The Lambert function method in qualitative analysis of fractional delay differential equations

F ac ional Calculus and Applied Analysis (2023) 26:1545–1565 h ps://doi.o g/10.1007/s13540-023-00176-x ORIGINAL PAPER The Lambe unc ion me hod in quali a i e analysis o ac ional delay di e en ial equa ions Jan ˇ Ce mák1 ·Tomáš Kisela1 ·Ludˇek Nech á al1 Recei ed: 13 Decembe 2022 / Re ised: 19 May 2023 / Accep ed: 25 May 2023 / Published online: 16 June 2023 © The Au ho (s) 2023 Abs ac We discuss an analy ical me hod o quali a i e in es iga ions o linea ac ional delay di e en ial equa ions. This me hod o igina es om he Lambe unc ion echnique ha is adi ionally used in s abili y analysis o o dina y delay di e en ial equa ions. Con a y o he exis ing esul s based on such a echnique, we show ha he me hod can esul in o ully explici s abili y c i e ia o a linea ac ional delay di e en ial equa ion, suppo ed by a p ecise desc ip ion o i s asymp o ics. As a by-p oduc o ou in es iga ions, we also s a e al e na e p oo s o some classical asse ions ha a e gi en in a mo e lucid o m compa ed o he exis ing p oo s. Keywo ds F ac ional delay di e en ial equa ion (p ima y) ·Lambe unc ion · S abili y ·Asymp o ic beha io Ma hema ics Subjec Classi ica ion (P ima y) 34K37 ·33E30 ·33E12 ·34K20 · 34K25 1 In oduc ion The pape discusses an analy ical me hod o quali a i e in es iga ions o ac ional delay di e en ial equa ions (FDDEs). These equa ions a e cu en ly e y in ensi ely s udied due o hei impo ance in a ious applica ion a eas, wi h a special emphasis o con ol heo y. Indeed, p esence o bo h he ime lag as well as non-in ege de i a i e BLudˇek Nech á al nech [email p o ec ed].cz Jan ˇ Ce mák [email p o ec ed].cz Tomáš Kisela [email p o ec ed].cz 1Ins i u e o Ma hema ics, B no Uni e si y o Technology, Technická 2896/2, 61669 B no, Czechia 123 1546 J. ˇ Ce mák e al. o de as con ol o unning pa ame e s in s udied models p o ides a e y e icien ool o a ious con ol p ocesses such as s abiliza ion o des abiliza ion o he pa icula solu ions o hese models ( o a pionee ing wo k in his di ec ion we e e o [17]). Sys ema ic in es iga ions o FDDEs we e ini ia ed in he pape [9]. He e, s abili y p ope ies o Dαx( )=λx( −τ), (1) whe e α, τ ∈R+,λ∈Rand Dαis a ac ional di e en ial ope a o , we e analyzed using he ac ha (1) is asymp o ically s able (i.e., any i s solu ion is e en ually ending o ze o) i and only i all he oo s o he cha ac e is ic equa ion sα−λexp(−sτ) =0(2) ha e nega i e eal pa s. To explo e such a loca ion o cha ac e is ic oo s wi h espec o he imagina y axis, he Lambe unc ion echnique was u ilized. The essence o he me hod consis s in a ep esen a ion o mula o cha ac e is ic oo s in e ms o app op ia e b anches o his mul i- alued unc ion ( o some p ecisions conce ning he co ec use o he Lambe unc ion echnique in s abili y analysis o (1), we e e also o [12]). A ce ain gene al disad an age o his app oach consis s in i s (seeming) disabili y o p o ide s abili y c i e ia in an explici o m depending on en y pa ame e s only (i.e., on