Volume 2008, Article ID 384910,16pages doi:10.1155/2008/384910
Research Article
On Singular Solutions of Linear Functional
Differential Equations with Negative Coefficients
Vita Pylypenko1and Andr ´as Ront ´o2
1National Technical University of Ukraine “Kyiv Polytechnic Institute”,
Faculty of Physics and Mathematics, 37 Peremohy Avenue, 03 056 Kyiv, Ukraine
2Institute of Mathematics, Academy of Sciences of Czech Republic, ˇZiˇzkova 22, 61662 Brno, Czech Republic
Correspondence should be addressed to Andr´as Ront ´o,[email protected]
Received 26 May 2008; Accepted 1 September 2008
Recommended by Alberto Cabada
The problem on solutions with specified growth for linear functional differential equations with negative coefficients is treated by using two-sided monotone iterations. New theorems on the existence and localisation of such solutions are established.
Copyrightq2008 V. Pylypenko and A. Ront ´o. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Problem setting and introduction
In this paper, we are concerned with the problem on the existence and localisation of solutions of singular linear functional differential equations. The class of equations under examination comprises, in particular, the differential equation with argument deviations
ut
n
k0
pktu
ωkt
ft, t∈a, b, 1.1
considered together with the additional condition
sup
t∈a,b
htut<∞, 1.2
whereh:a, b→Ris a certain given continuous nondecreasing function such thatht>0 fort∈a, band
lim
The argument deviations in1.1are given by arbitrary Lebesgue measurable functionsωk,
k 0,1, . . . , n, transforming the intervala, bto itselfthis setting does not lead one to any loss of generality, see, e.g., 1.
We are interested in conditions guaranteeing the existence of solutions with property 1.2in the case where the coefficients of1.1are nonpositive.
The class of solutions of1.1determined by condition 1.2includes, in particular, those possessing the property
lim
t→ahtut c, 1.4
wherec∈R. Problem1.1,1.4withhsatisfying1.3is referred to as the singular Cauchy problem 2. It is reduced, in a natural way, to the classical regular Cauchy problem ifhis equal identically to a nonzero constant. Regular and singular Cauchy problems for various classes of functional differential equations are treated, in particular, in 2–11; a problem on regular solutions possessing properties of type1.4is studied in 12.
By asolutionof problem1.1,1.2we mean a locally absolutely continuous function
u : a, b→Rsuch that hu ∈ L1a, b,Rwhich satisfies 1.1almost everywhere on the intervala, band possesses property1.2. One says that a functionu :a, b→Rislocally absolutely continuousif its restrictionu|aε,bto the interval aε, bis absolutely continuous for anyε∈0, b−a.
A solution of 1.1, 1.2 may have a nonintegrable singularity at the point a. For example, the functionut λt−4,t ∈ 0,1for any realλ andε ∈ 0,∞is a solution of the homogeneous problem
ut −4 t3u
t1/2, t∈0,1,
sup
ξ∈0,1
ξ4εuξ<∞ 1.5
in the sense of the above definition. It should be noted that the property of the uniqueness of a solution is not typical for such problems; in other words, the resonance usually takes place.
2. Notation
The following notation is used throughout the paper. 1R: −∞,∞.
2If−∞< a < b <∞andA⊆a, bis a measurable set, thenL1A,Ris the Banach space of all the Lebesgue integrable functionsu:A→Rwith the standard norm
L1A,Ru−→
A
utdt. 2.1
3Ba, b,Ris the Banach space of all the bounded functionsu:a, b→Rwith the standard norm
Ba, b,Ru−→ sup
t∈a,b
4L1; loca, b,R is the set of functions u : a, b→R such that u|aε,b ∈ L1 a
ε, b,Rfor anyε∈0, b−a.
5ACloca, b,R is the set of all the locally absolutely continuous functions u : a, b→R.
6ACloc;ha, b,R is the set of all the locally absolutely continuous functions u :
a, b→Rsuch thathu∈L1a, b,Rand supt∈a,bht|ut|<∞.
3. Existence of solutions with restricted growth
Without significant loss of generality we can assume that the functionh:a, b→Rinvolved in1.2has the following properties.
The function h:a, b−→0,∞is absolutely continuous and nondecreasing,
and the relation lim
t→aht 0 holds.
