The nonlinear Kneser problem for singular in phase variables second-order differential equations

Pa s ania and P˚uža Bounda y Value P oblems 2014, 2014:147 h p://www.bounda y aluep oblems.com/con en /2014/1/147 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. Pa s ania and P˚uža Bounda y Value P oblems 2014, 2014:147 Page 2 o 17 h p://www.bounda y aluep oblems.com/con en /2014/1/147 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 =. Pa s ania and P˚uža Bounda y Value P oblems 2014, 2014:147 Page 3 o 17 h p://www.bounda y aluep oblems.com/con en /2014/1/147 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. Pa s ania and P˚uža Bounda y Value P oblems 2014, 2014:147 Page 4 o 17 h p://www.bounda y aluep oblems.com/con en /2014/1/147 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> , (.) Pa s ania and P˚uža Bounda y Value P oblems 2014, 2014:147 Page 5 o 17 h p://www.bounda y aluep oblems.com/con en /2014/1/147 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; Pa s ania and P˚uža Bounda y Value P oblems 2014, 2014:147 Page 6 o 17 h p://www.bounda y aluep oblems.com/con en /2014/1/147 (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 (.). Pa s ania and P˚uža Bounda y Value P oblems 2014, 2014:147 Page 7 o 17 h p://www.bounda y aluep oblems.com/con en /2014/1/147 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 <+∞(.) Pa s ania and P˚uža Bounda y Value P oblems 2014, 2014:147 Page 8 o 17 h p://www.bounda y aluep oblems.com/con en /2014/1/147 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.(.) Pa s ania and P˚uža Bounda y Value P oblems 2014, 2014:147 Page 9 o 17 h p://www.bounda y aluep oblems.com/con en /2014/1/147 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(). Pa s ania and P˚uža Bounda y Value P oblems 2014, 2014:147 Page 16 o 17 h p://www.bounda y aluep oblems.com/con en /2014/1/147 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 Re e ences 1. Knese , A: Un e suchung und asymp o ische da s ellung de in eg ale gewisse diffe en ialgleichungen bei g ossen eellen we hen de a gumen s, I. J. Reine Angew. Ma h. 116, 173-212 (1896) 2. Fe mi, E: Un me odo s a is ico pe la de e minazione di alcune p op ie à dell’a omo. Rend. R. Accad. Naz. Lincei 6, 602-607 (1927) 3. Thomas, LH: The calcula ion o a omic fields. P oc. Camb. Philos. 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