Asian rainbow options provide investors with a new option solution as an effective tool for asset allocation and risk management. In this paper, we address the pricing problem of Asian rainbow options with stochastic interest rates that obey the Vasicek model. By introducing the Vasicek model as the change process of the stochastic interest rate, based on the non-arbitrage principle and the stochastic differential equation, the number of assets of the Asian rainbow option is expanded to n dimensions, and the pricing formulas of the Asian rainbow option with multiple (n) assets under the Vasicek interest rate model are obtained. The multi-asset pricing results under stochastic interest rates provide more possibilities for Asian rainbow options. Furthermore, Monte Carlo simulation experiments show that the pricing formula is accurate and efficient under double stochastic errors. Finally, we perform parameter sensitivity analysis to further justify the pricing model.
Citation: Yao Fu, Sisi Zhou, Xin Li, Feng Rao. Multi-assets Asian rainbow options pricing with stochastic interest rates obeying the Vasicek model[J]. AIMS Mathematics, 2023, 8(5): 10685-10710. doi: 10.3934/math.2023542
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Asian rainbow options provide investors with a new option solution as an effective tool for asset allocation and risk management. In this paper, we address the pricing problem of Asian rainbow options with stochastic interest rates that obey the Vasicek model. By introducing the Vasicek model as the change process of the stochastic interest rate, based on the non-arbitrage principle and the stochastic differential equation, the number of assets of the Asian rainbow option is expanded to n dimensions, and the pricing formulas of the Asian rainbow option with multiple (n) assets under the Vasicek interest rate model are obtained. The multi-asset pricing results under stochastic interest rates provide more possibilities for Asian rainbow options. Furthermore, Monte Carlo simulation experiments show that the pricing formula is accurate and efficient under double stochastic errors. Finally, we perform parameter sensitivity analysis to further justify the pricing model.
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)+K2supt−τ≤s≤tw(s),t≥a, |
where 0<K2<K1. Under these conditions, positive constants K3 and K4 exist such that
w(t)≤K3e−K4(t−a),t≥a. |
Halanay first introduced this inequality while studying the stability of a specific differential equation [10]
υ′(t)=−Aυ(t)+Bυ(t−τ),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]:
{x′i(t)=−cixi(t)+∑nj=1bijfj(xj(t−τ))+∑nj=1aijfj(xj(t))+Ii,t>0,xi(t)=ϕi(t),−τ≤t≤0, 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(x−s)ds,x≥0. |
The solutions exhibit exponential decay if the kernels satisfy the conditions
∫∞0eβsk(s)ds<∞, |
for some β>0, and
A(x)∫∞0k(s)ds≤B(x)−C,C>0,x∈R. |
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(t−υ)]≤−qw(t)+∫taw(r)k(t−r)dr, p>0, 0<α<1, υ,t>a,w(t)=ϖ(t),a−υ≤t≤a. | (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 Eα(−q(φ(t−υ)−φ(a))α)/Eα(−q(φ(t)−φ(a))α) is challenging for convergence analysis. The ideal decay rate would be Eα(−q(φ(t)−φ(a))α), but the neutral delay introduces new challenges, particularly near ν. Approximating with (φ(t)−φ(a))−α (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 [a,b] to be an infinite or finite interval, and φ to be an n- continuously differentiable function on [a,b] such that φ is increasing and φ′(ϰ)≠0 on [a,b].
Definition 2.1. The φ-Riemann-Liouville fractional integral of a function ω with respect to a function φ is defined as
