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Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 637142, 14 pages
Research Article

On Nonoscillation of Advanced Differential Equations with Several Terms

1Department of Mathematics, Ben-Gurion University of Negev, Beer-Sheva 84105, Israel
2Department 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: JosefΒ DiblΓ­k

Copyright Β© 2011 L. Berezansky and E. Braverman. 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.


Existence of positive solutions for advanced equations with several terms βˆ‘Μ‡π‘₯(𝑑)+π‘šπ‘˜=1π‘Žπ‘˜(𝑑)π‘₯(β„Žπ‘˜(𝑑))=0,β„Žπ‘˜(𝑑)β‰₯𝑑 is investigated in the following three cases: (a) all coefficients π‘Žπ‘˜ are positive; (b) all coefficients π‘Žπ‘˜ 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 Μ‡π‘₯(𝑑)βˆ’π‘Ž(𝑑)π‘₯(β„Ž(𝑑))+𝑏(𝑑)π‘₯(𝑑)=0,Μ‡π‘₯(𝑑)βˆ’π‘Ž(𝑑)π‘₯(𝑑)+𝑏(𝑑)π‘₯(𝑔(𝑑))=0,(1.1) where π‘Ž(𝑑)β‰₯0, 𝑏(𝑑)β‰₯0, β„Ž(𝑑)β‰₯𝑑, and 𝑔(𝑑)β‰₯𝑑.

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 π‘Ž, 𝑏, and β„Ž are equicontinuous on [0,∞), π‘Ž(𝑑)β‰₯0, 𝑏(𝑑)β‰₯0, β„Ž(𝑑)β‰₯𝑑, and limsupπ‘‘β†’βˆž[β„Ž(𝑑)βˆ’π‘‘]<∞, then the advanced equation Μ‡π‘₯(𝑑)+π‘Ž(𝑑)π‘₯(β„Ž(𝑑))+𝑏(𝑑)π‘₯(𝑑)=0(1.2) 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 π‘Ž, 𝑏, and β„Ž are equicontinuous on [0,∞), π‘Ž(𝑑)β‰₯0, 𝑏(𝑑)β‰₯0, β„Ž(𝑑)β‰₯𝑑, limsupπ‘‘β†’βˆž[β„Ž(𝑑)βˆ’π‘‘]<∞, and limsupπ‘‘β†’βˆžξ€œπ‘‘β„Ž(𝑑)π‘Žξ‚»ξ€œ(𝑠)expπ‘ β„Ž(𝑠)𝑏1(𝜏)π‘‘πœπ‘‘π‘ <𝑒,(1.3) then the advanced equation Μ‡π‘₯(𝑑)βˆ’π‘Ž(𝑑)π‘₯(β„Ž(𝑑))βˆ’π‘(𝑑)π‘₯(𝑑)=0(1.4) has a nonoscillatory solution.

Corollary 2.3 of the present paper extends Theorem B to the case of several coefficients π‘Žπ‘˜β‰₯0 and advanced arguments β„Žπ‘˜ (generally, 𝑏(𝑑)≑0); if ξ€œmaxπ‘˜β„Žπ‘˜π‘‘π‘š(𝑑)𝑖=1π‘Žπ‘–1(𝑠)𝑑𝑠≀𝑒,(1.5) then the equation Μ‡π‘₯(𝑑)+π‘šξ“π‘˜=1π‘Žπ‘˜ξ€·β„Ž(𝑑)π‘₯π‘˜ξ€Έ(𝑑)=0(1.6)

has an eventually positive solution. To the best of our knowledge, only the opposite inequality (with minπ‘˜β„Žπ‘˜(𝑑) rather than maxπ‘˜β„Žπ‘˜(𝑑) 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 0β‰€π‘Ž(𝑑)≀𝑏(𝑑) and β„Ž(𝑑)β‰₯𝑑, then there exists a nonoscillatory solution of the equation Μ‡π‘₯(𝑑)βˆ’π‘Ž(𝑑)π‘₯(β„Ž(𝑑))+𝑏(𝑑)π‘₯(𝑑)=0.(1.7)

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 Μ‡π‘₯(𝑑)+π‘Ž(𝑑)π‘₯(β„Ž(𝑑))βˆ’π‘(𝑑)π‘₯(𝑔(𝑑))=0,β„Ž(𝑑)β‰₯𝑑,𝑔(𝑑)≀𝑑,π‘Ž(𝑑)β‰₯0,𝑏(𝑑)β‰₯0(1.8)

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 equationΜ‡π‘₯(𝑑)βˆ’π‘šξ“π‘˜=1π‘Žπ‘˜ξ€·β„Ž(𝑑)π‘₯π‘˜ξ€Έ(𝑑)=0,(2.1) under the following conditions:(a1)π‘Žπ‘˜(𝑑)β‰₯0, π‘˜=1,…,π‘š, are Lebesgue measurable functions locally essentially bounded for 𝑑β‰₯0,(a2)β„Žπ‘˜βˆΆ[0,∞)→ℝ are Lebesgue measurable functions, β„Žπ‘˜(𝑑)β‰₯𝑑, π‘˜=1,…,π‘š.

