Existence of solutions converging to zero for nonlinear delayed differential systems

Rebenda and Šmarda Advances in Difference Equations (2015) 2015:349 DOI 10.1186/s13662-015-0687-0 RESEARCH Open Access Existence of solutions converging to zero for nonlinear delayed differential systems Josef Rebenda1and Zdenˇ ek Šmarda1,2* *Correspondence: [email protected] 1CEITEC BUT, Brno University of Technology, Technicka 3058/10, Brno, 61600, Czech Republic 2Department of Mathematics, Brno University of Technology, Brno, Czech Republic Abstract We present a result about an interesting asymptotic property of real two-dimensional delayed differential systems satisfying certain sufficient conditions. We employ two previous results, which were obtained using a Razumikhin-type modification of the Wa˙ zewski topological method for retarded differential equations and the method of a Lyapunov-Krasovskii functional. The result is illustrated by a nontrivial explanatory example. MSC: 34K12; 34K20 Keywords: differential system with delays; stability of solutions 1 Introduction Various properties of solutions of differential equations with delay were extensively studied recently.Amongotherswemention[–]andthereferencestherein.Theresultscontained in this paper are a generalization of previous research published in [–]and []. Our aim hereistostudy theasymptoticbehavior of solutions of thefollowingsystem of differential equations: x(t)=A(t)x(t)+ m  k= Bk(t)xθk(t)+ht,x(t),xθ(t),...,xθm(t),() where t–θk(t)≥ are bounded nonconstant delays satisfying limt→∞θk(t)=∞,θk(t)are real functions, h(t,x,y)=h(t,x,y,...,ym),h(t,x,y,...,ym) is a real vector function, where x=(x,x), yk=(yk,yk), and A(t)=aij(t),Bk(t)=bijk(t),i,j=,;k=,...,m, are real square matrices. Inthispaper,weintroduceaninterestingresult,whichisacombinationoftwotheorems presentedin[], oneregardingtheinstabilityofsolutions, theother onedealing withthe existence of bounded solutions. ©2015 Rebenda and Šmarda. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Rebenda and Šmarda Advances in Difference Equations (2015) 2015:349 Page 2 of 10 It is supposed that the function hsatisfies the Carathéodory conditions on [t,∞)× R(m+), the functions bijk are locally Lebesgue integrable on [t,∞), and the functions θk, aij are locally absolutely continuous on [t,∞). Since we study two-dimensional systems, we use a transformation into complex variables to simplify the system () into one equation with complex coefficients. Thecomplexvariablesaredefinedasz=x+ix,w=y +iy,...,wm=ym+iym.Using this transformation we get z(t)=a(t)z(t)+b(t)¯ z(t)+ m  k= Ak(t)zθk(t)+Bk(t)¯ zθk(t) +gt,z(t),zθ(t),...,zθm(t),() where we assume (J=[t,∞)): •Ak,Bk∈Lloc(J,C)(i.e. locally Lebesgue integrable complex-valued functions on J)for k=,...,m, •θk∈ACloc(J,R)(i.e. locally absolutely continuous real-valued functions on J)for k=,...,m, •a,b∈ACloc(J,C)(i.e. locally absolutely continuous complex-valued functions on J), •g∈K(J×Cm+,C)(i.e. a complex-valued function which satisfies the Carathéodory conditions on J×Cm+). Obviously, the function gis in general dependent on ¯ zaswellas onevery ¯ z(θk).However, thefactthatthefunctiongsatisfiestheCarathéodoryconditionsenablesustosignificantly simplify the notation by using only z, since the validity of the Carathéodory conditions is not violated by composing with continuous functions ¯ z,¯ wk,andθk. The relations between the functions are the following: a(t)= a(t)+a(t)+i a(t)–a(t), b(t)= a(t)–a(t)+i a(t)+a(t), Ak(t)= bk(t)+bk(t)+i