Sufficient Conditions for the Non-Existence of Global Solutions to Fractional Systems with Lower-Order Hadamard-Type Fractional Derivatives

Abstract

This study investigates the absence of global solutions for a system of fractional differential equations. The system features a nonlinear source term with nonlocal temporal behavior and involves two Hadamard fractional derivatives (HFDs) of varying orders. Compared to previous studies, this system offers greater generality. To establish our conclusions, we employ properties of fractional derivatives, the method of test function, and integral inequality techniques. Finally, illustrative examples showcase our findings.

1. Introduction

Variable-order derivatives have emerged as a result of the necessity of understanding the dependence of electric power on charge transfer distance, initially modeled as a non-integer quantity [1]. Insights derived from the classical theory of ordinary differential equations, developed by A.N. Kolmogorov and others, play a fundamental role in advancing the mathematical and theoretical understanding of real-world phenomena [2]. Consequently, the study of nonlinear fractional differential equations has gained significant relevance across various scientific and engineering disciplines due to their ability to model complex systems exhibiting memory and hereditary properties [3].

In physics and materials science, these equations are instrumental in describing anomalous diffusion and viscoelastic behavior, where traditional integer-order models fail to account for long-term dependencies [4,5,6,7]. In engineering, fractional-order controllers are widely implemented in control systems to enhance stability and performance, particularly in automation and robotics [8]. Similarly, in biomedical sciences, fractional calculus is applied for modeling tumor growth and analyzing biological signals, yielding more accurate representations of physiological processes [9]. Additionally, fractional derivatives provide effective modeling frameworks for anomalous relaxation in dielectrics, particularly in non-Debye relaxation behaviors. Well-established fractional-order models, such as the Cole–Cole and Havriliak–Negami equations, accurately capture the intricate polarization dynamics observed in dielectric materials, polymers, and biological tissues [10,11,12,13]. These diverse applications underscore the importance of investigating the existence and non-existence of solutions to such systems, which is a key focus of this study.

In this context, Hadamard [14] introduced Hadamard fractional calculus in 1892. Since then, numerous researchers have sought to develop fractional models based on the Hadamard-type fractional operator, aiming to generalize associated relations and introduce novel concepts across various fields.

Compared to Hadamard fractional calculus, Riemann–Liouville fractional calculus exhibits notable distinctions. One fundamental difference lies in the kernel functions: the Hadamard fractional calculus kernel is given by ${\left(log{\displaystyle \frac{r}{s}}\right)}^{(\alpha -1)}=K_H(r,s)$, whereas the Riemann–Liouville fractional calculus kernel is expressed as $K_R\left(r,s\right)={\left(r-s\right)}^{\alpha -1},$ where $r>s>0,\alpha >0$. Furthermore, while the Hadamard kernel satisfies the scaling property $K_H\left(\vartheta r,\vartheta s\right)=K_H\left(r,s\right)$, the Riemann–Liouville kernel exhibits a different behavior, as $K_R\left(\vartheta r,\vartheta s\right)={\vartheta }^{\alpha -1}{K}_R\left(r,s\right)\ne K_R\left(r,s\right)$, highlighting a fundamental distinction in their mathematical structures.

Hadamard and Riemann–Liouville fractional calculus provide distinct approaches, each suited to different applications. The Riemann–Liouville derivative, characterized by power-law kernels, is extensively used in physics and engineering, particularly in anomalous diffusion [4], viscoelasticity [15], and memory-dependent systems [16]. Its global nature makes it particularly valuable in signal processing and control theory [17].

In contrast, Hadamard fractional derivatives, which incorporate logarithmic kernels, are well suited for modeling scale-invariant processes, including fractal geometry [18] and differential equations in non-Euclidean spaces [19]. They have proven particularly effective in describing geological creep phenomena, as exemplified by the Lomnitz logarithmic creep law for igneous rocks [18,20,21,22]. Furthermore, Hadamard fractional calculus has found applications in fracture mechanics and material deformation analysis [23].

While Riemann–Liouville calculus is more widely employed in engineering and physics, Hadamard calculus plays a crucial role in mathematical analysis, material mechanics, and geophysics [19]. The choice between these fractional models depends on the specific properties and characteristics of the system under investigation.

Both approaches play a significant role in advancing the understanding of global solutions in fractional calculus, which are fundamental for ensuring stability and well-posedness across various disciplines. These solutions are particularly crucial in physics, where they enable accurate modeling of diffusion processes and wave propagation [15,17]. In control theory, they provide the foundation for the stability of fractional-order systems [16]. Similarly, in biological sciences, global solutions facilitate the modeling of memory effects in epidemiology [24]. The existence of such solutions is influenced by factors including system nonlinearity and initial conditions, and in certain cases, the absence of solutions may arise due to these constraints.

Motivated by these advancements, researchers have introduced various fractional derivatives to further explore and construct the theory of fractional-order differential equations. In particular, Hadamard-type fractional calculus has garnered considerable attention in recent years, leading to significant applications in viscoelasticity, control systems, economic growth, transportation, and biomedical signal processing. Over the past decade, research efforts have been dedicated to establishing the existence of solutions to Hadamard fractional differential equations, analyzing their stability properties, and developing numerical methods for their approximation [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39]. Moreover, the issue of non-existence of solutions has been explored in various studies [25,40,41,42,43,44,45,46,47,48,49,50], with extensive analyses provided in these works and their references.

For example, the following system was investigated in the research cited in [51]

$$\left\{\begin{array}{c}{\mu }^{\prime }\left(\varrho \right)+{}^{C}D_0^{{\beta }_1}\mu \left(\varrho \right)={\left|\nu \left(\varrho \right)\right|}^q,\varrho >0,0<{\beta }_1<1,q>1,\hfill \\ \\ {\nu }^{\prime }\left(\varrho \right)+{}^{C}D_0^{{\beta }_2}\nu \left(\varrho \right)={\left|\mu \left(\varrho \right)\right|}^p,\varrho >0,0<{\beta }_2<1,p>1,\hfill \\ \\ \mu \left(0\right)={\mu }_0,\nu \left(0\right)={\nu }_0,\hfill \end{array}\right.$$

(1)

where ${}^{C}D_0^{\kappa }$ is the Caputo FD of order $\kappa >0$. It appears that if ${\mu }_0>0$ and ${\nu }_0>0$ with $1-{\displaystyle \frac{1}{pq}}<{\beta }_2+{\displaystyle \frac{{\beta }_1}{p}}$ or $1-{\displaystyle \frac{1}{pq}}<{\beta }_1+{\displaystyle \frac{{\beta }_2}{q}},$ then the system (1) admits no global solutions.

In contrast, earlier research [41] has examined the following problem:

$$\left\{\begin{array}{c}{D}_0^{{\alpha }_1}\mu \left(\varrho \right)+D_0^{{\beta }_1}\mu \left(\varrho \right)\ge {\varrho }^{{\gamma }_1}{\left|\mu \left(\varrho \right)\right|}^q,\varrho >0,0<{\beta }_1<{\alpha }_1<1,q>1,\hfill \\ \\ I_0^{1-{\alpha }_1}\mu \left(0\right)={\mu }_0.\hfill \end{array}\right.$$

The authors of this research show that the problem has no non-trivial global solution for ${\gamma }_1>{\beta }_1,$ $1<q<{\displaystyle \frac{1+{\gamma }_1}{1-{\beta }_1}},$ where $D_0^{\kappa }$ is the Riemann–Liouville FD of order $\kappa >0$.

In [40], researchers analyzed the following problem:

$$\left\{\begin{array}{c}{}^{C}D_0^{{\alpha }_1+1}\mu \left(\varrho \right)+{}^{C}D_0^{{\beta }_1+1}\mu \left(\varrho \right)\ge {\varrho }^{{\gamma }_1}{\left|{}^{C}D_0^{\kappa }\mu \left(\varrho \right)\right|}^q,\varrho >0,q>1,\hfill \\ \\ {\mu }^{\left(i\right)}\left(0\right)={\mu }_i,i=0,1,\dots ,n,n=\left[-{\alpha }_1\right],\hfill \end{array}\right.$$

where ${\alpha }_1,{\beta }_1,\kappa \in \left(n-1,n\right),$ $n\in \mathbb{N}.$ They proved that there are no solutions if $q\left(\kappa -n\right)+n-{\beta }_1-1<{\gamma }_1<q-1$ and ${\mu }_i>0.$

