CONDITIONS ON THE QUALITATIVE BEHAVIOUR OF SOLUTIONS FOR A CERTAIN CLASS OF FIFTH-ORDER NON-LINEAR DIFFERENTIAL EQUATIONS

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1 ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I.CUZA IAŞI Tomul XLVI, s.i, Mtemtică,, f.. CONDITIONS ON THE QUALITATIVE BEHAVIOUR OF SOLUTIONS FOR A CERTAIN CLASS OF FIFTH-ORDER NON-LINEAR DIFFERENTIAL EQUATIONS BY A.U. AFUWAPE O.A. ADESINA Abstrct. In this pper, we initite the use of the frequency domin method to study the qulittive behviour of solutions for certin clss of fifth order non liner differentil equtions. In prticulr, we prove the existence of unique solution which is, s well s its first four derivtives, bounded, globlly exponentilly stble periodic or lmost periodic whenever the forcing term is periodic or lmost periodic. Key words: Exponentil stbility, bounded solutions, periodic solutions, lmost periodic solutions, dul systems frequency-domin. 1. Introduction. This pper initites study on the qulittive behviour of solutions for certin fifth order non liner differentil equtions with the use of the frequency domin method. This method is one of the powerful fruitful techniques tht hs over the yers, gined incresing significnce in studying qulittive behviour of solutions of differentil equtions. It reduces the conditions for the existence of Lypunovs function of the form qudrtic form plus the integrl of the non liner term to mtrix inequlity, hence overcoming the problems rising in the prcticl construction of Lypunovs functions. The results obtined by using the frequency domin method when pplicble re in wy stronger more effective thn those obtined by using Lypunovs method [11]. KALMAN [8], POPOV [9] YACUBOVICH [1] [15] hd, in their efforts to solve Lure s problem rising in utomtic controls developed the frequency domin method. Lter, BARBALAT HALANAY [7] mde some generliztions on the works of POPOV [9] YACUBOVICH [1-15]. AFUWAPE [1] [4] used these generliztions to discuss some qulittive pro-

2 78 A.U. AFUWAPE O.A. ADESINA perties of solutions for certin third fourth order non liner differentil equtions. The purpose of the present pper, is to extend erlier results in [3], [4], [5] to fifth-order non liner differentil equtions of the form: 1.1 x v + x iv + bx + cx + gx + hx = pt 1. x v + x iv + bx + cx + g 1 xx + hx = pt where the functions g, g 1, h p re continuous in their respective rguments while, b c re positive constnts. We shll prove under certin conditions tht the solutions of equtions using the frequency domin method hve the following properties: P α the existence of unique solution xt, which together with its first four derivtives x t, x t, x t x iv t is bounded in IR ; P β the existence of solution which is globlly exponentilly stble, together with its first four derivtives ; P γ the existence of solution which is periodiclmost periodic together with its first four derivtives. We shll now recll without proof, the generlized theorem of YACU- BOVICH [4] s will be pplied in this work. Theorem 1.1. [4] Consider the system: 1.3 X = AX Bϕσ + P t, σ = C X where A is n n n rel mtrix, B C re n m rel mtrices with C s the trnspose of C, ϕσ = colϕ j σ j, j = 1,,..., m P t is n n-vector. Suppose tht in 1.3, the following ssumptions re true: i A is stble mtrix; ii P t is bounded for ll t in IR iii for some constnts ˆµ j, j = 1,,..., m 1.4 ϕ jσ j ϕ j ˆσ j σ j ˆσ j ˆµ j, σ j ˆσ j