bk(t)–bk(t), Bk(t)= bk(t)–bk(t)+i bk(t)+bk(t), g(t,z,w,...,wm) =ht, (z+¯ z),  i(z–¯ z),  (w+¯ w),...,  i(wm–¯ wm) +iht, (z+¯ z),  i(z–¯ z),  (w+¯ w),  i(w–¯ w),...,  i(wm–¯ wm). Conversely, putting a(t)=Rea(t)+b(t),a(t)=Imb(t)–a(t), a(t)=Ima(t)+b(t),a(t)=Rea(t)–b(t), bk(t)=ReAk(t)+Bk(t),bk(t)=ImBk(t)–Ak(t), Rebenda and Šmarda Advances in Difference Equations (2015) 2015:349 Page 3 of 10 bk(t)=ImAk(t)+Bk(t),bk(t)=ReAk(t)–Bk(t), h(t,x,y,...,ym)=Reg(t,x+ix,y +iy,...,ym+iym), h(t,x,y,...,ym)=Img(t,x+ix,y +iy,...,ym+iym), equation() can be written in the real form () as well. The equivalence of the dynamical invariants and asymptotic properties of the solutions of the real system () and the complex equation ()isshownin[]forthesimplecase covering ordinary differential equations. In this paper we consider ()inthecasewhen liminf t→∞ a(t)–b(t)> () and study the behavior of the solutions of () under this assumption, which generally meansthatdetA(t)>fortsufficientlylarge.Thissituationcorrespondstothecasewhen the equilibrium point  of the autonomous homogeneous system x=Ax,() where Ais supposed to be a regular constant matrix, is a center, a focus or a node. Such a situation has some geometrical aspects, which are used in an analysis of the transformed equation(). See [] for more details. Further, we suppose that () satisfies the uniqueness property of solutions. 2Preliminaries Throughout this paper we will assume that liminf t→∞ a(t)–b(t)>, t≥θk(t)≥t–rfor t≥t+r,() where r>  is a constant, which means that the delays θkare bounded. This is the same caseasconsideredin[].Similarresultsforacasedifferentfrom()wereobtainedin[]. Then there are numbers T≥t+rand μ>suchthat a(t)>b(t)+μfor t≥T.() Denote c(t)=¯ a(t)b(t) |a(t)|,γ(t)=a(t)+a(t)–b(t)() and ϑ(t)=Re(γ(t)γ(t)–¯ c(t)c(t))–|γ(t)c(t)–γ(t)c(t)| γ(t)–|c(t)|, α(t)=–b(t) a(t)sgnRea(t). () Moreover, we assume the following conditions to be valid: Rebenda and Šmarda Advances in Difference Equations (2015) 2015:349 Page 4 of 10 (i) The numbers T≥t+rand μ>  satisfy condition (). (ii)Therearefunctions,κ,κk:[T,∞)→R,whereiscontinuouson[T,∞),suchthat γ(t)g(t,z,w,...,wm)+c(t)¯ g(t,z,w,...,wm) ≤κ(t)γ(t)z+c(t)¯ z+m  k= κk(t)γθk(t)wk+cθk(t)¯ wk+(t) for t≥T,z,wk∈C(k=,...,m). (iin)TherearenumbersRn≥ and functions κn,κnk :[T,∞)→Rsatisfying the inequality γ(t)g(t,z,w,...,wm)+c(t)¯ g(t,z,w,...,wm) ≤κn(t)γ(t)z+c(t)¯ z+m  k= κnk(t)γθk(t)wk+cθk(t)¯ wk for t≥τn≥T,|z|+m k= |wk|>Rn. (iii) The function β∈ACloc([T,∞),R –)issuchthat θ k(t)β(t)≤–λk(t)a.e.on[T,∞), () where λkis given for t≥Tby λk(t)=κk(t)+Ak(t)+Bk(t)γ(t)+|c(t)| γ(θk(t))–|c(θk(t))|.() (iiin) The function βn∈ACloc([T,∞),R –)issuchthat θ k(t)βn(t)≤–λnk(t)a.e.on[τn,∞), () where λnk is given for t≥Tby λnk(t)=κnk(t)+Ak(t)+Bk(t)γ(t)+|c(t)| γ(θk(t))–|c(θk(t))|.() (ivn) The function Λnis real locally Lebesgue integrable and the inequalities β n(t)≥ Λn(t)βn(t), Θn(t)≥Λn(t) are satisfied for almost all t∈[τn,∞), where Θnis given by Θn(t)=α(t)Rea(t)+ϑ(t)–κn(t)+mβn(t). Furthermore, denote Θ(t)=α(t)Rea(t)+ϑ(t)–κ(t). () 3 Main results First of all, we recall the two results from []. Lemma  Let the assumptions (i), (ii), (iii), (iv)be fulfilled for some τ≥T.Suppose there exist t≥τand ν∈(–∞,∞)such that inf t≥tt t Λ(s)ds–lnγ(t)+c(t)≥ν.