Definition 2.1. A locally absolutely continuous function π‘₯∢[𝑑0,∞)→𝑅 is called a solution of problem (2.1) if it satisfies (2.1) for almost all π‘‘βˆˆ[𝑑0,∞).

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

Theorem 2.2. Suppose that the inequality 𝑒(𝑑)β‰₯π‘šξ“π‘˜=1π‘Žπ‘˜ξ‚»ξ€œ(𝑑)expβ„Žπ‘˜π‘‘(𝑑)𝑒(𝑠)𝑑𝑠,𝑑β‰₯𝑑0(2.2) has a nonnegative solution which is integrable on each interval [𝑑0,𝑏], then (2.1) has a positive solution for 𝑑β‰₯𝑑0.

Proof. Let 𝑒0(𝑑) be a nonnegative solution of inequality (2.2). Denote 𝑒𝑛+1(𝑑)=π‘šξ“π‘˜=1π‘Žπ‘˜ξ‚»ξ€œ(𝑑)expβ„Žπ‘˜π‘‘(𝑑)𝑒𝑛(𝑠)𝑑𝑠,𝑛=0,1,…,(2.3) then 𝑒1(𝑑)=π‘šξ“π‘˜=1π‘Žπ‘˜ξ‚»ξ€œ(𝑑)expβ„Žπ‘˜π‘‘(𝑑)𝑒0ξ‚Ό(𝑠)𝑑𝑠≀𝑒0(𝑑).(2.4) By induction we have 0≀𝑒𝑛+1(𝑑)≀𝑒𝑛(𝑑)≀𝑒0(𝑑). Hence, there exists a pointwise limit 𝑒(𝑑)=limπ‘›β†’βˆžπ‘’π‘›(𝑑). By the Lebesgue convergence theorem, we have 𝑒(𝑑)=π‘šξ“π‘˜=1π‘Žπ‘˜ξ‚»ξ€œ(𝑑)expβ„Žπ‘˜π‘‘(𝑑)ξ‚Ό.𝑒(𝑠)𝑑𝑠(2.5) Then, the function 𝑑π‘₯(𝑑)=π‘₯0ξ€Έξ‚»ξ€œexp𝑑𝑑0𝑑𝑒(𝑠)𝑑𝑠foranyπ‘₯0ξ€Έ>0(2.6) is a positive solution of (2.1).

Corollary 2.3. If ξ€œmaxπ‘˜β„Žπ‘˜π‘‘π‘š(𝑑)𝑖=1π‘Žπ‘–1(𝑠)𝑑𝑠≀𝑒,𝑑β‰₯𝑑0,(2.7) then (2.1) has a positive solution for 𝑑β‰₯𝑑0.

Proof. Let 𝑒0βˆ‘(𝑑)=π‘’π‘šπ‘˜=1π‘Žπ‘˜(𝑑), then 𝑒0 satisfies (2.2) at any point 𝑑 where βˆ‘π‘šπ‘˜=1π‘Žπ‘˜(𝑑)=0. In the case when βˆ‘π‘šπ‘˜=1π‘Žπ‘˜(𝑑)β‰ 0, inequality (2.7) implies 𝑒0(𝑑)βˆ‘π‘šπ‘˜=1π‘Žπ‘˜ξ‚†βˆ«(𝑑)expβ„Žπ‘˜π‘‘(𝑑)𝑒0β‰₯𝑒(𝑠)𝑑𝑠0(𝑑)βˆ‘π‘šπ‘˜=1π‘Žπ‘˜(ξ‚†βˆ«π‘‘)expmaxπ‘˜β„Žπ‘˜π‘‘(𝑑)𝑒0(=π‘’βˆ‘π‘ )π‘‘π‘ π‘šπ‘˜=1π‘Žπ‘˜(𝑑)βˆ‘π‘šπ‘˜=1π‘Žπ‘˜(ξ‚†π‘’βˆ«π‘‘)expmaxπ‘˜β„Žπ‘˜π‘‘(𝑑)βˆ‘π‘šπ‘–=1π‘Žπ‘–ξ‚‡β‰₯π‘’βˆ‘(𝑠)π‘‘π‘ π‘šπ‘˜=1π‘Žπ‘˜(𝑑)βˆ‘π‘šπ‘˜=1π‘Žπ‘˜(𝑑)𝑒=1.(2.8) Hence, 𝑒0(𝑑) is a positive solution of inequality (2.2). By Theorem 2.2, (2.1) has a positive solution for 𝑑β‰₯𝑑0.