Further, it has been shown in [25] that the problem

$$\left\{\begin{array}{c}{\mathcal{D}}_{\mathfrak{a}}^{{\alpha }_1,{\beta }_1}\mu \left(\varrho \right)\ge {\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{{\gamma }_1}{\left|\mu \left(\varrho \right)\right|}^q,\varrho >\mathfrak{a}>0,0<{\alpha }_1<1,0\le {\beta }_1\le 1,q>1,\hfill \\ \\ {\mathcal{I}}_{\mathfrak{a}}^{\left(1-{\alpha }_1\right)\left(1-{\beta }_1\right)}\mu \left(\varrho \right){|}_{\varrho =\mathfrak{a}}={\mu }_0,\hfill \end{array}\right.$$

does not have global solutions when ${\mu }_0\ge 0$ and $q<{\displaystyle \frac{1+{\gamma }_1}{\left(1-{\alpha }_1\right)\left(1-{\beta }_1\right)}},$ where ${\mathcal{D}}_a^{{\alpha }_1,{\beta }_1}$ is the Hilfer–Hadamard fractional derivative

$${\mathcal{D}}_{\mathfrak{a}}^{{\alpha }_1,{\beta }_1}\mu \left(\varrho \right)={\mathcal{I}}_{\mathfrak{a}}^{\left(1-{\beta }_1\right)}\left(d{\displaystyle \frac{d}{d\varrho }}\right){\mathcal{I}}_{\mathfrak{a}}^{\left(1-{\beta }_1\right)\left(1-{\alpha }_1\right)}\mu \left(\varrho \right),0<{\alpha }_1<1,0\le {\beta }_1\le 1.$$

In [52], authors considered the problem

$$\left\{\begin{array}{c}{\mathcal{D}}_{\mathfrak{a}}^{{\alpha }_1}\mu \left(\varrho \right)+{\mathcal{D}}_{\mathfrak{a}}^{{\beta }_1}\mu \left(\varrho \right)\ge {\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{{\gamma }_1}{\left|\mu \left(\varrho \right)\right|}^q,\varrho >\mathfrak{a}>0,0<{\beta }_1<{\alpha }_1<1,q>1,\hfill \\ \\ {\mathcal{I}}_{\mathfrak{a}}^{1-{\alpha }_1}\mu \left(\varrho \right){|}_{\varrho =\mathfrak{a}}={\mu }_0.\hfill \end{array}\right.$$

(2)

They proved that if ${\gamma }_1>-q,$ then (2) has no global solutions when ${\mu }_0\ge 0.$

In the current work, we investigate the nonlinear fractional differential problem containing a polynomial nonlinearity with variable coefficient

$$\left\{\begin{array}{c}{\mathcal{D}}_{\mathfrak{a}}^{{\alpha }_1}\mu \left(\varrho \right)+{\mathcal{D}}_{\mathfrak{a}}^{{\beta }_1}\mu \left(\varrho \right)={\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{{\gamma }_1}{\left|\nu \left(\varrho \right)\right|}^q,\varrho >\mathfrak{a}>0\hfill \\ \\ {\mathcal{D}}_{\mathfrak{a}}^{{\alpha }_2}\nu \left(\varrho \right)+{\mathcal{D}}_{\mathfrak{a}}^{{\beta }_2}\nu \left(\varrho \right)={\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{{\gamma }_2}{\left|\mu \left(\varrho \right)\right|}^p,\varrho >\mathfrak{a}>0\hfill \\ \\ {\mathcal{I}}_{\mathfrak{a}}^{1-{\alpha }_1}\mu \left(\varrho \right){{|}_{\varrho =\mathfrak{a}}={\mu }_0,{\mathcal{I}}_{\mathfrak{a}}^{1-{\alpha }_2}\nu \left(\varrho \right)|}_{\varrho =\mathfrak{a}}={\nu }_0,\hfill \end{array}\right.$$

(3)

where ${\mathcal{I}}_a^{\delta }$ and ${\mathcal{D}}_a^{\kappa }$ are the Hadamard fractional integral (HFI) and Hadamard fractional derivative (HFD), respectively (see Definitions 3 and 4 below).

In light of this problem, we demonstrate under certain appropriate assumptions that there is no non-trivial solution on $q,$ $p,$ ${\gamma }_i,$ ${\alpha }_i,$ ${\beta }_i,$ $i=1,2,$ and the initial conditions ${\mu }_0,{\nu }_0$ in a suitable space that we will define. Initially, we present some foundational findings that will be leveraged in the main result of this work. The proof relies primarily on carefully manipulation of the fractional derivatives, fractional integrals, and the test function method improved by Pohozaev and Mitidieri [53]. To the best of our knowledge, this particular type of fractional derivative, as well as the function spaces and weak solution spaces considered, have not been utilized in the literature on fractional systems with Hadamard-type derivatives of any order.

The research paper is presented as follows: Section 2 presents relevant notation, definitions, lemmas, and properties that will be used in the proof. Section 3 states and proves the result of non-existence, which is then demonstrated with a further example.

2. Preliminaries

Below, we present some basic principles that will form the foundation of this investigation. For more information on the applications and theory of fractional derivatives and fractional integrals, we refer the reader to [17,19,54,55,56].

Definition 1 

([19]). The space $X_k^p\left(c,d\right),$ $k\in \mathbb{R}$ and $1\le p<\infty$, is defined by

$$X_k^p\left(c,d\right)=\left\{\varpi :\left(c,d\right)\to \mathbb{R};\mathit{measurable}\mathit{in}\left(c,d\right):{∥\varpi ∥}_{X_k^p}<\infty \right\},$$

where

$${∥\varpi ∥}_{X_k^p}={\left({\int }_c^d{\left|{\zeta }^k\varpi \left(\zeta \right)\right|}^p{\displaystyle \frac{d\zeta }{\zeta }}\right)}^{1/p}.$$

Definition 2 

([19]).  The space $C_{\gamma ,ln}\left[c,d\right]$ is defined by

$$C_{\gamma ,ln}\left[c,d\right]=\left\{\varpi :(c,d]\to \mathbb{R};{\left(ln{\displaystyle \frac{\varrho }{c}}\right)}^{\gamma }\varpi \left(\varrho \right)\in C\left[c,d\right]\right\},0\le \gamma <1,d>c.$$

Equipped with the norm

$${∥\varpi ∥}_{C_{\gamma ,ln}={\left|{\left(ln{\displaystyle \frac{t}{c}}\right)}^{\gamma }\varpi \left(t\right)\right|}_C,C_{0,ln}(c,d]=C(c,d].}$$

Definition 3 

([19]). The left–sided and right–sided Hadamard fractional integrals of order $\alpha >0$ on $\left(\mathfrak{a},\eta \right)$ are defined, respectively, by

$${\mathcal{I}}_{\mathfrak{a}}^{\alpha }\varpi \left(\varrho \right):={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{\mathfrak{a}}^{\varrho }{\left(ln{\displaystyle \frac{\varrho }{\zeta }}\right)}^{\alpha -1}{\displaystyle \frac{\varpi \left(\zeta \right)d\zeta }{\zeta }},\mathfrak{a}<\varrho <\eta ,$$

$${\mathcal{I}}_{\eta }^{\alpha }\varpi \left(\varrho \right):={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{\varrho }^{\eta }{\left(ln{\displaystyle \frac{\zeta }{\varrho }}\right)}^{\alpha -1}{\displaystyle \frac{\varpi \left(\zeta \right)d\zeta }{\zeta }},\mathfrak{a}<\varrho <\eta ,$$

on the condition the integrals on the right-hand sides exist. When $\alpha =0$, we set

$${\mathcal{I}}_{\mathfrak{a}}^0\varpi ={\mathcal{I}}_{\eta }^0\varpi =\varpi .$$

Definition 4 

([19]). The Hadamard right-sided and left-sided fractional derivatives of order $0\le \alpha <1$ on $\left(a,\eta \right)$ are defined by

$$\left({\mathcal{D}}_{\eta }^{\alpha }\varpi \right)\left(\varrho \right):={\left(-\delta \right)}^n\left({\mathcal{I}}_{\eta }^{n-\alpha }\varpi \right)\left(\varrho \right),\varrho <\eta ,$$

$${\mathcal{D}}_{\mathfrak{a}}^{\alpha }\varpi \left(\varrho \right):={\delta }^n\left({\mathcal{I}}_{\mathfrak{a}}^{n-\alpha }\varpi \right)\left(\varrho \right),\varrho >\mathfrak{a},$$

respectively, where ${\delta }^n={\left(\varrho {\displaystyle \frac{d}{d\varrho }}\right)}^n$ and $n=-\left[-\alpha \right].$

