We study the real, Euclidean, classical field equation 0$$]]> where φ: ℝ d →ℝ is suitably small at infinity. We study existence and regularity assuming that λ≧0, F ∈ C ∞ (ℝ), and aF ( a )≧0∀ a ∈∝. These hypotheses allow strongly nonlinear F and nonunique solutions for f ≠0. When F′ ≧0, we prove uniqueness, various contractivity properties, analytic dependence on the coupling constant λ, and differentiability in the external source f . For applications in the loop expansion in quantum field theory, it is useful to know that φ is in the Schwartz class L whenever f is, and we provide a proof of this fact. The technical innovations of the problem lie in treating the noncompactness of R d , the strong nonlinearity of F , and the polynomial weights in the seminorms defining L .