Generalization of some fractional versions of Hadamard inequalities via exponentially $(\alpha, h-m)$-convex functions

1.   Introduction

Time-fractional differential equations arise in the mathematical modeling of a variety of real-world phenomena in many areas of sciences and engineering, such as elasticity, heat transfer, circuits systems, continuum mechanics, fluid mechanics, wave theory, etc. For more details, we refer the reader to ^{[4,6,7,8,14,15,17,24]} and the references therein. Consequently, the study of time-fractional differential equations attracted much attention of many researchers (see e.g. ^{[1,5,9,10,19,22,23]} and the references therein).

Multi-time differential equations arise, for example, in analyzing frequency and amplitude modulation in oscillators, see Narayan and Roychowdhury ^[18]. Some methods for solving Multi-time differential equations can be found in ^[20,21].

The study of blowing-up solutions to time-fractional differential equations was initiated by Kirane and his collaborators, see e.g. ^[3,11,12,13]. In particular, Kirane et al. ^[11] considered the two-times fractional differential equation

where $p, m > 1$, $0 < \alpha, \beta < 1$, ${}^C\!D_{0|t}^\alpha$ is the Caputo fractional derivative of order $\alpha$ with respect to the first time-variable $t$, and ${}^C\!D_{0|s}^\beta$ is the Caputo fractional derivative of order $\beta$ with respect to the second time-variable $s$. Namely, the authors provided sufficient conditions for which any solution to (1.1) blows-up in a finite time. In the same reference, the authors extended their study to the case of systems.

In this paper, we investigate the nonexistence of global solutions to two-times-fractional differential inequalities of the form

where $p > 1$, $m\geq 1$, $\gamma, \sigma\in \mathbb{R}$, $a > 0$, $0 < \alpha, \beta < 1$, ${}^{HC}\!D_{a|t}^\alpha$ is the Hadamard-Caputo fractional derivative of order $\alpha$ with respect to the first time-variable $t$, and ${}^C\!D_{a|s}^\beta$ is the Caputo fractional derivative of order $\beta$ with respect to the second time-variable $s$. Using the test function method (see e.g. ^[16]) and a judicious choice of a test function, we establish sufficient conditions ensuring the nonexistence of global solutions to (1.2). Our obtained conditions depend on the parameters $\alpha, \beta, p, m, \gamma, \sigma$, and the initial values.

Our motivation for considering problems of type (1.2) is to study the combination effect of the two fractional derivatives of different nature ${}^{HC}\!D_{a|t}^\alpha$ and ${}^C\!D_{a|s}^\beta$ on the nonexistence of global solutions to (1.2). As far as we know, the study of nonexistence of global solutions for time fractional differential equations (or inequalities) involving both Hadamard-Caputo and Caputo fractional derivatives, was never considered in the literature.

The rest of the paper is organized as follows: In Section 2, we recall some concepts from fractional calculus and provide some useful lemmas. In Section 3, we state our main results and provide some examples. Section 4 is devoted to the proofs of our main results.

2.   Some preliminaries

Let $a, T\in \mathbb{R}$ be such that $0 < a < T$. The left-sided and right-sided Riemann-Liouville fractional integrals of order $\theta > 0$ of a function $\vartheta\in L^1([a, T])$, are defined respectively by (see ^[10])

and

for almost everywhere $t\in [a, T]$, where $\Gamma$ is the Gamma function.

Notice that, if $\vartheta\in C([a, T])$, then $I_a^\theta\vartheta, I_T^\theta\vartheta \in C([a, T])$ with

The Caputo fractional derivative of order $\theta\in (0, 1)$ of a function $\vartheta\in AC([a, \infty))$, is defined by (see ^[10])

for almost everywhere $t\geq a$.

Lemma 2.1. [see ^[10]]Let $\kappa > 0$, $p, q\geq 1$, and $\frac{1}{p}+\frac{1}{q}\leq 1+\kappa$ ($p\neq 1$, $q\neq 1$, in the case $\frac{1}{p}+\frac{1}{q} = 1+\kappa$). Let $\vartheta\in L^p([a, T]$ and $w\in L^q([a, T])$. Then

The left-sided and right-sided Hadamard fractional integrals of order $\theta > 0$ of a function $\vartheta\in L^1([a, T])$, are defined respectively by (see ^[10])

and

for almost everywhere $t\in [a, T]$.

