Oscillation criteria for two-dimensional system of non-linear ordinary differential equations

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1 Elecronic Journal of Qualiaive Theory of Differenial Equaions 216, No. 52, 1 17; doi: /ejqde hp:// Oscillaion crieria for wo-dimensional sysem of non-linear ordinary differenial equaions Zdeněk Oplušil Brno Universiy of Technology, 2 Technická, Brno, Czech Republic Received 19 April 216, appeared 2 July 216 Communicaed by Ivan Kiguradze Absrac. New oscillaion crieria are esablished for he sysem of non-linear equaions u = g( v 1 sgn v, v = p( u sgn u, where >, g :, +, +, and p :, + R are locally inegrable funcions. Moreover, we assume ha he coefficien g is non-inegrable on, + ]. Among ohers, presened oscillaory crieria generalize well-known resuls of E. Hille and Z. Nehari and complemen analogy of Harman Winner heorem for he considered sysem. Keywords: wo dimensional sysem of non-linear differenial equaions, oscillaory properies. 21 Mahemaics Subjec Classificaion: 34C1. 1 Inroducion On he half-line R + =, +, we consider he wo-dimensional sysem of nonlinear ordinary differenial equaions u = g( v 1 sgn v, v = p( u sgn u, (1.1 where > and p, g : R + R are locally Lebesgue inegrable funcions such ha g( for a. e. (1.2 and g(sds = +. (1.3 By a soluion of sysem (1.1 on he inerval J, + we undersand a pair (u, v of funcions u, v : J R, which are absoluely coninuous on every compac inerval conained in J and saisfy equaliies (1.1 almos everywhere in J. oplusil@fme.vubr.cz

2 2 Z. Oplušil I was proved by Mirzov in 1] ha all non-exendable soluions of sysem (1.1 are defined on he whole inerval, +. Therefore, when we are speaking abou a soluion of sysem (1.1, we assume ha i is defined on, +. Definiion 1.1. A soluion (u, v of sysem (1.1 is called non-rivial if u( + v( = for. We say ha a non-rivial soluion (u, v of sysem (1.1 is oscillaory if is each componen has a sequence of zeros ending o infiniy, and non-oscillaory oherwise. In 1, Theorem 1.1], i is shown ha a cerain analogue of Surm s heorem holds for sysem (1.1 if he funcion g is nonnegaive. Especially, under assumpion (1.2, if sysem (1.1 has an oscillaory soluion, hen any oher is non-rivial soluion is also oscillaory. Definiion 1.2. We say ha sysem (1.1 is oscillaory if all is non-rivial soluions are oscillaory. Oscillaion heory for ordinary differenial equaions and heir sysems is a widely sudied and well-developed opic of he qualiaive heory of differenial equaions. As for he resuls which are closely relaed o hose of his secion, we should menion 2, 4 9, 11 13]. Some crieria esablished in hese papers for he second order linear differenial equaions or for wo-dimensional sysems of linear differenial equaions are generalized o he considered sysem (1.1 below. Many resuls (see, e.g., survey given in 2] have been obained in oscillaion heory of he so-called half-linear equaion ( r( u q 1 sgn u + p( u q 1 sgn u = (1.4 (alernaively his equaion is referred as equaion wih he scalar q-laplacian. Equaion (1.4 is usually considered under he assumpions q > 1, p, r :, + R are coninuous and r is posiive. One can see ha equaion (1.4 is a paricular case of sysem (1.1. Indeed, if he funcion u, wih properies u C 1 and r u q 1 sgn u C 1, is a soluion of equaion (1.4, hen he vecor funcion (u, r u q 1 sgn u is a soluion of sysem (1.1 wih g( := r 1 q 1 ( for and := q 1. Moreover, he equaion u + 1 p( u u 1 sgn u = (1.5 is also sudied in he exising lieraure under he assumpions ], 1] and p : R + R is a locally inegrable funcion. I is menioned in 6] ha if u is a so-called proper soluion of (1.5 hen i is also a soluion of sysem (1.1 wih g 1 and vice versa. Some oscillaions and non-oscillaions crieria for equaion (1.5 can be found, e.g., in 6, 7]. Finally, we menion he paper 1], where a cerain analogy of Harman Winner s heorem is esablished (origin one can find in 3, 14], which allows us o derive oscillaion crieria of Hille Nehari s ype for sysem (1.1. Le f ( := g(ds for. In view of assumpions (1.2 and (1.3, here exiss g such ha f ( > for > g and f ( g =. We can assume wihou loss of generaliy ha g =, since we are ineresed in behaviour of soluions in he neighbourhood of +, i.e., we have f ( > for > (1.6

