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Saddle solutions for a class of systems of periodic and reversible semilinear elliptic equations

  • Received: 01 July 2018 Revised: 01 February 2019
  • 35J50, 35J47, 34C37, 34C25, 35B40

  • We study systems of elliptic equations $ -\Delta u(x)+F_{u}(x, u) = 0 $ with potentials $ F\in C^{2}({\mathbb{R}}^{n}, {\mathbb{R}}^{m}) $ which are periodic and even in all their variables. We show that if $ F(x, u) $ has flip symmetry with respect to two of the components of $ x $ and if the minimal periodic solutions are not degenerate then the system has saddle type solutions on $ {\mathbb{R}}^{n} $.

    Citation: Francesca Alessio, Piero Montecchiari, Andrea Sfecci. Saddle solutions for a class of systems of periodic and reversible semilinear elliptic equations[J]. Networks and Heterogeneous Media, 2019, 14(3): 567-587. doi: 10.3934/nhm.2019022

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  • We study systems of elliptic equations $ -\Delta u(x)+F_{u}(x, u) = 0 $ with potentials $ F\in C^{2}({\mathbb{R}}^{n}, {\mathbb{R}}^{m}) $ which are periodic and even in all their variables. We show that if $ F(x, u) $ has flip symmetry with respect to two of the components of $ x $ and if the minimal periodic solutions are not degenerate then the system has saddle type solutions on $ {\mathbb{R}}^{n} $.



    We generalize the classical Halanay inequality to encompass fractional-order systems with both discrete and distributed neutral delays. This inequality, originally formulated for integer-order systems, is now generalized to non-integer orders.

    Lemma 1.1. Consider a nonnegative function $ w(t) $ that satisfies the inequality

    $ w(t)K1w(t)+K2suptτstw(s),ta, $

    where $ 0 < K_{2} < K_{1}. $ Under these conditions, positive constants $ K_{3} $ and $ K_{4} $ exist such that

    $ w(t)K3eK4(ta),ta. $

    Halanay first introduced this inequality while studying the stability of a specific differential equation [10]

    $ υ(t)=Aυ(t)+Bυ(tτ),τ>0. $

    Since then, the inequality has been generalized to include variable coefficients and delays of varying magnitude, both bounded and unbounded [1,25,26]. These generalizations have found applications in Hopfield neural networks and the analysis of Volterra functional equations, particularly in the context of problems described by the following system [12,16,27]:

    $ {xi(t)=cixi(t)+nj=1bijfj(xj(tτ))+nj=1aijfj(xj(t))+Ii,t>0,xi(t)=ϕi(t),τt0, i=1,...,n. $

    Such problems arise in various fields, including parallel computing, cryptography, image processing, combinatorial optimization, signal theory, and geology [15,17,18].

    Additionally, a generalization of the Halanay inequality to systems with distributed delays is presented in [21]:

    $ w(x)B(x)w(x)+A(x)0k(s)w(xs)ds,x0. $

    The solutions exhibit exponential decay if the kernels satisfy the conditions

    $ 0eβsk(s)ds<, $

    for some $ \beta > 0, $ and

    $ A(x)0k(s)dsB(x)C,C>0,xR. $

    See also [22] for further details.

    This study broadens the scope of Halanay's inequality to encompass fractional-order systems. The justification for using fractional derivatives is provided in [2,3]. We also consider neutral delays, where delays appear in the leading derivative. Specifically, we analyze the stability of the following problem:

    $ {Dφ,αC[w(t)pw(tυ)]qw(t)+taw(r)k(tr)dr, p>0, 0<α<1, υ,t>a,w(t)=ϖ(t),aυta. $ (1.1)

    We establish sufficient conditions on the kernel $ k $ to guarantee Mittag-Leffler stability, ensuring that the solutions satisfy

    $ w(t)AEα(q[φ(t)φ(a)]α),t>a. $

    We provide examples of function families that satisfy our assumptions. As an application, we consider a fractional-order Cohen-Grossberg neural network system with neutral delays [9]. This system represents a more general form of the traditional Hopfield neural network.

    There is extensive research on the existence, stability, and long-term behavior of Cohen-Grossberg neural network systems. Our focus is on research that specifically addresses networks with time delays or fractional-order dynamics. For integer-order neutral Cohen-Grossberg systems, refer to [5,7,24]. The fractional case with discrete delays was explored in [14]. While the Halanay inequality has been adapted for fractional-order systems with discrete delays in [4,11,28], we are unaware of any work addressing our specific problem (1.1).

    The techniques used for integer-order systems are not directly applicable to the fractional-order case. For example, the Mittag-Leffler functions lack the semigroup property, and estimating the expression $ E_{\alpha }(-q\left(\varphi \left(t-\upsilon \right) -\varphi \left(a\right) \right) ^{\alpha })/E_{\alpha }(-q\left(\varphi \left(t\right) -\varphi \left(a\right) \right) ^{\alpha }) $ is challenging for convergence analysis. The ideal decay rate would be $ E_{\alpha }(-q\left(\varphi \left(t\right) -\varphi \left(a\right) \right) ^{\alpha }) $, but the neutral delay introduces new challenges, particularly near $ \nu $. Approximating with $ \left(\varphi \left(t\right) -\varphi \left(a\right) \right) ^{-\alpha } $ (using Mainardi's conjecture) does not fully resolve these issues.

    This paper is organized into eight sections, beginning with background information in Section 2. Section 3 presents our inequality for systems with discrete time delays, and Section 4 discusses two potential kernel functions. Section 5 investigates a fractional Halanay inequality in the presence of distributed neutral delays. Solutions of arbitrary signs for the problem in Section 3 are addressed in Section 6, and Section 7 applies our results to a Cohen-Grossberg system with neutral delays. Section 8 provides the conclusion, summarizing the findings and highlighting directions for future research.

    This section provides fundamental definitions and lemmas essential for the subsequent analysis. Throughout the paper, we consider $ \left[a, b\right] $ to be an infinite or finite interval, and $ \varphi $ to be an $ n $- continuously differentiable function on $ \left[a, b\right] $ such that $ \varphi $ is increasing and $ \varphi ^{\prime }\left(\varkappa \right) \neq 0 $ on $ \left[a, b\right] $.

    Definition 2.1. The $ \varphi $-Riemann-Liouville fractional integral of a function $ \omega $ with respect to a function $ \varphi $ is defined as

    $ Iφ,αω(z)=1Γ(α)za[φ(z)φ(s)]α1ω(s)φ(s)ds,α>0,z>a $

    provided that the right side exists.

    Definition 2.2. The $ \varphi $-Caputo derivative of order $ \alpha > 0 $ is defined by

    $ Dφ,αCω(ϰ)=Iφ,nα(1φ(ϰ)ddϰ)nω(ϰ), $

    which can be expressed equivalently as

    $ Dφ,αCω(ϰ)=1Γ(nα)ϰa[φ(ϰ)φ(τ)]nα1φ(τ)ω[n]φ(τ)dτ, ϰ>a, $

    where

    $ ω[n]φ(ϰ)=(1φ(ϰ)ddϰ)nω(ϰ), n=[α]. $

    Particularly, when $ 0 < \alpha < 1 $

    $ Dφ,αCω(ϰ)=Iφ,1α(1φ(ϰ)ddϰ)ω(ϰ)=1Γ(1α)ϰa[φ(ϰ)φ(τ)]αω(τ)dτ. $

    The Mittag-Leffler functions used in this context are defined as follows:

    $ Eα(y):=n=0ynΓ(1+αn), Re(α)>0, $

    and

    $ Eα,β(y):=n=0ynΓ(β+αn), Re(β)>0, Re(α)>0. $

    Lemma 2.1. [13] The Cauchy problem

    $ {Dφ,αCy(ζ)=λy(ζ),0<α1ζ>a,λRy(a)=ya, $ (2.1)

    has the solution

    $ y(ζ)=yaEα(λ[φ(ζ)φ(a)]α),ζa. $

    Lemma 2.2. [13] The Cauchy problem

    $ {Dφ,αCy(ζ)=λy(ζ)+h(ζ),0<α1λRζ>a,y(a)=yaR, $ (2.2)

    admits the solution for $ \zeta \geq a $

    $ y(ζ)=yaEα(λ[φ(ζ)φ(a)]α)+ζa[φ(ζ)φ(s)]α1Eα,α(λ[φ(ζ)φ(s)]α)φ(s)h(s)ds. $

    Lemma 2.3. For $ \lambda, \; \nu, \; \omega > 0, $ the following inequality is valid for all $ z > a $:

    $ za[φ(s)φ(a)]λ1[φ(z)φ(s)]ν1eω[φ(s)φ(a)]φ(s)dsC[φ(z)φ(a)]ν1, $

    where

    $ C=max{1,21ν}Γ(λ)[1+λ(λ+1)/ν]ωλ. $

    Proof. For $ z > a $, let

    $ I(z)=[φ(z)φ(a)]1νza[φ(s)φ(a)]λ1[φ(z)φ(s)]ν1eω[φ(s)φ(a)]φ(s)ds. $

    Set $ \xi \left[\varphi \left(z\right) -\varphi \left(a\right) \right] = \varphi \left(s\right) -\varphi \left(a\right). $ Then, $ \left[\varphi \left(z\right) -\varphi \left(a\right) \right] d\xi = \varphi ^{\prime }\left(s\right) ds $ and

    $ I(z)=[φ(z)φ(a)]λ10(1ξ)ν1ξλ1eωξ[φ(z)φ(a)]dξ, z>a. $

    As for $ 0\leq \xi < 1/2 $, we have $ \left(1-\xi \right) ^{\nu -1}\leq \max \left\{ 1, 2^{1-\nu }\right\}, $ therefore

    $ I(z)max{1,21ν}[φ(z)φ(a)]λ1/20ξλ1eωξ[φ(z)φ(a)]dξ+[φ(z)φ(a)]λ11/2(1ξ)ν1ξλ1eωξ[φ(z)φ(a)]dξ. $ (2.3)

