MISCELLANEOUS DYNAMIC EQUATIONS. Elvan Akın Bohner and Martin Bohner. 1. Introduction

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1 MISCELLANEOUS DYNAMIC EQUATIONS Elvan Akın Bohner and Marin Bohner We consider several dynamic equaions and resen mehods on how o solve hese equaions. Among hem are linear equaions of higher order, Euler Cauchy equaions of higher order, logisic equaions or Verhuls equaions), Bernoulli equaions, Riccai equaions, and Clairau equaions.. Inroducion The calculus on ime scales has been inroduced in order o unify he heories of differenial equaions and of difference equaions and in order o exend hose heories o a more general class of so-called dynamic equaions. For an inroducion ino he calculus of ime scales we refer o [2, 3, 4]. Here we jus would like o menion ha any nonemy and closed subse T of R is called a ime scale, and ha he forward jum oeraor σ : T T is defined by σ) inf{s T : s }. Similarly, a backward jum oeraor ρ is defined, and he graininess µ of T is given by µ) σ). In his aer we consider cerain classes of simle dynamic equaions and resen mehods on how o solve hem. Discree versions of some of hose equaions are sudied in [, Chaer 3]. Firs, in Secion 2, we consider higher order linear dynamic equaions wih consan coefficiens, i.e., equaions of he form.) α k y k. For a reader no familiar wih he ime scales calculus, i is helful before sudying [2]) o hink of he oeraor as he usual derivaive if T R and he usual forward difference oeraor if T Z. Of course, as in he heory of ordinary differenial equaions, he so-called characerisic olynomial lays a cenral rôle, and in aricular he case of mulile roos of he characerisic olynomial will yield some ineresing resuls. Using a similar mehod, in Secion 3 we carry he sudy of Euler Cauchy equaions from he second order case as resened in [2, Secion 3.7] o he higher order case. However, in he higher order case we are forced due o he fac ha forward jum oeraors are no necessarily differeniable o facor hose Euler Cauchy equaions aroriaely, namely as.2) α k M k y, where he oeraors M k are defined recursively by.3) M y y and M k+ y M k y) for k N.

2 2 ELVAN AKIN BOHNER AND MARTIN BOHNER As in he higher order linear case, an associaed characerisic olynomial is imoran, and as for linear equaions, he case of mulile roos of his characerisic olynomial urns ou o be of aricular ineres. Moivaed by he fac ha for ordinary differenial equaions) he recirocal of a soluion of a linear equaion is a soluion of a logisic equaion, we inroduce our wo versions wo, as here are wo versions of linear equaions denoe v σ v σ).4) u )u + f) and v )v σ + f) of logisic equaions. These logisic equaions will aear in he forms.5) y [ ) + f)y)] y and x [) f)x)] x, where is defined as follows: The se R of all regressive i.e., + µ)) for all T) and rd-coninuous i.e., coninuous a oins wih σ) and lef-sided limis exis a oins wih ρ) ) funcions is an Abelian grou he so-called regressive grou) under he addiion defined by q + q + µq, and is he addiive inverse of R, i.e., / + µ) see [2, Exercise 2.26]). Now, knowledge of soluions of he linear equaions.4) hen enables us o exhibi soluions of he logisic equaions.5). Nex, we are aiming o inroduce Bernoulli equaions in such a way ha heir soluions raised o he αh ower are soluions of logisic equaions.5). However, in order o do so, we firs need o use he chain rule [2, Theorem.9] for finding a simle form of he derivaive of y α. This leads o he inroducion of a circle do mulilicaion defined by.6) α )) : α) + µ))h) α dh, which hen may be used o sudy Bernoulli equaions of he form [ )].7) x ) α f)xα ) x, and soluions of Bernoulli equaions see Secion 6) hen have our above described desired roery, so ha hey can readily be found as we know soluions of he logisic equaions.5). Furhermore, he circle do roduc urns ou o be ineresing in iself, and we devoe Secion 5 of his aer o he sudy of his roduc. Ineresing roeries of his roduc are he formulas.8) x α ) x α α x x, + µα ) + µ)α, and e α e α. Noe ha for R, he unique soluion of y )y saisfying y ) y see [2, Theorem 2.33]) is denoed as e, ), and he hird formula in.8) nicely sulemens he known relaions see [2, Theorem 2.36], and his is essenially a consequence of he roduc rule q) q + σ q and he quoien rule /q) q q )/qq σ )).9) e e q e q and e e q. e q Moreover, he se of osiively regressive funcions R +, i.e., he se of all rdconinuous funcions ha saisfy + µ)) > for all T see [2, Definiion

