GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 3, 1999, 263-284

OSCILLATION AND NONOSCILLATION IN DELAY OR

ADVANCED DIFFERENTIAL EQUATIONS AND IN INTEGRODIFFERENTIAL EQUATIONS

I.-G. E. KORDONIS AND CH. G. PHILOS

Abstract. Some new oscillation and nonoscillation criteria are given for linear delay or advanced differential equations with variable coefficients and not (necessarily) constant delays or advanced arguments. Moreover, some new results on the existence and the nonexistence of positive solutions for linear integrodifferential equations are obtained.

1. Introduction and Preliminaries

With the past two decades, the oscillatory behavior of solutions of differential equations with deviating arguments has been studied by many authors. The problem of the oscillations caused by the deviating arguments (delays or advanced arguments) has been the subject of intensive investigations. Among numerous papers dealing with the study of this problem we choose to refer to the papers by Arino, Györi and Jawhari [1], Györi [2], Hunt and Yorke [3], Jaroš and Stavroulakis [4], Koplatadze and Chanturija [5], Kwong [6], Ladas [7], Ladas, Sficas and Stavroulakis [8, 9], Ladas and Stavroulakis [10], Li [11, 12], Nadareishvili [13], Philos [14, 15, 16], Philos and Sficas [17], Tramov [18], and Yan [19] and to the references cited therein; see also the monographs by Erbe, Kong and Zhang [20], Györi and Ladas [21], and Ladde, Lakshmikantham and Zhang [22] and the references therein. In particular, we mention the sharp oscillation results by Ladas [7] and Koplatadze and Chanturija [5] (see also Kwong [6]); for some very recent related results we refer to Jaroš and Stavroulakis [4], Li [11, 12], and Philos and Sficas [17] (see also the references cited therein). In the special case of an autonomous delay or advanced differential equation it is known that a necessary and sufficient condition for the oscillation of all solutions is that its characteristic equation have no real roots; such a result was proved

263 by Arino, Györi and Jawhari [1], Ladas, Sficas and Stavroulakis [8, 9], and Tramov [18] (see also Arino and Györi [23] for the general case of neutral differential systems and Philos, Purnaras and Sficas [24] and Philos and Sficas [25] for some general forms of neutral differential equations). Also, for a class of delay differential equations with periodic coefficients, a necessary and sufficient condition for the oscillation of all solutions is given by Philos [15] (in this case a characteristic equation is also considered). For the existence of positive solutions of delay differential equations we refer to the paper by Philos [26]. The reader is referred to the books by Driver [27], Hale [28], and Hale and Vertuyn Lunel [29] for the basic theory of delay differential equations.

1991 Mathematics Subject Classification. 34K15, 34C10.

Key words and phrases. Oscillation, nonoscillation, positive solution, delay differential equation, advanced differential equation, integrodifferential equation.

1072-947X/99/0500-0263$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

The literature is scarce concerning the oscillation and nonoscillation of solutions of integrodifferential equations. We mention the papers by Gopalsamy [30, 31, 32], Györi and Ladas [33], Kiventidis [34], Ladas, Philos and Sficas [35], Philos [36, 37, 38], and Philos and Sficas [39] dealing with the problem of the existence and the nonexistence of positive solutions of integrodifferential equations or of systems of such equations. Integrodifferential equations belong to the class of differential equations with unbounded delays; for a survey on equations with unbounded delays see the paper by Corduneanu and Lakshmikantham [40]. For the basic theory of integrodifferential equations (and, more generally, of integral equations) we refer to the books by Burton [41] and Corduneanu [42].

In this paper we deal with the oscillation and nonoscillation problem for first order linear delay or advanced differential equations as well as for first order linear integrodifferential equations. The discrete analogs of the results of this paper have recently been obtained by the authors [43] and the second author [44].

Consider the delay differential equation

x′(t) + ∑_{j∈J} p_{j}(t)x(t − τ_{j}(t)) = 0  (E_{1})

and the advanced differential equation

x′(t) − ∑_{j∈J} p_{j}(t)x(t + τ_{j}(t)) = 0,  (E_{2})

where J is an (nonempty) initial segment of natural numbers and for j ∈ J p_{j} and τ_{j} are nonnegative continuous real-valued functions on the interval [0,∞). For the delay equation (E_{1}) it will be supposed that the set J is necessarily finite and that the delays τ_{j} for j ∈ J satisfy

lim_{t→∞} [t − τ_{j}(t)] = ∞ for j ∈ J;

with respect to the advanced equation (E_{2}) the set J may be infinite.

Let t_{0} ≥ 0. By a solution on [t_{0}, ∞) of the delay differential equation (E_{1}) we mean a continuous real-valued function x defined on the interval [t_{−1}, ∞), where

t_{−1} = min_{j∈J} min_{t≥t_{0}} [t − τ_{j}(t)] ,

which is continuously differentiable on [t_{0}, ∞) and satisfies (E_{1}) for all t ≥ t_{0}. (Note that t_{−1} ≤ t_{0} and that t_{−1} depends on the delays τ_{j} for j ∈ J and the initial point t_{0}.) A solution on [t_{0}, ∞) of the advanced differential equation (E_{2}) is a continuously differentiable function x on the interval [t_{0}, ∞), which satisfies (E_{2}) for all t ≥ t_{0}.

As usual, a solution of (E_{1}) or (E_{2}) is said to be oscillatory if it has arbitrarily large zeros, and otherwise the solution is called nonoscillatory.

Consider also the integrodifferential equations

x′(t) + q(t) ∫_{0}^{t} K(t − s)x(s)ds = 0  (E_{3})

and

x′(t) + r(t) ∫_{−∞}^{t} K(t − s)x(s)ds = 0  (E_{4})

as well as the integrodifferential inequalities

y′(t) + q(t) ∫_{0}^{t} K(t − s)y(s)ds ≤ 0  (I_{1})

and

y′(t) + r(t) ∫_{−∞}^{t} K(t − s)y(s)ds ≤ 0,  (I_{2})

where the kernel K is a nonnegative continuous real-valued function on the interval [0, ∞), and the coefficients q and r are nonnegative continuous realvalued functions on the interval [0, ∞) and the real line ℝ, respectively.

