GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 1, 1999, 45-64

ON PERIODIC SOLUTIONS OF NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS

I. KIGURADZE AND B. PŮŽA

Abstract. Sufficient conditions are established for the existence and uniqueness of an ω-periodic solution of the functional differential equation

\frac{dx(t)}{dt} = f(x)(t),

where f is a continuous operator acting from the space of n-dimen-sional ω-periodic continuous vector functions into the space of ndimensional ω-periodic and summable on [0, ω] vector functions.

1. Statement of the Problem and Basic Notation

Let n be a natural number, ω > 0, and f : C_{ω}(R^{n}) → L_{ω}(R^{n}) be a continuous operator. Consider the vector functional differential equation

\frac{dx(t)}{dt} = f(x)(t).  (1.1)

A vector function x : R → R^{n} is called an ω-periodic solution of equation (1.1) if it is absolutely continuous, satisfies (1.1) almost everywhere on R and

x(t + ω) = x(t) for t ∈ R.

In the second section of this paper, using the principle of a priori boundedness we establish new sufficient conditions for the existence and uniqueness of an ω-periodic solution of equation (1.1). In the third section we give corollaries of the main results for the vector differential equation with deviating arguments

\frac{dx(t)}{dt} = f_{0}(t, x(t), x(τ_{1}(t)), …, x(τ_{m}(t))),  (1.2)

45 where f_{0} : R × R^{(m+1)n} → R^{n} satisfies the local Carathéodory conditions and is ω-periodic in the first argument, i.e., satisfies the equality

1991 Mathematics Subject Classification. 34C25, 34K15.

Key words and phrases. Functional differential equation, ω-periodic solution, principle of a priori boundedness.

1072-947X/99/0100-0045$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1999 Plenum Publishing Corporation

f_{0}(t + ω, x_{0}, x_{1}, …, x_{m}) = f_{0}(t, x_{0}, x_{1}, …, x_{m})  (1.3)

for almost all t ∈ R and for all x_{k} ∈ R^{n} (k = 0, 1, …, m). As for τ_{k} : R → R (k = 1, …, m), they are measurable and such that

(τ_{k}(t + ω) − τ_{k}(t))/ω (k = 1, …, m) are integer numbers. (1.4)

The above-mentioned propositions strengthen the earlier results on periodic solutions of systems of ordinary differential equations and functional differential equations of types (1.1) and (1.2) (see [1–23] and the references cited therein).

Throughout this paper, use will be made of the following notation: R^{n} is the space of all n-dimensional column vectors x = (x_{i})_{i=1}^{n} with the

elements x_{i} ∈ R (i = 1, …, n) and the norm

∥x∥ = ∑_{i=1}^{n} |x_{i}|.

R^{n×n} is the space of all n × n-matrices X = (x_{ik})_{i,k=1}^{n} with the elements x_{ik} ∈ R (i, k = 1, …, n) and the norm

∥X∥ = ∑_{i,k=1}^{n} |x_{ik}|.

R_{+}^{n} = {(x_{i})_{i=1}^{n} ∈ R^{n} : x_{i} ≥ 0 (i = 1, …, n)}.
R_{+}^{n×n} = {(x_{ik})_{i,k=1}^{n} ∈ R^{n×n} : x_{ik} ≥ 0 (i, k = 1, …, n)}.

If x, y ∈ R^{n} and X, Y ∈ R^{n×n}, then

x ≤ y ⟺ y − x ∈ R_{+}^{n}, X ≤ Y ⟺ Y − X ∈ R_{+}^{n×n}.

x · y is the scalar product of the vectors x and y ∈ R^{n}. If x = (x_{i})_{i=1}^{n} ∈ R^{n} and X = (x_{ik})_{i,k=1}^{n} ∈ R^{n×n}, then

|x| = (|x_{i}|)_{i=1}^{n}, |X| = (|x_{ik}|)_{i,k=1}^{n},
sgn(x) = (sgn x_{i})_{i=1}^{n}.

det(X) is the determinant of the matrix X. X^{−1} is the matrix inverse to X. r(X) is the spectral radius of the matrix X. E is the unit matrix. C([0, ω]; R^{n}) is the space of all continuous vector functions x : [0, ω] → R^{n}

with the norm

∥x∥_{C} = max {∥x(t)∥ : 0 ≤ t ≤ ω}.

C_{ω}(R^{n}) with ω > 0 is the space of all continuous ω-periodic vector functions x : R → R^{n} with the norm

∥x∥_{Cω} = max {∥x(t)∥ : 0 ≤ t ≤ ω};

if x = (x_{i})_{i=1}^{n} ∈ C_{ω}(R^{n}), then

|x|_{Cω} = (∥x_{i}∥_{Cω})_{i=1}^{n}.

L([0, ω]; R^{n}) is the space of all vector functions x : R → R^{n} with summable on [0, ω] elements and with the norm

∥x∥_{L} = ∫_{0}^{ω} ∥x(t)∥ dt.

L_{ω}(R^{n}) is the space of all ω-periodic vector functions x : R → R^{n} with summable on [0, ω] elements and with the norm

∥x∥_{Lω} = ∫_{0}^{ω} ∥x(t)∥ dt.

