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JOURNAL

OF THE TECHNICAL UNIVERSITY AT

PLOVDIV, BULGARIA

FUNDAMENTAL SCIENCES AND APPLICATIONS

VOL.11 2005-2006

Series A - Pure and Applied Mathematics

EDITOR-IN-CHIEF: PEYO STOILOV

Editors: V. Petrov & St. Yordanov & M. Petrov & S. Tabakova D. Katsov & V. Georgiev & M. Kolarov & B. Gargov

Scientific Secretary: M. Deneva

EDITORIAL BOARD

DRUMI BAINOV MATHEMATICS

GERARD PHILIPPIN

MATHEMATICS

BINGGEN ZHANG MATHEMATICS

MARIN NENCHEV

PHYSICS, QUANTUM AND OPTOELECTRONICS

TZVETAN PARASKOV MECHANICS

LJUDMIL GENOV ELECTRICAL ENGINEERING

ANGEL VACHEV

MECHANICAL ENGINEERING

NIKOLAJ VELCHEV PHYSICS

RUMEN DIMITROV

CHEMISTRY

2

JOURNAL OF THE TECHNICAL UNIVERSITY AT PLOVDIV

FUNDAMENTAL SCIENCES AND APPLICATIONS Series A - Pure and Applied Mathematics The journal is indexed and reviewed by Mathematical Reviews and Math. Scl, Current Math. Publications, Zentralblatt fϋr mathematik.

The manusripts should be sent to the Editor-in-Chief Peyo Stoilov, e-mail: [email protected] Department of Mathematics, Technical University, Tsanko Dyustabanov 25, Plovdiv, BULGARIA. Acceptance for publication will be based on a positive recommendation by a member of the Editorial Bord.

Copyright © 2006 by Technical University at Plovdiv Plovdiv, BULGARIA, e-mail: [email protected], Internet addres http://www.tu-plovdiv.bg ISSN 1310-8271

3

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VOL.11 2005-2006

Series A - Pure and Applied Mathematics

JOURNAL OF THE TECHNICAL UNIVERSITY AT PLOVDIV FUNDAMENTAL SCIENCES AND APPLICATIONS publishes

new and original results in the fields MATHEMATICS, MECHANICS, PHYSICS, CHEMISTRY, ECONOMICS AND THEIR APPLICATIONS IN TECHNICAL SCIENCES.

4

Copyright © 2006 by Technical University at Plovdiv Plovdiv, BULGARIA, e-mail: [email protected], Internet addres http://www.tu-plovdiv.bg

ISSN 1310-8271

5

CONTENTS Page

1. A Connection on Manifolds with a Nilpotent Structure

Asen Hristov 7

2. Some properties of one connection on spaces with an almost product structure

Iva Dokuzova 21

3. Lp- Equivalence between a Linear and Nonlinear perturbed impulsive differential

equations with a Generalized dichotomous linear part

Albena Kosseva, Stepan Kostadinov 29

4. Existence of Bounded and Periodic Solutions of Nonlinear Impulse Differential Equations of the Dissipative Type

Atanaska Georgieva, Stepan Kostadinov 43

5. Nonlinear Semigroup for Nonlinear Abstract Impulse Differential Equations

Atanaska Georgieva, Stepan Kostadinov 55 6. Construction of analytic functions, which

determine Bounded Toeplitz operators on

1Η and ∞Η

Peyo Stoilov 67

6

JOURNAL OF THE TECHNICAL UNIVERSITY AT PLOVDIV FUNDAMENTAL SCIENCES AND APPLICATIONS

VOL.11 2005-2006 Series A - Pure and Applied Mathematics

c©Journal of Technical University at PlovdivFundamental Sciences and Applications, Vol. 11, 2005-2006Series A-Pure and Applied MathematicsBulgaria, ISSN 1310-8271

A Connection on Manifolds with aNilpotent Structure

Asen Hristov

Abstract

All the connections, pure toward the nilpotent stucture, are found. Ex-amples of manifolds, for which the curvature tensor is pure or hybridous, aregiven. For a manifold of B-type a necessary and sufficient condition for purityof the curvature tensor is proved. It is verified that the conformal change ofthe metric of a B-manifold does not retain its puruty.

Keywords: semi-Riemannian geometry, curvature tensor, geodesics, Ein-stein field equations, Schwarzschild solution.

We suppose that B is a submanifold of E, where dim B = n, and dim E = m+n.We assume that

δ : E → B

is a submersion in E. If point p belongs to the base B then the set of points

δ(−1)(p) ⊂ E

are a layer over it. A priori we suppose a local triviality of δ. By TB we denote thetangential differentiation on the base B. If we designate by

(zi, zn+i, z2n+a), i = 1, 2, . . . , n; a = 1, 2, . . . ,m ,

01991 Mathematics Subject Classification: 30E20, 30D500Key words and phrases: semi-Riemannian geometry, curvature tensor, geodesics, Einstein field

equations, Schwarzschild solution.0Received September 15, 2006

7

8 Asen Hristov

then zi are coordinates of a point from B, zn+i are coordinates of a point in a layerfrom TB(π : TB → B, where π−1(p) is a tangential layer over a point p ∈ B), andz2n+a are coordinates of a point from σ−1(p). We can interpret the variables(

zi, zn+i, z2n+a)

as local coordinates of a point from a 2n + m - dimensional manifold M providedthat these coordinates are replaced by the following rule [2]:

zi = ϕi(z1, z2, . . . , zn)

zn+i =∑

k

∂ϕi(z1, z2, . . . , zn)

∂zk.zn+k (1)

z2n+a = Θa(z1, z2, . . . , zn; z2n+1, z2n+2, . . . , z2n+m)

It is proved in [2] that M tolerates an integratable nilpotent structure f , calledsemitangential by the author of this paper, which commutates with the Jacobian of(1). With regard to a suitable local basis on M, the matrix of f has the followingform (

fαβ

)=

E

,

E is a single matrix from row n, and the blank blocks are zero blocks.We assume that ∇ = gσ

αβ is a torsionless connection on M with connectioncoefficients gσ

αβ. As regards ∇, we impose a purity condition of the coefficientstoward f as well:

gλβσf

αβ = gα

βλfλσ = gα

λσfλβ , α, β, σ, λ, · · · = 1, 2, . . . ,m + 2n

A solution to the above condition for purity is

gkis = gn+k

i,n+s = gn+kn+i,s , gn+k

is and g2n+aβσ , gσ

β,2n+a (2)

In the special case m = 0, i.e. when M is the tangential differentiation of B, thecoefficients of the second series of solution (2) are

g2n+aβσ = gσ

β,2n+a = 0 .

In this case, if∇ = (gh

ik) is a torsionless connection on the base, then we denoteui = zi, yi = zn+i so therefore

ghik = gh

ik, gni,n+k = gh

n+i,k = gnn+i,n+k = gn+n

n+i,n+k = 0 ,

A Connection on Manifolds with a Nilpotent Structure 9

gn+nik = zs ∂gh

ik

∂us,

gn+hi,n+k = gn+h

n+i,k = ghik .

Thus gαβσ coincide with the coefficients of the complete lift of the connection

∇.

Lemma 1. If a connection has pure coefficients with respect to one upper and onelower index, and f is covariantly constant, then f is an integratable structure.

Proof. From the condition

∇σfαβ = ∂σf

αβ − gα

σλfλβ + gν

σβfαν = 0

follows that

∇σfαβ = ∂σf

αβ = 0 ,

which shows that fαβ = const.

Corollary 1. If f is an integratable structure and ∇f = 0, then we have purity ofthe coefficients gα

βσ with respect to one upper and one lower index.

Lemma 2. If f is integratable and Rσαβγ are the corresponding components of the

curvature tensor for ∇, then the condition ∇f = 0 is sufficient for Rσαβγ to be pure

with respect to one upper and one lower index.

Proof. From the Ricci identity

∇α∇βfγσ −∇β∇αfγ

σ = Rσαβγf

γσ −Rσ

αβγfγλ

the proof is obvious.

The purity of the Riemannian curvature tensor with respect to an upper anda lower index does not automatically mean purity with respect to two lower in-dices. An example of this are Riemannian manifolds, the functional tensor of whichconforms in a suitable way with the semitangential structure f .

Example 1. The Riemannian manifold M(g, f) with a metric tensor g = (gαβ)and an assigned integratiable nilpotent tensor field f of type (1,1) is called a manifoldof B-type, if for every two vector fields x and y there follows:

g(fx, y) = g(x, fy) .

10 Asen Hristov

We note down that the last condition is equivalent to the purity of gαβ, whereg = (gαλf

λβ ) is a symmetrical tensor field, and under certain conditions Rσ

αβγ is purewith respect to all indices.

Detailed information about this type of manifolds and research work, connectedwith them, can be found in the studies of E. V. Pavlov.

Example 2. Riemannian manifolds M(g, f) with a metric tensor g = gαβ andan assigned integratiable nilpotent tensor field f of type (1,1) is called a manifoldof Kahler type if for every two vector fields x and y there follows:

g(x, fy) = −g(fx, y)

The last equation is called a condition for the hybridity of g toward f . In thisexample the tensor field g = (gαλf

λβ = gβα) is antisymmetric, Rσ

αβγ is not pure withrespect to all indices.

Particular examples of this type of Kahler manifolds can be found in ([3], p.137).

Assertion 1. If M(g, f) is a manifold from type B, then the components Rγσαβ and

Rσαβγ are pure with respect to the two indices (β, γ). If M(g, f) is of Kahler type,then Rσβαγ and Rτ

σβα are hybrid with respect to (α, γ) and pure with respect to (α, τ)respectively.

Proof. The condition in both cases follows from Lemma 2. We have

Rλαβγ . fσ

α = Rσαβλ . fλ

γ

Hence follows thatRλ

αβγ.Fλσ = Rσαβλ.f

λγ ,

where Fλσ = gστfτλ , and taking the antisymmetrization of Fλσ into consideration,

we obtain

−Rλαβγ.f

λσ = Rλ

αβγ.fλγ .

When M(g, f) is of type B, the proof is analogical.

In connection with the B-type manifolds, two important theorems need to bementioned, which are proved in [4] and [5] respectively.

Theorem 1. If the components Rσαβλ of the curvature tensor of a B-type manifoldare pure with respect to two indices, then they are pure with respect to all doubleindices.


Theorem 2. The components Rσαβλ of the curvature tensor of a B-type manifoldare pure then and only then when the partial derivatives of the metric tensor arepure.

Corollary from Assertion 1. Tensor G, defined by means of the equation

G(x, y, v, w) = g(x, v)g(y, w)− g(x, w)g(y, v)

is curvature-related for M(g, f). At that the tensor∗G, for which

∗G(x, y, v, w) = G(x, fy, v, fw) = −g(x, fw)g(fy, v)

is pure or hybrid depending on M(g, f). In case of

x 6= ker f, G(x, fx, y, fy) = −[g(x, fy)]2 ,

from where follows:

a) If M(g, f) is of type B, then g(x, fx) 6= 0. In this case, on the basis ofAssertion 1 and Theorem 1, it is possible for M(g, f) to be flat on account of thefact that the holomorphic curvature in the direction of x, fx is zero.

b) If M(g, f) is of Kahler type. Now g(x, fx) = 0. Therefore the holomorphiccurvature in the direction of x, fx is indefinite.

Theorem 3. If M(g, f) is a B-type manifold, then the Riemannian connection,originating from g, possesses pure connection coefficients with respect to all indicesthen and only then, when the partial derivatives

∂σgαβ =∂

∂zσgαβ

are pure towards f .

Proof. We need to note down in advance that the symmetrical tensor field

g = (gαβ = gλβfλα)

is transferred parallely towards the Riemannian connection, originating from g. Be-cause of fα

β = const we have

∇σgαβ = ∂σgαβ − gλσαgλβ − gτ

σβ gατ = 0

Here we used the corollary from Lemma 1.

12 Asen Hristov

a) Let ∂σgαβ be pure toward f . In this case from the condition

2gσαβ = gσλ(∂αgβλ + ∂βgλα − ∂λgαβ) (3)

and the purity of gσλ, resulting from the equations

gλσgλγ = δσγ ⇒

⇒ gλσgλγfνσ = f ν

γ ⇒

⇒ gλσδβλf ν

σ = f νγ gγβ ⇒

⇒ gβσf νσ = gσβf ν

γ (the purity of gαβ) ,

there follows the purity of the connection coefficients with respect to all indices.

b) We assume that gσαβ are pure with respect to all indices. In this case from

∇σgαβ = 0 follow the conditions

∂σgαβ − gλσαgλβ − gν

σβ gαν = 0

and

f τσ∂τgαβ − gλ

σαgλβ − gνσβ gαν = 0 .

By means of their term-by-term subtraction we obtain

f τσ (∂τgαβ) = ∂σ(fλ

αgλβ) .

Definition 1. If for M(g, f) the partial derivatives of the components of g are pure,we say that M(g, f) is a B-manifold.

Theorem 4. If M(g, f) is a B-manifold and det(g + g) 6= 0, then the metrics g andg + g originate one and same Riemannian connection.

Proof. Let us take into consideration that the matrices (gαβ + gαβ) and (gαβ − gαβ)

are mutually inverse. Here we have denoted gαβ = gαλfβλ and used the purity of

gαβ, which was proved in the previous theorem. If gσαβ are the coefficients of the

Riemannian connection, originating from g+g, we apply formula (3) but in referenceto the metric g + g. In the course of the calculations we should keep in mind thatthe objects ∂σgαβ are pure. Thus there follows that gσ

αβ = gσαβ.


The theorem proved holds true in the most common case. In the special casewhere m = 0, i.e. when M(g, f) is a tangential differentiation on base B, a similartheorem is proved for the lifts of the base metric in ([7], p. 149). There g + g isdesignated as a metric I + II. On the other hand, the change g → g + g is thesimplest CH-change. On condition that f 2 = I (I is the identical transformation),the CH-change is studied in detail in [4]. To the question whether it is the conformalchange that needs to be investigated, or the generalization and the CH-change forM(g, f) with f 2 = 0, we obtain an answer by means of

Theorem 5. If h(z1, . . . , zm+2n) is a random manifold, and M(g, f) is a B-manifold, then M(hg, f) is a B-manifold then and only then, when h = const.

Proof. If M(hg, f) is a B-manifold, there follows that

∂λ(hgαβ)fλσ = ∂σ(hgλβfλ

α) .

Hence we obtain

fλσ hλδ

βα = hσf

βα , (hλ =

∂h

∂zλ)

This equation holds true for all values of the indices. For β = n + i , α =k , fn+i

k = δik in particular, from the specified equation we have

fλσ hλδ

n+ik = hσδ

ik

or hσ = 0 for every σ.In the opposite case, where h = const, it is obvious that M(hg, f) is a B-

manifold.

Geodesic and holomorphic plane curves. If we assume that on the manifoldM(f), which is provided with a pure connection with respect to f , the curve c : zα =zα(+) is geodesic, after changing the parameter t : t = h(q), the equation of c

δzα

dt=

d2zα

dt2+ γα

λβ

dzλ

dt.d2zβ

dt= 0

(zα = dzα

dtis the covariant differentiation toward the connection, which has the

γασβ coefficients) is equivalent to

δ

dq(dzα

dq) =

dh

dq.dzα

dq.

It is possible to transfer the field υ = (υα) parallely with respect to c

14 Asen Hristov

δυα

dq=

dυα

dq+ γα

λβ zλ dυβ

dt= 0

After changing υα → λ(t)υα we have

δ

dt(λ(t)υλ) = λ′(t)υα

In both of the examples given we will say that the directions dzα

dtand λυα are

transferred parallely with respect to c.