α,λand τin he case o (1)). As he o he pape s on s abili y and asymp o ic p ope ies o (1) ollowed, he Lambe unc ion me hod was eplaced by some al e na e classical ools o s abili y in es iga ions (such as D–pa i ion me hod o τ-decomposi ion me hod) modi ied o he ac ional case. Using hese app oaches, e ec i e and non-imp o able s abili y condi ions o (1), suppo ed by some asymp o ic bounds, we e de i ed in [16]( he case λ∈R,0<α<1), [6] ( he case λ∈C,0<α<1), and pa ially also in [7] ( he case λ∈C,α>0). Some o he men ioned esul s can be ex ended also o he case o a wo– e m FDDE Dαx( )=μx( )+λx( −τ). (3) In his espec , we e e o [2,5,15] ( he case μ, λ ∈R,0<α<1) and [8]( he case μ, λ ∈R,1<α<2). Following he in ege -o de case (see, e.g., [1]), (1) and (3) may se e as es equa ions o nume ical analysis o FDDEs. F om his poin o iew, i is e y impo an o desc ibe hei basic quali a i e p ope ies in he s onges possible o m. Then, when analyzing app op ia e nume ical schemes applied o hese es equa ions, he abili y o keep he key quali a i e p ope ies o he unde lying exac equa ions is o basic impo ance. Fo some o he ecen ad ances in quali a i e heo y o FDDEs, we e e , e.g., o [3,10,11,18–20,23]. Following he abo e ou lines, he aim o his pape is wo old. Fi s , we deepen he exis ing knowledge on some quali a i e p ope ies o (1) wi h he Capu o ac ional de i a i e. Second, pe haps a mo e impo an aspec o he pape consis s in he way how we aim o do i . We come back o he Lambe unc ion me hod used in [9] and show ha his app oach can o e mo e han o mulae depending on he use o 123 The Lambe unc ion me hod in quali a i e... 1547 suppo ing so wa e packages. In ac , his echnique can esul in o ac ually e ec i e s abili y and asymp o ic c i e ia. The pape is o ganized as ollows. Sec ion 2 ecalls some exis ing indings on (1) and essen ials o he Lambe unc ion heo y. In Sec . 3, we explo e he Lambe unc ion me hod in de ails. In pa icula , we gi e an al e na e p oo o he classical asse ion saying ha he cha ac e is ic oo gene a ed by he p incipal b anch o he Lambe unc ion has he la ges eal pa , and o mula e a c i e ion ha enables o localize alues o he p incipal b anch in he complex plane. Sec ion 4p esen s applica ions o hese esul s o (1). He e, we ex end he exis ing s abili y c i e ia o (1) o a bi a y (posi i e) eal alues o α, and o mula e sha p asymp o ic es ima es o he solu ions o (1). Some inal ema ks in Sec . 5conclude he pape . 2 Basic ma hema ical backg ound In his sec ion, we summa ize some known ac s ele an o ou nex in es iga ions. Fi s , we ecall a close ela ionship be ween s abili y and asymp o ic p ope ies o (1), and dis ibu ion o he cha ac e is ic oo s o (2). Then, we ecall some basics o he Lambe unc ion and i s use in s abili y analysis o FDDEs. I was shown in [7] ha any solu ion xo (1) wi h he Capu o ac ional de i a i e (and a gene ally complex λ) can be w i en using he Mi ag-Le le ype unc ion Gλ,τ α,β ( )= /τ−1  j=0 λj( −jτ)αj+β−1 (α j+β) ,α,β>0,(4) whe e · deno es he uppe in ege pa . Mo e p ecisely, i φis