OSCILLATION THEORY OF DYNAMIC EQUATIONS ON TIME SCALES

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1 Universiy of Nebraska - Lincoln DigialCommons@Universiy of Nebraska - Lincoln Disseraions, heses, and Suden Research Papers in Mahemaics Mahemaics, Deparmen of OSCILLAION HEORY OF DYNAMIC EQUAIONS ON IME SCALES Raegan J. Higgins Universiy of Nebraska a Lincoln, s-rhiggin4@mah.unl.edu Follow his and addiional works a: hp://digialcommons.unl.edu/mahsuden Par of he Science and Mahemaics Educaion Commons Higgins, Raegan J., "OSCILLAION HEORY OF DYNAMIC EQUAIONS ON IME SCALES" (2008). Disseraions, heses, and Suden Research Papers in Mahemaics. 3. hp://digialcommons.unl.edu/mahsuden/3 his Aricle is brough o you for free and open access by he Mahemaics, Deparmen of a DigialCommons@Universiy of Nebraska - Lincoln. I has been acceped for inclusion in Disseraions, heses, and Suden Research Papers in Mahemaics by an auhorized adminisraor of DigialCommons@Universiy of Nebraska - Lincoln.

2 OSCILLAION HEORY OF DYNAMIC EQUAIONS ON IME SCALES by Raegan J. Higgins A DISSERAION Presened o he Faculy of he Graduae College a he Universiy of Nebraska In Parial Fulfilmen of Requiremens For he Degree of Docor of Philosophy Major: Mahemaics Under he Supervision of Professors Lynn H. Erbe and Allan C. Peerson Lincoln, Nebraska May, 2008

3 OSCILLAION HEORY OF DYNAMIC EQUAIONS ON IME SCALES Raegan J. Higgins, Ph. D. Universiy of Nebraska, 2008 Advisers: Lynn H. Erbe and Allan C. Peerson In pas years mahemaical models of naural occurrences were eiher enirely coninuous or discree. hese models worked well for coninuous behavior such as populaion growh and biological phenomena, and for discree behavior such as applicaions of Newon s mehod and discreizaion of parial differenial equaions. However, hese models are deficien when he behavior is someimes coninuous and someimes discree. he exisence of boh coninuous and discree behavior creaed he need for a differen ype of model. his is he concep behind dynamic equaions on ime scales. For example, dynamic equaions can model insec populaions ha are coninuous while in season, die ou in, say, winer, while heir eggs are incubaing or dorman, and hen hach in a new season, giving rise o a nonoverlapping populaion. hroughou his work, we will be concerned wih cerain dynamic equaions on ime scales. We sar wih a brief inroducion o he ime scale calculus and some heory necessary for he new resuls. he main concern will hen be he oscillaory behavior of soluions o cerain second order dynamic equaions. In Chaper 3, an equaion of paricular ineres is one conaining boh advanced and delayed argumens. We will use he mehod of Riccai subsiuion o prove some oscillaion resuls of he soluions. In Chaper 4 we again sudy he oscillaory behavior of a second dynamic equaion. However, in his chaper, he equaion only has delayed argumens. In addiion o using Riccai subsiuion, we use he mehod of upper and lower soluions o develop necessary and sufficien condiions for oscillaory soluions. In he final chaper we are

4 ineresed in he exisence of nonoscillaory soluions of dynamic equaions on ime scales. he common heme among hese resuls is he use of he Riccai subsiuion echnique and he inegraion of dynamic inequaliies.

5 iv ACKNOWLEDGMENS I humbly ake his opporuniy o publicly hank all he individuals who have posiively conribued o my educaion. I firs give honor and praise o God. He has shown His omnipoence ye again. I give my sincere appreciaion o my parens, Reginald (Jacqueline) Higgins and Sharon (Gary) Reed. heir encouragemen, love, and pride were an incredible source of suppor for me in his process. I am proud o be heir daugher. I am graeful o my family in he Mahemaics Deparmen a he Universiy of Nebraska-Lincoln for endless suppor. In addiion, I am hankful o my advisors, Dr. Lynn Erbe and Dr. Allan Peerson. While Dr. Peerson assised me wih he fine deails of my work, Dr. Erbe helped me wih he big picure. heir paience and confidence in my abiliy o succeed were remarkable. I would also like o hank my officemaes for making his experience an unforgeable one. I would have no considered graduae school if i were no for he encouragemen of he mahemaics faculy and saff of Xavier Universiy of Louisiana. I would like o hank Dr. Vlajko Kocic for inroducing me o research and he field of difference equaions. I would also like o hank Mrs. Erica Houson for her houghful advice. I am graeful o all he people ha have enered my life and influenced me in counless ways. I will miss he love and suppor of he M. Zion Bapis Church family and he Lincoln Alumnae Chaper of Dela Sigma hea Sororiy, Inc. hanks o he Black Graduae Suden Associaion for eaching me how o embrace my heriage. hanks o ehia and April for making our house a home; love will forever dwell in Lasly, I have o say Asane Sana o Kamau Oginga Siwau. I am exremely blessed o have a mae ha is loving, supporive, paien, and unselfish. hanks for always being here.

6 v Conens Conens v 1 Inroducion 1 2 Preliminaries he Calculus on ime Scales Differeniaion Inegraion Oscillaion Crieria for Funcional Dynamic Equaions Oscillaion of Nonlinear Dynamic Equaions Oscillaion of a Linear Dynamic Equaion Oscillaion of a Nonlinear Dynamic Equaion wih Advanced and Delayed Argumens Oscillaion Crieria for Nonlinear Delay Dynamic Equaions Oscillaion of a Dynamic Equaion wih a Single Delay Oscillaion of a Dynamic Equaion wih Several Delays Asympoic Behavior for Funcional Dynamic Equaions Asympoic Behavior of Dynamic Equaions Asympoic Behavior of a Dynamic Equaion wih a Single Delay.. 72 Bibliography 76

7 1 Chaper 1 Inroducion he heory of ime scales is a new area of mahemaics ha unifies and exends discree and coninuous analysis. he ime scale calculus allows us o model siuaions in which he behavior is boh coninuous and discree. For example, i can model insec populaions ha are coninuous while in season, die ou in, say, winer, while heir eggs are incubaing or dorman, and hen hach in a new season, giving rise o a non overlapping populaion. In recen years here has been an increasing ineres in sudying he oscillaion and nonoscillaion of soluions of dynamic equaions on ime scales. Already many resuls concerning second order dynamic equaions have been esablished [3, 7, 15]. In his presen work we aim o exend he resuls of [11] and [16] o dynamic equaions on ime scales and o improve hose of [26]. For oscillaion of nonlinear delay dynamic equaions, Zhang and Shanliang [26] considered he equaion y () + q()f(y( τ)) = 0, (1.1) where τ R and τ, f : R R is coninuous and nondecreasing, and uf(u) > 0 for u 0. By using comparison heorems, hey proved ha he oscillaion of (1.1) is equivalen o ha of he nonlinear dynamic equaion y () + q()f(y σ ()) = 0, (1.2) where σ() is he nex poin in he ime scale, and esablished some sufficien condiions for oscillaion by applying he resuls esablished in [9] for (1.2) on unbounded above ime scales. In Chaper 3 we show ha he oscillaion of (p()y ()) + q()f(y(τ())) = 0, where τ() is a delay given by a funcion, τ, of, is equivalen o ha of (p()y ()) + q()f(y σ ()) = 0.

8 2 on an isolaed ime scale where sup =. In exending he resuls of [11] o dynamic equaions on ime scales, we esablish oscillaion crieria for he second order nonlinear dynamic equaion y + f (, y σ (), y(τ())) = 0 (1.3) wih rearded argumen in Chaper 4. In order o obain he resuls for (1.3), we improve and exend some resuls of [6] and [17]. In he final chaper, Chaper 5, we are ineresed in he asympoic behavior of soluions of dynamic equaions on ime scales. In [16], he auhor obains necessary and sufficien condiions for he exisence of a bounded nonoscillaory soluion of y + f(, y)g(y ) = 0 wih a prescribed limi a and necessary and sufficien condiions for a nonoscillaory soluion whose derivaive has a posiive limi a. We exend some of hese resuls o y + f(, y σ )g(y ) = 0.

