Existence of solutions converging to zero for nonlinear delayed differential systems

Rebenda and Šma da Ad ances in Diffe ence Equa ions (2015) 2015:349 DOI 10.1186/s13662-015-0687-0 RESEARCH Open Access Exis ence o solu ions con e ging o ze o o nonlinea delayed di e en ial sys ems Jose Rebenda1and Zdenˇ ek Šma da1,2* *Co espondence: sma da@ eec. u b .cz 1CEITEC BUT, B no Uni e si y o Technology, Technicka 3058/10, B no, 61600, Czech Republic 2Depa men o Ma hema ics, B no Uni e si y o Technology, B no, Czech Republic Abs ac We p esen a esul abou an in e es ing asymp o ic p ope y o eal wo-dimensional delayed diffe en ial sys ems sa is ying ce ain sufficien condi ions. We employ wo p e ious esul s, which we e ob ained using a Razumikhin- ype modifica ion o he Wa˙ zewski opological me hod o e a ded diffe en ial equa ions and he me hod o a Lyapuno -K aso skii unc ional. The esul is illus a ed by a non i ial explana o y example. MSC: 34K12; 34K20 Keywo ds: diffe en ial sys em wi h delays; s abili y o solu ions 1 In oduc ion Va ious p ope ies o solu ions o diffe en ial equa ions wi h delay we e ex ensi ely s ud- ied ecen ly.Amongo he swemen ion[–]and he e e ences he ein.The esul scon- ained in his pape a e a gene aliza ion o p e ious esea ch published in [–]and []. Ou aim he eis os udy heasymp o icbeha io o solu ions o he ollowingsys em o diffe en ial equa ions: x( )=A( )x( )+ m  k= Bk( )xθk( )+h ,x( ),xθ( ),...,xθm( ),() whe e –θk( )≥ a e bounded noncons an delays sa is ying lim →∞θk( )=∞,θk( )a e eal unc ions, h( ,x,y)=h( ,x,y,...,ym),h( ,x,y,...,ym) is a eal ec o unc ion, whe e x=(x,x), yk=(yk,yk), and A( )=aij( ),Bk( )=bijk( ),i,j=,;k=,...,m, a e eal squa e ma ices. In hispape ,wein oduceanin e es ing esul ,whichisacombina iono wo heo ems p esen edin[], one ega ding heins abili yo solu ions, heo he onedealing wi h he exis ence o bounded solu ions. ©2015 Rebenda and Šma da. This a icle is dis ibu ed unde he e ms o he C ea i e Commons A ibu ion 4.0 In e na- ional 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 you gi e app op ia e c edi o he o iginal au ho (s) and he sou ce, p o ide a link o he C ea i e Commons license, and indica e i changes we e made. Rebenda and Šma da Ad ances in Diffe ence Equa ions (2015) 2015:349 Page 2 o 10 I is supposed ha he unc ion hsa isfies he Ca a héodo y condi ions on [ ,∞)× R(m+), he unc ions bijk a e locally Lebesgue in eg able on [ ,∞), and he unc ions θk, aij a e locally absolu ely con inuous on [ ,∞). Since we s udy wo-dimensional sys ems, we use a ans o ma ion in o complex a i- ables o simpli y he sys em () in o one equa ion wi h complex coefficien s. Thecomplex a iablesa edefinedasz=x+ix,w=y +iy,...,wm=ym+iym.Using his ans o ma ion we ge z( )=a( )z( )+b( )¯ z( )+ m  k= Ak( )zθk( )+Bk( )¯ zθk( ) +g ,z( ),zθ( ),...,zθm( ),() whe e we assume (J=[ ,∞)): •Ak,Bk∈Lloc(J,C)(i.e. locally Lebesgue in eg able complex- alued unc ions on J) o k=,...,m, •θk∈ACloc(J,R)(i.e. locally absolu ely con inuous eal- alued unc ions on J) o k=,...,m, •a,b∈ACloc(J,C)(i.e. locally absolu ely con inuous complex- alued unc ions on J), •g∈K(J×Cm+,C)(i.e. a complex- alued unc ion which sa isfies he Ca a héodo y condi ions on J×Cm+). Ob iously, he unc ion gis in gene al dependen on ¯ zaswellas one e y ¯ z(θk).Howe e , he ac ha he unc iongsa isfies heCa a héodo ycondi ionsenablesus osignifican ly simpli y he no a ion by using only z, since he alidi y o he Ca a héodo y condi ions is no iola ed by composing wi h con inuous unc ions ¯ z,¯ wk,andθk. The ela ions be ween he unc ions a e he ollowing: a( )= a( )+a( )+i a( )–a( ), b( )= a( )–a( )+i a( )+a( ), Ak( )= bk( )+bk( )+i bk( )–bk( ), Bk( )= bk( )–bk( )+i bk( )+bk( ), g( ,z,w,...,wm) =h , (z+¯ z),  i(z–¯ z),  (w+¯ w),...,  i(wm–¯ wm) +ih , (z+¯ z),  i(z–¯ z),  (w+¯ w),  i(w–¯ w),...,  i(wm–¯ wm). Con e sely, pu ing a( )=Rea( )+b( ),a( )=Imb( )–a( ), a( )=Ima( )+b( ),a( )=Rea( )–b( ), bk( )=ReAk( )+Bk( ),bk( )=ImBk( )–Ak( ), Rebenda and Šma da Ad ances in Diffe ence Equa ions (2015) 2015:349 Page 3 o 10 bk( )=ImAk( )+Bk( ),bk( )=ReAk( )–Bk( ), h( ,x,y,...,ym)=Reg( ,x+ix,y +iy,...,ym+iym), h( ,x,y,...,ym)=Img( ,x+ix,y +iy,...,ym+iym), equa ion() can be w i en in he eal o m () as well. The equi alence o he dynamical in a ian s and asymp o ic p ope ies o he solu ions o he eal sys em () and he complex equa ion ()isshownin[] o hesimplecase co e ing o dina y diffe en ial equa ions. In his pape we conside ()in hecasewhen limin →∞ a( )–b( )> () and s udy he beha io o he solu ions o () unde his assump ion, which gene ally means ha de A( )> o sufficien lyla ge.Thissi ua ionco esponds o hecasewhen he equilib ium poin  o he au onomous homogeneous sys em x=Ax,() whe e Ais supposed o be a egula cons an ma ix, is a cen e , a ocus o a node. Such a si ua ion has some geome ical aspec s, which a e used in an analysis o he ans o med equa ion(). See [] o mo e de ails. Fu he , we suppose ha () sa isfies he uniqueness p ope y o solu ions. 2P elimina ies Th oughou his pape we will assume ha limin →∞ a( )–b( )>, ≥θk( )≥ – o ≥ + ,() whe e >  is a cons an , which means ha he delays θka e bounded. This is he same caseasconside edin[].Simila esul s o acasediffe en om()we eob ainedin[]. Then he e a e numbe s T≥ + and μ>such ha a( )>b( )+μ o ≥T.