a con inuous ini ial (complex- alued) unc ion on [−τ,0],φ0=φ(0)and φj,j=1,...,α−1, a e (complex) cons an s (conside ed when α>1), hen x( )=α−1  j=0 φjGλ,τ α, j+1( )+λ0 −τ Gλ,τ α,α( −τ−ξ)φ(ξ)dξ(5) is he solu ion o (1) sa is ying x( )=φ( ) o all ∈[−τ,0], and lim →0+x(j)( )= φj,j=1,...,α−1. Based on some asymp o ic esul s on (4), he solu ion (5) can be ew i en by he use o he cha ac e is ic oo s ha ing non-nega i e eal pa s. We ecall ha (2) admi s coun ably many oo s, and only a ini e numbe o hem is lying igh o any line (s)=p,p∈R( h oughou he pape , he symbol (z)and (z)s ands o he eal and imagina y pa o z∈C, espec i ely). I we deno e by S he se o all oo s o (2) ha ing non-nega i e eal pa s (no e ha Smus be a ini e se ), hen, o a non-in ege α,(5) can be ew i en as x( )= s∈S csexp(s )+O( j−α)as →∞ (6) 123 1548 J. ˇ Ce mák e al. whe e csa e complex coe icien s depending on α,τ,λ,φ, and j∈{−1,0,...,α− 1}( he pa icula alue o jdepends on limi beha io o φa =0). No ice ha j−α<0, i.e., he unc ion j−αalways ends o ze o. By (6), he oo s o (2) play an essen ial ole in quali a i e beha io o he solu- ions o (1). Following he classical in ege -o de pa e n, he au ho s in [9]used he ollowing chain o s eps sαexp(sτ) =λ→sexpτ αs=λ1 α→τ αsexpτ αs=τ αλ1 α(7) o exp ess he oo s o (2) ia he Lambe unc ion in oduced as he solu ion o W(z)exp(W(z)) =z,z∈C.(8) Be o e we ecall he oo o mula o (2) based on his special unc ion, some o i s basic p ope ies migh be collec ed. The Lambe unc ion is a mul i- alued unc ion (excep a z=0) wi h in ini ely many (single- alued) b anches Wk,k∈Z. Nei he o hem can be exp essed in e ms o elemen a y unc ions. In pa icula , W0is called a p incipal b anch. Fo any z∈C,(W0(z)) is be ween −πand π. The o he b anches a e numbe ed so ha (Wk(z)) is be ween (2k−2)π and (2k+1)π while (W−k(z)) is be ween −(2k+1)π and −(2k−2)π o any z∈Cand k=1,2,....Mo e p ecisely, he anges o W±kand W±(k+1),k=0,1,..., a e sepa a ed by he cu es {w=x+iy∈C:x=−yco (y), 2kπ<|y|<(2k+1)π} and he anges o W1and W−1a e sepa a ed by he hal -line {w=x+iy∈C:−∞<x≤−1,y=0}. These sepa a ing cu es co espond o he b anch cu s in he z-plane de ined as {z=ξ+iη∈C:−∞<ξ≤−exp(−1), η =0} in he case o W0, and {z=ξ+iη∈C:−∞<ξ≤0,η=0} in he case o Wk,k= 0. Con en ionally, he b anch cu (ha ing he a gumen π in he z-plane) is mapped by Wkon i s uppe bounda y in he w-plane. Only he b anches W0and W−1 ake on eal alues o a eal z∈[−exp(−1), ∞)and a eal z∈[−exp(−1), 0), espec i ely. Fu he de ails on he Lambe unc ion (including some his o ical ema ks) can be ound in [4], o o he commen s, see also [13] and [22]. Now, ollowing (7), all he oo s o (2) can be exp essed in he o m sk=α τWkτ αλ1 α,k∈Z.