OSCILLATION CONSTANT FOR MODIFIED EULER TYPE HALF-LINEAR EQUATIONS

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1 Elecronic Journal of Differenial Equaions, Vol. 205 (205), No. 220, pp. 4. ISSN: URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu fp ejde.mah.xsae.edu OSCILLATION CONSTANT FOR MODIFIED EULER TYPE HALF-LINEAR EQUATIONS PETR HASIL, MICHAL VESELÝ Absrac. Applying he modified half-linear Prüfer angle, we sudy oscillaion properies of he half-linear differenial equaion r() p Φ(x ) + s() log p Φ(x) = 0, Φ(x) = x p sgn x. We show ha his equaion is condiionally oscillaory in a very general case. Moreover, we idenify he criical oscillaion consan (he borderline depending on he funcions r and s which separaes he oscillaory and non-oscillaory equaions). Noe ha he used mehod is differen from he sandard mehod based on he half-linear Prüfer angle.. Inroducion For any p >, we analyze he oscillaion behavior of he second-order half-linear differenial equaion R()Φ(x ) + S()Φ(x) = 0, Φ(x) = x p sgn x, (.) where R is a posiive coninuous funcion and S is a coninuous funcion. In his paper, we focus on he so-called condiional oscillaion. To recall he noion of he condiional oscillaion in a simple form, we consider (.) wih S() = γc(), where C is a coninuous funcion and γ R, i.e., R()Φ(x ) + γc()φ(x) = 0. (.2) We say ha (.2) is condiionally oscillaory if here exiss a posiive consan Γ such ha (.2) is oscillaory for γ > Γ and non-oscillaory for γ < Γ. Such a consan Γ is called he criical oscillaion consan of (.2). Of course, once we know is value, we are able o formulae oscillaion resuls direcly for (.). For his reason, some resuls of his kind are presened in differen forms in he lieraure (see he below given resuls and references). Now we collec he mos relevan references of he reaed opic. The reader can find he fundamenal heory overview on half-linear equaions in books, 5. The condiional oscillaion of half-linear differenial equaions is sudied, e.g., in papers 4, 7, (see also 3, 8, 24, 25). Of course, he idenificaion of oscillaion 200 Mahemaics Subjec Classificaion. 34C0, 34C5. Key words and phrases. Half-linear equaions; Prüfer angle; oscillaion heory; condiional oscillaion; oscillaion consan. c 205 Texas Sae Universiy - San Marcos. Submied March 6, 205. Published Augus 24, 205.

2 2 P. HASIL, M. VESELÝ EJDE-205/220 consans is a subjec of research also in he field of difference equaions (see 26) and in he field of dynamic equaions on ime scales (see 4). In he linear case, we refer o 0, 2, 20, 28 (and also 6, 7, 8). Very general resuls abou he condiional oscillaion can be found in 3, 5, 27. In 3, 27, here are reaed equaions wih asympoically almos periodic coefficiens and coefficiens having mean values (for he definiions of he asympoic almos periodiciy and he mean value, see Definiions 4.5 and 4.7 below). In addiion, he resuls of 5 cover equaions whose coefficiens are unbounded. The main resuls of 5, 27 are reformulaed in Theorems. and.2 below. Noe ha he main resul of 27 is superior o he resuls of 3. To menion he basic moivaion explicily in Theorems. and.2, we use he sandard symbols R + := (0, ) and R a := a, ) for any a > 0 in he res of his aricle. Theorem. (27). Le r : R R + be a coninuous funcion, for which mean value M(r /( p) ) exiss and for which i holds 0 < inf r() sup r() <, R R and le s : R R be a coninuous funcion having mean value M(s). Le ( p ) pm ( ) Γ p := r p. p Consider he equaion r()φ(x ) + s() Φ(x) = 0. (.3) p Equaion (.3) is oscillaory if M(s) > Γ, and non-oscillaory if M(s) < Γ. Theorem.2 (5). Le r : R R + and s : R R be coninuous funcions such ha here exis δ > and ε (0, /2) for which + Consider he equaion r(τ) dτ < δ ε, + s(τ) dτ < δ ε, R. (.4) r p () Φ(x ) + s() Φ(x) = 0. (.5) p (i) If here exis, R, S R + saisfying ( p ) p R p S > p and + + r(τ) dτ R, s(τ) dτ S for all sufficienly large, hen (.5) is oscillaory. (ii) If here exis, R, S R + saisfying ( p ) p, R p S < p + r(τ) dτ R, + s(τ) dτ S for all sufficienly large, hen (.5) is non-oscillaory.

