Maximum Principle for Nonlinear Fractional Differential Equations with the Hilfer Derivative

Abstract

In this paper, two significant inequalities for the Hilfer fractional derivative of a function in the space $A{C}^{\gamma }([0,b],{\mathbb{R}}^n)$, $0\le \gamma \le 1$ are obtained. We first verified the extremum principle for the Hilfer fractional derivative. In addition, we estimated the Hilfer derivative of a function at its extreme points. Furthermore, we derived and proved a maximum principle for a nonlinear Hilfer fractional differential equation. Finally, we analyzed the solutions of a nonlinear Hilfer fractional differential equation. Our results generalize and extend some obtained theorems on this topic.

1. Introduction

Recently, the subject of fractional calculus has been extensively developed and studied by many academics due to its applications in numerous branches of applied sciences, technology, and engineering. Examples include engineering [1], ecology [2], biology [3], medicine [4], chemistry [5], animal science [6], finance [7], control theory [8], and other branches. In recent decades, Hilfer defined a generalized fractional derivative of Riemann–Liouville (R-L), which includes fractional derivatives of Caputo and R-L [9]. Later on, this derivative became known as the Hilfer fractional derivative and has been used widely in modeling in many fields of science, see, e.g., [10,11,12,13,14,15].

In recent years, maximum principles have been regarded as one of a few efficient ways for obtaining preliminary information about solutions of ordinary or partial differential equations without having a good understanding of the solutions themselves. Furthermore, the maximum principle is considered to be an essential analytical tool in the study of existence and uniqueness results for linear and nonlinear fractional differential equations, we refer to the books [16,17] and survey [18].

Numerous academics have recently focused on the formulation and implementation of maximum principles for diffusion equations. In [19], Luchko proposed and proved for the first time a maximum principle for a generalized fractional diffusion equation with the Caputo derivative over an open bounded domain ${\mathsf{\Omega }}_{\mathcal{T}}:=G\times (0,T),G\in {\mathbb{R}}^n$ in the form:

$$\begin{array}{c}\hfill {{}^{c}D}_{0^{+}}^{\rho }y(\nu ,\varrho )=\mathsf{\Delta }y(\nu ,\varrho )+\mathcal{F}(\nu ,\varrho ),(\nu ,\varrho )\in {\mathsf{\Omega }}_{\mathcal{T}},\end{array}$$

with the conditions

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{y}_{\varrho =0}=y_0\left(x\right),\hfill & x\in \overline{G},\hfill \\ y_S=v(x,\varrho ),\hfill & (x,\varrho )\in S\times [0,\mathcal{T}],\hfill \end{array}\right.\end{array}$$

where ${{}^{c}D}_{0^{+}}^{\rho }$ is the Caputo derivative of order $0<\rho \le 1$, $\mathcal{F}$ is a smooth function, S is open and bounded in ${\mathcal{R}}^n$, and $\mathsf{\Delta }y(\nu ,\varrho )$ is in fact a linear elliptic differential operator of the second order. Luchko [20] demonstrated the existence and uniqueness of solutions for initial-boundary value problems of the multidimensional time fractional diffusion equation, and also formulated and discussed the maximum principle for the generalized time fractional diffusion equations, including the multi-term diffusion equation and the diffusion equation of distributed order [21]. Al-Refai [22] applied the maximum principle to Caputo fractional boundary value problems of order $1<\rho <2$. Ye et al. [23] presented a maximum principle for multi-term time-space fractional differential equations utilizing the modified Riesz space-fractional derivative. In [24], Al-Refai and Luchko obtained weak and strong maximum principles for one-dimensional linear and non-linear fractional diffusion equations with the R-L time-fractional derivative

$${{}^{RL}D}_{0^{+}}^{\rho }y(\nu ,\varrho )=\mathsf{\Delta }y(\nu ,\varrho )+\mathcal{F}(\nu ,\varrho ),(\nu ,\varrho )\in (0,l)\times (0,\mathcal{T}],$$

where the initial and boundary conditions are the sides of a given rectangle, ${{}^{RL}D}_{0^{+}}^{\rho }$ is the R-L derivative of order $0<\rho \le 1$, and $\mathsf{\Delta }y(\nu ,\varrho )$ is a second order differential equation. Al-Refai and Luchko [25] investigated the initial boundary value problems for linear and nonlinear multi-term fraction diffusion equations with the R-L derivative. Cao et al. [26] proposed maximum and minimum principles for time-fractional Caputo–Katugampola diffusion equations. In [27], Al-Refai and Pal developed a maximum principle for a general fractional linear boundary value problem with a Caputo–Fabriaio fractional derivative. Al-Refai [28] established and verified weak and strong maximum principles for fractional nonlinear differential equations with R-L fractional derivatives of order $0<\rho <1$. Al-Refai and Baleanu [29] derived and proved a comparison principle for linear fractional differential equations via a non-local derivative in the form:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{{}^{NL}D}_{a,b}^{\rho }y\left(\varrho \right)=\mathcal{F}(y,\varrho ),\hfill & \varrho \in (a,b),\hfill \\ {{}^{NL}D}_{a,b}^{\rho }y\left(\varrho \right)+w\left(\varrho \right)y\left(\varrho \right)=r\left(\varrho \right),\hfill & \varrho \in (a,b),\hfill \\ y\left(a\right)=y_a,\hfill \\ y\left(b\right)=y_b,,\hfill \end{array}\right.\end{array}$$

where ${{}^{NL}D}_{a,b}^{\rho }$ is the non-local fractional derivative of order $0<\rho <1$ and $w,r\in C[a,b]$. Luchko et al. [30] proved a maximum principle for the general multi-term space-time fractional transport equation.