(9) 123 The Lambe unc ion me hod in quali a i e... 1549 By (6), a c ucial ole in analysis o (1) is played by he igh mos cha ac e is ic oo (i.e., he oo o (2) wi h he la ges eal pa ). The ollowing classical asse ion says ha his oo is jus s0. Lemma 1 Le z ∈C. Then W0(z)has he la ges eal pa (W0(z)) among all he o he eal pa s (Wk(z)),k∈Z. The o iginal p oo o Lemma 1is p e y long (see [22]). As a by-p oduc o ou nex p ocedu es, we a e going o p esen an al e na e (and mo e simple) way how o p o e his asse ion. Rema k 1 As poin ed ou in [12], he exp ession (9) is no qui e co ec o some complex alues o λ. Mo e p ecisely, (7) con ains aking he 1/α-powe which means ha he oo s gi en by (9) a e iden ical o hose o (2) only in he case |A g(λ)|≤απ (we ecall ha −π<A g(·)≤π). This inequali y is sa is ied i ially when α≥1 bu makes a es ic ion when 0 <α<1. In o he wo ds, i |A g(λ)|>απ, hen he ep esen a ion (9) can p oduce some supe luous oo s ha a e ac ually no he ue oo s o (2). As an example, we can conside , e.g., he case λ=−1, α=1/2 and τ=1 when (2) has he igh mos oo s0≈−0.4172 −i2.2651 (i.e., (1) is asymp o ically s able) while (9) p oduces s0≈0.4263 >0. On his accoun , we discuss quali a i e p ope ies o (1) o α>1. Commen s o he case 0 <α<1 a e p o ided in he inal sec ion. 3Somead anceson heLambe W unc ion This sec ion con ains se e al key esul s on he Lambe unc ion which p o ed o be use ul in quali a i e in es iga ions o (1). To ob ain an ac ually e ec i e and s ong asymp o ic desc ip ion o he solu ions o (1), we need o e ec i ely localize he posi ion o he igh mos cha ac e is ic oo in he complex plane. Mo e p ecisely, by (6) and (9), we need o de i e e ec i e exp essions o he eal and imagina y pa s o W0(z)in e ms o z. Thus, keeping in mind in ended s abili y and asymp o ic analysis o (1), we can pose he ollowing p oblems: Fo gi en p ∈Rand z ∈C,is i possible o cha ac e ize he p ope ies (W0(z)) < p and (W0(z)) =p di ec ly in e ms o z and p,i.e., wi hou an e alua ion o he p incipal b anch o he Lambe unc ion? Fu he , o gi en q ∈Rand z ∈C,is i possible o simila ly elabo a e on he p ope ies |(W0(z))|>q and |(W0(z))|=q? The ollowing esul yields an a i ma i e answe o hese ques ions. Theo em 1 Le p,q∈R,p>−1,0<q<π, and z ∈C,z= 0. Then (i) (W0(z)) < p i and only i ei he |z|<pexp(p)o |z|≥|p|exp(p)and a ccos pexp(p) |z|+|z|2−p2exp(2p) exp(p)<|A g(z)|; (10) 123 1550 J. ˇ Ce mák e al. (ii) (W0(z)) =p i and only i |z|≥|p|exp(p)and a ccos pexp(p) |z|+|z|2−p2exp(2p) exp(p)=|A g(z)|; (11) (iii) |(W0(z))|>q i and only i |A g(z)|>q and q sin(|A g(z)|−q)expqco (|A g(z)|−q)<|z|; (12) (i ) |(W0(z))|=q i and only i |A g(z)|>q and q sin(|A g(z)|−q)expqco (|A g(z)|−q)=|z|.(13) P oo (i) We w i e z=|z|exp(iA g(z)) and pu xk=(Wk(z)),yk=(Wk(z)) whe e Wk,k∈Za e pa icula b anches o he Lambe unc ion. Subs i u ion in o (8) yields exp(xk)(xkcos(yk)−yksin(yk)) =|z|cos(A g(z)), (14) exp(xk)(xksin(yk)+ykcos(yk)) =|z|sin(A g(z)). (15) I we sol e (14)–(15) wi h espec o unknowns xkexp(xk)and ykexp(xk), hen xkexp(xk)=|z|cos(A g(z)−yk), (16) ykexp(xk)=|z|sin(A g(z)−yk). (17) To show ha x0=(W0)(z)) < pwhene e |z|<pexp(p), we conside (16) implying x0exp(x0)≤|z|<pexp(p). Then he mono ony p ope y o he unc ion g(p)=pexp(p)on (−1,∞)ac ually implies x0<p. Now we assume ha |z|≥|p|exp(p). Squa ing and adding (16) and (17) we ge |z|2=((xk)2+(yk)2)exp(2xk), i.e., |yk|=|z|2−(xk)2exp(2xk) exp(xk).