3.1
The following statement on problem1.1,1.2is true.
Theorem 3.1. Lethin condition1.2possess properties3.1, the functionsωk,k0,1, . . . , n, are
Lebesgue measurable, and let the functionspk∈L1; loca, b,R,k0,1, . . . , n, be such that
pkt≤0, t∈a, b, k0,1, . . . , n, 3.2
max
k0,1,...,n
b
a
hs hωks
pksds <∞. 3.3
Moreover, let there exist a nonnegative functionϕ∈ACloc;ha, b,Rsuch that
ϕt≤
n
k0
pktϕ
ωkt
, t∈a, b. 3.4
Then for an arbitrary u0 ∈ ACloc;ha, b,R and any function f : a, b→R from
L1; loca, b,Rpossessing the property
hf ∈L1
a, b,R 3.5
and satisfying the estimate
ϕt−
n
k0
pktϕ
ωkt
≤ft−u0t
n
k0
pktu0
ωkt
≤0, t∈a, b, 3.6
problem1.1,1.2has a solutionu:a, b→Rsuch that
Remark 3.2. The set of functions f satisfying inequalities 3.6 is nonempty because ϕ is assumed to possess property3.4. Of course, it makes sense to consider only thoseϕwhich do not satisfy the corresponding homogeneous equation.
The next corollary deals with the case where the solution in question is bounded from below.
Corollary 3.3. Assume thathin condition1.2possesses properties3.1and the functionspk ∈
L1; loca, b,R, k 0,1, . . . , n, satisfy 3.2 and 3.3. Moreover, let the functional differential
inequality3.4have a nonnegative solutionϕ∈ACloc;ha, b,R.
Then for an arbitrary realλand any functionf:a, b→RfromL1; loca, b,Rpossessing
property3.5and satisfying the estimate
ϕt−
n
k0
pktϕ
ωkt
≤ft λ
n
k0
pkt≤0, t∈a, b, 3.8
problem1.1,1.2has a solutionu:a, b→Rsuch that
0≤ut−λ≤ϕt, t∈a, b. 3.9
For concrete classes of weight functionsh,Theorem 3.1allows one to obtain efficient results concerning the existence, localisation, and approximate construction of solutions of 1.1possessing property1.2. Let us consider the problem on the existence of solutions of 1.1possessing the property
sup
t∈a,b
t−aγut<∞, 3.10
whereγis a given positive constant.
Corollary 3.4. Let the functions pk ∈ L1; loca, b,R, k 0,1, . . . , n, be nonpositive almost
everywhere ona, b. Moreover, let
max
k0,1,...,n
b
a
s−a ωks−a
γ
pksds <∞. 3.11
Then for arbitrary functionu0 ∈ ACloc;ha, b,R, number δ ∈ 0, γ, and functionf :
a, b→RfromL1; loca, b,Rpossessing the properties3.5and
−γδt−a−γδ−1−
n
k0
pkt
ωkt−a
−γδ≤
ft−u0t
n
k0
pktu0
ωkt
≤0, t∈a, b,
3.12
problem1.1,1.2has a solutionu:a, b→Rsuch that
Remark 3.5. Under condition 3.11, the coefficient pk, k 0,1, . . . , n, may have a
nonin-tegrable singularity at the point a if the corresponding argument deviation ωk has the
property
ess inf
t∈a,b
t−a ωkt−a
0 3.14
and, in particular, if ess inft∈a,bωkt> a.
In particular, in the case of the problem on finding solutions of the equation
ut
n
k0
pktu
tβkft, t∈0,1, 3.15
satisfying the additional condition
sup
t∈0,1
tγut<∞, 3.16
whereγ >0,βk≥0,k0,1, . . . , n, we have the following.
Corollary 3.6. Let the functions pk ∈ L1; loc0,1,R, k 0,1, . . . , n, be nonpositive almost
everywhere on0,1and let the relation
n
k0 1
0
t1−βkγp
ktdt <∞ 3.17
be satisfied. Moreover, assume that there exists someδ∈0, γsuch that
ess sup
t∈0,1
n
k0
pkttγ−δ1−βk1≤γ−δ. 3.18
Then for an arbitrary u0 ∈ ACloc;h0,1,R and any function f : 0,1→R from
L1;loc0,1,Rpossessing the properties
hf ∈L1
0,1,R,
−γδt−γδ−1−
n
k0
pkttβk−γδ≤ft−u0t
n
k0
pktu0
tβk≤0, t∈0,1, 3.19
problem3.15,3.16has a solutionu:0,1→Rsuch that
Remark 3.7. Under the conditions ofCorollary 3.6, the coefficientpk,k 0,1, . . . , n, of3.15
may be nonintegrable if the corresponding exponentβksatisfies the inequalities 0< βk <1,
that is, if there is an advance of argument in thekth term.
