Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 13, 1-9; http://www.math.u-szeged.hu/ejqtde/

A note on a nonlinear functional differential system with feedback control^{∗}

Yuqing Wang Xiaoming He

College of Science, Minzu University of China, Beijing 100081, P.R. of China

Abstract In this note we apply Avery-Peterson multiple fixed point theorem to investigate the existence of multiple positive periodic solutions to the following nonlinear non-autonomous functional differential system with feedback control

\{ \begin{matrix}\frac{dx}{dt} = −r(t)x(t) + F (t, x_{t}, u(t − δ(t))), \\ \frac{du}{dt} = −h(t)u(t) + g(t)x(t − σ(t)). \end{matrix}

We prove the system above admits at least three positive periodic solutions under certain growth conditions imposed on F. 2000 Mathematics Subject Classification: 34K 13; 34C 10; 34C 27. Keywords: Functional differential equation; Feedback control; Fixed-point theorem; Positive periodic solution.

1. Introduction In this note, we obtain a new result for the existence of at least three positive

ω-periodic solutions of the following system with feedback control:

\{ \begin{matrix}\frac{dx}{dt} = −r(t)x(t) + F (t, x_{t}, u(t − δ(t))), \\ \frac{du}{dt} = −h(t)u(t) + g(t)x(t − σ(t)). \end{matrix}  (1)

where δ(t), σ(t) ∈ C(ℝ, ℝ), r(t), h(t), g(t) ∈ C(ℝ, (0, +∞)), all of the above functions are ω-periodic functions and ω > 0 is a constant. F (t, x_{t}, z) is a function defined on ℝ × BC × ℝ and F (t + ω, x_{t+ω}, z) = F (t, x_{t}, z), where BC denotes the Banach space of bounded continuous functions φ : ℝ → ℝ with the norm || φ|| = sup_{θ∈R} |φ(θ)|. If x ∈ BC, then x_{t} ∈ BC for any t ∈ ℝ is defined by x_{t}(τ) = x(t + τ) for τ ∈ ℝ.

Many special cases of (1) have been widely investigated by many authors, see, for example [10,8,4,6]. But most work is for autonomous. In real world, any biological or environmental parameters, however, are naturally subject to fluctuation in time, so it is reasonable to study the corresponding non-autonomous systems. System (1) includes many mathematical ecological models with feedback control, such as the multiplicative delay logistic model with feedback control studied in [7]. For more information about the population models concerned with (1), we refer to [12] and the references therein.

∗e-mail address: larawyq@126.com, xmhe923@126.com

Considering the biological and environmental periodicity, such as seasonal effects of weather, food supplies, mating habits, we shall focus on the existence of periodic solutions with strictly positive components of (1). Recently, some authors have studied the existence of at least one and two positive periodic solutions of (1) (cf. [9,3,11]). The methods used in the references are mainly the continuation theorem of Gaines and Mawhin’s coincidence degree theory [4] and the fixed point theorem in cones [5]. We note that in a recent paper [9], by using a fixed point theorem [1] involving three functionals on cones, Liu and Li studied problem (1) and obtained two positive periodic solutions with some monotonic properties, that is, x(t) ≥ 0 is ω−periodic and x(t)e^{∫_{0}^{t} r(s)ds} is no decreasing on [0, ω]. The main goal of this paper is to revisit problem (1) and obtain at least three positive periodic solutions of (1) by means of a three fixed-points theorem due to Avery and Peterson [2].

The paper is divided into three sections, including this section. In Section 2 we give some background knowledge and preliminary results. Section 3 is devoted to the proof of the main result.

2. Background material and results For the reader’s convenience, we shall summarize below a few concepts and results

from the theory of cones in Banach spaces [5].

Let E be a real Banach space and P ⊂ E a cone. The map α is said to be a nonnegative continuous concave functional on the cone P provided that α : P → [0, ∞) is continuous and

α(tx + (1 − t)y) ≥ tα(x) + (1 − t)α(y)

for all x, y ∈ P and 0 ≤ t ≤ 1. Similarly, we say the map γ is a nonnegative continuous convex functional on cone P provided that γ : P → [0, ∞) is continuous and

γ(tx + (1 − t)y) ≤ tγ(x) + (1 − t)γ(y)

for all x, y ∈ P and 0 ≤ t ≤ 1.

