Preamble

This work, published in 1967 (Volume III, No. 10), investigates the boundary value problem for the $n$-th order differential equation $y^{(n)} + f(t, y, \dots, y^{(n-1)}) = 0$. Building upon the foundational research of A. Ya. Khokhryakov \cite{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 18}, we consider the general equation $y^{(n)} + f(t, y) = 0$ and its second-order counterpart $y'' + f(t, y, y') = 0$. The boundary conditions are defined as $a_{11} y(a) + a_{12} y'(a) = 0$ and $a_{21} y(\beta) + a_{22} y'(\beta) = 0$, following the frameworks established in \cite{16} and \cite{17}.

Specifically, we examine the periodic boundary value problem:
$$y^{(n)} + f(t, y, \dots, y^{(\nu)}) = 0$$
$$y^{(k)}(a) - y^{(k)}(\beta) = 0, \quad (k = 0, \dots, \nu)$$
where $f$ is a function of $t, y, \dots, y^{(\nu)}$. This study extends the results presented in \cite{19} regarding the existence and uniqueness of solutions for equations of the form $L[y] = y^{(n)} + \sum g_k(t) y^{(k)} = 0$ under periodic conditions $y^{(k)}(a) - y^{(k)}(\beta) = 0$ for $k = 0, \dots, n-1$. We assume $g_k(t) = q_k$ and $a < \beta < T_0$. According to \cite{16}, the operator $L$ maps the space of functions on $[a, \beta]$ into a corresponding function space, where the Green's function $\Gamma(t, s)$ plays a critical role in establishing the existence of solutions.

1. Existence and Uniqueness Theorems

Consider the operator $L[y] = y^{(n)} + f(t, y, \dots, y^{(\nu)})$ in the region $R$, where $\nu < n$. We assume the function $f$ satisfies conditions similar to those in \cite{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 18}. Let $L_1$ and $L_2$ be linear operators such that the differential equation can be expressed as $y' + f(t, y) = 0$. We define a transformation $x = Ty$, where $T$ is an $n \times n$ matrix, leading to the transformed system:
$$x' + B(t)x = 0, \quad x(a) - x(\beta) = 0$$
The matrix $B(t)$ is related to the coefficients $g_k(t)$ of the original equation. For the system to possess a unique solution, we require $\det B(t) = g_0(t) > 0$. Furthermore, we assume the coefficients $g_i(t)$ satisfy the conditions:
$$b_k(t) \le g_k(t) \le 0, \quad (k = 0, \dots, n-2)$$
where $b_k(t)$ are defined by the structural properties of the operator $L$.

For the third-order case ($n=3$), the equation $y''' + g_2(t)y'' + g_1(t)y' + g_0(t)y = 0$ with periodic boundary conditions $y^{(k)}(a) - y^{(k)}(\beta) = 0$ ($k=0, 1, 2$) is solvable if the coefficients satisfy:
$$\begin{aligned}
b_0(t) &= g_0(t) - g_1(t) + g_2(t) - 1 \ge 0 \
b_1(t) &= g_1(t) - 2g_2(t) + 3 < 0 \
b_2(t) &= g_2(t) - 2g_0(t) > 0
\end{aligned}$$
Under these conditions, the Green's function $\Gamma(t, s)$ exists for all $t, s \in [a, \beta]$.

2. Comparison and Monotonicity

Let $z(t)$ and $u(t)$ be functions in $C^1[a, \beta]$ such that $z \ge u$. We define the operator $T$ such that $T(z - u) > 0$ for $t \in [a, \beta]$. If $y(t)$ is a solution to the boundary value problem (1.1), and there exist upper and lower solutions $z(t)$ and $u(t)$ satisfying:
$$\frac{dz}{dt} + A(t)z + f(t, z) - Q(t)z > 0$$
$$\frac{du}{dt} + A(t)u + f(t, u) - Q(t)u < 0$$
then the solution $y(t)$ is bounded by $u(t) \le y(t) \le z(t)$. This monotonicity property is essential for applying fixed-point theorems. Specifically, if the function $f$ satisfies a Lipschitz-like condition:
$$f(t, y_1) - f(t, y_2) < -Q(t)(y_1 - y_2)$$
then the operator $K$ defined by the Green's function is a contraction, ensuring the uniqueness of the solution $x(t) = Kx(t)$ in the interval $[Tu, Tz]$.

3. Convergence and Error Estimates

Consider the system (3.1) where $B(t)$ is an $n \times n$ matrix. Let $B_1 = \min B(t)$ and $B_2 = \max B(t)$ be matrices in the class $\Lambda^+$. The Green's functions $\Gamma_1(t, s)$ and $\Gamma_2(t, s)$ corresponding to these constant matrices provide bounds for the actual Green's function $\Gamma(t, s)$ of the time-varying system:
$$\Gamma_2(t, s) > \Gamma(t, s) > \Gamma_1(t, s)$$
The existence of a solution $y(t)$ to the nonlinear problem (3.1) can be established if the norm of the operator satisfies $B_{11} L \epsilon < \epsilon$, where $L$ is the Lipschitz constant of $f$ and $\epsilon = (1, \dots, 1)$. The error between two approximate solutions $y(t)$ and $\bar{y}(t)$ is bounded by:
$$|y(t) - \bar{y}(t)| \le \int_a^\beta |\Gamma(t, s)| L |y(s) - \bar{y}(s)| ds$$
This leads to the convergence criterion $A_1^{-1} L < 1$, where $A_1 = \min g_0(t)$.

4. Conclusion

The periodic boundary value problem for the $n$-th order equation $y^{(n)} + g(t)y + f(t, y) = 0$ possesses a unique solution if $g(t) > M(t) > 0$ and the nonlinear term $f$ satisfies the growth condition $|f(t, y)| \le M(t)|y| + \psi(t)$. The results presented here generalize several previous findings in the literature \cite{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18} and provide a robust framework for the qualitative analysis of higher-order differential equations with periodic constraints.