot e^{\nabla}_{q_{1}}(s,0), $$

and using Lemma 15 of [10], we get

$$ \bigl[F_{v_{1}^{1}}\bigl(s,v_{0},v_{1}^{1} \bigr)\cdot e_{q_{1}}(s,0)\bigr]^{\nabla}=F_{v_{0}} \bigl(s,v_{0},v_{1}^{1}\bigr) \cdot e_{q_{0}}^{\rho}(s,0)+F_{v_{1}^{1}}\bigl(s,v_{0},v_{1}^{1} \bigr)\cdot e_{q_{1}}^{ \nabla}(s,0). $$

It is concluded that

$$ F_{v_{0}}\bigl(s,v_{0},v_{1}^{1}\bigr) \cdot e_{q_{0}}^{\rho}(s,0)=F^{\nabla}_{v_{1}^{1}} \bigl(s,v_{0},v_{1}^{1}\bigr) \cdot e^{\rho}_{q_{1}}(s,0), $$

for $s \in [\rho ^{2}(a),b]_{\mathcal{K}}$. The result of Theorem 2.2 establishes. □

3 Variational approach to the high-order optimization problem

This section deals with the high-order optimization problem. We begin the discussion with the theorem as follows.

Theorem 3.1

(Leibniz Formula of nabla derivatives)

Denote $T^{(l)}_{j}$ to be the set including $\rho ^{j}\nabla ^{l-j}$. If $f^{\Lambda}$ persists for $\Lambda \in T^{(l)}_{j}$, then

$$ (fg)^{\nabla ^{l}}=\sum^{l}_{j=0} \biggl(\sum_{\Lambda \in T^{(l)}_{j}}f^{ \Lambda} \biggr)g^{\nabla ^{j}} $$

(3.1)

holds for all $l \in \mathbb{N}_{0}$.

Proof

With the convention that $\sum_{\Lambda \in \phi}f^{\Lambda}=f$, the formula (3.1) holds for $l=0$. We now show the formula by induction. First, if $l=1$, the formula (3.1) is $(fg)^{\nabla}=f^{\nabla}g+f^{\rho}g^{\nabla}$ (see [10], Theorem 4), then the formula is true. Next, we assume that the formula (3.1) is true for $l=k \in \mathbb{N}$. Then, we calculate

$$\begin{aligned} (fg)^{\nabla ^{k+1}}= {}& \sum^{k}_{j=0} \biggl\{ \biggl(\sum_{\Lambda \in T^{(k)}_{j}}f^{ \Lambda} \biggr)^{\nabla}g^{\nabla ^{j}}+\biggl(\sum_{\Lambda \in T^{(k)}_{j}}f^{ \Lambda} \biggr)^{\rho}g^{\nabla ^{j+1}}\biggr\} \\ = {}& \sum^{k}_{j=0}\biggl(\sum _{\Lambda \in T^{(k)}_{j}}f^{\Lambda \nabla}\biggr)g^{ \nabla ^{j}}+\sum ^{k+1}_{j=1}\biggl(\sum_{\Lambda \in T^{(k)}_{j-1}}f^{ \Lambda \rho} \biggr)g^{\nabla ^{j}} \\ = {}& \biggl(\sum_{\Lambda \in T^{(k)}_{0}}f^{\Lambda \nabla}\biggr)g+\sum ^{k}_{j=1}\biggl( \sum _{\Lambda \in T^{(k)}_{j}}f^{\Lambda \nabla}\biggr)g^{\nabla ^{j}} \\ &{}+\biggl( \sum _{\Lambda \in T^{(k)}_{k}}f^{\Lambda \rho}\biggr)g^{\nabla ^{k+1}}+ \sum ^{k}_{j=1}\biggl(\sum _{\Lambda \in T^{(k)}_{j-1}}f^{\Lambda \rho}\biggr)g^{ \nabla ^{j}} \\ = {}& \biggl(\sum_{\Lambda \in T^{(k+1)}_{0}}f^{\Lambda}\biggr)g+ \biggl(\sum_{\Lambda \in T^{(k+1)}_{k+1}}f^{\Lambda} \biggr)g^{\nabla ^{k+1}}+\sum^{k}_{j=1} \biggl( \sum_{\Lambda \in T^{(k)}_{j}}f^{\Lambda \nabla}+\sum _{\Lambda \in T^{(k)}_{j-1}}f^{ \Lambda \rho}\biggr)g^{\nabla ^{j}} \\ = {}& \sum^{k+1}_{j=0}\biggl(\sum _{\Lambda \in T^{(k+1)}_{j}}f^{\Lambda}\biggr)g^{ \nabla ^{j}}, \end{aligned}$$

which implies the formula (3.1) holds for $l=k+1$. Therefore, the formula (3.1) holds for all $m \in \mathbb{N}_{0}$. □

