Electronic Journal of Qualitative Theory of Differential Equations Proc. 7th Coll. QTDE, 2004, No. 10 1-11; http://www.math.u-szeged.hu/ejqtde/

ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF SOME

FUNCTIONAL DIFFERENTIAL EQUATIONS BY SCHAUDER’S THEOREM

Dedicated to Professor Lásló Hatvani on his 60th birthday

TETSUO FURUMOCHI ^{1}

Department of Mathematics, Shimane University

1060 Nishikawatsucho, Matsue 690-8504, Japan furumochi@math.shimane-u.ac.jp

ABSTRACT. In a series of papers (Burton-Furumochi [1-4]) we have studied stability properties of functional differential equations by means of fixed point theory. Here we obtain new stability and boundedness results for half-linear equations and integro-differential equations by using Schauder’s first theorem and a weighted norm, and show some examples.

1. INTRODUCTION

In Part I of Burton-Furumochi [4], we studied asymptotic stability in halflinear equations by using Schauder’s first theorem, Ascoli-Arzela like lemma with the concept of equi-convergence, and the uniform norm. In this paper, we generalize the stability results in [4], and obtain boundedness results for some functional differential equations by using Schauder’s first theorem and a weighted norm instead of the uniform norm.

Let r_{0} be a fixed nonnegative constant and let h : [−r_{0}, ∞) → [1, ∞) be any strictly increasing and continuous function with

h(−r_{0}) = 1

and

h(t) → ∞ as t → ∞.

For any t_{0} ∈ R^{+} := [0, ∞) let C_{t_{0}} be the space of continuous functions ϕ : [t_{0} − r_{0}, ∞) → R := (−∞, ∞) with

∥ϕ∥_{h} := sup \{\frac{|ϕ(t)|}{h(t − t_{0})} : t ≥ t_{0} − r_{0}\} < ∞.

Then, ∥ · ∥_{h} is a norm on C_{t_{0}}, and (C_{t_{0}}, ∥ · ∥_{h}) is a Banach space.

1This paper is in final form and no version of it will be submitted for publication elsewhere, and partly supported in part by Grant-in-Aid for Scientific Research (C), No. 14540158, Japan Society for the Promotion of Science.

1991 Mathematics Subject Classfication. 34D20, 34K20, 47H10.

Key words and phrases. Stability, boundedness, Schauder’s theorem.

First we state a lemma without proof (see Burton [5; p. 169]).

Lemma. If the set {ϕ_{k}(t)} of R-valued functions on [t_{0}−r_{0}, ∞) is uniformly bounded and equi-continuous, then there is a bounded and continuous function ϕ and a subsequence {ϕ_{k_{j}}} such that

∥ϕ_{k_{j}} − ϕ∥_{h} → 0 as j → ∞.

2. STABILITY IN HALF-LINEAR EQUATIONS

Consider the scalar half-linear equation

x′(t) = −a(t)x(t) − b(t)g(x(t − r(t))), t ∈ R^{+},  (1)

where a, b : R^{+} → R, g : R → R and r : R^{+} → R^{+} are continuous. Let α be any fixed positive number. We assume that there are constants β > 0, γ > 0 and r_{0} ≥ 0 so that

|g(x)| ≤ β|x| for x ∈ R with |x| ≤ α,  (2)

sup {e^{∫_{τ}^{t}(a(s)−βγ|b(s)|)ds} : t ∈ R^{+}} ≤ γ,  (3)

where τ = τ(t) := max (0, t − r(t)),

t − r(t) ≥ −r_{0},  (4)

σ = σ(t_{0}) := sup { ∫_{t_{0}}^{t} (βγ|b(s)| − a(s))ds : t ≥ t_{0}} < ∞,  (5)

and define a number η = η(t_{0}) by

η := αe^{−σ}.  (6)

Corresponding to Eq. (1), consider the scalar linear equation

q′ = (βγ|b(t)| − a(t))q, t ∈ R^{+}.  (7)

Let q : [t_{0} − r_{0}, ∞) → R^{+} be a continuous function such that

q(t) = η on [t_{0} − r_{0}, t_{0}],

and that q(t) is the unique solution of the initial value problem

q′ = (βγ|b(t)| − a(t))q, q(t_{0}) = η, t ≥ t_{0}.