The statements formulated above are consequences of the general theorem which will be established inSection 4.
4. A general theorem
Let us consider the functional differential equation
ut lut ft, t∈ a, b, 4.1
wherel:ACloca, b,Rn→L1; loca, b,Rnis a linear operator andf ∈L1; loca, b,Rnis a given locally integrable function. Assuming implicitly conditions3.1on the functionh, we pose the problem on finding solutions of4.1possessing property1.2.
Definition 4.1. An operator l : ACloca, b,Rn→L1; loca, b,Rn is said to be pointwise
negativeiflut≤0 for a.e.t∈a, bwhenever inft∈a,but≥0 andhu∈L1a, b,Rn. The following theorem holds.
Theorem 4.2. Let the operator l : ACloca, b,Rn→L1; loca, b,Rn in 4.1 be pointwise
negative such that
hl
1
h
∈L1
a, b,R. 4.2
Furthermore, let there exist some nonnegative absolutely continuous functiong:a, b→Rsuch that
h
g
h
∈L1
a, b,R, 4.3
and almost everywhere ona, b,
ht2l
g
h
t≥gtht−gtht, t∈a, b. 4.4
Then for an arbitraryu0 ∈ACloc;ha, b,Rand any functionf:a, b→RfromL1; loca,
b,Rpossessing the properties3.5and
gtht−gtht
ht2 −l
g
h
problem4.1,1.2has at least one solutionu:a, b→Rsuch that
0≤ut−u0t≤
gt
ht, t∈a, b. 4.6
It should be noted that the solution of problem4.1,1.2, the existence of which is stated in Theorem 4.2, can be found approximately by using a convergent monotone two-sided iteration procedure. Moreover, besides4.6, the solution indicated admits the estimate
gtht−gtht
ht2 ≤u
t−u
0t≤0, t∈a, b. 4.7
5. Auxiliary statements
In the sequel, we need an abstract theorem on operators in partially ordered normed spaces 13, Theorem 4.1. In order to state it, we first formulate definitions. We use 13as the main referencesee also 14,15.
5.1. General notions
Let E,· be a normed space overRand letK be a cone 13in E, that is, a nonempty closed subset of Epossessing the properties K ∩−K {0} and α1Kα2K ⊆ K for all {α1, α2} ⊂ 0,∞. A coneKgenerates a natural partial ordering ofE. As usual, we will write
uKvandvKuif and only ifv−u∈K.
Definition 5.1see 13. A sequence{uk|k≥0} ⊂Eis said to beorder-boundedif there exists
somev∈Esuch thatukKvfor anyk≥0. A sequence{uk|k≥0} ⊂Eis said to bemonotone
ifukKuk1for anyk≥0.
Definition 5.2see 13. A coneK⊂Eis said to beregularif every order-bounded monotone
sequence converges inE.
Definition 5.3 see 13. Given a cone K ⊂ E, a functionall : E→Ris said to be positive
with respect toKiflu≥ 0 for anyuK0. A positive functionall :E→Ris calledstrictly
nondecreasingon the coneKif
lim
n→ ∞l
u1u2. . .un
∞ 5.1
for any sequenceun∈K,n1,2, . . ., possessing the property infn≥1un>0.
A linear functionall:E→Ris said to beuniformly positiveonKif there exists a certain
λ∈0,∞such that
lx≥λx ∀x∈K. 5.2
Definition 5.4see 13. One says that an operatorT :E→Eismonotonewith respect toK
ifTuKTvfor anyuandvfromEsuch thatuKv.
Theorem 5.5see 13. LetT :E→Ebe a monotone and continuous operator. Let there exist some elements{u0, v0} ⊂Esuch thatu0Kv0and, moreover,
Tu0Ku0, 5.3
Tv0Kv0. 5.4
Moreover, assume that the coneKis regular. Then the operatorT has at least one fixed pointusuch thatu0KuKv0.
