Elliptic Quadratic Operator Equations

0.4 MB·

by  Rasul Ganikhodjaev

ELLIPTIC QUADRATIC OPERATOR EQUATIONS
RASULGANIKHODJAEV,FARRUKHMUKHAMEDOV,ANDMANSOORSABUROV
Abstract. In thepresent paper is devoted to the study of elliptic quadratic operator equations over
thefinitedimensionalEuclideanspace. Weprovidenecessaryandsufficientconditionsfortheexistence
7 ofsolutionsofellipticquadraticoperatorequations. TheiterativeNewton-Kantorovichmethodisalso
1 presented for stable solutions.
0
2 Mathematics Subject Classification 2010: 47H60, 47J05, 52Axx, 52Bxx.
Key words: Ellipticoperator; quadraticoperator; numberofsolutions; rankof ellipticoperator, stable
n
solution, Newton-Kantorovich method.
a
J

8

]
A Contents
F 1. Introduction 1
.
h 1.1. Hammerstein integral equations 1
at 1.2. Summary of the main results 3
m 2. A classification of quadratic operators 6
[ 3. Examples 9

4. The necessary condition for an existence of solutions 10
1
v 4.1. Auxiliary results 11
0 4.2. The proof of Theorem 4.1. 14
9
4.3. Some examples 15
9
5. The sufficient condition for an existence of solutions 15
1
0 5.1. The lower dimensional space 16
1. 5.2. The higher dimensional space 16
0 5.3. Some examples 20
7
6. An iterative method for stable solutions 22
1
6.1. The Newton-Kantorovich method 22
:
v
6.2. Some examples 24
i
X 7. The rank of elliptic quadratic operators 25
r 7.1. The rank of k 26
a
7.2. Some examples: Lower ranks 28
7.3. Some examples: Higher ranks 29
8. Elliptic quadratic operator equation of rank 1 30
References 35
1. Introduction
1.1. Hammerstein integral equations. A nonlinear Hammerstein integral equation appeared
(1.1) x(t)= K (t,s,u)x(s)x(u)dsdu+ K (t,s)x(s)ds+f(t)
1 2
Z Z Z
Ω Ω Ω
inseveralproblemsofastrophysics,mechanics,andbiology, whereK : Ω Ω Ω R,K : Ω Ω R,
1 2
and f : Ω R are given functions and x : Ω R is an unknown function×. G×ene→rally, in orde×r to→solve
→ →
1
2 RASULGANIKHODJAEV,FARRUKHMUKHAMEDOV,ANDMANSOORSABUROV
the nonlinear Hammerstein integral equation (1.1) over some functions space, one should imposesome
constrains for the functions K (, , ),K (, ), and f(). For instance, by using contraction methods,
1 2
· · · · · ·
some sufficient conditions were obtained for the existence of solutions of the integral equation (1.1)
over the space C(Ω) of continuous functions (see [14], [33, 34], [44], [53]). It is worth of noting that,
unlike a linear integral equation (i.e. K (t,s,u) 0), in general, the nonlinear Hammerstein integral
1
≡
equation (1.1) may have many solutions.
Particularly, if K and K are Goursat’s degenerate kernels, i.e.
1 2
n
(1.2) K (t,s,u) = a (s)b (u)c (t),
1 i j k
i,j,k=1
X
n
(1.3) K (t,s) = d (s)e (t),
2 i j
i,j=1
X
where a (), b (), c (), d (), e () are given functions then we have that
i i i i i
· · · · ·
n n
x(t) = a (s)x(s)ds b (u)x(u)du c (t)+ d (s)x(s)ds e (t)+f(t).
i j k i j
    
i,j,k=1 Z Z i,j=1 Z
X Ω Ω X Ω
    
Let
a (s)x(s)ds = x , b (s)x(s)ds = x , d (s)x(s)ds = x , i,j,k = 1,n.
i i j n+j k 2n+k
Z Z Z
Ω Ω Ω
In this setting, the solution of the nonlinear Hammerstein integral equation (1.1) takes the following
form
n n
x(t)= x x c (t)+ x e (t)+f(t),
i n+j k 2n+i j
i,j,k=1 i,j=1
X X
where x = (x , ,x ,x , ,x ,x , ,x ) R3n is a solution of the following quadratic
1 n n+1 2n 2n+1 3n
··· ··· ··· ∈
operator equation
3n 3n
(1.4) A x x + B x +C = 0, k = 1,3n.
ij,k i j ik i k
∀
i,j=1 i=1
X X
for suitable (A )3n , (B )3n , and (C )3n .
ij,k i,j,k=1 ik i,k=1 k k=1
Consequently, in order to find solutions of the nonlinear Hammerstein integral equation (1.1) with
Goursat’s degenerate kernels, we have to solve the quadratic operator equation (1.4) over R3m.
