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Boundary Value Problems
Figure 1 | Boundary Value Problems

Figure 1

From: Existence and uniqueness of finite beam deflection on nonlinear non-uniform elastic foundation with arbitrary well-posed boundary condition

Figure 1

Illustration for Theorem 1. The nonlinear non-uniform deflection \(\widetilde{\mathcal{L}}_{\mathbf{M}}[\mathbf{b},w,f]\) exists uniquely in \(\overline{B} ( \mathcal{L}_{\mathbf{M}}[\mathbf{b},w], r )\), depicted as the darker-shaded ball. The uniqueness of \(\widetilde{\mathcal{L}}_{\mathbf{M}}[\mathbf{b},w,f]\) is guaranteed up to the larger region \(\overline{B} ( \mathcal{L}_{\mathbf{M}}[\mathbf{b},w], R )\), depicted as the ball including the lighter-shaded region. Iteration process with Ψ starting from any \(u_{0}\) in \(\overline{B} ( \mathcal{L}_{\mathbf{M}}[\mathbf{b},w], R )\) converges uniformly to \(\widetilde{\mathcal{L}}_{\mathbf{M}}[\mathbf{b},w,f]\). The deflection horizon \(\overline{B} ( 0, \rho^{-1}(\sigma k) )\), depicted as the outermost ball, always contains \(\overline{B} ( \mathcal{L}_{\mathbf{M}}[\mathbf {b},w], R )\) and all the deflections in our analysis

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