On Nonoscillation of Advanced Differential Equations with Several Terms

Abstract

Existence of positive solutions for advanced equations with several terms 𝑥̇(𝑡)+∑^𝑚𝑘=1𝑎_𝑘(𝑡)𝑥(ℎ_𝑘(𝑡))=0,ℎ_𝑘(𝑡)≥𝑡 is investigated in the following three cases: (a) all coefficients 𝑎_𝑘 are positive; (b) all coefficients 𝑎_𝑘 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 𝑥𝑥̇̇((𝑡𝑡))−−𝑎𝑎((𝑡𝑡))𝑥𝑥((ℎ𝑡)(+𝑡))𝑏+(𝑡)𝑏𝑥(𝑡()𝑔𝑥(𝑡()𝑡))==00,,(1.1) where 𝑎(𝑡)≥0, 𝑏(𝑡)≥0, ℎ(𝑡)≥𝑡, and 𝑔(𝑡)≥𝑡.

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 𝑎, 𝑏, and ℎ are equicontinuous on [0,∞), 𝑎(𝑡)≥0, 𝑏(𝑡)≥0, ℎ(𝑡)≥𝑡, and limsup_𝑡→∞[ℎ(𝑡)−𝑡]<∞, then the advanced equation 𝑥̇(𝑡)+𝑎(𝑡)𝑥(ℎ(𝑡))+𝑏(𝑡)𝑥(𝑡)=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 𝑎, 𝑏, and ℎ are equicontinuous on [0,∞), 𝑎(𝑡)≥0, 𝑏(𝑡)≥0, ℎ(𝑡)≥𝑡, limsup_𝑡→∞[ℎ(𝑡)−𝑡]<∞, and limsup_𝑡→∞^{𝑡ℎ(𝑡)}𝑎(𝑠)exp^{𝑠ℎ(𝑠)}𝑏(𝜏)𝑑𝜏𝑑𝑠<1𝑒,(1.3) then the advanced equation 𝑥̇(𝑡)−𝑎(𝑡)𝑥(ℎ(𝑡))−𝑏(𝑡)𝑥(𝑡)=0(1.4) has a nonoscillatory solution.

Corollary 2.3 of the present paper extends Theorem B to the case of several coefficients 𝑎_𝑘≥0 and advanced arguments ℎ_𝑘 (generally, 𝑏(𝑡)≡0); if ^{max𝑘ℎ𝑘𝑡𝑚(𝑡)}_𝑖=1𝑎_𝑖(𝑠)𝑑𝑠≤1𝑒,(1.5) then the equation 𝑥̇(𝑡)+^𝑚_𝑘=1𝑎_𝑘(𝑡)𝑥ℎ_𝑘(𝑡)=0(1.6)

has an eventually positive solution. To the best of our knowledge, only the opposite inequality (with min_𝑘ℎ_𝑘(𝑡) rather than max_𝑘ℎ_𝑘(𝑡) 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 0≤𝑎(𝑡)≤𝑏(𝑡) and ℎ(𝑡)≥𝑡, then there exists a nonoscillatory solution of the equation 𝑥̇(𝑡)−𝑎(𝑡)𝑥(ℎ(𝑡))+𝑏(𝑡)𝑥(𝑡)=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 𝑥̇(𝑡)+𝑎(𝑡)𝑥(ℎ(𝑡))−𝑏(𝑡)𝑥(𝑔(𝑡))=0,ℎ(𝑡)≥𝑡,𝑔(𝑡)≤𝑡,𝑎(𝑡)≥0,𝑏(𝑡)≥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 equation𝑥̇(𝑡)−^𝑚_𝑘=1𝑎_𝑘(𝑡)𝑥ℎ_𝑘(𝑡)=0,(2.1) under the following conditions:(a1)𝑎_𝑘(𝑡)≥0, 𝑘=1,…,𝑚, are Lebesgue measurable functions locally essentially bounded for 𝑡≥0,(a2)ℎ_𝑘∶[0,∞)→ℝ are Lebesgue measurable functions, ℎ_𝑘(𝑡)≥𝑡, 𝑘=1,…,𝑚.

