Constructions of harmonic maps between Hadamard manifolds

1.1. LOCAL EXISTENCE OF SOLUTIONS TO (1.1.1) WITH (1.1.2) 15
1.1 Local existence of solutions to (1.1.1) with (1.1.2)
In this section, we consider the following ordinary differential equation which includes the
equation (1.1.1) as a special case:
(1.1.3) ¨r(t)+F (t)ṙ(t) − G(t, r(t)) = 0.
Here we assume that F and G satisfy the following conditions:
⎧
(C-1)
F ∈ C ∞ ((0, ∞)),
(C-2) tF ∈ C([0, ∞)), tF(t) − 1 ≥ 0 on [0, ∞),
⎪⎨ (C-3) G ∈ C ∞ ((0, ∞) × R), G(t, 0) = 0 for t>0,
(C-4) G(t, s) > 0 for (t, s) ∈ (0, ∞) × (0, ∞),
lim
t→0
t 2 G(t, s) > 0 for s>0, lim
t→0,s→0 t2 G(t, s) =0, and
⎪⎩ (C-5) s 1 ≤ s 2 =⇒ G(t, s 1 ) ≤ G(t, s 2 ) for any t ∈ (0, ∞).
Let f i and h i be functions satisfying the conditions (F-1) through (F-4) and (H-1) through
(H-3), respectively. Define F (t) and G(t, s) by
f
F (t) =p ˙ 1 (t)
f 1 (t) + q f ˙ 2 (t)
f 2 (t) ,
G(t, s) =µ 2 h 1(s)h ′ 1 (s)
f 1 (t) 2
+ ν 2 h 2(s)h ′ 2 (s)
f 2 (t) 2 .
Then it is easily verified that these functions, F and G, satisfy the conditions (C-1) through
(C-5).
We are going to prove the short time existence of a solution to the equation (1.1.3). Our
goal is to show the following
Theorem 1.1.1. Under the conditions (C-1) through (C-5), for any t 0 > 0 and r 0 > 0, there
exists a unique positive solution r = r(t) to the equation (1.1.3) on [0,t 0 ] which satisfies the