Nonoscillation of First-Order Dynamic Equations with Several Delays

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

(1.1)

where is the forward difference operator defined by , and the delay differential equation

(1.2)

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:

(1.3)

where , , for all , , is a delay function satisfying , , and for all . Let us denote

(1.4)

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

(1.5)

then every solution of the equation

(1.6)

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

(1.7)

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

(1.8)

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

(1.9)

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:

(2.1)

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

(2.2)

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:

(2.3)

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

(2.4)

defined by the right hand side in (2.3) solves (2.1). For , set and . Considering the definition of the fundamental solution , we have

(2.5)

for all . After making some arrangements, we get

(2.6)

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:

(2.7)

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

(2.8)

can be written in the form

(2.9)

see [4, Theorem ].

Next, we will apply the following result (see [6, page 2]).

Lemma 2.4 (see [6]).

If the delay dynamic inequality

(2.10)

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

(2.11)

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

(2.12)

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

(2.13)

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

(2.14)

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

(2.15)

3. Main Nonoscillation Results

Consider the delay dynamic equation

(3.1)

and the corresponding inequalities

(3.2)

(3.3)

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

(3.5)

Evidently . From (3.5), we see that satisfies the ordinary dynamic equation

(3.6)

From Lemma 2.4, we deduce that . Since on , then by [4, Theorem ] and (3.6), we have

(3.7)

Hence, using (3.7) in (3.2), for all , we obtain

(3.8)

Since , then by [4, Theorem ] we have

(3.9)

(iv) Let satisfy and (3.4) on , where is such that for all . Now, consider the initial value problem

(3.10)

Let be a solution of (3.10), and set for , then we see that also satisfies the following auxiliary equation

(3.11)

which has the unique solution

(3.12)

(see Example 2.3). Substituting (3.12) in (3.10), for all , we obtain

(3.13)

which can be rewritten as

(3.14)

Hence, we get

(3.15)

for all , where for ,

(3.16)

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:

(3.17)

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

(3.18)

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

(3.19)

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

(3.20)

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

(3.21)

Then, the following delay -difference equation:

(3.22)

where is defined by

(3.23)

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

(3.24)

Then, the following delay -difference equation:

(3.25)

where the -difference operator is defined by

(3.26)

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:

(4.1)

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

(4.2)

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:

(4.3)

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

(4.4)

It follows from the solution representation formula (2.3) that

(4.5)

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

(4.6)

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

(4.7)

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:

(4.8)

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

(4.9)

for all . In view of the solution representation formula (2.3), for all , we have

(4.10)

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:

(4.11)

where

(4.12)

and

(4.13)

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.

Figure 1

The graph of 30 iterates for the solutions of (4. 11) and (4.13) illustrates the result of Theorem 4.5, here for all .

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

(4.14)

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

(4.15)

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

(4.16)

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

(4.17)

which implies that (4.16) holds. The proof is therefore completed.

Corollary 4.10.

Suppose that there exist with and such that and

(4.18)

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

(4.19)

It is a well-known fact that the constant above is the best possible for difference equations since the difference equation

(4.20)

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

(4.21)

where the -difference operator is defined by (3.26). For simplicity of notation, we let

(4.22)

Then, we have

(4.23)

Letting , we can compute that

(4.24)

which implies that the regressivity condition in Corollary 4.10 holds. So that (4.21) has an eventually positive solution if

(4.25)

where , or equivalently .

Theorem 4.13.

Suppose that for all and (4.15) is true on . If and on , then for the solution of

(4.26)

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

(4.27)

for all . Let

(4.28)

By Corollary 3.3, we have for all . Then, solves

(4.29)

By Corollary 4.7, we know that given by

(4.30)

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

(4.31)

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

(4.32)

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

(4.33)

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:

(4.34)

where

(4.35)

and

(4.36)

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.

Figure 2

The graph of 7 iterates for the solutions of (4. 34) and (4.36) illustrates the result of Theorem 4.15, here for all .

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

(5.1)

for all and all . It follows from Lemma 2.2 that

(5.2)

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.