Notice that, if $\vartheta\in C([a, T])$, then $J_a^\theta\vartheta, J_T^\theta\vartheta \in C([a, T])$ with

The Hadamard-Caputo fractional derivative of order $\theta\in (0, 1)$ of a function $\vartheta\in AC([a, \infty))$, is defined by (see ^[2])

for almost everywhere $t\geq a$, where

We have the following integration by parts rule.

Lemma 2.2. Let $\kappa > 0$, $p, q\geq 1$, and $\frac{1}{p}+\frac{1}{q}\leq 1+\kappa$ ($p\neq 1$, $q\neq 1$, in the case $\frac{1}{p}+\frac{1}{q} = 1+\kappa$). If $\vartheta \circ \exp \in L^p([\ln a, \ln T])$ and $w \circ \exp \in L^q([\ln a, \ln T])$, then

Proof. Using the change of variable $x = \ln \tau$, we obtain

that is,

Similarly, we have

By (2.3), we obtain

Using the change of variable $x = \ln t$, we get

Since $\vartheta \circ \exp \in L^p([\ln a, \ln T])$ and $w \circ \exp \in L^q([\ln a, \ln T])$, by Lemma 2.1, we deduce that

Using again the change of variable $x = \ln t$, there holds

Then, by (2.4), the desired result follows.

By elementary calculations, we obtain the following properties.

Lemma 2.3. For sufficiently large $\lambda$, let

Let $\kappa\in (0, 1)$. Then

Lemma 2.4. For sufficiently large $\lambda$, let

Let $\kappa\in (0, 1)$. Then

3.   Main results

First, let us define global solutions to (1.2). To do this, we need to introduce the functional space

We say that $u$ is a global solution to (1.2), if $u\in X_a$ and $u$ satisfies the fractional differential inequality

for almost everywhere $t, s\geq a$, as well as the initial conditions

Now, we state our main results.

Theorem 3.1. Let $u_0\in L^1([a, \infty))$, $u_1\in L^m([a, \infty), \frac{1}{t}\, dt)$, and $u_1\not\equiv 0$.Let

If

then, for all $\alpha\in (0, 1)$, (1.2) admits no global solution.

Remark 3.1. Notice that by (3.1), the set of exponents $p$ satisfying (3.2) is nonempty.

Theorem 3.2. Let $u_0\in L^1([a, \infty))$, $u_1\in L^m([a, \infty), \frac{1}{t}\, dt)$, and $u_1\not\equiv 0$. Let

If

and

then (1.2) admits no global solution.

Remark 3.2. Notice that by (3.3), the set of real numbers $\gamma$ satisfying (3.4) is nonempty.

We illustrate our obtained results by the following examples.

Example 3.1. Consider the fractional differential inequality

where $a > 0$ and $0 < \alpha < 1$. Observe that (3.6) is a special case of (1.2) with

Moreover, we have

and $u_0\in L^1([a, \infty))$, $u_1\in L^m([a, \infty), \frac{1}{t}\, dt)$. Hence, condition (3.1) is satisfied. Then, by Theorem 3.1, we deduce that, if

that is,

then (3.6) admits no global solution.

Example 3.2. Consider the fractional differential inequality

where $a > 0$. Then (3.7) is a special case of (1.2) with

On the other hand, we have

which shows that condition (3.3) is satisfied. Moreover, we have

which shows that conditions (3.4) and (3.5) are satisfied. Then, by Theorem 3.2, we deduce that (3.7) admits no global solution.

4.   Proofs of the main results

In this section, $C$ denotes a positive constant independent on $T$, whose value may change from line to line.