9 3 Chaper 2 Preliminaries In his chaper we inroduce some basic conceps concerning he calculus on ime scales. Mos of hese resuls will be saed wihou proof. he proofs can be found in [2] and [5]. 2.1 he Calculus on ime Scales A ime scale is an arbirary nonempy closed subse of he real numbers. hus R, Z, N, N 0, i.e., he real numbers, he inegers, he naural numbers, and he nonnegaive inegers are examples of ime scales. Definiion Le be a ime scale. For, we define he forward jump operaor σ : by σ() = inf {s : s > }, and he backward jump operaor ρ : by ρ() = sup {s : s < }. In he case ha {s : s > } is empy, we pu inf = sup (i.e., σ() = if has a maximum ). Similarly, if {s : s < } is empy, we pu sup = inf (i.e., ρ() = if has a minimum ). If f : R is a funcion, we define he funcion f σ : R by f σ () = f(σ()) for all. Poins are classified as follows: If σ() >, we say is righ-scaered, while if ρ() < we say is lef-scaered. Also, if < sup and σ() =, hen is said o be righ-dense, and if > inf and ρ() =, hen is called lef-dense. Poins ha are righ-scaered and lef-scaered a he same ime are called isolaed, and poins ha are boh righ and lef dense are called dense.

10 4 Definiion he graininess funcion, µ : [0, ), is defined by µ() := σ(). he backward graininess funcion, ν : [0, ), is defined by ν() := ρ(). Definiion We also need below he se κ which is derived from he ime scale as follows: If he maximum, m, of is lef-scaered, hen κ = \ {m}. Oherwise, κ =. hroughou his work we make he blanke assumpion ha a and b are poins in. Ofen we assume a b. We hen define he inerval [a, b] in by [a, b] := { : a b}. 2.2 Differeniaion Now we consider a funcion f : R and define he so-called dela derivaive of f a a poin κ. By convenion we will define lim f(s) = f() if is an isolaed s poin. Definiion Assume f : R is a funcion and le κ. hen we define f () o be he number (provided i exiss) wih he propery ha given any ɛ > 0, here is a neighborhood U of such ha [f(σ()) f(s)] f ()[σ() s] ɛ σ() s for all s U. We call f () he dela (or Hilger) derivaive of f a. Moreover, we say ha f is dela differeniable (or in shor: differeniable) on κ provided f exiss for all κ. Some useful relaionships concerning he dela derivaive are now given. heorem Assume f, g : R are funcions and le κ. hen we have he following: (i) If f is differeniable a, hen f is coninuous a. (ii) If f is coninuous a and is righ-scaered, hen f is differeniable a wih f () = f(σ()) f(). µ()

11 5 (iii) If is righ-dense, hen f is differeniable a iff he limi exiss as a finie number. In his case, f() f(s) lim s s f f() f(s) () = lim. s s (iv) If f is differeniable a, hen f(σ()) = f() + µ()f (). Looking a properies (ii) and (iii) in he above heorem gives us a more inuiive undersanding of he derivaive ha canno be gained via he definiion alone. If is righ dense, hen he dela-derivaive behaves much he same way as he usual derivaive. I can be viewed as he slope of he angen line o he funcion a, alhough if is boh righ-dense and lef-scaered, he limi is a one-sided limi. On he oher hand, if is righ-scaered, hen f () is he slope of he line segmen conaining f() and f(σ()). In his insance, he behavior of he funcion o he lef of is irrelevan beyond he requiremen ha f is coninuous a. hus, he dela derivaive combines he discree behavior of he forward difference operaor and he coninuous behavior of he usual derivaive. We nex provide he heorem ha allows us o find he derivaive of sums, producs, and quoiens of differeniable funcions. heorem Assume f, g are differeniable a κ. hen: (i) he sum f + g : f : R is differeniable a wih (f + g) () = f () + g (). (ii) For any consan α, αf : R is differeniable a wih (αf) () = αf (). (iii) he produc fg : R is differeniable a wih (fg) () = f ()g() + f(σ())g () = f()g () + f ()g(σ()). (iv) If g()g(σ()) 0, hen f g is differeniable a and ( ) f () = f ()g() f()g (). g g()g(σ())

12 6 Finally, we presen a chain rule which calculaes (f g), where g : R and f : R R. his chain rule is due o Chrisian Pözsche, who derived i firs in heorem Le f : R R be coninuously differeniable and suppose g : R is dela differeniable. hen f g : R is dela differeniable on κ and he formula { 1 } (f g) () = f (g() + hµ()g ())dh g () holds κ Inegraion Of course, he calculus on ime scales would no be complee wihou a concep of inegraion o complemen he derivaive. In order o describe funcions ha are inegrable, we inroduce he following concep. Definiion A funcion f : R is called rd-coninuous provided i is coninuous a righ-dense poins in and is lef-sided limis exis (finie) a all lefdense poins in. he se of rd-coninuous funcions f : R will be denoed in his disseraion by C rd = C rd () = C rd (, R). he se of funcions f : R ha are differeniable and whose derivaive is rdconinuous is denoed by C 1 rd = C 1 rd() = C 1 rd(, R). Definiion A funcion F : R is called an aniderivaive of f : R provided F () = f() κ. hen we define he Cauchy inegral by b a f() = F (b) F (a), a, b. heorem Every rd-coninuous funcion has an aniderivaive. In paricular, if 0, hen F defined by F () := 0 f(τ) τ for

13 7 is an aniderivaive of f. he following heorem provides useful properies of dela inegrals. heorem If a,b,c, α R, and f,g C rd, hen (i) (ii) (iii) (iv) (v) b a b a b a b [f() + g()] = (αf)() = α f() = f() = c a b b a b f() + b a a f() ; f() ; f() + b a a c b a f() ; f σ ()g () = (fg)(b) (fg)(a) f ()g() where f, g can be inerchanged; g() ; b a (vi) If f() g() on [a, b), hen b (vii) if f() 0 for all a < b, hen a b f() g() ; b a a f() 0. he following resul provides useful properies of he dela inegral. heorem Le a, b and f C rd. (i) If = R, hen b f() = b a a f()d, where he inegral on he righ is he usual Riemann inegral from calculus. (ii) If [a, b] consiss of only isolaed poins, hen b [a,b) µ()f() if a < b f() = 0 if a = b a [b,a) µ()f() if a > b.

14 8 Chaper 3 Oscillaion Crieria for Funcional Dynamic Equaions 3.1 Oscillaion of Nonlinear Dynamic Equaions We shall consider he second order nonlinear funcional dynamic equaion and he second order nonlinear dynamic equaion (p()y ()) + q()f(y(τ())) = 0 (3.1) (p()y ()) + q()f(y σ ()) = 0 (3.2) on an isolaed ime scale wih sup =. We assume p, q, τ, and f saisfy he following Condiion (E): (i) p C rd (, (0, )) saisfies (ii) q C rd (, R + ). (iii) τ C rd (, ) saisfies 0 1 p() =,. lim τ() = and M > 0 such ha R() R(τ()) < M where R() = 0 1 p(s) s. (iv) f : R R is coninuous, increasing, and f( u) = f(u) for u R and uf(u) > 0 for u 0.

