Almost Periodic and Asymptotically Almost Periodic Solutions of Liénard Equations

Manusc ip submi ed o Websi e: h p://AIMsciences.o g AIMS’ Jou nals Volume 00, Numbe 0, Xxxx XXXX pp. 000–000 ALMOST PERIODIC AND ASYMPTOTICALLY ALMOST PERIODIC SOLUTIONS OF LI´ ENARD EQUATIONS TOM´ AS CARABALLO AND DAVID CHEBAN Abs ac . The aim o his pape is o s udy he almos pe iodic and asymp- o ically almos pe iodic solu ions on (0,+∞) o he Li´ena d equa ion x′′ + (x)x′+g(x) = F( ), whe e F:T→R(T=R+o R) is an almos pe iodic o asymp o ically almos pe iodic unc ion and g: (a, b)→Ris a s ic ly dec easing unc ion. We s udy also his p oblem o he ec o ial Li´ena d equa ion. We analyze his p oblem in he amewo k o gene al non-au onomous dy- namical sys ems (cocycles). We apply he gene al esul s ob ained in ou ea ly pape s [3, 7] o p o e he exis ence o almos pe iodic (almos au omo phic, ecu en , pseudo ecu en ) and asymp o ically almos pe iodic (asymp o - ically almos au omo phic, asymp o ically ecu en , asymp o ically pseudo ecu en ) solu ions o Li´ena d equa ions (bo h scala and ec o ial). 1. In oduc ion. In his pape we s udy he exis ence o almos pe iodic and asymp o ically almos pe iodic solu ions o he Li´ena d equa ion x′′ + (x)x′+g(x) = F( ),(1) whe e F:T→R(T=R+:= [0,+∞) o R:= (−∞,+∞)) is a con inuous o locally in eg able unc ion and , g : (a, b)→R(−∞ ≤ a < b ≤+∞) a e locally Lipschi z con inuous unc ions. We assume ha he ollowing condi ions a e ul illed: (i) gis s ic ly dec easing; (ii) (x)≥0 o all x∈(a, b); (iii) Fis almos pe iodic ( espec i ely, almos au omo phic, ecu en , pseudo ecu en ) o asymp o ically almos pe iodic ( espec i ely, asymp o ically e- cu en , asymp o ically pseudo ecu en ). The ypical equa ion o ype (1) is x′′ +cx′+1 xα=F( ), whe e c≥0, α > 0 and F:T→Ris an almos pe iodic o asymp o ically almos pe iodic unc ion. Da e: No embe 19, 2010. 2000 Ma hema ics Subjec Classi ica ion. p ima y: 34C11, 34C15, 34C27, 34C35, 34D05, 34D23, 34D45, 37C55, 37C60, 37C70, 37C75. Key wo ds and ph ases. Non-au onomous dynamical sys ems; skew-p oduc sys ems; cocy- cles; global a ac o ;con e gen sys ems; quasi-pe iodic, almos pe iodic, almos au omo phic, ecu en solu ions; asymp o ically almos pe iodic solu ions; Liena d equa ion. 1 2 TOM´ AS CARABALLO AND DAVID CHEBAN In he pe iodic case (i.e., when Fis pe iodic), he dynamics o equa ion (1) was in ensi ely s udied by P. Ma ´ınez-Amo es and P. J. To es [13] and J. Campos and P. J. To es [2]. Fo he almos pe iodic case (i.e. o almos pe iodic F) hese esul s we e gene alized by P. Cieu a in [8]. The almos au omo phic and asymp o ically almos au omo phic solu ions o equa ion (1) we e s udied by P. Cieu a e al. [9], while he exis ence o pseudo almos pe iodic solu ions o equa ion (1) was analyzed by El Hadi Ai Dads e al. [9]. Ou main esul in he p esen pape s a es ha , when he unc ion Fis τ– pe iodic ( espec i ely, quasi pe iodic, almos pe iodic, almos au omo phic, ecu - en , pseudo ecu en ), i