Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 65, 1-11; http://www.math.u-szeged.hu/ejqtde/

Lower Bounds and Positivity Conditions for Green’s Functions to Second Order Differential-Delay Equations

M. I. Gil’ Department of Mathematics Ben Gurion University of the Negev P.0. Box 653, Beer-Sheva 84105, Israel E-mail: gilmi@cs.bgu.ac.il

Abstract We consider the Cauchy problem on the positive half-line for the differential-delay

equation

ü(t) + 2c_{0}(t)u̇(t) + c_{1}(t)u̇(t − h) + d_{0}(t)u(t) + d_{1}(t)u(t − h) + d_{2}(t)u(t − 2h) = 0

where c_{k}(t), d_{j}(t) (t ≥ 0; k = 0, 1; j = 0, 1, 2) are continuous functions. Conditions providing the positivity of the Green function and a lower bound for that function are derived. Our results are new even in the case of ordinary differential equations. Applications of the obtained results to equations with nonlinear causal mappings are also discussed. Equations with causal mappings include ordinary differential and integro-differential equations. In addition, we establish positivity conditions for solutions of functional differential equations with variable and distributed delays.

Key words: differential-delay equation, Green’s function, positivity, causal mapping AMS (MOS) subject classification: 34K11, 34K06, 34K99

1 Introduction and statement of the main result

The theory of nonoscillations of solutions to functional differential equations was extensively developed. Mainly the first order equations were considered, see the well-known publications [1, 2, 12, 21, 23, 28], and references therein. Higher order equations were investigated in the very interesting papers [3, 4, 5, 22, 25].

It should be noted that the existence of nonoscillating solutions does not provide the positivity of the Green functions. Obtaining the positivity conditions for the Green functions requires additional efforts, but the positivity is very important for various applications. For instance, the positivity of Green’s function provides the existence of non-negative solutions to equations with various nonlinearities, in particular, with nonlinear causal mappings, cf. [17, 18]. Moreover, positive Green’s functions play an essential role in the stability theory, see [15, 16].

In this paper we consider the Cauchy problem for a nonautonomous second order differentialdelay equation and derive conditions that provide the positivity of the Green function. In addition, a lower bound for that function is established. Applications to equations with nonlinear causal mappings are also discussed. It should be noted that equations with causal mappings include such traditional equations as ordinary differential and integro-differential equations, cf. [7, 27]. Moreover, we establish positivity conditions for solutions of functional differential equations with variable and distributed delays.

Denote R_{+} = [0, ∞), C(ω) is the space of continuous scalar-valued functions defined on a real segment ω with the sup-norm. For t > 0, let us consider the equation

(1.1)  ü(t) + 2c_{0}(t)u̇(t) + c_{1}(t)u̇(t − h) + d_{0}(t)u(t) + d_{1}(t)u(t − h) + d_{2}(t)u(t − 2h) = 0

where u̇(t) = du(t)/dt, ü(t) = d^{2}u(t)/dt^{2}, c_{k}(.), d_{j}(.) ∈ C(R_{+}) (k = 0, 1; j = 0, 1, 2) and ċ_{0}(.) ∈ C(R_{+}).

The Green function G(t, s) to equation (1.1) is a function defined for t ≥ s − 2h (s ≥ 0), having the continuous first and second derivatives in t for t > s, satisfying that equation for all t > s ≥ 0 and the conditions

(1.2)  G(t, s) = 0 (s − 2h ≤ t ≤ s), \frac{∂G(t, s)}{∂t} = 0 (s − 2h ≤ t < s), lim_{t↓s} \frac{∂G(t, s)}{∂t} = 1.

Furthermore, put c_{0}(t) ≡ c_{0}(0) for −2h ≤ t ≤ 0 and

(1.3)  a_{1}(t) = c_{1}(t)e^{∫_{t−h}^{t} c_{0}(s)ds}, a_{2}(t) = d_{2}(t)e^{∫_{t−2h}^{t} c_{0}(s)ds} (t ≥ 0).

We need the following preliminary result.

