Iterative oscillation tests for differential equations with several non-monotone arguments

Sufficient oscillation conditions involving $\limsup $ and $\liminf $ for first-order differential equations with several non-monotone deviating arguments and nonnegative coefficients are obtained. The results are based on the iterative application of the Gr\"{o}nwall inequality. Examples illustrating the significance of the results are also given.


Definition 2.
A solution x(t) of (1.1) (or (1.2)) is oscillatory if it is neither eventually positive nor eventually negative. If there exists an eventually positive or an eventually negative solution, the equation is nonoscillatory. An equation is oscillatory if all its solutions oscillate.
In 1990, Zhou [25] proved that if σ i (t) − t ≤ σ 0 for some σ 0 > 0, 1 ≤ i ≤ m, and  [4] established the following theorem in the case that the argument τ (t) is non-monotone and g(t) is defined as Then all solutions of (1.6) oscillate.
In addition to purely mathematical interest, consideration of non-monotone arguments is important, since it approximates the natural phenomena described by equations of the type of (1.1) or (1.2). In fact, there are always natural disturbances (e.g. noise in communication systems) that affect all the parameters of the equation and therefore monotone arguments will generally become non-monotone. In view of this, it is interesting to consider the case where the arguments (delays and advances) are non-monotone. In the present paper we obtain sufficient oscillation conditions involving lim sup and lim inf.

