Preamble

In this section, we consider the asymptotic behavior of solutions to $n$-th order differential equations of the form:
$$u^{(n)} + a(x)u = f(x, u, u', \dots, u^{(n-1)}) \tag{1.1}$$
where the function $f(x, y_1, \dots, y_n)$ satisfies certain growth conditions relative to the linear part of the equation. We are interested in establishing the existence and uniqueness of solutions $u(x)$ that exhibit specific asymptotic properties as $x \to \infty$.

Consider the related linear equation:
$$v^{(n)} + a(x)v = 0 \tag{1.2}$$
Let $y = (y_1, \dots, y_m)$ and let $L(x, t, y)$ be an $n$-dimensional vector function. We define the operator $A_k$ acting on the space $S_k$ of continuous functions. Under the Carathéodory conditions, we assume there exist functions $\phi_j(x, t)$ and $\psi_j(x, t)$ such that:
$$|K_j(x, t, y)| \le \phi_j(x, t), \quad |L_j(x, y, z)| \le \psi_j(x, t) \tag{1.3}$$
Furthermore, we assume the following integrability conditions hold for $x > a > 0$:
$$\int_a^\infty \phi_1(x, t) dt < r_1, \quad \int_a^\infty \phi_2(x, t) dt < r_2 \tag{1.4}$$
$$\int_a^\infty \psi_1(x, t) dt + \int_a^\infty \psi_2(x, t) dt < \infty \tag{1.5}$$

We define the operator $A_k y(x)$ for $y(x) \in S_k$ as:
$$y(x) = \int_a^x L_1(t, y(t), z_1(t)) dt + \int_a^x L_2(t, y(t), z_2(t)) dt \tag{1.6}$$
where $z_1(t)$ and $z_2(t)$ are integral terms involving the kernels $K_1$ and $K_2$. Using the estimates in (1.3) and (1.4), one can show that $A_k$ maps the set $S_k$ into itself. To prove the existence of a fixed point, we examine the continuity and compactness of the operator $A_k$. Specifically, we consider a sequence ${y_j(x)} \subset S_k$ and evaluate the limit:
$$\lim_{j \to \infty} |A_k y_j(x_2) - A_k y_j(x_1)| = 0 \tag{1.8}$$
By applying the Schauder fixed-point theorem, we establish the existence of a solution $y_k(x)$ to the integral equation (1.11) on the interval $[a, a+k]$. By extending this interval and considering the limit as $k \to \infty$, we obtain a solution $y(x)$ defined for all $x \ge a$ that satisfies the required asymptotic conditions (1.5).

Section 2. Case with $a(x) = 0$

We first analyze the case where $a(x) = 0$. Suppose the nonlinear term satisfies the following inequality:
$$|x^{n-k} f(x, x^{k-1}y_1, \dots, x^{k-n}y_n)| \le \phi(x, |y|) \tag{2.1}$$
for $1 \le k \le n$. We consider the differential equation:
$$u^{(n)} = f(x, u, u', \dots, u^{(n-1)}) \tag{2.2}$$
with the asymptotic boundary conditions:
$$u^{(j-1)}(x) = c_j x^{k-j} + o(x^{k-j}), \quad j = 1, 2, \dots, n \tag{2.3}$$
where $c_j$ are constants. By defining appropriate kernels $K_v$ and operators $L_v$ as shown in (2.6) and (2.7), we can transform the differential equation into an equivalent system of integral equations.

Theorem 2.1. If the function $\phi(x, X)$ is monotonic and satisfies the integrability condition:
$$\int_a^\infty \phi(t, k|c| + 1) dt < \infty \tag{2.8}$$
then there exists a solution to (2.2) satisfying the asymptotic representation (2.3).

Theorem 2.2. For the linear case where $f(x, u, \dots) = \sum b_j(x) u^{(j-1)}$, if the coefficients satisfy:
$$x^{n-j} |b_j(x)| \le \phi(x) \tag{2.11}$$
then the equation $u^{(n)} = \sum b_j(x) u^{(j-1)}$ has a solution satisfying (2.13).

Theorem 2.3. Consider the inequality $|f(x, y_1, \dots, y_n)| \le \phi(x, |y_1|)$. Let $v(x)$ be a solution to the comparison equation. If the integral of $\phi$ converges, then the solution $u(x)$ to (2.2) is bounded by $v(x)$ as $x \to \infty$. This allows us to establish the stability of the asymptotic behavior (2.13) under perturbations of the initial conditions or the functional form of $f$.

Theorem 2.4. If a solution $u(x)$ to (2.2) exists on $[x_0, \infty)$ and satisfies the condition $x^{j-n}|u^{(j-1)}(x)| < X$, then it must satisfy the asymptotic property (2.13) provided the conditions of Theorem 2.1 hold.

Section 3. General Case for $a(x)$

In the general case, we assume $a(x)$ is a continuous function. Let $\sigma_k(x)$ be the fundamental system of solutions to the linear equation (1.2). We define:
$$\mu(x) = |a(x)|^{\frac{2-n}{2n}} \exp \left( \int |a(t)|^{1/n} dt \right) \tag{3.2}$$
Theorem 3.1. Suppose $f$ satisfies:
$$|f(x, G_{k1}y_1, \dots, G_{kn}y_n)| \le \phi(x, |y|) \tag{3.3}$$
where $G_{kj}$ are weight functions related to the linear solutions. Then equation (1.1) has a solution $u_k(x)$ such that:
$$u_k^{(j-1)}(x) = c \sigma_{kj}(x) + o(\sigma_{kj}(x)) \tag{3.5}$$
where $\sigma_{kj}$ represents the $j$-th derivative of the $k$-th fundamental solution.

Theorem 3.2. If the nonlinear term $f$ satisfies the growth condition (3.18), then any solution $u(x)$ to (1.1) can be represented as a linear combination of the fundamental solutions of the linear part plus a vanishing error term:
$$u^{(j-1)}(x) = \sum_{k=1}^n [c_k \sigma_{kj}(x) + o(\sigma_{kj}(x))] \tag{3.19}$$
This result is obtained by applying the method of variation of parameters and utilizing the estimates for the Wronskian $W(x)$ of the system. The convergence of the integrals in (3.20) ensures that the coefficients $c_k$ approach stable limits as $x \to \infty$.

Theorem 3.3. Under the conditions of Theorem 3.2, the mapping between the initial data at $x_0$ and the asymptotic coefficients $c_k$ is continuous. Furthermore, if the integral condition (3.24) is satisfied, the asymptotic behavior (3.19) is unique for a given set of coefficients.