L_{ω}(R_{+}) = {x ∈ L_{ω}(R) : x(t) ≥ 0 for t ∈ R};
L_{ω}(R_{−}) = {x ∈ L_{ω}(R) : x(t) ≤ 0 for t ∈ R}.

L_{ω}(R^{n×n}) is the space of all matrix functions X : R → R^{n×n} with elements from L_{ω}(R).

If Z : R → R^{n×n} is an ω-periodic continuous matrix function with columns z_{1}, …, z_{n}, and g : C_{ω}(R^{n}) → L_{ω}(R^{n}) is a linear operator, then by g(Z) we understand the matrix function with columns g(z_{1}), …, g(z_{n}).

2. Periodic Solutions of Equation (1.1)

Throughout this section, f : C_{ω}(R^{n}) → L_{ω}(R^{n}) is assumed to be a continuous operator such that

f^{∗}(·, ρ) ∈ L_{ω}(R_{+}) for ρ ∈]0, +∞[ ,

where

f^{∗}(t, ρ) = sup {∥f(x)(t)∥ : x ∈ C_{ω}(R^{n}), ∥x∥_{Cω} ≤ ρ}.

We introduce

Definition 2.1. Let β be a positive number. We say that an operator p : C_{ω}(R^{n})×C_{ω}(R^{n}) → L_{ω}(R^{n}) belongs to the class V_{ω}^{n}(β) if it is continuous and satisfies the following three conditions:

(i) p(x, ·) : C_{ω}(R^{n}) → L_{ω}(R^{n}) is a linear operator for any arbitrarily fixed x ∈ C_{ω}(R^{n});

(ii) there exists a nondecreasing in the second argument function α : R × R_{+} → R_{+} such that α(·, ρ) ∈ L_{ω}(R) for ρ ∈]0, +∞[ , and for any x, y ∈ C_{ω}(R) and for almost all t ∈ R the inequality

∥p(x, y)(t)∥ ≤ α(t, ∥x∥_{Cω})∥y∥_{Cω}

holds;

(iii) for any x ∈ C_{ω}(R^{n}) and q ∈ L_{ω}(R^{n}), an arbitrary ω-periodic solution y of the differential equation

\frac{dy(t)}{dt} = p(x, y)(t) + q(t)  (2.1)

admits the estimate

∥y∥_{Cω} ≤ β∥q∥_{Lω} .  (2.2)

Definition 2.2. We say that an operator p : C_{ω}(R^{n}) × C_{ω}(R^{n}) → L_{ω}(R^{n}) belongs to the set V_{ω}^{n} if there exists β > 0 such that p ∈ V_{ω}^{n}(β).

Theorem 2.1. Let there exist a positive number ρ_{0} and an operator p ∈ V_{ω}^{n} such that for any λ ∈]0, 1[ an arbitrary ω-periodic solution of the differential equation

\frac{dx(t)}{dt} = (1 − λ)p(x, x)(t) + λf(x)(t)  (2.3)

admits the estimate

∥x∥_{Cω} ≤ ρ_{0}.  (2.4)

Then equation (1.1) has at least one ω-periodic solution.

Proof. For arbitrary x ∈ C([0, ω]; R^{n}), we denote by v_{ω}(x) the vector function defined by the equality

v_{ω}(x)(t) = x(t − jω) + \frac{t − jω}{ω} [x(0) − x(ω)] for jω ≤ t < (j + 1)ω (2.5)
(j = 0, 1, −1, 2, −2, …),

and for any x and y ∈ C([0, ω]; R^{n}) we set

p̃(x, y)(t) = p(v_{ω}(x), v_{ω}(y))(t), l(x, y) = y(ω) − y(0),  (2.6)
f̃(x)(t) = f(v_{ω}(x))(t).

Obviously, v_{ω} : C([0, ω]; R^{n}) → C_{ω}(R^{n}) is a linear bounded operator, while f̃ : C([0, ω]; R^{n}) → L([0, ω]; R^{n}) and p̃ : C([0, ω]; R^{n}) × C([0, ω]; R^{n}) → L([0, ω]; R^{n}) are continuous operators. Moreover, the restrictions on [0, ω] of ω-periodic solutions of equations (1.1) and (2.3) are respectively solutions of the differential equations

\frac{dx(t)}{dt} = f̃(x)(t)  (2.7)

and

\frac{dx(t)}{dt} = (1 − λ)p̃(x, x)(t) + λf̃(x)(t)  (2.8)

with the periodic boundary condition

x(ω) = x(0),  (2.9)

and vice versa the periodic extension of an arbitrary solution of problem (2.7), (2.9) (problem (2.8), (2.9)) is an ω-periodic solution of equation (1.1) (equation (2.3)). Consequently, for any λ ∈]0, 1[ , an arbitrary solution of problem (2.8), (2.9) admits estimate (2.4).

On the other hand, it follows from the condition p ∈ V_{ω}^{n} and equalities (2.6) that the pair of operators (p̃, l) is compatible in the sense of Definition 1 from [14].

