Solvability of multi-point boundary value problem at resonance (III) - Liu, Bing; Yu, Jianshe (Applied mathematics and computation, 2002)
In this paper, we consider the following second-order ordinary differential equation (E) x″=f(t,x(t),x ′(t))+e(t), t∈(0,1), subject to one of the following boundary value conditions: (B1) x(0)= ∑ i=1 m−2α ix(ξ i), x(1)=βx(η), (B2) x(0)= ∑ i=1 m−2α ix(ξ i), x ′(1)=βx ′(η), (B3) x ′(0)= ∑ i=1 m−2α ix ′(ξ i), x(1)=βx(η), (B4) x ′(0)= ∑ i=1 m−2α ix ′(ξ i), x ′(1)=βx ′(η), where α i (1⩽i⩽m−2) , β∈ R, 0< ξ 1< ξ 2<⋯< ξ m−2 <1, 0< η<1. When all the α i 's have no the same sign, some existence results are given for (E) with boundary conditions (B 1), (B 2), (B 3), (B 4) at resonance case. We also give some examples to demonstrate our results.