Let γ and θ be nonnegative continuous convex functionals on P, α be a nonnegative continuous concave functional on P, and ψ be a nonnegative continuous functional on P. Then for positive real numbers a, b, c and d, we define the following convex sets:

P (γ, d) = {x ∈ P | γ(x) < d},
P (γ, α, b, d) = {x ∈ P | b ≤ α(x), γ(x) ≤ d},
P (γ, θ, α, b, c, d) = {x ∈ P | b ≤ α(x), θ(x) ≤ c, γ(x) ≤ d},

and a closed set R(γ, ψ, a, d) = {x ∈ P | a ≤ ψ(x), γ(x) ≤ d}.

The following fixed point theorem due to Avery and Peterson [2] is crucial in the proof of our main result.

Theorem 2.1 [2]. Let P be a cone in a real Banach space E. Let γ and θ be nonnegative continuous convex functionals on P, α be a nonnegative continuous concave functional on P, and ψ be a nonnegative continuous functional on P satisfying ψ(λx) ≤ λψ(x) for 0 ≤ λ ≤ 1, such that for some positive numbers M and d,

α(x) ≤ ψ(x) and ∥x∥ ≤ Mγ(x)  (∗)

for all x ∈ \overline{P (γ, d)}. Suppose T : \overline{P (γ, d)} → \overline{P (γ, d)} is completely continuous and there exist positive numbers a, b, and c with a < b such that

(S1) {x ∈ P (γ, θ, α, b, c, d)| α(x) > b} ≠ ∅ and α(T x) > b for x ∈ P (γ, θ, α, b, c, d);

(S2) α(T x) > b for x ∈ P (γ, α, b, d) with θ(T x) > c;

(S3) 0 ∉ R(γ, ψ, a, d) and ψ(T x) < a for x ∈ R(γ, ψ, a, d) with ψ(x) = a.

Then T has at least three fixed points x_{1}, x_{2}, x_{3} ∈ \overline{P (γ, d)}, such that

γ(x_{i}) ≤ d for i = 1, 2, 3, b < α(x_{1}),
a < ψ(x_{2}), with α(x_{2}) < b, and ψ(x_{3}) < a.

In order to apply Theorem 2.1 to establish the existence of multiple positive solutions of system (1), we shall define an operator on a cone in a suitable Banach space. To this end, we first transform system (1) into a single equation. By integrating the latter equation in system (1) from t to t + ω, we obtain

u(t) = ∫_{t}^{t+ω}k(t, s)g(s)x(s − σ(s))ds := (Φx)(t),  (2)

where

k(t, s) = \frac{exp\{∫_{t}^{s} h(v)dv\}}{exp\{∫_{0}^{ω} h(v)dv\} − 1}.

When x is an ω-periodic function, it is easy to see that k(t + ω, s + ω) = k(t, s), u(t + ω) = u(t), and

n := \frac{exp\{− ∫_{0}^{ω} h(v)dv\}}{exp\{∫_{0}^{ω} h(v)dv\} − 1} ≤ k(t, s) ≤ \frac{exp\{∫_{0}^{ω} h(v)dv\}}{exp\{∫_{0}^{ω} h(v)dv\} − 1} := m  (∗∗)

for(t, s) ∈ ℝ^{2}, where m, n are positive constants.

Therefore, the existence problem of ω-periodic solution of system (1) is equivalent to that of ω-periodic solution of the following equation:

\frac{dx}{dt} = −r(t)x(t) + F (t, x_{t}, (Φx)(t − δ(t))).  (3)

In what follows, we shall adopt the following notations for convenience: ℝ = (−∞, ∞), ℝ^{+} = [0, ∞), I = [0, ω], f^{∗} := max_{t∈I} f(t), f_{∗} := min_{t∈I} f(t), where f is a continuous positive periodic function with period ω. BC(X, Y) denotes the space of bounded continuous functions φ : X → Y.

Throughout the paper we always assume that

(H_{1}) F (t, x_{t}, (Φx)(t − δ(t))) is a continuous function of t for each x ∈ BC(ℝ, ℝ^{+}), and F (t, x_{t}, (Φx)(t − δ(t))) ≥ 0 for (t, x) ∈ ℝ × BC(ℝ, ℝ^{+}), where Φ is defined as (2).

(H_{2}) For any C > 0 and ε > 0, there exists μ > 0 such that for γ, ξ ∈ BC, ∥γ∥ ≤ C, ∥ξ∥ ≤ C, ∥γ − ξ∥ < μ and for 0 ≤ s ≤ ω imply

|F (s, γ_{s}, (Φγ)(s − δ(s))) − F (s, ξ_{s}, (Φξ)(x − δ(s)))| < ε.