Lemma 1.1 Let the conditions

(1.4)  −ċ_{0}(t) + c_{0}^{2}(t) + d_{0}(t) ≤ 0 and − c_{1}(t)c_{0}(t − h) + d_{1}(t) ≤ 0 (t ≥ 0)

hold and the Green function G_{0}(t, s) to the equation

(1.5)  ü(t) + a_{1}(t)u̇(t − h) + a_{2}(t)u(t − 2h) = 0 (t > 0)

be nonnegative. Then the Green function G(t, s) to equation (1.1) is also nonnegative. Moreover,

(1.6)  G(t, s) ≥ G_{0}(t, s) (t > s ≥ 0).

Proof: Substitute the equality

u(t) = w(t)e^{− ∫_{0}^{t} c_{0}(s)ds}

into (1.1). Then, taking into account that

\frac{d}{dt} ∫_{0}^{t−h} c_{0}(s)ds = \frac{d}{dt} ∫_{h}^{t} c_{0}(s_{1} − h)ds_{1} = c_{0}(t − h),

we have

e^{− ∫_{0}^{t} c_{0}(s)ds}[ẅ(t) − 2c_{0}(t)ẇ(t) + w(t)(−ċ_{0}(t) + c_{0}^{2}(t) + d_{0}(t))+
2(c_{0}(t)ẇ(t) − c_{0}^{2}(t)w(t))] + c_{1}(t)e^{− ∫_{0}^{t−h} c_{0}(s)ds}[−c_{0}(t − h)w(t − h) + ẇ(t − h)]+

d_{1}(t)e^{− ∫_{0}^{t−h} c_{0}(s)ds}w(t − h) + d_{2}(t)e^{− ∫_{0}^{t−2h} c_{0}(s)ds}w(t − 2h) = 0.

Or

(1.7)  ẅ(t) + a_{1}(t)ẇ(t − h) + m_{0}(t)w(t) + m_{1}(t)w(t − h) + a_{2}(t)w(t − 2h) = 0,

where

m_{0}(t) := −ċ_{0}(t) + c_{0}^{2}(t) + d_{0}(t); m_{1}(t) := e^{∫_{t−h}^{t} c_{0}(s)ds}[−c_{1}(t)c_{0}(t − h) + d_{1}(t)].

According to (1.4), m_{0}(t) ≤ 0, m_{1}(t) ≤ 0. Hence, by the comparison principle, cf. [8, Chapter 2, Section 2], it easily follows that if the Green function to equation (1.5) is nonnegative, then the Green function to equation (1.7) is also nonnegative and (1.6) holds. But (1.7) and (1.1) are equivalent. This proves the lemma. □

2 The main result

In the sequel it is assumed that c_{1}(t), c_{2}(t) are positive for all t ≥ 0. Therefore the functions a_{1}(t), a_{2}(t) defined by (1.3) are positive. Put

(2.1)  b_{1} := sup_{t≥0} a_{1}(t) and b_{2} := sup_{t≥0} a_{2}(t) + \frac{b_{1}}{2} (b_{1} − inf_{t≥0} a_{1}(t)),

and consider the equation

(2.2)  ü(t) + b_{1}u̇(t − h) + b_{2}u(t − 2h) = 0.

Denote by H(t) the Green function to equation (2.2). By the Laplace transform we easily have

H(t) = \frac{1}{2πi} ∫_{ν−i∞}^{ν+i∞} \frac{e^{λt}dλ}{λ^{2} + b_{1}λe^{−λh} + b_{2}e^{−2λh}}

where ν is a real constant. It is assumed that

(2.3)  inf_{t≥0} a_{1}(t) > \frac{b_{1}}{2} + \frac{2}{b_{1}} sup_{t≥0} a_{2}(t).

Then

b_{2} < sup_{t≥0} a_{2}(t) + \frac{b_{1}}{2} (b_{1} − \frac{b_{1}}{2} − \frac{2}{b_{1}} sup_{t≥0} a_{2}(t)) = \frac{b_{1}^{2}}{4} .

That is,

b_{1}^{2} > 4b_{2}.