(18) 123 The Lambe unc ion me hod in quali a i e... 1551 Fo he p incipal b anch, i holds x0≥−y0co (y0), |y0|<π, i.e., x0sin(y0)+ y0cos(y0)≥0 whene e y0≥0. Mul iplying his by exp(x0)and using (15), one ge s |z|sin(A g(z)) =exp(x0)(x0sin(y0)+y0cos(y0)) ≥0 which implies A g(z)≥0 o y0≥0. I y0<0, he same a gumen a ion leads o A g(z)≤0, hence A g(z)y0≥0, i.e., |A g(z)−y0|≤π. Then (16) wi h k=0is equi alen o a ccos(x0exp(x0)/|z|)=|A g(z)−y0|.(19) Mo eo e , sign analysis o (17) wi h espec o A g(z)y0≥0 yields |A g(z)|≥|y0|, i.e., |A g(z)−y0|=|A g(z)|−|y0|.(20) Then, using (18), (19) and (20), we a e able o se up an implici dependence be ween x0=(W0(z)) and zin he o m (x0,z)=0 whe e is de ined ia (p,z)=a ccos pexp(p) |z|−|A g(z)|+|z|2−p2exp(2p) exp(p) o all p>−1 and z∈Csuch ha |p|exp(p)≤|z|.Le zbe ixed. Then d dp(p,z)=−(2p+1)exp(3p)+|z|2exp(p) exp(2p)|z|2−p2exp(2p)≤− (p+1)2exp(p) |z|2−p2exp(2p)≤0, hence, is dec easing in pi |p|exp(p)≤|z|. The e o e, (p,z)< (x0,z)=0 whene e p>x0=(W0(z)) and |p|exp(p)≤|z|. (ii) The p ope y ollows di ec ly om he p oo o (i) using he ac ha (p,z)=0 i and only i p=x0due o mono ony o wi h espec o p. (iii) Since W0is symme ic in he sense W0(z)=W0(z) o all z∈Cexcep hose lying on he b anch cu along he nega i e eal axis be ween −∞ and −exp(−1),i su ices o assume he case y0=(W0(z)) > q>0. We ha e al eady obse ed ha A g(z)≥y0. In addi ion, a s onge p ope y holds, namely A g(z)>y0. Indeed, possible equali y A g(z)=y0implies y0=0 (due o (17)) which con adic s he assump ion y0>q>0. Hence, i mus be 0 <A g(z)−y0<πas well as 0<A g(z)−q<π.Wedi ide(16)by(17) and pu k=0 o ge x0=y0co (A g(z)−y0). (21) 123 1552 J. ˇ Ce mák e al. Taking he loga i hm o (17) wi h k=0, we also ha e x0=ln|z|sin(A g(z)−y0)−ln(y0). (22) Combining (21) and (22), we a i e a ln|z|sin(A g(z)−y0)−ln(y0)−y0co (A g(z)−y0)=0 ep esen ing again an implici dependence, now be ween (W0(z)) and z. I we deno e h(q,z)=ln|z|sin(A g(z)−q)−ln(q)−qco (A g(z)−q), hen we ha e dh dq(q,z)=−qsin(2(A g(z)−q)) −sin2(A g(z)−q)−q2 qsin2(A g(z)−q). While he denomina o is posi i e, he nume a o N(q,z)=−qsin(2(A g(z)−q)) −sin2(A g(z)−q)−q2 is nega i e o each 0 ≤q≤A g(z). Indeed, we ha e dN dq(q,z)=2q(cos2(A g(z)−q)−1)≤0 which implies ha N(·,z)is non-inc easing and, oge he wi h N(0,z)= −sin2(A g(z)) < 0, nega i e on [0,A g(z)]. Consequen ly, h(·,z)is dec easing and he e o e h(q,z)>h(y0,z)=0 whene e 0 <q<y0<A g(z). Taking in o accoun he abo e men ioned symme y, we a i e (a e some elemen a y algeb a) a (12). (i ) The equi ed p ope y is again a consequence o mono ony o he unc ion h om he p e ious pa .  Rema k 2 (a) The p ope ies (ii) and (i ) o Theo em 1p o ide a new ool o e alu- a ions o he p incipal b anch o he Lambe unc ion. Le z= 0 be a ixed complex numbe . Then he le -hand side o (11) is dec easing o all p∈[a,W0(|z|)](a=−1 i |z|≥exp(−1)and a=W0(−|z|)i |z|<exp(−1)) om π o he ze o alue. Hence, (11) has a unique oo p∗lying in his in e al, and his oo equals jus (W0(z)). Simila ly, he le -hand side o (13) is inc easing o all q∈(0,A g(z)) om he ze o alue o in ini y, i.e., (13) admi s a unique posi i e oo q∗which is jus (W0(z)). To illus a e his e alua ion echnique, we compu e W0(z) o z=1 2+i√3 2. Then |z|=1, A g(z)=π/3 and he s anda d New on me hod e u ns (z)=p∗≈0.4843 in 5 i e a ions wi h he ini ial alue p0=0.5 and he s opping c i e ion aken as |pk+1−pk|≤10−16. The same me hod gi es (z)=q∗≈0.3808 in 7 i e a ions wi h he ini ial alue q0=0.5 and he same p ecision as in he case o he eal pa . 