Selçuk J. Appl. Math. Selçuk Journal of Vol. 7. No. 1. pp. 25-31, 2005 Applied Mathematics
A Philos Criterion for Second-order Dynamic Equations Martin Bohner and Howard Warth
Department of Mathematics, Rolla Building, University of Missouri, Rolla e-mail: b ohner@ um r.edu, hlw6c2@ um r.edu
Received: November 17, 2005
Summary. A Philos type condition sufficient to guarantee oscillation of second-order dynamic equations is presented.
Introduction
Second-order ordinary linear differential equations have a long research history. Second-order dynamic equations, a generalized version of these equations, have been the object of more recent research. Ordinary differential equations are studied over intervals of the real line. Dynamic equations, on the other hand, are studied on arbitrary closed subsets of the real line. We will define oscillation and then show how oscillation conditions formulated for linear differential equations can be generalized and applied to second-order dynamic equations.
Preliminaries
To understand the theory of oscillation for second-order dynamic equations re-quires some background in both the theory of oscillation for linear differential equations and in the theory of dynamic equations on time scales. Necessary definitions and theory will now be introduced. First consider the second-order linear differential equation
(1) 00() + ()() ≥ 0
where is a continuous real-valued function on the interval [0 ∞) without
any restriction on its sign. A solution of (1) is said to be oscillatory if it has arbitrarily large zeros, and otherwise it is said to be nonoscillatory. Equation (1) is said to be oscillatory if all its solutions are oscillatory. Many authors have studied the differential equation (1) in order to determine criteria necessary to guarantee that it is oscillatory. Many oscillation criteria have been found which involve integrals containing the function . Important results along these lines
have been produced by Wintner [11], Hartman [5], Coles [4], and Willett [11]. Kamenev [7], in 1976, established a new integral criterion for the oscillation of the differential equation (1) which has the result of Wintner as a special case. In particular, Kamenev proved that
(2) lim sup →∞ 1 −1 Z 0 ( − )−1() = ∞
for some integer ≥ 2 is sufficient to guarantee the oscillation of equation (1). In 1986, Yan [13] found additional criteria that guarantee that equation (1) is oscillatory when Kamenev’s condition (2) does not hold. Subsequently, Kamenev’s criterion has been extended by Philos [8,9,19], Yan [12,13], and Yeh [14,15]. In this paper we build upon Philos’ extension of Kamenev’s criterion. In particular we will see that Philos’ criterion will be a particular case of our Corollary 1.
Next, we will provide a brief introduction to the time scales calculus and the theory of dynamic equations on time scales. For a thorough intoduction to the theory of dynamic equations on time scales see [2,3]. The calculus of time scales was initiated by Stefan Hilger in [6] in order to create a theory that could unify discrete and continuous calculus.
Definition 1. A time scale is an arbitrary closed subset of the real numbers. In this paper, the time scale is assumed to be unbounded above.
The two most familiar examples of time scales are the real numbers, T = R, and the integers, T = Z.
Definition 2. Let T be a time scale. For ∈ T we define the forward jump operator : T → T by () := inf { ∈ T : }. For a function : T → R we define () := (()) for all ∈ T.
Definition 3. Now we consider a function : T → R. We define the delta (or Hilger ) derivative of at a point ∈ T as follows. The delta derivative, ∆()
is the number (if it exists) such that given any ≥ 0, there is a neighborhood of such that
¯
¯[(()) − ()] − ∆
()[() − ]¯¯ ≤ |() − | ∈ Similarly, if we have a function ( ) of two variables, then we can define the partial delta derivative of with respect to , for example, as follows.
Definition 4.Assume : T2 → R and let ∈ T. Then we define ∆( )
to be the number, if it exists, with the property that given ≥ 0, there is a neighborhood of , with fixed, such that
¯
Definition 5. For : T → R we define ( ) = ( ()) for all ∈ T.
Consider now the second-order dynamic equation
(3) ¡()∆¢∆+ () = 0 [0 ∞)
where is an rd-continuous positive function on [0 ∞) and is an rd-continuous
function on [0 ∞) without any restriction on its sign.
Definition 6. A solution of equation (1) is said to be oscillatory if it has arbi-trarily large zeros, and otherwise it is said to be nonoscillatory. The equation (3) is said to be oscillatory if all its solutions are oscillatory.
Philos [10] (1989) proved the following theorem which generalizes Kamenev’s oscillation criterion (2) for equation (1).
Let : := {( ) : ≥ ≥ 0} → R be a continuous function such that
( ) = 0 ≥ 0 ( ) 0 ≥ 0
and has a continuous and nonpositive partial derivative on with respect to the second variable. Moreover, let : → R be a continuous function with
−( ) = ( )p( ) all ( ) ∈ Then equation (1) is oscillatory if
(4) lim sup →∞ 1 ( 0) Z 0 ∙ ( )() −142( )() ¸ = ∞
3. The Philos Criterion
Before we show how the Philos criterion (4) can be generalized to time scales we will need the following recent lemma (Akın-Bohner, Bohner, and Saker Lemma 3.1) from the time scales calculus.
