Pacific Journal of Mathematics

Transcription

1 Pacific Journal of Mathematic OSCILLAION AND NONOSCILLAION OF FORCED SECOND ORDER DYNAMIC EQUAIONS MARIN BOHNER AND CHRISOPHER C. ISDELL Volume 230 No. March 2007

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3 PACIFIC JOURNAL OF MAHEMAICS Vol. 230, No., 2007 OSCILLAION AND NONOSCILLAION OF FORCED SECOND ORDER DYNAMIC EQUAIONS MARIN BOHNER AND CHRISOPHER C. ISDELL Ocillation and nonocillation propertie of econd order Sturm Liouville dynamic equation on time cale for example, econd order elf-adjoint differential equation and econd order Sturm Liouville difference equation have attracted much interet. Here we conider a given homogeneou equation and a correponding equation with forcing term. We give new condition implying that the latter equation inherit the ocillatory behavior of the homogeneou equation. We alo give new condition that introduce ocillation of the inhomogeneou equation while the homogeneou equation i nonocillatory. Finally, we explain a gap in a reult given in the literature for the continuou and the dicrete cae. A more ueful reult i preented, improving the theory even for the correponding continuou and dicrete cae. Example illutrating the theoretical reult are upplied.. Introduction he theory of dynamic equation on time cale continue to be a rapidly growing area of reearch. Behind the main motivation for the ubject lie the key concept that dynamic equation on time cale repreent a way of unifying and extending continuou and dicrete analyi. In thi paper, we conider the econd order linear dynamic equation () (c(t)x ) + q(t)x σ = 0 together with an inhomogeneou equation of the form (2) (c(t)u ) + q(t)u σ = f (t). Equation () and (2) are o-called dynamic equation on a time cale. hroughout thi paper we aume that c, q, and f are rd-continuou real-valued function defined on the time cale uch that c(t) = 0 for all t and f i not eventually MSC2000: primary 34C0, 34K, 39A; econdary 39A0, 39A2. Keyword: dynamic equation, generalized zero, ocillation, nonocillation, inhomogeneou equation, time cale. Supported by the Autralian Reearch Council Dicovery Project (DP ). 59

4 60 MARIN BOHNER AND CHRISOPHER C. ISDELL identically equal to zero. No further aumption on the ign of thee function are impoed. Since we are intereted in ocillatory behavior of olution of () and (2), we aume that the time cale i unbounded above. he etup of thi paper i a follow. In Section 2, we give ome preliminarie concerning the time cale calculu. In Section 3, we introduce the Komkov tranformation and preent ome baic reult about equation () and (2). In Section 4, given that () i nonocillatory, we offer criteria that introduce ocillation in (2) and alo criteria that preerve nonocillation in (2). Finally, in Section 5, we explain a gap in a reult given in the continuou cae by Rankin [979, heorem ] and in the dicrete cae by Grace and El-Morhedy [997, heorem 2.]. A more ueful reult i preented, hence improving the theory even for the correponding continuou and dicrete cae. hroughout, relevant example illutrating the theoretical reult are upplied. 2. he time cale calculu In thi ection we preent ome definition and elementary reult connected to the time cale calculu. For further tudy we refer the reader to [Bohner and Peteron 200; 2003]. A time cale i an arbitrary nonempty cloed ubet of the real number. On we define the forward and backward jump operator by σ (t) := inf { : > t} and ρ(t) := up { : < t} for t. A point t with t > inf i aid to be left-dene if ρ(t) = t and right-dene if σ (t) = t, left-cattered if ρ(t) < t and right-cattered if σ (t) > t. Next, the grainine function µ i defined by µ(t) := σ (t) t for t. For a function f : the (delta) derivative f (t) at t i defined to be the number (provided it exit) with the property uch that for every ε > 0 there exit a neighborhood U of t with f (σ (t)) f () f (t)(σ (t) ) ε σ (t) for all U. A ueful formula i (3) f σ = f + µf, where f σ := f σ. We will ue the product rule and the quotient rule for the derivative of the product f g and the quotient f/g (if gg σ = 0) of two differentiable function f and g ( ) f (4) ( f g) = f g + f σ g = f g + f g σ and = f g g f g gg σ. For a, b and a function f :, the Cauchy integral of f i defined by (5) b a f (t) t = F(b) F(a),

