Gewoehnliche Differentialgleichungen by Harro Heuser
By Harro Heuser
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Example text
184) Set C0 J0 , E := y ∈ C J0 , E : y(0) = a . e. e. e. t ∈ J0 , A1 = t ∈ J : y(t) < y(t) ≤ y(t) , A2 = t ∈ J : y(t) ≤ y(t) < y(t) . 27. For each y ∈ C(J, E), the set S1F,τ y is nonempty. 1), there exists v ∈ S1F,y . 188) A3 = t ∈ J : y(t) ≤ y(t) ≤ y(t) . 189) where Then by decomposability w ∈ S1F,τ y . 7. Claim 1. G(y) is convex for each y ∈ C0 (J0 , E). This is obvious since S1F,τ y is convex (because F has convex values). Claim 2. G sends bounded sets into relatively compact sets in C0 (J0 , E).
Then for each h ∈ G(y) there exists v ∈ S1F,y such that t h(t) = T(t)a + 0 T(t − s)v(s)ds, t ∈ J0 . 152) . Claim 3. G sends bounded sets in C(J0 , E) into equicontinuous sets. Let u1 , u2 ∈ J0 , u1 < u2 , Br := { y ∈ C(J0 , E) : y ∞ ≤ r } be a bounded set in C0 (J0 , E) as in Claim 2 and y ∈ Br . 153) ≤ T u2 a − T u1 a u2 + 0 T u2 − s − T u1 − s v(s)ds u2 +M v(s) ds. 3), together with the Arzel´a-Ascoli theorem, we can conclude that G : C(J0 , E) → P (C(J0 , E)) is a compact multivalued map, and therefore, a condensing map.
92) i = 0, . . , m − 1. 93) f t, ri (t) dt ≤ ri wi − ri zi , i = 0, . . , m with zi < w i , zi , wi ∈ ti , ti+1 . 54) has at least one solution. e. t ∈ J, t = tk , k = 1, . . , m, Δy |t=tk = Ik y tk− , k = 1, . . 95) y(0) = y0 , where F : J × Rn → P (Rn ) is a multivalued map with nonempty compact values, y0 ∈ Rn , P (Rn ) is the family of all subsets of Rn , Ik ∈ C(Rn , Rn ) (k = 1, 2, . . , m), Δy |t=tk = y(tk+ ) − y(tk− ), y(tk− ) and y(tk+ ) represent the left and right limits of y(t) at t = tk , respectively.