On Global Solutions to a ψ-Caputo Fractional Inequality

Abstract

This paper investigates the absence of globally defined, non-trivial solutions for some fractional differential inequalities of the $\psi$-Caputo type, with polynomial sources involving fractional derivatives. We rigorously establish these results within an appropriate function space using properties of $\psi$-fractional integrals and derivatives and the test function technique. To demonstrate the applicability and enhance understanding, we provide three illustrative examples. Our findings broaden the scope of previous literature, encompassing existing results as special cases.

1. Introduction

Over the past three decades, fractional differential equations (FDEs) have attracted significant attention across many scientific and engineering disciplines, as they often model complex systems more accurately than classical integer-order equations. This enhanced modeling capability arises from the nonlocal character of fractional derivatives, which naturally incorporate memory and hereditary effects, providing more realistic descriptions of physical processes [1,2,3]. Applications span physics, biology, and engineering, including epidemiological modeling and control systems [4,5].

Among fractional operators, the $\psi$-Caputo derivative—defined with respect to a strictly increasing function $\psi$—generalizes the classical Caputo derivative by allowing nonuniform memory effects and time-scaling phenomena. It reduces to the standard Caputo derivative when $\psi \left(t\right)=t$. Key foundational results for Caputo-type derivatives and their applications can be found in [6,7,8], while extensions involving $\psi$-fractional operators are discussed in [9,10].

The $\psi$-Caputo framework is particularly useful for the study of fractional differential inequalities, which arise in qualitative analysis when establishing a priori estimates, stability, boundedness of solutions, and blow-up behavior in the presence of perturbations, uncertainties, or nonlinear source terms. Unlike their ordinary counterparts, $\psi$-Caputo fractional inequalities explicitly account for history-dependent evolution, making them suitable for modeling viscoelastic materials, anomalous diffusion, and control systems with memory.

Existence and uniqueness results for many classes of FDEs have been extensively studied. Early surveys and foundational works addressing the existence and stability of solutions for nonlinear fractional differential equations are reported in [11,12,13,14]. Later studies extended these results to Hilfer, Hadamard, and generalized fractional operators, including nonlocal initial conditions and pantograph-type equations [15,16,17,18,19]. More recent works have focused on asymptotic behavior, convergence, and global properties of solutions for generalized fractional systems [20,21,22,23].

In contrast, results on nonexistence and blow-up criteria are relatively scarce. This is due to the analytical challenges posed by the nonlocal nature of fractional operators and the subtleties of initial conditions. Important early contributions on blow-up and critical exponents are found in [24,25]. Nonexistence results for fractional Schrodinger equations, diffusion equations, and related systems have been studied in [26,27,28,29,30], while more recent investigations for Hilfer–Katugampola and $\psi$-type inequalities appear in [31,32,33].

Fractional differential inequalities of the Caputo type have proven to be powerful tools for extending classical qualitative results—such as comparison principles, Grönwall-type estimates, and blow-up criteria—from ordinary differential equations to systems with memory and hereditary effects. In practice, these inequalities capture how nonlinear source terms interact with accumulated past states, and they can indicate whether memory mechanisms stabilize the system or allow finite-time singularities.

Beyond abstract theory, the class of $\psi$-Caputo fractional inequalities studied here admits a meaningful applied interpretation. Such inequalities naturally arise in systems where the present evolution depends on the cumulative influence of past states. In this setting, lower-order fractional derivatives correspond to stronger memory effects, since they weight the system’s history more heavily through slowly decaying kernels. This persistent memory tends to slow the system’s response and can counteract rapid growth induced by nonlinear source terms, effectively acting as a damping mechanism. Conversely, higher-order fractional derivatives weaken memory effects and permit faster dynamics. This qualitative distinction provides insight into why the presence of lower-order fractional derivatives can narrow the range of parameters leading to finite-time blow-up—an effect that has no analogue in classical integer-order models.

Crucially, the conditions we derive for $\psi$-Caputo inequalities reduce to well-known ODE results when $\psi \left(t\right)=t$ and the derivative orders are integers. In this limit, our nonexistence and blow-up criteria coincide with classical criteria, confirming that the present framework is a consistent generalization: the extra restrictions that appear in the fractional case reflect genuine memory effects rather than artefacts of the method.

We now formulate the problem under investigation. Consider

$$\left\{\begin{array}{c}{}^{C}D_a^{\psi ,\kappa }y\left(t\right)+{}^{C}D_a^{\psi ,\upsilon }y\left(t\right)=f\left(t,y\left(t\right)\right),t>a,\hfill \\ y_{\psi }^{\left[i\right]}\left(a\right)=b_i,i=0,\dots ,n-1,\hfill \end{array}\right.$$

(1)

where ${}^{C}D_a^{\psi ,\sigma }$ denotes the $\psi$-Caputo derivative of order $\sigma >0$ (defined below). The parameters $\kappa ,$ $\upsilon \in \left(n,n+1\right),$ where $n=-\left[-\kappa \right]$ denotes the smallest integer greater than or equal to $\kappa$.

To study global solvability, we restrict to $\kappa =\alpha +1,$ $\upsilon =\beta +1$ with $\alpha ,$ $\beta \in \left(n-1,n\right)$ and $n=-\left[-\alpha \right].$ We assume the nonlinearity satisfies

$$f\left(t,y\left(t\right)\right)\ge {\left[\psi \left(t\right)-\psi \left(a\right)\right]}^{\rho }{\left|{}^{C}D_a^{\psi ,\gamma }y\left(t\right)\right|}^m,$$

for parameters $m>1$, $\rho \in \mathbb{R}$ and $\gamma \in \left(n-1,n\right).$ Under these hypotheses, Problem (1) reduces to the inequality problem:

$$\left\{\begin{array}{c}{}^{C}D_a^{\psi ,\alpha +1}y\left(t\right)+{}^{C}D_a^{\psi ,\beta +1}y\left(t\right)\ge {\left[\psi \left(t\right)-\psi \left(a\right)\right]}^{\rho }{\left|{}^{C}D_a^{\psi ,\gamma }y\left(t\right)\right|}^m,t>a,\hfill \\ \\ y_{\psi }^{\left[i\right]}\left(a\right)=b_i,i=0,1,\dots ,n,\hfill \end{array}\right.$$

(2)

where $\alpha ,$ $\beta ,$ $\gamma \in \left(n-1,n\right)$ and $n=-\left[-\alpha \right]\in \mathbb{N}.$

Our main goal is to determine parameter regimes for $\rho$ and m under which Problem (2) admits no nontrivial global solution. Such nonexistence conditions are important because they provide necessary constraints for existence results and clarify how memory and fractional orders affect blow-up dynamics.

To position our work with existing literature and to demonstrate the generality of the framework, we briefly review several classical and fractional problems that arise as special cases of Problem (1). These examples illustrate consistency with known results and clarify how previously studied models fit into our setting.

