Waźewski Topological Principle and V-bounded Solutions of Nonlinear Systems

0.16 MB·English

by  Volodymyr Lagoda

Wa´zewski Topological Principle
9
and V-bounded Solutions of Nonlinear Systems
0
0
2

n Volodymyr Lagoda, Igor Parasyuk
∗ †
a
J
January 2, 2009
2

]
A
C
. Abstract
h
t
a We use the Wa´zewski topological principle to establish a number of new sufficient
m
conditions for the existence of proper (defined on the entire time axis) solutions of
[
essentially nonlinear nonautonomous systems. The systems under consideration are

1 characterized by the monotonicity property with respect to a certain auxiliary guiding
v
functionW(t,x)dependingontimeandphasecoordinates. Anotherauxiliaryfunction
4
3 V(t,x),whichispositivelydefinedinthephasevariablesxforanyt,isusedtoestimate
2 the deviation of the proper solutions from the origin.
0
.
1
0 1 Introduction
9
0
: The goal of this paper is to lay down sufficient conditions under which the nonlinear
v
nonautonomous system of ODEs
i
X
r x˙ = f(t,x) (1)
a
where f : Ω Rn (Ω R1+n) has a solution x(t) extendable on the entire time
7→ ⊂
axis and possessing the property that a given positively definite (with respect to x-
variables) function V(t,x) is bounded along the graph of x(t). We especially focus on
getting estimates for the function V(t,x(t)). The main results are obtained by using
the Wa´zewski topological principle [1, 2, 3, 4], and some of them generalize the results
of V. M. Cheresiz [5].
∗National Taras Shevchenko Univesity of Kyiv, Volodymyrs’ka 64, Kyiv, 01033,Ukraine
†ibid.
1
It should be noted that the Wa´zewski topological principle was successfully explo-
ited for proving the existence of bounded solutions to some boundary value problems
in [6] and to quasihomogeneous systems in [7, 8] (see also a discussion in [9]).
To apply the the Wa´zewski principle, along with the function V which can be
naturally considered as an analogue of time-dependent norm, weuse another auxiliary
functionW(t,x). Ingeneralcase,thisfunctionisasign-changingone,butitmusthave
positively definite derivative by virtue of the system (1) in the domain where V v
0
≥
for some constant v > 0. We call V and W the estimating function and the guiding
0
function respectively and we say that together they form the V–W-pair of the system.
Note that the term ”guiding function” we borrow from [10] (originally — ”guiding
potential”). Basically topological method of guiding functions, which was developed
by M. A. Krasnosel’ski and A. I. Perov, is an effective tool for proving the existence
of bounded solutions of essentially nonlinear systems too (see the bibliography in
[10, 11]). But, except [9, 13], in all papers known to us only independent of time
guiding functions were used.
In [5], the role of V–W-pair plays the square of Euclidean norm together with
an indefinite nondegenerate quadratic form. It appears that in this case sufficient
conditions for the existence of bounded solutions as well as the estimates of their
norms coincide with those obtained by means of technique developed in [14, 15] for
indefinitely monotone (not necessarily finite dimensional) systems.
Weshallnotmention hereanotherinterestingapproaches instudyingtheexistence
problem of bounded solutions to nonlinear systems, because they have not been used
in this paper. For the corresponding information the reader is referred to [16, 17, 18,
19, 20, 21, 22, 23, 24, 25, 26, 27, 28].
This paper is organized as follows. Section 2 contains necessary definitions, in
particular, the notion of V–W-pair is introduced and some additional conditions im-
posed on estimating and guiding functions are described. In section 3 we prove two
main theorems concerning the existence and the uniqueness of V-bounded solution
to a nonlinear nonautonomous system possessing V–W-pair. Finally, in section 4 we
show how the results of section 3 can be applied in the case where the estimating and
guiding functions are nonautonomous quadratic forms. In this connection it should
be pointed out that guiding quadratic forms play an important role in the theory of
linear dichotomous systems with (integrally) bounded coefficients [29, 30, 31].
2 The definition of V–W-pair and the main
assumptions
Let Ω be a domain of R1+n = t R x Rn such that the projection of Ω on the
{ ∈ }×{ ∈ }
time axis t R covers all this axis, and let in the system (1) f() C(Ω Rn). It
{ ∈ } · ∈ 7→
2
will be always assumed that each solution of the system has the uniqueness property.
