Initial-final value problems for ordinary differential equations and applications to equivariant harmonic maps (Q1277285)

The following ordinary differential equation \[ \ddot r(t)+ \Biggl\{p{\dot f_1(t)\over f_1(t)}+ q{\dot f_2(t)\over f_2(t)}\Biggr\}\dot r(t)- \Biggl\{\mu^2 {h_1(r(t)) h_1'(r(t))\over f_1(t)^2}+ \nu^2{h_2(r(t)) h_2'(r(t))\over f_2(t)^2}\Biggr\}= 0,\;(0,\infty) \] with \(\lim_{t\to 0}(r(t))= 0\), where \(f_i\) and \(h_i\) are given functions defined on \((0,\infty)\) satisfying certain conditions (relaxed) and \(\dot r\) (resp. \(h'\)) means \({dr\over dt}\) (resp. \({dh\over dr}\)), \(\mu\) and \(\nu\) are nonnegative numbers, is studied in order to show the existence of equivariant maps between real hyperbolic spaces and from real hyperbolic spaces to real Euclidean spaces.