If t_{0} ≥ 0, by a solution on [t_{0}, ∞) of the integrodifferential equation (E_{3}) (resp. of the integrodifferential inequality (I_{1})) we mean a continuous real-valued function x [resp. y] defined on the interval [0, ∞), which is continuously differentiable on [t_{0}, ∞) and satisfies (E_{3}) [resp. (I_{1})] for all t ≥ t_{0}. In particular, a solution on [0, ∞) of (I_{1}) is a continuously differentiable real-valued function y on the interval [0, ∞) satisfying (I_{1}) for every t ≥ 0. Moreover, if t_{0} ∈ ℝ, then a solution on [t_{0}, ∞) of the integrodifferential equation (E_{4}) [resp. of the integrodifferential inequality (I_{2})] is a continuous real-valued function x [resp. y] defined on the real line ℝ, which is continuously differentiable on [t_{0}, ∞) and satisfies (E_{4}) [resp. (I_{2})] for all t ≥ t_{0}. Also, a continuously differentiable real-valued function y on the real line ℝ, which satisfies (I_{2}) for every t ∈ ℝ, is called a solution on ℝ of (I_{2}).

The results of the paper will be presented in Sections 2, 3, 4 and 5. Section 2 contains some results which provide sufficient conditions for the oscillation of all solutions of the delay differential equation (E_{1}) or of the advanced differential equation (E_{2}). Conditions which guarantee the existence of a positive solution of the delay equation (E_{1}) or of the advanced equation (E_{2}) will be given in Section 3. Section 4 deals with the nonexistence of positive solutions of the integrodifferential inequalities (I_{1}) and (I_{2}) (and, in particular, of the integrodifferential equations (E_{3}) and (E_{4})). More precisely, in Section 4 necessary conditions are given for (E_{3}) or, more generally, for (I_{1}) to have solutions on [t_{0}, ∞), where t_{0} ≥ 0, which are positive on [0, ∞); analogously, necessary conditions are derived for (E_{4}) or, more generally, for (I_{2}) to have solutions on [t_{0}, ∞), where t_{0} ∈ ℝ, which are positive on ℝ. In Section 5, sufficient conditions are obtained for the equation (E_{3}) to have a solution on [t_{0}, ∞), where t_{0} > 0, which is positive on [0, ∞) and tends to zero at ∞; similarly, sufficient conditions are given for the existence of a solution on [t_{0}, ∞), where t_{0} ∈ ℝ, of the equation (E_{4}) which is positive on ℝ and tends to zero at ∞.

2. Sufficient Conditions for the Oscillation of Delay or Advanced Differential Equations

In this section, we will give conditions which guarantee the oscillation of all solutions of the delay differential equation (E_{1}) (Theorem 2.1) or of the advanced differential equation (E_{2}) (Theorem 2.2).

To state Theorem 2.1, it is needed to consider the points T_{i} (i = 0, 1, …) defined as

T_{0} = 0

and for i = 1, 2, …

T_{i} = min {s ≥ 0 : min_{j∈J} min_{t≥s} [t − τ_{j}(t)] ≥ T_{i−1}}.

(It is clear that 0 ≡ T_{0} ≤ T_{1} ≤ T_{2} ≤ ….)

Theorem 2.1. Assume that

p ≡ inf_{t≥0} ∑_{j∈J_{0}} p_{j}(t) > 0 and τ ≡ min_{j∈J_{0}} inf_{t≥0} τ_{j}(t) > 0

for a nonempty set J_{0} ⊆ J. Moreover, suppose that there exists a nonnegative integer m such that

∫_{t^{⋆}−τ}^{t^{⋆}} P_{m}(s)ds > log \frac{4}{(pτ)^{2}} for a sufficiently large t^{⋆} ≥ T_{m} + τ,

where

P_{0}(t) = ∑_{j∈J} p_{j}(t) for t ≥ 0 ≡ T_{0}

and, when m > 0, for i = 0, 1, …, m − 1

P_{i+1}(t) = ∑_{j∈J} p_{j}(t) exp [ ∫_{t−τ_{j}(t)}^{t} P_{i}(s)ds] for t ≥ T_{i+1}.

Then all solutions of the delay differential equation (E_{1}) are oscillatory.

Proof. Let x be a nonoscillatory solution on an interval [t_{0}, ∞), t_{0} ≥ 0, of the delay differential equation (E_{1}). Without restriction of generality one can assume that x(t) > 0, t ∈ [0, ∞). Furthermore, there is no loss of generallity to suppose that x is positive on the whole interval [t_{−1}, ∞), where

t_{−1} = min_{j∈J} min_{t≥t_{0}} [t − τ_{j}(t)] .

(Clearly, −∞ < t_{−1} ≤ t_{0}.) Then it follows from (E_{1}) that x′(t) ≤ 0 for all t ≥ t_{0} and so x is decreasing on the interval [t_{0}, ∞).

Now we define

S_{0} = min {s ≥ 0 : min_{j∈J} min_{t≥s} [t − τ_{j}(t)] ≥ t_{0}}

and, provided that m > 0,

S_{i} = min {s ≥ 0 : min_{j∈J} min_{t≥s} [t − τ_{j}(t)] ≥ S_{i−1}} (i = 0, 1, …, m).

It is obvious that t_{0} ≤ S_{0} ≤ S_{1} ≤ … ≤ S_{m}. Moreover, we can immediately see that T_{i} ≤ S_{i} (i = 0, 1, …, m).

We will show that

x′(t) + P_{m}(t)x(t) ≤ 0 for every t ≥ S_{m}.  (2.1)

By the decreasing nature of x on [t_{0}, ∞) it follows from (E_{1}) that for t ≥ S_{0}

0 = x′(t) + ∑_{j∈J} p_{j}(t)x(t − τ_{j}(t)) ≥ x′(t) + [ ∑_{j∈J} p_{j}(t)]x(t),

i.e.,

x′(t) + P_{0}(t)x(t) ≤ 0 for every t ≥ S_{0}.  (2.2)

Hence (2.1) is satisfied if m = 0. Let us assume that m > 0. Then by (2.2) we obtain for j ∈ J and t ≥ S_{1}

log \frac{x(t − τ_{j}(t))}{x(t)} = − ∫_{t−τ_{j}(t)}^{t} \frac{x′(s)}{x(s)}ds ≥ ∫_{t−τ_{j}(t)}^{t} P_{0}(s)ds.