Definition 1. A function V() C1(R Rn R ) of variables t R, x Rn will be
+
called the estimating function·, i∈f for any×t R7→the function V () :=∈ V(t, )∈:Rn R
t +
∈ · · 7→
is positively definite, has a unique critical point, the origin, and satisfies the condition
lim V (x) = .
kxk→∞ t
∞
Note that, as is well known, for any t R and each c > 0 the set V−1([0,c]) :=
∈ t
x Rn : 0 V (x) c is compact, its boundary is a closed connected hypersurface
t
{ ∈ ≤ ≤ }
V−1(c) surrounding the origin, and in addition, if c > c , then the set V−1([0,c ]) is
t 2 1 t 1
a proper subset of the set V−1([0,c ]).
t 2
Definition 2. A global solution x(t), t I of the system (1) is said to be V-bounded
∈
if sup V(t,x(t)) < .
t∈I ∞
For U() C1(Ω R) we put
· ∈ 7→
∂V ∂V
U˙ := + f.
f
∂t ∂x ·
Definition 3. For the system (1), a function W() C1(Ω R) will be called the
· ∈ 7→
guiding function concordant with V if for some v > 0 such that Ω V−1 [v , ) = ∅
0 0
∩ ∞ 6
there exist functions
(cid:0) (cid:1)
a() C Ω V−1 [v , ) (0, ) ,
0
· ∈ ∩ ∞ 7→ ∞
G() C([v , ) (0, )), g() C([v , ) (0, ))
0 (cid:0) (cid:0) (cid:1) 0 (cid:1)
· ∈ ∞ 7→ ∞ · ∈ ∞ 7→ ∞
satisfying the inequalities
V˙ (t,x) a(t,x)G(V(t,x)) (t,x) Ω V−1 [v , ) , (2)
f 0
≤ ∀ ∈ ∩ ∞
(cid:12)W˙f(t,x(cid:12)) a(t,x)g(V(t,x)) (t,x) Ω V−1((cid:0)[v0, ))(cid:1), (3)
(cid:12) (cid:12) ≥ ∀ ∈ ∩ ∞
g(v) g(v ) > 0 v v . (4)
0 0
≥ ∀ ≥
Definition 4. For the system (1), the estimating function V and the concordant
guiding function W will be called the V–W-pair of this system.
Define
v
F(v) := g(u)/G(u) du.
Zv0
(cid:0) (cid:1)
On the half-line v v , this function is monotonically increasing and has the inverse
0
≥
F−1() : [0, ) [v , ).
0
· ∞ 7→ ∞
3
Denote by Π := t Rn the ”vertical” hyperplane in R1+n, and in so far suppose
t
{ }×
thatthesystem(1)hasV–W-pair whichsatisfiesthefollowing additionalconditions:
(A): lim F(v) = ;
v→∞
0 ∞∞
(B): α(s)ds = α(s)ds = , where
−∞ 0 ∞
R R
α(t) := inf a(t,x) : x Ω , V (x) > v ;
t t 0
{ ∈ }
(C): there exist numbers w+, w (w+ > w ) such that
− −
V−1 [0,v ) W−1 (w ,w+) , V−1(v ) Ω,
0 − 0
⊂ ⊂
andinaddition, for(cid:0)anyt (cid:1) Rthen(cid:0)umberw+(cid:1)belongstotherangeofW () :=
t
∈ ·
W(t, ) : Ω R where Ω := Π Ω.
t t t
· 7→ ∩
(D): the domain , which is defined as such a connected component of the set
W
W−1 w ,w+) that contains V−1 [0,v ) , has the property: for any suffi-
− 0
ciently large by absolute value negative t there exists a set [∂
t t t
(cid:0) (cid:0) (cid:1) M ⊂ W ∪ W ∩
W−1(w+)], where := Π , such that the set ∂ W−1(w+)= ∅
t Wt W∩ t Mt∩ Wt∩ t 6
is a retract of ∂ W−1(w+), but is not a retract of , and, besides,
∪s≥t Ws∩ s Mt
liminfsup V (x) : x = ν < .
t t
t→−∞ { ∈ M } ∞
Remark 1. If V−1 [0,v ] Ω and W(V−1(v )) [w ,w+], then one can redefine
0 0 −
⊂ ∈
the guiding function in the domain V−1 [0,v ) in such a way that V−1 [0,v )
0 0
(cid:0) (cid:1) ⊂
W−1 (w ,w+) .
