Pa s ania and P˚uža Bounda y Value P oblems 2014, 2014:147
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R E S E A R C H Open Access
The nonlinea Knese p oblem o singula
in phase a iables second-o de di e en ial
equa ions
Nino Pa s ania1,2* and Bedˇ
ich P˚uža3
Dedica ed o ou dea eache , P o . I an Kigu adze
*Co espondence: [email p o ec ed]
1A. Razmadze Ma hema ical
Ins i u e o I. Ja akhish ili Tbilisi
S a e Uni e si y, 6 Tama ash ili S .,
Tbilisi, 0177, Geo gia
2In e na ional Black Sea Uni e si y, 2
Da id Agmashenebeli Alley 13km,
Tbilisi, 0131, Geo gia
Full lis o au ho in o ma ion is
a ailable a he end o he a icle
Abs ac
Fo he singula in phase a iables diffe en ial equa ion
u = ( ,u,u),
sufficien condi ions a e ound o he exis ence o a solu ion sa is ying he condi ions
ϕ(u)=c,u( )>0, u( )<0 o >0,
whe e ϕ:C([0,a];R+)→R+is a con inuous nondec easing unc ional, c>0,anda>0.
MSC: 34B16; 34B40
Keywo ds: diffe en ial equa ion; second o de ; singula in phase a iables; Knese
solu ion; Knese p oblem; nonlinea
1 S a emen o he p oblem and o mula ion o he main esul s
Suppose
D=( ,x,y): >,x>,y<
,R+=[,+∞[,
and :D→R+is a con inuous unc ion. Conside he diffe en ial equa ion
u =  ,u,u. (.)
A con inuous unc ion u:R+→R+is said o be he Knese solu ion o Eq. (.)i i is
wice con inuously diffe en iable in he in e al ],+∞[, and in his in e al i sa isfies he
inequali ies
u( )>, u( )<
and he diffe en ial equa ion (.).
©2014 Pa s ania and P˚uža; licensee Sp inge . This is an Open Access a icle dis ibu ed unde he e ms o he C ea i e Commons
A ibu ion License (h p://c ea i ecommons.o g/licenses/by/4.0), which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion
in any medium, p o ided he o iginal wo k is p ope ly c edi ed.
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In he p esen pape , we in es iga e he p oblem on he exis ence o a Knese solu ion
o Eq. (.) sa is ying he condi ion
ϕ(u)=c,(.)
whe e ϕ:C([,a]; R+)→R+is a con inuous, nondec easing unc ional, a>,andc>.
I is na u al o name his p oblem he nonlinea Knese p oblem since i was fi s s udied
by Knese []in hecasewhe eEq.(.) and condi ion (.)ha e he o ms
u = ( ,u), (.)
u() = c,(.)
whe e :R+×R+→R+is a con inuous unc ion. Pa icula ly, in []i isp o ed ha i
is a nondec easing in he second a gumen unc ion sa is ying he local Lipschi z con-
di ion in his a gumen and ( ,)≡, hen o an a bi a ily fixed c> , he diffe en ial
equa ion (.) has a unique Knese solu ion sa is ying condi ion (.).
 yea s la e since Knese ’s pape was published, in hei s udy o he p oblem on he
dis ibu ion o elec ons in a hea y a om, Fe mi []andThomas[] had o in es iga e he
p oblem analogous o he Knese one o he conc e e second-o de diffe en ial equa ion
u = –
u
.(.)
In pa icula , hey ha e p o ed ha Eq. (.) has a unique solu ion sa is ying he bounda y
condi ions
u() = , lim
→+∞u( )=. (.)
I is easy o see ha a solu ion o p oblem (.), (.) is a Knese solu ion o Eq. (.)and
ice e sa, a Knese solu ion o ha equa ion, sa is ying he ini ial condi ion
u() = , (.)
is a solu ion o p oblem (.), (.). The e o e, p oblem (.), (.)isequi alen o he
Knese p oblem o Eq. (.) wi h he ini ial condi ion (.).
A e he pape s by Fe mi and Thomas we e published, many ma hema icians ha e been
in e es ed in he Knese - ype p oblems, and such p oblems ha e been in es iga ed in de-
ail o a wide class o diffe en ial equa ions and sys ems.