Kamocki [31] presented a new method of expressing the Hilfer derivative, which is defined as a modified R-L derivative, and then proved the importance of the expression in practical applications of fractional differential equations. Inspired by the above discussion of maximum principles, we shall formulate and prove an extremum principle for the Hilfer derivative as well as a maximum principle for the nonlinear Hilfer fractional system. In this paper, we consider the following nonlinear fractional system involving a Hilfer derivative of the form:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{{}^{H}D}_{0^{+}}^{\rho ,\eta }y(\nu ,\varrho )=\mathsf{\Delta }y(\nu ,\varrho )+\mathcal{F}(\nu ,\varrho ),\hfill & (\nu ,\varrho )\in (0,\pi )\times (0,\mathcal{T}]:={\mathsf{\Omega }}_{\mathcal{T}},\hfill \\ y(0,\varrho )=y(\pi ,\varrho )=0,\hfill & \varrho \in (0,\mathcal{T}],\hfill \\ I_{0^{+}}^{1-\gamma }y(\nu ,0)=y_0,\hfill & \nu \in [0,\pi ],\hfill \end{array}\right.\end{array}$$

(1)

where ${{}^{H}D}_{0^{+}}^{\rho ,\eta }$ stands for the Hilfer fractional derivative of order $0<\rho <1$ and type $0\le \eta \le 1$, the operator $\mathsf{\Delta }y(\nu ,\varrho )$ stands for a second order differential operator, the nonlinear term $\mathcal{F}(\nu ,\varrho ):{\mathsf{\Omega }}_{\mathcal{T}}\to {\mathbb{R}}^n$, and $\gamma =\rho +\eta -\rho \eta$, $0<\gamma \le 1$. We assume the operator $\mathsf{\Delta }y(\nu ,\varrho )$, defined as

$$\mathsf{\Delta }y(\nu ,\varrho )=p(\nu ,\varrho )y_{\nu \nu }(\nu ,\varrho )+q(\nu ,\varrho )y_{\nu }(\nu ,\varrho )+r(\nu ,\varrho )y(\nu ,\varrho ),(\nu ,\varrho )\in {\mathsf{\Omega }}_{\mathcal{T}}.$$

(2)

We assume the functions $p(\nu ,\varrho )$, $q(\nu ,\varrho )$, and $r(\nu ,\varrho )$ are continuous functions on ${\overline{\mathsf{\Omega }}}_b=[0,\pi ]\times [0,\mathcal{T}]$, and $p(\nu ,\varrho )>0$, $r(\nu ,\varrho )<0$ for $(\nu ,\varrho )\in {\overline{\mathsf{\Omega }}}_{\mathcal{T}}$.

This paper is organized as follows: some essential definitions are introduced in Section 2. In Section 3, we estimate the Hilfer fractional derivative of a function $f\in A{C}^{\gamma }([0,\mathcal{T}],{\mathbb{R}}^n)$ at its extreme points. Moreover, maximum and minimum principles are established and applied to system (1). Section 4 is devoted to the conclusion.

2. Preliminaries

In this section, we will review the R-L fractional integral and derivative and the Hilfer fractional derivative, all of which are used in the proof of the theorems in this paper.

We begin by introducing the R-L fractional integral and the R-L and Hilfer derivatives.

Definition 1

(see [14]). The R-L fractional integral of order $\rho >0$ for a function $f:[0,+\infty )\to \mathbb{R}$ is given by

$$I_{0^{+}}^{\rho }f\left(\varrho \right)=\frac{1}{\mathsf{\Gamma }\left(\rho \right)}{\int }_0^{\varrho }{(\varrho -s)}^{\rho -1}f\left(s\right)\mathrm{d}s.$$

Definition 2

(see [32]). The R-L fractional derivative of order $0<\rho <1$ for a function $f\in C^m([0,+\infty ),\mathbb{R})$ is given by

$${{}^{RL}D}_{0^{+}}^{\rho }f\left(\varrho \right)=\frac{1}{\mathsf{\Gamma }(1-\rho )}\frac{d^m}{d{\varrho }^m}{\int }_0^{\varrho }{(\varrho -s)}^{-\rho }f\left(s\right)\mathrm{d}s.$$

