Preamble

This work continues the investigations presented in [1, 2, 3]. Specifically, in [3], the concept of a $v$-stable solution was introduced. We consider the properties of such solutions and their relationship with various classes of functions. Following the methodologies established in [7, 8], we define the operator $h$ acting on the space $K(a)$ as described in [4]. Let $h$ be a parameter such that for $x \in K(a)$, the relation $x(t - h)$ satisfies the condition:
$$\langle \eta, x(t - h) \rangle \tag{1.1}$$
where $\eta$ is a functional in the space $K^$. We assume $h = 1$ and $t > 0$. Using the representation for the inner product:
$$\langle \eta, x \rangle = \int x(p^\alpha(t)) d\eta_\alpha(t) \quad (\alpha = 1, 2, \dots) \tag{1.2}$$
where $\eta_\alpha$ are components associated with the functional $\eta \in K^$. For $x \in K(a)$ and $t > 0$, the shift property holds:
$$\langle \eta, x(t - 1) \rangle = \langle \eta, x(t) \rangle$$
This implies that $\langle \eta, x(t - 1) \rangle \in K(a + 1)$. From (1.2), it follows that $x(t-1)$ can be represented as:
$$\langle \eta, x(t - 1) \rangle = \int x(p^{\alpha+1}(t-1)) d\eta_{\alpha+1}(t)$$
By setting $\tau = t - 1$, we obtain the following relation for the functional $X(t)$:
$$\langle X(t), x \rangle = \int X(p^{\alpha+1})(t) d\eta_{\alpha+1}(t) \tag{1.3}$$
The evolution of the measure $\eta_{\alpha+1}$ is governed by:
$$\eta_{\alpha+1}(t+1) = \eta_\alpha(t) + Q_{p_{\alpha+1}}(t) \tag{1.4}$$
for $0 < t < a$ ($\alpha = 1, 2, \dots$), where $Q_{p_{\alpha+1}}$ is a distribution on $K(a)$. Following the results in [2], we have:
$$\langle \eta, x \rangle = \int x^{(p_{\alpha+1})}(t) d[(-1)^{p_{\alpha+1}} \Delta^{p_{\alpha+1}} \eta_{\alpha+1}(t)]$$
The relationship between successive measures is given by:
$$\eta_{\alpha+1}(t) = (-1)^{\alpha} \eta_\alpha(t - \alpha) \tag{1.5}$$
where $\alpha = p_{\alpha+1} - p_\alpha \ge 2$. Thus, for $0 < t < a$, the measure satisfies:
$$\eta_{\alpha+1}(t) = \begin{cases} \eta_\alpha(t - 1) + Q_{p_{\alpha+1}}(t-1), & a < t < a+1 \ \dots \end{cases} \tag{1.6}$$
This construction ensures that for $1 < t < a$, the functional $\eta_{\alpha+1}(t)$ satisfies the required stability conditions.

2. Stability and Differential Equations

Consider the differential equation:
$$\dot{\mu} = A(t)\mu + P(\mu) + f(t) \tag{2.1}$$
where $A(t)$ is an $n \times n$ matrix and $P(\mu)$ is a nonlinear operator. We assume $P(\mu)$ satisfies a growth condition of the form:
$$\langle P(\mu), x \rangle = \int x^{(p)}(t) dR_p(\mu, t) \tag{2.2}$$
where $R_p$ is a kernel representing the nonlinear part. Following the approach in [5] and [2], we define the transition operators:
$$G_{\alpha+1}(t) = G_{\alpha+1}(e^\alpha m_{\alpha+1}, p_{\alpha+1})(t) \tag{2.3}$$
where $m_\alpha$ are the corresponding moments. The operator $R_\alpha$ is defined as:
$$R_\alpha(m, p_\alpha)(t) = \begin{cases} (-1)^q p_\alpha \dots, & 0 < t < a \ G_{\alpha+1}(e^\alpha m, p_{\alpha+1})(t), & a < t < a+1 \end{cases} \tag{2.4}$$
For $\mu \in K^*$, the solution $m(t)$ can be expressed via the integral equation:
$$m(t) = -A_p(t, 0)\mu_0 + \int_0^t A_p(t, s) R_p(m(s), s) ds - \int_0^t A_p(t, s) d\eta(s) \tag{2.7}$$
where $A_p(t, s)$ is the fundamental solution matrix satisfying:
$$A_p(t, s) = A_p(t+1, s+1) \tag{2.8}$$
and the exponential bound:
$$|A_p(t, s)| \le A_0 e^{-\alpha(t-s)} \tag{2.9}$$
for $0 < A_0$ and $\alpha > 0$. We assume the nonlinearity $R_p$ satisfies a Lipschitz condition:
$$|R_p(m_1, t) - R_p(m_2, t)|E \le L |m_1 - m_2|_E \tag{2.11}$$
Under these conditions, if the initial perturbation $|\mu_0|_E$ is sufficiently small, there exists a unique solution in the space $N$. Specifically, let $\chi = \alpha - L A_0 > 0$. Then the solution satisfies:
$$|m(t)|_E \le A_0 e^{-\alpha t} |\mu_0|_E + A_0 L e^{-\alpha t} \int_0^t e^{\alpha s} |m(s)|_E ds + A_0 e^{-\alpha t} \int_0^t e^{\alpha s} d(\text{var } \eta) \tag{2.14}$$
Applying Gronwall's inequality, we obtain:
$$|m(t)|_E \le (A_0 e^{-\lambda t} + \rho) e \tag{2.15}$$
where $\rho$ depends on the variation of the noise term $\eta$. This estimate demonstrates the asymptotic stability of the solution. For any two initial conditions $\mu$, the corresponding solutions satisfy:}$ and $\mu_{02
$$|m(\mu_{01}, T) - m(\mu_{02}, T)|E \le A_0 e^{-\lambda T} |\mu$$} - \mu_{02}|_E \tag{2.17
which implies the contractive property of the mapping for sufficiently large $T$.

3. Higher-Order Stability

If $p > q$, we consider the stability of the $v$-th order. Let $\Delta = \mu - \bar{\mu}$ be the deviation from the steady state. The equation for the deviation is:
$$\dot{\Delta} = A(t)\Delta + P(\Delta) + Y \tag{3.1}$$
where $P(\Delta) = P(\mu + \Delta) - P(\mu)$. The nonlinear term is represented as:
$$\langle P(\Delta), x \rangle = \int x^{(r-1)}(t) dR_r(f(s), s) \tag{3.3}$$
The kernel $R_r$ is constructed using the moments of the lower-order terms. This formulation allows us to extend the stability results to the space $K^*(q)$, ensuring that the system remains stable under perturbations in higher-order derivatives.

References

1. [Author Name], Journal of Computational Mathematics, Vol. 2, No. 7, 1966.
2. [Author Name], Journal of Computational Mathematics, Vol. 2, No. 7, 1966.
3. G. E. [Author Name], Journal of Computational Mathematics, Vol. 3, No. 2, 1967.
4. Gantmacher, F. R., Theory of Matrices, Moscow, 1959.
5. Gelfand, I. M., Shilov, G. E., Generalized Functions, Vol. 1, 1965.
6. Wexler, D., Revue de Math. Pures et Appl., Acad. RPR, 1965, v. X, No. 8, p. 1163–1199.
7. Wexler, D., Journal of Differential Equations, Academic Press, New York-London, v. 2, No. 1, 1966.