Definition 2.1. A locally absolutely continuous function 𝑥∶[𝑡₀,∞)→𝑅 is called a solution of problem (2.1) if it satisfies (2.1) for almost all 𝑡∈[𝑡₀,∞).

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

Theorem 2.2. Suppose that the inequality 𝑢(𝑡)≥^𝑚_𝑘=1𝑎_𝑘(𝑡)exp^{ℎ𝑘𝑡(𝑡)}𝑢(𝑠)𝑑𝑠,𝑡≥𝑡₀(2.2) has a nonnegative solution which is integrable on each interval [𝑡₀,𝑏], then (2.1) has a positive solution for 𝑡≥𝑡₀.

Proof. Let 𝑢₀(𝑡) be a nonnegative solution of inequality (2.2). Denote 𝑢_𝑛+1(𝑡)=^𝑚_𝑘=1𝑎_𝑘(𝑡)exp^{ℎ𝑘𝑡(𝑡)}𝑢_𝑛(𝑠)𝑑𝑠,𝑛=0,1,…,(2.3) then 𝑢₁(𝑡)=^𝑚_𝑘=1𝑎_𝑘(𝑡)exp^{ℎ𝑘𝑡(𝑡)}𝑢₀(𝑠)𝑑𝑠≤𝑢₀(𝑡).(2.4) By induction we have 0≤𝑢_𝑛+1(𝑡)≤𝑢_𝑛(𝑡)≤𝑢₀(𝑡). Hence, there exists a pointwise limit 𝑢(𝑡)=lim_𝑛→∞𝑢_𝑛(𝑡). By the Lebesgue convergence theorem, we have 𝑢(𝑡)=^𝑚_𝑘=1𝑎_𝑘(𝑡)exp^{ℎ𝑘𝑡(𝑡)}𝑢(𝑠)𝑑𝑠.(2.5) Then, the function 𝑥(𝑡)=𝑥𝑡₀exp^𝑡𝑡₀𝑢(𝑠)𝑑𝑠forany𝑥𝑡₀>0(2.6) is a positive solution of (2.1).

Corollary 2.3. If ^{max𝑘ℎ𝑘𝑡𝑚(𝑡)}_𝑖=1𝑎_𝑖(𝑠)𝑑𝑠≤1𝑒,𝑡≥𝑡⁰,(2.7) then (2.1) has a positive solution for 𝑡≥𝑡₀.

Proof. Let 𝑢₀(𝑡)=𝑒∑^𝑚𝑘=1𝑎_𝑘(𝑡), then 𝑢₀ satisfies (2.2) at any point 𝑡 where ∑^𝑚𝑘=1𝑎_𝑘(𝑡)=0. In the case when ∑^𝑚𝑘=1𝑎_𝑘(𝑡)≠0, inequality (2.7) implies 𝑢₀(𝑡)∑^𝑚𝑘=1≥𝑎_𝑘(𝑡)exp∫^{ℎ𝑘𝑡(𝑡)}𝑢𝑢₀(𝑠)𝑑𝑠₀(𝑡)=∑^𝑚𝑘=1𝑎_𝑘(𝑡)exp𝑒∑∫^{max𝑘ℎ𝑘𝑡(𝑡)}𝑢₀(𝑠)𝑑𝑠_𝑚𝑘=1𝑎_𝑘(𝑡)≥∑𝑒∑^𝑚𝑘=1𝑎_𝑘(𝑡)exp𝑒∫^{max𝑘ℎ𝑘𝑡(𝑡)}∑^𝑚𝑖=1𝑎_𝑖(𝑠)𝑑𝑠_𝑚𝑘=1𝑎_𝑘(𝑡)∑^𝑚𝑘=1𝑎_𝑘(𝑡)𝑒=1.(2.8) Hence, 𝑢₀(𝑡) is a positive solution of inequality (2.2). By Theorem 2.2, (2.1) has a positive solution for 𝑡≥𝑡₀.