So the polynomial z^{2} + b_{1}z + b_{2} has the real roots

r_{1} := −b_{1}/2 − \sqrt{b_{1}^{2}/4 − b_{2}}, r_{2} := −b_{1}/2 + \sqrt{b_{1}^{2}/4 − b_{2}}.

Now we are in a position to formulate our main result.

Theorem 2.1 Let the conditions (1.4), (2.3) and

(2.4)  −ehr_{k} < 1 (k = 1, 2)

hold. Then the Green function G(t, s) to equation (1.1) is nonnegative and

(2.5)  G(t, s) ≥ H(t − s) (t > s ≥ 0).

This theorem is proved below. To the best of our knowledge, Theorem 2.1 is new even in the case of ordinary differential equations, cf. [9, 10, 24].

To prove Theorem 2.1 we need the following result.

Lemma 2.2 Let conditions (2.3) and (2.4) hold. Then the Green function G_{0}(t, s) to equation (1.5) is nonnegative and

(2.6)  G_{0}(t, s) ≥ H(t − s) (t > s ≥ 0).

Proof: With a real number b, let us consider the equation

(2.7)  u̇(t) − bu(t − h) = 0 (h = const > 0; u̇ = du/dt, t ≥ 0).

The Green function H_{1}(b, t) to this equation is a solution of (2.7) with the initial conditions

(2.8)  u(0) = 1; u(t) = 0 (t < 0).

We need the following result: let the condition

(2.9)  −ehb < 1

hold. Then the Green function to equation (1.1) is positive. For the proof see for instance [16]. Clearly, for any b ≥ 0, condition (2.9) holds.

According to this result, under condition (2.4) with b = r_{k} we have H_{1}(r_{k}, t) ≥ 0 and thanks to the convolution property,

H(t) = \frac{1}{2πi} ∫_{ν−i∞}^{ν+i∞} \frac{e^{λt}dλ}{(λ − r_{1}e^{−λh})(λ − r_{2}e^{−λh})} = ∫_{0}^{t} H_{1}(r_{1}, t − s)H_{1}(r_{2}, s)ds ≥ 0.

Hence,

(2.10)  Ḣ(t) = \frac{1}{2πi} ∫_{ν−i∞}^{ν+i∞} \frac{λe^{λt}dλ}{(λ − r_{1}e^{−λh})(λ − r_{2}e^{−λh})}.

But

\frac{2λ}{(λ − e^{−λ}r_{1})(λ − r_{2}e^{−λ})} =

\frac{1}{λ − r_{1}e^{−λ}} + \frac{1}{λ − r_{2}e^{−λ}} + \frac{(r_{1} + r_{2})e^{−λ}}{(λ − r_{k}e^{−λ})(λ − r_{k}e^{−λ})}

Since r_{1} + r_{2} = −b_{1}, and H(t) ≥ 0, we have

Ḣ(t) = \frac{1}{2}[H_{1}(r_{1}, t) + H_{1}(r_{2}, t) − b_{1}H(t − h)] ≥ −\frac{b_{1}}{2} H(t − h).

Consequently

(2.11)  Ḣ(t − h) ≥ −\frac{b_{1}}{2} H(t − 2h).

Now consider the equation

(2.12) Eu(t) := ü(t) + a_{1}(t)u̇(t − h) + a_{2}(t)u(t − 2h) = f(t) (t > 0; f ∈ C(R_{+}), f(t) ≥ 0)

with the zero initial condition

(2.13)  u(t) = 0 (−2h ≤ t ≤ 0).

The derivatives at zero are understood as the right derivatives. Simultaneously, with the same zero initial condition, consider the equation

(2.14)  E_{0}v(t) := v̈(t) + b_{1}v̇(t − h) + b_{2}v(t − 2h) = f(t) (t > 0).

Then

v(t) := E_{0}^{−1}f(t) = ∫_{0}^{t} H(t − s)f(s)ds.

Put W = E_{0} − E. With the notation Δ_{k}(t) = b_{k} − a_{k}(t) (k = 1, 2), according to (2.11) and (2.1) we have

W H(t − s) = Δ_{1}(t)Ḣ(t − s − h) + Δ_{2}(t)H(t − t − 2h) ≥
(2.15)  H(t − s − 2h)(−b_{1}Δ_{1}(t)/2 + Δ_{2}(t)) ≥ 0.

