This paper studies the existence and uniqueness of solutions for the following nonlinear multi-point third-order boundary value problem (BVP):
\[ \begin{cases} x'''(t)+f(t,x(t),x'(t),x''(t))=0, ~0 < t < 1,\\ x(0)=0, ~g(x'(0), x''(0), x(\xi_1), \dots, x(\xi_{m-2}))=A,\\ ~h(x'(1), x''(1), x(\eta_1), \dots, x(\eta_{n-2}))=B, \end{cases} \tag{1.1} \]
where \( 0< \xi_i, \eta_j <1, ~i=1,2, \dots, m-2, ~ j=1,2, \dots, n-2, ~ A, B \in \mathbb{R},\) and \(f:[0,1] \times \mathbb{R}^3 \to \mathbb{R}, ~g: \mathbb{R}^m \to \mathbb{R}, ~h: \mathbb{R}^n \to \mathbb{R}\) are continuous functions. Assuming some monotonicity conditions on the functions \(f, g, h,\) the existence of a solution for the BVP (1.1) is proved by applying the method of lower and upper solutions, and Leray-Schauder degree theory.
In order to establish the uniqueness of the solution for BVP (1.1), the authors make use of the following auxiliary boundary value problem:
\[ \begin{cases} x'''(t)+a(t)x''(t)+b(t)x'(t)+c(t)x(t)=0, ~0 < t < 1,\\ x(0)=0, ~p_1x'(0)+q_1x''(0)+\sum _{i=1}^{m-2}r_{i}x(\xi_{i})=0,\\ ~~p_2x'(1)+q_2x''(1)+\sum _{j=1}^{n-2}R_{j}x(\eta_{j})=0, \end{cases} \]
where \(a(t), b(t), c(t) \in C[0,1], ~c(t)\geq 0, t \in [0,1], ~p_1, p_2, q_1, q_2, r_i, R_j \in \mathbb{R}\) with \(q_1 \leq 0, q_2 \geq 0, r_i \leq 0, R_j \leq 0.\) Some illustrative examples are also presented.