- Research Article
- Open access
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Nonoscillation of First-Order Dynamic Equations with Several Delays
Advances in Difference Equations volume 2010, Article number: 873459 (2010)
Abstract
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.
1. Introduction
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 [1], 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.
The objective of this paper is to unify some results obtained in [2, 3] for the delay difference equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ1_HTML.gif)
where is the forward difference operator defined by
, and the delay differential equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ2_HTML.gif)
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 [4] for an introduction to the time-scale calculus.
Let us now present some oscillation and nonoscillation results on delay dynamic equations, and from now on, we will without further more mentioning suppose that the time scale is unbounded from above because of the definition of oscillation. The object of the present paper is to study nonoscillation of the following delay dynamic equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ3_HTML.gif)
where ,
, for all
,
,
is a delay function satisfying
,
, and
for all
. Let us denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ4_HTML.gif)
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.
In the papers [6, 7], the authors studied oscillation of (1.3) and proved the following oscillation criterion.
Theorem A (see [6, Theorem ] and [7, Theorem
]).
Suppose that . If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ5_HTML.gif)
then every solution of the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ6_HTML.gif)
is oscillatory.
Theorem A is the generalization of the well-known oscillation results stated for and
in the literature (see [8, Theorems
and
]). In [9], 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 [11], Şahiner and Stavroulakis gave the generalization of a well-known oscillation criterion, which is stated below.
Theorem B (see [11, Theorem ]).
Suppose that and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ7_HTML.gif)
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 [7].
Theorem C (see [7, Theorem ]).
Suppose that and there exist a constant
and a point
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ8_HTML.gif)
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]).
Suppose that for all
and there exist a constant
and
such that
and for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ9_HTML.gif)
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
Consider now the following delay dynamic initial value problem:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ10_HTML.gif)
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
.
Definition 2.1.
Let , and
. The solution
of the initial value problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ11_HTML.gif)
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.
Lemma 2.2.
Let be a solution of (2.1), then
can be written in the following form:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ12_HTML.gif)
As functions are assumed to vanish out of their domains, if
for
.
Proof.
As the uniqueness for the solution of (2.1) was proven in [5], it suffices to show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ13_HTML.gif)
defined by the right hand side in (2.3) solves (2.1). For , set
and
. Considering the definition of the fundamental solution
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ14_HTML.gif)
for all . After making some arrangements, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ15_HTML.gif)
which proves that satisfies (2.1) for all
since
and
for each
. The proof is therefore completed.
Example 2.3.
Consider the following first-order dynamic equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ16_HTML.gif)
then the fundamental solution of (2.7) can be easily computed as for
provided that
(see [4, Theorem
]). Thus, the general solution of the initial value problem for the nonhomogeneous equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ17_HTML.gif)
can be written in the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ18_HTML.gif)
see [4, Theorem ].
Next, we will apply the following result (see [6, page 2]).
Lemma 2.4 (see [6]).
If the delay dynamic inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ19_HTML.gif)
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.
Lemma 2.5.
Let and
, and assume that
,
for all
,
for all
, and two functions
satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ20_HTML.gif)
Then, nonnegativity of on
implies the same for
.
Proof.
Assume for the sake of contradiction that is nonnegative but
becomes negative at some points in
. Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ21_HTML.gif)
We first prove that cannot be right scattered. Suppose the contrary that
is right scattered; that is,
, then we must have
for all
and
; otherwise, this contradicts the fact that
is maximal. It follows from (2.11) that after we have applied the formula for
-integrals, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ22_HTML.gif)
This is a contradiction, and therefore is right-dense. Note that every right-neighborhood of
contains some points for which
becomes negative; therefore,
for all
. It is well known that rd-continuous functions (more truly regulated functions) are bounded on compact subsets of time scales. Pick
, then for each
, we may find
such that
for all
and all
. Set
. Moreover, since
is right-dense and
is rd-continuous, we have
; hence, we may find
with
such that
and
. Note that
since
on
. Then, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ23_HTML.gif)
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 ]).
Assume that satisfies
, then one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ24_HTML.gif)
3. Main Nonoscillation Results
Consider the delay dynamic equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ25_HTML.gif)
and the corresponding inequalities
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ26_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ27_HTML.gif)
under the same assumptions which were formulated for (2.1). We now prove the following result, which plays a major role throughout the paper.
Theorem 3.1.
Suppose that for all ,
is a delay function and
. Then, the following conditions are equivalent.
-
(i)
Equation (3.1) has an eventually positive solution.
-
(ii)
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) -
(iv)
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
.
Proof.
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).