() Rebenda and Šmarda Advances in Difference Equations (2015) 2015:349 Page 5 of 10 If z(t)is any solution of () satisfying min θ(t)≤s≤tz(s)>R,Δ(t)>Re–ν, () where θ(t)= min k=,...,mθk(t), Δ(t)=γ(t)–c(t)z(t)+β(t)max θ(t)≤s≤tz(s)m  k= t θk(t)γ(s)+c(s)ds, then z(t)≥Δ(t) γ(t)+|c(t)|expt t Λ(s)ds() for all t ≥tfor which z(t)is defined. Lemma Let the conditions (i), (ii), (iii) be fulfilled and Λ,θ k(k=,...,m)be continuous functions suchthatinequalityΛ(t)≤Θ(t)holdsa.e.on [T,∞),where Θisdefinedby(). Suppose that ξ:[T–r,∞)→Ris a continuous function such that Λ(t)+β(t)m  k= θ k(t)exp–t θk(t)ξ(s)ds–ξ(t)>(t)C– exp–t Tξ(s)ds() for t ∈[T,∞]and some constant C >.Then there exists a t>T and a solution z(t)of () satisfying z(t)≤C γ(t)–|c(t)|expt Tξ(s)ds() for t ≥t. If we combine the previous two results, we are able to prove the following theorem, which is the fundamental result of this paper. Theorem  Assume that the hypotheses (i), (ii), (iin), (iii), (iiin), (ivn)are valid for T ≤τn, where <Rn,n∈N,infn∈NRn=.Suppose that θ k,Λare continuous functions such that inequalityΘ(t)≥Λ(t)issatisfiedalmosteverywhereon[T,∞),whereΘ(t)=α(t)Rea(t)+ ϑ(t)–κ(t).Let ξ:[T–r,∞)→Rbe a continuous function satisfying the inequality (t)C– exp–t Tξ(s)ds<Λ(t)+β(t)m  k= θ k(t)exp–t θk(t)ξ(s)ds–ξ(t)() for some constant C >and t ∈[T,∞). Assume inf τn≤s≤t<∞t sΛn(σ)dσ–lnγ(t)+c(t)≥ν,() Rebenda and Šmarda Advances in Difference Equations (2015) 2015:349 Page 6 of 10 limsup t→∞ t TΛn(s)–ξ(s)ds+ln γ(t)–|c(t)| γ(t)+|c(t)|=∞,() lim t→∞βn(t)max θ(t)≤s≤t exp[s Tξ(σ)dσ] γ(s)–|c(s)| m  k= t θk(t)γ(s)+c(s)ds=, () for n ∈N,where ν∈(–∞,∞)and θ(t)=mink=,...,mθk(t). Then there is a solution z(t)of () with the property lim t→∞ min θ(t)≤s≤tz(s)=. () Proof Using Lemma we obtain the existence of T≤tand a solution z(t)of()satisfyingfort≥tthe inequality z(t)≤C γ(t)–|c(t)|expt Tξ(s)ds.() From ()weget inf τ≤t<∞t τ ΛN(s)ds–lnγ(t)+c(t)≥ν>–∞. Lemma yields Ψ(τ) γ(t)+|c(t)|expt τ ΛN(s)ds≤z(t)() for τ≤t,whereΨis given 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 that () does not hold. This implies the existence of ε>  satisfying limsupt→∞minθ(t)≤s≤t|z(s)|>ε.WetakeN∈Nsuch that max{RN, μRNe–ν}<ε.Then maxRN, μRNe–ν<min θ(τ)≤s≤τz(s)() holds for some τ>max{T,τN,t}.Taking() into account we may assume that 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, with respect to (), (), (), (), (), and the nonpositiveness of βN,weobtain 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 Šmarda Advances in Difference Equations (2015) 2015:349 Page 7 of 10 The inequalities ()and() give the estimation Ψ(τ) γ(t)+|c(t)|expt τ ΛN(s)ds≤C γ(t)–|c(t)|expt Tξ(s)ds, which means that t TΛN(s)–ξ(s)ds+ln γ(t)–|c(t)| γ(t)+|c(t)|≤τ TΛN(s)ds–lnC–Ψ(τ) for τ≤t, which is in contradiction to (). The proof is complete.  Remark  Theorem  covers more general situations than Theorem  in [], where the different fundamental assumption liminft→∞(|Ima(t)|–|b(t)|)>issupposedtohold. Indeed, if we takefor example a(t)≡+iand b(t)≡i,thencondition()inthispaperis satisfied but the condition liminft→∞(|Ima(t)|–|b(t)|)> is not valid. Thefollowingnontrivialexamplewasconstructedtoillustrateanapplicationofthetheoretical result presented in Theorem . Example  Consider the two-dimensional system of the nonlinear delayed differential equations x(t)=x(t)+  tx(t)–x(t)+ m  k=  me–txt–e–kt+e–t,() x(t)=x(t)+x(t)+  tx(t)– m  k=  me–txt–e–kt.