3 Oscillaion crieria for wo-dimensional sysem of non-linear ODEs 3 and, moreover, For any λ,, we pu c (; λ := λ f λ ( lim f ( = +. (1.7 ( g(s s f λ +1 f λ (ξp(ξdξ ds for >. (s Now, we formulae an analogue (in a suiable form for us of he Harman Winner s heorem for he sysem (1.1 esablished in 1]. Theorem 1.3 (1, Corollary 2.5 (wih ν = 1 + λ]. Le condiions (1.2 and (1.3 hold, λ <, and eiher lim c (; λ = +, or < lim inf c (; λ < lim sup c (; λ. Then sysem (1.1 is oscillaory. One can see ha wo cases are no covered by Theorem 1.3, namely, he funcion c (; λ has a finie limi and lim inf c (; λ =. The aim of his Secion is o find oscillaion crieria for sysem (1.1 in he firs menioned case. Consequely, in wha follows, we assume ha lim c (; λ =: c (λ R. (1.8 2 Main resuls In his secion, we formulae main resuls and heirs corollaries. Theorem 2.1. Le λ, and (1.8 hold. Le, moreover, he inequaliy lim sup be saisfied. Then sysem (1.1 is oscillaory. f λ ( ( 1+ ln f ( (c (λ c (; λ > ( We inroduce he following noaions. For any λ, and µ ], +, we pu ( Q(;, λ := f λ ( c (λ p(s f λ (sds for >, ( 1 H(;, µ := f µ p(s f µ (sds for >, ( where he number c (λ is given by (1.8. Moreover, we denoe lower and upper limis of he funcions Q( ;, λ and H( ;, µ as follows Q (, λ := lim inf Q(;, λ, H Q (, λ := lim sup Q(;, λ, (, µ := lim inf H(;, µ, H (, µ := lim sup H(;, µ. Now we formulae wo corollaries of Theorem 2.1.

4 4 Z. Oplušil Corollary 2.2. Le λ,, µ ], +, and (1.8 hold. Le, moreover, Then sysem (1.1 is oscillaory. µ lim inf (Q(;, λ + H(;, µ > λ ( λ(µ ( (2.2 Corollary 2.3. λ,, µ ], +, and (1.8 hold. Le, moreover, eiher Q (, λ > 1 ( 1+, (2.3 λ 1 + or Then sysem (1.1 is oscillaory. H (, µ > 1 ( 1+. (2.4 µ 1 + Remark 2.4. Oscillaion crieria (2.3 and (2.4 coincide wih he well-known Hille Nehari s resuls for he second order linear differenial equaions esablished in 4, 12]. Theorem 2.5. Le λ,, µ ], +, and (1.8 hold. Le, moreover, lim sup (Q(;, λ + H(;, µ > 1 ( λ ( µ 1+. (2.5 λ 1 + µ 1 + Then sysem (1.1 is oscillaory. Now we give wo saemens complemening Corollary 2.3 in a cerain sense. Theorem 2.6. Le λ,, µ ], +, and (1.8 hold. Le, moreover, inequaliies ( γ γ 1+ Q (, λ 1 ( +1 (2.6 λ λ + 1 and be saisfied, where H (, µ > 1 ( µ 1+ γ A(, λ (2.7 µ 1 + γ := and A(, λ is he smalles roo of he equaion Then sysem (1.1 is oscillaory. ( λ ( x + γ 1+ x + ( λq (, λ γ =. (2.9 Theorem 2.7. Le λ,, µ ], +, and (1.8 hold. Le, moreover, inequaliies ( µ (1 + µ 1 + (µ (1 + H (, µ 1 ( 1+ (2.1 µ 1 + and Q (, λ > B(, µ + 1 ( λ 1+ (2.11 λ 1 + be saisfied, where B(, µ is he greaes roo of he equaion Then sysem (1.1 is oscillaory. x 1+ x + (µ H (, µ =. (2.12