    Let $ u = \omega \xi \left[\varphi \left(z\right) -\varphi \left(a\right) \right]. $ Then, $ d\xi = \left[\varphi \left(z\right) -\varphi \left(a\right) \right] ^{-1}\omega ^{-1}du $ and

    $ [φ(z)φ(a)]λ1/20ξλ1eωξ[φ(z)φ(a)]dξωλ0uλ1eudu=ωλΓ(λ). $ (2.4)

    If $ 1\leq \omega \xi \left[\varphi \left(z\right) -\varphi \left(a\right) \right] $, then

    $ eωξ[φ(z)φ(a)][ωξ[φ(z)φ(a)]]1+[λ]Γ([λ]+2)[ωξ[φ(z)φ(a)]]λΓ(λ+2). $

    Therefore, when $ 1/2 < \xi \leq 1, $

    $ ξλ1eωξ[φ(z)φ(a)]ξλ1Γ(2+λ)[ωξ[φ(z)φ(a)]]λ2ωλΓ(λ+2)[φ(z)φ(a)]λ, $

    and consequently

    $ [φ(z)φ(a)]λ11/2(1ξ)ν1ξλ1eωξ[φ(z)φ(a)]dξ[φ(z)φ(a)]λ11/2(1ξ)ν12ωλΓ(2+λ)[φ(z)φ(a)]λdξ=2ωλΓ(2+λ)11/2(1ξ)ν1dξ=21νωλΓ(λ+2)ν. $

    When $ \omega \xi \left[\varphi \left(z\right) -\varphi \left(a\right) \right] < 1, $ it implies that $ \left[\omega \xi \left[\varphi \left(z\right) -\varphi \left(a\right) \right] \right] ^{\lambda } < 1\leq e^{\omega \xi \left[\varphi \left(z\right) -\varphi \left(a\right) \right] }. $ Consequently,

    $ [φ(z)φ(a)]λ11/2ξλ1(1ξ)ν1eωξ[φ(z)φ(a)]dξ<[φ(z)φ(a)]λ11/2ξλ1(1ξ)ν1[ωξ[φ(z)φ(a)]]λdξ<2ωλ11/2(1ξ)ν1dξ=21νωλν. $ (2.5)

    Taking into account (2.3)–(2.5), we infer that

    $ I(z)max{1,21ν}ωλΓ(λ)+21νωλΓ(λ+2)νmax{1,21ν}ωλΓ(λ)(1+λ(λ+1)ν), z>a. $

    The proof is complete.

    Lemma 2.4. [8, (4.4.10), (4.9.4)] For $ \beta > 0, $ $ \nu > 0, $ and $ \lambda, \lambda ^{\ast }\in \mathbb{C}, $ $ \lambda \neq \lambda ^{\ast }, $ we have

    $ ϰ0zβ1Eα,β(λzα)(ϰz)ν1Eα,ν(λ(ϰz)α)dz=λEα,β+ν(λϰα)λEα,β+ν(λϰα)λλϰβ+ν1, $

    and for $ \sigma > 0, $ $ \gamma > 0, $

    $ Iσzγ1Eα,γ(pzα)(ϰ)=ϰσ+γ1Eα,σ+γ(pϰα). $

    Lemma 2.5. For $ \beta > 0, $ $ \nu > 0, $ and $ \lambda, \lambda ^{\ast }\in \mathbb{C}, $ $ \lambda \neq \lambda ^{\ast }, $ we have

    $ ϰaEα,β(λ[φ(z)φ(a)]α)[φ(ϰ)φ(z)]ν1[φ(z)φ(a)]β1×Eα,ν(λ[φ(ϰ)φ(z)]α)φ(z)dz=[φ(ϰ)φ(a)]β+ν1λEα,β+ν(λ[φ(ϰ)φ(a)]α)λEα,β+ν(λ[φ(ϰ)φ(a)]α)λλ, $

    and for $ \sigma > 0, $ $ \gamma > 0, $

    $ Iφ,σ[φ(z)φ(a)]γ1Eα,γ(p[φ(z)φ(a)]α)(ϰ)=[φ(ϰ)φ(a)]σ+γ1×Eα,σ+γ(p[φ(ϰ)φ(a)]α). $ (2.6)

    Proof. Let $ u = \varphi \left(\varkappa \right) -\varphi \left(z\right). $ Then,

    $ ϰaEα,β(λ[φ(z)φ(a)]α)[φ(ϰ)φ(z)]ν1[φ(z)φ(a)]β1×Eα,ν(λ[φ(ϰ)φ(z)]α)φ(z)dz=φ(ϰ)φ(a)0Eα,β(λ[φ(ϰ)φ(a)u]α)[φ(ϰ)φ(a)u]β1uν1Eα,ν(λuα)du. $

    At this point, we can utilize Lemma 2.4 to derive the following:

    $ ϰaEα,β(λ[φ(z)φ(a)]α)[φ(ϰ)φ(z)]ν1[φ(z)φ(a)]β1×Eα,ν(λ[φ(ϰ)φ(z)]α)φ(z)dz=[φ(ϰ)φ(a)]β+ν1λEα,β+ν(λ[φ(ϰ)φ(a)]α)λEα,β+ν(λ[φ(ϰ)φ(a)]α)λλ. $

    To prove the second formula in the lemma, we have

    $ Iφ,σ[φ(z)φ(a)]γ1Eα,γ(p[φ(z)φ(a)]α)(ϰ)=1Γ(σ)ϰaEα,γ(p[φ(z)φ(a)]α)[φ(ϰ)φ(z)]σ1[φ(z)φ(a)]γ1φ(z)dz. $

    From the first formula in the lemma, with $ \beta = \gamma, $ $ \nu = \sigma, $ $ \lambda = p $, $ \lambda ^{\ast } = 0, $ we obtain

    $ Iφ,σ[φ(z)φ(a)]γ1Eα,γ(p[φ(z)φ(a)]α)(ϰ)=1Γ(σ)ϰa[φ(z)φ(a)]γ1Eα,γ(p[φ(z)φ(a)]α)[φ(ϰ)φ(z)]σ1φ(z)dz=[φ(ϰ)φ(a)]γ+σ1Eα,γ+σ(p[φ(ϰ)φ(a)]α), $

    where we have used

    $ Eα,σ(λ[φ(ϰ)φ(z)]α)=1Γ(σ). $

    Mainardi's conjecture. [19] For fixed $ \gamma $ with $ 0 < \gamma < 1, $ the following holds:

    $ 11+qΓ(1γ)tγEγ(qtγ)1qΓ(1+γ)1tγ+1,q, t0. $ (2.7)

    This result was later established in [6,23].

    To start, we will introduce the concept of Mittag-Leffler stability.

    Definition 3.1. For $ 0 < \alpha < 1 $, a solution $ v(z) $ is defined as $ \alpha $ -Mittag-Leffler stable if there exist positive constants $ A $ and $ \gamma $ such that

    $ v(z)AEα(γ[φ(z)φ(a)]α),z>a, $

    where $ \left\Vert.\right\Vert $ represents a specific norm.

    Theorem 3.1. Let $ u(t) $ be a nonnegative function fulfilling the conditions

    $ Dφ,αC[u(t)pu(tυ)]qu(t)+tau(s)k(ts)ds,0<α<1,t>a, $ (3.1)

    with the initial condition

    $ u(t)=ϖ(t)0,aυta, $ (3.2)

    where $ k $ is a nonnegative function integrable over its domain, and $ q > 0 $. Assume $ p > 0, $ and that $ k $ satisfies the following inequality for some $ M > 0 $:

    $ taEα,α(q[φ(t)φ(s)]α)[φ(t)φ(s)]α1×(saEα(q[φ(σ)φ(a)]α)k(sσ)dσ)φ(s)dsMEα(q[φ(t)φ(a)]α),t>a. $ (3.3)

    Further, assume that the constant $ M $ satisfies

    $ M<11(φ(a+υ)φ(a))α(1q+Γ(1α)[φ(a+3υ)φ(a)]α)p, $ (3.4)

    with the additional condition

    $ 1(φ(a+υ)φ(a))α(1q+Γ(1α)[φ(a+3υ)φ(a)]α)p<1. $ (3.5)

    Then, $ u(t) $ exhibits Mittag-Leffler decay, i.e.,

    $ u(t)CEα(q[φ(t)φ(a)]α),t>a $

    for some constant $ C > 0. $

    Proof. Solutions of (3.1) and (3.2) will be compared to those of

    $ {Dφ,αC[w(t)pw(tυ)]=qw(t)+taw(s)k(ts)ds,0<α<1,t>a,w(t)=ϖ(t)0,aυta. $ (3.6)

    The equation presented in (3.6) can be expressed equivalently as

    $ Dφ,αC[w(t)pw(tυ)]=q[w(t)pw(tυ)]+tak(ts)w(s)dsqpw(tυ),t>a. $

    This permits to profit from the form

    $ w(t)pw(tυ)=[ϖ(a)pϖ(aυ)]Eα(q[φ(t)φ(a)]α)+ta[φ(t)φ(s)]α1Eα,α(q[φ(t)φ(s)]α)×(qpw(sυ)+sak(sσ)w(σ)dσ)φ(s)ds. $

    Capitalizing on the nonnegativity of the solution, we find for $ t > a $,

    $ w(t)ϖ(a)Eα(q(φ(t)φ(a))α)+pw(tυ)+taEα,α(q[φ(t)φ(s)]α)×[φ(t)φ(s)]α1(sak(sσ)w(σ)dσ)φ(s)ds. $ (3.7)