Rewrite (2.12) as E_{0}u − W u = f. With u = E_{0}^{−1}y we obtain

(2.16)  y − W E_{0}^{−1}y = f.

But

W E_{0}^{−1}f(t) = (b_{1} − a_{1}(t))\frac{d}{dt} ∫_{0}^{t} H(t − s − h)f(s)ds + (b_{2} − a_{2}(t)) ∫_{0}^{t} H(t − s − 2h)f(s)ds =

∫_{0}^{t}[Δ_{1}(t)Ḣ(t − s − h) + Δ_{2}(t)H(t − s − 2h)]f(s)ds,

since H(t) = 0, t ≤ 0. So thanks to (2.15), W E_{0}^{−1} is a positive Volterra operator. Consequently, for each positive T < ∞, the spectral radius of W E_{0}^{−1} in space C(0, T) is equal to zero. Hence,

y = ∑_{k=0}^{∞}(W E_{0}^{−1})^{k}f,

and therefore,

0 ≤ f ≤ y = (I − W E_{0}^{−1})^{−1}f (f ≥ 0).

Consequently

0 ≤ E_{0}^{−1}f ≤ x = E_{0}^{−1}y ≤ E_{0}^{−1}(I − W E_{0}^{−1})^{−1}f (f ≥ 0).

This proves the lemma. □

The assertion of Theorem 2.1 follows from Lemmas 1.1 and 2.2.

3 Equations with nonlinear causal mappings

For a positive T < ∞, let E be a Banach space of functions defined on [0, T] with the unit operator I. For all τ ∈ [0, T) and w ∈ E, let the projections P_{τ} be defined by

(P_{τ}w)(t) = \{ \begin{matrix}w(t) & if 0 ≤ t ≤ τ, \\ 0 & if τ < t ≤ T \end{matrix} .

In addition, P_{T} = I. A mapping F : E → E satisfying the condition

P_{τ}F P_{τ} = P_{τ}F (τ ∈ [0, T])

will be called a causal mapping (operator). This definition is somewhat different from the definition of the causal operator presented in [7]. In the linear case our definition coincides with the one accepted in [13].

Denote by B(0, T) the space of real bounded measurable functions defined on [0, T] with the sup-norm |.|_{B(0,T)}. Let us point an example of a causal mapping. To this end consider in B(0, T) the mapping

(F w)(t) = f(t, w(t)) + ∫_{0}^{t} k(t, s, w(s))ds (w ∈ B(0, T))

with a continuous kernel k, defined on [0, T]^{2} × ℝ and a continuous function f : [0, T] × ℝ → ℝ. For each τ ∈ [0, T), we have

(P_{τ}F w)(t) = f_{τ}(t, w(t)) + ∫_{0}^{τ} k_{τ}(t, s, w(s))ds

where

k_{τ}(t, s, w(s)) = \{ \begin{matrix}k(t, s, w(s)) & if 0 ≤ t ≤ τ, \\ 0 & if τ < t ≤ T \end{matrix}

(0 ≤ s ≤ t), and

f_{τ}(t, w(t)) = \{ \begin{matrix}f(t, w(t)) & if 0 ≤ t ≤ τ, \\ 0 & if τ < t ≤ T \end{matrix} .

Thus the considered mapping is causal. Note that, the integral operator

∫_{0}^{c} k(t, s, w(s))ds

with a fixed positive c ≤ T is not causal.

Denote by K_{+} the cone of nonnegative functions from B(0, T). For a positive number R ≤ ∞, put

K_{R} := {w ∈ K_{+} : 0 ≤ w(t) ≤ R, 0 ≤ t ≤ T }.

Everywhere below F is a continuous causal operator mapping K_{R} into K_{+}. Consider the equation

(3.1)  x(t) = f(t) + ∫_{0}^{t} Q(t, t_{1})(F x)(t_{1})dt_{1} (0 < t ≤ T < ∞),

where Q : [0 ≤ s ≤ t ≤ T] → [0, ∞) is a nonnegative measurable kernel and f ∈ K_{+} is given. A solution of (3.1) is a function x ∈ B(0, T) which satisfies (3.1) for all t ∈ [0, T].