Iφ,αω(z)=1Γ(α)∫za[φ(z)−φ(s)]α−1ω(s)φ′(s)ds,α>0,z>a |
provided that the right side exists.
Definition 2.2. The φ-Caputo derivative of order α>0 is defined by
Dφ,αCω(ϰ)=Iφ,n−α(1φ′(ϰ)ddϰ)nω(ϰ), |
which can be expressed equivalently as
Dφ,αCω(ϰ)=1Γ(n−α)∫ϰa[φ(ϰ)−φ(τ)]n−α−1φ′(τ)ω[n]φ(τ)dτ, ϰ>a, |
where
ω[n]φ(ϰ)=(1φ′(ϰ)ddϰ)nω(ϰ), n=−[−α]. |
Particularly, when 0<α<1
Dφ,αCω(ϰ)=Iφ,1−α(1φ′(ϰ)ddϰ)ω(ϰ)=1Γ(1−α)∫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<α≤1, ζ>a,λ∈Ry(a)=ya, | (2.1) |
has the solution
y(ζ)=yaEα(λ[φ(ζ)−φ(a)]α),ζ≥a. |
Lemma 2.2. [13] The Cauchy problem
{Dφ,αCy(ζ)=λy(ζ)+h(ζ),0<α≤1, λ∈R, ζ>a,y(a)=ya∈R, | (2.2) |
admits the solution for ζ≥a
y(ζ)=yaEα(λ[φ(ζ)−φ(a)]α)+∫ζa[φ(ζ)−φ(s)]α−1Eα,α(λ[φ(ζ)−φ(s)]α)φ′(s)h(s)ds. |
Lemma 2.3. For λ,ν,ω>0, the following inequality is valid for all z>a:
∫za[φ(s)−φ(a)]λ−1[φ(z)−φ(s)]ν−1e−ω[φ(s)−φ(a)]φ′(s)ds≤C[φ(z)−φ(a)]ν−1, |
where
C=max{1,21−ν}Γ(λ)[1+λ(λ+1)/ν]ω−λ. |
Proof. For z>a, let
I(z)=[φ(z)−φ(a)]1−ν∫za[φ(s)−φ(a)]λ−1[φ(z)−φ(s)]ν−1e−ω[φ(s)−φ(a)]φ′(s)ds. |
Set ξ[φ(z)−φ(a)]=φ(s)−φ(a). Then, [φ(z)−φ(a)]dξ=φ′(s)ds and
I(z)=[φ(z)−φ(a)]λ∫10(1−ξ)ν−1ξλ−1e−ωξ[φ(z)−φ(a)]dξ, z>a. |
As for 0≤ξ<1/2, we have (1−ξ)ν−1≤max{1,21−ν}, therefore
I(z)≤max{1,21−ν}[φ(z)−φ(a)]λ∫1/20ξλ−1e−ωξ[φ(z)−φ(a)]dξ+[φ(z)−φ(a)]λ∫11/2(1−ξ)ν−1ξλ−1e−ωξ[φ(z)−φ(a)]dξ. | (2.3) |
Let u=ωξ[φ(z)−φ(a)]. Then, dξ=[φ(z)−φ(a)]−1ω−1du and
[φ(z)−φ(a)]λ∫1/20ξλ−1e−ωξ[φ(z)−φ(a)]dξ≤ω−λ∫∞0uλ−1e−udu=ω−λΓ(λ). | (2.4) |
If 1≤ωξ[φ(z)−φ(a)], 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<ξ≤1,
ξλ−1e−ωξ[φ(z)−φ(a)]≤ξλ−1Γ(2+λ)[ωξ[φ(z)−φ(a)]]λ≤2ω−λΓ(λ+2)[φ(z)−φ(a)]λ, |
and consequently
[φ(z)−φ(a)]λ∫11/2(1−ξ)ν−1ξλ−1e−ωξ[φ(z)−φ(a)]dξ≤[φ(z)−φ(a)]λ∫11/2(1−ξ)ν−12ω−λΓ(2+λ)[φ(z)−φ(a)]λdξ=2ω−λΓ(2+λ)∫11/2(1−ξ)ν−1dξ=21−νω−λΓ(λ+2)ν. |
When ωξ[φ(z)−φ(a)]<1, it implies that [ωξ[φ(z)−φ(a)]]λ<1≤eωξ[φ(z)−φ(a)]. Consequently,
[φ(z)−φ(a)]λ∫11/2ξλ−1(1−ξ)ν−1e−ωξ[φ(z)−φ(a)]dξ<[φ(z)−φ(a)]λ∫11/2ξλ−1(1−ξ)ν−1[ωξ[φ(z)−φ(a)]]−λdξ<2ω−λ∫11/2(1−ξ)ν−1dξ=21−νω−λν. | (2.5) |
Taking into account (2.3)–(2.5), we infer that
I(z)≤max{1,21−ν}ω−λΓ(λ)+21−νω−λΓ(λ+2)ν≤max{1,21−ν}ω−λΓ(λ)(1+λ(λ+1)ν), z>a. |
The proof is complete.
Lemma 2.4. [8, (4.4.10), (4.9.4)] For β>0, ν>0, and λ,λ∗∈C, λ≠λ∗, we have
∫ϰ0zβ−1Eα,β(λzα)(ϰ−z)ν−1Eα,ν(λ∗(ϰ−z)α)dz=λEα,β+ν(λϰα)−λ∗Eα,β+ν(λ∗ϰα)λ−λ∗ϰβ+ν−1, |
and for σ>0, γ>0,
Iσzγ−1Eα,γ(pzα)(ϰ)=ϰσ+γ−1Eα,σ+γ(pϰα). |
Lemma 2.5. For β>0, ν>0, and λ,λ∗∈C, λ≠λ∗, we have
∫ϰaEα,β(λ[φ(z)−φ(a)]α)[φ(ϰ)−φ(z)]ν−1[φ(z)−φ(a)]β−1×Eα,ν(λ∗[φ(ϰ)−φ(z)]α)φ′(z)dz=[φ(ϰ)−φ(a)]β+ν−1λ∗Eα,β+ν(λ∗[φ(ϰ)−φ(a)]α)−λEα,β+ν(λ[φ(ϰ)−φ(a)]α)λ∗−λ, |
and for σ>0, γ>0,
Iφ,σ[φ(z)−φ(a)]γ−1Eα,γ(p[φ(z)−φ(a)]α)(ϰ)=[φ(ϰ)−φ(a)]σ+γ−1×Eα,σ+γ(p[φ(ϰ)−φ(a)]α). | (2.6) |
Proof. Let u=φ(ϰ)−φ(z). Then,
∫ϰaEα,β(λ[φ(z)−φ(a)]α)[φ(ϰ)−φ(z)]ν−1[φ(z)−φ(a)]β−1×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]β−1uν−1Eα,ν(λ∗uα)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)]ν−1[φ(z)−φ(a)]β−1×Eα,ν(λ∗[φ(ϰ)−φ(z)]α)φ′(z)dz=[φ(ϰ)−φ(a)]β+ν−1λ∗Eα,β+ν(λ∗[φ(ϰ)−φ(a)]α)−λEα,β+ν(λ[φ(ϰ)−φ(a)]α)λ∗−λ. |
To prove the second formula in the lemma, we have
Iφ,σ[φ(z)−φ(a)]γ−1Eα,γ(p[φ(z)−φ(a)]α)(ϰ)=1Γ(σ)∫ϰaEα,γ(p[φ(z)−φ(a)]α)[φ(ϰ)−φ(z)]σ−1[φ(z)−φ(a)]γ−1φ′(z)dz. |
From the first formula in the lemma, with β=γ, ν=σ, λ=p, λ∗=0, we obtain
Iφ,σ[φ(z)−φ(a)]γ−1Eα,γ(p[φ(z)−φ(a)]α)(ϰ)=1Γ(σ)∫ϰa[φ(z)−φ(a)]γ−1Eα,γ(p[φ(z)−φ(a)]α)[φ(ϰ)−φ(z)]σ−1φ′(z)dz=[φ(ϰ)−φ(a)]γ+σ−1Eα,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 γ with 0<γ<1, the following holds:
11+qΓ(1−γ)tγ≤Eγ(−qtγ)≤1qΓ(1+γ)−1tγ+1,q, t≥0. | (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<α<1, a solution v(z) is defined as α -Mittag-Leffler stable if there exist positive constants A and γ such that
‖v(z)‖≤AEα(−γ[φ(z)−φ(a)]α),z>a, |
where ‖.‖ represents a specific norm.