123 The Lambe unc ion me hod in quali a i e... 1553 In ac , he alue p∗+iq∗ma ches he alue p oduced by he MATLAB command lambe w(1/2+sq (3)/2*1i) o all he 15 digi s behind he decimal poin . S anda dly, he New on o Halley me hod is applied di ec ly o he equa ion wexp(w)− z=0 using he complex a i hme ic. MATLAB employs he la e me hod wi h some ad anced guess o he s a ing poin . Fo compu ing he alues o he Lambe unc ion wi h a bi a y p ecision, we e e o he ecen pape [14]. (b) Using a di e en app oach, he p ope y (i) o Theo em 1was also discussed in [21]. In he sequel, we cla i y o de ing o he eal as well as imagina y pa s o he pa icula b anches o he Lambe unc ion. This o de ing may be use ul in a deepe asymp o ic analysis o (1), and, mo eo e , esul s in o an al e na e p oo o Lemma 1. Following he p oo o Theo em 1, we in oduce he unc ions Gz(x,y)=xsin(y)+ycos(y)−|z|sin(A g(z)) exp(−x)and z(x)=|z|2exp(−2x)−x2. In iew o (15) and (18), he couples (xk,yk), whe e xk=(Wk(z)),yk=(Wk(z)), ha e o mee he ela ions Gz(x,y)=0 and y=± z(x), espec i ely. The ollowing asse ion speci ies o de ing o imagina y pa s o he b anches o he Lambe unc ion. Lemma 2 Le z ∈C {0}. Then (Wk(z)) ≤(Wk+1(z)) o all k ∈Z. In ac , all he inequali ies a e s ic wi h he only excep ion: I z ∈[−exp(−1), 0), hen we ha e (W−1(z)) =(W0(z)) =0. P oo Fo he sake o o mal simplici y, we iden i y complex numbe s w=x+iy wi h couples (x,y)∈R2.Fi s ,le z∈C {0}be such ha 0 ≤A g(z)≤πand de ine se s Sz j,j∈Z,as Sz j={(x,y)∈R2:Gz(x,y)=0,(2j−1)π < y<(2j+1)π} o j=1,2,...; Sz j={(x,y)∈R2:Gz(x,y)=0,0≤y<π} o j=0; Sz j={(x,y)∈R2:Gz(x,y)=0,−2π<y≤0} o j=−1; Sz j={(x,y)∈R2:Gz(x,y)=0,2jπ<y<(2j+2)π} o j=−2,−3,... (no e ha he equa ion Gz(x,y)=0 has no solu ion o y=(2j−1)π,j=1,2,..., and o y=2jπ,j=−1,−2,...). We wish o show ha Sz jis a pa o he ange o Wkjus when j=k. Le k≥1 be a bi a y. Then, by he de ini ion o Wk(see also Sec . 2), (2k−2)π < yk<(2k+1)π. (23) Le jbe such ha (xk,yk)∈Sz j. We dis inguish he ollowing cases wi h espec o j. 123 1560 J. ˇ Ce mák e al. as →∞, whe e c=cs0is he complex cons an om (6) co esponding o he igh mos cha ac- e is ic oo s0.I αis an in ege , hen he domina ing ole o s0in asymp o ic beha io o (1) is well known. In his case, he asse ion o (ii) holds as well.  