    Therefore, for $ t > a, $

    $ w(t)Eα(q(φ(t)φ(a))α)ϖ(a)+pEα(q(φ(t)φ(a))α)w(tυ)+1Eα(q(φ(t)φ(a))α)ta[φ(t)φ(s)]α1Eα,α(q[φ(t)φ(s)]α)×(sak(sσ)Eα(q(φ(σ)φ(a))α)w(σ)Eα(q(φ(σ)φ(a))α)dσ)φ(s)ds, $

    and

    $ w(t)Eα(q(φ(t)φ(a))α)ϖ(a)+pEα(q(φ(t)φ(a))α)w(tυ)+1Eα(q(φ(t)φ(a))α)ta[φ(t)φ(s)]α1×Eα,α(q[φ(t)φ(s)]α)×(sak(sσ)Eα(q(φ(σ)φ(a))α)dσ)φ(s)ds×supaσtw(σ)Eα(q(φ(σ)φ(a))α)ϖ(a)+pEα(q(φ(t)φ(a))α)w(tν)+Msupaσtw(σ)Eα(q(φ(σ)φ(a))α). $

    We will repeatedly utilize the following estimation:

    $ 1Eα(q(φ(t)φ(a))α)taEα,α(q[φ(t)φ(s)]α)[φ(t)φ(s)]α1×(sak(sσ)w(σ)dσ)φ(s)ds=1Eα(q(φ(t)φ(a))α)taEα,α(q[φ(t)φ(s)]α)[φ(t)φ(s)]α1×(saEα(q[φ(σ)φ(a)]α)k(sσ)w(σ)Eα(q[φ(σ)φ(a)]α)dσ)φ(s)dsMsupaσtw(σ)Eα(q(φ(σ)φ(a))α),t>a. $ (3.8)

    Then, for $ t > a, $ the following inequality holds:

    $ w(t)Eα(q[φ(t)φ(a)]α)ϖ(a)+pEα(q[φ(t)φ(a)]α)w(tυ)+Msupaσtw(σ)Eα(q[φ(σ)φ(a)]α). $ (3.9)

    This inequality will serve as our initial reference.

    For $ t\in \lbrack a, a+\upsilon], $ since $ E_{\alpha }(-q\left[\varphi \left(t\right) -\varphi \left(a\right) \right] ^{\alpha }) $ is decreasing, it follows that

    $ Eα(q[φ(t)φ(a)]α)Eα(q[φ(a+υ)φ(a)]α), $

    and hence

    $ w(t)Eα(q[φ(t)φ(a)]α)(1+pEα(q[φ(a+υ)φ(a)]α)supaυσaϖ(σ)+Msupaσtw(σ)Eα(q[φ(σ)φ(a)]α), $

    or

    $ (1M)w(t)Eα(q(φ(t)φ(a))α)(1+pEα(q(φ(a+υ)φ(a))α)supaυσaϖ(σ). $ (3.10)

    If $ t\in \lbrack a+\upsilon, a+2\upsilon], $ owing to relations (3.9) and (3.10), we find

    $ w(t)Eα(q(φ(t)φ(a))α)supaυσaϖ(σ)+p1M(1+pEα(q(φ(a+υ)φ(a))α)×Eα(q(φ(tυ)φ(a))α)Eα(q(φ(t)φ(a))α)supaνσaϖ(σ)+Msupaσtw(σ)Eα(q(φ(σ)φ(a))α). $

    Observe that

    $ Eα(q(φ(tυ)φ(a))α)Eα(q(φ(t)φ(a))α)1Eα(q(φ(t)φ(a))α)1Eα(q(φ(2υ+a)φ(a))α)1+qΓ(1α)(φ(2υ+a)φ(a))α=:A. $ (3.11)

    Therefore,

    $ w(t)Eα(q(φ(t)φ(a))α)[1+A(Eα(q(φ(υ+a)φ(a))α+p)pEα(q(φ(υ+a)φ(a))α(1M)]supaυσaϖ(σ)+Msupaσtw(σ)Eα(q(φ(σ)φ(a))α), $

    and consequently,

    $ w(t)Eα(q(φ(t)φ(a))α)(1M)[1+A(1M)p+AEα(q[φ(a+υ)φ(a)]α)(1M)p2]supaυσaϖ(σ). $ (3.12)

    Notice that we will write (3.12) as

    $ w(t)Eα(q(φ(t)φ(a))α)(1M)AEα(q(φ(υ+a)φ(a))α)×[1+p1M+(p1M)2]supaυσaϖ(σ). $ (3.13)

    When $ t\in \lbrack a+2\nu, a+3\nu], $ the estimations

    $ φ(t)φ(a)φ(tυ)φ(a)φ(a+3υ)φ(a)φ(a+υ)φ(a), $

    together with (2.7), imply for $ t\geq a+2\nu, $

    $ Eα(q(φ(tυ)φ(a))α)Eα(q(φ(t)φ(a))α)1+q(φ(t)φ(a))αΓ(1α)1+q(φ(tυ)φ(a))αΓ(1+α)11+q(φ(t)φ(a))αΓ(1α)qΓ(1+α)1(φ(tυ)φ(a))αΓ(1+α)q(φ(tυ)φ(a))α+Γ(1+α)(φ(t)φ(a))αΓ(1α)(φ(tυ)φ(a))αΓ(1+α)q(φ(a+υ)φ(a))α+(φ(a+3υ)φ(a))αΓ(1+α)Γ(1α)(φ(a+υ)φ(a))αΓ(1+α)(φ(a+υ)φ(a))α×(1q+Γ(1α)(φ(a+3υ)φ(a))α),  $ (3.14)

    Notice that $ \Gamma (1+\alpha) $ can be approximated by one.

    By virtue of relations (3.13) and (3.14), having in mind (3.9), we infer

    $ w(t)Eα(q(φ(t)φ(a))α)ϖ(a)+p1MEα(q(φ(tυ)φ(a))α)Eα(q(φ(t)φ(a))α)×AEα(q(φ(a+υ)φ(a))α)×[1+p1M+(p1M)2]supaυσaϖ(σ)+Msupaσtw(σ)Eα(q(φ(σ)φ(a))α), $

    or

    $ w(t)Eα(q[φ(t)φ(a)]α)(1M)supaυσaϖ(σ){1+p1MAVEα(q[φ(υ+a)φ(a)]α)×[1+p1M+(p1M)2]}, $ (3.15)

    where

    $ V:=1(φ(a+υ)φ(a))α(1q+Γ(1α)[φ(a+3υ)φ(a)]α). $

    As

    $ AVEα(q[φ(a+υ)φ(a)]α)>1, $

    we can rewrite Eq (3.15) as follows:

    $ (1M)w(t)Eα(q[φ(t)φ(a)]α)supaυσaϖ(σ)AEα(q[φ(υ+a)φ(a)]α)×{1+pV1M+(pV1M)2+(pV1M)3}. $

    We now make the following claim.

    Claim. For $ t\in \lbrack a+(n-1)\upsilon, a+n\upsilon], $

    $ (1M)w(t)Eα(q(φ(t)φ(a))α)AEα(q(φ(υ+a)φ(a))α)×nk=0(pV1M)ksupaνσaϖ(σ). $

    It is evident that the assertion is valid for the cases $ n = 1, $ $ 2, $ and $ 3 $. Assume that it holds for $ n, $ i.e., on $ [a+(n-1)\upsilon, a+n\upsilon]. $ Now, let $ t\in \lbrack a+n\upsilon, a+\upsilon (n+1)]. $ Utilizing (3.9), we derive

    $ w(t)Eα(q[φ(t)φ(a)]α)supaυσaϖ(σ)+pEα(q[φ(tυ)φ(a)]α)(1M)Eα(q[φ(t)φ(a)]α)×AEα(q[φ(a+υ)φ(a)]α)nk=0(pV1M)ksupaυσaϖ(σ)+Msupaσtw(σ)Eα(q[φ(σ)φ(a)]α). $

    and by (3.14)

    $ w(t)Eα(q(φ(t)φ(a))α)(1M)[1+Vp1MAEα(q(φ(a+υ)φ(a))α)nk=0(Vp1M)k]supaυσaϖ(σ)AEα(q(φ(a+υ)φ(a))α)[1+n+1k=1(Vp1M)k]supaυσaϖ(σ)=AEα(q[φ(a+υ)φ(a)]α)n+1k=0(Vp1M)ksupaυσaϖ(σ). $

    Therefore, the claim holds true. Then, for $ t > a, $

    $ w(t)[AEα(q[φ(a+υ)φ(a)]α)(1M)k=0(pV1M)ksupaυσaϖ(σ)]×Eα(q(φ(t)φ(a))α). $ (3.16)

    The series in (3.16) converges due to (3.4) and (3.5). The proof is complete.

    In this section, we identify two classes of functions that satisfy the conditions of the theorem.