−
(cid:0) (cid:1) (cid:0) (cid:1)
Rema(cid:0)rk 2. The(cid:1)condition (D) is fulfilled if for any negative sufficiently large by abso-
lute value t there exists a finite collection of compact manifolds with border
t,j
{M }
such that: ∂ ∂ = ∅, j = k; the interior of belongs to ; the set
t,j t,k t,j t
M ∩ M 6 M W
∂ is a retract of ∂ W−1(w+) and
∪j≥1 Mt,j ∪s≥t Ws∩ s
liminfmax V (x) :x = ν < .
t j≥1 t,j
t→−∞ { ∈∪ M } ∞
In fact, in this case, taking into account that any compact manifold can not be re-
tracted to its border, it is sufficient to put := .
t j≥1 t,j
M ∪ M
Remark 3. Since the set W−1(w+) is not empty and in any point of this set the
Mt∩ t
function V takes values not less than v , we get the inequality ν v .
0 0
≥
4
3 The existence and the uniqueness
of V-bounded solution
The lemma given below open the door to estimation of solutions of the system (1) by
means of functions V in the presence of V–W-pair.
Lemma 1. Suppose that the system (1) has V–W-pair satisfying the condition (A).
Let this system has a global solution x(t), t I R, such that
∈ ⊆
W∗ := supW(t,x(t)) < , W := inf W(t,x(t)) >
∗
t∈J ∞ t∈J −∞
J := t I :V(t,x(t)) > v .
0
{ ∈ }
Then for any t I, in the case where V(t ,x(t )) v and J = ∅, the following
0 0 0 0
∈ ≤ 6
inequality holds true
V(t,x(t)) F−1 W∗ W t I [t , ) (5)
0 0
≤ − ∀ ∈ ∩ ∞
where (cid:0) (cid:1)
W = inf W(t,x(t)) : t I, V(t,x(t)) = v .
0 0
{ ∈ }
If V(t ,x(t )) > v , then in the case where V(t,x(t)) > v for all t I [t , ),
0 0 0 0 0
∈ ∩ ∞
we have
V(t,x(t)) F−1 F V(t ,x(t )) +W∗ W(t ,x(t )) , (6)
0 0 0 0
≤ −
and otherwise (cid:0) (cid:0) (cid:1) (cid:1)
V(t,x(t)) max F−1 F V(t ,x(t )) +W0 W(t ,x(t )) ,F−1 W∗ W ,
0 0 0 0 0
≤ − −
(7)
(cid:8) (cid:0) (cid:0) (cid:1) (cid:1) (cid:0) (cid:1)(cid:9)
where
W0 = sup W(t,x(t)) : t I, V(t,x(t)) = v .
0
{ ∈ }
In addition, if [t ,t∗] I is such a segment that (t ,t∗) J and
∗ ∗
⊂ ⊂
V t ,x(t ) = V t∗,x(t∗) = v ,
∗ ∗ 0
then (cid:0) (cid:1) (cid:0) (cid:1)
1
V(t,x(t)) F−1 W t∗,x(t∗) W t ,x(t ) t [t ,t∗]. (8)
∗ ∗ ∗
≤ 2 − ∀ ∈
(cid:18) (cid:19)
h i
(cid:0) (cid:1) (cid:0) (cid:1)
If the condition (B) is fulfilled and [t ,θ] I, where θ is determined by the equality
0
⊂
θ
α(s)ds = (W∗ W )/g(v ), (9)
∗ 0
−
Zt0
then there exists τ [t ,θ] for which V(τ,x(τ)) v .
0 0
∈ ≤
5
Proof. Letthecondition (A) is fulfilled. Throughoutthis proof,putv(t) := V(t,x(t)).
Then in view of (2), (3) we have
d g(v(t))v˙(t) d
F(v(t)) = | | W(t,x(t)) t J. (10)
dt G(v(t)) ≤ dt ∀ ∈
(cid:12) (cid:12)
(cid:12) (cid:12)
If v(t0) v0 (cid:12)(cid:12)and J = ∅(cid:12)(cid:12), then there exists an interval (t∗,T) J [t0, ) such
≤ 6 ⊂ ∩ ∞
that v(t ) = v . Since F(v(t )) = 0, then, as a consequence of (10) and inequality
∗ 0 ∗
W(t ,x(t )) W , we have
∗ ∗ 0
≥
F(v(t)) W(t,x(t)) W(t ,x(t )) W∗ W t [t ,T],
∗ ∗ 0 ∗
≤ − ≤ − ∀ ∈
and from this it follows that v(t) F−1(W∗ W ) for all t [t ,T]. Taking into
0 ∗
≤ − ∈
account that v = F−1(0) F−1(W∗ W ) and the function F−1() is monotonically
0 0
≤ − ·
increasing, one ascertains that (5) is true for all t [t , ) I.
