The main aim of this paper is the investigation of the stability problem for
ordinary delay differential equations. More precisely, we would like to study
the following problem. Assume that for a continuous function a given delay
differential equation is fulfilled only approximately. Is it true that in this
case this function is close to an exact solution of this delay differential
equation?Comment: 11 page

Gselmann, Eszter

Kelemen, Anna

Miskolc Mathematical Notes

arXiv

Repository of the Academy's Library

Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 17 (2016), No. 1, pp. 281–291 DOI: 10.18514/MMN.2016.1148STABILITY IN THE CLASS OF FIRST ORDER DELAYDIFFERENTIAL EQUATIONSESZTER GSELMANN AND ANNA KELEMENReceived 27 February, 2014Abstract. The main aim of this paper is the investigation of the stability problem for ordinarydelay differential equations. More precisely, we would like to study the following problem.Assume that for a continuous function a given delay differential equation is fulfilled only ap-proximately. Is it true that in this case this function is close to an exact solution of this delaydifferential equation?2010 Mathematics Subject Classification: 39B82; 34K20Keywords: delay differential equation, functional-differential equation, stability, continuous de-pendence1. INTRODUCTION AND PRELIMINARIESWhenever we would like to analyze mathematically a problem, arising in the ’real’world, first we have to select a mathematical model for formulation of our problem.This means that we have to choose or invent some mathematical problem to presentour ’real world problem’. The mathematical models chosen are often differentialequations. Mostly this differential equations are rather special, more precisely, inthem the unknown functions and its derivatives are all evaluated at the same instantt .A more general type of differential equations, called functional differential equa-tions, are those in which the unknown functions occur with different arguments. Suchtype of equations are rather natural for example in control systems, since any systeminvolving feedback control will certainly involve time delays. These arise because afinite time is required to sense information and then react to it.We remark an other problem which comes from electrodynamics, the two-bodyproblem. Let us consider two electrons and try to describe the interaction betweenthem. We have to take it into account that the interactions between the two particlesThis research has been supported by the Hungarian Scientific Research Fund (OTKA) Grant NK111651 and by the TA´MOP 4.2.4.A/2-11-1-2012-0001 (Nemzeti Kiva´lo´sa´g Program – Hazai hallgato´i,illetve kutato´i szeme´lyi ta´mogata´st biztosı´to´ rendszer kidolgoza´sa e´s mu˝ko¨dtete´se konvergencia pro-gram) project implemented through the EU and Hungary co-financed by the European Social Fund.c 2016 Miskolc University Press282 ESZTER GSELMANN AND ANNA KELEMENtravel not instantaneously, but with finite speed c (the speed of light). Let x1.t/ andx2.t/ represent the positions of the two particles at time t . The the fields reachingx1.t/ at time t from the second particle were generated at t r , where the delay mustsatisfycr D jx1.t/ x2.t   r/j :Clearly, the (unknown) delay r is not a constant but it depends on t . In terms of thesedelays one can obtain the following system for the motionx001.t/ D f1 x1.t/ x2.t   r21.t//;x01.t/;x02.t   r21.t//x002.t/ D f2 x2.t/ x1.t   r12.t//;x02.t/;x01.t   r12.t//;with certain maps f1;f2WR3! R.In a real world applications, initial conditions are almost never known exactly.Rather, experimental and physical errors will only allow us to say that their valuesare approximately equal to those in our mathematical model. Thus, to retain physicalrelevance, we need to be sure that small errors in our initial measurements do not in-duce a large change in the solution. A similar argument can be made for any physicalparameters, e.g., masses, charges, spring stiffnesses, frictional coefficients, etc., thatappear in the differential equation itself. A slight change in the parameters should nothave a dramatic effect on the solution. Such type of results are referred to continuousdependence on the initial values.It can also however happen that not only the initial value but also the equation issatisfied ’approximately’. These type of investigations began in the 1990s. For ex-ample in [1] C. Alsina and R. Ger studied the following problem. Let " > 0 be given,I  R be a nonvoid open interval and assume that for the differentiable