For $\psi \left(t\right)=t,$ $f\left(t,y\right)=2t^{\rho }{y}^m\left(t\right)$, $a=0$ and $\kappa =\upsilon =1$, Problem (1) reduces to

$$\left\{\begin{array}{c}{y}^{\prime }\left(t\right)=t^{\rho }{\left|y\left(t\right)\right|}^m,\hfill \\ y\left(0\right)=b,\hfill \end{array}\right.$$

whose explicit solution for $\rho \ne -1$ is

$$y\left(t\right)={\left[{\displaystyle \frac{1-m}{1+\rho }}{t}^{\rho +1}+b^{1-m}\right]}^{1/\left(1-m\right)},t\ge 0.$$

For $\rho >-1$ and $m>1$, this yields finite-time blow-up at

$$t^{*}={\left[{\displaystyle \frac{b^{1-m}\left(1+\rho \right)}{m-1}}\right]}^{1/\left(1+\rho \right)}.$$

In case $f\left(t,y\right)=y^m\left(t\right),$ $\psi \left(t\right)=t$, $a=0,$ $\kappa =1$ and $\upsilon =0$ in (1), we recognize the so-called Bernoulli differential problem

$$\left\{\begin{array}{c}{y}^{\prime }\left(t\right)+y\left(t\right)=y^m\left(t\right),\hfill \\ y\left(0\right)=b,\hfill \end{array}\right.$$

whose solution is given by

$$y\left(t\right)={\left[\left(b^{1-m}-1\right)\exp \left(m-1\right)t+1\right]}^{1/\left(1-m\right)}.$$

It becomes evident that $y\left(t\right)$ experiences a blow-up, or singularity, at the finite time given by $t^{*}={\displaystyle \frac{1}{1-m}}\ln \left(1-b^{1-m}\right)$ when $m,b>1.$

By focusing on the specific case $\psi \left(t\right)=t$, $a=0$ and $\kappa =\upsilon ,$ Problem (1) with $f\left(t,y\right)\ge 2t^{\gamma }{\left|y\left(t\right)\right|}^m$ is reduced to a problem involving one single non-integer order:

$$\left\{\begin{array}{c}{D}_0^{\alpha }y\left(t\right)\ge t^{\gamma }{\left|y\left(t\right)\right|}^m,t>0,\hfill \\ I_0^{1-\alpha }y\left(t\right){|}_{t=0}=b.\hfill \end{array}\right.$$

(3)

Problem (3) has been meticulously explored by Laskri and Tatar [24]. Their findings reveal that if $1<m\le {\displaystyle \frac{\gamma +1}{1-\alpha }}$ and $\gamma >-\alpha$, then Problem (3) does not possess any global nontrivial solution when $b\ge 0$.

When $f\left(t,y\right)\ge t^{\rho }{\left|y\left(t\right)\right|}^m,$ $\psi \left(t\right)=t$ and $a=0,$ Problem (1) becomes

$$\left\{\begin{array}{c}{}^{C}D_0^{\kappa }y\left(t\right)+{}^{C}D_0^{\upsilon }y\left(t\right)\ge t^{\rho }{\left|y\left(t\right)\right|}^m,t>0,\hfill \\ y^{\left(k\right)}\left(0\right)=b_k,k=0,\dots ,n-1,\hfill \end{array}\right.$$

(4)

which was analyzed in [27]. Under suitable parameter constraints, the authors obtain nonexistence results consistent with the fractional effects we describe.

Setting $\psi \left(t\right)=t$, $a=0,$ $0<\kappa ,\upsilon ,\gamma <1$ and $\rho =0$ in (2) yields

$$\left\{\begin{array}{c}{}^{C}D_0^{\kappa +1}y\left(t\right)+{}^{C}D_0^{\upsilon +1}y\left(t\right)\ge {\left|{}^{C}D_0^{\gamma }y\left(t\right)\right|}^m,t>0,\hfill \\ y\left(0\right)=b_0,y^{\prime }\left(0\right)=b_1,\hfill \end{array}\right.$$

(5)

a problem recently treated by Jleli and Samet [32], who obtained nonexistence conditions in agreement with our general conclusions.

For $\psi \left(t\right)=\ln t$ and $f\left(t,y\right)\ge {\left(\ln {\displaystyle \frac{t}{a}}\right)}^{\gamma }{\left|y\left(t\right)\right|}^m$, one obtains

$$\left\{\begin{array}{c}{D}_a^{\kappa }y\left(t\right)+D_a^{\upsilon }y\left(t\right)\ge {\left(\ln {\displaystyle \frac{t}{a}}\right)}^{\gamma }{\left|y\left(t\right)\right|}^m,t>a,\hfill \\ I_a^{1-\alpha }y\left(t\right){|}_{t=a}=b,\hfill \end{array}\right.$$

(6)

which was considered in [33]. Under the condition $\gamma >-m$ (or $\gamma >-1$ in certain formulations), the nonexistence of global nontrivial solutions is obtained.

The above reductions demonstrate that many classical and recently studied fractional models are contained within our framework. Moreover, they clarify how our hypotheses compare with those in the existing literature.

In contrast to most existing works, which focus on single-order fractional derivatives or fixed kernels, this paper addresses $\psi$-Caputo inequalities involving two distinct fractional orders under minimal assumptions on the function $\psi$.

Motivated by the influence of lower-order derivatives on blow-up phenomena, the present work investigates $\psi$-Caputo fractional inequalities involving two distinct fractional orders, a setting that goes beyond most existing studies, which are often restricted to single fractional derivatives or to fixed kernels. By allowing an arbitrary increasing function $\psi$ and wide parameter ranges, we derive nonexistence results in a significantly more general framework; several known results appear as special cases of our theorems, which illustrates the genuine extension achieved by our analysis.

The paper is organized as follows. Section 2 introduces essential definitions, notations, and auxiliary lemmas. Section 3 presents the test-function method and related technical results. Section 4 contains the main nonexistence theorem and discusses its implications for blow-up behavior. Finally, Section 5 provides illustrative examples supporting the theoretical findings.

2. Preliminaries

Below we introduce some essential definitions and lemmas that will be utilized in our results. Throughout the paper, we consider $\left(a,b\right)$ to be an infinite or finite interval, and $\psi$ to be an n-continuously differentiable function on $\left(a,b\right)$ such that $\psi$ is increasing and ${\psi }^{\prime }\left(\varkappa \right)\ne 0$ on $\left(a,b\right)$.

Definition 1.

We introduce the following standard function spaces:

$AC\left[a,b\right]:$ The space of all functions that are absolutely continuous on the interval $\left[a,b\right].$

$A{C}^n\left[a,b\right]:$ The space of those functions f that meet the following criteria:

• Their $\left(n-1\right)-th$ derivatives, denoted by $f^{\left(n-1\right)}$, belong to $AC\left[a,b\right]$.
• They possess continuous derivatives up to order $n-1$ on the interval $\left[a,b\right]$.

