On an Analogy of Fermat’s Theorem for the 1st Level General Fractional Derivative and Its Applications

1. Introduction

Fermat’s theorem, also known as the interior extrema theorem, states that the first order derivative of a differentiable function vanishes at its local extrema. Despite its simple formulation, this result plays a fundamental role in the derivation of many important findings in both mathematical analysis, differential equations, and optimization. In particular, the Fermat theorem is one of the foundations of the maximum principle that is commonly used to study the qualitative behavior of solutions to boundary- and initial-boundary-value problems for differential equations, including those involving fractional derivatives.

In this paper, we formulate and prove an analogy of Fermat’s theorem for the 1st level general fractional derivative (1st level GFD). This derivative is a very general operator that contains many fractional derivatives in use such as Caputo, Riemann–Liouville, Hilfer, and general fractional derivatives with Sonin kernels among its particular cases. A key observation is that the analogies of the Fermat theorem formulated for the fractional derivatives take the form of inequalities rather than equalities. The exact form of these inequalities varies for different fractional derivatives.

Another important distinction from integer-order derivatives is that, for many fractional derivatives, their values at the local extrema may be both positive and negative. An exception to this behavior is provided by fractional derivatives of Caputo type with the order $\alpha$ between zero and one, which are non-negative at maximum points and non-positive at minimum points. For the first time, a result of this kind was derived in [1]: If a function f attains its maximum over the interval $[0,T]$ at the point $t_0\in (0,T]$ then the inequality

$${(}^C{D}_{0+}^{\alpha }f)\left(t_0\right)\ge 0,0<\alpha <1,$$

(1.1)

holds true, where ${}^{C}D_{0+}^{\alpha }$ is the Caputo fractional derivative of order $\alpha (0<\alpha <1)$ defined by the equation

$${(}^C{D}_{0+}^{\alpha }f)\left(t\right)=\left(I_{0+}^{1-\alpha }{f}^{\prime }\right)\left(t\right),$$

(1.2)

and notation $I_{0+}^{\alpha }$ stands for the Riemann-Liouville fractional integral of the order $\alpha ,\alpha >0$:

$$\left(I_{0+}^{\alpha }f\right)\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-\tau )}^{\alpha -1}f\left(\tau \right)d\tau .$$

(1.3)

Please note that the inequality (1.1) as well as other Fermat-type inequalities mentioned in Introduction were derived for functions from certain spaces of functions. For details concerning these spaces, we refer to the original publications.

In [2], the inequality (1.1) was specified and extended to the case of the Riemann-Liouville fractional derivative: If a function f attains its maximum over the interval $[0,T]$ at the point $t_0\in (0,T]$ then the inequalities

$${(}^C{D}_0^{\alpha }f)\left(t_0\right)\ge {\displaystyle \frac{t_0^{-\alpha }}{\mathsf{\Gamma }(1-\alpha )}}[f\left(t_0\right)-f\left(0\right)]\ge 0,0<\alpha <1,$$

(1.4)

$${(}^{RL}{D}_{0+}^{\alpha }f)\left(t_0\right)\ge {\displaystyle \frac{t_0^{-\alpha }}{\mathsf{\Gamma }(1-\alpha )}}f\left(t_0\right),0<\alpha <1$$

(1.5)

hold true, where ${}^{RL}D_{0+}^{\alpha }$ is the Riemann-Liouville fractional derivative of order $\alpha (0<\alpha <1)$ defined by the equation

$${(}^{RL}{D}_{0+}^{\alpha }f)\left(t\right)={\displaystyle \frac{d}{dt}}\left(I_{0+}^{1-\alpha }f\right)\left(t\right).$$

(1.6)

In [3], these results were extended to the case of the Caputo and Riemann-Liouville fractional derivatives of arbitrary order $\alpha (\alpha >0)$ and in [4], the Fermat theorems for the Caputo and Riemann-Liouville fractional derivatives of order $\alpha (0<\alpha <1)$ were proved under weaker conditions compared to those formulated in [2].

Recently, the so-called general fractional integrals and derivatives with Sonin kernels have been introduced, and related fractional differential equations have been investigated (see, e.g., [5,6,7,8,9]). The functions are called Sonin kernels if they satisfy the Sonin condition ([10]):

$$(\kappa *k)\left(t\right)=1,t>0,$$

(1.7)

where * stands for the Laplace convolution. For a given Sonin kernel $\kappa$ the kernel k is referred to as its associated Sonin kernel.