Then q(t) can be expressed in two ways as

q(t) = ηe^{− ∫_{t_{0}}^{t} a(s)ds} + βγ ∫_{t_{0}}^{t} e^{− ∫_{s}^{t} a(u)du}|b(s)|q(s)ds

= ηe^{∫_{t_{0}}^{t} (βγ|b(s)|−a(s))ds}, t ≥ t_{0},  (8)

which together with (5) and (6), implies

0 < q(t) ≤ ηe^{σ} = α, t ≥ t_{0}.  (9)

Concerning the stability of the zero solution of Eq. (1), we have the following theorem.

Theorem 1. Suppose that the solutions of Eq. (1) are uniquely determined by continuous initial functions, and that (2)-(5) hold. Then we have:

(i) The zero solution of Eq. (1) is stable. (ii) If we have

σ^{∗} := sup {σ(t) : t ∈ R^{+}} < ∞,

then the zero solution of Eq. (1) is uniformly stable.

(iii) If we have

∫_{0}^{t} (a(s) − βγ|b(s)|)ds → ∞ as t → ∞,  (10)

then the zero solution of Eq. (1) is asymptotically stable.

(iv) In addition to σ^{∗} < ∞, if we have

∫_{t_{0}}^{t} (a(s) − βγ|b(s)|)ds → ∞ uniformly for t_{0} ∈ R^{+} as t → ∞,  (11)

then the zero solution of Eq. (1) is uniformly asymptotically stable.

Proof. (i) Assumption (5) implies that the zero solution of Eq. (7) is stable. Thus, for any ϵ ∈ (0, α] and t_{0} ∈ R^{+}, there is a δ = δ(ϵ, t_{0}) ∈ (0, η] such that for any q_{0} with |q_{0}| ≤ δ, we have

|q(t, t_{0}, q_{0})| < ϵ for all t ≥ t_{0}.

For the t_{0}, let (C_{t_{0}}, ∥ · ∥_{h}) be the Banach space of continuous functions ϕ : [t_{0} − r_{0}, ∞) → R with the weighted norm ∥ · ∥_{h}. For a continuous function ψ : [t_{0} − r_{0}, ∞) → R^{+} with

sup {|ψ(θ)| : −r_{0} ≤ θ ≤ 0} ≤ δ,

let S be a set of continuous functions ϕ : [t_{0} − r_{0}, ∞) → R such that

ϕ(t) = ψ(t − t_{0}) for t_{0} − r_{0} ≤ t ≤ t_{0},
|ϕ(t)| ≤ q(t) for t ≥ t_{0},

and

|ϕ(t_{1}) − ϕ(t_{2})| ≤ L|t_{1} − t_{2}| for t_{1}, t_{2} ∈ R^{+} with t_{0} ≤ τ_{1} ≤ t_{1}, t_{2} ≤ τ_{2},

where q(t) is defined by (8) with η = δ, and where L : R^{+} × R^{+} → R^{+} is defined by

L = L(τ_{1}, τ_{2}) := max {(|a(t)| + βγ|b(t)|)α : τ_{1} ≤ t ≤ τ_{2}}.  (12)

Since we have (9), we obtain

|q′(t)| ≤ (|a(t)| + βγ|b(t)|)α, t ≥ t_{0}.

Thus the function ξ(t) defined by

ξ(t) := \{ \begin{matrix}ψ(t − t_{0}), & t_{0} − r_{0} ≤ t ≤ t_{0}, \\ \frac{ψ(0)q(t)}{δ}, & t > t_{0}, \end{matrix}

is an element of S, and from Lemma, S is a compact convex nonempty subset of C_{t_{0}}.

