Comparison principles for a class of general integro-differential inequalities with applications

Abstract

Comparison principles for fractional differential equations have been investigated in many papers using different types of fractional integral and derivative operators. We here prove the strongest such results so far, for a very broad class of operators that is even more general than those with Sonine kernels. Starting from inequalities valid at global extrema, we obtain comparison principles for these general operators, which are applied to prove bounds on solutions to related integro-differential equations. Many results in the literature will be considered as particular cases of the current study.

1 Introduction

Fractional calculus began with the idea of generalising nth-order derivatives and integrals to cases where $n\not \in {\mathbb {Z}}$: first, rational values of n (hence the term “fractional”), then real and even complex values (Oldham and Spanier 1974; Samko et al. 1993). Nowadays, the term “fractional calculus” has expanded to include operators that merely interpolate between a function and its first derivative (without caring about including other values of n), and even more general operators which no longer have any direct connection with the classical derivative and integral operators. Some of the wealth of operators that have been considered as part of fractional calculus are catalogued in recent overview papers (Hilfer and Luchko 2019; Baleanu and Fernandez 2019; Teodoro et al. 2019).

In mathematics, any useful result should be proved in the most general possible setting, to maximise its usefulness. (This is in contrast to applied sciences, where very abstract and general results may not be directly applicable to any given real-world problem, except via particular cases.) Therefore, a recent trend in fractional calculus has been towards considering general classes of operators (Baleanu and Fernandez 2019), such as conjugated operators (Fernandez and Fahad 2022) and nth-level operators (Luchko 2020a), among others.

Among the wealth of fractional-calculus operators readily available in the literature, many are essentially convolution operators, acting by convolution with some suitable kernel function. These include the original operators with which fractional calculus began, those now called Riemann–Liouville operators and defined using power-function kernels, as well as many others. There have been several recent efforts to define some broad classes of kernels, such as analytic kernels (Fernandez et al. 2019) and Sonine kernels (Luchko 2020b, c). In this paper, we will try to consider the most general possible kernel function to get the conclusion that we are aiming for: in other words, we will impose as few assumptions as possible on our kernel functions and the corresponding operators.

The types of results that we consider in this paper are called comparison principles. Essentially, by considering fractional differential inequalities and comparing their solutions, it is possible to prove bounds on solutions to fractional differential equations. Such results have previously been considered for various different fractional-calculus operators (Al-Refai and Luchko 2022; Al-Refai et al. 2021; Yakar 2011), and the aim of the current work is to extend those previous results to a much more general setting.

In particular, we consider general integro-differential operators of convolution type defined as follows, respectively of Riemann–Liouville and Caputo types:

$$\begin{aligned} \left( {}^{RL}_{0}D_ku\right) (t)&=\frac{{\textrm{d}}}{{\textrm{d}}t} \int _0^t k(t-\tau ) u(\tau )\,{\textrm{d}}\tau , \quad t>0, \end{aligned}$$

(1)

$$\begin{aligned} \left( {}^{C}_{0}{D}_ku\right) (t)&=\int _0^t k(t-\tau ) u' (\tau )\,{\textrm{d}}\tau , \quad t>0, \end{aligned}$$

(2)

and correspondingly, a general convolution-type integral operator defined as follows:

$$\begin{aligned} \left( {}_{0}I_\psi u\right) (t)=\int _0^t \psi (t-s) u(s) {\textrm{d}}s, \quad t>0, \end{aligned}$$

(3)

acting on suitable functions u defined on $(0,\infty ).$ These are essentially general convolution operators combined, in the case of (1) and (2), with simple derivatives. To restrict the scope of the problem to a scale where we will be able to prove our desired results, we make assumptions on the kernel functions k and $\psi$ as follows.

Assumption 1

The kernel function $k:(0,\infty )\rightarrow {\mathbb {R}}$ for the operators (1) and (2) is assumed to satisfy the following conditions:

• (a) $k\in C^1(0,\infty ).$

• (b) $k\ge 0$ and $k'\le 0$ everywhere.

• (c) $t k(t)\rightarrow 0$ as $t\rightarrow 0^+.$

• (d) k has a well-defined Laplace transform $\widehat{k},$ with $s\widehat{k}(s)\rightarrow 0$ as $s\rightarrow 0$ and $s\widehat{k}(s)\rightarrow \infty$ as $s\rightarrow \infty .$

Assumption 2

The kernel function $\psi :(0,\infty )\rightarrow {\mathbb {R}}$ for the operator (3) is assumed to satisfy the following conditions:

• (a) $\psi \in C^1(0,\infty ).$

• (b) $\psi \ge 0$ and $\psi '\le 0$ everywhere.

