Gliklikh Y.E. Global and Stochastic Analysis with Applications to Mathematical Physics

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286 12 Accessible Points and Sub-Manifolds of Mechanical Systems. Controllability

¨x = −y − 2K·x,

¨y = x − 2K·y, (12.2)

¨z = −2K·z.

where

K = 

¯r

/2=(˙x

+˙y

+˙z

)/2

is the kinetic energy. To obtain these equations, one applies d’Alembert’s

principle (see, e.g., [190]) to the holonomic constraint F (¯r)=x

+ y

+ z

Denote the North and South Pole of the sphere by N =(0, 0, 1) and

S =(0, 0, −1), respectively. Let ¯r(t)=



x(t),y(t),z(t)



be the trajectory of

the system such that ¯r(t

)=S for some t

and

¯r(t

)=V =0.Notethatif

V =0,then¯r(t) ≡ S. It is clear that V ∈ T

must have the form (X, Y, 0).

We claim that the kinetic energy increases along ¯r(t)until¯r hits the North

or South Pole. By (12.2)wehave

K(¯r(t)) = ˙xy +˙yx and

K(¯r(t)) = x

+ y

Note also that

K(¯r(t

)) = 0. This means that

K(¯r(t)) > 0, unless ¯r(t)=S

or ¯r(t)=N. In fact, the derivative

K(¯r(t)) is also increasing. Since

K(N)=

K(S)=0,wehave

K(¯r(t)) = N for any t>t

To clarify the geometric picture, consider the function z(t)=z(¯r(t)). Let

be such that ˙z(t

)=0andz(t) is increasing on [t

]. The last

equation in (12.2) implies that z(t

) > 0 and, as a consequence, ¨z(t

) < 0,

i.e., z(t

) is a local maximum of z(t). Since K is increasing along ¯r(t), we see

that z(t

) < 1. In the same way, one may show that

signz(t

)=(−1)

i+1

and |z(t

)| > |z(t

i+1

for all points t

<... such that ˙z(t

)=0.

Therefore the trajectory ¯r(t) heads to the equator of S

and oscillates near

it. In particular, the trajectory never reaches the point N =(0, 0, 1).

Example 12.3. Let us replace the ﬁeld ¯α in the system of Example 12.2 by

the force ﬁeld

Ω (¯r,

¯r)=

[

¯r, ¯r]

1+

¯r

where [·, ·] is the vector product in R

. The equation of motion of the me-

chanical system is as follows:

¯r = Ω (¯r,

¯r) − 2K·¯r. (12.3)

A straightforward calculation shows that

K =(Ω (¯r,

¯r) ,

¯r) − 2K·(¯r,

¯r)=0

12.2 Examples of Points that Cannot be Connected by a Trajectory 287

along a solution of (12.3) (i.e., the force is always orthogonal to the velocity)

and

b =0,where

b =[

¯r,

¯r]=−



¯r

· ¯r

1+

¯r

−



¯r





¯r, ¯r



Therefore, the kinetic energy K = 

¯r

/2 is constant along the trajectory

r(t)andr(t) lies in the plane orthogonal to the constant vector

b.Inother

words, the trajectory is the circle



¯r(t),



= const on S

. Assume that there

is a trajectory passing through two antipodal points. Then it must be a great

circle on S

and, therefore,



¯r(t),



=0.

Let α be the angle between ¯r(t)and

b. A straightforward calculation (based

on the explicit formulas for 

b and (¯r(t),

b) and on the equality ¯r(t)≡1)

shows that

cos α =

φ (

¯r)



¯r

where φ:[0, ∞) → R is a bounded function. Hence, (¯r(t),

b) tends to zero

as K→∞, assuming non-zero values only. This means that there is no

trajectory in the system passing through two antipodal points. Note also that

any two points which are not antipodal can be connected by a trajectory with

suﬃciently high kinetic energy.

Example 12.4. Replace the force Ω (¯r,

¯r) of the preceding example by the

gyroscopic force A(¯r,

¯r)=[

¯r, ¯r]. The equation of motion of the new system is

¯r =[

¯r, ¯r] − 2K

¯r.