Thus we have shown that for problem (2.7), (2.9), all the conditions of Theorem 1 from [14] are fulfilled, which guarantees the solvability of this problem. However, according to the above-said, the existence of an ωperiodic solution of equation (1.1) follows from the solvability of problem (2.7), (2.9).

Corollary 2.1. Let there exist β > 0 and p ∈ V_{ω}^{n}(β) such that for any x ∈ C_{ω}(R^{n}) almost everywhere on R the inequality

∥f(x)(t) − p(x, x)(t)∥ ≤ γ(t, ∥x∥_{Cω})  (2.10)

holds, where γ(·, ρ) ∈ L_{ω}(R_{+}) for 0 < ρ < +∞, and

lim sup_{ρ→+∞} \frac{1}{ρ} ∫_{a}^{b} γ(t, ρ) dt < \frac{1}{β} .  (2.11)

Then equation (1.1) has at least one ω-periodic solution.

Proof. By (2.11) there exists ρ_{0} > 0 such that

β ∫_{0}^{ω} γ(t, ρ) dt < 1 for ρ ≥ ρ_{0}.  (2.12)

Let x be an ω-periodic solution of (2.3) for some λ ∈]0, 1[ . Assume

δ(t) = f(x)(t) − p(x, x)(t).

Then

\frac{dx(t)}{dt} = p(x, x)(t) + λδ(t)

and, as follows from (2.10), the vector function δ satisfies almost everywhere on R the inequality

∥δ(t)∥ ≤ γ(t, ∥x∥_{Cω}),

whence, owing to p ∈ V_{ω}^{n}(β), we have

∥x∥_{Cω} ≤ β ∫_{0}^{ω} ∥δ(t)∥ dt ≤ β ∫_{0}^{ω} γ(t, ∥x∥_{Cω}) dt.

From this inequality, by virtue of (2.12), follows estimate (2.4).

If now we take into account Theorem 2.1 the validity of the corollary will become obvious.

Corollary 2.2. Let for any x ∈ C_{ω}(R^{n}), inequality (2.10) be fulfilled almost everywhere on R, where γ(·, ρ) ∈ L_{ω}(R_{+}) for 0 < ρ < +∞, and p : C_{ω}(R^{n}) × C_{ω}(R^{n}) → L_{ω}(R^{n}) is a continuous operator such that p(x, ·) :

C_{ω}(R^{n}) → L_{ω}(R^{n}) is linear and ∫_{0}^{ω} p(x, E)(s) ds is a nonsingular matrix for

an arbitrarily fixed x ∈ C_{ω}(R^{n}). Let, moreover, there exist matrices A and B ∈ R_{+}^{n×n} such that

r(A + BA^{2}) < 1,  (2.13)
∫_{0}^{ω} |p(x, y)(s)| ds ≤ A|y|_{Cω} , |[ ∫_{0}^{ω} p(x, E)(s) ds]^{−1}| ≤ B (2.14)

for any x and y ∈ C_{ω}(R^{n}) and the function γ satisfies condition (2.11), where β = ∥(E − A − BA^{2})^{−1}(E + BA)∥. Then equation (1.1) has at least one ω-periodic solution.

Proof. By virtue of Corollary 2.1, to prove Corollary 2.2 it suffices to establish that for any x ∈ C_{ω}(R^{n}) and q ∈ L_{ω}(R^{n}) an arbitrary ω-periodic solution y of equation (2.1) admits estimate (2.2).

By (2.5), from (2.1) we have

y(t) = y(0) + p^{1}(x, y)(t) + q_{0}(t),  (2.15)

where

p^{1}(x, y)(t) = ∫_{0}^{t} p(x, v_{ω}(y))(s) ds, q_{0}(t) = ∫_{0}^{t} q(s) ds.  (2.16)

Therefore

y(t) = y(0) + p^{1}(x, E)(t)y(0) + p^{1}(x, p^{1}(x, y))(t) + p^{1}(x, q_{0})(t),

whence because of the ω-periodicity of y and the nonsingularity of the matrix

p^{1}(x, E)(ω) = ∫_{0}^{ω} p(x, E)(s) ds

we obtain

y(0) = −[ ∫_{0}^{ω} p(x, E)(s) ds]^{−1}[p^{1}(x, p^{1}(x, y))(ω) + p^{1}(x, q_{0})(ω)].

According to (2.14) and (2.16), the latter representation results in

|y(0)| ≤ B(A|v_{ω}(p^{1}(x, y))|_{Cω} + A|v_{ω}(q_{0})|_{Cω} ) ≤
≤ B(A|p^{1}(x, y)|_{Cω} + A|q_{0}|_{Cω} ) ≤ BA^{2}|y|_{Cω} + BA|q|_{Lω} .

Taking this estimate into account, from (2.15) we find that

|y|_{Cω} ≤ BA^{2}|y|_{Cω} + BA|q|_{Lω} + A|y|_{Cω} + |q|_{Lω}

and

(E − A − BA^{2})|y|_{Cω} ≤ (E + BA)|q|_{Lω} .

Hence by (2.13) we have

|y|_{Cω} ≤ (E − A − BA^{2})^{−1}(E + BA).

Thus estimate (2.2) is valid.