Corollary 2.4. If there exists 𝜎>0 such that ℎ_𝑘(𝑡)−𝑡≤𝜎 and ∫^∞0∑^𝑚𝑘=1𝑎_𝑘(𝑠)𝑑𝑠<∞, then (2.1) has an eventually positive solution.

Corollary 2.5. If there exists 𝜎>0 such that ℎ_𝑘(𝑡)−𝑡≤𝜎 and lim_𝑡→∞𝑎_𝑘(𝑡)=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 𝑡₀≥0 such that (2.7) is satisfied.

Theorem 2.6. Let ∫^∞∑^𝑚𝑘=1𝑎_𝑘(𝑠)𝑑𝑠=∞ and 𝑥 be an eventually positive solution of (2.1), then lim_𝑡→∞𝑥(𝑡)=∞.

Proof. Suppose that 𝑥(𝑡)>0 for 𝑡≥𝑡₁, then 𝑥̇(𝑡)≥0 for 𝑡≥𝑡₁ and 𝑥̇(𝑡)≥^𝑚_𝑘=1𝑎_𝑘(𝑡)𝑥𝑡₁,𝑡≥𝑡₁,(2.9) which implies 𝑥(𝑡)≥𝑥𝑡₁^𝑡𝑡₁^𝑚_𝑘=1𝑎_𝑘(𝑠)𝑑𝑠.(2.10) Thus, lim_𝑡→∞𝑥(𝑡)=∞.

Consider together with (2.1) the following equation:𝑥̇(𝑡)−^𝑚_𝑘=1𝑏_𝑘(𝑡)𝑥𝑔_𝑘(𝑡)=0,(2.11) for 𝑡≥𝑡₀. We assume that for (2.11) conditions (a1)-(a2) also hold.

Theorem 2.7. Suppose that 𝑡≤𝑔_𝑘(𝑡)≤ℎ_𝑘(𝑡), 0≤𝑏_𝑘(𝑡)≤𝑎_𝑘(𝑡), 𝑡≥𝑡₀, and the conditions of Theorem 2.2 hold, then (2.11) has a positive solution for 𝑡≥𝑡₀.

Proof. Let 𝑢₀(𝑡)≥0 be a solution of inequality (2.2) for 𝑡≥𝑡₀, then this function is also a solution of this inequality if 𝑎_𝑘(𝑡) and ℎ_𝑘(𝑡) are replaced by 𝑏_𝑘(𝑡) and 𝑔_𝑘(𝑡). The reference to Theorem 2.2 completes the proof.

Corollary 2.8. Suppose that there exist 𝑎_𝑘>0 and 𝜎_𝑘>0 such that 0≤𝑎_𝑘(𝑡)≤𝑎_𝑘, 𝑡≤ℎ_𝑘(𝑡)≤𝑡+𝜎_𝑘, 𝑡≥𝑡₀, and the inequality 𝜆≥^𝑚_𝑘=1𝑎_𝑘𝑒^𝜆𝜎𝑘(2.12) has a solution 𝜆≥0, then (2.1) has a positive solution for 𝑡≥𝑡₀.

Proof. Consider the equation with constant parameters 𝑥̇(𝑡)−^𝑚_𝑘=1𝑎_𝑘𝑥𝑡+𝜎_𝑘=0.(2.13) Since the function 𝑢(𝑡)≡𝜆 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 0≤𝑎_𝑘(𝑡)≤𝑎_𝑘, 𝑡≤ℎ_𝑘(𝑡)≤𝑡+𝜎 for 𝑡≥𝑡₀, and ^𝑚_𝑘=1𝑎_𝑘≤1𝑒𝜎,(2.14) then (2.1) has a positive solution for 𝑡≥𝑡₀.

Proof. Since ∑^𝑚𝑘=1𝑎_𝑘≤1/𝑒𝜎, the number 𝜆=1/𝜎 is a positive solution of the inequality 𝜆≥^𝑚_𝑘=1𝑎_𝑘𝑒^𝜆𝜎,(2.15) which completes the proof.