3 EJDE-205/220 OSCILLATION FOR MODIFIED EULER TYPE EQUATIONS 3 In his paper, moivaed by Theorems. and.2, our aim is o exend he family of condiionally oscillaory equaions. More precisely, we idenify he criical oscillaion consan for he Euler ype equaions in he form r() p Φ(x ) + s() log p Φ(x) = 0, (.6) where r : R e R + and s : R e R are coninuous funcions. We inroduce a new modificaion of he half-linear Prüfer angle in combinaion wih he Riccai ransformaion. We remark ha he resuls proven in his paper are sronger han he ones known for (.5) (compare (3.) and (3.2) wih (.4)). To he bes of our knowledge, he presened resuls are new even for (.6) wih periodic coefficiens r, s. The aricle is organized as follows. The noion of he Prüfer angle is menioned in he nex secion. Auxiliary resuls are colleced in Secion 3. The conen of Secions 2 and 3 gives he descripion of our mehod which is used in he proofs of he main resuls in Secion 4. Secion 4 is finished by corollaries and examples. 2. Equaion for Prüfer angle In his secion, we derive he equaion for he modified half-linear Prüfer angle which will be fundamenal for our invesigaion. To do his, we have o begin wih he well-known noion of he Riccai equaion corresponding o he general half-linear equaion R()Φ(x ) + S()Φ(x) = 0, Φ(x) = x p sgn x, (2.) where p > and R : R e R + and S : R e R are coninuous funcions. The Riccai equaion w () + S() + (p )R p () w() p p = 0 (2.2) can be obained applying he ransformaion ( x () ) w() = R()Φ x() (2.3) o (2.), where x is a non-rivial soluion of (2.). Of course, he ransformaion is valid whenever x() 0. For more deails including he used calculaions, see 5, Secion..4. To inroduce he modified half-linear Prüfer ransformaion, we should shorly recall he half-linear rigonomeric funcions. The half-linear sine and cosine funcions, denoed by sin p and cos p, are defined via he iniial problem Φ(x ) + (p )Φ(x) = 0, x(0) = 0, x (0) =. (2.4) More precisely, sin p is given as he odd exension of he soluion of (2.4) wih period 2π p := 4π p sin π. p Then, cos p is inroduced as he derivaive of sin p. The funcions sin p and cos p have many properies similar o he common sine and cosine funcions. In his paper, we will use only he Pyhagorean ideniy, whose half-linear version is sin p x p + cos p x p =, x R. (2.5)

4 4 P. HASIL, M. VESELÝ EJDE-205/220 For more deails concerning he half-linear rigonomeric funcions, we refer o 5, Secion..2. We inroduce he modified Prüfer ransformaion in he form x() = ρ() sin p ϕ(), R q ()x () = ρ() log cos p ϕ() (2.6) ogeher wih he subsiuion v() = (log ) p/q w(), R e, (2.7) where log sands for he naural logarihm and q is he number conjugaed wih p, i.e., p + q = pq. (2.8) Noe ha subsiuions similar o (2.7) can be used also in he Riccai equaion (2.2). This approach leads o he so-called adaped (or weighed) Riccai equaion. Neverheless, we use his process only parially (see below) and we have o ake ino consideraion he modified Prüfer ransformaion (2.6) as well. Using (2.7), (2.3), (2.6), and (2.8) successively, we obain ( x v() = (log ) p/q w() = (log ) p/q () ) R()Φ x() = (log ) p/q R() Φ(R q () cos p ϕ()) Φ(log sin p ϕ()) ( cosp ϕ() ) = Φ. sin p ϕ() (2.9) One can easily verify ha ( cosp ) ˆv() := Φ (2.0) sin p solves he equaion ˆv () + p + (p ) ˆv() q = 0. (2.) Equaion (2.) is he Riccai equaion associaed o he equaion in (2.4). Hence, due o (2.5), (2.8), (2.0), and (2.), we have v () = ˆv(ϕ()) = p + (p ) ˆv(ϕ()) q ϕ () = ( p) + cos p ϕ() p ϕ () sin p ϕ() p = sin p ϕ() p ϕ (). On he oher hand, considering (2.7) ogeher wih (2.2), we have v () = p q (log ) p q w() + (log ) p/q w () = p q v() log + (log )p/q S() (p )R q () w() q = p q v() log (log )p/q S() (p )R q () v() q log. We combine (2.2) and (2.3). This leads o p sin p ϕ() p ϕ () = p q v() log (log )p/q S() (p )R q () v() q log. (2.2) (2.3)

5 EJDE-205/220 OSCILLATION FOR MODIFIED EULER TYPE EQUATIONS 5 Taking ino accoun (2.9), we obain ( p)ϕ () = p q log Φ(cos p ϕ()) sin p ϕ() (log ) p/q S() sin p ϕ() p (p )R q () cos p ϕ() p which gives us he desired equaion for he Prüfer angle as ϕ () = R q () cos p ϕ() p log log log Φ(cos (log )p/q p ϕ()) sin p ϕ() + S() sin p ϕ() p. p (2.4) In his aricle, we apply (2.4) o he sudy of he half-linear equaions wih he coefficiens in he form Thus, we will analyze he equaions R() = r() p, S() = s() log p, R e. (2.5) r() p Φ(x ) + s() log p Φ(x) = 0, (2.6) where r : R e R + and s : R e R are coninuous funcions. We poin ou ha he form of (2.4) for he coefficiens in (2.5) is which can be simply verified. ϕ () = r q () cos p ϕ() p log Φ(cos p ϕ()) sin p ϕ() + s() sin p ϕ() p p, (2.7) 3. Prüfer angle of average funcion In his secion, we assume ha he coefficiens r : R e R + and s : R e R in (2.6) are such ha lim lim + r q (τ) dτ log = 0, (3.) + s(τ) dτ log = 0 (3.2) hold for some R +. For his number, we define he funcion ψ which deermines he average value of an arbirarily given soluion ϕ of (2.7) over inervals of he lengh ; i.e., we pu ψ() := + ϕ(τ) dτ, R e, where ϕ is a soluion of (2.7) on R e. We formulae and prove auxiliary resuls concerning he funcion ψ.

6 6 P. HASIL, M. VESELÝ EJDE-205/220 Lemma 3.. I holds uniformly wih respec o s, +. lim log ϕ(s) ψ() = 0 (3.3) Proof. For s, +, i is seen ha lim sup log ϕ(s) ψ() + lim sup log ϕ (τ) dτ + = lim sup log r q (τ) cos p ϕ(τ) p τ log τ Φ(cos p ϕ(τ)) sin p ϕ(τ) + s(τ) sin p ϕ(τ) p dτ p + lim sup log r q (τ) cos p ϕ(τ) p log + Φ(cos p ϕ(τ)) sin p ϕ(τ) + s(τ) sin p ϕ(τ) p dτ. p Since (see direcly (2.5)) and, consequenly, we have sin p x p, cos p x p, x R, (3.4) Φ(cos p x) sin p x = cos p x p sin p x, x R, (3.5) 0 lim sup log ϕ(s) ψ() + log lim sup Using (3.) and (3.2), we obain 0 lim inf r q (τ) + + s(τ) p dτ, log ϕ(s) ψ() lim sup uniformly wih respec o s, +. Lemma 3.2. I holds ψ () = cosp ψ() p log + sin p ψ() p (p ) + + s, +. log ϕ(s) ψ() = 0 r q (τ) dτ Φ(cos p ψ()) sin p ψ() s(τ) dτ + Ψ() for all > e, where Ψ : R e R is a coninuous funcion such ha lim Ψ() = 0. Proof. For any > mahrme, we have ψ () = + r q (τ) cos p ϕ(τ) p τ log τ