5 Oscillaion crieria for wo-dimensional sysem of non-linear ODEs 5 Finally, we formulae an asserion for he case, when boh condiions (2.6 and (2.1 are fulfilled. In his case we can obain beer resuls han in Theorems 2.6 and 2.7. Theorem 2.8. Le λ,, µ ], +, and (1.8 hold. Le, moreover, condiions (2.6 and (2.1 be saisfied and lim sup (Q(;, λ + H(;, µ > B(, µ A(, λ + Q (, λ + H (, µ γ, (2.13 where he number γ is defined by (2.8, A(, λ is he smalles roo of equaion (2.9, and B(, µ is he greaes roo of equaion (2.12. Then sysem (1.1 is oscillaory. Remark 2.9. Presened saemens generalize resuls saed in 2,4 9,11 13] concerning sysem (1.1 as well as equaions (1.4 and (1.5. In paricular, if we pu = 1, λ =, and µ = 2, hen we obain oscillaory crieria for linear sysem of differenial equaions presened in 13]. Moreover, he resuls of 6] obained for equaion (1.5 are in a compliance wih hose above, where we pu g 1, λ =, and µ = 1 +. Observe also ha Corollary 2.3 and Theorems 2.6 and 2.7 exend oscillaion crieria for equaion (1.5 saed in 7], where he coefficien p is suppose o be non-negaive. In he monograph 2], i is noed ha he assumpion p( for large enough can be easily relaxed o p(sds > for large. I is worh menioning here ha we do no require any assumpion of his kind. Finally we show an example, where we can no apply oscillaory crieria from he above menioned papers, bu we can use Theorem 2.1 succesfully. Example 2.1. Le = 2, g( 1, λ =, and ( 2 p( := cos 2 I is clear ha he funcion p and is inegral ( 2 p(sds = sin ( for. 1 2( for change heirs sign in any neighbourhood of +. Therefore neiher of resuls menioned in Remark 2.9 can be applied. On he oher hand, we have c 2 (; = 2 ( s 2 s (ξ cos ξ (ξ dξ ds = cos ln( ( + 1 and hus, he funcion c 2 (, has he finie limi Moreover, lim sup c ( = lim c 2(; = 1 2. ( 2 2 cos 2 ln (c 2 ( c 2 (; = lim sup 3 ln( ln ln Consequenly, according o Theorem 2.1, sysem (1.1 is oscillaory. for > 1 = 1. ( + 1 ln

6 6 Z. Oplušil 3 Auxiliary lemmas We firs formulae wo lemmas esablished in 1], which we use in his secion. Lemma 3.1 (1, Lemma 3.1]. Le > and ω. Then he inequaliy ωx x 1+ ( ω is saisfied for all x R. Lemma 3.2 (1, Lemma 3.2]. Le >. Then x + y 1+ y 1+ + (1 + x y 1 sgn y for x, y R. Remark 3.3. One can easily verify (see he proofs of Lemma 4.2 and Corollary 2.5 in 1] ha if (u, v is a soluion of sysem (1.1 saisfying wih > and he funcion c ( ; λ has a finie limi (1.8, hen u( = for (3.1 c u (λ = f λ ( ρ( + f λ (sp(sds + g(s f λ 1 (sh(sds, 1+ (γ γ λ 1 f λ ( (3.2 where he number γ is defined by (2.8, and h( := f (ρ( + γ 1+ (1 + f (ρ(γ 1 γ 1+ for, (3.3 ρ( := Moreover, according o Lemma 3.2, we have v( u( sgn u( 1 ( λ f for. (3.4 ( 1 + h( for (3.5 and one can show (see Lemma 4.1 and he proof of Corollary 2.5 in 1] ha g(s f λ 1 (sh(sds < +. (3.6 Lemma 3.4. Le λ,, (1.8 and (2.6 hold, where he number γ is defined by (2.8. Then every non-oscillaory soluion (u, v of sysem (1.1 saisfies ( f (v( lim inf u( sgn u( γ where A(, λ denoes he smalles roo of equaion (2.9. A(, λ, (3.7