Rema k 6 (a) The asymp o ic o mula om Theo em 3(ii) immedia ely implies x( )=O(exp(u0 )) as →∞ (32) o any solu ion xo (1), and he cons an u0is non-imp o able. Mo eo e , o la ge , he oo s o he eal and imagina y pa s o x end o he oo s o cos( 0 )and sin( 0 ), espec i ely. In bo h he cases, he dis ance be ween he subsequen oo s ends o π/ 0. These p ope ies a e illus a ed by Example 1. (b) The asymp o ic beha io o (1) signi ican ly depends on s abili y o (1). In pa icula , he exponen ial e ms in (6) a e anishing in he asymp o ically s able case (s0)<0. Howe e , he si ua ion changes in he limi case α=1 when, in acco dance wi h he i s -o de heo y, he igh mos cha ac e is ic oo s0de e mines an exponen ial decay a e o he solu ions also in he asymp o ically s able case. Since he abo e a gumen a ion can be ex ended o his p oblem as well, ou esul s p o ide a con ibu ion also o he co esponding classical i s -o de heo y. Example 1 Le α=1.2, τ=1, and conside (1) along wi h he ini ial condi ions φ( )=1(−1≤ ≤0), φ0=φ(0)=1, and φ1=lim →0+x( )=0. We compa e he co esponding (nume ical) solu ions o (1) o wo dis inc alues o λ, namely λ1=−2+i and λ2=−3+i0.1. As indica ed by Fig. 2, bo h he alues λ1,λ2lie in he ins abili y egion. In pa icula , he eal pa s u0o he co esponding igh mos oo s a e app oxima ely 0.4721 and 0.4917, and hei imagina y pa s 0a e 1.2321 and 1.5844, espec i ely. The eal pa s o he solu ions o (1) wi h wo abo e speci ied se s o en ies, along wi h he g ow h- a e unc ions exp(u0 ), a e depic ed in Figs. 3and 4. The g aphs sugges ha he modulus o cons an cin oduced in Theo em 3(ii) is less han one o λ=λ1, and g ea e han one o λ=λ2. To illus a e beha io o he solu ions xin be e de ail, Figs. 5and 6depic he a io (x( ))/ exp(u0 ) o λ1and λ2, espec i ely. The esul ing unc ions a e bounded, bu do no end o ze o which is a consequence o non-imp o abili y o he cons an u0in (32). As men ioned in Rema k 6(a), he dis ance be ween he subsequen oo s o (x( )) ends o π/ 0.Figs.7and 8illus a e his ac . We can see ha while in he case o λ1 he con e gence is a he as and he dis ance seems o be somewha s abilized a ound he se en h oo , in he case o λ2, he s abiliza ion occu s a ound he hund ed h oo . 5 Concluding ema ks The aim o he pape was o de elop he Lambe unc ion heo y, and hen apply he ob ained esul s in quali a i e in es iga ions o (1). Using his app oach, we we e able 123 The Lambe unc ion me hod in quali a i e... 1561 Fig. 3 The eal pa o he solu ion xo (1) o α=1.2, τ=1andλ1=−2+i, along wi h he co esponding g ow h- a e unc ions ±exp(0.4721 ) Fig. 4 The eal pa o he solu ion xo (1) o α=1.2, τ=1andλ2=−3+i0.1, along wi h he co esponding g ow h- a e unc ions ±exp(0.4917 ) o o mula e a p ecise asymp o ic desc ip ion o he solu ions o (1). Pa icula ly, in addi ion o an algeb aic decay a e o he solu ions in he s able case (desc ibed in some ea lie pape s), we could obse e an exponen ial g ow h o he solu ions in he 123 1562 J. ˇ Ce mák e al. Fig. 5 The eal pa o he solu ion xo (1) o α=1.2, τ=1andλ1=−2+i, di ided by i s g ow h- a e unc ion exp(0.4721 ) Fig. 6 The eal pa o he solu ion xo (1) o α=1.2, τ=1andλ2=−3+i0.1, di ided by i s g ow h- a e unc ion exp(0.4917 ) uns able case; he a e o his g ow h was de e mined as a (unique) eal oo o an auxilia y anscenden al equa ion. 123 The Lambe unc ion me hod in quali a i e... 1563 Fig. 7 The dis ance be ween he subsequen oo