Introduction

This work investigates the stability and asymptotic behavior of solutions to a class of nonlinear integro-differential equations. We consider systems of the form:
$$x'(t) = f(t) - b(t) - A(t)\phi(x(t), t) - \int_0^t K(t, s)\psi(x(s), s) \, ds \tag{1.1}$$
where $x(t)$ and $f(t)$ are $n$-dimensional vectors, and $A(t)$ and $K(t, s)$ are $n \times n$ matrices. This formulation generalizes several models previously studied in the literature \cite{1, 3}, specifically addressing cases where the kernel $K(t, s)$ and the nonlinearities $\phi$ and $\psi$ satisfy certain monotonicity and positivity conditions.

1.1 Fundamental Assumptions and Definitions

We assume that the functions $f(t), b(t), \phi(x, t), \psi(x, t), A(t)$, and $K(t, s)$ are continuous for $0 \le s \le t < \infty$. Let $D_1$ and $D_2$ denote the partial derivatives with respect to the first and second arguments of the kernel, respectively, such that $D_1 K(t, s) = \frac{\partial}{\partial t} K(t, s)$ and $D_2 K(t, s) = \frac{\partial}{\partial s} K(t, s)$. We define $D = D_1 + D_2$. The following conditions are imposed on the system components:

• The matrix $A(t)$ is positive definite, satisfying $(A(t)x, x) > 0$ for $x \neq 0$, and there exists a constant $M$ such that $|A(t)| \le M$.
• The kernel $K(t, s)$ satisfies $K(t, 0) > 0$, $D_1 K(t, 0) < 0$, and $D_2 K(t, s) \ge 0$. Furthermore, we assume the mixed derivative condition $D_1 D_2 K(t, s) \le 0$.
• The nonlinear functions $\phi(x, t)$ and $\psi(x, t)$ are such that $(\phi(x, t), \psi(x, t)) > 0$ for $x \neq 0$. We also require that $\psi(x, t)$ is the gradient of some scalar potential $\Psi(x, t)$, i.e., $\psi(x, t) = \nabla_x \Psi(x, t)$, where $\Psi(x, t) = \int_0^x \psi(\xi, t) \, d\xi$.

1.2 Stability Analysis via Lyapunov Functionals

To analyze the stability of the equilibrium solution $x(t) = 0$, we construct a Lyapunov functional $V(t)$. Let $F(t) = \int_0^t |f(s)| \, ds$. We define the functional as:
$$V(t) = \left[ 1 + E(t) \right] \exp(-K_1 F(t)) \tag{1.3}$$
where $E(t)$ is a quadratic-like form involving the integral of the nonlinearity:
$$E(t) = \int_0^x \psi(\xi, t) \, d\xi + \frac{1}{2} \left( K(t, 0) \int_0^t \psi(x(s), s) \, ds, \int_0^t \psi(x(s), s) \, ds \right) + \dots \tag{1.2}$$
By differentiating $V(t)$ along the trajectories of (1.1), we obtain:
$$\begin{aligned} V'(t) = & -K_1 |f(t)| V(t) + \exp(-K_1 F(t)) \Bigl[ (f(t), \psi(x(t), t)) \ & - (A(t)\phi(x(t), t), \psi(x(t), t)) + \dots \Bigr] \end{aligned} \tag{1.4}$$
Under the prescribed conditions, specifically $(A(t)\phi(x, t), \psi(x, t)) > 0$ and the monotonicity of the kernel, it can be shown that $V'(t) \le 0$. This implies the boundedness of the solution $x(t)$ for all $t \ge 0$.

2. Asymptotic Behavior

We further investigate the conditions under which the solution $x(t)$ tends to zero as $t \to \infty$. Suppose that the kernel $K(t, s)$ is of the convolution type $K(t - s)$. If the integral $\int_0^\infty |K(t)| \, dt$ is bounded and the external forcing $f(t)$ vanishes asymptotically, we can apply Barbalat's Lemma.

2.1 Linearized System and Resolvents

Consider the linear operator associated with (1.1). Let $W(t, s)$ be the resolvent kernel satisfying:
$$\frac{\partial}{\partial t} W(t, s) + A(t)W(t, s) + \int_s^t M(t, \tau)W(\tau, s) \, d\tau = 0 \tag{2.6}$$
where $M(t, s) = -D_2 L(t, s)$ and $L(t, s) = A(s) + \int_s^t K(\tau, s) \, d\tau$. If the matrix $A(0) + \int_0^\infty K(s, 0) \, ds$ is positive definite, the solution can be expressed via the variation of constants formula:
$$u(t) = \left[ W(t, 0) + \int_0^t W(t, s)F(s) \, ds \right] B x_0 \tag{2.10}$$
where $B$ is a constant matrix. As $t \to \infty$, if $W(t, s) \to 0$, then the state $x(t)$ converges to the equilibrium.

3. Systems with Time Delays

The analysis extends to systems with a constant time delay $L > 0$:
$$x'(t) = z(x(t), t) - \int_{t-L}^t K(t, s)\psi(x(s), s) \, ds \tag{4.1}$$
For such systems, the initial condition is defined on the interval $[-L, 0]$. We assume the kernel $K(t, s)$ vanishes for $s = t - L$. By constructing a modified Lyapunov functional:
$$V(t) = (1 + E(t)) \exp(-K_1 F(t)) \tag{4.6}$$
where $E(t)$ accounts for the history of the state $\psi(x(s), s)$ over the interval $[t-L, t]$, we can establish similar stability criteria. Specifically, if the mixed derivatives of the kernel satisfy $D_1 D_2 K(t, s) \le 0$ and the potential function $\Psi(x, t)$ is positive definite, the solution $x(t)$ and its derivative $x'(t)$ both approach zero as $t \to \infty$.

Conclusion

The results presented here provide sufficient conditions for the global stability and asymptotic convergence of nonlinear integro-differential systems. These conditions rely on the passivity of the nonlinear components and the monotonicity of the integral kernel, consistent with previous findings in \cite{1, 2, 3}. The use of exponential weighting in the Lyapunov functional allows for the inclusion of non-vanishing perturbations $f(t)$, provided they are integrable.