Corollary 2.4. If there exists 𝜎>0 such that β„Žπ‘˜(𝑑)βˆ’π‘‘β‰€πœŽ and ∫∞0βˆ‘π‘šπ‘˜=1π‘Žπ‘˜(𝑠)𝑑𝑠<∞, then (2.1) has an eventually positive solution.

Corollary 2.5. If there exists 𝜎>0 such that β„Žπ‘˜(𝑑)βˆ’π‘‘β‰€πœŽ and limπ‘‘β†’βˆžπ‘Žπ‘˜(𝑑)=0, then (2.1) has an eventually positive solution.

Proof. Under the conditions of either Corollary 2.4 or Corollary 2.5, obviously there exists 𝑑0β‰₯0 such that (2.7) is satisfied.

Theorem 2.6. Let βˆ«βˆžβˆ‘π‘šπ‘˜=1π‘Žπ‘˜(𝑠)𝑑𝑠=∞ and π‘₯ be an eventually positive solution of (2.1), then limπ‘‘β†’βˆžπ‘₯(𝑑)=∞.

Proof. Suppose that π‘₯(𝑑)>0 for 𝑑β‰₯𝑑1, then Μ‡π‘₯(𝑑)β‰₯0 for 𝑑β‰₯𝑑1 and Μ‡π‘₯(𝑑)β‰₯π‘šξ“π‘˜=1π‘Žπ‘˜ξ€·π‘‘(𝑑)π‘₯1ξ€Έ,𝑑β‰₯𝑑1,(2.9) which implies 𝑑π‘₯(𝑑)β‰₯π‘₯1ξ€Έξ€œπ‘‘π‘‘1π‘šξ“π‘˜=1π‘Žπ‘˜(𝑠)𝑑𝑠.(2.10) Thus, limπ‘‘β†’βˆžπ‘₯(𝑑)=∞.

Consider together with (2.1) the following equation:Μ‡π‘₯(𝑑)βˆ’π‘šξ“π‘˜=1π‘π‘˜ξ€·π‘”(𝑑)π‘₯π‘˜ξ€Έ(𝑑)=0,(2.11) for 𝑑β‰₯𝑑0. We assume that for (2.11) conditions (a1)-(a2) also hold.

Theorem 2.7. Suppose that π‘‘β‰€π‘”π‘˜(𝑑)β‰€β„Žπ‘˜(𝑑), 0β‰€π‘π‘˜(𝑑)β‰€π‘Žπ‘˜(𝑑), 𝑑β‰₯𝑑0, and the conditions of Theorem 2.2 hold, then (2.11) has a positive solution for 𝑑β‰₯𝑑0.

Proof. Let 𝑒0(𝑑)β‰₯0 be a solution of inequality (2.2) for 𝑑β‰₯𝑑0, then this function is also a solution of this inequality if π‘Žπ‘˜(𝑑) and β„Žπ‘˜(𝑑) are replaced by π‘π‘˜(𝑑) and π‘”π‘˜(𝑑). The reference to Theorem 2.2 completes the proof.

Corollary 2.8. Suppose that there exist π‘Žπ‘˜>0 and πœŽπ‘˜>0 such that 0β‰€π‘Žπ‘˜(𝑑)β‰€π‘Žπ‘˜, π‘‘β‰€β„Žπ‘˜(𝑑)≀𝑑+πœŽπ‘˜, 𝑑β‰₯𝑑0, and the inequality πœ†β‰₯π‘šξ“π‘˜=1π‘Žπ‘˜π‘’πœ†πœŽπ‘˜(2.12) has a solution πœ†β‰₯0, then (2.1) has a positive solution for 𝑑β‰₯𝑑0.

Proof. Consider the equation with constant parameters Μ‡π‘₯(𝑑)βˆ’π‘šξ“π‘˜=1π‘Žπ‘˜π‘₯𝑑+πœŽπ‘˜ξ€Έ=0.(2.13) Since the function 𝑒(𝑑)β‰‘πœ† 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 0β‰€π‘Žπ‘˜(𝑑)β‰€π‘Žπ‘˜, π‘‘β‰€β„Žπ‘˜(𝑑)≀𝑑+𝜎 for 𝑑β‰₯𝑑0, and π‘šξ“π‘˜=1π‘Žπ‘˜β‰€1,π‘’πœŽ(2.14) then (2.1) has a positive solution for 𝑑β‰₯𝑑0.

Proof. Since βˆ‘π‘šπ‘˜=1π‘Žπ‘˜β‰€1/π‘’πœŽ, the number πœ†=1/𝜎 is a positive solution of the inequality ξƒ©πœ†β‰₯π‘šξ“π‘˜=1π‘Žπ‘˜ξƒͺπ‘’πœ†πœŽ,(2.15) which completes the proof.