So we have

x(t − τ_{j}(t)) ≥ x(t) exp [ ∫_{t−τ_{j}(t)}^{t} P_{0}(s)ds] for j ∈ J and t ≥ S_{1}.

Thus (E_{1}) gives for t ≥ S_{1}

0=x′(t)+∑_{j∈J} p_{j}(t)x(t−τ_{j}(t)) ≥ x′(t)+{ ∑_{j∈J} p_{j}(t) exp [ ∫_{t−τ_{j}(t)}^{t}P_{0}(s)ds]}x(t),

i.e.,

x′(t) + P_{1}(t)x(t) ≤ 0 for every t ≥ S_{1}.  (2.3)

This means that (2.1) is fulfilled when m = 1. Let us consider the case where m > 1. Then it follows from (2.3) that

x(t − τ_{j}(t)) ≥ x(t) exp [ ∫_{t−τ_{j}(t)}^{t} P_{1}(s)ds] for j ∈ J and t ≥ S_{2}

and so (E_{1}) yields

x′(t) + P_{2}(t)x(t) ≤ 0 for every t ≥ S_{2}.  (2.4)

Thus (2.1) holds if m = 2. If m > 2, we can use (2.4) and (E_{1}) to obtain an inequality similar to (2.4) with P_{3} in place of P_{2} and S_{3} in place of S_{2}. Following the same procedure in the case where m > 3, we can finally arrive at (2.1).

Next, it follows from (2.1) that for t ≥ S_{m} + τ

log \frac{x(t − τ)}{x(t)} = − ∫_{t−τ}^{t} \frac{x′(s)}{x(s)}ds ≥ ∫_{t−τ}^{t} P_{m}(s)ds

and so we have

x(t − τ) ≥ x(t) exp [ ∫_{t−τ}^{t} P_{m}(s)ds] for all t ≥ S_{m} + τ.  (2.5)

On the other hand, by the decreasing character of x on [t_{0}, ∞), from (E_{1}) we obtain for t ≥ S_{0}

0 = x′(t) + ∑_{j∈J} p_{j}(t)x(t − τ_{j}(t)) ≥ x′(t) + ∑_{j∈J_{0}} p_{j}(t)x(t − τ_{j}(t)) ≥

≥ x′(t) + [ ∑_{j∈J_{0}} p_{j}(t)]x(t − τ) ≥ x′(t) + px(t − τ),

i.e.,

x′(t) + px(t − τ) ≤ 0 for every t ≥ S_{0}.  (2.6)

Following the same arguments used in the proof of Lemma in [8] (see also Lemma 1.6.1 in [21]), from (2.6) it follows that

x(t − τ) ≤ \frac{4}{(pτ)^{2}} x(t) for all t ≥ S_{0} + τ/2.  (2.7)

Combining (2.5) and (2.7), we get

exp [ ∫_{t−τ}^{t} P_{m}(s)ds] ≤ \frac{4}{(pτ)^{2}} for all t ≥ S_{m} + τ

or, equivalently,

∫_{t−τ}^{t} P_{m}(s)ds ≤ log \frac{4}{(pτ)^{2}} for every t ≥ S_{m} + τ.

This is a contradiction, since t^{⋆} is sufficiently large and so it can be supposed that t^{⋆} ≥ S_{m} + τ.

Theorem 2.2. Let J_{0} be a nonempty subset of J and assume that p > 0 and τ > 0, where p and τ are defined as in Theorem 2.1. Moreover, suppose that there exists a nonnegative integer m such that

∫_{t^{⋆}}^{t^{⋆}+τ} P_{m}(s)ds > log \frac{4}{(pτ)^{2}} for a sufficiently large t^{⋆} ≥ 0,

where

P_{0}(t) = ∑_{j∈J} p_{j}(t) for t ≥ 0

and, when m > 0, for i = 0, 1, …, m − 1

P_{i+1}(t) = ∑_{j∈J} p_{j}(t) exp [ ∫_{t}^{t+τ_{j}(t)} P_{i}(s)ds] for t ≥ 0.

Then all solutions of the advanced differential equation (E_{2}) are oscillatory.

Proof. Assume, for the sake of contradiction, that the advanced differential equation (E_{2}) has a nonoscillatory solution x on an interval [t_{0}, ∞), where t_{0} ≥ 0. Without loss of generality, we can suppose that x is eventually positive. Furthermore, we may (and do) assume that x is positive on the whole interval [t_{0}, ∞). Then (E_{2}) gives x′(t) ≥ 0 for every t ≥ t_{0} and so the solution x is increasing on the interval [t_{0}, ∞).

We will prove that

x′(t) − P_{m}(t)x(t) ≥ 0 for every t ≥ t_{0}.  (2.8)

By taking into account the fact that x is increasing on [t_{0}, ∞), from (E_{2}) we obtain for t ≥ t_{0}

0 = x′(t) − ∑_{j∈J} p_{j}(t)x(t + τ_{j}(t)) ≤ x′(t) − [ ∑_{j∈J} p_{j}(t)]x(t)

and consequently

x′(t) − P_{0}(t)x(t) ≥ 0 for every t ≥ t_{0}.  (2.9)

Thus, (2.8) holds when m = 0. Let us consider the case where m > 0. Then we can use (2.9) to derive for j ∈ J and t ≥ t_{0}

log \frac{x(t + τ_{j}(t))}{x(t)} = ∫_{t}^{t+τ_{j}(t)} \frac{x′(s)}{x(s)}ds ≥ ∫_{t}^{t+τ_{j}(t)} P_{0}(s)ds.

This gives

x(t + τ_{j}(t)) ≥ x(t) exp [ ∫_{t}^{t+τ_{j}(t)} P_{0}(s)ds] for j ∈ J and t ≥ t_{0}.