We say that the curve zα(t) from M(f) is a holomorphic plane curve (a PH-curve)for M(f), if zα(t) are the solutions to the differential equation

δzα

dt=

d2zα

dt2+ γα

λβ

dzλ

dt.d2zβ

dt= a(t)

dzλ

dt+ b(t)(fλ

ν

dzν

dt) .

Here a(t) and b(t) are functions. We will adopt the designation

dzα

dt= fλ

ν

dzν

dt.

There exists a special case, in which the geodesic curves and the PH-curves coin-cide. That is when M(f) is projective-Euclidean or PH-Euclidean. This peculiarityis illustrated best about the three-dimensional projective space B3 (an extension ofthe corresponding Euclidian one). In B3 the above curves coincide with the abso-lute straight line ω, which has a common point with each straight line from theabsolutely congruent straight lines [8]. The absolute straight line ω is the set of allpoints belonging to ker f .

Assertion 2. If zα(t) is geodesic for M(f) and zα∈ ker f , then zα(t) is a PH-curve.

Proof. Let us recall that f is covariantly constant and the objects γαλβ are pure with

regard to f . In this case, from the differential equation of the geodesic curve therefollows

δ

dt˜zα = 0 .

Let us consider the linear combination υα = h(t)zα + l(t)˜zα for certain functionsh(t) and l(t). For the vector field υ = (υα) we have

δυα

dt= [h(t)]′zα + [l(t)]′ ˜zα ,


which shows that a random vector from the holomorphic vector space zα, ˜zα,at a parallel transfer with respect to zα(t), remains in zα, ˜zα, i.e. zα(t) is a PH-curve.

Assertion 3. If zα(t) is geodesic for M(f) and zα ∈ ker f , then zα(t) is a PH-curveand it belongs to the special plane (straight line) ω in the biaxial space Bn(B3).

Proof. The assertion is obvious because

δ

dt˜zα = 0 .

Assertion 4. If zα(t) is a PH-curve and zα ∈ ker f , then zα(t) is geodesic.

Proof. From the differential equation of PH-curves follows that

δzα

dt= a(t)zα .

The tangential direction is transferred parallely with respect to zα(t), therefore zα(t)is geodesic.

We note down that in this case all curves

zα(t) = (c1, . . . , cn, zn+1(t), . . . , z2n(t), z2n+1(t), . . . , z2n+m(t))

possess one and the same tangential vector field. The special case, where n = 1, m =0 in the two-dimensional affine plane 0xy, is interesting. There zα(t) = (c, z(t)). Thecorresponding geodesic curve is a straight line, which is parallel to the axis 0y.

Let us consider a particular case, where M(f) is a tangential differentiation ofthe B manifold.

Definition 2. We say that the pair of functions p(u, v), q(u, v) is holomorphic if

∂p

∂u=

∂q

∂v

and∂p

∂v= 0 .

16 Asen Hristov

On the basis of this definition of the two-dimensional surface

SH ⊂ M(f) : zi(u); zn+i(u, v) = vdzi

du ,

we will say that it is holomorphic in M(f).

Assertion 5. As regards a holomorphic two-dimensional surface SH , with the changeof the parameters u = h(u), v = t(u, v) in such a way that for the differentiable func-tions h(u) and t(u, v) there holds

u = h(u)

anddh

du=

∂t

∂v,

the holomorphicity of SH is preserved.

Proof. We substitute the values in the parametric equation of

SH : zi = zi(u), zn+i = vdzi

du(u, v)

with their equals. In this case zi depend only on the variables u, and at that

dzi

du=

dzi

du.dh

du;

∂zn+i

∂v=

∂zn+1

∂v.dt

dv=

∂

∂v(v

dzi

du).

∂t

∂v=

dzi

du.dh

du=

dzi

du, i.e.

∂zn+i

∂v=

dzi

du.

Theorem 6. Every geodesic curve β(u) = (u = u, v = const) (u - line) from thetwo-dimensional holomorphic surface SH ⊂ M(f) is a PH-curve for M(f).


Proof. Let γ(v) = (u = const, v = v) be the v-lines for SH , and

β = (zi; vzi; 0, . . . , 0)

andγ = (0, . . . ; zi; 0, . . . , 0)

are the corresponding velocities of β(u) and γ(u). Obviously

γ = fβ .

Besides,δβα

du=

dβα

du+ gα

λν βλβν = 0 .

Taking into consideration the purity of the objects gαλν from the last two equations,

we obtain

δγα

du=

d

du(fα

λ βλ) + gανσf

νλ βρβσ =

=d

du(fα

λ βλ) + gλρσf

αλ βρβσ =

=[ d

duβλ + gλ

ρσβρβσ

]fα

λ = 0

Therefore γ is transferred parallely with respect to the u-lines β, i.e. β is a PH-curvefor M(f).

We note down that the theorem proved above is a confirmation of Assertion 2.Here the object of consideration is a holomorphic two-dimensional surface SH fromM(f), and we can always consider that on it there is set a geodesic vector field,which does not belong to ker f . As it is recorded in [9], SH is interpreted as a real

model of a geodesic curve on the manifold∗

M over the algebra

R(ε) = a + εb; ε2 = 0; a, b ∈ R .

There the surfaces SH are called analytical.

If q = qi is a form on M(f), there can be found a lot of connections, which arepure with respect to f . The tensor of the affine deformation T , through which theseconnections are obtained, is

T nik = δn

i qk + fni qk + δn

k qi + fnk qi .

18 Asen Hristov

In the above equation qi := qsfsi .

Obviously T is pure with respect to all indices. Therefore any connection

Γα

βγ = Γαβγ + Tα

βγ

is pure too.

Theorem 7. The connection ∇ with coefficients Γα

βσ possesses the following prop-erties:

(1) it has symmetrical and pure coefficients,(2) ∇f = 0 ,(3) ∇ and ∇ have common PH-curves.

Proof.

(1) The integrated tensor of T is

Tn

is = T ninf

ns = fn

i qs + fns qi .

It hence follows that the connection with coefficients Γα

βγ is also pure.

(2) Let us assume that ∇f = 0. Then

∇fni = Γ

n

sjfji − Γ

p

sifnp =

= Γnsjf

ji + T n

sjfji − Γp

sifnp − T p

sifnp =

= ∇f + T nsi − T n

si = 0 .

(3) We denote with δ the covariant differential as regards ∇. Since

δ

dtxn = a(t)xn + b(t)˜xn ,

there follows that

δ

dtxn = x + [Γn

ik + δni pk + δn

k pi + fni qk + fn

k qi]xixn =

=δ

dtxn + xnp(x) + xnp(x) + ˜xnq(x) + ˜xnq(x) .

Here we have designated pk := qk. Finally we obtain

δ

dtxn = (a + 2p(x))xn + (b + 2q(x))˜xn .


References

[1] E. V. Pavlov, A. H. Hristov, Metric Manifolds With a Semi-Tangential Struc-ture. (in Russian)

[2] V. V. Vishnevskiy. On the Geometric Model of a Semi-Tangential Structure.Higher School Bulletin, Mathematics, issue 3, 1983, pp. 73-75. (in Russian)

[3] V. V. Vishnevskiy, A. P. Shirokov, V. V. Shurigin. Spaces Over Algebras.Kazan University Press, 1988. (in Russian)

[4] Pavlov E. V., Manifolds With an Algebraic Structure and CH-Mapping. Re-search work qualifying for an academic degree. Plovdiv, 2003. (in Bulgarian)

[5] Vishnevskiy V. V. Certain Properties of Differential Geometric StructuresThat are Determined by Algebras., Scientific works of Plovdiv University, v.10, b. 1, 1972 - mathematics. (in Russian)

[6] K. Yano. Differential Geometry on Complex and Almost Complex Spaces. TheMacmillan Company, New York, 1965.

[7] Yano Kenturo, Shigera Ishihara, Tangent and Cotangent Bundles - New York,1973.

[8] N.V. Talantova, A Biaxial Space of a Parabolic Type, Higher School Bulletin,Mathematics, 1959. (in Russian)

[9] V. S. Talapin, A-Planar Transformation of a Connection in Real Realizationof Manifolds Over Algebras, Higher School Bulletin, Mathematics, 1980. (inRussian)

Asen HristovFaculty of Mathematics and InformaticsUniversity of Plovdiv ”P. Hilendarski”24, Tzar Assen Str.4000 Plovdiv, BULGARIAE-mail: [email protected]

.

20 Asen Hristov


Some properties of one connection onspaces with an almost product structure

Iva Dokuzova

Abstract

In a space M with a metric g and an almost product structure J weintroduce an affine connection by using the Levi-Civita connection ∇ of g.We get some properties of the obtained transformation.

Keywords: semi-Riemannian geometry, curvature tensor, geodesics, Ein-stein field equations, Schwarzschild solution.

1 Introduction

We consider a space M(dim M = n) with a metric g and an almost product structureJ , which preserves the scalar product. So

J2 = id (J 6= id), g(Jx, Jy) = g(x, y), x, y ∈ χM. (1)

Let ∇ be the Levi-Civita connection of g and R be the curvature tensor field of ∇.If ∇ satisfies

∇J = 0, (2)

then M is a locally decomposable Riemannian space [2], [3], i.e. M = M1 × M2,where M1, M2 are both Riemannian spaces. We note that trJ = dim M1 − dim M2,n = dim M1 + dim M2.

02000 Mathematics Subject Classification: 53C15, 53B05,53B30.0Key words and phrases: semi-Riemannian geometry, curvature tensor, geodesics, Einstein field

equations, Schwarzschild solution.0Received November 10, 2006

22 Iva Dokuzova

Now we consider a space M , which satisfies (1). For brevity, we call a linearconnection ∇ on M an FAPJ-connection, if ∇ satisfies

∇iJjk +∇jJki +∇kJij = 0, Jij = Jsi gsj. (3)

In the above identity Jsi and gsj are the local coordinates of the structure and of the

metric, respectively.We say that the space M , which satisfies (1) is in the class FAP , if the Levi-

Civita connection ∇ of g is a FAPJ-connection [1]. Obviously, if M is a locallydecomposable space, then M is in the class FAP .

2 ∇ connection

Let M be in the class FAP , ∇ be the Levi-Civita connection of g, and Γkij be the

Christoffel symbols of ∇. If a and b are arbitrary smooth vector fields on M , thenthe connection ∇, defined by the relation:

Γk

ij = Γkij + gija

k + Jijbk − 1

2Jk

i aj −1

2δki bj −

1

2Jk

j ai −1

2δkj bi, (4)

where ai = J ti at, bi = J t

i bt, is a symmetric FAPJ-connection.

Proof. We see that Γk

ij = Γk

ji, thus ∇ is a symmetric connection. Now we find the

covariant derivative ∇ of Jij by using the well known formula:

∇iJjk = ∂iJjk − Γa

ijJak − Γa

ikJaj

and (4). We get

∇iJjk = ∇iJjk −1

2gij ak −

1

2gikaj + gjkai −

1

2Jij bk −

1

2Jikbj + Jjkbi.

After direct calculations we obtain ∇iJjk +∇jJki +∇kJij = 0.

Let M be in the class FAP , also ∇ and ∇ satisfy (4). If ∇J = 0, then ∇J = 0.

Proof. From

∇iJkj = ∂iJ

kj − Γ

a

ijJka + Γ

k

iaJaj

and by using (4), we get

∇iJkj = ∇iJ

kj + gij(b

k − ak) + Jij(ak − bk) +

1

2δki (aj − bj) +

1

2Jk

i (bj − aj).

Some properties of one connection on spaces with an almost product structure 23

Let us assume that ∇iJkj = 0. Then the above identity implies

∇iJkj = gij(a

k − bk) + Jij(bk − ak) +

1

2δki (bj − aj) +

1

2Jk

i (aj − bj). (5)

In (5) we contract with k = i and we obtain

∇iJij =

n

2(bj − aj) +

trJ

2(aj − bj). (6)

The connection ∇ satisfies equation like (3), thus we have ∇iJik = 0. From the last

equality and (6) we get the system:

n(bj − aj) + trJ(aj − bj) = 0.

n(bj − aj) + trJ(aj − bj) = 0.

The only solution of this system is bj = aj, aj = bj. After substituting the lastresult in (5), we obtain ∇iJ

kj = 0.

3 The case a = 0

Let M be in the class FAP , also ∇ and ∇ satisfy (4). If ak = 0 (consequentlyak = 0), then (4) has the form:

Γk

ij = Γkij + T k

ij, T kij = Jijb

k − 1

2δki bj −

1

2δkj bi. (7)

For the curvature tensor fields R of ∇ and R of ∇ it is well known the identity:

Rh

ijk = Rhijk +∇jT

hik −∇kT

hij + T s

ikThsj − T s

ijThsk. (8)

From (7) and (8) we obtain

Rh

ijk = Rhijk + Jik(∇jb

h + bjbh − 1

2δhj bsbs)

− Jij(∇kbh + bkb

h − 1

2δhk bsbs)

− 1

2δhi (∇j bk −∇kbj)−

1

4δhk (2∇j bi + bibj)

+1

4δhj (2∇kbi + bibk)− bh(∇kJij −∇jJik).

(9)

Let M be in the class FAP , also ∇ and ∇ satisfy (7). Then ∇ is an equiaffineconnection, if and only if, the vector field b is gradient.

24 Iva Dokuzova

Proof. We know ∇ is an equiaffine connection, i.e. Riijk = 0. That’s why contracting

(9) with h = i, we get

Ri

ijk = (n

2+

1

2)(∇kbj −∇j bk).

Then Ri

ijk = 0, if and only if, ∇kbj = ∇j bk. The last condition implies the vector

field b is gradient.

Let M be a locally decomposable space and ∇ be the Levi-Civita connection ofg. Also, let b 6= 0 be a smooth vector field on M and ∇ be a locally flat FAPJ-connection, defined by the relation (7). Then the vector field b is an isotropic and∇ is a locally flat connection.

Proof. We consider a locally decomposable space M . We denote

Pkh = ∇kbh + bkbh −ϕ

2gkh, ϕ = bsbs, Pkh = J t

hPkt. (10)

From (2) it is follows

Pkh = ∇kbh + bkbh −ϕ

2Jkh. (11)

We assume ∇ is a locally flat connection and it is necessary and sufficient thatR = 0. From Ri

ijk = 0 we get ∇kbj = ∇j bk. So b is a gradient vector. By using theabove conditions, (2), (10), (11) and lowering the index h in (9), we have

Rhijk = JijPkh − JikPjh +1

4gkhQji −

1

4gjhQki, (12)

whereQij = 2Pij − bibj + ϕJij. (13)

If we denote Qij = Jsj Qis, then from (2), (10), (11), and (13) we have

Qij = 2Pji − bibj + ϕgij. (14)

With the help of the identities Rhijk = Rjkhi, Qij = Qji, and (12) we find

4(JikPjh − JkiPhj − JijPkh + JkhPij) = gkhQij − gijQkh. (15)

We exchange k and h in (15) and by using the relation Qij = Qji, we get

Jik(Phj − Pjh) + Jij(Pkh − Phk) + Jih(Pjk − Pkj) = 0.

In the last equation we contract with J ij and we obtain

Pkh = Phk. (16)


Let us denote Q = Qss, P = P s

s , Q = Qss, P = P s

s . We substitute (16) in (15) andby contracting with gkh, and then with Jkh we get

nQij −Qgij = 4((trJ)Pij − PJij);

(trJ)Qij − Qgij = 4(nPij − P Jij).