Lemma 1. Let : T → R and : T → R \ {0} be two functions. Then (∆)2= ∆ µ 2 ¶∆ + ∙³ ´∆¸2
Using this lemma, with ( ·) and replacing and , we can now prove the following main result of this paper.
Theorem 1. Let : := {( ) ∈ T2 : ≥ ≥
0} → R be a continuous
function such that
and has a continuous and nonpositive partial derivative on with respect to the second variable, i.e., ∆( ) ≤ 0. Then (3) is oscillatory if
(5) lim sup →∞ 1 2( 0) Z 0 h (( ))2() −¡∆( )¢2() i ∆ = ∞
Proof. Let be a nonoscillatory solution of the dynamic equation (3) and 0 ≥ 0 be such that () 6= 0 for all ≥ 0. Define the Riccati substitution
= ∆ on [
0 ∞). After making the Riccati substitution and then using
the quotient rule, it follows from (3) that = −∆−(
∆)2
[0 ∞)
Thus, for every with ≥ ≥ 0, we obtain, using integration by parts,
Z (( ))2()∆ = Z 2( ()) ∙ −4() −()( 4())2 ()4() ¸ ∆ = −2( )()¯¯= + Z ¡ 2¢∆ ( )()∆ − Z 2( ())()( ∆())2 ()() ∆ = − Z ()∆() ()() h 2( ())∆() −¡2¢∆( )()i∆ +2( )( ) = − Z ()∆() ()() h 2( )∆() −¡2¢∆ ( )()i∆ +2( )( ) = Z ()∆() µ 2( ·) ¶∆ ()∆ + 2( )( ) = − Z () ⎡ ⎣()() (µ ( ·) ¶∆ () )2 −¡∆( )¢2 ⎤ ⎦ ∆ +2( )( ) ≤ 2( )( ) + Z ¡ ∆( )¢2()∆
where for the last equal sign we have used Lemma 1. Therefore, for every ≥ 0,
Z 0 h (( ))2() −¡∆( )¢2()i∆ ≤ 2( 0)(0) ≤ 2( 0)|(0)| ≤ 2( 0)|(0)|
Thus, Z 0 h (( ))2() −¡4( )¢2() i ∆ = Z 0 0 h (( ))2 () −¡∆( )¢2()i∆ + Z 0 h (( ))2() −¡∆( )¢2() i ∆ ≤ Z 0 0 2( ())|()|∆ + 2( 0)|(0)| ≤ 2( 0) "Z 0 0 |()|∆ + |(0)| #
for all ≥ 0. This gives
lim sup →∞ 1 2( 0) Z 0 £ (( ))2 () − (∆( ))2()¤ ∆ ≤ Z 0 0 |()|∆+|(0)| which contradicts (5).
Example 1. Assume that the conditions of Theorem 2 hold. Let T = R. Then ∆( ) =
( ), and the oscillation condition (5) becomes
lim sup →∞ 1 2( 0) Z 0 " 2( )() − µ ( ) ¶2 () # = ∞
Example 2. Assume that the conditions of Theorem 2 hold. Let T = Z. Let ∆( ) be the usual forward difference operator with respect to the second
variable. Then ∆( ) = ∆
( ), and the oscillation condition (5) becomes
lim sup →∞ 1 2( 0) −1 X =0 h 2( + 1)() − (∆( ))2() i = ∞
Corollary 1. Let : → R be a continuous function which is such that ( ) = 0 ≥ 0 ( ) 0 ≥ 0
and has a continuous and nonpositive partial delta derivative on with respect to the second variable. Moreover, let : → R be a continuous function with
−∆( ) =( )
2 hp
Then equation (3) is oscillatory if (6) lim sup →∞ 1 ( 0) Z 0 ∙ ( )() −1 4 2( )() ¸ ∆ = ∞
Proof. Define the function ( ) such that ( ) =p( ), where ( ) is defined as in Theorem 2. Then ( ) = 0 for ≥ 0 and ( ) ≥ 0 for
≥ 0. By the time scales identity for taking the delta derivative of a square
root, ∆( ) = ∆( ) p ( ) +p( ()) = − ( ) 2
Clearly, ( ) = p( ). Substituting the above expressions for
and ∆ into condition (5) yields the desired result.
Example 3. If, in Corollary 1, we let T = R, then −∆( ) = −
( ) = ( ) p
( )
and ( ) = ( ). Therefore condition (6) reduces to Philos’ oscillation
criterion (1)
Example 3. If, in Corollary 1, we let T = Z, then ( ) = ( + 1) and
∆( ) = −∆( ) = − [( + 1) − ( )] = ( ) 2 hp ( ) +p( + )i and so condition (6) becomes
lim sup →∞ 1 ( 0) −1 X =0 ∙ ( + 1)() −142( )() ¸ = ∞ 4. References
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