functionxWI ! R inequality ˇˇx0.t/ x.t/ˇˇ< " .t 2 I / :Is it true that there exist constants C; such thatjx.t/ C exp.t/j< "is fulfilled for any x 2 I . In the above mentioned paper (among others) this problemis answered in the affirmative. Roughly speaking this result expresses the following.Assume that for the differentiable function xWI ! R differential equationx0.t/D x.t/ .t 2 I /is fulfilled only approximately. Is it true that in such a situation there exists a solutionto this differential equationexWI !R such that the functions x andex are close to eachother?Later, in [9] the first order equationx0.t/D x.t/was considered for Banach space valued functions, and the same question was answeredaffirmatively.STABILITY IN THE CLASS OF ... 283Recently the study of such problems became an intensively developing area, seefor instance Jung [6], [7] and Andra´s–Kolumba´n [2]. For example, in the latter workusing the so-called Perov fixed point theorem, the authors studied the following sys-temx0.t/D f1.t;x.t/;y.t//y0.t/D f2.t;x.t/;y.t// almost everywherex.0/D ˛Œxy.0/D ˇŒyon bounded as well as on unbounded subset of 0;C1Œ. Nevertheless, the use ofPerov fixed point theorem, allowed only that the functions f1;f2 are Lipschitz func-tions with small Lipschitz constant.Furthermore, in [8] the stability problem for a rather special delay differentialequations was investigated, namelyx.t/D x.t   /As we will see later, the stability problem of this equation will be a particular case ofours with the choice mD 1;g1.t/D t    andf .t;x.g1.t///D x.g1.t//D x.t   /:For further results concerning stability of functional as well as ordinary differ- entialequations we refer also to [4, 5].Therefore the main aim of this paper is to investigate and to extend the abovepresented problem for a rather large class of first order ordinary delay differentialequations.First of all, we will list the notation and the terminology that will be used sub-sequently. While doing so we will relay on the two basic monographs Driver [3](concerning delay differential equations) and Walter [10] (concerning integral in-equalities).Let J D Œt0;ˇŒ R be an interval and D  Rn be an open set. Let further f WJ Dm! Rn be a mapping and g1; : : : ;gmWJ ! R be functions so that  gj .t/ t .t 2 Œt0;ˇŒ/for all j D 1; : : : ;mwith a certain real number  . Finally, let  W Œ; t0!D be a giveninitial function. Consider the system of delay differential equationsx0.t/D f .t;x.g1.t//; : : : ;x.gm.t/// :Given t 2 Œt0;ˇŒ and any function W Œ; t !D defineF.t;t /D f .t;x.g1.t//; : : : ;x.gm.t/// :284 ESZTER GSELMANN AND ANNA KELEMENWith this notation, our main aim is to investigate the Hyers–Ulam stability of thefollowing problem.x0.t/D F.t;xt / t  t0x.t/D .t/ t 2 Œ; t0: (P)A solution of this problem (P) is a continuous function xW Œ;ˇ1Œ! D for someˇ1 2 Œ; t0 such that(i) for all t 2 Œ;ˇ1 we have .t;xt / 2 Œt0;ˇ1ŒDm(ii) for any t 2 Œt0;ˇ1Œx0.t/D F.t;xt /is fulfilled.(iii) x.t/D .t/ for all t 2 Œ; t0.Henceforth we will assume that f and g1; : : : ;gm are continuous. Thus if x iscontinuous, then the mappingŒ;ˇ1Œ3 t 7 ! F.t;xt /is continuous, as well. In such a situation howeverx.t/D8<:.t/; if t 2 Œ; t0.t0/CZ tt0F.s;xs/ds; if t 2 Œt0;ˇ1Œ: (I )The following lemma will be utilized during our main results, see also Walter[10, Theorem III. (Gronwall’s inequality), page 14].Lemma 1. Let J  R be an interval, g;vWJ ! R be continuous functions andhWJ ! R be a nonnegative, Lebesgue integrable function such thatv.t/ g.t/CZ tt0h.s/v.s/ds .t 2 J / :Thenv.t/ g.t/CZ tt0g.s/h.s/eH.t/ H.s/ds .t 2 J / :Furthermore, if g is absolutely continuous, thenv.t/ eH.t/g.t0/CZ tt0g0.s/eH.s/dsholds, whereH.t/DZ tt0h.s/ds .t 2 J / :STABILITY IN THE CLASS OF ... 285Finally we remark the without the loss of generality it is enough to restrict ourselvesto first order delay differential equations. Indeed, let J D Œt0;ˇŒ be an interval,D  Rn be an open set. Let F WJ Dm ! R be a continuous function. Let fur-ther g1; : : : ;gmWJ ! R be continuous functions so that  gj .t/ t .t 2 Œt0;ˇŒ/for all j D 1; : : : ;m with a certain real number  . Finally, let  W Œ; t0! R be .n 1/ times continuously differentiable initial function. Assume that the .n  1/ timescontinuously differentiable function xW Œ;ˇŒ! R is a solution of the problemx.n/.t/D Ft;x.g1.t//; : : : ;x.n 1/.gm.t//.t 2 Œt0;ˇŒ/x.t/D .t/ .t 2 Œ; t0/Define the function yW Œ;ˇŒ! Rn byy.t/D0BBB@x.t/x0.t/:::x.n 1/.t/1CCCA .t 2 Œ;ˇŒ/ :Then the continuous function y is a solution of the following first order delay systemy01.t/D