Definition 3

(see [15]). The Hilfer fractional derivative (left-sided) ${{}^{H}D}_{a^{+}}^{\rho ,\eta }$ of order $0<\rho <1$, type $0\le \eta \le 1$ for a function $f\in L^1([0,\mathcal{T}],{\mathbb{R}}^n)$ is defined as

$$\begin{array}{cc}\hfill {{}^{H}D}_{0^{+}}^{\rho ,\eta }f\left(\varrho \right)=& \left(I_{0^{+}}^{\eta (1-\rho )}D\left(I_{0^{+}}^{(1-\eta )(1-\rho )}f\right)\right)\left(\varrho \right)\hfill \\ \hfill =& \left(I_{0^{+}}^{\eta (1-\rho )}D{I}_{0^{+}}^{1-\gamma }f\right)\left(\varrho \right)\hfill \\ \hfill =& \left(I_{0^{+}}^{\eta (1-\rho )}{{}^{RL}D}_{0^{+}}^{\gamma }f\right)\left(\varrho \right),\hfill \end{array}$$

where $D:=\frac{\mathrm{d}}{\mathrm{d}\varrho }$ and $\gamma =\rho +\eta -\rho \eta$.

3. Maximum Principle

In this section, we first present some remarks that are helpful in the proofs of our theorems. Moreover, to show that a function in the representation (3) has maximum points, two examples are presented. Then, we estimate the Hilfer derivative of a function in the space $f\in A{C}^{\gamma }([0,\mathcal{T}],{\mathbb{R}}^n)$, $0\le \gamma \le 1$ at its extreme points. Finally, we derive and prove the maximum principle for system (1).

Remark 1.

We denote $A{C}^{\mu }([0,\mathcal{T}],{\mathbb{R}}^n)$, $\mu \in (0,1)$, in short, $A{C}^{\mu }$ is the set of all functions $f:[0,\mathcal{T}]\to {\mathbb{R}}^n$ that have the following representation,

$$f\left(\varrho \right)=\frac{c}{\mathsf{\Gamma }\left(\mu \right)}{\varrho }^{\mu -1}+\left(I_{0^{+}}^{\mu }\psi \right)\left(\varrho \right),a.e.\varrho \in [0,\mathcal{T}],$$

(3)

where $c\in {\mathbb{R}}^n$ and $\psi \in L^1([0,\mathcal{T}],{\mathbb{R}}^n)$. The integral representation (3) was given in [14,31].

Remark 2.

In the following examples, we show that the function f in Remark 1 has a maximum value at the maximum point ${\varrho }_0\in (0,\mathcal{T}]$. Let us choose $\mathcal{T}=1$.

Example 1.

Let $\psi \left(\varrho \right)=1$ for all $\varrho \in (0,1]$, and $c=-1$. Then, by applying the convolution integrals to $\psi \left(\varrho \right)$ and $g_{\mu }\left(\varrho \right)=\frac{{\varrho }^{\mu -1}}{\mathsf{\Gamma }\left(\mu \right)}$, $\varrho \in (0,1]$, $0<\mu <1$, we have

$$f\left(\varrho \right)=\frac{-1}{\mathsf{\Gamma }\left(\mu \right)}{\varrho }^{\mu -1}+g_{1+\mu }\left(\varrho \right)=\frac{{\varrho }^{\mu }}{\mathsf{\Gamma }(1+\mu )}-\frac{{\varrho }^{\mu -1}}{\mathsf{\Gamma }\left(\mu \right)}.$$

Since f is non-decreasing function on $(0,1]$, then one can easily see that f attains its maximum at the right boundary of $(0,1]$, that is, $f_{max}=f\left(1\right)=\frac{1-\mu }{\mathsf{\Gamma }(1+\mu )}.$

Example 2.

Let $\psi \left(\varrho \right)=2\varrho -e^{\varrho }$ for all $\varrho \in (0,1]$, and $c=-1$, we have

$$f\left(\varrho \right)=I_{0^{+}}^{\mu }(2\varrho -e^{\varrho })-\frac{1}{\mathsf{\Gamma }\left(\mu \right)}{\varrho }^{\mu -1}.$$

(4)

Due to difficulty in obtaining the exact values of the function f in (4) manually, particularly the term $I_{0^{+}}^{\mu }{e}^{\varrho }$, we have computed the maximum point numerically by using a MATLAB tool called Fractional Trapezoidal Formula to obtain that f attains its maximum point in $(0,1]$.

Remark 3.

We note that the function f in (3) might have no maximum points when $c>0$. Thus, we assume that the initial point $c=y_0<0$, where $y_0$ is the initial state of system (1). However, we later will impose a condition on $y_0$ in the proof of our maximum and minimum principles, that is, $y_0<0$ and $y_0>0$, respectively.

Next, we present the new expression of the Hilfer derivative that was presented in [31].

