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: 1.55). 02/2009; 208(1):23-30. DOI: 10.1016/j.amc.2008.11.026
Source: DBLP


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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    • "When L is linear, as is known, the coincidence degree theory of Mawhin[19]has played an important role in dealing with the existence of solutions for these problems. For more recent results, we refer the reader to[3,5,6,6,8,9,14,20,22,24,25]and the references therein. Moreover boundary value problems on the half line arise in many applications in physics such that in modeling the unsteady flow of a gas through semi-infinite porous media, in plasma physics, in determining the electrical potential in an isolated neutral atom, or in combustion theory. "
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