### Some functional inequalities and their Baire category properties.

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We consider the problem of the vanishing of non-negative continuous solutions ψ of the functional inequalities (1) ψ(f(x)) ≤ β(x,ψ(x)) and (2) α(x,ψ(x)) ≤ ψ(f(x)) ≤ β(x,ψ(x)), where x varies in a fixed real interval I. As a consequence we obtain some results on the uniqueness of continuous solutions φ :I → Y of the equation (3) φ(f(x)) = g(x,φ(x)), where Y denotes an arbitrary metric space.

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