Let Q :Rn Rm be a quadratic operator
→
n n n
Q(x) = a x x , a x x , , a x x ,
ij,1 i j ij,2 i j ij,m i j
···
(cid:18)i,j=1 i,j=1 i,j=1 (cid:19)
X X X
where a R are structural coefficients and x = (x , ,x ) Rn. Without loss of generality, one
ij,k 1 n
∈ ··· ∈
can assume that a = a for any i,j = 1,n and k = 1,m. Let A = (a )n be a symmetric
ij,k ji,k k ij,k i,j=1
matrix for k = 1,m. In this case, the quadratic operator can be written in the following form
Q(x) = ((A x,x),(A x,x), ,(A x,x))
1 2 m
···
Let H (Q) be a real linear span of symmetric matrices A , ,A . We say that H (Q) is
n,m 1 m n,m
···
positive definite (resp. positive semidefinite) if there exists a positive definite(resp. a nonzero positive
semidefinite but not positive definite) matrix in it. We say that H (Q) is indefinite if every nonzero
n,m
matrix in it is indefinite. Let R (Q) = Q(x) : x Rn and W (Q) = Q(x) : x = 1 be
n,m n,m 2
the images of Rn and B(0) = x Rn : x{ = 1 , re∈spect}ively, under the qu{adratic kopkerator.}Let
2
Ker (Q) = x Rn : Q(x)={0 ∈be a kekrnkel of th}e quadratic operator.
n,m
{ ∈ }
ELLIPTIC QUADRATIC OPERATOR EQUATIONS 3
The study of convexity of the sets R (Q), W (Q) and on the relationship between the sets
n,m n,m
H (Q) and Ker (Q) are traced back to O. Toeplitz [54], F. Hausdorff [36], P. Halmos [35], C.A.
n,m n,m
Berger [27], R. Westwick [57], P. Finsler [31, 32], G.A. Bliss [55], W.T. Reid [51], A.A. Albert [3], E.J.
McShane [45], M. Hestenes [37, 38, 39], F. John [43], L. Dines [28, 29, 30], and many others (see also
[15]-[19]).
Let us consider the quadratic operator equation
(1.5) Q(x)+Ax+b = 0, x Rn
∈
where Q : Rn Rm is a quadratic operator and A : Rn Rm is a linear operator, and b Rm is a
→ → ∈
vector.
1.2. Summary of the main results. The main goal is to study the structure of the set
X (Q,A,b) = x Rn :Q(x)+Ax+b = 0
n,m
{ ∈ }
of solutions of quadratic operator equation.
Definition 1.1. A quadratic operator Q : Rn Rm is called
→
(i) elliptic (in short EQO) if there exists a linear continuous functional f : Rm R such that
→
f(Q(x)) is a positive definite quadratic form;
(ii) parabolic (in short PQO) if there exists a nonzero linear continuous functional f : Rm R
→
such that f(Q(x)) is a positive semidefinite but no positive definite quadratic form;
(iii) hyperbolic (in short HQO) if for any nonzero linear continuous functional f : Rm R the
→
quadratic form f(Q(x)) is indefinite.
Remark 1.1. It is clear that the quadratic operator Q : Rn Rm is elliptic (resp. parabolic,
→
hyperbolic) if and only if H (Q) is positive definite (resp. positive semidefinite, indefinite).
n,m
Consequently, by means of Dines’s result, we can fully describe all elliptic, parabolic, hyperbolic
quadratic operators. Namely, we have the following result.
Theorem 1.2. [28, 29, 30] Let Q : Rn Rm be a quadratic operator. The following statements hold
→
true:
(i) Q is elliptic if and only if tr(SA ) = 0 with S = ST for all i= 1,m implies that S is indefinite.
i
(ii) Q is parabolic if and only if there exists positive semidefinite S = ST with tr(SA ) = 0 for all
i
i= 1,m, but no such positive definite S.