Property 1 

([19]). If $\alpha ,$ $\beta >0$ and $0<a<\eta <\infty$, then

$$\left({\mathcal{I}}_{\mathfrak{a}}^{\alpha }{\left(ln{\displaystyle \frac{\zeta }{\mathfrak{a}}}\right)}^{\beta -1}\right)\left(\varrho \right)={\displaystyle \frac{\Gamma \left(\beta \right)}{\Gamma \left(\alpha +\beta \right)}}{\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{\beta +\alpha -1},\varrho >\mathfrak{a},$$

$$\left({\mathcal{I}}_{\eta }^{\alpha }{\left(ln{\displaystyle \frac{\eta }{\zeta }}\right)}^{\beta -1}\right)\left(\varrho \right)={\displaystyle \frac{\Gamma \left(\beta \right)}{\Gamma \left(\alpha +\beta \right)}}{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\beta +\alpha -1},\varrho <\eta ,$$

$$\left({\mathcal{D}}_{\mathfrak{a}}^{\alpha }{\left(ln{\displaystyle \frac{\zeta }{\mathfrak{a}}}\right)}^{\beta -1}\right)\left(\varrho \right)={\displaystyle \frac{\Gamma \left(\beta \right)}{\Gamma \left(\beta -\alpha \right)}}{\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{\beta -\alpha -1},\varrho >\mathfrak{a},$$

and

$$\left({\mathcal{D}}_{\eta }^{\alpha }{\left(ln{\displaystyle \frac{\eta }{\zeta }}\right)}^{\beta -1}\right)\left(\varrho \right)={\displaystyle \frac{\Gamma \left(\beta \right)}{\Gamma \left(\beta -\alpha \right)}}{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\beta -\alpha -1},\varrho <\eta .$$

Lemma 1 

([57]). Let $1\le p\le \infty$ and $\alpha >0$. If $\phi \in X_{-1/p}^q[0,\infty )$ and $\psi \in L_p[0,\infty ),$ then

$${\int }_0^{\infty }\psi \left(\varrho \right){\mathcal{I}}_{+}^{\alpha }\phi \left(\varrho \right){\displaystyle \frac{d\varrho }{\varrho }}={\int }_0^{\infty }\phi \left(\varrho \right){\mathcal{I}}_{-}^{\alpha }\psi \left(\varrho \right){\displaystyle \frac{d\varrho }{\varrho }},$$

where ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$.

Lemma 2 

([58,59]). For positive values of ς, τ, we have the following inequalities

$$2^{\chi -1}\left({\varsigma }^{\chi }+{\tau }^{\chi }\right)\le {\left(\varsigma +\tau \right)}^{\chi }\le {\tau }^{\chi }+{\varsigma }^{\chi },0\le \chi \le 1,$$

and

$${\varsigma }^{\chi }+{\tau }^{\chi }\le {\left(\varsigma +\tau \right)}^{\chi }\le 2^{\chi -1}\left({\varsigma }^{\chi }+{\tau }^{\chi }\right),\chi \ge 1.$$

Lemma 3. 

If $h\in C\left[a,\eta \right]$, then

$$\underset{\varrho \to {\mathfrak{a}}^{+}}{lim}{\mathcal{I}}_{\mathfrak{a}}^{\alpha }h\left(\varrho \right)={\mathcal{I}}_{\mathfrak{a}}^{\alpha }h\left(\mathfrak{a}\right)=0,\alpha >0$$

and

$$\underset{\varrho \to {\eta }^{-}}{lim}{\mathcal{I}}_{\eta }^{\alpha }h\left(\varrho \right)={\mathcal{I}}_{\eta }^{\alpha }h\left(\eta \right)=0,\alpha >0.$$

Proof. 

If $h\in C\left[\mathfrak{a},\eta \right]$, then

$$\left|h\left(\varrho \right)\right|<C,C>0.$$

Therefore,

$$\left|{\mathcal{I}}_{\mathfrak{a}}^{\alpha }h\left(\varrho \right)\right|\le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{\mathfrak{a}}^{\varrho }{\left(ln{\displaystyle \frac{\varrho }{\zeta }}\right)}^{\alpha -1}\left|h\left(\zeta \right)\right|{\displaystyle \frac{d\zeta }{\zeta }}$$

$$\le {\displaystyle \frac{C}{\Gamma \left(\alpha \right)}}{\int }_{\mathfrak{a}}^{\varrho }{\left(ln{\displaystyle \frac{\varrho }{\zeta }}\right)}^{\alpha -1}{\displaystyle \frac{d\zeta }{\zeta }}$$

$$\le {\displaystyle \frac{C}{\alpha \Gamma \left(\alpha \right)}}{\left[-{\left(ln{\displaystyle \frac{\varrho }{\zeta }}\right)}^{\alpha }\right]}_{\zeta =\mathfrak{a}}^{\varrho }={\displaystyle \frac{C}{\Gamma \left(1+\alpha \right)}}{\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{\alpha },\varrho >\mathfrak{a}.$$

When $\alpha >0,$ we obtain

$${\mathcal{I}}_{\mathfrak{a}}^{\alpha }h\left(\mathfrak{a}\right)=\underset{\varrho \to {\mathfrak{a}}^{+}}{lim}{\mathcal{I}}_{\mathfrak{a}}^{\alpha }h\left(\varrho \right)=0.$$

The second part is proved similarly. □

We use the test function

$$\begin{array}{cc}\phi \left(\varrho \right):=\hfill & \left\{\begin{array}{c}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\kappa },\kappa >0,\varrho \in \left[\mathfrak{a},\eta \right],\hfill \\ \\ 0,\varrho >\eta .\hfill \end{array}\right.\hfill \end{array}$$

(4)

In the lemmas below, we provide our initial estimate.

Lemma 4. 

Let $\alpha \ge 0$ and $\phi \left(\varrho \right)$ be as in (4). Then,

$${\mathcal{I}}_{\eta }^{\alpha }\phi \left(\varrho \right)={\displaystyle \frac{\Gamma \left(1+\kappa \right)}{\Gamma \left(1+\kappa +\alpha \right)}}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\kappa +\alpha },\varrho \in \left(\mathfrak{a},\eta \right),$$

and

$${\mathcal{D}}_{\eta }^{\alpha }\phi \left(\varrho \right)={\displaystyle \frac{\Gamma \left(1+\kappa \right)}{\Gamma \left(1+\kappa -\alpha \right)}}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\kappa -\alpha },\varrho \in \left(\mathfrak{a},\eta \right).$$

Proof. 

The proof follows directly from Property 1. □

Lemma 5. 

Let $\alpha \ge 0$ and $\phi \left(\varrho \right)$ be as in (4). Then,

$${\mathcal{I}}_{\eta }^{\alpha }\left(\delta \phi \right)\left(\varrho \right)=-{\displaystyle \frac{\Gamma \left(1+\kappa \right)}{\Gamma \left(\kappa +\alpha \right)}}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\kappa +\alpha -1},\varrho \in \left(\mathfrak{a},\eta \right),\delta =\varrho {\displaystyle \frac{d}{d\varrho }}.$$

Proof. 

We have

$$\left(\delta \phi \right)\left(\varrho \right)=\varrho {\displaystyle \frac{d}{d\varrho }}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\kappa }=-\kappa {\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\kappa -1},\varrho \in \left(\mathfrak{a},\eta \right).$$

Then,

$$\begin{array}{ccc}\hfill {\mathcal{I}}_{\eta }^{\alpha }\left(\delta \phi \right)\left(\varrho \right)& =& {\mathcal{I}}_{\eta }^{\alpha }\left(-\kappa {\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\kappa -1}\right)=-\kappa {\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }{\mathcal{I}}_{\eta }^{\alpha }{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\kappa -1}\hfill \\ & =& -\kappa {\displaystyle \frac{\Gamma \left(\kappa \right)}{\Gamma \left(\kappa +\alpha \right)}}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\alpha +\kappa -1}\hfill \\ & =& -{\displaystyle \frac{\Gamma \left(\kappa +1\right)}{\Gamma \left(\kappa +\alpha \right)}}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\alpha +\kappa -1},\varrho \in \left(\mathfrak{a},\eta \right).\hfill \end{array}$$

□

Lemma 6. 

Let $0<\alpha <1$ and $\phi \left(\varrho \right)$ be as in (4). If $\varpi \in AC\left[a,\eta \right],$ where $AC\left[a,\eta \right]$ is the space of real-valued absolutely continuous functions on $\left[a,\eta \right]$, then

$${\int }_{\mathfrak{a}}^{\eta }\phi \left(\varrho \right){\mathcal{D}}_{\mathfrak{a}}^{\alpha }\varpi \left(\varrho \right){\displaystyle \frac{d\varrho }{\varrho }}=-{\mathcal{I}}_{\mathfrak{a}}^{1-\alpha }\varpi \left(\mathfrak{a}\right)+{\displaystyle \frac{\Gamma \left(\kappa +1\right)}{\Gamma \left(\kappa +1-\alpha \right)}}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }{\int }_{\mathfrak{a}}^{\eta }{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\kappa -\alpha }\varpi \left(\varrho \right){\displaystyle \frac{d\varrho }{\varrho }}.$$

Proof. 