The general fractional integral (GFI), the general fractional derivative (GFD), and the regularized general fractional derivative (RGFD) with the Sonin kernels $\kappa$ and k are defined as follows, respectively:

$$\left({\mathbb{I}}_{\left(\kappa \right)}f\right)\left(t\right):=(\kappa *f)\left(t\right)={\int }_0^t\kappa (t-\tau )f\left(\tau \right)d\tau ,t>0,$$

(1.8)

$$\left({\mathbb{D}}_{\left(k\right)}f\right)\left(t\right):={\displaystyle \frac{d}{dt}}(k*f)\left(t\right)={\displaystyle \frac{d}{dt}}\left({\mathbb{I}}_{\left(k\right)}f\right)\left(t\right),t>0,$$

(1.9)

$${(}_{*}{\mathbb{D}}_{\left(k\right)}f)\left(t\right):=\left({\mathbb{I}}_{\left(k\right)}{f}^{\prime }\right)\left(t\right),t>0.$$

(1.10)

The GFI (1.8), the GFD (1.9), and the RGFD (1.10) with the power law Sonin kernels

$$\kappa \left(t\right)=h_{\alpha }\left(t\right),k\left(t\right)=h_{1-\alpha }\left(t\right),h_{\alpha }\left(t\right):={\displaystyle \frac{t^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}},0<\alpha <1$$

are reduced to the Riemann-Liouville fractional integral and the Riemann-Liouville and Caputo fractional derivatives of order $\alpha$ ($0<\alpha <1$), respectively. It is also worth mentioning that fractional differential equations with the GFDs and RGFDs have been actively employed for modeling of various physical processes and systems, see, e.g., [11,12,13,14,15].

Fermat-type results for the GFD (1.9) and RGFD (1.10) with Sonin kernels were first established in [7]. If a function f attains its maximum over the interval $[0,T]$ at the point $t_0,t_0\in (0,T]$, then the following inequalities hold true:

$$\left({\mathbb{D}}_{\left(k\right)}f\right)\left(t_0\right)\ge k\left(t_0\right)f\left(t_0\right),$$

(1.11)

$${(}_{*}{\mathbb{D}}_{\left(k\right)}f)\left(t_0\right)\ge k\left(t_0\right)(f\left(t_0\right)-f\left(0\right))\ge 0.$$

(1.12)

In [16], the inequality (1.12) was proved for the kernels k that satisfy some other conditions compared to those assumed in [7].

Finally, we mention that in [17], another Fermat-type result for the GFD (1.9) was derived: under the conditions $f\left(t\right)\ge 0\forall t\in [0,t_0]$ and $f\left(t_0\right)=0forafixed{t}_0\in (0,T)$, the inequality

$$\left({\mathbb{D}}_{\left(k\right)}f\right)\left(t_0\right)\le 0$$

(1.13)

holds true. In this paper, we derive an inequality of type (1.13) for the 1st level general fractional derivative and demonstrate some of its applications.

The rest of the paper is organized as follows. In Section 2, the 1st level general fractional derivative (1st level GFD) and its basic properties are discussed. This fractional derivative is a very general operator that has both the GFD and the RGFD with Sonin kernels among its particular cases. Section 3 contains our main finding in form of an analogy of Fermat’s theorem of type (1.13) for the 1st level GFD. Section 4 deals with some applications of this result including derivation of a comparison principle for fractional differential inequalities involving the 1st level GFDs as well as a priori estimates for solutions of initial-value problems for the fractional differential equations with the 1st level GFDs.

2. Definition and Basic Properties of the 1st Level GFD

The 1st level GFD and the corresponding 1st level GFI with the generalized Sonin kernels were defined and investigated for the first time in [18]. In this section, we recall their definitions and basic properties needed in further discussions.

Let the functions k, $k_1$, and $k_2$ satisfy the generalized Sonin condition

$$(k*k_1*k_2)\left(t\right)=1,t>0,$$

(2.1)

where * stands for the Laplace convolution. These functions will be referred to as the 1st level Sonin kernels.

The 1st level GFD with the 1st level Sonin kernels $k_1$ and $k_2$ is defined by the relation

$${(}^{1L}{\mathbb{D}}_{(k_1,k_2)}f)\left(t\right)=\left({\mathbb{I}}_{\left(k_1\right)}{\displaystyle \frac{d}{dt}}{\mathbb{I}}_{\left(k_2\right)}f\right)\left(t\right),t>0,$$

(2.2)

where ${\mathbb{I}}_{\left(k_1\right)}$ and ${\mathbb{I}}_{\left(k_2\right)}$ are the GFIs with the kernels $k_1$ and $k_2$, respectively.

We note that the 1st level Sonin kernels $k_1$ and $k_2$ also satisfy the Sonin condition (1.7) with the associated kernels $k*k_2$, and $k*k_1$, respectively, i.e., they are also the Sonin kernels.