3 3 FIFTH-ORDER NON-LINEAR DIFFERENTIAL EQUATIONS 79 iv there exists digonl mtrix D >, such tht the frequency domin inequlity 1.5 πω = MD + ReDGiω > holds for ll ω in IR, where Giω = C iωi A 1 B is the trnsfer function 1 M = dig, j = 1,,..., m. ˆµ j Then, system 1.3 hs properties P α, P β P γ. The pln of this pper is s follows: In Section, we give some nottions ssumptions. Section 3 is devoted to the min results of this pper. In Section 4 we give some preliminry results tht re of importnce to the proofs of the min results. Section 5 is dedicted to the proof of Theorem 3.1 its cosequences. The fct tht our equtions re of fifth order with two non liner terms mkes our clcultions more difficult does not give ny trivil generliztion to the erlier results in [3,4,5]. As will be seen lter in the work, crucil prt in the nlysis of our min results is to prove tht the frequency domin inequlity is strictly positive.. Nottions ssumptions. The following re the nottions dopted in this work: The liner homogeneous eqution of equtions is given s.1 x v + x iv + bx + cx + dx + ex = The Routh Hurwitz conditions for stbility of solutions of.1 re:. >, b c >, b cc d >, d >, b cc d + e b bc + e e >, e > the consequences of these conditions re:.3 b >, c >, bc d >, e d >, b 4d >, c 4e > Thus, equtions v vc + e = v vb + d = hve two rel positive roots given by v 1, v v 3, v 4 respectively, where v 1 = 1 [c c 4e 1 ].4 v = 1 [c + c 4e 1 ]

4 8 A.U. AFUWAPE O.A. ADESINA 4.5 v 3 = 1 [b b 4d 1 ] v 4 = 1 [b + b 4d 1 ] such tht < v 1 < v 3 < v < v 4. We shll ssume s bsic throughout this pper tht the following ssumptions hold: Assumptions. i the functions gx, g 1 x, hx pt re continuous in their respective rguments, ii g = g 1 = h = iii pt is bounded in IR iv there exist constnts d >, µ such tht for z, z in IR,.6 d gz g z z z d + µ, z z, v there exist constnts d >, µ such tht for z in IR, z,.7 d 1 z z g 1 sds d + µ, vi there exist constnts e >, µ 1 such tht for z, z in IR,.8 e hz h z z z e + µ 1, z z, vii the prmeters d, e, µ 1, µ in iv-vi stisfy the inequlity.9 µ 1 µ 4µ d µ 1 c < 4e, µ viii the prmeters, b, c d stisfy the following inequlities b > µ 1 µ d c > µ

5 5 FIFTH-ORDER NON-LINEAR DIFFERENTIAL EQUATIONS 81 ix pt is periodic lmost periodic. 3. Min result. The min results of this pper re the following: Theorem 3.1. Consider eqution 1.1 where g, h p stisfy ssumptions i viii. Then, eqution 1.1 hs properties P α P β. If in ddition, ssumption ix holds, then, it hs property P γ. Remrk 3.. If hx = ex in eqution 1.1, then, we hve the following independent result. Theorem 3.3. Suppose tht in the eqution: 3.1 x v + x iv + bx + cx + gx + ex = pt where, b, c e re positive constnts with g p stisfying ssumptions i iv vi trivilly stisfied such tht inequlity b b 4d 1 3. µ > M > c b + d e holds, where M is constnt depending on d. Then, eqution 3.1 hs properties P α P β. If in ddition, ssumption ix holds, then, it hs property P γ. Remrk 3.4. If gx = dx, d > in eqution 1.1 or g 1 xx = dx in eqution 1., then, the following independent result holds. Theorem 3.5. Suppose tht in the eqution: 3.3 x v + x iv + bx + cx + dx + hx = pt where, b, c d re positive constnts, with functions, h p stisfying ssumptions i, ii, iii {iv trivilly stified} such tht inequlity 3.4 µ 1 M 1 b c c + c 4e 1 + d e is stisfied, where M 1 is constnt depending on e. Then, eqution 3.1 hs properties P α P β. If in ddition, ssumption ix holds, then, it hs property P γ.