(iii) Q is hyperbolic if and only if there exists positive definite S = ST with tr(SA ) = 0 for all
i
i= 1,m.
There is a strong relation between the convexity of the sets R (Q), W (Q) and the uniqueness
n,m n,m
of the set Ker (Q) whenever Q : Rn Rm is the elliptic quadratic operator and n m (see [1, 2],
n,m
→ ≥
[4]-[13], [20]-[26], [40, 41, 42], [46]-[50], [52],[56],[58]). In this survey paper, we are going to describe
the set X (Q,L,b) whenever Q :Rn Rm is the elliptic quadratic operator and n = m.
n,m
Let Q : Rn Rn be the elliptic quad→ratic operator. We know that, the quadratic form f(Q(x)) is
positive defini→te in Rn if and only if there exists a positive number α > 0 such that
f(Q(x)) α x 2,
≥ ·k k2
for any x Rn. Thus, Q : Rn Rn is the elliptic quadratic operator if and only if there exist a
continuous∈linear functional f : R→n R and a number α > 0 such that
→
f(Q(x)) α x 2, x Rn.
≥ ·k k2 ∀ ∈
Let K′ be a set of all continuous linear functionals f :Rn R such that the quadratic form f(Q(x))
Q →
is positive defined, i.e.,
K′ = f : α > 0, f(Q(x)) α x 2, x Rn .
Q ∃ f ≥ fk k2 ∀ ∈
It is clear that KQ′ 6= ∅. (cid:8) (cid:9)
4 RASULGANIKHODJAEV,FARRUKHMUKHAMEDOV,ANDMANSOORSABUROV
Proposition 1.3. If Q : Rn Rn is the elliptic quadratic operator then K′ is an open convex
→ Q
cone. Moreover, for any given minihedral cone K Rn there exists the elliptic quadratic operator
⊂
Q : Rn Rn such that K′ = K.
→ Q
For every f K′ , we define an ellipsoid
∈ Q
= x Rn :f(Q(x)+Ax+b) 0 ,
f
E { ∈ ≤ }
corresponding to f K′ . We define the following set
∈ Q
(Q,A,b) .
n f
E ≡ E
f∈K′
\Q
Theorem 1.4. If the equation (1.5) is solvable then (Q,A,b) = .
n
E 6 ∅
The following theorem gives more accurate description of the set X (Q,A,b).
n
Theorem 1.5. If the equation (1.5) is solvable then X (Q,A,b) Extr( (Q,A,b)).
n n
⊂ E
The following theorem gives a solvability criterion for the elliptic operator equation (1.5).
Theorem 1.6. The elliptic operator equation (1.5) is solvable if and only if ∂ = . Moreover,
f
f∈K′ E 6 ∅
Q
T
if the elliptic operator equation (1.5) is solvable then X (Q,A,b) = ∂ .
n f
f∈K′ E
Q
T
Let K be the set of extreme rays of the closed cone K′ . We define a set
Q
Π (K) = x Rn : f(Q(x)+Ax+b) 0, f K .
f
{ ∈ ≤ ∈ }
Proposition 1.7. One has that Π (K)= (Q,A,b).
f n
f∈K E
T
Theorem 1.8. Every vertex of (Q,A,b) is a solution of the elliptic operator equation (1.5).
n
E
A solution of the elliptic operator equation (1.5) which is the vertex of (Q,L,b) has a special
n
E
property among other solutions. We know that the set of all elliptic quadratic operators are closed
under the small perturbation.
Definition 1.2. A solution x of the elliptic operator equation (1.5) is called stable if for any ǫ > 0
0
there exists δ > 0 such that the perturbed elliptic operator equation Q˜(x)+A˜x+˜b = 0 has a solution
x˜ such that x˜ x < ǫ whenever Q˜ Q < δ, A˜ A < δ, ˜b b < δ.
0 0
k − k k − k k − k k − k
Theorem 1.9. A solution of the elliptic operator equation (1.5) is stable if and only if it is a vertex
of (Q,L,b).
n
E
We could speak more about the stable solutions of the elliptic operator equation (1.5).
Theorem 1.10. An elliptic operator equation (1.5) has an even (possibly, zero) number of stable
solutions.