() Deno e c( )=¯ a( )b( ) |a( )|,γ( )=a( )+a( )–b( )() and ϑ( )=Re(γ( )γ( )–¯ c( )c( ))–|γ( )c( )–γ( )c( )| γ( )–|c( )|, α( )=–b( ) a( )sgnRea( ). () Mo eo e , we assume he ollowing condi ions o be alid: Rebenda and Šma da Ad ances in Diffe ence Equa ions (2015) 2015:349 Page 4 o 10 (i) The numbe s T≥ + and μ>  sa is y condi ion (). (ii)The ea e unc ions,κ,κk:[T,∞)→R,whe eiscon inuouson[T,∞),such ha γ( )g( ,z,w,...,wm)+c( )¯ g( ,z,w,...,wm) ≤κ( )γ( )z+c( )¯ z+m  k= κk( )γθk( )wk+cθk( )¯ wk+( ) o ≥T,z,wk∈C(k=,...,m). (iin)The ea enumbe sRn≥ and unc ions κn,κnk :[T,∞)→Rsa is ying he in- equali y γ( )g( ,z,w,...,wm)+c( )¯ g( ,z,w,...,wm) ≤κn( )γ( )z+c( )¯ z+m  k= κnk( )γθk( )wk+cθk( )¯ wk o ≥τn≥T,|z|+m k= |wk|>Rn. (iii) The unc ion β∈ACloc([T,∞),R –)issuch ha θ k( )β( )≤–λk( )a.e.on[T,∞), () whe e λkis gi en o ≥Tby λk( )=κk( )+Ak( )+Bk( )γ( )+|c( )| γ(θk( ))–|c(θk( ))|.() (iiin) The unc ion βn∈ACloc([T,∞),R –)issuch ha θ k( )βn( )≤–λnk( )a.e.on[τn,∞), () whe e λnk is gi en o ≥Tby λnk( )=κnk( )+Ak( )+Bk( )γ( )+|c( )| γ(θk( ))–|c(θk( ))|.() (i n) The unc ion Λnis eal locally Lebesgue in eg able and he inequali ies β n( )≥ Λn( )βn( ), Θn( )≥Λn( ) a e sa isfied o almos all ∈[τn,∞), whe e Θnis gi en by Θn( )=α( )Rea( )+ϑ( )–κn( )+mβn( ). Fu he mo e, deno e Θ( )=α( )Rea( )+ϑ( )–κ( ). () 3 Main esul s Fi s o all, we ecall he wo esul s om []. Lemma  Le he assump ions (i), (ii), (iii), (i )be ulfilled o some τ≥T.Suppose he e exis ≥τand ν∈(–∞,∞)such ha in ≥   Λ(s)ds–lnγ( )+c( )≥ν.() Rebenda and Šma da Ad ances in Diffe ence Equa ions (2015) 2015:349 Page 5 o 10 I z( )is any solu ion o () sa is ying min θ( )≤s≤ z(s)>R,Δ( )>Re–ν, () whe e θ( )= min k=,...,mθk( ), Δ( )=γ( )–c( )z( )+β( )max θ( )≤s≤ z(s)m  k=  θk( )γ(s)+c(s)ds, hen z( )≥Δ( ) γ( )+|c( )|exp  Λ(s)ds() o all ≥  o which z( )is defined. Lemma Le he condi ions (i), (ii), (iii) be ulfilled and Λ,θ k(k=,...,m)be con inuous unc ions such ha inequali yΛ( )≤Θ( )holdsa.e.on [T,∞),whe e Θisdefinedby(). Suppose ha ξ:[T– ,∞)→Ris a con inuous unc ion such ha Λ( )+β( )m  k= θ k( )exp– θk( )ξ(s)ds–ξ( )>( )C– exp– Tξ(s)ds() o ∈[T,∞]and some cons an C >.Then he e exis s a >T and a solu ion z( )o () sa is ying z( )≤C γ( )–|c( )|exp Tξ(s)ds() o ≥ . I we combine he p e ious wo esul s, we a e able o p o e he ollowing heo em, which is he undamen al esul o his pape . Theo em  Assume ha he hypo heses (i), (ii), (iin), (iii), (iiin), (i n)a e alid o T ≤τn, whe e <Rn,n∈N,in n∈NRn=.Suppose ha θ k,Λa e con inuous unc ions such ha inequali yΘ( )≥Λ( )issa isfiedalmos e e ywhe eon[T,∞),whe eΘ( )=α( )Rea( )+ ϑ( )–κ( ).Le ξ:[T– ,∞)→Rbe a con inuous unc ion sa is ying he inequali y ( )C– exp– Tξ(s)ds<Λ( )+β( )m  k= θ k( )exp– θk( )ξ(s)ds–ξ( )() o some cons an C >and ∈[T,∞). Assume in τn≤s≤ <∞ sΛn(σ)dσ–lnγ( )+c( )≥ν,() Rebenda and Šma da Ad ances in Diffe ence Equa ions (2015) 2015:349 Page 6 o 10 limsup →∞  TΛn(s)–ξ(s)ds+ln γ( )–|c( )| γ( )+|c( )|=∞,() lim →∞βn( )max θ( )≤s≤ exp[s Tξ(σ)dσ] γ(s)–|c(s)| m  k=  θk( )γ(s)+c(s)ds=, () o n ∈N,whe e ν∈(–∞,∞)and θ( )=mink=,...,mθk( ). Then he e is a solu ion z( )o () wi h he p ope y lim →∞ min θ( )≤s≤ z(s)=. () P oo Using Lemma we ob ain he exis ence o T≤ and a solu ion z( )o ()sa is- ying o ≥  he inequali y z( )≤C γ( )–|c( )|exp Tξ(s)ds.