# 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:**This paper deals with the existence of solutions for the following nth order multi-point boundary value problem at resonancewhere f:[0,1]×Rn⟶R is a continuous function, e∈L1[0,1], αi∈R(1⩽i⩽m-2), 0<ξ1<ξ2<⋯<ξm-2<1 and 0<η<1. An existence theorem is obtained by using the coincidence degree theory of Mawhin.Applications of Mathematics 05/2005; 177(1-177):55-65. DOI:10.1016/j.cam.2004.08.003 · 0.15 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The structure of eigenvalues of −y″+q(x)y=λy, y(0)=0, and y(1)=∑k=1mαky(ηk), will be studied, where q∈L1([0,1],ℝ), α=(αk)∈ℝm, and 0<η1<⋯<ηm<1. Due to the nonsymmetry of the problem, this equation may admit complex eigenvalues. In this paper, a complete structure of all complex eigenvalues of this equation will be obtained. In particular, it is proved that this equation has always a sequence of real eigenvalues tending to +∞. Moreover, there exists some constant Aq>0 depending on q, such that when α satisfies ‖α‖≤Aq, all eigenvalues of this equation are necessarily real.Advances in Difference Equations 01/2010; DOI:10.1155/2010/381932 · 0.63 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We show that it is important to allow the nonlinear term to change sign when discussing existence of a positive solution for multipoint, or more general nonlocal, boundary value problems in the resonant case. When the nonlinear term has a fixed sign we obtain simple necessary and sufficient conditions for the existence of positive solutions.Applied Mathematics and Computation 03/2010; 216(2):497–500. DOI:10.1016/j.amc.2010.01.056 · 1.60 Impact Factor