Definition 2.

Let $p\ge 1$. The space of p-summable functions with respect to the function ψ is defined as follows:

$$L_{\psi }^p\left(a,b\right)=\left\{\omega :\left[a,b\right]\to \mathbb{R}:{\int }_a^b{\left|\omega \left(z\right)\right|}^p{\psi }^{\prime }\left(z\right)dz<\infty \right\}.$$

Definition 3

([6]). The ψ-Riemann–Liouville left-sided and right-sided fractional integrals of a given function ω with respect to another fixed function ψ are defined as

$$I_a^{\psi ,\alpha }\omega \left(z\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^z{\left[\psi \left(z\right)-\psi \left(s\right)\right]}^{\alpha -1}\omega \left(s\right){\psi }^{\prime }\left(s\right)ds,\alpha >0,z>a$$

and

$$I_b^{\psi ,\alpha }\omega \left(z\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_z^b{\left[\psi \left(s\right)-\psi \left(z\right)\right]}^{\alpha -1}\omega \left(s\right){\psi }^{\prime }\left(s\right)ds,\alpha >0,z<b$$

respectively, as long as the right-hand sides exist. When $\alpha =0$, we define $I_a^{\psi ,0}\omega =I_b^{\psi ,0}\omega =\omega$.

Definition 4

([9]). The ψ-Caputo left-sided and right-sided fractional derivatives of order $\varrho >0$ are defined by

$$\begin{array}{ccc}\hfill {}^{C}D_a^{\psi ,\varrho }\omega \left(\varkappa \right)& =& I_a^{\psi ,n-\varrho }{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(\varkappa \right)}}{\displaystyle \frac{d}{d\varkappa }}\right)}^n\omega \left(\varkappa \right)\hfill \\ & & \\ & =& {\displaystyle \frac{1}{\Gamma \left(n-\varrho \right)}}{\int }_a^{\varkappa }{\left[\psi \left(\varkappa \right)-\psi \left(\tau \right)\right]}^{n-\varrho -1}{\psi }^{\prime }\left(\tau \right){\omega }_{\psi }^{\left[n\right]}\left(\tau \right)d\tau ,\varkappa >a,\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {}^{C}D_b^{\psi ,\varrho }\omega \left(\varkappa \right)& =& I_b^{\psi ,n-\varrho }{\left(-{\displaystyle \frac{1}{{\psi }^{\prime }\left(\varkappa \right)}}{\displaystyle \frac{d}{d\varkappa }}\right)}^n\omega \left(\varkappa \right)\hfill \\ & & \\ & =& {\displaystyle \frac{1}{\Gamma \left(n-\varrho \right)}}{\int }_{\varkappa }^b{\left[\psi \left(\tau \right)-\psi \left(\varkappa \right)\right]}^{n-\varrho -1}{\psi }^{\prime }\left(\tau \right){\left(-1\right)}^n{\omega }_{\psi }^{\left[n\right]}\left(\tau \right)d\tau ,\varkappa <b,\hfill \end{array}$$

respectively, where

$$n=-\left[-\varrho \right],{\omega }_{\psi }^{\left[n\right]}\left(\varkappa \right)={\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(\varkappa \right)}}{\displaystyle \frac{d}{d\varkappa }}\right)}^n\omega \left(\varkappa \right).$$

Particularly, when $0<\varrho <1$

$$\begin{array}{ccc}\hfill {}^{C}D_a^{\psi ,\varrho }\omega \left(\varkappa \right)& =& I_a^{\psi ,1-\varrho }\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(\varkappa \right)}}{\displaystyle \frac{d}{d\varkappa }}\right)\omega \left(\varkappa \right)\hfill \\ & & \\ & =& {\displaystyle \frac{1}{\Gamma \left(1-\varrho \right)}}{\int }_a^{\varkappa }{\left[\psi \left(\varkappa \right)-\psi \left(\tau \right)\right]}^{-\varrho }{\omega }^{\prime }\left(\tau \right)d\tau ,\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {}^{C}D_b^{\psi ,\varrho }\omega \left(\varkappa \right)& =& I_b^{\psi ,1-\varrho }\left(-{\displaystyle \frac{1}{{\psi }^{\prime }\left(\varkappa \right)}}{\displaystyle \frac{d}{d\varkappa }}\right)\omega \left(\varkappa \right)\hfill \\ & & \\ & =& {\displaystyle \frac{-1}{\Gamma \left(1-\varrho \right)}}{\int }_{\varkappa }^b{\left[\psi \left(\tau \right)-\psi \left(\varkappa \right)\right]}^{-\varrho }{\omega }^{\prime }\left(\tau \right)d\tau .\hfill \end{array}$$

Next, we shall recall some lemmas.

Lemma 1

([10]). Let $\alpha >0,$ $q,$ $p\ge 1$ and ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}\le 1$. If $\Psi \in L_{\psi }^q\left(a,b\right)$ and $\Phi \in L_{\psi }^p\left(a,b\right),$ then

$${\int }_a^b\Phi \left(t\right)I_a^{\psi ,\alpha }\Psi \left(t\right){\psi }^{\prime }\left(t\right)dt={\int }_a^b\Psi \left(t\right)I_b^{\psi ,\alpha }\Phi \left(t\right){\psi }^{\prime }\left(t\right)dt.$$

(7)

Lemma 2

([9]). If $\alpha \ge 0$ and $\beta >0,$ then

$$I_T^{\psi ,\alpha }{\left[\psi \left(T\right)-\psi \left(t\right)\right]}^{\beta }={\displaystyle \frac{\Gamma \left(\beta +1\right)}{\Gamma \left(\beta +\alpha +1\right)}}{\left[\psi \left(T\right)-\psi \left(t\right)\right]}^{\beta +\alpha },$$

$$D_T^{\psi ,\alpha }{\left[\psi \left(T\right)-\psi \left(t\right)\right]}^{\beta }={\displaystyle \frac{\Gamma \left(\beta +1\right)}{\Gamma \left(\beta -\alpha +1\right)}}{\left[\psi \left(T\right)-\psi \left(t\right)\right]}^{\beta -\alpha },$$

$${}^{C}D_T^{\psi ,\alpha }{\left[\psi \left(T\right)-\psi \left(t\right)\right]}^{\beta }={\displaystyle \frac{\Gamma \left(\beta +1\right)}{\Gamma \left(\beta -\alpha +1\right)}}{\left[\psi \left(T\right)-\psi \left(t\right)\right]}^{\beta -\alpha }.$$

Lemma 3

([9]). Let $\alpha >0$ and $\beta >0.$ If $h\in L_{\psi }^p\left(a,b\right),$ then

$$I_a^{\psi ,\beta }{I}_a^{\psi ,\alpha }h\left(t\right)=I_a^{\psi ,\beta +\alpha }h\left(t\right),t>a,$$

$$I_b^{\psi ,\beta }{I}_b^{\psi ,\alpha }h\left(t\right)=I_b^{\psi ,\beta +\alpha }h\left(t\right),t<b.$$

Lemma 4

([9]). Given a function $h\in C^n\left[a,b\right]$ and $\alpha >0,$ the identity below holds:

$$I_a^{\psi ,\alpha }{}^{C}D_a^{\psi ,\alpha }h\left(t\right)=h\left(t\right)-\sum _{k=0}^{n-1}{\displaystyle \frac{h_{\psi }^{\left[k\right]}\left(a\right)}{k!}}{\left[\psi \left(t\right)-\psi \left(a\right)\right]}^k,t\ge a.$$

Lemma 5.