5 FORCED SECOND ORDER DYNAMIC EQUAIONS 6 where F i an antiderivative of f, i.e., F = f hold. he function f : i called rd-continuou if it i continuou in right-dene point and if the leftided limit exit in left-dene point. Hilger main exitence theorem [Bohner and Peteron 200, heorem.74] ay that rd-continuou function poe antiderivative. If p : i rd-continuou and regreive (i.e., + µ(t)p(t) = 0 for all t ), then another exitence theorem ay that the initial value problem y = p(t)y, y(t 0 ) = (where t 0 ) poee a unique olution e p (, t 0 ). Example. Note that in the cae = we have and in the cae = we have σ (t) = t, µ(t) 0, f (t) = f (t), σ (t) = t +, µ(t), f (t) = f (t) = f (t + ) f (t). Another important time cale i = q 0 := {q k : k 0 } with q >, for which σ (t) = qt, µ(t) = (q )t, f (t) = f (qt) f (t), (q )t and thi time cale give rie to o-called q-difference equation. 3. Generalized zero and the Komkov tranformation We ay that a olution x of () (or (2)) ha a generalized zero in [t, σ (t)] if c(t)x(t)x(σ (t)) 0. Next, x i called ocillatory provided [, ) contain infinitely many zero for each. Otherwie we ay that x i nonocillatory. he equation () (or (2)) i called ocillatory if all olution of () (or (2)) are ocillatory. Otherwie we ay that () (or (2)) i nonocillatory. It i a well-known fact that () i ocillatory if and only if it ha an ocillatory olution. he proof i eay: Suppoe x i a nonocillatory olution of (), i.e., cxx σ > 0 on [, ) for ome > 0. Let x be any olution of () uch that x and x are linearly independent. hen ( x/x) = W (x, x)/(cxx σ ) by the quotient rule (4), where W (x, x) := c( x x x x), the Wronkian, i actually equal to a nonzero contant (ue the product rule (4) to verify thi). Hence x/x i eventually trictly monotone, and therefore it i eventually of one ign. hu (c x x σ )/(cxx σ ) = ( x/x)( x σ /x σ ) i eventually poitive, and hence c x x σ > 0 eventually, meaning that x i nonocillatory a well. In contrat to (), it i not true that (2) i ocillatory if and only if it ha an ocillatory olution. We upply the following example. Example 2. Suppoe x olve () uch that x(t) for all t and uch that for all t there exit t, t 2 t with x(t ) = and x(t 2 ) =. hen u := +x/2

6 62 MARIN BOHNER AND CHRISOPHER C. ISDELL i a nonocillatory olution of (2) with f = q while u 2 := + 2x i an ocillatory olution. Example 3. Conider the econd order linear dynamic equation (6) ( µ(t)u ) + 4 µ(t) uσ = 4 µ(t) on an iolated time cale (i.e., each point i left-cattered and right-cattered). A olution of the correponding homogeneou equation i x = e 2/µ (, t 0 ), where t 0. Since x σ = x, thi olution i ocillatory. Hence the correponding homogeneou equation i ocillatory. However, u := + x/2 i a nonocillatory olution of (6) while u 2 := + 2x i an ocillatory olution of (6). Hence (6) poee both ocillatory and nonocillatory olution. he tranformation u = xy, where x olve () and u olve (2), wa tudied by Komkov [972] and ha been uccefully applied, for example, in [Grace and El-Morhedy 997; Patula 979; Rankin 979]. Our reult given in thi paper mainly rely on the following eay but ueful identity. We abbreviate the operator (cx ) + qx σ by Lx. Lemma. If u = xy, then (7) W (x, u) = cxx σ y and [W (x, u)] = x σ Lu u σ Lx. In particular, if x olve () and u olve (2), then (8) [W (x, u)] = (cxx σ y ) = f x σ, and if in addition x(t) = 0 for all t, then (9) y(t) = y( ) + c( )x( )x(σ ( ))y ( ) + c()x()x(σ ()) c()x()x(σ ()) f (τ)x(σ (τ)) τ for all t. Proof. We apply the product rule (4) to u = xy to find u = x y + x σ y and cxx σ y = cxu cxx y = cu x cx u = W (x, u). hen, uing the product rule again, we find (cxx σ y ) = (cu ) x σ + cu x (cx ) u σ cx u = ( Lu qu σ ) x σ ( Lx qx σ ) u σ = x σ Lu u σ Lx.