• (c) $\psi$ has a well-defined Laplace transform ${\mathcal {L}}(\psi ).$

Remark 1

Some of the conditions imposed in Assumptions 1 and 2 are largely standard, similar to those imposed in for example (Luchko and Yamamoto 2018), and some of them will be needed at specific points in our proofs below (these are noted in each case). On the other hand, it is worth noting that we have made certain stylistic choices in these Assumptions, regarding the generality of the operators to be considered. There is ongoing debate around which operators can reasonably be called “fractional derivatives/integrals” and which cannot (Hilfer and Luchko 2019; Teodoro et al. 2019), which we do not wish to enter into; however, we make the following notes.

The assumption that $\psi '\le 0$ everywhere [the second part of Assumption 2(b)] is not actually used in any of the main results; thus, it could be safely removed and our work would still be valid, but it is always valid in all of the examples which are of interest to fractional calculus. We have chosen to keep this condition, because it is usually imposed for kernels that are to be used for fractional derivatives. In this paper, we only use integral (not derivative) operators with kernel $\psi ,$ hence the unnecessity of this part of the assumption.

On the other hand, we have not imposed a Sonine condition, something that is often done when such general kernels are considered in fractional calculus. Having a fundamental theorem of calculus, to provide a suitable connection between (1)–(2) as fractional derivatives and (3) as a fractional integral, would require that the functions k and $\psi$ are Sonine kernels (Sonine 1884), connected by the relation $k*\psi =1.$ Integro-differential operators with Sonine kernels have been studied extensively in recent years (Al-Refai and Luchko 2022; Kochubei 2011; Luchko 2020b). To make our results as general as possible, we have chosen not to impose a Sonine condition; however, as it does not conflict with any of the assumptions we have imposed, the reader is free to add this as an additional assumption to connect the operators with each other and make them more closely aligned with the traditional properties of fractional calculus.

To ensure that the integro-differential operators (1)–(3) are well defined, we shall assume that u is in the following function space:

$$\begin{aligned} CW^1[0,T]=\{u\in C[0,T]\cap C^1(0,T]:\ u'\in L^1(0,T)\}. \end{aligned}$$

(4)

In this paper, we shall consider a nonlinear initial value problem of the following form:

$$\begin{aligned} \left( {}^{C}_{0}D_ku\right) (t)&=f\Big (t,u(t), \left( {}_{0}I_\psi u\right) (t)\Big ), \quad t\in (0,T]; \end{aligned}$$

(5)

$$\begin{aligned} u(0)&=u_0. \end{aligned}$$

(6)

This is a general version of many fractional initial value problems considered in the literature, in which a single Caputo-type fractional derivative of the unknown function u(t) is assumed to depend on the variable t,  the function u(t),  and a non-local fractional integral term. Various special cases of such fractional integro-differential problems have been studied in the literature, and several have found real-world applications (Butkovskii et al. 2013a, b; Tarasov 2009), so it is worth investigating the problem in the general case to get stronger results. We will first present some useful inequalities related to maximum and minimum values, which will help us to analyse the above problem and qualitative properties of its solutions.

2 Main results

We start with the following preliminary results that will be employed to derive comparison principles for related integro-differential inequalities.

Lemma 1

Let $u\in CW^1[0,T]$ and let k be a kernel satisfying all the conditions in Assumption 1.

• (i) If u attains a global maximum at $t_0\in (0,T],$ then

  $$\begin{aligned} \left( {}^{C}_{0}D_ku\right) (t_0)\ge k(t_0)\big (u(t_0)-u(0)\big )\ge 0. \end{aligned}$$

• (ii) If u attains a global minimum at $t_0\in (0,T],$ then

  $$\begin{aligned} \left( {}^{C}_{0}D_ku\right) (t_0)\le k(t_0)\big (u(t_0)-u(0)\big )\le 0. \end{aligned}$$

Proof

We note first that the result of (i) is very similar to Luchko and Yamamoto (2018, Theorem 3.1). The conditions they imposed on f (our u) are precisely equivalent to being in the space $CW^1[0,T],$ and the imposed conditions on the kernel function k are different only in the following ways:

• 1. In Luchko and Yamamoto (2018), they require $k>0$ and $k'<0,$ strict inequalities rather than our $k\ge 0$ and $k'\le 0.$ However, this difference is not important, as everything in the proof of Luchko and Yamamoto (2018, Theorem 3.1) would work in the same way with the non-strict inequalities.

• 2. In Luchko and Yamamoto (2018), they include an explicit requirement that $k\in L^1_{\text {loc}}({\mathbb {R}}^+).$ However, this is a consequence of $k\in C^1({\mathbb {R}}^+)$ together with tk(t) going to zero as $t\rightarrow 0^+,$ so it is not an independent assumption and does not need to be stated separately.