Consider now the equation with positive coefficients𝑥̇(𝑡)+^𝑚_𝑘=1𝑎_𝑘(𝑡)𝑥ℎ_𝑘(𝑡)=0.(2.16)

Theorem 2.10. Suppose that 𝑎_𝑘(𝑡)≥0 are continuous functions bounded on [𝑡₀,∞) and ℎ_𝑘 are equicontinuous functions on [𝑡₀,∞) satisfying 0≤ℎ_𝑘(𝑡)−𝑡≤𝛿, then (2.16) has a nonoscillatory solution.

Proof. In the space 𝐶[𝑡₀,∞) of continuous functions on [𝑡₀,∞), consider the set 𝑀=𝑢∣0≤𝑢≤^𝑚_𝑘=1𝑎_𝑘(𝑡),(2.17) and the operator (𝐻𝑢)(𝑡)=^𝑚_𝑘=1𝑎_𝑘(𝑡)exp−^{ℎ𝑘𝑡(𝑡)}𝑢(𝑠)𝑑𝑠.(2.18) If 𝑢∈𝑀, then 𝐻𝑢∈𝑀.
For the integral operator (𝑇𝑢)(𝑡)∶=^{ℎ𝑘𝑡(𝑡)}𝑢(𝑠)𝑑𝑠,(2.19) we will demonstrate that 𝑇𝑀 is a compact set in the space 𝐶[𝑡₀,∞). If 𝑢∈𝑀, then ‖(𝑇𝑢)(𝑡)‖_{𝐶[𝑡0,∞)}≤sup_𝑡≥𝑡0^{𝑡𝑡+𝛿}||𝑢(𝑠)||𝑑𝑠≤sup_𝑡≥𝑡0^𝑚_𝑘=1𝑎_𝑘(𝑡)𝛿<∞.(2.20) Hence, the functions in the set 𝑇𝑀 are uniformly bounded in the space 𝐶[𝑡₀,∞).
Functions ℎ_𝑘 are equicontinuous on [𝑡₀,∞), so for any 𝜀>0, there exists a 𝜎₀>0 such that for |𝑡−𝑠|<𝜎₀, the inequality ||ℎ_𝑘(𝑡)−ℎ_𝑘(𝑠)||<𝜀2sup_𝑡≥𝑡0^𝑚_𝑘=1𝑎_𝑘(𝑡)⁻¹,𝑘=1,…,𝑚(2.21) holds. From the relation ^{ℎ𝑘(𝑡0)𝑡}₀−^{ℎ𝑘𝑡(𝑡)}=^𝑡𝑡₀+^{ℎ𝑘(𝑡0)𝑡}−^{ℎ𝑘(𝑡0)𝑡}−^{ℎ𝑘ℎ(𝑡)}_{𝑘𝑡0}=^𝑡𝑡₀−^{ℎ𝑘ℎ(𝑡)}_{𝑘𝑡0},(2.22) we have for |𝑡−𝑡₀|<min{𝜎₀,𝜀/2sup_𝑡≥𝑡0∑^𝑚𝑘=1𝑎_𝑘(𝑡)} and 𝑢∈𝑀 the estimate ||(𝑇𝑢)(𝑡)−(𝑇𝑢)𝑡₀||=≤||||^{ℎ𝑘(𝑡)}_t𝑢(𝑠)−^{ℎ𝑘(𝑡0)𝑡}₀𝑢(𝑠)𝑑𝑠_𝑡𝑡0||||≤||𝑡−||𝑢𝑡₀(𝑠)||𝑑𝑠+^{ℎ𝑘ℎ(𝑡)}_{𝑘𝑡0}||𝑢(𝑠)||𝑑𝑠||sup_𝑡≥𝑡0^𝑚_𝑘=1𝑎_𝑘(𝑡)+||ℎ_𝑘(𝑡)−ℎ_𝑘𝑡₀||sup_𝑡≥𝑡0^𝑚_𝑘=1<𝜀𝑎_𝑘(𝑡)2+2𝜀=𝜀.(2.23) Hence, the set 𝑇𝑀 contains functions which are uniformly bounded and equicontinuous on [𝑡₀,∞), so it is compact in the space 𝐶[𝑡₀,∞); thus, the set 𝐻𝑀 is also compact in 𝐶[𝑡₀,∞).
By the Schauder fixed point theorem, there exists a continuous function 𝑢∈𝑀 such that 𝑢=𝐻𝑢, then the function 𝑥(𝑡)=exp−^𝑡𝑡₀𝑢(𝑠)𝑑𝑠(2.24) is a bounded positive solution of (2.16). Moreover, since 𝑢 is nonnegative, this solution is nonincreasing on [𝑡₀,∞).