The 1st level GFI corresponding to the 1st level GFD defined by the equation (2.2) is the GFI with the 1st level Sonin kernel k satisfying the generalized Sonin condition (2.1):

$$\left({\mathbb{I}}_{\left(k\right)}f\right)\left(t\right)=(k*f)\left(t\right)={\int }_0^{t}k(t-\tau )f\left(\tau \right)d\tau ,t>0.$$

(2.3)

Evidently, the 1st level Sonin kernel k is also a Sonin kernel because it satisfies the Sonin condition (1.7) with the associated kernel $k_1*k_2$.

It is worth mentioning that the 1st level GFD (2.2) generalizes both the GFD and the RGFD. Indeed, formally setting ${\mathbb{I}}_{\left(k_1\right)}=Id$ in equation (2.2) ($k_1=\delta$ in the generalized Sonin condition (2.1)) leads to the GFD with the Sonin kernel $k_2$:

$${(}^{1L}{\mathbb{D}}_{(\delta ,k_2)}f)\left(t\right)={\displaystyle \frac{d}{dt}}\left({\mathbb{I}}_{\left(k_2\right)}f\right)\left(t\right)=\left({\mathbb{D}}_{\left(k_2\right)}f\right)\left(t\right),t>0,$$

(2.4)

whereas the case ${\mathbb{I}}_{\left(k_2\right)}=Id$ in equation (2.2) ($k_2=\delta$ in the generalized Sonin condition (2.1)) corresponds to the RGFD with the kernel $k_1$:

$${(}^{1L}{\mathbb{D}}_{(k_1,\delta )}f)\left(t\right)=\left({\mathbb{I}}_{\left(k_1\right)}{\displaystyle \frac{d}{dt}}f\right)\left(t\right)={(}_{*}{\mathbb{D}}_{\left(k_1\right)}f)\left(t\right),t>0.$$

(2.5)

Thus, the 1st level GFD unifies the GFD and the RGFD in one formula. On the other hand, the 1st level GFD is a far reaching generalization of the Hilfer fractional derivative (see, e.g., [19,20]) of order $\alpha \in (0,1)$ and type $\gamma \in (0,1-\alpha )$ in the parametrization introduced in [18]:

$$\left(D_{0+}^{(\alpha ,\gamma )}f\right)\left(t\right)=\left(I_{0+}^{\gamma }{\displaystyle \frac{d}{dt}}{I}_{0+}^{1-\alpha -\gamma }f\right)\left(t\right),t>0.$$

(2.6)

Because the functions

$$k\left(t\right)=h_{\alpha }\left(t\right),k_1\left(t\right)=h_{\gamma }\left(t\right),k_2\left(t\right)=h_{1-\alpha -\gamma }\left(t\right),0<\alpha <1,0<\gamma <1-\alpha ,t>0$$

(2.7)

are the 1st level Sonin kernels, the Hilfer fractional derivative (2.6) is a particular case of the 1st level GFD generated by the power law kernels $k_1$ and $k_2$ defined as in the formula (2.7):

$$\left(D_{0+}^{(\alpha ,\gamma )}f\right)\left(t\right)={(}^{1L}{\mathbb{D}}_{(h_{\gamma },h_{1-\alpha -\gamma })}f)\left(t\right),t>0.$$

(2.8)

In [18], the 1st level GFD and the 1st level GFI were shown to build a kind of generalized fractional calculus (GFC) because they satisfy fractional analogies of the 1^st and the 2^nd fundamental theorems of calculus on appropriately defined functional spaces. The 1^st fundamental theorem of the GFC says that the 1st level GFD (2.2) with the kernels $k_1$ and $k_2$ is a left-inverse operator to the 1st level GFI (2.3) with the kernel k ([18]):

$${(}^{1L}{\mathbb{D}}_{(k_1,k_2)}{\mathbb{I}}_{\left(k\right)}f)\left(t\right)=f\left(t\right),t>0,f\in {\mathbb{I}}_{\left(k_1\right)}\left(C_{-1}(0,T)\right),$$

(2.9)

where

$${\mathbb{I}}_{\left(k\right)}\left(C_{-1}(0,T)\right):=\{f:f\left(t\right)=\left({\mathbb{I}}_{\left(k\right)}\varphi \right)\left(t\right),\varphi \in C_{-1}(0,T)\}$$

(2.10)

and

$$C_{-1}(0,T):=\left\{f\right|f\left(t\right)=t^{p}g\left(t\right),t>0,p>-1,g\in C[0,T]\}.$$

(2.11)