Mos o he esul s on he sol abili y and unique sol abili y o he Knese p oblem
o second-o de nonlinea diffe en ial equa ions, ob ained un il he beginning o s o
he las cen u y, a e eflec ed in he monog aph by Sansone []. F om u he in es i-
ga ions, fi s o all he pape by Ha man and Win ne [], whe e he Knese p oblem
o Eq. (.) was s udied in he case when :R+×R→R+is a con inuous unc ion,
should be no ed. Kigu adze [, ] s udied he same p oblem in he case when he unc-
ion :],+∞[×R→Rhas a nonin eg able singula i y in he fi s a gumen a he poin
=.
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The Knese p oblem o singula in a ime a iable highe -o de nonlinea diffe en ial
equa ions fi s was s udied by Kigu adze in [], whe e he op imal condi ions a e es ab-
lished o he sol abili y o he abo e-men ioned p oblem (see, [, Sec . ] as well). Anal-
ogous esul s we e ob ained by Kigu adze and Rachůnko á [] o heKnese p oblem
wi h a nonlinea ini ial condi ion.
Sufficien condi ions o he sol abili y o he Knese - ype p oblems o nonlinea di -
e en ial sys ems we e ob ained by Chan u ia [], Coffman [], Ha man and Win ne
[], Kigu adze and Rachůnko á [], and Rachůnko á [, ].
In all he abo e-men ioned wo ks, diffe en ial equa ions and sys ems, no ha ing singu-
la i ies in phase a iables, a e conside ed. The Knese p oblem o he diffe en ial equa-
ion wi h a singula i y in one o he phase a iables fi s was in es iga ed by Kigu adze [].
Howe e , in his pape we conside no he gene al diffe en ial equa ion bu he Emden-
Fowle ype highe -o de diffe en ial equa ion
u(n)=p( )u–λ.
As o he gene al diffe en ial equa ion (.) wi h singula i ies in phase a iables, o i
he Knese p oblem has been p ac ically uns udied so a . The aim o he p esen pape is
o fill his gap.
In wha ollows i is assumed ha he unc ion sa isfies he inequali y
g( )≤xλ|y|μ ( ,x,y)≤g( )(.)
in he domain D.He eλand μa e nonnega i e cons an s, λ+μ>,andgi:],+∞[→R+
(i= ,) a e con inuous unc ions, no equal iden ically o ze o in an a bi a y neighbo -
hood o +∞,i.e., he e exis s a sequence o posi i e numbe s ( k)+∞
k= such ha
lim
k→+∞ k=+∞,gi( k)> (i=,;k=,,...). (.)
Consequen ly, Eq. (.) has singula i ies in phase a iables since ei he
lim
x→ ( k,x,y)=+∞ o y<(k=,,...)
o
lim
y→ ( k,x,y)=+∞ o x>(k=,,...).
Th oughou he pape , he ollowing no a ion and defini ions a e used.
ν=+λ+μ
+μ. (.)
C([,a]; R) is he Banach space o con inuous unc ions u:[,a]→Rwi h he no m
uC=maxu( ):≤ ≤a,
C[,a]; R+=u∈C[, a];R:u( )≥ o ≤ ≤a.
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A unc ionalϕ:C([,a];R+)→R+is said o be nondec easing i o any u∈C([,a];R+)
and u∈C([,a]; R+) heinequali yϕ(u+u)≥ϕ(u)holds.
Fo any x∈R+,wepu ϕ(x)=ϕ(u), whe e u( )≡x.
AKnese solu ionuo Eq. (.) is called anishing a infini y i lim →+∞u( )=,andi
is called emo e om ze o i lim →+∞u( )>.
Theo em . I Eq.(.)has a Knese solu ion u, hen
+∞
g(s)ds <+∞ o >, +∞
+∞
g(s)ds
+μ
d <+∞, (.)
and
u( )> ( ;δ) o ≥, (.)
whe e
( ;δ)=δν+(+μ)
+μν+∞
+∞
s
g(x)dx
+μ
ds
ν
, (.)
δ=lim
→+∞u( ). (.)
Co olla y . I condi ion (.)holds and
c<ϕ (·;), (.)
hen p oblem (.), (.)has no Knese solu ion.
Theo em . I
+∞
g(s)ds <+∞ o >, +∞
+∞
g(s)ds
+μ
d <+∞, (.)
hen o any posi i e numbe δEq.(.)has a leas one Knese solu ion sa is ying equali y
(.).
Theo em . I along wi h (.) he condi ion
+∞
g(s)
λ
(s;)ds <+∞ o >, +∞
+∞
g(s)
λ
(s;)ds
+μ
d <+∞(.)
is sa isfied, hen Eq.(.)has a leas one anishing a infini y Knese solu ion.
