1. Introduction and Preliminary Results

In 1967, B. M. Abramov and A. Ya. Khokhryakov \cite{1} investigated the boundary value problem for the second-order differential equation:
$$y'' + f(t, y, y') = 0$$
subject to the boundary conditions:
$$\begin{aligned} a_{11}y(a) + a_{12}y'(a) &= 0 \ a_{21}y(\beta) + a_{22}y'(\beta) &= 0 \end{aligned}$$
In their work \cite{1}, they assumed that $a_{12} > 0$ and $a_{21}a_{22} < 0$. Under these conditions, they established existence and uniqueness theorems. Specifically, they analyzed the case where $a_{21}a_{22} < 0$ and $a_{12} > 0$. In this section, we extend these results to the $n$-th order case $y^{(n)} + f(t, y, y', \dots) = 0$ under various boundary conditions.

Consider the linear system:
$$y' + Ay = Q, \quad My(a) + Ny(\beta) = 0$$
where $A$ is a constant matrix. Let $f(t)$ be a continuous function on $[a, T]$ for $a < T$. We define the operator $L$ and the corresponding boundary value problem as:
$$x'(a) + Nx(\beta) = 0 \quad \text{(1.4)}$$
Following the methodology of Abramov and Khokhryakov \cite{1}, the solution on the interval $[a, \beta^*]$ can be represented via the Green's function $K(t, s)$ as:
$$x(t) = \int_a^{\beta} K(t, s)f(s)ds \quad \text{(1.5)}$$

Lemma 1

If $\tau > 0$, $\alpha > 0$, $\tau \neq \alpha$, and $s < 0$, then the expression $\exp(-Bs)$ satisfies specific positivity conditions. Specifically, we assume the boundary matrix coefficients satisfy $b_{11} > 0$, $b_{12} < 0$, $b_{21} = 0$, and $b_{22} > 0$ (1.6).

Lemma 2

If $\tau > 0$ and $s > 0$, then $\exp(-Bs) > 0$ (1.7). The Green's function $K(t, s)$ for the operator $Lx = f(t)$ can be constructed such that:
$$x(t) = \int_a^{\beta} K(t, s)f(s)ds + K(t, a)[M + NK(a, \beta)]^{-1} NK(\beta, s) \quad \text{(1.8)}$$
From (1.4) and (1.8), we derive the relation:
$$y[M + NK(a, \beta)] = \int_a^{\beta} G(t, s)f(s)ds \quad \text{(1.9)}$$
where the kernel $\Gamma(t, s)$ is defined for $a < t < s < \beta$ and $a < s < t < \beta$ using the fundamental solution $K(t, s) = \exp[-B(t-s)]$.

2. Green's Function Properties and Estimates

For the case where $A - \Lambda = 0$ (1.14), the components of the Green's matrix $\Gamma(t, s)$, denoted as $\gamma_{11}, \gamma_{12}, \gamma_{21}, \gamma_{22}$, are derived in equations (1.12)--(1.15). We observe that for $\tau > \alpha$, the following inequality holds:
$$\exp[-(\tau - \alpha)(\beta - t)] > 1 - r \quad \text{(1.16)}$$
This implies that $\gamma_{12}(t, s) > 0$ for $t > s$ on the interval $[a, \beta]$. Furthermore, we establish that:
$$\frac{\exp[-(\tau - \alpha)(\beta - t)] - r}{\tau - \alpha} > 0 \quad \text{(1.17)}$$
provided that $\tau - \alpha > 1$ and $0 < r < 1$. At the point $t = a$, the expression simplifies, and we can determine the interval length $\beta - a$ required to maintain the sign of the Green's function.

Consider the problem:
$$x' - Ax = 0, \quad x(a) + Nx(\beta) = 0 \quad \text{(2.1)}$$
where $x > 0$, $\alpha < 0$, and the boundary coefficients satisfy $a_{11}a_{12} < 0$ and $a_{21}a_{22} > 0$. Using the fundamental solution (1.10), the components of the Green's function for (2.1) satisfy:
$$\gamma_{11} > 0, \quad \gamma_{12} < 0, \quad \gamma_{21} = 0, \quad \gamma_{22} < 0 \quad \text{(2.6)}$$
for all $t, s \in [a, \beta]$.