Let $\alpha \in \left(n-1,n\right),$ $n\in \mathbb{N},$ and $g\in C^n\left[a,b\right].$ Then, the initial value problem

$$\left\{\begin{array}{c}{}^{C}D_a^{\psi ,\alpha }y\left(t\right)=g\left(t\right),t>a,\\ y_{\psi }^{\left[i\right]}\left(a\right)=d_i,i=0,\dots ,n-1\end{array}\right.$$

(8)

admits the solution

$$y\left(t\right)=d_0+d_1\left[\psi \left(t\right)-\psi \left(a\right)\right]+\dots +{\displaystyle \frac{d_{n-1}}{\left(n-1\right)!}}{\left[\psi \left(t\right)-\psi \left(a\right)\right]}^{n-1}+I_a^{\psi ,\alpha }g\left(t\right),t\ge a.$$

Proof. 

First, we apply $I_a^{\psi ,\alpha }$ to both sides of (8) to obtain

$$I_a^{\psi ,\alpha }{}^{C}D_a^{\psi ,\alpha }y\left(t\right)=I_a^{\psi ,\alpha }g\left(t\right).$$

The next identity is a direct consequence of Lemma 4:

$$y\left(t\right)-\sum _{k=0}^{n-1}{\displaystyle \frac{y_{\psi }^{\left[k\right]}\left(a\right)}{k!}}{\left[\psi \left(t\right)-\psi \left(a\right)\right]}^k=I_a^{\psi ,\alpha }g\left(t\right),$$

which can be equivalently expressed as

$$y\left(t\right)=\sum _{k=0}^{n-1}{\displaystyle \frac{d_k}{k!}}{\left[\psi \left(t\right)-\psi \left(a\right)\right]}^k+I_a^{\psi ,\alpha }g\left(t\right),t\ge a.$$

 □

3. The Test Function

We strategically employ the test function $\phi \left(t\right)$, defined by

$$\phi \left(t\right)=\left\{\begin{array}{c}{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{-\lambda }{\left[\psi \left(T\right)-\psi \left(t\right)\right]}^{\lambda },a\le t\le T,\lambda >0,\hfill \\ \\ 0,t>T.\hfill \end{array}\right.$$

(9)

The aforementioned function possesses several notable properties. Namely, we report the ones below.

Lemma 6.

Let $\rho \ge 0$ and φ be as in (9). Then,

$$I_T^{\psi ,\rho }\phi \left(t\right)={\displaystyle \frac{\Gamma \left(\lambda +1\right)}{\Gamma \left(\rho +\lambda +1\right)}}{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{-\lambda }{\left[\psi \left(T\right)-\psi \left(t\right)\right]}^{\lambda +\rho },a\le t\le T,$$

$$I_T^{\psi ,\rho }\phi \left(a\right)={\displaystyle \frac{\Gamma \left(\lambda +1\right)}{\Gamma \left(\rho +\lambda +1\right)}}{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{\rho }\mathit{and}{I}_T^{\psi ,\rho }\phi \left(T\right)=0,$$

$${\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^2{I}_T^{\psi ,\rho }\phi \left(t\right)={\displaystyle \frac{\Gamma \left(\lambda +1\right)}{\Gamma \left(\rho +\lambda -1\right)}}{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{-\lambda }{\left[\psi \left(T\right)-\psi \left(t\right)\right]}^{\lambda +\rho -2},a\le t\le T.$$

Proof. 

These relations can be easily derived from Lemma 2. □

Lemma 7.

The function $\phi \left(t\right)$ is assumed to be as in (9) with $\lambda +p\left(\beta +\alpha -2\right)+1>0$, $\gamma \left(1-p\right)+1>0$, $p>1$ and $\beta ,$ $\alpha \ge 0$. Then,

$$\begin{array}{ccc}& & {\int }_a^T{\phi }^{1-p}\left(t\right){\left[\psi \left(t\right)-\psi \left(a\right)\right]}^{\gamma \left(1-p\right)}{\left(I_T^{\psi ,\beta }{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^2{I}_T^{\psi ,\alpha }\phi \left(t\right)\right)}^p{\psi }^{\prime }\left(t\right)dt\hfill \\ & =& K_{\beta ,\alpha }^{\gamma ,p,\lambda }{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{p\left(\beta +\alpha -2\right)+\gamma \left(1-p\right)+1},\hfill \end{array}$$

where

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

Proof. 

By virtue of Lemmas 6 and 2, it appears that

$$\begin{array}{ccc}\hfill I_T^{\psi ,\beta }{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^2{I}_T^{\psi ,\alpha }\phi \left(t\right)& =& {\displaystyle \frac{\Gamma \left(\lambda +1\right)}{\Gamma \left(\alpha +\lambda -1\right)}}{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{-\lambda }{I}_T^{\psi ,\beta }{\left[\psi \left(T\right)-\psi \left(t\right)\right]}^{\lambda +\alpha -2}\hfill \\ & =& {\displaystyle \frac{\Gamma \left(\lambda +1\right)}{\Gamma \left(\beta +\alpha +\lambda -1\right)}}{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{-\lambda }{\left[\psi \left(T\right)-\psi \left(t\right)\right]}^{\beta +\lambda +\alpha -2}.\hfill \end{array}$$

Then,

$${\int }_a^T{\phi }^{1-p}\left(t\right){\left[\psi \left(t\right)-\psi \left(a\right)\right]}^{\gamma \left(1-p\right)}{\left(I_T^{\psi ,\beta }{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^2{I}_T^{\psi ,\alpha }\phi \left(t\right)\right)}^p{\psi }^{\prime }\left(t\right)dt$$

$$\begin{array}{ccc}& =& {\left[{\displaystyle \frac{\Gamma \left(\lambda +1\right)}{\Gamma \left(\beta +\alpha +\lambda -1\right)}}\right]}^p{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{-p\lambda }{\int }_a^T{\left[\psi \left(t\right)-\psi \left(a\right)\right]}^{\gamma \left(1-p\right)}\hfill \\ & & \times {\left[{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{-\lambda }{\left[\psi \left(T\right)-\psi \left(t\right)\right]}^{\lambda }\right]}^{1-p}{\left[\psi \left(T\right)-\psi \left(t\right)\right]}^{p\left(\beta +\lambda +\alpha -2\right)}{\psi }^{\prime }\left(t\right)dt\hfill \\ & =& {\left[{\displaystyle \frac{\Gamma \left(\lambda +1\right)}{\Gamma \left(\beta +\alpha +\lambda -1\right)}}\right]}^p{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{-\lambda }\hfill \\ & & \times {\int }_a^T{\left[\psi \left(t\right)-\psi \left(a\right)\right]}^{\gamma \left(1-p\right)}{\left[\psi \left(T\right)-\psi \left(t\right)\right]}^{\lambda +p\left(\beta +\alpha -2\right)}{\psi }^{\prime }\left(t\right)dt.\hfill \end{array}$$