y2.t/y02.t/D y3.t/:::y0n 1.t/D yn.t/y0n.t/D F.t;y.g1.t//; : : : ;y.gm.t//.t 2 Œt0;ˇŒ/ ;furthermorey.t/D0BBB@.t/ 0.t/::: .n 1/.t/1CCCA .t 2 Œ; t0/ ;where yi W Œ;ˇŒ! R denotes the i th coordinate function of the function y for alli D 1; : : : ;n. Furthermore, the converse implication is also valid. Namely, if thefunction yW Œ;ˇŒ! Rn is a solution of the latter system, then the first coordinatefunction y1W Œ;ˇŒ! R is a solution of the above nth order delay equation.2. MAIN RESULTSIn this section, our main results will be divided into four parts. First we will provea basic lemma that will be utilized later. After that (with the aid of Lemma 2) we willpresent a uniqueness theorem. Afterwards a continuous dependence on the initial286 ESZTER GSELMANN AND ANNA KELEMENfunction type result will follow. Finally, we will end our paper with a stability typeresult.2.1. A useful lemmaLemma 2. Let J D Œt0;ˇŒ R be an interval and D  Rn be an open set. Letfurther f;ef WJ Dm! Rn be mappings and g1; : : : ;gmWJ ! R be functions so that  gj .t/ t .t 2 Œt0;ˇŒ/for all j D 1; : : : ;m with a certain real number  . Finally, let ;e W Œ; t0! D begiven initial functions. Assume that the functions x;exW Œ;ˇŒ! D are solutions toproblemsx0.t/D F.t;xt / t  t0x.t/D .t/ t 2 Œ; t0:and ex0.t/D eF .t;ext / t  t0ex.t/De.t/ t 2 Œ; t0:respectively. Suppose further that we also have the following.(i) there exists a continuous nonnegative function hWJ ! R such that for all t 2Œ;ˇŒ and for any ´ 2DmF.t;´/ eF .t;´/ h.t/k´k :(ii) there exists a continuous nonnegative function kWJ ! R such thatF.t;´/ F.t;´0/ k.t/´ ´0for all t 2 Œ;ˇŒ and for any ´;´0 2Dm.Thenkx.t/ ex.t/k    eexpZ tt0h.s/Ck.s/ds.t 2 J / :Proof. Applying integral inequality I , we immediately get thatx.t/D .t0/CZ tt0F.s;xs/dsand ex.t/De.t0/CZ tt0F.s;exs/dsholds for all t 2 J . Thereforekx.t/ ex.t/kSTABILITY IN THE CLASS OF ... 287D.t0/CZ tt0F.s;xs/ds e.t0/CZ tt0F.s;exs/ds.t0/ e.t0/CZ tt0F.s;xs/ eF .s;ess/ds.t0/ e.t0/CZ tt0F.s;xs/ F.s;exs/dsCZ tt0F.s;exs/ eF .s;exs/ds.t0/ e.t0/CZ tt0k.s/ supskx./ ex./kdsCZ tt0h.s/ supskx./ ex./kds.t0/ e.t0/CZ tt0.k.s/Ch.s// supskx./ ex./kdsDefine the function vW Œ;ˇŒ! R byv.t/D suptkx./ ex./k .t 2 Œ;ˇŒ/ :Thenkx.t/ ex.t/k  v.t0/CZ tt0.k.s/Ch.s//v.s/ds .t 2 J /;or, since .t/ e.t/ v.t0/ .t 2 Œ; t0/holds, we havev.t/ v.t0/CZ tt0.k.s/Ch.s//v.s/dsfor all t 2 J . Now, Lemma 1 implies thatkx.t/ ex.t/k    eexpZ tt0k.s/Ch.s/dsfor all t 2 J , which ends the proof. 2.2. CorollariesA uniqueness theoremBased on the previous results, now we are able to prove the following uniquenesstheorem.Theorem 1. Let J D Œt0;ˇŒ be an interval, D  Rn be an open set. Let F WJ Dm! Rn be a function so thatF.t;´/ F.t;´0/ k.t/´ ´0  t 2 J;´ 2Dm288 ESZTER GSELMANN AND ANNA KELEMENis fulfilled with a certain continuous function kWJ ! R. Let further g1; : : : ;gmWJ !R be continuous functions so that  gj .t/ t .t 2 Œt0;ˇŒ/for all j D 1; : : : ;m with a certain real number  . Finally, let  W Œ; t0! R be agiven continuous initial function. Then problem (P) has at most one solution on anyinterval Œ;ˇ1Œ, where t0 < ˇ1  ˇ.Proof. Suppose on the contrary that there are two different solutions, say x andex.Since x.t/Dex.t/ on Œ; t0, there exists t 2t0;ˇ1Œ such that x.t/¤ex.t/. Lett1 D infft 2t0;ˇ1Œ jx.t/¤ex.t/g :Then t1 2t0;ˇ1Œ and due to the continuity of x andex we havex.t/Dex.t/for all t 2 Œ; t1Œ. Since t1 2t0;ˇ1Œ and for all j D 1; : : : ;m x.gj .t1// belongs to theopen set D, there are positive real numbers 1;2 such thatŒt1 ; t1Ct0;ˇ1Œ and Aj D˚ 2 Rn j x.gj .t//   2	D:LetK D supt2Œt1 1;t1C1k.t/:Since k is continuous on the compact interval Œt1 1; t1C1 we have K 2 R.Again, due to the continuity of x and g1; : : : ;gm, there exists ˇ2 2t1; t1C1Œ suchthat x.gj .t//;ex.gj .t// 2Aj for all j D 1; : : : ;m and t 2t1;ˇ2Œ. Let now t 2 Œt1;ˇ2Œ,thenkx.t/ ex.t/k D Z tt1F.s;xs/ F.s;exs/dsK Z tt1supskx./ ex./kds:Definev.t/D suptkx./ ex./k .t 2 Œ;ˇ1Œ/ :Thenkx.t/ ex.t/k K Z tt1v.s/ds .t 2 Œt1;ˇ2Œ/ ;or, since x ex on Œ; t1 we havev.t/KZ tt1v.s/ds .t 2 Œt1;ˇ2Œ/ :In view of Lemma 1., this implies that v.t/D 0 for all t 2 Œt1;ˇ2Œ, that is,x.t/Dex.t/ .t 2 Œt1;ˇ1Œ/ ;which contradicts to the definition of t1. STABILITY IN THE CLASS OF ... 