Define a mapping P on S by (P ϕ)(t) := ψ(t − t_{0}) for t_{0} − r_{0} ≤ t ≤ t_{0}, and

(P ϕ)(t) := ψ(0)e^{− ∫_{t_{0}}^{t} a(s)ds} − ∫_{t_{0}}^{t} e^{− ∫_{s}^{t} a(u)du}b(s)g(ϕ(s − r(s)))ds, t > t_{0},

where ϕ ∈ S. Then we have

(P ϕ)(t) = ψ(t − t_{0}) for t_{0} − r_{0} ≤ t ≤ t_{0},

and from (2) and (8) we obtain

|(P ϕ)(t)| ≤ δe^{− ∫_{t_{0}}^{t} a(s)ds} + β ∫_{t_{0}}^{t} e^{− ∫_{s}^{t} a(u)du}|b(s)|q(s − r(s))ds

≤ δe^{− ∫_{t_{0}}^{t} a(s)ds} + βγ ∫_{t_{0}}^{t} e^{− ∫_{s}^{t} a(u)du}|b(s)|q(s)ds = q(t), t ≥ t_{0}.

Moreover, we have

(P ϕ)′(t) = −a(t)(P ϕ)(t) − b(t)g(ϕ(t − r(t)), t > t_{0},

which implies

|(P ϕ)′(t)| ≤ |a(t)|q(t) + β|b(t)|q(t − r(t))
≤ (|a(t)| + βγ|b(t)|)q(t) ≤ (|a(t)| + βγ|b(t)|)α, t > t_{0},

and hence, P maps S into S. In addition, P is continuous. Thus, by Schauder’s first theorem, P has a fixed point ϕ in S and that is the solution x(t, t_{0}, ψ) of Eq. (1) which satisfies

|x(t, t_{0}, ψ)| ≤ q(t) = q(t, t_{0}, δ) < ϵ, t ≥ t_{0},

and hence, the zero solution of Eq. (1) is stable.

(ii)-(iv) If σ^{∗} < ∞, then the zero solution of Eq. (7) is uniformly stable. Next, Assumption (10) implies that q(t) → 0 as t → ∞, and hence, the zero solution of Eq. (7) is asymptoticaly stable. Moreover, Assumptions σ^{∗} < ∞ and (11) imply that the zero solution of Eq. (7) is uniformly asymptotically stable. Thus, the uniform stability, the asymptotic stability and the uniform asymptotic stability of the zero solution of Eq. (1) can be similarly proved as in the proof of (i). So we omit the details.

Now we show two examples.

Example 1. Let g(x) ≡ x on R, and define functions a, b, r : R^{+} → R^{+} by

a(t) := 2 + |t sin t|, t ∈ R^{+},

b(t) := \frac{max(1, 1 + 2t sin t)}{25}, t ∈ R^{+},

and

r(t) := \frac{1}{t + 1}, t ∈ R^{+}.

Then, (2)-(5) hold with β = 1, γ = 25 and r_{0} = 1, and σ^{∗} = ∞. Thus, concerning the stability of the zero solution of the equation

x′(t) = −a(t)x(t) − b(t)x(t − \frac{1}{t + 1}), t ∈ R^{+},  (13)

Theorem 1 does not assure uniform stability, but assures stability.

Example 2. Let a : R^{+} → R be a 13-periodic function satisfying

a(t) := \{ \begin{matrix}−1, & 0 ≤ t < 1, \\ 6t − 7, & 1 ≤ t < 2, \\ 5, & 2 ≤ t < 12, \\ 77 − 6t, & 12 ≤ t ≤ 13, \end{matrix}

and let r(t) ≡ r (0 ≤ r ≤ (ln 2)/5), b(t) ≡ B (0 < B ≤ 53/26), (2) hold with β = 1. Then (3)-(5) hold with β = 1, γ = 2 and r_{0} = r, and σ^{∗} < ∞. Moreover, if 0 < B < 53/26, then (11) holds. Thus, by Theorem 1, the zero solution of Eq. (1) is uniformly stable. Moreover, if 0 < B < 53/26, then the zero solution of Eq. (1) is uniformly asymptotically stable.

If g(x) ≡ x on R and r(t) ≡ r on R^{+}, then Eq. (1) becomes

x′(t) = −a(t)x(t) − b(t)x(t − r), t ∈ R^{+}.  (14)

In Hale [6; p. 108], under the assumption

a(t) ≥ δ > 0, |b(t)| ≤ θδ, θ < 1,  (15)

where δ and θ are constants, the uniform asymptotic stability of the zero solution of Eq. (14) is discussed by using a Liapunov functional.