7 Oscillaion crieria for wo-dimensional sysem of non-linear ODEs 7 Proof. Le (u, v be a non-oscillaory soluion of sysem (1.1. Then here exiss > such ha (3.1 holds. Define he funcion ρ by (3.4. Then we obain from (1.1 ha ρ ( = p( g( ρ( + γ f ( 1+ + γ g( f 1+ ( for a. e.. (3.8 Muliplaying he las equaliy by f λ ( and inegraing i from o, we ge f λ (sρ (sds = g(s f λ 1 (s ρ(s f (s + γ 1+ + γ g(s f λ 1 (sds ds f λ (sp(sds for. (3.9 Inegraing he lef-hand side of (3.9 by pars, we obain f λ (ρ( = ( γ γ 1+ + f λ ( ρ( g(s f λ 1 (sds where he funcion h is defined in (3.3. Hence, f λ (sp(sds g(s f λ 1 (sh(sds for, f λ (ρ( = δ( ( γ γ 1+ λ f λ (sp(sds 1 f λ ( g(s f λ 1 (sh(sds for, (3.1 where ( u δ( := f λ ( ρ( + f λ (sp(sds + γ γ 1+ λ Therefore, in view of relaions (3.2 and (3.6, i follows from (3.1 ha 1 f λ (. f λ (ρ( = c (λ ( γ γ 1+ λ f λ (sp(sds + 1 f λ ( g(s f λ 1 (sh(sds for. (3.11 Hence, Pu f (ρ( = Q(;, λ + f λ ( g(s f λ 1 (sh(sds ( γ γ 1+ λ for. (3.12 m := lim inf f (ρ(. (3.13 I is clear ha if m = +, hen (3.7 holds. Therefore, we suppose ha m < +.

8 8 Z. Oplušil In view of (2.6, (3.5, and (3.13, relaion (3.12 yields ha If Q (, λ = λ m Q (, λ ( γ γ 1+. (3.14 λ 1+ (γ γ, hen is a roo of equaion (2.9. Moreover, in view of Lemma 3.2 and he assumpion λ <, we see ha he funcion x x + γ 1+ x γ 1+ is posiive on ],. Consequenly, by virue of noaions (3.4, (3.13 and relaion (3.14, desired esimae (3.7 holds. Now suppose ha Q (, λ > λ arbirary. According o (3.14, i is clear ha Choose ε such ha Then i follows from (3.12 ha (γ γ 1+. Le ε ], Q (, λ λ 1+ (γ γ be m > ε. (3.15 f (ρ( m ε and Q(;, λ Q (, λ ε for ε. (3.16 f (ρ( Q (, λ ε + f λ ( g(s f λ 1 (sh(sds ( γ γ 1+ for ε. λ (3.17 On he oher hand, he funcion x x + γ 1+ (1 + xγ 1 γ 1+ is non-decreasing on, +. Therefore, by virue of (3.5, (3.15, and (3.16, one ges from (3.17 ha f (ρ( Q (, λ ε + (m ε + γ 1+ γ λ(m ε λ for ε, which implies 1+ (m ε + γ γ λ(m ε m Q (, λ ε +. λ Since ε was arbirary, he laer relaion leads o he inequaliy m + γ 1+ m + Q (, λ( λ γ. (3.18 x + Q (, λ( λ γ is decreasing on ], ( 1+ γ] and increasing on ( 1+ γ, +. Therefore, in view of One can easily derive ha he funcion y : x x + γ 1+ assumpion (2.6, he funcion y is non-posiive a he poin ( 1+ γ, which ogeher wih (3.4, (3.13, and (3.18 implies desired esimae (3.7. Lemma 3.5. Le µ ], + and (2.1 hold. Then every non-oscillaory soluion (u, v of sysem (1.1 saisfies f (v( lim sup u( B(, µ, (3.19 sgn u( where B(, µ is he greaes roo of equaion (2.12.