equa ion (1) admi s a solu ion which is bounded on R+, hen i has a unique τ–pe iodic ( espec i ely, quasi pe iodic, almos pe iodic, al- mos au omo phic, ecu en , pseudo ecu en ) solu ion, and e e y solu ion o (1), bounded on R+, is asymp o ically τ–pe iodic ( espec i ely, asymp o ically quasi pe- iodic, asymp o ically almos pe iodic, asymp o ically almos au omo phic, asymp- o ically ecu en , asymp o ically pseudo ecu en ). We ob ain also an analog o his esul when he unc ion Fis asymp o ically τ–pe iodic ( espec i ely, asymp- o ically quasi pe iodic, asymp o ically almos pe iodic, asymp o ically almos au- omo phic, asymp o ically ecu en , asymp o ically pseudo ecu en ). These e- sul s a e new and con ain, as pa icula cases, some o he esul s ci ed abo e. We p esen ou esul s in he amewo k o gene al non-au onomous dynamical sys ems (cocycles) and we apply ou abs ac heo y de eloped in [3, 7] o Li´ena d di e en ial equa ions (bo h scala and ec o ial). The pape is o ganized as ollows. In Sec ion 2, we collec some no ions (global a ac o , minimal se , poin /compac dissipa i i y, non-au onomous dynamical sys ems wi h con e gence, quasi pe iod- ici y, Le i an/Boh almos pe iodici y, almos au omo phy, ecu ence, pseudo e- cu ence, Poisson s abili y, e c) and ac s om he heo y o dynamical sys ems which will be necessa y in his pape . We gi e he e also some esul s conce ning a special class o non-au onomous dynamical sys em (NDS): he so-called NDS wi h weak con e gence. We gi e a gene aliza ion o he no ion o con e gen NDS. On he one hand, his ype o NDS is e y close o NDS wi h con e gence (because hey conse e some p ope ies o con e gen sys ems) and la ge han ha o con e gen sys ems. On he o he hand, we analyze he class o compac dissipa i e NDS wi h non i ial Le inson cen e . Sec ion 3 is de o ed o he exis ence o almos pe iodic (almos au omo phic, ecu en , pseudo ecu en ) and asymp o ically almos pe iodic (asymp o ically almos au omo phic, asymp o ically ecu en , asymp o ically pseudo ecu en ) solu ions o Li´ena d equa ion (1). In Sec ions 4 we p esen some esul s abou Sp– asymp o ically almos pe i- odic (asymp o ically almos pe iodic in he sense o S epano ) solu ions o Li´ena d equa ion (1). Finally, Sec ions 5 is de o ed o s udy he p oblem o almos pe iodici y ( e- spec i ely, almos au omo phy, ecu ence, pseudo ecu ence) and asymp o ically almos pe iodici y ( espec i ely, asymp o ically almos au omo phy, asymp o ically ecu ence, asymp o ically pseudo ecu ence) o solu ions o he ec o ial Li´ena d equa ion. ALMOST PERIODIC SOLUTIONS OF LI´ ENARD EQUATIONS 3 2. Nonau onomous Dynamical Sys ems wi h Con e gence. Le us s a by ecalling some concep s and no a ions abou he heo y o non-au onomous dynamical sys ems which will be necessa y o ou analysis. 