We can also approximate the stable solutions of elliptic operator equation (1.5) by the Newton-
Kantorovich method. It is easy to check that the set
D= Rn Π (K)
f
\
(cid:18)f∈K (cid:19)
[
is an open set. Let D be a connected component of D and D be its closure.
0 0
Let P :Rn Rn be a mapping defined as P(x) := Q(x)+Ax+b for any x Rn.
→ ∈
ELLIPTIC QUADRATIC OPERATOR EQUATIONS 5
Theorem 1.11. If there exists x D such that D does not contain any straight line passing through
0 0 0
x then there exists a stable soluti∈on x of the elliptic operator equation (1.5) which belongs to D .
0 ∗ 0
Moreover, the inverse [P′(x )]−1 of the mapping P′(x ) exists and the sequence x ∞ defined as
0 0 { k}k=1
follows
x = x [P′(x )]−1P(x ), k = 0,1,...
k+1 k k k
−
converges to the stable solution x .
∗
We are aiming to classify the set of elliptic operators based on their ranks.
Let Q :Rn Rn be an elliptic quadratic operator. Let K be the set of extremal rays of K′ . Then,
Q
→
due to Krein-Milman theorem, we have that conv(K) = K′ , where conv(K) is a convex hull of K. Let
Q
rg Q stand for the rank of the quadratic form f(Q(x)). It is clear that the rank rg Q of the quadratic
f f
form f(Q(x)) is equal to the rank of the associated symmetric matrix A. Due to the construction of
the set K′ , one has that rg Q = n whenever f K′ and rg Q < n whenever f ∂K′ .
Q f ∈ Q f ∈ Q
Definition 1.3. The number
rgQ = maxrg Q
f
f∈K
is called a rank of the elliptic quadratic operator Q :Rn Rn.
→
It is clear that 1 rgQ n 1 for any elliptic quadratic operator. Moreover, if A,B are invertible
≤ ≤ −
matrices such that AQ(B()) also is an elliptic quadratic operator then rg(AQ(B)) = rgQ.
·
Definition 1.4. An elliptic quadratic operator Q : Rn Rn is called homogeneous of rank k, if one
has that rg Q = k for any f K. →
f
∈
′
We can describe the cone K for a homogeneous elliptic quadratic operators of order k.
Q
Theorem 1.12. Let Q be a homogeneous elliptic quadratic operator. One has that rgQ = 1 if and
only if K′ is a miniedral cone, i.e, K contains exactly n extremal rays. Moreover, if rgQ = 1 then
Q
there exist invertible matrices A,B such that AQ(Bx)= (x2,x2,...,x2).
1 2 n
Theorem 1.13. Let Q be a homogeneous elliptic quadratic operator. If rgQ 2 then K is an infinite
set. Moreover, if rgQ = n 1 then K = ∂K′ . ≥
Q
−
′
In general, it is a tedious work to describethe cone K of homogeneous elliptic quadratic operators
Q
with rank 2 rgQ n 2. It can be observed in some examples.
≤ ≤ −
We can provide some explicit sufficient conditions for the solvability of elliptic rank-1 operator
equation (1.5). Let Q : Rn Rn be an elliptic operator of the rank-1. Then, the elliptic operator
→
equation (1.5) can be written as follows
n
(1.6) x2 = a x +b ; k = 1,n.
k ki i k
i=1
X
Theorem 1.14. Let A = (a )n be a matrix such that a a 0 for all i ,i ,j = 1,n. If one
ij i,j=1 i1j · i2j ≥ 1 2
has that
n 2
(1.7) min a +4 min b > 0
ij i
i=1,n| | i=1,n
(cid:18)j=1 (cid:19)
X
then the elliptic operator equation (1.6) has at least two stable solutions.
Remark 1.15. In the case n = 1, the condition (1.7) coincides with the positivity of the discriminant
of the quadratic equation x2 = ax+b. Hence, the condition (1.7) isa necessary and sufficient condition
for the existence of two stable solutions whenever n= 1.
6 RASULGANIKHODJAEV,FARRUKHMUKHAMEDOV,ANDMANSOORSABUROV
2. A classification of quadratic operators
In this section we are going to classify quadratic operators into three classes and to study their
properties. In what follows, we shall consider quadratic operators on the finite dimensional Euclidian
space Rn.
Let B : Rn Rn Rn be a symmetric bilinear operator. A quadratic operator Q : Rn Rn is
× → →
defined as follows
Q(x) = B(x,x), x Rn.