Remark 4

(see [31]). Let $f\in L^1([0,\mathcal{T}],{\mathbb{R}}^n)$, $\mu \in (0,1)$, then the function f has the left-sided R-L derivative of order μ if and only if $f\in A{C}^{\mu }([0,\mathcal{T}];{\mathbb{R}}^n)$. Moreover, for $\psi \in L^1([0,\mathcal{T}];{\mathbb{R}}^n)$, $c\in {\mathbb{R}}^n$, we have

$$\left({{}^{RL}D}_{0^{+}}^{\mu }f\right)\left(\varrho \right)=\psi \left(\varrho \right),a.e.\varrho \in [0,\mathcal{T}],and\left(I_{0^{+}}^{1-\mu }f\right)\left(0\right)=c.$$

Remark 5

(see [31]). Let $f\in L^1([0,\mathcal{T}];{\mathbb{R}}^n)$, $\gamma =\rho +\eta -\rho \eta$, $0<\gamma \le 1$, $0<\rho <1$, $0<\eta \le 1$, f has the left-sided Hilfer fractional derivative of order ρ and type η if and only if $f\in A{C}^{\gamma }([0,\mathcal{T}],{\mathbb{R}}^n)$. Moreover,

$$\left({{}^{H}D}_{0^{+}}^{\rho ,\eta }f\right)\left(\varrho \right)=\left(I_{0^{+}}^{\eta (1-\rho )}\psi \right)\left(\varrho \right),a.e.\varrho \in [0,\mathcal{T}]and\left(I_{0^{+}}^{1-\gamma }f\right)\left(0\right)=c,c\in {\mathbb{R}}^n.$$

(5)

Lemma 1

(see [31]). Let $f\in A{C}^{\gamma }([0,\mathcal{T}],{\mathbb{R}}^n)$, then f has the left-sided Hilfer fractional derivative of order ρ and type η as follows:

$$\left({{}^{H}D}_{0^{+}}^{\rho ,\eta }f\right)\left(\varrho \right)=\left({{}^{RL}D}_{0^{+}}^{\rho }f\right)\left(\varrho \right)-\frac{\left(I_{0^{+}}^{1-\gamma }f\right)\left(0\right)}{\mathsf{\Gamma }(\gamma -\rho )}{\varrho }^{\gamma -\rho -1},a.e.\varrho \in [0,\mathcal{T}].$$

Proof of Lemma 1.

The proof was given in [31], which is summarized here because it is required for establishing Theorem 1. Let $f\in A{C}^{\gamma }([0,\mathcal{T}],{\mathbb{R}}^n)$ be introduced as the same as in Remark 1 with $\mu =\gamma$. Then, $c=\left(I_{0^{+}}^{1-\gamma }f\right)\left(0\right)$ and $\psi \left(\varrho \right)=\frac{d}{d\varrho }$$\left(I_{0^{+}}^{1-\gamma }f\right)\left(\varrho \right)$, $I_{0^{+}}^{1-\gamma }f\in A{C}^{\gamma }([0,\mathcal{T}],{\mathbb{R}}^n)$, we get

$$\left(I_{0^{+}}^{1-\gamma }f\right)\left(\varrho \right)=\left(I_{0^{+}}^{1-\gamma }f\right)\left(0\right)+{\int }_0^{\varrho }\frac{d}{ds}\left. [\left(I_{0^{+}}^{1-\gamma }f\right)\left(\varrho \right)\right]\mathrm{d}\varrho ,\varrho \in [0,\mathcal{T}].$$

Thus, we get

$${\int }_0^{\varrho }\frac{d}{d\varrho }\left. [\left(I_{0^{+}}^{1-\gamma }f\right)\left(\varrho \right)\right]\mathrm{d}\varrho =\left(I_{0^{+}}^{1-\gamma }f\right)\left(\varrho \right)-\left(I_{0^{+}}^{1-\gamma }f\right)\left(0\right).$$

Now, utilizing Definition 3 and the semigroup property, we have

$$\begin{array}{cc}\hfill \left({{}^{H}D}_{0^{+}}^{\rho ,\eta }f\right)\left(\varrho \right)=& \left(I_{0^{+}}^{\eta (1-\rho )}D{I}_{0^{+}}^{1-\gamma }f\right)\left(\varrho \right)\hfill \\ \hfill =& \left(D{I}_{0^{+}}^{\eta (1-\rho )}{I}_{0^{+}}^{1}D{I}_{0^{+}}^{1-\gamma }f\right)\left(\varrho \right)\hfill \\ \hfill =& D{I}_{0^{+}}^{\eta (1-\rho )}\left(I_{0^{+}}^{1-\gamma }f-\left(I_{0^{+}}^{1-\gamma }f\right)\left(0\right)\right)\left(\varrho \right)\hfill \\ \hfill =& \left({{}^{RL}D}_{0^{+}}^{\rho }f\right)\left(\varrho \right)-\frac{\left(I_{0^{+}}^{1-\gamma }f\right)\left(0\right)}{\mathsf{\Gamma }(\gamma -\rho )}{\varrho }^{\gamma -\rho -1}.\hfill \end{array}$$

□

In the following, let us recall the extremum principle for the R-L derivative.