Theorem 3.1. Let u(t) be a nonnegative function fulfilling the conditions
Dφ,αC[u(t)−pu(t−υ)]≤−qu(t)+∫tau(s)k(t−s)ds,0<α<1,t>a, | (3.1) |
with the initial condition
u(t)=ϖ(t)≥0,a−υ≤t≤a, | (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)]α−1×(∫saEα(−q[φ(σ)−φ(a)]α)k(s−σ)dσ)φ′(s)ds≤MEα(−q[φ(t)−φ(a)]α),t>a. | (3.3) |
Further, assume that the constant M satisfies
M<1−1(φ(a+υ)−φ(a))α(1q+Γ(1−α)[φ(a+3υ)−φ(a)]α)p, | (3.4) |
with the additional condition
1(φ(a+υ)−φ(a))α(1q+Γ(1−α)[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(t−υ)]=−qw(t)+∫taw(s)k(t−s)ds,0<α<1,t>a,w(t)=ϖ(t)≥0,a−υ≤t≤a. | (3.6) |
The equation presented in (3.6) can be expressed equivalently as
Dφ,αC[w(t)−pw(t−υ)]=−q[w(t)−pw(t−υ)]+∫tak(t−s)w(s)ds−qpw(t−υ),t>a. |
This permits to profit from the form
w(t)−pw(t−υ)=[ϖ(a)−pϖ(a−υ)]Eα(−q[φ(t)−φ(a)]α)+∫ta[φ(t)−φ(s)]α−1Eα,α(−q[φ(t)−φ(s)]α)×(−qpw(s−υ)+∫sak(s−σ)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(t−υ)+∫taEα,α(−q[φ(t)−φ(s)]α)×[φ(t)−φ(s)]α−1(∫sak(s−σ)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(t−υ)+1Eα(−q(φ(t)−φ(a))α)∫ta[φ(t)−φ(s)]α−1Eα,α(−q[φ(t)−φ(s)]α)×(∫sak(s−σ)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(t−υ)+1Eα(−q(φ(t)−φ(a))α)∫ta[φ(t)−φ(s)]α−1×Eα,α(−q[φ(t)−φ(s)]α)×(∫sak(s−σ)Eα(−q(φ(σ)−φ(a))α)dσ)φ′(s)ds×supa≤σ≤tw(σ)Eα(−q(φ(σ)−φ(a))α)≤ϖ(a)+pEα(−q(φ(t)−φ(a))α)w(t−ν)+Msupa≤σ≤tw(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)]α−1×(∫sak(s−σ)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)]α−1×(∫saEα(−q[φ(σ)−φ(a)]α)k(s−σ)w(σ)Eα(−q[φ(σ)−φ(a)]α)dσ)φ′(s)ds≤Msupa≤σ≤tw(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(t−υ)+Msupa≤σ≤tw(σ)Eα(−q[φ(σ)−φ(a)]α). | (3.9) |
This inequality will serve as our initial reference.
For t∈[a,a+υ], since Eα(−q[φ(t)−φ(a)]α) 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)supa−υ≤σ≤aϖ(σ)+Msupa≤σ≤tw(σ)Eα(−q[φ(σ)−φ(a)]α), |
or
(1−M)w(t)Eα(−q(φ(t)−φ(a))α)≤(1+pEα(−q(φ(a+υ)−φ(a))α)supa−υ≤σ≤aϖ(σ). | (3.10) |
If t∈[a+υ,a+2υ], owing to relations (3.9) and (3.10), we find
w(t)Eα(−q(φ(t)−φ(a))α)≤supa−υ≤σ≤aϖ(σ)+p1−M(1+pEα(−q(φ(a+υ)−φ(a))α)×Eα(−q(φ(t−υ)−φ(a))α)Eα(−q(φ(t)−φ(a))α)supa−ν≤σ≤aϖ(σ)+Msupa≤σ≤tw(σ)Eα(−q(φ(σ)−φ(a))α). |
Observe that
Eα(−q(φ(t−υ)−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(1−α)(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(1−M)]supa−υ≤σ≤aϖ(σ)+Msupa≤σ≤tw(σ)Eα(−q(φ(σ)−φ(a))α), |
and consequently,
w(t)Eα(−q(φ(t)−φ(a))α)(1−M)≤[1+A(1−M)p+AEα(−q[φ(a+υ)−φ(a)]α)(1−M)p2]supa−υ≤σ≤aϖ(σ). | (3.12) |
Notice that we will write (3.12) as
w(t)Eα(−q(φ(t)−φ(a))α)(1−M)≤AEα(−q(φ(υ+a)−φ(a))α)×[1+p1−M+(p1−M)2]supa−υ≤σ≤aϖ(σ). | (3.13) |
When t∈[a+2ν,a+3ν], the estimations
φ(t)−φ(a)φ(t−υ)−φ(a)≤φ(a+3υ)−φ(a)φ(a+υ)−φ(a), |
together with (2.7), imply for t≥a+2ν,