∀ ∈
It is well-known that every quadratic operator Q : Rn Rn uniquely defines the symmetrical bilinear
operator B : Rn Rn Rn associated with the given→quadratic operator Q
× →
1 1
B(x,y) = [Q(x+y) Q(x y)]= [Q(x+y) Q(x) Q(y)].
4 − − 2 − −
Moreover, every quadratic operator Q : Rn Rn can be written in the coordinate form as follows
→
n n n
Q(x) = a x x , a x x , , a x x ,
ij,1 i j ij,2 i j ij,n i j
···
(cid:18)i,j=1 i,j=1 i,j=1 (cid:19)
X X X
where a R are structural coefficients and x = (x ,...,x ) Rn. Without loss any generality, one
ij,k 1 n
can assume∈that a = a . We denote the set of all quadrati∈c operators acting on Rn by Q .
ij,k ji,k n
Any quadratic operator Q : Rn Rn is bounded, i.e., there exists a positive number M > 0 such
→
that
Q(x) M x 2, x Rn,
k k ≤ ·k k ∀ ∈
and it is continuous. Let us define the norm of the quadratic operator Q by
Q = sup Q(x) .
k k k k
kxk≤1
It is clear that Q(x) Q x 2 for any x Rn.
k k ≤ k k·k k ∈
One can see that the set Q forms the n2(n+1) dimensional normed space with the quadratic
n 2 −
operator norm. We are going to classify quadratic operators into three classes.
Definition 2.1. A quadratic operator Q : Rn Rm is called
→
(i) elliptic (in short EQO) if there exists a linear continuous functional f : Rm R such that
→
f(Q(x)) is a positive definite quadratic form;
(ii) parabolic (in short PQO) if there exists a nonzero linear continuous functional f : Rm R
→
such that f(Q(x)) is a positive semidefinite but no positive definite quadratic form;
(iii) hyperbolic (in short HQO) if for any nonzero linear continuous functional f : Rm R the
→
quadratic form f(Q(x)) is indefinite.
We denote the sets of all elliptic quadratic, parabolic quadratic, and hyperbolicquadratic operators
acting on Rn by EQ , PQ , and HQ , respectively.
n n n
We know that, the quadratic form f(Q(x)) is positive defined in a finite dimensional vector space
if and only if there exists a positive number α> 0 such that
f(Q(x)) α x 2,
≥ ·k k
for any x Rn. Thus Q :Rn Rn is an EQO if and only if there exist a continuous linear functional
f : Rn ∈R and a number α>→0 such that
→
f(Q(x)) α x 2, x Rn.
≥ ·k k ∀ ∈
First of all, we shall study some basic properties of quadratic operators.
Let Q :Rn Rn be an EQO and K′ be a set of all continuous linear functionals f : Rn R such
→ Q →
that the quadratic form f(Q(x)) is positive defined, i.e.,
(2.1) K′ = f : α > 0, f(Q(x)) α x 2, x Rn .
Q f f
∃ ≥ k k ∀ ∈
(cid:8) (cid:9)
ELLIPTIC QUADRATIC OPERATOR EQUATIONS 7
Due to the definition of EQO we have that K′ = .
Q 6 ∅
We recall that a set K Rn is called a cone if λK K for any λ >0 and K ( K)= .
⊂ ⊂ ∩ − ∅
Proposition 2.1. If Q : Rn Rn is an EQO and K′ is defined by (2.1) then K′ is an open convex
→ Q Q
cone.
Proof. Let us prove that K′ is an open set. If f K′ then there exists α > 0 such that f (Q(x))
Q 0 ∈ Q 0 0 ≥
α x 2 for any x Rn. Since Q is bounded, we have
0
k k ∈
Q(x) M x 2,
k k ≤ ·k k
for some M > 0 and for any x Rn. If we take ε = α0 then for any linear functional f : Rn R
∈ 2M →
with f f < ε we get
0
k − k
f(Q(x)) = f (Q(x))+(f f )(Q(x)) α x 2 f f Q(x)
0 0 0 0
− ≥ k k −k − k·k k
α α
α x 2 0 M x 2 = 0 x 2.
0
≥ k k − 2M · ·k k 2 k k
This means that f K′ , and K′ is an open set.
