R E S E A R C H
Open Access
Second-order initial value problems with
singularities
Petio Kelevedjiev
1*and Nedyu Popivanov
2Dedicated to Professor Ivan Kiguradze.
*Correspondence: keleved@mailcity.com 1Department of Mathematics, Technical University of Sofia, Branch Sliven, Sliven, Bulgaria
Full list of author information is available at the end of the article
Abstract
Using barrier strip arguments, we investigate the existence of
C[0,T]∩C2(0,T]-solutions to the initial value problemx=f(t,x,x),x(0) =A, limt→0+x(t) =B, which may be singular atx=Aandx=B.
MSC: 34B15; 34B16; 34B18
Keywords: initial value problem; singularities; existence; monotone and positive solutions; barrier strips
1 Introduction
In this paper we study the solvability of initial value problems (IVPs) of the form
x=ft,x,x, (.)
x() =A, lim
t→+x
(t) =B, B> . (.)
Here the scalar functionf(t,x,p) is defined on a set of the form (Dt×Dx×Dp)\(SA∪SB), whereDt,Dx,Dp⊆R,SA=T× {A} ×P,SB=T×X × {B},Ti⊆Dt,i= , ,X ⊆Dx,
P⊆Dp, and so it may be singular atx=Aandp=B. IVPs of the form
ϕ(t)x(t)=ϕ(t)fx(t),
x() =A, x() = ,
have been investigated by Rachůnková and Tomeček [–]. For example in [], the authors have discussed the set of all solutions to this problem with a singularity att= . Here
A< ,ϕ∈C[,∞)∩C(,∞) withϕ() = ,ϕ(t) > fort∈(,∞) andlim
t→∞ϕϕ((tt))= ,f is locally Lipschitz on (–∞,L] with the propertiesf(L) = andxf(x) < forx∈(–∞, )∪ (,L), whereL> is a suitable constant.
Agarwal and O’Regan [] have studied the problem
x=ϕ(t)ft,x,x, t∈(,T],
x() =x() = ,
wheref(t,x,p) may be singular atx= and/orp= . The obtained results give a positive
C[,T]∩C(,T]-solution under the assumptions thatϕ∈C[,T],ϕ(t) > fort∈(,T],
f : [,T]×(,∞)→(,∞) is continuous and
f(t,x,p)≤g(x) +h(x)r(p) +w(p) for (t,x,p)∈[,T]×(,∞),
whereg,h,r, andware suitable functions. IVPs of the form
x(t) =ft,x(t),x(t), <t< ,
x() =x() = ,
wheref(t,x,p)∈C((, )×(,∞)), maybe singular att= ,t= , x= orp= , have been studied by Yang [, ]. The solvability inC[, ] andC[, ]∩C(, ) is established in these works, respectively, under the assumption that
<f(t,x,p)≤k(t)F(x)G(y) for (t,x,p)∈(, )×(,∞),
wherek,F, andGare suitable functions.
The solvability of various IVPs has been studied also by Bobisud and O’Regan [], Bobisud and Lee [], Cabada and Heikkilä [], Cabadaet al. [, ], Cid [], Maagli and Masmoudi [], and Zhao []. Existence results for problem (.), (.) with a singularity at the initial value ofxhave been reported in Kelevedjiev-Popivanov [].
Here, as usual, we use regularization and sequential techniques. Namely, we proceed as follows. First, by means of the topological transversality theorem [], we prove an exis-tence result guaranteeingC[a,T]-solutions to the nonsingular IVP for equations of the form (.) with boundary conditions
x(a) =A, x(a) =B.
Moreover, we establish the neededa prioribounds by the barrier strips technique. Fur-ther, the obtained existence theorem assuresC[,T]-solutions for each nonsingular IVP included in the family
x=ft,x,x,
x() =A+n–, x() =B–n–,
(.)
wheren∈Nis suitable. Finally, we apply the Arzela-Ascoli theorem on the sequence{xn} ofC[,T]-solutions thus constructed to (.) to extract a uniformly convergent subse-quence and show that its limit is aC[,T]∩C(,T]-solution to singular problem (.), (.). In the caseA≥,B≥ we establishC[,T]∩C(,T]-solutions with important properties - monotony and positivity.
solutions to the BVP
gt,x,x,x= , t∈(, ),
x() = , x() =B, B> ,
which may be singular atx= . Note that despite the more general equation of this prob-lem, the conditions imposed here as well as the results obtained are not consequences of those in [].
