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We consider the boundary value problem for the deflection of a finite beam on an elastic foundation subject to vertical loading. We construct a one-to-one correspondence $\Gamma$ from the set of equivalent well-posed two-point boundary conditions to $\mathrm{gl}(4,\mathbb{C})$. Using $\Gamma$, we derive eigenconditions for the integral operator $\mathcal{K}_\mathbf{M}$ for each well-posed two-point boundary condition represented by $\mathbf{M} \in \mathrm{gl}(4,8,\mathbb{C})$. Special features of our eigenconditions include; (1) they isolate the effect of the boundary condition $\mathbf{M}$ on $\mathrm{Spec}\,\mathcal{K}_\mathbf{M}$, (2) they connect $\mathrm{Spec}\,\mathcal{K}_\mathbf{M}$ to $\mathrm{Spec}\,\mathcal{K}_{l,\alpha,k}$ whose structure has been well understood. Using our eigenconditions, we show that, for each nonzero real $\lambda \not \in \mathrm{Spec}\,\mathcal{K}_{l,\alpha,k}$, there exists a real well-posed boundary condition $\mathbf{M}$ such that $\lambda \in \mathrm{Spec}\,\mathcal{K}_\mathbf{M}$. This in particular shows that the integral operators $\mathcal{K}_\mathbf{M}$ arising from well-posed boundary conditions, may not be positive nor contractive in general, as opposed to $\mathcal{K}_{l,\alpha,k}$.
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참고문헌 (15)
참고문헌 신청
F. W. Beaufait / 1980 / Analysis of elastic beams on nonlinear foundations / Comput. Struct 12(5):669~676
Alberto Cabada / 2013 / Extremal solutions to fourth order discontinuous functional boundary value problems / Mathematische Nachrichten 286(17-18):1744~1751
최성우 / 2015 / ON POSITIVENESS AND CONTRACTIVENESS OF THE INTEGRAL OPERATOR ARISING FROM THE BEAM DEFLECTION PROBLEM ON ELASTIC FOUNDATION / 대한수학회보 52(4):1225~1240
2015 / Spectral analysis of the integral operator arising from the beam deflection problem on elastic foundation II: eigenvalues / Bound. Value Probl 2015:6
2020 / Existence and uniqueness of finite beam deflection on nonlinear non-uniform elastic foundation with arbitrary well-posed boundary condition / Bound. Value Probl 2020:23
Alberto Cabada / 2013 / Extremal solutions to fourth order discontinuous functional boundary value problems / Mathematische Nachrichten 286(17-18):1744~1751
최성우 / 2015 / ON POSITIVENESS AND CONTRACTIVENESS OF THE INTEGRAL OPERATOR ARISING FROM THE BEAM DEFLECTION PROBLEM ON ELASTIC FOUNDATION / 대한수학회보 52(4):1225~1240
2015 / Spectral analysis of the integral operator arising from the beam deflection problem on elastic foundation II: eigenvalues / Bound. Value Probl 2015:6
2020 / Existence and uniqueness of finite beam deflection on nonlinear non-uniform elastic foundation with arbitrary well-posed boundary condition / Bound. Value Probl 2020:23
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