Hence from (E_{2}) it follows that for t ≥ t_{0}

0=x′(t)−∑_{j∈J} p_{j}(t)x(t+τ_{j}(t))≤x′(t)−{ ∑_{j∈J} p_{j}(t) exp [ ∫_{t}^{t+τ_{j}(t)} P_{0}(s)ds]}x(t)

i.e.,

x′(t) − P_{1}(t)x(t) ≥ 0 for every t ≥ t_{0}.  (2.10)

So (2.8) is satisfied if m = 1. Let us suppose that m > 1. Then, using the same arguments as above with (2.10) in place of (2.9), we can obtain

x′(t) − P_{2}(t)x(t) ≥ 0 for every t ≥ t_{0}.

Thus (2.8) is fulfilled when m = 2. Repeating the above procedure if m > 2, we can finally arrive at (2.8).

Now from (2.8) we get for t ≥ t_{0}

log \frac{x(t + τ)}{x(t)} = ∫_{t}^{t+τ} \frac{x′(s)}{x(s)}ds ≥ ∫_{t}^{t+τ} P_{m}(s)ds

and consequently

x(t + τ) ≥ x(t) exp [ ∫_{t}^{t+τ} P_{m}(s)ds] for all t ≥ t_{0}.  (2.11)

Next, taking into account the fact that x is increasing on [t_{0}, ∞), from (E_{2}) we derive for t ≥ t_{0}

0 = x′(t) − ∑_{j∈J} p_{j}(t)x(t + τ_{j}(t)) ≤ x′(t) − ∑_{j∈J_{0}} p_{j}(t)x(t + τ_{j}(t)) ≤

≤ x′(t) − [ ∑_{j∈J_{0}} p_{j}(t)]x(t + τ) ≤ x′(t) − px(t + τ)

and so

x′(t) − px(t + τ) ≥ 0 for all t ≥ t_{0}.  (2.12)

As in the proof of Lemma 1.6.1 in [21], (2.12) gives

x(t + τ) ≤ \frac{4}{(pτ)^{2}} x(t) for every t ≥ t_{0}.  (2.13)

A combination of (2.11) and (2.13) yields

∫_{t}^{t+τ} P_{m}(s)ds ≤ log \frac{4}{(pτ)^{2}} for all t ≥ t_{0}.

The point t^{⋆} is sufficiently large and so we can assume that t^{⋆} ≥ t_{0}. We have thus arrived at a contradiction. This contradiction completes the proof of the theorem.

3. Existence of Positive Solutions of Delay or Advanced Differential Equations

Our results in this section are Theorems 3.1 and 3.2 below. Theorem 3.1 provides conditions under which the delay differential equation (E_{1}) has a positive solution; analogously, the conditions which ensure the existence of a positive solution of the advanced differential equation (E_{2}) are established by Theorem 3.2.

Let us consider the delay differential inequality

y′(t) + ∑_{j∈J} p_{j}(t)y(t − τ_{j}(t)) ≤ 0  (H_{1})

and the advanced differential inequality

y′(t) − ∑_{j∈J} p_{j}(t)y(t + τ_{j}(t)) ≥ 0,  (H_{2})

which are associated with the delay differential equation (E_{1}) and the advanced differential equation (E_{2}), respectively. For the delay inequality (H_{1}) it will be assumed that J is finite and that lim_{t→∞} [t − τ_{j}(t)] = ∞ for j ∈ J, while for the advanced inequality (H_{2}) the set J may be infinite.

Let t_{0} ≥ 0 and define t_{−1} = min_{j∈J} min_{t≥t_{0}} [t − τ_{j}(t)] . (Clearly, −∞ < t_{−1} ≤ t_{0}.) By a solution on [t_{0}, ∞) of the delay differential inequality (H_{1}) we mean a continuous real valued function y defined on the interval [t_{−1}, ∞), which is continuously differentiable on [t_{0}, ∞) and satisfies (H_{1}) for all t ≥ t_{0}. A solution on [t_{0}, ∞) of the delay inequality (H_{1}) or, in particular, of the delay equation (E_{1}) will be called positive if it is positive on the whole interval [t_{−1}, ∞).

Let again t_{0} ≥ 0. A solution on [t_{0}, ∞) of the advanced differential inequality (H_{2}) is a continuously differentiable function y on the interval [t_{0}, ∞), which satisfies (H_{2}) for all t ≥ t_{0}. A solution on [t_{0}, ∞) of the advanced inequality (H_{2}) or, in particular, of the advanced equation (E_{2}) is said to be positive if all its values for t ≥ t_{0} are positive numbers.

In order to prove Theorems 3.1 and 3.2 we need Lemmas 3.1 and 3.2 below, respectively. These lemmas guarantee that if there exists a positive solution of the delay inequality (H_{1}) or of the advanced inequality (H_{2}), then the delay equation (E_{1}) or the advanced equation (E_{2}), respectively, also has a positive solution.

Lemma 3.1 below is similar to Lemma in [16] concerning the particular case of constant delays. The method of proving Lemma 3.1 is similar to that of Lemma in [16] (see also the proof of the Lemma in [14] and the proof of Theorem 1 in [26].

Lemma 3.1. Let t_{0} ≥ 0 and let y be a positive solution on [t_{0}, ∞) of the delay differential inequality (H_{1}). Set

t_{1} = min {s ≥ 0 : min_{j∈J} min_{t≥s} [t − τ_{j}(t)] ≥ t_{0}}

and assume that t_{1} > t_{0}. (Clearly, we have min_{j∈J} min_{t≥t_{1}} [t − τ_{j}(t)] = t_{0}.) Moreover, suppose that there exists a nonempty subset J_{0} of J such that the functions τ_{j} for j ∈ J_{0}, and ∑_{j∈J_{0}} p_{j} are positive on [t_{1}, ∞).

Then there exists a positive solution x on [t_{1}, ∞) of the delay differential equation (E_{1}) with lim_{t→∞} x(t) = 0 and such that x(t) ≤ y(t) for all t ≥ t_{0}.