Corollary 2.2 deals with the case where

sup \{\frac{1}{1 + ∥x∥_{Cω}} ∫_{0}^{ω} ∥f(x)(t)∥ dt : x ∈ C_{ω}(R^{n})\} < +∞,

whereas Corollary 2.1 covers the class of equations of type (1.1) for which the last condition is violated. As an example, consider the integro-differential equation

\frac{dx(t)}{dt} = p_{1}(t, x(t)) ∫_{0}^{ω} p_{2}(s, x(s))x(s) dφ(s) + p_{0}(t, x(t)), (2.17)

where the functions p_{i} : R×R → R (i = 0, 1) satisfy the local Carathéodory conditions and are ω-periodic in the first argument, p_{2} : [0, ω] × R → R is continuous and φ : [0, ω] → R is nondecreasing.

Corollary 2.3. Let on [0, ω] × R the inequalities

σ_{i}p_{i}(t, x) ≥ δ_{i}(t) (i = 1, 2)  (2.18)

be fulfilled, where σ_{i} ∈ {−1, 1} (i = 1, 2), δ_{1} : [0, ω] → R_{+} is a summable function and δ_{2} : [0, ω] → R_{+} is a continuous function such that

δ = ( ∫_{0}^{ω} δ_{1}(s) ds)( ∫_{0}^{ω} δ_{2}(s) dφ(s)) > 0.  (2.19)

Let further

lim sup_{ρ→+∞} (\frac{1}{ρ} ∫_{0}^{ω} γ(t, ρ) dt) < \frac{δ}{1 + 3δ} ,

where

γ(t, ρ) = max {|p_{0}(t, x)| : |x| ≤ ρ}.

Then equation (2.17) has at least one ω-periodic solution.

Proof. Suppose

p(x, y)(t) = p_{1}(t, x(t)) ∫_{0}^{ω} p_{2}(s, x(s))y(s) dφ(s).

By virtue of Corollary 2.1, to prove Corollary 2.3 it suffices to establish that p ∈ V_{ω}^{1}(\frac{1+3δ}{δ}).

Let x ∈ C_{ω}(R), q ∈ L_{ω}(R) and let y be an arbitrary ω-periodic solution of the equation

\frac{dy(t)}{dt} = p_{1}(t, x(t)) ∫_{0}^{ω} p_{2}(s, x(s))y(s) dφ(s) + q(t).

Suppose

∫_{0}^{ω} p_{2}(s, x(s))y(s) dφ(s) = c.  (2.20)

Then

y(t) = y(0) + c ∫_{0}^{t} p_{1}(s, x(s)) ds + ∫_{0}^{t} q(s) ds.  (2.21)

Therefore

c ∫_{0}^{ω} p_{1}(s, x(s)) ds + ∫_{0}^{ω} q(s) ds = 0

and consequently

c = −( ∫_{0}^{ω} p_{1}(s, x(s)) ds)^{−1}( ∫_{0}^{ω} q(s) ds).  (2.22)

If we substitute (2.21) in (2.20) and calculate y(0), then we obtain

y(0) =

= c( ∫_{0}^{ω} p_{2}(s, x(s)) dφ(s))^{−1}[1 −∫_{0}^{ω} p_{2}(s, x(s))( ∫_{0}^{s} p_{1}(ξ, x(ξ)) dξ) dφ(s)] −

−( ∫_{0}^{ω} p_{2}(s, x(s)) dφ(s))^{−1} ∫_{0}^{ω} p_{2}(s, x(s))( ∫_{0}^{s} q(ξ) dξ) dφ(s). (2.23)

Introduce the function

η(t) = ( ∫_{0}^{ω} p_{1}(s, x(s)) ds)^{−1} ∫_{0}^{t} p_{1}(s, x(s)) ds.  (2.24)

Then from (2.21) and (2.22) we get

y(t) = y(0) + ∫_{0}^{t}(1 − η(t))q(s) ds − η(t) ∫_{t}^{ω} q(s) ds.

On the other hand, taking into account (2.18) and (2.19), from (2.22)–(2.24) we find

|y(0)| ≤ (\frac{1}{δ} + 2)∥q∥_{Lω} , 0 ≤ η(t) ≤ 1 for 0 ≤ t ≤ ω.

Therefore

∥y∥_{Lω} ≤ |y(0)| + ∥q∥_{Lω} ≤ \frac{1 + 3δ}{δ} ∥q∥_{Lω} ,

which because of the arbitrariness of x ∈ C_{ω}(R) and q ∈ L_{ω}(R) results in p ∈ V_{ω}^{1}(\frac{1+3δ}{δ}).

Theorem 2.2. Let for any x ∈ C_{ω}(R^{n}) almost everywhere on R the inequality

f(x)(t) · sgn(σx(t)) ≤ p_{0}(t)∥x(t)∥ + γ(t, ∥x∥_{Cω})  (2.25)

be fulfilled, where σ ∈ {−1, 1}, p_{0} ∈ L_{ω}(R), γ(·, ρ) ∈ L_{ω}(R_{+}) for 0 < ρ < +∞,

∫_{0}^{ω} p_{0}(s) ds < 0  (2.26)

and

lim sup_{ρ→+∞} \frac{1}{ρ}( ∫_{t}^{t+ω}exp (σ ∫_{s}^{t} p_{0}(ξ) dξ)γ(s, ρ) ds) <

< | exp (−σ∫_{0}^{ω}p_{0}(ξ) dξ) − 1| uniformly with respect to t ∈ [0, ω]. (2.27)

Then equation (1.1) has at least one ω-periodic solution.

