Solvability of multi-point boundary value problem at resonance (III)

• Published in: Applied mathematics and computation Vol. 129; no. 1; pp. 119 - 143
• Main Authors: Liu, Bing, Yu, Jianshe
• Format: Journal Article
• Language: English
• Published: New York, NY Elsevier Inc 15.06.2002; Elsevier
• Subjects: Boundary value problem; Coincidence degree theory; Exact sciences and technology; Mathematical analysis; Mathematics; Ordinary differential equations; Resonance; Sciences and techniques of general use; 34B15; Resonance; Boundary value problem; Coincidence degree theory; 34B10; Second order equation; Kernel function; Differential equation; Isomorphism; Multiple point boundary value problem; Equation system
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/S0096-3003(01)00036-4

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.