283
lim y l(02Cl')(x, y, z)1 = 0 uniformly in x E M and z E [ii, (0)
!/-oo
for some y > 0;
lim z l(03a)(x, y, z)1 = 0 uniformly in x E M and y E [ii, (0);
<--+00
(A3) g E C 2 (R+), g(O) = 0, infg' > 0, and supg' < 00.
As before, an initial condition for (22) is a temperature history, i.e. a function in
C+([ - T, 0] x Ml, and we can use comparison equations similar to (18) and (19) in
order to establish global unique solvability for nonnegative initial data. We refer to
[Hetzer (1995)] for an outline. Thus, let us denote by u = u( t, X; f.l, 17) the unique
global solution of (22), u(O,·):c;: 17, for 17 E C+([-T,O] x M) and p E R+. The timeI-map II = D(p,17) maps now R+ x C+([-T,O] x M) into e+([-T, 0] x M) and is
defined by II(f.l,t'})(s,x) = u(1 + s,X;f.l,v) for (p,v) E R+ x C+([-T,O] x M) and
(s,x) E [-T,O] x M. Again, the I-periodic solutions of (22) for a given f.l are in an
one-to-one correspondence with the fixed points of II(p,'), so we introduce:
~:= ((f.l,17) E R+ x e+([-T,O] x M) : II(f.l, 17) =c: 17},
I:J3 principal branch of ~ (i.e. the component containing (0,0)).
We want to derive the S-shapedness of 1:J3, but already applying the Leray-Schauder
continuation method requires some extra care compared with what was done previously.
Let us just mention that II is not completely continuous since the time-lag T is ~ 1. A
standard procedure to overcome this obstacle consists in subtracting a I-shift operator
S
{
{}(I+s,x)-17(O,x) for-TSsS-l,xEM
J9(s,x) := o
for -1 < s S 0, x E M.
Since that approach does not leave C+ ([ - T, OJ x M) invariant, we first extend II
appropriately to e([ - T, OJ x M) by selecting the odd extension for g and setting
a(x,y,z) = G(x,y+,z+) for X E M, y,z E R. We refer for details to [Hetzer (1995)]
and state:
Lemma 4. Let (AI) - (A3) be fulfilled, then the projection prl(l:J3) = R+.
Next, we want to describe the structural assumptions. To this end, we consider the
Frechet derivative (D2II)(f.l,17) for a point (p,{)) E I:J3 \ {(O,O)}, which exists thanks to
lim y l(02Cl')(x, y, z)1 = 0 uniformly in x E M and z E [ii, (0)
!/-oo
for some y > 0;
lim z l(03a)(x, y, z)1 = 0 uniformly in x E M and y E [ii, (0);
<--+00
(A3) g E C 2 (R+), g(O) = 0, infg' > 0, and supg' < 00.
As before, an initial condition for (22) is a temperature history, i.e. a function in
C+([ - T, 0] x Ml, and we can use comparison equations similar to (18) and (19) in
order to establish global unique solvability for nonnegative initial data. We refer to
[Hetzer (1995)] for an outline. Thus, let us denote by u = u( t, X; f.l, 17) the unique
global solution of (22), u(O,·):c;: 17, for 17 E C+([-T,O] x M) and p E R+. The timeI-map II = D(p,17) maps now R+ x C+([-T,O] x M) into e+([-T, 0] x M) and is
defined by II(f.l,t'})(s,x) = u(1 + s,X;f.l,v) for (p,v) E R+ x C+([-T,O] x M) and
(s,x) E [-T,O] x M. Again, the I-periodic solutions of (22) for a given f.l are in an
one-to-one correspondence with the fixed points of II(p,'), so we introduce:
~:= ((f.l,17) E R+ x e+([-T,O] x M) : II(f.l, 17) =c: 17},
I:J3 principal branch of ~ (i.e. the component containing (0,0)).
We want to derive the S-shapedness of 1:J3, but already applying the Leray-Schauder
continuation method requires some extra care compared with what was done previously.
Let us just mention that II is not completely continuous since the time-lag T is ~ 1. A
standard procedure to overcome this obstacle consists in subtracting a I-shift operator
S
{
{}(I+s,x)-17(O,x) for-TSsS-l,xEM
J9(s,x) := o
for -1 < s S 0, x E M.
Since that approach does not leave C+ ([ - T, OJ x M) invariant, we first extend II
appropriately to e([ - T, OJ x M) by selecting the odd extension for g and setting
a(x,y,z) = G(x,y+,z+) for X E M, y,z E R. We refer for details to [Hetzer (1995)]
and state:
Lemma 4. Let (AI) - (A3) be fulfilled, then the projection prl(l:J3) = R+.
Next, we want to describe the structural assumptions. To this end, we consider the
Frechet derivative (D2II)(f.l,17) for a point (p,{)) E I:J3 \ {(O,O)}, which exists thanks to