Acco ding o Co olla y ., o small cp oblem (.), (.) has no Knese solu ion. Thus
we can expec he sol abili y o ha p oblem only o la ge c.
Suppose ha condi ion (.) holds. Then ob iously condi ion (.) is sa isfied as well.
We in oduce he unc ion
( ;δ)=δ++∞
( + μ)+∞
s
g(x)
λ
(x;δ)dx
+μ
ds o ≥,δ> , (.)
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and he numbe
c=in ϕ (·;δ):δ>
. (.)
Theo em . Le he unc ion gsa is y condi ion (.), and
lim
x→+∞ϕ(x)=+∞. (.)
I ,mo eo e ,
c>c, (.)
hen p oblem (.), (.)has a leas one Knese solu ion.
Rema k . In he case, whe e condi ions (.)holdand
c∈ϕ (·;),c,
he ques ion on he exis ence o a Knese solu ion o p oblem (.), (.) emains open.
Conside now hecasewhe e
g( )≡g( ), =cons ≥, (.)
i.e., hecasewhe einequali y(.)has he o m
g( )≤xλ|y|μ ( ,x,y)≤g( ).
F om Theo ems .,.,and. we immedia ely ha e he ollowing co olla y.
Co olla y . Le he unc ion gsa is y iden i y (.), and le he unc ional ϕsa is y
condi ion (.). Then he ollowing asse ions a e equi alen :
(i) he unc ion gsa isfies condi ions (.);
(ii) Eq.(.)has a leas one emo e om ze o Knese solu ion;
(iii) o any δ>,p oblem (.), (.)has a leas one Knese solu ion;
(i ) o any sufficien ly la ge c>,p oblem (.), (.)has a leas one Knese solu ion.
The ollowing s a emen is also alid.
Co olla y . Le he unc ion gsa is y iden i y (.), and he unc ional ϕsa is y con-
di ion (.). Le ,mo eo e , he e exis numbe s αand βsuch ha
lim in
→ αg( )>, lim sup
→ αg( )<+∞, (.)
lim in
→+∞ βg( )>, lim sup
→+∞ βg( )<+∞. (.)
Then he ollowing asse ions a e equi alen :
(i) α<+μ,β>+μ;
(ii) Eq.(.)has a leas one emo e om ze o Knese solu ion;

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(iii) Eq.(.)has a leas one anishing a infini y Knese solu ion;
(i ) o any δ>,p oblem (.), (.)has a leas one Knese solu ion;
( ) o any sufficien ly la ge c>,p oblem (.), (.)has a leas one Knese solu ion.
Rema k . In Theo em . and i s co olla ies i can be assumed, o example, ha
ϕ(u)=a

ψu(s)dσ(s),
whe e ψ:R+→R+is a con inuous, nondec easing unc ion, and σ:[,a]→Ris a non-
dec easing unc ion such ha
lim
x→+∞ψ(x)=+∞,σ(a)–σ() > .
Rema k . The abo e- o mula ed heo ems and hei co olla ies co e he case, whe e
he unc ion has a nonin eg able singula i y in a ime a iable a he poin = . Indeed,
i condi ions (.)hold,whe e<α<+μ, hen


(s,x,y)ds =+∞ o ( ,x,y)∈D.
2 Auxilia y p oposi ions
2.1 Lemmas on ap io ies ima es
Conside he diffe en ial inequali ies
u( )
μu( )≥g( )u–λτ( )(.)
and
g( )u–λτ( )≤u( )
μu( )≤g( )u–λτ( ). (.)
E e ywhe e in his sec ion i is assumed ha λand μa e nonnega i e cons an s, τ:R+→
R+is a con inuous unc ion such ha
τ( )≥ o ∈R+,(.)
and gi:],+∞[→R+(i= , ) a e con inuous unc ions, no equal iden ically o ze o in
an a bi a y neighbo hood o +∞,i.e., he e exis s a sequence o posi i e numbe s ( k)+∞
k=
such ha condi ion (.)issa isfied.
A con inuous unc ion u:R+→],+∞[issaid obe he Knese solu ion o he diffe en ial
inequali y (.)(o he diffe en ial inequali y (.)) i i is wice con inuously diffe en iable
in he in e al ],+∞[ and in his in e al along wi h he inequali y u( ) <  sa isfies he
diffe en ial inequali y (.) ( he diffe en ial inequali y (.)).
Lemma . I he diffe en ial inequali y (.)has a Knese solu ion u, hen he unc ion
gsa isfies condi ion (.), and u admi s es ima e (.), whe e is he unc ion gi en by
equali y (.), and ν,δa e numbe s gi en by equali ies (.)and (.).