Another basic property of the 1st level GFC operators is the formula ([18])

$$\left({\mathbb{I}}_{\left(k\right)1L}{\mathbb{D}}_{(k_1,k_2)}f\right)\left(t\right)=f\left(t\right)-\left({\mathbb{I}}_{\left(k_2\right)}f\right)\left(0\right)(k_1*k)\left(t\right),t>0,f\in C_{-1,\left(k_2\right)}^1(0,T),$$

(2.12)

where

$$C_{-1,\left(k_2\right)}^1(0,T)=\left\{f\in C_{-1}(0,T):{\mathbb{D}}_{\left(k_2\right)}f\in C_{-1}(0,T)\right\}.$$

(2.13)

The formula (2.12) is referred to as the 2^nd fundamental theorem of the GFC for the 1st level GFD.

3. Fermat-Type Theorem for the 1st Level GFD

In this section, we derive a Fermat-type theorem for the 1st level GFD with the kernels $k_1$ and $k_2$ that satisfy the following conditions:

$\left(C_1\right)k_1,k_2\in C_{-1}(0,T)$ are nonnegative for $t>0$,

$\left(C_2\right)k_1*k_2\in C^1(0,T)\cap L_1^{loc}(0,T)$, $k_1*k_2$ decreases for $t>0$, $(k_1*k_2)\left(t\right)=o\left(t^{-1}\right),t\to 0$.

The main result of this section is formulated in the following theorem:

Theorem 3.1.

Let a function $u\in C{W}^1[0,T]$ satisfy the conditions

$$u\left(t\right)\ge 0,t\in [0,t_{*}],u\left(t_{*}\right)=0,forsome{t}_{*}\in (0,T),$$

where the space $C{W}^1[0,T]$ is defined as follows:

$$C{W}^1[0,T]=\{f\in C[0,T]:f^{\prime }\in C^1(0,T]\cap L^1(0,T)\}.$$

Then the inequality

$${(}^{1L}{\mathbb{D}}_{(k_1,k_2)}u)\left(t_{*}\right)\le 0$$

holds true.

Proof. 

To prove the theorem, we need a special representation of the 1st level GFD of the function u at the point $t_{*}$. Employing the first and second fundamental theorems of calculus and the commutativity property of the GFIs, we first get

$$\begin{array}{ccc}\hfill {(}^{1L}{\mathbb{D}}_{(k_1,k_2)}u)\left(t_{*}\right)& =& \left({\mathbb{I}}_{\left(k_1\right)}{\displaystyle \frac{d}{dt}}{\mathbb{I}}_{\left(k_2\right)}u\right)\left(t_{*}\right)\hfill \\ & =& \left({\displaystyle \frac{d}{dt}}{\mathbb{I}}_{\left(h_1\right)}{\mathbb{I}}_{\left(k_1\right)}{\displaystyle \frac{d}{dt}}{\mathbb{I}}_{\left(k_2\right)}u\right)\left(t_{*}\right)\hfill \\ & =& \left({\displaystyle \frac{d}{dt}}{\mathbb{I}}_{\left(k_1\right)}{\mathbb{I}}_{\left(h_1\right)}{\displaystyle \frac{d}{dt}}{\mathbb{I}}_{\left(k_2\right)}u\right)\left(t_{*}\right)\hfill \\ & =& \left({\displaystyle \frac{d}{dt}}{\mathbb{I}}_{\left(k_1\right)}({\mathbb{I}}_{\left(k_2\right)}u-\left({\mathbb{I}}_{\left(k_2\right)}u\right)\left(0\right))\right)\left(t_{*}\right)\hfill \\ & =& \left({\displaystyle \frac{d}{dt}}{\mathbb{I}}_{(k_1*k_2)}u\right)\left(t_{*}\right)-\left({\mathbb{I}}_{\left(k_2\right)}u\right)\left(0\right)\left({\displaystyle \frac{d}{dt}}{\mathbb{I}}_{\left(k_1\right)}1\right)\left(t_{*}\right).\hfill \end{array}$$

(3.1)

The formula

$$\left({\displaystyle \frac{d}{dt}}{\mathbb{I}}_{\left(k_1\right)}1\right)\left(t\right)=\left({\displaystyle \frac{d}{dt}}{\mathbb{I}}_{\left(h_1\right)}{k}_1\right)\left(t\right)=k_1\left(t\right)$$

and the identity ([7])

$$\left({\displaystyle \frac{d}{dt}}{\mathbb{I}}_{(k_1*k_2)}u\right)\left(t\right)={\int }_0^t{(k_1*k_2)}^{\prime }(t-s)\left(u\left(s\right)-u\left(t\right)\right)ds+u\left(t\right)(k_1*k_2)\left(t\right)$$