The solution of the above system is as follows:

Qij =4

tr2J − n2[(Pn− P trJ)Jij +

1

4(QtrJ −Qn)gij]; (17)

Pij =1

n2 − tr2J[(P n− P trJ)Jij +

1

4(QtrJ − Qn)gij]. (18)

Equation (17) implies Qij = Qji.On the other hand from (14) and (16) we get

bibj = bjbi. (19)

Let us denote φ = bibi. We contract (19) with bi. So

ϕbj = φbj. (20)

Then we contract with bj in (20). Thus we have ϕ = ±φ. We find two solutions.First of them is φ = ϕ = 0, and second is bj = ±bj ( ϕ 6= 0).

Transvecting (17) with J ij, we obtain

QtrJ − Qn = 4(Pn− P trJ). (21)

From (12), (17), (18), and (21) we get

Rhijk =1

n2 − tr2J((P trJ − P n)(JikJjh − JijJkh)

+1

4(QtrJ −Qn)(gikgjh − gkhgij)).

(22)

Let S be the Ricci tensor of ∇. Transvecting with gij in (22), we obtain

Skh =1

n2 − tr2J[JkhtrJ(P n− P trJ)

+ gkh((P trJ − P n) +1

4(QtrJ −Qn)(1− n))].

(23)

For the scalar curvature τ = Sijgij we have

τ =1

n2 − tr2J((tr2J − n)(P n− P trJ) +

n− n2

4(QtrJ −Qn)). (24)

26 Iva Dokuzova

If (M, g, J) is a locally decomposable space, then RhijkJij = SahJ

ak . Thus it

follows RhijkJijJhk = τ . Then from (22) we find

τ =1

n2 − tr2J((n2 − n)(P n− P trJ) +

n− tr2J

4(QtrJ −Qn)). (25)

Collecting the system (24), (25), we get

Qn− QtrJ = 4(P n− P trJ). (26)

From (10), (11), (13), and (14) we obtain

P = ∇kbk + (1− n

2)ϕ

P = ∇kbk + φ− trJ

2ϕ (27)

Q = 2P − φ + ϕtrJ = 2∇kbk + φ

Q = 2P + (n− 1)ϕ = 2∇kbk + ϕ.

At first we consider the case φ = ϕ = 0.Remark Obviously, everywhere in our considerations we suppose that g is un-

defined metric. If we accept that g is defined, then we see that (7) is trivial trans-formation.

Thus the vector field b is isotropic one and (27) has the form:

P = ∇kbk, P = ∇kb

k, Q = 2∇kbk, Q = 2∇kb

k. (28)

From (21), (26), and the last system we get

Pn− P trJ = 0

P n− P trJ = 0.

It’s only solution is P = P = Q = Q = 0. So we get R = 0. We prove that ∇ is alocally flat connection if φ = ϕ = 0.

Now, let us consider the case bj = bj (φ = ϕ). We have ∇kbj = ∇kbj and from(27) we find

P = ∇kbk + (1− n

2)ϕ

P = ∇kbk + (1− trJ

2)ϕ + φ (29)

Q = Q = 2∇kbk + φ.

From (21), (26), and the last system we get P = P . Then we have ϕ = 0. This casereduces to the previous case. Analogously if bj = −bj (φ = −ϕ) we get ϕ = 0. Sothe theorem is proved.


Let ∇ and ∇ satisfy the conditions of Theorem 3, and α = b, b be the two-dimensional section in TpM , p ∈ M . Then for two arbitrary vectors x, y ∈ α wehave g(x, x) = g(x, y) = g(y, y) = 0.

References

[1] I. R. Dokuzova. A note on Schur’s theorem in Riemannian manifolds with analmost product structure. Journal of TU, Fundamental Sciences and Applications- Plovdiv, vol.10, 2002–2003, 65–71.

[2] A. M. Naveira. A classification of Riemannian almost product manifolds.Rend. Math. Roma, no. 3, 1983, 577–592.

[3] K. Yano. Differential geometry on complex and almost complex spaces. NewYork, Pergamont Press, 1965

Iva DokuzovaDepartment of Mathematics and PhysicsUniversity of Plovdiv ”P. Hilendarski”Branch ”Ljuben Karavelov”-Kardjali26, Belomorski Boul.Kardjali, BULGARIAE-mail:[email protected]


Lp - Equivalence between a Linear andNonlinear perturbed impulsive

differential equations with a Generalizeddichotomous linear part

Albena Kosseva, Stepan Kostadinov

Abstract

By help of Schauder-Tychonoff’s fixed point theorem sufficient conditionsfor Lp - equivalence between a linear and nonlinear perturbed impulsivedifferential equations with the some linear part are obtained.

1 Introduction

We consider linear and nonlinear perturbed impulsive differential equations withthe some linear part. The linear equatin is supposed to be generalized dichotomous.That means that there exist two nonnegative functions h(t) and k(t) such thatfollowing relations

‖U(t)P1U−1(s)‖ ≤Mh(t)h−1(s), (0 ≤ s < t);

‖U(t)P2U−1(s)‖ ≤Mk−1(t)k(s), (0 ≤ t ≤ s)

are satisfied (Pi : X → X, i = 1, 2 are projectors with P1 + P2 = I , g−1(t) = 1g(t)

.It

may be noted that for ordinary differential equations the notion of (h, k)-dichotomywas introduced in [3].

02000 Mathematics Subject Classification: 34A37, 34G10, 34D09 .0Key words and phrases: Impulse differential equations; Lp - equivalence; (h, k) - dichotomous.0Received September 15, 2006

29

30 Albena Kosseva, Stepan Kostadinov

By help of the Schauder-Tychonoff’s fixed point theorem we prove that everybounded solution of the linear equation induces a bounded solution of the nonlinearequation and vice versa. The difference of both the solutions lies in the space Lp.

2 Problem statement

Let X be a real Banach space with norm ‖.‖ and identity operator I.We consider the following differential equation in X

dy

dt= A(t)y ; t 6= tn (n = 1, 2, 3, ...) (1)

y(t+n ) = Qny(tn) ; (n = 1, 2, 3, ...) (2)

anddx

dt= A(t)x+ f(t, x) ; t 6= tn (n = 1, 2, 3, ...) (3)

x(t+n ) = Qnx(tn) +Rnx(tn) ; (n = 1, 2, 3, ...) (4)

where the function A(t) is piecewise continuous from the left with discontinuitiesof the first kind at the points tn, n = 1, 2, 3, ... and Qn : X → X , (n = 1, 2, 3, ...)are bounded operators with bounded inverse operators Q−1

n .We notiece by U(t) , t ∈ [0,∞) the evolutionary operator of (1), (2) (see for

example [1]).Let h(t), k(t) (t ≥ 0) be two positive functions.

Definition 1. . The equation (1), (2) is said to be (h, k) - dichotomous if thereexist constant M > 0, and projector Pi : X → X (i = 1, 2) such that

‖U(t)P1U−1(s)‖ ≤Mh(t)h−1(s), (0 ≤ s < t); (5)

‖U(t)P2U−1(s)‖ ≤Mk−1(t)k(s), (0 ≤ t ≤ s). (6)

where P1 + P2 = I.

We introduce the following spaces:

Lp(h) = g(t) : R+ → X : supt≥0

h(t)

∫ t

0

h−1(s)‖g(s)‖pds <∞

Lp(k) = g(t) : R+ → X : supt≥0

k−1(t)

∫ ∞

t

k(s)‖g(s)‖pds <∞

lp(h) = gn∞n=1 ⊂ X : supt≥0

h(t)∑tn<t

h−1(t+n )‖gn‖p <∞

Lp - Equivalence between a Linear and Nonlinear perturbed impulsive... 31

lp(k) = gn∞n=1 ⊂ X : supt≥0

k−1(t)∑tn≥t

k(t+n )‖gn‖p <∞

with norms

‖g‖Lp(h) = supt≥0

(h(t)

∫ t

0

h−1(s)‖g(s)‖pds)1p

‖g‖Lp(k) = supt≥0

(k−1(t)

∫ ∞

t

k(s)‖g(s)‖pds)1p

‖g‖lp(h) = supt≥0

(h(t)∑tn<t

h−1(t+n )‖gn‖p)1p

‖g‖lp(k) = supt≥0

(k−1(t)∑tn≥t

k(t+n )‖gn‖p)1p .

For X = R1 we shall write L1p(h), L

1p(k), l

1p(h), l

1p(k).

Definition 2. . The impulsive equations (1), (2) and (3), (4) are called Lp(h, k)-equivalent if to any bounded solution y(t) of (1), (2) corresponds at least on boundedsolution x(t) of (3), (4) such that x(t)− y(t) ∈ Lp(h) ∩ Lp(k).

We introduce the conditions:H1. Let there exist constants M1 > 0 and M2 > 0 such that

h(t)

∫ t

0

h−1(s)ds ≤M1 , k−1(t)

∫ ∞

t

k(s)ds ≤M2, t ≥ 0.

H2. Let there exist constants M1 > 0 and M2 > 0 such that

h(t)∑tn<t

h−1(t+n ) ≤ M1 , k−1(t)∑tn≥t

k(t+n ) ≤ M2, t ≥ 0.

3 Main results

Theorem 1. . Let the following conditions be satisfied:1. The function A(t) is piecewise continuous from the left with discontinuities of

the first kind at points tn , n = 1, 2, 3, ....2. The operators Qn have bounded inverse operators Q−1

n , n = 1, 2, 3, ...3. The impulsive equation (1), (2) is (h, k) -dichotomous.4. Conditions (H1) and (H2) hold.Then for any function f ∈ Lp(h)∩Lp(k) and for any sequence rn∞n=1 ∈ lp(h)∩

lp(k) the equation

dx

dt= A(t)x+ f(t) ; t 6= tn (n = 1, 2, 3, ...) (7)


x(t+n ) = Qnx(tn) + rn ; t = tn (n = 1, 2, 3, ...) (8)

has a bounded solution x(t) for which the following formula is valid

x(t) = U(t)ξ +∫ t

0U(t)P1U

−1(s)f(s)ds−∫∞

tU(t)P2U

−1(s)f(s)ds+

+∑tn<t

U(t)P1U−1(t+n )rn −

∑tn≥t

U(t)P2U−1(t+n )rn ,

(9)

where ξ ∈ X1 (X1 = ξ ∈ X : supt≥0

‖U(t)ξ‖ <∞).

Proof. We shall estimate the norm of the integrals and sums in (9).

‖∫ t

0

U(t)P1U−1(s)f(s)ds‖ ≤

∫ t

0

‖U(t)P1U−1(s)‖‖f(s)‖ds

≤M

∫ t

0

h(t)h−1(s)‖f(s)‖ds ≤M

∫ t

0

(h(t)h−1(s))1q (h(t)h−1(s))

1p‖f(s)‖ds

≤M(

∫ t

0

h(t)h−1(s)ds)1q (

∫ t

0

h(t)h−1(s)‖f(s)‖pds)1p ≤MM

1q

1 ‖f‖Lp(h)

‖∫ ∞

t

U(t)P2U−1(s)f(s)ds‖ ≤

∫ ∞

t

‖U(t)P2U−1(s)‖‖f(s)‖ds

≤M(

∫ ∞

t

k−1(t)k(s)ds)1q (

∫ ∞

t

k−1(t)k(s)‖f(s)‖pds)1p ≤MM

1q

2 ‖f‖Lp(k)

‖∑tn<t

U(t)P1U−1(t+n )rn‖ ≤M

∑tn<t

h(t)h−1(t+n )‖rn‖

≤M(∑tn<t

h(t)h−1(t+n ))1q (∑tn<t

h(t)h−1(t+n )‖rn‖p)1p

≤MM1

1q ‖rn‖lp(h)

‖∑tn≥t

U(t)P2U−1(t+n )rn‖ ≤M

∑tn≥t

k−1(t)k(t+n )‖rn‖

≤MM2

1q ‖rn‖lp(k).

It is immediately verified that the function x(t) is a solution of the nonhomoge-neous impulsive equation (7), (8).


We define the operator G with the formula

Gf(t) =

∫ ∞

0

g(t, s)f(s)ds ; (10)

where

g(t, s) =

U(t)P1U

−1(s) ; 0 ≤ s < t−U(t)P2U

−1(s) ; 0 ≤ t ≤ s(11)

For ξ ∈ Lp(h) ∩ Lp(k) we set |ξ|Lp(h)∩Lp(k) = max‖ξ‖Lp(h), ‖ξ‖Lp(k).By L∞ = L∞([0,∞), X) we denote the space of all essential bounded functions

f : [0,∞) → X.

Lemma 1. . Let 1 ≤ p ≤ ∞. Let the following conditions be satisfied:1. The impulsive equation (1), (2) is (h, k) - dichotomous.2. Condition (H1) holds.Then the operator G defined with (10) maps Lp(h) ∩ Lp(k) into Lp(h) ∩ Lp(k) ∩

L∞([0,∞), X) and for f ∈ Lp(h) ∩ Lp(k) the following estimates are valid

‖Gf‖L∞ ≤M(M1q

1 +M1q

2 )|f |Lp(h)∩Lp(k) (12)

‖Gf‖Lp(h) ≤ 21qMM1(1 + (

M2

M1

)1q )|f |Lp(h)∩Lp(k) (13)

‖Gf‖Lp(k) ≤ 21qMM2(1 + (

M1

M2

)1q )|f |Lp(h)∩Lp(k) (14)

where 1p

+ 1q

= 1.

Proof. We shall prove that G : Lp(h) ∩ Lp(k) → L∞([0,∞), X)and also theestimate (12). We estimate ‖Gf(t)‖ by means of Holder’s inequality

‖Gf(t)‖ ≤∫ ∞

0

‖g(t, s)f(s)‖ds ≤M

∫ t

0

h(t)h−1(s)‖f(s)‖ds+M∫ ∞

t

k−1(t)k(s)‖f(s)‖ds

≤M(

∫ t

0

h(t)h−1(s)ds)1q (

∫ t

0

h(t)h−1(s)‖f(s)‖pds)1p

+M(

∫ ∞

t

k−1(t)k(s)ds)1q (

∫ ∞

t

k−1(t)k(s)‖f(s)‖pds)1p

≤MM1q

1 ‖f‖Lp(h) +MM1q

2 ‖f‖Lp(k) ≤M(M1q

1 +M1q

2 )max‖f‖Lp(h), ‖f‖Lp(k)We have

‖Gf(t)‖ ≤ α1(t) + α2(t),where

α1(t) = M

∫ t

0

h(t)h−1(s)‖f(s)‖ds and α2(t) = M

∫ ∞

t

k−1(t)k(s)‖f(s)‖ds.