• 3. We have imposed an extra requirement in Assumption 1(d), but it is only used later and is not necessary for this result.

Therefore, part (i) follows from (2018, Theorem 3.1). Then, applying the result of (i) for the function $-u(t)$ yields the result of (ii). $\square$

Remark 2

Estimates of fractional derivatives at their extreme points have been considered in a number of settings: first, by Luchko (2009) and the first author Al-Refai (2012) for the Caputo and Riemann–Liouville derivatives, and later for several fractional differential operators with various different kernels and under different types of conditions (Al-Refai and Luchko 2014; Al-Refai 2021; Borikhanov et al. 2018; Borikhanov and Torebek 2018), including for the Prabhakar derivative (Al-Refai et al. 2022). The results in Lemma 1 include most of these as particular cases, due to the generality of the presented operators.

Lemma 2

Fix $\lambda \in {\mathbb {R}}$ and $\gamma \in {\mathbb {R}}^+,$ and let k and $\psi$ be two functions satisfying Assumptions 1 and 2, respectively. If $u\in CW^1[0,T]$ is a solution to the following initial value problem : 

$$\begin{aligned} \left( {}^{C}_{0}D_ku\right) (t)&=\lambda u(t)+\gamma \left( {}_{0}I_\psi u\right) (t), \quad t\in (0,T], \end{aligned}$$

(7)

$$\begin{aligned} u(0)&= 1, \end{aligned}$$

(8)

then it satisfies $u(t)>0$ for all $t\in [0,T].$

Proof

Assume the result is untrue. Because $u(0)=1>0$ and $u\in C[0,T],$ there exists $t_0>0$ such that $u(t)>0$ for all $t\in [0,t_0)$ and $u(t_0)=0,$ which means that u attains a global minimum on $[0,t_0]$ at $t_0.$ Applying the result in Lemma 1, we have

$$\begin{aligned} \left( {}^{C}_{0}D_ku\right) (t_0)\le k(t_0)\big (u(t_0)-u(0)\big )=-k(t_0)<0. \end{aligned}$$

Because $u(t)\ge 0$ for $t\in [0,t_0],$ we have $\left( {}_{0}I_\psi u\right) (t_0) \ge 0.$ But now, we have $\left( {}^{C}_{0}D_ku\right) (t_0)<0$ and $\lambda u(t_0)+\gamma ({}_0I_\psi u)(t_0)\ge 0,$ which contradicts Eq. (7), and completes the proof. $\square$

The solution of the initial value problem (7)–(8) can be obtained via the Laplace transform. Applying the Laplace transform to Eq. (7), and using hat notation for the transformed functions, we have

$$\begin{aligned} \widehat{k}(s)\Big (s\widehat{u}(s)-u(0)\Big ) =\lambda \widehat{u}(s)+\gamma \widehat{u}(s)\widehat{\psi }(s). \end{aligned}$$

Using $u(0)=1,$ this yields

$$\begin{aligned} \widehat{u}(s)=H(s):=\frac{\widehat{k}(s)}{s\widehat{k}(s) -\gamma \widehat{\psi }(s)-\lambda }. \end{aligned}$$

(9)

Because $s \widehat{k}(s)\rightarrow \infty$ as $s\rightarrow \infty ,$ we have $H(s)\rightarrow 0$ as $s\rightarrow \infty .$ Assuming that H has a finite number of singularities, then by Jordan’s lemma, the unique solution of the initial value problem (7)–(8) is given by $u(t)={\mathcal {L}}^{-1}(H)(s).$

2.1 Comparison principles for linear integro-differential inequalities

Following Lemma 2 concerning solutions to integro-differential equations, we now prove two further results concerning integro-differential inequalities.

Lemma 3

Let k and $\psi$ be two functions satisfying Assumptions 1 and 2, respectively. If $u\in CW^1[0,T]$ satisfies the following inequalities : 

$$\begin{aligned} \left( {}^{C}_{0}D_ku\right) (t) +s(t) u(t)+h(t) \left( {}_{0}I_\psi u\right) (t)&\le 0, \quad t\in (0,T], \end{aligned}$$

(10)

$$\begin{aligned} u(0)&< 0, \end{aligned}$$

(11)

where s, h are bounded on [0, T] and $h(t)\le 0$ on [0, T],  then $u(t)< 0$ for all $t\in [0,T].$