    First class: Consider the set of functions $ k $ that fulfill the following inequality for all $ s\geq a: $

    $ saEα(q[φ(σ)φ(a)]α)k(sσ)dσC1[φ(s)φ(a)]λ1,C1,λ>0. $ (4.1)

    The family of functions $ k(t-s) $ defined as

    $ k(ts)C2[φ(t)φ(s)]αeb[φ(s)φ(a)]φ(s) $

    satisfies the specified relation when the constants $ b $ and $ C_{2} $ are carefully chosen. Indeed, since

    $ Eα(qtα)11+qtαΓ(1+α)=Γ(1+α)Γ(1+α)+qtαΓ(1+α)qtα,t>0, $ (4.2)

    it follows that

    $ saEα(q[φ(σ)φ(a)]α)k(sσ)dσC2Γ(1+α)qsa[φ(σ)φ(a)]α[φ(s)φ(σ)]αeb[φ(σ)φ(a)]φ(σ)dσ2αC2Γ(1+α)Γ(1α)[3α]bα1q[φ(s)φ(a)]α,s>a. $

    Therefore, (4.1) holds with

    $ C1:=2αC2Γ(1+α)Γ(1α)[3α]bα1q,λ:=1α. $

    By applying formula (2.6), we obtain

    $ taEα,α(q[φ(t)φ(s)]α)[φ(t)φ(s)]α1×(sak(sσ)Eα(q[φ(σ)φ(a)]α)dσ)φ(s)dsC1taEα,α(q[φ(t)φ(s)]α)[φ(t)φ(s)]α1[φ(s)φ(a)]αφ(s)dsC1Γ(α)Eα,1(q[φ(t)φ(a)]α). $ (4.3)

    To ensure that assumption (3.4) is met, we can select $ C_{1} $ (or $ C_{2} $ for the specific example) such that

    $ \begin{equation*} C_{1}\Gamma (\alpha ) < 1-\frac{1}{\left( \varphi \left( a+\upsilon \right) -\varphi \left( a\right) \right) ^{\alpha }}\left( \frac{1}{q}+\Gamma (1-\alpha )\left[ \varphi \left( a+3\upsilon \right) -\varphi \left( a\right) \right] ^{\alpha }\right) p. \end{equation*} $

    Second class: Assume that $ k(t-s)\leq C_{3}\left[\varphi \left(t\right) -\varphi \left(s\right) \right] ^{\alpha -1}E_{\alpha, \alpha }(-b\left[\varphi \left(t\right) -\varphi \left(s\right) \right] ^{\alpha })\varphi ^{\prime }\left(s\right) $ for some $ b > 0 $ and $ C_{3} > 0 $ to be determined. A double use of (2.6) and (4.2) gives

    $ \begin{eqnarray} &&C_{3}\int_{a}^{t}\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha -1}E_{\alpha , \alpha }\left( -q\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha }\right) \\ &&\times \left( \int_{a}^{s}\left[ \varphi \left( s\right) -\varphi \left( \sigma \right) \right] ^{\alpha -1}E_{\alpha , \alpha }(-b\left[ \varphi \left( s\right) -\varphi \left( \sigma \right) \right] ^{\alpha })E_{\alpha }(-q\left[ \varphi \left( \sigma \right) -\varphi \left( a\right) \right] ^{\alpha })\varphi ^{\prime }\left( \sigma \right) d\sigma \right) \varphi ^{\prime }\left( s\right) \, ds \\ &\leq &\frac{C_{3}\Gamma (1+\alpha )}{q}\int_{a}^{t}E_{\alpha , \alpha }\left( -q\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha }\right) \left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha -1} \\ &&\times \left( \int_{a}^{s}\left[ \varphi \left( s\right) -\varphi \left( \sigma \right) \right] ^{\alpha -1}E_{\alpha , \alpha }(-b\left[ \varphi \left( s\right) -\varphi \left( \sigma \right) \right] ^{\alpha })\left[ \varphi \left( \sigma \right) -\varphi \left( a\right) \right] ^{-\alpha }\varphi ^{\prime }\left( \sigma \right) d\sigma \right) \varphi ^{\prime }\left( s\right) \, ds \\ &\leq &\frac{C_{3}\Gamma (\alpha )\Gamma (1+\alpha )}{q}\int_{a}^{t}E_{ \alpha , \alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha }E_{\alpha , 1}(-b\left[ \varphi \left( s\right) -\varphi \left( a\right) \right] ^{\alpha }\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha -1}\varphi ^{\prime }\left( s\right) \, ds \\ &\leq &\frac{C_{3}\Gamma ^{2}(1+\alpha )\Gamma (\alpha )}{qb} \int_{a}^{t}E_{\alpha , \alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha }\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha -1}\left[ \varphi \left( s\right) -\varphi \left( a\right) \right] ^{-\alpha }\varphi ^{\prime }\left( s\right) \, ds \\ &\leq &\frac{C_{3}\Gamma ^{2}(1+\alpha )\Gamma ^{2}(\alpha )}{qb}E_{\alpha , 1}(-q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }. \end{eqnarray} $ (4.4)

    Clearly, $ M = \frac{C_{3}\Gamma ^{2}(1+\alpha)\Gamma ^{2}(\alpha)}{qb}. $ It suffices now to impose the condition on $ C_{3} $ and/or the constant $ b $ in order to fulfill the condition on $ M $.

    In this section, we will examine the inequality that arises when the neutral delay is distributed,

    $ \begin{equation} \left\{ \begin{array}{l} D_{C}^{\varphi , \alpha }\left[ u(t)-p\int_{a}^{t}u(s)g(t-s)\, ds\right] \leq -qu(t)+\int_{a}^{t}u(s)k(t-s)\, ds, \text{ }t, \upsilon > a, \;0 < \alpha < 1, \text{ }p > 0, \\ \\ u(t) = u_{0}\geq 0, \;t\in \lbrack a-\upsilon , a], \end{array} \right. \end{equation} $ (5.1)

    which we will contrast with

    $ \begin{equation} \left\{ \begin{array}{l} D_{C}^{\varphi , \alpha }\left[ w(t)-p\int_{a}^{t}w(s)g(t-s)\, ds\right] = -qw(t)+\int_{a}^{t}w(s)k(t-s)\, ds, \;t, \upsilon > a, \;0 < \alpha < 1, \text{ }p > 0, \\ w(t) = w_{0} = u_{0}\geq 0, \;t\in \lbrack a-\upsilon , a]. \end{array} \right. \end{equation} $ (5.2)

    We assume $ g $ is a continuous function (to be determined later) and that the solutions are nonnegative.

    Let us reformulate this as

    $ \begin{equation*} \left\{ \begin{array}{l} D_{C}^{\varphi , \alpha }\left[ w(t)-p\int_{a}^{t}w(s)g(t-s)\, ds\right] = -q \left[ w(t)-p\int_{a}^{t}w(s)g(t-s)\, ds\right] \\ -qp\int_{a}^{t}w(s)g(t-s)\, ds+\int_{a}^{t}w(s)k(t-s)\, ds, \;t, \upsilon > a, \;0 < \alpha < 1, \text{ }p > 0 \\ w(t) = w_{0}\geq 0, \;t\in \lbrack a-\upsilon , a]. \end{array} \right. \end{equation*} $

    Therefore,

    $ \begin{eqnarray*} w(t)-p\int_{a}^{t}w(s)g(t-s)\, ds & = &E_{\alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha })w_{0} \\ &&+\int_{a}^{t}\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha -1}E_{\alpha , \alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha }) \\ &&\times \left( -qp\int_{a}^{s}g(s-\sigma )w(\sigma )\, d\sigma +\int_{a}^{s}k(s-\sigma )w(\sigma )\, d\sigma \right) \varphi ^{\prime }\left( s\right) ds, \end{eqnarray*} $

    and, for $ t > a, $

    $ \begin{eqnarray} w(t) &\leq &E_{\alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha })w_{0}+p\int_{a}^{t}g(t-s)\, w(s)ds+\int_{a}^{t}E_{\alpha , \alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha }) \\ &&\times \left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha -1}\left( \int_{a}^{s}k(s-\sigma )w(\sigma )\, d\sigma \right) \varphi ^{\prime }\left( s\right) ds. \end{eqnarray} $ (5.3)

    Dividing both sides of (5.3) by $ E_{\alpha }(-q\left[\varphi \left(t\right) -\varphi \left(a\right) \right] ^{\alpha }), $ we find

    $ \begin{eqnarray*} \frac{w(t)}{E_{\alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha })} & = &w_{0}+\frac{p}{E_{\alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha })} \int_{a}^{t}w(s)\, g(t-s)ds \\ &&+\frac{1}{E_{\alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha })}\int_{a}^{t}\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha -1} \\ &&\times E_{\alpha , \alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha }) \\ &&\times \left( \int_{a}^{s}k(s-\sigma )E_{\alpha }(-q\left[ \varphi \left( \sigma \right) -\varphi \left( a\right) \right] ^{\alpha })d\sigma \right) \varphi ^{\prime }\left( s\right) ds \\ &&\times \sup\limits_{a\leq \sigma \leq t}\frac{w(\sigma )}{E_{\alpha }(-q\left[ \varphi \left( \sigma \right) -\varphi \left( a\right) \right] ^{\alpha })}, \end{eqnarray*} $

    or, for $ t > a, $

    $ \begin{eqnarray*} \frac{w(t)}{E_{\alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha })} &\leq &w_{0}+\frac{p}{E_{\alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha })} \int_{a}^{t}g(t-s)E_{\alpha }(-q\left[ \varphi \left( s\right) -\varphi \left( a\right) \right] ^{\alpha }) \\ &&\times \left( \frac{w(s)}{E_{\alpha }(-q\left[ \varphi \left( s\right) -\varphi \left( a\right) \right] ^{\alpha })}\right) \, ds \\ &&+M\sup\limits_{a\leq \sigma \leq t}\frac{w(\sigma )}{E_{\alpha }(-q\left[ \varphi \left( \sigma \right) -\varphi \left( a\right) \right] ^{\alpha })}. \end{eqnarray*} $

    The relation

    $ \begin{equation*} \frac{p}{E_{\alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha })}\int_{a}^{t}g(t-s)E_{\alpha }(-q\left[ \varphi \left( s\right) -\varphi \left( a\right) \right] ^{\alpha })\, ds\leq M^{\ast }, \end{equation*} $

    is assumed for some $ M^{\ast } > 0. $ Then,

    $ \begin{equation*} \frac{w(t)}{E_{\alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha })}\leq w_{0}+\left( M^{\ast }+M\right) \sup\limits_{a\leq \sigma \leq t}\frac{w(\sigma )}{E_{\alpha }(-q\left[ \varphi \left( \sigma \right) -\varphi \left( a\right) \right] ^{\alpha })}, \;t > a, \end{equation*} $

    and

    $ \begin{equation*} w(t)\leq \frac{w_{0}}{1-M^{\ast }-M}E_{\alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }), \;t > a, \end{equation*} $

    in the case that

    $ \begin{equation*} M^{\ast }+M < 1. \end{equation*} $

    Example. Take $ k $ as above, and select $ g $ fulfilling

    $ \begin{equation*} g(t-s)\leq C_{4}\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha -1}E_{\alpha , \alpha }(-c\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha })\varphi ^{\prime }\left( s\right), \end{equation*} $

    for some $ C_{4}, c > q. $ Then,

    $ \begin{eqnarray*} \int_{a}^{t}E_{\alpha }(-q\left[ \varphi \left( s\right) -\varphi \left( a\right) \right] ^{\alpha })g(t-s)ds &\leq &\frac{\Gamma (1+\alpha )}{q} \int_{a}^{t}\left[ \varphi \left( s\right) -\varphi \left( a\right) \right] ^{-\alpha }g(t-s)ds \\ &\leq &\frac{C_{4}\Gamma (1+\alpha )}{q}\int_{a}^{t}E_{\alpha , \alpha }(-c \left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha }) \\ &&\times \left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha -1}\left[ \varphi \left( s\right) -\varphi \left( a\right) \right] ^{-\alpha }\varphi ^{\prime }\left( s\right) ds \\ &\leq &C_{4}\frac{\Gamma (1+\alpha )\Gamma (\alpha )}{q}E_{\alpha , 1}(-q \left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }), \;t > a. \end{eqnarray*} $

    A value for $ M^{\ast } $ would be

    $ \begin{equation*} M^{\ast } = \frac{C_{4}p\Gamma (1+\alpha )\Gamma (\alpha )}{q}. \end{equation*} $

    Therefore, we have proved the following theorem.