that is valid under the condition $\left(C_2\right)$ lead to the desired representation

$$\begin{array}{ccc}\hfill {(}^{1L}{\mathbb{D}}_{(k_1,k_2)}u)\left(t_{*}\right)& =& {\int }_0^{t_{*}}{(k_1*k_2)}^{\prime }(t_{*}-s)\left(u\left(s\right)-u\left(t_{*}\right)\right)ds\hfill \\ & +& u\left(t_{*}\right)(k_1*k_2)\left(t\right)-\left({\mathbb{I}}_{\left(k_2\right)}u\right)\left(0\right)k_1\left(t\right)\hfill \\ & =& {\int }_0^{t_{*}}{(k_1*k_2)}^{\prime }(t_{*}-s)u\left(s\right)ds-\left({\mathbb{I}}_{\left(k_2\right)}u\right)\left(0\right)k_1\left(t\right).\hfill \end{array}$$

(3.2)

Inequality $u\left(t\right)\ge 0,t\in [0,t_{*}]$ and conditions $\left(C_1\right)$ and $\left(C_2\right)$ implicate inequalities

$$-\left({\mathbb{I}}_{\left(k_2\right)}u\right)\left(0\right)k_1\left(t\right)\le 0,$$

$${\int }_0^{t_{*}}{(k_1*k_2)}^{\prime }(t_{*}-s)u\left(s\right)ds\le 0,$$

and the statement of the theorem immediately follows from the representation (3.2). □

Remark 3.1.

As already mentioned, the 1st level GFD ${}^{1L}\mathbb{D}_{(k_1,k_2)}$ includes the GFD and the RGFD as particular cases. Thus, Theorem 3.1 generalizes the results derived in [17] for GFD and RGFD.

Remark 3.2.

One can easily check that several known pairs of Sonin kernels possess the following property: if the kernels are non-negative and one of them is decreasing, then the other kernel is also decreasing. Our conjecture is that this statement holds for any pair of non-negative Sonin kernels. However, we could not prove or disprove this conjecture and leave it as an open problem for further research. If this conjecture is true, then the property that $k_1*k_2$ decreases for $t>0$ (see the condition $\left(C_1\right)$) can be reformulated as decreasing of their associated non-negative Sonin kernel κ.

4. Applications of Fermat’s Theorem for the Inequalities and Equations Involving the 1st Level GFDs

In this section, we present some important consequences from the Fermat theorem for the 1st level GFDs. In particular, comparison properties for solutions to the fractional differential inequalities and norm estimates for solutions to the fractional differential equations involving the 1st level GFDs are formulated and proved.

We start with the case of the strict non-linear fractional differential inequalities involving the 1st level GFDs provided in the next theorem.

Theorem 4.1.

Let the functions $u_1,u_2\in C{W}^1[0,T]$ satisfy the fractional differential inequality and the inequality for the initial values

$$\begin{array}{ccc}\hfill {(}^{1L}{\mathbb{D}}_{(k_1,k_2)}{u}_1)\left(t\right)+g(t,u_1)& <& {(}^{1L}{\mathbb{D}}_{(k_1,k_2)}{u}_2)\left(t\right)+g(t,u_2),t\in (0,T],\hfill \end{array}$$

(4.1)

$$\begin{array}{ccc}\hfill u_1\left(0\right)& <& u_2\left(0\right),\hfill \end{array}$$

(4.2)

where the function g can be arbitrary.

Then the functions $u_1$ and $u_2$ satisfy the inequalities

$$u_1\left(t\right)<u_2\left(t\right),t\in [0,T),u_1\left(T\right)\le u_2\left(T\right).$$

(4.3)

Proof. 

We start with the proof of the inequality $u_1\left(t\right)<u_2\left(t\right),t\in [0,T)$. Assume by contradiction that this inequality is not true. Because $u_1\left(0\right)<u_2\left(0\right)$ and $u_1,u_2\in C[0,T]$, there exists a point $t_0\in (0,T)$ such that

$$u_1\left(t_0\right)=u_2\left(t_0\right),u_1\left(t\right)\le u_2\left(t\right),t\in [0,t_0].$$

Now we introduce the auxiliary function $v\left(t\right)=u_2\left(t\right)-u_1\left(t\right)$ and rewrite the above conditions in the form

$$v\left(t\right)\ge 0,t\in [0,t_0],v\left(t_0\right)=0.$$

For the function v, Theorem 3.1 states that

$${(}^{1L}{\mathbb{D}}_{(k_1,k_2)}v)\left(t_0\right)={(}^{1L}{\mathbb{D}}_{(k_1,k_2)}{u}_2)\left(t_0\right)-{(}^{1L}{\mathbb{D}}_{(k_1,k_2)}{u}_1)\left(t_0\right)\le 0.$$