To prove this theorem, it is necessary to establish an a priori estimate of nonnegative ω-periodic solutions of the differential inequality

σu′(t) ≤ p_{0}(t)u(t) + γ(t, ∥u∥_{Cω}),  (2.28)

where σ ∈ {−1, 1}, p_{0} ∈ L_{ω}(R), and γ(·, ρ) ∈ L_{ω}(R_{+}) for 0 < ρ < +∞.

Note that by an ω-periodic solution of inequality (2.28) we mean an absolutely continuous ω-periodic function u : R → R satisfying this inequality almost everywhere on R.

Lemma 2.1. Let inequality (2.26) be fulfilled, and let there exist a nonnegative constant ρ_{0} such that

∫_{t}^{t+ω}exp (σ ∫_{s}^{t} p_{0}(ξ) dξ)γ(s, ρ) ds <

< | exp ( − σ ∫_{0}^{ω} p_{0}(s) ds) − 1|ρ for 0 ≤ t ≤ ω, ρ > ρ_{0}. (2.29)

Then an arbitrary nonnegative ω-periodic solution u of (2.28) admits the estimate

∥u∥_{Cω} ≤ ρ_{0}.  (2.30)

Proof. Let u be an arbitrary ω-periodic solution of the differential inequality (2.28). Suppose

q(t) = u′(t) − σp_{0}(t)u(t).

Then by Theorem 6.4 from [11] we find

u(t) = [ exp ( −σ∫_{0}^{ω}p_{0}(s) ds) − 1]^{−1} ∫_{t}^{t+ω}exp (σ∫_{s}^{t}p_{0}(ξ) dξ)q(s) ds. (2.31)

On the other hand, owing to (2.28), the inequality

σq(t) ≤ γ(t, ∥x∥_{Cω})

holds almost everywhere on R. If along with this inequality we take into consideration inequality (2.26), then from (2.31) we obtain

u(t) ≤ | exp ( − σ ∫_{0}^{ω} p_{0}(s) ds) − 1|^{−1} ×

× ∫_{t}^{t+ω}exp (σ ∫_{s}^{t} p_{0}(ξ) dξ)γ(s, ∥x∥_{Cω}) ds for 0 ≤ t ≤ ω. (2.32)

Suppose now that estimate (2.30) is not valid. Then there exists a point t_{0} ∈ [0, ω] such that

∥u∥_{Cω} = u(t_{0}) > ρ_{0}.

Taking into account this fact and condition (2.29), from (2.32) we obtain the contradiction

∥u∥_{Cω} < ∥u∥_{Cω} ,

which proves the lemma.

Proof of Theorem 2.2. First of all it should be noted that by condition (2.27), there exists a positive number ρ_{0} such that inequality (2.29) is fulfilled.

For any x and y ∈ C_{ω}(R^{n}) suppose

p(x, y)(t) = σp_{0}(t)y(t).

Then by Theorem 6.4 from [11], inequality (2.26) guarantees the condition

p ∈ V_{ω}^{n}.

According to this fact and Theorem 2.1, to prove Theorem 2.2 it suffices to establish that for any λ ∈]0, 1[ an arbitrary ω-periodic solution of the differential equation

\frac{dx(t)}{dt} = (1 − λ)σp_{0}(t)x(t) + λf(x)(t)  (2.33)

admits estimate (2.4).

Indeed, let x be such a solution. Suppose

u(t) = ∥x(t)∥.

Then by (2.25) from (2.33) we find

σu′(t) = x′(t) · sgn(σx(t)) =
= (1 − λ)p_{0}(t)∥x(t)∥ + λf(x)(t) · sgn(σx(t)) ≤
≤ p_{0}(t)∥x(t)∥ + γ(t, ∥x∥_{Cω}) = p_{0}(t)u(t) + γ(t, ∥u∥_{Cω}).

Consequently u is a nonnegative ω-periodic solution of the differential inequality (2.28). This function by Lemma 2.1 admits estimate (2.30). Therefore x admits estimate (2.4).

Theorem 2.3. Let for any x and y ∈ C_{ω}(R^{n}), almost everywhere on R the condition

[f(x)(t) − f(y)(t)] · sgn [σ(x(t) − y(t))] ≤
≤ p_{0}(t)∥x(t) − y(t)∥ + γ_{0}(t)∥x − y∥_{Cω}  (2.34)

be fulfilled, where σ ∈ {−1, 1}, the function p_{0} ∈ L_{ω}(R) satisfies inequality (2.26) and the function γ_{0} ∈ L_{ω}(R_{+}) satisfies the inequality

∫_{t}^{t+ω}exp (σ ∫_{s}^{t} p_{0}(ξ) dξ)γ_{0}(s) ds <

< | exp ( − σ ∫_{0}^{ω} p_{0}(ξ) dξ) − 1| for 0 ≤ t ≤ ω.  (2.35)

Then equation (1.1) has one and only one ω-periodic solution.