By integrating equation (3) from t to t + ω, we have

x(t) = ∫_{t}^{t+ω} G(t, s)F (s, x_{s}, (Φx)(s − δ(s)))ds,

where

G(t, s) = \frac{exp\{∫_{t}^{s} r(v)dv\}}{exp\{∫_{0}^{ω} r(v)dv\} − 1}.  (4)

It is clear that G(t + ω, s + ω) = G(t, s) for all (t, s) ∈ ℝ^{2} and

p := \frac{exp\{− ∫_{0}^{ω} r(v)dv\}}{exp\{∫_{0}^{ω} r(v)dv\} − 1} ≤ G(t, s) ≤ \frac{exp\{∫_{0}^{ω} r(v)dv\}}{exp\{∫_{0}^{ω} r(v)dv\} − 1} := q

for all s ∈ [t, t + ω], where p, q are positive constants.

We denote by β := \frac{p}{q}. In order to use Theorem 2.1, we let E be the set E = \{x ∈ C(ℝ, ℝ) : x(t + ω) = x(t), t ∈ ℝ} with the norm ∥x∥ = sup_{t∈I} |x(t)|; then E ⊂ BC is a Banach space. Also we define P as P = {x ∈ E : x(t) ≥ β∥x∥, t ∈ I}. One may readily verify that P is a cone in E.

Define an operator T : P → P by

(T x)(t) = ∫_{t}^{t+ω} G(t, s)F (s, x_{s}, (Φx)(s − δ(s)))ds  (5)

for x ∈ P, t ∈ ℝ, where G(t, s) is defined by (4) and Φ is defined by (2). Under the conditions (H_{1}), (H_{2}), we known that T : P → P is well defined, completely continuous [9], and each positive ω-periodic solution of (1) is a fixed point of the operator T on P.

From now on, we fix 0 ≤ η ≤ l ≤ ω, τ ∈ ℝ and let the nonnegative continuous concave functional α, the nonnegative continuous convex functionals θ, γ, and the nonnegative continuous functional ψ be defined on the cone P by

α(x) = min_{η≤t≤l} |x_{t}(τ)|,

ψ(x) = θ(x) = max_{0≤t≤ω} |x_{t}(τ)| = max_{0≤t≤ω} |x(t)|,

γ(x) = max_{0≤t≤ω} |(Φx)(t − δ(t))|.

The following lemma is useful in the proof of our main result.

Lemma 2.1. For x ∈ P, there exists a constant M > 0 such that

max_{0≤t≤ω} |x_{t}(τ)| ≤ M max_{0≤t≤ω} |(Φx)(t − δ(t))|.

Proof. For x ∈ P, we have

max_{0≤t≤ω} |(Φx)(t − δ(t))| = max_{0≤t≤ω}(Φx)(t − δ(t))

= max_{0≤t≤ω} ∫_{t−δ(t)}^{t−δ(t)+ω} k(t − δ(t), s)g(s)x(s − σ(s))ds

= max_{0≤t≤ω} ∫_{0}^{ω} k(t − δ(t), s)g(s)x(s − σ(s))ds

≥ β∥x∥ max_{0≤t≤ω} ∫_{0}^{ω} k(t − δ(t), s)g(s)ds = Lβ max_{0≤t≤ω} |x_{t}(τ)|,

(6)

where L := max_{0≤t≤ω} ∫_{0}^{ω} k(t−δ(t), s)g(s)ds. Setting M := \frac{1}{Lβ} , we complete the proof.

By Lemma 2.1, the functionals defined above satisfy relations

α(x) ≤ θ(x) = ψ(x), ∀x ∈ P.  (7)

Moreover, for each x ∈ P, there holds

∥x∥ = max_{0≤t≤ω} |x_{t}(τ)| ≤ \frac{max_{0≤t≤ω} |(Φx)(t − δ(t))|}{Lβ} = Mγ(x).  (8)

We also find that ψ(λx) = λψ(x) for ∀λ ∈ [0, 1] and ∀x ∈ P. Therefore, the condition (∗) of Theorem 2.1 is satisfied.