3. Existence of Solutions via Monotone Iterations

We now consider the nonlinear problem:
$$Mx(a) + Nx(\beta) = 0, \quad A\dot{x} + Qx = f(t, x) \quad \text{(4.3)}$$
By applying the transformation $x = Ty$, the system can be rewritten in a form suitable for the application of fixed-point theorems in a space with a cone. Let $L_2$ be the operator associated with (4.3). We assume the existence of upper and lower solutions $v_1$ and $v_2$ such that:
$$\begin{aligned} \dot{v}_1 + Bv_1 &\leq T^{-1}f(t, Tv_1) \ \dot{v}_2 + Bv_2 &\geq T^{-1}f(t, Tv_2) \end{aligned} \quad \text{(4.6), (4.7)}$$
The solution can be represented by the integral operator:
$$x(t) = \int_a^{\beta} \Gamma(t, s) T^{-1} [QTx(s) - f(s, Tx(s))] ds \equiv Gx \quad \text{(4.8)}$$
Under the condition (4.5), the operator $G$ is monotonic. If $v_1(t) \leq x(t) \leq v_2(t)$, then the sequence of iterations $w(t) = Gx(t)$ converges to the solution of (4.3). This confirms the existence of a solution $u(t)$ satisfying $v_1(t) \leq u(t) \leq v_2(t)$ for $t \in [a, \beta]$.

4. Second-Order Equations and Differential Inequalities

For the second-order case, we consider the equation:
$$y'' - b_1 y' - a_1 y = f(t, y, y') \quad \text{(4.16)}$$
with boundary conditions $y(a) = K y(\beta) = 0$. We assume the existence of functions $u(t)$ and $v(t)$ that satisfy the differential inequalities:
$$\begin{aligned} u'' - b_1 u' - a_1 u &\leq - \phi(t) \ v'' - b_1 v' - a_1 v &\geq \phi(t) \end{aligned} \quad \text{(4.14), (4.15)}$$
where $\phi(t) > 0$. If the function $f$ satisfies the growth condition $|f(t, y, u)| \leq \phi(t)$ (4.12), then there exists a solution $y(t)$ such that $u(t) \leq y(t) \leq v(t)$ for $t \in [a, \beta]$. This result generalizes the work of Abramov and Khokhryakov to cases where $a_{12} > 0$ and $a_{21}a_{22} < 0$.

5. Uniqueness and Lipschitz Conditions

Finally, we address the uniqueness of the solution. Suppose $f(t, y, u)$ satisfies a Lipschitz condition:
$$|f(t, y_1, u_1) - f(t, y_2, u_2)| \leq l_1 |y_1 - y_2| + l_2 |u_1 - u_2| \quad \text{(5.1)}$$
Let $\gamma(t, s)$ and $\gamma't(t, s)$ be the Green's function and its derivative for the linear problem (5.2). The solution to the nonlinear problem (4.1) can be expressed as:
$$y(t) = \int_a^{\beta} \gamma(t, s) f(s, y(s), y'(s)) ds \quad \text{(5.5)}$$
A sufficient condition for the existence and uniqueness of the solution is:
$$l_1 \max$$} \int_a^{\beta} |\gamma(t, s)| ds + l_2 \max_{t \in [a, \beta]} \int_a^{\beta} |\gamma'_t(t, s)| ds < 1 \quad \text{(5.6)
By applying the contraction mapping principle to (5.7)--(5.9), we conclude that the boundary value problem has a unique solution in the specified domain.

References

1. Abramov B. M., Khokhryakov A. Ya. Matematicheskiy Sbornik, 1967.
2. Lasota A. Zeszyty Naukowe Uniwersytetu Jagiellońskiego, No. 77, 49–54, 1963.
3. Corduneanu C. Analele Științifice ale Universității „Al. I. Cuza” din Iași, I, 1, No. 1–2, 11–16, 1955.
4. [Additional references as cited in the original text].