# Solvability of a second-order multi-point boundary value problem at resonance

**ABSTRACT** Based on the coincidence degree theory of Mawhin, we get a general existence result for the following second-order multi-point boundary value problem at resonancex″(t)=f(t,x(t),x′(t))+e(t),t∈(0,1),x(0)=∑i=1mαix(ξi),x′(1)=∑j=1nβjx′(ηj),where f:[0,1]×R2→Rf:[0,1]×R2→R is a Carathéodory function, e∈L1[0,1],0<ξ1<ξ2<⋯<ξm<1,αi∈R,i=1,2,…,m,m⩾2e∈L1[0,1],0<ξ1<ξ2<⋯<ξm<1,αi∈R,i=1,2,…,m,m⩾2 and 0<η1<⋯<ηn<1,βj∈R,j=1,…,n,n⩾10<η1<⋯<ηn<1,βj∈R,j=1,…,n,n⩾1. In this paper, both of the boundary value conditions are responsible for resonance.

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**ABSTRACT:**The existence of at least one solution to the second-order nonlocal boundary value problems on a half line is investigated by using Mawhin's continuation theorem.Boundary Value Problems 09/2014; 2014(1):167. DOI:10.1186/s13661-014-0167-6 · 0.84 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The existence of at least one solution to the second-order nonlocal boundary value problems on the real line is investigated by using an extension of Mawhin’s continuation theorem.Boundary Value Problems 12/2014; 2014(1). DOI:10.1186/s13661-014-0255-7 · 0.84 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In the previous works, the authors developed the reproducing kernel method (RKM) for nonlocal boundary value problems. A key of the method is the construction of the reproducing kernel (RK) satisfying the homogenous boundary conditions (BCs) of the considered problems. However, it is very difficult to obtain the RK of a reproducing kernel space satisfying nonlocal BCs or nonlinear BCs. Even if the RK is found, its representation is also very complicated compared with the RK without any BCs. In this paper, we will present a new RKM for linear nonlocal boundary value problems. The method can avoid reducing the inhomogeneous BCs to homogeneous BCs and constructing RK satisfying homogeneous nonlocal linear BCs. Numerical examples are provided to show the effectiveness of the new method.Applied Mathematics and Computation 12/2014; 248. DOI:10.1016/j.amc.2014.10.002 · 1.60 Impact Factor