3 MISCELLANEOUS DYNAMIC EQUATIONS ]), ogeher wih addiion and real) scalar mulilicaion, urns ou o be a vecor sace, which we call he regressive vecor sace. In Secion 7 we also consider Riccai equaions of he form ) 2 z.) z + q) + q)z σ + ), ) where he circle square of x R is defined o be x 2 x) x). Proeries of he circle square are given in [2, Theorem 2.9], and we use hem o relae soluions of he Riccai equaion o soluions of he logisic equaions.5), rovided one aricular soluion of he Riccai equaion is known. Finally, in Secion 8 we sudy Clairau equaions aearing as.) y y + fy ), where f : R R may be any coninuously differeniable funcion. 2. Linear Equaions We consider homogeneous linear dynamic equaions of higher order wih consan coefficiens.), where y k are defined recursively by y y and y k+ y k) for k N and α k R for all k n wih α n. We call 2.) ϕλ) α k λ k he associaed characerisic olynomial. Theorem 2.. If λ R is a zero of 2.), hen e λ, ) solves.). Proof. According o our assumions, λ R, and hence y) e λ, ) is well defined. We find y k λ k y for all k N, and herefore α k y k α k λ k y ϕλ)y. Hence, if ϕλ), hen y solves.). Definiion 2.2. Equaion.) is called regressive if λ R for all zeros λ of ϕ. Theorem 2.3. Equaion.) is regressive if and only if 2.2) α k µ)) n k for all T. Proof. To show his, le λ, λ 2,..., λ n be he zeros of ϕ couning muliliciies. Then n ϕλ) α k λ k α n λ λ ).

4 4 ELVAN AKIN BOHNER AND MARTIN BOHNER The condiion λ R for all n is equivalen o n + µ)λ ). If n µ), hen α n + µ)λ ) α n n α k µ)) n k. If µ), hen n n [ α n + µ)λ ) α n µ)) n ) ] λ µ) µ)) n ϕ ) µ) n µ)) n α k ) k µ) α k µ)) n k. This roves our resul. Remark 2.4. Le λ, λ 2,..., λ n be he zeros of he characerisic olynomial 2.). Then i is easy o see ha.) is regressive iff n n λ i R, where λ i : λ λ 2... λ n. i Examle 2.5. Suose T 2 N and consider he equaion i 2.3) y 3y + 2y. The characerisic olynomial for 2.3) is λ 2 3λ + 2 and hence has zeros and 2. The regressiviy condiion 2.2) for 2.3) reads 2µ 2 ) + 3µ) + for all T, and i is clearly saisfied. Hence wo soluions of 2.3) are y ) e, ) log k ) and y 2 ) e 2, ) log k ). Now we consider linear equaions in he mulile roo case. The following lemma collecs some ideniies, ha are useful when rying o solve his roblem by he variaion of arameers mehod. Lemma 2.6. Suose λ R, le be differeniable, and assume ha here exiss a sequence { k } k N saisfying 2.4) and k k+ + λµ) for k N. Le y) )e λ, ). Then we have { k } k 2.5) y k ) λ k e λ for all k N and 2.6) { n } ϕ ) λ) α k y k e λ.! k