15 9 By a soluion of (3.1) we mean a nonrivial real-valued funcion y saisfying (3.1) for 0 a, where a > 0. A soluion y of (3.1) is said o be oscillaory if i is neiher evenually posiive nor evenually negaive; oherwise, i is nonoscillaory. Equaion (3.1) is said o be oscillaory if all is soluions are oscillaory. Our aenion is resriced o hose soluions of (py ) + q()f(y()) = 0 which exis on some half line [ y, ) and saisfy sup { y() : > 0 } > 0 for any 0 y. Definiion A nonempy closed subse K on a Banach space X is called a cone if i possess he following properies: (i) if α R + and x K, hen αx K. (ii) if x, y K, hen x + y K. (iii) if x K \ {0}, hen x K. Le X be a Banach space and K be a cone wih nonempy inerior. hen we define a parial ordering on X by x y if and only if y x K. Our main resul, which follows, is an exension of heorem 2.1 of [26]. heorem Assume (E) holds and µ() is bounded. We furher assume p() τ() σ() for all or τ() σ() for all. hen he oscillaion of he second order nonlinear dynamic equaion (p()y ()) + q()f(y σ ()) = 0 (3.2) is equivalen o he oscillaion of he second order nonlinear funcional dynamic equaion (p()y ()) + q()f(y(τ())) = 0. (3.1) We will need he following fixed-poin heorem [10]. heorem (Knaser s Fixed-Poin heorem) Le X be a parially ordered Banach space wih ordering. Le Ω be a subse of X wih he following properies: he infimum of Ω belongs o Ω and every nonempy subse of Ω has a supremum which belongs o Ω. If S : Ω Ω is an increasing mapping, hen S has a fixed poin in Ω. In order o prove heorem 3.1.2, we will need o begin wih he following lemmas. Lemma Assume ha (E) holds. A necessary and sufficien condiion for equaion (3.2) o be oscillaory is ha, he inequaliy (p()y ()) + q()f(y σ ()) 0, (3.3)

16 10 has no evenually posiive soluions. Proof. SUFFICIENCY. Assume (3.3) has no evenually posiive soluions. hen neiher does (3.2), and so (p()y ()) + q()f(y σ ()) = 0 is oscillaory. If y is an evenually negaive soluion of (3.2), hen le x = y. hen x is evenually posiive and (px ) + qf(x σ ) = (py ) qf(y σ ) = [ (px ) + qf(x σ ) ] = 0 for sufficienly large by Condiion (E) (iv). hus x is an evenually posiive soluion of (3.3), which is a conradicion. Hence, (p()y ()) + q()f(y σ ()) = 0 is oscillaory. NECESSIY. Suppose ha (3.2) is oscillaory, and by way of conradicion, assume ha (3.3) has an evenually posiive soluion y, namely, here exiss 0 ( 0 a) such ha y() > 0 for 0. As σ() for all, σ() 0 for all [ 0, ). hen y σ () > 0 for 0. Using his fac along wih he sign condiion on f in (E), we have [p()y ()] 0 for 0, and so p()y () is decreasing on [ 0, ). We claim ha y () > 0 for all large. If no, hen for some 1 [ 0, ), we have y ( 1 ) 0. I follows ha p()y () 0, [ 1, ). Now, if y ( 2 ) < 0 for some 2 1, hen y() y( 2 ) = = 2 y (s) s 2 p(s)y (s) s p(s) p( 2 )y s ( 2 ) 2 p(s) as, which is a conradicion o our assumpion ha y() > 0 for 0. Hence i follows ha y () 0 on [ 1, ), and so (p()y ()) 0 and q()f(y σ ()) > 0, which is conradicory. Consequenly, here exiss ( 0 ) such ha y() > 0, y () > 0, and (p()y ()) 0 for all. Since p()y () is coninuous, he inegrals below are well-defined. Inegraing (p()y ()) + q()f(y σ ()) 0 from o s yields i.e., p(s)y (s) p()y () + s q(u)f(y σ (u)) u 0, for s, and s, p()y () p(s)y (s) + s q(u)f(y σ (u)) u. (3.4)

17 11 Since p()y () > 0 is decreasing for, lim p()y () = k 0 exiss. Leing s in (3.4) we obain y () k p() + 1 p() q(u)f(y σ (u)) u 1 p() q(u)f(y σ (u)) u. (3.5) Since q(u)f(y σ (u)) u exiss and is coninuous, inegraing (3.5) from o yields y() y( ) + 1 p(s) s q(u)f(y σ (u)) u s,. (3.6) Define X o be he Banach space of all coninuous funcions on [a, ) saisfying x() =, where is defined by lim x := max x() for all x X. [a, ) Le Ω := { } ω C([ 0, ), R + ) : 0 ω() 1 and lim ω() = for 0, which is endowed wih he usual poinwise ordering : ω 1 ω 2 ω 1 () ω 2 () for 0. One can show ha for any nonempy subse N of Ω sup N Ω and inf Ω Ω. Define a mapping S on Ω by { 1, if 0, (Sω)() = ( 1 y( ) + ) 1 q(u)f(y σ (u)ω σ (u)) u s, if. y() p(s) s We claim ha SΩ Ω and S is monoone increasing. For any ω Ω, (Sω)() is cerainly coninuous and for, q()f(y σ ()ω σ ()) q()f(y σ ()) since 0 ω σ () 1 and f is nondecreasing. herefore, from (3.6), i follows ha 0 (Sω)() 1 for, and so S(ω) Ω. Moreover, if ω 1 ω 2, ω 1, ω 2 Ω, hen, since f is nondecreasing, f(y σ (u)ω 1 (u)) f(y σ (u)ω 2 (u)) and so (Sω 1 )() (Sω 2 )(). herefore, by Knaser s Fixed Poin heorem, here exiss ω Ω such ha S ω = ω. Hence, ω() = 1 y() ( y( ) + 1 p(u) u ) q(v)f(y σ (v) ω σ (v)) v u, for.

18 12 Observe ha ω() y( ) y() > 0 for. Se z() := ω()y(). hen z() > 0 is coninuous and z() = y( ) + 1 p(u) u q(v)f(z σ (v)) v u, for. As z () = 1 q(u)f(z σ (u)) u and (p()z ()) = q()f(z σ ()), p() (p()z ()) +q()f(z σ ()) = 0 has a posiive soluion, which is a conradicion o he assumpion ha all soluions of (3.2) are oscillaory. his complees he proof. Lemma Assume ha (E) holds. hen, every soluion of he second order nonlinear funcional dynamic equaion (p()y ()) + q()f(y(τ())) = 0 oscillaes if and only if he inequaliy has no evenually posiive soluions. (p()y ()) + q()f(y(τ())) 0 he proof is similar o ha of Lemma and so we omi i. We can now prove heorem Proof of heorem Since µ µ() is bounded, here exiss N > 0 such ha p p() N for all. Le K := M + N, where M > 0 is such ha R() R(τ()) < M where R() = 0 1 p(s) s. SUFFICIENCY. he oscillaion of (3.2) implies ha of (3.1). Suppose, o he conrary, ha y is a nonoscillaory soluion of (3.1). We will only consider he case where here exiss 0 such ha y() > 0 for 0, since he oher case is similar. From equaion (3.1) and Condiion (E), here exiss 1 ( 1 0 ) such ha y() > 0, (py )() > 0, (py ) () 0, y(τ()) > 0, 1 as in he proof of Lemma Hence, since p()y () > 0 is decreasing for 1, lim p()y () = L 0 exiss. We will disinguish several cases. (I) Assume σ() τ() for all. increasing. Consequenly, I follows ha y(τ()) y σ () > 0 as y is (p()y ()) + q()f(y σ ()) (p()y ()) + q()f(y(τ())) = 0,

19 13 and so (3.3) has an evenually posiive soluion. By Lemma 3.1.4, equaion (3.2) has a nonoscillaory soluion, which is a conradicion. (II) Suppose nex ha τ() σ() for all. (a) Assume L > 0. hen here exiss 2 wih 2 1 such ha p()y () L + 1, for all 2. Since lim τ() =, here is a 3 2 such ha τ() 2 for 3. herefore, if 3, we have y σ () y(τ()) = Consequenly, σ() τ() (L + 1) p(s)y (s) s p(s) σ() τ() s p(s) = (L + 1)[R σ () R() + R() R(τ())] [ ] σ() s (L + 1) + R() R(τ()) p(s) [ ] µ() (L + 1) p() + M. y(τ()) y σ () (L + 1)K, 3. Le z() = y() (L + 1)K. Noe ha for all large enough, p()y () L. By inegraing boh sides from 0 o we obain 1 y() y( 0 ) L 0 p(s) s. By leing, we see ha z() > 0 for large enough. Hence, for all sufficienly large, z() > 0, z σ () y(τ()), and (p()z ()) + q()f(z σ ()) 0. Hence, (3.3) has an evenually posiive soluion. By Lemma 3.1.4, we have ha (p()y ()) + q()f(y σ ()) = 0 is nonoscillaory, which is a conradicion. (b) Assume L = 0. Since boh y () and y() are posiive, here exiss ɛ 0 > 0 and 2 1 such ha y() > Mɛ 0 for all 2. Corresponding o his ɛ 0, here exiss 3 1 such ha p()y () ɛ 0 for all 3. Now, if