For the functions α1(t) and α2(t) the estimates

α1(t) = M

∫ t

0

h(t)h−1(s)‖f(s)‖ds ≤MM1q

1 ‖f‖Lp(h)

and

α2(t) = M

∫ ∞

t

k−1(t)k(s)‖f(s)‖ds ≤MM1q

2 ‖f‖Lp(k).

are valid. We shall prove that Gf ∈ Lp(h)

(h(t)

∫ t

0

h−1(s)‖Gf(s)‖pds)1p ≤ (h(t)

∫ t

0

h−1(s)(α1(s) + α2(s))pds)

1p

≤ 21q (h(t)

∫ t

0

h−1(s)αp1(s)ds)

1p + 2

1q (h(t)

∫ t

0

h−1(s)αp2(s)ds)

1p

≤ 21q (h(t)

∫ t

0

h−1(s)MpMpq

1 ‖f‖pLp(h)ds)

1p + 2

1q (h(t)

∫ t

0

h−1(s)MpMpq

2 ‖f‖pLp(k)ds)

1p

≤ 21qMM

1q

1 (h(t)

∫ t

0

h−1(s)ds)1p‖f‖Lp(h) + 2

1qMM

1q

2 (h(t)

∫ t

0

h−1(s)ds)1p‖f‖Lp(k)

≤ 21qMM

1q

1 M1p

1 ‖f‖Lp(h) + 21qMM

1q

2 M1p

1 ‖f‖Lp(k)

≤ 21qMM1‖f‖Lp(h) + 2

1qMM

1q

2 M1− 1

q

1 ‖f‖Lp(k).

Hence

‖Gf‖Lp(h) ≤ 21qMM1(1 + (

M2

M1

)1q )max‖f‖Lp(h), ‖f‖Lp(k)

which implies estimate (13).We shall prove that Gf ∈ Lp(k).

(k−1(t)

∫ ∞

t

k(s)‖Gf(s)‖pds)1p ≤ (k−1(t)

∫ ∞

t

k(s)(α1(s) + α2(s))pds)

1p

≤ 21q (k−1(t)

∫ ∞

t

k(s)αp1(s)ds)

1p + 2

1q (k−1(t)

∫ ∞

t

k(s)αp2(s)ds)

1p

≤ 21qMM

1q

1 (k−1(t)

∫ ∞

t

k(s)ds)1p‖f‖Lp(h) + 2

1qMM

1q

2 (k−1(t)

∫ ∞

t

k(s)ds)1p‖f‖Lp(k)

≤ 21qMM

1q

1 M1p

2 ‖f‖Lp(h) + 21qMM

1q

2 M1p

2 ‖f‖Lp(k)

≤ 21qMM

1q

1 M1− 1

q

2 ‖f‖Lp(h) + 21qMM2‖f‖Lp(k).


Hence

‖Gf‖Lp(k) ≤ 21qMM2(1 + (

M1

M2

)1q )max‖f‖Lp(h), ‖f‖Lp(k)

which implies estimate (14).

We define the operator G in the set lp(h) ∩ lp(k) by means of the equality

Gr(t) =∞∑

n=0

gn(t)rn , (15)

where

gn(t) =

U(t)P1U

−1(t+n ) ; tn < t−U(t)P2U

−1(t+n ) ; t ≤ tn.(16)

For ξ ∈ lp(h) ∩ lp(k) we set |ξ|lp(h)∩lp(k) = max‖ξ‖lp(h); ‖ξ‖lp(k).

Lemma 2. . Let 1 ≤ p ≤ ∞. Let the following conditions be satisfied:1. The impulsive equation (1), (2) is (h, k) - dichotomous.2. Conditions (H1) and (H2) hold.Then the operator G defined by (15) maps lp(h) ∩ lp(k) into Lp(h) ∩ Lp(k) ∩

L∞([0,∞), X) and for r ∈ lp(h) ∩ lp(k) the following estimates are valid

‖Gr‖L∞ ≤M(M1

1q + M2

1q )|rn|lp(h)∩lp(k) (17)

‖Gr‖Lp(h) ≤ 21qMM

1p

1 (M1

1q + M2

1q )|rn|lp(h)∩lp(k) (18)

‖Gr‖Lp(k) ≤ 21qMM

1p

2 (M1

1q + M2

1q )|rn|lp(h)∩lp(k) (19)

where 1p

+ 1q

= 1.

Proof. We shall prove that G : lp(h) ∩ lp(k) → L∞([0,∞), X) and also theestimate (17). Let r = rn∞n=1 ∈ lp(h) ∩ lp(k).Then

‖Gr(t)‖ ≤ α1(t) + α2(t),where

α1(t) = M∑tn<t

h(t)h−1(t+n )‖rn‖ and α2(t) = M∑tn≥t

k−1(t)k(t+n )‖rn‖.

For ˜α1(t) and ˜α2(t) we obtaine the estimates

α1(t) = M∑tn<t

(h(t)h−1(t+n ))1q (h(t)h−1(t+n ))

1p‖rn‖

≤M(∑tn<t

h(t)h−1(t+n ))1q (∑tn<t

h(t)h−1(t+n )‖rn‖p)1p ≤MM1

1q ‖rn‖lp(h)


α2(t) = M∑tn≥t

(k−1(t)k(t+n ))1q (k−1(t)k(t+n ))

1p‖rn‖

≤M(∑tn≥t

k−1(t)k(t+n ))1q (∑tn≥t

k−1(t)k(t+n )‖rn‖p)1p ≤MM2

1q ‖rn‖lp(k).

Hence

‖Gr(t)‖ ≤M(M1

1q + M2

1q )max‖rn‖lp(h); ‖rn‖lp(k).

We shall prove that G : lp(h) ∩ lp(k) → Lp(h) and also the estimate (18).

(h(t)

∫ t

0

h−1(s)‖Gr(s)‖pds)1p ≤ (h(t)

∫ t

0

h−1(s)(α1(s) + α2(s))pds)

1p

≤ 21q (h(t)

∫ t

0

h−1(s)α1p(s)ds)

1p + 2

1q (h(t)

∫ t

0

h−1(s)α2p(s)ds)

1p

≤ 21q (h(t)

∫ t

0

h−1(s)MpM1

pq ‖rn‖p

lp(h)ds)1p + 2

1q (h(t)

∫ t

0

h−1(s)MpM2

pq ‖rn‖p

lp(k)ds)1p

≤ 21qMM1

1q (h(t)

∫ t

0

h−1(s)ds)1p‖rn‖lp(h) + 2

1qMM2

1q (h(t)

∫ t

0

h−1(s)ds)1p‖rn‖lp(k)

≤ 21qMM1

1qM

1p

1 ‖rn‖lp(h) + 21qMM2

1qM

1p

1 ‖rn‖lp(k)

≤ 21qMM

1p

1 (M1

1q + M2


We shall prove that G : lp(h) ∩ lp(k) → Lp(k) and also the estimate (19).

(k−1(t)

∫ ∞

t

k(s)‖Gr(s)‖pds)1p ≤ (k−1(t)

∫ ∞

t

k(s)(α1(s) + α2(s))pds)

1p

≤ 21qMM1

1q (k−1(t)

∫ ∞

t

k(s)ds)1p‖rn‖lp(h) + 2

1qMM2

1q (k−1(t)

∫ ∞

t

k(s)ds)1p‖rn‖lp(k)

≤ 21qMM1

1qM

1p

2 ‖rn‖lp(h) + 21qMM2

1qM

1p

2 ‖rn‖lp(k)

≤ 21qMM

1p

2 (M1

1q + M2


Lemma 3. . Let the following conditions be satisfied:1. 1 ≤ p ≤ ∞.2. The function A(t) is piecewise continuous from the left with discontinuities of

the first kind at points tn , n = 1, 2, 3, ....3. The equation (1),(2) is (h, k) - dichotomous.Then for any nonnegative function w ∈ L1

p(h) ∩ L1p(k) and for any sequence of

nonegative numbers wn∞n=1 ∈ l1p(h) ∩ l1p(k) the set of functions

A = Gf + Grn : ‖f(t)‖ ≤ w(t) ; ‖rn‖ ≤ wn , n = 1, 2, 3, ...

is uniformly bounded and equicontinuous on each interval (tn, tn+1] (n = 1, 2, 3, ...).


By C = C([0,∞), X) we denote the space of all functions f : [0,∞) → X whichare continuous for t 6= tn and such that at the points t = tn they have discontinuitiesof the first kind and are continuous from the left. With respect to the metric

ρ(x, y) = ‖|x− y‖|

where

‖|z‖| =∞∑

n=1

2−n

suptn<t≤tn+1

‖z(t)‖

1 + suptn<t≤tn+1

‖z(t)‖

the space C is locally convex.

Lemma 4. . Let the following conditions be satisfied:1. Conditions 1, 2 and 3 of Lemma 3 hold.2. The set K is centrally symetric convex compact subset of X.Then for any nonnegative function w ∈ L1

p(h) ∩ L1p(k) and for any sequence of

nonegative numbers wn∞n=1 ∈ l1p(h) ∩ l1p(k) the set of functions

A(K) = Gf + Grn : w−1(t)f(t) ∈ K (0 ≤ t <∞) , w−1n rn ∈ K(n = 1, 2, 3, ...)

is compact in C.

For the proof of Lemma 3 and Lemma 4 see [1].

Theorem 2. . Let the following conditions be satisfied:1. The function A(t) is piecewise continuous from the left with discontinuities of

the first kind at points tn , n = 1, 2, 3, ....2. The impulsive equation (1),(2) is (h, k) - dichotomous.3. Conditions (H1) and (H2) hold.4. The inequality

‖f(t, x)‖ ≤ ψr(t) (‖x‖ ≤ r, 0 ≤ t <∞) (20)

is valid, where ψr ∈ L1p(h) ∩ L1

p(k) and the set

K(r) = ψ−1r (t)f(t, x) : 0 ≤ t <∞, ‖x‖ ≤ r (21)

is relativly compact.5. The sequence Rn∞n=1 satisfies the inequalities

‖Rnx‖ ≤ χn(r), (‖x‖ ≤ r, n = 1, 2, 3, ...) (22)

where χn(r) ∈ l1p(h) ∩ l1p(k) and the set

R(r) = χ−1n (r)Rnx : n = 1, 2, 3, ..., ‖x‖ ≤ r (23)


is relatively compact in X.6. The inequality

(M1q

1 +M1q

2 )|ψr+ρ|L1p(h)∩L1

p(k) + (M1

1q + M2

1q )|χn(r + ρ)|l1p(h)∩l1p(k) ≤

ρ

M(24)

holds for some ρ > 0.Then the linear impulsive equation (1), (2) is Lp(h, k) - equivalent to the non-

linear impulsive differential equation (3),(4).

Proof. Each bounded solution y(t) of the linear impulsive equation (1), (2) hasthe form

y(t) = U(t)ξ (25)

where ξ ∈ X1 (the subspace X1 consists of all ξ for which the function U(t)ξ isbounded for 0 ≤ t < ∞). Let x(t) (0 ≤ t < ∞) be a bounded solution of (3),(4). Then x(t) is a solution of the nonhomogeneous linear equation (7), (8) forf(t) = f(t, x(t)) and rn = Rnx(tn).Hence the function x(t) satisfies the nonlinearintegral equation

x(t) = U(t)ξ +Gf(t, x(t)) + G(Rnx(tn))(t). (26)

Conversely, each bounded solution of the nonlinear integral equation (26) is a solu-tion of the nonlinear impulsive equation (3), (4).

We setz(t) = x(t)− U(t)ξ (27)

and rewrite equation (27) in the form

z(t) = Gf(t, U(t)ξ + z(t)) + G(Rn(U(tn)ξ + z(tn)))(t). (28)

We shall show that for ξ ∈ X1 equation (28) has a bounded solution z(t). Forthis purpose we shall consider the operator defined by the formula

Fz(t) = Gf(t, U(t)ξ + z(t)) + G(Rn(U(tn)ξ + z(tn)))(t) (29)

on the setD(ρ) = z(t) ∈ C([0,∞), X), ‖z(t)‖ ≤ ρ, 0 ≤ t <∞. (30)

In view of Lemmas 1 and 2 and conditions 3 and 4 of Theorem 2 for ‖U(t)ξ‖ ≤ rand z ∈ D(ρ) we obtain the following estimates forFz

‖Fz(t)‖ ≤ ‖Gf(t, U(t)ξ + z(t))‖+ ‖G(Rn(U(tn)ξ + z(tn)))(t)‖

≤M(M1q

1 +M1q

2 )|ψr+ρ|L1p(h)∩L1

p(k) +M(M1

1q + M2

1q )| χn(r + ρ)|l1p(h)∩l1p(k)


From Lemma 3 and 4 and condition 5 of Theorem 2 we obtain the estimate

‖Fz(t)‖ ≤ ρ (0 ≤ t <∞, z ∈ D(ρ)).

Hence the set D(ρ) is invariant with respect to the operator F . By means of theSchauder-Tychonoff theorem we shall show that the operator F has a fixed point inthe set D(ρ). The set D(ρ) is a closed and convex subset of the space C([0,∞), X).From Lemmas 3 and 4, and the compactness of the sets K(r) and R(r) definedrespectivly by (21) and (23) there follows the compactness of the set FD(ρ) in thespace C([0,∞), X).

We shall establish the continuity of the operator F . Let zk(t)∞k=1 ⊂ D(ρ) bea sequence tending to z(t) ∈ D(ρ) in the space C([0,∞), X). Then the sequinceuk(t) = f(t, U(t)ξ + zk(t))∞k=1 tends to f(t, U(t)ξ + z(t)) for any t, and the se-quence vk = Rn(U(tn)ξ+zk(tn))∞n=1 tends to the sequence Rn(U(tn)ξ+z(tn))∞n=1

coordinate-wise (n = 1, 2, 3, ...). From (20) and (22 ) the inequalities

‖uk(t)‖ ≤ ψr+ρ(t) (0 ≤ t <∞)

‖vk(t)‖ ≤ χn(r + ρ) (n = 1, 2, 3, ...).

follow.Using Lebesgue theorem to pass to the limit under the sign of the integral,

we obtain that the sequence of functions Guk(t) tends to the function Gu(t) fort ∈ [0,∞). By the analogue Lebesgue theorem for series we obtain that the sequenceof functions Gvk(n) tends to the function Gv(n). Since the functions Guk(t)+Gvk(t)lie in a compact set, they tend to the function Gu(t) + Gv(t) in the metric of thespace C([0,∞), X) too.

Let z∗(t) be a fixed point of the operator F . Then from (20) and (22) it follows

that the function u∗ and sequence v(n)∗ ∞n=1 defined respectively by the formula

u∗(t) = f(t, U(t)ξ + z∗(t)) , v(n)∗ = Rn(u(tn)ξ + z∗(tn))

lie in Lp(h) ∩ Lp(k) and lp(h) ∩ lp(k) respectively.From Lemmas 3 and 4 it follows that the function z∗(t) = Gu∗(t) + G(v∗(n))(t)

also belongs to the set Lp(h)∩Lp(k) . Thus the differance z∗(t) between the boundedsolution x∗(t) = U(t)ξ + z∗(t) of the nonlinear impulsive equation (3), (4) and thebounded solution U(t)ξ of the linear impulsive equation lies in the set Lp(h)∩Lp(k). Example.

Let dimX = 2 and t ≥ 12, tn = n (n = 1, 2, 3, ...). We consider the equation

dx

dt= A(t)x ; t 6= n (n = 1, 2, 3, ...) (31)


x(n+) = Qnx(n) ; (n = 1, 2, 3, ...) (32)

where

x(1

2) =

(ξ1ξ2

)

x(t) =

(x1(t)

x2(t)

), A(t) =

( −α 0

0 α

), Qn =

( nn+1

0

0 1

)and α > 0, n = 1, 2, 3, ....

Then the evolutionary operator V (t) of the ordinary equation (31)is

V (t) =

(e−αt 0

0 eαt

).

Let the projectors P1 and P2 are

P1 =

(1 00 0

); P2 =

(0 00 1

).