5 MISCELLANEOUS DYNAMIC EQUATIONS 5 Proof. We show 2.5) by inducion. Firs y y e λ e λ, and so 2.5) holds for k. If 2.5) holds for some k N, hen use he roduc rule) { k } ) k y k+ ) λ k e λ k ) k λ k { e λ + e σ } λ k ) k λ k { λe λ + } + µλ) e λ { k { ) k λ k+ + k k [ ) k k λ k+ + + } k + ) λ k+ { k+ } k ) + λ k e λ e λ )] λ k+ + k+ } so ha 2.5) holds for k +. This roves ha 2.5) holds for all k N. Now { k } k α k y k α k ) λ k e λ { n } k α k ) λ k e λ k { n } k! α k! k )! λk e λ k { n }! ϕ) λ) e λ, and his roves 2.6). Theorem 2.7. Suose λ R, and assume ha λ is a zero of 2.) wih muliliciy a leas m N. Pu m, k for all k m, and recursively k ) k+τ) for m 2 k. + λµτ) Then y) )e λ, ) is a soluion of.), where. Proof. Noe ha he sequence { k } k N defined as above is well defined only rdconinuous funcions are inegraed) and clearly saisfies 2.4). Since ϕ ) λ) for all < m, 2.6) from Lemma 2.6 imlies ha y) )e λ, ) solves.). e λ

6 6 ELVAN AKIN BOHNER AND MARTIN BOHNER Examle 2.8. If λ is a rile zero of ϕ, hen e λ, ), e λ, ) + λµτ), and e λ, ) τ s +λµs) + λµτ). are hree soluions of.). Noe ha for T R we have indeed e.g., + λµτ) dτ. 3. Euler Equaions Throughou his secion we assume ha T is a ime scale wih T, ). Moivaed by he fac ha soluions of Euler differenial equaions i.e., for T R) are of he form y) λ wih y ) λ y), we wan o look for soluions of he form y) e λ, ) also in he case of an Euler dynamic equaion on an arbirary ime scale. Noe ha for such y we have y ) λ y) and y ) λλ ) y), σ) and unforunaely y may no be differeniable see [2, Examle.56]) again for a general ime scale. Bu noe ha y ) λy), y )) λ2 y), y )) λ 2 y), y )) ) λ3 y),, and hence we wish o consider equaions of he form.2), where he oeraors M k are defined recursively by.3). We call.2) an Euler equaion of nh order, and as before 2.) is called he associaed characerisic olynomial. Theorems 3. and 3.3 below may be shown in a way comleely analogous o he roofs of Theorems 2. and 2.3, and hence we omi heir roofs. Theorem 3.. If λ/ R and λ is a zero of 2.), hen e λ, ) solves.2). Definiion 3.2. Equaion.2) is called regressive if λ Theorem 3.3. Equaion.2) is regressive if and only if 3.) α k k µ)) n k for all T. R for all zeros λ of ϕ. Examle 3.4. If n 2, hen.2) can be rewrien as ) α 2 y ) + α y + α y α 2 [σ)y α + + y + α ] y. α 2 α 2 Hence.2) is equivalen o 3.2) σ)y + ay + by wih a α α 2 +, b α α 2.

7 MISCELLANEOUS DYNAMIC EQUATIONS 7 The characerisic olynomial of 3.2) has he same zeros as λ 2 + a )λ + b. The regressiviy condiion 3.) for 3.2) is equivalen o σ) aµ) + bµ)) 2 for all T. Examle 3.5. If n 3, hen.2) can be exanded as in Examle 3.4 if T is a ime scale wih differeniable forward jum σ. E.g., if T R, hen 3.3) α 3 y ) ) + α 2 y ) + α y + α y, which is equivalen o 3.4) 3 y +a 2 y +by +cy wih a 3+ α 2 α 3, b + α α 3 + α 2 α 3, c α α 3. Noe ha he characerisic equaion for 3.4) reads λ 3 + a 3)λ 2 + b a + 2)λ + c while he characerisic equaion for 3.3) reads in a sense more naural and suggesive simly α 3 λ 3 + α 2 λ 2 + α λ + α. This remark alies o Euler equaions of any order and on any ime scale: Characerisic equaions for Euler equaions in facored form are easier o remember han hose for he equaions in exanded form. Examle 3.6. If ϕ) or ϕ), hen we have on any ime scale) e, ) and e, ). Examle 3.7. I is easy o verify e λ, ) ) λ if T R, Γ + λ) Γ ) e λ, ) Γ) Γ + λ) where Γ is he gamma funcion, and if T Z, e λ, ) log q [+q )λ] if T q N wih q >. Now we consider Euler equaions in he mulile roo case. The following lemma collecs some ideniies ha are useful when rying o solve his roblem by he variaion of arameers mehod. Is roof is comleely analogous o he roof of Lemma 2.6, and herefore we omi i. Lemma 3.8. Suose λ R, le be differeniable, and assume ha here exiss a sequence { k } k N saisfying 3.5) and k k+ + λµ) for k N. Le y) )e λ, ). Then we have { k 3.6) M k y } k ) λ k e λ