Lemma 2

(see [24]). The R-L derivative of the function $f\in A{C}^{\gamma }([0,\mathcal{T}],{\mathbb{R}}^n)$ at the maximum point ${\varrho }_0\in (0,\mathcal{T}]$ satisfies the following inequality

$$\left({{}^{RL}D}_{0^{+}}^{\rho }f\right)\left({\varrho }_0\right)\ge \frac{{\varrho }_0^{-\rho }}{\mathsf{\Gamma }(1-\rho )}f\left({\varrho }_0\right).$$

Proof of Lemma 2.

The proof is almost identical to that in [24]; thus, it is omitted here. □

Now we are able to demonstrate the extremum principle for the Hilfer derivative and our maximum principle as well.

Theorem 1

(Extremum principle). Let $f\in A{C}^{\gamma }([0,\mathcal{T}],{\mathbb{R}}^n)$ attain its maximum at a point ${\varrho }_0\in (0,\mathcal{T}]$, $\gamma =\rho +\eta -\rho \eta$, $0<\gamma \le 1$, $0<\rho <1$, $0\le \eta \le 1$, $\gamma \ge \rho$, $\gamma >\eta$ and $1-\gamma <1-\eta (1-\rho )$. Then, the following inequality holds,

$$\left({{}^{H}D}_{0^{+}}^{\rho ,\eta }f\right)\left({\varrho }_0\right)\ge \frac{{\varrho }_0^{-\rho }}{\mathsf{\Gamma }(1-\rho )}f\left({\varrho }_0\right)-\frac{\left(I_{0^{+}}^{1-\gamma }f\right)\left(0\right)}{\mathsf{\Gamma }(\gamma -\rho )}{\varrho }_0^{\gamma -\rho -1}.$$

Proof of Theorem 1.

It follows from Lemmas 1 and 2 that

$$\begin{array}{cc}\hfill \left({{}^{H}D}_{0^{+}}^{\rho ,\eta }f\right)\left({\varrho }_0\right)=& \left({{}^{RL}D}_{0^{+}}^{\rho }f\right)\left({\varrho }_0\right)-\frac{\left(I_{0^{+}}^{1-\gamma }f\right)\left(0\right)}{\mathsf{\Gamma }(\gamma -\rho )}{\varrho }_0^{\gamma -\rho -1}\hfill \\ \hfill \ge & \frac{{\varrho }_0^{-\rho }}{\mathsf{\Gamma }(1-\rho )}f\left({\varrho }_0\right)-\frac{\left(I_{0^{+}}^{1-\gamma }f\right)\left(0\right)}{\mathsf{\Gamma }(\gamma -\rho )}{\varrho }_0^{\gamma -\rho -1}.\hfill \end{array}$$

□

Now, we will demonstrate our maximum principle.

Theorem 2

(Maximum principle). Let $y(\nu ,\varrho )\in A{C}^{\gamma }([0,\mathcal{T}],{\mathbb{R}}^n)$ such that $y_0<0$ and

$$P_{\rho ,\eta }y(\nu ,\varrho )={{}^{H}D}_{0^{+}}^{\rho ,\eta }y(\nu ,\varrho )-\mathsf{\Delta }y(\nu ,\varrho )\le 0,\forall (\nu ,\varrho )\in {\mathsf{\Omega }}_{\mathcal{T}}.$$

Then, we get

$$\underset{(\nu ,\varrho )\in {\overline{\mathsf{\Omega }}}_{\mathcal{T}}}{max}y(\nu ,\varrho )\le max\left\{\underset{(\nu ,\varrho )\in \partial {\mathsf{\Omega }}_{\mathcal{T}}}{max}y(\nu ,\varrho ),0\right\},$$

(6)

where $\partial {\mathsf{\Omega }}_{\mathcal{T}}$ is the boundary of the domain ${\mathsf{\Omega }}_{\mathcal{T}}$.

Proof of Theorem2.

If y attains its maximum at the boundary $\partial {\mathsf{\Omega }}_{\mathcal{T}}$, then the inequality (6) holds. On the other hand, we will consider the case where y attains its maximum in the interior of ${\mathsf{\Omega }}_{\mathcal{T}}$.

Let us assume y attains its positive maximum, which is $y({\nu }_0,{\varrho }_0)$, $({\nu }_0,{\varrho }_0)\in {\mathsf{\Omega }}_{\mathcal{T}}$, then we have the following inequality

$$y({\nu }_0,{\varrho }_0)>max\left\{\underset{(\nu ,\varrho )\in \partial {\mathsf{\Omega }}_{\mathcal{T}}}{max}y(\nu ,\varrho ),0\right\}=M\ge 0.$$

Assume $0<ϵ:=y({\nu }_0,{\varrho }_0)-M$. Since $y_{\nu }({\nu }_0,{\varrho }_0)=0$, $y_{\nu \nu }({\nu }_0,{\varrho }_0)\le 0$, $p({\nu }_0,{\varrho }_0)>0$, and $r({\nu }_0,{\varrho }_0)<0$, we have

$$\mathsf{\Delta }y({\nu }_0,{\varrho }_0)=p({\nu }_0,{\varrho }_0)y_{\nu \nu }({\nu }_0,{\varrho }_0)+q({\nu }_0,{\varrho }_0)y_{\nu }({\nu }_0,{\varrho }_0)+r({\nu }_0,{\varrho }_0)y({\nu }_0,{\varrho }_0).$$