0
∈ ∞ ∩
If now v(t ) > v , then until v(t) > v we have
0 0 0
F(v(t)) F(v(t )) W(t,x(t)) W(t ,x(t )).
0 0 0
− ≤ −
In the case where v(t) > v for all t [t , ) I, we obtain the inequality (6).
0 0
∈ ∞ ∩
Otherwise there exists the nearest to t moment t > t such that v(t ) = 0. Then
0 ∗ 0 ∗
W(t ,x(t )) W0, and on the segment [t ,t ], we arrive at
∗ ∗ 0 ∗
≤
F(v(t)) F(v(t )) W0 W(t ,x(t ))
0 0 0
− ≤ − ⇔
v(t) F−1 F(v(t ))+W0 W(t ,x(t )) .
0 0 0
≤ −
(cid:0) (cid:1)
Taking into account the estimate obtained above for v(t) on the segment [t ,T], one
∗
ascertains that the inequality (7) holds true.
Now let us estimate v(t) on [t ,t∗] I under the condition that (t ,t∗) J and
∗ ∗
⊂ ⊂
v(t ) = v(t∗) = v . Let tˆbe a point at which v(t) reaches its maximum on [t ,t∗].
∗ 0 ∗
Then from the inequality (10) it follows that
tˆg(v(t))v˙(t) t∗ g(v(t))v˙(t)
W(t∗,x(t∗)) W(t ,x(t )) | | dt+ | | dt
∗ ∗
− ≥ Zt∗ G(v(t)) Ztˆ G(v(t)) ≥
2F(v(tˆ)) F(v(t )) F(v(t∗)) = 2F(v(tˆ)) 2F(v(t)) t [t ,t∗],
∗ ∗
− − ≥ ∀ ∈
and we obtain the inequality (8).
Next, let the condition (B) is fulfilled and [t ,θ] I. Let us prove that there
0
⊂
exists a number τ belonging to [t ,θ] for which v(τ) v . Obviously, it is sufficient
0 0
≤
to consider the case where v(t ) > v . If we suppose the contrary, i.e. that v(t) > v
0 0 0
for all t [t ,θ], then we can find such a small ǫ > 0 that the inequality v(t) > v and
0 0
∈
6
thus the inequality dW(t,x(t)) α(t)g(v ) > 0 holds true for all t [t ,θ+ǫ] J.
dt ≥ 0 ∈ 0 ⊂
From this in virtue of (4) we arrive at inequality
θ θ
W∗ W(θ+ǫ,x(θ+ǫ)) > g(v ) α(s)ds+W(t ,x(t )) g(v ) α(s)ds+W
0 0 0 0 ∗
≥ ≥
Zt0 Zt0
which contradicts the definition of θ. Hence, there do exists a number τ [t ,θ] with
0
∈
the required property.
Remark 4. If it is impossible to find F−1() explicitly, then in order to obtain efficient
·
estimatesofsolutionsonecanreplacethefunctionF(v)byanotherappropriatestrictly
monotonic function F (v), which satisfies the inequality F (v) F(v) for all v > v
1 1 0
≤
and tends to infinity when v .
→ ∞
Put
w0(t) := max W (x) : x V−1(v ) ,
t ∈ t 0
w0(t) := min (cid:8)Wt(x) : x∈ Vt−1(v0)(cid:9),
ω := liminfw (t),
0 (cid:8) 0 (cid:9)
t→−∞
ω˜ := liminf inf W (x) : x , V (x) v .
t t t 0
t→−∞ { ∈ M ≥ }
(cid:0) (cid:1)
It is clear that the inequalities
w ω w+, w ω˜ w+
− 0 −
≤ ≤ ≤ ≤
holds true once the condition (D) is satisfied.
Now we are in position to prove the following statement.
Theorem 1. Assume that the system (1) has V–W-pair satisfying the conditions (A)-
(D). Let there exists a number V∗ such that
V∗ > max F−1 F(ν)+w+ ω˜ ,F−1(w+ ω ) , (11)
0
− −
and the set cls V−1 [0,V∗)(cid:8) (cid:0) (here cls mea(cid:1)ns the closure op(cid:9)eration) belongs to
∩W
the domain Ω. Then the system (1) has a V-bounded solution x (t), t R, which
∗
(cid:0) (cid:0) (cid:1) (cid:1) ∈
satisfies the inequality
1 w+ w
V(t,x(t)) F−1 supw0(s) infw (s) F−1 − − =:v t R.