Remark 5.6. The assertion of Theorem 5.5is established in 13under the assumption that
TK⊆K. The positivity ofT, however, is not used in the proof.
5.2. The spaceACloc;ha, b,R
We assume that the functionhinvolved in the nonlocal condition1.2possesses properties 3.1.
Lemma 5.7. The setACloc;ha, b,Ris a Banach space with respect to the norm
ACloc;h
a, b,Ru−→ u:
b
a
hsusds sup
ξ∈a,b
hξuξ. 5.5
Proof. Let us assume that{um :m≥ 1}is a Cauchy sequence in the spaceACloc;ha, b,R.
Then, in view of5.5, the sequence{hum:m≥1}is a Cauchy sequence inL1a, b,Rand, therefore, there exists some Lebesgue integrable functionv:a, b→Rsuch that
lim
m→∞ b
a
hsums−vsds0. 5.6
According to the definition of the setACloc;ha, b,R, every functionhum:a, b→R,m
1,2, . . ., is bounded in a neighbourhood of the pointa, whence, in view of the assumption, we find that the sequence{hum:m≥1}is a Cauchy sequence in the complete spaceBa, b,R.
Therefore, one can specify a bounded functionw:a, b→Rsuch that
lim
m→∞tmax∈ a,bhsums−ws0. 5.7
Let us put
u: w
fix an arbitraryδ∈0, b−a, and construct the sequence of functions
zm,δ :−
b
aδ
hsums−us
dshbumb−ub
−haδumaδ−uaδ
, m1,2, . . . .
5.9
Since the numberδis positive, we see from assumption3.1thath/h ∈ L1 aδ, b,R, whence, in view of5.7and the Lebesgue dominated convergence theorem, we obtain
lim
m→∞
b
aδ
hsums−us
ds0. 5.10
By virtue of assumption 3.1, ht/0 for a < t ≤ b. It then follows from 5.7 that limm→∞umt utfor anyt∈a, b. By virtue of5.10, this yields
lim
m→∞zm,δ 0
∀δ∈0, b−a. 5.11
On the other hand, it follows immediately from5.9that
zm,δ
b
aδ
hs us hsumsds−hb ub haδuaδ, 5.12
whence, in view of5.6,
b
aδ
vsds−
b
aδ
hs usdshb ub−haδ uaδ 5.13
for any 0< δ < b−a. Recalling thatv∈L1a, b,Randhu∈Ba, b,R, using the relation −∞ < limδ→0abδvsds abvsds < ∞, and passing to the limit asδ→0 in5.13, we
find that
b
a
hs usds−
b
a
vsdshb ub− lim
ξ→ahξ uξ 5.14
and, in particular,hu∈L1a, b,R.
It follows from5.13that the functionsvandusatisfy the relation
htut −
b
t
Moreover, the producthuhas a derivative almost every where ona, b. Indeed, it follows
from5.14that, for almost everyt∈a, band anyη∈a−t, b−t,
1
η
htη utη−ht ut−1 η
b
tη
vs hsusds−
b
t
vs hs usds
1
η
tη
t
vs hs usds.
5.16
Therefore, limη→01/ηhtηutη−ht ut vt ht ut,which means that
htutvt ht ut 5.17
for a.e.t ∈ a, b. Relation5.17implies, in particular, thatuexists almost everywhere on a, band, moreover,
ht ut vt 5.18
for a.e.t∈a, b. Using5.5and5.18, we get
um−u
b
a
hsums−usds sup ξ∈a,b
hξumξ−uξ−→0 5.19
asm→ ∞, that is, the sequence{um:m≥1}converges touinACloc;ha, b,R.
Lemma 5.8. The set
K:
u∈ACloc;h
a, b,R: inf
t∈a,but≥0,ess supt∈a,b u
t≤0 5.20
is a regular cone in the spaceACloc;ha, b,R.