Eα(−q(φ(t−υ)−φ(a))α)Eα(−q(φ(t)−φ(a))α)≤1+q(φ(t)−φ(a))αΓ(1−α)1+q(φ(t−υ)−φ(a))αΓ(1+α)−1≤1+q(φ(t)−φ(a))αΓ(1−α)qΓ(1+α)−1(φ(t−υ)−φ(a))α≤Γ(1+α)q(φ(t−υ)−φ(a))α+Γ(1+α)(φ(t)−φ(a))αΓ(1−α)(φ(t−υ)−φ(a))α≤Γ(1+α)q(φ(a+υ)−φ(a))α+(φ(a+3υ)−φ(a))αΓ(1+α)Γ(1−α)(φ(a+υ)−φ(a))α≤Γ(1+α)(φ(a+υ)−φ(a))α×(1q+Γ(1−α)(φ(a+3υ)−φ(a))α), | (3.14) |
Notice that Γ(1+α) 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)+p1−MEα(−q(φ(t−υ)−φ(a))α)Eα(−q(φ(t)−φ(a))α)×AEα(−q(φ(a+υ)−φ(a))α)×[1+p1−M+(p1−M)2]supa−υ≤σ≤aϖ(σ)+Msupa≤σ≤tw(σ)Eα(−q(φ(σ)−φ(a))α), |
or
w(t)Eα(−q[φ(t)−φ(a)]α)(1−M)≤supa−υ≤σ≤aϖ(σ){1+p1−MAVEα(−q[φ(υ+a)−φ(a)]α)×[1+p1−M+(p1−M)2]}, | (3.15) |
where
V:=1(φ(a+υ)−φ(a))α(1q+Γ(1−α)[φ(a+3υ)−φ(a)]α). |
As
AVEα(−q[φ(a+υ)−φ(a)]α)>1, |
we can rewrite Eq (3.15) as follows:
(1−M)w(t)Eα(−q[φ(t)−φ(a)]α)≤supa−υ≤σ≤aϖ(σ)AEα(−q[φ(υ+a)−φ(a)]α)×{1+pV1−M+(pV1−M)2+(pV1−M)3}. |
We now make the following claim.
Claim. For t∈[a+(n−1)υ,a+nυ],
(1−M)w(t)Eα(−q(φ(t)−φ(a))α)≤AEα(−q(φ(υ+a)−φ(a))α)×∑nk=0(pV1−M)ksupa−ν≤σ≤aϖ(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)υ,a+nυ]. Now, let t∈[a+nυ,a+υ(n+1)]. Utilizing (3.9), we derive
w(t)Eα(−q[φ(t)−φ(a)]α)≤supa−υ≤σ≤aϖ(σ)+pEα(−q[φ(t−υ)−φ(a)]α)(1−M)Eα(−q[φ(t)−φ(a)]α)×AEα(−q[φ(a+υ)−φ(a)]α)∑nk=0(pV1−M)ksupa−υ≤σ≤aϖ(σ)+Msupa≤σ≤tw(σ)Eα(−q[φ(σ)−φ(a)]α). |
and by (3.14)
w(t)Eα(−q(φ(t)−φ(a))α)(1−M)≤[1+Vp1−MAEα(−q(φ(a+υ)−φ(a))α)∑nk=0(Vp1−M)k]supa−υ≤σ≤aϖ(σ)≤AEα(−q(φ(a+υ)−φ(a))α)[1+∑n+1k=1(Vp1−M)k]supa−υ≤σ≤aϖ(σ)=AEα(−q[φ(a+υ)−φ(a)]α)∑n+1k=0(Vp1−M)ksupa−υ≤σ≤aϖ(σ). |
Therefore, the claim holds true. Then, for t>a,
w(t)≤[AEα(−q[φ(a+υ)−φ(a)]α)(1−M)∑∞k=0(pV1−M)ksupa−υ≤σ≤aϖ(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≥a:
∫saEα(−q[φ(σ)−φ(a)]α)k(s−σ)dσ≤C1[φ(s)−φ(a)]λ−1,C1,λ>0. | (4.1) |
The family of functions k(t−s) defined as
k(t−s)≤C2[φ(t)−φ(s)]−αe−b[φ(s)−φ(a)]φ′(s) |
satisfies the specified relation when the constants b and C2 are carefully chosen. Indeed, since
Eα(−qtα)≤11+qtαΓ(1+α)=Γ(1+α)Γ(1+α)+qtα≤Γ(1+α)qtα,t>0, | (4.2) |
it follows that
∫saEα(−q[φ(σ)−φ(a)]α)k(s−σ)dσ≤C2Γ(1+α)q∫sa[φ(σ)−φ(a)]−α[φ(s)−φ(σ)]−αe−b[φ(σ)−φ(a)]φ′(σ)dσ≤2αC2Γ(1+α)Γ(1−α)[3−α]bα−1q[φ(s)−φ(a)]−α,s>a. |
Therefore, (4.1) holds with
C1:=2αC2Γ(1+α)Γ(1−α)[3−α]bα−1q,λ:=1−α. |
By applying formula (2.6), we obtain
∫taEα,α(−q[φ(t)−φ(s)]α)[φ(t)−φ(s)]α−1×(∫sak(s−σ)Eα(−q[φ(σ)−φ(a)]α)dσ)φ′(s)ds≤C1∫taEα,α(−q[φ(t)−φ(s)]α)[φ(t)−φ(s)]α−1[φ(s)−φ(a)]−αφ′(s)ds≤C1Γ(α)Eα,1(−q[φ(t)−φ(a)]α). | (4.3) |
To ensure that assumption (3.4) is met, we can select C1 (or C2 for the specific example) such that
C1Γ(α)<1−1(φ(a+υ)−φ(a))α(1q+Γ(1−α)[φ(a+3υ)−φ(a)]α)p. |
Second class: Assume that k(t−s)≤C3[φ(t)−φ(s)]α−1Eα,α(−b[φ(t)−φ(s)]α)φ′(s) for some b>0 and C3>0 to be determined. A double use of (2.6) and (4.2) gives