7 EJDE-205/220 OSCILLATION FOR MODIFIED EULER TYPE EQUATIONS 7 We can replace ψ () by + log Φ(cos p ϕ(τ)) sin p ϕ(τ) + s(τ) sin p ϕ(τ) p p dτ. r q (τ) cos p ϕ(τ) p Φ(cos p ϕ(τ)) sin p ϕ(τ) + s(τ) sin p ϕ(τ) p because we can easily esimae (see (3.), (3.2), (3.4), (3.5)) lim sup log + + s(τ) sin p ϕ(τ) p p + s(τ) sin p ϕ(τ) p τ log τ dτ p r q (τ) cos p ϕ(τ) p Φ(cos p ϕ(τ)) sin p ϕ(τ) + dτ, r q (τ) cos p ϕ(τ) p Φ(cos p ϕ(τ)) sin p ϕ(τ) log dτ p log + lim sup log r q (τ) + + s(τ) τ log τ p log + lim sup log ( + ) log( + ) r q (τ) + + s(τ) dτ p log lim sup + r q (τ) + + s(τ) dτ ( + ) log p + lim sup r q (τ) + + s(τ) dτ = 0. + p dτ Applying he uniform coninuiy of he half-linear rigonomeric funcions and (3.3) in Lemma 3., we see ha ( lim Φ(cos p ψ()) sin p ψ() + ) Φ(cos p ϕ(τ)) sin p ϕ(τ) dτ = 0. (3.6) In addiion, he half-linear rigonomeric funcions are coninuously differeniable and periodic (see, e.g., 5, Secion..2). Hence, hey have he Lipschiz propery on R and, consequenly, here exiss a consan L R + for which cosp x p cos p y p L x y, x, y R, (3.7) sinp x p sin p y p L x y, x, y R. (3.8) Therefore, from (3.), Lemma 3., and (3.7), i follows lim sup cos p ψ() p lim sup lim sup + + r q (τ) dτ + r q (τ) cosp ψ() p cos p ϕ(τ) p dτ log + r q (τ)l ψ() ϕ(τ) dτ = 0. log r q (τ) cos p ϕ(τ) p dτ (3.9)

8 8 P. HASIL, M. VESELÝ EJDE-205/220 Analogously, from (3.2), Lemma 3., and (3.8), we have lim sup sin p ψ() p + s(τ) dτ + s(τ) sin p ϕ(τ) p dτ (p ) p (p ) lim sup lim sup (p ) + s(τ) sinp ψ() p sin p ϕ(τ) p dτ log + s(τ) L ψ() ϕ(τ) dτ = 0. log (3.0) Finally, he saemen of he lemma direcly comes from he combinaion of (3.6), (3.9), and (3.0). The coninuiy of Ψ is obvious. 4. Oscillaion consan A firs, we recall known resuls concerning he sudied equaions wih consan coefficiens. Theorem 4.. If A, B R + saisfy B/A > q p, hen he equaion is oscillaory. A p Φ(x ) + B log p Φ(x) = 0 Proof. See, e.g., 5, Theorem.4.4 (or direcly 6 and 7). Theorem 4.2. If C, D R + saisfy D/C < q p, hen he equaion is non-oscillaory. C p Φ(x ) + Proof. See 5, Theorem.4.4 (or 6, 7). D log p Φ(x) = 0 Now we can prove he announced resuls which idenify he criical oscillaion consan for he analyzed equaions wih very general coefficiens. Theorem 4.3. Le, R, S R + be such ha (3.2) is valid and R p S > q p. If here exiss T > e wih he propery ha + hen (2.6) is oscillaory. r q (τ) dτ R, + s(τ) dτ S, R T, Proof. We sudy he equaion for he Prüfer angle corresponding o he average funcion ψ. I is well-known ha he non-oscillaion of soluions of (2.6) is equivalen o he boundedness from above of he Prüfer angle ϕ given by (2.7). We refer, e.g., o 3, 4, 9, 24. I also suffices o consider direcly (2.6) and (2.7) when sin p ϕ() = 0. Based on Lemma 3., we know ha he boundedness (from above) of ϕ is equivalen o he boundedness (from above) of ψ. Hence, we will show ha ψ is unbounded from above. A firs, we assume ha (3.) is rue. Taking ino accoun Lemma 3.2, we have ψ () = cosp ψ() p + r q (τ) dτ Φ(cos p ψ()) sin p ψ() log