() F om ()wege in τ≤ <∞ τ ΛN(s)ds–lnγ( )+c( )≥ν>–∞. Lemma yields Ψ(τ) γ( )+|c( )|exp τ ΛN(s)ds≤z( )() o τ≤ ,whe eΨis gi en by Ψ(τ)=γ(τ)–c(τ)z(τ)+βN(τ)max θ(τ)≤s≤τz(s)m  k= τ θk(τ)γ(s)+c(s)ds. Assume ha () does no hold. This implies he exis ence o ε>  sa is ying limsup →∞minθ( )≤s≤ |z(s)|>ε.We akeN∈Nsuch ha max{RN, μRNe–ν}<ε.Then maxRN, μRNe–ν<min θ(τ)≤s≤τz(s)() holds o some τ>max{T,τN, }.Taking() in o accoun we may assume ha βN(τ)Cmax θ(τ)≤s≤τ exp[s Tξ(σ)dσ] γ(s)–|c(s)| m  k= τ θk(τ)γ(s)+c(s)ds< RNe–ν.() Hence, wi h espec o (), (), (), (), (), and he nonposi i eness o βN,weob ain γ(τ)–c(τ)z(τ)+βN(τ)max θ(τ)≤s≤τz(s)m  k= τ θk(τ)γ(s)+c(s)ds ≥γ(τ)–c(τ)z(τ)+βN(τ)Cmax θ(τ)≤s≤τ exp[s Tξ(σ)dσ] γ(s)–|c(s)| ×m  k= τ θk(τ)γ(s)+c(s)ds≥μ μRNe–ν– RNe–ν>RNe–ν. Rebenda and Šma da Ad ances in Diffe ence Equa ions (2015) 2015:349 Page 7 o 10 The inequali ies ()and() gi e he es ima ion Ψ(τ) γ( )+|c( )|exp τ ΛN(s)ds≤C γ( )–|c( )|exp Tξ(s)ds, which means ha  TΛN(s)–ξ(s)ds+ln γ( )–|c( )| γ( )+|c( )|≤τ TΛN(s)ds–lnC–Ψ(τ) o τ≤ , which is in con adic ion o (). The p oo is comple e.  Rema k  Theo em  co e s mo e gene al si ua ions han Theo em  in [], whe e he diffe en undamen al assump ion limin →∞(|Ima( )|–|b( )|)>issupposed ohold. Indeed, i we ake o example a( )≡+iand b( )≡i, hencondi ion()in hispape is sa isfied bu he condi ion limin →∞(|Ima( )|–|b( )|)> is no alid. The ollowingnon i ialexamplewascons uc ed oillus a eanapplica iono he he- o e ical esul p esen ed in Theo em . Example  Conside he wo-dimensional sys em o he nonlinea delayed diffe en ial equa ions x( )=x( )+  x( )–x( )+ m  k=  me– x –e–k +e– ,() x( )=x( )+x( )+  x( )– m  k=  me– x –e–k .() This sys em can be w i en in ma ix o m (), whe e A( )=+  – +  ,Bk( )=  me–  e – ,and h ,x( ),xθ( ),...,xθm( )=e– . Following ou app oach, we use a ans o ma ion in o he complex plane and ob ain he delayed diffe en ial equa ion () wi h complex- alued coefficien s, z( )=(+i)z( )+i¯ z( )+  z( )+ m  k=  me– ¯ z –e–k +e– ,() whe e a( )≡+i,b( )≡i,Ak( )≡, Bk( )≡, θk( )= –e –k o k= ,...,m, g( ,z,w,...,wm)=  z+m k=  me– wk+e– . Ob iously –≤θk( )≤ and θ k( )=+ke–k ≥> o ≥. Suppose =andT≥. Then γ( )=|a( )|+|a( )|–|b( )|≡+ √, c( )= ¯ a( )b( )/|a( )|≡+i . Rebenda and Šma da Ad ances in Diffe ence Equa ions (2015) 2015:349 Page 8 o 10 Fu he , γ( )g( ,z,w,...,wm)+c( )¯ g( ,z,w,...,wm) ≤γ+|c| γ–|c| γ( )z+c( )¯ z+γ+|c| γ–|c| m  k=  me– γθk( )wk+cθk( )¯ wk+e– =√ √  γ( )z+c( )¯ z+√ √ e– m m  k= γθk( )wk+cθk( )¯ wk+e– . Thus condi ions (i) and (ii) a e ulfilled wi h κ( )≡√ √,κk( )=e– √ m√,and( )=e– .To mee condi ion (iin), we es ima e o Rn= n,τn=n,and|z|>Rn γ( )g( ,z,w,...,wm)+c( )¯ g( ,z,w,...,wm) ≤γ+|c| γ–|c| +( ) Rnγ( )z+c( )¯ z +γ+|c| γ–|c| m  k=  me– γθk( )wk+cθk( )¯ wk =√ √ +ne– γ( )z+c( )¯ z+√ √ e– m m  k= γθk( )wk+cθk( )¯ wk, whe e κn( )=√ √[ +ne– ]andκnk( )=√ √e– m. Condi ion (iii) holds wi h –λk( )θ k( )– =–e– √ m√  +ke–k ≥–e– √ m√=β( ). Condi ion (iiin)issa isfied o βn( )=–e–( –) m≤–e– √ m√=–λnk( )θ k( )–. We ge condi ion (i n) by se ing n( )=Θn( )= –√ √ +ne– –e–( –) >. Fu he , we pu ( )=Θ( )= –√ √. Then condi ion ()holds o ξ( )= –√  √–√ √ e– m–e– and ξ( )> o ≥T=. Nowi isno difficul o e i ycondi ions()and(),sincen( )–ξ( )>andn( )>  o n∈N.In es iga ing he ac o so hep oduc in pa en hesesin (),we come o he Rebenda and Šma da Ad ances in Diffe ence Equa ions (2015) 2015:349 Page 9 o 10 conclusion ha βn=Oe– ,exps Tξ(σ)dσ=Oe.  and m  k=  θk( )γ(s)+c(s)ds=Om  k= e–k =Oe– . Consequen ly, he p oduc o hese ac o s is asymp o ically equal o O(e–δ ), whe e δ> and huscondi ion ()is sa isfied. All assump ionso Theo em a e ulfilled and we can conclude ha he e exis >  and a solu ion z( )o ()sa is ying() o ≥ . Rema k  This esul is sligh ly su p ising. Howe e , i is in good ag eemen wi h he well-known ac ha in oducing delay in o an uns able sys em wi hou delay can cause a changeo beha io o hesys em.Suchsi ua ionsa edesc ibedandco esponding esul s a e o mula ed e.g.in []. 4Conclusion We p o ed an in e es ing esul abou he s abili y o wo-dimensional sys ems wi h boundeddelays.Sufficien condi ions o hes abili yo ano iginallyuns ablesys emwe e p esen ed. The esul is in pe ec ag eemen wi h he esul s s a ed and p o ed in he es- ablished li e a u e. An example showed how his esul can be used in p ac ice. 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 The au ho s ha e made con ibu ions o he same significance. All au ho s ead and app o ed he final manusc ip . Acknowledgemen s The wo k o he au ho s was ealized in CEITEC - Cen al Eu opean Ins i u e o Technology wi h esea ch in as uc u e suppo ed by he p ojec CZ.1.05/1.1.00/02.0068 financed om Eu opean Regional De elopmen Fund and he second au ho was suppo ed by he p ojec FEKT-S-11-2-921 o Facul y o Elec ical Enginee ing and Communica ion, B no Uni e si y o Technology. This suppo is g a e ully acknowledged. Recei ed: 15 Sep embe 2015 Accep ed: 5 No embe 2015 Re e ences 1. Baculíko á, B, Džu ina, J, Rogo chenko, YV: Oscilla ion o hi d o de inomial delay diffe en ial equa ions. Appl. Ma h. Compu . 218(13), 7023-7033 (2012) 2. Be ezansky, L, B a e man, E: On nonoscilla ion and s abili y o sys ems o diffe en ial equa ions wi h a dis ibu ed delay. Au oma ica 48(4), 612-618 (2012) 3. Be ezansky, L, Diblík, J, S oboda, Z, Šma da, Z: Simple uni o m exponen ial s abili y condi ions o a sys em o linea delayed diffe en ial equa ions. Appl. Ma h. Compu . 250(1), 605-614 (2012) 4. 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