3. Main result

Denote by λ_{ω} = max_{0≤t≤ω} ∫_{0}^{ω} k(t − δ(t), s)g(s)ds × max_{0≤t≤ω} ∫_{0}^{ω} G(t − σ(t), s)ds, and

N = min_{η≤t≤l} ∫_{0}^{ω} G(t+τ, s)ds, D = max_{0≤t≤ω} ∫_{0}^{ω} G(t+τ, s)ds, L = max_{0≤t≤ω} ∫_{0}^{ω} k(t, s)g(s)ds.

Our main result is the following

Theorem 3.1. Let 0 < a < b < \frac{b}{β} ≤ \frac{d}{L}, and suppose that F satisfies the following conditions:

(C1) F (t, u, v) ≤ d/λ_{ω}, for (t, u, v) ∈ [0, ω] × [0, Md] × [0, d];

(C2) F (t, u, v) > b/N, for (t, u, v) ∈ [η, l] × [b, b/β] × [0, d];

(C3) F (t, u, v) < a/D, for (t, u, v) ∈ [0, ω] × [0, a] × [0, d].

Then equation (3) admits at least three positive solutions x_{1}, x_{2} and x_{3} such that

max_{0≤t≤ω} |(Φx_{i})(t − δ(t))| ≤ d; b < min_{η≤t≤l} |x_{1}(t + τ)|, a < max_{0≤t≤ω} |x_{2}(t + τ)| with min_{η≤t≤l} |x_{2}(t +

τ)| < b, and max_{0≤t≤ω} |x_{3}(t + τ)| < a.

Proof. Problem (1) has a solution x = x(t) if and only if x solves the operator equation x = T x. Thus we set out to verify that the operator T satisfies the Avery-Peterson fixed point theorem which will prove the existence of three fixed points of T which satisfy Theorem 2.1.

For x ∈ \overline{P (γ, d)}, there is γ(x) = max_{0≤t≤ω} |(Φx)(t−δ(t))| ≤ d. By Lemma 2.1, we have

max_{0≤t≤ω} |x_{t}(τ)| = max_{0≤t≤ω} x(t+τ) ≤ Md. Then condition (C1) implies F (t, x_{t}(τ), (Φx)(t−

δ(t)) ≤ d/λ_{ω}. On the other hand, for x ∈ \overline{P (γ, d)}, there is T x ∈ P and

(T x)(t − σ(t)) = ∫_{t−σ(t)}^{t−σ(t)+ω} G(t − σ(t), s)F (s, x_{s}, (Φx)(s − δ(s)))ds
= ∫_{0}^{ω} G(t − σ(t), s)F (s, x_{s}, (Φx)(s − δ(s)))ds
≤ max_{0≤t≤ω} ∫_{0}^{ω} G(t − σ(t), s)ds × \frac{d}{λ_{ω}} ,

we have

γ(T x) = max_{0≤t≤ω} |(Φ(T x))(t − δ(t))|

= max_{0≤t≤ω} ∫_{t−δ(t)}^{t−δ(t)+ω} k(t − δ(t), s)g(s)(T x)(s − σ(s))ds
= max_{0≤t≤ω} ∫_{0}^{ω} k(t − δ(t), s)g(s)(T x)(s − σ(s))ds
≤ max_{0≤t≤ω} ∫_{0}^{ω} k(t − δ(t), s)g(s)ds × max_{0≤t≤ω} ∫_{0}^{ω} G(t − σ(t), s)ds\frac{d}{λ_{ω}} ≤ d.

Therefore, T : \overline{P (γ, d)} → \overline{P (γ, d)}.

To check condition (S1) of Theorem 2.1, we take x(t) = b/β. It is easy to see that x(t) = b/β ∈ P (γ, θ, α, b, b/β, d) and α(x) = b/β > b, and so {x ∈ P (γ, θ, α, b, b/β, d)| α(x) > b} ≠ ∅.

Hence, for x ∈ P (γ, θ, α, b, b/β, d), there is

min_{η≤t≤l} |x_{t}(τ)| ≥ b, max_{0≤t≤ω} |x_{t}(τ)| ≤ b/β, max_{0≤t≤ω} |(Φx)(t − δ(t))| ≤ d

such that b ≤ x_{t}(τ) ≤ b/β, 0 ≤ (Φx)(t − δ(t)) ≤ d, t ∈ [η, l].