9 Oscillaion crieria for wo-dimensional sysem of non-linear ODEs 9 Proof. Le (u, v be a non-oscillaory soluion of sysem (1.1. Then here exiss > such ha (3.1 holds. Define he funcion ρ by (3.4. Then from (1.1 we obain he equaliy (3.8, where he number γ is defined by (2.8. Muliplying (3.8 by f µ ( and inegraing i from o, we obain f µ (sρ (sds = + γ f µ (sp(sds g(s f µ 1 (sds for. g(s f µ 1 (s ρ(s f (s + γ 1+ ds Inegraing he lef-hand side of he las equaliy by pars, we ge f (ρ( = f µ ( g(s f µ 1 (s + δ( f µ ( H(;, µ + γ µ µ f (sρ(s ρ(s f (s + γ 1+ for, ] ds (3.2 where u δ( := f µ ( ρ( + f µ (sp(sds γ µ f µ (. (3.21 According o Lemma 3.1, i follows from (3.2 ha f (ρ( δ 1 ( f µ ( H(;, µ + 1 ( µ 1+ γ for, (3.22 µ 1 + where δ 1 ( := δ( f µ ( µ ( ( µ 1+ µγ. ( Pu M := lim sup ( f (ρ( + γ. (3.24 Obviously, if M = hen (3.19 holds. Therefore, suppose ha By virue of (1.7, inequaliy (3.22 yields M >. M H (, µ + 1 ( µ 1+. (3.25 µ 1 + If H (, µ = ( µ (1+ µ µ 1+, hen i is no difficul o verify ha ( (µ (1+ 1+ is a roo of he equaion (2.12 and he funcion x x 1+ x + (µ H (, µ is posiive on ]( µ 1+, +. Consequenly, i follows from (3.24 and (3.25 ha (3.19 is saisfied. Now suppose ha H (, µ > Using he laer inequaliy in (3.25, we ge ( µ (1 + µ 1 + (µ (1 +. M < ( µ. 1 +

10 1 Z. Oplušil Le ε ], ( µ 1+ M be arbirary and choose ε such ha γ + f (ρ( M + ε, H(;, µ H (, µ ε for ε. (3.26 Observe ha he funcion x µx x 1+ is non-decreasing on ], ( µ 1+ ] and hus, using relaions (3.26 and M + ε < ( µ 1+, from (3.2 we ge f (ρ( δ 2 ( f µ ( H (, µ + ε + + f µ ( γ µ µγ µ g(s f µ 1 (s µ (M + ε M + ε 1+ ] ds for ε, where Consequenly, u δ 2 ( := f µ ( ρ( + f µ (sp(sds + γ f µ (. f (ρ( + γδ 3 ( f µ ( H (, µ + ε + 1+ µ (M + ε M + ε µ for ε, where δ 3 ( := δ 2 ( 1+ µ (M + ε M + ε µ f µ (, which, by virue of he assumpion < µ and condiion (1.7 and (3.24, yields ha M H (, µ + ε + Since ε was arbirary, he laer inequaliy leads o One can easily derive ha he funcion y : x x µ (M + ε M + ε µ M 1+ M + (µ H (, µ. (3.27 x + H (, µ(µ is decreasing on ], ( 1+ ] and increasing on ( 1+, +. Therefore, in view of assumpion (2.1, he funcion y is non-posiive a he poin ( 1+, which ogeher wih (3.4, (3.24, and (3.27 implies desired esimae ( Proofs of main resuls Proof of Theorem 2.1. Assume on he conrary ha sysem (1.1 is no oscillaory, i.e., here exiss a soluion (u, v of sysem (1.1 saisfying relaion (3.1 wih >. Analogously o he proof of Lemma 3.4 we show ha equaliy (3.11 holds, where he funcions h, ρ and he number γ are defined by (3.3, (3.4, and (2.8. Moreover, condiions (3.5 and (3.6 are saisfied.