Consider now the equation with positive coefficientsΜ‡π‘₯(𝑑)+π‘šξ“π‘˜=1π‘Žπ‘˜ξ€·β„Ž(𝑑)π‘₯π‘˜ξ€Έ(𝑑)=0.(2.16)

Theorem 2.10. Suppose that π‘Žπ‘˜(𝑑)β‰₯0 are continuous functions bounded on [𝑑0,∞) and β„Žπ‘˜ are equicontinuous functions on [𝑑0,∞) satisfying 0β‰€β„Žπ‘˜(𝑑)βˆ’π‘‘β‰€π›Ώ, then (2.16) has a nonoscillatory solution.

Proof. In the space 𝐢[𝑑0,∞) of continuous functions on [𝑑0,∞), consider the set 𝑀=π‘’βˆ£0β‰€π‘’β‰€π‘šξ“π‘˜=1π‘Žπ‘˜ξƒ°,(𝑑)(2.17) and the operator (𝐻𝑒)(𝑑)=π‘šξ“π‘˜=1π‘Žπ‘˜ξ‚»βˆ’ξ€œ(𝑑)expβ„Žπ‘˜π‘‘(𝑑)ξ‚Ό.𝑒(𝑠)𝑑𝑠(2.18) If π‘’βˆˆπ‘€, then π»π‘’βˆˆπ‘€.
For the integral operator ξ€œ(𝑇𝑒)(𝑑)∢=β„Žπ‘˜π‘‘(𝑑)𝑒(𝑠)𝑑𝑠,(2.19) we will demonstrate that 𝑇𝑀 is a compact set in the space 𝐢[𝑑0,∞). If π‘’βˆˆπ‘€, then β€–(𝑇𝑒)(𝑑)‖𝐢[𝑑0,∞)≀sup𝑑β‰₯𝑑0ξ€œπ‘‘π‘‘+𝛿||||𝑒(𝑠)𝑑𝑠≀sup𝑑β‰₯𝑑0π‘šξ“π‘˜=1π‘Žπ‘˜(𝑑)𝛿<∞.(2.20) Hence, the functions in the set 𝑇𝑀 are uniformly bounded in the space 𝐢[𝑑0,∞).
Functions β„Žπ‘˜ are equicontinuous on [𝑑0,∞), so for any πœ€>0, there exists a 𝜎0>0 such that for |π‘‘βˆ’π‘ |<𝜎0, the inequality ||β„Žπ‘˜(𝑑)βˆ’β„Žπ‘˜||<πœ€(𝑠)2sup𝑑β‰₯𝑑0π‘šξ“π‘˜=1π‘Žπ‘˜ξƒͺ(𝑑)βˆ’1,π‘˜=1,…,π‘š(2.21) holds. From the relation ξ€œβ„Žπ‘˜(𝑑0)𝑑0βˆ’ξ€œβ„Žπ‘˜π‘‘(𝑑)=ξ€œπ‘‘π‘‘0+ξ€œβ„Žπ‘˜(𝑑0)π‘‘βˆ’ξ€œβ„Žπ‘˜(𝑑0)π‘‘βˆ’ξ€œβ„Žπ‘˜β„Ž(𝑑)π‘˜ξ€·π‘‘0ξ€Έ=ξ€œπ‘‘π‘‘0βˆ’ξ€œβ„Žπ‘˜β„Ž(𝑑)π‘˜ξ€·π‘‘0ξ€Έ,(2.22) we have for |π‘‘βˆ’π‘‘0|<min{𝜎0,πœ€/2sup𝑑β‰₯𝑑0βˆ‘π‘šπ‘˜=1π‘Žπ‘˜(𝑑)} and π‘’βˆˆπ‘€ the estimate ||𝑑(𝑇𝑒)(𝑑)βˆ’(𝑇𝑒)0ξ€Έ||=||||ξ€œβ„Žπ‘˜(𝑑)tξ€œπ‘’(𝑠)βˆ’β„Žπ‘˜(𝑑0)𝑑0||||β‰€ξ€œπ‘’(𝑠)𝑑𝑠𝑑𝑑0||||ξ€œπ‘’(𝑠)𝑑𝑠+β„Žπ‘˜β„Ž(𝑑)π‘˜ξ€·π‘‘0ξ€Έ||||≀||𝑒(𝑠)π‘‘π‘ π‘‘βˆ’π‘‘0||sup𝑑β‰₯𝑑0π‘šξ“π‘˜=1π‘Žπ‘˜||β„Ž(𝑑)+π‘˜(𝑑)βˆ’β„Žπ‘˜ξ€·π‘‘0ξ€Έ||sup𝑑β‰₯𝑑0π‘šξ“π‘˜=1π‘Žπ‘˜<πœ€(𝑑)2+πœ€2=πœ€.(2.23) Hence, the set 𝑇𝑀 contains functions which are uniformly bounded and equicontinuous on [𝑑0,∞), so it is compact in the space 𝐢[𝑑0,∞); thus, the set 𝐻𝑀 is also compact in 𝐢[𝑑0,∞).
By the Schauder fixed point theorem, there exists a continuous function π‘’βˆˆπ‘€ such that 𝑒=𝐻𝑒, then the function ξ‚»βˆ’ξ€œπ‘₯(𝑑)=exp𝑑𝑑0𝑒(𝑠)𝑑𝑠(2.24) is a bounded positive solution of (2.16). Moreover, since 𝑒 is nonnegative, this solution is nonincreasing on [𝑑0,∞).