Proof

Assume the result is untrue. Because $u(0)< 0$ and $u\in C[0,T],$ there exists $t_0>0$, such that $u(t_0)=0$ and $u(t)\le 0$ for all $t\in [0,t_0],$ which means that $\left( {}_{0}I_\psi u\right) (t_0)\le 0$ and also that u attains a global maximum on $[0,t_0]$ at $t_0.$ By virtue of the result in Lemma 1, we have

$$\begin{aligned} 0&\ge \left( {}^{C}_{0}D_ku\right) (t_0) +s(t_0) u(t_0)+h(t_0)\left( {}_{0}I_\psi u\right) (t_0) \\&\ge \left( {}^{C}_{0}D_ku\right) (t_0) \\&\ge k(t_0)\big (u(t_0)-u(0)\big )=-k(t_0) u(0)>0, \end{aligned}$$

which is a contradiction. $\square$

We can strengthen the result of Lemma 3 by showing that the above comparison principle holds true even when the strict inequality in the initial condition is replaced by a non-strict inequality. Note that the above proof technique will not work with this non-strict inequality.

Lemma 4

Let k and $\psi$ be two functions satisfying Assumptions 1 and 2, respectively. If $u\in CW^1[0,T]$ satisfies the following inequalities : 

$$\begin{aligned} \left( {}^{C}_{0}D_ku\right) (t) +s(t) u(t)+h(t) \left( {}_{0}I_\psi u\right) (t)&\le 0, \quad t\in (0,T], \end{aligned}$$

(12)

$$\begin{aligned} u(0)&\le 0, \end{aligned}$$

(13)

where s, h are bounded on [0, T] and $h(t)\le 0$ on [0, T],  then $u(t)\le 0$ for all $t\in [0,T].$

Proof

Let us first define real constants $m,\mu$ by

$$\begin{aligned} m=\min _{t\in [0,T]} \{s(t)\}, \quad \mu =\max _{t\in [0,T]}\{ -h(t)\} \ge 0, \end{aligned}$$

and choose real positive constants $L, \varepsilon >0$ such that $L>m.$ Now, let $u_\varepsilon =u-\varepsilon g,$ where g is the unique solution of the following initial value problem:

$$\begin{aligned} \left( {}^{C}_{0}D_kg\right) (t)&=(L-m) g(t)+\mu \left( {}_{0}I_\psi g\right) (t),\\ g(0)&=1. \end{aligned}$$

Using the result of Lemma 2, we have $g(t)>0$ for all $t\in [0,T],$ and thus, $u(t)\ge u_\varepsilon (t)$ for all $t\in [0,T].$ Then

$$\begin{aligned} 0&\ge \left( {}^{C}_{0}D_ku\right) (t)+s(t) u(t)+h(t) \left( {}_{0}I_\psi u\right) (t)\nonumber \\&=\left( {}^{C}_{0}D_k\left( u_\varepsilon +\varepsilon g\right) \right) (t) +s(t) \left( u_\varepsilon +\varepsilon g\right) (t)+h(t) \left( {}_{0} I_\psi \left( u_\varepsilon +\varepsilon g\right) \right) (t)\nonumber \\&=\left( {}^{C}_{0}D_ku_\varepsilon \right) (t)+ s(t) u_\varepsilon (t) +h(t) \left( {}_{0}I_\psi u_\varepsilon \right) (t)+\varepsilon r(t), \end{aligned}$$

(14)

where the function r is as follows:

$$\begin{aligned} r(t)&:=\left( {}^{C}_{0}D_kg\right) (t)+s(t) g(t)+h(t) \left( {}_{0}I_\psi g\right) (t) \nonumber \\&=(L-m) g(t)+\mu \left( {}_{0}I_\psi g\right) (t)+s(t) g(t)+h(t) \left( {}_{0}I_\psi g\right) (t) \nonumber \\&=L g(t)+\big (s(t)-m\big ) g(t)+\big (\mu +h(t)\big ) \left( {}_{0}I_\psi g\right) (t)\ge 0, \end{aligned}$$

(15)

using the definition of g and then the definitions of L,  m,  and $\mu$ and the fact that $g(t)>0$ for all $t\in [0,T].$

Combining inequalities (14) and (15), we have

$$\begin{aligned} \left( {}^{C}_{0}D_ku_\varepsilon \right) (t)+ s(t) u_\varepsilon (t)+h(t) \left( {}_{0}I_\psi u_\varepsilon \right) (t)\le 0. \end{aligned}$$

(16)

Because $u_\varepsilon \in CW^1[0,T]$ and $u_\varepsilon (0)=-\varepsilon <0,$ we can now use the result of Lemma 3 to get $u_\varepsilon (t)=u(t)-\varepsilon g(t) < 0$ for all $t\in [0,T].$ Since this result holds for arbitrary $\varepsilon >0,$ we have $u(t)\le 0$ for all $t\in [0,T],$ which proves the result. $\square$

Remark 3

There are many choices for the positive constant L. For instance, one can choose $L=|m|+1.$

Remark 4

The result in the above lemma not only generalises many existing ones in the literature, but also it is a stronger result. For the particular case with $h(t)=0,$ the condition on s(t) to be only bounded is a weaker condition than others found in the literature, where it is assumed that s(t) is non-negative or bounded below (Al-Refai 2021; Al-Refai et al. 2021, 2022).