8 8 ELVAN AKIN BOHNER AND MARTIN BOHNER for all k N and 3.7) { n α k M k y } ϕ ) λ)! As in Theorem 2.7, we now may use Lemma 3.8 o obain he following resul. Theorem 3.9. Suose λ R, and assume ha λ is a zero of 2.) wih muliliciy a leas m N. Pu m, k for all k m, and recursively k ) e λ. k+τ) for m 2 k. τ + λµτ) Then y) )e λ, ) is a soluion of.2), where. Examle 3.. If λ is a rile zero of ϕ, hen e λ, ), e λ, ) τ + λµτ), and e, λ ) τ are hree soluions of.2). Noe ha for T R we have indeed e.g., τ + λµτ) dτ τ log. s s+λµs) τ + λµτ) Examle 3.. Suose λ is a double zero of a characerisic olynomial 2.) for T q N where q > ), and le. Then wo soluions of he corresonding Euler q-difference equaion.2) are given by y ) log q [+q )λ] see Examle 3.7) and y 2 ) [ + q )λ] y ) τ + λµτ) y ) τ. Noe ha q m τ m σq k ) q k τ m µq k m ) q )q k q k q k mq ), and so y 2 ) q ) log q [+q )λ] log q. In aricular, if q 2, hen y ) log 2 +λ) and y 2 ) log 2 +λ) log 2. Examle 3.2. Le T N 2 and consider he equaion + ) 2 y y + y. Here, σ) + ) 2 and µ) + 2. The characerisic olynomial is λ 2 2λ +. Hence wo soluions are given by see Examle 3.6) y ) e, ) and y 2) y ) Noe ha m 2 m στ) k σk 2 ) k 2 m στ) k τ + µτ) µk 2 m ) σk 2 ) m k στ). ) 2 k, k +

9 MISCELLANEOUS DYNAMIC EQUATIONS 9 and hence y 2 ) k ) 2 k. k + 4. Logisic Equaions Le us firs recall he following resuls from [2, Theorems 2.77 and 2.74]. Theorem 4. Variaion of Consans). Suose R and f C rd. Le T. i) Le u R. The unique soluion u saisfying he lef equaion in.4) and u ) u is given by u) e, )u + e, στ))fτ). ii) Le v R. The unique soluion v saisfying he righ equaion in.4) and v ) v is given by v) e, )v + e, τ)fτ). Moivaed by he fac ha for differenial equaions) u /y solves a linear equaion if y is a soluion of he logisic equaion, we assume ha u is a soluion of he firs linear equaion in.4), and hen y /u saisfies ) y u u uu σ u + f uu σ + fy)y σ [ + fy)] y. Therefore we call he firs equaion in.5) wih + fy R) a logisic dynamic equaion or Verhuls equaion). Similarly, we could sar wih a soluion v of he second linear equaion in.4) and find ha x /v saisfies ) x v v vv σ vσ f vv σ fx σ )x [ fx] x. We also call he second equaion in.5) wih fx R) a logisic dynamic equaion. Throughou we assume R and f C rd. Noe ha if u, hen y /u solves he firs equaion in.5) and auomaically saisfies + fy R, as can be seen from he calculaion + µ + fy) u + µu + f) u u + µu u uσ u. Similarly, if v, hen x /v solves he second equaion in.5) and saisfies + µfx v + µf v vσ + µf v ) v vσ + µv σ v vσ + µ) v so ha fx R follows. Using Theorem 4., i is now easy o solve logisic equaions as follows. Theorem 4.2. Suose R and f C rd.