To obtain further results, we also need the following several lemmas. In the rest of the paper, we respectively denote $e_{q}(\rho (s),0)\cdot \varphi (\rho (s))$ and $(e_{q}(s,0)\cdot \varphi (s))^{\nabla}$ as $(e_{q}\cdot \varphi )^{\rho}(s)$ and $(e_{q}\cdot \varphi )^{\nabla}(s)$, for $\varphi \in C^{t}([a,b],\mathbb{R})$, and the following hypothesis is valid

$$ (H):\quad \text{for each } s\in \mathbb{T}, (t-1) \bigl(\rho (s)-b_{1}s-b_{0} \bigr)=0, \text{for some } b_{1}\in \mathbb{R^{+}}, b_{0}\in \mathbb{R}. $$

Lemma 3.1

For $s\in [a,b]_{\mathcal{K}^{t}}$ and $i=1,2,\ldots ,t$,

$$\begin{aligned} (e_{q}\cdot \varphi )^{\rho \nabla ^{i}}(s)=b_{1}^{i}(e_{q} \cdot \varphi )^{\nabla ^{i}\rho}(s) \end{aligned}$$

(3.2)

is tenable.

Proof

We use mathematical induction to prove the lemma. Here we provide an expression:

$$\begin{aligned} \varphi ^{\rho}(s)=\varphi (s)-\nu (s)\varphi ^{\nabla}(s), \end{aligned}$$

(3.3)

which will be applied multiple times later. When $i=1$,

$$\begin{aligned} (e_{q}\cdot \varphi )^{\rho \nabla}(s)= {}& \bigl[(e_{q} \cdot \varphi ) (s)- \nu (s) (e_{q}\cdot \varphi )^{\nabla}(s) \bigr]^{\nabla}\quad (\text{by (3.3)}) \\ = {}& (e_{q}\cdot \varphi )^{\nabla}(s)-\nu ^{\nabla}(s) (e_{q}\cdot \varphi )^{\nabla \rho}(s)-\nu (s) (e_{q} \cdot \varphi ) ^{\nabla ^{2}}(s)\quad (\text{by Theorem 3.1}) \\ ={} & (e_{q}\cdot \varphi )^{\nabla \rho}(s)-\nu ^{\nabla}(s) (e_{q} \cdot \varphi )^{\nabla \rho}(s) \quad (\text{by (3.3)}) \\ ={} & b_{1}(e_{q}\cdot \varphi )^{\nabla \rho}(s)\quad \bigl( \text{by } 1- \nu ^{\nabla}(s)=b_{1}\bigr), \end{aligned}$$

which indicates (3.2) holds.

Next, suppose that $(e_{q}\cdot \varphi )^{\rho \nabla ^{k}}(s)=b_{1}^{k}(e_{q}\cdot \varphi )^{\nabla ^{k}\rho}(s)$ holds. When $i=k+1$, we have