(7)

Suppose that

$$w(\nu ,\varrho )=y(\nu ,\varrho )+\frac{ϵ}{2}\frac{(\mathcal{T}-\varrho )}{\mathcal{T}},(\nu ,\varrho )\in {\overline{\mathsf{\Omega }}}_{\mathcal{T}},$$

(8)

we can easily get

$$w_{\nu }(\nu ,\varrho )=y_{\nu }(\nu ,\varrho ),\mathrm{and}{w}_{\nu \nu }(\nu ,\varrho )=y_{\nu \nu }(\nu ,\varrho ),(\nu ,\varrho )\in {\overline{\mathsf{\Omega }}}_{\mathcal{T}}.$$

(9)

Also, we obtain

$$w(\nu ,\varrho )\le y(\nu ,\varrho )+\frac{ϵ}{2},(\nu ,\varrho )\in {\overline{\mathsf{\Omega }}}_{\mathcal{T}}.$$

Then,

$$w({\nu }_0,{\varrho }_0)>y({\nu }_0,{\varrho }_0)=M+ϵ\ge ϵ+y(\nu ,\varrho )\ge w(\nu ,\varrho )+\frac{ϵ}{2},(\nu ,\varrho )\in \partial {\mathsf{\Omega }}_{\mathcal{T}}.$$

Thus, w cannot attain its maximum at $\partial {\mathsf{\Omega }}_{\mathcal{T}}$. Therefore, we presume w attains its maximum at the point $({\nu }_1,{\varrho }_1)\in {\mathsf{\Omega }}_{\mathcal{T}}$, then one can see that

$$w({\nu }_1,{\varrho }_1)\ge w({\nu }_0,{\varrho }_0)>M+ϵ\ge ϵ>0.$$

We derive the Hilfer derivative for Equation (8) as follows:

$$\begin{array}{cc}\hfill {{}^{H}D}_{0^{+}}^{\rho ,\eta }w(\nu ,\varrho )& ={{}^{H}D}_{0^{+}}^{\rho ,\eta }y(\nu ,\varrho )+{{}^{H}D}_{0^{+}}^{\rho ,\eta }\frac{ϵ}{2}\frac{(\mathcal{T}-\varrho )}{\mathcal{T}}\hfill \\ \hfill & ={{}^{H}D}_{0^{+}}^{\rho ,\eta }y(\nu ,\varrho )+\frac{ϵ}{2\mathsf{\Gamma }(1-\rho )}{\varrho }^{-\rho }-\frac{ϵ}{2\mathcal{T}}\frac{{\varrho }^{1-\rho }}{\mathsf{\Gamma }(2-\rho )}\hfill \\ \hfill & ={{}^{H}D}_{0^{+}}^{\rho ,\eta }y(\nu ,\varrho )-\frac{ϵ{\varrho }^{-\rho }}{2\mathsf{\Gamma }(1-\rho )}\left(\frac{\varrho }{\mathcal{T}(1-\rho )}-1\right).\hfill \end{array}$$

(10)

In view of the semigroup property and applying the convolution to Equation (8), we have

$$\begin{array}{cc}\hfill {\left.I_{0^{+}}^{1-\gamma }w(\nu ,\varrho )\right|}_{\varrho =0}& ={\left.I_{0^{+}}^{1-\gamma }\left(y(\nu ,\varrho )+\frac{ϵ}{2}\frac{(\mathcal{T}-\varrho )}{\mathcal{T}}\right)\left(\varrho \right)\right|}_{\varrho =0}\hfill \\ \hfill & ={\left.y_0+I_{0^{+}}^{1-\gamma }\left(\frac{ϵ}{2}\frac{(\mathcal{T}-\varrho )}{\mathcal{T}}\right)\right|}_{\varrho =0}\hfill \\ \hfill & ={\left.y_0+\left(I_{0^{+}}^{1-\gamma }\frac{ϵ}{2}\right)\left(\varrho \right)\right|}_{\varrho =0}-{\left.\left(I_{0^{+}}^{1-\gamma }\frac{ϵ\varrho }{2\mathcal{T}}\right)\left(\varrho \right)\right|}_{\varrho =0}\hfill \\ \hfill & =y_0.\hfill \end{array}$$

Calculating (10) at the maximum point $({\nu }_1,{\varrho }_1)$ yields

$$\begin{array}{cc}\hfill {{}^{H}D}_{0^{+}}^{\rho ,\eta }y({\nu }_1,{\varrho }_1)& ={{}^{H}D}_{0^{+}}^{\rho ,\eta }w({\nu }_1,{\varrho }_1)+\frac{ϵ{\varrho }_1^{-\rho }}{2\mathsf{\Gamma }(1-\rho )}\left(\frac{{\varrho }_1}{\mathcal{T}(1-\rho )}-1\right).\hfill \end{array}$$