C3∫ta[φ(t)−φ(s)]α−1Eα,α(−q[φ(t)−φ(s)]α)×(∫sa[φ(s)−φ(σ)]α−1Eα,α(−b[φ(s)−φ(σ)]α)Eα(−q[φ(σ)−φ(a)]α)φ′(σ)dσ)φ′(s)ds≤C3Γ(1+α)q∫taEα,α(−q[φ(t)−φ(s)]α)[φ(t)−φ(s)]α−1×(∫sa[φ(s)−φ(σ)]α−1Eα,α(−b[φ(s)−φ(σ)]α)[φ(σ)−φ(a)]−αφ′(σ)dσ)φ′(s)ds≤C3Γ(α)Γ(1+α)q∫taEα,α(−q[φ(t)−φ(s)]αEα,1(−b[φ(s)−φ(a)]α[φ(t)−φ(s)]α−1φ′(s)ds≤C3Γ2(1+α)Γ(α)qb∫taEα,α(−q[φ(t)−φ(s)]α[φ(t)−φ(s)]α−1[φ(s)−φ(a)]−αφ′(s)ds≤C3Γ2(1+α)Γ2(α)qbEα,1(−q[φ(t)−φ(a)]α. | (4.4) |
Clearly, M=C3Γ2(1+α)Γ2(α)qb. It suffices now to impose the condition on C3 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,
{Dφ,αC[u(t)−p∫tau(s)g(t−s)ds]≤−qu(t)+∫tau(s)k(t−s)ds, t,υ>a,0<α<1, p>0,u(t)=u0≥0,t∈[a−υ,a], | (5.1) |
which we will contrast with
{Dφ,αC[w(t)−p∫taw(s)g(t−s)ds]=−qw(t)+∫taw(s)k(t−s)ds,t,υ>a,0<α<1, p>0,w(t)=w0=u0≥0,t∈[a−υ,a]. | (5.2) |
We assume g is a continuous function (to be determined later) and that the solutions are nonnegative.
Let us reformulate this as
{Dφ,αC[w(t)−p∫taw(s)g(t−s)ds]=−q[w(t)−p∫taw(s)g(t−s)ds]−qp∫taw(s)g(t−s)ds+∫taw(s)k(t−s)ds,t,υ>a,0<α<1, p>0w(t)=w0≥0,t∈[a−υ,a]. |
Therefore,
w(t)−p∫taw(s)g(t−s)ds=Eα(−q[φ(t)−φ(a)]α)w0+∫ta[φ(t)−φ(s)]α−1Eα,α(−q[φ(t)−φ(s)]α)×(−qp∫sag(s−σ)w(σ)dσ+∫sak(s−σ)w(σ)dσ)φ′(s)ds, |
and, for t>a,
w(t)≤Eα(−q[φ(t)−φ(a)]α)w0+p∫tag(t−s)w(s)ds+∫taEα,α(−q[φ(t)−φ(s)]α)×[φ(t)−φ(s)]α−1(∫sak(s−σ)w(σ)dσ)φ′(s)ds. | (5.3) |
Dividing both sides of (5.3) by Eα(−q[φ(t)−φ(a)]α), we find
w(t)Eα(−q[φ(t)−φ(a)]α)=w0+pEα(−q[φ(t)−φ(a)]α)∫taw(s)g(t−s)ds+1Eα(−q[φ(t)−φ(a)]α)∫ta[φ(t)−φ(s)]α−1×Eα,α(−q[φ(t)−φ(s)]α)×(∫sak(s−σ)Eα(−q[φ(σ)−φ(a)]α)dσ)φ′(s)ds×supa≤σ≤tw(σ)Eα(−q[φ(σ)−φ(a)]α), |
or, for t>a,
w(t)Eα(−q[φ(t)−φ(a)]α)≤w0+pEα(−q[φ(t)−φ(a)]α)∫tag(t−s)Eα(−q[φ(s)−φ(a)]α)×(w(s)Eα(−q[φ(s)−φ(a)]α))ds+Msupa≤σ≤tw(σ)Eα(−q[φ(σ)−φ(a)]α). |
The relation
pEα(−q[φ(t)−φ(a)]α)∫tag(t−s)Eα(−q[φ(s)−φ(a)]α)ds≤M∗, |
is assumed for some M∗>0. Then,
w(t)Eα(−q[φ(t)−φ(a)]α)≤w0+(M∗+M)supa≤σ≤tw(σ)Eα(−q[φ(σ)−φ(a)]α),t>a, |
and
w(t)≤w01−M∗−MEα(−q[φ(t)−φ(a)]α),t>a, |
in the case that
M∗+M<1. |
Example. Take k as above, and select g fulfilling
g(t−s)≤C4[φ(t)−φ(s)]α−1Eα,α(−c[φ(t)−φ(s)]α)φ′(s), |
for some C4,c>q. Then,
∫taEα(−q[φ(s)−φ(a)]α)g(t−s)ds≤Γ(1+α)q∫ta[φ(s)−φ(a)]−αg(t−s)ds≤C4Γ(1+α)q∫taEα,α(−c[φ(t)−φ(s)]α)×[φ(t)−φ(s)]α−1[φ(s)−φ(a)]−αφ′(s)ds≤C4Γ(1+α)Γ(α)qEα,1(−q[φ(t)−φ(a)]α),t>a. |
A value for M∗ would be
M∗=C4pΓ(1+α)Γ(α)q. |
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(t), g(t)≥0 for all t such that
∫taEα,α(−q[φ(t)−φ(s)]α)[φ(t)−φ(s)]α−1×(∫saEα(−q[φ(σ)−φ(a)]α)k(s−σ)dσ)φ′(s)ds≤MEα(−q[φ(t)−φ(a)]α),t>a, |
p∫tag(t−s)Eα(−q[φ(s)−φ(a)]α)ds≤M∗Eα(−q[φ(t)−φ(a)]α),t>a, |
hold for some M, M∗>0 with
M∗+M<1. |
Then, we can find a positive constant C such that
w(t)≤CEα(−q[φ(t)−φ(a)]α),t>a. |
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