9 EJDE-205/220 OSCILLATION FOR MODIFIED EULER TYPE EQUATIONS 9 + sin p ψ() p (p ) log + s(τ) dτ + Ψ() R cos p ψ() p Φ(cos p ψ()) sin p ψ() + S sin p ψ() p p + Ψ() for all R T and for some coninuous funcion Ψ : R T R saisfying lim Ψ() = 0. (4.) Le ε > 0 be arbirary. From (2.5) and (4.), we have ε ( cos p x p + sin p x p ) > Ψ() (4.2) p for all x R and for all large R T. Thus, ψ () > (R ε) cos p ψ() p Φ(cos p ψ()) sin p ψ() log + (S ε) sin p ψ() p (4.3) p for all large R T. Le ε be so small ha (R ε) p (S ε) > q p and R ε > 0. Using Theorem 4. for A = (R ε) q and B = S ε, we know ha he equaion (R ε) q p Φ(x ) + S ε log p Φ(x) = 0 is oscillaory; i.e., any soluion ˆϕ : R T R of he equaion ˆϕ () = (R ε) cos p ˆϕ() p Φ(cos p ˆϕ()) sin p ˆϕ() log + (S ε) sin p ˆϕ() p (4.4) p has he propery ha lim sup ˆϕ() =. Indeed, one can simply compue ha B/A = (R ε) p (S ε) > q p. Considering (4.3) wih (4.4) and he 2π p -periodiciy of sin p and cos p, we see ha lim sup ˆϕ() = implies lim sup ψ() =. To finish he proof, we have o consider he case when (3.) is no valid. Evidenly, here exiss a coninuous funcion r : R e R + wih he properies and r q () r q (), lim + + We acually know ha he equaion r() p Φ(x ) + r q (τ) dτ R, R T, r q (τ) dτ log = 0. (4.5) s() log p Φ(x) = 0 is oscillaory (cf. (3.) and (4.5)). Since r() r() for all R T, he Surmian half-linear comparison heorem (see, e.g., 5, Theorem.2.4) gives he oscillaion of (2.6).

10 0 P. HASIL, M. VESELÝ EJDE-205/220 Theorem 4.4. Le, R, S R + be such ha (3.2) is valid and R p S < q p. If here exiss T > e wih he propery ha + r q (τ) dτ R, hen (2.6) is non-oscillaory. + s(τ) dτ S, R T, (4.6) Proof. We proceed analogously as in he proof of Theorem 4.3. In his proof, we show ha ψ is bounded from above. Noe ha (3.) is valid (see he firs inequaliy in (4.6)). From Lemma 3.2, we obain ψ () log R cos p ψ() p Φ(cos p ψ()) sin p ψ() + S sin p ψ() p p + Ψ() for any R T and for a coninuous funcion Ψ : R T R saisfying (4.). In addiion, using (4.2), we obain ψ () < (R + ε) cos p ψ() p Φ(cos p ψ()) sin p ψ() log + (S + ε) sin p ψ() p (4.7) p for any ε > 0 and all sufficienly large R T. We choose ε in such a way ha (R + ε) p (S + ε) < q p. We pu C = (R + ε) q and D = S + ε in Theorem 4.2. Since D/C = (R + ε) p (S + ε) < q p, we know ha he equaion (R + ε) q p Φ(x ) + S + ε log p Φ(x) = 0 is non-oscillaory. This fac means ha any soluion ˇϕ : R T R of he equaion ˇϕ () = (R + ε) cos p ˇϕ() p Φ(cos p ˇϕ()) sin p ˇϕ() log + (S + ε) sin p ˇϕ() p (4.8) p has he propery ha lim sup ˇϕ() <. Finally, considering (4.7) ogeher wih (4.8) and considering he 2π p -periodiciy of he generalized rigonomeric funcions, we have he inequaliy lim sup ψ() <. Therefore, he saemen of he heorem is proven. Now we menion definiions which enable us o formulae he below given Corollaries 4.8 and 4.0 (moivaed by he resuls of 3, 27 recalled in Inroducion). Definiion 4.5. Le f : R e R be a coninuous funcion. The mean value M(f) of f is defined as a+ M(f) := lim f(τ) dτ a if he limi is finie and if i exiss uniformly wih respec o a R e. Definiion 4.6. A coninuous funcion f : R R is called almos periodic if, for all ε > 0, here exiss l(ε) > 0 such ha any inerval of lengh l(ε) of he real line conains a leas one poin s for which f( + s) f() < ε, R.