7 FORCED SECOND ORDER DYNAMIC EQUAIONS 63 Hence, if Lx = 0 and Lu = f, (8) follow. By uing the definition (5) of the integral, we conclude that (9) hold. Concluding thi ection, we ue Lemma to derive the following reult. For the continuou verion ee [Rankin 979, Equation (3 )] and for the dicrete verion ee [Patula 979, heorem 6] and [Grace and El-Morhedy 997, Lemma 2.]. heorem. Suppoe x olve () and u olve (2). If W (x, u) i eventually of one ign (either poitive or negative), then x ocillate if and only if u ocillate. Proof. Uing (7), we know that W (x, u) = cxx σ y i eventually of one ign, where u = xy. Firt uppoe x i not ocillatory, i.e., cxx σ > 0 eventually. Hence y i eventually of one ign. herefore y i eventually of one ign. Hence 0 < yy σ = u u σ x x σ = cuuσ cxx σ eventually, o eventually cuu σ > 0, i.e., u i not ocillatory. Similarly (by conidering the tranformation x = uỹ) we may how that if u i not ocillatory, then x i not ocillatory either. Corollary. Suppoe c(t) > 0 for all t. If () i nonocillatory and f i eventually of one ign, then (2) i nonocillatory. Proof. Let x be any (nonocillatory) olution of () o x i eventually of one ign. Suppoe u i any olution of (2) and let y = u/x. By (8), [W (x, u)] i eventually of one ign, and hence W (x, u) i eventually of one ign. hu u i nonocillatory according to heorem. Corollary 2. Suppoe () i ocillatory (nonocillatory). If there exit a olution x of () uch that f (t)x(σ (t)) t = then (2) i ocillatory (nonocillatory). or f (t)x(σ (t)) t =, Proof. Suppoe u i a olution of (2) and define y by u = xy. By (8), W (x, u)(t) = W (x, u)( ) + f ()x(σ ()). Hence W (x, u) i eventually of one ign, and the claim follow with heorem. Example 4. Conider the Fibonacci difference equation x(t+2) = x(t+)+x(t), i.e., ( ( ) t+ x(t) ) +( ) t+ x(t+) = 0, t. If a = ( + 5)/2, then x(t) = a t i a olution of thi equation. Since c(t)x(t)x(t + ) = ( a)( a 2 ) t,

8 64 MARIN BOHNER AND CHRISOPHER C. ISDELL the equation i ocillatory. Now, ince τ=0 aτ+ =, Corollary 2 implie that x(t + 2) = x(t + ) + x(t) + ( ) t, i.e., ( ( ) t+ x(t) ) + ( ) t+ x(t + ) =, i alo ocillatory. 4. Ocillation and nonocillation criteria he next theorem generalize a reult due to Rankin for = [Rankin 979, heorem 2] and a reult due to Grace and El-Morhedy for = [Grace and El-Morhedy 997, heorem 2.2]. heorem 2. Suppoe x i an eventually nonocillatory olution of (). If for ome ufficiently large, (0) () and (2) lim inf t lim up t then (2) i ocillatory. c()x()x(σ ()) t c(t)x(t)x(σ (t)) <, c()x()x(σ ()) f (τ)x(σ (τ)) τ =, f (τ)x(σ (τ)) τ =, Proof. Suppoe u i an eventually nonocillating olution of (2) uch that y = u/x i eventually of one ign (note that yy σ = (cuu σ )/(cxx σ ) > 0). But (9) together with (0), (), and (2) enure that lim inf t y(t) = and lim up t y(t) =. hi i a contradiction, and therefore there cannot exit an eventually nonocillating olution of (2). hu (2) i ocillatory. Example 5. Let q > and conider the q-difference equation (ee Example ) (3) u = ( ) log q t, t q 0 := { q k : k 0 }. One olution of the correponding homogeneou equation i x(t) = t, o the homogeneou equation i nonocillatory and (0) i atified ince c()x()x(σ ()) = t a t. Some calculation now how that q 2 qτ( ) log q τ (q ) 2 τ = q 3 + q 2 + q + t( )log q t,