On the other hand, in Burton-Furumochi [1], under the assumption

∫_{0}^{t} e^{− ∫_{s}^{t} a(u)du}|b(s)|ds ≤ η < 1 on R^{+}, ∫_{0}^{t} a(s)ds → ∞ as t → ∞, (16)

where η is a constant, the asymptotic stability of the zero solution of Eq. (14) is discussed by using the contraction principle.

But the functions a(t) and b(t) ≡ 1 in Example 2 satisfy neither (15) nor (16).

Next consider the scalar integro-differential equation

x′(t) = −a(t)x(t) − ∫_{t−r(t)}^{t} b(t, s)g(x(s))ds, t ∈ R^{+},  (17)

where a, r : R^{+} → R, b : R^{+} × R → R and g : R → R are continuous. Let α be any fixed positive number. We assume that there are constants β > 0, γ > 0 and r_{0} ≥ 0 so that (2) and (4) hold and

sup_{t∈R^{+}} { sup_{τ≤v≤t} e^{∫_{v}^{t}(a(s)−βγ ∫_{s−r(s)}^{s} |b(s,u)|du)ds}} ≤ γ,  (18)

where τ = τ(t) := max (0, t − r(t)), and

σ = σ(t_{0}) := sup_{t∈R^{+}} ∫_{t_{0}}^{t} (βγ ∫_{s−r(s)}^{s} |b(s, u)|du − a(s))ds < ∞.  (19)

For this σ, define a number δ = δ(t_{0}) by δ := αe^{−σ}.

Corresponding to Eq. (17), consider the scalar linear equation

q′ = (βγ ∫_{t−r(t)}^{t} |b(t, s)|ds − a(t))q, t ∈ R^{+}.  (20)

Let q : [t_{0} − r_{0}, ∞) → R be a continuous function such that

q(t) = δ on [t_{0} − r_{0}, t_{0}],

and that q(t) is the unique solution of the initial value problem

q′ = (βγ ∫_{t−r(t)}^{t} |b(t, s)|ds − a(t))q, q(t_{0}) = δ, t ≥ t_{0}.

Then q(t) can be expressed as

q(t) = δe^{∫_{t_{0}}^{t} (βγ ∫_{s−r(s)}^{s} |b(s,u)|du−a(s))ds}, t ≥ t_{0},

which together with (19), implies (9) with η = δ.

Concerning the stability of the zero solution of Eq. (17), we have the following theorem.

Theorem 2. Suppose that the solutions of Eq. (17) are uniquely determined by continuous initial functions, and that (2), (4), (18) and (19) hold. Then we have:

(i) The zero solution of Eq. (17) is stable.

(ii) If σ^{∗} := sup{σ(t) : t ∈ R^{+}} < ∞, then the zero solution of Eq. (17) is uniformly stable.

(iii) If we have

∫_{0}^{t} (a(s) − βγ ∫_{s−r(s)}^{s} |b(s, u)|du)ds → ∞ as t → ∞,  (21)

then the zero solution of Eq. (17) is asymptotically stable.

(iv) In addition to σ^{∗} < ∞, if we have

∫_{t_{0}}^{t} (a(s) − βγ ∫_{s−r(s)}^{s} |b(s, u)|du)ds → ∞

uniformly for t_{0} ∈ R^{+} as t → ∞,  (22)

then the zero solution of Eq. (17) is uniformly asymptotically stable.

This theorem can be easily proved by taking the set S in the proof of Theorem 1 for the above function q(t) and a function L = L(τ_{1}, τ_{2}) with

(|a(t)| + βγ ∫_{t−r(t)}^{t} |b(t, s)|ds)α ≤ L for τ_{1} ≤ t ≤ τ_{2},

and by defining a mapping P on S by (P ϕ)(t) := ψ(t − t_{0}) for t_{0} − r_{0} ≤ t ≤ t_{0}, and

(P ϕ)(t) := ψ(0)e^{− ∫_{t_{0}}^{t} a(s)ds} + ∫_{t_{0}}^{t} e^{− ∫_{s}^{t} a(u)du} ∫_{s−r(s)}^{s} b(s, u)g(ϕ(u))duds

for t > t_{0}, where ϕ ∈ S. So we omit the details of the proof.