Theorem 2.11. Suppose that the conditions of Theorem 2.10 hold, ξ€œβˆžπ‘‘0π‘šξ“π‘˜=1π‘Žπ‘˜(𝑠)𝑑𝑠=∞,(2.25) and π‘₯ is a nonoscillatory solution of (2.16), then limπ‘‘β†’βˆžπ‘₯(𝑑)=0.

Proof. Let π‘₯(𝑑)>0 for 𝑑β‰₯𝑑0, then Μ‡π‘₯(𝑑)≀0 for 𝑑β‰₯𝑑0. Hence, π‘₯(𝑑) is nonincreasing and thus has a finite limit. If limπ‘‘β†’βˆžπ‘₯(𝑑)=𝑑>0, then π‘₯(𝑑)>𝑑 for any 𝑑, and thus βˆ‘Μ‡π‘₯(𝑑)β‰€βˆ’π‘‘π‘šπ‘˜=1π‘Žπ‘˜(𝑑) which implies limπ‘‘β†’βˆžπ‘₯(𝑑)=βˆ’βˆž. This contradicts to the assumption that π‘₯(𝑑) is positive, and therefore limπ‘‘β†’βˆžπ‘₯(𝑑)=0.

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

Example 2.12. Consider the equation Μ‡π‘₯(𝑑)+π‘₯(β„Ž(𝑑))=0, where β„Ž(𝑑)=𝑑𝑑ln𝑑, 𝑑β‰₯3, π‘₯(3)=1/ln3. Then, π‘₯(𝑑)=1/(ln𝑑) is the solution which tends to zero slower than π‘‘βˆ’π‘Ÿ for any π‘Ÿ>0.

Consider now the advanced equation with positive and negative coefficientsΜ‡π‘₯(𝑑)βˆ’π‘šξ“π‘˜=1ξ€Ίπ‘Žπ‘˜ξ€·β„Ž(𝑑)π‘₯π‘˜ξ€Έ(𝑑)βˆ’π‘π‘˜ξ€·π‘”(𝑑)π‘₯π‘˜(𝑑)ξ€Έξ€»=0,𝑑β‰₯0.(2.26)

