Nonoscillation of First-Order Dynamic Equations with Several Delays
© Elena Braverman and Başak Karpuz. 2010
Received: 18 February 2010
Accepted: 21 July 2010
Published: 2 August 2010
For dynamic equations on time scales with positive variable coefficients and several delays, we prove that nonoscillation is equivalent to the existence of a positive solution for the generalized characteristic inequality and to the positivity of the fundamental function. Based on this result, comparison tests are developed. The nonoscillation criterion is illustrated by examples which are neither delay-differential nor classical difference equations.
Oscillation of first-order delay-difference and differential equations has been extensively studied in the last two decades. As is well known, most results for delay differential equations have their analogues for delay difference equations. In , Hilger revealed this interesting connection, and initiated studies on a new time-scale theory. With this new theory, it is now possible to unify most of the results in the discrete and the continuous calculus; for instance, some results obtained separately for delay difference equations and delay-differential equations can be incorporated in the general type of equations called dynamic equations.
Although we further assume familiarity of readers with the notion of time scales, we would like to mention that any nonempty, closed subset of is called a time scale, and that the forward jump operator is defined by for , where the interval with a subscript is used to denote the intersection of the real interval with the set . Similarly, the backward jump operator is defined to be for , and the graininess is given by for . The readers are referred to  for an introduction to the time-scale calculus.
then is finite, since asymptotically tends to infinity. By a solution of (1.3), we mean a function such that and (1.3) is satisfied on identically. For a given function , (1.3) admits a unique solution satisfying on (see [5, Theorem 3.1]). As usual, a solution of (1.3) is called eventually positive if there exists such that on , and if is eventually positive, then is called eventually negative. A solution, which is neither eventually positive nor eventually negative, is called oscillatory, and (1.3) is said to be oscillatory provided that every solution of (1.3) is oscillatory.
Theorem A is the generalization of the well-known oscillation results stated for and in the literature (see [8, Theorems and ]). In , Bohner et al. used an iterative method to advance the sufficiency condition in Theorem A, and in [10, Theorem ] Agwo extended Theorem A to (1.3). Further, in , Şahiner and Stavroulakis gave the generalization of a well-known oscillation criterion, which is stated below.
Theorem B (see [11, Theorem ]).
Then every solution of (1.6) is oscillatory.
The present paper is mainly concerned with the existence of nonoscillatory solutions. So far, only few sufficient nonoscillation conditions have been known for dynamic equations on time scales. In particular, the following theorem, which is a sufficient condition for the existence of a nonoscillatory solution of (1.3), was proven in .
Theorem C (see [7, Theorem ]).
where satisfies for all . Then, (1.6) has a nonoscillatory solution.
In [10, Theorem , and Corollary 3.3], Agwo extended Theorem C to (1.3).
Theorem D (see [10, Corollary 3.3]).
where on . Then, (1.3) has a nonoscillatory solution.
As was mentioned above, there are presently only few results on nonoscillation of (1.3); the aim of the present paper is to partially fill up this gap. To this end, we present a nonoscillation criterion; based on it, comparison theorems on oscillation and nonoscillation of solutions to (1.3) are obtained. Thus, solutions of two different equations and/or two different solutions of the same equation are compared, which allows to deduce oscillation and nonoscillation results.
The paper is organized as follows. In Section 2, some important auxiliary results, definitions and lemmas which will be needed in the sequel are introduced. Section 3 contains a nonoscillation criterion which is the main result of the present paper. Section 4 presents comparison theorems. All results are illustrated by examples on "nonstandard" time scales (which lead to neither differential nor classical difference equations).
2. Definitions and Preliminaries
where , is the initial point, is the initial value, is the initial function such that has a finite left-sided limit at the initial point provided that it is left-dense, is the forcing term, and is the coefficient corresponding to the delay function for all . We assume that for all , , is a delay function satisfying , and for all . We recall that is finite, since for all .
For convenience in the notation and simplicity in the proofs, we suppose that functions vanish out of their specified domains, that is, let be defined for some , then it is always understood that for , where is the characteristic function of defined by for and for .
which satisfies , is called the fundamental solution of (2.1).
The following lemma (see [5, Lemma ]) is extensively used in the sequel; it gives a solution representation formula for (2.1) in terms of the fundamental solution.
As functions are assumed to vanish out of their domains, if for .
which proves that satisfies (2.1) for all since and for each . The proof is therefore completed.
see [4, Theorem ].
Next, we will apply the following result (see [6, page 2]).
Lemma 2.4 (see ).
where and is a delay function, has a solution which satisfies for all for some fixed , then the coefficient satisfies , where satisfies for all .
The following lemma plays a crucial role in our proofs.
Then, nonnegativity of on implies the same for .
which yields the contradiction by canceling the negative terms on both sides of the inequality. This completes the proof.
The following lemma will be applied in the sequel.
Lemma 2.6 (see [6, Lemma ]).
3. Main Nonoscillation Results
under the same assumptions which were formulated for (2.1). We now prove the following result, which plays a major role throughout the paper.
Equation (3.1) has an eventually positive solution.
Inequality (3.2) has an eventually positive solution and/or (3.3) has an eventually negative solution.
- (iii)There exist a sufficiently large and such that and for all(3.4)
The fundamental solution is eventually positive; that is, there exists a sufficiently large such that holds on for any ; moreover, if (3.4) holds for all for some fixed , then holds on for any .