11 EJDE-205/220 OSCILLATION FOR MODIFIED EULER TYPE EQUATIONS As a direc generalizaion of he almos periodiciy, we consider he noion of he so-called asympoic almos periodiciy. Definiion 4.7. We say ha a coninuous funcion f : R e R is asympoically almos periodic if f can be expressed in he form f() = f ()+f 2 (), R e, where f is almos periodic and f 2 has he propery ha lim f 2 () = 0. Concerning coefficiens wih mean values, we obain a new resul which reads as follows. Corollary 4.8. Le coninuous funcions r : R e R + and s : R e R be such ha he mean values M(r q ) R +, M(s) R exis and le (3.2) be valid for some R +. Le M(r q ) p M(s) q p. Then, (2.6) is oscillaory if and only if M(r q ) p M(s) > q p. Proof. The corollary follows from Theorems 4.3 and 4.4. Le ε > 0 be arbirary. The exisence of M(r q ) and M(s) implies he exisence of n N such ha n +n r q (τ) dτ M(r q ) < ε, n +n s(τ) dτ M(s) < ε for all R e. If M(r q ) p M(s) > q p, hen i suffices o choose ε so ha M(r q ) ε > 0, M(r q ) ε p M(s) ε > q p, o pu R = M(r q ) ε and S = M(s) ε, and o replace by n in Theorem 4.3. Obviously, (3.2) is rue also for n. If M(r q ) p M(s) < q p, hen we choose ε so ha M(s) + ε > 0, M(r q ) + ε p M(s) + ε < q p, we consider R = M(r q ) + ε and S = M(s) + ε, and we replace by n in Theorem 4.4. Remark 4.9. We poin ou ha he requiremen abou he validiy of (3.2) for some R + canno be omied in he saemen of Corollary 4.8. Indeed, he exisence of M(s) does no imply (3.2). We remark ha, from he same reason, Theorem. does no follow from Theorem.2. For asympoically almos periodic coefficiens, we ge a new resul as well. Again, we formulae i explicily. Corollary 4.0. Le funcions r q : R e R + and s : R e R be asympoically almos periodic and le M(r q ) p M(s) R + \{q p }. Then, (2.6) is oscillaory if and only if M(r q ) p M(s) > q p. Proof. Since any asympoically almos periodic funcion has mean value and i is bounded (see, e.g., 2, 9), his corollary is a consequence of Corollary 4.8. Remark 4.. Le us pay our aenion o (.3). We repea ha he resul abou (.3), which corresponds o Corollary 4.8, is proven in 27 and ha he one, which corresponds o Corollary 4.0, is proven in 3. Remark 4.2. In Corollaries 4.8 and 4.0, he case M(r q ) p M(s) = q p canno be solved as oscillaory or non-oscillaory for general coefficiens (which have mean values or which are asympoically almos periodic). We conjecure ha his case is no possible o solve even for general almos periodic coefficiens. Our conjecure is based on consrucions of almos periodic funcions menioned in 22 (see also 2, 23).