Let $\psi \left(t\right)-\psi \left(a\right)=s\left[\psi \left(T\right)-\psi \left(a\right)\right].$ The relation above becomes

$${\int }_a^T{\phi }^{1-p}\left(t\right){\left[\psi \left(t\right)-\psi \left(a\right)\right]}^{\gamma \left(1-p\right)}{\left(I_T^{\psi ,\beta }{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^2{I}_T^{\psi ,\alpha }\phi \left(t\right)\right)}^p{\psi }^{\prime }\left(t\right)dt$$

$$\begin{array}{ccc}& =& {\left[{\displaystyle \frac{\Gamma \left(\lambda +1\right)}{\Gamma \left(\beta +\alpha +\lambda -1\right)}}\right]}^p{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{\gamma \left(1-p\right)+p\left(\beta +\alpha -2\right)+1}{\int }_0^1{s}^{\gamma \left(1-p\right)}{\left(1-s\right)}^{\lambda +p\left(\beta +\alpha -2\right)}ds\hfill \\ & =& K_{\beta ,\alpha }^{\gamma ,p,\lambda }{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{\gamma \left(1-p\right)+p\left(\beta +\alpha -2\right)+1},\hfill \end{array}$$

having in mind the definition of the Beta function

$$B\left(\alpha ,\beta \right)={\int }_0^1{s}^{\alpha -1}{\left(1-s\right)}^{\beta -1}ds={\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{\Gamma \left(\alpha +\beta \right)}},\alpha ,\beta >0.$$

 □

Remark 1.

This paper relies on the equivalent relationships below, which hold true for m and $m^{\prime }$ when both are greater than 1 and satisfy ${\displaystyle \frac{1}{m}}+{\displaystyle \frac{1}{m^{\prime }}}=1$:

(a) 

$m^{\prime }={\displaystyle \frac{m}{m-1}}.$

(b) 

${\displaystyle \frac{m^{\prime }}{m}}=m^{\prime }-1.$

(c) 

$m(\alpha -1)+1>0⟺m^{\prime }\alpha >1,$ for $\alpha >0.$

4. Nonexistence Result

We now turn our attention to the following fractional differential problem:

$$\left\{\begin{array}{c}{}^{C}D_a^{\psi ,\alpha +1}y\left(t\right)+{}^{C}D_a^{\psi ,\beta +1}y\left(t\right)\ge {\left[\psi \left(t\right)-\psi \left(a\right)\right]}^{\rho }{\left|{}^{C}D_a^{\psi ,\gamma }y\left(t\right)\right|}^m,t>a,m>1,\hfill \\ \\ {\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{i}y\left(a\right)=y_{\psi }^{\left[i\right]}\left(a\right)=b_i,i=0,\dots ,n,n=-\left[-\alpha \right],\hfill \end{array}\right.$$

(10)

where $\alpha ,$ $\beta ,$ $\gamma \in \left(n-1,n\right),$ $n=-\left[-\alpha \right]\in \mathbb{N}$ and ${}^{C}D_a^{\psi ,\sigma }$ denotes the $\psi$-Caputo fractional derivative of order $\sigma >0.$

Theorem 1.

Under the assumptions that $m>1$, $m\left(\gamma -n\right)+n-\beta -1<\rho <m-1$ and $b_n>0$, Problem (10) does not possess any global nontrivial solution within the function space $A{C}^{n+1}[a,\infty )$.

Proof. 

We commence by assuming the existence of a global solution $y\in A{C}^{n+1}[a,\infty )$ to Problem (10). Let $\phi$ be the test function defined in (9), with $\lambda >{\displaystyle \frac{m}{m-1}}\left(\alpha +1-\gamma \right)-1$. Multiplying both sides of the equation in Problem (10) by $\phi \left(t\right){\psi }^{\prime }\left(t\right)$ and subsequently integrating over the interval $[a,T]$, it is easy to see that

$$I={\int }_a^T\phi \left(t\right){\left[\psi \left(t\right)-\psi \left(a\right)\right]}^{\rho }{\left|{}^{C}D_a^{\psi ,\gamma }y\left(t\right)\right|}^m{\psi }^{\prime }\left(t\right)dt\le I_1+I_2,$$

(11)

where

$$I_1={\int }_a^T\phi \left(t\right){}^{C}D_a^{\psi ,\alpha +1}y\left(t\right){\psi }^{\prime }\left(t\right)dt,$$

and

$$I_2={\int }_a^T\phi \left(t\right){}^{C}D_a^{\psi ,\beta +1}y\left(t\right){\psi }^{\prime }\left(t\right)dt.$$

Next, applying the definition of ${}^{C}D_a^{\gamma }y\left(t\right)$ and invoking Lemma 1 leads to

$$I_1={\int }_a^T\phi \left(t\right)I_a^{\psi ,n-\alpha }{y}_{\psi }^{\left[n+1\right]}\left(t\right){\psi }^{\prime }\left(t\right)dt={\int }_a^T{y}_{\psi }^{\left[n+1\right]}\left(t\right)I_T^{\psi ,n-\alpha }\phi \left(t\right){\psi }^{\prime }\left(t\right)dt.$$

(12)

In addition, an integration by parts with the help of Lemma 6 yields

$$I_1=y_{\psi }^{\left[n\right]}\left(t\right)I_T^{\psi ,n-\alpha }\phi \left(t\right){|}_{t=a}^T-{\int }_a^T{y}_{\psi }^{\left[n\right]}\left(t\right){\displaystyle \frac{d}{dt}}{I}_T^{\psi ,n-\alpha }\phi \left(t\right)dt$$

$$=-b_n{I}_T^{\psi ,n-\alpha }\phi \left(a\right)-{\int }_a^T{\psi }^{\prime }\left(t\right)y_{\psi }^{\left[n\right]}\left(t\right){\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}{I}_T^{\psi ,n-\alpha }\phi \left(t\right)dt.$$

(13)