11 Oscillaion crieria for wo-dimensional sysem of non-linear ODEs 11 Muliplying of (3.11 by g( f 1 λ ( and inegraing i from o, one ges Observe ha g(s f 1 (sρ(sds = c (λ ( g(s + f 1+λ (s = f λ ( λ f λ ( λ Hence, i follows from (4.1 ha f λ ( (c (λ c (; λ = + s g(s f 1+λ (s ds ( g(s s f 1+λ f λ (ξp(ξdξ ds (s ( g(s + f 1+λ g(ξ f λ 1 (ξh(ξdξ (s s ( γ γ 1+ λ g(ξ f λ 1 (ξh(ξdξ ds g(s f (s ds for, g(s f λ 1 (sh(sds + 1 g(s λ f (s h(sds g(s f λ 1 (sh(sds for. g(s f (s + f λ ( ( ( λ f (sρ(s h(s + c (λ c ( ; λ + γ γ 1+ ds ] ds ] g(s f λ 1 (sh(sds f λ ( g(s f λ 1 (sh(sds for. On he oher hand, according o (2.8, (3.3, and Lemma 3.1 wih ω :=, he esimae ( ( λ f (sρ(s h(s + γ γ 1+ = ( f (sρ(s + γ f (sρ(s + γ 1+ holds for s. Moreover, in view of (1.2, (1.6, and (3.5, i is clear ha f λ ( g(s f λ 1 (sh(sds for. ( Consequenly, by virue of he las inequaliy and (4.3, i follows from (4.2 ha (4.1 (4.2 (4.3 ( f λ ( c (λ c (; λ] f λ ( Hence, in view of (1.7, we ge 1+ ln f ( f ( c (λ c ( ; λ + ] g(s f λ 1 (sh(sds for. which conradics (2.1. lim sup f λ ( 1+ ( ln f ( c (λ c (; λ], 1 +

12 12 Z. Oplušil Proof of Corollary 2.2. Observe ha for >, we have f λ ( ln f ( (c (λ c (; λ = λ ln f ( g(s Q(s;, λds (4.4 f (s and Q(;, λ + H(;, µ = (µ λ f µ ( g(s f µ 1 (sq(s;, λds. (4.5 Moreover, i is easy o show ha g(s f (s Q(s;, λds = f µ ( g(s f µ 1 (sq(s;, λds ( s + (µ g(s f µ 1 (s g(ξ f µ 1 (ξq(ξ;, λdξ ds for >. (4.6 On he oher hand, by virue of (2.2, from relaion (4.5 one ges lim inf ( +1 f µ ( g(s f µ 1 1 (sq(s;, λds > + 1 ( λ(µ. Therefore, in view of relaion (1.7, i follows from (4.6 ha lim inf 1 ln f ( ( g(s +1 1 Q(s;, λds > f (s + 1 λ. (4.7 Now, equaliy (4.4 and inequaliy (4.7 guaranee he validiy of condiion (2.1 and hus, he asserion of he corollary follows from Theorem 2.1. Proof of Corollary 2.3. If assumpion (2.3 holds, hen i follows from (4.4 ha condiion (2.1 is saisfied and hus, he asserion of he corollary follows from Theorem 2.1. Le now assumpion (2.4 be fulfilled. Observe ha f (sp(sds = H(;, µ + (µ Therefore, in view of (2.4, we obain lim inf On he oher hand, i is clear ha Hence, we have c (; λ = ( λ2 g( f 1+ λ = g(s H(s;, µds for >. f (s 1 ( +1 f (sp(sds >. (4.8 ln f ( + 1 ( λg( + f ( ( λg( f λ+1 ( τ c (τ; λ c (; λ = ( λ ( s g(s f λ 1 (s f λ (ξp(ξdξ ds f λ (sp(sds f (sp(sds for >. ( g(s s f λ+1 f (ξp(ξdξ ds τ > (s

