Abstract and Applied AnalysisVolume 2011 (2011), Article ID 637142, 14 pagesdoi:10.1155/2011/637142

Research Article

On Nonoscillation of Advanced Differential Equations with Several Terms

L. Berezansky¹ and E. Braverman²

¹Department of Mathematics, Ben-Gurion University of Negev, Beer-Sheva 84105, Israel²Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, AB, Canada, T2N 1N4

Received 16 December 2010; Accepted 27 January 2011

Academic Editor: Josef Diblík

Copyright © 2011 L. Berezansky and E. Braverman. L. Berezansky and . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Existence of positive solutions for advanced equations with several terms is investigated in the following three cases: (a) all coefficients are positive; (b) all coefficients are negative; (c) there is an equal number of positive and negative coefficients. Results on asymptotics of nonoscillatory solutions are also presented.

1. Introduction

This paper deals with nonoscillation properties of scalar advanced differential equations. Advanced differential equations appear in several applications, especially as mathematical models in economics; an advanced term may, for example, reflect the dependency on anticipated capital stock [1, 2].

It is not quite clear how to formulate an initial value problem for such equations, and existence and uniqueness of solutions becomes a complicated issue. To study oscillation, we need to assume that there exists a solution of such equation on the halfline. In the beginning of 1980s, sufficient oscillation conditions for first-order linear advanced equations with constant coefficients and deviations of arguments were obtained in [3] and for nonlinear equations in [4]. Later oscillation properties were studied for other advanced and mixed differential equations (see the monograph [5], the papers [6–12] and references therein). Overall, these publications mostly deal with sufficient oscillation conditions; there are only few results [7, 9, 12] on existence of positive solutions for equations with several advanced terms and variable coefficients, and the general nonoscillation theory is not complete even for first-order linear equations with variable advanced arguments and variable coefficients of the same sign. The present paper partially fills up this gap. We obtain several nonoscillation results for advanced equations using the generalized characteristic inequality [13]. The main method of this paper is based on fixed point theory; thus, we also state the existence of a solution in certain cases.

In the linear case, the best studied models with advanced arguments were the equations of the types where , , , and .

Let us note that oscillation of higher order linear and nonlinear equations with advanced and mixed arguments was also extensively investigated, starting with [14]; see also the recent papers [15–19] and references therein.

For equations with an advanced argument, the results obtained in [20, 21] can be reformulated as Theorems A–C below.

Theorem A (see [20]). If , , and are equicontinuous on , , , , and , then the advanced equation has a nonoscillatory solution.

In the present paper, we extend Theorem A to the case of several deviating arguments and coefficients (Theorem 2.10).

Theorem B (see [20]). If , , and are equicontinuous on , , , , , and then the advanced equation has a nonoscillatory solution.

Corollary 2.3 of the present paper extends Theorem B to the case of several coefficients and advanced arguments (generally, ); if then the equation

has an eventually positive solution. To the best of our knowledge, only the opposite inequality (with rather than in the upper bound) was known as a sufficient oscillation condition. Coefficients and advanced arguments are also assumed to be of a more general type than in [20]. Comparison to equations with constant arguments deviations, and coefficients (Corollary 2.8) is also outlined.

For advanced equations with coefficients of different sign, the following result is known.

Theorem C (see [21]). If and , then there exists a nonoscillatory solution of the equation

This result is generalized in Theorem 2.13 to the case of several positive and negative terms and several advanced arguments; moreover, positive terms can also be advanced as far as the advance is not greater than in the corresponding negative terms.

We also study advanced equations with positive and negative coefficients in the case when positive terms dominate rather than negative ones; some sufficient nonoscillation conditions are presented in Theorem 2.15; these results are later applied to the equation with constant advances and coefficients. Let us note that analysis of nonoscillation properties of the mixed equation with a positive advanced term

was also more complicated compared to other cases of mixed equations with positive and negative coefficients [21].

