The Asymptotic Behavior of Nonoscillatory Solutions of Some Nonlinear Dynamic Equations on Time Scales

Transcription

1 Advances in Dynamical Sysems and Applicaions. ISSN Volume 1 Number 1 (2006, pp c Research India Publicaions hp:// The Asympoic Behavior of Nonoscillaory Soluions of Some Nonlinear Dynamic Equaions on Time Scales Xiangyun Shi 1, Xueyong Zhou 1 and Weiping Yan 2 1 Deparmen of Mahemaics, Xinyang Normal Universiy, Xinyang , Henan, P.R. China Corresponding auhor: xiangyunshi@126.com 2 Deparmen of Mahemaics, Shanxi Universiy, Taiyuan , Shanxi, P.R. China Absrac In his paper, he asympoic behavior of nonoscillaory soluions of he nonlinear dynamic equaion on ime scales r(g(y ( ] f(, y( = 0, 0 is considered under he condiion 0 ( m1 = for m 1 0. r( Three sufficien and necessary condiions are obained, which include and improve M.R.S. Kulenović and Ć. Ljubović s recen resuls in he coninuous case 5] and provide some new resuls in he discree case, as well as oher more general siuaions. AMS subjec classificaion: 39A10. Keywords: Time scales, dynamic equaion, asympoic behavior, nonoscillaory soluions. Received December 15, 2005; Acceped March 12, 2006

2 104 Xiangyun Shi, Xueyong Zhou and Weiping Yan 1. Inroducion The sudy of dynamic equaions on ime scales is an area of mahemaics ha recenly has received a lo of aenion. I has been creaed in order o unify he sudy of differenial equaions and difference equaions, and we refer he reader o he paper 3] for a comprehensive reamen of he subjec. Much recen research has been given o he asympoic properies of soluions of differenial equaion r(g(y ( ] p(f(y( = 0, R. We refer he reader o he papers 4, 5]. Bu i s discree counerpar r(g( y( ] p(f(y( = 0, N has few resuls. In his paper, we consider he dynamic equaion r(g(y ( ] f(, y( = 0, 0 (1.1 under he condiions (A 1 r C 1 rd (0,, (0, ]. (A 2 f C(T R, R, f is coninuously increasing wih respec o he second variable and yf(, y > 0, for y 0. (A 3 g C 1 R, R], g is a sricly increasing differeniable funcion on R and yg(y > 0, for y 0. If T = R, f(, y( = p(y(, hen equaion (1.1 reduces o he differenial equaion in 5]. If T = N, he difference equaion r(g( y( ] f(, y( = 0, N is anoher special case of (1.1. The aim of his paper, on he one hand, is o revisi he proofs of all heorems in 5] which use he Knaser Tarski fixed-poin heorem under decreasing mapping, on he oher hand, is o exend he resuls ha we have obained o he discree case as well as o more general siuaions. The paper is organized as follows: In he nex secion we presen some basic definiions concerning he calculus on ime scales. In Secion 3, we give hree sufficien and necessary condiions for he asympoic behavior of every nonoscillaory soluion of (1.1. In he final secion, we also apply our resuls o discree sysems by wo examples.