289We remark that the case whenk.t/D k .t 2 J /can be found in the monograph of Driver, see [3, Theorem A., p. 259].Continuous dependence on the initial functionUsing Lemma 2. with the choice h 0, the following result can be derived imme-diately.Theorem 2. Let J D Œt0;ˇŒ be an interval, D  Rn be an open set. Let F WJ Dm! Rn be a function so thatF.t;´/ F.t;´0/ k.t/´ ´0  t 2 J;´ 2Dmis fulfilled with a certain continuous function kWJ ! R. Let further g1; : : : ;gmWJ !R be continuous functions so that  gj .t/ t .t 2 Œt0;ˇŒ/for all j D 1; : : : ;m with a certain real number  . Finally, let ;e W Œ; t0! R be agiven continuous initial functions. If the functions x;exW Œ;ˇŒ! R are solutions to theproblems(x0.t/D F.t;xt / for t  t0x.t/D .t/ t 2 Œ; t0 and(ex0.t/D F.t;ex t / for t  t0ex.t/De.t/ t 2 Œ; t0 :Then for all t 2 Œt0;ˇŒkx.t/ ex.t/k    eexpZ tt0k.s/ds:A stability resultFinally, we end this paper with the following stability type result.Theorem 3. Let " > 0 be arbitrarily fixed, J D Œt0;ˇŒ be an interval and D  Rnbe an open set. Let F WJ Dm! Rn be a function so thatF.t;´/ F.t;´0/ k.t/´ ´0  t 2 J;´ 2Dmis fulfilled with a certain continuous function kWJ ! R. Let further g1; : : : ;gmWJ !R be continuous functions so that  gj .t/ t .t 2 Œt0;ˇŒ/for all j D 1; : : : ;m with a certain real number  . Finally, let  W Œ; t0!R be a givencontinuous initial function. Assume that the continuous initial function xW Œt0ˇŒ! Rfulfills x0.t/ F.t;xt / " .t 2 Œt0;ˇŒ/x.t/D .t/ .t 2 Œ; t0/ :290 ESZTER GSELMANN AND ANNA KELEMENThen there exists a uniquely determined solutionexW Œt0;ˇŒ! R of the problemx0.t/D F.t;xt / .t 2 Œt0;ˇŒ/x.t/D .t/ .t 2 Œ; t0/ :such thatkx.t/ ex.t/k  ".t   t0/expZ tt0k.s/dsfor all t 2 Œt0;ˇŒ.Proof. Assume that for all t 2 Œt0;ˇŒ we havex0.t/ F.t;xt / ":Equivalently, this means that x is a solution to the problemx0.t/D eF .t;xt / .t 2 Œt0;ˇŒ/x.t/D .t/ .t 2 Œ; t0/ ;where eF .t;´/D F.t;´/Cb.t/  t 2 Œt0;ˇŒ;´ 2Dm ;with a certain bounded function bWJ ! R.LetexWJ ! R be the uniquely determined solution of the problemex0.t/D F.t;ex t / .t 2 Œt0;ˇŒ/ex.t/D .t/ .t 2 Œ; t0/ :In this case howeverkx.t/ ex.t/k D Z tt0eF .s;xs/ F.s;exs/dsZ tt0k.s/ supskx.s/ ex.s/kdsC ".t   t0/for any t 2 Œt0;ˇŒ. Similarly as above, define the function vW Œ;ˇŒ! R throughv.t/D suptkx.t/ ex.t/k .t 2 Œ;ˇŒ/ :Then the previous inequality simply yield thatv.t/ ".t   t0/CZ tt0k.s/v.s/ds .t 2 Œ;ˇŒ/ :Thus, in virtue of Lemma 1, for all t 2 Œt0;ˇŒkx.t/ ex.t/k  ".t   t0/expZ tt0k.s/ds:STABILITY IN THE CLASS OF ... 291AcknowledgementThis paper is dedicated to Professor A´rpa´d Sza´z (University of Debrecen) on theoccasion of this 70th birthday.REFERENCES[1] C. Alsina and R. Ger, “On some inequalities and stability results related to the exponential func-tion,” J. Inequal. Appl., vol. 2, no. 4, pp. 373–380, 1998, doi: 10.1155/s102558349800023x.[2] S. Andra´s and J. Kolumba´n, “On the Ulam-Hyers stability of first order differential sys-tems with nonlocal initial conditions,” Nonlinear Anal., vol. 82, no. 1–11, 2013, doi:10.1016/j.na.2012.12.008.[3] R. D. Driver, Ordinary and delay differential equations, ser. Applied Mathematical Sciences.New York: Springer-Verlag, 1977, vol. 20, doi: 10.1007/978-1-4684-9467-9.[4] R. Ger and W. Fechner, “Some stability results for equations and inequalities connected withthe exponential functions,” Functional Equations and Difference Inequalities and Ulam StabilityNotions (F.U.N.), Nova Science Publishers, pp. 37–46, 2010.[5] S.-M. Jung, “Hyers-Ulam stability of linear differential equations of first order,” Appl. Math. Lett.,vol. 17, no. 10, pp. 1135–1140, 2004, doi: 10.1016/j.aml.2003.11.004.[6] S.-M. Jung, “A fixed point approach to the stability of differential equations y0 D F.x;y/,” Bull.Malays. Math. Sci. Soc. (2), vol. 33, no. 1, pp. 47–56, 2010.[7] S.-M. Jung, “Approximate solutions of a linear differential equation of third order,” Bull. Malays.Math. Sci. Soc. (2), vol. 35, no. 4, pp. 1063–1073, 2012.[8] S.-M. Jung and J. Brzde¸k, “Hyers-Ulam stability of the delay equation y0.t/D y.t   /,” Abstr.Appl. Anal., pp. Article ID: ID 372 176, 10 pages, 2010.[9] S.-E. Takahasi, T. Miura, and S. Miyajima, “On the Hyers-Ulam stability of the Banach space-valued differential equation y0 D y,” Bull. Korean Math. Soc., vol. 39, no. 2, pp. 309–315, 2002,doi: 10.4134/BKMS.2002.39.2.309.[10] W. Walter, Differential- und Integral-Ungleichungen und ihre Anwendung bei Abscha¨tzungs- undEindeutigkeits-problemen, ser. Springer Tracts in Natural Philosophy. Berlin: Springer-Verlag,1964, vol. 2, doi: 10.1007/978-3-662-35247-2.Authors’ addressesEszter GselmannDepartment on Analysis, Institute of Mathematics, University of Debrecen, H–4010 Debrecen P.O.Box:12E-mail address: gselmann@science.unideb.huAnna KelemenFaculty of Informatics, University of Debrecen, H–4010 Debrecen P.O.Box:12E-mail address: pandora0303@hotmail.com