2.1. Compac Global A ac o s o Dynamical Sys ems. Le (X, ρ) be a me ic space, Rbe he g oup o eal numbe s, R+be he semi-g oup o nonnega i e eal numbe s, Tbe one o he wo se s Ro R+. Adynamical sys em is a iple (X, T, π), whe e π:T×X→Xis a con- inuous mapping sa is ying he ollowing condi ions: π(0, x) = x(∀x∈X) and π(s, π( , x)) = π(s+ , x) (∀ , τ ∈Tand x∈X). When T=R+( espec i ely, R), he dynamical sys em (X, T, π) is called a semi- low ( espec i ely, low). The unc ion π(·, x) : T→Xis called a mo ion passing h ough he poin xa he momen = 0 and he se Σx:= π(T, x) is called he ajec o y o his mo ion. A nonemp y se M⊆Xis called posi i ely in a ian (nega i ely in a ian , in a ian ) wi h espec o he dynamical sys em (X, T, π) o , simply, posi i ely in a ian (nega i ely in a ian , in a ian ), i π( , M)⊆M(M⊆π( , M), π( , M) = M) o e e y ∈T. A closed posi i ely in a ian se , which does no con ain any own closed posi i ely in a ian subse , is called minimal. I is easy o see ha e e y posi i ely in a ian minimal se is in a ian . The dynamical sys em (X, T, π) is called: −poin dissipa i e i he e exis s a nonemp y compac subse K⊆Xsuch ha o e e y x∈X lim →+∞ρ(π( , x), K) = 0; (2) −compac dissipa i e i he e exis s a nonemp y compac subse K⊆Xsuch ha lim →+∞ρ(π( , x), K) = 0 uni o mly wi h espec o xon compac subse s o X. Le (X, T, π) be compac dissipa i e and Kbe a compac se a ac ing e e y compac subse o X. Le us se J:= ω(K) := ≥0[ τ≥ π(τ, K).(3) I can be shown [5, Ch.I] ha he se Jde ined by equali y (3) does no depend on he choice o he a ac ing se K, bu is cha ac e ized only by he p ope ies o he dynamical sys em (X, T, π) i sel . The se Jis called he Le inson cen e o he compac dissipa i e dynamical sys em (X, T, π). 2.2. Non-Au onomous Dynamical Sys ems wi h Con e gence. Recall ha gi en wo dynamical sys ems (X, T1, π) and (Y, T2, σ), a iple h(X, T1, π),(Y, T2, σ), hi, whe e his a homomo phism om (X, T1, π) on o (Y, T2, σ), is called a non- au onomous dynamical sys em. The non-au onomous dynamical sys em h(X, T1, π), (Y, T2, σ), hiis said o be con e gen (see [5]) i he ollowing condi ions a e ul illed: (i) he dynamical sys ems (X, T1, π) and (Y, T2, σ) a e compac dissipa i e; (ii) he se JXTXycon ains no mo e han one poin o all y∈JY, whe e Xy:= h−1(y) := {x∈X|h(x) = y}and JX( espec i ely, JY) is he Le inson cen e o he dynamical sys em (X, T1, π) ( espec i ely, (Y, T2, σ)). 4 TOM´ AS CARABALLO AND DAVID CHEBAN Thus, a non-au onomous dynamical sys em h(X, T1, π),(Y, T2, σ), hiis con e - gen , i he sys ems (X, T1, π) and (Y, T2, σ) a e compac dissipa i e wi h Le inson cen e s JXand JY espec i ely, and JXpossesses “ i ial” sec ions, i.e., JXTXy consis s o a single poin o all y∈JY. In his case he Le inson cen e JXo he dynamical sys em h(X, T1, π) is a copy (an homeomo phic image) o he Le inson cen e JYo he dynamical sys em (Y, T2, σ). Thus, he dynamics on JXis he same as on JY. Rema k 2.1. We no e ha con e gen sys ems a e in some sense he simples dissipa i e dynamical sys ems. I Yis compac , in a ian , T2=R,h(X, T1, π), (Y, T2, σ), hiis a con e gen non-au onomous dynamical sys em and JXis he Le inson cen e o (X, T1, π), hen (JX,T2, π) and (Y, T2, σ) a e homeomo phic. Al hough he Le inson cen e o a con e gen sys em can be comple ely desc ibed, i may be su icien ly complica ed. Recall [7] ha he poin x∈Xis called asymp o ically τ–pe iodic ( espec i ely, asymp o ically quasi pe iodic,asymp o ically Boh almos pe iodic,asymp o ically ecu en ,asymp o ically pseudo ecu en ), i he e exis s a τ-pe iodic ( espec i ely, quasi pe iodic, Boh almos pe iodic, ecu en , pseudo ecu en ) poin p∈Xsuch ha lim →+∞ρ(π( , x), π( , p)) = 0. 2.3. Non-Au onomous Dynamical Sys ems wi h Weak Con e gence. In his sec ion we will s udy a class o non-au onomous dynamical sys ems which is e y close o con e gen sys ems, bu possessing a non- i ial global a ac o . This means ha his class o non-au onomous sys ems will conse e almos all p ope ies o con e gen sys ems, bu will ha e a “non i ial” global a ac o JX, i.e., he e exis s a leas one poin y∈JYsuch ha he se JXTXycon ains mo e han one poin . A non-au onomous dynamical sys em h(X, T1, π),(Y, T2, σ), hiis said o be weak con e gen , i he ollowing condi ions hold: (i) he dynamical sys ems (X, T1, π) and (Y, T2, σ) a e compac dissipa i e wi h Le inson cen e s JXand JY espec i ely; (ii) i ollows ha lim →+∞ρ(π( , x1), π( , x2)) = 0, o all x1, x2∈JXwi h h(x1) = h(x2). Gi en x∈X, le us deno e by Mx:= {{ n} ⊆ T: such ha he sequence {π( n, x)}con e ges}and Lx:= {{ n} ∈ Mx: n→+∞}. Simila ly, o any y∈Y, deno e by My:= {{ n} ⊆ T: such ha he sequence {σ( n, y)}con e ges} and Ly:= {{ n} ∈ My: n→+∞}. Rema k 2.2. 1. Recall ha he poin x∈Xis called [14]-[16] compa able ( e- spec i ely, uni o mly compa able) by he cha ac e o ecu ence wi h he poin y∈Y, i Ny⊆Nx( espec i ely, My⊆Mx), whe e Nx:= {{ n} ∈ Mx,such ha {π( n, x} → x}}. 2. The no ions o compa abili y and uni o m compa abili y o mo ions by he cha ac e o ecu ence play a e y impo an ole [14]-[16] in he s udy o s abili y in he sense o Poisson (in pa icula , pe iodici y, quasi-pe iodici y, almos pe iod- ici y, almos au omo phy, ecu ence, e c) o solu ions o di e en ial equa ions wi h Poisson s able coe icien s. ALMOST PERIODIC SOLUTIONS OF LI´ ENARD EQUATIONS 5 Recall [7] ha he poin x∈Xis called compa able wi h y∈Yby he cha ac e o ecu ence in in ini y i Ly⊆Lx. Rema k 2.3. No e ha he no ion o compa abili y by he cha ac e o ecu ence in in ini y plays an impo an ole [7] in he p oblem o exis ence o asymp o i- cally almos pe iodic solu ions o di e en ial equa ions wi h asymp o ically almos pe iodic coe icien s. The nex heo em con ains su icien condi ions ensu ing asymp o ical s a iona - i y (asymp o ical pe iodici y, asymp o ical almos pe iodici y, e c) o poin s which a e compa able by he cha ac e o ecu ence in in ini y. Theo em 2.4. [7] Suppose ha he ollowing condi ions hold: (i) (X, T1, π)and (Y, T2, σ)a e wo dynamical sys ems; (ii) he poin y∈Yis asymp o ically s a iona y ( espec i ely, asymp o ically τ– pe iodic, asymp o ically quasi-pe iodic, asymp o ically almos pe iodic, asymp- o ically