Proof. From (2.34) and (2.35) we arrive at conditions (2.25) and (2.27), where

γ(t, ρ) = γ_{0}(t)ρ + ∥f(t, 0, …, 0)∥.

Therefore by Theorem 2.2, equation (1.1) has at least one ω-periodic solution x.

To complete the proof of the theorem, it remains to show that an arbitrary ω-periodic solution y of equation (1.1) coincides with x. Suppose

u(t) = ∥x(t) − y(t)∥.

Then by (2.34), u is a nonnegative ω-periodic solution of the differential inequality (2.28), where

γ(t, ρ) = γ_{0}(t)ρ.

On the other hand, owing to (2.35), the function γ satisfies inequality (2.29), where ρ_{0} = 0, whence by Lemma 2.1 it follows that u(t) ≡ 0. Consequently x(t) ≡ y(t).

Example 2.1. Consider the integro-differential equation

\frac{dx(t)}{dt} = σp_{0}(t)x(t) + σ_{0}[|p_{0}(t)| ∫_{−∞}^{+∞}p_{1}(s)|x(s)| ds + p_{2}(t)], (2.36)

where σ and σ_{0} ∈ {−1, 1}, p_{1} : R → R_{+} is a summable function, while p_{0} ∈ L_{ω}(R_{−}) and p_{2} ∈ L_{ω}(R_{+}) are functions different from zero on a set of positive measure. Because of the restrictions imposed on p_{0} and p_{2},

∫_{t}^{t+ω}|g(t, s)| |p_{0}(s)| ds = 1  (2.37)

and

∫_{t}^{t+ω}|g(t, s)|p_{2}(s) ds ≥ δ,  (2.38)

where

g(t, s) = [ exp ( − σ ∫_{0}^{ω} p_{0}(ξ) dξ) − 1]^{−1} exp (σ ∫_{s}^{t} p_{0}(ξ) dξ),

and δ is a positive constant. On the other hand, the operator

f(x)(t) = σp_{0}(t)x(t) + σ_{0}[|p_{0}(t)| ∫_{−∞}^{+∞}p_{1}(s)|x(s)| ds + p_{2}(t)]

satisfies condition (2.34), where

γ_{0}(t) = |p_{0}(t)| ∫_{−∞}^{+∞}p_{1}(s) ds.

This and equality (2.37) imply that if

∫_{−∞}^{+∞}p_{1}(s) ds < 1,  (2.39)

then inequality (2.35) is fulfilled. In that case, by Theorem 2.3 equation (2.36) has a unique ω-periodic solution.

Show that if

∫_{−∞}^{+∞}p_{1}(s) ds ≥ 1,  (2.40)

then equation (2.36) has no ω-periodic solution. Assume to the contrary that it has a solution x. Then

x(t) = σ_{0} ∫_{t}^{t+ω}g(t, s)[|p_{0}(s)| ∫_{−∞}^{+∞}p_{1}(ξ)|x(ξ)| dξ + p_{2}(s)] ds.

Hence, taking into account (2.37), (2.38) and (2.40), we find

μ ≥ μ ∫_{−∞}^{+∞}p_{1}(ξ) dξ + δ ≥ μ + δ,

where μ = min{|x(t)| : t ∈ R}. The obtained contradiction shows that equation (2.36) has no ω-periodic solution when condition (2.40) is fulfilled.

The example under consideration shows that in Theorem 2.2 (Theorem 2.3) it is impossible to replace the strict inequality (2.27) (the strict inequality (2.35)) by the nonstrict one.

Theorem 2.4. Let there exist ρ_{0} ∈]0, +∞[ , δ ∈]0, 1[ and σ ∈ {−1, 1} such that for any x ∈ C_{ω}(R^{n}) satisfying

∥x∥_{Cω} > ρ_{0},  (2.41)

almost everywhere on the set

{t ∈ R : ∥x(t)∥ > (1 − δ)∥x∥_{Cω} }  (2.42)

the inequality

f(x)(t) · sgn(σx(t)) ≤ 0  (2.43)

is fulfilled. Then equation (1.1) has at least one ω-periodic solution.

Proof. By Theorem 2.1, to prove Theorem 2.4 it suffices to establish that for any λ ∈]0, 1[ , an arbitrary ω-periodic solution of the differential equation

\frac{dx(t)}{dt} = −σ(1 − λ)x(t) + λf(x)(t)  (2.44)

admits estimate (2.4).

Assume to the contrary that for some λ ∈]0, 1[ equation (2.44) has an ω-periodic solution x satisfying (2.41). Then there exist t_{0} ∈]0, +∞[ , t_{∗} ∈ ]0, t_{0}[ and t^{∗} ∈]t_{0}, +∞[ such that

∥x(t_{0})∥ = ∥x∥_{Cω} , ∥x(t)∥ > (1 − δ)∥x∥_{Cω} for t_{∗} ≤ t ≤ t^{∗}.

