JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2024, Vol. 59 ›› Issue (4): 38-44.doi: 10.6040/j.issn.1671-9352.0.2023.078

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Existence of solutions for a class of third-order multi-point boundary value problems with dim Ker L=3 at resonance on a half-line

DU Ruijuan   

  1. School of Artificial Intelligence, Gansu University of Political Science and Law, Lanzhou 730070, Gansu, China
  • Published:2024-04-12

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Abstract: The author considers the solvability for a class of third-order multi-point boundary value problems at resonance with dim Ker L=3{ x(t)=f(t,x(t),x'(t),x″(t))+e(t), t∈［0,+∞),x(0)=∑mi=1αix(ξi), x(1)=∑nj=1βjx(ηj), limt→+∞x'(t)=∑lk=1γkx″(ζk)where f:［0,1］×R→R satisfies S-Carathéodory conditions, e∈L1［0,∞), αi, βj, γk∈R, 0<ξ1<ξ2<…<ξm<+∞, 0<η1<η2<…<ηn<+∞, 0<ζ1<ζ2<…<ζl<+∞(m,n,l∈Z+), and satisfy the following conditions:(C1)∑mi=1αi=1, ∑mi=1αiξi=0, ∑mi=1αiξ2i=0, ∑nj=1βj=1, ∑nj=1βjηj=1, ∑nj=1βjη2j=1, ∑lk=1γk=1; 山 东 大 学 学 报 (理 学 版)第59卷- 第4期杜睿娟:一类半直线上三阶多点边值问题在dim Ker L=3共振情形下解的存在性 \=- (C2)Δ=|Q1e-t Q2e-t Q3e-tQ1te-t Q2te-t Q3te-tQ1t2e-t Q2t2e-t Q3t2e-t|:=|a11 a12 a13a21 a22 a23a31 a32 a33|≠0;whereQ1y=∑mi=1αi∫ξi0∫s0∫τ0y(v)dvdτds, Q2y=∑nj=1βj∫ηj0∫s0∫τ0y(v)dvdτds, Q3y=∑lk=1γk∫+∞γky(s)ds.

Key words: multi-point boundary value problem, resonance, half-line, Fredholm operator

CLC Number: 

  • O175.8
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