{Dφ,αC[u(t)−pu(t−υ)]≤−qu(t)+∫tak(t−s)u(s)ds, p>0,0<α<1,t,υ>a,u(t)=ϖ(t)≥0,a−υ≤t≤a, |
with |ϖ(s)|≤w0Eα(−q(φ(s+υ)−φ(a))α) for s∈[a−υ,a], w0>0. To clarify these concepts, let us suppose that 1>p>0, and examine the following expression:
w(t)−pw(t−υ)=[ϖ(a)−pϖ(a−υ)]Eα(−q[φ(t)−φ(a)]α)+∫ta[φ(t)−φ(s)]α−1Eα,α(−q[φ(t)−φ(s)]α)×(−qpw(s−υ)+∫sak(s−σ)w(σ)dσ)φ′(s)ds. |
Then, for t>a
|w(t)|≤2w0Eα(−q[φ(t)−φ(a)]α)+p|w(t−υ)|+qp∫taEα,α(−q[φ(t)−φ(s)]α)[φ(t)−φ(s)]α−1|w(s−υ)|φ′(s)ds+∫taEα,α(−q[φ(t)−φ(s)]α)[φ(t)−φ(s)]α−1×(∫sak(s−σ)|w(σ)|dσ)φ′(s)ds. | (6.1) |
For t∈[a,a+υ],
|w(t)|Eα(−q[φ(t)−φ(a)]α)≤3w0+qpw0Eα(−q[φ(t)−φ(a)]α)∫ta[φ(t)−φ(s)]α−1×Eα,α(−q[φ(t)−φ(s)]α)Eα(−q[φ(s)−φ(a)]α)φ′(s)ds+Msupa≤σ≤tw(σ)Eα(−q[φ(σ)−φ(a)]α), |
where M is defined as in Eq (3.3). Again, as
∫taEα,α(−q[φ(t)−φ(s)]α)[φ(t)−φ(s)]α−1Eα(−q[φ(s)−φ(a)]α)φ′(s)ds≤Γ(1+α)q∫taEα,α(−q[φ(t)−φ(s)]α)[φ(t)−φ(s)]α−1[φ(s)−φ(a)]−αφ′(s)ds≤Γ(1+α)Γ(α)qEα,1(−q[φ(t)−φ(a)]α), | (6.2) |
we can write
|w(t)|Eα(−q[φ(t)−φ(a)]α)≤3w0+w0Γ(1+α)Γ(α)p+Msupa≤σ≤tw(σ)Eα(−q[φ(σ)−φ(a)]α), |
or
(1−M)|w(t)|Eα(−q[φ(t)−φ(a)]α)≤3w0+w0Γ(1+α)Γ(α)p. | (6.3) |
If t∈[a+υ,a+2υ], we first observe that
|w(t−υ)|≤3w0+w0Γ(1+α)Γ(α)p(1−M)×Eα(−q[φ(t−υ)−φ(a)]α)Eα(−q[φ(t)−φ(a)]α)Eα(−q[φ(t)−φ(a)]α)≤A3w0+w0Γ(1+α)Γ(α)p(1−M)Eα(−q(φ(t)−φ(a))α), |
where A is as in (3.11). Using the fact that
w0≤A3w0+w0Γ(1+α)Γ(α)p1−M, |
and relations (6.1) and (6.3), we get
|w(t)|≤2w0Eα(−q[φ(t)−φ(a)]α)+pA3w0+w0Γ(1+α)Γ(α)p(1−M)Eα(−q(φ(t)−φ(a))α)+qpA3w0+w0Γ(1+α)Γ(α)p(1−M)×∫taEα,α(−q[φ(t)−φ(s)]α)[φ(t)−φ(s)]α−1×Eα(−q(φ(s)−φ(a))α)φ′(s)ds+∫taEα,α(−q[φ(t)−φ(s)]α)[φ(t)−φ(s)]α−1×(∫sak(s−σ)|w(σ)|dσ)φ′(s)ds. |
Next, in view of (6.2), we find
|w(t)|≤2w0Eα(−q[φ(t)−φ(a)]α)+pA3w0+w0Γ(α)Γ(α+1)p(1−M)Eα(−q(φ(t)−φ(a))α)+qpA3w0+w0Γ(1+α)Γ(α)p(1−M)×Γ(1+α)Γ(α)qEα(−q[φ(t)−φ(a)]α)+∫ta[φ(t)−φ(s)]α−1Eα,α(−q[φ(t)−φ(s)]α)×(∫sak(s−σ)|w(σ)|dσ)φ′(s)ds. |
or
(1−M)|w(t)|Eα(−q[φ(t)−φ(a)]α)≤2w0+pAw0[1+Γ(1+α)Γ(α)](1−M)×(3+Γ(1+α)Γ(α)p)≤2w0+3Aw0[1+Γ(1+α)Γ(α)](1−M)p+Aw0[1+Γ(1+α)Γ(α)]2(1−M)p2. | (6.4) |
For t∈[a+2υ,a+3υ], by virtue of (3.14),
Eα(−q(φ(t−υ)−φ(a))α)Eα(−q(φ(t)−φ(a))α)≤1(φ(a+υ)−φ(a))α×(1q+Γ(1−α)(φ(a+3υ)−φ(a))α)=:V>1, |
and therefore
|w(t)|≤2w0Eα(−q[φ(t)−φ(a)]α)+pV(1−M)×[2w0+3Aw0[1+Γ(1+α)Γ(α)](1−M)p+Aw0[1+Γ(1+α)Γ(α)]2(1−M)p2]×Eα(−q(φ(t)−φ(a))+pVΓ(1+α)Γ(α)(1−M)×[2w0+3Aw0[1+Γ(1+α)Γ(α)](1−M)p+Aw0[1+Γ(1+α)Γ(α)]2(1−M)p2]×Eα(−q(φ(t)−φ(a))+∫ta[φ(t)−φ(s)]α−1Eα,α(−q[φ(t)−φ(s)]α)(∫sak(s−σ)|w(σ)|dσ)φ′(s)ds. |
So,
(1−M)|w(t)|Eα(−q[φ(t)−φ(a)]α)≤2w0+pV(1−M)[2w0+3Aw0[1+Γ(1+α)Γ(α)](1−M)p+Aw0[1+Γ(1+α)Γ(α)]2(1−M)p2]+pVΓ(1+α)Γ(α)(1−M)×[2w0+3Aw0[1+Γ(1+α)Γ(α)](1−M)p+Aw0[1+Γ(1+α)Γ(α)]2(1−M)p2], |
or
(1−M)|w(t)|Eα(−q[φ(t)−φ(a)]α)≤2w0+2w0pV[1+Γ(1+α)Γ(α)]1−M+3w0Ap2V[1+Γ(1+α)Γ(α)]2(1−M)2+Aw0V[1+Γ(1+α)Γ(α)]3(1−M)2p3. | (6.5) |
Writing (6.5) in the form