From Definition 4, we obtain

$${\int }_{\mathfrak{a}}^{\eta }\phi \left(\varrho \right){\mathcal{D}}_{\mathfrak{a}}^{\alpha }\varpi \left(\varrho \right){\displaystyle \frac{d\varrho }{\varrho }}={\int }_{\mathfrak{a}}^{\eta }\phi \left(\varrho \right)\left(\varrho {\displaystyle \frac{d}{d\varrho }}\right){\mathcal{I}}_{\mathfrak{a}}^{1-\alpha }\varpi \left(\varrho \right){\displaystyle \frac{d\varrho }{\varrho }}={\int }_{\mathfrak{a}}^{\eta }\phi \left(\varrho \right){\displaystyle \frac{d}{d\varrho }}{\mathcal{I}}_{\mathfrak{a}}^{1-\alpha }\varpi \left(\varrho \right)d\varrho .$$

(5)

An integration by parts in (5) yields

$${\int }_{\mathfrak{a}}^{\eta }\phi \left(\varrho \right){\mathcal{D}}_{\mathfrak{a}}^{\alpha }\varpi \left(\varrho \right){\displaystyle \frac{d\varrho }{\varrho }}={\left[\phi \left(\varrho \right){\mathcal{I}}_{\mathfrak{a}}^{1-\alpha }\varpi \left(\varrho \right)\right]}_{\eta =\mathfrak{a}}^{\eta }-{\int }_{\mathfrak{a}}^{\eta }{\phi }^{\prime }\left(\varrho \right){\mathcal{I}}_{\mathfrak{a}}^{1-\alpha }\varpi \left(\varrho \right)d\varrho .$$

Since $\phi \left(\eta \right)=0$ and $\phi \left(\mathfrak{a}\right)=1,$ the last equality becomes

$$\begin{array}{ccc}\hfill {\int }_{\mathfrak{a}}^{\eta }\phi \left(\varrho \right){\mathcal{D}}_{\mathfrak{a}}^{\alpha }\varpi \left(\varrho \right){\displaystyle \frac{d\varrho }{\varrho }}& =& -{\mathcal{I}}_{\mathfrak{a}}^{1-\alpha }\varpi \left(\mathfrak{a}\right)-{\int }_{\mathfrak{a}}^{\eta }{\phi }^{\prime }\left(\varrho \right){\mathcal{I}}_{\mathfrak{a}}^{1-\alpha }\varpi \left(\varrho \right)d\varrho \hfill \\ & =& -{\mathcal{I}}_{\mathfrak{a}}^{1-\alpha }\varpi \left(\mathfrak{a}\right)-{\int }_{\mathfrak{a}}^{\eta }\left(\varrho {\displaystyle \frac{d}{d\varrho }}\right)\phi \left(\varrho \right){\mathcal{I}}_{\mathfrak{a}}^{1-\alpha }\varpi \left(\varrho \right){\displaystyle \frac{d\varrho }{\varrho }}\hfill \\ & =& -{\mathcal{I}}_{\mathfrak{a}}^{1-\alpha }\varpi \left(\mathfrak{a}\right)-{\int }_{\mathfrak{a}}^{\eta }\left(\delta \phi \right)\left(\varrho \right){\mathcal{I}}_{\mathfrak{a}}^{1-\alpha }\varpi \left(\varrho \right){\displaystyle \frac{d\varrho }{\varrho }}.\hfill \end{array}$$

By using Lemmas 1 and 5, we find

$$\begin{array}{ccc}\hfill {\int }_{\mathfrak{a}}^{\eta }\phi \left(\varrho \right){\mathcal{D}}_{\mathfrak{a}}^{\alpha }\varpi \left(\varrho \right){\displaystyle \frac{d\varrho }{\varrho }}& =& -{\mathcal{I}}_{\mathfrak{a}}^{1-\alpha }\varpi \left(\mathfrak{a}\right)-{\int }_{\mathfrak{a}}^{\eta }\varpi \left(\varrho \right){\mathcal{I}}_{\eta }^{1-\alpha }\delta \phi \left(\varrho \right){\displaystyle \frac{d\varrho }{\varrho }}\hfill \\ & =& -{\mathcal{I}}_{\mathfrak{a}}^{1-\alpha }\varpi \left(\mathfrak{a}\right)+{\displaystyle \frac{\Gamma \left(\kappa +1\right)}{\Gamma \left(\kappa +1-\alpha \right)}}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }{\int }_{\mathfrak{a}}^{\eta }{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\kappa -\alpha }\varpi \left(\varrho \right){\displaystyle \frac{d\varrho }{\varrho }}.\hfill \end{array}$$

□

Lemma 7. 

Let $\phi \left(\varrho \right)$ be defined as in (4) with $\kappa +1>\beta$ and $\alpha ,\beta \ge 0$. Then,

$${\mathcal{I}}_{\eta }^{\alpha }{\mathcal{D}}_{\eta }^{\beta }\phi \left(\varrho \right)={\displaystyle \frac{\Gamma \left(\kappa +1\right)}{\Gamma \left(1+\kappa +\alpha -\beta \right)}}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\alpha +\kappa -\beta }.$$

Proof. 

By using Lemma 4, we obtain

$${\mathcal{D}}_{\eta }^{\beta }\phi \left(\varrho \right)={\displaystyle \frac{\Gamma \left(1+\kappa \right)}{\Gamma \left(1+\kappa -\beta \right)}}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\kappa -\beta }.$$

By virtue of Property 1, we obtain

$$\begin{array}{ccc}\hfill {\mathcal{I}}_{\eta }^{\alpha }{\mathcal{D}}_{\eta }^{\beta }\phi \left(\varrho \right)& =& {\displaystyle \frac{\Gamma \left(1+\kappa \right)}{\Gamma \left(1+\kappa -\beta \right)}}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }{\mathcal{I}}_{\eta }^{\alpha }{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\kappa -\beta }\hfill \\ & =& {\displaystyle \frac{\Gamma \left(1+\kappa \right)}{\Gamma \left(1+\kappa +\alpha -\beta \right)}}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\alpha +\kappa -\beta }.\hfill \end{array}$$

□

Lemma 8. 

Let $\phi \left(\varrho \right)$ be as in (4) with $\kappa >p\beta -1,$ $p>1$ and $\beta >0$. Then,

$${\int }_{\mathfrak{a}}^{\eta }{\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{\gamma \left(1-p\right)}{\phi }^{1-p}\left(\varrho \right){\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{p\left(\kappa -\beta \right)}{\displaystyle \frac{d\varrho }{\varrho }}=K_{\kappa ,\beta }^{\gamma ,p}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{\gamma +1+p\left(\kappa -\gamma -\beta \right)},$$

where

$$\gamma \left(1-p\right)+1>0\mathit{and}{K}_{\kappa ,\beta }^{\gamma ,p}={\displaystyle \frac{\Gamma \left(\kappa +1-\beta p\right)\Gamma \left(\gamma \left(1-p\right)+1\right)}{\Gamma \left(\gamma \left(1-p\right)+\kappa -\beta p+2\right)}}.$$

Proof. 

We find from (4)

$$\begin{array}{ccc}\hfill {\phi }^{1-p}\left(\varrho \right){\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{p\left(\kappa -\beta \right)}& =& {\left[{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\kappa }\right]}^{1-p}{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{p\left(\kappa -\beta \right)}\hfill \\ & =& {\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa \left(1-p\right)}{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\kappa -\beta p}.\hfill \end{array}$$

Then,

$$\begin{array}{ccc}& & {\int }_{\mathfrak{a}}^{\eta }{\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{\gamma \left(1-p\right)}{\phi }^{1-p}\left(\varrho \right){\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{p\left(\kappa -\beta \right)}{\displaystyle \frac{d\varrho }{\varrho }}\hfill \\ & =& {\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa \left(1-p\right)}{\int }_{\mathfrak{a}}^{\eta }{\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{\gamma \left(1-p\right)}{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\kappa -\beta p}{\displaystyle \frac{d\varrho }{\varrho }}.\hfill \end{array}$$

Let $\zeta ln{\displaystyle \frac{\eta }{a}}=ln{\displaystyle \frac{\varrho }{a}}.$ Therefore,

$$\begin{array}{ccc}& & {\int }_{\mathfrak{a}}^{\eta }{\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{\gamma \left(1-p\right)}{\phi }^{1-p}\left(\varrho \right){\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{p\left(\kappa -\beta \right)}{\displaystyle \frac{d\varrho }{\varrho }}\hfill \\ & =& {\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{\gamma \left(1-p\right)+\kappa -\kappa \left(1-p\right)-\beta p+1}{\int }_0^1{\zeta }^{\gamma \left(1-p\right)}{\left(1-\zeta \right)}^{\kappa -\beta p}d\zeta \hfill \\ & =& K_{\kappa ,\beta }^{\gamma ,p}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{\gamma +1+p\left(\kappa -\beta -\gamma \right)}.\hfill \end{array}$$

□

Lemma 9. 

Let $\phi \left(\varrho \right)$ be as in (4) with $\kappa >p\beta -1,$ $p>1$ and $\beta >0$. Then,

$${\int }_{\mathfrak{a}}^{\eta }{\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{\gamma \left(1-p\right)}{\phi }^{1-p}\left(\varrho \right){\left[{\mathcal{D}}_{\eta }^{\beta }\phi \left(\varrho \right)\right]}^p{\displaystyle \frac{d\varrho }{\varrho }}=C_{\kappa ,\beta }^{\gamma ,p}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{\gamma \left(1-p\right)-\beta p+1},$$

where

$$C_{\kappa ,\beta }^{\gamma ,p}={\displaystyle \frac{\Gamma \left(1+\kappa -\beta p\right)\Gamma \left(1+\gamma \left(1-p\right)\right)}{\Gamma \left(2+\gamma \left(1-p\right)+\kappa -\beta p\right)}}{\left[{\displaystyle \frac{\Gamma \left(\kappa +1\right)}{\Gamma \left(\kappa -\beta +1\right)}}\right]}^p,\gamma \left(1-p\right)>-1.$$

Proof. 