Thus, by condition (C2) of this theorem, we have F (t, x_{t}, (Φx)(t − δ(t))) > b/N and

α(T x) = min_{η≤t≤l} |(T x)(t + τ)| = min_{η≤t≤l} ∫_{t+τ}^{t+τ+ω} G(t + τ, s)F (s, x_{s}, (Φx)(s − δ(s)))ds
> \frac{b}{N} min_{η≤t≤l} ∫_{0}^{ω} G(t + τ, s)ds = b,

i.e. α(T x) > b for allx ∈ P (γ, θ, α, b, b/β, d). This show that condition (S1) of Theorem 2.1 is satisfied.

Secondly, with (7), (8), we have α(T x) ≥ βθ(T x) > β × \frac{b}{β} = b for all x ∈ P (γ, α, b, d) with θ(T x) > \frac{b}{β} . Thus, condition (S2) of Theorem 2.1 is satisfied.

Finally, we show that condition (S3) of Theorem 2.1 also holds. Clearly, as ψ(0) = 0 < a, there holds 0 ∉ R(γ, ψ, a, d). Suppose that x ∈ R(γ, ψ, a, d) with ψ(x) = a. This implies that for t ∈ [0, ω], there is max_{0≤t≤ω} |x_{t}(τ)| = a, max_{0≤t≤ω} |(Φx)(t − δ(t))| ≤ d.

Hence,

0 ≤ x_{t}(τ) ≤ a, 0 ≤ (Φx)(t − δ(t)) ≤ d, t ∈ [0, ω],

and by the condition (C3) of this theorem,

ψ(T x) = max_{0≤t≤ω} |(T x)_{t}(τ)| = max_{0≤t≤ω} ∫_{t+τ}^{t+τ+ω} G(t + τ, s)F (s, x_{s}, (Φx)(s − δ(s)))ds
< \frac{a}{D} max_{0≤t≤ω} ∫_{0}^{ω} G(t + τ, s)ds = a.

So, the condition (S3) of Theorem 2.1 is satisfied. On the other hand, for x ∈ P, (7), (8) holds. Therefore, an application of Theorem 2.1 implies (3) has at least three positive solutions.

Remark 3.1. Under the assumptions (H_{1}), (H_{2}) and the following assumption

(H_{0}) mg^{∗} > \frac{1}{ω}, ng_{∗} < \frac{1}{ω},

where m, n are given in (∗∗) and g^{∗} := max_{t∈I} g(t), g_{∗} := min_{t∈I} g(t) are defined as in Section 2, Liu and Li [9] obtained two positive periodic solutions for problem (1). But in this paper we completely remove assumption (H_{0}). Moreover, we obtain at least three positive periodic solutions for (1) with more general nonlinearity F.

References

[1] R.I. Avery, J. Henderson, Two positive fixed points of nonlinear operators on ordered Banach spaces, Comm. Appl. Nonlinear Anal. 8(2001), 27-36.

[2] R.I. Avery, A.C. Peterson, Three positive fixed points of nonlinear operators on ordered Banach spaces, Computers Math. Appl. 42(2001), 313-322.

[3] S. Cheng, G. Zhang, Existence of positive periodic solutions for non-autonomous functional differential equations, Electronic J. Diff. Eqns. 59(2001), 1-8.

[4] R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer, Berlin, 1977.

[5] D.J. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Orlando, FL, 1988.

[6] W.S.C. Gurney, S.P. Blathe, R.M. Nishet, Nicholson’s blowflies revisited, Nature 287(1980), 17-21.

[7] H.F. Huo, W.T. Li, Positive solutions of a class of delay differential system with feedback control, Appl. Math. Comput. 148(2004), 35-46.

[8] Y. Kuang, Delay Differential Equations with Application in Population Dynamics, Academic Press, New York,1993.

[9] P. Liu, Y.K. Li, Multiple positive periodic solutions of nonlinear functional differential system with feedback control, J. Math. Anal. Appl. 288(2003), 819-832.

[10] S.H. Saker, S. Agarwal, Oscillation and global attractivity in a periodic Nicholson’s blowflies model, Math. Comput. Modelling 35(2002), 719-731.

[11] A. Wan, D.Q. Jiang, X.J. Xu, A new existence theory for positive periodic solutions to functional differential equations, Compuers Math. Appl. 47(2004), 1257-1262.

[12] D. Ye, M. Fan, W.P. Zhang, Periodic solutions of density dependent predatorprey systems with Holling Type 2 functional response and infinite delays, ZAMM 85(2005), 213-221.

(Received December 15, 2008)