20 14 := max { 2, 3 }, we have y σ () y(τ()) = σ() τ() p(s)y (s) p(s) s σ() s ɛ 0 τ() p(s) [ ] σ() s ɛ 0 + R() R(τ()) p(s) [ ] µ() ɛ 0 p() + M. Consequenly, y(τ()) y σ () ɛ 0 K,. Again, we se z() := y() ɛ 0 K. hen for sufficienly large z() > 0, z σ () y(τ()), and (p()z ()) + q()f(z σ ()) 0. Hence, (3.3) has an evenually posiive soluion. Again by Lemma 3.1.4, (p()y ()) + q()f(y σ ()) = 0 is nonoscillaory, which is a conradicion. NECESSIY. he oscillaion of (3.1) implies ha of equaion (3.2). Suppose ha here is a nonoscillaory soluion y() of (3.2) and wihou loss of generaliy, we assume here exiss 1 such ha y() > 0, p()y () > 0, and (p()y ()) 0, 1. Since p()y () > 0 is decreasing for 1, lim p()y () = L 0 exiss. disinguish several cases. We (I) Assume τ() for all. As y is increasing, y σ () y(τ()). Furhermore, as f is increasing, we have (p()y ()) + q()f(y(τ())) (p()y ()) + q()f(y σ ()) = 0. So y() is an evenually posiive soluion of (p()y ()) + q()f(y(τ())) 0. By Lemma 3.1.5, equaion (3.1) is nonoscillaory, which is a conradicion. (II) Suppose τ() σ() for all. (a) Assume L > 0. I follows ha here exiss 2 wih 2 1 such ha p()y () L + 1 for all 2. Since lim τ() =, here is a 3 2 such

21 15 ha τ() 2 for 3. herefore, if 3, we have y(τ()) y σ () = τ() σ() p(s)y (s) p(s) s (L + 1)[R(τ()) R(σ())] (L + 1)[R(τ()) R() + R() R(σ())] (L + 1)[M + N], which leads o y σ () y(τ()) (L + 1)K, 3. Le z() := y() (L + 1)K. hen for sufficienly large, we have z() > 0, z(τ()) y σ (), and (p()z ()) + q()f(z(τ())) 0. his leads o a conradicion as in par (I) above. (b) Assume L = 0. Since y () > 0 and y() > 0, here is an ɛ 0 > 0 and a 2 1 such ha y() > Mɛ 0 for all 2. Corresponding o his ɛ 0, here exiss 3 1 such ha p()y () ɛ 0 for all 3. Now, if := max { 2, 3 }, we have y(τ()) y σ () = τ() σ() ɛ 0 τ() σ() p(s)y (s) p(s) s p(s) ɛ 0 [ M + µ() p() and so y σ () y(τ()) ɛ 0 K for. Now se z() := y() ɛ 0 K. hen for sufficienly large z() > 0, z(τ()) y σ (), and (p()z ()) + q()f(z σ ()) 0, which again leads o a conradicion. his complees he proof. Remark Under he assumpions heorem we see ha he funcional τ in equaion (3.1) has no influence on is oscillaion. As a corollary o heorem we have he following: ], s

22 16 Corollary Le = Z and τ : Z Z. Assume q : N 0 N 0 is coninuous and f, τ, and p saisfy (E). hen, he oscillaion of he wo equaions (p() y()) + q()f(y( + 1)) = 0 and is equivalen. (p() y()) + q()f(y(τ())) = 0 Proof. Since µ() = 1 Z and y σ () = y( + 1), he resul follows from heorem Remark One can prove analogous resuls when considering and (p()y ()) + q 1 ()f 1 (y(τ 1 ())) + q 2 ()f 2 (y(τ 2 ())) = 0 (p()y ()) + q 1 ()f 1 (y σ ()) + q 2 ()f 2 (y σ ()) = 0 and heir corresponding inequaliies. Le r R and assume ha p r is a differeniable funcion. Assume ha (E 1 ) here exiss M > 0 such ha r()e r (, 0 ) M for all large. (E 2 ) Condiion (E) holds, µ() is bounded, and f(u) K u for u 0 for some p() K > 0, and define he auxiliary funcions H 1 () = µ() H 1 (, 0 ) := 1 + p() 0 H 2 () = H 2 (, 0 ) := 1 + µ()r() p()e r (, 0 ), s p(s) H 3 () = [ H 3 (, 0 ) := e r (σ(), 0 ) Kp() + 1 ] 2 r () + r2 (), 4H 1 () H 4 () = r()(1 + µ()r()) H 4 (, 0 ) := r(), H 1 () for > 0, for some 0. By combining heorem and heorem 3.1 in [9], we obain he following resul: heorem Assume ha (E 1 ) and (E 2 ) hold. Furhermore, assume ha here exiss r R + such ha p r is differeniable and such ha for any 0 a here exiss,

23 17 a 1 > 0 so ha where lim sup H(s) s =, 1 H() = H(, 0 ) = H 3 () (H 4()) 2 H 1 (), 4H 2 () for > 0. hen equaion (3.1) is oscillaory on [a, ). We end his secion wih comparing (p()y ) + q()f(y(τ())) o (p()y ()) + q()g(y( τ())) = 0, (3.7) on a ime scale where q, g, and τ saisfy condiion (E) and µ is bounded. p From heorem we see ha he oscillaion of (3.7) is equivalen o ha of We ge he following resul. (p()y ()) + q()g(y σ ()) = 0. (3.8) heorem Assume condiion (E) holds and µ is bounded on. p Furher assume ha q() q() for all large and g(u) f(u) for u > 0. hen, he oscillaion of equaion (3.7) implies ha of equaion (3.1). Proof. Oherwise, wihou loss of generaliy, we assume ha (3.1) has an evenually posiive soluion. From heorem 3.1.2, equaion (3.2) also has an evenually posiive soluion y(). hen (p()y ()) + q()g(y σ ())) (p()y ()) + q()f(y σ ())) = 0, which implies (3.8) has an evenually posiive soluion. So, equaion (3.7) also has an evenually posiive soluion, which is a conradicion. 3.2 Oscillaion of a Linear Dynamic Equaion In his secion we give wo heorems abou he oscillaory behavior of (p()y ()) + q()y σ () = 0 (3.9) on a ime scale where sup =, p C rd (, (0, )) and q C rd (, R). hese are heorems and

24 18 We impose he following condiion a 1 p(s) s = and a q(s) s < for some a. (E 3 ) o prove our main resul, we need he following lemma. Lemma [14] Assume for all large, and lim inf τ q(s) 0 and 0 (E 4 ) 1 p(s) s =. (E 5) If y is a soluion of (3.9) such ha y() > 0, for [, ), hen here exiss S [, ) such ha y () > 0 for [S, ). Proof. he proof is by conradicion. We consider wo cases: (a) Suppose ha y () < 0 for [, ). Define Q(, ) = q(s) s. We may assume, by condiion (E 4 ), ha is such ha Q(, ) 0 for [, ). Indeed, if no such exiss, hen for [τ, ) fixed bu arbirary, we define { 1 = 1 ( ) := sup > : } q(s) s < 0. If 1 =, hen choosing n such ha Q( n, ) < 0 for all n, we obain a conradicion o (E 4 ). Hence, we mus have 1 is finie, which implies ha Q(, 1 ) 0 for [ 1, ). Now an inegraion by pars gives (wih 1 = ) q(s)y σ (s) s = Q (s, )y σ (s) s = Q(, )y() 0. Inegraing (3.9) we have, from his las esimae, Q(s, )y (s) s y () p( )y ( ) p() (3.10) for [, ). Inegraing (3.10) for we see ha y() by (E 5 ), a conradicion. herefore, y () < 0 canno hold for all large.