We obtain from (4) and Lemma 4:

$$\begin{array}{ccc}\hfill {\phi }^{1-p}\left(\varrho \right){\left[{\mathcal{D}}_{\eta }^{\beta }\phi \left(\varrho \right)\right]}^p& =& {\left[{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\kappa }\right]}^{1-p}\hfill \\ & & \times {\left[{\displaystyle \frac{\Gamma \left(\kappa +1\right)}{\Gamma \left(\kappa +1-\beta \right)}}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\kappa -\beta }\right]}^p\hfill \\ & =& {\left[{\displaystyle \frac{\Gamma \left(\kappa +1\right)}{\Gamma \left(\kappa +1-\beta \right)}}\right]}^p{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\kappa -\beta p}.\hfill \end{array}$$

Then,

$$\begin{array}{ccc}\hfill {\int }_{\mathfrak{a}}^{\eta }{\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{\gamma \left(1-p\right)}{\phi }^{1-p}\left(\varrho \right){\left[{\mathcal{D}}_{\eta }^{\beta }\phi \left(\varrho \right)\right]}^p{\displaystyle \frac{d\varrho }{\varrho }}& =& {\left[{\displaystyle \frac{\Gamma \left(\kappa +1\right)}{\Gamma \left(1+\kappa -\beta \right)}}\right]}^p{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }\hfill \\ & & \times {\int }_{\mathfrak{a}}^{\eta }{\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{\gamma \left(1-p\right)}{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\kappa -\beta p}{\displaystyle \frac{d\varrho }{\varrho }}.\hfill \end{array}$$

Let $ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}=\zeta ln{\displaystyle \frac{\eta }{\mathfrak{a}}}.$ Therefore,

$$\begin{array}{ccc}\hfill {\int }_{\mathfrak{a}}^{\eta }{\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{\kappa \left(p-1\right)}{\phi }^{1-p}\left(\varrho \right){\left[{\mathcal{D}}_{\eta }^{\beta }\phi \left(\varrho \right)\right]}^p{\displaystyle \frac{d\varrho }{\varrho }}& =& {\left[{\displaystyle \frac{\Gamma \left(1+\kappa \right)}{\Gamma \left(\kappa -\beta +1\right)}}\right]}^p{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{1+\gamma \left(1-p\right)-\beta p}\hfill \\ & & \times {\int }_0^1{\zeta }^{\gamma \left(1-p\right)}{\left(1-\zeta \right)}^{\kappa -\beta p}d\zeta \hfill \\ & =& C_{\kappa ,\beta }^{\gamma ,p}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{\gamma \left(1-p\right)-\beta p+1}.\hfill \end{array}$$

□

3. Non-Existence Result

This section is devoted to investigating the non-existence of non-trivial solutions for the following nonlinear system

$$\left\{\begin{array}{c}{\mathcal{D}}_{\mathfrak{a}}^{{\alpha }_1}\mu \left(\varrho \right)+{\mathcal{D}}_{\mathfrak{a}}^{{\beta }_1}\mu \left(\varrho \right)={\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{{\gamma }_1}{\left|v\left(\varrho \right)\right|}^q,\varrho >\mathfrak{a}>0,\\ \\ {\mathcal{D}}_{\mathfrak{a}}^{{\alpha }_2}\nu \left(\varrho \right)+{\mathcal{D}}_{\mathfrak{a}}^{{\beta }_2}\nu \left(\varrho \right)={\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{{\gamma }_2}{\left|\mu \left(\varrho \right)\right|}^p,\varrho >\mathfrak{a}>0\end{array}\right.$$

(6)

subject to the initial conditions

$${\mathcal{I}}_{\mathfrak{a}}^{1-{\alpha }_1}\mu \left(\varrho \right){{|}_{\varrho =a}={\mu }_0\mathrm{and}{\mathcal{I}}_{\mathfrak{a}}^{1-{\alpha }_2}\nu \left(\varrho \right)|}_{\varrho =\mathfrak{a}}={\nu }_0,$$

(7)

where ${\mathcal{D}}_{\mathfrak{a}}^{\kappa }$ is the HFD of order $\kappa >0,$ $0<{\beta }_i<{\alpha }_i<1$, $i=1,2,$ $p,$ $q>1$ and ${\gamma }_i,$ $i=1,2,$ are constants.

Theorem 1. 

Assume that ${\mathcal{I}}_{\mathfrak{a}}^{1-{\alpha }_1}\mu ,$ ${\mathcal{I}}_{\mathfrak{a}}^{1-{\alpha }_2}\nu \in AC\left[\mathfrak{a},\eta \right],$ $\eta >a,$ ${\gamma }_1<q-1$, ${\gamma }_2<p-1$, $\left(1+q\right)p>-\left({\gamma }_2+p{\gamma }_1\right)$ or $\left(1+p\right)q>-\left({\gamma }_1+q{\gamma }_2\right).$ Then, the system (6) does not admit non-trivial solutions when ${\mu }_0\ge 0$, ${\nu }_0\ge 0$.

Proof. 

We argue by contradiction. Assume $\left(\mu ,\nu \right)$ is a global solution of (6) and (7). Let $\phi \left(\varrho \right)$ be as in (4) with $\kappa >max\left\{{\displaystyle \frac{p{\alpha }_1}{p-1}}-1,{\displaystyle \frac{q{\alpha }_2}{q-1}}-1\right\}$. Multiplying (6) by $\phi \left(\varrho \right)/\varrho$ and integrating over $\left(a,\eta \right)$, we get

$${\int }_{\mathfrak{a}}^{\eta }\phi \left(\varrho \right){\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{{\gamma }_1}{\left|\nu \left(\varrho \right)\right|}^q{\displaystyle \frac{d\varrho }{\varrho }}={\int }_{\mathfrak{a}}^{\eta }\phi \left(\varrho \right){\mathcal{D}}_{\mathfrak{a}}^{{\alpha }_1}\mu \left(\varrho \right){\displaystyle \frac{d\varrho }{\varrho }}+{\int }_{\mathfrak{a}}^{\eta }\phi \left(\varrho \right){\mathcal{D}}_{\mathfrak{a}}^{{\beta }_1}\mu \left(\varrho \right){\displaystyle \frac{d\varrho }{\varrho }},$$

(8)

$${\int }_{\mathfrak{a}}^{\eta }\phi \left(\varrho \right){\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{{\gamma }_2}{\left|\mu \left(\varrho \right)\right|}^p{\displaystyle \frac{d\varrho }{\varrho }}={\int }_{\mathfrak{a}}^{\eta }\phi \left(\varrho \right){\mathcal{D}}_{\mathfrak{a}}^{{\alpha }_2}\nu \left(\varrho \right){\displaystyle \frac{d\varrho }{\varrho }}+{\int }_{\mathfrak{a}}^{\eta }\phi \left(\varrho \right){\mathcal{D}}_{\mathfrak{a}}^{{\beta }_2}\nu \left(\varrho \right){\displaystyle \frac{d\varrho }{\varrho }}.$$

(9)

Let

$$I_1={\int }_{\mathfrak{a}}^{\eta }\phi \left(\varrho \right){\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{{\gamma }_1}{\left|\nu \left(\varrho \right)\right|}^q{\displaystyle \frac{d\varrho }{\varrho }},$$

(10)

$$I_2={\int }_{\mathfrak{a}}^{\eta }\phi \left(\varrho \right){\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{{\gamma }_2}{\left|\mu \left(\varrho \right)\right|}^p{\displaystyle \frac{d\varrho }{\varrho }},$$

(11)

$$I_3={\int }_{\mathfrak{a}}^{\eta }\phi \left(\varrho \right){\mathcal{D}}_{\mathfrak{a}}^{{\alpha }_1}\mu \left(\varrho \right){\displaystyle \frac{d\varrho }{\varrho }},$$

(12)

$$I_4={\int }_{\mathfrak{a}}^{\eta }\phi \left(\varrho \right){\mathcal{D}}_{\mathfrak{a}}^{{\beta }_1}\mu \left(\varrho \right){\displaystyle \frac{d\varrho }{\varrho }},$$

(13)

$$I_5={\int }_{\mathfrak{a}}^{\eta }\phi \left(\varrho \right){\mathcal{D}}_{\mathfrak{a}}^{{\alpha }_2}\nu \left(\varrho \right){\displaystyle \frac{d\varrho }{\varrho }},$$