Stability in the class of first order delay differential equations

DEA University of Debrecen Electronic Archive

STABILITY IN THE CLASS OF FIRST ORDER DELAYDIFFERENTIAL EQUATIONSESZTER GSELMANN AND ANNA KELEMENAbstract. The main aim of this paper is the investigation of the stabilityproblem for ordinary delay differential equations. More precisely, we wouldlike to study the following problem. Assume that for a continuous functiona given delay differential equation is fulfilled only approximately. Is it truethat in this case this function is close to an exact solution of this delaydifferential equation?1. Introduction and preliminariesWhenever we would like to analyze mathematically a problem, arising in the’real’ world, first we have to select a mathematical model for formulation ofour problem. This means that we have to choose or invent some mathematicalproblem to present our ’real world problem’. The mathematical models chosenare often differential equations. Mostly this differential equations are ratherspecial, more precisely, in them the unknown functions and its derivatives areall evaluated at the same instant t.A more general type of differential equations, called functional differentialequations, are those in which the unknown functions occur with different ar-guments. Such type of equations are rather natural for example in controlsystems, since any system involving feedback control will certainly involvetime delays. These arise because a finite time is required to sense informationand then react to it.We remark an other problem which comes from electrodynamics, the two-body problem. Let us consider two electrons and try to describe the interactionbetween them. We have to take it into account that the interactions betweenthe two particles travel not instantaneously, but with finite speed c (the speed2010 Mathematics Subject Classification. Primary 39B82 and Secondary 34K20.Key words and phrases. delay differential equation, functional-differential equation, sta-bility, continuous dependence.This research has been supported by the Hungarian Scientific Research Fund (OTKA)Grant NK 81402 and by the TA´MOP 4.2.4.A/2-11-1-2012-0001 (Nemzeti Kiva´lo´sa´g Pro-gram – Hazai hallgato´i, illetve kutato´i szeme´lyi ta´mogata´st biztos´ıto´ rendszer kidolgoza´sae´s mu˝ko¨dtete´se konvergencia program) project implemented through the EU and Hungaryco-financed by the European Social Fund.12 ESZTER GSELMANN AND ANNA KELEMENof light). Let x1(t) and x2(t) represent the positions of the two particles attime t. The the fields reaching x1(t) at time t from the second particle weregenerated at t− r, where the delay must satisfycr = |x1(t)− x2(t− r)| .Clearly, the (unknown) delay r is not a constant but it depends on t. In termsof these delays one can obtain the following system for the motionx′′1(t) = f1 (x1(t)− x2(t− r21(t)), x′1(t), x′2(t− r21(t)))x′′2(t) = f2 (x2(t)− x1(t− r12(t)), x′2(t), x′1(t− r12(t))) ,with certain maps f1, f2 : R3 → R.In a real world applications, initial conditions are almost never known ex-actly. Rather, experimental and physical errors will only allow us to say thattheir values are approximately equal to those in our mathematical model.Thus, to retain physical relevance, we need to be sure that small errors in ourinitial measurements do not induce a large change in the solution. A similar ar-gument can be made for any physical parameters, e.g., masses, charges, springstiffnesses, frictional coefficients, etc., that appear in the differential equationitself. A slight change in the parameters should not have a dramatic effect onthe solution. Such type of results are referred to continuous dependence onthe initial values.It can also however happen that not only the initial value but also theequation is satisfied ’approximately’. These type of investigations began inthe 1990s. For example in [1] C. Alsina and R. Ger studied the followingproblem. Let ε > 0 be given, I ⊂ R be a nonvoid open interval and assumethat for the differentiable function x : I → R inequality∣∣x′(t)− x(t)∣∣ < ε (t ∈ I) .Is it true that there exist constants C, κ such that|x(t)− C exp(t)| < κεis fulfilled for any x ∈ I. In the above mentioned paper (among others) thisproblem is answered in the affirmative. Roughly speaking this result expressesthe following. Assume that for the differentiable function x : I → R differentialequationx′(t) = x(t) (t ∈ I)is fulfilled only approximately. Is it true that in such a situation there exists asolution to this differential equation x˜ : I → R such that the functions x andx˜ are close to each other?Later, in [9] the first order equationx′(t) = λx(t)STABILITY IN THE CLASS OF ... 