Proof. The set 5.20is obviously closed, and the axioms of a cone see5.1 are verified
directly. If a monotonous strictly nondecreasing functional is defined on the coneK, then the coneKis regularsee, e.g., 13, Theorem 1.11. In our case, the functional
ϕu:−
b
a
hsusds sup
ξ∈a,b
hξuξ u∈ACloc;h
a, b,R 5.21
Lemma 5.9. If the operatorland functionhsatisfy condition4.2andhis nondecreasing, then
sup
t∈a,b
ht
b
t
l
1
h
sds <∞. 5.22
Proof. Property5.22is a consequence of the estimate
ht
b
t
l
1
h
sds≤
b
t
hsl
1
h
sds, 5.23
which is valid for anyt∈a, bbecausehis nondecreasing.
5.3. The operatorT
Let us fix a certain locally integrable functionf :a, b→Rwith properties3.5and put
Tut:ub−
b
t
lus fsds, t∈a, b, 5.24
for anyufromACloc;ha, b,R.
Lemma 5.10. Iflis pointwise negative and3.1holds, then
1T is a well-defined mapping fromACloc;ha, b,RtoACloc;ha, b,R;
2T is continuous with respect to norm5.5;
3T is monotone with respect to cone5.20.
Proof. 1Letube an arbitrary function fromACloc;ha, b,R. Then there exists a constant
μu>0 such that
ut≤ μu
ht, 5.25
and hence
− μu
ht≤ −ut≤ut≤ut≤ μu
ht, t∈a, b. 5.26
Relation5.26, due to the pointwise negativityseeDefinition 4.1and linearity ofl, yields
l
−μu
h
t≥l− |u|t≥lut≥l|u|t≥l
μ
u
h
t, 5.27
−μul
1
h
t≥ −l|u|t≥lut≥l|u|t≥μul
1
h
Thus, we have the estimate
lut≤μul1
h
t, t∈a, b, 5.29
and therefore, due to assumption4.2,
lim
t→a b
t
hslusds <∞. 5.30
Sincef is assumed to satisfy3.5, this guarantees that the function hw, wherew : Tu, belongs toL1a, b,R.
Furthermore, using condition3.5and arguing similarly to the proof ofLemma 5.9, we find that
sup
t∈a,b
ht
b
t
fsds <∞. 5.31
Then, by virtue of3.1and5.27,
sup
t∈a,bht
Tut sup
t∈a,bht ub−
b
t
lus fsds
≤hbub sup
t∈a,b
ht
b
t
lusfsds
≤hbub μusup t∈a,b
ht
b t l 1 h
tds sup
t∈a,b
ht
b
t
fsds,
5.32
whence, in view of assumptions 3.1, 4.2, 5.31 and Lemma 5.9, it follows that supt∈a,bht|Tut| < ∞.Thus, T is well defined and maps the space ACloc;ha, b,R
into itself.
2The continuity ofTfollows from the properties ofl. Indeed, letuandvbe arbitrary functions fromACloc;ha, b,Rand letε ∈0,∞be given. According to5.5and5.24,
in view of the linearity ofl, we have
Tv−Tu
b
a
htlv−utdt sup
t∈a,b
htvb−ub−
b
t
lv−usds
≤ b
a
htlv−utdt sup
t∈a,b
htvb−ub sup
t∈a,b
ht
b
t
lv−usds.
Sinceuandvbelong toACloc;ha, b,R, it follows that
−βu,v
ht ≤vt−ut≤ βu,v
ht, t∈a, b, 5.34
whereβu,v : supt∈a,bht|vt−ut| < ∞.By assumption,l is pointwise negative, and
therefore
lv−ut≤ −βu,vl
1
h
t, t∈a, b. 5.35
By virtue of condition4.2, it follows from5.35that
b
a
htlv−utdt ≤βu,vC, 5.36
whereC:−abhtl1/htdt <∞note thatCis positive.
Due toLemma 5.9, assumption4.2ensures the validity of relation5.22. Using5.22 and 5.36, taking estimate5.23from the proof of Lemma 5.9into account, and arguing analogously, we obtain
sup
t∈a,bht b
t
lv−usds≤ sup
t∈a,b b
t
hslv−usds
≤ sup
t∈a,b b
t
hslv−usds≤βu,vC.
5.37
Sincev−u ≥ βu,v, we see from5.33,5.36, and5.37 thatTv−Tu < ε whenever
v−u < ε12C−1. This, in view of the arbitrariness ofε, proves the continuity of the mappingT.