Proof. It follows from the inequality (H_{1}) that for t̃≥ t ≥ t_{0}

y(t) ≥ y(t̃) + ∫ _{t} ; t∑_{j∈J} p_{j}(s)y(s − τ_{j}(s))ds > ∫ _{t} ; t∑_{j∈J} p_{j}(s)y(s − τ_{j}(s))ds.

Thus, as t̃→ ∞, we obtain

y(t) ≥ ∫_{t}^{∞} ∑_{j∈J} p_{j}(s)y(s − τ_{j}(s))ds for every t ≥ t_{0}.  (3.1)

Let X be the space of all nonnegative continuous real-valued functions x on the interval [t_{0}, ∞) with x(t) ≤ y(t) for every t ≥ t_{0}. Then using (3.1) we can easily show that the formulae

(Lx)(t) = ∫_{t}^{∞} ∑_{j∈J} p_{j}(s)x(s − τ_{j}(s))ds, if t ≥ t_{1}

and

(Lx)(t) = ∫_{t_{1}}^{∞} ∑_{j∈J} p_{j}(s)x(s − τ_{j}(s))ds +

+ ∫_{t}^{t_{1}} ∑_{j∈J} p_{j}(s)y(s − τ_{j}(s))ds, if t_{0} ≤ t < t_{1}

are meaningful for any function x ∈ X and that, by these formulae, an operator L : X → X is defined. Furthermore, we see that, for any pair of functions x_{1} and x_{2} in X such that x_{1}(t) ≤ x_{2}(t) for t ≥ t_{0}, we have (Lx_{1})(t) ≤ (Lx_{2})(t) for t ≥ t_{0}. This means that the operator L is monotone. Next, we set

x_{0} = y|[t_{0}, ∞) and x_{ν} = Lx_{ν−1} (ν = 1, 2, …).

Clearly, (x_{ν})_{ν≥0} is a decreasing sequence of functions in X. (Note that the decreasing character of this sequence is considered with the usual pointwise ordering in X.) Define

x = lim_{ν→∞} x_{ν} pointwise on [t_{0}, ∞).

By the Lebesgue dominated convergence theorem, we obtain x = Lx, i.e.,

x(t) = ∫_{t}^{∞} ∑_{j∈J} p_{j}(s)x(s − τ_{j}(s))ds, if t ≥ t_{1}  (3.2)

and

x(t) = ∫_{t_{1}}^{∞} ∑_{j∈J} p_{j}(s)x(s − τ_{j}(s))ds +

+ ∫_{t}^{t_{1}} ∑_{j∈J} p_{j}(s)y(s − τ_{j}(s))ds, if t_{0} ≤ t < t_{1}.  (3.3)

Equation (3.2) gives

x′(t) = − ∑_{j∈J} p_{j}(t)x(t − τ_{j}(t)) for all t ≥ t_{1},

which means that the function x is a solution on [t_{1}, ∞) of the delay equation (E_{1}). Clearly, we have 0 ≤ x(t) ≤ y(t) for every t ≥ t_{0}. Moreover, from (3.2) it follows that x tends to zero at ∞. Hence it remains to show that x is positive on the whole interval [t_{0}, ∞). From (3.3) we obtain for any t ∈ [t_{0}, t_{1})

x(t) ≥ ∫_{t}^{t_{1}} ∑_{j∈J} p_{j}(s)y(s − τ_{j}(s))ds ≥

≥ [min_{j∈J} min_{t_{0}≤s≤t_{1}} y(s − τ_{j}(s))] ∫_{t}^{t_{1}} ∑_{j∈J} p_{j}(s)ds.

Thus, by taking into account the facts that y is positive on the interval [t_{−1}, t_{1}], where t_{−1} = min_{j∈J} min_{t≥t_{0}} [t − τ_{j}(t)] (clearly, −∞ < t_{−1} ≤ t_{0}), and that ∑_{j∈J} p_{j}(t_{1}) ≥ ∑_{j∈J_{0}} p_{j}(t_{1}) > 0, we conclude that x is positive on the interval [t_{0}, t_{1}). We claim that x is also positive on the interval [t_{1}, ∞).

Otherwise, there exists a point T ≥ t_{1} such that x(T) = 0, and x(t) > 0 for t ∈ [t_{0}, T). Then (3.2) gives

0 = x(T) = ∫_{T}^{∞} ∑_{j∈J} p_{j}(s)x(s − τ_{j}(s))ds

and so

∑_{j∈J} p_{j}(s)x(s − τ_{j}(s)) = 0 for all s ≥ T.

Taking into account the fact that x is positive on [t_{0}, T) as well as the fact that τ_{j}(T) > 0 for j ∈ J_{0} and that ∑_{j∈J_{0}} p_{j}(T) > 0, we have

0 = ∑_{j∈J} p_{j}(T)x(T − τ_{j}(T)) ≥ ∑_{j∈J_{0}} p_{j}(T)x(T − τ_{j}(T)) ≥

≥ [min_{j∈J_{0}} x(T − τ_{j}(T))] ∑_{j∈J_{0}} p_{j}(T) > 0.

But, this is a contradiction and so our claim is proved.

Theorem 3.1. Set

t_{0} = min {s ≥ 0 : min_{j∈J} min_{t≥s} [t − τ_{j}(t)] ≥ 0}.

(Clearly, t_{−1} ≡ min_{j∈J} min_{t≥t_{0}} [t − τ_{j}(t)] = 0). Suppose that there exist positive real numbers γ_{j} for j ∈ J such that

exp [ ∑_{i∈J} γ_{i} ∫_{t−τ_{j}(t)}^{t} p_{i}(s)ds] ≤ γ_{j} for all t ≥ t_{0} and j ∈ J.

Also, define

t_{1} = min {s ≥ 0 : min_{j∈J} min_{t≥s} [t − τ_{j}(t)] ≥ t_{0}}

and assume that t_{1} > t_{0}. (Obviously, min_{j∈J} min_{t≥t_{1}} [t − τ_{j}(t)] = t_{0}.) Moreover, suppose that there exists a nonempty subset J_{0} of J such that the functions τ_{j} for j ∈ J_{0}, and ∑_{j∈J_{0}} p_{j} are positive on [t_{1}, ∞).