The analysis of this example is quite similar to that of Example 2. First, we

prove that

K =0and

b =0,where

b =[

¯r,

¯r]. This implies that the trajectory

lies in the plane orthogonal to

b. If the trajectory were a great circle, so that

(¯r,

b) = 0, then this would give us the equality [¯r,

¯r] = 0, which is impossible.

The author is grateful to Evgenii I. Yakovlev for pointing out Exam-

ple 12.4.

Example 12.5. Replace the force Ω (¯r,

¯r)

of the preceding example by

α (¯r,

¯r)=[¯r,

¯r] 

¯r.

By d’Alembert’s principle, as above, the equation of motion with a constraint

takes the form:

¯r =[¯r,

¯r] 

¯r−2T ¯r where the kinetic energy T =

¯r

.Since

the acceleration is everywhere orthogonal to the velocity, it is obvious that

T = 0. Then, as above, we prove that

b =0for

b =[

¯r,

¯r]. Direct calculations

yield

b = 0. This means that any trajectory satisﬁes the relation (

b, ¯r) = const

(the parentheses denote the inner product in R

), i.e., it is a circle on the

sphere that also lies in a plane orthogonal to the constant vector

b. Antipodal

points are joined by a great circle, i.e., (

b, ¯r) = 0. From this we get the equality

288 12 Accessible Points and Sub-Manifolds of Mechanical Systems. Controllability

for the mixed product (¯r,

¯r,

¯r) = 0, which is impossible. Thus antipodal points

on the sphere cannot be connected by a trajectory.

12.3 Existence of Solutions

In the examples of Section 12.2, the points which could not be connected

by a trajectory were conjugate along all geodesics of the Levi-Civit´a connec-

tion. In this section, we prove that if two points are not conjugate along a

geodesic, then there exists a trajectory joining the points if the force has less

than quadratic growth in velocity, or quadratic growth under some additional

assumption. We consider the general case where the force ﬁeld ¯α(t, m, X)is

discontinuous or includes a control parameter (see Section 11.7.) Thus the

trajectories are, in fact, solutions of the diﬀerential inclusion (11.17) with a

set-valued vector force ﬁeld a(t, m, X) which can be either upper or lower

semicontinuous and has convex closed values. This general result yields, as a

simple corollary, the existence of such a trajectory for a mechanical system

with continuous ¯α [88, 90]. The case of bounded force, which is a special

case of force with sub-quadratic growth in velocity, has many speciﬁc conse-

quences.

The requisite deﬁnitions from set-valued analysis can be found in Sec-

tion 4.1.

We start with the following technical statement.

Lemma 12.6 Let a real number δ satisfy the inequality 0 <δ<

(ε+C)

Then there exists a suﬃciently small positive number ϕ such that (εt

−1

−ϕ) >

0 and the inequality δ((εt

−1

− ϕ)+Ct

−1

)

<εt

−2

− ϕt

−1

holds.

Proof. For δ satisfying the hypothesis of the Lemma we get δ(εt

−1

+Ct

−1

)

εt

−2

. From the continuity of both sides of this inequality it follows that there

exists a suﬃciently small ϕ>0 such that (εt

−1

− ϕ) > 0 and the inequality

δ((εt

−1

− ϕ)+Ct

−1

)

< (εt

−1

− ϕ)t

−1

= εt

−2

− ϕt

−1

holds. 

For the remainder of this section, M is a complete Riemannian manifold

and by ·we denote the norm in a tangent space generated by the Rie-

mannian metric. Introduce a norm on the set a(t, m, X) ∈ T

M by the usual

formula a(t, m, X) =sup

y∈a(t,m,X)

y.

Deﬁnition 12.7. We say that a(t, m, X)hasless than quadratic growth in

X if for any compact Θ ⊂ M and any ﬁnite interval [0,l] the relation

lim

X→∞

a(t, m, X)

X

holds uniformly in t ∈ [0,l]andm ∈ Θ.

12.3 Existence of Solutions 289

Deﬁnition 12.8. We say that a(t, m, X)hasaquadratic bound in X if for

any compact Θ ⊂ M and any ﬁnite interval [0,l] the relation

lim

X→∞

a(t, m, X)

X

= a(t, m)

holds uniformly in t ∈ [0,l]andm ∈ Θ,wherea(t, m) ≥ 0 is a real bounded

function on [0,l] ×Θ that is not identical zero.