2.2 Comparison principles for nonlinear integro-differential inequalities

In this subsection, we will study some more advanced integro-differential inequalities, which are no longer linear, so the right-hand sides may depend on the functions in more complicated ways. To get a handle on these dependences, we introduce the following assumption.

Assumption 3

The function f(t, u, z) for the right-hand side of the integro-differential inequalities is assumed to satisfy the following conditions.

• (a) f(t, u, z) is defined for $t\in (0,T]$ and $u,z\in {\mathbb {R}}.$

• (b) f has continuous partial derivatives to first order.

• (c) $\frac{\partial f}{\partial z} \ge 0$ everywhere, that is, f is non-decreasing with respect to the third parameter.

• (d) f satisfies the following one-sided Lipschitz condition. There exist $L_1,L_2\ge 0$ such that

  $$\begin{aligned} f(t,u_1,z_1)-f(t,u_2,z_2)\le L_1(u_1-u_2)+L_2(z_1-z_2) \end{aligned}$$

  whenever $u_1\ge u_2$ and $z_1\ge z_2.$

Theorem 1

Let k,  $\psi ,$ and f be functions satisfying Assumptions 1, 2 and 3, respectively. If $v,w\in CW^1[0,T]$ satisfy the following inequalities : 

$$\begin{aligned} \left( {}^{C}_{0}D_kv\right) (t)&\le f\Big (t, v(t), \left( {}_{0}I_\psi v\right) (t)\Big ), \quad t\in (0,T], \end{aligned}$$

(17)

$$\begin{aligned} \left( {}^{C}_{0}D_kw\right) (t)&\ge f\Big (t, w(t), \left( {}_{0}I_\psi w\right) (t)\Big ), \quad t\in (0,T], \end{aligned}$$

(18)

$$\begin{aligned} v(0)&\le w(0), \end{aligned}$$

(19)

then $v(t)\le w(t)$ for all $t\in [0,T].$

Proof

Let $w_\varepsilon =w+\varepsilon g$ where $\varepsilon >0$ and g is the unique solution of the following initial value problem:

$$\begin{aligned} \left( {}^{C}_{0}D_kg\right) (t)&=(L_1+1) g(t)+L_2 \left( {}_{0}I_\psi g\right) (t), \quad t\in (0,T], \\ g(0)&=1. \end{aligned}$$

Then, $g(t)>0$ for all $t\in [0,T]$ by virtue of Lemma 2, and thus, $w_\varepsilon (t)>w(t)$ for all $t\in [0,T],$ which means $\left( {}_{0}I_\psi w_\varepsilon \right) (t)\ge \left( {}_{0}I_\psi w\right) (t)$ for all $t\in [0,T].$ Applying the one-sided Lipschitz condition from Assumption 3(d), we have

$$\begin{aligned}&f\Big (t,w_\varepsilon (t),\left( {}_{0}I_\psi w_\varepsilon \right) (t)\Big )-f\Big (t,w(t),\left( {}_{0}I_\psi w\right) (t)\Big )\nonumber \\&\quad \le L_1\big (w_\varepsilon (t)-w(t)\big )+L_2\left( \left( {}_{0} I_\psi \big (w_\varepsilon -w\big )\right) (t)\right) =\varepsilon L_1 g(t)+\varepsilon L_2 \left( {}_{0}I_\psi g\right) (t). \end{aligned}$$

(20)

Combining inequalities (18) and (20), we have

$$\begin{aligned} \left( {}^{C}_{0}D_kw_\varepsilon \right) (t)&=\left( {}^{C}_{0}D_kw\right) (t)+\varepsilon \left( {}^{C}_{0}D_kg\right) (t) \nonumber \\&\ge f\Big (t,w(t),\left( {}_{0}I_\psi w\right) (t)\Big ) +\varepsilon \Big ((L_1+1) g(t)+L_2 \left( {}_{0}I_\psi g\right) (t) \Big ) \nonumber \\&\ge f\Big (t,w_\varepsilon (t),\left( {}_{0}I_\psi w_\varepsilon \right) (t)\Big )+ \varepsilon g(t) \nonumber \\&> f\Big (t,w_\varepsilon (t),\left( {}_{0}I_\psi w_\varepsilon \right) (t)\Big ). \end{aligned}$$