In nonoscillation theory, results on asymptotic properties of nonoscillatory solutions are rather important; for example, for equations with several delays and positive coefficients, all nonoscillatory solutions tend to zero if the integral of the sum of coefficients diverges; under the same condition for negative coefficients, all solutions tend to infinity. In Theorems 2.6 and 2.11, the asymptotic properties of nonoscillatory solutions for advanced equations with coefficients of the same sign are studied.

The paper is organized as follows. Section 2 contains main results on the existence of nonoscillatory solutions to advanced equations and on asymptotics of these solutions: first for equations with coefficients of the same sign, then for equations with both positive and negative coefficients. Section 3 involves some comments and open problems.

2. Main Results

Consider first the equation under the following conditions:(a1), , are Lebesgue measurable functions locally essentially bounded for ,(a2) are Lebesgue measurable functions, , .

Definition 2.1. A locally absolutely continuous function is called a solution of problem (2.1) if it satisfies (2.1) for almost all .

The same definition will be used for all further advanced equations.

Theorem 2.2. Suppose that the inequality has a nonnegative solution which is integrable on each interval , then (2.1) has a positive solution for .

Proof. Let be a nonnegative solution of inequality (2.2). Denote then By induction we have . Hence, there exists a pointwise limit . By the Lebesgue convergence theorem, we have Then, the function is a positive solution of (2.1).

Corollary 2.3. If then (2.1) has a positive solution for .

Proof. Let , then satisfies (2.2) at any point where . In the case when , inequality (2.7) implies Hence, is a positive solution of inequality (2.2). By Theorem 2.2, (2.1) has a positive solution for .

Corollary 2.4. If there exists such that and , then (2.1) has an eventually positive solution.

Corollary 2.5. If there exists such that and , then (2.1) has an eventually positive solution.

Proof. Under the conditions of either Corollary 2.4 or Corollary 2.5, obviously there exists such that (2.7) is satisfied.

Theorem 2.6. Let and be an eventually positive solution of (2.1), then .

Proof. Suppose that for , then for and which implies Thus, .

Consider together with (2.1) the following equation: for . We assume that for (2.11) conditions (a1)-(a2) also hold.

Theorem 2.7. Suppose that , , , and the conditions of Theorem 2.2 hold, then (2.11) has a positive solution for .

Proof. Let be a solution of inequality (2.2) for , then this function is also a solution of this inequality if and are replaced by and . The reference to Theorem 2.2 completes the proof.

Corollary 2.8. Suppose that there exist and such that , , , and the inequality has a solution , then (2.1) has a positive solution for .

Proof. Consider the equation with constant parameters Since the function is a solution of inequality (2.2) corresponding to (2.13), by Theorem 2.2, (2.13) has a positive solution. Theorem 2.7 implies this corollary.

Corollary 2.9. Suppose that , for , and then (2.1) has a positive solution for .

Proof. Since , the number is a positive solution of the inequality which completes the proof.

Consider now the equation with positive coefficients

Theorem 2.10. Suppose that are continuous functions bounded on and are equicontinuous functions on satisfying , then (2.16) has a nonoscillatory solution.

Proof. In the space of continuous functions on , consider the set and the operator If , then .For the integral operator we will demonstrate that is a compact set in the space . If , then Hence, the functions in the set are uniformly bounded in the space .Functions are equicontinuous on , so for any , there exists a such that for , the inequality holds. From the relation we have for and the estimate Hence, the set contains functions which are uniformly bounded and equicontinuous on , so it is compact in the space ; thus, the set is also compact in .By the Schauder fixed point theorem, there exists a continuous function such that , then the function is a bounded positive solution of (2.16). Moreover, since is nonnegative, this solution is nonincreasing on .

Theorem 2.11. Suppose that the conditions of Theorem 2.10 hold, and is a nonoscillatory solution of (2.16), then .

Proof. Let for , then for . Hence, is nonincreasing and thus has a finite limit. If , then for any , and thus which implies . This contradicts to the assumption that is positive, and therefore .

Let us note that we cannot guarantee any (exponential or polynomial) rate of convergence to zero even for constant coefficients , as the following example demonstrates.

Example 2.12. Consider the equation , where , , . Then, is the solution which tends to zero slower than for any .