$$\begin{aligned} (e_{q}\cdot \varphi )^{\rho \nabla ^{k+1}}(s)= {}&\bigl[(e_{q} \cdot \varphi )^{ \rho \nabla ^{k}}(s)\bigr]^{\nabla} \\ = {}& \bigl[b_{1}^{k}(e_{q}\cdot \varphi )^{\nabla ^{k}\rho}(s)\bigr]^{\nabla} \\ ={} & b_{1}^{k}\bigl[(e_{q}\cdot \varphi )^{\nabla ^{k}}(s)-\nu (s) (e_{q} \cdot \varphi )^{\nabla ^{k+1}}(s) \bigr]^{\nabla}\quad (\text{by (3.3)}) \\ = {}& b_{1}^{k}\bigl[(e_{q}\cdot \varphi )^{\nabla ^{k+1}}(s)-\nu ^{\nabla}(s) (e_{q} \cdot \varphi )^{\nabla ^{k+1}\rho}(s)-\nu (s) (e_{q}\cdot \varphi )^{ \nabla ^{k+2}}(s) \bigr] \\ &\quad {} (\text{by Theorem 3.1}) \\ = {}& b_{1}^{k}\bigl[(e_{q}\cdot \varphi )^{\nabla ^{k+1}\rho}(s)-\nu ^{ \nabla}(s) (e_{q}\cdot \varphi )^{\nabla ^{k+1}\rho}(s)\bigr] \quad (\text{by (3.3)}) \\ = {}& b_{1}^{k+1}(e_{q}\cdot \varphi )^{\nabla ^{k+1}\rho}(s). \end{aligned}$$

The conclusion of Lemma 3.1 establishes. □

Lemma 3.2

Suppose that $\varphi ^{\nabla ^{i}}(b)$ vanishes, for $i\in \{0,1,\ldots ,t\}$, and therefore $(e_{q_{i-1}}\cdot \varphi )^{\rho \nabla ^{i-1}}(b)$ also disappears.

Proof

When $i=t$, it is deduced that

$$\begin{aligned} (e_{q_{t-1}}\cdot \varphi )^{\rho \nabla ^{t-1}}(b)=\sum ^{t-1}_{j=0}\biggl( \sum_{\Lambda \in S^{(t-1)}_{j}}e_{q_{t-1}}^{\rho \Lambda} \biggr)\cdot \bigl( \varphi ^{\rho}\bigr)^{\nabla ^{j}}(b),\quad (\text{by Theorem 3.1}) \end{aligned}$$

and

$$\begin{aligned} \bigl(\varphi ^{\rho}\bigr)^{\nabla ^{j}}(b)=0, \quad \text{for } j=0,1, \ldots ,t-1, \quad (\text{by Lemma 12 of [10]}). \end{aligned}$$

Therefore, $(e_{q_{t-1}}\cdot \varphi )^{\rho \nabla ^{t-1}}(b)$ vanishes.

Similarly, for $i=t-1,\ldots ,1$, we have

$$\begin{aligned} &(e_{q_{t-2}}\cdot \varphi )^{\rho \nabla ^{t-2}}(b)=0, \quad \text{for } i=t-1, \\ & \vdots \\ & (e_{q_{0}}\cdot \varphi )^{\rho}(b)=0, \quad \text{for } i=1. \end{aligned}$$

From the above discussion, $(e_{q_{i-1}}\cdot \varphi )^{\rho \nabla ^{i-1}}(b)=0$ holds, $i\in \{1,\ldots ,t\}$. □

Lemma 3.3

When the assumption $\varphi ^{\nabla ^{i}}(\sigma ^{t}(a))=0$ for $i\in \{0,1,\ldots ,t\}$ is tenable, then

$$ (e_{q_{i}}\cdot \varphi )^{\nabla ^{i}}\bigl(\sigma ^{i}(a) \bigr)=0, $$

founds, for $i\in \{0,1,\ldots ,t\}$.

Proof

Consider the case $i=t$. By Theorem 3.1, it is easy to know

$$\begin{aligned} (e_{q_{t}}\cdot \varphi )^{\nabla ^{t}}\bigl(\sigma ^{t}(a) \bigr)= {}&\sum^{t}_{j=0}\biggl( \sum _{\Lambda \in T^{(t)}_{j}}e_{q_{t}}^{\Lambda}\biggr)\cdot \varphi ^{ \nabla ^{j}}\bigl(\sigma ^{t}(a)\bigr) \\ = {}& 0, \quad \text{for } j=0,1,\ldots ,t. \end{aligned}$$

For the cases $i=0,1,\ldots ,t-1$, the deductions are similar to those of [10, Lemma 13], so we overlook the details. □