Theorem 2.13. Suppose that π‘Žπ‘˜(𝑑) and π‘π‘˜(𝑑) are Lebesgue measurable locally essentially bounded functions, π‘Žπ‘˜(𝑑)β‰₯π‘π‘˜(𝑑)β‰₯0, β„Žπ‘˜(𝑑) and π‘”π‘˜(𝑑) are Lebesgue measurable functions, β„Žπ‘˜(𝑑)β‰₯π‘”π‘˜(𝑑)β‰₯𝑑, 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 𝑒0 be a nonnegative solution of (2.2) and denote 𝑒𝑛+1(𝑑)=π‘šξ“π‘˜=1ξ‚΅π‘Žπ‘˜ξ‚»ξ€œ(𝑑)expβ„Žπ‘˜π‘‘(𝑑)𝑒𝑛(𝑠)π‘‘π‘ βˆ’π‘π‘˜ξ‚»ξ€œ(𝑑)expπ‘”π‘˜π‘‘(𝑑)𝑒𝑛(𝑠)𝑑𝑠,𝑑β‰₯𝑑0,𝑛β‰₯0.(2.27) We have 𝑒0β‰₯0, and by (2.2), 𝑒0β‰₯π‘šξ“π‘˜=1π‘Žπ‘˜ξ‚»ξ€œ(𝑑)expβ„Žπ‘˜π‘‘(𝑑)𝑒0ξ‚Όβ‰₯(𝑠)π‘‘π‘ π‘šξ“π‘˜=1ξ‚΅π‘Žπ‘˜ξ‚»ξ€œ(𝑑)expβ„Žπ‘˜π‘‘(𝑑)𝑒0ξ‚Ό(𝑠)π‘‘π‘ βˆ’π‘π‘˜ξ‚»ξ€œ(𝑑)expπ‘”π‘˜π‘‘(𝑑)𝑒0(𝑠)𝑑𝑠=𝑒1(𝑑).(2.28) Since π‘Žπ‘˜(𝑑)β‰₯π‘π‘˜(𝑑)β‰₯0 and π‘‘β‰€π‘”π‘˜(𝑑)β‰€β„Žπ‘˜(𝑑), then 𝑒1(𝑑)β‰₯0.
Next, let us assume that 0≀𝑒𝑛(𝑑)β‰€π‘’π‘›βˆ’1(𝑑). The assumptions of the theorem imply 𝑒𝑛+1β‰₯0. Let us demonstrate that 𝑒𝑛+1(𝑑)≀𝑒𝑛(𝑑). This inequality has the form π‘šξ“π‘˜=1ξ‚΅π‘Žπ‘˜ξ‚»ξ€œ(𝑑)expβ„Žπ‘˜π‘‘(𝑑)𝑒𝑛(𝑠)π‘‘π‘ βˆ’π‘π‘˜ξ‚»ξ€œ(𝑑)expπ‘”π‘˜π‘‘(𝑑)𝑒𝑛≀(𝑠)π‘‘π‘ ξ‚Όξ‚Άπ‘šξ“π‘˜=1ξ‚΅π‘Žπ‘˜ξ‚»ξ€œ(𝑑)expβ„Žπ‘˜π‘‘(𝑑)π‘’π‘›βˆ’1ξ‚Ό(𝑠)π‘‘π‘ βˆ’π‘π‘˜ξ‚»ξ€œ(𝑑)expπ‘”π‘˜π‘‘(𝑑)π‘’π‘›βˆ’1,(𝑠)𝑑𝑠(2.29) which is equivalent to π‘šξ“π‘˜=1ξ‚»ξ€œexpβ„Žπ‘˜π‘‘(𝑑)π‘’π‘›π‘Ž(𝑠)π‘‘π‘ ξ‚Όξ‚΅π‘˜(𝑑)βˆ’π‘π‘˜ξ‚»βˆ’ξ€œ(𝑑)expβ„Žπ‘˜π‘”(𝑑)π‘˜(𝑑)𝑒𝑛≀(𝑠)π‘‘π‘ ξ‚Όξ‚Άπ‘šξ“π‘˜=1ξ‚»ξ€œexpβ„Žπ‘˜π‘‘(𝑑)π‘’π‘›βˆ’1π‘Ž(𝑠)π‘‘π‘ ξ‚Όξ‚΅π‘˜(𝑑)βˆ’π‘π‘˜ξ‚»βˆ’ξ€œ(𝑑)expβ„Žπ‘˜π‘”(𝑑)π‘˜(𝑑)π‘’π‘›βˆ’1.(𝑠)𝑑𝑠(2.30) This inequality is evident for any 0≀𝑒𝑛(𝑑)β‰€π‘’π‘›βˆ’1(𝑑), π‘Žπ‘˜(𝑑)β‰₯0, and π‘π‘˜β‰₯0; thus, we have 𝑒𝑛+1(𝑑)≀𝑒𝑛(𝑑).
By the Lebesgue convergence theorem, there is a pointwise limit 𝑒(𝑑)=limπ‘›β†’βˆžπ‘’π‘›(𝑑) satisfying 𝑒(𝑑)=π‘šξ“π‘˜=1ξ‚΅π‘Žπ‘˜ξ‚»ξ€œ(𝑑)expβ„Žπ‘˜π‘‘(𝑑)𝑒(𝑠)π‘‘π‘ βˆ’π‘π‘˜ξ‚»ξ€œ(𝑑)expπ‘”π‘˜π‘‘(𝑑)𝑒(𝑠)𝑑𝑠,𝑑β‰₯𝑑0,(2.31)𝑒(𝑑)β‰₯0, 𝑑β‰₯𝑑0. Then, the function 𝑑π‘₯(𝑑)=π‘₯0ξ€Έξ‚»ξ€œexp𝑑𝑑0𝑒(𝑠)𝑑𝑠,𝑑β‰₯𝑑0(2.32) is a positive nondecreasing solution of (2.26) for any π‘₯(𝑑0)>0 and is a negative nonincreasing solution of (2.26) for any π‘₯(𝑑0)<0.

Corollary 2.14. Let π‘Žπ‘˜(𝑑) and π‘π‘˜(𝑑) be Lebesgue measurable locally essentially bounded functions satisfying π‘Žπ‘˜(𝑑)β‰₯π‘π‘˜(𝑑)β‰₯0, and let β„Žπ‘˜(𝑑) and π‘”π‘˜(𝑑) be Lebesgue measurable functions, where β„Žπ‘˜(𝑑)β‰₯π‘”π‘˜(𝑑)β‰₯𝑑. Assume in addition that inequality (2.7) holds. Then, (2.26) has a nonoscillatory solution.