3 Nonoscillaory Soluions of Some Nonlinear Dynamic Equaions Some Definiions on Time Scales A ime scale T is an arbirary nonempy closed subse of real numbers R. Assume ha T has he opology ha i inheris from he sandard opology on R. We define he forward and backward jump operaors σ, ρ : T T by σ( := inf{s T : s > } and ρ( := inf{s T : s < }. The poin T is called righ-scaered, righ-dense, lef-scaered, lef-dense if σ( >, σ( =, ρ( <, ρ( = holds, respecively. The se T κ is derived from he ime scale T as follows. If T has a lef-scaered maximum, hen T κ = T { }. Oherwise, T κ = T. For a, b T wih a b, define he closed inerval a, b] in T by a, b] = { T : a b}. Oher open, half-open inervals in T can be similarly defined. Definiion 2.1. If f : T R is a funcion and T κ, hen he -derivaive of f a he poin is defined o be he number f ( wih he propery ha for each ɛ > 0, here is a neighborhood U of such ha (f(σ( f(s f ((σ( s ɛ σ( s for all s U. The funcion f is called -differeniable on T if f ( exiss for all T κ. Definiion 2.2. If F = f holds on T κ, hen we define he inegral of f by s f(τ τ = F ( F (s, s, T κ. We refer o 1,2,6] for addiional deails concerning he calculus on ime scales. By a soluion of (1.1, we mean a nonrivial real valued funcion x saisfying equaion (1.1 for 0. A soluion x of (1.1 is said o be oscillaory if i is neiher evenually posiive nor negaive; oherwise, i is nonoscillaory. 3. Main Resuls and Proof The following resul provides useful informaion on he global asympoic behavior of nonoscillaory soluions of (1.1. Lemma 3.1. Le (A 1 (A 3 be saisfied. If ( (A 4 m1 r( = for m 1 0 0

4 106 Xiangyun Shi, Xueyong Zhou and Weiping Yan holds, hen every soluion y of (1.1 evenually saisfies eiher y( K 1 or y( ( K 2 + M1 s, where K 1, K 2 and M 1 are posiive consans. Furhermore, r(s every posiive (negaive nondecreasing (nonincreasing soluion ends o + (. Proof. Wihou loss of generaliy, we can assume y( > 0 evenually, ha is y( > 0 for 1 0. Equaion (1.1 implies r(g(y ( ] > 0 for 1. Then here exiss 2 > 1 such ha r(g(y ( has consan sign for 2. If r(g(y ( < 0 for 2, hen (A 1 and (A 3 imply y ( < 0 for 2, ha is, y( is an evenually posiive decreasing funcion, so here exis consans K 1 > 0 and 3 2 such ha y( K 1 for 3. If r(g(y ( > 0 for 2, hen y ( > 0 for 2. Now, r(g(y ( is an evenually posiive increasing funcion, so here exis consans M 1 ( > 0, and 2 such ha r(g(y ( M 1 holds for. Then y ( M1 holds for. Inegraing his inequaliy from r( o > we obain The proof is complee. y( y( + ( M1 s, y( = K 2 > 0. r(s Now we are ready o presen he main resuls of his paper. Theorem 3.2. Assume ha (A 1 (A 4 hold. Then every, in absolue value, nondecreasing soluion y of (1.1 saisfies lim y( <, if and only if here exiss a consan C 0 such ha ( 1 f(s, C s <. (3.1 0 r( 0 Proof. Wihou loss of generaliy, we can suppose ha y( > 0 evenually. Firs assume ha (3.1 holds. Then 0 and C > 0 can be chosen such ha ( 1 f(s, C s C r( 2. If we can prove ha y is he soluion of he dynamic equaion y( = C ( 1 s 2 + f(u, y(u u s, r(s hen we can see ha y is he desired soluion of (1.1. Now we consruc he sequence {x m }: x 0 ( = C 2, x m ( = C ( r(s f(u, x m 1 (u u s, m = 1, 2. s

5 Nonoscillaory Soluions of Some Nonlinear Dynamic Equaions 107 Thus C 2 x 0 x 1 x 2 x m 1 x m C. Then he limi of he sequence {x m } exiss. We denoe i by x, i.e., Furher, lim x m( = x (. m f(s, x m f(s, C, ( 1 f(s, x m s C r( 2 hold. By Lebesgue s dominaed convergence heorem, we have lim x m( = C ( 1 s m 2 + f(u, lim r(s x m 1(u u s. T m 0 Tha is, x ( = C ( r(s f(u, x (u u s. So x saisfies (1.1. Furhermore, he limi of x ( as exiss. Conversely, le y > 0 be a bounded nondecreasing soluion of (1.1. Then s lim y( = m (0,. Le 0 be such ha y( m 2 for. Inegraing (1.1 from o we have which yields r( g(y ( = r( g(y ( + f(s, y(s s ( s, m s, 2 f f(s, y(s s ( 1 ( y ( f s, m s. r( 2 Inegraing his inequaliy from o >, and leing, we obain ( 1 ( s, m s <. r( 2 f The proof is complee.