It is assumed that there is a linear positive causal operators A in B(0, T), such that

(3.2)  F v ≤ Av (v ∈ K_{R}).

That is, the operator V defined on B(0, T) by

v → (V v)(t) := ∫_{0}^{t} Q(t, t_{1})(Av)(t_{1})dt_{1}

is a majorant for the mapping

∫_{0}^{t} Q(t, t_{1})(F v)(t_{1})dt_{1} (0 < t ≤ T).

Assume that V is compact and

(3.3)  b(s) := sup_{s≤t≤T} Q(t, s) ∈ L^{1}(0, T)

where L^{1}(0, T) is the space of scalar integrable functions defined on [0, T]. We need the following result proved in [18].

Theorem 3.1 Let V be compact and the conditions (3.2), (3.3) and

(3.4)  e^{|A|_{B(0,T)} ∫_{0}^{T} b(s)ds}|f|_{B(0,T)} ≤ R

hold. Then (3.1) has a solution x ∈ K_{R}, satisfying the inequality

|x|_{B(0,T)} ≤ e^{|A|_{B(0,T)} ∫_{0}^{T} b(s)ds}|f|_{B(0,T)}.

Now let us consider the equation

(3.5)  (Eu)(t) = (F u)(t)

(0 < t ≤ T < ∞) where E is the linear operator defined by (2.12). Take the initial conditions

(3.6)  u(t) = ϕ_{0}(t) (−2h ≤ t ≤ 0); u̇(t) = ϕ_{1}(t) (−h ≤ t ≤ 0)

with given continuous functions ϕ_{0}, ϕ_{1}. Equation (3.5) is equivalent to the equation

(3.7)  x(t) = y(t) + ∫_{0}^{t} G(t, t_{1})(F x)(t_{1})dt_{1} (t ∈ (0, T)),

where G is the Green function to equation (1.1) and y(t) is a solution of problem (1.1),(3.6). A continuous solution of (3.7) will be called a mild solution of problem (3.5), (3.6). Now Theorems 2.1 and 3.1 imply

Corollary 3.2 Under the hypothesis of Theorem 2.1, let condition (3.2) hold for R = ∞. Then equation (3.5) has a nonnegative mild solution.

4 Functional differential equations with variable and distributed delays

Consider the equation

(4.1)  u(t) = f(t) + ∫_{0}^{t} K(t, s) ∫_{0}^{2h} u(s − τ)R(s, τ)dr(τ)ds

where f(t) and R(t, τ) are continuous functions, and r(τ) (t ≥ 0; 0 ≤ τ ≤ 2h) is nondecreasing. Take the zero initial conditions

(4.2)  u(t) = u̇(t) = 0 (t ≤ 0).

Lemma 4.1 Let K(t, s), f(t) and R(s, τ) be non-negative functions. Then a solution of the problem (4.1), (4.2) is non-negative. Moreover,

(4.3)  u(t) ≥ f(t), t > 0.

Proof: Put

V u(t) := ∫_{0}^{t} K(t, s) ∫_{0}^{2h} u(s − τ)R(s, τ)dr(τ) ds.

We have

V u(t) = ∫_{0}^{2h} ∫_{0}^{t} K(t, τ)u(s − τ)R(s, τ)ds dr(τ) =
∫_{0}^{2h} ∫_{0}^{t−τ} K(t, s_{1} + τ)u(s_{1})R(s_{1} + τ, τ)ds_{1}dr(τ).

V is a nonnegative operator in C(R_{+}). Moreover, for any finite T,

|V u(t)| ≤ M_{T} ∫_{0}^{t} |u(s_{1})|ds_{1} (0 ≤ t ≤ T),

where

M_{T} := sup_{0≤t,s_{1}≤T} ∫_{0}^{2h} K(t, s_{1} + τ)R(s_{1} + τ, τ)dr(τ).

