- Research Article
- Open Access
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.
- Difference Equation
- Fundamental Solution
- Comparison Theorem
- Delay Differential Equation
- Delay Function
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 ]).
In [10, Theorem , and Corollary 3.3], Agwo extended Theorem C to (1.3).
Theorem D (see [10, Corollary 3.3]).
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).
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 .
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.
see [4, Theorem ].
Next, we will apply the following result (see [6, page 2]).
Lemma 2.4 (see ).
The following lemma plays a crucial role in our proofs.
The following lemma will be applied in the sequel.
Lemma 2.6 (see [6, Lemma ]).
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.
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 ]).
The proof is therefore completed.
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.
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).
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.
As an application of Theorem 4.5, we give a simple example on a nonstandard time scale below.
Then, (3.1) has an eventually positive solution.
which implies that (4.16) holds. The proof is therefore completed.
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 .
Is the following proposition valid?
(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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