# Download New Developments in Differential Equations by Wiktor Eckhaus PDF

By Wiktor Eckhaus

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1, one concludes that (vi) is equivalent to F (y, x)ρ′− (y, x) ≤ [F (y, x)]2 ≤ F (y, x)ρ′+ (y, x) in the case F (y, x) > 0, and also to the inequality with opposite signs in the other case. Both cases give finally ρ′− (y, x) ≤ F (y, x) ≤ ρ′+ (y, x). g. [Acz´el (1966); Kuczma (1985); Acz´el and Dhombres (1989)]). Its conditional form described below deserves further study. 1) where ⊥ denotes an orthogonal relation defined on X. For instance, in an inner product space (X, ·, · ) the functional X ∋ x → x, x ∈ R is orthogonally additive (Pythagoras theorem).

Now we turn our attention to the James orthogonality. 2 Let (X, · ) be a real normed linear space with dim X ≥ 2. s. whose inner product is F . Proof. Assume that F satisfies the above mentioned conditions. 3). 10) whence u + λv 2 − u 2λ 2 = u + λ 2 F (u,v) u 2 u−v 2 − u 2 2λ . So, taking limit when λ tends to zero from the right, we have ρ′+ (u, v) = ρ′+ u, 2 F (u, v) u−v u 2 =2 F (u, v) u u 2 2 − ρ′− (u, v) and therefore F (u, v) = (ρ′+ (u, v) + ρ′− (u, v))/2. 10) dividing by λ > 0 and taking limit when λ tends to infinity, one obtains v = v− ρ′+ (u, v) + ρ′− (u, v) u , u, v ∈ X, u = 0.

3) is satisfied with ⊥ := ⊥ρ . The question is: What about spaces which are not smooth? Assume that (X, · ) is a normed linear space with dim X ≥ 2. We will show that the relation ⊥ρ satisfies the four properties of the orthogonality space (see [Alsina, Sikorska and Tom´as (2007)]. The first three are easy to check. In order to check the fourth, we need some auxiliary results. 1 For any two vectors x and w in X, we have lim ρ′± (x + tw, w) = ρ′± (x + t0 w, w). t→t0 Proof. 2, we can write lim ρ′± (x + tw, w) = lim ρ′± (x + t0 w + sw, w) = ρ′± (x + t0 w, w).