    Theorem 5.1. Let $ u(t) $ be a nonnegative solution of (5.1), where $ q $ and $ p $ are positive and $ k $ and $ g $ are continuous functions with $ k\left(t\right), $ $ g\left(t\right) \geq 0 $ for all $ t $ such that

    $ \begin{eqnarray*} &&\int_{a}^{t}E_{\alpha , \alpha }\left( -q\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha }\right) \left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha -1} \\ &&\times \left( \int_{a}^{s}E_{\alpha }\left( -q\left[ \varphi \left( \sigma \right) -\varphi \left( a\right) \right] ^{\alpha }\right) k\left( s-\sigma \right) d\sigma \right) \, \varphi ^{\prime }\left( s\right) ds \\ &\leq &ME_{\alpha }\left( -q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }\right) , \;t > a, \end{eqnarray*} $
    $ \begin{equation*} p\int_{a}^{t}g(t-s)E_{\alpha }(-q\left[ \varphi \left( s\right) -\varphi \left( a\right) \right] ^{\alpha })\, ds\leq M^{\ast }E_{\alpha }\left( -q \left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }\right) , \;t > a, \end{equation*} $

    hold for some $ M, $ $ M^{\ast } > 0 $ with

    $ \begin{equation*} M^{\ast }+M < 1. \end{equation*} $

    Then, we can find a positive constant $ C $ such that

    $ \begin{equation*} w(t)\leq CE_{\alpha }\left( -q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }\right) , \;t > a. \end{equation*} $

    Before delving into applications, it is important to note that previous research on Halanay inequalities, including our earlier work, often assumes that solutions are non-negative. This supposition is sufficient for applications like neural networks without time delays. To determine the stability of the equilibrium solution, we can simplify the problem by shifting the equilibrium point to the origin using a variable transformation and then analyzing the magnitude of the solutions. However, when dealing with systems that have time delays, this approach becomes more complex. Directly proving stability for solutions that can be positive or negative presents new challenges, as time delays now appear within convolution integrals. The necessary estimations are more intricate and require careful analysis.

    Now, we return to

    $ \begin{equation*} \left\{ \begin{array}{l} D_{C}^{\varphi , \alpha }\left[ u(t)-pu(t-\upsilon )\right] \leq -qu(t)+\int_{a}^{t}k(t-s)u(s)\, ds, \text{ }p > 0, \;0 < \alpha < 1, \;t, \upsilon > a, \\ u(t) = \varpi (t)\geq 0, \;a-\upsilon \leq t\leq a, \end{array} \right. \end{equation*} $

    with $ \left\vert \varpi (s)\right\vert \leq w_{0}E_{\alpha }(-q(\varphi \left(s+\upsilon \right) -\varphi \left(a\right))^{\alpha }) $ for $ s\in \lbrack a-\upsilon, a], $ $ w_{0} > 0. $ To clarify these concepts, let us suppose that $ 1 > p > 0, $ and examine the following expression:

    $ \begin{eqnarray*} w(t)-pw(t-\upsilon ) & = &\left[ \varpi (a)-p\varpi (a-\upsilon )\right] E_{\alpha }\left( -q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }\right) \\ &&+\int_{a}^{t}\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha -1}E_{\alpha , \alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha }) \\ &&\times \left( -qpw(s-\upsilon )+\int_{a}^{s}k(s-\sigma )w(\sigma )d\sigma \right) \, \varphi ^{\prime }\left( s\right) ds. \end{eqnarray*} $

    Then, for $ t > a $

    $ \begin{eqnarray} \left\vert w(t)\right\vert &\leq &2w_{0}E_{\alpha }\left( -q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }\right) +p\left\vert w(t-\upsilon )\right\vert \\ &&+qp\int_{a}^{t}E_{\alpha , \alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha })\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha -1}\left\vert w(s-\upsilon )\right\vert \varphi ^{\prime }\left( s\right) ds \\ &&+\int_{a}^{t}E_{\alpha , \alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha })\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha -1} \\ &&\times \left( \int_{a}^{s}k(s-\sigma )\left\vert w(\sigma )\right\vert d\sigma \right) \, \varphi ^{\prime }\left( s\right) ds. \end{eqnarray} $ (6.1)

    For $ t\in \lbrack a, a+\upsilon], $

    $ \begin{eqnarray*} \frac{\left\vert w(t)\right\vert }{E_{\alpha }\left( -q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }\right) } &\leq &3w_{0}+ \frac{qpw_{0}}{E_{\alpha }\left( -q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }\right) }\int_{a}^{t}\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha -1} \\ &&\times E_{\alpha , \alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha })E_{\alpha }(-q\left[ \varphi \left( s\right) -\varphi \left( a\right) \right] ^{\alpha })\, \varphi ^{\prime }\left( s\right) ds \\ &&+M\sup\limits_{a\leq \sigma \leq t}\frac{w(\sigma )}{E_{\alpha }\left( -q\left[ \varphi \left( \sigma \right) -\varphi \left( a\right) \right] ^{\alpha }\right) }, \end{eqnarray*} $

    where $ M $ is defined as in Eq (3.3). Again, as

    $ \begin{eqnarray} &&\int_{a}^{t}E_{\alpha , \alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha })\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha -1}E_{\alpha }(-q\left[ \varphi \left( s\right) -\varphi \left( a\right) \right] ^{\alpha })\, \varphi ^{\prime }\left( s\right) ds \\ &\leq &\frac{\Gamma (1+\alpha )}{q}\int_{a}^{t}E_{\alpha , \alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha })\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha -1}\left[ \varphi \left( s\right) -\varphi \left( a\right) \right] ^{-\alpha }\, \varphi ^{\prime }\left( s\right) ds \\ &\leq &\frac{\Gamma (1+\alpha )\Gamma (\alpha )}{q}E_{\alpha , 1}\left( -q \left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }\right), \end{eqnarray} $ (6.2)

    we can write

    $ \begin{eqnarray*} \frac{\left\vert w(t)\right\vert }{E_{\alpha }\left( -q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }\right) } &\leq &3w_{0}+w_{0}\Gamma (1+\alpha )\Gamma (\alpha )p \\ &&+M\sup\limits_{a\leq \sigma \leq t}\frac{w(\sigma )}{E_{\alpha }\left( -q\left[ \varphi \left( \sigma \right) -\varphi \left( a\right) \right] ^{\alpha }\right) }, \end{eqnarray*} $

    or

    $ \begin{equation} (1-M)\frac{\left\vert w(t)\right\vert }{E_{\alpha }\left( -q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }\right) }\leq 3w_{0}+w_{0}\Gamma (1+\alpha )\Gamma (\alpha )p. \end{equation} $ (6.3)

    If $ t\in \lbrack a+\upsilon, a+2\upsilon], $ we first observe that

    $ \begin{eqnarray*} \left\vert w(t-\upsilon )\right\vert &\leq &\frac{3w_{0}+w_{0}\Gamma (1+\alpha )\Gamma (\alpha )p}{(1-M)} \\ &&\times \frac{E_{\alpha }\left( -q\left[ \varphi \left( t-\upsilon \right) -\varphi \left( a\right) \right] ^{\alpha }\right) }{E_{\alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha })} E_{\alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }) \\ &\leq &A\frac{3w_{0}+w_{0}\Gamma (1+\alpha )\Gamma (\alpha )p}{(1-M)} E_{\alpha }(-q\left( \varphi \left( t\right) -\varphi \left( a\right) \right) ^{\alpha }), \end{eqnarray*} $

    where $ A $ is as in (3.11). Using the fact that

    $ \begin{equation*} w_{0}\leq A\frac{3w_{0}+w_{0}\Gamma (1+\alpha )\Gamma (\alpha )p}{1-M}, \end{equation*} $

    and relations (6.1) and (6.3), we get

    $ \begin{eqnarray*} \left\vert w(t)\right\vert &\leq &2w_{0}E_{\alpha }\left( -q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }\right) \\ &&+pA\frac{3w_{0}+w_{0}\Gamma (1+\alpha )\Gamma (\alpha )p}{(1-M)}E_{\alpha }(-q\left( \varphi \left( t\right) -\varphi \left( a\right) \right) ^{\alpha }) \\ &&+qpA\frac{3w_{0}+w_{0}\Gamma (1+\alpha )\Gamma (\alpha )p}{(1-M)} \\ &&\times \int_{a}^{t}E_{\alpha , \alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha })\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha -1} \\ &&\times E_{\alpha }(-q\left( \varphi \left( s\right) -\varphi \left( a\right) \right) ^{\alpha })\varphi ^{\prime }\left( s\right) ds \\ &&+\int_{a}^{t}E_{\alpha , \alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha })\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha -1} \\ &&\times \left( \int_{a}^{s}k(s-\sigma )\left\vert w(\sigma )\right\vert d\sigma \right) \, \varphi ^{\prime }\left( s\right) ds. \end{eqnarray*} $

