Abstract
We discuss a third-order multipoint boundary value problem under some appropriate resonance conditions. By using the coincidence degree theory, we establish the existence result of solutions. The emphasis here is that the dimension of the linear operator is equal to two. Our results supplement other results.
Highlights
IntroductionIf the linear equation Lx = x(t) = 0 with boundary conditions (2) has nontrivial solutions, that is, dimKer L ≥ 1, the BVP (1)-(2) is called a resonance problem
For any l ∈ N, there exists kl ∈ {l(m − 2) + 1, . . . , (l + 1)(m − 2)}, such that ∑mi=−12 αiξikl+2 ≠ 0
It is clear that Ker L = {x ∈ dom L : x = a + ct2, a = ((βη2 − 1)/(1 − β))c, c ∈ R}
Summary
If the linear equation Lx = x(t) = 0 with boundary conditions (2) has nontrivial solutions, that is, dimKer L ≥ 1, the BVP (1)-(2) is called a resonance problem. In [2, 3], the authors established the existence results for resonance boundary value problems with the case of dimKer L = 2. Du et al [1] studied the existence results of BVP (1)(2) under the resonance conditions (C2) and (C4), that is, dimKer L = 1, but they did not discuss the other three cases. Under the resonance conditions, (C1), (C3), or (C5), we could imply dimKer L = 2; we supplement the results in [1].
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