(ii)(iii) Let
be an eventually positive solution of (3.2), the case where
is an eventually negative solution to (3.3) is equivalent, and thus we omit it. Let us assume that there exists
such that
and
for all
and all
. It follows from (3.2) that
holds on
, that is,
is nonincreasing on
. Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ29_HTML.gif)
Evidently . From (3.5), we see that
satisfies the ordinary dynamic equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ30_HTML.gif)
From Lemma 2.4, we deduce that . Since
on
, then by [4, Theorem
] and (3.6), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ31_HTML.gif)
Hence, using (3.7) in (3.2), for all , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ32_HTML.gif)
Since , then by [4, Theorem
] we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ33_HTML.gif)
(iv) Let
satisfy
and (3.4) on
, where
is such that
for all
. Now, consider the initial value problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ34_HTML.gif)
Let be a solution of (3.10), and set
for
, then we see that
also satisfies the following auxiliary equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ35_HTML.gif)
which has the unique solution
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ36_HTML.gif)
(see Example 2.3). Substituting (3.12) in (3.10), for all , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ37_HTML.gif)
which can be rewritten as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ38_HTML.gif)
Hence, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ39_HTML.gif)
for all , where
for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ40_HTML.gif)
Applying Lemma 2.5 to (3.15), we learn that nonnegativity of on
implies nonnegativity of
on
, and nonnegativity of
on
implies the same for
on
by (3.12). On the other hand, by Lemma 2.2,
has the following representation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ41_HTML.gif)
Since is eventually nonnegative for any eventually nonnegative function
, we infer that the kernel
of the integral on the right-hand side of (3.17) is eventually nonnegative. Indeed, assume the contrary that
on
but
is not nonnegative, then we may pick
and find
such that
. Then, letting
for
, we are led to the contradiction
, where
is defined by (3.17). To prove eventual positivity of
, set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ42_HTML.gif)
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.
Remark 3.2.
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
.
Corollary 3.3.
If ,
satisfies (3.4) on
and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ43_HTML.gif)
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.
Example 3.4 (see [2, Theorem ] and [8, Section
]).
Let , and suppose that there exist
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ44_HTML.gif)
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
.
Example 3.5 (see [3, Theorem ] and [8, Section
]).
Let ,
, and suppose that there exist
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ45_HTML.gif)
Then, the following delay -difference equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ46_HTML.gif)
where is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ47_HTML.gif)
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.
Example 3.6.
Let for
, and suppose that there exist
and
, where
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ48_HTML.gif)
Then, the following delay -difference equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ49_HTML.gif)
where the -difference operator
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ50_HTML.gif)
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
In this section, we state comparison results on oscillation and nonoscillation of delay dynamic equations. To this end, consider (3.1) together with the following equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ51_HTML.gif)
where ,
and
is a delay function for all
. Let
be the fundamental solution of (4.1).
Theorem 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.
Proof.
By Theorem 3.1, there exist a sufficiently large and
with
such that (3.4) holds on
. Note that
and
imply that
is nondecreasing in
, hence
is nonincreasing in
(see [4, Theorem
]). Without loss of generality, we may suppose that
and
hold on
for all
. Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ52_HTML.gif)
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.
Corollary 4.2.
Assume that all the conditions of Theorem 4.1 hold. If (4.1) is oscillatory, then so is (3.1).
For the following result, we do not need the coefficient to be nonnegative for all
; consider (3.1) together with the following equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ53_HTML.gif)
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.
Theorem 4.3.
Suppose that ,
on
for all
and some fixed
, and that
on
for any
. Then,
holds on
for any
.
Proof.
From (4.3), any fixed and all
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ54_HTML.gif)
It follows from the solution representation formula (2.3) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ55_HTML.gif)
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.
Corollary 4.4.
Suppose that the delay differential inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ56_HTML.gif)
where for
and
are same as in (3.1) for all
, has an eventually positive solution, then so does (3.1).
Proof.
By Theorem 3.1, we know that the fundamental solution of the corresponding differential equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ57_HTML.gif)
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.
We now compare two solutions of (2.1) and the following initial value problem:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ58_HTML.gif)
where ,
and
for all
are the same as in (2.1) and
for all
.
Theorem 4.5.
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).
Proof.
By Theorems 3.1 and 4.3, we have on
for any
. Rearranging (2.1), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ59_HTML.gif)
for all . In view of the solution representation formula (2.3), for all
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ60_HTML.gif)
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.
Example 4.6.