For the case of set-valued vector force ﬁelds we modify the deﬁnitions from

Section 4.1 as follows:

Deﬁnition 12.9. We say that a(t, m, X) satisﬁes the upper Carath´eodory

condition if:

1) for every (m, X) ∈ TM the map F (·,m,X):I  T

M is measurable;

2) for almost all t ∈ I the map F (t, ·, ·):TM  TM is upper semicontin-

uous.

Deﬁnition 12.10. Let I =[0,l] ⊂ R. The set-valued force ﬁeld a : I×TM 

TM is said to be almost lower semicontinuous if there exists a countable

sequence of disjoint compact sets {I

}, I

⊂ I, such that:

(i) I\∪

has measure zero;

(ii) the restriction of a on each I

× TM is lower semicontinuous.

Theorem 12.11 Let a(t, m, X) satisfy the upper Carath´eodory condition,

have convex closed bounded values and have less than quadratic growth in X.

Let the points m

and m

be non-conjugate along a geodesic g of the Levi-

Civit´a connection. Then there exists a positive number L(m

,g) such that

if 0 <t

<L(m

,g) there exists a solution m(t) of (11.17) for which

m(0) = m

and m(t

)=m

Proof. For a C

-curve γ(t)=Sv(t), v(·) ∈ C

(I,T

M), consider the set-

valued vector ﬁeld a(t, γ(t), ˙γ(t)). Denote by Γ the operator of parallel trans-

lation of vectors along γ(·) at the point γ(0) = m

. Apply the operator Γ

to all sets a(t, γ(t), ˙γ(t)) along γ(·). As a result, for any v ∈ C

(I,T

we obtain a set-valued map Γ aSv :[0,l]  T

M that has convex values.

Itisshownin[125] that the map Γ aS : C

([0,l],T

M) × [0,l]  T

satisﬁes the upper Carath´eodory condition. Denote by PΓ aSv the set of all

measurable selectors of Γ aSv :[0,l]  T

M (such selectors exist by [31]).

Deﬁne on C

([0,t

],T

M) the set-valued operator



PΓ aS by the formula



PΓ aSv =





f(τ )dτ



f(·) ∈ PΓ aSv



Lemma 12.12 The map



PΓ aS sends bounded subsets of C

(I,T

M) to

compact sets.

290 12 Accessible Points and Sub-Manifolds of Mechanical Systems. Controllability

Proof. Since the metric ·, · is complete, for any ball U

in C

(I,T

the union of curves {(γ, ˙γ)|γ ∈ U

} lies, by Lemma 3.53, in a compact

subset of TM. Then for those curves ˙γ(t) is uniformly bounded. Hence,

since Deﬁnition 12.7 is fulﬁlled for a, all sets a(t, γ, ˙γ), where γ ∈SU

,are

uniformly bounded. As a consequence, since parallel translations preserve

the norm, the sets (ΓaSv)(t)forv ∈ U

are also uniformly bounded, and so

are all their measurable selectors PΓ aSv. Thus, the continuous curves u ∈



v∈U

(



PΓ aS)v are uniformly bounded and equicontinuous. The lemma

follows. 

Lemma 12.13 The map



PΓ aS is upper semicontinuous and has convex

values.

Proof. It suﬃces to prove that the set-valued map



PΓ

◦Shas a closed

graph. In other words, that v

→ v

and u

→ u

,whereu

∈ (



PΓ aS)v

implies that u

∈ (



PΓ aS)v

, i.e., ˙u

∈ (Γ aSv

)(t) for almost all t.Since

the map



PΓ aS sends bounded sets to compact sets, the map is upper

semicontinuous provided that it has a closed graph (see [31]). Recall that the

sets (Γ aSv

)(t) are convex and the map (Γ aSv)(t) is upper semicontinuous

in v and t.Asaresult,wehave ˙u

∈ (Γ aSv

)(t). 