By invoking Lemma 3, we derive the following sequence of identities:

$$\begin{array}{ccc}\hfill {\psi }^{\prime }\left(t\right)y_{\psi }^{\left[n\right]}\left(t\right)& =& {\displaystyle \frac{d}{dt}}{\int }_a^t{\psi }^{\prime }\left(s\right)y_{\psi }^{\left[n\right]}\left(s\right)ds={\displaystyle \frac{d}{dt}}{I}_a^{\psi ,1}{y}_{\psi }^{\left[n\right]}\left(t\right)\hfill \\ & =& {\displaystyle \frac{d}{dt}}\left(I_a^{\psi ,\gamma +1-n}{I}_a^{\psi ,n-\gamma }{y}_{\psi }^{\left[n\right]}\left(t\right)\right)={\displaystyle \frac{d}{dt}}\left(I_a^{\psi ,\gamma +1-n}{D}_a^{\psi ,\gamma }y\left(t\right)\right).\hfill \end{array}$$

Subsequently,

$$I_1=-b_n{I}_T^{\psi ,n-\alpha }\phi \left(a\right)-{\int }_a^T{\displaystyle \frac{d}{dt}}\left(I_a^{\psi ,\gamma +1-n}{}^{C}D_a^{\psi ,\gamma }y\left(t\right)\right){\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}{I}_T^{\psi ,n-\alpha }\phi \left(t\right)dt.$$

Moreover, performing an integration by parts, having in mind Lemmas 1 and 6, it holds that

$$\begin{array}{ccc}\hfill I_1& =& -b_n{I}_T^{\psi ,n-\alpha }\phi \left(a\right)-I_a^{\psi ,\gamma +1-n}{}^{C}D_a^{\psi ,\gamma }y\left(t\right){\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}{I}_T^{\psi ,n-\alpha }\phi \left(t\right){|}_{t=a}^T\hfill \\ & & +{\int }_a^T{I}_a^{\psi ,\gamma +1-n}{}^{C}D_a^{\psi ,\gamma }y\left(t\right){\displaystyle \frac{d}{dt}}\left[{\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}{I}_T^{\psi ,n-\alpha }\phi \left(t\right)\right]dt\hfill \\ & =& -b_n{I}_T^{\psi ,n-\alpha }\phi \left(a\right)+{\int }_a^T{I}_a^{\psi ,\gamma +1-n}{}^{C}D_a^{\psi ,\gamma }y\left(t\right){\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^2\left[I_T^{\psi ,n-\alpha }\phi \left(t\right)\right]{\psi }^{\prime }\left(t\right)dt\hfill \\ & =& -b_n{I}_T^{\psi ,n-\alpha }\phi \left(a\right)+{\int }_a^T{}^{C}D_a^{\psi ,\gamma }y\left(t\right)I_T^{\psi ,\gamma +1-n}\left[{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^2{I}_T^{\psi ,n-\alpha }\phi \left(t\right)\right]{\psi }^{\prime }\left(t\right)dt.\hfill \end{array}$$

(14)

Note that, in view of Lemma 6,

$${\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}{I}_T^{\psi ,n-\alpha }\phi \left(t\right){|}_{t=T}=0,\lambda >1+\alpha -n,$$

and by the fundamental theorem of calculus,

$$\begin{array}{ccc}& & I_a^{\psi ,\gamma +1-n}{}^{C}D_a^{\psi ,\gamma }y\left(t\right){{|}_{t=a}=I_a^{\psi ,\gamma +1-n}{I}_a^{\psi ,n-\gamma }{y}_{\psi }^{\left[n\right]}\left(t\right)|}_{t=a}\hfill \\ & =& I_a^{\psi ,1}{y}_{\psi }^{\left[n\right]}\left(t\right){{|}_{t=a}=\left(y_{\psi }^{\left[n-1\right]}\left(t\right)-y_{\psi }^{\left[n-1\right]}\left(a\right)\right)|}_{t=a}=0.\hfill \end{array}$$

Now, we proceed by inserting the expression

$${\left[\psi \left(t\right)-\psi \left(a\right)\right]}^{\rho /m}{\left[\psi \left(t\right)-\psi \left(a\right)\right]}^{-\rho /m}\phi {\left(t\right)}^{1/m}\phi {\left(t\right)}^{-1/m}$$

within the integral in (14)

$$\begin{array}{ccc}\hfill I_1& =& -b_n{I}_T^{\psi ,n-\alpha }\phi \left(a\right)+{\int }_a^T{}^{C}D_a^{\psi ,\gamma }y\left(t\right){\left[\psi \left(t\right)-\psi \left(a\right)\right]}^{\rho /m}\phi {\left(t\right)}^{1/m}{\psi }^{\prime }{\left(t\right)}^{1/m}\hfill \\ & & \times {\left[\psi \left(t\right)-\psi \left(a\right)\right]}^{-\rho /m}\phi {\left(t\right)}^{-1/m}{I}_T^{\psi ,\gamma +1-n}{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^2\left[I_T^{\psi ,n-\alpha }\phi \left(t\right)\right]{\psi }^{\prime }{\left(t\right)}^{1/m^{\prime }}dt.\hfill \end{array}$$

The $\epsilon$-Young inequality with $0<\epsilon <1/2$ allows us to write

$$\begin{array}{ccc}\hfill I_1& \le & -b_n{I}_T^{\psi ,n-\alpha }\phi \left(a\right)+\epsilon {\int }_a^T{\left|{}^{C}D_a^{\psi ,\gamma }y\left(t\right)\right|}^m\phi \left(t\right){\left[\psi \left(t\right)-\psi \left(a\right)\right]}^{\rho }{\psi }^{\prime }\left(t\right)dt\hfill \\ & & +K\left(\epsilon ,m\right){\int }_a^T\phi {\left(t\right)}^{-m^{\prime }/m}{\left[\psi \left(t\right)-\psi \left(a\right)\right]}^{-\rho m^{\prime }/m}\hfill \\ & & \times {\left(I_T^{\psi ,\gamma +1-n}{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^2{I}_T^{\psi ,n-\alpha }\phi \left(t\right)\right)}^{m^{\prime }}{\psi }^{\prime }\left(t\right)dt,\hfill \end{array}$$

where $K\left(\epsilon ,m\right)>0.$ Thanks to Lemmas 6 and 7, we entail

$$\begin{array}{ccc}\hfill I_1& \le & -b_n{\displaystyle \frac{\Gamma \left(\lambda +1\right)}{\Gamma \left(n-\alpha +\lambda +1\right)}}{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{n-\alpha }\hfill \\ & & +\epsilon {\int }_a^T{\left|{}^{C}D_a^{\psi ,\gamma }y\left(t\right)\right|}^m\phi \left(t\right){\left[\psi \left(t\right)-\psi \left(a\right)\right]}^{\rho }{\psi }^{\prime }\left(t\right)dt\hfill \\ & & +K\left(\epsilon ,m\right)K_{\gamma +1-n,n-\alpha }^{\rho ,m^{\prime },\lambda }{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{\rho \left(1-m^{\prime }\right)+m^{\prime }\left(\gamma -\alpha -1\right)+1}.\hfill \end{array}$$