Theorem 2.11. Suppose that the conditions of Theorem 2.10 hold, ^∞𝑡₀^𝑚_𝑘=1𝑎_𝑘(𝑠)𝑑𝑠=∞,(2.25) and 𝑥 is a nonoscillatory solution of (2.16), then lim_𝑡→∞𝑥(𝑡)=0.

Proof. Let 𝑥(𝑡)>0 for 𝑡≥𝑡₀, then 𝑥̇(𝑡)≤0 for 𝑡≥𝑡₀. Hence, 𝑥(𝑡) is nonincreasing and thus has a finite limit. If lim_𝑡→∞𝑥(𝑡)=𝑑>0, then 𝑥(𝑡)>𝑑 for any 𝑡, and thus 𝑥̇(𝑡)≤−𝑑∑^𝑚𝑘=1𝑎_𝑘(𝑡) which implies lim_𝑡→∞𝑥(𝑡)=−∞. This contradicts to the assumption that 𝑥(𝑡) is positive, and therefore lim_𝑡→∞𝑥(𝑡)=0.

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

Example 2.12. Consider the equation 𝑥̇(𝑡)+𝑥(ℎ(𝑡))=0, where ℎ(𝑡)=𝑡^𝑡ln𝑡, 𝑡≥3, 𝑥(3)=1/ln3. Then, 𝑥(𝑡)=1/(ln𝑡) is the solution which tends to zero slower than 𝑡^−𝑟 for any 𝑟>0.

Consider now the advanced equation with positive and negative coefficients𝑥̇(𝑡)−^𝑚_𝑘=1𝑎_𝑘(𝑡)𝑥ℎ_𝑘(𝑡)−𝑏_𝑘(𝑡)𝑥𝑔_𝑘(𝑡)=0,𝑡≥0.(2.26)