2 Topological transversality theorem
In this short section we state our main tools - the topological transversality theorem and a theorem giving an important property of the constant maps.
So, letXbe a metric space andYbe a convex subset of a Banach spaceE. LetU⊂Ybe open inY. The compact mapF:U→Yis calledadmissibleif it is fixed point free on∂U. We denote the set of all such maps byL∂u(U,Y).
A mapFinL∂u(U,Y) isessentialif every mapGinL∂u(U,Y) such thatG|∂U=F|∂U
has a fixed point inU. It is clear, in particular, every essential map has a fixed point inU.
Theorem .([, Chapter I, Theorem .]) Let p∈U be fixed and F∈L∂u(U,Y)be the constant map F(x) =p for x∈U.Then F is essential.
We say that the homotopy{Hλ:X→Y}, ≤λ≤,is compact if the mapH(x,λ) :X×
[, ]→Y given byH(x,λ)≡Hλ(x)for(x,λ)∈X×[, ]is compact.
Theorem .([, Chapter I, Theorem .]) Let Y be a convex subset of a Banach space E and U⊂Y be open.Suppose:
(i) F,G:U→Yare compact maps.
(ii) G∈L∂U(U,Y)is essential.
(iii) H(x,λ),λ∈[, ],is a compact homotopy joiningFandG,i.e.
H(x, ) =F(x) and H(x, ) =G(x). (iv) H(x,λ),λ∈[, ],is fixed point free on∂U.
Then H(x,λ), λ∈[, ],has at least one fixed point in U and in particular there is a x∈U such that x=F(x).
3 Nonsingular problem
Consider the IVP
x=f(t,x,x),
x(a) =A, x(a) =B, B≥, (.)
wheref :Dt×Dx×Dp→R,Dt,Dx,Dp⊆R.
We include this problem into the following family of regular IVPs constructed forλ∈
[, ]
x=λf(t,x,x),
x(a) =A, x(a) =B, (.)
(R) There exist constantsT>a,m,m,M,M, and a sufficiently smallτ> such that
m≥, M–τ≥M≥B≥m≥m+τ,
[a,T]⊆Dt, [A–τ,M+τ]⊆Dx, [m,M]⊆Dp, whereM=A+M(T–a),
f(t,x,p)∈C[a,T]×[A–τ,M+τ]×[m–τ,M+τ]
,
f(t,x,p)≤ for(t,x,p)∈[a,T]×Dx×[M,M], (.)
f(t,x,p)≥ for(t,x,p)∈[a,T]×DM×[m,m],
whereDM=Dx∩(–∞,M].
Our first result ensures bounds for the eventualC-solutions to (.). We need them to prepare the application of the topological transversality theorem.
Lemma . Let(R)hold.Then each solution x∈C[a,T]to the family(.)λ,λ∈[, ],
satisfies the bounds
A≤x(t)≤M, m≤x(t)≤M, m≤x(t)≤M for t∈[a,T],
where m=min
f(t,x,p) : (t,x,p)∈[a,T]×[A,M]×[m,M]
,
M=max
f(t,x,p) : (t,x,p)∈[a,T]×[A,M]×[m,M]
.
Proof Suppose that the set
S–=
t∈[a,T] :M<x(t)≤M
is not empty. Then
x(a) =B≤M and x∈C[a,T]
imply that there exists an interval [α,β]⊂S–such that
x(α) <x(β).
This inequality and the continuity ofx(t) guarantee the existence of someγ ∈[α,β] for which
x(γ) > .
Since x(t), t∈[a,T], is a solution of the differential equation, we have (t,x(t),x(t))∈ [a,T]×Dx×Dp. In particular forγ we have
Thus, we apply (R) to conclude that
x(γ) =λfγ,x(γ),x(γ)≤,
which contradicts the inequalityx(γ) > . This has been established above. Thus,S–is empty and as a result
x(t)≤M fort∈[a,T].
Now, by the mean value theorem for eacht∈(a,T] there exists aξ∈(a,t) such that
x(t) –x(a) =x(ξ)(t–a),
which yields
x(t)≤M fort∈[a,T].