    Next, in view of (6.2), we find

    $ \begin{eqnarray*} \left\vert w(t)\right\vert &\leq &2w_{0}E_{\alpha }\left( -q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }\right) \\ &&+pA\frac{3w_{0}+w_{0}\Gamma (\alpha )\Gamma (\alpha +1)p}{(1-M)}E_{\alpha }(-q\left( \varphi \left( t\right) -\varphi \left( a\right) \right) ^{\alpha }) \\ &&+qpA\frac{3w_{0}+w_{0}\Gamma (1+\alpha )\Gamma (\alpha )p}{(1-M)} \\ &&\times \frac{\Gamma (1+\alpha )\Gamma (\alpha )}{q}E_{\alpha }\left( -q \left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }\right) \\ &&+\int_{a}^{t}\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha -1}E_{\alpha , \alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha }) \\ &&\times \left( \int_{a}^{s}k(s-\sigma )\left\vert w(\sigma )\right\vert d\sigma \right) \, \varphi ^{\prime }\left( s\right) ds. \end{eqnarray*} $

    or

    $ \begin{eqnarray} \left( 1-M\right) \frac{\left\vert w(t)\right\vert }{E_{\alpha }\left( -q \left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }\right) } &\leq &2w_{0}+\frac{pAw_{0}\left[ 1+\Gamma (1+\alpha )\Gamma (\alpha )\right] }{(1-M)} \\ &&\times \left( 3+\Gamma (1+\alpha )\Gamma (\alpha )p\right) \\ &\leq &2w_{0}+\frac{3Aw_{0}\left[ 1+\Gamma (1+\alpha )\Gamma (\alpha )\right] }{(1-M)}p \\ &&+\frac{Aw_{0}\left[ 1+\Gamma (1+\alpha )\Gamma (\alpha )\right] ^{2}}{(1-M) }p^{2}. \end{eqnarray} $ (6.4)

    For $ t\in \lbrack a+2\upsilon, a+3\upsilon], $ by virtue of (3.14),

    $ \begin{eqnarray*} \frac{E_{\alpha }(-q\left( \varphi \left( t-\upsilon \right) -\varphi \left( a\right) \right) ^{\alpha })}{E_{\alpha }(-q\left( \varphi \left( t\right) -\varphi \left( a\right) \right) ^{\alpha })} &\leq &\frac{1}{\left( \varphi \left( a+\upsilon \right) -\varphi \left( a\right) \right) ^{\alpha }} \\ \times \left( \frac{1}{q}+\Gamma (1-\alpha )\left( \varphi \left( a+3\upsilon \right) -\varphi \left( a\right) \right) ^{\alpha }\right) & = &:V > 1, \end{eqnarray*} $

    and therefore

    $ \begin{eqnarray*} \left\vert w(t)\right\vert &\leq &2w_{0}E_{\alpha }\left( -q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }\right) +\frac{pV }{\left( 1-M\right) } \\ &&\times \left[ 2w_{0}+\frac{3Aw_{0}\left[ 1+\Gamma (1+\alpha )\Gamma (\alpha )\right] }{(1-M)}p+\frac{Aw_{0}\left[ 1+\Gamma (1+\alpha )\Gamma (\alpha )\right] ^{2}}{(1-M)}p^{2}\right] \\ &&\times E_{\alpha }(-q\left( \varphi \left( t\right) -\varphi \left( a\right) \right) \\ &&+\frac{pV\Gamma (1+\alpha )\Gamma (\alpha )}{\left( 1-M\right) } \\ &&\times \left[ 2w_{0}+\frac{3Aw_{0}\left[ 1+\Gamma (1+\alpha )\Gamma (\alpha )\right] }{(1-M)}p+\frac{Aw_{0}\left[ 1+\Gamma (1+\alpha )\Gamma (\alpha )\right] ^{2}}{(1-M)}p^{2}\right] \\ &&\times E_{\alpha }(-q\left( \varphi \left( t\right) -\varphi \left( a\right) \right) \\ &&+\int_{a}^{t}\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha -1}E_{\alpha , \alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha })\left( \int_{a}^{s}k(s-\sigma )\left\vert w(\sigma )\right\vert d\sigma \right) \, \varphi ^{\prime }\left( s\right) ds. \end{eqnarray*} $

    So,

    $ \begin{eqnarray*} &&\left( 1-M\right) \frac{\left\vert w(t)\right\vert }{E_{\alpha }\left( -q \left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }\right) } \\ &\leq &2w_{0}+\frac{pV}{\left( 1-M\right) }\left[ 2w_{0}+\frac{3Aw_{0}\left[ 1+\Gamma (1+\alpha )\Gamma (\alpha )\right] }{(1-M)}p+\frac{Aw_{0}\left[ 1+\Gamma (1+\alpha )\Gamma (\alpha )\right] ^{2}}{(1-M)}p^{2}\right] \\ &&+\frac{pV\Gamma (1+\alpha )\Gamma (\alpha )}{\left( 1-M\right) } \\ &&\times \left[ 2w_{0}+\frac{3Aw_{0}\left[ 1+\Gamma (1+\alpha )\Gamma (\alpha )\right] }{(1-M)}p+\frac{Aw_{0}\left[ 1+\Gamma (1+\alpha )\Gamma (\alpha )\right] ^{2}}{(1-M)}p^{2}\right], \end{eqnarray*} $

    or

    $ \begin{eqnarray} \left( 1-M\right) \frac{\left\vert w(t)\right\vert }{E_{\alpha }\left( -q \left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }\right) } &\leq &2w_{0}+2w_{0}\frac{pV\left[ 1+\Gamma (1+\alpha )\Gamma (\alpha )\right] }{1-M} \\ &&+3w_{0}A\frac{p^{2}V\left[ 1+\Gamma (1+\alpha )\Gamma (\alpha )\right] ^{2} }{\left( 1-M\right) ^{2}} \\ &&+\frac{Aw_{0}V\left[ 1+\Gamma (1+\alpha )\Gamma (\alpha )\right] ^{3}}{ (1-M)^{2}}p^{3}. \end{eqnarray} $ (6.5)

    Writing (6.5) in the form

    $ \begin{eqnarray} \frac{\left\vert w(t)\right\vert }{E_{\alpha }\left( -q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }\right) }\left( 1-M\right) &\leq &2w_{0}+2w_{0}\frac{pV\left[ 1+\Gamma (1+\alpha )\Gamma (\alpha )\right] }{1-M} \\ &&+3w_{0}A\left( \frac{pV\left[ 1+\Gamma (1+\alpha )\Gamma (\alpha )\right] }{\left( 1-M\right) }\right) ^{2} \\ &&+w_{0}A\left( \frac{pV\left[ 1+\Gamma (1+\alpha )\Gamma (\alpha )\right] }{ 1-M}\right) ^{3} \\ &\leq &3w_{0}A\left[ 1+\frac{pV\left[ 1+\Gamma (1+\alpha )\Gamma (\alpha ) \right] }{1-M}\right. \\ &&+\left( \frac{pV\left[ 1+\Gamma (1+\alpha )\Gamma (\alpha )\right] }{1-M} \right) ^{2} \\ &&\left. +\left( \frac{pV\left[ 1+\Gamma (1+\alpha )\Gamma (\alpha )\right] }{1-M}\right) ^{3}\right] \end{eqnarray} $ (6.6)

    provides the basis for our next claim.

    Claim. On the interval $ [a+(n-1)\upsilon, a+n\upsilon], $ it is clear that

    $ \begin{equation*} \frac{\left\vert w(t)\right\vert }{E_{\alpha }\left( -q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }\right) }\left( \frac{1-M }{w_{0}}\right) \leq 3A\sum\nolimits_{k = 0}^{n}\left( \frac{Vp\left[ 1+\Gamma (1+\alpha )\Gamma (\alpha )\right] }{1-M}\right) ^{k}. \end{equation*} $

    The validity of the claim for $ n = 1, 2, $ and $ 3 $ is established by Eqs (6.3), (6.4), and (6.6). Let $ t\in \lbrack a+n\upsilon, a+(n+1)\upsilon]. $ Then from (6.1),

    $ \begin{eqnarray*} \left\vert w(t)\right\vert &\leq &2w_{0}E_{\alpha }\left( -q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }\right) \\ &&+3ApV\frac{w_{0}}{1-M}\sum\nolimits_{k = 0}^{n}\left( \frac{Vp\left[ \Gamma (\alpha )\Gamma (1+\alpha )+1\right] }{1-M}\right) ^{k}E_{\alpha }\left( -q \left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }\right) \\ &&+3Ap\Gamma (1+\alpha )\Gamma (\alpha )\frac{w_{0}V}{1-M} \sum\nolimits_{k = 0}^{n}\left( \frac{Vp\left[ \Gamma (\alpha )\Gamma (1+\alpha )+1\right] }{1-M}\right) ^{k} \\ &&\times E_{\alpha }\left( -q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }\right) \\ &&+\int_{a}^{t}E_{\alpha , \alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha })\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha -1}\left( \int_{a}^{s}k(s-\sigma )\left\vert w(\sigma )\right\vert d\sigma \right) \, \varphi ^{\prime }\left( s\right) ds, \end{eqnarray*} $

    or

    $ \begin{eqnarray*} \left( \frac{1-M}{w_{0}}\right) \frac{\left\vert w(t)\right\vert }{E_{\alpha }\left( -q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }\right) } &\leq &2+\frac{3VpA}{1-M}\sum\nolimits_{k = 0}^{n}\left( \frac{Vp\left[ \Gamma (1+\alpha )\Gamma (\alpha )+1\right] }{1-M}\right) ^{k} \\ &&+\frac{3VpA}{1-M}\Gamma (1+\alpha )\Gamma (\alpha ) \\ &&\times \sum\nolimits_{k = 0}^{n}\left( \frac{Vp\left[ \Gamma (1+\alpha )\Gamma (\alpha )+1\right] }{1-M}\right) ^{k}. \end{eqnarray*} $