Using Theorem 1 and the second equation of (1), we get

$$\begin{array}{cc}\hfill {{}^{H}D}_{0^{+}}^{\rho ,\eta }y({\nu }_1,{\varrho }_1)\ge & \frac{{\varrho }_1^{-\rho }}{\mathsf{\Gamma }(1-\rho )}w({\nu }_1,{\varrho }_1)-\frac{I_{0^{+}}^{1-\gamma }w({\nu }_1,0)}{\mathsf{\Gamma }(\gamma -\rho )}{\varrho }_1^{\gamma -\rho -1}\hfill \\ \hfill & +\frac{ϵ{\varrho }_1^{-\rho }}{2\mathsf{\Gamma }(1-\rho )}\left(\frac{{\varrho }_1}{\mathcal{T}(1-\rho )}-1\right)\hfill \\ \hfill =& \frac{{\varrho }_1^{-\rho }}{\mathsf{\Gamma }(1-\rho )}w({\nu }_1,{\varrho }_1)-\frac{y_0}{\mathsf{\Gamma }(\gamma -\rho )}{\varrho }_1^{\gamma -\rho -1}\hfill \\ \hfill & +\frac{ϵ{\varrho }_1^{-\rho }}{2\mathsf{\Gamma }(1-\rho )}\left(\frac{{\varrho }_1}{\mathcal{T}(1-\rho )}-1\right)\hfill \\ \hfill \ge & \frac{ϵ{\varrho }_1^{-\rho }}{\mathsf{\Gamma }(1-\rho )}-\frac{y_0}{\mathsf{\Gamma }(\gamma -\rho )}{\varrho }_1^{\gamma -\rho -1}\hfill \\ \hfill & +\frac{ϵ{\varrho }_1^{-\rho }}{2\mathsf{\Gamma }(1-\rho )}\left(\frac{{\varrho }_1}{\mathcal{T}(1-\rho )}-1\right).\hfill \end{array}$$

(11)

On the other hand, we have

$$\begin{array}{cc}\hfill \mathsf{\Delta }y({\nu }_1,{\varrho }_1)=& \mathsf{\Delta }\left(w({\nu }_1,{\varrho }_1)-\frac{ϵ}{2}\frac{(\mathcal{T}-{\varrho }_1)}{\mathcal{T}}\right)\hfill \\ \hfill =& p({\nu }_1,{\varrho }_1)w_{\nu \nu }({\nu }_1,{\varrho }_1)+q({\nu }_1,{\varrho }_1)w_{\nu }({\nu }_1,{\varrho }_1)\hfill \\ \hfill & +r({\nu }_1,{\varrho }_1)\left(w({\nu }_1,{\varrho }_1)-\frac{ϵ}{2}\frac{(\mathcal{T}-{\varrho }_1)}{\mathcal{T}}\right)\hfill \\ \hfill \le & r({\nu }_1,{\varrho }_1)\left(w({\nu }_1,{\varrho }_1)-\frac{ϵ}{2}\frac{(\mathcal{T}-{\varrho }_1)}{\mathcal{T}}\right)\hfill \\ \hfill \le & r({\nu }_1,{\varrho }_1)\left(ϵ-\frac{ϵ}{2}\frac{(\mathcal{T}-{\varrho }_1)}{\mathcal{T}}\right).\hfill \end{array}$$

(12)

Noting that $({\nu }_1,{\varrho }_1)\in {\mathsf{\Omega }}_b$, by using the results in (11) and (12), we have

$$\begin{array}{cc}\hfill P_{\rho ,\eta }y({\nu }_1,{\varrho }_1)=& {{}^{H}D}_{0^{+}}^{\rho ,\eta }y({\nu }_1,{\varrho }_1)-\mathsf{\Delta }y({\nu }_1,{\varrho }_1)\hfill \\ \hfill \ge & \frac{ϵ{\varrho }_1^{-\rho }}{\mathsf{\Gamma }(1-\rho )}-\frac{y_0}{\mathsf{\Gamma }(\gamma -\rho )}{\varrho }_1^{\gamma -\rho -1}+\frac{ϵ{\varrho }_1^{-\rho }}{2\mathsf{\Gamma }(1-\rho )}\left(\frac{{\varrho }_1}{\mathcal{T}(1-\rho )}-1\right)\hfill \\ \hfill & -r({\nu }_1,{\varrho }_1)\left(ϵ-\frac{ϵ}{2}\frac{(\mathcal{T}-{\varrho }_1)}{\mathcal{T}}\right)\hfill \\ \hfill =& \frac{ϵ{\varrho }_1^{-\rho }}{\mathsf{\Gamma }(1-\rho )}-\frac{y_0}{\mathsf{\Gamma }(\gamma -\rho )}{\varrho }_1^{\gamma -\rho -1}+\frac{ϵ{\varrho }_1^{1-\rho }}{2\mathcal{T}\mathsf{\Gamma }(2-\rho )}-\frac{ϵ{\varrho }_1^{-\rho }}{2\mathsf{\Gamma }(1-\rho )}\hfill \\ \hfill & -r({\nu }_1,{\varrho }_1)ϵ+r({\nu }_1,{\varrho }_1)\frac{ϵ}{2}\frac{(\mathcal{T}-{\varrho }_1)}{\mathcal{T}}\hfill \\ \hfill \ge & \frac{ϵ{\varrho }_1^{-\rho }}{2\mathsf{\Gamma }(1-\rho )}-\frac{y_0}{\mathsf{\Gamma }(\gamma -\rho )}{\varrho }_1^{\gamma -\rho -1}+\frac{ϵ{\varrho }_1^{1-\rho }}{2\mathcal{T}\mathsf{\Gamma }(2-\rho )}\hfill \\ \hfill & -r({\nu }_1,{\varrho }_1)ϵ+r({\nu }_1,{\varrho }_1)\frac{ϵ}{2}\frac{(\mathcal{T}-{\varrho }_1)}{\mathcal{T}}\hfill \\ \hfill \ge & \frac{ϵ{\varrho }_1^{-\rho }}{2\mathsf{\Gamma }(1-\rho )}-\frac{y_0}{\mathsf{\Gamma }(\gamma -\rho )}{\varrho }_1^{\gamma -\rho -1}+\frac{ϵ{\varrho }_1^{1-\rho }}{2\mathcal{T}\mathsf{\Gamma }(2-\rho )}-r({\nu }_1,{\varrho }_1)\frac{ϵ}{2}>0,\hfill \end{array}$$