(15)

Similarly,

$$\begin{array}{ccc}\hfill I_2& \le & -b_n{\displaystyle \frac{\Gamma \left(\lambda +1\right)}{\Gamma \left(n-\beta +\lambda +1\right)}}{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{n-\beta }\hfill \\ & & +\epsilon {\int }_a^T{\left|{}^{C}D_a^{\psi ,\gamma }y\left(t\right)\right|}^m\phi \left(t\right){\left[\psi \left(t\right)-\psi \left(a\right)\right]}^{\rho }{\psi }^{\prime }\left(t\right)dt\hfill \\ & & +K\left(\epsilon ,m\right)K_{\gamma +1-n,n-\beta }^{\rho ,m^{\prime },\lambda }{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{\rho \left(1-m^{\prime }\right)+m^{\prime }\left(\gamma -\beta -1\right)+1}.\hfill \end{array}$$

(16)

Hence, we infer from (11), (15) and (16) that

$$\begin{array}{ccc}& & \left(1-2\epsilon \right)I+\Gamma \left(\lambda +1\right)b_n{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{n-\beta }\left({\displaystyle \frac{{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{\beta -\alpha }}{\Gamma \left(n-\alpha +\lambda +1\right)}}+{\displaystyle \frac{1}{\Gamma \left(n-\beta +\lambda +1\right)}}\right)\hfill \\ & \le & K\left(\epsilon ,m\right)K_{\gamma +1-n,n-\alpha }^{\rho ,m^{\prime },\lambda }{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{\rho \left(1-m^{\prime }\right)+m^{\prime }\left(\gamma -\alpha -1\right)+1}\hfill \\ & & +K\left(\epsilon ,m\right)K_{\gamma +1-n,n-\beta }^{\rho ,m^{\prime },\lambda }{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{\rho \left(1-m^{\prime }\right)+m^{\prime }\left(\gamma -\beta -1\right)+1}\hfill \\ & =& K\left(\epsilon ,m\right){\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{\rho \left(1-m^{\prime }\right)+m^{\prime }\left(\gamma -1\right)+1}\hfill \\ & & \times \left\{K_{\gamma +1-n,n-\alpha }^{\rho ,m^{\prime },\lambda }{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{-\alpha m^{\prime }}+K_{\gamma +1-n,n-\beta }^{\rho ,m^{\prime },\lambda }{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{-\beta m^{\prime }}\right\}.\hfill \end{array}$$

(17)

Since $b_n>0$ and $0<\epsilon <1/2,$ obviously

$$\begin{array}{ccc}& & \left(1-2\epsilon \right){\int }_a^T{\left|{}^{C}D_0^{\gamma }y\left(t\right)\right|}^m\phi \left(t\right){\left[\psi \left(t\right)-\psi \left(a\right)\right]}^{\rho }{\psi }^{\prime }\left(t\right)dt\hfill \\ & & +\Gamma \left(\lambda +1\right)b_n{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{n-\beta }\left({\displaystyle \frac{{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{\beta -\alpha }}{\Gamma \left(n-\alpha +\lambda +1\right)}}+{\displaystyle \frac{1}{\Gamma \left(n-\beta +\lambda +1\right)}}\right)\hfill \\ & \ge & \Gamma \left(\lambda +1\right)b_n{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{n-\beta }\left({\displaystyle \frac{{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{\beta -\alpha }}{\Gamma \left(n-\alpha +\lambda +1\right)}}+{\displaystyle \frac{1}{\Gamma \left(n-\beta +\lambda +1\right)}}\right)\hfill \\ & \ge & {\displaystyle \frac{\Gamma \left(\lambda +1\right)}{\Gamma \left(n-\beta +\lambda +1\right)}}{b}_n{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{n-\beta }.\hfill \end{array}$$

Therefore,

$$\begin{array}{ccc}\hfill C_1{b}_n{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{n-\beta }& \le & C_2{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{\rho \left(1-m^{\prime }\right)+m^{\prime }\left(\gamma -\alpha -1\right)+1}\hfill \\ & & +C_3{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{\rho \left(1-m^{\prime }\right)+m^{\prime }\left(\gamma -\beta -1\right)+1},\hfill \end{array}$$

where

$$C_1={\displaystyle \frac{\Gamma \left(\lambda +1\right)}{\Gamma \left(\lambda +1+n-\beta \right)}},C_2=K\left(\epsilon ,m\right)K_{\gamma +1-n,n-\alpha }^{\rho ,m^{\prime },\lambda },C_3=K\left(\epsilon ,m\right)K_{\gamma +1-n,n-\beta }^{\rho ,m^{\prime },\lambda }$$

or

$$\begin{array}{ccc}\hfill b_n& \le & {\displaystyle \frac{1}{C_1}}{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{\beta -n}\hfill \\ & & \times \left(C_2{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{\rho \left(1-m^{\prime }\right)+m^{\prime }\left(\gamma -\alpha -1\right)+1}+C_3{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{\rho \left(1-m^{\prime }\right)+m^{\prime }\left(\gamma -\beta -1\right)+1}\right)\hfill \\ & =& {\displaystyle \frac{1}{C_1}}{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{\beta -n+\rho \left(1-m^{\prime }\right)+m^{\prime }\left(\gamma -\beta -1\right)+1}\left(C_2{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{m^{\prime }\left(\beta -\alpha \right)}+C_3\right).\hfill \end{array}$$

As $\beta -n+\rho \left(1-m^{\prime }\right)+m^{\prime }\left(\gamma -\beta -1\right)+1<0$ and $m^{\prime }\left(\beta -\alpha \right)<0,$ we see that

$${\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{\beta -n+\rho \left(1-m^{\prime }\right)+m^{\prime }\left(\gamma -\beta -1\right)+1},{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{m^{\prime }\left(\beta -\alpha \right)}\to 0\mathrm{as}T\to \infty .$$

Whereby

$$b_n\le 0,$$

which contradicts the initial assumption that $b_n>0$. □

Theorem 2.

Under the constraints $m>1,$ $b_n=0,$ and $m\left(\gamma -\beta \right)-1<\rho <m-1,$ the sole global solution to Problem (10) possesses the form

$$y\left(t\right)=b_0+b_1\left[\psi \left(t\right)-\psi \left(a\right)\right]+\dots +{\displaystyle \frac{b_{n-1}}{\left(n-1\right)!}}{\left[\psi \left(t\right)-\psi \left(a\right)\right]}^{n-1},t>a.$$

Proof. 