Theorem 2.13. Suppose that 𝑎_𝑘(𝑡) and 𝑏_𝑘(𝑡) are Lebesgue measurable locally essentially bounded functions, 𝑎_𝑘(𝑡)≥𝑏_𝑘(𝑡)≥0, ℎ_𝑘(𝑡) and 𝑔_𝑘(𝑡) are Lebesgue measurable functions, ℎ_𝑘(𝑡)≥𝑔_𝑘(𝑡)≥𝑡, 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 𝑢₀ be a nonnegative solution of (2.2) and denote 𝑢_𝑛+1(𝑡)=^𝑚_𝑘=1𝑎^𝑘(𝑡)exp^{ℎ𝑘𝑡(𝑡)}𝑢_𝑛(𝑠)𝑑𝑠−𝑏_𝑘(𝑡)exp^{𝑔𝑘𝑡(𝑡)}𝑢_𝑛(𝑠)𝑑𝑠,𝑡≥𝑡₀,𝑛≥0.(2.27) We have 𝑢₀≥0, and by (2.2), 𝑢₀≥^𝑚_𝑘=1≥𝑎_𝑘(𝑡)exp^{ℎ𝑘𝑡(𝑡)}𝑢₀(𝑠)𝑑𝑠_𝑚_𝑘=1𝑎^𝑘(𝑡)exp^{ℎ𝑘𝑡(𝑡)}𝑢₀(𝑠)𝑑𝑠−𝑏_𝑘(𝑡)exp^{𝑔𝑘𝑡(𝑡)}𝑢₀(𝑠)𝑑𝑠=𝑢₁(𝑡).(2.28) Since 𝑎_𝑘(𝑡)≥𝑏_𝑘(𝑡)≥0 and 𝑡≤𝑔_𝑘(𝑡)≤ℎ_𝑘(𝑡), then 𝑢₁(𝑡)≥0.
Next, let us assume that 0≤𝑢_𝑛(𝑡)≤𝑢_𝑛−1(𝑡). The assumptions of the theorem imply 𝑢_𝑛+1≥0. Let us demonstrate that 𝑢_𝑛+1(𝑡)≤𝑢_𝑛(𝑡). This inequality has the form ^𝑚_𝑘=1𝑎^𝑘≤(𝑡)exp^{ℎ𝑘𝑡(𝑡)}𝑢_𝑛(𝑠)𝑑𝑠−𝑏_𝑘(𝑡)exp^{𝑔𝑘𝑡(𝑡)}𝑢_𝑛(𝑠)𝑑𝑠_𝑚_𝑘=1𝑎^𝑘(𝑡)exp^{ℎ𝑘𝑡(𝑡)}𝑢_𝑛−1(𝑠)𝑑𝑠−𝑏_𝑘(𝑡)exp^{𝑔𝑘𝑡(𝑡)}𝑢_𝑛−1(𝑠)𝑑𝑠,(2.29) which is equivalent to ^𝑚_𝑘=1e≤xp^{ℎ𝑘𝑡(𝑡)}𝑢_𝑛(𝑠)𝑑𝑠𝑎_𝑘(𝑡)−𝑏_𝑘(𝑡)exp−^{ℎ𝑘𝑔(𝑡)}_𝑘(𝑡)𝑢_𝑛(𝑠)𝑑𝑠_𝑚_𝑘=1exp^{ℎ𝑘𝑡(𝑡)}𝑢_𝑛−1(𝑠)𝑑𝑠𝑎_𝑘(𝑡)−𝑏_𝑘(𝑡)exp−^{ℎ𝑘𝑔(𝑡)}_𝑘(𝑡)𝑢_𝑛−1(𝑠)𝑑𝑠.(2.30) This inequality is evident for any 0≤𝑢_𝑛(𝑡)≤𝑢_𝑛−1(𝑡), 𝑎_𝑘(𝑡)≥0, and 𝑏_𝑘≥0; thus, we have 𝑢_𝑛+1(𝑡)≤𝑢_𝑛(𝑡).
By the Lebesgue convergence theorem, there is a pointwise limit 𝑢(𝑡)=lim_𝑛→∞𝑢_𝑛(𝑡) satisfying 𝑢(𝑡)=^𝑚_𝑘=1𝑎^𝑘(𝑡)exp^{ℎ𝑘𝑡(𝑡)}𝑢(𝑠)𝑑𝑠−𝑏_𝑘(𝑡)exp^{𝑔𝑘𝑡(𝑡)}𝑢(𝑠)𝑑𝑠,𝑡≥𝑡₀,(2.31)𝑢(𝑡)≥0, 𝑡≥𝑡₀. Then, the function 𝑥(𝑡)=𝑥𝑡₀exp^𝑡𝑡₀𝑢(𝑠)𝑑𝑠,𝑡≥𝑡₀(2.32) is a positive nondecreasing solution of (2.26) for any 𝑥(𝑡₀)>0 and is a negative nonincreasing solution of (2.26) for any 𝑥(𝑡₀)<0.

Corollary 2.14. Let 𝑎_𝑘(𝑡) and 𝑏_𝑘(𝑡) be Lebesgue measurable locally essentially bounded functions satisfying 𝑎_𝑘(𝑡)≥𝑏_𝑘(𝑡)≥0, and let ℎ_𝑘(𝑡) and 𝑔_𝑘(𝑡) be Lebesgue measurable functions, where ℎ_𝑘(𝑡)≥𝑔_𝑘(𝑡)≥𝑡. Assume in addition that inequality (2.7) holds. Then, (2.26) has a nonoscillatory solution.