Lemma 3.4

If $f_{0},f_{1},\ldots ,f_{t}\in C^{t}([a,b],\mathbb{R})$, and

$$ \int ^{b}_{\sigma ^{t-1}(a)}\Biggl[\sum ^{t}_{j=0}f_{j}(s) (e_{q_{j}} \varphi )^{\rho ^{t-j}\nabla ^{j}}(s)\Biggr]\nabla s=0, $$

when $\varphi ,\varphi ^{\nabla},\ldots ,\varphi ^{\nabla ^{t-1}}$ vanish at $\sigma ^{t-1}(a)$ and b, therefore

$$ \sum^{t}_{j=0}(-1)^{j}\biggl( \frac{1}{b_{1}}\biggr)^{\frac{j(j-1)}{2}}f^{\nabla ^{j}}_{j}(s) \cdot e^{\rho ^{t}}_{q_{j}}(s)=0,\quad s\in [a,b]_{\mathcal{K}^{t}}, $$

sets up.

Proof

When $t=1$, the known condition

$$\begin{aligned} & \int ^{b}_{a}\bigl[f_{0}(e_{q_{0}} \varphi )^{\rho}(s)+f_{1}(e_{q_{1}} \varphi )^{\nabla}(s)\bigr]\nabla s \\ &\quad = \int ^{b}_{a}f_{0}(e_{q_{0}} \varphi )^{\rho}(s)\nabla s +\bigl[f_{1}(s) (e_{q_{1}} \varphi ) (s)\bigr]^{b}_{a} - \int ^{b}_{a}f^{\nabla}_{1}(e_{q_{1}} \varphi )^{ \rho}(s)\nabla s \\ &\quad = \int ^{b}_{a}f_{0}(e_{q_{0}} \varphi )^{\rho}(s)\nabla s - \int ^{b}_{a}f^{ \nabla}_{1}(e_{q_{1}} \varphi )^{\rho}(s)\nabla s\quad \bigl(\text{by } \varphi (a)=\varphi (b)=0 \bigr) \\ &\quad = \int ^{b}_{a}\bigl[f_{0}e_{q_{0}}^{\rho}(s,0) - f^{\nabla}_{1}e_{q_{1}}^{ \rho}(s,0)\bigr] \varphi ^{\rho}(s) \nabla s \\ &\quad = 0 \end{aligned}$$

holds. By virtue of Lemma 2.1 of [10], $f_{0}e_{q_{0}}^{\rho}(s,0) - f^{\nabla}_{1}e_{q_{1}}^{\rho}(s,0)=0$ is tenable.

For $t>1$, assume this lemma is correct. The correctness of the result will be validated below for $t+1$. If

$$ \int ^{b}_{\sigma ^{t}(a)}\Biggl[\sum ^{t+1}_{j=0}f_{j}(s) (e_{q_{j}} \varphi )^{\rho ^{t+1-j}\nabla ^{j}}(s)\Biggr]\nabla s=0 $$

for $\varphi \in C^{t+1}([a,b],\mathbb{R})$, s.t. $\varphi (\sigma ^{t}(a))=0$, $\varphi (b)=0, \ldots , \varphi ^{ \nabla ^{t}}(\sigma ^{t}(a))=0$, $\varphi ^{\nabla ^{t}}(b)=0$, then

$$ \sum^{t+1}_{j=0}(-1)^{j}\biggl( \frac{1}{b_{1}}\biggr)^{\frac{j(j-1)}{2}}f^{ \nabla ^{j}}_{j}\cdot e^{\rho ^{t+1}}_{q_{j}}(s)=0,\quad s\in [a,b]_{ \mathcal{K}^{t+1}}, $$

holds.

Note that

$$\begin{aligned} \int ^{b}_{\sigma ^{t}(a)}\Biggl[\sum ^{t+1}_{j=0}f_{j}(s) (e_{q_{j}} \varphi )^{\rho ^{t+1-j}\nabla ^{j}}(s)\Biggr]\nabla s= {}& \int ^{b}_{ \sigma ^{t}(a)}\Biggl[\sum ^{t}_{j=0}f_{j}(s) (e_{q_{j}} \varphi )^{\rho ^{t+1-j} \nabla ^{j}}(s)\Biggr]\nabla s \\ &{}+ \int ^{b}_{\sigma ^{t}(a)}f_{t+1}(s)\cdot \bigl[(e_{q_{t+1}}\varphi )^{ \nabla ^{t}}\bigr]^{\nabla}(s)\nabla s. \end{aligned}$$