13 Oscillaion crieria for wo-dimensional sysem of non-linear ODEs 13 and consequenly, by virue of assumpion (1.8 and condiion (4.8, we ge ( c g(s ln f (s 1 s (λ c (; λ = ( λ f λ+1 f (ξp(ξdξ ds for >. (4.9 (s ln f (s In view of (4.8, here exis ε > and ε > such ha Hence, i follows from (4.9 ha 1 ( +1 f (sp(sds + ε for ε. ln f ( + 1 c (λ c (; λ ( λ ( ( +1 g(s ln f (s + ε + 1 f λ+1 (s for ε. Since ε >, by virue of (1.7, from he las relaion we derive inequaliy (2.1. Therefore, he asserion of he corollary follows from Theorem 2.1. Proof of Theorem 2.5. Assume on he conrary ha sysem (1.1 is no oscillaory, i.e., here exiss a soluion (u, v of sysem (1.1 saisfying relaion (3.1 wih >. Analogously o he proofs of Lemmas 3.4 and 3.5 we derive equaliies (3.11 and (3.2, where he numbers γ, δ( and he funcions h, ρ are given by (2.8, (3.21 and (3.3, (3.4. I follows from (3.11 and (3.2 ha Q(;, λ + H(;, µ = f λ ( + ( λ + f µ ( γ γ 1+ g(s f λ 1 (sh(sds + γ µ + δ( f µ ( g(s f µ 1 (s µ f (sρ(s ρ(s f (s + γ 1+ ] ds (4.1 is saisfied for. Moreover, according o Lemma 3.1 wih ω := µ, i is clear ha µ ( f (ρ( + γ ρ( f ( + γ 1+ ( µ 1+ for. ( Therefore, using (2.8, (3.5, and (4.11 in relaion (4.1, we ge Q(;, λ + H(;, µ 1 ( λ ( µ 1+ + δ( f µ ( λ 1 + µ 1 + for, (4.12 where δ( := δ( ( µ 1+ µγ] 1 + f µ ( µ. Consequenly, by virue of (1.7, relaion (4.12 leads o a conradicion wih assumpion (2.5.

14 14 Z. Oplušil Proof of Theorem 2.6. Suppose on he conrary ha sysem (1.1 is no oscillaory. Then here exiss a soluion (u, v of sysem (1.1 saisfying relaion (3.1 wih >. Analogously o he proof of Lemma 3.5 one can show ha relaion (3.22 holds, where he numbers γ, δ 1 ( and he funcion ρ are given by (2.8, (3.23, and (3.4. On he oher hand, according o Lemma 3.4, esimae (3.7 is fulfilled, where A(, λ is he smalles roo of equaion (2.9. Le ε > be arbirary. Then here exiss ε such ha Hence, i follows from (3.22 ha f (ρ( A(, λ ε for ε. H(;, µ δ 1 ( f µ ( A(, λ + ε + 1 ( µ 1+ γ for ε. µ 1 + Since ε was arbirary, in view of (1.7, from he laer inequaliy we ge which conradics assumpion (2.7. H (, µ 1 ( µ 1+ γ A(, λ, γ, µ 1 + Proof of Theorem 2.7. Assume on he conrary ha sysem (1.1 is no oscillaory, i.e., here exiss a soluion (u, v of sysem (1.1 saisfying relaion (3.1 wih >. Analogously o he proof of Lemma 3.4 we show ha equaliy (3.12 holds, where he number γ and he funcions h, ρ are defined by (2.8, (3.3, and (3.4. On he oher hand, according o Lemma 3.5, esimae (3.19 is fulfilled, where B(, µ is he greaes roo of equaion (2.12. Le ε > be arbirary. Then here exiss ε such ha f (ρ( + γ B(, µ + ε for ε. In view of he las inequaliy, (1.2, (1.6 and (3.5, i follows from (3.12 ha Q(;, λ B(, µ + ε γ + ( γ γ 1+ λ for ε. Since ε was arbirary, we ge which conradics (2.11. Q (, λ B(, µ + γ 1+ λ, Proof of Theorem 2.8. Suppose on he conrary ha sysem (1.1 is no oscillaory. Then here exiss a soluion (u, v of sysem (1.1 saisfying relaion (3.1 wih >. Pu m := A(, λ, M := B(, µ, (4.13 i.e., m denoes he smalles roo of equaion (2.9 and M is he greaes roo of equaion (2.12. According o Lemmas 3.4 and 3.5, we have lim inf f (ρ( m, lim sup ( f (ρ( + γ M, (4.14 where he funcion ρ and he number γ are defined in (3.4 and (2.8.