(14)

and

$$I_6={\int }_{\mathfrak{a}}^{\eta }\phi \left(\varrho \right){\mathcal{D}}_{\mathfrak{a}}^{{\beta }_2}\nu \left(\varrho \right){\displaystyle \frac{d\varrho }{\varrho }}.$$

(15)

Next, we shall estimate $I_3,$ $I_4$, $I_5$ and $I_6.$

From Lemma 6, we can write $I_3$ as follows

$$\begin{array}{ccc}\hfill I_3& =& {\int }_{\mathfrak{a}}^{\eta }\phi \left(\varrho \right){\mathcal{D}}_{\mathfrak{a}}^{{\alpha }_1}\mu \left(\varrho \right){\displaystyle \frac{d\varrho }{\varrho }}\hfill \\ & =& -{\mathcal{I}}_{\mathfrak{a}}^{1-{\alpha }_1}\mu \left(\mathfrak{a}\right)+{\displaystyle \frac{\Gamma \left(\kappa +1\right)}{\Gamma \left(\kappa +1-{\alpha }_1\right)}}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }{\int }_{\mathfrak{a}}^{\eta }\mu \left(\varrho \right){\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\kappa -{\alpha }_1}{\displaystyle \frac{d\varrho }{\varrho }}.\hfill \end{array}$$

(16)

Next, we multiply by ${\phi }^{1/p}\left(\varrho \right){\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{{\gamma }_2/p}{\phi }^{-1/p}\left(\varrho \right){\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{-{\gamma }_2/p}$ inside the integral of (16), we get

$$\begin{array}{ccc}\hfill I_3& =& -{\mu }_0+{\displaystyle \frac{\Gamma \left(\kappa +1\right)}{\Gamma \left(\kappa +1-{\alpha }_1\right)}}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }\hfill \\ & & \times {\int }_{\mathfrak{a}}^{\eta }{\phi }^{1/p}\left(\varrho \right)\mu \left(\varrho \right){\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{{\gamma }_2/p}{\phi }^{-1/p}\left(\varrho \right){\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{-{\gamma }_2/p}{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\kappa -{\alpha }_1}{\displaystyle \frac{d\varrho }{\varrho }}.\hfill \end{array}$$

(17)

By using the Hölder’s inequality, we find

$$\begin{array}{ccc}\hfill I_3& \le & -{\mu }_0+{\displaystyle \frac{\Gamma \left(\kappa +1\right)}{\Gamma \left(\kappa +1-{\alpha }_1\right)}}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }{\left({\int }_{\mathfrak{a}}^{\eta }\phi \left(\varrho \right){\left|\mu \left(\varrho \right)\right|}^p{\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{{\gamma }_2}{\displaystyle \frac{d\varrho }{\varrho }}\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & & \times {\left({\int }_{\mathfrak{a}}^{\eta }{\phi }^{-p^{\prime }/p}\left(\varrho \right){\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{-{\gamma }_2{p}^{\prime }/p}{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{p^{\prime }\left(\kappa -{\alpha }_1\right)}{\displaystyle \frac{d\varrho }{\varrho }}\right)}^{{\displaystyle \frac{1}{p^{\prime }}}}\hfill \\ & \le & -{\mu }_0+{\displaystyle \frac{\Gamma \left(\kappa +1\right)}{\Gamma \left(\kappa +1-{\alpha }_1\right)}}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }{I}_2^{{\displaystyle \frac{1}{p}}}\hfill \\ & & \times {\left({\int }_{\mathfrak{a}}^{\eta }{\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{{\gamma }_2\left(1-p^{\prime }\right)}{\phi }^{1-p^{\prime }}\left(\varrho \right){\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{p^{\prime }\left(\kappa -{\alpha }_1\right)}{\displaystyle \frac{d\varrho }{\varrho }}\right)}^{{\displaystyle \frac{1}{p^{\prime }}}},\hfill \end{array}$$

(18)

where p and $p^{\prime }$ satisfy $p+p^{\prime }=p{p}^{\prime }.$ By Lemma 8, we have

$$I_3\le -{\mu }_0+{\displaystyle \frac{\Gamma \left(\kappa +1\right){\left(K_{\kappa ,{\alpha }_1}^{{\gamma }_2,p^{\prime }}\right)}^{1/p^{\prime }}}{\Gamma \left(\kappa +1-{\alpha }_1\right)}}{\left(ln{\displaystyle \frac{\eta }{a}}\right)}^{-{\gamma }_2-{\alpha }_1+\left(1+{\gamma }_2\right)/p^{\prime }}{I}_2^{{\displaystyle \frac{1}{p}}}.$$

(19)

Since ${\mu }_0>0,$ then

$$I_3<{\displaystyle \frac{\Gamma \left(\kappa +1\right){\left(K_{\kappa ,{\alpha }_1}^{{\gamma }_2,p^{\prime }}\right)}^{1/p^{\prime }}}{\Gamma \left(\kappa +1-{\alpha }_1\right)}}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\gamma }_2-{\alpha }_1+\left(1+{\gamma }_2\right)/p^{\prime }}{I}_2^{{\displaystyle \frac{1}{p}}}.$$

(20)

Now, we turn to $I_4$. First, note that

$${\mathcal{I}}_{\mathfrak{a}}^{1-{\beta }_1}\mu ={\mathcal{I}}_{\mathfrak{a}}^{{\alpha }_1-{\beta }_1}{\mathcal{I}}^{1-{\alpha }_1}\mu \in AC\left[\mathfrak{a},\eta \right],$$

since ${\mathcal{I}}^{1-{\alpha }_1}\mu \in AC\left[a,\eta \right]$. Therefore,

$${\mathcal{I}}_{\mathfrak{a}}^{1-{\beta }_1}\mu \left(\mathfrak{a}\right)=\underset{\varrho \to a}{lim}{\mathcal{I}}_{\mathfrak{a}}^{1-{\beta }_1}\mu \left(\varrho \right)=\underset{\varrho \to \mathfrak{a}}{lim}{\mathcal{I}}_{\mathfrak{a}}^{{\alpha }_1-{\beta }_1}{\mathcal{I}}_{\mathfrak{a}}^{1-{\alpha }_1}\mu \left(\varrho \right)=0.$$

(21)

Now

$$\begin{array}{ccc}\hfill I_4& =& {\int }_{\mathfrak{a}}^{\eta }\phi \left(\varrho \right){\mathcal{D}}_{\mathfrak{a}}^{{\beta }_1}\mu \left(\varrho \right){\displaystyle \frac{d\varrho }{\varrho }}=-{\mathcal{I}}_{\mathfrak{a}}^{1-{\beta }_1}\mu \left(a\right)\hfill \\ & & +{\displaystyle \frac{\Gamma \left(\kappa +1\right)}{\Gamma \left(\kappa +1-{\beta }_1\right)}}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }{\int }_{\mathfrak{a}}^{\eta }\mu \left(\varrho \right){\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\kappa -{\beta }_1}{\displaystyle \frac{d\varrho }{\varrho }}\hfill \\ & =& {\displaystyle \frac{\Gamma \left(\kappa +1\right)}{\Gamma \left(\kappa +1-{\beta }_1\right)}}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }{\int }_a^{\eta }\mu \left(\varrho \right){\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\kappa -{\beta }_1}{\displaystyle \frac{d\varrho }{\varrho }}.\hfill \end{array}$$

(22)

Next, we multiply by ${\phi }^{1/p}\left(\varrho \right){\left(ln{\displaystyle \frac{\varrho }{a}}\right)}^{{\gamma }_2/p}{\phi }^{-1/p}\left(\varrho \right){\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{-{\gamma }_2/p}$ inside the integral of (22), we get

$$\begin{array}{ccc}\hfill I_4& =& {\displaystyle \frac{\Gamma \left(\kappa +1\right)}{\Gamma \left(\kappa +1-{\beta }_1\right)}}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }\hfill \\ & & \times {\int }_{\mathfrak{a}}^{\eta }\mu \left(\varrho \right){\phi }^{1/p}\left(\varrho \right){\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{{\gamma }_2/p}{\phi }^{-1/p}\left(\varrho \right){\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{-{\gamma }_2/p}{\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{\kappa -{\beta }_1}{\displaystyle \frac{d\varrho }{\varrho }}.\hfill \end{array}$$

Using the Hölder inequality, we find

$$\begin{array}{ccc}\hfill I_4& \le & {\displaystyle \frac{\Gamma \left(\kappa +1\right)}{\Gamma \left(\kappa +1-{\beta }_1\right)}}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-\kappa }{\left({\int }_{\mathfrak{a}}^{\eta }\phi \left(\varrho \right){\left|\mu \left(\varrho \right)\right|}^p{\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{{\gamma }_2}{\displaystyle \frac{d\varrho }{\varrho }}\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & & \times {\left({\int }_{\mathfrak{a}}^{\eta }{\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{{\gamma }_2\left(1-p^{\prime }\right)}{\phi }^{1-p^{\prime }}\left(\varrho \right){\left(ln{\displaystyle \frac{\eta }{\varrho }}\right)}^{p^{\prime }\left(\kappa -{\beta }_1\right)}{\displaystyle \frac{d\varrho }{\varrho }}\right)}^{{\displaystyle \frac{1}{p^{\prime }}}}.\hfill \end{array}$$