Using the formula of integration by parts, the second term on the right-hand side can be rewritten as

$$\begin{aligned}& \int ^{b}_{\sigma ^{t}(a)}f_{t+1}(s)\cdot \bigl[(e_{q_{t+1}}\varphi )^{ \nabla ^{t}}\bigr]^{\nabla}(s)\nabla s \\& \quad = \bigl\{ f_{t+1}(s)\cdot \bigl[(e_{q_{t+1}} \varphi )^{\nabla ^{t}} \bigr]\bigr\} _{\sigma ^{t}(a)}^{b}- \int ^{b}_{\sigma ^{t}(a)}f_{t+1}^{ \nabla}(s) \cdot \bigl[(e_{q_{t+1}}\varphi )^{\nabla ^{t}}\bigr]^{\rho}(s) \nabla s. \end{aligned}$$

In virtue of $\varphi ^{\nabla ^{t}}(\sigma ^{t}(a))=0$, $\varphi ^{\nabla ^{t}}(b)=0$, Lemma 3.3 and Theorem 3.1, we can conclude that

$$ \int ^{b}_{\sigma ^{t}(a)}f_{t+1}(s)\cdot \bigl[(e_{q_{t+1}}\varphi )^{ \nabla ^{t}}\bigr]^{\nabla}(s)\nabla s=- \int ^{b}_{\sigma ^{t}(a)}f_{t+1}^{ \nabla}(s) \cdot \bigl[(e_{q_{t+1}}\varphi )^{\nabla ^{t}}\bigr]^{\rho}(s) \nabla s. $$

By Lemma 3.1, we have

$$\begin{aligned} (e_{q_{t+1}}\cdot \varphi )^{\nabla ^{t}\rho}(s)= &\biggl(\frac{1}{b_{1}} \biggr)^{t}(e_{q_{t+1}} \cdot \varphi )^{\rho \nabla ^{t}}(s), \quad s\in [a,b]_{\mathcal{K}^{t+1}}. \end{aligned}$$

Therefore, from what has been discussed above, it follows that

$$ \int ^{b}_{\sigma ^{t}(a)}f_{t+1}(s)\cdot \bigl[(e_{q_{t+1}}\varphi )^{ \nabla ^{t}}\bigr]^{\nabla}(s)\nabla s=- \int ^{b}_{\sigma ^{t}(a)}f_{t+1}^{ \nabla}(s) \cdot \biggl[\biggl(\frac{1}{b_{1}}\biggr)^{t}(e_{q_{t+1}}\cdot \varphi )^{ \rho \nabla ^{t}}(s) \biggr]\nabla s, $$

and

$$\begin{aligned} & \int ^{b}_{\sigma ^{t}(a)}\Biggl[\sum ^{t+1}_{j=0}f_{j}(s) (e_{q_{j}} \varphi )^{\rho ^{t+1-j}\nabla ^{j}}(s)\Biggr]\nabla s \\ &\quad = \int ^{b}_{\sigma ^{t}(a)}\Biggl[\sum ^{t}_{j=0}f_{j}(s) (e_{q_{j}} \varphi )^{\rho ^{t+1-j}\nabla ^{j}}(s)\Biggr]\nabla s - \int ^{b}_{\sigma ^{t}(a)}f_{t+1}^{ \nabla}(s) \cdot \biggl[\biggl(\frac{1}{b_{1}}\biggr)^{t}(e_{q_{t+1}}\cdot \varphi )^{ \rho \nabla ^{t}}(s) \biggr]\nabla s \\ &\quad = \int ^{b}_{\sigma ^{t}(a)}\Biggl\{ \sum ^{t-1}_{j=0}f_{j}(s)\cdot \bigl[(e_{q_{j}} \varphi )^{\rho}\bigr]^{\rho ^{t-j}\nabla ^{j}}(s) + f_{t}(s)\cdot (e_{q_{t}} \cdot \varphi )^{\rho \nabla ^{t}}(s) \\ &\qquad {}-f_{t+1}^{\nabla}(s) \cdot \biggl( \frac{1}{b_{1}}\biggr)^{t} (e_{q_{t+1}}\cdot \varphi )^{\rho \nabla ^{t}}(s) \Biggr\} \nabla s \\ &\quad = \int ^{\sigma ^{t+1}(a)}_{\sigma ^{t}(a)}\Biggl\{ \sum ^{t-1}_{j=0}f_{j}(s) \cdot \bigl[(e_{q_{j}}\varphi )^{\rho}\bigr]^{\rho ^{t-j}\nabla ^{j}}(s) \\ &\qquad {}+ f_{t}(s)\cdot (e_{q_{t}}\cdot \varphi )^{\rho \nabla ^{t}}(s)-f_{t+1}^{ \nabla}(s) \cdot \biggl(\frac{1}{b_{1}}\biggr)^{t} (e_{q_{t+1}}\cdot \varphi )^{ \rho \nabla ^{t}}(s) \Biggr\} \nabla s \\ &\qquad {}+ \int ^{b}_{\sigma ^{t+1}(a)}\Biggl\{ \sum ^{t-1}_{j=0}f_{j}(s)\cdot \bigl[(e_{q_{j}} \varphi )^{\rho}\bigr]^{\rho ^{t-j}\nabla ^{j}}(s) \\ &\qquad {}+ f_{t}(s)\cdot (e_{q_{t}}\cdot \varphi )^{\rho \nabla ^{t}}(s)-f_{t+1}^{ \nabla}(s) \cdot \biggl(\frac{1}{b_{1}}\biggr)^{t} (e_{q_{t+1}}\cdot \varphi )^{ \rho \nabla ^{t}}(s) \Biggr\} \nabla s. \end{aligned}$$