This allows us to use (.) to show similarly to above that the set
S+=
t∈[a,T] :m≤x(t) <m
is empty. Hence,
≤m≤x(t) fort∈[a,T] and so
A≤x(t) fort∈[a,T].
To estimatex(t), we observe firstly that (R) implies in particular
f(t,x,M)≤ for (t,x)∈[a,T]×[A,M] and
f(t,x,m)≥ for (t,x)∈[a,T]×[A,M],
which yieldm≤ andM≥. Multiplying both sides of the inequalityλ≤ bymand
M, we get, respectively,m≤λmandλM≤M. On the other hand, we have estab-lished
x(t)∈[A,M] and x(t)∈[m,M] fort∈[a,T]. Thus,
m≤λm≤λf
and eachλ∈[, ] and so
x(t)∈[m,M] fort∈[a,T]. Let us mention that some analogous results have been obtained in Kelevedjiev []. For completeness of our explanations, we present the full proofs here.
Now we prove an existence result guaranteeing the solvability of IVP (.).
Theorem . Let(R)hold.Then nonsingular problem(.)has at least one non-decreasing solution in C[a,T].
Proof Preparing the application of Theorem ., we define first the set
U=x∈CI[a,T] :A–τ<x<M+τ,m–τ<x<M+τ,m–τ<x<M+τ
,
whereCI[a,T] ={x∈C[a,T] :x(a) =A,x(a) =B}. It is important to notice that according to Lemma . allC[a,T]-solutions to family (.) are interior points ofU. Further, we introduce the continuous maps
j:CI[a,T]→C[a,T] byjx=x,
V:CI[a,T]→C[a,T] byVx=x, and fort∈[a,T] andx(t)∈j(U) the map
:C[a,T]→C[a,T] by (x)(t) =ft,x(t),x(t).
Clearly, the mapis also continuous since, by assumption, the functionf(t,x(t),x(t)) is continuous on [a,T] if
x(t)∈[m–τ,M+τ] and x(t)∈[m–τ,M+τ] fort∈[a,T].
In addition we verify thatV–exists and is also continuous. To this aim we introduce the linear map
W:CI[a,T]→C[a,T],
defined byWx=x, whereCI[a,T] ={x∈C[a,T] :x(a) = ,x(a) = }. It is one-to-one because each functionx∈C
I[a,T] has a unique image, and each functiony∈C[a,T] has
a unique inverse image which is the unique solution to the IVP
x=y, x(a) = , x(a) = .
It is not hard to see thatWis bounded and so, by the bounded inverse theorem, the map
W–exists and is linear and bounded. Thus, it is continuous. Now, usingW–, we define
where(t) =B(t–a) +Ais the unique solution of the problem
x= , x(a) =A, x(a) =B.
Clearly,V–is continuous sinceW–is continuous. We already can introduce a homotopy
H :U×[, ]→CI[a,T], defined by
H(x,λ)≡Hλ(x)≡λV–j(x) + ( –λ).
It is well known thatjis completely continuous, that is,jmaps each bounded subset ofC
I[a,T] into a compact subset ofC[a,T]. Thus, the imagej(U) of the bounded setU is compact. Now, from the continuity ofandV–it follows that the sets(j(U)) and
V–((j(U))) are also compact. In summary, we have established that the homotopy is compact. On the other hand, for its fixed points we have
λV–j(x) + ( –λ)=x
and
Vx=λj(x),
which is the operator form of family (.). So, each fixed point of Hλis a solution to (.), which, according to Lemma ., lies inU. Consequently, the homotopy is fixed point free on∂U.
Finally, H(x) is a constant map mapping each functionx∈Uto(t). Thus, according to Theorem ., H(x) =is essential.
So, all assumptions of Theorem . are fulfilled. Hence H(x) has a fixed point inUwhich means that the IVP of (.) obtained forλ= (i.e.(.)) has at least one solutionx(t) in
C[a,T]. From Lemma . we know that
x(t)≥m≥ fort∈[a,T],
from which its monotony follows.
The validity of the following results follows similarly.
Theorem . Let B> and let(R)hold for m> .Then problem(.)has at least one
strictly increasing solution in C[a,T].
Theorem . Let A> (A= )and let(R)hold for m= .Then problem(.)has at least
one positive(nonnegative)non-decreasing solution in C[a,T].
Theorem . Let A≥,B> and let(R)hold for m> .Then problem(.)has at least
4 A problem singular atxandx
In this section we study the solvability of singular IVP (.), (.) under the following as-sumptions.