Hence for the evolutionary operator U(t) of the equations (31), (32) is fulfilled

‖U(t)P1U−1(s)‖ = ‖Qn...Qm+1V (t)P1V

−1(s)‖ ≤ ‖P1V (t, s)‖ ≤ 3e−α(t−s);

1

2≤ s < t (

1

2≤ m < s ≤ m+ 1 < n < t ≤ n+ 1)

‖U(t)P2U−1(s)‖ = ‖Q−1

n+1...Q−1m V (t)P2V

−1(s)‖ ≤ ‖Q−1n+1‖...‖Q−1

m ‖‖P2V (t, s)‖ ≤

≤ m+ 1

n+ 1e−α(s−t) ≤ 3

s+ 1

t+ 1e−α(s−t);

1

2≤ t < s (

1

2≤ n < t ≤ n+ 1 < m < s ≤ m+ 1)

and M = 3, h(t) = e−αt, k(t) = (t+ 1)e−αt.

Then M1 = 1α, M2 = α+1

α2 , M1 = 11−e−α and M2 = eα

1−e−α ( 11−e−α + 1).

The equation (31), (32) is not exponentially dichotomous, but (h, k)-dichotomous.We consider and the equation

dx

dt= A(t)x+ f(t, x) ; t 6= tn (n = 1, 2, 3, ...) (33)

x(t+n ) = Qnx(tn) +Rnx(tn) ; (n = 1, 2, 3, ...) (34)


where

f(t, x) =

(e−tx1sin

2x1

e−tx2cos2x2

), Rnx = e−n

(2− sinx1

2− sinx2

).

Then ‖f(t, x)‖ ≤ e−tr, (‖x‖ ≤ r) , ‖Rnx‖ ≤ e−n.3, ψr(t) = e−tr , χn = 3e−n , ψr ∈L1

p(h) ∩ L1p(k) and χn ∈ l1p(h) ∩ l1p(k).

From Theorem 2 follows that the equation (31), (32) and (33), (34) are Lp(h, k))-equivalence.

References

[1] Bainov D. D.,Kostadinov S. I., Zabreiko P. P., Lp-equivalence of a linear anda nonlinear impulsive differential equations in a Banach space, Proceeding ofthe Edinburgh Mathematical Society, 36 (1992), 17-33.

[2] Bainov D. D.,Kostadinov S. I., Myshkis A. D., Asymptotic Equivalence of Ab-stract Impulsive Differential Equations , International Journal of TheoreticalPhysics, vol.36, 2 (1996), 383-393

[3] Pinto M., Dichotomies and asymptotic formulas for the solutions of differintialequations, Journal of Mathematical Analysis and Applications, 195 (1995) 16-31.

A. Kosseva S. KostadinovDepartment of Mathematics Department of Mathematics and InformaticsTechnical University University of Plovdiv ”P. Hilendarski”25, Tzanko Djustabanov Str. 24, Tzar Assen Str.4000 Plovdiv,BULGARIA 4000 Plovdiv,BULGARIAE-mail: [email protected] E-mail: [email protected]

.



Existence of Bounded and PeriodicSolutions of Nonlinear Impulse

Differential Equations of the DissipativeType

At. Georgieva, St. Kostadinov

Abstract

In an arbitrary Banach space nonlinear impulse differential equations ofgeneralized dissipative type are considered. Sufficient condi-tions for the existence of bounded and periodic solutions are founded.

1 Introduction

Impulse differential equations are mathematical models of evolution processes, whichchange state in theire development at certain moments by jumps. The qualitativetheory of these equations begin with the work of Mil’man and Mishkis (1960)[7]. Inrecent years, this theory has become the focus of numerous investigations [1], [3], [4].In the present paper, we consider nonlinear impulse differential equations under theassumption that the right hand side of the corresponding ordinary equation is ofdissipative type. We denote that a special class of dissipative conditions for impulsedifferential equations in a Hilbert space is considered in [4]. Some results for ordinarydifferential equations are obtained in [6].

02000 Mathematics Subject Classification: 34A37, 34G10, 34C25 .0Key words and phrases: Impulse differential equations, periodic solutions, dissipative condition.0Received September 30, 2006

43

44 At. Georgieva, St. Kostadinov

2 Statement of the problem

Let X be an arbitrary real Banach space with norm |.| and identity operator I. Weconsider the impulse differential equation

dx

dt= f(t, x), (t 6= tn) (1)

x(t+n ) = Qn(x(tn)), (2)

where f : [a, b]×X → X is a continuous function, the operatorsQn : X → X (n = 1, 2, ...p) are continuous and a < t1 < t2 < ... < tp ≤ b. Weassume that all functions are left continuity at tn.

Let X∗ be the dual space of X and

Fx = x∗ ∈ X∗ : x∗(x) = |x|2 = |x∗|2 (x ∈ X).

Using F we introduce the semi-scalar product [6], [8]

(x, y)− = infy∗(x) : y∗ ∈ Fy. (3)

In [6] many well-known properties of the semi-scalar product are shown. Here,we remark only that if X is a Hilbert space, then the semi-scalar product in (3) isequal to the ordinary scalar product.

Remark 1. If X is a complex Banach space, we can introduce a semi-scalar productby the formula

(x, y)− = Re(x, y)− + iIm(x, y)−,

whereRe(x, y)− = infRe y∗(x) : y∗(y) ∈ R and y∗(y) = |y|2,

Im(x, y)− = infIm y∗(x) : y∗(y) ∈ R and y∗(y) = |y|2.

It is easy to show that Lemma 3.2 [6] is valid in the complex case too.

We setc = sup

a≤t≤b|f(t, 0)|.

Definition 1. [6] We say that X is a strictly convex space if x 6= y and|x| = |y| = 1 imply |λx+ (1− λ)y| < 1 for λ ∈ (0, 1).

Let ω : (a, b]× R+ → R be an arbitrary function.We introduce the conditions.H1. The inequality ω(t, 0) ≥ 0 (a < t ≤ b) holds.

Existence of Bounded and Periodic Solutions of Nonlinear Impulse ... 45

H2. The inequality

x′(t) ≤ ω(t, x) +x

2+c2

2x

has for any p > a on [p, b] only bounded solutions, where z′ = D(z) denotes the leftupper derivate of the function z at the point t ([5], [6]).

H3. (dissipative condition) [6]

(f(t, x)− f(t, y), x− y)− ≤ ω(t, |x− y|)|x− y| (t ∈ (a, b]; x, y ∈ X).

Remark 2. It may be noted that the functions ω(t, x) = κx (κ-constant) orω(t, x) = x

t, t ∈ (a, b]. satisfy the conditions H1 and H2.

The aim of the present paper is the search for sufficient conditions for the ex-istence of bounded and periodic solutions of the impulse equation (1), (2) if theconditions H1−H3 are fulfilled.

3 Main results

Theorem 1. Let the conditions H1 − H3 be fulfilled. Then the impulse equation(1), (2), x(a) = ξ has for any ξ ∈ X an unique solution ϕ bounded on [a, b], forwhich ϕ(a) = ξ.

Proof. It is sufficient to show the assertion for an ordinary equation, i.e. Qn =I (n = 1, 2, ...p).

For any k ∈ N there exists a function xk(t) such that

Dxk(t) = f(t, xk(t)) + εk(t), |εk(t)| ≤1

k(a ≤ t ≤ b).

For arbitrary m,n ∈ N we set

ϕm,n(t) = |xm(t)− xn(t)|2, ψm,n(t) =√ϕm,n(t), |εk| =

1

k.

Now from Lemma 3.2 [6], we obtain

ψ′m,nψm,n ≤ (x′m − x′n, xm − xn)− = (f(t, xm)− εm − f(t, xn) + εn, xm − xn)− ≤≤ (f(t, xm)− f(t, xn), xm − xn)− + (εm − εn, xm − xn)− ≤≤ ω(t,

√ϕm,n)

√ϕm,n + |εn − εm|

√ϕm,n

and

ϕ′m,n ≤ 2ω(t,√ϕm,n)

√ϕm,n + 2(

1

m+

1

n)√ϕm,n. (4)


From conditionH1 it follows that exists a constant b1 ∈ (a, b] such that the inequality

y′ ≤ ω(t, y) + δ (t ∈ (a, b])

has for any δ > 0 a solution yδ for which

limδ→0

yδ(t) = 0 (t ∈ (a, b1]).

Hence ϕm,n(t) convergens uniformly on a subinterval [a, b1] of [a, b]. In what followswe shall prove that b1 = b. Assume b1 < b. Then

b1∫a

|f(t, x(t))|dt = ∞.

We have

(f(t, u), u)− = (f(t, u)− f(t, 0) + f(t, 0), u)− ≤≤ (f(t, u)− f(t, 0), u)− + (f(t, 0), u)− ≤

≤ ω(t, |u|)|u|+ |f(t, 0)||u| ≤ ω(t, |u|)|u|+ 12|u|2 + c2

2.

(5)

Set ψ(t) = |x(t)|2. Then from Lemma 3.2 [6] and (5) we get

ψ′ ≤ 2(f(t, x), x)− ≤ 2ω(t, |x|)|x|+ |x|2 + c2,

i.e.

ψ′ ≤ 2ω(t,√ψ)

√ψ + ψ + c2. (6)

Condition H2 yields that the function ψ(t) is bounded on [a, b]. Hence b1 = b.The uniqueness of the solution x(t) can be obtained by standard verification.

Remark 3. The conditions H2 and H3 are necessary only for the proof of theextension of the function ψ(t) on the whole interval [a, b]. If the function f(t, x)isbounded on every bounded subset of [a, b] × X, then the assertion of Theorem 1 istrue without condition H2. The dissipative condition H3 does not guarantee theboundness of f(t, x). Also, in the very special case when X is a Hilbert space, fdoes not depend on t and ω(t, x) = κx (κ− constant) [9]. It must be noted that evenwithout impulses, Theorem 1 is a generalization of Theorem 3.2 [6].

Let the functions ϕ1, ϕ2 : [a, b] → X be differentiable for t 6= tn. Furthermore,let tn be discontinuous points of first kind at which ϕ1 and ϕ2 are continuous fromthe left.


Lemma 1. Let the following conditions be fulfilled:1. The function ω(t, x) has the form ω(t, x) = α(t)x, where α : R → R, is integrablyon every bounded subinterval of R.2. Condition H3 holds.3. There are constants pn and hn, such that

|ϕ1(t+n )− ϕ2(t

+n )| ≤ pn|ϕ1(tn)− ϕ2(tn)|+ hn.

4. The space X is strictly convex.Then the following estimate is valid

|ϕ1(t)− ϕ2(t)| ≤ |ϕ1(a)− ϕ2(a)|∏

a<tk<t

pke

t∫a

α(s)ds+

+t∫

a

|b(s)|∏

s<tk<t

pke

t∫s

α(τ)dτds+

∑a<tk<t

hk

∏tk<ti<t

pie

t∫tk

|b(τ)|dτ

,

(7)

where b(t) = ϕ′1(t)− f(t, ϕ1(t))− ϕ′2(t) + f(t, ϕ2(t)).

Proof: For t 6= tn it follows by [6]

u(t)u′(t) ≤ (ϕ′1(t)− ϕ′2(t), ϕ1(t)− ϕ2(t))−, (8)

where u(t) = |ϕ1(t) − ϕ2(t)|. From (8), condition H3 and condition 4 of Lemma 1we obtain

u′(t) ≤ (ϕ′1(t)− ϕ′2(t),ϕ1(t)−ϕ2(t)

u(t))−≤

≤ (b(t), ϕ1(t)−ϕ2(t)u(t)

)−

+ (f(t, ϕ1)− f(t, ϕ2),ϕ1(t)−ϕ2(t)

u(t))−≤

≤ |b(t)|+ α(t)u(t).

(9)

For t = tn condition 3 of Lemma 1 yields

u(t+n ) ≤ pnu(tn) + hn. (10)

The assertion of Lemma 1 follows from (9), (10) and [2].

Corollary 1. Let the conditions H1, H2 and the conditions of Lemma 1 be fulfilled.Then for the solution x(t) of the impulse differential equation (1), (2) we have thefollowing estimate

|x(t)| ≤ |x(a)|∏

a<tk<t

pke

t∫a

α(s)ds+

t∫a

|f(s, 0)|∏

s<tk<t

pke

t∫s

α(τ)dτds+

+∑

a<tk<t

hk

∏tk<ti<t

pie

t∫tk

|f(τ,0)|dτ

.


The proof of Corollary 1 follows immediately from (7) for ϕ1(t) = x(t),ϕ2(t) ≡ 0 (t ∈ [a, b]).

Now we consider the impulse differential equation (1), (2) on the whole axis andsuppose that tn < tn+1 (n ∈ Z) and lim

n→±∞tn = ±∞. Furthermore let the operators

Qn : X → X (n ∈ Z) be continuous.

Definition 2. We say that the impulse differential equation (1), (2) is (T, r) periodicif there exist T > 0 and r ∈ N, such that

f(t, x) = f(t+ T, x), t ∈ R, x ∈ X and tn+r = tn + T, Qn+r = Qn (n ∈ Z).

Theorem 2. Let the following conditions be fulfilled.1. The conditions H1−H3 holds.2. The impulse equation (1), (2) is (T, r) periodic.3. ω(t, x) = α(t)x, where α : R → R is a integrably function.4. |Qnx−Qny| ≤ qn|x− y| (x, y ∈ X, n = 1, 2, ...p, where p is the number of pointstn in the interval [a, a+ T ] a ∈ R).

5.∏

a≤tk<a+T

qke

a+T∫a

α(s)ds< 1.

Then the impulse equation (1), (2) has a unique T−periodic solution ϕ0(t).If in addition the space X∗ is strictly convex and

6. |Qnx| ≤ pn|x|+ hn (x ∈ X, n = 1, 2, ...p), then for the functionψ(t) = |ϕ0(t)| the following estimate is valid

ψ(t) ≤ ψ(a)∏

a<tk<t

pke

t∫a

α(s)ds+ c

t∫a

∏s<tk<t

pke

t∫s

α(τ)dτds+

+∑

a<tk<t

hk

∏tk<ti<t

pie

t∫tk

α(τ)dτ

,

(11)

where c = supa≤t≤a+T

|f(t, 0)|.

Proof: From Theorem 1 it follows that for any ξ ∈ X there exists a boundedsolution x(t, ξ) of the Cauchy problem on [a, a+ T ] such that x(a, ξ) = ξ.

We consider the operator P : X → X defined by

Pξ = x(a+ T, ξ).

Let x(t), y(t) be two solutions of (1), (2) with initial values ξ1 and ξ2 respectively.We set ϕ(t) = |x(t)− y(t)|2. From Lemma 3.2 [6] for t 6= tn we obtain

ϕ′ = 2(f(t, x(t))− f(t, y(t)), x(t)− y(t))−


and hence for t 6= tn the relations

ϕ′ ≤ 2ω(t, |x− y|)|x− y| = 2α(t)|x− y|2 = 2α(t)ϕ(t) (12)

hold. Condition 4 of Theorem 2 yields

ϕ(t+n ) = |x(t+n )− y(t+n )|2 = |Qnx(tn)−Qny(tn)|2 ≤ q2n|x(tn)− y(tn)|2,

i.e.ϕ(t+n ) ≤ q2

nϕ(tn). (13)

From (12) and (13) we obtain the estimate

ϕ(t) ≤ ϕ(a)∏

a≤tn<t

q2ne

2t∫

aα(s)ds

(14)

from which, for t = a+ T , it follows that

ϕ(a+ T ) ≤ ϕ(a)∏

a≤tn<a+T

q2ne

2a+T∫a

α(s)ds.