9 FORCED SECOND ORDER DYNAMIC EQUAIONS 65 o () and (2) are atified. Hence, by heorem 2, (3) i ocillatory. Now we preent an improvement of heorem. For =, ee [Grace and El-Morhedy 997, heorem 2.3]. heorem 3. Suppoe x olve () uch that x(t) = 0 for all t and (4) If (5) c()x()x(σ ()) t c(t)x(t)x(σ (t)) i bounded above or below. f ()x(σ ()) t =, then () and (2) either are both ocillatory or both nonocillatory. Proof. Let u be a olution of (2). By heorem, we may aume that W (x, u) i ocillating. Firt uppoe that the integral in (4) i bounded below. Let and D > 0 be uch that W (x, u)( ) 0 hen, by (9), and y(t) y( ) DW (x, u)( ) + D for all t. c()x()x(σ ()) c()x()x(σ ()) f (τ)x(σ (τ)) τ, o y(t) a t by (5). Hence y > 0 eventually and cuu σ ha eventually the ame ign a cxx σ. If, however, the integral in (4) i bounded above, then we pick and E > 0 uch that W (x, u)( ) 0 and E for all t. c()x()x(σ ()) In thi cae the concluion now follow a in the previou cae. he lat reult in thi ection i a nonocillation criterion. We refer to [Grace and El-Morhedy 997, heorem 3.] for =. he following auxiliary reult i needed. Lemma 2. Suppoe () i nonocillatory. hen there exit a olution x of () atifying (0). Proof. Let x be any (nonocillatory) olution of (). If x atifie (0), then we are done. If not, then /(c()x()x(σ ())) =. Let x be any olution of () uch that x and x are linearly independent, i.e., W (x, x) k = 0. hen

10 66 MARIN BOHNER AND CHRISOPHER C. ISDELL ( x/x) = k/(cxx σ ), o ( x/x)(t) ± a t and hence (x/ x)(t) 0 a t. hu (x/ x) = k/(c x x σ ) and therefore k c() x() x(σ ()) = x( ) x( ) x(t) x(t) x( ) x( ) a t. Hence x olve () and atifie (0). Below we ue for α the notation α + = max{0, α} and α = min{0, α}. heorem 4. Suppoe c(t) > 0 and q(t) > 0 for all t. Suppoe that all olution of () are nonocillatory and bounded. If (6) and (7) c() f + (τ) τ = f (τ) τ >, then (2) i nonocillatory. Proof. Let λ > 0 be uch that the integral in (7) i bounded below by λ. By Lemma 2, there exit a olution x of () atifying (0). Since all olution of () are bounded, there exit M > 0 uch that x(t) < M for all t. We will how that (5) i atified. hen heorem 3 i employed to complete the proof. Firt, putting z = cx, we ee that z = qx σ i eventually of one ign. hu z i eventually of one ign. Hence x i eventually of one ign. herefore x i eventually increaing or eventually decreaing. Since x i nonocillatory, it i either eventually poitive or eventually negative. So there are the following four poibilitie: (i) x i eventually poitive and increaing; (ii) x i eventually poitive and decreaing; (iii) x i eventually negative and increaing; (iv) x i eventually negative and decreaing. If (iii) or (iv) hold, then we may replace x by x, which i alo a olution of () that atifie (0) and (i) or (ii). hu it i ufficient to dicu the cae (i) and (ii). If (i) hold, then f (τ)x(σ (τ)) τ c()x()x(σ ()) x(σ ( )) t M 2 f + (τ) τ λm c() c()x()x(σ ()) (6) a t