almos au omo phic, asymp o ically ecu en ); (iii) he poin xis compa able wi h y∈Yby he cha ac e o ecu ence in in ini y. Then, he poin xis also asymp o ically s a iona y ( espec i ely, asymp o ically τ–pe iodic, asymp o ically quasi-pe iodic, asymp o ically almos pe iodic, asymp o - ically almos au omo phic, asymp o ically ecu en ). Le (X, T, π) be a dynamical sys em. Deno e by ΩX:= S{ωx|x∈X},whe e ωxis he ω-limi se o he poin x. The ollowing esul s, which ha e been p o ed in [3, 4], ensu e he exis ence o compac minimal se s o poin dissipa i e non- au onomous dynamical sys ems, as well as he exis ence o s a iona y (asymp o i- cally pe iodic, asymp o ically quasi-pe iodic, asymp o ically almos pe iodic, e c...) poin s in he space X. Co olla y 2.5. [3, 4] Le h(X, T1, π),(Y, T2, σ), hibe a non-au onomous dynamical sys em such ha he ollowing condi ions hold: (i) he dynamical sys ems (X, T1, π)and (Y, T2, σ)a e poin dissipa i e; (ii) ΩYis a compac minimal se ; (iii) lim →+∞ρ(π( , x1), π( , x2)) = 0 holds o all x1, x2∈Xwi h h(x1) = h(x2); (i ) o e e y y∈ΩY, he se L˜ XT˜ Xycon ains a mos one poin , whe e L˜ X:= {x∈˜ X: he e exis s a leas one en i e mo ion γ(·) = π(·, x) h ough he poin xsuch ha γ(R)⊆˜ Xand γ(R)is ela i ely compac }, whe e ˜ X:= h−1(ΩY). Then, he e exis s a unique compac minimal se M⊆Xsuch ha (i) he sec ion MTXyo he se Mconsis s o a single poin my o all y∈Y; (ii) ΩX=M; (iii) e e y poin x∈Xis compa able wi h h(x)by he cha ac e o ecu ence in in ini y. Co olla y 2.6. [3, 4] Le h(X, T1, π),(Y, T2, σ), hibe a non-au onomous dynamical sys em such ha he ollowing condi ions hold: (i) he dynamical sys em (X, T, π)is poin dissipa i e; (ii) he e exis s a poin y0∈Ysuch ha Y:= H+(y0) := {σ( , y0) : ∈T+}; 6 TOM´ AS CARABALLO AND DAVID CHEBAN (iii) he poin y0is asymp o ically s a iona y ( espec i ely, asymp o ically τ–pe iodic, asymp o ically quasi-pe iodic, asymp o ically almos pe iodic, asymp o ically almos au omo phic, asymp o ically ecu en ); (i ) lim →+∞ρ(π( , x1), π( , x2)) = 0 holds o all x1, x2∈Xwi h h(x1) = h(x2); ( ) o e e y y∈ΩY he se L˜ XT˜ Xycon ains a mos one poin , whe e he se s L˜ Xand ˜ Xya e he ones de ined in he p e ious co olla y. Then, he e exis s a unique compac minimal se M⊆Xsuch ha (i) he sec ion MTXyo he se Mconsis s o a single poin my o all y∈Y; (ii) ΩX=M; (iii) e e y poin x∈Xis asymp o ically s a iona y ( espec i ely, asymp o ically τ– pe iodic, asymp o ically quasi-pe iodic, asymp o ically almos pe iodic, asymp- o ically almos au omo phic, asymp o ically ecu en ). 