3was considered for Banach space valued functions, and the same question wasanswered affirmatively.Recently the study of such problems became an intensively developing area,see for instance Jung [6], [7] and Andra´s–Kolumba´n [2]. For example, in thelatter work using the so-called Perov fixed point theorem, the authors studiedthe following systemx′(t) = f1(t, x(t), y(t))y′(t) = f2(t, x(t), y(t)) almost everywherex(0) = α[x]y(0) = β[y]on bounded as well as on unbounded subset of ]0,+∞[. Nevertheless, theuse of Perov fixed point theorem, allowed only that the functions f1, f2 areLipschitz functions with small Lipschitz constant.Furthermore, in [8] the stability problem for a rather special delay differen-tial equations was investigated, namelyx(t) = λx(t− τ)As we will see later, the stability problem of this equation will be a particularcase of ours with the choice m = 1, g1(t) = t− τ andf (t, x(g1(t))) = λx(g1(t)) = λx(t− τ).For further results concerning stability of functional as well as ordinary differ-ential equations we refer also to [4, 5].Therefore the main aim of this paper is to investigate and to extend theabove presented problem for a rather large class of first order ordinary delaydifferential equations.First of all, we will list the notation and the terminology that will be usedsubsequently. While doing so we will relay on the two basic monographsDriver [3] (concerning delay differential equations) and Walter [10] (concerningintegral inequalities).Let J = [t0, β[⊂ R be an interval and D ⊂ Rn be an open set. Let furtherf : J ×Dm → Rn be a mapping and g1, . . . , gm : J → R be functions so thatγ ≤ gj(t) ≤ t (t ∈ [t0, β[)for all j = 1, . . . ,m with a certain real number γ. Finally, let θ : [γ, t0] → Dbe a given initial function. Consider the system of delay differential equationsx′(t) = f (t, x(g1(t)), . . . , x(gm(t))) .Given t ∈ [t0, β[ and any function χ : [γ, t]→ D defineF (t, χt) = f (t, x(g1(t)), . . . , x(gm(t))) .4 ESZTER GSELMANN AND ANNA KELEMENWith this notation, our main aim is to investigate the Hyers–Ulam stabilityof the following problem.x′(t) = F (t, xt) t ≥ t0x(t) = θ(t) t ∈ [γ, t0]. (P)A solution of this problem (P) is a continuous function x : [γ, β1[→ D forsome β1 ∈ [γ, t0] such that(i) for all t ∈ [γ, β1] we have (t, xt) ∈ [t0, β1[×Dm(ii) for any t ∈ [t0, β1[x′(t) = F (t, xt)is fulfilled.(iii) x(t) = θ(t) for all t ∈ [γ, t0].Henceforth we will assume that f and g1, . . . , gm are continuous. Thus if xis continuous, then the mapping[γ, β1[3 t 7−→ F (t, xt)is continuous, as well. In such a situation howeverx(t) =θ(t), if t ∈ [γ, t0]θ(t0) +∫ tt0F (s, xs)ds, if t ∈ [t0, β1[.(I )The following lemma will be utilized during our main results, see also Walter[10, Theorem III. (Gronwalls inequality), page 14].Lemma 1. Let J ⊂ R be an interval, g, v : J → R be continuous functionsand h : J → R be a nonnegative, Lebesgue integrable function such thatv(t) ≤ g(t) +∫ tt0h(s)v(s)ds (t ∈ J) .Thenv(t) ≤ g(t) +∫ tt0g(s)h(s)eH(t)−H(s)ds (t ∈ J) .Furthermore, if g is absolutely continuous, thenv(t) ≤ eH(t)(g(t0) +∫ tt0g′(s)eH(s)ds)holds, whereH(t) =∫ tt0h(s)ds (t ∈ J) .STABILITY IN THE CLASS OF ... 5Finally we remark the without the loss of generality it is enough to restrictourselves to first order delay differential equations. Indeed, let J = [t0, β[ bean interval, D ⊂ Rn be an open set. Let F : J × Dm → R be a continuousfunction. Let further g1, . . . , gm : J → R be continuous functions so thatγ ≤ gj(t) ≤ t (t ∈ [t0, β[)for all j = 1, . . . ,m with a certain real number γ. Finally, let θ : [γ, t0] → Rbe (n− 1) times continuously differentiable initial function. Assume that the(n− 1) times continuously differentiable function x : [γ, β[→ R is a solution ofthe problemx(n)(t) = F(t, x(g1(t)), . . . , x(n−1)(gm(t)))(t ∈ [t0, β[)x(t) = θ(t) (t ∈ [γ, t0])Define the function y : [γ, β[→ Rn byy(t) =x(t)x′(t)...x(n−1)(t) (t ∈ [γ, β[) .Then the continuous function y is a solution of the following first order delaysystemy′1(t) = y2(t)y′2(t) = y3(t)...y′n−1(t) = yn(t)y′n(t) = F (t, y(g1(t)), . . . , y(gm(t))(t ∈ [t0, β[) ,furthermorey(t) =θ(t)θ′(t)...