which contradicts the fact that $P_{\rho ,\eta }y(\nu ,\varrho )\le 0$ for all $(\nu ,\varrho )\in {\mathsf{\Omega }}_{\mathcal{T}}$. Consequently, (6) holds. One can note that the last inequality holds only when $y_0<0$ and $r({\nu }_1,{\varrho }_1)<0$. □

Remark 6.

An analogous result that follows from Theorem 2 can be obtained for the Hilfer fractional derivative at minimum points by replacing y with $-y$. Specifically, introducing the minimum principle, this can be illustrated in the following theorem.

Theorem 3

(Minimum principle). Let $y(\nu ,\varrho )\in A{C}^{\gamma }([0,\mathcal{T}],{\mathbb{R}}^n)$ such that $y_0>0$ and

$$P_{\rho ,\eta }y(\nu ,\varrho )={{}^{H}D}_{0^{+}}^{\rho ,\eta }y(\nu ,\varrho )-\mathsf{\Delta }y(\nu ,\varrho )\ge 0,\forall (\nu ,\varrho )\in {\mathsf{\Omega }}_{\mathcal{T}}.$$

Then,

$$\underset{(\nu ,\varrho )\in {\overline{\mathsf{\Omega }}}_{\mathcal{T}}}{min}y(\nu ,\varrho )\ge min\left\{\underset{(\nu ,\varrho )\in \partial {\mathsf{\Omega }}_{\mathcal{T}}}{min}y(\nu ,\varrho ),0\right\},$$

thus, the minimum principle is established.

The following theorems are the direct results of Theorems 2 and 3.

Theorem 4.

Suppose that $\mathcal{F}(\nu ,\varrho )\le 0$ for every $(\nu ,\varrho )\in {\mathsf{\Omega }}_{\mathcal{T}}$, $y_0<0$. If $y(\nu ,\varrho )\in A{C}^{\gamma }([0,\mathcal{T}],{\mathbb{R}}^n)$ is a solution of system (1), then

$$y(\nu ,\varrho )\le 0,(\nu ,\varrho )\in {\overline{\mathsf{\Omega }}}_{\mathcal{T}}.$$

Therefore, the nonlinear Hilfer fractional system (1) has no positive solutions in $A{C}^{\gamma }([0,\mathcal{T}],{\mathbb{R}}^n)$.

Proof of Theorem 4.

We can simply obtain the proof using Theorem 2. □

Theorem 5.

Suppose that $\mathcal{F}(\nu ,\varrho )\ge 0$ for any $(\nu ,\varrho )\in {\mathsf{\Omega }}_{\mathcal{T}}$, $y_0>0$. If $y(\nu ,\varrho )\in A{C}^{\gamma }([0,\mathcal{T}],{\mathbb{R}}^n)$ is a solution of system (1), then

$$y(\nu ,\varrho )\ge 0,(\nu ,\varrho )\in {\overline{\mathsf{\Omega }}}_{\mathcal{T}}.$$

Consequently, the nonlinear Hilfer fractional system (1) has no negative solutions in $A{C}^{\gamma }([0,\mathcal{T}],{\mathbb{R}}^n)$.

Proof of Theorem 5.

One can directly obtain the proof of this theorem by applying Theorem 3. □

4. Conclusions

In this paper, we first have proved the extremum principle for the Hilfer fractional derivative. Then, we have derived and proved maximum and minimum principles for a nonlinear fractional differential equation with the Hilfer derivative. Moreover, we have used this maximum principle to prove the existence of the solutions to system (1). For future research, we shall use the results of this paper to investigate existence results for Hilfer fractional integrodifferential equations with fractional Brownian motion.