Because $u_1\left(t_0\right)=u_2\left(t_0\right)$ by the definition of the point $t_0$, the above inequality yields

$${(}^{1L}{\mathbb{D}}_{(k_1,k_2)}{u}_2)\left(t_0\right)+g\left(t_0,u_2\left(t_0\right)\right)\le {(}^{1L}{\mathbb{D}}_{(k_1,k_2)}{u}_1)\left(t_0\right)+g\left(t_0,u_1\left(t_0\right)\right).$$

The last inequality contradicts the fractional differential inequality (4.1) and thus the assumption made at the beginning of the proof is false, and we have proved the inequality $u_1\left(t\right)<u_2\left(t\right),t\in [0,T).$

The inequality $u_1\left(T\right)\le u_2\left(T\right)$ immediately follows from the inequality $u_1\left(t\right)<u_2\left(t\right),t\in [0,T)$ as $u_1,u_2\in C[0,T]$. This completes the proof of the theorem. □

Now we proceed with a bit complicated case of non-strict non-linear fractional differential inequalities involving the 1st level GFDs.

Theorem 4.2.

Let the functions $u_1,u_2\in C{W}^1[0,T]$ satisfy the non-strict non-linear fractional differential inequality and the inequality for the initial values

$$\begin{array}{ccc}\hfill {(}^{1L}{\mathbb{D}}_{(k_1,k_2)}{u}_1)\left(t\right)+g(t,u_1)& \le & {(}^{1L}{\mathbb{D}}_{(k_1,k_2)}{u}_2)\left(t\right)+g(t,u_2),t\in (0,T],\hfill \end{array}$$

(4.4)

$$\begin{array}{ccc}\hfill u_1\left(0\right)& \le & u_2\left(0\right),\hfill \end{array}$$

(4.5)

where the function $g=g(t,u)$ satisfies the one-sided Lipschitz condition with respect to its second argument

$$-L(u-v)\le g(t,u)-g(t,v),wheneveru\ge v$$

(4.6)

with constant L such that

$$L={\displaystyle \frac{(k_1*k_2)\left(T\right)}{2}}>0.$$

(4.7)

Then the functions $u_1$ and $u_2$ satisfy the inequality

$$u_1\left(t\right)\le u_2\left(t\right),t\in [0,T].$$

Proof. 

First, we show that the constant function $h\left(t\right)=ϵ,t>0$ is a solution to the fractional initial value problem

$${(}^{1L}{\mathbb{D}}_{(k_1,k_2)}h)\left(t\right)=ϵ(k_1*k_2)\left(t\right),\left({\mathbb{I}}_{\left(k_2\right)}h\right)\left(0\right)=0.$$

(4.8)

By direct calculations, we have

$$\begin{array}{ccc}\hfill {(}^{1L}{\mathbb{D}}_{(k_1,k_2)}h)\left(t\right)& =& \left({\mathbb{I}}_{\left(k_1\right)}{\displaystyle \frac{d}{dt}}{\mathbb{I}}_{\left(k_2\right)}h\right)\left(t\right)=\left({\mathbb{I}}_{\left(k_1\right)}{\displaystyle \frac{d}{dt}}{\mathbb{I}}_{\left(k_2\right)}ϵ\right)\left(t\right)\hfill \\ & =& \left({\mathbb{I}}_{\left(k_1\right)}{\displaystyle \frac{d}{dt}}{k}_2*ϵ\right)\left(t\right)=\left({\mathbb{I}}_{\left(k_1\right)}ϵk_2\right)\left(t\right)\hfill \\ & =& ϵ\left(k_1*k_2\right)\left(t\right).\hfill \end{array}$$

(4.9)

Because the function h is continuous, then $\left({\mathbb{I}}_{\left(k_2\right)}h\right)\left(0\right)=0$, which shows that it is a solution to the initial value problem (4.8).

Now, for $ϵ>0$, we define the auxiliary function $u_{ϵ}\left(t\right)=u_2\left(t\right)+h\left(t\right)=u_2\left(t\right)+ϵ$. Applying the one sided Lipschitz condition (4.6) for the functions $u_{ϵ}$ and $u_2$, we arrive at the inequality

$$g(t,u_{ϵ})-g(t,u_2)\ge -L(u_{ϵ}-u_2)=-Lϵ.$$

(4.10)