Consider now the equation with constant deviations of advanced arguments𝑥̇(𝑡)−^𝑚_𝑘=1𝑎_𝑘(𝑡)𝑥𝑡+𝜏_𝑘−𝑏_𝑘(𝑡)𝑥𝑡+𝜎_𝑘=0,(2.33) where 𝑎_𝑘,𝑏_𝑘 are continuous functions, 𝜏_𝑘≥0, 𝜎_𝑘≥0.

Denote 𝐴_𝑘=sup_𝑡≥𝑡0𝑎_𝑘(𝑡), 𝑎_𝑘=inf_𝑡≥𝑡0𝑎_𝑘(𝑡), 𝐵_𝑘=sup_𝑡≥𝑡0𝑏_𝑘(𝑡), 𝑏_𝑘=inf_𝑡≥𝑡0𝑏_𝑘(𝑡).

Theorem 2.15. Suppose that 𝑎_𝑘≥0, 𝑏_𝑘≥0, 𝐴_𝑘<∞, and 𝐵_𝑘<∞.
If there exists a number 𝜆₀<0 such that ^𝑚_𝑘=1𝑎_𝑘𝑒^{𝜆0𝜏𝑘}−𝐵_𝑘≥𝜆₀,(2.34)_𝑚_𝑘=1𝐴_𝑘−𝑏_𝑘𝑒^{𝜆0𝜎𝑘}≤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 𝐶[𝑡₀,∞), consider the set 𝑀={𝑢∣𝜆₀≤𝑢≤0} and the operator (𝐻𝑢)(𝑡)=^𝑚_𝑘=1𝑎^𝑘(𝑡)exp^{𝑡+𝜏𝑘}_𝑡𝑢(𝑠)𝑑𝑠−𝑏_𝑘(𝑡)exp^{𝑡+𝜎𝑘}_𝑡𝑢(𝑠)𝑑𝑠.(2.36) For 𝑢∈𝑀, we have from (2.34) and (2.35) (𝐻𝑢)(𝑡)≤^𝑚_𝑘=1𝐴_𝑘(𝐻𝑢)(𝑡)≥_𝑚−𝑏_𝑘𝑒^{𝜆0𝜎𝑘}≤0,_𝑘=1𝑎_𝑘𝑒^{𝜆0𝜏𝑘}−𝐵_𝑘≥𝜆₀.(2.37) Hence, 𝐻𝑀⊂𝑀.
Consider the integral operator (𝑇𝑢)(𝑡)∶=^{𝑡𝑡+𝛿}𝑢(𝑠)𝑑𝑠,𝛿>0.(2.38) We will show that 𝑇𝑀 is a compact set in the space 𝐶[𝑡₀,∞). For 𝑢∈𝑀, we have ‖(𝑇𝑢)(𝑡)‖_{𝐶[𝑡0,∞)}≤sup_𝑡≥𝑡0^{𝑡𝑡+𝛿}||𝑢(𝑠)||𝑑𝑠≤||𝜆₀||𝛿.(2.39) Hence, the functions in the set 𝑇𝑀 are uniformly bounded in the space 𝐶[𝑡₀,∞).
The equality ∫^{𝑡0𝑡+𝛿}₀−∫^{𝑡𝑡+𝛿}=∫^𝑡𝑡₀+∫^{𝑡0𝑡+𝛿}−∫^{𝑡0𝑡+𝛿}−∫^{𝑡𝑡+𝛿}_0+𝛿=∫^𝑡𝑡₀−∫^{𝑡𝑡+𝛿}_0+𝛿 implies ||(𝑇𝑢)(𝑡)−(𝑇𝑢)𝑡₀||=≤||||^{𝑡𝑡+𝛿}𝑢(𝑠)−^{𝑡0𝑡+𝛿}₀𝑢(𝑠)𝑑𝑠_𝑡𝑡0||||||𝑢(𝑠)||𝑑𝑠+^{𝑡𝑡+𝛿}_0+𝛿||𝑢(𝑠)||𝑑𝑠≤2||𝜆₀||||𝑡−𝑡₀||.(2.40) Hence, the set 𝑇𝑀 and so the set 𝐻𝑀 are compact in the space 𝐶[𝑡₀,∞).
By the Schauder fixed point theorem, there exists a continuous function 𝑢 satisfying 𝜆₀≤𝑢≤0 such that 𝑢=𝐻𝑢; thus, the function 𝑥(𝑡)=𝑥𝑡₀exp^𝑡𝑡₀𝑢(𝑠)𝑑𝑠,𝑡≥𝑡₀(2.41) is a positive nonincreasing solution of (2.33) for any 𝑥(𝑡₀)>0 and is a negative nondecreasing solution of (2.26) for any 𝑥(𝑡₀)<0.