In the same way as above, replacing ${\alpha }_1$ with ${\beta }_1,$ we see that

$$I_4\le {\displaystyle \frac{\Gamma \left(\kappa +1\right){\left(K_{\kappa ,{\beta }_1}^{{\gamma }_2,p^{\prime }}\right)}^{1/p^{\prime }}}{\Gamma \left(\kappa +1-{\beta }_1\right)}}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\gamma }_2-{\beta }_1+\left(1+{\gamma }_2\right)/p^{\prime }}{I}_2^{{\displaystyle \frac{1}{p}}}.$$

(23)

Also, we can estimate $I_5$ and $I_6$ by

$$\begin{array}{ccc}\hfill I_5& \le & -{\nu }_0+{\displaystyle \frac{\Gamma \left(\kappa +1\right){\left(K_{\kappa ,{\alpha }_2}^{{\gamma }_1,q^{\prime }}\right)}^{1/q^{\prime }}}{\Gamma \left(\kappa +1-{\alpha }_2\right)}}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\gamma }_1-{\alpha }_2+\left({\gamma }_1+1\right)/q^{\prime }}{I}_1^{{\displaystyle \frac{1}{q}}}\hfill \end{array}$$

(24)

$$\begin{array}{ccc}& \le & {\displaystyle \frac{\Gamma \left(\kappa +1\right){\left(K_{\kappa ,{\alpha }_2}^{{\gamma }_1,q^{\prime }}\right)}^{1/q^{\prime }}}{\Gamma \left(\kappa +1-{\alpha }_2\right)}}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\gamma }_1-{\alpha }_2+\left({\gamma }_1+1\right)/q^{\prime }}{I}_1^{{\displaystyle \frac{1}{q}}},\hfill \end{array}$$

(25)

and

$$I_6\le {\displaystyle \frac{\Gamma \left(\kappa +1\right){\left(K_{\kappa ,{\beta }_2}^{{\gamma }_1,q^{\prime }}\right)}^{1/q^{\prime }}}{\Gamma \left(\kappa +1-{\beta }_2\right)}}{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\gamma }_1-{\beta }_2+\left({\gamma }_1+1\right)/q^{\prime }}{I}_1^{{\displaystyle \frac{1}{q}}}.$$

(26)

We use (20), (23), (25) and (26), to write (8) and (9) in the form

$$I_1\le K_2\left[{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\gamma }_2-{\alpha }_1+\left({\gamma }_2+1\right)/p^{\prime }}+{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\gamma }_2-{\beta }_1+\left({\gamma }_2+1\right)/p^{\prime }}\right]I_2^{{\displaystyle \frac{1}{p}}},$$

(27)

$$I_2\le K_3\left[{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\gamma }_1-{\alpha }_2+\left({\gamma }_1+1\right)/q^{\prime }}+{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\gamma }_1-{\beta }_2+\left({\gamma }_1+1\right)/q^{\prime }}\right]I_1^{{\displaystyle \frac{1}{q}}},$$

(28)

with

$$K_2=max\left[{\displaystyle \frac{\Gamma \left(\kappa +1\right){\left(K_{\kappa ,{\alpha }_1}^{{\gamma }_2,p^{\prime }}\right)}^{1/p^{\prime }}}{\Gamma \left(\kappa +1-{\alpha }_1\right)}},{\displaystyle \frac{\Gamma \left(\kappa +1\right){\left(K_{\kappa ,{\beta }_1}^{{\gamma }_2,p^{\prime }}\right)}^{1/p^{\prime }}}{\Gamma \left(\kappa +1-{\beta }_1\right)}}\right],$$

$$K_3=max\left[{\displaystyle \frac{\Gamma \left(\kappa +1\right){\left(K_{\kappa ,{\alpha }_2}^{{\gamma }_1,q^{\prime }}\right)}^{1/q^{\prime }}}{\Gamma \left(\kappa +1-{\alpha }_2\right)}},{\displaystyle \frac{\Gamma \left(\kappa +1\right){\left(K_{\kappa ,{\beta }_2}^{{\gamma }_1,q^{\prime }}\right)}^{1/q^{\prime }}}{\Gamma \left(\kappa +1-{\beta }_2\right)}}\right].$$

Consequently, (27) and (28) become

$$\begin{array}{ccc}\hfill I_1^{1-{\displaystyle \frac{1}{pq}}}& \le & K_2{K}_3^{1//p}\left[{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\gamma }_2-{\alpha }_1+\left({\gamma }_2+1\right)/p^{\prime }}+{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\gamma }_2-{\beta }_1+\left({\gamma }_2+1\right)/p^{\prime }}\right]\hfill \\ & & \times {\left[{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\gamma }_1-{\alpha }_2+\left({\gamma }_1+1\right)/q^{\prime }}+{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\gamma }_1-{\beta }_2+\left({\gamma }_1+1\right)/q^{\prime }}\right]}^{{\displaystyle \frac{1}{p}}},\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill I_2^{1-{\displaystyle \frac{1}{pq}}}& \le & K_3{K}_2^{1/q}\left[{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\gamma }_1-{\alpha }_2+\left({\gamma }_1+1\right)/q^{\prime }}+{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\gamma }_1-{\beta }_2+\left({\gamma }_1+1\right)/q^{\prime }}\right]\hfill \\ & & \times {\left[{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\gamma }_2-{\alpha }_1+\left({\gamma }_2+1\right)/p^{\prime }}+{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\gamma }_2-{\beta }_1+\left({\gamma }_2+1\right)/p^{\prime }}\right]}^{{\displaystyle \frac{1}{q}}}.\hfill \end{array}$$

(30)

Using Lemma 2 with $0\le \chi \le 1$, we estimate the inequalities (29) and (30) as

$$\begin{array}{ccc}\hfill I_1^{1-{\displaystyle \frac{1}{pq}}}& \le & K_2{K}_3^{1//p}\left[{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\alpha }_1-{\gamma }_2+\left({\gamma }_2+1\right)/p^{\prime }}+{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\gamma }_2-{\beta }_1+\left({\gamma }_2+1\right)/p^{\prime }}\right]\hfill \\ & & \times \left[{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\gamma }_1/p-{\alpha }_2/p+\left({\gamma }_1+1\right)/p{q}^{\prime }}+{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\gamma }_1/p-{\beta }_2/p+\left({\gamma }_1+1\right)/p{q}^{\prime }}\right],\hfill \end{array}$$

$$\begin{array}{ccc}\hfill I_2^{1-{\displaystyle \frac{1}{pq}}}& \le & K_3{K}_2^{1/q}\left[{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\gamma }_1-{\alpha }_2+\left({\gamma }_1+1\right)/q^{\prime }}+{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\gamma }_1-{\beta }_2+\left({\gamma }_1+1\right)/q^{\prime }}\right]\hfill \\ & & \times \left[{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\gamma }_2/q-{\alpha }_1/q+\left({\gamma }_2+1\right)/q{p}^{\prime }}+{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\gamma }_2/q-{\beta }_1/q+\left({\gamma }_2+1\right)/q{p}^{\prime }}\right],\hfill \end{array}$$

or

$$I_1^{1-{\displaystyle \frac{1}{pq}}}\le K_2{K}_3^{1//p}\left[{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{s_1}+{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{s_2}+{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{s_3}+{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{s_4}\right],$$

$$I_2^{1-{\displaystyle \frac{1}{pq}}}\le K_3{K}_2^{1/q}\left[{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{s_5}+{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{s_6}+{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{s_7}+{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{s_8}\right],$$

where

$$s_1=-{\gamma }_2-{\alpha }_1-{\alpha }_2/p-{\gamma }_1/p+\left(1+{\gamma }_2\right)/p^{\prime }+\left(1+{\gamma }_1\right)/p{q}^{\prime },$$

$$s_2=-{\gamma }_2-{\alpha }_1-{\beta }_2/p-{\gamma }_1/p+\left(1+{\gamma }_2\right)/p^{\prime }+\left(1+{\gamma }_1\right)/p{q}^{\prime },$$

$$s_3=-{\gamma }_2-{\beta }_1-{\alpha }_2/p-{\gamma }_1/p+\left(1+{\gamma }_2\right)/p^{\prime }+\left(1+{\gamma }_1\right)/p{q}^{\prime },$$

$$s_4=-{\gamma }_2-{\beta }_1-{\beta }_2/p-{\gamma }_1/p+\left(1+{\gamma }_2\right)/p^{\prime }+\left(1+{\gamma }_1\right)/p{q}^{\prime },$$

$$s_5=-{\gamma }_1-{\alpha }_2-{\alpha }_1/q-{\gamma }_2/q+\left(1+{\gamma }_1\right)/q^{\prime }+\left(1+{\gamma }_2\right)/q{p}^{\prime },$$

$$s_6=-{\gamma }_1-{\alpha }_2-{\beta }_1/q-{\gamma }_2/q+\left(1+{\gamma }_1\right)/q^{\prime }+\left(1+{\gamma }_2\right)/q{p}^{\prime },$$

$$s_7=-{\gamma }_1-{\beta }_2-{\alpha }_1/q-{\gamma }_2/q+\left(1+{\gamma }_1\right)/q^{\prime }+\left(1+{\gamma }_2\right)/q{p}^{\prime },$$

$$s_8=-{\gamma }_1-{\beta }_2-{\beta }_1/q-{\gamma }_2/q+\left(1+{\gamma }_1\right)/q^{\prime }+\left(1+{\gamma }_2\right)/q{p}^{\prime }.$$