Hence it is clear that [t_{∗}, t^{∗}] is included in (2.42). Therefore inequality (2.43) is fulfilled almost everywhere on [t_{∗}, t^{∗}]. Consequently

σ \frac{d∥x(t)∥}{dt} = −(1 − λ)∥x(t)∥ + λf(x)(t) · sgn(σx(t)) < 0

for almost all t ∈ [t_{∗}, t^{∗}].

Hence for σ = 1 (σ = −1) it follows that

∥x(t_{∗})∥ > ∥x(t_{0})∥ ( ∥x(t^{∗})∥ > ∥x(t_{0})∥).

But this is impossible because ∥x(t_{0})∥ = ∥x∥_{Cω}. The obtained contradiction proves the theorem.

As an example, consider the nonlinear differential system

\frac{dx_{i}(t)}{dt} = −σl_{0}(t)|x_{i}(t)|^{λ} sgn x_{i}(t) + σf_{1i}(t, x_{1}(t), …, x_{n}(t)) +
+ f_{2i}(t, l_{1}(x_{1})(t), …, l_{n}(x_{n})(t)) (i = 1, …, n),  (2.45)

where σ ∈ {−1, 1}, λ ∈]0, +∞[ , l_{0} ∈ L_{ω}(R_{+}), f_{1i} and f_{2i} : R × R^{n} → R (i = 1, …, n) are ω-periodic in the first argument functions satisfying the local Carathéodory conditions, and l_{i} : C_{ω}(R) → C_{ω}(R) (i = 1, …, n) are linear bounded operators with norms ∥l_{1}∥, …, ∥l_{n}∥.

Set

μ(λ) = \begin{cases}λ − 1 & for λ > 1 \\ 0 & for λ ≤ 1 \end{cases} .

Theorem 2.4 implies

Corollary 2.4. Let on R × R^{n} the inequalities

∑_{i=1}^{n} f_{1i}(t, x_{1}, …, x_{n}) sgn x_{i} ≤ l_{0}(t)[η_{1}( ∑_{i=1}^{n} |x_{i}|)^{λ} + η_{0}],  (2.46)

∑_{i=1}^{n} |f_{2i}(t, x_{1}, …, x_{n})| ≤ l_{0}(t)[η_{2}( ∑_{i=1}^{n} |x_{i}|)^{λ} + η_{0}]  (2.47)

be fulfilled, where η_{i} (i = 0, 1, 2) are positive constants such that

η_{1} + ∥l_{i}∥^{λ}η_{2} < n^{−μ(λ)} (i = 1, …, n).  (2.48)

Then system (2.45) has at least one ω-periodic solution.

Proof. If we assume x(t) = (x_{i}(t))_{i=1}^{n},

f_{i}(x)(t) = −σl_{0}(t)|x_{i}(t)|^{λ} sgn x_{i}(t) + σf_{1i}(t, x_{1}(t), …, x_{n}(t)) +
+ f_{2i}(t, l_{1}(x_{1})(t), …, l_{n}(x_{n})(t)) (i = 1, …, n)

and

f(x)(t) = (f_{i}(x)(t))_{i=1}^{n},

then system (2.45) will take form (1.1). On the other hand, taking into account (2.46) and (2.47), the operator f satisfies the condition

f(x)(t) · sgn(σx(t)) ≤

≤ −l_{0}(t) ∑_{i=1}^{n} |x_{i}(t)|^{λ} + l_{0}(t)[(η_{1} + ∥l_{i_{0}}∥^{λ}η_{2})∥x∥λ_{Cω} + 2η_{0}],

where ∥l_{i_{0}}∥ = max{∥l_{1}∥, …, ∥l_{n}∥}. Hence in view of the inequality

∥x∥^{λ} ≤ n^{μ(λ)} ∑_{i=1}^{n} |x_{i}|^{λ}

we obtain

f(x)(t) · sgn(σx(t)) ≤
≤ −l_{0}(t)[n^{−μ(λ)}∥x(t)∥^{λ} − (η_{1} + ∥l_{i_{0}}∥^{λ}η_{2})∥x∥λ_{Cω} − 2η_{0}]. (2.49)

By virtue of (2.48), there exists δ ∈]0, 1[ such that

ε = (1 − δ)^{λ}n^{−μ(λ)} − η_{1} − ∥l_{i_{0}}∥^{λ}η_{2} > 0.

Set

ρ_{0} = (2η_{0}/ε)^{\frac{1}{λ}} .

Let x ∈ C_{ω}(R^{n}) be an arbitrary vector function satisfying (2.41). Then by (2.49) inequality (2.43) holds almost everywhere on set (2.42). Therefore all the conditions of Theorem 2.4 are fulfilled, which guarantees the existence of at least one ω-periodic solution of (2.45).

3. Periodic Solutions of Equation (1.2)

Throughout this section f_{0} : R×R^{(m+1)n} → R^{n} is assumed to be a vector function satisfying the local Carathéodory conditions and also condition (1.3), while τ_{k} : R → R (k = 1, …, m) are assumed to be measurable functions satisfying condition (1.4).

For any x ∈ C_{ω}(R^{n}) we assume that

f(x)(t) = f_{0}(t, x(t), x(τ_{1}(t)), …, x(τ_{m}(t))).