(21)

In other words, the perturbed function $w_\varepsilon$ satisfies the same integro-differential inequality (18) as the function w. We now aim to prove that $v(t)<w_\varepsilon (t)$ for all $t\in [0,T].$

Assume the contrary, and let $r:=v-w_\varepsilon .$ We know that $r(0)<0$ and $r\in CW^1[0,T]\subset C[0,T],$ so there exists $t_0\in (0,T]$ such that $r(t_0)=0$ and $r(t)\le 0$ for all $t\in [0,t_0],$ which means that $\left( {}_{0}I_\psi r\right) (t_0)\le 0$ and that $r(t_0)$ is a global maximum on $[0,t_0].$ By Lemma 1

$$\begin{aligned} 0&<k(t_0)\big (r(t_0)-r(0)\big )\le \left( {}^{C}_{0}D_kr\right) (t_0) =\left( {}^{C}_{0}D_kv\right) (t_0)-\left( {}^{C}_{0} D_kw_\varepsilon \right) (t_0)\nonumber \\&\le f\Big (t_0, v(t_0), \left( {}_{0}I_\psi v\right) (t_0)\Big ) -f\Big (t_0, w_\varepsilon (t_0), \left( {}_{0}I_\psi w_\varepsilon \right) (t_0)\Big )\nonumber \\&=f\Big (t_0, v_0, \left( {}_{0}I_\psi v\right) (t_0)\Big ) -f\Big (t_0, v_0, \left( {}_{0}I_\psi w_\varepsilon \right) (t_0)\Big ), \end{aligned}$$

(22)

where we have written $v_0=v(t_0)=w_\varepsilon (t_0)$ for simplicity, recalling that these two numbers are equal. On the other hand, applying the mean value theorem to f,  we have that there exists $\xi _2$ between $\left( {}_{0}I_\psi v\right) (t_0)$ and $\left( {}_{0}I_\psi w_\varepsilon \right) (t_0)$, such that

$$\begin{aligned}&f\Big (t_0, v_0, \left( {}_{0}I_\psi v\right) (t_0)\Big ) -f\Big (t_0, v_0, \left( {}_{0}I_\psi w_\varepsilon \right) (t_0)\Big )\nonumber \\&\quad =\frac{\partial f}{\partial z}(\xi _2) \cdot \left( {}_{0}I_\psi \left( v-w_\varepsilon \right) \right) (t_0) =\frac{\partial f}{\partial z}(\xi _2)\cdot \left( {}_{0}I_\psi r\right) (t_0)\le 0. \end{aligned}$$

(23)

Combining the inequalities (22) and (23), we reach a contradiction. Thus, we have proved the claim that $v(t)<w_\varepsilon (t)=w(t)+\varepsilon g(t)$ for all $t \in [0,T].$ Because this holds for arbitrary $\varepsilon >0,$ we, therefore, have $v(t)\le w(t)$ for all $t\in [0,T],$ which proves the theorem. $\square$

Remark 5

A special case of the above result was discussed in Kutlay and Yakar (2022) for the particular kernel functions

$$\begin{aligned} k(t)=h_\alpha (t), \quad \psi (t)=h_{1-\alpha }(t), \quad 0<\alpha <1, \end{aligned}$$

where $h_\alpha (t)=\frac{t^{\alpha -1}}{\Gamma (\alpha )}.$ These are the Sonine kernels that produce the classical Riemann–Liouville fractional integral and Caputo fractional derivative. It is worth mentioning that the proof in Kutlay and Yakar (2022) is not correct as $v(t_0)=w_\varepsilon (t_0)$ does not imply $\left( {}_{0}I_\psi v\right) (t_0)=\left( {}_{0}I_\psi w_\varepsilon \right) (t_0);$ hence, our extra Assumption 3(c) was needed to complete the proof rigorously.

Let us consider another special case, where

$$\begin{aligned} k(t)=h_\alpha (t), \quad \psi (t)=h_{\beta }(t), \quad 0<\alpha ,\beta <1. \end{aligned}$$

These will also give a classical Caputo derivative and Riemann–Liouville integral, but of two incommensurate orders. As $\beta \rightarrow 0,$ we have ${}_{0}I_\psi$ becoming the identity operator, in which case the problem (5)–(6) reduces to

$$\begin{aligned} \left( {}^{C}_{0}D^{\alpha }u\right) (t)&=f\big (t,u(t),u(t)\big ), \quad t>0, \\ u(0)&=u_0, \end{aligned}$$

where ${}^{C}_{0}D^{\alpha }$ denotes the standard Caputo derivative of order $\alpha .$ This particular case was studied in Yakar (2011) for the Riemann–Liouville derivative, and for different initial conditions in Al-Refai (2020).