    Then,

    $ \begin{eqnarray*} &&\left( \frac{1-M}{w_{0}}\right) \frac{\left\vert w(t)\right\vert }{ E_{\alpha }\left( -q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }\right) } \\ &\leq &3A\left\{ 1+\left[ \Gamma (1+\alpha )\Gamma (\alpha )+1\right] \frac{ pV}{1-M}\sum\nolimits_{k = 0}^{n}\left( \frac{Vp\left[ \Gamma (1+\alpha )\Gamma (\alpha )+1\right] }{1-M}\right) ^{k}\right\}, \end{eqnarray*} $

    i.e.,

    $ \begin{equation*} \left( \frac{1-M}{w_{0}}\right) \frac{\left\vert w(t)\right\vert }{E_{\alpha }\left( -q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }\right) }\leq 3A\left\{ 1+\sum\nolimits_{k = 0}^{n}\left( \frac{Vp \left[ \Gamma (1+\alpha )\Gamma (\alpha )+1\right] }{1-M}\right) ^{1+k}\right\} . \end{equation*} $

    Thus,

    $ \begin{equation*} \left( \frac{1-M}{w_{0}}\right) \frac{\left\vert w(t)\right\vert }{E_{\alpha }\left( -q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }\right) }\leq 3A\sum\nolimits_{k = 0}^{n+1}\left( \frac{Vp\left[ \Gamma (1+\alpha )\Gamma (\alpha )+1\right] }{1-M}\right) ^{k}, \end{equation*} $

    demonstrating that the assertion holds. Moreover, the series converges if the following condition is satisfied:

    $ \begin{equation*} \frac{1+\Gamma (1+\alpha )\Gamma (\alpha )}{1-M}Vp < 1. \end{equation*} $

    We have just proved the following result.

    Theorem 6.1. Suppose that $ u(t) $ is a solution of

    $ \begin{equation*} \left\{ \begin{array}{l} D_{C}^{\varphi , \alpha }\left[ u(t)-pu(t-\upsilon )\right] \leq -qu(t)+\int_{0}^{t}k(t-s)u(s)\, ds, \;t, \upsilon > a, \mathit{\text{}}p > 0, \;0 < \alpha < 1, \\ u(t) = \varpi (t), \;a-\upsilon \leq t\leq a, \end{array} \right. \end{equation*} $

    with $ \left\vert \varpi (t)\right\vert \leq E_{\alpha }(-q(\varphi \left(t+\upsilon \right) -\varphi \left(a\right))^{\alpha }), $ $ a-\upsilon \leq t\leq a, $ $ q > 0, $ $ p > 0, $ and $ k $ is a nonnegative function verifying

    $ \begin{eqnarray*} &&\int_{a}^{t}\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha -1}E_{\alpha , \alpha }\left( -q\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha }\right) \\ &&\times \left( \int_{a}^{s}E_{\alpha }(-q\left[ \varphi \left( \sigma \right) -\varphi \left( a\right) \right] ^{\alpha })k(s-\sigma )d\sigma \right) \, \varphi ^{\prime }\left( s\right) ds \\ &\leq &ME_{\alpha }\left( -q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }\right) , \;t > a, \end{eqnarray*} $

    for some $ M $ such that

    $ \begin{equation*} M < 1-\left[ \Gamma (1+\alpha )\Gamma (\alpha )+1\right] Vp, \end{equation*} $

    with

    $ \begin{equation*} \left[ \Gamma (1+\alpha )\Gamma (\alpha )+1\right] Vp < 1. \end{equation*} $

    Then,

    $ \begin{equation*} \left\vert w(t)\right\vert \leq CE_{\alpha }\left( -q\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha }\right) , \;t > a, \end{equation*} $

    where $ C > 0 $ is a positive constant.

    Neural networks are a fundamental part of artificial intelligence and are widely used to address complex problems in various fields. In this work, we utilize our findings to analyze the behavior of Cohen-Grossberg neural networks. Specifically, we consider the following problems:

    $ \begin{equation*} \left\{ \begin{array}{l} D_{C}^{\varphi , \alpha }\left[ x_{i}(t)-px_{i}(t-\upsilon )\right] = -h_{i}\left( x_{i}\left( t\right) \right) \left[ g_{i}\left( x_{i}\left( t\right) \right) -\sum\limits_{j = 1}^{n}a_{ij}f_{j}\left( x_{j}\left( t\right) \right) -\sum\limits_{j = 1}^{n}b_{ij}l_{j}\left( x_{j}\left( t-\tau \right) \right) \right. \\ \left. -\sum\limits_{j = 1}^{n}d_{ij}\int\nolimits_{a}^{\infty }k_{j}\left( s\right) \Phi _{j}\left( x_{j}\left( t-s\right) \right) ds-I_{i}\right] , { \ } t, \upsilon > a, \text{ }p > 0 \\ x_{i}\left( t\right) = x_{i0}\left( t\right) , \;t\in \lbrack a-\upsilon , a], \text{ }i = 1, 2, ..., n, \end{array} \right. \end{equation*} $

    and

    $ \begin{equation*} \left\{ \begin{array}{l} D_{C}^{\varphi , \alpha }\left[ x_{i}(t)-p\int\nolimits_{a}^{t}x_{i}(s)\psi _{i}(t-s)ds\right] = -h_{i}\left( x_{i}\left( t\right) \right) \left[ g_{i}\left( x_{i}\left( t\right) \right) -\sum\limits_{j = 1}^{n}b_{ij}l_{j}\left( x_{j}\left( t-\tau \right) \right) \right. \\ \left. -\sum\limits_{j = 1}^{n}a_{ij}f_{j}\left( x_{j}\left( t\right) \right) -\sum\limits_{j = 1}^{n}d_{ij}\int\nolimits_{a}^{\infty }\Phi _{j}\left( x_{j}\left( t-s\right) \right) k_{j}\left( s\right) ds-I_{i}\right] , { \ }t > a, \text{ }p > 0, \\ x_{i}\left( 0\right) = x_{i0}\left( t\right) , \text{ }t\leq a, \text{ } i = 1, 2, ..., n, \end{array} \right. \end{equation*} $

    where $ x_{i.}\left(t\right) $ stands the state of the $ i $th neuron, $ n $ is the number of neurons, $ g_{i} $ is a suitable function, $ h_{i} $ represents an amplification function, $ b_{ij}, $ $ a_{ij}, $ $ d_{ij} $ represent the weights or strengths of the connections from the $ j $th neuron to the $ i $th neuron, $ I_{i} $ is the external input to the $ i $th neuron, $ \psi _{i} $ are the neutral delay kernels, $ f_{j}, l_{j}, \Phi _{j} $ denote the signal transmission functions, $ \upsilon $ is the neutral delay, $ \tau $ corresponds to the transmission delay, $ \phi _{i} $ is the history of the $ i $ th state, and $ k_{j} $ denotes the delay kernel function. These systems represent a general class of Cohen-Grossberg neural networks with both continuously distributed and discrete delays. To streamline our analysis and highlight our key findings, we have opted to examine simpler systems with fixed time delays. More complex scenarios involving variable delays or multiple delays can be explored in future research. To simplify our analysis, let us examine the simpler case

    $ \begin{equation} \left\{ \begin{array}{l} D_{C}^{\varphi , \alpha }\left[ x_{i}(t)-px_{i}(t-\upsilon )\right] = -h_{i}\left( x_{i}\left( t\right) \right) \left[ g_{i}\left( x_{i}\left( t\right) \right) -\sum\limits_{j = 1}^{n}d_{ij}\int\nolimits_{a}^{\infty }k_{j}\left( s\right) f_{j}\left( x_{j}\left( t-s\right) \right) ds-I_{i} \right] , \\ x_{i}\left( t\right) = x_{i0}\left( t\right) , \;t\in \lbrack a-\upsilon , a], \text{ }i = 1, 2, ..., n, \end{array} \right. \end{equation} $ (7.1)

    for $ t, \upsilon > a, $ $ p > 0. $

    We adopt the following standard assumptions.

    (A1) The functions $ f_{i} $ are assumed to satisfy the Lipschitz condition

    $ \begin{equation*} \left\vert f_{i}\left( x\right) -f_{i}\left( y\right) \right\vert \leq L_{i}\left\vert x-y\right\vert \text{ for every }x, y\in \mathbb{R} \text{ and for each }i = 1, 2, ..., n, \end{equation*} $

    where $ L_{i} $ denotes the Lipschitz constant corresponding to the function $ f_{i}. $

    (A2) The delay kernel functions $ k_{j} $ are nonnegative and exhibit piecewise continuity. Additionally, each $ k_{j} $ has a finite integral over its domain, expressed as $ \kappa _{j} = \int\nolimits_{a}^{\infty }k_{j}\left(s\right) ds < \infty $, for $ j = 1, ..., n. $

    (A3) The functions $ g_{i} $ have derivatives that are uniformly bounded by a constant $ G $. Specifically,

    $ \begin{equation*} \left\vert g_{i}^{\prime }\left( z\right) \right\vert \leq G, \text{ for all } z\in \mathbb{R} \text{ and for each }i = 1, 2, ..., n, \end{equation*} $

    where $ G > 0 $ is a fixed constant.