2.4. Pseudo Recu en Dynamical Sys ems wi h Con e gence. A non- au onomous dynamical sys em h(X, T1, π),(Y, T2, σ), hiis said o be uni o mly s able in he posi i e di ec ion on compac subse s o Xi , o a bi a y ε > 0 and compac subse K⊂X, he e is δ=δ(ε, K)>0 such ha inequali y ρ(x1, x2)< δ (x1, x2∈K, h(x1) = h(x2)) implies ha ρ(π( , x1), π( , x2)) < ε o ∈T+ 1,whe e T+ 1:= { ∈T1: ≥0}. Deno e by X˙ ×X={(x1, x2)∈X×X|h(x1) = h(x2)}. I he e exis s a unc ion V:X˙ ×X→R+wi h he ollowing p ope ies: (i) Vis con inuous; (ii) Vis posi i e de ined, i.e., V(x1, x2) = 0 i and only i x1=x2; (iii) V(π( , x1), π( , x2)) ≤V(x1, x2) o all (x1, x2)∈X˙ ×Xand ∈T+ 1, hen he non-au onomous dynamical sys em h(X, T1, π),(Y, T2, σ), hiis called (see [5, 6], [12], and [17]) V–mono one. Le (X, h, Y ) be a ibe space, i.e., Xand Ybe wo me ic spaces and h:X→Y be a homomo phism om Xon o Y. The subse M⊆Xis said o be condi ionally ela i ely compac , i he p e-image h−1(Y′)TMo e e y ela i ely compac subse Y′⊆Yis a ela i ely compac subse o X, in pa icula , My:= h−1(y)TMis ela i ely compac o e e y y. The se Mis called condi ionally compac i i is closed and condi ionally ela i ely compac . Example 2.7. Le Kbe a compac space, X:= K×Y,h=p 2:X→Y, hen he iple (X, h, Y )is a ibe space. The space Xis condi ionally compac , bu no compac . Deno e by K:= {a∈C(R+,R+)|a(0) = 0, a is s ic ly inc easing}. Recall ha he dynamical sys em (X, T1, π) is called asymp o ically compac i o e e y posi i ely in a ian bounded subse M⊆X he e exis s a nonemp y compac subse K⊆Xsuch ha lim →+∞β(π( , M), K) = 0, whe e β(A, B) := sup a∈A ρ(a, B) and ρ(a, B) := in b∈Bρ(a, b). Now, we s a e wo esul s p o ed in [3, 4] which p o ide some su icien condi ions ensu ing he con e gence cha ac e o a non-au onomous dynamical sys ems, as well ALMOST PERIODIC SOLUTIONS OF LI´ ENARD EQUATIONS 7 as he exis ence o pe iodic ( espec i ely, quasi-pe iodic, almos pe iodic, e c) poin s in he ibe o a pe iodic ( espec i ely, quasi-pe iodic, almos pe iodic, e c) poin . Theo em 2.8. Le h(X, T, π),(Y, R, σ), hibe a non-au onomous dynamical sys em sa is ying he ollowing condi ions: 1. he dynamical sys em (Y, R, σ)is pseudo ecu en ; 2. he dynamical sys em (X, T, π)is asymp o ically compac ; 3. he e exis s a poin x0∈Xy0wi h ela i ely compac posi i e semi- ajec o y Σ+ x0:= {π( , x0) : ≥0}; 4. he non-au onomous dynamical sys em h(X, T, π),(Y, R, σ), hiis V–mono one; 5. o all (x1, x2)∈LX˙ ×LX ∆X(whe e ∆X:= {(x, x) : x∈X}) he e exis s a posi i e numbe 0= 0(x1, x2)∈Tsuch ha V(π( 0, x1), π( 0, x2)) < V(x1, x2); 6. he e a e unc ions a, b ∈ K such ha Im(a) = Im(b)and a(ρ(x1, x2)) ≤ V(x1, x2)≤b(ρ(x1, x2)) o all (x1, x2)∈X˙ ×X. Then, he ollowing s a emen s ake place: (i) he NDS h(X, T1, π),(Y, R, σ), hiis con e gen ; (ii) JX=ωx0; (iii) h(JX) = Y. Co olla y 2.9. Le h(X, T, π),(Y, R, σ), hibe a non-au onomous dynamical sys em such ha : (i) he dynamical sys em (Y, R, σ)is ansi i e, i.e., he e exis s a poin y0∈Y such ha H(y0) = Y; (ii) he poin y0is τ–pe iodic ( espec i ely, quasi pe iodic, Boh almos pe iodic, ecu en , pseudo ecu en ); (iii) he dynamical sys em (X, T, π)is asymp o ically compac ; (i ) he e exis s a poin x0∈Xy0wi h ela i ely compac posi i e semi- ajec o y Σ+ x0:= {π( , x0) : ≥0}; ( ) he non-au onomous dynamical sys em h(X,T,π),(Y,R,σ),hiis V–mono one; ( i) o all (x1, x2)∈LX˙ ×LX ∆X(whe e ∆X:= {(x, x) : x∈X}) he e exis s a posi i e numbe 0= 