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By Hokkanen, Morosanu

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Of course, if CG−1 C/p(1) = C/p(1) then the two notions of solution are identical. 3), if G is, in addition, a single-valued mapping. 1 Let p(r) = q(r) = r, C ≥ 0, and let G, K ⊂ IR × IR be defined by  if ξ < 0, ξ K(ξ) = ξ, G(ξ) = [0, 1] if ξ = 0,  ξ + 1 if ξ > 0. 4) are all satisfied. 23) 0 1 r r s s u(0) + 0 u (σ) dσ ds . 23)) u is nondecreasing in [0, 1]. Similarly, if u(0) < 0, then u is nonpositive and nonincreasing in [0, 1]. 23) shows that u is identically zero. 23) implies that u (0) = 0.

2) with (u0 , f ) := (u0n , fn ). 8) to conclude the proof. 3 (H. Br´ezis). 2) has a unique strong solution u such that t → t 2 u (t) belongs to L2 (0, T ; H), t → ψ u(t) is integrable on [0, T ] and absolutely continuous on [δ, T ], for all δ ∈ (0, T ). If, in addition, u0 ∈ D(ψ), then u ∈ L2 (0, T ; H), t → ψ u(t) is absolutely continuous on [0, T ], and ψ u(t) ≤ ψ(u0 ) + 1 2 T f (s) 2 H ds for all t ∈ [0, T ]. 3 Let C be a nonempty closed subset of H. A continuous semigroup of contractions on C is a family of operators S(t): C → C, t ≥ 0, satisfying: (A1) S(0)x = x for all x ∈ C; (A2) S(t + s)x = S(t)S(s)x for all x ∈ C, t, s ≥ 0; (A3) for every x ∈ C, the mapping t → S(t)x is continuous on [0, ∞); (A4) S(t)x − S(t)y H ≤ x−y H for all x, y ∈ C, t ≥ 0.

22) are identical: u (1) = C − 1. 1 below. In fact, we can find the explicit solution in this case, namely u(r) = C −1 e r − e −r , 0 ≤ r ≤ 1. 30) Observe that we have the same solution uC for C = 1 and C = 2. 29) that r0 depends continuously on C ∈ [1, 2]. 30), we can deduce that uC depends continuously on C. We recommend that the reader also examines the same example but with p(r) = q(r) = r. This is a multivalued and degenerate problem and similar aspects can be observed. Of course, in this case the boundary condition at r = 0 is automatically satisfied, so it is superfluous.

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Functional Methods For Differential Equaqtions by Hokkanen, Morosanu


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