θ(n−1)(t) (t ∈ [γ, t0]) ,where yi : [γ, β[→ R denotes the ith coordinate function of the function y forall i = 1, . . . , n. Furthermore, the converse implication is also valid. Namely,if the function y : [γ, β[→ Rn is a solution of the latter system, then the firstcoordinate function y1 : [γ, β[→ R is a solution of the above nth order delayequation.6 ESZTER GSELMANN AND ANNA KELEMEN2. Main resultsIn this section, our main results will be divided into four parts. First wewill prove a basic lemma that will be utilized later. After that (with the aidof Lemma 2) we will present a uniqueness theorem. Afterwards a continuousdependence on the initial function type result will follow. Finally, we will endour paper with a stability type result.2.1. A useful lemma.Lemma 2. Let J = [t0, β[⊂ R be an interval and D ⊂ Rn be an open set. Letfurther f, f˜ : J ×Dm → Rn be mappings and g1, . . . , gm : J → R be functionsso thatγ ≤ gj(t) ≤ t (t ∈ [t0, β[)for all j = 1, . . . ,m with a certain real number γ. Finally, let θ, θ˜ : [γ, t0] →D be given initial functions. Assume that the functions x, x˜ : [γ, β[→ D aresolutions to problemsx′(t) = F (t, xt) t ≥ t0x(t) = θ(t) t ∈ [γ, t0].andx˜′(t) = F˜ (t, x˜t) t ≥ t0x˜(t) = θ˜(t) t ∈ [γ, t0].respectively. Suppose further that we also have the following.(i) there exists a continuous nonnegative function h : J → R such that forall t ∈ [γ, β[ and for any z ∈ Dm∥∥∥F (t, z)− F˜ (t, z)∥∥∥ ≤ h(t) ‖z‖ .(ii) there exists a continuous nonnegative function k : J → R such that∥∥F (t, z)− F (t, z′)∥∥ ≤ k(t) ∥∥z − z′∥∥for all t ∈ [γ, β[ and for any z, z′ ∈ Dm.Then‖x(t)− x˜(t)‖ ≤∥∥∥θ − θ˜∥∥∥ exp(∫ tt0h(s) + k(s)ds)(t ∈ J) .Proof. Applying integral inequality I , we immediately get thatx(t) = θ(t0) +∫ tt0F (s, xs)dsandx˜(t) = θ˜(t0) +∫ tt0F (s, x˜s)dsSTABILITY IN THE CLASS OF ... 7holds for all t ∈ J . Therefore‖x(t)− x˜(t)‖=∥∥∥∥θ(t0) + ∫ tt0F (s, xs)ds−(θ˜(t0) +∫ tt0F (s, x˜s)ds)∥∥∥∥≤∥∥∥θ(t0)− θ˜(t0)∥∥∥+ ∥∥∥∥∫ tt0F (s, xs)− F˜ (s, s˜s)ds∥∥∥∥≤∥∥∥θ(t0)− θ˜(t0)∥∥∥+ ∥∥∥∥∫ tt0F (s, xs)− F (s, x˜s)ds∥∥∥∥+ ∥∥∥∥∫ tt0F (s, x˜s)− F˜ (s, x˜s)ds∥∥∥∥≤∥∥∥θ(t0)− θ˜(t0)∥∥∥+ ∫ tt0k(s) supγ≤ξ≤s‖x(ξ)− x˜(ξ)‖ ds+∫ tt0h(s) supγ≤ξ≤s‖x(ξ)− x˜(ξ)‖ ds≤∥∥∥θ(t0)− θ˜(t0)∥∥∥+ ∫ tt0(k(s) + h(s)) supγ≤ξ≤s‖x(ξ)− x˜(ξ)‖ dsDefine the function v : [γ, β[→ R byv(t) = supγ≤ξ≤t‖x(ξ)− x˜(ξ)‖ (t ∈ [γ, β[) .Then‖x(t)− x˜(t)‖ ≤ v(t0) +∫ tt0(k(s) + h(s))v(s)ds (t ∈ J) ,or, since ∥∥∥θ(t)− θ˜(t)∥∥∥ ≤ v(t0) (t ∈ [γ, t0])holds, we havev(t) ≤ v(t0) +∫ tt0(k(s) + h(s))v(s)dsfor all t ∈ J . Now, Lemma 1 implies that‖x(t)− x˜(t)‖ ≤∥∥∥θ − θ˜∥∥∥ exp(∫ tt0k(s) + h(s)ds)for all t ∈ J , which ends the proof. 2.2. Corollaries.8 ESZTER GSELMANN AND ANNA KELEMENA uniqueness theorem. Based on the previous results, now we are able to provethe following uniqueness theorem.Theorem 1. Let J = [t0, β[ be an interval, D ⊂ Rn be an open set. LetF : J ×Dm → Rn be a function so that∥∥F (t, z)− F (t, z′)∥∥ ≤ k(t)∥∥z − z′∥∥ (t ∈ J, z ∈ Dm)is fulfilled with a certain continuous function k : J → R. Let further g1, . . . , gm: J → R be continuous functions so thatγ ≤ gj(t) ≤ t (t ∈ [t0, β[)for all j = 1, . . . ,m with a certain real number γ. Finally, let θ : [γ, t0] → Rbe a given continuous initial function. Then problem (P) has at most onesolution on any interval [γ, β1[, where t0 < β1 ≤ β.Proof. Suppose on the contrary that there are two different solutions, say xand x˜. Since x(t) = x˜(t) on [γ, t0], there exists t ∈]t0, β1[ such that x(t) 6= x˜(t).Lett1 = inf {t ∈]t0, β1[ |x(t) 6= x˜(t)} .Then t1 ∈]t0, β1[ and due to the continuity of x and x˜ we havex(t) = x˜(t)for all t ∈ [γ, t1[. Since t1 ∈]t0, β1[ and for all j = 1, . . . ,m x(gj(t1)) belongsto the open set D, there are positive real numbers ρ1, ρ2 such that[t1 − ρ, t1 + ρ] ⊂]t0, β1[ and Aj = {ξ ∈ Rn | ‖x(gj(t))− ξ‖ ≤ ρ2} ⊂ D.LetK = supt∈[t1−ρ1,t1+ρ1]k(t).Since k is continuous on the compact interval [t1−ρ1, t1 +ρ1] we have K ∈ R.Again, due to the continuity of x and g1, . . . , gm, there exists β2 ∈]t1, t1+ρ1[such that x(gj(t)), x˜(gj(t)) ∈ Aj for all j = 1, . . . ,m and t ∈]t1, β2[. Let nowt ∈ [t1, β2[, then‖x(t)− x˜(t)‖ =∥∥∥∥∫ tt1F (s, xs)− F (s, x˜s)ds∥∥∥∥ ≤ K ∫ tt1supγ≤ξ≤s‖x(ξ)− x˜(ξ)‖ ds.Definev(t) = supγ≤ξ≤t‖x(ξ)− x˜(ξ)‖ (t ∈ [γ, β1[) .Then‖x(t)− x˜(t)‖ ≤ K∫ tt1v(s)ds (t ∈ [t1, β2[) ,STABILITY IN THE CLASS OF ... 