|w(t)|Eα(−q[φ(t)−φ(a)]α)(1−M)≤2w0+2w0pV[1+Γ(1+α)Γ(α)]1−M+3w0A(pV[1+Γ(1+α)Γ(α)](1−M))2+w0A(pV[1+Γ(1+α)Γ(α)]1−M)3≤3w0A[1+pV[1+Γ(1+α)Γ(α)]1−M+(pV[1+Γ(1+α)Γ(α)]1−M)2+(pV[1+Γ(1+α)Γ(α)]1−M)3] | (6.6) |
provides the basis for our next claim.
Claim. On the interval [a+(n−1)υ,a+nυ], it is clear that
|w(t)|Eα(−q[φ(t)−φ(a)]α)(1−Mw0)≤3A∑nk=0(Vp[1+Γ(1+α)Γ(α)]1−M)k. |
The validity of the claim for n=1,2, and 3 is established by Eqs (6.3), (6.4), and (6.6). Let t∈[a+nυ,a+(n+1)υ]. Then from (6.1),
|w(t)|≤2w0Eα(−q[φ(t)−φ(a)]α)+3ApVw01−M∑nk=0(Vp[Γ(α)Γ(1+α)+1]1−M)kEα(−q[φ(t)−φ(a)]α)+3ApΓ(1+α)Γ(α)w0V1−M∑nk=0(Vp[Γ(α)Γ(1+α)+1]1−M)k×Eα(−q[φ(t)−φ(a)]α)+∫taEα,α(−q[φ(t)−φ(s)]α)[φ(t)−φ(s)]α−1(∫sak(s−σ)|w(σ)|dσ)φ′(s)ds, |
or
(1−Mw0)|w(t)|Eα(−q[φ(t)−φ(a)]α)≤2+3VpA1−M∑nk=0(Vp[Γ(1+α)Γ(α)+1]1−M)k+3VpA1−MΓ(1+α)Γ(α)×∑nk=0(Vp[Γ(1+α)Γ(α)+1]1−M)k. |
Then,
(1−Mw0)|w(t)|Eα(−q[φ(t)−φ(a)]α)≤3A{1+[Γ(1+α)Γ(α)+1]pV1−M∑nk=0(Vp[Γ(1+α)Γ(α)+1]1−M)k}, |
i.e.,
(1−Mw0)|w(t)|Eα(−q[φ(t)−φ(a)]α)≤3A{1+∑nk=0(Vp[Γ(1+α)Γ(α)+1]1−M)1+k}. |
Thus,
(1−Mw0)|w(t)|Eα(−q[φ(t)−φ(a)]α)≤3A∑n+1k=0(Vp[Γ(1+α)Γ(α)+1]1−M)k, |
demonstrating that the assertion holds. Moreover, the series converges if the following condition is satisfied:
1+Γ(1+α)Γ(α)1−MVp<1. |
We have just proved the following result.
Theorem 6.1. Suppose that u(t) is a solution of
{Dφ,αC[u(t)−pu(t−υ)]≤−qu(t)+∫t0k(t−s)u(s)ds,t,υ>a,p>0,0<α<1,u(t)=ϖ(t),a−υ≤t≤a, |
with |ϖ(t)|≤Eα(−q(φ(t+υ)−φ(a))α), a−υ≤t≤a, q>0, p>0, and k is a nonnegative function verifying
∫ta[φ(t)−φ(s)]α−1Eα,α(−q[φ(t)−φ(s)]α)×(∫saEα(−q[φ(σ)−φ(a)]α)k(s−σ)dσ)φ′(s)ds≤MEα(−q[φ(t)−φ(a)]α),t>a, |
for some M such that
M<1−[Γ(1+α)Γ(α)+1]Vp, |
with
[Γ(1+α)Γ(α)+1]Vp<1. |
Then,
|w(t)|≤CEα(−q[φ(t)−φ(a)]α),t>a, |
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:
{Dφ,αC[xi(t)−pxi(t−υ)]=−hi(xi(t))[gi(xi(t))−n∑j=1aijfj(xj(t))−n∑j=1bijlj(xj(t−τ))−n∑j=1dij∫∞akj(s)Φj(xj(t−s))ds−Ii], t,υ>a, p>0xi(t)=xi0(t),t∈[a−υ,a], i=1,2,...,n, |
and
{Dφ,αC[xi(t)−p∫taxi(s)ψi(t−s)ds]=−hi(xi(t))[gi(xi(t))−n∑j=1bijlj(xj(t−τ))−n∑j=1aijfj(xj(t))−n∑j=1dij∫∞aΦj(xj(t−s))kj(s)ds−Ii], t>a, p>0,xi(0)=xi0(t), t≤a, i=1,2,...,n, |
where xi.(t) stands the state of the ith neuron, n is the number of neurons, gi is a suitable function, hi represents an amplification function, bij, aij, dij represent the weights or strengths of the connections from the jth neuron to the ith neuron, Ii is the external input to the ith neuron, ψi are the neutral delay kernels, fj,lj,Φj denote the signal transmission functions, υ is the neutral delay, τ corresponds to the transmission delay, ϕi is the history of the i th state, and kj 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
{Dφ,αC[xi(t)−pxi(t−υ)]=−hi(xi(t))[gi(xi(t))−n∑j=1dij∫∞akj(s)fj(xj(t−s))ds−Ii],xi(t)=xi0(t),t∈[a−υ,a], i=1,2,...,n, | (7.1) |
for t,υ>a, p>0.