10 ELVAN AKIN BOHNER AND MARTIN BOHNER i) Le y. If u) e, ) + e, στ))fτ) for all T, y hen y) u) e, ) y + e στ), )fτ) solves he firs equaion in.5) and saisfies y ) y. ii) Le x. If hen v) e, ) x + e, τ)fτ) for all T, x) v) e, ) x + e τ, )fτ) solves he second equaion in.5) and saisfies x ) x. In alicaions e.g., oulaion dynamics) one ofen assumes ha here exiss a consan N such ha 4.) ) Nf) for all T. For he remainder of his secion we assume 4.). Then we can evaluae he inegral in he denominaor of he soluion given in Theorem 4.2 ii) exlicily as e τ, )fτ) e τ, ) τ) N N [e, ) ]. Hence we obain he following corollary from Theorem 4.2 ii). Corollary 4.3. Suose R and le N be a consan. Le x. If hen 4.2) x) saisfies x ) x and solves x N + e, ) for all T, N e, ) x N + e,) N 4.3) x x x N + µ N x ) N + wih x N x N R. ) e, ) Remark 4.4. As is common in oulaion dynamics, we call he funcion from Corollary 4.3 he inrinsic growh funcion, while we refer o N as he sauraion level or environmenal carrying caaciy. Noe also ha he funcions x ) and x 2 ) N are soluions of 4.3), he so-called equilibrium soluions or criical soluions). If he saring value of x is beween hese wo soluions, i.e., x, N), hen /x /N > and hence x N + e, ) > e, ) N N

11 MISCELLANEOUS DYNAMIC EQUATIONS so ha < x) < e, ) N e, ) N, rovided R+. Noe ha R + ensures ha e, ) is osiive see [2, Theorem 2.48]). x > N, hen /x /N < and hence x) > N, rovided R +. Assume now ha ) > for all T. Then R +. If x, N), hen < x) < N and ) x x x N + µ N x > so ha x is sricly increasing. Similarly, if x > N, hen x is sricly decreasing. In any case, if x and if T is unbounded above, hen lim x) N, rovided lim e, ). Examle 4.5. Le T R, le α be a consan, and consider he coninuous) logisic equaion 4.4) x [α αx)] x, i.e., x αx x). Everyhing in Remark 4.4 alies o 4.4) here ) α and N ), and by Corollary 4.3 he soluion of 4.4) wih xs) z where s R and z R \ {}) is x) e α s) z + eα s) + z ) e. αs ) Examle 4.6. Le T Z and consider he discree) logisic equaion 4.5) x [ x)] x, i.e., x x x) + x. Everyhing in Remark 4.4 alies o 4.5) here ) and N ), and by Corollary 4.3 he soluion of 4.5) wih xs) z where s Z and z R \ {}) is x)! s! z +! s! + z ). s!! Remark 4.7. One ofen finds an equaion of he form 4.6) y n+ αy n y ) n N referred o as a logisic difference equaion. Observe ha his equaion and our logisic difference equaion inroduced in his secion are no he same. Our equaion has he advanage ha i is easily accessible wih he mehods used o sudy he normal logisic differenial equaion. Even hough 4.6) aears as a naural analogue of he logisic differenial equaion, he resuls given in his secion indicae ha our equaion deserves he name logisic equaion raher han 4.6) which is of course also worhy of sudy for is richness in dynamics). If