9or, since x ≡ x˜ on [γ, t1] we havev(t) ≤ K∫ tt1v(s)ds (t ∈ [t1, β2[) .In view of Lemma 1., this implies that v(t) = 0 for all t ∈ [t1, β2[, that is,x(t) = x˜(t) (t ∈ [t1, β1[) ,which contradicts to the definition of t1. We remark that the case whenk(t) = k (t ∈ J)can be found in the monograph of Driver, see [3, Theorem A., p. 259].Continuous dependence on the initial function. Using Lemma 2. with thechoice h ≡ 0, the following result can be derived immediately.Theorem 2. Let J = [t0, β[ be an interval, D ⊂ Rn be an open set. LetF : J ×Dm → Rn be a function so that∥∥F (t, z)− F (t, z′)∥∥ ≤ k(t) ∥∥z − z′∥∥ (t ∈ J, z ∈ Dm)is fulfilled with a certain continuous function k : J → R. Let further g1, . . . , gm: J → R be continuous functions so thatγ ≤ gj(t) ≤ t (t ∈ [t0, β[)for all j = 1, . . . ,m with a certain real number γ. Finally, let θ, θ˜ : [γ, t0]→ Rbe a given continuous initial functions. If the functions x, x˜ : [γ, β[→ R aresolutions to the problems{x′(t) = F (t, xt) for t ≥ t0x(t) = θ(t) t ∈ [γ, t0]and{x˜′(t) = F (t, x˜t) for t ≥ t0x˜(t) = θ˜(t) t ∈ [γ, t0].Then for all t ∈ [t0, β[‖x(t)− x˜(t)‖ ≤∥∥∥θ − θ˜∥∥∥ exp(∫ tt0k(s)ds).A stability result. Finally, we end this paper with the following stability typeresult.Theorem 3. Let ε > 0 be arbitrarily fixed, J = [t0, β[ be an interval andD ⊂ Rn be an open set. Let F : J ×Dm → Rn be a function so that∥∥F (t, z)− F (t, z′)∥∥ ≤ k(t) ∥∥z − z′∥∥ (t ∈ J, z ∈ Dm)is fulfilled with a certain continuous function k : J → R. Let further g1, . . . , gm: J → R be continuous functions so thatγ ≤ gj(t) ≤ t (t ∈ [t0, β[)10 ESZTER GSELMANN AND ANNA KELEMENfor all j = 1, . . . ,m with a certain real number γ. Finally, let θ : [γ, t0] → Rbe a given continuous initial function. Assume that the continuous initialfunction x : [t0β[→ R fulfills‖x′(t)− F (t, xt)‖ ≤ ε (t ∈ [t0, β[)x(t) = θ(t) (t ∈ [γ, t0]) .Then there exists a uniquely determined solution x˜ : [t0, β[→ R of the problemx′(t) = F (t, xt) (t ∈ [t0, β[)x(t) = θ(t) (t ∈ [γ, t0]) .such that‖x(t)− x˜(t)‖ ≤ ε(t− t0) exp(∫ tt0k(s)ds)for all t ∈ [t0, β[.Proof. Assume that for all t ∈ [t0, β[ we have∥∥x′(t)− F (t, xt)∥∥ ≤ ε.Equivalently, this means that x is a solution to the problemx′(t) = F˜ (t, xt) (t ∈ [t0, β[)x(t) = θ(t) (t ∈ [γ, t0]) ,whereF˜ (t, z) = F (t, z) + b(t) (t ∈ [t0, β[, z ∈ Dm) ,with a certain bounded function b : J → R.Let x˜ : J → R be the uniquely determined solution of the problemx˜′(t) = F (t, x˜t) (t ∈ [t0, β[)x˜(t) = θ(t) (t ∈ [γ, t0]) .In this case however‖x(t)− x˜(t)‖ =∥∥∥∥∫ tt0F˜ (s, xs)− F (s, x˜s)ds∥∥∥∥≤∫ tt0k(s) supγ≤ξ≤s‖x(s)− x˜(s)‖ ds+ ε(t− t0)for any t ∈ [t0, β[. Similarly as above, define the function v : [γ, β[→ R throughv(t) = supγ≤ξ≤t‖x(t)− x˜(t)‖ (t ∈ [γ, β[) .Then the previous inequality simply yield thatv(t) ≤ ε(t− t0) +∫ tt0k(s)v(s)ds (t ∈ [γ, β[) .STABILITY IN THE CLASS OF ... 11Thus, in virtue of Lemma 1, for all t ∈ [t0, β[‖x(t)− x˜(t)‖ ≤ ε(t− t0) exp(∫ tt0k(s)ds).Acknowledgement. This paper is dedicated to Professor A´rpa´d Sza´z (Univer-sity of Debrecen) on the occasion of this 70th birthday.References[1] Claudi Alsina and Roman Ger. On some inequalities and stability results related to theexponential function. J. Inequal. Appl., 2(4):373–380, 1998.[2] Szila´rd Andra´s and Jo´zsef J. Kolumba´n. On the Ulam-Hyers stability of first orderdifferential systems with nonlocal initial conditions. Nonlinear Anal., 82:1–11, 2013.[3] Rodney D. Driver. Ordinary and delay differential equations. Springer-Verlag, New York,1977. Applied Mathematical Sciences, Vol. 20.[4] Roman Ger and W lodzimierz Fechner. Some stability results for equations and inequal-ities connected with the exponential functions, volume Functional Equations and Dif-ference Inequalities and Ulam Stability Notions (F.U.N.), pages 37–46. Nova SciencePublishers, 2010.[5] Soon-Mo Jung. Hyers-Ulam stability of linear differential equations of first order. Appl.Math. Lett., 17(10):1135–1140, 2004.[6] Soon-Mo Jung. A fixed point approach to the stability of differential equations y′ =F (x, y). Bull. Malays. Math. Sci. Soc. (2), 33(1):47–56, 2010.[7] Soon-Mo Jung. Approximate solutions of a linear differential equation of third order.Bull. Malays. Math. Sci. Soc. (2), 35(4):1063–1073, 2012.[8] Soon-Mo Jung and Janusz Brzde¸k. Hyers-Ulam stability of the delay equation y′(t) =λy(t− τ). Abstr. Appl. Anal., pages Art. ID 372176, 10, 2010.[9] Sin-Ei Takahasi, Takeshi Miura, and Shizuo Miyajima. On the Hyers-Ulam stabilityof the Banach space-valued differential equation y′ = λy. Bull. Korean Math. Soc.,39(2):309–315, 2002.[10] Wolfgang Walter. Differential- und Integral-Ungleichungen und ihre Anwendung bei Ab-scha¨tzungs- und Eindeutigkeits-problemen. Springer Tracts in Natural Philosophy, Vol.2. Springer-Verlag, Berlin, 1964.Department on Analysis, Institute of Mathematics, University of Debrecen,H–4010 Debrecen P.O.Box:12E-mail address: gselmann@science.unideb.huFaculty of Informatics, University of Debrecen, H–4010 Debrecen P.O.Box:12E-mail address: pandora0303@hotmail.com

Stability in the class of first order delay differential equations

Abstract

Similar works

Full text

Available Versions

Repository of the Academy's Library

DEA University of Debrecen Electronic Archive

Crossref