Condition 5 of Theorem 2 and the last inequality yield that the operator P is acontraction and has a unique fixed point ξ. Hence the impulse equation (1), (2) hasa unique T−periodic solution ϕ0(t).

The proof of the estimate (11) is analogous to the proof of Corollary 1.

Corollary 2. Let the conditions of Theorem 2 be fulfilled and let for any t0 ∈ R

limτ→−∞

∏τ<tn<t0

q2ne

t∫τ

α(s)ds= 0. (15)

Then the periodic solution ϕ0(t) is the unique bounded solution of (1), (2) on R.

Proof: Let x(t) and y(t) be two solutions of (1), (2) which are bounded on R.Then from (14) we obtain the inequality

|x(t)− y(t)| ≤∏

τ≤tn<t

q2ne

t∫τ

α(s)ds|x(τ)− y(τ)|, τ ≤ t.

For τ → −∞ condition (15) implies that x(t) = y(t).

Corollary 3. Let the conditions of Theorem 2 be fulfilled and letQn0 = 0 (n = 1, 2, ...p). Then for c = 0 it follows that ϕ0 = 0.


Proof: From (12) and the relations

ψ(t+n ) = |ϕ0(t+n )| = |Qnϕ0(tn)| ≤ qn|ϕ0(tn)| = qnψ(tn)

we obtain the estimate

ψ(t) ≤ ψ(a)∏

a<tn<t

qne

t∫a

α(s)ds. (16)

For t = a+ T it follows that

ψ(a+ T ) = ψ(t) ≤ ψ(a)∏

a<tn<a+T

qne

a+T∫a

α(s)ds

and from condition 5 of Theorem 2 we obtain ψ(t) ≡ 0 (a ≤ t ≤ a+ T ).

Corollary 4. In case of α(t) = κ (κ− constant), after a partial integration of theintegral on the right-hand side in (12) we get the estimate

ψ(t) ≤ (ψ(a) +c

κ)

∏a<tn<t

qneκt − c

κ, (a ≤ t ≤ a+ +T ). (17)

Corollary 5. From (17) and for qn = 1 (n = 1, 2, ...p) we obtain

ψ(t) ≤ − cκ.

Proof: From

ψ(t) ≤ (ψ(a) +c

κ)eκt − c

κ

for t = a+ T it follows that

ψ(a+ T ) ≤ (ψ(a) +c

κ)eκ(a+T ) − c

κ.

Remark 4. It is easy to show that the assertions of Theorem 1 and Theorem 2 arevalid if the space X is a Hilbert space and the dissipative condition H3 is replacedby the following condition:

(f(t, x)− f(t, y), U(x− y))− ≤ ω(t, |x− y|)|x− y| (t ∈ (a, b]; x, y ∈ X),

where U is a linear bounded operator for which inf(Ux, x) : |x| = 1 > 0.

In this case it is sufficient to consider the equivalent norm |x|u =√

(Ux, x) andthe scalar product (x, y)u = (Ux, y).


4 Examples

Example 1. Let X be an arbitrary real Bahach space and

f(t, x) =| sin t|

| sin t|+ cx (c > 0), ω(t, x) =

x

t.

We take a = 1, ti = i+ 1 (i = 1, 2, ..., 6), tn+6 = tn + 2π (n = 1, 2, ...) and

Qnξ = pnξ + an cos |ξ|+ qn (ξ ∈ X),

where the sequences pn, an, and qn satisfy the conditions

pn+6 = pn, an+6 = an, qn+6 = qn (n = 1, 2, ...).

The conditions H1, H2 are fulfilled. From the properties of the semi-scalar product(see [6], [8]), it follows that the condition H3 is fulfilled too with c ≤ 1 + 2π. Itis not hard to check that the operators Qn are Lipschitz continuous with constantsqn = pn + an. Condition 5 of Theorem 2 is fulfilled for

6∏i=1

(pn + an) <1

1 + 2π

The example 1 can be easy generalized in many directions.

The strongest condition of Theorem 2 is condition 5. The following exampleshows, that condition 5 cannot be omitted.

Example 2. Let X = R. We consider the impulse equation

x′ = −x, (t 6= 2n− 1) (18)

x(t+n ) = Qx(tn), (n ∈ Z) (19)

where

Qx =px

1 + x2, p > 0.

Let T ∈ (1, 2) be an arbitrary number. Then the dissipative condition H3 is fulfilledfor ω(t, x) = κx, κ = −1. Condition 5 of Theorem 2 is fulfilled for p < eT

and hence the trivial solution is the unique T−periodic solution of (18), (19). Byx(1−) = x0e

−1 we have

x(1+) = Q(x0e−1) =

px0e−1

1 + x20e−2,


i.e.

x(t) =px0e

−1

1 + x20e−2e−(t−1) =

px0e−t

1 + x20e−2

(1 ≤ t ≤ ω).

For t = T we obtain

x(T ) =px0e

−T

1 + x20e−2

= x0. (20)

From (20), it follows that for p < eT the equation (18), (19) has a unique T−periodicsolution, but for p > eT we can get three T−periodic solutions.

If Qn = I, i.e. we have an ordinary equation, condition 5 of Theorem 2 is fulfilledif

T∫0

α(s)ds < 0.

In general, the properties of the impulse differential equations (1), (2) are differentfrom the properties of the ordinary one. In particular, if the ordinary equation hasa periodic solution, the impulse equation lacks one. It is interesting that the inversesituation is also possible.

To show that we consider the following ordinary equation in R

x′ = 1. (21)

The equation (21) does not have periodic solutions.We consider the impulse equation

x′ = 1, t 6= 2n+ 1

2(22)

x(t+n ) = f(x(tn)), (n = 0, 1, 2, ...) (23)

where the function f is Lipschitz continuous with constant q ∈ (0, 1). The solutionof (22), (23) with 1

2< t ≤ 1 has the form

x(t) = t− 1

2+ f(

1

2+ x0).

Must be validx(1) = x0

for (22), (23) to have a 1−periodic solution, i.e.

1

2+ f(

1

2+ x0) = x0. (24)

From the fixed point theorem of Banach it follows that this equation has a uniquesolution, i.e. the impulse equation (22), (23) has exactly one periodic solution.


Example 3. We consider the case when

f(t, x) = Ax+ g(t, x), Qnx = Pnx+ hn(x),

where A : D(A) → X, Pn : D(Pn) → X are linear (D(A), D(Pn) lies dense in X,n = 1, 2, ...).

Let the following conditions be fulfilled.1. The impulse equation (1), (2) is (T, r) periodic.2. For some σ ∈ R, ((A− σI)z, z)− ≤ 0 is valid.

3. (g(t, x)− g(t, y), x− y)− ≤ α1(t)|x− y|2 for some nonnegative function α1(t),x, y ∈ X.

4. |Pn(x− y)| ≤ pn|x− y|, |hn(x)− hn(y)| ≤ qn|x− y| (x, y ∈ X),where ∏

a≤tk<a+T

(pk + qk)e

a+T∫a

(α1(t)+σ)dt< 1.

Then the conditions of Theorem 2 are fulfilled.We will show that the condition 2 is fulfilled for the following operator. Let Ω be

a bounded domain with smooth boundary in Rn, 1 < p <∞ and let

A(x,D)u = −n∑

k,l=1

∂

∂xk

(ak,l(x)∂u

∂xl

),

where |ak,l(x)| ≤M for 1 ≤ k, l ≤ n and x ∈ Ω.We take Au = σu− Apu, where Apu = A(x,D)u, D(Ap) =

= W 2,p(Ω) ∩W 1,p0 (Ω) (see [8]).

References

[1] Bainov D.D., Kostadinov S.I., Abstract Impulse Differential Equations,Descartes Press Co., Koriyama, (1996), pp.170.

[2] Bainov D.D., Simeonov P., Integral Inequalities and Applications, Kluwer Aca-demic Publishers, (1992), pp.245.

[3] Bainov D.D., Kostadinov S.I., Myshkis A.D., Bounded and periodic solutionsof differential equations with impulse effect in a Banach space, Differential andIntegral Equations, V1, N2, (1986), pp.223-230.

[4] Bainov D.D., Kostadinov S.I., Zabreiko P.P., Monotonic impulsive differentialequations, Indian Journal pure Appl. Math., V26, N4, (1995), pp.315-320.


[5] Cartan H., Calcul Differential, Formes Differentiales, Herman, Paris, (1967).

[6] Deimling K., Ordinary Differential Equations in Banach Spaces, Lecture Notesin Mathematics, Springer Verlag, (1977), pp.136.

[7] Mil’man V.D., Myshkis A.D., On the stability of motion in the presence ofimpulses, Sib. Math. J.1, N2, (1960), pp.233-237 (in Russian).

[8] Pazy A., Semigroups of Linear Operators and Applications to Partial Differ-ential Equations, Springer Verlag, (1983), pp.279.

[9] Trubnikov Ju.V., Perov A.I., Differential Equations with Monoton Unlineari-ties, Minsk, Nauka i Technika, (1986), pp.199, (in Russian).

A. Georgieva S. KostadinovDepartment of Mathematics Department of Mathematics and InformaticsUniversity of Food Technologies University of Plovdiv ”P. Hilendarski”26, Maritza Blvd. 24, Tzar Assen Str.4002 Plovdiv,BULGARIA 4000 Plovdiv,BULGARIAE-mail: [email protected] E-mail: [email protected]


Nonlinear Semigroup for NonlinearAbstract Impulse Differential Equations

At. Georgieva, St. Kostadinov

Abstract

A semigroup induced by the Poincare map of the nonlinear abstract im-pulse differential equation

dx

dt= A(t)x + f(t, x), (t 6= tn)

x(t+n ) = Qnx(tn) + hn(x(tn)), (t = tn)

with possibly unbounded linear operators A(t) and Qn is considered.

1 Introduction

Impulse equations are useful mathematical models of many processes in all branchesof the physics, which suddenly change their state at certain moments. In the math-ematical simulation is often possibly the change to neglecte and assume that thechange takes place by jumps.

The qualitative theory of these equations has been intensively developing in thelast 5 years (see for examples [2]-[7]). In this paper are considered nonlinear abstractimpulse differential equations and semigroups induced by the the Poincare map.

02000 Mathematics Subject Classification: 34A37, 34G10, 34C25 .0Key words and phrases: Impulse differential equations, nonlinear semigroups.0Received June 15, 2006

55


2 Statement of the problem

Let X be a Banach space with norm ‖.‖ and identity I. By Lp(X),

1 ≤ p < ∞ we denote the space of all functions g : R → X for which∞∫−∞

‖g(t)‖pdt <

∞ with norm ‖g‖p = (∞∫−∞

‖g(t)‖pdt)

1p

.

We define for every continuous function k(., .) : R× R → R+ and each h > 0

Lhp,k = g : R → X;

∞∫−∞

kp(t, t− h)‖g(t)‖pdt < ∞,

‖g‖p,k = (

∞∫−∞

kp(t, t− h)‖g(t)‖pdt)

1p

.

Definition 1. A family X(t, s); t, s ∈ R, t ≥ s of (possibly nonlinear) operatorsacting on X is called an evolutionary process if it satisfies the following conditions

1. X(s, s)x = x for all s ∈ R, x ∈ X,2. X(t, s)X(s, r) = X(t, r) for all t ≥ s > r.Let k(., .) : R× R → R+ be a continuous function.The evolutionary process X(t, s); t, s ∈ R, t ≥ s is called k(t, s)-continuous if

it satisfies the conditions.3. For all t ≥ s and x, y ∈ X is valid

‖X(t, s)x−X(t, s)y‖ ≤ k(t, s)‖x− y‖ (1)

4. The function X(t, s)x is continuously continuable from(tj−1, tj)× (tk−1, tk)×X to [tj−1, tj]× [tk−1, tk]×X for every j, k = 0,±1,±2, . . . .

Remark 1. It may be noted that the ordinary case (i.e. Qn = I, hn = 0) fork(t, s) = Keω(t−s) (K > 0, ω ∈ R, t ≥ s) is detailedly considered in [1].

Every evolutionary process X(t, s); t, s ∈ R, t ≥ s induces the so-called evolu-tion semigroup T h; h > 0 defined by the relation

(T hv)(t) = X(t, t− h)v(t− h) (2)

for all t ∈ R, where v belong to a suitable space of functions, (see [1]).We consider the following impulse differential equation

dx

dt= A(t)x + f(t, x) for t 6= tn (3)

Nonlinear Semigroup for Nonlinear Abstract Impulse Differential Equations 57

x(t+n ) = Qnx(tn) + hn(x(tn)) for t = tn, (4)

where A(t) : D(A(t)) → X (t ∈ R) and Qn : D(Qn) → D(A(tn)) are linear (possiblyunbounded) operators (by D(T ) we denote the domain of the operator T ). The func-tions f(., .) : R × X → X and hn : X → D(A(tn))(n = 0,±1,±2, . . .) are continuous. The points of jump tn satisfy the followingconditions tn < tn+1 (n = 0,±1,±2, . . .), lim

n→±∞tn = ±∞.

Furthermore we assume that all considered functions are continuous on the left.Obviously the impulse equation (3), (4) induces an evolutionary process by the

aim of the Poincare map, i.e.

X(t, s)ξ = x(t; s, ξ), (5)

where x(t; s, ξ) is the solution of (3), (4) starting in ξ for t = s.Let U(t, s) be the linear evolutionary process induced by the ordinary equation.

dx

dt= A(t)x (6)

We setV (t, s) = U(t, tn)QnU(tn, tn−1) . . . QkU(tk, s) (7)

for tk−1 < s ≤ tk ≤ tn < t ≤ tn+1.

Lemma 1. The function x(t) = V (t, s)ξ for all ξ ∈ D(A(s)) satisfies the impulseCauchy problem

dx

dt= A(t)x for t 6= tn, (8)

x(t+n ) = Qnx(tn) for t = tn, (9)

x(s) = ξ

(n = 0,±1,±2, . . .).

Proof: Let t 6= tn. Then

∂V (t, s)

∂t= lim

h→0+

V (t + h, t)− I

hV (t, s) = A(t)V (t, s).

Hencedx

dt=

d

dt(V (t, s)ξ) = A(t)V (t, s)ξ.

Let t = tn. Then

V (t+n , s) = U(t+n , tn)Qn . . . QkU(tk, s) = QnV (tn, s).

Thereforex(t+n ) = V (t+n , s)ξ = QnV (tn, s)ξ.


Remark 2. The operator V (t, s) is bounded if one of the following conditions holds1. QnU(tn, tn−1) is a bounded operator (n = 0,±1,±2, . . .),2. U(tn+1, tn)Qn is a bounded operator (n = 0,±1,±2, . . .).

Definition 2. The linear impulse equation (8), (9)is said to be well posed if for everys ∈ R and ξ ∈ D(A(s)) the function x(t) = V (t, s)ξ is the uniquely solution of theimpulse equation (8), (9) satisfying x(s) = ξ.

Lemma 2. Suppose the linear impulse equation (8), (9)is well posed. Then everysolution x(t) of the integral equation

x(t) = V (t, s)ξ +

t∫s

V (t, η)f(η, x(η))dη +∑

s<tn<t

V (t, t+n )hn(x(tn)) (10)

for t ≥ s is a solution of the impulse equation (3), (4) starting fromξ ∈ D(A(s)) at t = s.

Lemma 2 can be proved by straightforward verification.

