Singular perturbations for third-order nonlinear multi-point boundary value problem

This paper is devoted to study the following third-order multi-point singularly perturbed boundary value problemɛx′′′(t)+f(t,x(t),x′(t),x′′(t),ɛ)=0,0⩽t⩽1,0<ɛ≪1,x(0,ɛ)=0,ax′(0,ɛ)-bx′′(0,ɛ)+∑i=1n-2αix(ξi,ɛ)=A,cx′(1,ɛ)+dx′′(1,ɛ)+∑i=1n-2βix(ηi,ɛ)=B,where a,b,c,d⩾0,A,B∈R,a+b>0,c+d>0, αi⩽0,βi⩽0, i=1,2,…,n-2, 0<ξ1<ξ2<⋯<ξn-2<1, and 0<η1<η2<⋯<ηn-2<1. The existence, uniqueness and asymptotic estimates of solutions of the boundary value problem are give by using priori estimates, differential inequalities technique and Leray-Schauder degree theory.