Consider the numbers ε and C from Lemma 3.48 constructed for the points

and m

and the geodesic g.LetΞ be the compact set from Lemma 3.51,

and let [0,l] be some interval. Choose a positive number δ<

(ε+C)

.Since

a satisﬁes Deﬁnition 12.7, one can easily see that there exists a Q>0such

that for X≥Q the inequality

max

(t,m)∈I×Ξ

a(t, m, Y ) <δX

(12.4)

holds for all Y  < X. For suﬃciently small t

> 0wegett

∈ [0,l]and

−1

ε − ϕ>Qwhere ϕ is as in Lemma 12.6.LetL(m

,g) be the upper

bound of t

such that the above relations hold and let 0 <t

<L(m

,g).

For this t

denote by K the corresponding number t

−1

ε −ϕ.

By construction, t

−1

ε>Kand so by Lemma 3.50 the operator B(v)=



PΓ aS(v + C

) that sends U

to C

([0,t

],T

M) is well-deﬁned. Like



PΓ aS, this operator is upper semicontinuous, has convex values and maps

bounded sets from C

([0,t

],T

M) to compact sets.

For v ∈ U

⊂ C

([0,t

],T

M), since parallel translation preserves the

norm of a vector, from the construction of the operator S,from(12.4)and

from Lemma 12.6 it follows that



a(t, S(v(t)+C

S(v(t)+C

))



<δ(t

−1

ε −ϕ + Ct

−1

)

< (t

−2

ε −t

−1

ϕ).

Since parallel translation is norm-preserving, from the last inequality it fol-

lows that

12.3 Existence of Solutions 291

Z(v + C

) =





PΓ aS(v(τ)+C

)



([0,t

],T

< (t

−1

ε −ϕ)=K.

Thus B sends the ball U

into itself and from the Glicksberg-Ky Fan The-

orem (see, e.g., [31]) it follows that B has a ﬁxed point u

∗

∈ U

, i.e.

∗

∈Bu

∗

.Letusshowthatm(t)=S(u

∗

(t)+C

∗

) is the desired solu-

tion. By construction we have m(0) = m

and m(t

)=m

, m(t)isa

-curve and ˙m(t) is absolutely continuous. Note that ˙u

∗

is a selector of

Γ a(t, S(u

∗

+ C

∗

S(u

∗

+ C

∗

)) since u

∗

is a ﬁxed point of Z.Inother

words, the inclusion ˙u

∗

(t) ∈ Γa(t, S(u

∗

+ C

∗

S(u

∗

+ C

∗

)) holds for all

points t at which the derivative exists. Using the properties of the covari-

ant derivative and the deﬁnition of u

∗

, one can show that ˙u

∗

(t) is parallel

˙m(t) along m(·)andΓ a(t, S(u

∗

+ C

∗

S(u

∗

+ C

∗

)) is parallel to

a(t, m(t), ˙m(t)). Hence,

˙m(t) ∈ a(t, m(t), ˙m(t)). 

Theorem 12.14 Let a(t, m, X) be almost lower semicontinuous, have closed

bounded values and have less than quadratic growth in X. Let the points m

and m

be non-conjugate along a geodesic g of the Levi-civit´a connection.

Then there exists a positive number L(m

,g) such that if 0 <t

L(m

,g) there exists a solution m(t) of (11.17) for which m(0) = m

and m(t

)=m

Proof. Here we use the same notation as in the proof of Theorem 12.11. Notice

that from the condition of less than quadratic growth for a it follows that for

all v ∈ C

([0,l],T

M) the curves from PΓ aSv are integrable. Hence, the set-

valued map PΓ aS sends C

([0,l],T

M)intoL

(([0,l], A,μ),T

M), where

A is the Borel σ-algebra and μ is the normalized Lebesgue measure. Since

a is almost lower semicontinuous, by analogy with [155] one can easily show

that PΓ aS : C

([0,l],T

M) → L

(([0,l], a,μ),T

M) is lower semicontin-

uous and has decomposable values. Then by the Bressan-Colombo Theorem

(Theorem 4.9) it has a continuous selector, which we denote by pΓ aS.