Now we show an example.

Example 3. Let a : R^{+} → R be the function defined in Example 2, and let r(t) ≡ r (r > 0), b(t, s) ≡ B (0 < B ≤ 53/(26r), (5−2Br)r ≤ ln 2), and β = 1. Then (2), (4), (18) and (19) hold with β = 1, r_{0} = r and γ = 2, and σ^{∗} < ∞. Moreover, if 0 < B < 53/(26r), then (22) holds. Thus, by Theorem 2, the zero solution of Eq. (17) is uniformly stable. Moreover, if 0 < B < 53/(26r), then the zero solution of Eq. (17) is uniformly asymptotically stable.

In Burton-Furumochi [1], under the assumption

there is an η < 1 with ∫_{0}^{t} e^{− ∫_{s}^{t} a(u)du} ∫_{s−r(s)}^{s} |b(s, u)|duds ≤ η,  (23)

the asymptotic stability of the zero solution of Eq. (17) is discussed by using the contraction principle. But the functions a(t) and b(t, s) ≡ B in Example 3 do not satisfy (23) if Br = 2.

3. BOUNDEDNESS IN HALF-LINEAR EQUATIONS

First we discuss the boundedness of solutions of Eq. (1). In order to do so, we replace Assumption (2) by

|g(x)| ≤ β|x| for x ∈ R. (2^{∗})

Then, concerning the boundedness of the solutions of Eq. (1), we have the following theorem.

Theorem 3. Suppose that the solutions of Eq. (1) are uniquely determined by continuous initial functions, and that (2^{∗}) and (3)-(5) hold. Then we have:

(i) The solutions of Eq. (1) are equi-bounded.

(ii) If σ^{∗} < ∞, then the solutions of Eq. (1) are uniformly bounded.

(iii) If we have (10), then the solutions of Eq. (1) are equi-ultimately bounded for any bound A > 0.

(iv) If σ^{∗} < ∞, and if we have (11), then the solutions of Eq. (1) are uniformly ultimately bounded for any bound A > 0.

Proof. (i) Assumption (5) implies that the solutions of Eq. (7) are equibounded. Thus, for any α > 0 and t_{0} ∈ R^{+}, there is an A = A(α, t_{0}) > 0 such that for any q_{0} with |q_{0}| ≤ α, we have

|q(t, t_{0}, q_{0})| < A for all t ≥ t_{0}.

For the t_{0}, let (C_{t_{0}}, ∥ · ∥_{h}) be the Banach space as in the proof of Theorem 1(i). For a continuous function ψ : [−r_{0}, 0] → R with sup{|ψ(θ)| : −r_{0} ≤ θ ≤ 0} ≤ α, let S be a set of continuous functions ϕ : [t_{0} − r_{0}, ∞) → R such that

ϕ(t) = ψ(t − t_{0}) for t_{0} − r_{0} ≤ t ≤ t_{0},
|ϕ(t)| ≤ q(t) for t ≥ t_{0},

and

|ϕ(t_{1}) − ϕ(t_{2})| ≤ L|t_{1} − t_{2}| for t_{1}, t_{2} with t_{0} ≤ τ_{1} ≤ t_{1}, t_{2} ≤ τ_{2},

where q(t) is defined by (8) with η = α, and where L = L(τ_{1}, τ_{2}) is a function given in (12) with α = A. Then q(t) satisfies

0 < q(t) < A for all t ≥ t_{0}.

As in the proof of Theorem 1, S is a compact convex nonempty subset of C_{t_{0}}. Let P be the mapping defined in the proof of Theorem 1. Then P maps S into S continuously. Thus, by Schauder’s first theorem, P has a fixed point ϕ and that is the solution x(t, t_{0}, ψ) of Eq. (1) which satisfies

|x(t, t_{0}, ψ)| ≤ q(t) = q(t, t_{0}, α) < A, t ≥ t_{0},

and hence, the solutions of Eq. (1) are equi-bounded.