12 2 ELVAN AKIN BOHNER AND MARTIN BOHNER 5. The Regressive Vecor Sace For α R, we wan o define he Bernoulli equaion in such a way, ha he subsiuion x x α ransforms a soluion x of he Bernoulli equaion ino a soluion x of he logisic equaion. To do so, we now find a formula for he derivaive of x α. This ask leads us in his secion o he inroducion of a circle do mulilicaion, which urns ou o make R +,, ) ino a vecor sace, he so-called regressive vecor sace. To begin wih, we use he chain rule [2, Theorem.9] o calculae If x), hen x ) x α ) ) x ) 5.) x ) x) x ) x) α x) + µ)x )h ) α dh. ) α α + µ) x ) x) h dh. In order o have everyhing well defined, we wan o assume ha, if α R \ N, 5.2) + µ) x ) h > for all T and all h [, ]. x) I is easy o see ha x /x R + is sufficien for 5.2). Le us inroduce he noaion { R if α N Rα) : R + if α R \ N. Noe ha R + imlies ha + µ))h > for all T and all h [, ]. Definiion 5.. For α R and Rα) we define α by.6). Remark 5.2. Noe ha α α rovided T R. Theorem 5.3. Le α R. If α N, suose ha x) for all T. If α N, suose ha x)x σ ) > for all T. Then he firs formula in.8) holds. Proof. Firs noe ha +µx /x x σ /x imlies x /x Rα). Then he saemen follows from 5.). Examle 5.4. I is easy o see ha 2, Remark 5.5. The hree formulas µ, and 2 ) 2. xy) x xy x y y, x/y) x x/y x y y, x α ) x α α x x could be used o define a logarihm as log f, ) f τ) fτ),

13 MISCELLANEOUS DYNAMIC EQUATIONS 3 bu hen he usual logarihm rules would no hold as we had formulas involving an addiional inegral, e.g., log xy, ) log x, ) + log y, ) + µτ)x τ)y τ). xτ)yτ) I is sill an oen roblem o define a logarihm ha saisfies somehow smooher logarihm rules. Theorem 5.6. Le α R. If Rα), hen α R. More recisely, he second formula in.8) holds. Proof. Le α R and Rα). Then + µα ) + µα + µh) α dh + +µ where we used he subsiuion s + µh. Therefore, α R. αs α ds + µ) α, The following resul emhasizes again he imorance of our circle do mulilicaion inroduced in Definiion 5.. Theorem 5.7. If α R and Rα), hen he hird formula in.8) holds. Proof. Firs noe ha Rα) imlies α R by Theorem 5.6. Now we le T and u y e α, ). Then y ) and by Theorem 5.3 we ski he second argumen ) ) y e α ) Hence y solves he iniial value roblem α e e e α α )y. y α ))y, y ). Bu his iniial value roblem has exacly one soluion, namely e α, ). We will use Theorem 5.7 o show ha R +,, ) saisfies he axioms of a vecor sace. To do so, he following easy auxiliary resul is helful. Lemma 5.8. Suose, q R. If e e q, hen q. Proof. Differeniae boh sides of e e q o obain his resul. Now we are ready o rove he main resul of his secion. Theorem 5.9. R +,, ) is a real vecor sace. Proof. As is known see e.g., [2, Lemma 2.47]), R +, ) is an Abelian grou. We now rove he remaining four vecor sace roeries 5.3) α β ) αβ) for all α, β R, R + ; 5.4) for all R + ; 5.5) α q) α ) α q) for all, q R +, α R; 5.6) α + β) α ) β ) for all α, β R, R +.