Let Q, L(m

,g), 0 <t

<L(m

,g)andK be as in the proof of

Theorem 12.11. Then on the ball U

⊂ C

([0,t

],T

M) the operator

Bv =



pΓ aS(v(s)+C

S(v(s)+C

))ds : U

→ C

([0,t

],T

is well-deﬁned.

Lemma 12.15 The mapping B : C

(I,T

M) → C

(I,T

M) is com-

pletely continuous.

Proof. The fact that B sends bounded sets to compact sets is proved by

analogy with the argument in the proof of Lemma 12.13.

From the properties of the operators S and Γ (see Section 3.2) it follows

that the operator B : C

(I,T

M) → L

((I,A,μ),T

M) is continuous.

Since C

continuously depends on v (see Theorem 3.47), this means that the

292 12 Accessible Points and Sub-Manifolds of Mechanical Systems. Controllability

vector



pΓ a(s, S(v(·)+C

S(v(·)+C

))ds ∈ T

M is continuous in

v ∈ C

(I,T

M). A very simple modiﬁcation of the above argument shows

that for any t

∗

∈ I the map sending v(·) ∈ C

(I,T

M) to the restriction

of pΓ a(t, S(v(·)+C

S(v(·)+C

)) on [0,t

∗

] is continuous as a map from

(I,T

M)toL

(([0,t

∗

], A,μ),T

M), hence we obtain that the vector



∗

pΓ a(s, S(v(·)+C

S(v(·)+C

))ds is jointly continuous in t and v for

any t

∗

∈ I. Thus for any ε>0, v ∈ C

(I,T

M)andt

∗

∈ I there exists a

δ = δ(ε, v, t

∗

) > 0 such that if v(·) −v

(·)

(I,T

δ and |t−t



| <

δ,

then





pΓ a(s, S(v(·)),

S(v(·)))ds

−





pΓ a(s, S(v

(·)),

S(v

(·)))ds



<ε.

Since I is compact, for a given v we can ﬁnd a unique δ = δ(ε, v) for all t ∈ I.

This completes the proof of continuity of B. 

Since parallel translation preserves the norm of a vector, from the con-

struction of S for any u ∈ U

with given a we get

Bv =





pΓ a(s, S(v(s)+C

S(v(s)+C

))ds



([0,t

],T

< (t

−1

ε −ϕ)=K.

Hence B sends U

into itself and hence, by the classical Schauder principle,

it has a ﬁxed point u

∗

∈ U

. Using the same argument as in the proof of

Theorem 12.11, one can easily prove that m(t)=S(u

∗

+ C

∗

)(t) is a solution

of (11.17) such that m(0) = m

and m(t

)=m

. 

Theorem 12.16 Let a(t, m, X) either satisfy the upper Carath´eodory condi-

tion and have convex closed bounded values or be almost lower semicontinuous

and have closed bounded values. Let also a(t, m, X) have a quadratic bound in

X and the points m

and m

be non-conjugate along a certain geodesic g of

the Levi-civit´a connection. Assume in addition that for t ∈ [0,l] and m ∈ Ξ,

where [0,l] is some interval and Ξ is the compact set from Lemma 3.51,for

the function a(t, m) from Deﬁnition 12.8 there exists a real number δ such that

the estimate a(t, m) <δ<

(ε+C)

holds. Then there exists a positive number

L(m

,g) such that if 0 <t

<L(m

,g) there exists a solution m(t)

of (11.17) for which m(0) = m

and m(t

)=m

The proof of Theorem 12.16 follows the same argument as that for Theo-

rems 12.11 and 12.14. The only modiﬁcation here is that for a with a quadratic

bound in X we assume the existence of a δ such that a(t, m) <δ<

(ε+C)

12.3 Existence of Solutions 293

while in the proof of Theorems 12.11 and 12.14 an analogous δ is shown to

exist for any a with less than quadratic growth in X.

It is worth noting that if there is more than one geodesic along which m

and m

are not conjugate, then any of these geodesics can be used in the

proof. Naturally, diﬀerent geodesics can give rise to diﬀerent solutions and

constants L.