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P oo In iew o (.)and he ac ha uis a Knese solu ion, om (.)wefind
u( )
+μ=(+μ)+∞
uλτ(x)u(x)
μu(x)u–λτ(x)dx
>(+μ)u–λ( )+∞
uλτ(x)u(x)
μu(x)dx ≥( + μ)u–λ( )+∞
g(x)dx,
and, consequen ly,
–u( )u( )λ
+μ>( + μ)+∞
g(x)dx
+μ
o >.
I we in eg a e his inequali y om o +∞, hendue oequali ies(.)and(.), we
ob ain
u( )ν>δν+(+μ)
+μν+∞
+∞
s
g(x)dx
+μ
ds o ≥.
The e o e condi ion (.)issa isfiedand he unc ionuadmi s es ima e (.). 
Le b∈],+∞[, and along wi h (.)le heinequali y
τ( )≤b o  ≤ ≤b(.)
be ulfilled. A con inuous unc ion u:[,b]→],+∞[issaid obea Knese solu ion o
he diffe en ial inequali y (.)(o he diffe en ial inequali y (.)) in he in e al [, b]i
i is con inuously diffe en iable in he in e al ], b] and in his in e al along wi h he in-
equali y u( ) <  sa isfies he diffe en ial inequali y (.) ( he diffe en ial inequali y (.)).
The ollowing lemma can be p o ed analogously o Lemma ..
Lemma . Le inequali y (.)be ulfilled and he diffe en ial inequali y (.)in he in-
e al [,b]ha e a Knese solu ion u.Then
b
b
s
g(x)dx
+μ
ds <+∞,(.)
and ha solu ion admi s he es ima e
u( )>w( ;δ,b) o ≤ ≤b,(.)
whe e δ=u(b)and
w( ;δ,b)=δν+(+μ)
+μνb
b
s
g(x)dx
+μ
ds
ν
o ≤ ≤b.(.)
Lemma . Le along wi h (.) he condi ion
b
b
s
g(x)dx
+μ
ds <+∞(.)
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be ulfilled,and le he diffe en ial inequali y (.)in he in e al [,b]ha e a Knese so-
lu ion u.Then his solu ion along wi h (.)admi s he es ima es
u( )≤w( ;δ,ε,b) o ≤ ≤b,(.)
ε+μ+w–λ(;δ,ε,b)b
g(s)ds
+μ
≤–u( )
≤–w( ;δ,ε,b) o < <b,(.)
whe e δ=u(b), ε=|u(b)|,and
w( ;δ,ε,b)=δ+b
ε+μ+(+μ)b
s
g(x)dx
wλ
(τ(x);δ,b)
+μ
ds (.)
o ≤ ≤b.
P oo Fi s no e ha he alidi y o condi ion (.) gua an ees he alidi y o condi ion
(.). On he o he hand, by Lemma . he unc ion uadmi s es ima e (.). By i ue o
his es ima e and he ac ha uis a Knese solu ion, (.)and(.)yield
u( )
+μ=ε+μ+(+μ)b
u(x)
μu(x)dx
≥ε+μ+b
g(x)u–λτ(x)dx
≥ε+μ+u–λ()b
g(x)dx o  < ≤b,(.)
u( )
+μ≤ε+μ+(+μ)b
g(x)u–λ(x)dx
≤ε+μ+(+μ)b
g(x)
wλ
(x;δ,b)dx =–w( ;δ,ε,b)+μ,
and, consequen ly,
–u( )≤–w( ;δ,ε,b) o < ≤b. (.)
In eg a ion o his inequali y om o b esul s in es ima e (.).
I along wi h (.) we ake in o accoun inequali ies (.)and(.), hen he alidi y o
es ima e (.) becomes e iden . 
2.2 Lemma on he sol abili y o a nonlinea Knese p oblem on a fini e in e al
Le b>a. Conside he p oblem on he exis ence o a Knese solu ion o Eq. (.)in he
in e al [, b] sa is ying condi ion (.).
I condi ion (.) holds, hen o any δ>,ε>,and ∈[,b], we assume
w( ;δ,ε,b)=δ+b
ε+μ+(+μ)b
s
g(x)dx
wλ
(x;δ,b)
+μ
ds.(.)
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Lemma . Le condi ion (.)be ulfilled and he e exis numbe s ε>,δ>,and
δ∗>δsuch ha
ϕδ∗>c(.)
and
ϕw(·;δ,ε,b)<c.(.)