The above special cases, already investigated in the literature, indicate that the problem in (5)–(6) is worth investigating for operators of Riemann–Liouville type as well.

Remark 6

In the literature, the definitions of v and w in inequalities (17)–(19) stand for the concepts of lower and upper solutions to the problem (5)–(6). The result in Theorem 1 indicates that any lower and upper solutions are ordered. This result is essential to establish an existence and uniqueness result to the problem via the successive method of lower and upper solutions.

2.3 Applications to integro-differential equations

The comparison principles obtained above for integro-differential inequalities can be utilised to study related integro-differential equations analytically. To illustrate some applications, we consider the following linear initial value problem:

$$\begin{aligned} \left( {}^{C}_{0}D_ku\right) (t)+s(t)u(t)+h(t) \left( {}_{0}I_\psi u\right) (t)&=f(t), \quad t\in (0,T], \end{aligned}$$

(24)

$$\begin{aligned} u(0)&= u_0, \end{aligned}$$

(25)

where s, h, f are bounded functions and $h(t)\le 0$ for all $t\in [0,T].$

Theorem 2

Let k and $\psi$ be two functions satisfying Assumptions 1 and 2, respectively,  and let s, h, f be bounded functions on [0, T], such that $h(t)\le 0$ for all $t\in [0,T].$

If $u\in CW^1[0,T]$ is a possible solution to the initial value problem (24)–(25),  and if $q(t):=s(t)+h(t)\left( {}_{0}I_\psi 1\right) (t) \ge 0$ for all $t\in [0,T],$ then the solution function u can be bounded as follows : 

$$\begin{aligned} \sup _{t\in [0,T]}\big |u(t)\big |\le M:=\sup _{t\in [0,T]} \left\{ \left| \frac{f(t)}{q(t)}\right| , |u_0| \right\} , \end{aligned}$$

(26)

provided that the constant M exists.

Proof

By the assumption that $q(t)\ge 0,$ and the definition of M,  we have

$$\begin{aligned} -q(t) M \le f(t) \le q(t) M, \quad t\in [0,T]. \end{aligned}$$

Now, let $w_1(t)=u(t)-M,$ so that

$$\begin{aligned}&\left( {}^{C}_{0}D_kw_1\right) (t)+s(t) w_1(t)+h(t) \left( {}_{0}I_\psi w_1\right) (t) \\&\quad =\left( {}^{C}_{0}D_ku\right) (t) +s(t) u(t)+h(t) \left( {}_{0}I_\psi u\right) (t)-M s(t)-Mh(t) \left( {}_{0}I_\psi 1\right) (t)\\&\quad =f(t)-M q(t) \le 0. \end{aligned}$$

This, together with $w_1(0)\le 0,$ proves that $w_1(t)\le 0$ for all $t\in [0,T],$ by virtue of the result in Lemma 4. Therefore

$$\begin{aligned} u(t)-M\le 0,\quad t\in [0,T], \end{aligned}$$

(27)

which already proves half of the desired result.

For the second half, let $w_2(t)=-u(t)-M,$ so that

$$\begin{aligned}&\left( {}^{C}_{0}D_kw_2\right) (t) +s(t) w_2(t)+h(t) \left( {}_{0}I_\psi w_2\right) (t) \\&\quad =-\left( {}^{C}_{0}D_ku\right) (t) +s(t) u(t) +h(t)\left( {}_{0}I_\psi u\right) (t)-M s(t)-Mh(t) \left( {}_{0}I_\psi 1\right) (t) \\&\quad =-f(t)-M q(t) \le 0. \end{aligned}$$

This, together with $w_2(0)\le 0,$ proves that $w_2(t)\le 0$ for all $t\in [0,T],$ by virtue of the result in Lemma 4. Therefore

$$\begin{aligned} -u(t)-M\le 0. \end{aligned}$$

(28)

The result is then obtained by combining inequalities (27) and (28). $\square$

Remark 7

A uniqueness result can also be obtained for the initial value problem (24)–(25), using linearity and applying the norm estimate of solutions from (26).