We adopt the following standard assumptions.
(A1) The functions fi are assumed to satisfy the Lipschitz condition
|fi(x)−fi(y)|≤Li|x−y| for every x,y∈R and for each i=1,2,...,n, |
where Li denotes the Lipschitz constant corresponding to the function fi.
(A2) The delay kernel functions kj are nonnegative and exhibit piecewise continuity. Additionally, each kj has a finite integral over its domain, expressed as κj=∫∞akj(s)ds<∞, for j=1,...,n.
(A3) The functions gi have derivatives that are uniformly bounded by a constant G. Specifically,
|g′i(z)|≤G, for all z∈R and for each i=1,2,...,n, |
where G>0 is a fixed constant.
(A4) The functions hi are strictly positive and continuous, and they satisfy the following bounds:
0<β_i≤hi(z)≤¯βi, for all z∈R and i=1,2,...,n. |
For simplicity, we suppose that the initial values xi0(t) are all zero for times before a.
Definition 7.1. The point x∗=(x∗1,x∗2,...,x∗n)T is said to be an equilibrium if, for each i=1,2,...,n, it satisfies the equation
gi(x∗i)=n∑j=1aijfj(x∗j)+n∑j=1dij∫∞akj(s)fj(x∗j)ds+Ii=n∑j=1(aij+dijκj)fj(x∗j)+Ii, t>a. |
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(t)−x∗=y(t). This leads to the following:
{Dφ,αC[yi(t)−pyi(t−υ)]=−hi(x∗i+yi(t))[gi(yi(t)+x∗i) −n∑j=1dij∫tafj(x∗j+yj(t−s))kj(s)ds−Ii], t>a, i=1,...,n,yi(t)=ψi(t):=ϕi(t)−x∗i, t∈[a−υ,a], i=1,...,n, |
or
{Dφ,αC[yi(t)−pyi(t−υ)]=−Hi(yi(t))[Gi(yi(t))−n∑j=1dij∫t0Fj(yj(t−s))kj(s)ds], t>a, i=1,...,n,yi(t)=ψi(t):=ϕi(t)−x∗i, t∈[a−υ,a], i=1,...,n, |
where
Fi(yi(t))=fi(yi(t)+x∗i)−fi(x∗i), Gi(yi(t))=gi(yi(t)+x∗i)−gi(x∗)Hi(yi(t))=hi(yi(t)+x∗i), t>a, i=1,...,n. |
Using the mean value theorem, the following inequality can be established:
Dφ,αC|yi(t)−pyi(t−υ)|≤sgn[yi(t)−pyi(t−υ)]Dφ,αC[yi(t)−pyi(t−υ)]=−sgn[yi(t)−pyi(t−υ)]Hi(yi(t))[g′i(ˉxi(t))yi(t)−n∑j=1dij∫∞aFj(yj(t−s))kj(s)ds]. |
By subtracting and adding the term pg′i(ˉxi(t))yi(t−υ), we obtain
Dφ,αC|yi(t)−pyi(t−υ)|≤−sgn[yi(t)−pyi(t−υ)]Hi(yi(t))[g′i(ˉxi(t))[yi(t)−pyi(t−υ)]+pg′i(ˉxi(t))yi(t−υ)−n∑j=1dij∫∞aFj(yj(t−s))kj(s)ds], t>a, i=1,...,n, |
or
Dφ,αC|yi(t)−pyi(t−υ)|≤−Hi(yi(t))G|yi(t)−pyi(t−υ)|+pGHi(yi(t))|yi(t−υ)|+Hi(yi(t))n∑j=1dij∫∞akj(s)Lj|yj(t−s)|ds, t>a, i=1,2,...,n. |
Therefore,
Dφ,αC|yi(t)−pyi(t−υ)|≤−Gβ_i|yi(t)−pyi(t−υ)|+pG¯βi|yi(t−υ)|+¯βin∑j=1Ljdij∫∞akj(s)|yj(t−s)|ds, t>a, i=1,...,n. |
Finally, we consider the equation for wi and rewrite it in the following form:
|wi(t)−pwi(t−υ)|=Eα(−Gβ_i[φ(t)−φ(a)]α)|Φi(a)−pΦi(a−υ)|+∫ta[φ(t)−φ(s)]α−1Eα,α(−q[φ(t)−φ(s)]α)×(pG¯βi|wi(t−υ)|+¯βin∑j=1Ljdij∫∞0kj(s)|wj(t−s)|ds)φ′(s)ds, t>a, i=1,2,...,n. |
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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