Let us prove the implications as follows: (i) (ii) (iii) (iv) (i).
(i) (ii) This part is trivial, since any eventually positive solution of (3.1) satisfies (3.2) too, which indicates that its negative satisfies (3.3).
where is an arbitrarily fixed number, and substitute (3.18) into (3.10), to see that satisfies (3.10) with a nonnegative forcing term . Hence, as is proven previously, we infer that is nonnegative on . Consequently, we have on for any (see [4, Theorem ]).
(iv) (i) Clearly, is an eventually positive solution of (3.1).
The proof is therefore completed.
Note that Theorem 3.1 for (1.6) includes Theorem C, by letting for , where satisfies . And Theorem 3.1 reduces to Theorem D, by letting for , where satisfies .
is a positive solution of (3.2), and is a negative solution to (3.3).
The following three examples are special cases of the above result, and the first two of them are corollaries for the cases and , which are well known in literature, and the third one, for with , has not been stated thus far yet.
Then, the delay-differential equation (1.2) has an eventually positive solution, and the fundamental solution satisfies on for any because we may let for .
has an eventually positive solution, and the fundamental solution satisfies on because we may let for . Notice that if for all and all , and are constants, then (3.21) reduces to an algebraic inequality.
has an eventually positive solution, and the fundamental solution satisfies on because we may let for . Notice that if for all and all , and are constants, then (3.24) becomes an algebraic inequality.
4. Comparison Theorems
where , and is a delay function for all . Let be the fundamental solution of (4.1).
Suppose that , and on for all and some fixed . If the fundamental solution of (3.1) is eventually positive, then the fundamental solution of (4.1) is also eventually positive.
for all . Thus, by Theorem 3.1 we have on for any , and equivalently, (4.1) has an eventually positive solution, which completes the proof.
The following result is an immediate consequence of Theorem 4.1.
Assume that all the conditions of Theorem 4.1 hold. If (4.1) is oscillatory, then so is (3.1).
where for all , and is the same delay function as in (3.1). Let and be the fundamental solutions of (3.1) and (4.3), respectively.
Suppose that , on for all and some fixed , and that on for any . Then, holds on for any .
for all . Lemma 2.5 implies nonnegativity of since on and the kernels of the integrals in (4.5) are nonnegative. Then dropping the nonnegative integrals on the right-hand side of (4.5), we get for all . The proof is hence completed.
where for and are same as in (3.1) for all , has an eventually positive solution, then so does (3.1).
is eventually positive, applying Theorem 4.3, we learn that the fundamental solution of (3.1) is also eventually positive since holds on for all . The proof is hence completed.
where , and for all are the same as in (2.1) and for all .
Suppose that for all and on , and on for any . Let be a solution of (2.1) with on , then holds on , where is a solution of (4.8).
which implies on . Therefore, the proof is completed.
As an application of Theorem 4.5, we give a simple example on a nonstandard time scale below.
Suppose that for all and on for any . Let be solutions of (3.1), (3.2) and (3.3), respectively. If on and on , then one has on .
and on be a solution of this equation. If holds on , then holds on .
Then, (3.1) has an eventually positive solution.
which implies that (4.16) holds. The proof is therefore completed.
where satisfies for all . Then, (3.1) has an eventually positive solution.
In this present case, we may let for to obtain (4.15).
The following example illustrates Corollary 4.10 for the nonstandard time scale .
where , or equivalently .
we have on .
cannot exceed the solution of (4.26) which has representation (4.27). Thus, on because of on , and on , which completes the proof.
is positive. If and on , then the solution of (4.26) is positive on .
for all since and on . Hence, holds on . Thus, the proof is completed.
If and on , then we have on .
The proof is similar to that of Theorem 4.13.
We give the following example as an application of Theorem 4.15.
In this paper, we have extended to equations on time scales most results obtained in [2, 3]: nonoscillation criteria, comparison theorems, and efficient nonoscillation conditions. However, there are some relevant problems that have not been considered.
(P1) In , it was demonstrated that equations with positive coefficients has slowly oscillating solutions only if it is oscillatory. The notion of slowly oscillating solutions can be easily extended to equations on time scales in such a way that it generalizes the one discussed in .
A solution of (3.1) is said to be slowly oscillating if it is oscillating and for every there exist with and for all such that on and for some .
Is the following proposition valid?
Suppose that for all , is a delay function and . If (3.1) is nonoscillatory, then the equation has no slowly oscillating solutions.
(P2) In Section 4, oscillation properties of equations with different coefficients, delays and initial functions were compared, as well as two solutions of equations with the same delays and initial conditions. Can any relation be deduced between nonoscillation properties of the same equation on different time scales?
(P3) The results of the present paper involve nonoscillation conditions for equations with positive and negative coefficients: if the relevant equation with positive coefficients only is nonoscillatory, so is the equation with coefficients of both signs. Is it possible to obtain efficient nonoscillation conditions for equations with positive and negative coefficients when the relevant equation with positive coefficients only is oscillatory?
for all . Since the integrand is nonnegative and not identically zero by (5.1), we learn that the right-hand side of (5.2) is negative on ; that is, on . Hence, is nonoscillatory, which is the contradiction justifying the proposition.
Thus, under the assumptions of Proposition 5.2 existence of a slowly oscillating solution of (3.1) implies oscillation of all solutions.
E. Braverman was partially supported by NSERC research grant.
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