Alternating series in terms of Riemann zeta function and Dirichlet beta function

1.   Introduction

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

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

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

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]:

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]:

The solutions exhibit exponential decay if the kernels satisfy the conditions

for some $\beta > 0,$ and

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:

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

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.

2.   Preliminaries

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

provided that the right side exists.

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

which can be expressed equivalently as

where

Particularly, when $0 < \alpha < 1$

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

and

Lemma 2.1. ^[13] The Cauchy problem

has the solution

Lemma 2.2. ^[13] The Cauchy problem

admits the solution for $\zeta \geq a$

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

where

Proof. For $z > a$, let

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

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

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

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

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

and consequently

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,

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

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

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

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

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

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

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

To prove the second formula in the lemma, we have

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

where we have used

□

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

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

3.   Halanay inequality for fractional-order systems with both discrete neutral delays and distributed delays

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

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

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

with the initial condition

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 $M > 0$:

Further, assume that the constant $M$ satisfies

with the additional condition

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

for some constant $C > 0.$

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

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

This permits to profit from the form

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

Therefore, for $t > a,$

and

We will repeatedly utilize the following estimation:

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

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

and hence

or

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

Observe that

Therefore,

and consequently,

Notice that we will write (3.12) as

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

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

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

or

where

As

we can rewrite Eq (3.15) as follows:

We now make the following claim.

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

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

and by (3.14)

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

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

4.   Examples

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 $s\geq a:$

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

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

it follows that

Therefore, (4.1) holds with

By applying formula (2.6), we obtain

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

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

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 $M$.

5.   Fractional Halanay inequality with both distributed neutral delays and distributed delays

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

which we will contrast with

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

Let us reformulate this as

Therefore,

and, for $t > a,$

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

or, for $t > a,$

The relation

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

and

in the case that

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

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

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

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

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

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

6.   Solutions that may be positive or negative

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

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:

Then, for $t > a$

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

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

we can write

or

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

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

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

Next, in view of (6.2), we find

or

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

and therefore

So,

or

Writing (6.5) in the form

provides the basis for our next claim.

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

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),

or

Then,

i.e.,

Thus,

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

We have just proved the following result.

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

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

for some $M$ such that

with

Then,

where $C > 0$ is a positive constant.

7.   Application in neural network theory

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:

and

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

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

We adopt the following standard assumptions.

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

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,

where $G > 0$ is a fixed constant.

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

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

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:

or

where

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

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

or

Therefore,

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

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

8.   Conclusions

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.

Use of Generative-AI tools declaration

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

Acknowledgments

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

Conflict of interest

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