25 19 (b) Nex, if y () 0 evenually, hen for every (large) [τ, ) here exiss 0 in [, ) such ha y ( 0 ) 0 and we may suppose ha lim inf q(s) s 0. 0 Since y() > 0 for [, ), he funcion z() := p()y () saisfies he Riccai y() equaion z () + q() + z 2 () p() + µ()z() = 0 for [, ) wih p() + µ()z() > 0. Inegraing he Riccai equaion from 0 o gives z() = z( 0 ) 0 q(s) s 0 z 2 (s) p(s) + µ(s)z(s) s. herefore i follows ha lim sup z() < 0, using he facs ha z( 0 ) 0, z() is evenually nonrivial, and (E 4 ) holds. Hence here exiss 2 [, ) such ha z() < 0 for [ 2, ) and so y () < 0 for [ 2, ), a conradicion o par (a). he proof is complee. Before we sae heorem 3.2.2, we need he following definions. A 0 () = A 1 () = A 0 () +. A n () = A 0 () + q(s) s, A 2 0(s) p(s) + µ(s)a 0 (s) s, A 2 n 1(s) p(s) + µ(s)a 0 (s) s, if he inegrals on he righ-hand side exis. Our firs resul is a generalizaion of heorem 3.1 of [26]. heorem Assume (E 3 ) and (E 4 ) hold, and one of he following wo condiions holds: (i) here exiss some posiive ineger m such ha A n is well defined for n = 0, 1, 2,..., m 1, and A 2 lim m 1(s) s =. a p(s) + µ(s)a m 1 (s)

26 20 (ii) A n is well defined for n = 0, 1, 2,..., and here exiss ( 0 ) such ha for all n. hen he second order dynamic equaion is oscillaory. lim A n( ) = n (p()y ()) + q()y σ () = 0 (3.9) Proof. If no, wihou loss of generaliy, we assume (3.9) has an evenually posiive soluion y(). From Lemma 3.2.1, we ge ha here exiss 1 ( 1 0 ) such ha Define he funcion z by y() > 0 and y () > 0 for all 1. z() = p()y () y() for 1. (3.11) hen z() > 0 and p() + µ()z() = p() + µ() p()y () y() for 1. From (3.11) we ge ha = p()y() + p()µ()y () y() > 0, z () = (p()y ()) y() (p()y ())y () y()y σ () = (p()y ()) p()(y ()) 2 p() y σ () y()y σ () p() y2 () y 2 () ( ) p()y 2 () y() = q() y() p()y σ () = q() z 2 y() () p()(y() + µ()y ()) ( ) = q() z 2 () y() 1 y() p()(1 + µ() y () ) y() z 2 () = q() p() + µ()z()

27 21 for 1. Hence, z is a soluion of he Riccai equaion z () = q() Inegraing boh sides of (3.12) from 1 o we ge z() z( 1 ) + hen, as z() > 0, 1 1 z 2 () p() + µ()z(), 1. (3.12) z 2 (s) p(s) + µ(s)z(s) s = q(s) s, 1. 1 z 2 (s) p(s) + µ(s)z(s) s z( 1) q(s) s z( 1 ), 1. 1 Leing we have ha z 2 (s) lim s <. 1 p(s) + µ(s)z(s) Inegraing (3.12) from o s we obain z() = z(s) + > s s q(τ) τ + q(τ) τ + s s for s, and s 1. Leing s we have z() q(s) s + z 2 (τ) p(τ) + µ(τ)z(τ) τ z 2 (τ) p(τ) + µ(τ)z(τ) τ. z 2 (s) p(s) + µ(s)z(s) s, 1. (3.13) Assume Condiion (i) holds and m = 1. From (3.13) we obain ha z() A 0 () for all 1. u 2 Observe ha F (u) = c 1 + c 2 u is increasing for u > 0, where c 1, c 2 0 are consans. I follows ha A 2 0(s) p(s) + µ(s)a 0 (s) s his conradics (i). If m > 1, we have z() q(s) s + z 2 (s) s <. p(s) + µ(s)z(s) A 2 0(s) p(s) + µ(s)a 0 (s) s = A 1(), for 1.

28 22 Repeaing he above procedure, we ge ha z() A m 1 () for all 1, and A 2 m 1(s) p(s) + µ(s)a m 1 (s) s z 2 (s) s <, p(s) + µ(s)z(s) which conradics Condiion (i). Assume ha Condiion (ii) holds. Similar o he above proof, we obain A n () z() for n = 0, 1, 2,.... hen, as y() > 0, lim A n( ) z( ) <, n which gives a conradicion o Condiion (ii). he proof is complee. Remark If = R and p() = 1 for all, hen heorem is he same as Yan s resul for second order linear differenial equaions [25]. o prove he nex resul, we need he following lemmas: Lemma [2, heorem 4.61] Assume a, p > 0, and le ω := sup. If ω <, hen we assume ρ(ω) = ω. If (py ) () + q()y σ () = 0 has a posiive soluion on [a, ω), hen here is a posiive soluion u, called a recessive soluion a ω, such ha for any second linearly independen soluion v, called a dominan soluion a ω, u() lim ω v() = 0, ω a p()u()u σ () =, and ω b p()v()v σ () <, where b < ω is sufficienly close. Furhermore p()v () v() > p()u () u() for < ω sufficienly close. Lemma [2, heorem 4.55] Assume z is a soluion of he he Riccai equaion Rz = 0, where Rz() := z () + q() + z 2 () p() + µ()z() on [a, σ 2 (b)] wih p() + µ()z() > 0 on [a, σ 2 (b)]. Le u be a coninuous funcion on [a, σ 2 (b)] whose derivaive is piecewise righ-dense coninuous wih

29 23 u(a) = u(σ 2 (b)) = 0. hen we have for all [a, σ 2 (b)], (zu 2 ) () = p()[u ()] 2 q()u 2 (σ()) { z()u σ () 2 p() + µ()z()u ()}. p() + µ()z() Using he previous lemmas, we have he following heorem which was proven for differenial equaions by Kelley and Peerson in [21]. heorem Assume I = [a, ). If a p() = and here is a 0 a and a u Crd 1 [ 0, ) such ha u() > 0 on [ 0, ) and {q()[u σ ()] 2 p()[u ()] 2 } =, 0 hen he second-order dynamic equaion is oscillaory on I. (p()y ()) + q()y σ () = 0 (3.9) Proof. We prove his heorem by conradicion. So assume (3.9) is nonoscillaory on I. Lemma 3.2.4, here is a dominan soluion y a such ha for 1 a, sufficienly large, p()y()y σ () <, 1 and we may assume y() > 0 on [ 1, ). Le 0 and u be as in he saemen of his heorem. Le =max{ 0, 1 }; hen le I follows ha z() := p()y (),. y() z () = (p()y ()) y() p()(y ()) 2 y()y σ () ( ) = q()yσ () p()y 2 () y() y σ () y() p()y σ () = q() z 2 y() () p()[y() + µ()y ()] z 2 () = q() p() + µ()z()

30 24 and hen by Lemma 3.2.5, we have for p() + µ()z() > 0 for all. (zu 2 ) () { = p()[u ()] 2 q()u 2 z()u(σ()) (σ()) p() + µ()z()u () p() + µ()z() p()[u ()] 2 q()u 2 (σ()). Inegraing from o, we obain which implies z()u 2 () z( )u 2 ( ) { q()u 2 (σ()) p()[u ()] 2} lim z()u2 () =. However, hen here is a 1 such ha for 1 } 2 z() = p()y () y() < 0. his implies ha y () < 0 for 1, and hence y is decreasing on [ 1, ). However, 1 1 which is a conradicion. p(s) s = y( 1)y σ ( 1 ) y( 1 )y σ ( 1 ) <, p(s)y( 1 )y σ ( 1 ) s p(s)y(s)y σ (s) s We conclude his secion wih an example ha shows how heorem can be used o obain oscillaion crieria. Example If a > 0 and a σ α ()q() =, where 0 < α < 1, hen y + q()y σ = 0 is oscillaory on [a, ). We will show ha his follows from heorem In he Pözsche Chain Rule [2, heorem 1.90], le g() = and f() = α 2, for 0 < α < 1. hen wih