6 8 A.U. AFUWAPE O.A. ADESINA 6 The next min result is on eqution 1.. Theorem 3.6. Consider eqution 1. where the functions g, h p stisfy ssumptions i iii v viii. Then, eqution 1. hs properties P α P β. If in ddition, ssumption ix holds, then, it hs property P γ. Remrk 3.7. If in eqution 1., hx = ex, then, the following independent result holds. Theorem 3.8. Consider the eqution: 3.5 x v + x iv + bx + cx + g 1 xx + ex = pt where, b, c e re positive constnts with g 1 p stisfying ssumptions i iii v with {vi trivilly stisfied} in such wy tht the inequlity 3. holds. Then, eqution 3.5 hs properties P α P β. If in ddition, ssumption ix holds, then, it hs property P γ. 4. Preliminry results to the proofs of the min results. In order to give the proofs of the min results, we shll reduce eqution 1.1 to the vector form so tht we cn pply Theorem 1.1: 4.1 X = AX Bϕσ + P t, σ = C X where x 1 1 x 1 X = x 3 ; A = 1 x 4 1 x 5 e d c b 1 1 B = ; C = ; P t = 1 1 pt σ = C x1 X = x

7 7 FIFTH-ORDER NON-LINEAR DIFFERENTIAL EQUATIONS 83 ϕσ = ĥx1 ĝx On evlution of Giω = C iωi A 1 B, we obtin for system 4.1, Giω = 1 iω iω where = ω 4 ω c + e + iωω 4 bω + d To get the frequency-domin inequlity 1.5, we shll choose τ 1 τ to be strictly positive numbers define mtrices D = digτ j M = 1 = dig j = 1,. After some clcultions, we get for ll ω ɛ IR, µ j π11 π 4.3 πω = 1 > π 1 π where 4.4 π 11 = τ 1 µ ω4 ω c + e 4.5 π 1 = τ 1 iωτ = π π = τ µ 1 + ω ω4 ω b + d with s the complex conjugte of s the product of. We shll now stte prove series of lemms which shll be used in proving our min results. Lemm 4.1. Let S 1 v = v + vc e + vv vb + d v + vc e

8 84 A.U. AFUWAPE O.A. ADESINA 8 where ω = v.then, π 11 ω is positive for ll v > provided tht µ 1 > M 1 d, e = S 1 v = mx v 1 <v<v S 1 v S 1 v > S 1 v 3 = 1 c b b b 4d 1 + d e where v is the unique rel root of A 1 v = M 1 d, e is the mximum vlue of S 1 v ttinble t sy v = v. Proof of Lemm 4.1. For π 11 ω to be positive, we shll hve: 4.7 µ 1 < v + vc e + vv vb + d v + vc e Let µ 1 < S 1 v = v + vc e + vv vb + d v + vc e On differentiting the right h side of inequlity 4.7, we get A 1 v = S 1v. v + vc e = c { v + vc e vv vb + d } +v vb + d5v 3vb + d v + vc e Thus, S 1 v cn be zero in the intervl v 1, v if A 1 v = 3 + 1v b b 5cv 5 +b 8 c + b + b 3d b 8bc + c d 5ev 4 +4 c + 6c + 8 e + e 3b c + c bc 5cd bd 8bev 3 +3b e 4c 8ce d + 4bcd + be c 3 ce 5dev +4c 4ce e + cd + 4de + e v + d e ce = On sketching the grph of S 1 v ginst v, we note tht there re symptotes t v 1 v. Furthermore, 4.8 S 1 v 3 = 1 c b b b 4d 1 + d e 4.9 S 1 v 4 = 1 c b b + b 4d 1 + d e