Proof of Theorem 3.1. Suppose that $u\in X_a$ is a global solution to (1.2). For sufficiently large $T$ and $\lambda$, let

where $\phi_1$ and $\phi_2$ are defined respectively by (2.5) and (2.8). Multiplying the inequality in (1.2) by $\frac{1}{t}\varphi$ and integrating over $\Omega_T: = (a, T)\times (a, T)$, we obtain

On the other hand, using Lemma 2.2, integrating by parts, using the initial conditions, and taking in consideration (2.2), we obtain

Integrating over $(a, T)$, we get

Similarly, using Lemma 2.1, integrating by parts, using the initial conditions, and taking in consideration (2.1), we obtain

Integrating over $(a, T)$, there holds

It follows from (4.1)–(4.3) that

On the other hand, by Young's inequality, we have

Similarly, since $p > m$, we have

Hence, combining (4.4)–(4.6), we deduce that

where

and

By the definition of the function $\varphi$, we have

Thus, using (2.6), we obtain

Integrating over $(a, T)$, we get

Similarly, by the definition of the function $\varphi$, we have

Thus, using (2.9), we obtain

Integrating over $(a, T)$, we get

Combining (4.8) with (4.9), there holds

Since $u_0\in L^1([a, \infty))$, $u_1\in L^m([a, \infty), \frac{1}{t}\, dt)$, and $u_1\not\equiv 0$, by the dominated convergence theorem, we deduce that for sufficiently large $T$,

Now, we shall estimate the terms $K_i$, $i = 1, 2$. By the definition of the function $\varphi$, the term $K_1$ can be written as

Next, by (2.8), we obtain

On the other hand, by (3.1) and (3.2), it is clear that $\gamma < p-1$. Thus, we deduce that

By (2.5) and (2.7), we have

Notice that by (3.1) and (3.2), we have $\sigma < p-1$. Thus, we get

Hence, it follows from (4.11)–(4.13) that

Similarly, we can write the term $K_2$ as

By (2.5), we have

Notice that by (3.2), we have $\sigma m < p-m$. Thus, we get

On the other hand, by (2.8) and (2.10), we have

Notice that by (3.2), we have $p > m(\gamma+1)$. Therefore, we obtain

Combining (4.16) with (4.17), there holds

Hence, it follows from (4.14) and (4.18) that

Thus, by (4.7), (4.10), and (4.19), we deduce that

Notice that by (3.1) and (3.2), we have

Hence, passing to the limit as $T\to \infty$ in (4.20), we obtain a contradiction with $u_1\not\equiv 0$. Consequently, (1.2) admits no global solution. The proof is completed.

Proof of Theorem 3.2. Suppose that $u\in X_a$ is a global solution to (1.2). Notice that in the proof of Theorem 3.1, to obtain (4.20), we used that

On the other hand, by (3.3)–(3.5), it can be easily seen that the above conditions are satisfied. Thus, (4.20) holds. Hence, taking $p = 1+\frac{\gamma}{\beta}$ in (4.20), we obtain

On the other hand, by (3.3)–(3.5), we have

Hence, passing to the limit as $T\to \infty$ in (4.21), we obtain a contradiction with $u_1\not\equiv 0$. This shows that (1.2) admits no global solution. The proof is completed.

5.   Conclusions

The two-times fractional differential inequality (1.2) is investigated. Namely, using the test function method and a judicious choice of a test function, sufficient conditions ensuring the nonexistence of global solutions to (1.2) are obtained. Two cases are discussed separately: $1 < p < 1+\frac{\gamma}{\beta}$ (see Theorem 3.1) and $p = 1+\frac{\gamma}{\beta}$ (see Theorem 3.2). In the first case, no assumption is imposed on the fractional order $\alpha\in (0, 1)$ of the Hadamard-Caputo fractional derivative, while in the second case, it is supposed that $\alpha > 1-\frac{1}{m}$. About the initial conditions, in both cases, it is assumed that $u_0\in L^1([a, \infty))$, $u_1\in L^m([a, \infty), \frac{1}{t}\, dt)$, and $u_1\not\equiv 0$.

Finally, it would be interesting to extend this study to two-times fractional evolution equations. For instance, the tow-times fractional semi-linear heat equation

deserves to be studied.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University for funding this work through Research Group no. RG-21-09-02.

Conflict of interest

The authors declare that they have no competing interests.