s o (x( )) o α=1.2, τ=1andλ1=−2+i is ending o π/1.2321 Fig. 8 The dis ance be ween he subsequen oo s o (x( )) o α=1.2, τ=1andλ2=−3+i0.1is ending o π/1.5844 Howe e , he impac o he p esen ed esul s is no limi ed o he heo y o FDDEs only. Ou app oach o e s an al e na e way how o p o e (and also s eng hen) some classical asse ions o he Lambe unc ion heo y. Mo eo e , o he bes o ou knowl- edge, he de i ed asymp o ic o mulae a e new also in he i s -o de case. He e, 123 1564 J. ˇ Ce mák e al. con a y o he ac ional case, ou esul s can be applied also in he s able case whe e a (non-imp o able) a e o exponen ial decay o he solu ions can be de e mined. Since we ha e o mula ed ou esul s o (1) wi h a complex coe icien λ, hei ex ension o he ec o case is nea ly s aigh o wa d p o ided he eigen alues o a ( eal) sys em ma ix a e simple. Rega ding eigen alues wi h highe mul iplici ies, some addi ional a gumen a ion seems o be necessa y. Based on ela ed cases discussed in ea lie pape s, one can expec a sligh modi ica ion o he solu ions g ow h, bu no impac on he asymp o ic equency. Ou inal ema k conce ns he case 0 <α<1 no in ol ed among he assump ions o he asse ions o Sec . 4. The p ocedu e o compu ing he cha ac e is ic oo s uses he law o exponen s which is, in gene al, no alid o complex numbe s. Thus, some supe luous oo s o he cha ac e is ic equa ion may appea i 0 <α<1(as illus a ed ia a coun e example in Rema k 1). In his case, ou s abili y and asymp o ic o mulae emain basically ue, bu we canno con i m hei s ic ness. In pa icula , we canno claim ha he abo e desc ibed a e o exponen ial g ow h o solu ions is non-imp o able. Ne e heless, we conjec u e ha a mo e ho ough analysis o he co esponding b anches o a complex powe can o e come his p oblem, and hus achie e he s ic asymp o ic esul s o all α>0. Such an analysis p o ides ano he possible opic o he nex esea ch. Acknowledgemen s The esea ch has been suppo ed by he G an GA20-11846S o he Czech Science Founda ion. Funding Open access publishing suppo ed by he Na ional Technical Lib a y in P ague. Decla a ions Con lic o in e es The au ho s decla e ha hey ha e no con lic o in e es . Open Access This a icle is licensed unde a C ea i e Commons A ibu ion 4.0 In e na ional License, which pe mi s use, sha ing, adap a ion, dis ibu ion and ep oduc ion in any medium o o ma , as long as 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 licence, and indica e i changes we e made. The images o o he hi d pa y ma e ial in his a icle a e included in he a icle’s C ea i e Commons licence, unless indica ed o he wise in a c edi line o he ma e ial. I ma e ial is no included in he a icle’s C ea i e Commons licence and you in ended use is no pe mi ed by s a u o y egula ion o exceeds he pe mi ed use, you will need o ob ain pe mission di ec ly om he copy igh holde . To iew a copy o his licence, isi h p://c ea i ecommons.o g/licenses/by/4.0/. Re e ences 1. Bellen, A., Zenna o, M.: Nume ical Me hods Fo Delay Di e en ial Equa ions. 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