(S) There are constantsT> ,m,mand a sufficiently smallν> such that
m> , B>m≥m+ν,
[,T]⊆Dt, (A,M˜+ν]⊆Dx, [m,B)⊆Dp, whereM˜=A+BT+ ,
f(t,x,p)∈C[,T]×(A,M˜+ν]×[m–ν,B)
, (.)
f(t,x,p)≤ for(t,x,p)∈[,T]×Dx×[m,B)
\SA (.)
and
f(t,x,p)≥ for(t,x,p)∈[,T]×DM˜×[m,m]
\SA,
whereDM˜= (–∞,M˜]∩Dx.
(S) For someα∈(,T]andμ∈(m,B)there exists a constantk< such thatkα+B>μ and
f(t,x,p)≤k< for(t,x,p)∈[,α]×(A,M˜]×[μ,B), whereT,mandM˜are as in (S).
Now, forn≥nα,μ, wherenα,μ>max{α–, (B+kα–μ)–}, andα,μ, andkare as in (S), we construct the following family of regular IVPs:
x=f(t,x,x),
x() =A+n–, x() =B–n–. (.) Notice, forn≥nα,μ, that we haveB–n–>μ–kα>μ>m> .
Lemma . Let(S)and(S)hold and let xn∈C[,T],n≥nα,μ,be a solution to(.)
such that
A<xn(t)≤ ˜M and m≤xn(t) <B for t∈[,T].
Then the following bound is satisfied for each n≥nα,μ:
xn(t) <φα(t) <B for t∈(,T],
whereφα(t) =kktα++BB,, tt∈∈([,α,Tα],].
Proof Since for eachn≥nα,μwe have
we will consider the proof for an arbitrary fixedn≥nα,μ, considering two cases. Namely,
xn(t) >μfort∈[,α] is the first case and the second one isxn(t) >μfort∈[,β) with
xn(β) =μfor someβ∈(,α].
Case. Fromμ<xn(t)≤B,t∈[,α], and (S) we have
xn(t) =ft,xn(t),xn(t)≤k fort∈[,α],
i.e. xn(t)≤kfort∈[,α]. Integrating the last inequality from totwe get
xn(t) –xn()≤kt, t∈[,α], which yields
xn(t)≤kt+B–n– fort∈[,α]. Nowm≤xn(t) <B,t∈[,T], and (.) imply
xn(t) =ft,xn(t),xn(t)
≤ fort∈[,T].
In particularxn(t)≤ fort∈[α,T], thus
xn(t)≤xn(α)≤kα+B–n– fort∈(α,T].
Case. As in the first case, we derive
xn(t)≤kt+B–n– fort∈[,β].
On the other hand, sincem≤xn(t) <Bfort∈[β,T], again from (.) it follows that
xn(t) =ft,xn(t),xn(t)
≤ fort∈[β,T],
which yields
xn(t)≤xn(β) =μ<kα+B–n–≤kt+B–n– fort∈[β,α] and
xn(t) <kα+B–n– fort∈(α,T]. So, as a result of the considered cases we get
xn(t)≤
kt+B–n–, t∈[,α],
kα+B–n–, t∈(α,T] <φα(t) fort∈[,T] andn≥nα,μ,
from which the assertion follows immediately.
Theorem . Let(S)and(S)hold.Then singular IVP(.), (.)has at least one strictly
increasing solution in C[,T]∩C(,T]such that
mt+A≤x(t)≤Bt+A for t∈[,T], m≤x(t) <B for t∈(,T].
Proof For each fixedn≥nα,μintroduceτ=min{(n)–,ν},
M=B–n–, M=B– (n)– and M=
B–n–T+A+ <M˜ having the properties
M–τ>M=B–n–>μ–kα>μ>m≥m+τ, [,T]⊆Dt,
A+n––τ,M+τ
⊆(A,M˜+τ]⊆Dx
and [m,M]⊆DpsinceM=B– (n)–<B. Besides,
f(t,x,p)≤ for (t,x,p)∈[,T]×Dx×[M,M]
\SA,
f(t,x,p)≥ for (t,x,p)∈[,T]×Dx×(–∞,M]
×[m,m]
\SA
and, in view of (.),
f(t,x,p)∈C[,T]×A+n––τ,M+ν
×[m–τ,M+τ]
.