Then there exists a positive solution on [t_{1}, ∞) of the delay differential equation (E_{1}), which tends to zero at ∞.

Proof. Define

y(t) = exp [ − ∑_{i∈J} γ_{i} ∫_{0}^{t} p_{i}(s)ds] for t ≥ 0

and observe that y is positive on the interval [0, ∞). By Lemma 3.1 it suffices to show that y is a solution on [t_{0}, ∞) of the delay differential inequality (H_{1}). To this end we have for every t ≥ t_{0}

y′(t) + ∑_{j∈J} p_{j}(t)y(t − τ_{j}(t)) =

= − [ ∑_{i∈J} γ_{i}p_{i}(t)]y(t) + { ∑_{j∈J} p_{j}(t) exp [ ∑_{i∈J} γ_{i} ∫_{t−τ_{j}(t)}^{t} p_{i}(s)ds]}y(t) =

=( ∑_{j∈J} p_{j}(t){ − γ_{j} + exp [ ∑_{i∈J} γ_{i} ∫_{t−τ_{j}(t)}^{t} p_{i}(s)ds]})y(t) ≤ 0.

Lemma 3.2. Let t_{0} ≥ 0 and let y be a positive solution on [t_{0}, ∞) of the advanced differential inequality (H_{2}).

Then there exists a positive solution x on [t_{0}, ∞) of the advanced differential equation (E_{2}) such that x(t) ≤ y(t) for all t ≥ t_{0}.

Proof. It follows from (H_{2}) that

y(t) ≥ y(t_{0}) + ∫_{t_{0}}^{t} ∑_{j∈J} p_{j}(s)y(s + τ_{j}(s))ds for all t ≥ t_{0}.  (3.4)

Consider the set X of all continuous real-valued functions x on the interval [t_{0}, ∞) such that 0 < x(t) ≤ y(t) for every t ≥ t_{0}. Then by (3.4) we can see that the formula

(Lx)(t) = y(t_{0}) + ∫_{t_{0}}^{t} ∑_{j∈J} p_{j}(s)x(s + τ_{j}(s))ds for t ≥ t_{0}

is meaningful for any function x in X and that this formula defines an operator L of X into itself. This operator is monotone in the sense that, if x_{1} and x_{2} are two functions in X with x_{1}(t) ≤ x_{2}(t) for t ≥ t_{0}, then we also have (Lx_{1})(t) ≤ (Lx_{2})(t) for t ≥ t_{0}. Next, we define x_{0} = y and x_{ν} = Lx_{ν−1} (ν = 1, 2, …). Clearly, x_{0}(t) ≥ x_{1}(t) ≥ x_{2}(t) ≥ ⋯ holds for every t ≥ t_{0} and so we can define x(t) = lim_{ν→∞} x_{ν}(t) for t ≥ t_{0}. Then applying the Lebesgue dominated convergence theorem, we have x = Lx, i.e.,

x(t) = y(t_{0}) + ∫_{t_{0}}^{t} ∑_{j∈J} p_{j}(s)x(s + τ_{j}(s))ds for every t ≥ t_{0}.

This ensures that x is a solution on [t_{0}, ∞) of the advanced equation (E_{2}), which is positive (on [t_{0}, ∞)) and such that x(t) ≤ y(t) for t ≥ t_{0}.

Theorem 3.2. Suppose that there exist positive real numbers δ_{j} for j ∈ J such that

exp [ ∑_{i∈J} δ_{i} ∫_{t}^{t+τ_{j}(t)} p_{i}(s)ds] ≤ δ_{j} for all t ≥ 0 and j ∈ J

and, when J is infinite,

∑_{i∈J} δ_{i} ∫_{0}^{t} p_{i}(s)ds < ∞ for every t ≥ 0.

Then there exists a positive solution on [0, ∞) of the advanced differential equation (E_{2}).

Proof. The function y defined by

y(t) = exp [ ∑_{i∈J} δ_{i} ∫_{0}^{t} p_{i}(s)ds] for t ≥ 0

is clearly positive on the interval [0, ∞). Moreover, for every t ≥ 0 we obtain

y′(t) − ∑_{j∈J} p_{j}(t)y(t + τ_{j}(t)) =

=[ ∑_{i∈J} δ_{i}p_{i}(t)]y(t) − { ∑_{j∈J} p_{j}(t) exp [ ∑_{i∈J} δ_{i} ∫_{t}^{t+τ_{j}(t)} p_{i}(s)ds]}y(t) =

=( ∑_{j∈J} p_{j}(t){δ_{j} − exp [ ∑_{i∈J} δ_{i} ∫_{t}^{t+τ_{j}(t)} p_{i}(s)ds]})y(t) ≥ 0

and hence y is a solution on [0, ∞) of the advanced inequality (H_{2}). So, the proof can be completed by applying Lemma 3.2.

4. Necessary Conditions for the Existence of Positive Solutions of Integrodifferential Equations and Inequalities

In this section the problem of the nonexistence of positive solutions of the integrodifferential equations (E_{3}) and (E_{4}) (or, more generally, of the integrodifferential inequalities (I_{1}) and (I_{2})) will be treated. The main results here are Theorems 4.1 and 4.2 below.

Theorem 4.1. Let t_{0} ≥ 0. Assume that

A ≡ inf_{t≥t_{0}+τ_{1}} [q(t) ∫_{τ_{0}}^{t−t_{0}} K(s)ds] > 0

for two points τ_{0} and τ_{1} with 0 < τ_{0} < τ_{1}. Moreover, suppose that there exists a nonnegative integer m such that

∫_{t^{⋆}−τ_{0}}^{t^{⋆}} U_{m}(s)ds > log \frac{4}{(Aτ_{0})^{2}} for some t^{⋆} ≥ t_{0} + τ_{1} + τ_{0}/2,

where

U_{0}(t) = q(t) ∫_{0}^{t−t_{0}} K(s)ds for t ≥ t_{0}

and, when m > 0, for i = 0, 1, …, m − 1

U_{i+1}(t) = q(t) ∫_{0}^{t−t_{0}} K(s) exp [ ∫_{t−s}^{t} U_{i}(ξ)dξ]ds for t ≥ t_{0}.