31 25 u() = (f g)() = α 2, we have u () = (f g) () = = α 2 α 2 { 1 α = α 2 α 2 2 } α 2 [ + hµ() 1] 2 dh 1 ( + hµ()) α 2 2 dh α 2 2 dh since α 2 < 0. herefore, i follows ha (u ()) 2 α2 4 α 2 for all. Hence, a { q()[u σ ()] 2 p()[u ()] 2} since 0 < α < 1 implies a a α 2 <. } {q()σ α () α2 4 α 2 hus y + q()y σ = 0 is oscillaory on [a, ) by heorem = 3.3 Oscillaion of a Nonlinear Dynamic Equaion wih Advanced and Delayed Argumens In his secion we esablish several oscillaion resuls (heorems ) for he second order nonlinear funcional dynamic equaion y + f (, y σ (), y τ 1 (), y τ 2 (), y ξ 1 (), y ξ 2 () ) = 0 (3.14) on a ime scale [ 0, ) where f C( R 5, R). We shall assume τ i () σ() ξ i () for all and τ i, ξ i C rd (, ) for i = 1, 2. We also assume lim τ i() = = lim ξ i () for i = 1, 2. Here we use he noaion y τ () = y(τ()) and y ξ () = y(ξ()). Our goal is o esablish some new oscillaion and nonoscillaion resuls for his equaion. We apply resuls from he heory of lower and upper soluions for relaed dynamic equaions along wih some addiional esimaes on he posiive soluions.

32 26 Concerning he funcion f = f(, u, v 1, v 2, w 1, w 2 ), we will always assume ha f saisfies he following Condiion (A): and f(, u, v 1, v 2, w 1, w 2 ) = f(, u, v 1, v 2, w 1, w 2 ) f(, u, v 1, v 2, w 1, w 2 ) > 0 if u, v 1, v 2, w 1, w 2 > 0,. We begin wih he following preliminary lemmas. Lemma Le y C 2 rd [ 0, ) saisfy y() > 0, y () > 0, y () 0 for 0. hen for each 0 < k < 1 here exiss k 0 such ha he following hold: (i) y τ () := y(τ()) ky σ () τ() σ(), k, and (ii) y ξ () := y(ξ()) y σ () ξ() kσ(), k. Proof. (i) For > 0 we have y σ () y τ () = as y is decreasing, and so σ() τ() y (s) s y (τ())(σ() τ()) y σ () y τ () + y (τ())(σ() τ()). (3.15) Also we have and hence which implies y τ () y( ) = y (τ()) y τ () τ() y (s) s y (τ())(τ() ) (3.16) y τ () y (τ()) y( ) + (τ() ) (3.17) y (τ()) 1 (τ() ) + y( ) y (τ()) < 1 τ(). (3.18)

33 27 herefore, (3.15) and (3.18) imply y σ () y τ () 1 + y (τ()) (σ() τ()) y τ () σ() τ() 1 + τ() = σ() τ(). (3.19) Now given any 0 < k < 1, here exiss k such ha Consequenly, we have from (3.19) and (3.20) σ() τ() < 1 σ() k τ(), k. (3.20) y τ () ky σ () τ() σ(), k and his complees he proof of (i). he proof of (ii) is similar. We have for < σ() ξ() and so we have Also we have so ha y ξ () y σ () = ξ() σ() y (s) s y (σ())(ξ() σ()) y ξ () y σ () 1 + y (σ()) (ξ() σ()). (3.21) y σ () y σ () y( ) + y (σ())(σ() ) y σ () y (σ()) y( ) y (σ()) + (σ() ) kσ(), k, 0 < k < 1. Hence, from (3.21) we have y ξ () y σ () 1 + ξ() σ() kσ() = (k 1)σ() + ξ() kσ() ξ() kσ(), k. his complees he proof of he lemma. We coninue wih he following resul for he case when f(, u, v 1, v 2, w 1, w 2 )

34 28 saisfies he following Condiion (B): For each fixed, f is nonincreasing in w 1, w 2 > 0 for fixed u, v 1, v 2 > 0, f is nondecreasing in v 1, v 2 > 0 for fixed u, w 1, w 2 > 0, and f is nondecreasing in u > 0 for fixed v 1, v 2, w 1, w 2 > 0. We inroduce he funcions g i (), h i () defined by g i () := τ i() σ(), h i() := ξ i() σ(), (3.22) where i = 1, 2. In order o prove our main resuls, we need a mehod for sudying boundary value problems (BVP). Namely we will define funcions called upper and lower soluions ha, no only imply he exisence of a BVP bu also provide bounds on he locaion of he soluion. Consider he second-order equaion where f is coninuous on [a, b] R. y = f(, y σ ) (3.23) Definiion [2, Definiion 6.53] We say ha α Crd 2 (3.23) on [a, σ 2 (b)] provided is a lower soluion of α () f(, α σ ()) for all [a, b]. Similarly, β C 2 rd is called an upper soluion of (3.23) on [a, σ2 (b)] provided β () f(, β σ ()) for all [a, b]. heorem [2, heorem 6.54] Le f be coninuous on [a, b] R. Assume ha here exis a lower soluion α and an upper soluion β of (3.23) wih α(a) A β(a) and α(σ 2 (b)) B β(σ 2 (b)) such ha hen he BVP α() β() for all [a, σ 2 (b)]. y = f(, y σ ) on [a, b], y(a) = A, y(σ 2 (b)) = B has a soluion y wih α() y() β() for all [a, σ 2 (b)]. he following is a generalizaion of heorem 7.4 of [20].

35 29 heorem Le f be coninuous on [a, b] R. Assume ha here exis a lower soluion α and an upper soluion β of (3.23) wih α() β() for all [a, ). hen for any α(a) c β(a) he BVP has a soluion y wih y = f(, y σ ), y(a) = c (3.24) α() y() β() for all [a, ). Proof. I follows from heorem ha for each n 1 here is a soluion y n () of [a, a + n] wih y n (a) = c, y n (a + n) = β(a + n) and α() y n () β() on [a, a + n]. hus, for any fixed n 1, y m () is a soluion on [a, a + n] saisfying α() y m () β() for all m n. Hence, for m n, he sequence y m () is poinwise bounded on [a, a + n]. We claim ha {y m ()} is equiconinuous on [a, a + n] for any fixed n 1. Since f is coninuous and y m () β() for all [a, a + n], here is consan K > 0 such ha ym () = f(, ym()) σ K for all [a, a + n]. I follows ha which gives ha Consequenly, y m() y m(a) = a a ym (s) s K s = K( a) K(a + n a) = Kn y m() y m(a) + Kn =: L. y m () y m (s) = s y m s L s < ɛ for all, s [a, a + n] provided s < δ = ɛ. Hence he claim holds. L So by Ascoli-Arzela and a sandard diagonalizaion argumen, {y m ()} conains a subsequence which converges uniformly on all compac subinervals [a, a + n] of [a, ) o a soluion y(), which is he desired soluion of he (3.24) ha saisfies α() y() β() for all [a, ). In he resuls ha follow by f (, u, v 1 v 2, w 1, w 2 ) =

36 30 we mean for some sufficienly large. Our firs main resul in his secion is: f (, u, v 1 v 2, w 1, w 2 ) < heorem Assume condiions (A) and (B) hold. hen all bounded soluions of y + f (, y σ (), y τ 1 (), y τ 2 (), y ξ 1 (), y ξ 2 () ) = 0 are oscillaory in case f (, α, αkg 1 (), αkg 2 (), α k h 1(), α ) 2 k h 2() 2 = (3.25) for all α 0 and for some k (0, 1), where g i (), h i () for i = 1, 2 are given by (3.22). Proof. If no, le u() be a bounded nonoscillaory soluion which we may assume saisfies u() > 0, u τ i () > 0, 0, i = 1, 2. Consequenly, u () = f(, u σ (), u τ 1 (), u τ 2 (), u ξ 1 (), u ξ 2 ()) < 0 for and so u () is decreasing for. I follows ha u () > 0 for. Indeed, if u ( 1 ) 0 for some 1, hen u () 0 for all 1. Now if u ( 2 ) < 0 for some 2 1, hen u() u( 2 ) = 2 u (s) s u ( 2 )( 2 ) as, which is a conradicion o our assumpion ha u() > 0 for 0. Also, if u ( 1 ) = 0, hen u () 0 on [ 1, ), and so u () 0 = f(, u σ (), u τ 1 (), u τ 2 (), u ξ 1 (), u ξ 2 ()), which is again a conradicion. Hence, we conclude ha for all u() > 0, u () > 0, u τ i () > 0 for i = 1, 2. From Lemma 3.3.1, given 0 < k < 1, here exiss 1 k such ha u τ 1 () kg 1 ()u σ () and u ξ 1 () 1 k h 1()u σ () for 1 k and here exiss 2 k such ha u τ 2 () kg 2 ()u σ () and u ξ 2 () 1 k h 2()u σ ()