12 2 P. HASIL, M. VESELÝ EJDE-205/220 A he end, we give some examples o illusrae he proven resuls. Example 4.3. Le us consider consans u > 0, v R, and w 0 and he funcion h : R R given by he formula v + nw2 n ( n), n, n + 2 ), n N; n h() := v + nw2 n (n ), n + n 2, n + 2 n 2 ), n N; n v, n + 2 2, n + ), n N. n We analyze he equaion (x ) 3 Hence, we deal wih (2.6), where p = 4 and One can verify ha u + h() log 4 x3 = 0. (4.9) r() = u, s() = h(), R e. M(h) = v, M(r q ) = M(r 3 ) = M(u 3 ) = u 3 > 0. Therefore, applying Corollary 4.8 (condiion (3.2) is rivially valid for all R + ), we know ha (4.9) is oscillaory when 4 4 uv > 3 4 and non-oscillaory when 4 4 uv < 3 4. Noe ha he second coefficien h has mean value, bu i is no asympoically almos periodic (i suffices o consider ha lim sup h() = ). Example 4.4. For a, b, c > 0, we consider he equaion ( 2 x ) 5 sin(a) + arcansin(b) + cos(b) sin cos c( + )log 6 x 5 = 0 (4.0) which is in he form of (2.6) wih ( ) 5, r() = Re, + + sin cos ( sin(a) + arcansin(b) + cos(b)) s() = c( +, R e, ) and p = 6. Since r q and s are asympoically almos periodic funcions (see, e.g., 2, 9), we can use direcly Corollary 4.0. We have ( ) M(r q ) = M(r /5 ) = M + + sin cos =, cm(s) = M( sin(a) + arcansin(b) + cos(b)) = M( sin(a) ) = 2 π. Therefore, (4.0) is oscillaory for ( 6 ) 6 2 c < Γ := 5 π and non-oscillaory for c > Γ. Example 4.5. Le p = 3/2 and le f, g :, R + be coninuous funcions. We find he oscillaion consan for he equaion f(sin ) Φ(x ) + g(sin ) log p Φ(x) = 0. (4.)

13 EJDE-205/220 OSCILLATION FOR MODIFIED EULER TYPE EQUATIONS 3 Evidenly, he funcions r() = f(sin ) and s() = g(sin ) are periodic and M(r 2 ) = 2π π π π d f(sin ), M(s) = 2π We can apply, e.g., Corollary 4.0. If Γ := π g(sin ) d 2π 2π π π π π g(sin ) d. d f(sin ) > 3 3/2, hen (4.) is oscillaory. If Γ < 3 3/2, hen (4.) is non-oscillaory. Acknowledgemens. The firs auhor is suppored by Gran P20/0/032 of he Czech Science Foundaion. The second auhor is suppored by he projec Employmen of Bes Young Scieniss for Inernaional Cooperaion Empowermen (CZ..07/2.3.00/ ) co-financed from European Social Fund and he sae budge of he Czech Republic. References R. P. Agarwal, A. R. Grace, D. O Regan; Oscillaion heory for second order linear, halflinear, superlinear and sublinear dynamic equaions. Kluwer Academic Publishers, Dordrech, C. Corduneanu; Almos periodic oscillaions and waves. Springer, New York, O. Došlý, H. Funková; Euler ype half-linear differenial equaion wih periodic coefficiens. Absrac Appl. Anal. 203 (203), ar. ID 74263, pp O. Došlý, P. Hasil; Criical oscillaion consan for half-linear differenial equaions wih periodic coefficiens. Ann. Ma. Pura Appl. 90 (20), no. 3, pp O. Došlý, P. Řehák; Half-linear differenial equaions. Elsevier, Amserdam, Á. Elber; Asympoic behaviour of auonomous half-linear differenial sysems on he plane. Sudia Sci. Mah. Hungar. 9 (984), no. 2-4, pp Á. Elber; Oscillaion and nonoscillaion heorems for some nonlinear ordinary differenial equaions. Ordinary and parial differenial equaions (Dundee, 982), pp , Lecure Noes in Mah., vol. 964, Springer, Berlin, Á. Elber, A. Schneider; Perurbaions of half-linear Euler differenial equaion. Resuls Mah. 37 (2000), no. -2, pp A. M. Fink; Almos periodic differenial equaions. Springer-Verlag, Berlin, F. Geszesy, M. Ünal; Perurbaive oscillaion crieria and Hardy-ype inequaliies. Mah. Nachr. 