Then the operator f : C_{ω}(R^{n}) → L_{ω}(R^{n}) is continuous. Therefore from Corollary 2.2 and Theorems 2.2–2.4 we obtain the following propositions.

Corollary 3.1. Let the inequality

∥f_{0}(t, x_{0}, x_{1}, …, x_{m}) − ∑_{k=0}^{m} P_{k}(t, x_{0}, x_{1}, …, x_{m})x_{k}∥ ≤

≤ γ(t, ∥x_{0}∥, ∥x_{1}∥, …, ∥x_{m}∥)

be fulfilled on R × R^{(m+1)n}, where P_{k} : R × R^{(m+1)n} → R^{n×n} (k = 0, 1, …, m) are ω-periodic in the first argument matrix functions satisfying the local Carathéodory conditions, and γ : R×R_{+}^{m+1} → R_{+} is nondecreasing in the last n arguments and ω-periodic in the first argument. Let, moreover, there exist matrices A and B ∈ R_{+}^{n×n} such that

r(A + BA^{2}) < 1,
lim sup_{ρ→+∞} \frac{1}{ρ} ∫_{0}^{ω} γ(s, ρ, …, ρ) ds < ∥(E − A − BA^{2})^{−1}(E + BA)∥

and for any x ∈ C_{ω}(R^{n}) the matrix

∑_{k=0}^{m} ∫_{0}^{ω} P_{k}(s, x(s), x(τ_{1}(s)), …, x(τ_{m}(s))) ds

is nondegenerate,

∑_{k=0}^{m} ∫_{0}^{ω} |P_{k}(s, x(s), x(τ_{1}(s)), …, x(τ_{m}(s)))| ds ≤ A,

and

|[ ∑_{k=0}^{m} ∫_{0}^{ω} P_{k}(s, x(s), x(τ_{1}(s)), …, x(τ_{m}(s))) ds]^{−1}| ≤ B.

Then equation (1.2) has at least one ω-periodic solution.

Corollary 3.2. Let the inequality

f_{0}(t, x_{0}, x_{1}, …, x_{m}) · sgn(σx_{0}) ≤ p_{0}(t)∥x_{0}∥ + γ(t, ∥x_{0}∥, …, ∥x_{m}∥)

be fulfilled on R × R^{(m+1)n}, where σ ∈ {−1, 1}, p_{0} ∈ L_{ω}(R), ∫_{0}^{ω} p_{0}(s) ds < 0,

62 I. KIGURADZE AND B. PŮŽA

γ(·, ρ, …, ρ) ∈ L_{ω}(R_{+}) for 0 < ρ < +∞, and

lim sup_{ρ→+∞} (\frac{1}{ρ} ∫_{t}^{t+ω}exp (σ ∫_{s}^{t} p_{0}(ξ) dξ)γ(s, ρ, …, ρ) ds) <

< | exp ( − σ ∫_{0}^{ω} p_{0}(ξ) dξ) − 1| uniformly with respect to t ∈ [0, ω].

Then equation (1.2) has at least one ω-periodic solution.

Corollary 3.3. Let the condition

[f_{0}(t, x_{0}, x_{1}, …, x_{m}) − f_{0}(t, y_{0}, y_{1}, …, y_{m})] · sgn (σ(x_{0} − y_{0})) ≤

≤ p_{0}(t)∥x_{0} − y_{0}∥ + ∑_{k=0}^{m} γ_{k}(t)∥x_{k} − y_{k}∥

be fulfilled on R × R^{(m+1)n}, where σ ∈ {−1, 1}, p_{0} ∈ L_{ω}(R), ∫_{0}^{ω} p_{0}(s) ds < 0,

γ_{k} ∈ L_{ω}(R_{+}) (k = 1, …, m) and

∑_{k=1}^{m} ∫_{t}^{t+ω}exp (σ ∫_{s}^{t} p_{0}(ξ) dξ)γ_{k}(s) ds <

< | exp ( − σ ∫_{0}^{ω} p_{0}(ξ) dξ) − 1| for 0 ≤ t ≤ ω.

Then equation (1.2) has one and only one ω-periodic solution.

Corollary 3.4. Let there exist ρ ∈]0, +∞[ , δ ∈]0, 1[ and σ ∈ {−1, 1} such that on the set

{(t, x_{0}, x_{1}, …, x_{m}) ∈ R × R^{(m+1)n} : ∥x_{0}∥ ≥ ρ,

(1 − δ)∥x_{k}∥ ≤ ∥x_{0}∥ (k = 1, …, m)}

the inequality

f_{0}(t, x_{0}, x_{1}, …, x_{m}) · sgn(σx_{0}) ≤ 0

is fulfilled. Then equation (1.2) has at least one ω-periodic solution.

Acknowledgement

This work was supported by Grant 201/96/0410 of the Grant Agency of the Czech Republic (Prague) and by Grant 619/1996 of the Development Fund of Czech Universities.

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(Received 12.09.1997)

Authors’ addresses:

I. Kiguradze A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia

B. Půža Department of Mathematics Masaryk University Janáčkovo nám. 2a, 66295 Brno Czech Republic