Consider now the equation with constant deviations of advanced argumentsΜ‡π‘₯(𝑑)βˆ’π‘šξ“π‘˜=1ξ€Ίπ‘Žπ‘˜ξ€·(𝑑)π‘₯𝑑+πœπ‘˜ξ€Έβˆ’π‘π‘˜ξ€·(𝑑)π‘₯𝑑+πœŽπ‘˜ξ€Έξ€»=0,(2.33) where π‘Žπ‘˜,π‘π‘˜ are continuous functions, πœπ‘˜β‰₯0, πœŽπ‘˜β‰₯0.

Denote π΄π‘˜=sup𝑑β‰₯𝑑0π‘Žπ‘˜(𝑑), π‘Žπ‘˜=inf𝑑β‰₯𝑑0π‘Žπ‘˜(𝑑), π΅π‘˜=sup𝑑β‰₯𝑑0π‘π‘˜(𝑑), π‘π‘˜=inf𝑑β‰₯𝑑0π‘π‘˜(𝑑).

Theorem 2.15. Suppose that π‘Žπ‘˜β‰₯0, π‘π‘˜β‰₯0, π΄π‘˜<∞, and π΅π‘˜<∞.
If there exists a number πœ†0<0 such that π‘šξ“π‘˜=1ξ€·π‘Žπ‘˜π‘’πœ†0πœπ‘˜βˆ’π΅π‘˜ξ€Έβ‰₯πœ†0,(2.34)π‘šξ“π‘˜=1ξ€·π΄π‘˜βˆ’π‘π‘˜π‘’πœ†0πœŽπ‘˜ξ€Έβ‰€0,(2.35) then (2.33) has a nonoscillatory solution; moreover, it has a positive nonincreasing and a negative nondecreasing solutions.

Proof. In the space 𝐢[𝑑0,∞), consider the set 𝑀={π‘’βˆ£πœ†0≀𝑒≀0} and the operator (𝐻𝑒)(𝑑)=π‘šξ“π‘˜=1ξ‚΅π‘Žπ‘˜ξ‚»ξ€œ(𝑑)exp𝑑+πœπ‘˜π‘‘ξ‚Όπ‘’(𝑠)π‘‘π‘ βˆ’π‘π‘˜ξ‚»ξ€œ(𝑑)exp𝑑+πœŽπ‘˜π‘‘.𝑒(𝑠)𝑑𝑠(2.36) For π‘’βˆˆπ‘€, we have from (2.34) and (2.35) (𝐻𝑒)(𝑑)β‰€π‘šξ“π‘˜=1ξ€·π΄π‘˜βˆ’π‘π‘˜π‘’πœ†0πœŽπ‘˜ξ€Έβ‰€0,(𝐻𝑒)(𝑑)β‰₯π‘šξ“π‘˜=1ξ€·π‘Žπ‘˜π‘’πœ†0πœπ‘˜βˆ’π΅π‘˜ξ€Έβ‰₯πœ†0.(2.37) Hence, π»π‘€βŠ‚π‘€.
Consider the integral operator ξ€œ(𝑇𝑒)(𝑑)∢=𝑑𝑑+𝛿𝑒(𝑠)𝑑𝑠,𝛿>0.(2.38) We will show that 𝑇𝑀 is a compact set in the space 𝐢[𝑑0,∞). For π‘’βˆˆπ‘€, we have β€–β€–(𝑇𝑒)(𝑑)𝐢[𝑑0,∞)≀sup𝑑β‰₯𝑑0ξ€œπ‘‘π‘‘+𝛿||||||πœ†π‘’(𝑠)𝑑𝑠≀0||𝛿.(2.39) Hence, the functions in the set 𝑇𝑀 are uniformly bounded in the space 𝐢[𝑑0,∞).
The equality βˆ«π‘‘0𝑑+𝛿0βˆ’βˆ«π‘‘π‘‘+𝛿=βˆ«π‘‘π‘‘0+βˆ«π‘‘0𝑑+π›Ώβˆ’βˆ«π‘‘0𝑑+π›Ώβˆ’βˆ«π‘‘π‘‘+𝛿0+𝛿=βˆ«π‘‘π‘‘0βˆ’βˆ«π‘‘π‘‘+𝛿0+𝛿 implies ||𝑑(𝑇𝑒)(𝑑)βˆ’(𝑇𝑒)0ξ€Έ||=||||ξ€œπ‘‘π‘‘+π›Ώξ€œπ‘’(𝑠)βˆ’π‘‘0𝑑+𝛿0||||β‰€ξ€œπ‘’(𝑠)𝑑𝑠𝑑𝑑0||||ξ€œπ‘’(𝑠)𝑑𝑠+𝑑𝑑+𝛿0+𝛿||||||πœ†π‘’(𝑠)𝑑𝑠≀20||||π‘‘βˆ’π‘‘0||.(2.40) Hence, the set 𝑇𝑀 and so the set 𝐻𝑀 are compact in the space 𝐢[𝑑0,∞).
By the Schauder fixed point theorem, there exists a continuous function 𝑒 satisfying πœ†0≀𝑒≀0 such that 𝑒=𝐻𝑒; thus, the function 𝑑π‘₯(𝑑)=π‘₯0ξ€Έξ‚»ξ€œexp𝑑𝑑0𝑒(𝑠)𝑑𝑠,𝑑β‰₯𝑑0(2.41) is a positive nonincreasing solution of (2.33) for any π‘₯(𝑑0)>0 and is a negative nondecreasing solution of (2.26) for any π‘₯(𝑑0)<0.