11 FORCED SECOND ORDER DYNAMIC EQUAIONS 67 (ue f = f + + f ), while if (ii) hold, then f (τ)x(σ (τ)) τ c()x()x(σ ()) f + (τ)x(σ ()) τ λm c()x()x(σ ()) f + (τ) τ λm M c() (6) a t. c()x()x(σ ()) c()x()x(σ ()) Hence (5) hold in either cae and the proof i complete. 5. Remark on the reult of Rankin, Grace and El-Morhedy he continuou verion of the following reult wa proved by Rankin [979, heorem ], while it dicrete verion wa given by Grace and El-Morhedy in [997, heorem 2.]. heorem 5. Suppoe x i an eventually nonocillatory olution of (). If for ufficiently large and ome M > 0, (8) (9) and (20) lim inf t f ()x(σ ()) =, c()x()x(σ ()) for all t, then (2) i ocillatory. t c(t)x(t)x(σ (t)) =, lim up t f (τ)x(σ (τ)) τ M f ()x(σ ()) =, c()x()x(σ ()) Below we how that the aumption of heorem 5 are never atified. hi mean that, although heorem 5 i true, the reult i not meaningful. heorem 6. he aumption of heorem 5 are never atified. Proof. Note firt that (2) ha a olution. Let u be any olution of (2) and define y by u = xy. Let uch that c(t)x(t)x(σ (t)) 0 for all t. By (8), we have c(t)x(t)x(σ (t))y (t) = c( )x( )x(σ ( ))y ( ) + o (9) implie f (τ)x(σ (τ)) τ, (2) lim inf c(t)x(t)x(σ t (t))y (t) =, lim up c(t)x(t)x(σ (t))y (t) =. t

12 68 MARIN BOHNER AND CHRISOPHER C. ISDELL By the firt relation in (2), there exit with c( )x( )x(σ ( ))y ( ) < 2M. hu, uing (8), we find c(t)x(t)x(σ (t))y (t) = c( )x( )x(σ ( ))y ( ) + < 2M + f (τ)x(σ (τ)) τ and hence y(t) < y( ) 2M (20) t y( ) M f (τ)x(σ (τ)) τ t c()x()x(σ ()) + f (τ)x(σ (τ)) τ c()x()x(σ ()) (8) a t. c()x()x(σ ()) By the econd relation in (2), there exit with c( )x( )x(σ ( ))y ( ) > 2M. hu, uing (8), we find c(t)x(t)x(σ (t))y (t) = c( )x( )x(σ ( ))y ( ) + > 2M + f (τ)x(σ (τ)) τ and hence y(t) > y( )+2M (20) t y( ) + M f (τ)x(σ (τ)) τ t c()x()x(σ ()) + f (τ)x(σ (τ)) τ c()x()x(σ ()) (8) a t. c()x()x(σ ()) hi i a contradiction, a y(t) and y(t) at the ame time for t. Example 6. Rankin [979, Example 2] tated that heorem 5 for the cae = can be ued to how that (22) u = t in t, t i ocillatory. Here we let x(t). Clearly, condition (8) and (9) are atified. A imple calculation how that τ in τ d τd (2 + 4)(t ),