We now prove that

$$\begin{aligned} &\int ^{\sigma ^{t+1}(a)}_{\sigma ^{t}(a)}\Biggl\{ \sum ^{t-1}_{j=0}f_{j}(s) \cdot \bigl[(e_{q_{j}}\varphi )^{\rho}\bigr] ^{\rho ^{t-j}\nabla ^{j}}(s) + f_{t}(s) \cdot (e_{q_{t}}\cdot \varphi )^{\rho \nabla ^{t}}(s) \\ &\quad {}-f_{t+1}^{\nabla}(s) \cdot \biggl(\frac{1}{b_{1}}\biggr)^{t} (e_{q_{t+1}}\cdot \varphi )^{\rho \nabla ^{t}}(s) \Biggr\} \nabla s \end{aligned}$$

(3.4)

is equal to zero. Utilizing the property of ∇-integral, (3.4) reduces to

$$\begin{aligned} &\Biggl\{ \sum^{t-1}_{j=0}f_{j} \bigl(\sigma ^{t+1}(a)\bigr)\cdot \bigl[(e_{q_{j}}\varphi )^{ \rho}\bigr] ^{\rho ^{t-j}\nabla ^{j}}\bigl(\sigma ^{t+1}(a)\bigr) + f_{t}\bigl(\sigma ^{t+1}(a)\bigr) \cdot (e_{q_{t}}\cdot \varphi )^{\rho \nabla ^{t}}\bigl(\sigma ^{t+1}(a)\bigr) \\ &\quad {}- f_{t+1}^{\nabla}\bigl(\sigma ^{t+1}(a)\bigr)\cdot \biggl(\frac{1}{b_{1}}\biggr)^{t} (e_{q_{t+1}} \cdot \varphi )^{\rho \nabla ^{t}}\bigl(\sigma ^{t+1}(a)\bigr) \Biggr\} \cdot \nu \bigl( \sigma ^{t+1}(a)\bigr) . \end{aligned}$$

By Lemma 3.1 and Lemma 3.3, for each $j\in \{0,1,\ldots ,t\}$, we have

$$\begin{aligned} \bigl[(e_{q_{j}}\varphi )^{\rho}\bigr] ^{\rho ^{t-j}\nabla ^{j}}\bigl( \sigma ^{t+1}(a)\bigr)= {}&(e_{q_{j}} \varphi )^{\rho ^{t+1-j}\nabla ^{j}} \bigl(\sigma ^{t+1}(a)\bigr) \\ = {}& (b_{1})^{j(t+1-j)}(e_{q_{j}}\varphi )^{\nabla ^{j}\rho ^{t+1-j}} \bigl( \sigma ^{t+1}(a)\bigr) \\ = {}& (b_{1})^{j(t+1-j)}(e_{q_{j}}\varphi )^{\nabla ^{j}} \bigl(\sigma ^{j}(a)\bigr) \\ = {}& 0, \end{aligned}$$