0(x1, x2)∈Tsuch ha V(π( 0, x1), π( 0, x2)) < V(x1, x2); ( ii) he e a e unc ions a, b ∈ K such ha Im(a) = Im(b)and a(ρ(x1, x2)≤ V(x1, x2)≤b(ρ(x1, x2)) o all (x1, x2)∈X˙ ×X. Then, (i) he e exis s a unique τ–pe iodic ( espec i ely, quasi pe iodic, Boh almos pe- iodic, ecu en , pseudo ecu en ) poin x0∈Xy0:= {x∈X:h(x) = y0}; (ii) e e y poin x∈Xis asymp o ically τ–pe iodic ( espec i ely, asymp o ically quasi pe iodic, asymp o ically Boh almos pe iodic, asymp o ically ecu en , asymp o ically pseudo ecu en ). 3. Almos pe iodic and asymp o ically almos pe iodic solu ions o Li´ena d equa ions. Conside he ollowing Li´ena d equa ion x′′ + (x)x′+g(x) = F( ),(4) whe e F:T→Ris a con inuous unc ion and , g :I→R(I:= (a, b) wi h −∞ ≤ a < b ≤+∞) a e locally Lipschi z con inuous unc ions. We assume ha he unc ions , g and Fsa is y he ollowing condi ions: 8 TOM´ AS CARABALLO AND DAVID CHEBAN (i) gis s ic ly dec easing; (ii) (x)≥0 o all x∈I; (iii) sup ∈T |F( )|<+∞. As we al eady poin ed ou , he ypical example o equa ion (4) is gi en by x′′ +cx′+1 xα=F( ), whe e cis a nonnega i e cons an , α > 0 and F:R→Ris a pe iodic ( espec i ely, quasi pe iodic, almos pe iodic, almos au omo phic, ecu en ) unc ion. Recall ha a unc ion ϕ:T→Iis said o be bounded i Q:= ϕ(T) is a compac subse o I. Rema k 3.1. A con inuous unc ion ϕ:T→Iis bounded i and only i [mϕ, Mϕ]⊂ I, whe e mϕ:= in ∈Tϕ( ) and Mϕ:= sup ∈T ϕ( ). Deno e by Cb(T,R) he se o all con inuous unc ions F:T→Rwi h no m ||F|| := sup ∈T |F( )|<+∞. The ollowing esul s a e well known. Lemma 3.2. [8] Le I:= ( 0,+∞)wi h 0=−∞ o 0∈Rand F∈Cb(T,R). I ϕ( )is a solu ion o equa ion (4) which is bounded on R+( espec i ely, bounded on R), hen he de i a i es ϕ′( )and ϕ′′( )a e also bounded on R+( espec i ely, bounded on R). Deno e by ϕ( , u, , F ) he unique solu ion o equa ion (4) sa is ying he ini ial condi ions ϕ(0, u, , F ) = uand ϕ′(0, u, , F ) = . We ha e he ollowing heo em. Theo em 3.3. [1, 8] The ollowing s a emen s esul o be ue: (i) i F∈Cb(R+,R), hen o any pai o solu ions ϕ( , ui, i, F )(i= 1,2) o equa ion (4), which a e bounded on R+, we ha e lim →+∞(|ϕ( , u1, 1, F )−ϕ( , u2, 2, F)|+|ϕ′( , u1, 1, F )−ϕ′( , u2, 2, F)|) = 0; (ii) i F∈Cb(R,R), hen equa ion (4) admi s a mos one solu ion which is bounded on R. Rema k 3.4. No e ha Theo em 3.3 emains ue i we eplace he condi ion F∈Cb(T,R) (T=R+ o i em (i) and T=R o i em (ii)) by F∈Sp(T,R), whe e Sp(T,R) is he space o all unc ions ϕ∈Lp loc(T,R) sa is ying he condi ion |ϕ|Sp:= sup ∈T (R +1 |ϕ(s)|pds)1/p <+∞and p≥1. This s a emen may be p o ed wi h a sligh modi ica ion o he p oo o Theo em 3.3. Deno e by C(T,R) he se o all con inuous unc ions F:T→Rendowed wi h he compac -open opology, by Fτ he τ–shi o F(τ∈T), ha is, Fτ( ) := F( +τ) o all ∈T, and (C(T,R),T, σ) he shi dynamical sys em (Bebu o ’s dynamical sys em), i.e., σ(τ, F ) := Fτ o all τ∈Tand F∈C(T,R). I is said ha he unc ion F∈C(T,R)possesses he p ope y (S)( o example, pe iodici y, almos pe iodici y, ecu ence, asymp