89 (998), pp P. Hasil; Condiional oscillaion of half-linear differenial equaions wih periodic coefficiens. Arch. Mah. 44 (2008), no. 2, pp P. Hasil, M. Veselý; Criical oscillaion consan for difference equaions wih almos periodic coefficiens. Absrac Appl. Anal. 202 (202), ar. ID 47435, pp P. Hasil, M. Veselý; Oscillaion of half-linear differenial equaions wih asympoically almos periodic coefficiens. Adv. Differ. Equ. 203 (203), no. 22, pp P. Hasil, J. Víovec; Condiional oscillaion of half-linear Euler-ype dynamic equaions on ime scales. Elecron. J. Qual. Theory Differ. Equ. 205 (205), no. 6, pp J. Jaroš, M. Veselý; Condiional oscillaion of Euler ype half-linear differenial equaions wih unbounded coefficiens. Submied o Sudia Sci. Mah. Hungar. 6 H. Krüger, G. Teschl; Effecive Prüfer angles and relaive oscillaion crieria. J. Differ. Equ. 245 (2008), no. 2, pp H. Krüger, G. Teschl; Relaive oscillaion heory for Surm Liouville operaors exended. J. Func. Anal. 254 (2008), no. 6, pp H. Krüger, G. Teschl; Relaive oscillaion heory, weighed zeros of he Wronskian, and he specral shif funcion. Commun Mah. Phys. 287 (2009), no. 2, pp K. M. Schmid; Criical coupling consan and eigenvalue asympoics of perurbed periodic Surm-Liouville operaors. Commun Mah. Phys. 2 (2000), pp

14 4 P. HASIL, M. VESELÝ EJDE-205/ K. M. Schmid; Oscillaion of perurbed Hill equaion and lower specrum of radially periodic Schrödinger operaors in he plane. Proc. Amer. Mah. Soc. 27 (999), pp M. Veselý; Almos periodic homogeneous linear difference sysems wihou almos periodic soluions. J. Differ. Equ. Appl. 8 (202), no. 0, pp M. Veselý; Consrucion of almos periodic funcions wih given properies. Elecron. J. Differ. Equ. 20 (20), no. 29, pp M. Veselý; Consrucion of almos periodic sequences wih given properies. Elecron. J. Differ. Equ (2008), no. 26, pp M. Veselý, P. Hasil; Condiional oscillaion of Riemann Weber half-linear differenial equaions wih asympoically almos periodic coefficiens. Sudia Sci. Mah. Hungar. 5 (204), no. 3, pp M. Veselý, P. Hasil; Non-oscillaion of half-linear differenial equaions wih periodic coefficiens. Elecron. J. Qual. Theory Differ. Equ. 205 (205), no., pp M. Veselý, P. Hasil; Oscillaion and non-oscillaion of asympoically almos periodic halflinear difference equaions. Absrac Appl. Anal. 203 (203), ar. ID , pp M. Veselý, P. Hasil, R. Mařík; Condiional oscillaion of half-linear differenial equaions wih coefficiens having mean values. Absrac Appl. Anal. 204 (204), ar. ID 25859, pp J. Víovec; Criical oscillaion consan for Euler-ype dynamic equaions on ime scales. Appl. Mah. Compu. 243 (204), pp Per Hasil Deparmen of Mahemaics and Saisics Faculy of Science, Masaryk Universiy Kolářská 2, CZ 6 37 Brno, Czech Republic address: hasil@mail.muni.cz Michal Veselý (corresponding auhor) Deparmen of Mahemaics and Saisics Faculy of Science, Masaryk Universiy Kolářská 2, CZ 6 37 Brno, Czech Republic address: michal.vesely@mail.muni.cz