Definition 3. A solution x(t) of the integral equation (10) will be called a mildsolution of the impulse equation (3), (4) starting from ξ ∈ D(A(s)) at t = s.

Lemma 3. ([3], [5], [6]) Let the function ϕ : [t0,∞) → R be nonnegative, piecewisecontinuous with discontinuities of the first kind at the points ti,v : [t0,∞) → R be a continuous nonnegative function, and c ≥ 0, αi ≥ 0 beconstants. Let the following inequality holds

ϕ(t) ≤ c +∑

t0<ti<t

βiϕ(ti) +

t∫t0

ϕ(τ)v(τ)dτ (t0 ≥ 0)

Then the following estimate is valid

ϕ(t) ≤ c∏

t0<ti<t

(1 + βi)e

t∫t0

v(τ)dτ

In this paper we will consider the evolutionary process induced by the nonlinearimpulse differential equation (3), (4). We will find a generator for the semigroup T h

as a sum of the generator A of the evolutionary process associated with the linearimpulse equation (8), (9) and a nonlinear operator F , acting in the functional spaceinduced by the function f(., .).


3 Examples

3.1 Let X = R. It is not hard to check that the evolutionary process induced by theordinary equation

dx

dt= 2tx (11)

is not k(t, s) = Keω(t−s)-continuous. In fact, in this case we have

|X(t, s)ξ −X(t, s)η| = et2−s2|ξ − η|

But the evolutionary process induced by the impulse equation

dx

dt= 2tx, t 6= n

x(n+) = Qnx(n) , (n = 0,±1,±2, . . .)

forQnξ = e−ω(n+1)−(n+1)2+ω(n−1)−en

ξ = e−2ω−(n+1)2−en

ξ, (n = 0, 1, 2, . . .)

andQnξ = e−ω(n+1)−n2+ω(n−1)−en

ξ = e−2ω−n2−en

ξ, (n = −1,−2, . . .)

is k(t, s) = Ke−ω(t−s)-continuous (ω > 0) for K = K(ω) large enough. Really for0 ≤ k < s ≤ k + 1 < n < t ≤ n + 1 and each ξ, η ∈ R we have

|X(t, s)ξ −X(t, s)η| == et2−n2

e−2ω−(n+1)2−enen2−(n−1)2 . . . e(k+2)2−(k+1)2e−2ω−(k+2)2−ek+1

e(k+1)2−s2|ξ − η| ≤≤ et2−s2

e−ω(n+1)−(n+1)2+ω(n−1)−ene−ω(k+2)−(k+2)2+ωk−ek+1|ξ − η| ≤

≤ et2−ω(n+1)−(n+1)2eω(n−1)−ene−s2+ωke−ω(k+2)−(k+2)2−ek+1|ξ − η| ≤

≤ K(ω)e−ω(t−s)|ξ − η|

The case k < s ≤ k + 1 < n < t ≤ n + 1 ≤ 0 can be considered analogously.For 0 ≤ n− 1 < s ≤ n < t ≤ n + 1 we have

|X(t, s)ξ −X(t, s)η| == et2−n2

e−ω(n+1)−(n+1)2+ω(n−1)−enen2−s2|ξ − η| ≤

≤ et2−ω(n+1)−(n+1)2e−s2+ω(n−1)|ξ − η| ≤ K(ω)e−ω(t−s)|ξ − η|

The case n− 1 < s ≤ n < t ≤ n + 1 ≤ 0 is analogously considered.At last for −1 < s ≤ 0 < t ≤ 1 we get

|X(t, s)ξ −X(t, s)η| = et2e−2ω−2e−s2|ξ − η| == et2−1−ωe−s2−1−ω|ξ − η| ≤ K(ω)e−ω(t−s)|ξ − η|


3.2 Let be again X = R. We will show that the evolutionary process induced by theordinary equation

dx

dt= x +

√x (12)

does not satisfy the condition (1) and following is not k(t, s)-continuous.Indeed, in this case we have

X(t, s)ξ = (et−s2

√ξ + e

t−s2 − 1)

2

and

|X(t, s)ξ −X(t, s)η| = |(e t−s2

√ξ + e

t−s2 − 1)

2− (e

t−s2√

η + et−s2 − 1)

2| =

= |et−s(ξ − η) + 2et−s2 (e

t−s2 − 1)(

√ξ −√η)|

Hence, there does not exist a function k(t, s) with

|X(t, s)ξ −X(t, s)η| ≤ k(t, s)|ξ − η| (ξ, η ∈ R).

We will show that there are not numbers qn and functions hn(.) : R → R suchthat the impulse equation

dx

dt= x +

√x for t 6= tn,

x(t+n ) = qnx(tn) + hn(x(tn)), (n = 0,±1,±2, . . .)

does not satisfy the condition (1).Really, in this case for t0 < s ≤ t1 < t ≤ t2 and ξ, η ∈ R we have

|X(t, s)ξ −X(t, s)η| =

= |(e t−t12

√x(t+1 ) + e

t−t12 − 1)

2− (e

t−t12

√y(t+1 ) + e

t−t12 − 1)

2| =

= |et−t1(x(t+1 )− y(t+1 )) + 2et−t1

2 (et−t1

2 − 1)(√

x(t+1 )−√

y(t+1 ))|,

where x(t) = x(t; s, ξ), y(t) = x(t; s, η). It is clear, that even in this case thecondition (1) is not fulfilled.3.3 Analogously consideretions are also valid for the equation.

dx

dt= 2tx + t

√x (13)

although the right-hand side is t-dependent. In this case for t ≥ s and ξ ∈ R wehave

X(t, s) = (et2−s2

2

√ξ +

1

2e

t2−s2

2 − 1

2)2


4 Main results

We begin with a theorem which guarantees the k(t, s)-continuity of the evolutionaryprocess induced by the impulse differential equation (3), (4).

Let k(., .) : R × R → R+ be an arbitrary function with inf−∞<s≤t<∞

k(t, s) =

= N > 0 and k(t, t) ≥ 1, (t ∈ R).

Theorem 1. Let the following conditions are satisfied.1. The linear equation (8), (9) is well posed.2. The function f(t, x) is Lipschitz continuous w.r. to x i.e.,

‖f(t, x)− f(t, x)‖ ≤ L(t)‖x− y‖, (t ∈ R, x, y ∈ X).

Then there exist linear operators Qn : D(Qn) → D(A(tn)) and Lipschitz contin-uous functions hn : X → D(A(tn)), (n = 0,±1,±2, . . .) i.e.,

‖hn(x)− hn(y)‖ ≤ Hn‖x− y‖, (x, y ∈ X)

such that the impulse equation (3), (4) induces for every sequence tn∞n=−∞ a k(t, s)-continuous evolutionary process.

Proof: We consider first the case f(t, x) = 0 (t ∈ R, x ∈ X).Let ‖U(t, s)‖ ≤ M(t, s), where M(., .) : R × R → R+ is continuous function

(−∞ < s < t < ∞) and Mn = maxtn−1≤s≤t≤tn+1

M(t, s), (n = 0,±1,±2, . . .).

In case 0 < N ≤ 1 we set

Qnη =N

(Mn + 1)2 η, (η ∈ X).

Then for tk−1 < s ≤ tk ≤ tn < t ≤ tn+1

V (t, s) = U(t, tn)N

(Mn + 1)2U(tn, tn−1)N

(Mn−1 + 1)2 . . .N

(Mk + 1)2U(tk, s).

Hence for x, y ∈ D(A(s)) we have

‖V (t, s)x− V (t, s)y‖ ≤≤ M(t, tn) N

(Mn+1)2M(tn, tn−1)

N(Mn−1+1)2

. . . N(Mk+1)2

M(tk, s)‖x− y‖ =

= M(t,tn)Mn+1

NM(tn,tn−1)

(Mn+1)(Mn−1+1)N . . .

M(tk−1,tk)

(Mk−1+1)(Mk+1)N M(tk,s)

Mk+1‖x− y‖ ≤

≤ Nn−k+1‖x− y‖ ≤ N‖x− y‖ < k(t, s)‖x− y‖


In case N > 1 we set

Qnη =N−1

(Mn + 1)2 η (η ∈ X)

Then for x, y ∈ D(A(s)) and tk−1 < s ≤ tk ≤ tn < t ≤ tn+1 we have

‖V (t, s)x− V (t, s)y‖ ≤

≤ M(t,tn)Mn+1

N−1 M(tn,tn−1)

(Mn+1)(Mn−1+1)N−1 . . .

M(tk−1,tk)

(Mk−1+1)(Mk+1)N−1 M(tk,s)

Mk+1‖x− y‖ ≤

≤ N−(n−k+1)‖x− y‖ ≤ N‖x− y‖

Let f(t, x) 6= 0. We denote

maxtn−1≤s≤t≤tn+1

N

t∫s

L(ξ)dξ = pn > 0, (n = 0,±1,±2, . . .)

In case 0 < N ≤ 1 we set

Qnη =N

2(Mn + 1)2 e−pn η, (η ∈ X).

Let hn : X → D(A(tn)) are arbitrary Lipschitz continuous functions with Lipschitzconstants Hn < 1

Nepn , (n = 0,±1,±2, . . .).

Then for tk−1 < s ≤ tk ≤ tn < t ≤ tn+1 we have

‖V (t, s)x‖ ≤ (N

2)n−k+1

e−pn−...−pk‖x‖ < Ne−pn−...−pk‖x‖ < N‖x‖

and hence for x, y ∈ D(A(s)) we obtain

‖X(t, s)x−X(t, s)y‖ ≤ ‖V (t, s)(x− y)‖+

+t∫

s

‖V (t, ξ)(f(ξ, x(ξ))− f(ξ, y(ξ)))‖dξ+

+∑

s<tn<t

‖V (t, t+n )(hn(x(tn))− hn(y(tn)))‖ ≤

≤ (N2)n−k+1

e−pn−...−pk‖x− y‖+ Nt∫

s

L(ξ)‖X(ξ, s)x−X(ξ, s)y‖dξ+

+Ne−pn−...−pk∑

s<tn<t

Hn‖X(tn, s)x−X(tn, s)y‖.


Applying Lemma 3 we obtain

‖X(t, s)x−X(t, s)y‖ ≤

≤ (N2)n−k+1

e−pn−...−pk‖x− y‖∏

s<tn<t

(1 + Ne−pn−...−pkHn)eN

t∫s

L(ξ)dξ≤

≤ (N2)n−k+1‖x− y‖

∏s<tn<t

(1 + Ne−pn−...−pk epn

N)×

×e−pn+N

t∫tn−1

L(ξ)dξ

. . . e−pk+N

tk∫s

L(ξ)dξ≤

≤ (N2)n−k+1‖x− y‖2n−k+1 ≤ N‖x− y‖ < k(t, s)‖x− y‖.

In case N > 1 we set

Qnη =N−1

2(Mn + 1)2 e−pn η, (n = 0,±1,±2, . . .)

hn are some as in the case 0 < N ≤ 1.Proceeding analougosly we get

‖X(t, s)x−X(t, s)y‖ < N−(n−k+1)‖x− y‖ < k(t, s)‖x− y‖.

Theorem 1 is proved.

Theorem 2. Let X(t, s); t ≥ s be a k(t, s)-continuous evolutionary process withX(t, s)0 = 0 for all t ≥ s. Then T h : Lh

p,k → Lp (h ≥ 0) and the maps T h (h ≥ 0)are strongly continuous.

Proof: Let v ∈ Lhp,k. Then

(∞∫−∞

‖(T hv)(t)‖pdt)

1p

= (∞∫−∞

‖X(t, t− h)v(t− h)‖pdt)

1p

≤

≤ (∞∫−∞

kp(t, t− h)‖v(t− h)‖pdt)

1p

< ∞.

The futher proof of Theorem 2 is a modification of the proof of Proposition 1[1].

Theorem 3. Let the following conditions are satisfied.1. The linear equation (8), (9) is well posed.2. ‖V (t, s)ξ‖ ≤ k1(t)k2(s)‖ξ‖ (ξ ∈ X and t ≥ s),where k1(.), k2(.) : R → R+ are continuous functions and there is a constant k such


that 1 ≤ k1(t)k2(t) ≤ k (t ∈ R).3. The function f(t, x) is Lipschitz continuous w.r. to x i.e.,

‖f(t, x)− f(t, y)‖ ≤ L‖x− y‖, (t ∈ R, x, y ∈ X).

4. The functions hn : X → D(A(tn)) are Lipschitz continuous i.e.,

‖hn(x)− hn(y)‖ ≤ Hn‖x− y‖ (x, y ∈ X, n = 0,±1,±2, . . .).

5. X(t, s)0 = 0 (t ≥ s).Then the impulse equation (3), (4) induces a k1(t)k2(s)-continuous evolutionary

process, which associated evolutionary semigroup T h : h ≥ 0 is strongly contin-uous and has an infinitezimal generator of the evolutionary process associated withthe linear impulse equation (8), (9) and F is a operator which maps every v ∈ Lh

p,k1k2

into the function t → f(t, v(t)).

Proof: It is not hard to check that the conditions 1 and 2 of Definition 1 arefulfilled. Using the results of [8] it can proved that the function X(t, s)x is continu-ously continiable from every set (tj−1, tj)×(tk−1, tk)×X to [tj−1, tj]××[tk−1, tk]×X(j, k = 0,±1,±2, . . .).

We shall prove that X(t, s) satisfies the condition (1).Let x, y ∈ D(A(s)) and t ≥ s. Then

‖X(t, s)x−X(t, s)y‖ ≤ ‖V (t, s)x− V (t, s)y‖+

+t∫

s

‖V (t, ξ)(f(ξ, x(ξ))− f(ξ, y(ξ)))‖dξ+

+∑

s<tn<t

‖V (t, t+n )(hn(x(tn))− hn(y(tn)))‖ ≤

≤ k1(t)k2(s)‖x− y‖+ k1(t)t∫

s

k2(ξ)L‖X(ξ, s)x−X(ξ, s)y‖dξ+

+k1(t)∑

s<tn<t

k2(t+n )Hn‖X(tn, s)x−X(tn, s)y‖.

From Lemma 3 we get

‖X(t, s)x−X(t, s)y‖ ≤ k1(t)k2(s)eLk(t−s)

∏s<tn<t

(1 + kHn)‖x− y‖.

Moreover

k1(t)k2(t)∏

t<tn<t

(1 + kHn) ≥ 1.

The futher proof of Theorem 2 is a modification of the proof of Proposition 2 [1].


References

[1] Aulbach B., Minh N., Nonlinear semigroups and the existence and stabilityof solutions of semilinear nonautonomous evolution equations, Abstract andApplied Analysis, Vol.1, No.4, (1996), pp.397-426.

[2] Bainov D.D., Kostadinov S.I., Abstract Impulsive Equations, SCT Publishing,(1994), pp.195.

[3] Bainov D.D., Kostadinov S.I., Minh N.V., Dichotomies and Integral Manifoldsof Impulsive Differential Equations, SCT Publishing, (1994), pp.110.

[4] Bainov D., Kostadinov S., Zabreiko P., Lp equivalence of a linear and a nonlin-ear impulsive differential equation in a Banach space, Journal of MathematicalAnalysis and Applications, 159 (1991), N2, 389-405.

[5] Bainov D., Kostadinov S., Zabreiko P., Lp-equivalence of impulsive equations,International Journal of Theoretical Physics, 27 (1988), N11, 1411-1424.

[6] Bainov D.D., Simeonov P.S., Systems with Impulse Effect. Stability, Theoryand Applications, Ellis Horwood, (1989), pp.255.

