Let (X, ρ) be a metric space and ↓USCC(X) and ↓CC(X) be the families of the regions below all upper semi-continuous compact-supported maps and below all continuous compact-supported maps from X to I = [0, 1], respectively. With the Hausdorff-metric, they are topological spaces. In this paper, we prove that, if X is an infinite compact metric space with a dense set of isolated points,Combining this statement with a result in our previous paper,if X is an infinite compact metric space. We also prove that, for a metric space X, (↓USCC(X), ↓CC(X)) ≈ (Σ, c0) if and only if X is non-compact, locally compact, non-discrete and separable.
In this paper, we develop an efficient spectral method for numerically solving the nonlinear Volterra integral equation with weak singularity and delays. Based on the symmetric collocation points, the spectral method is illustrated, and the convergence results are obtained. In the end, two numerical experiments are carried out to confirm the theoretical results.
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