0 ∗
≤ (cid:18)2 s≥t −s≤t (cid:19) ≤ (cid:18) 2 (cid:19) ∀ ∈
h i
(12)
7
Proof. By the conditions (C) and (D) the set ∂ W−1(w+) does not intersect the
W ∩
set V−1([0,v )). Then from the definition of guiding function it follows that the set
0
∂ W−1(w+) coincides with the set of exit points of integral curves of the system
W ∩
(1) from the domain and it consists of the strict exit points only.
W
By the condition (D) we can choose a sequence of moments t , j ,
j
→ −∞ → ∞
and a sequence of sets [∂ W−1(w+)] in such a way that
Mtj ⊂ Wtj ∪ Wtj ∩ tj
sup V (x) : x ν +δ, inf W (x) :x , V (x) v ω˜ δ,
{ tj ∈ Mtj} ≤ { tj ∈Mtj tj ≥ 0} ≥ −
V∗ > max F−1(F(ν +δ)+w+ ω˜ +δ),F−1(w+ ω ) ν +δ > v
0 0
{ − − } ≥
for sufficiently small δ > 0 and for all j, and the intersection of each with
Mtj
∂ W−1(w+) be the retract for the set of exit points from (t,x) : t t but
j
W ∩ W ∩{ ≥ }
there does not exist a retraction of on ∂ W−1(w+). Then by Wa´zewski
Mtj Mtj∩ W∩
principle for any j there exists a point (t ,x ) such that the nonextendable
j 0j ∈ Mtj
solutionx (t), t I ,whichsatisfiestheinitialconditionx (t ) = x hastheproperty
j j j j 0j
∈
(t,x (t)) t [t , ) I .
j j j
∈ W ∀ ∈ ∞ ∩
Observe that V(t ,x ) ν +δ, and thus by the lemma 1 setting I = [t , ) I ,
j 0j j j
≤ ∞ ∩
v (t) = V(t,x (t)) we obtain v (t) < V∗, t I. Hence, taking into account the
j j j
∈
condition of the theorem we have
(t,x (t)) cls V−1 [0,V∗) Ω, t [t , ) I .
j j j
∈ ∩W ⊂ ∈ ∞ ∩
In view of this we conclude t(cid:0)hat [(cid:0)t , ) (cid:1)I . (cid:1)
j j
∞ ⊂
Next, applying the lemma 1 again, we can find τ for which v (τ ) v , and from
j j j 0
≤
(B) it follows that τ , j . Besides, if there exists at least one t τ for
j j
→ −∞ → −∞ ≥
which v (t) > v , then there exist moments t , t∗ such that τ t < t, t < t and
j 0 ∗ j ∗ ∗
≤
v (t ) = v (t∗) = v , but v (t) > v for t (t ,t∗). Then in virtue of inequality (8),
j ∗ j 0 j 0 ∗
∈
for any pair of such moments we have
w0(t∗) w (t )
v (t) F−1 − 0 ∗ t [t ,t∗], (13)
j ∗
≤ 2 ∀ ∈
(cid:18) (cid:19)
and thus v (t) v for all t τ .
j ∗ j
≤ ≥
Now one can provetheexistence of V-boundedsolution x (t)by theknownscheme
∗
(see, e.g., [5, 8, 10]. Namely, if we denote by x(t,t ,x ) the solution which for t = t
0 0 0
takes the value x , then setting ξ := x (0), we obtain the equalities
0 j j
x (t) = x(t,0,x (0)) = x(t,0,ξ ), t [t , ).
j j j j
∈ ∞
Having selected from the sequence ξ cls V−1([0,v ]) Ω a subsequence
j ∈ 0 ∗ ∩W0 ⊂ 0
converging to x , put x (t) := x(t,0,x ). Using the reductio ad absurdum reasoning
∗ ∗ ∗
(cid:0) (cid:1)
8
it is easy to show that on the maximal existence interval I of this solution we have
the inclusion
(t,x (t)) cls(V−1([0,v ]) ).
∗ ∗
∈ ∩W
Therefore I = R and V(t,x (t)) v for all t R. Finally, taking into account (13),
∗ ∗
≤ ∈
we arrive at the inequality (12).
Remark 5. As is easily seen from the proof of the Theorem 1, it is sufficient to require
that the inequalities (2),(3) hold true on the set cls V−1 [v ,V∗) only, and the
0
∩W
inequalities (4) — for v [v ,V∗] only.