Then p oblem (.), (.)has a Knese solu ion u in he in e al [, b]such ha
δ≤u(b)≤δ∗,u(b)=–ε.(.)
P oo Fo a bi a ily fixed δ∈[δ,δ∗] and na u al numbe k, we conside he Cauchy p ob-
lem
u( )=u( )
–μ  ,uτk( ),uτk( ),(.)
u(b)=δ,u(b)=–ε,(.)
whe e
( ,x,y)=|y|μ ( ,x,y), (.)
τk( )=⎧
⎨
⎩
+b
k o  ≤ ≤b–b
k,
b o b–b
k< ≤b.(.)
By i ue o condi ions (.), (.) and equali ies (.), (.), p oblem (.), (.)has
auniquesolu ionin hein e al[,b], which is a Knese solu ion o he diffe en ial in-
equali y (.) as well, whe e τ(x)≡τk(x). Deno e his solu ion by uk( ;δ). I is clea ha
he unc ion ( ,δ)→uk( ;δ) is con inuous on [,b]×[δ,δ∗], and on ],b[×[δ,δ∗]i sa -
isfies he inequali ies
uk( ;δ)>δ,u
k( ;δ)≤–ε.(.)
On he o he hand, by Lemma .,on],b]×[δ,δ∗] his unc ion admi s he es ima es
uk( ;δ)≤wk( ;δ,ε,b)≤ ,(.)
–u
k( ;δ)≤–w
k( ;δ,ε,b)≤–w
( ), (.)
whe e
wk( ;δ,ε,b)=δ+b
ε+μ+(+μ)b
s
g(x)dx
wλ
(τk(x);δ,b)
+μ
ds,
w( )=δ∗+b
ε+μ+(+μ)δ–λ
b
s
g(x)dx
+μ
ds,
=w().
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Then due o condi ion (.), he e exis s a sequence o posi i e numbe s (εk)+∞
k= sa is ying
(.) and he inequali ies
ϕw(·;δ,εk,bk)<c(k=,,...). (.)
By Lemma . and condi ions (.), (.), o any na u al k,p oblem(.), (.)hasa
Knese solu ion ukin he in e al [, bk]such ha
δ≤δk≤δ∗,u(bk)=–εk,
whe e δk=u(bk).
Wi hou loss o gene ali y, we can assume ha he sequence (δk)+∞
k= is con e ging. Pu
δ=lim
k→+∞δk.
By Lemma ., hesequence(uk)+∞
k= con ains a uni o mly con e ging on e e y fini e
in e al om R+subsequence (ukm)+∞
m= such ha he unc ion, defined by he equali y
u( )= lim
m→+∞ukm( ) o ≥,
is a Knese solu ion o p oblem (.), (.). On he o he hand, i in he equali y ϕ(ukm)=c
we pass o he limi as m→+∞, hen i becomes clea ha usa isfies condi ion (.)as
well. Thus uis a Knese solu ion o p oblem (.), (.). 
To con ince ou sel es ha Co olla y . is alid, i suffices o no e ha i condi ions
(.)-(.) a e ulfilled, hen each o condi ions (.), (.), (.)issa isfiediffα<+μ
and β>+μ.
Compe ing in e es s
The au ho s decla e ha hey ha e no compe ing in e es s.
Au ho s’ con ibu ions
All au ho s ead and app o ed he final manusc ip .
Au ho de ails
1A. Razmadze Ma hema ical Ins i u e o I. Ja akhish ili Tbilisi S a e Uni e si y, 6 Tama ash ili S ., Tbilisi, 0177, Geo gia.
2In e na ional Black Sea Uni e si y, 2 Da id Agmashenebeli Alley 13km, Tbilisi, 0131, Geo gia. 3Facul y o Business and
Managemen , B no Uni e si y o Technology, Kolejní 2906/4, B no, 612 00, Czech Republic.
Acknowledgemen s
Fo he fi s au ho his wo k is suppo ed by he Sho a Rus a eli Na ional Science Founda ion (P ojec #
FR/317/5-101/12), and o he second au ho his wo k is suppo ed by he In e nal G an Agency a B no Uni e si y o
Technology (P ojec # FP-S-13-2148).
Recei ed: 11 Feb ua y 2014 Accep ed: 29 May 2014
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doi:10.1186/s13661-014-0147-x
Ci e his a icle as: Pa s ania and P˚uža: The nonlinea Knese p oblem o singula in phase a iables second-o de
di e en ial equa ions. Bounda y Value P oblems 2014 2014:147.