37 31 for k 2. By he monooniciy assumpion on f we have 0 = u () + f (, u σ (), u τ 1 (), u τ 2 (), u ξ 1 (), u ξ 2 () ) (3.26) ( u () + f, u σ (), kg 1 ()u σ (), kg 2 ()u σ (), 1 k h 1()u σ (), 1 ) k h 2()u σ () for k := max{k 1, k 2 }. Now, if we se ( F (, u σ ()) := f, u σ (), kg 1 ()u σ (), kg 2 ()u σ (), 1 k h 1()u σ (), 1 ) k h 2()u σ (), hen (3.26) shows ha β() := u() is an upper soluion for he dynamic equaion u + F (, u σ ()) = 0. Also, he consan funcion α() := u( k ) saisfies he inequaliy α () + F (, α σ ()) 0, and so α() is a lower soluion. herefore, by heorem 3.3.4, he BVP y + F (, y σ ()) = 0, y( k ) = u( k ) has a soluion y() wih u( k ) y() u(), k. I follows ha y() > 0 and y () 0. herefore y () > 0. Now, since y() is bounded, we have ha lim y() := L > 0 exiss. Inegraion for k < s < implies Leing we obain y ( ) y (s) + y (s) s s F (r, y σ (r)) r = 0. F (r, y σ (r)) r,

38 32 and so inegraing again for k < <, we obain y() y( ) = = = From (3.27) we have s r y() y( ) + y (s) s F (r, y σ (r)) r s F (r, y σ (r)) s r + (r )F (r, y σ (r)) r + (r )F (r, y σ (r)) r. (r )F (r, y σ (r)) r > F (r, y σ (r)) s r (3.27) ( )F (r, y σ (r)) r (r )F (r, y σ (r)) r. Since y() u() M for some M > 0 and funcion of, i follows ha (r )F (r, y σ (r)) r is an increasing (r )F (r, y σ (r)) r <. Since 2r for r sufficienly large, i follows ha rf (r, y σ (r)) r <. he same 0 < k < 1 as in he firs par of he proof, we may assume ha we have y() kl for k k. Since y() y σ () L, using he monooniciy of f, ( f, y σ (), kg 1 ()y σ (), kg 2 ()y σ (), 1 k h 1()y σ (), 1 ) k h 2()y σ () ( f, y σ (), kg 1 ()y σ (), kg 2 ()y σ (), L k h 1(), L ) k h 2() ( f, kl, k 2 g 1 ()L, k 2 g 2 ()L, L k h 1(), L ) k h 2(). herefore, wih α := kl, i follows ha k rf ( r, α, αkg 1 (r), αkg 2 (r), α k h 1(r), α ) 2 k h 2(r) r <, 2 a conradicion o our assumpion (3.25). For

39 33 his complees he proof. he nex resul shows ha a converse of heorem is rue under an addiional assumpion. heorem Assume f saisfies condiions (A) and (B) and ha lim inf g i() := m i > 0 and lim sup h i () := M i <, (3.28) where g i (), h i (), 1 i 2, are defined in (3.22). bounded. hen, if Also, assume ha σ()/ is y + f (, y σ (), y τ 1 (), y τ 2 (), y ξ 1 (), y ξ 2 () ) = 0 (3.14) has a bounded nonoscillaory soluion, i follows ha ( σ()f, α, αk m, αk m, α M k, α M ) 2 k 2 < (3.29) for some α 0 and for any 0 < k < 1, where m and M saisfy m < m i M > M i for i = 1, 2. and Proof. Noe ha for any β σ()f(, β,..., β) < if, and only if, f(, β,..., β) < since σ()/ is bounded on. Assume (3.28) holds. hen for ɛ > 0 wih ɛ < min{m 1, m 2 }, here exiss i 0 such ha g i () m i ɛ := m i provided i and here exiss i 0 such ha h i () M i + ɛ := M i provided i, i = 1, 2. I follows ha for α > 0 αg i () α m i α m where m := min{ m 1, m 2 } m i < m i, and 1 α h i() 1 α M i 1 α M where M := max{ M 1, M2 } M i > M i, i = 1, 2. Assume (3.14) has a bounded

40 34 nonoscillaory soluion. hen by heorem 3.3.5, ( f, α, αkg 1 (), αkg 2 (), α k h 1(), α ) 2 k h 2() 2 < for some α 0 and for all 0 < k < 1. By he monooniciy assumpion of f, we have ( f, α, αk m, αk m, α M k, α M ) <, 2 k 2 which proves he resul. he previous resul says ha condiion (3.28) is sufficien in order o replace he auxiliary funcions g i (), h i () for i = 1, 2, given by (3.22) wih upper bounds. Our nex resul gives a sufficien condiion for y + f (, y σ (), y τ 1 (), y τ 2 (), y ξ 1 (), y ξ 2 () ) = 0 (3.14) o have bounded nonoscillaory soluions. heorem Assume f saisfies condiions (A) and (B). If ( σ()f, α, α, α, α 2, α ) 2 <, (3.30) for all α 0, hen (3.14) has a bounded nonoscillaory soluion. Proof. If (3.30) holds, assume o be specific ha α > 0 and le 0 < β < α. Choose 1 such ha τ 1 (), τ 2 () 1 for and such ha σ()f (, α, α, α, α 2, α ) < β 2 2. Define y 0 () β for 0 and { β y n+1 () = (σ(s) )f(s, yσ n(s), y τ 1 n (s), y τ 2 n (s), y ξ 1 n (s), y ξ 2 n (s)) s, <, β (σ(s) )f(s, y σ n(s), y τ 1 n (s), y τ 2 n (s), y ξ 1 n (s), y ξ 2 n (s)) s, Observe 1 τ i () σ() ξ i () for all and i = 1, 2. We claim ha β 2 y n() β and all n 0. (3.31) By consrucion he claim holds for y 0 (). Noice ha when τ i () < for any i = 1, 2, y n (τ i ()) < β as y () 0 for all less han. Assume he inequaliy

41 35 holds for y m (), 1 m n. hen for y m+1 () = β β β > β β 2 = β 2. (σ(s) )f(s, ym(s), σ y τ 1 m(s), y τ 2 m(s), y ξ 1 m(s), y ξ 2 m(s)) s σ(s)f(s, ym(s), σ y τ 1 m(s), y τ 2 m(s), y ξ 1 m(s), y ξ 2 m(s)) s ( σ(s)f s, α, α, α, α 2, α ) s 2 Furhermore, since s, we have ym(s), σ y τ 1 m(s), y τ 2 m(s), y ξ 1 m(s), y ξ 2 m(s) are all posiive. Hence by condiion (A) (σ(s) )f(s, ym(s), σ y τ 1 m(s), y τ 2 m(s), y ξ 1 m(s), y ξ 2 m(s)) 0 for s. Consequenly, y m+1 () β for. herefore, by inducion, (3.31) holds. I remains o show ha he se {y n ()} n=0 is equiconinuous. o do his, we show ha { yn () } is uniformly bounded. I follows ha n=0 [ yn () = 0 f(s, yn(s), σ y τ 1 n (s), y τ 2 n (s), y ξ 1 n (s), y ξ 2 n (s)) s = < β 2. + (σ() σ())f(, y σ n(), y τ 1 n (), y τ 2 n (), y ξ 1 n (), y ξ 2 n ())] f(s, y σ n(s), y τ 1 n (s), y τ 2 n (s), y ξ 1 n (s), y ξ 2 f(s, β, β, β, β/2, β/2) s σ(s)f(s, β, β, β, β/2, β/2) s n (s)) s herefore, he Ascoli-Arzela heorem along wih a sandard diagonalizaion argumen yields a subsequence of {y n ()} n=0 which converges uniformly on compac subinervals of [, ) o a soluion y() of (3.14) saisfying β/2 y() < β,. his proves he heorem. We nex inroduce he following condiion which replaces he nonincreasing assumpion for he funcion f in he w 1, w 2 variables by assuming f is nondecreasing