If a geodesic, along which m

and m

are not conjugate, is length mini-

mizing, the constant C characterizes the Riemannian distance between these

points. C and ε together provide certain characteristics of the Riemannian

geometry on M in a neighborhood of m

.Theorem12.16 establishes an in-

terrelation between C, ε and the quadratic bounds of (11.17), under which

the two-point boundary value problem for non-conjugate points m

and m

is always solvable.

Note that the case of uniformly bounded force is a particular case of force

with less than quadratic growth in velocity. Assume that the conﬁguration

space M is compact, the metric ·, · has a non-negative sectional curvature

and the force is uniformly bounded. Then there are no conjugate points on

M. Recall that in the conditions of Theorems 12.11 and 12.14 there exists a

constant L>0 such that any two points can be connected by a trajectory

m(t) with t ∈ [0,t

] for any t

<L. If the force is bounded and M is ﬂat and

possibly non-compact, one may take L = ∞ (see Remark 3.52) In particular,

one may take L = ∞ if M is a Euclidean space. This means that, in such M,

the corresponding two-point boundary-value problem has a solution on any

time interval.

Unlike the case of bounded forces, for force ﬁelds of less than quadratic

growth in velocity (and consequently for those with a quadratic bound) even

on ﬂat conﬁguration spaces a trajectory joining the points generally exists

only on small time intervals. Nevertheless there is a subclass of forces with

quadratic bound that has the following property: if a ﬁeld and a pair of points

satisfy the conditions of Theorem 12.16, there exists a trajectory joining the

points on every ﬁnite time interval. This is the class satisfying the estimate

a(t, m, X) <a(t, m)X

, (12.5)

where a(t, m) > 0 is a continuous real-valued function on I × M.Evidently

the force satisfying (12.5) also satisﬁes the Deﬁnition 12.8 of a quadratic

bound.

It should be pointed out that the existence of the above-mentioned solution

on an arbitrary ﬁnite time interval was previously known for single-valued

quadratic ﬁelds a on manifolds that correspond to vector ﬁelds of geodesic

sprays of connections on tangent bundles. In the latter case, applying a linear

change of time along the solution on a given time interval, one obtains a so-

lution on another time interval and by this method a solution on an arbitrary

ﬁnite interval can be constructed. This approach cannot be extended to the

general set-valued case with estimate (12.5).

294 12 Accessible Points and Sub-Manifolds of Mechanical Systems. Controllability

We begin the proof of the above-mentioned property for inclusions satis-

fying (12.5) with two technical statements.

Lemma 12.17 For 0 <δ<

(ε+C)

and for any t>0 both roots K

1,2

of the

equation δ(Kt + Ct

−1

)

= K are positive.

Proof. Transform the equation δ(Kt + Ct

−1

)

= K into the form (δt

)K +

(2Cδ−1)K+C

−2

δ = 0. Its discriminant is equal to D =1−4Cδ. This means

that for δ<

the roots are real and take the form K

1,2

1−2Cδ±

√

1−4Cδ

2δt

Since (1 − 2Cδ) >

√

1 −4Cδ2δt

,wehaveK

1,2

> 0. But, as pointed out in

Remark 3.49, ε<C and so

(ε+C)

. 

Lemma 12.18 For 0 <δ<

(ε+C)

and for all t>0 the inequality t

−1

ε>

1−2Cδ−

√

1−4Cδ

2δt

holds.

Proof. In order to prove this statement, consider the following system



δ<

1−2Cδ−

√

1−4Cδ

2δ

<ε.

By means of elementary transformations, taking into account Remark 3.49,

this system can be transformed into the following form

⎧

⎨

⎩



δ<

+2Cε+C

δ ≥

2(ε+C)

δ<

Since by Remark 3.49 ε<C, we obtain that δ<

(ε+C)

. 