Consider now the advanced equation with positive and negative coefficients

Theorem 2.13. Suppose that and are Lebesgue measurable locally essentially bounded functions, , and are Lebesgue measurable functions, , and inequality (2.2) has a nonnegative solution, then (2.26) has a nonoscillatory solution; moreover, it has a positive nondecreasing and a negative nonincreasing solutions.

Proof. Let be a nonnegative solution of (2.2) and denote We have , and by (2.2), Since and , then .Next, let us assume that . The assumptions of the theorem imply . Let us demonstrate that . This inequality has the form which is equivalent to This inequality is evident for any , , and ; thus, we have .By the Lebesgue convergence theorem, there is a pointwise limit satisfying , . Then, the function is a positive nondecreasing solution of (2.26) for any and is a negative nonincreasing solution of (2.26) for any .

Corollary 2.14. Let and be Lebesgue measurable locally essentially bounded functions satisfying , and let and be Lebesgue measurable functions, where . Assume in addition that inequality (2.7) holds. Then, (2.26) has a nonoscillatory solution.

Consider now the equation with constant deviations of advanced arguments where are continuous functions, , .

Denote , , , .

Theorem 2.15. Suppose that , , , and . If there exists a number such that then (2.33) has a nonoscillatory solution; moreover, it has a positive nonincreasing and a negative nondecreasing solutions.

Proof. In the space , consider the set and the operator For , we have from (2.34) and (2.35) Hence, .Consider the integral operator We will show that is a compact set in the space . For , we have Hence, the functions in the set are uniformly bounded in the space .The equality implies Hence, the set and so the set are compact in the space .By the Schauder fixed point theorem, there exists a continuous function satisfying such that ; thus, the function is a positive nonincreasing solution of (2.33) for any and is a negative nondecreasing solution of (2.26) for any .

Let us remark that (2.35) for any implies .

Corollary 2.16. Let , , and for the inequality holds, then (2.33) has a bounded positive solution.

Proof. The negative number defined in (2.42) is a solution of both (2.34) and (2.35); by definition, it satisfies (2.35), and (2.43) implies (2.34).

Example 2.17. Consider the equation with constant advances and coefficients where , , . Then, is the minimal value of for which inequality (2.35) holds; for (2.44), it has the form .Inequality (2.34) for (2.44) can be rewritten as where the function decreases on if and for any negative if ; besides, . Thus, if for some , then for any . Hence, the inequality is necessary and sufficient for the conditions of Theorem 2.15 to be satisfied for (2.44).Figure 1 demonstrates possible values of advances and , such that Corollary 2.16 implies the existence of a positive solution in the case . Then, (2.46) has the form , which is possible only for and for these values is equivalent to

Figure 1: The domain of values satisfying inequality (2.47). If the values of advances and are under the curve, then (2.44) has a positive solution.

3. Comments and Open Problems

In this paper, we have developed nonoscillation theory for advanced equations with variable coefficients and advances. Most previous nonoscillation results deal with either oscillation or constant deviations of arguments. Among all cited papers, only [8] has a nonoscillation condition (Theorem 2.11) for a partial case of (2.1) (with ), which in this case coincides with Corollary 2.4. The comparison of results of the present paper with the previous results of the authors was discussed in the introduction.

Finally, let us state some open problems and topics for research.(1)Prove or disprove: if (2.1), with , has a nonoscillatory solution, then (2.26) with positive and negative coefficients also has a nonoscillatory solution.

As the first step in this direction, prove or disprove that if and the equation has a nonoscillatory solution, then the equation also has a nonoscillatory solution, where .

If these conjectures are valid, obtain comparison results for advanced equations.(2)Deduce nonoscillation conditions for (2.1) with oscillatory coefficients. Oscillation results for an equation with a constant advance and an oscillatory coefficient were recently obtained in [22].(3)Consider advanced equations with positive and negative coefficients when the numbers of positive and negative terms do not coincide.(4)Study existence and/or uniqueness problem for the initial value problem or boundary value problems for advanced differential equations.

Acknowledgments

L. Berezansky was partially supported by the Israeli Ministry of Absorption. E. Braverman was partially supported by NSERC.

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