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

School of Mathematical Sciences, Xuzhou Normal University, Xuzhou, Jiangsu 221116, People’s Republic of China
Applied Mathematics and Computation (Impact Factor: 1.6). 02/2009; 208:23-30. DOI: 10.1016/j.amc.2008.11.026
Source: DBLP

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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