which indicates the integral (3.4) is zero. Moreover, it is concluded that

$$\begin{aligned} &\int ^{b}_{\sigma ^{t}(a)}\Biggl[\sum ^{t+1}_{j=0}f_{j}(s) (e_{q_{j}} \varphi )^{\rho ^{t+1-j}\nabla ^{j}}(s)\Biggr]\nabla s \\ &\quad = \int ^{b}_{ \sigma ^{t+1}(a)}\Biggl\{ \sum ^{t-1}_{j=0}f_{j}(s)\cdot \bigl[(e_{q_{j}}\varphi )^{ \rho}\bigr]^{\rho ^{t-j}\nabla ^{j}}(s) \\ &\qquad {}+ f_{t}(s)\cdot (e_{q_{t}}\cdot \varphi )^{\rho \nabla ^{t}}(s)-f_{t+1}^{ \nabla}(s) \cdot \biggl(\frac{1}{b_{1}}\biggr)^{t} (e_{q_{t+1}}\cdot \varphi )^{ \rho \nabla ^{t}}(s) \Biggr\} \nabla s. \end{aligned}$$

By Lemma 3.1 and Lemma 3.2, it is easy to get

$$\begin{aligned}& \varphi ^{\rho}\bigl(\sigma ^{t+1}(a)\bigr)=\varphi \bigl(\sigma ^{t}(a)\bigr)=0, \\& \bigl(\varphi ^{\rho}\bigr)^{\nabla}\bigl(\sigma ^{t+1}(a)\bigr)=b_{1}\varphi ^{\nabla}\bigl( \sigma ^{t}(a)\bigr)=0, \\& \vdots \\& \bigl(\varphi ^{\rho}\bigr)^{\nabla ^{t-1}}\bigl(\sigma ^{t+1}(a)\bigr)=(b_{1})^{t-1} \varphi ^{\nabla ^{t-1}} \bigl(\sigma ^{t}(a)\bigr)=0, \end{aligned}$$

and

$$\begin{aligned}& \varphi ^{\rho}(b)=0, \\& \bigl(\varphi ^{\rho}\bigr)^{\nabla}(b)=0, \\& \vdots \\& \bigl(\varphi ^{\rho}\bigr)^{\nabla ^{t-1}}(b)=0. \end{aligned}$$

According to the mathematical inductive method, it is led to the conclusion that

$$\begin{aligned} &\sum^{t-1}_{j=0}(-1)^{j}\biggl( \frac{1}{b_{1}}\biggr)^{\frac{j(j-1)}{2}}f^{ \nabla ^{j}}_{j}(s)\cdot e^{\rho ^{t+1}}_{q_{j}}(s) \\ &\quad {}+(-1)^{t}\biggl( \frac{1}{b_{1}} \biggr)^{\frac{t(t-1)}{2}}\biggl\{ f_{t}^{\nabla ^{t}}(s)\cdot e_{q_{t}}^{ \rho ^{t+1}}-\biggl[f_{t+1}^{\nabla}(s) \biggl( \frac{1}{b_{1}}\biggr)^{t}\biggr]^{\nabla ^{t}} \cdot e_{q_{t+1}}^{\rho ^{t+1}}\biggr\} =0, \end{aligned}$$

$s\in [a,b]_{\mathcal{K}^{t+1}}$, which is equivalent to

$$ \sum^{t+1}_{j=0}(-1)^{j}\biggl( \frac{1}{b_{1}}\biggr)^{\frac{j(j-1)}{2}}f^{ \nabla ^{j}}_{j}(s)\cdot e^{\rho ^{t+1}}_{q_{j}}(s)=0,\quad s\in [a,b]_{ \mathcal{K}^{t+1}}. $$

 □

Theorem 3.2

If $z_{*}$ is a solution of problem (1.1), then $z_{*}$ fulfils the Euler–Lagrange equation

$$\begin{aligned}& \sum^{t}_{j=0}(-1)^{j}\biggl( \frac{1}{b_{1}}\biggr)^{\frac{j(j-1)}{2}}F^{\nabla ^{j}}_{v_{j}} \bigl(s,(e_{q_{0}}z)^{ \rho ^{t}}(s),(e_{q_{1}}z)^{\rho ^{t-1}\nabla}(s), \ldots ,(e_{q_{t-1}}z)^{ \rho \nabla ^{t-1}}(s),(e_{q_{t}}z)^{\nabla ^{t}}(s) \bigr)\cdot e^{\rho ^{t}}_{q_{j}} \\& \quad =0, \end{aligned}$$

for $s \in [a,b]_{\mathcal{K}^{2t}}$.

