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By the Schauder Fixed Point theorem, T possesses a fixed point v in C. 7) s v(t) = α − 0 0 q(τ ) f (v(τ ))dτ q(s) 1/(m−1) ds. 1. 5) has a semi–classical solution on [0, τ ) for τ > 0 sufficiently small. 7). 2. 5) is unique as long as it exists and remains in (β, γ). Proof. 5), with v1 (0) = v2 (0) = α ∈ (β, γ), whose values lie in (β, γ). 7). Put ω(t) = ρm−1 (t) − ρm−1 (t). 7), so that ω(t) = 1 q(t) t q(s)[f (v1 (s)) − f (v2 (s))]ds, 0 ≤ t < T, 0 where [0, T ) is the maximal interval in which both v1 and v2 exist and remain in (β, γ).

3) is satisfied. Several remarks can be added. 3) admit the possibility that p > 0 and s < 0, though obviously not at the same time. 4) f (u) = −cup + dus , where c, d are positive constants, can be transformed by the change of variable u = ηv, η = (c/d)1/(s−p) , to the form f (v) = c˜(−v p + v s ), c˜ = cη p = dη s > 0. 3) holds. 2) to hold. On the other hand, when m = 2 we have Ψ (1) = 0. 3), that is, in this case, −1 < p < s ≤ 1, p ≤ 1 − s, are both necessary and sufficient for the validity of (F 1)–(F 5).

GARC´IA-HUIDOBRO, R. MANASEVICH, AND J. SERRIN and in turn T [v] ∈ C. Hence T (C) ⊂ C. Let (vk )k be a sequence in C and let s, t be two points in [0, t0 ]. Then 2 ¯m 1/(m−1) t M |t − s|. |T [vk ](t) − T [vk ](s)| ≤ m By the Ascoli-Arzel`a theorem this means that T maps bounded sequences into relatively compact sequences with limit points in C, since C is closed. 6), and so T [vk ] tends to T [v] pointwise in [0, t0 ] as k → ∞. By the above argument, it is obvious that T [vk ] − T [v] ∞ → 0 as k → ∞ as claimed.