42 36 in w 1, w 2 for w 1, w 2 > 0 and for fixed and u, v 1, v 2 > 0. he funcion f is said o saisfy Condiion ( B) if for each fixed, f is nondecreasing in w 1, w 2 > 0 for fixed u, v 1, v 2 > 0, f is nondecreasing in v 1, v 2 > 0 for fixed u, w 1, w 2 > 0, and f is nondecreasing in u > 0 for fixed v 1, v 2, w 1, w 2 > 0. Of course, if f is independen of w 1, w 2 > 0, hen condiions (B) and ( B) coincide. Now we have he following: heorem Assume condiions (A) and ( B) hold. hen all soluions of y + f (, y σ (), y τ 1 (), y τ 2 (), y ξ 1 (), y ξ 2 () ) = 0 (3.14) are oscillaory in case f(, α, αkg 1 (), αkg 2 (), α, α) = (3.32) for all α 0 and for some k (0, 1), where g i () for i = 1, 2 is given by (3.22). Proof. If u is an evenually posiive soluion of (3.14), as in he proof of heorem 3.3.5, we conclude ha for all 1 0 u() > 0, u (), u τ 1 () > 0, u τ 2 () > 0. Hence, by Lemma given 0 < k < 1, here exiss k = max{ 1, 2 } 1 so ha u τ i () kg i ()u σ () for all k and i = 1, 2. Furhermore, since u () > 0 for 1 and i = 1, 2, u ξ i () u σ (). herefore, by he monooniciy assumpion on f from ( B), i follows ha 0 = u () + f (, u σ (), u τ 1 (), u τ 2 (), u ξ 1 (), u ξ 2 () ) u () + f(, u σ (), kg 1 ()u σ (), kg 2 ()u σ (), u σ (), u σ ()). If we se F (, u σ ) := f(, u σ (), kg 1 ()u σ (), kg 2 ()u σ (), u σ (), u σ ()), hen he remainder of he proof is similar o ha of heorem If we replace assumpion (B) by ( B) in heorem 3.3.6, hen we may give a necessary and sufficien condiion for he exisence of a bounded nonoscillaory soluion. In his case, we only need o assume he firs par of (3.28). heorem Assume f saisfies (A) and ( B) and ha lim inf g i() := m i > 0, (3.33) where g i () := τ i(), 1 i 2. Assume furher ha σ()/ is bounded. hen equaion σ()

43 37 (3.14) has a bounded nonoscillaory soluion, if and only if, σ()f(, α, α, α, α, α) < (3.34) for some α 0. Proof. Assume (3.14) has a bounded nonoscillaory soluion. By condiion (3.33), we have for any ɛ > 0 wih ɛ < min{m i 1 i 2} and any δ > 0, here exiss i 0 such ha δg i () δ(m i ɛ) = δ m i where m i < m i, i = 1, 2. By definiion of g i (), we have 0 m i g i () 1 for i, i = 1, 2. Hence δm δ m i < δm i δ for i = 1, 2, where m := min{ m 1, m 2 }. hen by heorem and he monooniciy of f, we have f(, δm, δm, δm, δm, δm) f(, δm, δm, δm, δ, δ) < f(, δm, δ m 1, δ m 2, δ, δ) f(, δm, δm 1, δm 2, δ, δ) <. f(, δ, δg 1 (), δg 2 (), δ, δ) By leing α = δm, we obain (3.34) is necessary for he exisence of a bounded nonoscillaory soluion. Conversely, if (3.34) holds, assume o be specific ha α > 0 and le 0 < β < α. Choose 1 0 such ha τ 1 (), τ 2 () 1 for such ha f(, α, α, α, α, α) < β 2. If we define y 0 () β for 0 and { β y n+1 () = (σ(s) )f(s, yσ n(s), y τ 1 n (s), y τ 2 n (s), y ξ 1 n (s), y ξ 2 n (s)) s, <, β (σ(s) )f(s, y σ n(s), y τ 1 n (s), y τ 2 n (s), y ξ 1 n (s), y ξ 2 n (s)) s,, hen he remainder of he proof is similar o ha of heorem o exend heorems and o unbounded soluions, we inroduce he class Φ of funcions φ such ha φ(u) denoes a coninuous nondecreasing funcion of

44 38 u saisfying uφ(u) > 0, u 0 wih ± ±1 du φ(u) <. We will say ha f(, u, v 1, v 2, w 1, w 2 ) saisfies Condiion (H ) provided for some φ Φ here exiss c 0 and 0 < α < 1 such ha for all for some posiive consan k. f(, u, αg 1 ()u, αg 2 ()u, 1 inf h α 1()u, 1 h α 2()u) u c φ(u) k (, f c, αg 1 ()c, αg 2 ()c, 1 α h 1()c, 1 2()c) α h We may now prove he following resul: heorem Suppose φ Φ. Assume f saisfies condiions (A), (B), and (H). hen all soluions of y + f (, y σ (), y τ 1 (), y τ 2 (), y ξ 1 (), y ξ 2 () ) = 0 (3.14) are oscillaory in case ( f, α, αkg 1 (), αkg 2 (), α k h 1(), α ) 2 k h 2() 2 = (3.25) holds for all α 0, where k is he consan appearing in condiion (H). Proof. If (3.25) holds for all α 0, assume u() be a nonoscillaory soluion of (3.14) wih u() > 0, u(τ 1 ()) > 0, u(τ 2 ()) > 0 for. As in he proof of heorem 3.3.5, given 0 < α < 1 from condiion (H) here exiss α such ha ( u + f, u σ (), αg 1 ()u σ (), αg 2 ()u σ () 1 α h 1()u σ (), 1 ) α h 2()u σ () 0, α. (3.35) Hence we obain a soluion y() of ( y + f, y σ (), αg 1 ()y σ (), αg 2 ()y σ (), 1 α h 1()y σ (), 1 ) α h 2()y σ () = 0 (3.36) wih 0 < u( α ) y() u(), α. We nex define he coninuously differeniable

45 39 real-valued funcion G(u) := u u 0 ds φ(s), where u 0 := y( α ) > 0. Observe ha G (u) = 1/φ(u). By he Pözche Chain Rule, ( 1 (G(y()) = 0 ) ( dh 1 y () φ(y h ()) 0 ) dh y () = φ(y σ () y () φ(y σ ()) where y h () := y() + hµ()y () y σ (). Since φ is nondecreasing, we have 1 φ(y h ()) 1 φ(y σ ()). Consequenly, (G(y())) y () φ(y σ ())) (3.37) Furhermore, since y() > 0 and y () is nonincreasing, lim y () = L 1 wih 0 L 1 <. Now inegraing (3.36) for α gives 0 = y () y ) ( + f s, y σ (s), αg 1 (s)y σ (s), αg 2 (s)y σ (s), 1 α h 1(s)y σ (s), 1 ) α h 2(s)y σ (s) s and leing in he above, we obain y ( ) ( = L 1 + f s, y σ (s), αg 1 (s)y σ (s), αg 2 (s)y σ (s), 1 α h 1(s)y σ (s), 1 ) α h 2(s)y σ (s) s ( f s, y σ (s), αg 1 (s)y σ (s), αg 2 (s)y σ (s), 1 α h 1(s)y σ (s), 1 ) α h 2(s)y σ (s) s ( > f s, y σ (s), αg 1 (s)y σ (s), αg 2 (s)y σ (s), 1 α h 1(s)y σ (s), 1 ) α h 2(s)y σ (s) s.