Let , and consider the following initial value problems:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ61_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ62_HTML.gif)
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ63_HTML.gif)
Denoting by and
the solutions of (4.11) and (4.13), respectively. Then,
on
by Theorem 4.5. For the graph of
iterates, see Figure 1.
Corollary 4.7.
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
.
Corollary 4.8.
Let be a solution of (3.1), and
on
for any
be the fundamental solution of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ64_HTML.gif)
and on
be a solution of this equation. If
holds on
, then
holds on
.
Theorem 4.9.
Suppose that there exist and
such that
and for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ65_HTML.gif)
Then, (3.1) has an eventually positive solution.
Proof.
By Corollary 4.4, it suffices to prove that (4.6) has an eventually positive solution. For this purpose, by Theorem 3.1, it is enough to demonstrate that satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ66_HTML.gif)
Note that and
imply that
is nondecreasing in
, hence
is nonincreasing in
(see [4, Theorem
]). From (4.15) and Lemma 2.6, for all
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ67_HTML.gif)
which implies that (4.16) holds. The proof is therefore completed.
Corollary 4.10.
Suppose that there exist with
and
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ68_HTML.gif)
where satisfies
for all
. Then, (3.1) has an eventually positive solution.
Proof.
In this present case, we may let for
to obtain (4.15).
Remark 4.11.
Particularly, letting and
in Corollary 4.10, we learn that (3.1) admits a nonoscillatory solution if
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ69_HTML.gif)
It is a well-known fact that the constant above is the best possible for difference equations since the difference equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ70_HTML.gif)
where , is nonoscillatory if and only if
(see [3, 12]).
The following example illustrates Corollary 4.10 for the nonstandard time scale .
Example 4.12.
Let ,
for
and
. We consider the following
-difference equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ71_HTML.gif)
where the -difference operator
is defined by (3.26). For simplicity of notation, we let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ72_HTML.gif)
Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ73_HTML.gif)
Letting , we can compute that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ74_HTML.gif)
which implies that the regressivity condition in Corollary 4.10 holds. So that (4.21) has an eventually positive solution if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ75_HTML.gif)
where , or equivalently
.
Theorem 4.13.
Suppose that for all
and (4.15) is true on
. If
and
on
, then for the solution
of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ76_HTML.gif)
we have on
.
Proof.
As in the proof of Theorem 4.9, we deduce that there exists satisfying (3.4). Hence,
on
for any
. By the solution representation formula (2.3), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ77_HTML.gif)
for all . Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ78_HTML.gif)
By Corollary 3.3, we have for all
. Then,
solves
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ79_HTML.gif)
By Corollary 4.7, we know that given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ80_HTML.gif)
cannot exceed the solution of (4.26) which has representation (4.27). Thus,
on
because of
on
, and
on
, which completes the proof.
Theorem 4.14.
Suppose that for all
,
on
for any
, and the solution
of the initial value problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ81_HTML.gif)
is positive. If and
on
, then the solution
of (4.26) is positive on
.
Proof.
Solution representation formula (2.3) implies for a solution of (4.31) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ82_HTML.gif)
for all since
and
on
. Hence,
holds on
. Thus, the proof is completed.
Theorem 4.15.
Suppose that ,
, (3.4) has a solution
with
,
is a solution of (4.26) and
is a positive solution of the following initial value problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ83_HTML.gif)
If and
on
, then we have
on
.
Proof.
The proof is similar to that of Theorem 4.13.
We give the following example as an application of Theorem 4.15.
Example 4.16.
Let , and consider the following initial value problems:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ84_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ85_HTML.gif)
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ86_HTML.gif)
If and
are the unique solutions of (4.34) and (4.36), respectively, then we have the graph of
iterates, see Figure 2, where
by Theorem 4.15.
5. Discussion
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 [2], 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 [2].
Definition 5.1.
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?
Proposition 5.2.
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?
We will only comment affirmatively on the proof of the proposition in Problem (P1). Really, let us assume the contrary that (3.1) is nonoscillatory but is a slowly oscillating solution of this equation. By Theorem 3.1, the fundamental solution
of (3.1) is positive on
for some
. There exist
and
with
for all
such that
on
and
on
. Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ87_HTML.gif)
for all and all
. It follows from Lemma 2.2 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F873459/MediaObjects/13662_2010_Article_1349_Equ88_HTML.gif)
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.
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Acknowledgment
E. Braverman was partially supported by NSERC research grant.
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Braverman, E., Karpuz, B. Nonoscillation of First-Order Dynamic Equations with Several Delays. Adv Differ Equ 2010, 873459 (2010). https://doi.org/10.1155/2010/873459
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DOI: https://doi.org/10.1155/2010/873459