3Letuandvbe arbitrary elements ofACloc;ha, b,Rsuch thatvKuwith respect
to cone5.20, that is,
−1ivit−uit≥0, t∈a, b, i0,1. 5.38
Sincelis assumed to be pointwise negative, in view of5.38, we have
Tvt−Tut vb−ub−
b
t
lv−usds≥0, t∈a, b,
Tvt−Tut lv−ut≤0, t∈a, b,
5.39
which means thatTvKTu.
Lemma 5.11. A functionu:a, b→RfromACloc;ha, b,Ris a solution of problem4.1,1.2
Proof. To obtain this assertion, it is sufficient to take into account5.24and the definition of the setACloc;ha, b,R.
6. Proofs
Let us now turn to the proofs of the statements formulated in Sections3and4.
6.1. Proof ofTheorem 3.1
It is clear that1.1can be represented in the form of4.1, where the operatorlis defined by the equality
lut
n
k0
pktu
ωkt
, t∈a, b. 6.1
Since the functionspk ∈ L1; loca, b,R,k 0,1, . . . , n are nonpositive, it follows that the operatorlis pointwise negative. Moreover, in view of property3.3, the operatorlsatisfies condition4.2. Furthermore, a nonnegative functionϕ∈ACloc;ha, b,Rcan be written in
the form
ϕ g
h, 6.2
wheregis a nonnegative absolutely continuous function ona, b. Then it follows from3.4 and3.6that inequalities4.4and4.5are true.
Thus, all the conditions ofTheorem 4.2hold, and therefore problem1.1,1.2has a solutionu:a, b→Rbelonging toACloc;ha, b,Rand satisfying estimate3.9.
6.2. Proof ofCorollary 3.3
The statement is an immediate consequence ofTheorem 3.1withu0t:λ,t∈a, b.
6.3. Proof ofCorollary 3.4
It is sufficient to applyTheorem 3.1in the case wherehhas the formht t−aγ,t∈a, b,
withγ > 0, and the functionϕ is given by the formulaϕt t−a−γδ,t ∈ a, b, where δ∈0, γ.
6.4. Proof ofCorollary 3.6
It is sufficient to applyCorollary 3.4 with a 0, b 1, and ωkt tβk,t ∈ 0,1, k
6.5. Proof ofTheorem 4.2
We are going to use Theorem 5.5. Lemmas 5.7 and 5.8guarantee that the set K given by 5.20is a regular cone in the Banach spaceACloc;ha, b,R. ByLemma 5.10, the operatorT :
ACloc;ha, b,R→ACloc;ha, b,Rdefined by formula5.24is continuous and monotone
with respect to the coneK. Therefore, in order to be able to applyTheorem 5.5, we need to specify a pair of elements u0 and v0 inACloc;ha, b,Rsuch that u0Kv0 and5.3,5.4 hold.
Let us putv0:u0ϕ, where
ϕ: g
h 6.3
andu0is the function from the formulation of the theorem. Note thatϕ ∈ ACloc;ha, b,R
in view of assumption 4.3 and the absolute continuity ofg. Moreover, −1iϕit ≥ 0,
t ∈ a, b,i 0,1, because, by assumption,g is a nonnegative function and the functional differential inequality4.4is satisfied. Hence,ϕ∈K.
The functions indicated satisfy relations5.3and5.4with respect to cone5.20, that is,
−1iTu
0
it−ui 0 t
≥0, t∈a, b, i0,1, 6.4
−1iTv
0
it−vi
0 t
≤0, t∈a, b, i0,1. 6.5
Indeed, it follows immediately from5.24that relations6.4and6.5withi1 are satisfied if and only if
−u0t−ϕt lu0
t lϕt≥ −ft≥ −u0t lu0
t, t∈a, b. 6.6
Therefore, it suffices to notice that6.6holds in view of4.5and6.3, whereas relations 6.4and6.5withi0 are obtained from6.6by integrating fromt∈a, btob.
In view of4.4, the function6.3satisfies the functional differential inequality
ϕt≤lϕt, t∈a, b, 6.7
and hence the set of functionsf ∈ACloc;ha, b,Rsatisfying condition6.6is nonempty.
ApplyingTheorem 5.5and usingLemma 5.11, we complete the proof ofTheorem 4.2.
Acknowledgments
References
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