[7] Samoilenko A.M., Perestyik N.A., Differential Equations with Impulse Effect,Kiev, Visha Skola, (1987), pp.285.

[8] Segal I., Non-linear semi-groups, Ann. Math. 78, (1963), 339-364.

A. Georgieva S. KostadinovDepartment of Mathematics Department of Mathematics and InformaticsUniversity of Food Technologies University of Plovdiv ”P. Hilendarski”26, Maritza Blvd. 24, Tzar Assen Str.4002 Plovdiv,BULGARIA 4000 Plovdiv,BULGARIAE-mail: [email protected] E-mail: [email protected]

.



Construction of analytic functions, whichdetermine bounded Toeplitz operators on

H1 and H∞

Peyo Stoilov

Abstract

For f ∈ H∞ we denote by Tf the Toeplitz operator on Hp, defined by

Tfh =∫

T

f(ζ)h(ζ)1− ζz

dm(ζ), h ∈ Hp.

In this paper we prove some sufficient conditions for the sequences of num-bers α = (αn)n≥0 in which the functions

f ∗ αdef=

∑n≥1

f(n) αn zn

determine bounded Toeplitz operators Tf∗α on H1 and H∞ for all f ∈ H∞.

1 Introduction

Let A be the class of all functions analytic in the unit disk D = ζ : |ζ| < 1 , m(ζ) -normalized Lebesgue measure on the circle T = ζ : |ζ| = 1 . Let Hp (0 < p ≤ ∞)is the space of all functions analytic in D and satisfying

‖f‖pHp = sup

0< r< 1

∫T|f(rζ)|p dm(ζ) < ∞, 0 < p < ∞,

01991 Mathematics Subject Classification: Primary 30E20, 30D500Key words and phrases: Analytic function, Toeplitz operators, Cauchy integrals, multipliers.0Received June 15, 2005.

67

68 Peyo Stoilov

‖f‖H∞ = supz∈D

|f(z)| < ∞, p = ∞.

Let M is the space of all finite, complex Borel measures on T with the usualvariation norm.

For µ ∈ M , the analytic function on D

Kµ(z) =

∫T

1

1− ζzdµ(ζ)

is called the Cauchy transforms of µ and the set of functions

K = f ∈ A : f = Kµ, µ ∈ M

is called the space of Cauchy transforms.For dµ(ζ) = ϕ(ζ)dm(ζ), ϕ ∈ Lp , 1 ≤ p ≤ ∞ , we denote Kµ(z) = Kϕ(z) and

Kp = f ∈ A : f = Kϕ, ϕ ∈ Lp , 1 ≤ p ≤ ∞.

By the theorem of M. Riez Kp = Hp for 1 < p < ∞ , however H1 K1 ,H∞ K∞ .

We note that K∞ = BMOA (the space of analytic functions of bounded meanoscillation )[1].

For f ∈ H∞ we denote by Tf the Toeplitz operator on Hp , defined by

Tfh = Kfh(z) =

∫T

f(ζ)h(ζ)

1− ζzdm(ζ), h ∈ Hp.

By the theorem of M. Riez for 1 < p < ∞ the operator Tf is bounded on Hp

for all f ∈ H∞. But if p = 1 and p = ∞ not every function f ∈ H∞ gives rise tobounded Toeplitz operator Tf on H1 and H∞ .

There is also an interesting connection between multipliers of the spaces K andKp , p = 1, ∞ and the Toeplitz operators.

Let M and Mp be the class to all multipliers of the spaces K and Kp :

M = f ∈ A : f g ∈ K, ∀g ∈ K ,

Mp = f ∈ A : f g ∈ Kp, ∀g ∈ Kp .

Since Kp = Hp for 1 < p < ∞ , then Mp = H∞ for 1 < p < ∞.However

M = M1 H∞, M∞$H∞

and

M = M1 =f ∈ H∞ : ‖Tf‖H∞ < ∞

[3],

Construction of analytic functions, which determine Toeplitz operators 69

M∞ =f ∈ H∞ : ‖Tf‖H1 < ∞

[2].

Let’s note, that more information, bibliography and review of results for thespaces K and M contains the new monograph [5] .

Since M = M1 H∞ , M∞ H∞ i.e. not all function f ∈ H∞ give rise tobounded Toeplitz operators on H1 and H∞, then naturally arises the followingtask:

To describe thesequencesof numbers α = (αn)n≥0 , for whichthe functions

f ∗ αdef=

∑n≥1

f(n) αn zn , z ∈ D

give rise to bounded Toeplitz operators Tf∗α on H1 and H∞ for all f ∈ H∞.

In this paper we prove some sufficient conditions for the sequences α = (αn)n≥0

in which Toeplitz operator Tf∗α is bounded on H1 and H∞ for all f ∈ H∞ .Further we will use the following important theorem:

Theorem of Smirnov.Let 0 < p < q , f ∈ Hp and has Lq boundary values ( f ∈ Lq(T) ). Then

f ∈ Hq .

We include also its proof for convenience of the reader.Proof. Since f ∈ Hp, then f = Bg , where B is a Blaschke product, g ∈ Hp and

g 6= 0 in D .The function gp ∈ H1 and applying the formula of Poisson to the function gp we

have

gp(z) =

∫Tgp(ζ)Pz(ζ) dm(ζ), Pz(ζ) =

1− |z|2

|ζ − z|2, ζ ∈ T, z ∈ D.

From this formula, taking into account that|f(z)| ≤ |g(z)| in D , |f(ζ)| = |g(ζ)| for almostevery ζ ∈ T ,follows

|f(z)|p ≤∫

T|f(ζ)|p Pz(ζ) dm(ζ).

If q = ∞ , then f ∈ L∞(T) and ‖f‖H∞ ≤ ‖f‖L∞(T) < ∞.If q < ∞ , then applying the Holder’s inequalitywe have

|f(z)|p ≤∫

T|f(ζ)|p (Pz(ζ)) p/q (Pz(ζ))1−p/q dm(ζ) ≤

70 Peyo Stoilov

≤(∫

T|f(ζ)|q Pz(ζ)dm(ζ)

)p/q (∫T

Pz(ζ)dm(ζ)

)1−p/q

=

=

(∫T|f(ζ)|q Pz(ζ)dm(ζ)

)p/q

⇒

|f(z)|q ≤∫

T|f(ζ)|q Pz(ζ)dm(ζ).

Integrating on the circle |z| = r, 0 < r < 1 we obtain

∫T|f(rη)|q dm(η) ≤

∫T

∫T|f(ζ)|q 1− r2

|ζ − rη|2dm(ζ)dm(η) ≤ ‖f‖Lq(T) < ∞.

Consequently f ∈ Hq.

2 Main results

Let N is the class of all functions f ∈ H∞ for which

Λ(f)def= ess sup

η∈T

∫T

|f(ζ)− f(η)||ζ − η|

dm(ζ) < ∞ .

For f ∈ N we denote ‖f‖N

def= ‖f‖H∞ + Λ(f).

Theorem 1. If f ∈ N , then Toeplitz operator Tf is bounded on Hp (p =1, ∞) and

‖Tf‖Hp ≤ ‖f‖N .

Proof. The case p = ∞ is proved in [3,4] and is generalized in [6] for themultipliers of the integrals of Cauchy-Stieltjes type in domains with closed Jordancurve.

We shall prove the case p = 1 .Let f ∈ N , h ∈ H1 . Let E be a subset with total measure (m (E) = 1) lying

on T so that

‖f‖H∞ = supη∈E

|f(η)| .

Then

‖Tfh‖H1 = sup0<r<1

∫T

∣∣∣∣∫T

f(ζ)h(ζ)

ζ − rηζdm(ζ)

∣∣∣∣ dm(η) =


= sup0<r<1

∫T

∣∣∣∣∫T

f(ζ)− f(rη)

ζ − rηh(ζ)ζdm(ζ) + f(rη)

∫T

1

ζ − rηh(ζ)ζdm(ζ)

∣∣∣∣ dm(η) ≤

≤ sup0<r<1

∫T

∫T

∣∣∣∣f(ζ)− f(rη)

ζ − rη

∣∣∣∣ |h(ζ)| dm(ζ) dm(η) +

∫T

∣∣f(rη)h(rη)∣∣ dm(η)

≤

≤ sup0 < r < 1ζ ∈ E

(∫T

∣∣∣∣f(ζ)− f(rη)

ζ − rη

∣∣∣∣ dm(η) + ‖f‖H∞

)‖h‖H1 .

We denote for ζ ∈ E

Fζ(z) =f(ζ)− f(z)

ζ − z, z ∈ D.

Then

‖Tfh‖H1 ≤ supζ∈E

(‖Fζ‖H1 + ‖f‖H∞

)‖h‖H1 .

To end the proof is necessary to show

f ∈ N ⇒ supζ∈E

‖Fζ‖H1 < ∞.

Since

1

ζ − z∈ Hp (0 < p < 1)

and f ∈ H∞ , then Fζ(z) ∈ Hp (0 < p < 1) .Furthermore

f ∈ N ⇒ supζ∈E

‖Fζ‖L1(T) ≤ Λ(f) < ∞

and according to the Theorem of Smirnov

Fζ(z) ∈ H1 , ‖Fζ‖H1 = ‖Fζ‖L1(T) ≤ Λ(f) < ∞ .

Consequently

‖Tf‖H1 ≤ supζ∈E

(‖Fζ‖H1 + ‖f‖H∞

)≤ Λ(f) + ‖f‖H∞ = ‖f‖N < ∞.

72 Peyo Stoilov

Remark. We note that from the Theorem of Stegenga [2] characterizing a classof bounded Toeplitz operators on H1 does not follow Theorem 1 for p = 1.

Lemma 1.[3] If pn is a polynomial of degree n , then

‖pn‖N ≤ 3 ‖pn‖H∞ log(n + 2).

Definition. A sequence α = (αn)n≥0 of positive numbers is called concave if

αn+2 − αn+1 ≥ αn+1 − αn ⇔ αn − 2αn+1 + αn+2 ≥ 0.

Theorem 2. Let α = (αn)n≥0 be a monotone decreasing, concave sequence ofpositive numbers and

‖α ‖ def=

∑n≥0

αn

n + 1< ∞.

Then f ∗α ∈ N , Toeplitz operator Tf∗α is bounded on H1 and H∞ for all f ∈ H∞

and‖Tf∗α‖Hp ≤ ‖f ∗ α‖N ≤ 12 ‖f‖H∞ ‖α ‖ , p = 1,∞.

Proof. Using Abel’s formula two times we obtain

∑n≥0

αn

n + 1=

∑n≥0

(αn − αn+1)n∑

k=0

1

k + 1≥

≥∑n≥0

(αn − αn+1) log(n + 2) =

=∑n≥0

(αn − 2αn+1 + αn+2)n∑

k=0

log(k + 2).

Since

n∑k=0

log(k + 2) ≥n∑

k=[n/2]

log(k + 2) ≥ (n/2 + 1) log([n/2] + 2) ≥ 1

4(n + 1) log(n + 2),

then

4∑n≥0

αn

n + 1≥

∑n≥0

(αn − 2αn+1 + αn+2)(n + 1) log(n + 2).

Further let f ∈ H∞ and


Sn(f) =n∑

k=0

f(k) zk; σn(f) =1

n + 1

n∑k=0

Sk(f).

Applying the Abel’s formula we obtain

f ∗ α =∑n≥0

f(n) αn zn =∑n≥0

( αn − αn+1) Sn(f) =

=∑n≥0

(αn − 2αn+1 + αn+2)(n + 1) σn(f).

Since by Lemma 1.

‖σn(f)‖N ≤ 3 ‖σn(f)‖H∞ log(n + 2) ≤ 3 ‖f‖H∞ log(n + 2),

then

‖f ∗ α‖ N ≤∑n≥0

(αn − 2αn+1 + αn+2)(n + 1) ‖σn(f)‖N ≤

≤ 3 ‖f‖H∞

∑n≥0

(αn − 2αn+1 + αn+2)(n + 1) log(n + 2) ≤

≤ 12 ‖f‖H∞

∑n≥0

αn

n + 1= 12 ‖f‖H∞ ‖α‖ < ∞.

The following proposition follows at once from Theorem 2.

Theorem 3. Let α denote one of the sequences (ε > 0) :(1

(n + 1)ε

)n≥0

;

(1

log1+ε(n + 2)

)n≥0

;

(1

log(n + 2) log1+ε log(n + 3)

)n≥0

, ..............................

Then f ∗ α ∈ N , Toeplitz operator Tf∗α is bounded on H1 and H∞ for allf ∈ H∞.

74 Peyo Stoilov

Remark. Theorem 3 was proved by another method in [3] ( Theorem 7. ) forthe bounded Toeplitz operators Tf∗α on H∞.

Theorem 4. Let the sequence α = (αn)n≥0 satisfythe conditions of Theorem 3.If the sequence a = (an)n≥0 ∈ `2 , then there existsa function f ∈ N , satisfying∣∣∣f(n)

∣∣∣ ≥ αn |an| , ‖f‖N ≤ c0 ‖α‖ ‖a‖`2 ,

where c0 is an absolute constant.

Proof. By the Theorem of Kislyakov [7] if a = (an)n≥0 ∈ `2 , then there exists afunction f ∈ H∞, satisfying

|g(n)| ≥ |an| , ‖g‖H∞ ≤ B ‖a‖`2 ,

where B is an absolute constant. By Theorem 2.3 f = g ∗ α ∈ N and

‖f‖N ≤ 12 ‖α‖ ‖g‖H∞ ≤ 12B ‖α‖ ‖a‖`2 .

References

[1] J. B. Garnett. Bounded analytic functions. Academic Press, Inc., New York-London, 1981. MR0628971 (83g:30037)

[2] D. A. Stegenga. Bounded Toeplitz operators on H1 and applications of theduality between H1 and the functions of bounded mean oscillation. Amer. J.Math., 98, 1976,no. 3, 573-589. MR0420326 (54 #8340)

[3] S. A. Vinogradov. Properties of multipliers of Cauchy - Stieltjes integrals andsome factorization problems for analytic functions. Amer. Math. Sos. Transl.(2) vol. 115, 1980, 1-32. MR0586560 (58 #28518)

[4] S. V. Hruscev, S. A. Vinogradov. Inner functions and multipliers of Cauchytype integrals. Ark. mat, 19, 1981, 23-42. MR0625535 (83c:30027)

[5] J. A. Cima, A. L. Matheson, T. W. Ross. The Cauchy transform. AmericanMathematical Society, Providence, RI,2006.MR2215991 (2006m: 30003)

[6] P. Stoilov. Multipliers of integrals of Cauchy - Stieltjes type. Mathematics andmathematical education, Publ. House Bulgar. Acad. Sci., Sofia, 1986, 316 -319. (Russian) MR0872936 (88e:30104)


[7] S. V. Kislyakov.Fourier coefficients of boundary values of functions that areanalytic in the disc and bidisc.Trudy Mat. Inst. Steklov. vol. 155,1981, 77–94.(Russian) MR0615566 (83a:42005)

Department of MathematicsTechnical University at Plovdiv25, Tsanko Dyustabanov,Plovdiv, Bulgariae-mail: [email protected]

.

76 Peyo Stoilov

77

JOURNAL

OF THE TECHNICAL UNIVERSITY AT PLOVDIV

FUNDAMENTAL SCIENCES AND APPLICATIONS

ISSN 1310-8271

Editor-in-Chief Peyo Stoilov

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