Proof

Suppose that $z_{*}$ is a weak extremum point for problem (1.1). Let $\varphi , \varphi ^{\nabla}, \ldots , \varphi ^{\nabla ^{t-1}}$ vanish at $\sigma ^{t-1}(a)$ and b. Setting $\Upsilon (\xi )\triangleq M[z_{*}+\xi \varphi ]$, it is obvious that ϒ has an extremum point at $\xi =0$, and therefore,

$$ \Upsilon ^{\prime }(\xi )=0. $$

Since

$$\begin{aligned} \Upsilon (\xi )= {}& M[z_{*}+\xi \varphi ] \\ = {}& \int ^{b}_{\sigma ^{t-1}(a)}F\bigl(s,(e_{q_{0}}z_{*})^{\rho ^{t}} (s)+ \xi (e_{q_{0}}\varphi )^{\rho ^{t}} (s), (e_{q_{1}}z_{*})^{\rho ^{t-1} \nabla}(s)+ \xi (e_{q_{1}}\varphi )^{\rho ^{t-1}\nabla}(s), \\ & {}\ldots , (e_{q_{t-1}}z_{*})^{\rho \nabla ^{t-1}} (s)+\xi (e_{q_{t-1}} \varphi )^{\rho \nabla ^{t-1}} (s), (e_{q_{t}}z_{*})^{\nabla ^{t}}(s)+ \xi (e_{q_{t}}\varphi )^{\nabla ^{t}}(s) \bigr)\nabla s, \end{aligned}$$

then

$$\begin{aligned} &\Upsilon ^{\prime }(0)=0 \\ &\quad \Leftrightarrow\quad \int ^{b}_{\sigma ^{t-1}(a)}\Biggl[\sum ^{t}_{j=0}F_{v_{j}}(\cdot ) \cdot (e_{q_{j}}\varphi )^{\rho ^{t-j}\nabla ^{j}}(s)\Biggr]\nabla s=0, \end{aligned}$$

where $F_{v_{j}}$ denotes the partial derivative of $F(s,v_{0},v_{1},\ldots ,v_{t})$ with respect to $v_{j}$ and we write, to keep it simple,

$$ (\cdot )=\bigl(s,(e_{q_{0}}z_{*})^{\rho ^{t}}(s),(e_{q_{1}}z_{*})^{\rho ^{t-1} \nabla}(s), \ldots , (e_{q_{t-1}}z_{*})^{\rho \nabla ^{t-1}}(s),(e_{q_{t}}z_{*})^{ \nabla ^{t}}(s) \bigr). $$

It is concluded by Lemma 3.4 that

$$ \sum^{t}_{j=0}(-1)^{j}\biggl( \frac{1}{b_{1}}\biggr)^{\frac{j(j-1)}{2}}F^{\nabla ^{j}}_{v_{j}} (\cdot ) \cdot e^{\rho ^{t}}_{q_{j}}=0, $$

$s \in ([a,b]_{\mathcal{K}^{t}})_{\mathcal{K}^{t}} = [a,b]_{\mathcal{K}^{2t}}$, and the intended result is proved. □

4 Conclusions

We introduce exponential functions and their nabla derivatives into first-order and high-order univariant variational problems, and extend the application of variational approach to the optimization problem on time scales. The first-order variational problems are two special cases of problem (1.1), and Euler–Lagrange equations for first-order variational problems are certificated by two different methods, and utilizing Lemma 15 of [10] to prove Theorem 2.2 is a new method. Furthermore, to address high-order variational problem (1.1), we demonstrate Leibniz Formula with nabla derivatives and four lemmas by taking advantage of mathematical induction and Leibniz Formula. The future research direction is to consider high-order multivariate variational problems, which will provide theoretical support for a wider range of extreme value applications.