All this implies that for eachn≥nα,μthe corresponding IVP of family (.) satisfies (R). Thus, we apply Theorem . to conclude that (.) has a solutionxn∈C[,T] for each
n≥nα,μ. We can use also Lemma . to conclude that for eachn≥nα,μandt∈[,T] we have
A<A+n–≤xn(t)≤M<M˜ (.) and
m≤xn(t)≤B–n–<B.
Now, these bounds allow the application of Lemma . from which one infers that for each
n≥nα,μandt∈[,T] the bounds
m≤xn(t) <φα(t)≤B (.)
hold. For later use, integrating the least inequality from tot,t∈(,T], we get
mt+A+n–≤xn(t) <Bt+A+n– fort∈[,T] (.) andn≥nα,μ.
We consider firstly the sequence{xn}ofC[,T]-solutions of (.) only for eachn≥nα,μ. Clearly, for eachn≥nα,μwe have in particular
which together with (.) gives
xn(t)≥xn(α)≥mα+A+n–>A>A fort∈[α,T],
whereA=mα+A. On combining the last inequality and (.) we obtain
A<xn(t) <M˜ fort∈[α,T],n≥nα,μ. (.)
From (.) we have in addition
m≤xn(t) <φα(α) =B+kα fort∈[α,T],n≥nα,μ. (.)
Now, using the fact that (.) implies continuity off(t,x,p) on the compact set [α,T]× [A,M˜]×[m,φα(α)] and keeping in mind that for eachn≥nα,μ
xn(t) =ft,xn(t),xn(t)
fort∈[α,T],
we conclude that there is a constantM, independent ofn, such that
xn(t) ≤M fort∈[α,T] andn≥nα,μ.
Using the obtaineda prioribounds forxn(t),xn(t) andxn(t) on the interval [α,T], we apply the Arzela-Ascoli theorem to conclude that there exists a subsequence{xnk},k∈N,
nk≥nα,μ, of{xn}and a functionxα∈C[α,T] such that
xnk–xα→ on the interval [α,T],
i.e., the sequences{xnk}and{xnk}converge uniformly on the interval [α,T] toxαandx α, respectively. Obviously, (.) and (.) are valid in particular for the elements of{xnk}and
{xn
k}, respectively, from which, lettingk→ ∞, one finds
A≤xα(t)≤ ˜M fort∈[α,T],
m≤xα(t)≤φα(α) <B fort∈[α,T].
Clearly, the functionsxnk(t),k∈N,nk≥nα,μ, satisfy integral equations of the form
xn
k(t) =x nk(α) +
t
α
fs,xnk(s),x nk(s)
ds, t∈(α,T].
Now, since f(t,x,p) is uniformly continuous on the compact set [α,T]× [A,M˜]× [m,φα(α)], from the uniform convergence of{xnk}it follows that the sequence{f(s,xnk(s),
xn
k(s))},nk≥nα,μis uniformly convergent on [α,T] to the functionf(s,xα(s),x
α(s)), which means
lim
k→∞
t
α
fs,xnk(s),x nk(s)
ds=
t
α
fs,xα(s),xα(s)
for eacht∈(α,T]. Returning to the integral equation and lettingk→ ∞yield
xα(t) =xα(α) +
t
α
fs,xα(s),xα(s)
ds, t∈(α,T],
which implies thatxα(t) is aC(α,T]-solution to the differential equationx=f(t,x,x) on (α,T]. Besides, (.) implies
mt+A≤xα(t)≤Bt+A fort∈[α,T].
Further, we observe that if the condition (S) holds for someα> , then it is true also for an arbitraryα∈(,α). We will use this fact considering a sequence{αi} ⊂(,α),i∈N, with the properties
αi+<αi fori∈Nand lim i→∞αi= .
For eachi∈Nwe consider sequences
{xi,nk}, nk≥ni+,μ,k∈N,ni+,μ>max
αi–+, (B+kαi+–μ)–
,
on the interval [αi+,T]. Thus, we establish that each sequence{xi,nk}has a subsequence
{xi+,nk},k∈N,nk≥ni+,μ, converging uniformly on the interval [αi+,T] to any function
xαi+(t),t∈[αi+,T], that is,
xi+,nk–xαi+→ on [αi+,T], (.)
which is aC(αi+,T]-solution to the differential equationx(t) =f(t,x(t),x(t)) on (αi+,T] and
mt+A≤xαi+(t)≤Bt+A fort∈[αi+,T],
m≤xαi+(t)≤φα(αi+) <B fort∈[αi+,T],
xαi+(t) =xαi(t) and x
αi+(t) =x
αi(t) fort∈[αi,T].