15 Oscillaion crieria for wo-dimensional sysem of non-linear ODEs 15 Analogously o he proof of Theorem 2.5 we show ha relaion (4.1 holds for, where he number δ( and he funcion h are defined by (3.21 and (3.3. In view of (2.6, one can easily show ha he funcion y : x x + γ 1+ x + Q (, λ( λ γ is posiive on ], and here exiss x, + such ha y( x, which yields ha m. On he oher hand, in view of (2.1, one can easily verify ha he funcion z : x x 1+ x + (µ H (, µ is posiive on ] ( µ ( 1+, + and here exiss x µ 1+ such ha z( x. Consequenly, we have M ( µ. 1+ We firs assume ha m > and M < ( µ. { ( 1+ Le ε µ } ], min m, 1+ M be arbirary. Then, by virue of (4.14, here exiss ε such ha f (ρ( m ε, f (ρ( + γ M + ε for ε. (4.15 The funcion x x + γ 1+ (1 + xγ 1 is non-decreasing on, +. Therefore, in view of (3.3 and (4.15, we ge f λ ( g(s f λ 1 (sh(sds 1+ m ε + γ λ(m ε γ 1+ λ (4.16 for ε. Moreover, he funcion x µx x 1+ is non-decreasing on ], ( µ 1+ and hus, in view of (4.15, we obain f µ ( g(s f µ 1 (s µ f (sρ(s ρ(s f (s + γ 1+ ] ds ε µ(m + ε M + ε 1+ µγ µ for ε. (4.17 Now i follows from (4.1, (4.16, and (4.17 ha Q(;, λ + H(;, µ M + ε + H (, µ (m ε + Q (, λ γ + (M + ε M + ε 1+ (µ H (, µ µ 1+ m ε + γ + δ( ε f µ ( for ε, (m ε + ( λq (, λ γ λ (4.18 where ε δ( ε := δ( + g(s f µ 1 (s µ f (sρ(s ρ(s f (s + γ 1+ Since ε was arbirary, in view of (1.7 and (4.13, inequaliy (4.18 yields ha ] ds. lim sup (Q(;, λ + H(;, µ B(, µ A(, λ, γ + Q (, λ + H (, µ γ, (4.19 which conradics assumpion (2.13. If m = hen, in view of (3.5, i is clear ha 1+ f λ ( g(s f λ 1 m + γ λm γ 1+ (sh(sds = λ (4.2

16 16 Z. Oplušil for. On he oher hand, if M = ( µ 1+ hen, using Lemma 3.1 wih ω := µ, one can show ha f µ ( g(s f µ 1 (s µ f (sρ(s ρ(s f (s + γ 1+ ] ds = ( µ µγ f µ ( µ f µ ( 1+ µm M µγ f µ ( µ f µ ( ( ( µ µγ µ ( ( µ µγ µ for. (4.21 Consequenly, if m = (resp. M = ( µ 1+, hen we derive from (4.1, he inequaliy (4.19 similarly as above, bu we use (4.2 insead of (4.16 (resp. (4.21 insead of (4.17. Acknowledgemens The published resuls were suppored by Gran No. FSI-S Modern mehods of applied mahemaics in engineering. References 1] M. Dosoudilová, A. Lomaidze, J. Šremr, Oscillaory properies of soluions o cerain wo-dimensional sysems of non-linear ordinary differenial equaions, Nonlinear Anal. 12(215, MR ] O. Došlý, P. Řehák, Half-linear differenial equaions, Norh-Holland Mahemaics Sudies, Vol. 22, Elsevier, Amserdam, 25. MR ] P. Harman, Ordinary differenial equaions, John Wiley & Sons, Inc., New York London Sydney, MR ] E. Hille, Non-oscillaion heorems, Trans. Amer. Mah. Soc. 64(1948, No. 2, MR ] T. Chanladze, N. Kandelaki, A. Lomaidze, Oscillaion and nonoscillaion crieria for a second order linear equaion, Georgian Mah. J. 6(1999, MR ] N. Kandelaki, A. Lomaidze, D. Ugulava, On oscillaion and nonoscillaion of a second order half-linear equaion, Georgian Mah. J. 7(2, No. 2, MR ] A. Lomaidze, Oscillaion and nonoscillaion of Emden Fowler ype equaion of second-order, Arch. Mah. (Brno 32(1996, No. 3, MR ] A. Lomaidze, Oscillaion and nonoscillaion crieria for second order linear differenial equaion, Georgian Mah. J. 4(1997, No. 2, MR ] A. Lomaidze, N. Parsvania, Oscillaion and nonoscillaion crieria for wodimensional sysems of firs order linear ordinary differenial equaions, Georgian Mah. J. 6(1999, No. 3, MR

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