Then there is no solution on [t_{0}, ∞) of the integrodifferential inequality (I_{1}) (and, in particular, of the integrodifferential equation (E_{3})), which is positive on [0, ∞).

Proof. Assume, for the sake of contradiction, that the integrodifferential inequality (I_{1}) admits a solution y on [t_{0}, ∞), which is positive on [0, ∞). Then (I_{1}) guarantees that y′(t) ≤ 0 for every t ≥ t_{0} and so the solution y is decreasing on the interval [t_{0}, ∞).

We first prove that

y′(t) + U_{m}(t)y(t) ≤ 0 for all t ≥ t_{0}.  (4.1)

To this end, using the decreasing character of y on [t_{0}, ∞), from (I_{1}) we obtain for any t ≥ t_{0}

0 ≥ y′(t) + q(t) ∫_{0}^{t} K(t − s)y(s)ds = y′(t) + q(t) ∫_{0}^{t} K(s)y(t − s)ds ≥

≥ y′(t) + q(t) ∫_{0}^{t−t_{0}} K(s)y(t − s)ds ≥ y′(t) + q(t)[ ∫_{0}^{t−t_{0}} K(s)ds]y(t)

and so we have

y′(t) + U_{0}(t)y(t) ≤ 0 for all t ≥ t_{0}.  (4.2)

Thus (4.1) is satisfied when m = 0. Let us assume that m > 0. Then it follows from (4.2) that for t ≥ t_{0} and 0 ≤ s ≤ t − t_{0}

log \frac{y(t − s)}{y(t)} = − ∫_{t−s}^{t} \frac{y′(ξ)}{y(ξ)}dξ ≥ ∫_{t−s}^{t} U_{0}(ξ)dξ

and consequently

y(t − s) ≥ y(t) exp [ ∫_{t−s}^{t} U_{0}(ξ)dξ] for t ≥ t_{0} and 0 ≤ s ≤ t − t_{0}. (4.3)

Furthermore, in view of (4.3), inequality (I_{1}) yields for t ≥ t_{0}

0 ≥ y′(t) + q(t) ∫_{0}^{t} K(t − s)y(s)ds = y′(t) + q(t) ∫_{0}^{t} K(s)y(t − s)ds ≥
≥ y′(t) + q(t) ∫_{0}^{t−t_{0}} K(s)y(t − s)ds ≥
≥ y′(t) + q(t){ ∫_{0}^{t−t_{0}} K(s) exp [ ∫_{t−s}^{t} U_{0}(ξ)dξ]ds}y(t).

Therefore

y′(t) + U_{1}(t)y(t) ≤ 0 for all t ≥ t_{0}.  (4.4)

Hence (4.1) is proved when m = 1. In the case where m > 1, we can repeat the above procedure with (4.4) in place of (4.2) to conclude that (4.1) is finally satisfied.

Now from (4.1) we obtain for t ≥ t_{0} + τ_{0}

log \frac{y(t − τ_{0})}{y(t)} = − ∫_{t−τ_{0}}^{t} \frac{y′(s)}{y(s)}ds ≥ ∫_{t−τ_{0}}^{t} U_{m}(s)ds

and hence

y(t − τ_{0}) ≥ y(t) exp [ ∫_{t−τ_{0}}^{t} U_{m}(s)ds] for every t ≥ t_{0} + τ_{0}.  (4.5)

Next, taking into account the fact that y is decreasing on [t_{0}, ∞), from (I_{1}) we derive for t ≥ t_{0} + τ_{1}

0 ≥ y′(t) + q(t) ∫_{0}^{t} K(s)y(t − s)ds ≥ y′(t) + q(t) ∫_{τ_{0}}^{t−t_{0}} K(s)y(t − s)ds ≥

≥ y′(t) + [q(t) ∫_{τ_{0}}^{t−t_{0}} K(s)ds]y(t − τ_{0}) ≥ y′(t) + Ay(t − τ_{0}),

i.e.,

y′(t) + Ay(t − τ_{0}) ≤ 0 for all t ≥ t_{0} + τ_{1}.  (4.6)

As in the proof of the Lemma in [8] (see also Lemma 1.6.1 in [21]), it follows from (4.6) that

y(t − τ_{0}) ≤ \frac{4}{(Aτ_{0})^{2}} y(t) for every t ≥ t_{0} + τ_{1} + τ_{0}/2.  (4.7)

A combination of (4.5) and (4.7) leads to

∫_{t−τ_{0}}^{t} U_{m}(s)ds ≤ log \frac{4}{(Aτ_{0})^{2}} for all t ≥ t_{0} + τ_{1} + τ_{0}/2,

which is a contradiction.

Theorem 4.2. Let t̂_{0} ∈ ℝ and set t_{0} = max{0, t̂_{0}}. Moreover, let the assumptions of Theorem 4.1 be satisfied with r in place of q.

Then there is no solution on [t̂_{0}, ∞) of the integrodifferential inequality (I_{2}) (and, in particular, of the integrodifferential equation (E_{4})), which is positive on ℝ.

Proof. Obviously, t_{0} ≥ 0. Assume that there exists a solution y on [t̂_{0}, ∞) of the integrodifferential inequality (I_{2}), which is positive on ℝ. Then, for every t ≥ t_{0}, we have

0 ≥ y′(t) + r(t) ∫_{−∞}^{t} K(t − s)y(s)ds = y′(t) + r(t) ∫_{−∞}^{0} K(t − s)y(s)ds +

+ r(t) ∫_{0}^{t} K(t − s)y(s)ds ≥ y′(t) + r(t) ∫_{0}^{t} K(t − s)y(s)ds.

This means that the function y|[0, ∞) is a solution on [t_{0}, ∞) of the integrodifferential inequality

y′(t) + r(t) ∫_{0}^{t} K(t − s)y(s)ds ≤ 0,

which is positive on [0, ∞). By Theorem 4.1, this is a contradiction and hence our proof is complete.

