In this paper, the authors analyze the existence and asymptotic behavior of the solutions of the third-order singularly perturbed differential equation \[ \varepsilon\frac{d^3y}{dx^3}-\frac1xf(x,y)\frac{d^2y}{dx^2}+g_1(x,y)\frac{dy}{dx}+g_2(x,y)=0, \ x\in(0,1) \] subject to the three-point boundary condition \[ y(0)=\mu y(\eta), \ p_1y'(0)-p_2y''(0)=A, \ y(1)=B, \] where \(\varepsilon\geq0\) is sufficiently small parameter, \(p_1>0\), \(p_2\geq0\), \(\eta\in(0,1)\), \(\mu<0\) and \(A\), \(B\) are given constants; the functions \(f>0\), \(g_1<0\) and \(g_2\) are \(C^2\) on \([0,1]\times\mathbb{R}\).
Combining the Green’s function approach to boundary value problems and the Schauder fixed point theorem, the appropriate lower and upper bounds on the solutions are constructed demonstrating the occurrence of boundary layer at the end-point \(x=1\) of the interval \([0,1]\).