o ically almos pe iodici y and so on), i he mo ion σ(τ, F ),gene a ed by he unc ion Fin he shi dynamical sys em (C(T,R),T, σ), possesses his p ope y. ALMOST PERIODIC SOLUTIONS OF LI´ ENARD EQUATIONS 9 The solu ion ϕ( ) o equa ion (4) is called [7, 14, 15, 16] compa ible ( espec i ely, uni o m compa ible) by he cha ac e o ecu ence, i he mo ion σ( , (ϕ, ϕ′)), gen- e a ed by by unc ion (ϕ, ϕ′)∈C(R,R)×C(R,R) is compa able ( espec i ely, uni o m compa able) by he cha ac e o ecu ence wi h he mo ion σ(τ, F ), i.e., NF⊆N(ϕ,ϕ′)( espec i ely, MF⊆M(ϕ,ϕ′)), MF:= {{ n}: he sequence {σ( n, F )} is con e gen },LF:= {{ n} ∈ MF: such ha n→+∞as n→ ∞} and ϕ′is he de i a i e o he unc ion ϕ. Example 3.5. Deno e by y=x′, hen equa ion (4) can be educed o he ollowing equi alen i s o de sys em x′=y y′=−g(x)− (x)y+F( ).(5) Along wi h sys em (5), conside i s H–class, i.e., he amily o sys ems x′=y y′=−g(x)− (x)y+G( ),(6) whe e G∈H(F). Recall ha we deno e by ϕ( , u, , F) he unique solu ion o equa ion (4) sa - is ying he ini ial condi ions ϕ(0, u, , F ) = uand ϕ′(0, u, , F ) = and de ined on R+(o on R). Then, (ϕ( , u, , F ), ϕ′( , u, , F )) is he unique solu ion o sys em (5) wi h he ini ial da a (ϕ(0, u, , F ), ϕ′(0, u, , F )) = (u, )∈R2. Le Y=H(F) := {σ( , F ) : ∈R}and le (Y, R, σ) be he shi dynamical sys- em on H(F), induced by Bebu o ’s dynamical sys em (C(R,R),R, σ). We se W:= R2×H(F),˜ X:= {((u, ), G)∈W: he e exis s a unique solu ion ϕ( , u, ) o equa ion (4) h ough he poin (u, )∈R2a he ini ial momen = 0 and de ined on R+},π( , ((u, ), G)) := (ϕ( , u, , G), ϕ′( , u, , G)) o all ∈R+and ((u, ), G)∈˜ X, whe e (ϕ( , u, , G), ϕ′( , u, , G)) is he unique solu ion o sys em (6) wi h ini ial da a (ϕ(0, u, , G), ϕ′(0, u, , G)) = (u, ). Le now ϕ( , u, , F ) be a solu ion o equa ion (4) bounded on R+, and we deno e by X=H+((u, , F )) := {(ϕ(τ, u, , F ), ϕ′(τ, u, , F ), Fτ) : τ∈R+},whe e Fτ:= F(·+τ). F om Lemma 3.2 i ollows ha he se Xis shi in a ian , i.e., π( , X)⊆X o all ∈R+and X⊆˜ X. Le h=p 2:X7→ Ybe he second p ojec ion o Xon o Y, hen he iple h(X, R+, π),(Y, R, σ), hiis a non-au onomous dynamical sys em, gene a ed by equa ion (4) (o sys em o equa ions (5)) and he solu ion ϕ( , u, , F ). A solu ion ϕo equa ion (4) is called compa ible by ecu ence in in ini y [7], i LF⊆L(ϕ,ϕ′), whe e ϕ′is he de i a i e o he unc ion ϕand Lϕ:= {{ n} ∈ Mϕ: n→+∞}. Theo em 3.6. Suppose ha F∈C(R+,R)and Fis asymp o ically ecu en . Then, e e y solu ion ϕ( , u, )o equa ion (4), which is bounded on R+, is compa - ible by ecu ence in in ini y. P oo . Conside he non-au onomous dynamical sys em h(X, R+, π),(Y, R, σ), hi gene a ed by equa ion (4) and i s solu ion ϕ( , u, , F ) (see Example 3.5). Since Fis asymp o ically ecu en , he dynamical sys em (Y, R, σ) is compac dissipa- i e and i s Le inson cen e JY=ωFis a compac minimal se , whe e ωFis he ω–limi se o he poin F∈C(R,R) in he shi dynamical sys em (Y, R, σ). By Lemma 3.2, he se X=H+((u, , F)) ⊆R2×C(R,R) is compac . Le now