Taking $b_n=0$ in (17), it is obvious that

$$\left(1-2\epsilon \right)I\le C_2{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{\rho \left(1-m^{\prime }\right)+m^{\prime }\left(\gamma -\alpha -1\right)+1}+C_3{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{\rho \left(1-m^{\prime }\right)+m^{\prime }\left(\gamma -\beta -1\right)+1},$$

or

$$I\le C_4{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{\rho \left(1-m^{\prime }\right)+m^{\prime }\left(\gamma -\beta -1\right)+1}\left(C_2{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{m^{\prime }\left(\beta -\alpha \right)}+C_3\right),$$

(18)

where $C_4={\displaystyle \frac{1}{1-2\epsilon }}$ and $0<\epsilon <1/2.$ By employing (9) and (11), we derive the inequality

$$\begin{array}{ccc}& & {\int }_a^T{\left(1-{\displaystyle \frac{\psi \left(t\right)-\psi \left(a\right)}{\psi \left(T\right)-\psi \left(a\right)}}\right)}^{\lambda }{\left[\psi \left(t\right)-\psi \left(a\right)\right]}^{\rho }{\left|{}^{C}D_a^{\psi ,\gamma }y\left(t\right)\right|}^m{\psi }^{\prime }\left(t\right)dt\hfill \\ & \le & C_4{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{\rho \left(1-m^{\prime }\right)+m^{\prime }\left(\gamma -\beta -1\right)+1}\left(C_2{\left[\psi \left(T\right)-\psi \left(a\right)\right]}^{m^{\prime }\left(\beta -\alpha \right)}+C_3\right).\hfill \end{array}$$

(19)

Since $\beta \le \alpha$ and $m\left(\gamma -\beta \right)-1<\rho$, it is evident that $m^{\prime }\left(\beta -\alpha \right)\le 0$ and $\rho \left(1-m^{\prime }\right)+m^{\prime }\left(\gamma -\beta -1\right)+1<0.$ Passing to the limit as $T\to \infty$ in (19) and utilizing Fatou’s lemma, we end up with

$${\int }_a^T{\left[\psi \left(t\right)-\psi \left(a\right)\right]}^{\rho }{\left|{}^{C}D_a^{\psi ,\gamma }y\left(t\right)\right|}^m{\psi }^{\prime }\left(t\right)dt=0,$$

which yields

$${}^{C}D_a^{\psi ,\gamma }y\left(t\right)=0,t>a.$$

Finally, it suffices to recall Lemma 5, to attain the expression for $y:\left(t\right)$

$$y\left(t\right)=b_0+b_1\left[\psi \left(t\right)-\psi \left(a\right)\right]+\dots +{\displaystyle \frac{b_{n-1}}{\left(n-1\right)!}}{\left[\psi \left(t\right)-\psi \left(a\right)\right]}^{n-1},t>a.$$

 □

5. Examples

This section comprises illustrative examples to solidify the understanding of the preceding results of theoretical nature.

Example 1.

Consider the following fractional differential problem

$$\left\{\begin{array}{c}{}^{C}D_a^{\psi ,1.7}y\left(t\right)+{}^{C}D_a^{\psi ,1.4}y\left(t\right)\ge {\left[\psi \left(t\right)-\psi \left(a\right)\right]}^{0.5}{\left|{}^{C}D_a^{\psi ,0.6}\right|}^2,t>a,\hfill \\ \\ y\left(a\right)=0,y_{\psi }^{\left[1\right]}\left(a\right)=1.\hfill \end{array}\right.$$

(20)

This problem can be viewed as a specific instance of Problem (10), where $\alpha =0.7$, $\beta =0.4$, $\rho =0.5$, $\gamma =0.6$, $m=2$, $b_0=0$, and $b_1=1$. Invoking Theorem 1, we conclude that the fractional differential problem (20) does not possess any nontrivial global solution within the function space $A{C}^2[a,\infty )$.

Example 2.

The fractional differential problem,

$$\left\{\begin{array}{c}{}^{C}D_a^{\psi ,2.6}y\left(t\right)+{}^{C}D_a^{\psi ,2.3}y\left(t\right)\ge {\left[\psi \left(t\right)-\psi \left(a\right)\right]}^{-0.4}{\left|{}^{C}D_a^{\psi ,1.5}y\left(t\right)\right|}^3,t>a,\hfill \\ \\ y\left(a\right)=1,y_{\psi }^{\left[1\right]}\left(a\right)=-2,y_{\psi }^{\left[2\right]}\left(a\right)=3,\hfill \end{array}\right.$$

(21)

is a particular case of (10) where $\alpha =1.6$, $\beta =1.3$, $\rho =-0.4$, $\gamma =1.5$, $m=3$, $b_0=1$, $b_1=-2$ and $b_2=3$. According to Theorem 1, the fractional differential problem (21) does not enjoy any nontrivial global solution within the framework of $A{C}^3[a,\infty )$.

Example 3.

Consider the following fractional differential problem:

$$\left\{\begin{array}{c}{}^{C}D_a^{\psi ,3.7}y\left(t\right)+{}^{C}D_a^{\psi ,3.5}y\left(t\right)\ge {\left[\psi \left(t\right)-\psi \left(a\right)\right]}^{0.8}{\left|{}^{C}D_a^{\psi ,2.6}y\left(t\right)\right|}^3,t>a,\hfill \\ \\ y\left(a\right)=5,y_{\psi }^{\left[1\right]}\left(a\right)=1,y_{\psi }^{\left[2\right]}\left(a\right)=2,y_{\psi }^{\left[3\right]}\left(a\right)=0.\hfill \end{array}\right.$$

(22)

This problem falls within the scope of Problem (10) when $\alpha =2.7$, $\beta =2.5$, $\rho =0.8$, $\gamma =2.6$, $m=3$, $b_0=5$, $b_1=1$, $b_2=2$ and $b_3=0$. Theorem 2 applies for this case and provides the solution

$$y\left(t\right)=5+\left[\psi \left(t\right)-\psi \left(a\right)\right]+{\left[\psi \left(t\right)-\psi \left(a\right)\right]}^2,t>a.$$

6. Conclusions

In this paper, we investigated the global solvability of certain $\psi$-Caputo-type fractional differential inequalities involving polynomial sources and two distinct fractional derivatives of orders $\kappa$ and $\upsilon$. By carefully applying the properties of $\psi$-fractional integrals and derivatives and employing the test function technique, we established rigorous nonexistence results within an appropriate function space. We showed that the lower-order derivative, of order $\upsilon$, plays a critical role in determining the range of parameter values that lead to finite-time blow-up, effectively narrowing the blow-up region and introducing a dissipative effect that competes with the blow-up term. This highlights the strong influence of the lower-order derivative on blow-up dynamics and the overall behavior of solutions.

Our results extend the scope of known nonexistence criteria in the literature, encompassing several existing problems as special cases and demonstrating the generality of our approach. Beyond theoretical insights, these findings have practical implications for systems with memory effects, such as viscoelastic materials, anomalous diffusion processes, and control systems with hereditary dynamics. Understanding how the lower-order derivative constrains blow-up can inform the design of materials or controllers to prevent undesirable finite-time singularities. Future work could explore numerical simulations, experimental validation, or optimization strategies, thereby bridging the gap between fractional theory and practical applications.