0
∈ (cid:0) (cid:0) (cid:1) (cid:1)
Theorem 2. Let Ω˜ be a subdomain of the domain Ω and let
Ω∗ := (t,z) R Rn : z = x y, (t,x) Ω˜, (t,y) Ω˜ .
{ ∈ × − ∈ ∈ }
Suppose that there exist functions U() C1(Ω∗ R), H(),h() C(R R ),
+ +
· ∈ 7→ · · ∈ 7→
b(),β() C(R (0, )) such that:
· · ∈ 7→ ∞
1) the function h() is nondecreasing, the function H() is strictly monotonically
· ·
increasing, and in addition,
1 t u
limsup β(t)h H−1 dt =
b(t) ◦ b(t) ∞
t→±∞ (cid:12)Z0 (cid:18) (cid:19) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
for any u> 0;
(cid:12) (cid:12)
2) for all (t,x),(t,y) Ω˜, the following inequalities hold true
∈
U(t,x y) b(t)H(V(t,x y)),
| − |≤ −
U′(t,x y)+U′(t,x y) (f(t,x) f(t,y)) β(t)h(V(t,x y)),
t − x − · − ≥ −
where V(t,x) is an estimating function. Then the system (1) cannot have two different
nonextendable solutions x(t), y(t), t R, whose graphs lie in Ω˜ and which have the
∈
property
supV(t,x(t) y(t)) < . (14)
t∈R − ∞
Proof. Suppose that the system (1) has a pair of solutions x(t), y(t), t R such that
∈
(t,x(t)), (t,y(t)) Ω˜ for all t R. Let us show that for these solutions the condition
∈ ∈
(14) fails.
Consider the functions u(t) := U(t,x(t) y(t)), v(t) := V(t,x(t) y(t)). By
− −
condition, the function u() does not decrease. Hence, there exist (either finite or
·
infinite) limits u = lim u(t), u∗ = lim u(t). If we suppose that x(t) y(t),
∗ t→−∞ t→∞
6≡
then there exists t such that x(t ) = y(t ), from whence v(t )> 0 and u˙(t ) > 0. For
0 0 0 0 0
6
this reason, u∗ >u(t ) > u .
0 ∗
9
First suppose that u 0. Then u(t) > u(t ) > 0 for t > t . Since
∗ 0 0
≥
v(t) H−1(u(t )/b(t)),
0
≥
then
t u(t )
u(t) u(t )+ β(s)h H−1 0 ds
0
≥ ◦ b(s)
Zt0 (cid:18) (cid:19)
and
1 t u(t )
H(v(t)) β(s)h H−1 0 ds, t t .
0
≥ b(t) ◦ b(s) ≥
Zt0 (cid:18) (cid:19)
Thus, limsup v(t) = .
t→∞ ∞
Now suppose that u < 0. Then there exists t′ such that u(t′) < 0. Then u(t)
∗
≤
u(t′) for all t < t′ and v(t) H−1(u(t′)/b(t)) for t < t′. Then
≥ | |
t′ u(t′)
u(t′) u(t) β(s)h H−1 | | ds, t t′,
− ≥ ◦ b(s) ≤
Zt (cid:18) (cid:19)
from whence, as above, we have limsup v(t) = .
t→−∞ ∞
4 Studying V-bounded solutions by means of
pair of quadratic forms
Consider the case where the V–W-pair of the system (1) is a pair of quadratic forms
V(t,x) = B(t)x,x , W(t,x) = C(t)x,x , (15)
h i h i
where , isascalarproductinRn, B(t) and C(t) arefamiliesofsymmetric
h· ·i { }t∈R { }t∈R
nondegenerate operators in Rn smoothly depending on parameter t and satisfying the
conditions:
(a): for any t R, the operator B(t) is positively definite;
∈
(b): for any t R, there exist projectors P (t),P (t) on corresponding invariant
+ −
∈
subspaces L (t),L (t) of operator C(t) such that the restriction of C(t) on
+ −
L (t) (on L (t)) is a positively definite (negatively definite) operator.
+ −
Observe that since the subspaces L (t), L (t) are mutually orthogonal then the
+ −
projectors P (t), P (t) are symmetric.
+ −
From C(t)-invariance of these subspaces it follows that P (t)C(t) = C(t)P (t)
± ±
and, as a consequence, we have the representation
C(t)= (P (t)+P (t))C(t)(P (t)+P (t)) = P (t)C(t)P (t)+P (t)C(t)P (t).
+ − + − + + − −
10