6 108 Xiangyun Shi, Xueyong Zhou and Weiping Yan The nex resul gives a characerizaion of anoher ype of asympoic soluion. Theorem 3.3. If (A 1 (A 4 are saisfied, g(y = y α, for every y 0, α = p q (p, q are boh odd (3.2 and α > 1, hen equaion (1.1 has an, in absolue value, nondecreasing soluion y, such ha y( MR(, 0, for some consan M > 1, if and only if f(, ±mr(, 0 < (3.3 for some m > 1, where R(, 0 = 0 0 ( 1 s. r(s Proof. Wihou loss of generaliy, we assume y( > 0 evenually. Firs assume (3.3 holds. Then we can find m 1 > 0, 0 such ha f(, mr(, m 1, and (1 + m 1 1/α m. If we can prove ha y is he soluion of he dynamic equaion ( 1 y( = r(s + 1 s f(u, y(u u s, r(s hen we can see ha he funcion y is a nondecreasing soluion of (1.1. Now we consruc he sequence {x m }: Then x 0 = R(,, ( 1 x m = r(s + 1 r(s f(u, x m 1 (u u s. s R(, x 0 x 1 x m 1 x m mr(,. Obviously, he limi of he sequence {x m } exiss. We denoe i by x. Applying Lebesgue s dominaed convergence heorem, we have ( 1 x ( = r(s + 1 s f(u, x (u u s. r(s Then y saisfies (3.1 and when, y( mr(, 0. Conversely, if y( > 0, 1 0, le y be a nondecreasing soluion wih he propery lim y( = MR(, 0. Then here exiss L > 0 such ha MR(, 0 y( LR(, 0, and r(g(y ( > 0.

7 Nonoscillaory Soluions of Some Nonlinear Dynamic Equaions 109 We will show ha lim r(g(y ( <. Obviously, he limi exiss because r(g(y ( is an evenually posiive and increasing funcion. If lim r(g(y ( =, hen ( here exiss such ha r(g(y ( > M for. This implies y ( M and y( M 1/α R(, 0, and consequenly, r( y( R(, 0 M 1/α > M, which is a conradicion. Therefore, he limi of r(g(y ( exiss, say m, and we have 2m r(g(y ( = r( g(y ( + f(s, LR(s, s 0 s, which shows ha (3.3 is saisfied. The proof is complee. f(s, y(s s Remark 3.4. If T = R, hen y( = O(R(, 0, ha is, y( and R(, 0 are infiniy of he same order. Nex we characerize bounded soluions under he condiion (A 4. Theorem 3.5. Consider (1.1 under condiions (A 1 (A 4 and (3.2. Le y be an, in absolue value, nonincreasing soluion of (1.1. Then lim y( = K (0, if and only if here is a consan C 0 such ha ( 1 f(s, C s r( <. (3.4 0 Proof. Wihou loss of generaliy, we can assume y( > 0 evenually. If (3.4 holds, hen 0 and C > 0 can be chosen such ha ( 1 f(s, C s < C r( 2. Since g is an odd funcion, if y is he soluion of dynamic equaion y( = C ( f(u, C u s, r(s hen y is a nonincreasing soluion of (1.1. Now we consruc he sequence {x m (}: x 0 ( = C 2, x m ( = C 2 + ( 1 r(s s s f(u, x m 1 u s.