14 4 ELVAN AKIN BOHNER AND MARTIN BOHNER Clearly, 5.4) follows direcly from Definiion 5.. For 5.3), 5.5), and 5.6) we use Theorem 5.7, Lemma 5.8, he rules.9) for he exonenial funcion, and he normal rules of exonens as follows: imlies 5.3), e α β ) e α β e α ) β e αβ e αβ) imlies 5.5), and imlies 5.6). e α q) e α q e e q ) α e α e α q e α e α q e α ) α q) e α+β) e α+β e α e β e α e β e α ) β ) Remark 5.. Noe ha 5.4) of course holds for each R. Furhermore, if α, β N, hen 5.3), 5.5), and 5.6) also hold for, q R. 6. Bernoulli Equaions Now we le α R \ {} and consider he Bernoulli equaion.7). Noe ha.7) is in he form of he second equaion in.5) if α. Inroducing x x α, we find if x never vanishes and solves.7), hen x x xα ) x α α x x α [ )] α fxα α ) f x, where we used Theorem 5.3 and he vecor sace roeries 5.3) and 5.5). Hence x solves he logisic equaion 6.) x [α ) f x)] x. Equaion 6.) is of he form.5), so ha we may aly Theorem 4.2 ii) use also Theorem 5.7) o find x as e α, ) x) x + e α τ, )fτ). I is hence easy o show he following resul. Theorem 6.. Suose α R \ {}, Rα), and f C rd. Le x. If hen x α + e α τ, )fτ) > for all T, x) [ x α solves he Bernoulli equaion.7). e, ) + e α τ, )fτ) Examle 6.2. Consider he iniial value roblem 6.2) x x + x 4, x). ] /α

15 MISCELLANEOUS DYNAMIC EQUATIONS 5 Noe ha he differenial equaion in 6.2) can be rewrien as a dynamic wih T R) Bernoulli equaion [ )] x α fxα x wih ), f) 3, α 3. According o Theorem 6. wih and x ), he soluion of 6.2) is x) [ x α e, ) + e α τ, )fτ) ] /α Examle 6.3. Consider he iniial value roblem e [ ) e 3 ] /3. 6.3) x 5x x3 + 5x + x 2 + x x 2, x). Noe ha 5 x x 2 + x x 5 x 2 ) x x2 imlies ha he difference equaion in 6.3) can be rewrien as a dynamic wih T Z) Bernoulli equaion [ )] x α fxα x wih ) 5, f), α 2. According o Theorem 6. wih and x ), he soluion of 6.3) is x) [ x α e, ) + ] /α e α τ, )fτ) 7. Riccai Equaions In his secion we consider he Riccai equaion 7.) z + q) + r)z σ + z 2 ) + µ)z, where, q, r are assumed o be rd-coninuous such ha ) for all T. Noe ha 7.) can be wrien in he form.). We now assume ha z is a known soluion of.). We ick any oher soluion z of.) and consider he difference y z z. I follows ha z y q rz σ [ z ) 2 ry σ + ) 2 + q + rz σ + ) ] 2 z. ) 2 z To coninue his calculaion, he following lemma is useful and can be roved by a direc comuaion. Lemma 7.. For, q R we have 2 q 2 [ q)] q ).

16 6 ELVAN AKIN BOHNER AND MARTIN BOHNER We may use Lemma 7. o coninue our calculaion from above as follows: [ z ) 2 ) ] 2 z y ry σ + [ ry σ z + z )] z z ) [ ry σ z + y z )]. ) Denoing s z z, we coninue o find y ry σ + ys ry rµy + ys ys r) rµy and herefore solve his las equaion for y ) 7.2) y y s r ys r). + µr Now, aly he formula o find where a b + c) a + b + c + µab + µac a b) + c + µa) s r) z z r z [ 2 z ) ] r 7.3) f ) + µ r z r z + y and [ + µ ) + y z r Hence, using 7.2), we find ha y saisfies he equaion 7.4) y [ g) + f)y)] y. )] g r 2 z ). g + fy, Equaion 7.4) is a logisic equaion as in.5). Hence he following resul holds. Theorem 7.2. Suose z is a soluion of he Riccai equaion.). Define f and g by 7.3) and le y be a soluion of he logisic equaion 7.4). Then z z +y is also a soluion of he Riccai equaion.). Corollary 7.3. Suose z is a soluion of he Riccai equaion.). Define f and g by 7.3). Then a soluion z of.) saisfying z ) z is given by e g, ) z) z ) + z + z ) e g, στ))fτ), rovided none of he denominaors is zero. Proof. This follows from Theorem 4.2, where soluions of.5) are given.