∈ Q Q
Let us show that K′ is a convex set. If f ,f K′ then one has f (Q(x)) α x 2 for any x Rn
Q 1 2 ∈ Q i ≥ ik k ∈
where α > 0, i = 1,2. Let f = λf +(1 λ)f , 0 λ 1. We then get
i λ 1 2
− ≤ ≤
f (Q(x)) = λf (Q(x))+(1 λ)f (Q(x)) (λα +(1 λ)α ) x 2 min α ,α x 2.
λ 1 2 1 2 1 2
− ≥ − ·k k ≥ { }·k k
Hence, f K′ for any 0 λ 1, i.e., K′ is a convex set.
λ ∈ Q ≤ ≤ Q
It immediately follows from the definition of the set K′ that λK′ K′ for any λ > 0 and
Q Q ⊂ Q
K′ ( K′ )= . This means that K′ is a cone. (cid:3)
Q∩ − Q ∅ Q
Remark 2.2. It is worth mentioning that a closure K′ of K′ may not be a cone. For example, let
Q Q
us consider the following EQO on R2
Q(x) = (x2+x2,x2+x2).
1 2 1 2
Then K′ = (α,β) : α+β > 0 . However, the set K′ = (α,β) : α+β 0 is a semi-plane which
Q { } Q { ≥ }
is not a cone.
Recall, given a cone K Rn, we can define a partial ordering with respect to K by x y if
K K
y x K. The cone K is⊂called minihedral if sup x,y exists for≤any x,y Rn, where the sup≤remum
is−take∈n with respect to the partial ordering .{It is}well-known that K∈ Rn is a minihedral cone
K
≤ ⊂
if and only if it is a conical hull of n linear independent vectors, i.e.,
n
K = cone z ,...,z = x :x = λ z , λ 0 .
1 n i i i
{ } ≥
(cid:26) i=1 (cid:27)
X
Proposition 2.3. For any given minihedral cone K Rn∗ there exists an EQO such that K′ = K.
Q
⊂
Proof. Let
C = x Rn : f(x) 0, f K
{ ∈ ≥ ∀ ∈ }
be a dual cone to the given minihedral cone K. It is known that C is also a minihedral cone. Without
loss of generality, we may suppose that
C = cone e , ,e ,
1 n
{ ··· }
where e =(δ ,...,δ ) and
i 1i ni
1 if i =j
δ =
ij 0 if i =j.
(cid:26) 6
We define the quadratic operator Q :Rn Rn as follows
→
Q(x) =(x2,...,x2),
1 n
8 RASULGANIKHODJAEV,FARRUKHMUKHAMEDOV,ANDMANSOORSABUROV
n
where x = x e .
i i
i=1
Let f = (Pλ1,...,λ2) be a linear functional. Then a quadratic form
n
f(Q(x)) = λ x2,
i i
i=1
X
is positive defined if and only if λ > 0,...,λ > 0. Consequently,
1 n
K′ = (λ ,...,λ ) :λ > 0,...,λ > 0
Q 1 n 1 n
{ }
and
′
K = (λ ,...,λ ) :λ 0,...,λ 0 = cone f ,...,f ,
Q 1 n 1 n 1 n
{ ≥ ≥ } { }
where f = (δ ,...,δ ). Since f (e ) = δ we hence have K′ = K. (cid:3)
i 1i ni i j ij Q
Lemma 2.4. If K and K are open cones in Rn, n 2 then there exists a minihedral cone K such
1 2
≥
that
K intK = , K intK = ,
1 2
∩ 6 ∅ ∩ 6 ∅
where intK is an interior of K.
Proof. LetK and K beopencones. Thenwe can take y K ,i = 1,2 such that y and y arelinear
1 2 i i 1 2
independent. We complete these vectors y ,y up to a b∈ase y ,y , ,y of Rn. Then, it is easy
1 2 1 2 n
to see that the minihedral cone K = cone{y ,y}, ,y satisfi{es all c·o·n·ditio}ns of the lemma. (cid:3)
1 2 n
{ ··· }
Proposition 2.5. The set EQ is a path connected subset of Q whenever n 2.
n n
≥
Proof. Let Q , i = 0,1 be elliptic operators and K′ , i = 0,1 be the corresponding open cones. Due
i Qi
to Lemma 2.4 we can choose the minihedral cone K such that K′ intK = , i = 0,1. According
Qi ∩ 6 ∅
′
to Proposition 2.3 we can construct an elliptic operator Q such that K = K. We define a quadratic
Q
operator Q :Rn Rn as follows
λ
→
2λQ+(1 2λ)Q if 0 λ 1
Qλ = 2(1 λ)Q−+(2λ 0 1)Q if ≤1 <≤λ2 1.