Next, we have the representation

$$\begin{array}{ccc}\hfill {(}^{1L}{\mathbb{D}}_{(k_1,k_2)}{u}_{ϵ})\left(t\right)+g(t,u_{ϵ})& =& {(}^{1L}{\mathbb{D}}_{(k_1,k_2)}{u}_2)\left(t\right)+{(}^{1L}{\mathbb{D}}_{(k_1,k_2)}h)\left(t\right)+g(t,u_{ϵ})\hfill \\ & =& {(}^{1L}{\mathbb{D}}_{(k_1,k_2)}{u}_2)\left(t\right)+ϵ(k_1*k_2)\left(t\right)+g(t,u_{ϵ})\hfill \\ & =& {(}^{1L}{\mathbb{D}}_{(k_1,k_2)}{u}_2)\left(t\right)+g(t,u_2)+ϵ(k_1*k_2)\left(t\right)\hfill \\ & +& g(t,u_{ϵ})-g(t,u_2).\hfill \end{array}$$

Applying the inequality (4.10), the one-sided Lipschitz condition (4.6) with the constant L that satisfies the inequality (4.7), and using the fact that the function $k_1*k_2$ is decreasing (see the condition $\left(C_2\right)$ in Section 4), we have the following chain of equalities and inequalities:

$$\begin{array}{ccc}\hfill {(}^{1L}{\mathbb{D}}_{(k_1,k_2)}{u}_{ϵ})\left(t\right)+g(t,u_{ϵ})& \ge & {(}^{1L}{\mathbb{D}}_{(k_1,k_2)}{u}_2)\left(t\right)+g(t,u_2)+ϵ(k_1*k_2)\left(t\right)-Lϵ\hfill \\ & =& {(}^{1L}{\mathbb{D}}_{(k_1,k_2)}{u}_2)\left(t\right)+g(t,u_2)+ϵ\left[(k_1*k_2)\left(t\right)-L\right]\hfill \\ & \ge & {(}^{1L}{\mathbb{D}}_{(k_1,k_2)}{u}_2)\left(t\right)+g(t,u_2)+ϵ{\displaystyle \frac{(k_1*k_2)\left(T\right)}{2}}\hfill \\ & >& {(}^{1L}{\mathbb{D}}_{(k_1,k_2)}{u}_2)\left(t\right)+g(t,u_2)\hfill \\ & \ge & {(}^{1L}{\mathbb{D}}_{(k_1,k_2)}{u}_1)\left(t\right)+g(t,u_1)\hfill \end{array}$$

(4.11)

Thus,

$$\begin{array}{ccc}\hfill {(}^{1L}{\mathbb{D}}_{(k_1,k_2)}{u}_{ϵ})\left(t\right)+g(t,u_{ϵ})& >& {(}^{1L}{\mathbb{D}}_{(k_1,k_2)}{u}_1)\left(t\right)+g(t,u_1).\hfill \end{array}$$

(4.12)

The last inequality along with the inequality

$$u_{ϵ}\left(0\right)=u_2\left(0\right)+ϵ\ge u_1\left(0\right)+ϵ>u_1\left(0\right)$$

allows us to employ Theorem 4.1 and thus we arrive at the inequality $u_1\left(t\right)<u_{ϵ}\left(t\right)=u_2\left(t\right)+ϵ,t\in [0,T)$. Since the last inequality holds for an arbitrarily small $ϵ>0$, we conclude that $u_1\left(t\right)\le u_2\left(t\right),t\in [0,T)$.

The inequality $u_1\left(T\right)\le u_2\left(T\right)$ immediately follows from the inequality $u_1\left(t\right)\le u_2\left(t\right),t\in [0,T)$ as $u_1,u_2\in C[0,T]$, which completes the proof of the theorem. □

In the rest of this section, we derive some important properties of solutions to the following initial-value problem for the linear fractional differential equation involving the 1st level GFD:

$$\begin{array}{ccc}\hfill {(}^{1L}{\mathbb{D}}_{(k_1,k_2)}u)\left(t\right)& +& r\left(t\right)u\left(t\right)=g\left(t\right),t\in (0,T],\hfill \end{array}$$

(4.13)

$$\begin{array}{ccc}\hfill u\left(0\right)& =& u_0,\hfill \end{array}$$

(4.14)

where inclusions $r,g\in C[0,T]$ hold.

In the following two theorems, we present a priori estimate for the norm of solutions to problem (4.13)-(4.14) and prove the uniqueness of its solution.

Theorem 4.3.

Let the function $u\in C{W}^1[0,T]$ be a solution to the initial-value problem (4.13)-(4.14).