9 9 FIFTH-ORDER NON-LINEAR DIFFERENTIAL EQUATIONS 85 From equtions , it is obvious tht the mximum S 1 v cn be ttined will lwys be negtive. Hence inequlity 4.7 holds. Lemm 4.. Let S v = v + vb d + v vc + e v v + vb d where ω = v. Then, π ω is positive for ll v >, provided tht µ < M d, e = S v = min v 3 <v<v 4 S v S v = S v = 1 b c c + c 4e 1 d e where v is the unique rel root of A v = with v 3 < v < v 4 M d, e is the minimum of S v ttinble t sy v = v. Furthermore, if v b b, then, S v > S if v = b with e = bc b 4 ε < bb 4d b c b, ε >, then, S v > S. Proof of Lemm 4.. For π ω to be positive for ll ω ɛ IR, the following inequlity must be vlid; ω = v 4.1 µ < v + vb d + v vc + e Let v v + vc d µ < S v = v + vb d + v vc + e v v + vc d On differentiting S v, we hve; A v = S v.v v + vb d = b v{v v + vb d vv vc + e +v vc + ev vc e v + vb d Obviously, S v cn be zero in the intervl v 3, v 4 if A v = v bv 6 + 4b 6 b 1c + 4dv 5 +5 d b 3 + 5c + 1bc e 4bdv b d 4cd + 4bc + 4ce + 5d v 3. +4d 3c d de bce 3ev be + d 3 cdev + de =

10 86 A.U. AFUWAPE O.A. ADESINA 1 We lso note tht, on sketching the grph of S v ginst v, there re symptotes t v 3 v 4. On substituting v = b into S v, we shll hve; S b Similrly, we obtin = b 4d 4 + bb bc + 4e b 4d 4.1 S v 1 = 1 b c c c 4e 1 d e 4.13 S v = 1 b c c + c 4e 1 d e Let us consider the following cses with the reltion: e = bc b 4 Cse I. If v = b, then, 4.14 S v = b 4d 4 b Therefore, S v = S. = S b Cse II. If v > b, then for some ε >, v = b + ε. Then, 4.15 S v = b 4d S b = b 4d 4 + b c ε + ε bb 4d Cse III. If v < b, then for some ε >, v = b ε. Thus, 4.17 S v = b 4d 4 b c ε

11 11 FIFTH-ORDER NON-LINEAR DIFFERENTIAL EQUATIONS S b = b 4d 4 + ε bb 4d On choosing ε < bb 4d b c we obtin the following inequlity b S v > S Hence, M d, e = Sv S v with A v =. This completes the proof of Lemm 4.. Lemm 4.3. Let be prmeter such tht λ = τ 1 τ > µ 1 µ 4µ d µ 1 c < λ < 4e µ Then, det.πω is positive for ll rel ω, provided tht the product µ 1 µ is non-negtive. Proof of Lemm 4.3. From the definition of πω in eqution 4.3, we obtin fter some simplifictions tht 1 det.πω = τ 1 τ + 1 ω µ 1 µ ω 4 ω b + d+ µ µ ω 4 ω c + e τ 1 + ω τ 4τ 1 τ Thus πω will be positive for ll ω in IR, if 4. + ω 6 µ + ω 4 µ 1 µ b + ω µ d µ 1 c µ 1µ τ 4τ 1 + µ 1 e µ 1µ τ 1 4τ > +

12 88 A.U. AFUWAPE O.A. ADESINA 1 The minimum of the left h side of inequlity 4. is e + µ 1 e µ 1µ τ 1 4τ From Lemms , inequlity 4., we hve the following s conditions for the positiveness of det.πω ee + µ 1 µ 1 µ > τ 1 τ 4µ d µ 1 c µ 1 µ > τ τ µ 1 µ b > 4.4 µ d µ 1 c > from inequlities 4.1, , we hve the following inequlity: 4.5 µ 1 µ 4µ d µ 1 c < λ < 4e µ 5. Conclusions to the proofs of Theorems 3.1, We shll, in this Section conclude the proofs of the bove nmed Theorems. We shll strt with Theorem 3.1, followed by Theorems in tht order. Proof of Theorem 3.1. Let gz = dz + ĝz hz = ez + ĥz Then, we cn rewrite inequlities.6.8 respectively for z z s 5.1 ĝz ĝ z z z µ

