Preamble

In this section, we consider the asymptotic behavior of the solution $x(t, \lambda)$ to a class of nonlinear integral equations as the parameter $\lambda \to 0$. Building upon the foundational work in \cite{2, 3, 5, 4, 1, 6, 7, 8}, we investigate the case where the solution exhibits a singularity of the form $x(t) = \lambda^{-1} \psi_{-1}(t) + \psi_0(t) + \dots$. Specifically, we analyze the equation:

$$
\begin{aligned}
x(t) = &\lambda \int_0^1 [A_1(t, s)x(s - \tau) + A_2(t, s)x(s - \tau) + A_3(t, s)x^2(s)] ds \
&+ \lambda^2 \int_0^1 [B_1(t, s)x(s) + B_2(t, s)x(s - \tau) + B_3(t, s)x(s - \tau)x(s - \tau)] ds
\end{aligned}
\tag{1}
$$

where the initial condition is given by $x(t, \lambda) = \phi(t, \lambda)$ on the interval $E_0 = [-\tau, 0]$. We assume that the function $\phi(t, \lambda)$ can be expanded as:

$$ \phi(t, \lambda) = \lambda^{-1} \phi_{-1}(t) + \phi_0(t) + \lambda \phi_1(t) + \dots \tag{2} $$

We seek a solution in the form of a formal power series:

$$ x(t, \lambda) = \sum_{k=-1}^{\infty} \lambda^k \psi_k(t) \tag{4} $$

Substituting expansion (4) into equation (1) and equating coefficients of like powers of $\lambda$, we obtain a system of recurrence relations for the functions $\psi_k(t)$. For the leading term $\psi_{-1}(t)$, we have:

$$ \psi_{-1}(t) = \lambda_0 \int_0^1 A_3(t, s) \psi_{-1}^2(s) ds, \quad \psi_{-1}(t) = \phi_{-1}(t) \text{ on } E_0 \tag{5_{-1}} $$

For subsequent terms $\psi_k(t)$ where $k = 0, 1, 2, \dots$, the equations take the form:

$$ \psi_k(t) = \int_0^1 2 A_3(t, s) \psi_{-1}(s) \psi_k(s) ds + F_k(\lambda_0, \psi_{-1}, \psi_0, \dots, \psi_{k-1}) \tag{5_k} $$

where $F_k$ are known functions determined by the preceding terms of the expansion. Specifically, $F_0$ depends on the linear operators $A_1$ and $A_2$ acting on the delayed components of $\psi_{-1}$.

Solvability and Asymptotic Convergence

To ensure the existence of the expansion, we introduce the linear operator $L \psi = \psi(t) - \int_0^1 2 A_3(t, s) \psi_{-1}(s) \psi(s) ds$. If $\lambda_0$ is not an eigenvalue of the kernel $2 A_3(t, s) \psi_{-1}(s)$, the functions $\psi_k(t)$ are uniquely determined. We define the remainder of the series after $K$ terms as $R_K(t, \lambda)$. Using the method of successive approximations and applying the contraction mapping principle in a suitable Banach space, we can establish the convergence of the series.

The error estimate for the $K$-th order approximation is governed by the functional $U(X, Y, \lambda)$, which satisfies:
$$ |R_K(t, \lambda)| \leq C \lambda^{K+1} $$
where $C$ is a constant independent of $\lambda$. This confirms that (4) is indeed an asymptotic expansion of the solution to the original problem (1)–(2).

Bifurcation and Branching of Solutions

In cases where the linear operator associated with $(5_k)$ is singular (i.e., the Fredholm alternative condition is triggered), we encounter branching points. Let $w(t)$ be the eigenfunction corresponding to the kernel $2 A_3(t, s) \psi_{-1}(s)$, and $v(t)$ be the eigenfunction of the adjoint operator. The solvability condition for $\psi_0(t)$ requires:

$$ \int_0^1 \left[ \phi_0(0) + \lambda_0 \int_0^1 F_0(z, s) ds \right] v(t) dt = 0 \tag{12} $$

If this condition is met, the general solution for $\psi_0(t)$ is given by $\psi_0(t) = C_0 w(t) + u_0(t)$, where $C_0$ is a constant to be determined from the solvability condition of the next equation in the hierarchy $(7_1)$. This leads to a quadratic equation for $C_0$:

$$ P_0 C_0^2 + Q_0 C_0 + T_0 = 0 \tag{13} $$

The roots of this equation determine the possible branches of the asymptotic solution. If $Q_1 \neq 0$, we obtain distinct branches $C_{01}$ and $C_{02}$, leading to two different asymptotic expansions.

Conclusion

We have established the following results:
1. If $\lambda_0$ is not a characteristic value, the problem (1)–(2) possesses a unique asymptotic expansion of the form (4).
2. If $\lambda_0$ is a characteristic value and the solvability conditions are satisfied, the solution may branch. The coefficients $C_k$ of these branches are determined uniquely by the subsequent equations in the hierarchy.
3. The formal series constructed are shown to be valid asymptotic representations of the actual solution $x(t, \lambda)$ as $\lambda \to 0$, with the error terms vanishing at the appropriate rates.