Preamble

This section addresses the existence and uniqueness of solutions for a class of nonlinear integral equations. We consider the equation:
$$u(x, t) = \int_0^1 k(x, y, u(y, t), \dots, u^{(k)}(y, t), t) \, dy$$
subject to the boundary conditions $u(x, 0) = 0$ and specific constraints on the derivatives at the boundaries. Following the methodology established by T. M. [3] and further developed in [1, 2], we analyze the operator properties in the space $C[R]$. We assume the kernel $k(x, y, \dots)$ is sufficiently smooth and satisfies the Lipschitz conditions:
$$\begin{aligned} |u_1| < \rho_1, \quad |u_2| < \rho_2, \quad |t| < \rho \end{aligned}$$
where $\rho$ defines the domain of interest.

§ 1. Existence and Uniqueness of Solutions

Let the kernel be represented by the expansion $k(x, y, u, t) = \sum k_{i,k,j}(x, y) u^i (u')^k t^j$. By applying the resolvent method to the linearized part of the equation, we transform the original problem into an equivalent integral form:
$$u(x, t) = \int_0^1 \left[ B_{0,0,j}(x, y) t^j + B_{1,k,0}(x, y) u^{(k)}(y, t) + \dots \right] dy \tag{7}$$
where the coefficients $B_{i,k,j}(x, y)$ are determined by the kernel $k(x, y)$ and the associated Fredholm resolvent $\Gamma(x, y)$.

To find the solution $u(x, t)$, we seek a power series expansion in terms of $t$:
$$u(x, t) = \sum_{i=1}^{\infty} u_i(x) t^i \tag{9}$$
Substituting this expansion into the integral equation and equating coefficients of like powers of $t$, we obtain a recursive system for $u_i(x)$:
$$u_i(x) = f_i(x, u_1, \dots, u_{i-1}) \tag{12}$$
For $i=1$, the first term is given by $u_1(x) = \int_0^1 B_{0,0,1}(x, y) \, dy$. For higher orders $i \ge 2$, the functions $f_i$ depend on the previously determined components.

We establish the convergence of this series by the method of majorants. Let $A$ be a constant such that $|B_{i,k,j}(x, y)| \le A$ for all $0 \le x, y \le 1$. We define a sequence of constants $a_i$ such that $|u_i(x)| \le a_i$. By constructing a majorizing scalar equation, we show that the series $\sum a_i t^i$ converges for $|t| < \rho$, which implies the uniform convergence of the series (9) in the space $C[R]$.

Furthermore, we prove the uniqueness of this solution. Suppose there exists another solution $\omega(x, t)$ satisfying the same conditions. By considering the difference $v(x, t) = u(x, t) - \omega(x, t)$ and applying the derived estimates, we show that $v(x, t)$ must vanish identically. Specifically, as $t \to 0$, the norm of the difference is bounded by a term that approaches zero faster than any power of $t$, leading to $\omega(x, t) = 0$ in the limit. Thus, the problem defined by (1) and (4) has a unique solution in $C[R]$.

§ 2. Extension to Parameter-Dependent Kernels

In this section, we generalize the results to cases where the kernel depends on an additional parameter or functional form. We consider the modified equation:
$$\frac{\partial u}{\partial t} + \int_0^1 E(x, y) \frac{\partial u}{\partial y} dy = \xi(t) \phi(x) + \int_0^1 \mathcal{K}(x, y, u, u', t) \, dy \tag{26}$$
where $\xi(t)$ is a given function and $\phi(x)$ represents the spatial distribution. Using the transformation $B_{i,k,j}(x, y) = k_{i,k,j}(x, y) + \int \Gamma(x, y_1) k_{i,k,j}(y_1, y) dy_1$, we reduce the problem to a form similar to § 1.

We seek a solution in the form of a generalized series:
$$u(x, t) = \sum_{i=1}^{\infty} u_i(x, \xi, \xi', \dots, \xi^{(i-1)}) t^i \tag{31}$$
The coefficients $u_i$ are determined by solving the system of equations (32) and (33). For the case $k_0 = 1$, the first-order term is:
$$u_1(x, \xi) = \xi(t) \phi(x) + \int_0^1 B_{0,0,1}(x, y) dy$$
For $n \ge 2$, the terms $u_n$ are calculated recursively. For example, the second-order term $u_2$ involves the derivatives of the parameter $\xi$ and the nonlinear interactions of $u_1$:
$$u_2(x, \xi, \xi') = \frac{1}{2} \left[ \int_0^1 (B_{1,0,0} u_1 + B_{0,1,1} u_1' + \dots) dy \right] \tag{41}$$

The convergence of the series (31) is guaranteed under the condition that the majorant series for the coefficients $a_i$ converges. This allows us to state Theorem 2: if the kernel $k(x, y)$ satisfies the smoothness conditions and the parameter $\xi(t)$ is sufficiently regular, the problem (1), (4) possesses a unique solution represented by the series (31).

As a practical example, consider the case where $E(x, y) = x/y$ and $\phi(x) = x$. Substituting these into the recursive formulas, we obtain the specific components $u_1, u_2, \dots$, which demonstrate the application of the method to linear and nonlinear boundary value problems. The results confirm that the solution depends continuously on the initial parameters and the boundary conditions.