13 FORCED SECOND ORDER DYNAMIC EQUAIONS 69 o (20) i atified if M i allowed to depend on. However, if M = M( ), then the proof of heorem 5 (and heorem 6) break down. Furthermore, the equation (22) i in fact not ocillatory: Clearly, u (t) = 4 2 co(t) + t(2 in t) and u 2 (t) = t in t 2 co t both are olution of (22), and u i nonocillatory while u 2 i ocillatory. Example 7. We note that Grace and El-Morhedy [997] did not upply an example to illutrate heorem 5 for the cae =. Conider the difference equation (23) 2 u = ( ) t+ (2t + ), t. Here we let x(t). Clearly, condition (8) and (9) are atified. A imple calculation how that t ( ) τ+ (2τ + ) 2 (t ) = τ= for, o (20) i atified if M i allowed to depend on. Furthermore, the equation (22) i in fact not ocillatory: Clearly, [ ] [ ] t t u (t) = t + ( ) t+ and u 2 (t) = ( ) t+, 2 2 where [x] denote the larget integer le than or equal to x, both are olution of (22), and u i nonocillatory while u 2 i ocillatory. We now preent the following reult. heorem 7. Let. Aume x i any olution of () with c(t)x(t)x(σ (t)) > 0 for all t. If (8) hold and if there exit M > 0 uch that (20) i atified, then (2) i not ocillatory. Proof. Define y(t) := 2M t c()x()x(σ ()) + f (τ)x(σ (τ)) τ. c()x()x(σ ()) Uing the product and the quotient rule (4), it i eay to check that u defined by u := yx i a olution of (2). However, (20) enure that lim t y(t) =, and therefore u i a nonocillatory olution of (2). hu (2) cannot be ocillatory. heorem 8. Let. Aume x i any olution of () with c(t)x(t)x(σ (t)) > 0 for all t. If (8) hold, if there exit M > 0 uch that (20) i atified, and if () and (2) hold, then (2) ha both ocillatory and nonocillatory olution.

14 70 MARIN BOHNER AND CHRISOPHER C. ISDELL Proof. Define y (t) := 2M t c()x()x(σ ()) + c()x()x(σ ()) f (τ)x(σ (τ)) τ and y 2 (t) := f (τ)x(σ (τ)) τ. c()x()x(σ ()) A in the proof of heorem 7, it i eay to check that u and u 2 defined by u := y x and u 2 := y 2 x both are olution of (2). While u i nonocillatory, u 2 i ocillatory. Hence (2) indeed ha both ocillatory and nonocillatory olution. Our next reult can be checked eaily a in the proof of heorem 7. heorem 9. Suppoe that the olution x of () atifie x(t) = 0 for all t. hen the olution of (2) atifying the initial condition u( ) = α and u ( ) = β i given by u := yx, where y(t) = γ + δ c()x()x(σ ()) + c()x()x(σ ()) where γ = α/x( ) and δ = c( )(βx( ) αx ( )). he following generalization of heorem 2 now become apparent. f (τ)x(σ (τ)) τ, heorem 0. Suppoe x i a nonocillatory olution of (). If for ome, and lim inf t lim up t then (2) i ocillatory. c()x()x(σ ()) c()x()x(σ ()) c()x()x(σ ()) f (τ)x(σ (τ)) τ c()x()x(σ ()) Reference f (τ)x(σ (τ)) τ = =, [Bohner and Peteron 2003] M. Bohner and A. Peteron (editor), Advance in dynamic equation on time cale, Birkhäuer, Boton, MR 2004d:34003 Zbl [Bohner and Peteron 200] M. Bohner and A. Peteron, Dynamic equation on time cale: An introduction with application, Birkhäuer, Boton, 200. MR 2002c:34002 Zbl [Grace and El-Morhedy 997] S. R. Grace and H. A. El-Morhedy, Ocillation and nonocillation theorem for certain econd-order difference equation with forcing term, J. Math. Anal. Appl. 26:2 (997), MR 99d:39005 Zbl

15 FORCED SECOND ORDER DYNAMIC EQUAIONS 7 [Komkov 972] V. Komkov, A technique for the detection of ocillation of econd order ordinary differential equation, Pacific J. Math. 42 (972), MR 47 #234 Zbl [Patula 979] W.. Patula, Growth, ocillation and comparion theorem for econd-order linear difference equation, SIAM J. Math. Anal. 0 (979), MR 80j:39004 Zbl [Rankin 979] S. M. Rankin, Ocillation reult for a nonhomogeneou equation, Pacific J. Math. 80: (979), MR 80h:34044 Zbl Received June 7, MARIN BOHNER DEPARMEN OF MAHEMAICS UNIVERSIY OF MISSOURI ROLLA, MO 6540 UNIED SAES bohner@umr.edu CHRISOPHER C. ISDELL SCHOOL OF MAHEMAICS AND SAISICS HE UNIVERSIY OF NEW SOUH WALES SYDNEY, NSW 2052 AUSRALIA cct@math.unw.edu.au

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