Theorem 12.19 Let a(t, m, X) have convex closed bounded values satisfy the

upper Carath´eodory condition and the estimate (12.5) for some continuous

function a(t, m) > 0. Let the points m

and m

be non-conjugate along a

geodesic g(·) of the Levi-Civit´a connection and let the estimate a(t, m) <δ

hold on I × Ξ, where the compact set Ξ is as in Lemma 3.51 and δ>0

satisﬁes the inequality δ<

(ε+C)

. Then for any t

> 0, t

∈ I there exists a

solution m(t) of the inclusion (11.17) for which m(0) = m

and m(t

)=m

Proof. For a C

-curve γ(t)=Sv(t), v(·) ∈ C

(I,T

M), consider the set-

valued vector ﬁeld a(t, γ(t), ˙γ(t)). Denote by Γ the operator of parallel trans-

lation of vectors along γ(·) at the point γ(0) = m

. Apply the operator Γ

to all sets a(t, γ(t), ˙γ(t)) along γ(·). As a result for any v ∈ C

(I,T

M)we

obtain a set-valued map Γ aSv :[0,l]  T

M that has convex values. As in

the proof of Theorem 12.11 the map ΓaS : C

([0,l],T

M) ×[0,l]  T

satisﬁes the upper Carath´eodory condition. Consider the operator



PΓ aS

from the proof of Theorem 12.11. This operator is upper semicontinuous, has

convex values and sends bounded sets from C

([0,t

],T

M)tocompact

sets.

12.3 Existence of Solutions 295

Let 0 <t

<l. Taking into account the hypotheses of the Theorem,

we obtain from Lemma 12.17 that K =

1−2Cδ−

√

1−4Cδ

2δt

is positive and from

Lemma 12.18 that t

−1

ε>Kt

. Consider the ball U

of radius Kt

centered

at the origin in the Banach space C

([0,t

],T

M). Since t

−1

ε>Kt

,by

Lemma 6.27 for any v(·) ∈ U

the vector C

is well-deﬁned. Thus we can

introduce the operator B from the proof of Theorem 12.11.Thisoperator

is also upper semi-continuous, convex-valued and sends bounded sets from

([0,t

],T

M) to compacts sets.

Since parallel translation preserves the norms of vectors, from the con-

struction of S and from the hypothesis we derive that for any v ∈ U

and

t ∈ [0,t

] the estimate



a(t, S(v(t)+C

dτ

S(v(t)+C

))



<δv(t)+C



holds. By construction δv(t)+C



≤ δ(Kt

+ Ct

−1

)

= K. Since parallel

translation is norm-preserving, for any curve u(t) ∈Zv(t)andforanyt ∈

[0,t

] the inequality u(t)≤Kt ≤ Kt

holds. Thus B sends the ball U

into itself and from the Bohnenblust-Karlin ﬁxed point theorem it follows

that B has a ﬁxed point u

∗

∈ U

, i.e. u

∗

∈Bu

∗

. The fact that that m(t)=

S(u

∗

(t)+C

∗

) is the desired solution is proved by analogy with the proof of

Theorem 12.11. 

Theorem 12.20 Let a(t, m, X) be almost lower semicontinuous, have closed

bounded values and satisfy (12.5) with a continuous function a(t, m) > 0.Let

the points m

and m

be non-conjugate along a geodesic g(·) of the Levi-Civit´a

connection and let the estimate a(t, m) <δhold on I ×Ξ, where the compact

set Ξ is as in Lemma 3.51 and δ>0 satisﬁes the inequality δ<

(ε+C)

.Then

for any t

> 0, t

∈ I there exists a solution m(t) of the inclusion (11.17)

for which m(0) = m

and m(t

)=m

Proof. Here we use the same notation as in the proof of Theorem 12.19.From

the hypothesis it follows that for all v ∈ C

([0,l],T

M) the curves from

PΓ aSv are integrable. Hence the set-valued map PΓ aS sends C

([0,l],T

into L

(([0,l], A,μ),T

M), as in the proof of Theorem 12.14,andtheoper-

ator

PΓ aS : C

([0,l],T

M)  L

(([0,l], A,μ),T

is lower semicontinuous and has decomposable values. Then by the Bressan-

Colombo Theorem (Theorem 4.9) it has a continuous selector, which we de-

note by pΓ aS.

Let t

and K be as in the proof of Theorem 12.19. Then on the ball

⊂ C

([0,t

],T

M) the operator B from the proof of Theorem 12.14 is

well-deﬁned and is completely continuous. Since parallel translation preserves

the norm of a vector, from the construction of S, from Lemma 12.17 and from

Lemma 12.18, for any u ∈ U

with given a we get