As a particular example, let us consider the pair of Sonine kernels, where $\alpha \in (0,1)$ and $r\ge 0$ are fixed and $t>0$ is the independent variable

$$\begin{aligned} k(t)&=\frac{t^{-\alpha }}{\Gamma (1-\alpha )} e^{-rt}, \\ \psi (t)&=\frac{t^{\alpha -1}}{\Gamma (\alpha )} e^{-r t} +r \int _0^t \frac{s^{\alpha -1}}{\Gamma (\alpha )} e^{- rs}\,{\textrm{d}}s. \end{aligned}$$

Here, the Laplace transform of k is $\widehat{k}(s)=\frac{1}{(s+r)^{1-\alpha }},$ which means that the Laplace transform of $\psi$ is $\widehat{\psi }(s)=\frac{1}{s\widehat{k}(s)}=\frac{(s+r)^{1-\alpha }}{s}.$ Substituting the above results into (9) yields

$$\begin{aligned} \widehat{u}(s)=H(s)=\frac{s}{s^2-\gamma (s+r)^{2(1-\alpha )} -\lambda s(s+r)^{1-\alpha }}. \end{aligned}$$

The inverse Laplace transform of this function, if it exists, will give the solution to the problem (7)–(8) with the above choice of kernel functions.

Specifically, if we take $r=0,$ then we end up with the standard power law kernels, $k(t)=\frac{t^{-\alpha }}{\Gamma (1-\alpha )}$ and $\psi (t)=\frac{t^{\alpha -1}}{\Gamma (\alpha )},$ and therefore

$$\begin{aligned} \widehat{u}(s)=H(s)=\frac{s^{2\alpha -1}}{s^{2\alpha } -\lambda s^{\alpha }-\gamma }=\frac{s^{-1}}{1-\lambda s^{-\alpha }-\gamma s^{-2\alpha }}. \end{aligned}$$

On the one hand, the inverse Laplace transform of such a function is given by Fernandez et al. (2020, Theorem 3) as a bivariate Mittag-Leffler function

$$\begin{aligned} u(t)=E_{\alpha ,2\alpha ,1}^1\big (\lambda t^\alpha ,\gamma t^{2\alpha }\big ), \end{aligned}$$

where this bivariate Mittag-Leffler function is defined in Fernandez et al. (2020, Definition 1). On the other hand, letting $\lambda _1,\lambda _2$ be the two distinct real solutions to the quadratic equation $X^2-\lambda X-\gamma =0,$ we can use a partial fractions approach

$$\begin{aligned} \widehat{u}(s)=H(s)=\frac{1}{\lambda _1-\lambda _2} \left( \frac{s^{2\alpha -1}}{s^{\alpha }-\lambda _1} -\frac{s^{2\alpha -1}}{s^{\alpha }-\lambda _2}\right) , \end{aligned}$$

and therefore express the solution function in terms of univariate Mittag-Leffler functions

$$\begin{aligned} u(t)=\frac{1}{\lambda _1-\lambda _2}\big ( t^{-\alpha } E_{\alpha ,1-\alpha }(\lambda _1 t^{\alpha })-t^{-\alpha } E_{\alpha ,1-\alpha }(\lambda _2 t^{\alpha })\big ). \end{aligned}$$

Thus, in the process of solving this problem, we have uncovered an interesting identity between the classical two-parameter Mittag-Leffler function and the bivariate Mittag-Leffler function, namely

$$\begin{aligned} E_{\alpha ,2\alpha ,1}^1\big ((\lambda _1+\lambda _2)t^\alpha , -\lambda _1\lambda _2 t^{2\alpha }\big )=\frac{1}{\lambda _1 -\lambda _2}\big ( t^{-\alpha }E_{\alpha ,1-\alpha } (\lambda _1 t^{\alpha })-t^{-\alpha }E_{\alpha ,1-\alpha } (\lambda _2 t^{\alpha })\big ), \end{aligned}$$

valid for $\lambda _1,\lambda _2\in {\mathbb {R}}$ distinct and $0<\alpha <1.$ This identity between Mittag-Leffler functions can also be proved directly by manipulating power series, although such a proof would be more lengthy and require some combinatorial computations.

3 Conclusions

We have formulated and proved new comparison principles for integro-differential inequalities using very general operators. The presented results include, as particular cases, several results already shown in the literature on fractional calculus; moreover, we obtain, in some cases, stronger results, as indicated in Remark 4. Therefore, our results indicate the power of the essential approach used to prove comparison principles, and how it can be applied in a very general setting, with arbitrary kernel functions satisfying just a few key conditions as given in Assumptions 1 and 2. If desired, extra conditions could also be imposed, such as a Sonine condition to ensure a fundamental theorem of fractional calculus between the integro-differential operators used, to embed them more firmly into the theory of fractional calculus. Such results lay the path for more techniques from fractional calculus to be extended to the most general possible setting, to avoid wasting of resources by reproducing proofs in many particular cases.