5. Sufficient Conditions for the Existence of Positive Solutions of Integrodifferential Equations

Theorems 5.1 and 5.2 below are the main results in this last section. Theorem 5.1 establishes conditions which guarantee the existence of positive solutions of the integrodifferential equation (E_{3}); similarly, Theorem 5.2 provides sufficient conditions for the existence of positive solutions of the integrodifferential equation (E_{4}).

To prove Theorems 5.1 and 5.2 we will apply Theorems A and B, respectively, which are known.

Theorem A (Philos [38]). Let y be a positive solution on [0, ∞) of the integrodifferential inequality (I_{1}). Moreover, let t_{0} > 0 and suppose that K is not identically zero on [0, t_{0}] and q is positive on [t_{0}, ∞).

Then there exists a solution x on [t_{0}, ∞) of the integrodifferential equation (E_{3}), which is positive on [0, ∞) and such that

x(t) ≤ y(t) for every t ≥ t_{0}, lim_{t→∞} x(t) = 0

and

x′(t) + q(t) ∫_{0}^{t} K(t − s)x(s)ds ≤ 0 for 0 ≤ t < t_{0}.

Theorem B (Philos [38]). Assume that K is not identically zero on [0, ∞). Let y be a positive solution on ℝ of the integrodifferential inequality (I_{2}). Moreover, let t_{0} ∈ ℝ and suppose that r is positive on [t_{0}, ∞).

Then there exists a solution x on [t_{0}, ∞) of the integrodifferential equation (E_{4}), which is positive on ℝ and such that

x(t) ≤ y(t) for every t ∈ ℝ, lim_{t→∞} x(t) = 0

and

x′(t) + r(t) ∫_{−∞}^{t} K(t − s)x(s)ds ≤ 0 for t < t_{0}.

We will now state and prove Theorems 5.1 and 5.2.

Theorem 5.1. Let λ be a positive continuous real-valued function on the interval [0, ∞) such that

exp { ∫_{t−s}^{t} q(ξ)[ ∫_{0}^{ξ} λ(σ)K(σ)dσ]dξ} ≤ λ(s) for all t ≥ 0 and 0 ≤ s ≤ t.

Moreover, let t_{0} > 0 and suppose that K is not identically zero on [0, t_{0}] and q is positive on [t_{0}, ∞).

Then there exists a solution on [t_{0}, ∞) of the integrodifferential equation (E_{3}), which is positive on [0, ∞) and tends to zero at ∞.

Proof. Define

y(t) = exp { − ∫_{0}^{t} q(ξ)[ ∫_{0}^{ξ} λ(σ)K(σ)dσ]dξ} for t ≥ 0.

Clearly, y is positive on the interval [0, ∞). By Theorem A it is enough to verify that y is a solution on [0, ∞) of the integrodifferential inequality (I_{1}). For this purpose we have, for every t ≥ 0,

y′(t) + q(t) ∫_{0}^{t} K(t − s)y(s)ds = y′(t) + q(t) ∫_{0}^{t} K(s)y(t − s)ds =

= − q(t)[ ∫_{0}^{t} λ(σ)K(σ)dσ]y(t) +

+ q(t)[ ∫_{0}^{t} K(s) exp { ∫_{t−s}^{t} q(ξ)[ ∫_{0}^{ξ} λ(σ)K(σ)dσ]dξ}ds]y(t) =

=q(t)[ − ∫_{0}^{t} λ(s)K(s)ds +
+ ∫_{0}^{t} K(s) exp { ∫_{t−s}^{t} q(ξ)[ ∫_{0}^{ξ} λ(σ)K(σ)dσ]dξ}ds]y(t) =

=q(t)( ∫_{0}^{t}K(s)[ − λ(s)+exp {∫_{t−s}^{t} q(ξ)[ ∫_{0}^{ξ}λ(σ)K(σ)dσ]dξ}]ds)y(t)≤
≤ 0.

Theorem 5.2. Assume that K is not identically zero on [0, ∞). Assume also that

∫_{−∞}^{0} r(ξ)dξ < ∞

and let μ be a positive continuous real-valued function on the interval [0, ∞) such that

∫_{0}^{∞} μ(σ)K(σ)dσ < ∞

and

exp {[ ∫_{0}^{∞} μ(σ)K(σ)dσ] ∫_{t−s}^{t} r(ξ)dξ} ≤ μ(s) for all t ∈ ℝ and s ≥ 0.

Moreover, let t_{0} ∈ ℝ and suppose that r is positive on [t_{0}, ∞).

Then there exists a solution on [t_{0}, ∞) of the integrodifferential equation (E_{4}), which is positive on ℝ and tends to zero at ∞.

Proof. Set

y(t) = exp { − [ ∫_{0}^{∞} μ(σ)K(σ)dσ] ∫_{−∞}^{t} r(ξ)dξ} for t ∈ ℝ.

We observe that y is positive on the real line ℝ. So by Theorem B it suffices to show that y is a solution on ℝ of the integrodifferential inequality (I_{2}). To this end we obtain, for every t ∈ ℝ,

y′(t) + r(t) ∫_{−∞}^{t} K(t − s)y(s)ds = y′(t) + r(t) ∫_{0}^{∞} K(s)y(t − s)ds =

= − r(t)[ ∫_{0}^{∞} μ(σ)K(σ)dσ]y(t) +

+ r(t)[ ∫_{0}^{∞} K(s) exp {[ ∫_{0}^{∞} μ(σ)K(σ)dσ] ∫_{t−s}^{t} r(ξ)dξ}ds]y(t) =

=r(t)[ − ∫_{0}^{∞} μ(s)K(s)ds +
+ ∫_{0}^{∞} K(s) exp {[ ∫_{0}^{∞} μ(σ)K(σ)dσ] ∫_{t−s}^{t} r(ξ)dξ}ds]y(t) =
=r(t)( ∫_{0}^{∞}K(s)[ − μ(s)+exp {[ ∫_{0}^{∞}μ(σ)K(σ)dσ] ∫_{t−s}^{t}r(ξ)dξ}]ds)y(t)≤
≤ 0.

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(Received 29.05.1997)

Authors’ address: Department of Mathematics University of Ioannina P.O. Box 1186 451 10 Ioannina Greece