The properties of the functions from{xαi},i∈N, imply that there exists a functionx(t) which is aC(,T]-solution to the equationx=f(t,x,x) on the interval (,T] and is such that
mt+A≤x(t)≤Bt+A fort∈(,T],
hencelimt→+x(t) =A,
m≤x(t)≤φα(t) <B fort∈(,T],
x(t) =xαi(t) fort∈[αi,T] andi∈N, (.)
x(t) =xα
We have to show also that
lim
t→+x
(t) =B. (.)
Reasoning by contradiction, assume that there exists a sufficiently smallε> such that for everyδ> there is at∈(,δ) such that
x(t) <B–ε.
In other words, assume that for every sequence{δj} ⊂(,T],j∈N, withlimj→∞δj= , there exists a sequence{tj}having the propertiestj∈(,δj),limj→∞tj= and
x(tj) <B–ε. (.)
It is clear that every interval (,δj),j∈N, contains a subsequence of{tj}converging to . Besides, from (.) and (.) it follows that for everyj∈Nthere areij,nj∈Nsuch that
αij<δjand
xi,n
k–x
→ on [αi,δj) (.)
for alli>ijand allnk≥max{ni,μ,nj}. Moreover, since the accumulation point of{tj}is , for each sufficiently largej∈Nthere is atj∈[αi,δj) wherei>ij. In summary, for every sufficiently largej∈N, that is, for every sufficiently smallδj> , there areij,nj∈Nsuch that for alli>ijandnk≥max{ni,μ,nj}from (.) and (.) we have
xi,nk(tj) <B–ε,
which contradicts to the fact thatxi,n
k() =B–n –
k andxi,nk∈C[,T]. This contradiction proves that (.) is true.
Now, it is easy to verify that the function
x(t) =
A, t= ,
x(t), t∈(,T],
is aC[,T]∩C(,T]-solution to (.), (.). This function is strictly increasing because
x(t) =x(t)≥m> fort∈(,T], and the bounds forx(t) andx(t) follows immediately from the corresponding bounds forx(t) andx(t). The following results provide information about the presence of other useful properties of the assured solutions. Their correctness follows directly from Theorem ..
Theorem . Let A≥and let(S)and(S)hold.Then the singular IVP(.), (.)has at
least one strictly increasing solution in C[,T]∩C(,T]with positive values for t∈(,T].
5 Examples
Example . Consider the IVP
x=
√
b–x
√
c–tPk
x,
whereb,c∈(,∞), and the polynomialPk(p),k≥, has simple zeroespandpsuch that <p<B<p.
Let us note that hereDt= (–c,c),Dx= [–b,b] andDp=R. Clearly, there is a sufficiently smallθ> such that
<p–θ, p+θ≤B≤p–θ
andPk(p)= forp∈[p–θ,p)∪(p,p+θ]∪[p–θ,p)∪(p,p+θ]. We will show that all assumptions of Theorem . are fulfilled in the case
Pk(p) > forp∈[p–θ,p) and Pk(p) < forp∈(p,p+θ];
the other cases as regards the sign ofPk(p) aroundpandpmay be treated similarly. For this case chooseτ =θ/,m=p> andM=p. Next, using the requirement [A–τ,M+
τ]⊆[–b,b],i.e.[–θ/,pT+θ/]⊆[–b,b], we get the following conditions forθandT: –θ/≥–b and pT+θ/≤b,
which yieldθ ∈(, b] andT≤b–θ
p . Besides, [,T]⊆(–c,c) yieldsT<C. Thus, <T<
min{c,b–θ
p }. Now, choosing
m=p–θ and M=p+θ,
we really can apply Theorem . to conclude that the considered problem has a strictly increasing solutionx∈C[,T] withx(t) > ont∈(,T] for eachT<min{c,b–θ
p }.
Example . Consider the IVP
x= (x
– )( –x) (x– )(x– ),
x() = , lim
t→+x
(t) = .