Hence it easily follows that V is a Volterra operator and its spectral radius is equal to zero. By the above mentioned comparison principle, inequality (4.3) holds, as claimed. □

Now let us consider the equation

ü(t) + 2c_{0}(t)u̇(t) + c_{1}(t)u̇(t − h) + d_{0}(t)u(t) + d_{1}(t)u(t − h) + d_{2}(t)u(t − 2h) =

(4.4)  f_{0}(t) + ∫_{0}^{2h} u(t − τ)R(t, τ)dr(τ) (f_{0} ∈ C(R_{+}); t > 0).

This equation under condition (4.2) is equivalent to (4.1) with

(4.5)  K(t, s) = G(t, s), f(t) = ∫_{0}^{t} G(t, s)f_{0}(s)ds.

Now the previous lemma and Theorem 2.1 imply Corollary 4.2 Under the hypothesis of Theorem 2.1, let f_{0} and R(t, τ) be non-negative. Then a solution u(t) of problem (4.4), (4.2) is non-negative and (4.3) holds with f defined by (4.5).

Furthermore, consider the equation

(4.6)  u(t) = f(t) + ∫_{0}^{t} K(t, s)m(s)u(s − v(s))ds

where m(t) ∈ C(R_{+}); v(t) is non-negative, continuously differentiable, v(t) ≤ 2h, and

(4.7)  0 ≤ β := inf_{t≥0} v̇(t) < 1.

Lemma 4.3 Let K(t, s), m(t) and f(t) be non-negative functions and condition (4.7) holds. Then a solution of problem (4.6), (4.2) is non-negative. Moreover, inequality (4.3) holds.

Proof: Put

V_{0}u(t) := ∫_{0}^{t} K(t, s)u(s − v(s))ds.

Let w(t) ≥ 0. Then

V_{0}w(t) ≥ \frac{1}{1 − β} ∫_{0}^{t} K(t, s)w(s − v(s))(1 − v̇(s))ds = \frac{1}{1 − β} ∫_{0}^{t−v(t)} K(t, ψ(s_{1}))w(s_{1})ds_{1}

where ψ(s) is inverse to the function s → s − v(s). Clearly, V_{0} is a nonnegative operator in C(R_{+}). Moreover, for any finite T,

|V_{0}u(t)| ≤ N_{T} ∫_{0}^{t} |u(s_{1})|ds_{1} (0 ≤ t ≤ T),

where

N_{T} := \frac{1}{1 − β} sup_{0≤t,s_{1}≤T} K(t, ψ(s_{1})).

Hence it easily follows that V_{0} is a Volterra operator and its spectral radius is equal to zero. By the above mentioned comparison principle, inequality (4.3) holds, as claimed. □

Now let us consider the equation

ü(t) + 2c_{0}(t)u̇(t) + c_{1}(t)u̇(t − h) + d_{0}(t)u(t) + d_{1}(t)u(t − h) + d_{2}(t)u(t − 2h) =

(4.8)  f_{0}(t) + m(t)u(t − v(t)) (m(t) ∈ C(R_{+}), t ≥ 0).

This equation under condition (4.2) is equivalent to (4.6) with (4.5) taken into account. Now the previous lemma and Theorem 2.1 imply

Corollary 4.4 Under the hypothesis of Theorem 2.1, let f_{0}(t) and m(t) be non-negative. Then a solution u(t) of problem (4.8), (4.2) is non-negative and (4.3) holds with f defined by (4.5).

References

[1] Agarwal, R. P., Grace, S. R. and O’Regan, D., Oscillation Theory for Difference and Functional Differential Equations. Kluwer, Dordrecht, 2000.

[2] Agarwal, R. P., O’Regan, D. and Wong, P. J.Y. , Positive Solutions of Differential, Difference and Integral equations, Kluwer, Dordrecht, 1999.