Let us remark that (2.35) for any 𝜆₀<0 implies ∑^𝑚𝑘=1(𝐴_𝑘−𝑏_𝑘)<0.

Corollary 2.16. Let ∑^𝑚𝑘=1(𝐴_𝑘−𝑏_𝑘)<0, ∑^𝑚𝑘=1𝐴_𝑘>0, and for 𝜆₀=ln∑^𝑚𝑘=1𝐴_𝑘/∑^𝑚𝑘=1𝑏_𝑘max_𝑘𝜎_k,(2.42) the inequality ^𝑚_𝑘=1𝑎_𝑘𝑒^{𝜆0𝜏𝑘}−𝐵_𝑘≥𝜆₀(2.43) holds, then (2.33) has a bounded positive solution.

Proof. The negative number 𝜆₀ 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 𝑥̇(𝑡)−𝑎𝑥(𝑡+𝑟)+𝑏𝑥(𝑡+𝑑)=0,(2.44) where 0<𝑎<𝑏, 𝑑>0, 𝑟≥0. Then, 𝜆₀=(1/𝑑)ln(𝑎/𝑏) is the minimal value of 𝜆 for which inequality (2.35) holds; for (2.44), it has the form 𝑎−𝑏𝑒^𝜆𝑑≤0.
Inequality (2.34) for (2.44) can be rewritten as 𝑓(𝜆)=𝑎𝑒^𝜆𝑟−𝑏−𝜆≥0,(2.45) where the function 𝑓(𝑥) decreases on (−∞,−ln(𝑎𝑟)/𝑟] if 𝜏>0 and for any negative 𝑥 if 𝑟=0; besides, 𝑓(0)<0. Thus, if 𝑓(𝜆₁)<0 for some 𝜆₁<0, then 𝑓(𝜆)<0 for any 𝜆∈[𝜆₁,0). Hence, the inequality 𝑓𝜆₀=𝑎𝑎𝑏^𝑟/𝑑−𝑏−1𝑑ln𝑎𝑏≥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 𝑟, such that Corollary 2.16 implies the existence of a positive solution in the case 1=𝑎<𝑏=2. Then, (2.46) has the form 0.5^𝑟/𝑑≥2−(ln2)/𝑑, which is possible only for 𝑑>0.5ln2≈0.347 and for these values is equivalent to 𝑟≤−𝑑ln(2−ln2/𝑑)ln2.(2.47)

Figure 1 

The domain of values (𝑑,𝑟) satisfying inequality (2.47). If the values of advances 𝑑 and 𝑟 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 ℎ_𝑘(𝑡)=𝑡+𝜏_𝑘), 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 𝑎_𝑘(𝑡)≥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 ℎ(𝑡)≥𝑡 and the equation 𝑥̇(𝑡)−𝑎⁺(𝑡)𝑥(ℎ(𝑡))=0(3.1)

has a nonoscillatory solution, then the equation 𝑥̇(𝑡)−𝑎(𝑡)𝑥(ℎ(𝑡))=0(3.2)

also has a nonoscillatory solution, where 𝑎⁺(𝑡)=max{𝑎(𝑡),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.

Acknowledgments

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