17 MISCELLANEOUS DYNAMIC EQUATIONS 7 Examle 7.4. We consider he equaion 7.5) z + [ )] z σ + µ)z. Equaion 7.5) is a Riccai equaion of he form 7.) wih coefficiens ), q), and r) ). By examinaion, z ) solves 7.5). We have z ) ) and r z ) ) [ )] ) so ha f and g from 7.3) are given by f) g). Hence, by Corollary 7.3, we find ha anoher soluion of 7.5) is given by z 2 z) ), z e ), ) rovided none of he demoninaors is zero. This soluion z saisfies z ) z. 8. Clairau Equaions The Clairau dynamic equaion aears as in.), where f : R R is some coninuously differeniable funcion. We roceed o rea.) as is coninuous analogue y y + fy ) and erform he subsiuion v y. Then 8.) y) v) + fv)). Differeniaing 8.) and using [2, Theorem.9], we conclude so ha v) y ) σ)v ) + v) + v ) { v ) σ) + Hence we obain he following resul. Theorem 8.. For any consan c R, 8.2) y) c + fc) f v) + hµ)v ) ) dh f v) + hµ)v ) ) } dh. is a soluion of.). Furher soluions of.) may be obained by solving 8.3) Examle 8.2. Consider he equaion f v) + hµ)v ) ) dh σ), y v. 8.4) y y + y ) 2, which is of he form.) wih fx) x 2. By 8.2) of Theorem 8., we find ha y) c + c 2 is a soluion of 8.4), where c R is an arbirary consan. Wih v y, he lef hand side of 8.3) becomes f v) + hµ)v ) ) dh 2 v) + hµ)v ) ) dh 2v) + µ)v ).

18 8 ELVAN AKIN BOHNER AND MARTIN BOHNER Hence we mus solve 8.5) 2v + µ)v σ), i.e., v + v σ σ). If T R, hen 8.5) becomes 2v), and herefore y) 2 4 of 8.4). Now we assume ha 8.6) T { k : k N } wih < < 2 <.... Then we may divide 8.5) by µ) o arrive a v 2 µ) vσ + σ) µ), is also a soluion which is an equaion as in.4) wih 2/µ and f σ/µ. By Theorem 4., a soluion is given by noe ha 2/µ) 2/µ) We find v) e 2/µ, )v + e 2/µ, τ) στ) µτ). m v m ) ) m v + ) m k k+ ) m c ) [α m + ] m ) k k, where we relaced v by c α. The value of he above sum deends on he ime scale, and we will do he calculaion for a few ime scales ha saisfy 8.6). Firs, m m T N imlies ) k k ) m 2 ) 4 4, k so we choose α /4 and find ha y) ) ) c ) c2 k c ) ) is also a soluion of 8.4) for any c R see also [, Examle 3..]). Nex, m T q N imlies ) k k ) m qm+ q + q q +, k so we choose α q/q + ) and derive ha y) q + q c )log q q2 q + ) 2 + c2 c ) log q is also a soluion of 8.4) for any c R. Finally, we noe ha m T N 2 imlies ) k m mm + ) k ), 2 k and hence we choose α and find ha y) c ) c2 is also a soluion of 8.4) for any c R. c ) ) 2 q ) 2 2q + ) 4 2 ) 2 2 4

19 MISCELLANEOUS DYNAMIC EQUATIONS 9 References [] R. P. Agarwal. Difference Equaions and Inequaliies. Marcel Dekker, Inc., New York, 992. [2] M. Bohner and A. Peerson. Dynamic Equaions on Time Scales: An Inroducion wih Alicaions. Birkhäuser Boson Inc, Boson, MA, 2. [3] S. Hilger. Analysis on measure chains a unified aroach o coninuous and discree calculus. Resuls Mah., 8:8 56, 99. [4] V. Lakshmikanham, B. Kaymakçalan, and S. Sivasundaram. Dynamic Sysems on Measure Chains, volume 37 of Mahemaics and is Alicaions. Kluwer Academic Publishers Grou, Dordrech, 996. Received Ocober 8, 22 Universiy of Missouri Rolla, Dearmen of Mahemaics and Saisics, Rolla, MO 654 address: akine@umr.edu, bohner@umr.edu h:// akine, h:// bohner