() This system can be written in matrix form (), where A(t)=+  t– +  t,Bk(t)=  me–t e –t,and ht,x(t),xθ(t),...,xθm(t)=e–t . Following our approach, we use a transformation into the complex plane and obtain the delayed differential equation () with complex-valued coefficients, z(t)=(+i)z(t)+i¯ z(t)+  tz(t)+ m  k=  me–t¯ zt–e–kt+e–t,() where a(t)≡+i,b(t)≡i,Ak(t)≡, Bk(t)≡, θk(t)=t–e –kt for k= ,...,m, g(t,z,w,...,wm)=  tz+m k=  me–twk+e–t. Obviously t–≤θk(t)≤tand θ k(t)=+ke–kt ≥>fort≥. Suppose t=andT≥. Then γ(t)=|a(t)|+|a(t)|–|b(t)|≡+ √, c(t)= ¯ a(t)b(t)/|a(t)|≡+i . Rebenda and Šmarda Advances in Difference Equations (2015) 2015:349 Page 8 of 10 Further, γ(t)g(t,z,w,...,wm)+c(t)¯ g(t,z,w,...,wm) ≤γ+|c| γ–|c| tγ(t)z+c(t)¯ z+γ+|c| γ–|c| m  k=  me–tγθk(t)wk+cθk(t)¯ wk+e–t =√ √  tγ(t)z+c(t)¯ z+√ √ e–t m m  k= γθk(t)wk+cθk(t)¯ wk+e–t. Thus conditions (i) and (ii) are fulfilled with κ(t)≡√ t√,κk(t)=e–t√ m√,and(t)=e–t.To meet condition (iin), we estimate for Rn= n,τn=n,and|z|>Rn γ(t)g(t,z,w,...,wm)+c(t)¯ g(t,z,w,...,wm) ≤γ+|c| γ–|c| t+(t) Rnγ(t)z+c(t)¯ z +γ+|c| γ–|c| m  k=  me–tγθk(t)wk+cθk(t)¯ wk =√ √ t+ne–tγ(t)z+c(t)¯ z+√ √ e–t m m  k= γθk(t)wk+cθk(t)¯ wk, where κn(t)=√ √[ t+ne–t]andκnk(t)=√ √e–t m. Condition (iii) holds with –λk(t)θ k(t)– =–e–t√ m√  +ke–kt ≥–e–t√ m√=β(t). Condition (iiin)issatisfiedfor βn(t)=–e–(t–) m≤–e–t√ m√=–λnk(t)θ k(t)–. We get condition (ivn) by setting n(t)=Θn(t)= –√ √ t+ne–t–e–(t–) >. Further, we put (t)=Θ(t)= –√ t√. Then condition ()holdsfor ξ(t)= –√ t√–√ √ e–t m–e–t and ξ(t)>fort≥T=. Nowitisnotdifficulttoverifyconditions()and(),sincen(t)–ξ(t)>andn(t)> forn∈N.Investigatingthefactorsoftheproductin parenthesesin (),we cometothe Rebenda and Šmarda Advances in Difference Equations (2015) 2015:349 Page 9 of 10 conclusion that βn=Oe–t,exps Tξ(σ)dσ=Oe.t and m  k= t θk(t)γ(s)+c(s)ds=Om  k= e–kt=Oe–t. Consequently, the product of these factors is asymptotically equal to O(e–δt), where δ> andthuscondition ()is satisfied. All assumptionsof Theorem arefulfilled and we can conclude that there exist t>  and a solution z(t)of()satisfying()fort≥t. Remark  This result is slightly surprising. However, it is in good agreement with the well-known fact that introducing delay into an unstable system without delay can cause a changeofbehaviorofthesystem.Suchsituationsaredescribedandcorrespondingresults are formulated e.g.in []. 4Conclusion We proved an interesting result about the stability of two-dimensional systems with boundeddelays.Sufficientconditionsforthestabilityofanoriginallyunstablesystemwere presented. The result is in perfect agreement with the results stated and proved in the established literature. An example showed how this result can be used in practice. Competing interests The authors declare that they have no competing interests. Authors’ contributions The authors have made contributions of the same significance. All authors read and approved the final manuscript. Acknowledgements The work of the authors was realized in CEITEC - Central European Institute of Technology with research infrastructure supported by the project CZ.1.05/1.1.00/02.0068 financed from European Regional Development Fund and the second author was supported by the project FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication, Brno University of Technology. This support is gratefully acknowledged. Received: 15 September 2015 Accepted: 5 November 2015 References 1. Baculíková, B, Džurina, J, Rogovchenko, YV: Oscillation of third order trinomial delay differential equations. Appl. Math. Comput. 218(13), 7023-7033 (2012) 2. Berezansky, L, Braverman, E: On nonoscillation and stability for systems of differential equations with a distributed delay. Automatica 48(4), 612-618 (2012) 3. Berezansky, L, Diblík, J, Svoboda, Z, Šmarda, Z: Simple uniform exponential stability conditions for a system of linear delayed differential equations. Appl. Math. 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