13 13 FIFTH-ORDER NON-LINEAR DIFFERENTIAL EQUATIONS ĥz ĥ z z z µ 1 By setting x = x 1, eqution 1.1 reduces to the equivlent form: x 1 = x x = x 3 x 3 = x 4 x 4 = x 5 x 5 = vector form ex 1 dx cx 3 bx 4 x 5 ĝx ĥx 1 + pt 5.3 X = AX Bϕσ + P t, σ = C X with X, A, B, C, P ϕσ s given in Section 4. The frequency-domin condition reduces to the mtrix inequlity 4.3 which is stisfied for ll ω in IR. This is vlid by using Lemms 4.1, The conclusions of Theorem 3.1 thus follow from the generlized theorem of Ycubovich Theorem 1.1. Remrk 5.1. Inequlities 4.5 is condition for the existence of unique, bounded solution, which is globlly exponentilly stble. Proof of Theorem 3.3. Let us rewrite inequlity.6 s inequlity 5.1 with gz = dz + ĝz. Then, we cn reduce eqution 3.1 to the equvlent form: x 1 = x x = x 3 x 3 = x 4 x 4 = x 5 x 5 = ex 1 dx cx 3 bx 4 x 5 ĝx + pt

14 9 A.U. AFUWAPE O.A. ADESINA 14 in vector form, to the system 5.3 with X A s given in Section 4 1 B = ; C = ; P t = 1 pt The trnsfer function where σ = C X = x ; ϕσ = ĝx Giω = C iωi A 1 B = iω = ω 4 ω c + e + iωω 4 ω b + d s the complex conjugte of with =. Choosing D = τ, τ > M = 1 µ, µ >, we obtin the frequency-domin inequlity 5.4 πω = τ µ 1 + ω ω 4 ω b + d = π in inequlity 4.3. By Lemm 4., inequlity 5.4 holds for ll ω in IR. The conclusions of the proof thus follow from Theorem 1.1. Proof of Theorem 3.5. Let hz = ez + ĥz Then, we cn rewrite inequlity.8 s inequlity 5.. Eqution 3.3 is reduced to its equivlent form s: x 1 = x > x = x 3 x 3 = x 4 x 4 = x 5 x 5 = ex 1 dx cx 3 bx 4 x 5 ĥx 1 + pt

15 15 FIFTH-ORDER NON-LINEAR DIFFERENTIAL EQUATIONS 91 in vector form X = AX Bϕσ + P t, σ = C X with X A, s given in Section 4, 1 B = ; C = ; P t = 1 pt σ = C X = x 1 ; ϕσ = ĥx 1. Simple clcultions give the trnsfer function s Giω = where re s given in the proof of Corollry 3.3. By defining D = τ 1, τ 1 > M = 1 µ 1, µ 1 >, we note tht inequlity 4.3 is equivlent to 5.5 πω = τ 1 µ ω4 ω c + e > By Lemm 4.1, inequlity 5.5 holds for ll ω inir. The conclusions of the proof of Corollry 3.5 follow from Theorem Proofs of Theorems In this Section we shll introduce the Dulity Principle see [5] to prove Theorems Definition 6.1. Following [5], we shll sy tht, the system: 6.1 X = A 1 X B 1 ϕˆσ 1 + P 1 t, ˆσ 1 = C 1 X is dul to system 1.3, if A 1 = A, B 1 = C, C 1 = B P 1 t = T P t, where T is non-singulr mtrix trnsformtion A is the trnspose of A.