[3] Berezansky, L and E. Braverman, Some oscillation problems for a second order linear delay differential equations. J. Math. An. Appl., 220, (1998) 719-740

[4] Berezansky, L. and E. Braverman, Nonoscillation of a second order linear delay differential equation with a middle term Functional Differential Equations, 6, (1999) 233-249

[5] Berezansky, L. and E. Braverman, On oscillation of a second order delay differential equation with a middle term Applied Mathematical Letters, 13, (2000) 21-25

[6] Chen, Shihua; Wang, Feng; Young, Todd, Positive periodic solution of two-species ratio-dependent predator–prey system with time delay in two-patch environment. Appl. Math. Comput., 150, No. 3, 737-748 (2004).

[7] Corduneanu, C., Functional Equations with Causal Operators, Taylor and Francis, London, 2002.

[8] Daleckii, Yu L. and Krein, M. G. Stability of Solutions of Differential Equations in Banach Space, Amer. Math. Soc., Providence, R. I., 1971.

[9] Elias, U., Oscillation Theory of Two-term Differential Equations. Dordrecht: Kluwer Academic Publishers, 1997.

[10] Eloe, P.W. and J. Henderson, Positive solutions for higher order ordinary differential equations. Electron. J. Differ. Equ., 1995/03 (1995).

[11] Eloe, P.W., Raffoul, Y., Reid, D.T. and Yin, K.C. Positive solutions of nonlinear functional difference equations. Comput. Math. Appl., 42, No. 3-5, 639-646 (2001).

[12] Erbe L. H., Qingkai Kong and B.G. Zhang. Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, 1996.

[13] Feintuch, A. and Saeks, R. System Theory. A Hilbert Space Approach. Ac. Press, New York, 1982.

[14] Ferreira, J. M.; Pinelas, S., Nonoscillations in retarded systems. J. Math. Anal. Appl., 308, No. 2, 714-729 (2005).

[15] Gil’ M. I. On Aizerman-Myshkis problem for systems with delay, Automatica, 36, (2000), 1669-1674.

[16] Gil’ M. I., Boundedness of solutions of nonlinear differential delay equations with positive green functions and the Aizerman - Myshkis problem, Nonlinear Analysis, TMA, 49, (2002), 1065-168

[17] Gil, M.I. The Aizerman-Myshkis problem for functional-differential equations with causal nonlinearities, Functional Differential Equations, 11, No 1-2, (2005) 445-457

[18] Gil, M.I. Positive solutions of equations with nonlinear causal mappings Positivity, (2007) accepted for publication

[19] Han, Fei; Wang, Quanyi. Positive periodic solution for a class of differential equation with state-dependent delays. Ann. Differ. Equations, 21, No. 3, 290-293 (2005).

[20] Krasnosel’skii, M.A. and Zabreiko P.P. Geometrical Methods of Nonlinear Analysis, Springer-Verlag, Berlin, 1984.

[21] Li, Horng-Jaan, Qualitative analysis of positive solutions of first-order functional differential equations of neutral type. Nonlinear Anal., Theory Methods Appl. 39, No.7(A), 891-907 (2000).

[22] Li, Wan-Tong; Zhong, Cheng-Kui, Unbounded positive solutions to higher-order nonlinear functional-differential equations. Appl. Math. Lett. 14, No.7, 825-830 (2001).

[23] Lin, Shizhong; Yu, Yuanhong, Oscillation for first order neutral delay differential equations with positive and negative coefficients, J. Math. Study 38, No. 3, 277-280 (2005).

[24] Naito, M. and K. Yano, Positivity solutions of higher order ordinary differential equations, J. Math. Anal. Appl. 250 , 27-48 (2000).

[25] Tanaka, Satoshi, Existence of positive solutions for a class of higher order neutral functional differential equations, Czech. Math. J. 51, No. 3, 573-583 (2001).

[26] Tsalyuk, Vadim, Stability of linear functional differential systems with multivalued delay feedback. Electron. J. Differ. Equ. 2007, Paper No. 36, 14 p., electronic only (2007).

[27] Vath, M. Volterra and Integral Equations of Vector Functions, Marcel Dekker, 2000.

[28] Yin, Fuqi; Fan, Guihong; Li, Yongkun, The existence of positive solutions for the quasilinear functional delay differential equations, J. Math. Study 35, No. 4, 364-370 (2002).

(Received September 25, 2009)