Notice that here
SA=R× {} ×
(–∞, )∪(,∞), SB=R×
(–∞, )∪(,∞)× {}.
It is easy to check that (S) holds, for example, form= ,m= ,ν= ., and an arbitrary fixedT> , moreover,M˜= T+ . Besides, fork= –/(T+ ),α=T/ andμ= , for example, we have
kα+B= –T/(T+ )+ > =μ
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Technical University of Sofia, Branch Sliven, Sliven, Bulgaria.2Faculty of Mathematics, ‘St. Kl. Ohridski’ University of Sofia, Sofia, Bulgaria.
Acknowledgements
The work is partially supported by the Sofia University Grant 158/2013 and by the Bulgarian NSF under Grant DCVP -02/1/2009.
Received: 30 January 2014 Accepted: 13 June 2014 References
1. Rach˚unková, I, Tomeˇcek, J: Bubble-type solutions of non-linear singular problem. Math. Comput. Model.51, 658-669 (2010)
2. Rach˚unková, I, Tomeˇcek, J: Homoclinic solutions of singular nonautonomous second-order differential equations. Bound. Value Probl.2009, Article ID 959636 (2009)
3. Rach˚unková, I, Tomeˇcek, J: Strictly increasing solutions of a nonlinear singular differential equation arising in hydrodynamics. Nonlinear Anal.72, 2114-2118 (2010)
4. Agarwal, RP, O’Regan, D: Second-order initial value problems of singular type. J. Math. Anal. Appl.229, 441-451 (1999) 5. Yang, G: Minimal positive solutions to some singular second-order differential equations. J. Math. Anal. Appl.266,
479-491 (2002)
6. Yang, G: Positive solutions of some second-order nonlinear singular differential equations. Comput. Math. Appl.45, 605-614 (2003)
7. Bobisud, LE, O’Regan, D: Existence of solutions to some singular initial value problems. J. Math. Anal. Appl.133, 215-230 (1988)
8. Bobisud, LE, Lee, YS: Existence of monotone or positive solutions of singular second-order sublinear differential equations. J. Math. Anal. Appl.159, 449-468 (1991)
9. Cabada, A, Heikkilä, S: Extremality results for discontinuous explicit and implicit diffusion problems. J. Comput. Appl. Math.143, 69-80 (2002)
10. Cabada, A, Cid, JA, Pouso, RL: Positive solutions for a class of singular differential equations arising in diffusion processes. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal.12, 329-342 (2005)
11. Cabada, A, Nieto, JJ, Pouso, RL: Approximate solutions to a new class of nonlinear diffusion problems. J. Comput. Appl. Math.108, 219-231 (1999)
12. Cid, JA: Extremal positive solutions for a class of singular and discontinuous second order problems. Nonlinear Anal. 51, 1055-1072 (2002)
13. Maagli, H, Masmoudi, S: Existence theorem of nonlinear singular boundary value problem. Nonlinear Anal.46, 465-473 (2001)
14. Zhao, Z: Positive solutions of nonlinear second order ordinary differential equations. Proc. Am. Math. Soc.121, 465-469 (1994)
15. Kelevedjiev, P, Popivanov, N: On the solvability of a second-order initial value problem. Paper presented at the 40th international conference on the applications of mathematics in engineering and economics, Technical University of Sofia, Sozopol, 8-13 June 2014
16. Granas, A, Guenther, RB, Lee, JW: Nonlinear boundary value problems for ordinary differential equations. Diss. Math. 244, 1-128 (1985)
17. Grammatikopoulos, MK, Kelevedjiev, PS, Popivanov, N: On the solvability of a singular boundary-value problem for the equationf(t,x,x,x) = 0. J. Math. Sci.149, 1504-1516 (2008)
18. Kelevedjiev, P, Popivanov, N: Second order boundary value problems with nonlinear two-point boundary conditions. Georgian Math. J.7, 677-688 (2000)
19. Palamides, P, Kelevedjiev, P, Popivanov, N: On the solvability of a Neumann boundary value problem for the differential equationf(t,x,x,x) = 0. Bound. Value Probl.2012, 77 (2012). doi:10.1186/1687-2770-2012-77
20. Kelevedjiev, P: Positive solutions of nonsingular and singular second order initial value problems. Int. Electron. J. Pure Appl. Math.2, 117-127 (2010)
doi:10.1186/s13661-014-0161-z
