Abstract:

The author considers the solvability for a class of third-order multi-point boundary value problems at resonance with dim Ker L=3 $\left\{\begin{array}{l}x^{\prime \prime \prime}(t)=f\left(t, x(t), x^{\prime}(t), x^{\prime \prime}(t)\right)+e(t), \quad t \in[0, +\infty), \\x(0)=\sum\limits_{i=1}^m \alpha_i x\left(\xi_i\right), \quad x(1)=\sum\limits_{j=1}^n \beta_j x\left(\eta_j\right), \quad \lim\limits_{t \rightarrow+\infty} x^{\prime}(t)=\sum\limits_{k=1}^l \gamma_k x^{\prime \prime}\left(\zeta_k\right)\end{array}\right.$ where f : [0, 1]× R→R satisfies S-Carathéodory conditions, e ∈ L¹[0, ∞), α_i, β_j, γ_k ∈ R, 0 < ξ₁ < ξ₂ < … < ξ_m < +∞, 0 < η₁ < η₂ < … < η_n < +∞, 0 < ζ₁ < ζ₂ < … < ζ_l < +∞(m, n, l ∈ Z⁺), and satisfy the following conditions:(C₁) $\sum\limits_{i=1}^m \alpha_i=1, \sum\limits_{i=1}^m \alpha_i \xi_i=0, \sum\limits_{i=1}^m \alpha_i \xi_i^2=0, \sum\limits_{j=1}^n \beta_j=1, \sum\limits_{j=1}^n \beta_j \eta_j=1, \sum\limits_{j=1}^n \beta_j \eta_j^2=1, \sum\limits_{k=1}^l \gamma_k=1$;(C₂) $\mathit{\Delta }=\left|\begin{array}{ccc}Q_1 e^{-t} & Q_2 e^{-t} & Q_3 e^{-t} \\Q_1 t e^{-t} & Q_2 t e^{-t} & Q_3 t e^{-t} \\Q_1 t^2 e^{-t} & Q_2 t^2 e^{-t} & Q_3 t^2 e^{-t}\end{array}\right|:=\left|\begin{array}{ccc}a_{11} & a_{12} & a_{13} \\a_{21} & a_{22} & a_{23} \\a_{31} & a_{32} & a_{33}\end{array}\right| \neq 0$;where $Q_1 y=\sum\limits_{i=1}^m \alpha_i \int_0^{\xi_i} \int_0^s \int_0^\tau y(v) \mathrm{d} v \mathrm{~d} \tau \mathrm{d} s, \quad Q_2 y=\sum\limits_{j=1}^n \beta_j \int_0^{\eta_j} \int_0^s \int_0^\tau y(v) \mathrm{d} v \mathrm{~d} \tau \mathrm{d} s, \quad Q_3 y=\sum\limits_{k=1}^l \gamma_k \int_{\gamma_k}^{+\infty} y(s) \mathrm{d} s.$

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