(cid:26) − − 1 2 ≤
Let us show that Q is elliptic for any λ [0,1]. If f K′ intK and 0 λ 1 then one has
λ ∈ 0 ∈ Q0 ∩ ≤ ≤ 2
f (Q (x)) α x 2, α > 0 and f (Q(x)) β x 2, β > 0. Hence
0 0 0 0 0 0 0
≥ k k ≥ k k
f (Q (x)) = f (2λQ(x)+(1 2λ)Q (x)) (2λα +(1 2λ)β ) x 2 min α ,β x 2.
0 λ 0 0 0 0 0 0
− ≥ − ·k k ≥ { }·k k
Similarly for f K′ intK and 1 < λ 1 we get that f (Q (x)) min α ,β x 2 where
1 ∈ Q1 ∩ 2 ≤ 1 λ ≥ { 1 1}·k k
α > 0,β > 0 such that f (Q (x)) α x 2, f (Q(x)) β x 2. This completes the proof. (cid:3)
1 1 1 1 1 1 1
≥ k k ≥ k k
Remark 2.6. It is worth noting that in the case n = 1, Lemma 2.4 and Proposition 2.5 are not true.
Indeed, in this case any quadratic operator has a form Q(x)= ax2, and the elipticity means that a = 0.
Thus, the set of all elliptical operators in one dimensional setting is R1 0 , which is not connec6ted.
\{ }
Proposition 2.7. The set EQ is an open subset of Q .
n n
Proof. LetQ : Rn Rn beEQO,then thereis alinearfunctionalf : Rn Rsuchthatf (Q (x))
0 0 0 0
→ → ≥
α x 2 for some α > 0. We then want to show that
0 0
k k
α
0
Q : Q Q < EQ .
0 n
{ k − k 2 f } ⊂
0
k k
Indeed, if Q : Rn Rn is a quadratic operator with
→
α
Q(x) Q (x) < 0 x 2, x R2,
0
k − k 2 f ·k k ∀ ∈
0
k k
ELLIPTIC QUADRATIC OPERATOR EQUATIONS 9
then one has that
α α
f (Q(x)) = f (Q (x))+f (Q(x) Q (x)) α x 2 f 0 x 2 = 0 x 2.
0 0 0 0 0 0 0
− ≥ k k −k k· 2 f ·k k 2 ·k k
0
k k
This means that Q is the EQO and EQ is the open subset of Q . (cid:3)
n n
Analogously, one can prove the following statement.
Proposition 2.8. The following statements hold true:
(i) The set PQ is a closed and path connected subset of Q with empty interior;
n n
(ii) The set HQ is an open subset of Q .
n n
3. Examples
We are going to provide some examples for quadratic operators.
Example 3.1 (Theclassification ofquadraticoperatorsonR2). Let us consider the quadratic operator
acting on R2, i.e.,
Q(x) =(a x2+2b x x +c x2,a x2+2b x x +c x2),
1 1 1 1 2 1 2 2 1 2 1 2 2 2
where x = (x ,x ) R2. We denote by
1 2
∈
2
a c a b b c
1 1 1 1 1 1
∆ = 4 .
(cid:12) a2 c2 (cid:12) − (cid:12) a2 b2 (cid:12)·(cid:12) b2 c2 (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
If we avoid the case a1 = b1 = c(cid:12)1 then we(cid:12) have(cid:12)the follow(cid:12)in(cid:12)g: (cid:12)
a2 b2 c(cid:12)2 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(i) Q is elliptic if and only if ∆ > 0;
(ii) Q is parabolic if and only if ∆ = 0;
(iii) Q is hyperbolic if and only if ∆ < 0.
Using this argument one can construct concrete examples:
(a) Q (x) = (x2,x2) is elliptic;
1 1 2
(b) Q (x) = (x2,x x ) is parabolic;
2 1 1 2
(c) Q (x) = (x2 x2,x x ) is hyperbolic.