In this case, if $1-{\displaystyle \frac{1}{pq}}<{\beta }_1+{\displaystyle \frac{{\beta }_2}{p}}+{\displaystyle \frac{{\gamma }_2}{p}}+{\displaystyle \frac{{\gamma }_1}{pq}},$ then $s_i<0,$ $i=1,2,3,4$ or if $1-{\displaystyle \frac{1}{pq}}<{\beta }_2+{\displaystyle \frac{{\beta }_1}{q}}+{\displaystyle \frac{{\gamma }_1}{q}}+{\displaystyle \frac{{\gamma }_2}{pq}},$ then $s_k<0,$ $k=5,6,7,8,$ and therefore ${\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{s_i}\to 0$ as $\eta \to \infty$. Then,

$$\underset{\eta \to \infty }{lim{I}_1=}\underset{\eta \to \infty }{lim{I}_2=}0.$$

This is a contradiction, since $\left(\mu ,\nu \right)$ is a non-trivial solution. □

Theorem 2. 

Assume that ${\mathcal{I}}_a^{1-{\alpha }_1}\mu ,$ ${\mathcal{I}}_{\mathfrak{a}}^{1-{\alpha }_2}\nu \in AC\left[\mathfrak{a},\eta \right],$ $\eta >\mathfrak{a},$ $\left(1-{\beta }_2\right)q-1<{\gamma }_1<q-1$, $\left(1-{\beta }_1\right)p-1<{\gamma }_2<p-1$, $\left(1+q\right)p\ge -\left({\gamma }_2+p{\gamma }_1\right)$ or $\left(1+p\right)q\ge -\left({\gamma }_1+q{\gamma }_2\right).$ Then, the system (6) does not admit non-trivial solutions when ${\mu }_0>0$, ${\nu }_0>0$.

Proof. 

We found from Theorem 1 that

$$I_1^{1-{\displaystyle \frac{1}{pq}}}\le K_2{K}_3^{1//p}\left[{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{s_1}+{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{s_2}+{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{s_3}+{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{s_4}\right],$$

and

$$I_2^{1-{\displaystyle \frac{1}{pq}}}\le K_3{K}_2^{1/q}\left[{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{s_5}+{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{s_6}+{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{s_7}+{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{s_8}\right].$$

In this case, if $1-{\displaystyle \frac{1}{pq}}\le {\beta }_1+{\displaystyle \frac{{\beta }_2}{p}}+{\displaystyle \frac{{\gamma }_2}{p}}+{\displaystyle \frac{{\gamma }_1}{pq}},$ then $s_i\le 0,$ $i=1,2,3,4$ or if $1-{\displaystyle \frac{1}{pq}}\le {\beta }_2+{\displaystyle \frac{{\beta }_1}{q}}+{\displaystyle \frac{{\gamma }_1}{q}}+{\displaystyle \frac{{\gamma }_2}{pq}},$ then $s_j\le 0,$ $j=5,6,7,8,$ and therefore $I_1$ and $I_2$ are bounded.

Now, from (19), (23), (24) and (26),

$${\mu }_0\le K_2\left[{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\gamma }_2-{\alpha }_1+\left({\gamma }_2+1\right)/p^{\prime }}+{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\gamma }_2-{\beta }_1+\left({\gamma }_2+1\right)/p^{\prime }}\right]I_2^{{\displaystyle \frac{1}{p}}},$$

$${\nu }_0\le K_3\left[{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\gamma }_1-{\alpha }_2+\left({\gamma }_1+1\right)/q^{\prime }}+{\left(ln{\displaystyle \frac{\eta }{\mathfrak{a}}}\right)}^{-{\gamma }_1-{\beta }_2+\left({\gamma }_1+1\right)/q^{\prime }}\right]I_1^{{\displaystyle \frac{1}{q}}}.$$

When $\eta \to \infty$, we obtain the contradiction $0<{\mu }_0,{\nu }_0\le 0.$ □

Example 1. 

The fractional differential system

$$\left\{\begin{array}{c}{\mathcal{D}}_{\mathfrak{a}}^{0.5}\mu \left(\varrho \right)+{\mathcal{D}}_{\mathfrak{a}}^{0.4}\mu \left(\varrho \right)={\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{-0.5}{\left|\nu \left(\varrho \right)\right|}^2,\varrho >\mathfrak{a}>0,\hfill \\ \\ {\mathcal{D}}_{\mathfrak{a}}^{0.6}\nu \left(\varrho \right)+{\mathcal{D}}_{\mathfrak{a}}^{0.3}\nu \left(\varrho \right)={\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{0.5}{\left|\mu \left(\varrho \right)\right|}^2,\varrho >a>0,\hfill \\ \\ {\mathcal{I}}_{\mathfrak{a}}^{1-{\alpha }_1}\mu \left(\varrho \right){{|}_{\varrho =\mathfrak{a}}=1,{\mathcal{I}}_{\mathfrak{a}}^{1-{\alpha }_2}\nu \left(\varrho \right)|}_{\varrho =\mathfrak{a}}=2,\hfill \end{array}\right.$$

(31)

is a special case of (3) when ${\alpha }_1=0.5,{\alpha }_2=0.6,$ ${\beta }_1=0.4,$ ${\beta }_2=0.3$, ${\gamma }_1=-0.5,$ ${\gamma }_2=0.5,$ ${\mu }_0=1$, ${\nu }_0=2$ and $p=q=2$. Consequently, Theorem 1, implies that the system (31) does not possess a non-trivial global solution.

Example 2. 

The fractional differential system

$$\left\{\begin{array}{c}{\mathcal{D}}_{\mathfrak{a}}^{{\alpha }_1}\mu \left(\varrho \right)+{\mathcal{D}}_{\mathfrak{a}}^{{\beta }_1}\mu \left(\varrho \right)={\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{{\gamma }_1}{\left|v\left(\varrho \right)\right|}^q,\varrho >\mathfrak{a}>0,\\ \\ {\mathcal{D}}_{\mathfrak{a}}^{{\alpha }_2}\nu \left(\varrho \right)+{\mathcal{D}}_{\mathfrak{a}}^{{\beta }_2}\nu \left(\varrho \right)={\left(ln{\displaystyle \frac{\varrho }{\mathfrak{a}}}\right)}^{{\gamma }_2}{\left|\mu \left(\varrho \right)\right|}^p,\varrho >\mathfrak{a}>0,\\ \\ {\mathcal{I}}_{\mathfrak{a}}^{1-{\alpha }_1}\mu \left(\varrho \right){{|}_{\varrho =a}={\mu }_0,{\mathcal{I}}_{\mathfrak{a}}^{1-{\alpha }_2}\nu \left(\varrho \right)|}_{\varrho =\mathfrak{a}}={\nu }_0,\end{array}\right.$$

(32)

is a special case of (3) when ${\alpha }_1=0.8,{\alpha }_2=0.7,{\beta }_1=0.4,{\beta }_2=0.6,{\gamma }_1=0.5,$ ${\gamma }_2=0.3,p=2,q=3,{\mu }_0=1,{\nu }_0=2.$ Thus, Theorem 2 shows that system (32) does not possess a non-trivial global solution.

4. Conclusions

In this study, we investigated the non-existence of global solutions for a system of fractional differential equations involving Hadamard fractional derivatives (HFDs) of varying orders. The system incorporated a nonlinear source term with nonlocal temporal behavior. Our findings were established using the test function method, properties of fractional derivatives, and integral inequality techniques. Notably, we proved that under specific conditions on the parameters ($q,$ $p,$ ${\gamma }_i,$ ${\alpha }_i,$ ${\beta }_i,$ $i=1,2$) and the initial values (${\mu }_0,{\nu }_0$), the system admits no non-trivial global solutions. This result contributes to a deeper understanding of the dynamics of fractional systems with Hadamard-type derivatives. The authors suggest several potential directions for future research based on this work:

• Investigate the case of higher-order Hadamard-type fractional derivatives, which may require different techniques.
• Consider fractional differential equations with more general nonlinear terms, such as those involving discontinuous nonlinearities or nonlocal conditions.
• Investigate whether the sufficient conditions for the non-existence of solutions can be relaxed while maintaining the core result.