    (A4) The functions $ h_{i} $ are strictly positive and continuous, and they satisfy the following bounds:

    $ \begin{equation*} 0 < \underline{\beta }_{i}\leq h_{i}(z)\leq \overline{\beta }_{i}\text{, for all }z\in \mathbb{R} \text{ and }i = 1, 2, ..., n. \end{equation*} $

    For simplicity, we suppose that the initial values $ x_{i0}\left(t\right) $ are all zero for times before $ a $.

    Definition 7.1. The point $ x^{\ast } = \left(x_{1}^{\ast }, x_{2}^{\ast }, ..., x_{n}^{\ast }\right) ^{T} $ is said to be an equilibrium if, for each $ i = 1, 2, ..., n $, it satisfies the equation

    $ \begin{eqnarray*} g_{i}\left( x_{i}^{\ast }\right) & = &\sum\limits_{j = 1}^{n}a_{ij}f_{j}\left( x_{j}^{\ast }\right) +\sum\limits_{j = 1}^{n}d_{ij}\int\nolimits_{a}^{\infty }k_{j}\left( s\right) f_{j}\left( x_{j}^{\ast }\right) ds+I_{i} \\ & = &\sum\limits_{j = 1}^{n}\left( a_{ij}+d_{ij}\kappa _{j}\right) f_{j}\left( x_{j}^{\ast }\right) +I_{i}, \text{ }t > a. \end{eqnarray*} $

    Previous studies have shown that an equilibrium exists and is unique. To facilitate our analysis, we translate the equilibrium point to the origin of the coordinate system by using the substitution $ x\left(t\right) -x^{\ast } = y\left(t\right) $. This leads to the following:

    $ \begin{equation*} \left\{ \begin{array}{l} D_{C}^{\varphi , \alpha }\left[ y_{i}(t)-py_{i}(t-\upsilon )\right] = -h_{i}\left( x_{i}^{\ast }+y_{i}\left( t\right) \right) \left[ \frac{{}}{{}} g_{i}\left( y_{i}\left( t\right) +x_{i}^{\ast }\right) \right. \\ { \ \ \ \ \ \ \ \ \ \ \ \ }\left. -\sum\limits_{j = 1}^{n}d_{ij}\int\nolimits_{a}^{t}f_{j}\left( x_{j}^{\ast }+y_{j}\left( t-s\right) \right) k_{j}\left( s\right) ds-I_{i}\right] , { \ }t > a, \text{ }i = 1, ..., n, \\ y_{i}\left( t\right) = \psi _{i}\left( t\right) : = \phi _{i}\left( t\right) -x_{i}^{\ast }, \text{ }t\in \lbrack a-\upsilon , a], \text{ }i = 1, ..., n, \end{array} \right. \end{equation*} $

    or

    $ \begin{equation*} \left\{ \begin{array}{l} D_{C}^{\varphi , \alpha }\left[ y_{i}(t)-py_{i}(t-\upsilon )\right] = -H_{i}\left( y_{i}\left( t\right) \right) \left[ G_{i}\left( y_{i}\left( t\right) \right) \right. \\ \left. -\sum\limits_{j = 1}^{n}d_{ij}\int\nolimits_{0}^{t}F_{j}\left( y_{j}\left( t-s\right) \right) k_{j}\left( s\right) ds\right] , { \ }t > a, \text{ }i = 1, ..., n, \\ y_{i}\left( t\right) = \psi _{i}\left( t\right) : = \phi _{i}\left( t\right) -x_{i}^{\ast }, \text{ }t\in \lbrack a-\upsilon , a], \text{ }i = 1, ..., n, \end{array} \right. \end{equation*} $

    where

    $ \begin{equation*} \begin{array}{c} F_{i}\left( y_{i}\left( t\right) \right) = f_{i}\left( y_{i}\left( t\right) +x_{i}^{\ast }\right) -f_{i}\left( x_{i}^{\ast }\right) , \text{ }G_{i}\left( y_{i}\left( t\right) \right) = g_{i}\left( y_{i}\left( t\right) +x_{i}^{\ast }\right) -g_{i}\left( x^{\ast }\right) \\ H_{i}\left( y_{i}\left( t\right) \right) = h_{i}\left( y_{i}\left( t\right) +x_{i}^{\ast }\right) , { \ }t > a, \text{ }i = 1, ..., n. \end{array} \end{equation*} $

    Using the mean value theorem, the following inequality can be established:

    $ \begin{equation*} \begin{array}{l} D_{C}^{\varphi , \alpha }\left\vert y_{i}(t)-py_{i}(t-\upsilon )\right\vert \leq sgn\left[ y_{i}(t)-py_{i}(t-\upsilon )\right] D_{C}^{\varphi , \alpha } \left[ y_{i}(t)-py_{i}(t-\upsilon )\right] \\ = -sgn\left[ y_{i}(t)-py_{i}(t-\upsilon )\right] H_{i}\left( y_{i}\left( t\right) \right) \left[ g_{i}^{\prime }\left( \bar{x}_{i}\left( t\right) \right) y_{i}\left( t\right) -\sum\limits_{j = 1}^{n}d_{ij}\int\nolimits_{a}^{\infty }F_{j}\left( y_{j}\left( t-s\right) \right) k_{j}\left( s\right) ds\right] . \end{array} \end{equation*} $

    By subtracting and adding the term $ pg_{i}^{\prime }\left(\bar{x}_{i}\left(t\right) \right) y_{i}(t-\upsilon), $ we obtain

    $ \begin{equation*} \begin{array}{c} D_{C}^{\varphi , \alpha }\left\vert y_{i}(t)-py_{i}(t-\upsilon )\right\vert \leq -sgn\left[ y_{i}(t)-py_{i}(t-\upsilon )\right] H_{i}\left( y_{i}\left( t\right) \right) \left[ g_{i}^{\prime }\left( \bar{x}_{i}\left( t\right) \right) \left[ y_{i}(t)-py_{i}(t-\upsilon )\right] \right. \\ \left. +pg_{i}^{\prime }\left( \bar{x}_{i}\left( t\right) \right) y_{i}(t-\upsilon )-\sum\limits_{j = 1}^{n}d_{ij}\int\nolimits_{a}^{\infty }F_{j}\left( y_{j}\left( t-s\right) \right) k_{j}\left( s\right) ds\right] , { \ }t > a, \text{ }i = 1, ..., n, \end{array} \end{equation*} $

    or

    $ \begin{equation*} \begin{array}{c} D_{C}^{\varphi , \alpha }\left\vert y_{i}(t)-py_{i}(t-\upsilon )\right\vert \leq -H_{i}\left( y_{i}\left( t\right) \right) G\left\vert y_{i}(t)-py_{i}(t-\upsilon )\right\vert +pGH_{i}\left( y_{i}\left( t\right) \right) \left\vert y_{i}(t-\upsilon )\right\vert \\ +H_{i}\left( y_{i}\left( t\right) \right) \sum\limits_{j = 1}^{n}d_{ij}\int\nolimits_{a}^{\infty }k_{j}\left( s\right) L_{j}\left\vert y_{j}\left( t-s\right) \right\vert ds, { \ }t > a, \text{ } i = 1, 2, ..., n. \end{array} \end{equation*} $

    Therefore,

    $ \begin{equation*} \begin{array}{c} D_{C}^{\varphi , \alpha }\left\vert y_{i}(t)-py_{i}(t-\upsilon )\right\vert \leq -G\underline{\beta }_{i}\left\vert y_{i}(t)-py_{i}(t-\upsilon )\right\vert +pG\overline{\beta }_{i}\left\vert y_{i}(t-\upsilon )\right\vert \\ +\overline{\beta }_{i}\sum\limits_{j = 1}^{n}L_{j}d_{ij}\int\nolimits_{a}^{ \infty }k_{j}\left( s\right) \left\vert y_{j}\left( t-s\right) \right\vert ds, { \ }t > a, \text{ }i = 1, ..., n. \end{array} \end{equation*} $

    Finally, we consider the equation for $ w_{i} $ and rewrite it in the following form:

    $ \begin{equation*} \begin{array}{c} \left\vert w_{i}(t)-pw_{i}(t-\upsilon )\right\vert = E_{\alpha }(-G\underline{ \beta }_{i}\left[ \varphi \left( t\right) -\varphi \left( a\right) \right] ^{\alpha })\left\vert \Phi _{i}(a)-p\Phi _{i}(a-\upsilon )\right\vert \\ +\int_{a}^{t}\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha -1}E_{\alpha , \alpha }(-q\left[ \varphi \left( t\right) -\varphi \left( s\right) \right] ^{\alpha }) \\ \times \left( pG\overline{\beta }_{i}\left\vert w_{i}(t-\upsilon )\right\vert +\overline{\beta }_{i}\sum\limits_{j = 1}^{n}L_{j}d_{ij}\int \nolimits_{0}^{\infty }k_{j}\left( s\right) \left\vert w_{j}\left( t-s\right) \right\vert ds\right) \, \varphi ^{\prime }\left( s\right) ds, { \ }t > a, \text{ }i = 1, 2, ..., n. \end{array} \end{equation*} $

    The Mittag-Leffler stability of this problem follows directly from our earlier result.

    We have investigated a general Halanay inequality of fractional order with distributed delays, incorporating delays of neutral type. General sufficient conditions were established to guarantee the Mittag-Leffler stability of the solutions, supported by illustrative examples. The rate of stability obtained appears to be the best achievable, consistent with previous findings in fractional-order problems.

    Furthermore, we applied our theoretical results to a practical problem, demonstrating their applicability. Our analysis suggests that these results can be extended to more general cases, such as variable delays or systems involving additional terms. It is worth noting that the conditions on the various parameters within the system could potentially be improved, as we did not focus on optimizing the estimations and bounds. In this regard, exploring optimal bounds for the delay coefficient $ p $ and the kernel $ k $ would be an interesting direction for future research.

    The author declares that have not used Artificial Intelligence (AI) tools in the creation of this article.

    The author sincerely appreciates the financial support and facilities provided by Imam Abdulrahman Bin Faisal University.

    The author declares that there is no conflict of interest regarding the publication of this paper.



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