16 9 A.U. AFUWAPE O.A. ADESINA 16 Remrk 6.. [5] If the conditions in Definition 6.1 re stisfied, then, the frequency domin inequlity for systems re equivlent. Proof of Theorem 3.6. We shll reduce eqution 1. to system which will be dul in the sense of Definition 6.1 to system 6.1. Let x g 1 sds = dx + ĝ 1 x hx = dx + ĥx. Then, inequlities.7 cn be written s 6. ĝ1x x µ 1, x futhermore, eqution 1. is reduced to the following system: x 1 = ex 5 ĥx 5 + pt x = x 1 dx 5 ĝx 5 x 3 = x cx 5 x 4 = x 3 bx 5 x 5 = x 4 x 5 which in vector form is the system 6.1 with A 1 = A, B 1 = C, C 1 = B P 1 t = T P t, where A, B, C P re s given in system 4.1 with 1 1 T = non singulr mtrix. This implies tht eqution 1. is dul to eqution 1.. Moreover, by Remrk 6., the frequency domin inequlity for both systems re equivlent the conclusions to Theorem 3.6 follow from Theorem 1.1.

17 17 FIFTH-ORDER NON-LINEAR DIFFERENTIAL EQUATIONS 93 Proof of Theorem 3.8. Following the proof of Theorem 3.6, we shll reduce eqution 3.5 to system tht will be dul to system 5.3. The equivlent system of eqution 3.5 is given s: x 1 = ex 5 x = x 1 dx 5 ĝx 5 + pt x 3 = x cx 5 x 4 = x 3 bx 5 x 5 = x 4 x 5 which is dul to system 5.3. Applictions of Lemm 6.1 Theorem 1.1 conclude the proof of Corollry 3.8. REFERENCES 1. AFUWAPE, A.U. Solutions of some third order non liner differentil equtions: Frequency domin method, An. Şt. Univ. Al.I. Cuz, Işi, s.i- Mt., t. XXV 1979, AFUWAPE, A.U. Conditions on the behviour of solutions for certin clss of third order non liner differentil equtions, An. Şt. Univ. Al.I. Cuz, Işi, s.i- Mt., t. XXV 1979, AFUWAPE, A.U. On some properties of solutions for certin fourth order non liner differentil equtions, Anlysis 5, 1985,. 4. AFUWAPE, A.U. Frequency domin pproch to non liner oscilltions of some third order differentil equtions, Nonliner Anlysis, Theory, Methods Applictions, vol. 1, No.1, 1986, BARBALAT, I. HALANAY, A. Applictions of the frequency-domin method to forced non-liner oscilltions, Mth. Nchr. 44, 197, BARBALAT, I. HALANAY, A. Conditions de comportment Presque li néire dns l théorie des oscilltions, Rev. Roum. Sci. Techn. Electrote chn. Energ. 9, 1974, BARBALAT, I. Conditions pour un bon comportment de certines equtions différentielles du troisième et du qutrième ordre, Equtions différentielles et fonctionnelles non linéires Edited by P. Jnssens, J. Mwhin N. Rouche, Hermnn, Pris 1973, 8-91.

18 94 A.U. AFUWAPE O.A. ADESINA KALMAN, R.E. Lypunov functions for the problem of Lurie in utomtic control, Proc. Ntn. Acd. Sci. U.S.A. 49, 1963, POPOV, V.M. Absolute stbility of non liner control systems, Aut. Rem. Control, 196, REISSIG, R., SANSONE, G. CONTI, R. Non liner differentil equtions of higher order, Noordhoff, Groninger ROUCHE, N., HABETS, P. LALOY, M. Stbility theory by Lipunov s direct method, Springer Verlg, New York, Heidelbreg, Berlin YACUBOVICH, V.A. Solutions of some mtrix inequlities occuring in the theory of utomtic control, Soviet Mth. Dokl. 4, 196, YACUBOVICH, V.A. The mtrix method in the theory of the stbility of non-liner control systems, Aut. Rem. Control 5, 1964, YACUBOVICH, V.A. Frequency-domin conditions for bsolute stbility dissiptivity of control systems with one differentible non linerity, Soviet Mth. Dokl. 6, 1965, YACUBOVICH, V.A. Periodic lmost periodic limit sttes of controlled systems with severl, in generl discotinuous non linerities, Soviet Mth. Dokl. 7, 1966, Received: 1.IX.1999 Deprtment of Mthemtics Obfemi Awolowo University Ile-Ife NIGERIA