By Robert C. Gunning, Hugo Rossi

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T x ||≥ f (T x) f (T x) ≥ f (εc || x || u) = εc || x || f (u) (xǫK∈ ) || T x ||≥ εc || x || f (u) ≥ εe f (x) f (u) (6) Nonlinear mappings in cones 49 ≥ δε f (u) (x ∈ Kε ) with δ = εc f (u) > 0. Let Vǫ be the mapping defined by taking Vǫ (0) = 0, Vε (x) =|| x || . || x + 2ε || x || u ||−1 (x + 2ε || x || u), x 0 V is well defined since || x + 2ǫ || x || u ||≥|| x || −2ε || x || || u || =|| x || (1 − 2ε) > 0 if || x || 0. Plainly Vǫ is continuous in E and || Vǫ x ||=|| x || (7) x ∈ C, || x ||= 1 Vε x ∈ Kε (8) Also For f (Vε x) = ||x||||x + 2ε||x||u||−1 { f (x) + 2ε||x|| f (u)} ≤ 2ǫ f (u) ≥ ε f (u) 1 + 2ǫ Let Aε be the mapping defined on Kǫ by Aε = Vε LT x x Lx = ||x|| when 58 0 Then by (6) and (8) Aε Kε ⊂ Kε By (6), Vε L is continuous in T Kε , and Aε continuously into a compact subset of Kε .

There exists u ∈ K ∩ u˜ , and we have T˜ u˜ = T u. , q(T u − u) = 0 It follows that p(T u − u) = 0, which contradicts (1) since u ∈ K. Problem . 2). It is not known whether the following proposition is true. Q. s. E, and let T be a continuous mapping of K into a compact subset of K. Then T has a fixed point in K. It is obvious that if T maps K into a compact convex subset H of K, then T has a fixed point. 2 to H instead of K. In particular, Q will 49 hold if every compact subset of K is contained in compact convex subset of K.

Schaeffer. 1. A subset C of a vector space E over R is called a positive cone if it satisfies (i) x, y ∈ C ⇒ x + y ∈ C (ii) x ∈ C, α ≥ 0 ⇒ α ∈ C (iii) x, −x ∈ C ⇒ x = 0 (iv) C contains non-zero vectors. A vector space E over R with a specified positive cone is called a partially ordered vector space , and we write x ≤ y( or y ≥ x) to denote that y − x ∈ C. , (v) x ≤ x (x ∈ E), (vi) x ≤ y, y ≤ z ⇒ x ≤ z, 43 Nonlinear mappings in cones 44 (vii) x ≤ y, y ≤ x ⇒ x = y. 51 Also the partial ordering and the linear structure are related by the properties: (viii) xi ≤ yi (i = 1, 2) ⇒ x1 + x2 ≤ y1 + y2 , (ix) x ≤ y, 0 ≤ α ≤ β ⇒ αx ≤ βy.