8 110 Xiangyun Shi, Xueyong Zhou and Weiping Yan Then C 2 = x 0( x 1 ( x m ( C. Clearly, he limi of he sequence {x m } exiss. We denoe i by x. Applying Lebesgue s dominaed convergence heorem, we have x ( = C ( f(u, x (u u s. r(s Then equaion (1.1 has an, in absolue value, nonincreasing soluion. Conversely, le y > 0 be a bounded nonincreasing soluion of (1.1 and lim y( = K (0,. Then y ( < 0 evenually and y( K for large enough. Equaion 2 (1.1 implies lim r(g(y ( = L (, 0. Now, we will show L = 0. Oherwise, L < 0 and so evenually ( ( L 1 r(g(y ( L y ( = L 1/α. r( r( Inegraing his relaion from o, we obain y( y( + L 1/α s ( 1 s, r(s which by (A 4 implies ha lim y( =. This is an immediae conradicion. Thus, L = 0. Inegraing (1.1 from o, we ge so ha r(g(y ( = g(y ( ( 1 r( f(s, y(s s Inegraing his inequaliy from o we obain f(s, K/2 s, f(s, K/2 s. y( K + ( 1 r(s Then (3.4 is saisfied. The proof is complee. s f(u, K 2 u s. In he presen paper, we will apply he resuls of he above heorems o wo discree cases. Example 3.6. If T = N, hen (1.1 reduces o he difference equaion r((g( y( ] f(, y( = 0, N. (3.5

9 Nonoscillaory Soluions of Some Nonlinear Dynamic Equaions 111 Le r( 1, g(u = u α = u α sgnu, f(u 1, u 2 = (u 1 +u 2 β = u 1 +u 2 β sgnu 2. (3.6 Now (3.5 can be wrien as ( y(n α (n + y(n β = 0, n N. (3.7 n 1 ] 1/α In his special case, (3.1 becomes (l + C β < for some C > 0. By n=n 0 l=n 0 Theorem 3.2, if β < α 1, (3.7 has a nondecreasing bounded soluion. In fac n 1 ] 1/α (l + C β (n n0 (n 1 + C β] 1/α (n 1 + C β+1 α, n=n 0 l=n 0 n=n 0 n=n 0 if β < α 1, he progression (n + C β+1 α is convergen, so he progression n=n 0 n 1 ] 1/α (l + C β is also convergen. l=n 0 n=n 0 Example 3.7. If T = hn, h > 0, and (3.6 hold, hen (1.1 can be wrien as (y ( α ] ( + y( β = 0, hn. (3.8 In his case, condiion (3.1 becomes h h n=n 0 /h n 1 l=n 0 /h (hl + C β 1/α <, where α = p/q (p and q are boh odd. By Theorem 3.2, if β < α 1, hen equaion (3.8 has a nondecreasing bounded posiive soluion. In fac, h n=n 0 /h h n 1 l=n 0 /h (hl + C β 1/α < h n=n 0 /h Thus, if β < α 1, he progression is convergen. References (h n 0 (hn h + C β ] 1/α. 1] Ravi Agarwal, Marin Bohner, Donal O Regan, and Allan Peerson. Dynamic equaions on ime scales: a survey. J. Compu. Appl. Mah., 141(1-2:1 26, 2002.

10 112 Xiangyun Shi, Xueyong Zhou and Weiping Yan 2] Marin Bohner and Allan Peerson. Dynamic equaions on ime scales. Birkhäuser Boson Inc., Boson, MA, ] Sefan Hilger. Analysis on measure chains a unified approach o coninuous and discree calculus. Resuls Mah., 18(1-2:18 56, ] M. R. S. Kulenović. Asympoic behavior of soluions of cerain differenial equaions and inequaliies of second order. Rad. Ma., 1(2: , ] M. R. S. Kulenović and Ć. Ljubović. The asympoic behavior of nonoscillaory soluions of some nonlinear differenial equaions. Nonlinear Anal., 42(5, Ser. A: Theory Mehods: , ] B. G. Zhang and Xinghua Deng. Oscillaion of delay differenial equaions on ime scales. Mah. Compu. Modelling, 36(11-13: , 2002.