(ii)-(iv) Under the assumptions in (ii)-(iv), the solutions of Eq. (7) are uniformly bounded, equi-bounded and equi-ultimately bounded for any bound A > 0, and uniformly bounded and uniformly ultimately bounded for any bound A > 0, respectively. Thus, the uniform boundedness, the equi-ultimate boundedness for any bound A > 0 and the uniform ultimate boundedness for any bound A > 0 of the solutions of Eq. (1) can be similarly proved as in the proof of (i). So, we omit the details.

Now we revisit Examples 1 and 2.

Example 1^{∗}. Let α = ∞. Then, (2^{∗}) and (3)-(5) hold with β = 1, γ = 25 and r_{0} = 1, and σ^{∗} = ∞. Thus, concerning the boundedness of the solutions of Eq. (13), Theorem 3 does not assure uniform boundedness, but assures equiboundedness.

Example 2^{∗}. Let α = ∞. Then, (2^{∗}) and (3)-(5) hold with β = 1, γ = 2 and r_{0} = 1, and σ^{∗} < ∞. Moreover, if 0 < B < 53/26, then (11) holds. Thus, by Theorem 3, the solutions of Eq. (1) are uniformly bounded. Moreover, if 0 < B < 53/26, then the solutions of Eq. (1) are uniformly ultimately bounded for any bound A > 0.

Next we revisit the scalar integro-differential equation

x′(t) = −a(t)x(t) − ∫_{t−r(t)}^{t} b(t, s)g(x(s))ds, t ∈ R^{+}, (17)

where we assume (2^{∗}) instead of (2). Concerning the boundedness of the solutions of Eq. (17), we have the following theorem.

Theorem 4. Suppose that the solutions of Eq. (17) are uniquely determined by continuous initial functions, and that (2^{∗}), (4), (18) and (19) hold. Then we have:

(i) The solutions of Eq. (17) are equi-bounded.

(ii) If σ^{∗} < ∞, then the solutions of Eq. (17) are uniformly bounded.

(iii) If we have (21), then the solutions of Eq. (17) are equi-ultimately bounded for any bound A > 0.

(iv) If we have σ^{∗} < ∞ and (22), then the solutions of Eq. (17) are uniformly ultimately bounded for any bound A > 0.

Since this theorem can be proved by a similar method used in the proof of Theorem 3, we omit the proof.

Finally we revisit Example 3.

Example 3^{∗}. Let a : R^{+} → R be the function defined in Example 2, and let r(t) ≡ r (r > 0) and b(t, s) ≡ B (0 < B ≤ 53/(26r), (5 − 2Br)r ≤ ln 2), and β = 1. Then (2^{∗}), (4), (18) and (19) hold with β = 1, r_{0} = r and γ = 2, and σ^{∗} < ∞. Moreover, if 0 < B < 53/(26r), then (22) holds. Thus, by Theorem 4, the solutions of Eq. (17) are uniformly bounded. Moreover, if 0 < B < 53/(26r), then the solutions of Eq. (17) are uniformly ultimately bounded for any bound A > 0.

REFERENCES

[1] T. A. Burton and T. Furumochi, Fixed points and problems in stability theory for ordinary and functional differential equations, Dynamic Systems and Applications 10(2001), 89-116.

[2] T. A. Burton and T. Furumochi, A note on stability by Schauder’s theorem, Funkcialaj Ekvacioj 44(2001), 73-82.

[3] T. A. Burton and T. Furumochi, Krasnoselskii’s fixed point theorem and stability, Nonlinear Analysis 49(2002), 445-454.

[4] T. A. Burton and T. Furumochi, Asymptotic behavior of solutions of functional differential equations by fixed point theorems, Dynamic Systems and Applications 11(2002), 499-521.

[5] T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Academic Press, Orlando, FL, 1985.

[6] J. K. Hale, Theory of Functional Differential Equations, Springer, New York, 1977.

(Received August 12, 2003)