Let us remark that (2.35) for any πœ†0<0 implies βˆ‘π‘šπ‘˜=1(π΄π‘˜βˆ’π‘π‘˜)<0.

Corollary 2.16. Let βˆ‘π‘šπ‘˜=1(π΄π‘˜βˆ’π‘π‘˜)<0, βˆ‘π‘šπ‘˜=1π΄π‘˜>0, and for πœ†0=ξ€·βˆ‘lnπ‘šπ‘˜=1π΄π‘˜/βˆ‘π‘šπ‘˜=1π‘π‘˜ξ€Έmaxπ‘˜πœŽk,(2.42) the inequality π‘šξ“π‘˜=1ξ€·π‘Žπ‘˜π‘’πœ†0πœπ‘˜βˆ’π΅π‘˜ξ€Έβ‰₯πœ†0(2.43) holds, then (2.33) has a bounded positive solution.

Proof. The negative number πœ†0 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 Μ‡π‘₯(𝑑)βˆ’π‘Žπ‘₯(𝑑+π‘Ÿ)+𝑏π‘₯(𝑑+𝑑)=0,(2.44) where 0<π‘Ž<𝑏, 𝑑>0, π‘Ÿβ‰₯0. Then, πœ†0=(1/𝑑)ln(π‘Ž/𝑏) is the minimal value of πœ† for which inequality (2.35) holds; for (2.44), it has the form π‘Žβˆ’π‘π‘’πœ†π‘‘β‰€0.
Inequality (2.34) for (2.44) can be rewritten as 𝑓(πœ†)=π‘Žπ‘’πœ†π‘Ÿβˆ’π‘βˆ’πœ†β‰₯0,(2.45) where the function 𝑓(π‘₯) decreases on (βˆ’βˆž,βˆ’ln(π‘Žπ‘Ÿ)/π‘Ÿ] if 𝜏>0 and for any negative π‘₯ if π‘Ÿ=0; besides, 𝑓(0)<0. Thus, if 𝑓(πœ†1)<0 for some πœ†1<0, then 𝑓(πœ†)<0 for any πœ†βˆˆ[πœ†1,0). Hence, the inequality π‘“ξ€·πœ†0ξ€Έξ‚€π‘Ž=π‘Žπ‘ξ‚π‘Ÿ/𝑑1βˆ’π‘βˆ’π‘‘ξ‚€π‘Žln𝑏β‰₯0(2.46) is necessary and sufficient for the conditions of Theorem 2.15 to be satisfied for (2.44).
Figure 1 demonstrates possible values of advances 𝑑 and π‘Ÿ, such that Corollary 2.16 implies the existence of a positive solution in the case 1=π‘Ž<𝑏=2. Then, (2.46) has the form 0.5π‘Ÿ/𝑑β‰₯2βˆ’(ln2)/𝑑, which is possible only for 𝑑>0.5ln2β‰ˆ0.347 and for these values is equivalent to π‘Ÿβ‰€βˆ’π‘‘ln(2βˆ’ln2/𝑑).ln2(2.47)

Figure 1: The domain of values (𝑑,π‘Ÿ) satisfying inequality (2.47). If the values of advances 𝑑 and π‘Ÿ 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 β„Žπ‘˜(𝑑)=𝑑+πœπ‘˜), 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 π‘Žπ‘˜(𝑑)β‰₯0, 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 β„Ž(𝑑)β‰₯𝑑 and the equation Μ‡π‘₯(𝑑)βˆ’π‘Ž+(𝑑)π‘₯(β„Ž(𝑑))=0(3.1)

has a nonoscillatory solution, then the equation Μ‡π‘₯(𝑑)βˆ’π‘Ž(𝑑)π‘₯(β„Ž(𝑑))=0(3.2)

also has a nonoscillatory solution, where π‘Ž+(𝑑)=max{π‘Ž(𝑑),0}.

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.


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


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