3 1− 2 1 2
Example 3.2 (The Stein–Ulam operator). Let us consider the following quadratic operator acting on
R3:
Q(x) = (x2+2x x ,x2+2x x ,x2+2x x ).
1 1 2 2 2 3 3 1 3
For a linear functional f(x) = λ x +λ x +λ x we have the following quadratic form
1 1 2 2 3 3
(3.1) f(Q(x)) = λ x2+2λ x x +λ x2+2λ x x +λ x2+2λ x x .
1 1 1 1 2 2 2 2 2 3 3 3 3 1 3
The matrix of this quadratic form is
λ λ λ
1 1 3
A = λ λ λ .
 1 2 2 
λ λ λ
 3 2 3 
 
Due to Silvester’s criterion, the quadratic form (3.1) is positive defined if and only if
λ > 0, λ > 0, λ > 0, λ > λ , λ > λ ,λ > λ .
1 2 3 2 1 3 2 1 3
However, this system of inequalities has no solutions. Therefore, this quadratic operator is not elliptic.
On the other hand if we take the linear functional f :R3 R as f(x)= x +x +x then we have
1 2 3
f(Q(x)) = (x +x +x )2 0. Consequently, Q : R3 R3→is the parabolic quadratic operator.
1 2 3
≥ →
10 RASULGANIKHODJAEV,FARRUKHMUKHAMEDOV,ANDMANSOORSABUROV
Example 3.3. Let us consider the following quadratic operator Q :Rn Rn
→
Q(x) = (x2+x2,x2+x2, ,x2 +x2,2x (x + +x )),
1 n 2 n ··· n−1 n n 1 ··· n−1
where x = (x , ,x ). Then for the linear functional f(x) = λ x + +λ x we get the following
1 n 1 1 n n
··· ···
quadratic form
n−1 n−1
(3.2) f(Q(x))= λ x2+ λ x2 +2λ x (x + +x ).
i i i n n n 1 n−1
···
!
i=1 i=1
X X
The matrix of this quadratic form (3.2) is
λ 0 0 0 λ
1 n
···
0 λ 0 0 λ
 2 n 
···
A=  · · · ··· · · .
 
 0 0 0 λ λ 
 ··· n−1 n 
 
 n−1 
 λ λ λ λ λ 
 n n n ··· n i 
 i=1 
 P 
Due to Silvester’s criterion, the quadratic form (3.2) is positive defined if and only if
n−1 n−1
1
λ λ2 > 0, λ > 0, i =1,n 1.
i − n λ i −
i
i=1 i=1
X X
Therefore, we obtain that
λ +λ + +λ
K′ = f = (λ , ,λ ) : λ > 0, i = 1,n 1, λ2 < 1 2 ··· n−1 .
Q 1 ··· n i − n 1 + + 1
(cid:26) λ1 ··· λn−1 (cid:27)
Consequently, the given quadratic operator Q is elliptic.
One can easily prove the following statement.
Proposition 3.4. Let Q :Rn Rn be a quadratic operator. Then the following assertions hold true:
→
(i) If one of matrices A , ,A is positive defined then Q is an EQO.
1 n
···
(ii) If matrices A , ,A are linear independent in the matrix algebra and commute each other
1 n
···
then Q is an EQO and K′ is a minihedral cone.
Q
4. The necessary condition for an existence of solutions
In this section, we will consider elliptic operator equation and we will provide some necessary
conditions for the existence of its solution.
The following equation
(4.1) P(x) Q(x)+Ax+b = 0, x Rn,
≡ ∈
is called an elliptic quadratic operator equation, where Q : Rn Rn is an elliptic quadratic operator,
A: Rn Rn is a linear operator and b Rn is a given vector.→
→ ∈
Let K′ be an open convex cone given by (2.1) associated with an elliptical operator Q. For every
Q
f K′ we denote by
∈ Q
= x Rn :f(Q(x)+Ax+b) 0 ,
f
E { ∈ ≤ }
and it is called an ellipsoid corresponding to f K′ . It is obvious that if = for some linear
∈ Q Ef ∅
functional f K′ then the elliptic operator equation (4.1) does not have any solutions.
∈ Q
Therefore, the necessary condition for the solvability of the elliptic operator equation (1.5) is that
= for any f K′ . Throughout this paper, we always assume that = for any f K′ .
Ef 6 ∅ ∈ Q Ef 6 ∅ ∈ Q