Then the following norm estimate of u holds true:

$$\left|\right|u\left|\right|=\underset{0\le t\le T}{max}\left|u\left(t\right)\right|\le M:=\underset{0\le t\le T}{max}\{{\displaystyle \frac{\left|g\right(t\left)\right|}{(k_1*k_2)\left(T\right)+r\left(t\right)}},\left|u\left(0\right)\right|\}$$

(4.15)

under the condition

$$r\left(t\right)\ge -{\displaystyle \frac{(k_1*k_2)\left(T\right)}{2}},t\in [0,T].$$

Proof. 

Because the function $k_1*k_2$ is non-increasing (see condition $\left(C_2\right)$ in Section 3) and ${(}^{1L}{\mathbb{D}}_{(k_1,k_2)}1)\left(t\right)=(k_1*k_2)\left(t\right),$ we have the inequality

$${(}^{1L}{\mathbb{D}}_{(k_1,k_2)}1)\left(t\right)+r\left(t\right)\ge (k_1*k_2)\left(T\right)+r\left(t\right)\ge {\displaystyle \frac{(k_1*k_2)\left(T\right)}{2}}>0.$$

Thus,

$$\tilde{M}:=\underset{0\le t\le T}{max}\left\{{\displaystyle \frac{\left|g\right(t\left)\right|}{(k_1*k_2)\left(T\right)+r\left(t\right)}}\right\}\ge {\displaystyle \frac{\left|g\right(t\left)\right|}{{(}^{1L}{\mathbb{D}}_{(k_1,k_2)}1)\left(t\right)+r\left(t\right)}}.$$

The last inequality can be rewritten in the form

$$-\tilde{M}\left({(}^{1L}{\mathbb{D}}_{(k_1,k_2)}1)\left(t\right)+r\left(t\right)\right)\le g\left(t\right)\le \tilde{M}\left({(}^{1L}{\mathbb{D}}_{(k_1,k_2)}1)\left(t\right)+r\left(t\right)\right).$$

(4.16)

According to the formulation of the theorem, $r\left(t\right)\ge -{\displaystyle \frac{(k_1*k_2)\left(T\right)}{2}}$ for $t\in [0,T]$ and thus the function $f(t,u)=r\left(t\right)u$ satisfies the one-sided Lipschitz condition (4.6) with the constant L that satisfies the inequality (4.7).

By the definition of M, the inequality $u\left(0\right)\le M$ is true. Moreover, we have the estimate

$${(}^{1L}{\mathbb{D}}_{(k_1,k_2)}u)\left(t\right)+r\left(t\right)u\left(t\right)=g\left(t\right)\le {(}^{1L}{\mathbb{D}}_{(k_1,k_2)}M)\left(t\right)+r\left(t\right)M$$

that allows us to apply Theorem 4.2 and get the inequality

$$u\left(t\right)\le M,t\in [0,T].$$

(4.17)

Analogously, we use the inequalities

$${(}^{1L}{\mathbb{D}}_{(k_1,k_2)}M)\left(t\right)+r\left(t\right)M\ge -g\left(t\right)={(}^{1L}{\mathbb{D}}_{(k_1,k_2)}(-M))\left(t\right)+r\left(t\right)(-M)$$

and $M\ge -u\left(0\right)$ and apply Theorem 4.2 to obtain the estimate

$$M\ge -u\left(t\right),t\in [0,T].$$

(4.18)

By combining the inequalities (4.17) and (4.18), we get the desired estimate $\left|u\right(t\left)\right|\le M,t\in [0,T]$ that completes the proof of the theorem. □

The following uniqueness result is a direct consequence of the solution norm estimate obtained in Theorem 4.3.

Theorem 4.4.

Under the condition

$$r\left(t\right)\ge -{\displaystyle \frac{(k_1*k_2)\left(T\right)}{2}},t\in [0,T]$$

the initial-value problem (4.13)-(4.14) possesses at most one solution from the space $C{W}^1[0,T].$

Proof. 

Let us assume that the functions $u_1,u_2\in C{W}^1[0,T]$ are two different solutions to the initial-value problem (4.13)-(4.14). Then the function $u=u_1-u_2,0\le t\le T$ is a solution to the initial-value problem

$$\begin{array}{ccc}\hfill {(}^{1L}{\mathbb{D}}_{(k_1,k_2)}u)\left(t\right)& +& r\left(t\right)u\left(t\right)=0,t\in (0,T],\hfill \end{array}$$

(4.19)

$$\begin{array}{ccc}\hfill u\left(0\right)& =& 0.\hfill \end{array}$$

(4.20)

Theorem 4.3 applied to the initial value problem (4.19)-(4.20) leads to the norm estimate of u in the form $\left|\right|u\left|\right|\le M=0.$ This implicates identity $u\left(t\right)=u_1\left(t\right)-u_2\left(t\right)\equiv 0,t\in [0,T]$ and completes the proof of the theorem. □