Main Results
In this section, we establish sufficient oscillation conditions for (1.1) and (1.2) satisfying (1.3) and (1.4), respectively. The method we apply is based on the iterative construction of solution estimates and repetitive application of the Grönwall inequality. It also uses some ideas of [13], where some oscillation results for a differential equation with a single delay were established. and As follows from their definitions, the functions g i (t), 1 ≤ i ≤ m and g(t) are non-decreasing Lebesgue measurable functions satisfying g(t) ≤ t, g i (t) ≤ t, 1 ≤ i ≤ m for all t ≥ 0.
The following lemma provides an estimation for a rate of decay for a positive solution. Such estimates are a basis for most oscillation conditions. Lemma 1. Assume that x(t) is a positive solution of (1.1). Denote  p i (ζ)a r (ζ, τ i (ζ)) dζ , r ∈ N.
Proof. The function x(t) is a positive solution of (1.1) for any t, so which means that the solution x(t) is monotonically decreasing. Thus x(τ i (t)) ≥ x(t) and Applying the Grönwall inequality, we obtain that is, estimate (2.5) is valid for r = 1.
Next, let us proceed to the induction step: assume that (2.5) holds for some r > 1, then Substituting (2.6) into (1.1) leads to the estimate Again, applying the Grönwall inequality, we have which completes the induction step and the proof of the lemma.
Let us illustrate how the estimate developed in Lemma 1 works in the case of autonomous equations. The series of estimates is evaluated using computer tools, which recently became an efficient tool in computer-assisted proofs [2]. We suggest that, similarly, a computer algebra can be used to construct the estimate iterates and, ideally, the limit estimate. The example below illustrates the procedure.
The equation has an exact nonoscillatory solution e −αt . For α = 0.5 the exact rate of decay (up to the sixth digit after the decimal point) is The largest value of the coefficient of 1/e is attained at α = 1; it is well known that it is the maximal coefficient when the equation is still nonoscillatory. The decay of the estimate then all solutions of (1.1) oscillate.
Proof. Assume, for the sake of contradiction, that there exists a nonoscillatory solution x(t) of (1.1). Since −x(t) is also a solution of (1.1), we can consider only the case when the solution x(t) is eventually positive. Then there exists t 1 > 0 such that which means that x(t) is an eventually non-increasing positive function.
Integrating (1.1) from g(t) to t, and using the fact that the function x is non-increasing, while the function g defined by (2.2) is non-decreasing, and taking into account that τ i (t) ≤ g(t) and x(τ i (s)) ≥ x(g(t))a r (g(t), τ i (s)), we obtain, for sufficiently large t, The last inequality contradicts (2.7), and the proof is complete. 5 The following example illustrates the significance of the condition lim Example 2. Consider the delay differential equation (1.6) with By (2.2), we find which means that (2.7) is satisfied for any r. However, equation (1.6) has a nonoscillatory solution which illustrates the significance of the condition lim t→∞ τ (t) = ∞ in Theorem 4.
Based on inequality (2.9), we establish the following theorem.
then all solutions of (1.1) oscillate.
Proof. Assume, for the sake of contradiction, that there exists a nonoscillatory solution x(t) of (1.1). Then, as in the proof of Theorem 4, we obtain, for sufficiently large t, .
Taking into account that (2.9) holds, the last inequality leads to which contradicts condition (2.10). The proof of the theorem is complete.
Next, let us proceed to an oscillation condition involving lim inf.
Proof. Assume, for the sake of contradiction, that there exists a nonoscillatory solution x(t) of (1.1). Similarly to the proof of Theorem 4, we can confine our discussion only to the case of x(t) being eventually positive. Then there exists t 1 > 0 such that x(t) > 0 and x (τ i (t)) > 0 for all t ≥ t 1 . Thus, from (1.1) we have which means that x(t) is an eventually non-increasing positive function. For t ≥ t 1 , (1.1) can be rewritten as x (t) = 0, for all t ≥ t 1 .
Integrating from g(t) to t gives ln Since g(t) ≥ τ i (ζ), by Lemma 1 we have x(τ i (ζ)) ≥ a r (g(t), τ i (ζ))x(g(t)), and therefore ln In view of x(g(t)) ≥ x(ζ), the last inequality becomes ln Also, from (2.11) it follows that there exists a constant c > 0 such that for some For a fixed s, the function a r (t, s) is non-decreasing it t, g is also non-decreasing, therefore for ζ ≤ t, a r (g(t), τ i (ζ)) ≥ a r (g(ζ), τ i (ζ)). Hence (2.14) Combining inequalities (2.12) and (2.14), we obtain ln Thus ). Repeating the above argument leads to a new estimate x(g(t))/x(t) > (ec) 2 , for t large enough. Continuing by induction, we get x(g(t)) x(t) ≥ (ec) k , for sufficiently large t, where ec > 1. As ec > 1, there is k ∈ N satisfying k > 2(ln(2) − ln(c))/(1 + ln(c)) such that for t large enough Further, integrating (1.1) from g(t) to t yields Inequality (2.5) in Lemma 1 used in the above equality leads to the differential inequality The strict inequality is valid if we omit x(t) > 0 in the left-hand side: From (2.13), for large enough t, Taking the integral on [g(t), t] which is not less than c, we split the interval into two parts where integrals are not less than c/2, let t m ∈ (g(t), t) be the splitting point: Since g(ζ) ≤ g(t m ) in the first integral, and g(ζ) ≤ g(t) in the second one, we obtain (2.17) Integrating (1.1) from t m to t, along with incorporating the inequality x(τ i (ζ)) ≥ a r (g(t), τ i (ζ))x(g(t)), gives Together with the second inequality in (2.17), this implies Similarly, integration of (1.1) from g(t) to t m with a later application of Lemma 1 leads to which together with the first inequality in (2.17) yields that Inequalities (2.18) and (2.19) imply , which contradicts (2.15). Thus, all solutions of (1.1) oscillate.
As non-oscillation of (1.1) is equivalent to existence of a positive or a negative solution of the relevant differentiation inequalities (see, for example, [1, Theorem 2.1, p. 25]), Theorems 4, 5 and 6 lead to the following result.
Theorem 7. Assume that all the conditions of anyone of Theorems 4, 5 and 6 hold. Then (i) the differential inequality has no eventually positive solutions; (ii) the differential inequality has no eventually negative solutions.

Advanced
Equations. Similar oscillation theorems for the (dual) advanced differential equation (1.2) can be derived easily. The proofs of these theorems are omitted, since they are quite similar to the proofs for the delay equation (1.1). Denote and Clearly, the functions ρ(t), ρ i (t), 1 ≤ i ≤ m, are Lebesgue measurable non-decreasing and ρ(t) ≥ t, ρ i (t) ≥ t, 1 ≤ i ≤ m for all t ≥ 0.
We would like to mention that Lemma 2 can be extented to the advanced type differential equation (1.2) (cf. [8, Section 2.6.6]).

25)
and x(t) is an eventually positive solution of (1.2). Then Based on the above inequality, we establish the following theorem.
then all solutions of (1.2) oscillate.
A slight modification in the proofs of Theorems 8, 9 and 10 leads to the following result about advanced differential inequalities.
Theorem 11. Assume that all the conditions of anyone of Theorems 8, 9 and 10 hold. Then (i) the differential inequality  Figure 1: The graphs of a) τ i (t) and b) g i (t) has no eventually positive solutions; (ii) the differential inequality has no eventually negative solutions.

Examples
In this section we provide two examples illustrating Theorems 4 and 8.
Similarly, examples to illustrate the other main results of the paper can be constructed.