Abstract
For the system \begin{equation}
x(t)=\int_0^tK(t,s,x(s))\,ds+f(t)\tag{1},
\label{1}
\end{equation}, where the vector function $K(t,s,x)$ has the form $$K(t,s,x)=K_1(t,s,x)-K_2(t,s,x),$$ and $K_j(t,s,x)$ ($j=1,2$) are continuous and non-decreasing with respect to $x$ in the domain $0<s\le t\le b$, $|x|\le a$ ($a,b<\infty$), criteria are provided that ensure the non-uniqueness of the solution to system (1).
For example, if a solution $x(t)$ of system (1) exists and the following condition is satisfied: $$q(s)=\sum_{j=1}^n(\xi_1^j-\xi_2^j)\le K_1^i(t,s,\xi_1^1,\dots,\xi_1^i,\dots,\xi_1^n)-K_1^i(t,s,\xi_2^1,\dots,\xi_2^i,\dots,\xi_2^n)\le[q(s)+p(s)]\sum_{j=1}^n(\xi_1^j-\xi_2^j),\K_2^i(t,s,\xi_1^1,\dots,\xi_1^i,\dots,\xi_1^n)-K_2^i(t,s_2,\xi_2^1,\dots,\xi_2^i,\dots,\xi_2^n)\le r(s)\sum_{j=1}^n(\xi_1^j-\xi_2^j),\quad \xi_1^j>\xi_2^j\quad (j=1,\dots,n),$$ where $q(t)$ is a non-negative function continuous on $[0,b]$ and non-integrable on $[0,b]$, and $p(t)$ and $r(t)$ are non-negative and integrable for $t\in[0,b]$, then system (1) has at least two solutions. Bibliography: 5 items.
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Preamble
This section examines the system of nonlinear integral equations of the form:
$$x(t) = \int_0^t K(t, s, x(s)) ds + f(t)$$
where $x(t) = {x_1(t), \dots, x_n(t)}$ and $f(t) = {f_1(t), \dots, f_n(t)}$ are vector functions defined on the interval $[0, b]$, and the kernel $K(t, s, \xi)$ is a matrix-valued function. We assume the initial condition $f(0) = 0$. Following the methodology established in \cite{1, 2, 3, 4, 5}, we investigate the stability and error bounds of solutions for such systems.
1. Comparison Theorems and Stability
Let $z_1(t)$ and $z_2(t)$ be two continuous functions on $[0, b]$ such that $||z_j|| \leq a$ and $z_1(t) > z_2(t)$. We consider the integral operators $K_1$ and $K_2$ associated with the system (1). For $t \in [0, b]$, we define the relationship between these functions through the integral inequality:
$$z_1(t) - z_2(t) \geq \int_0^t [K_1(t, s, z_1(s)) - K_2(t, s, z_2(s))] ds + f(t)$$
If there exists a positive function $\tau(t)$ such that $\tau(0) = 0$ and $\tau(t) > 0$ for $t > 0$, we can establish bounds for the difference between the approximate and exact solutions. Specifically, if $z_1(t) - \tau(t) > z_2(t)$, then the following inequality holds:
$$z_1(t) - \tau(t) > \int_0^t [K_1(t, s, z_1(s) - \tau(s)) - K_2(t, s, z_2(s))] ds + f(t)$$
This implies that the solution remains within a defined neighborhood of the reference trajectory, provided the kernels satisfy certain Lipschitz-type conditions.
2. Error Estimation and Convergence
To derive practical error estimates, we assume the kernels satisfy the following growth conditions:
$$|K_i(t, s, \xi_1, \dots, \xi_n)| \leq q_i(s) \sum_{j=1}^n |\xi_j|$$
where $q_i(s)$ are integrable functions on $[0, b]$. Let $z_1(t)$ and $z_2(t)$ be approximate solutions. We define the error functions as:
$$\begin{aligned} z_1(t) &= \int_0^t [q_1(s) z_1(s) - q_2(s) z_2(s)] ds + |f(t)| \ z_2(t) &= \int_0^t [q_2(s) z_2(s) - q_1(s) z_1(s)] ds - |f(t)| \end{aligned}$$
Under these conditions, the stability of the system can be analyzed using the Gronwall-Bellman inequality. If the integral $\int_0^b [q_1(s) + q_2(s)] ds$ is bounded, then the solutions $z_i(t)$ are uniquely determined and continuous. Furthermore, if we define a majorizing function $\phi(t)$, the error can be bounded by:
$$\tau(t) \leq \int_0^t \exp\left( -\int_s^t p(\sigma) d\sigma \right) \phi(s) ds$$
where $p(s)$ is a function related to the partial derivatives of the kernels.
3. Existence of Solutions in a Given Domain
Consider the existence of a solution $x(t)$ for the system (1) such that $z_2(t) < x(t) < z_1(t)$, where $z_1(0) = z_2(0) = 0$. Suppose there exist operators $Q_1$ and $Q_2$ that satisfy:
$$Q_1(t, s, \xi - \eta) \leq K_1(t, s, \xi) - K_1(t, s, \eta) \leq Q_2(t, s, \xi - \eta)$$
If we can find functions $\tau_1(t)$ and $\tau_2(t)$ such that $\tau_1(0) = \tau_2(0) = 0$ and $\tau_1(t) > \tau_2(t)$, satisfying the system of integral inequalities:
$$\begin{aligned} \tau_1(t) &> \int_0^t [Q_1(t, s, \tau_1(s)) - Q_2(t, s, \tau_2(s))] ds \ \tau_2(t) &< \int_0^t [Q_1(t, s, \tau_2(s)) - Q_2(t, s, \tau_1(s))] ds \end{aligned}$$
then the solution $x(t)$ is guaranteed to exist and remain within the bounds $x(t) + \tau_2(t) < z(t) < x(t) + \tau_1(t)$. This result is critical for ensuring the convergence of iterative numerical methods applied to nonlinear Volterra equations.
References
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