Mem. Differential Equations Math. Phys. 10(1997), 134–137

I. Kiguradze

ON PERIODIC SOLUTIONS OF FIRST ORDER NONLINEAR DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

(Reported on September 9–16, 1996)

In the present note, we establish sufficient conditions for the existence and uniqueness of a periodic solution of the differential equation

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

where f : R × R^{m} → R is a function satisfying the local Carathéodory conditions and τ_{k} : R → R (k = 1, …, m) are measurable functions. In what follows, the function f is assumed to be periodic in the first argument with the period ω > 0, i.e., on R × R^{m} the equality

f(t + ω, x_{1}, …, x_{m}) = f(t, x_{1}, …, x_{m}).  (2)

is fulfilled. As for the functions τ_{k} (k = 1, 2, …), they are such that

τ_{k}(t + ω) = μ_{k}(t)ω + τ_{k}(t) for t ∈ R (k = 1, …, m),  (3)

where μ_{k} (k = 1, …, m) are the functions admitting only integer values.

For any k ∈ {1, …, m} and t ∈ R, we denote by ν_{k}(t) the integer part of the number \frac{τ_{k}(t)}{ω} and assume that

τ_{0k}(t) = τ_{k}(t) − ν_{k}(t)ω (k = 1, …, m).

Then

0 ≤ τ_{0k}(t) < ω for t ∈ R (k = 1, …, m).

On the other hand, due to the conditions (2) and (3), the set of restrictions of all ωperiodic solutions of the equation (1) on the segment [0, ω] coincides with the set of solutions of the boundary value problem

\frac{dx(t)}{dt} = f(t, x(τ_{01}(t)), …, x(τ_{0m}(t))),  (4)
x(0) = x(ω).  (5)

Hence the equation (1) has at least one ω-periodic (a unique ω-periodic) solution if and only if the problem (4), (5) is solvable (uniquely solvable).

1991 Mathematics Subject Classification. 34K10.

Key words and phrases. Nonlinear first order differential equation with deviating arguments, periodic solution, existence theorem, uniqueness theorem.

Theorem 1. Let for some σ ∈ {−1, 1} the equalities

|f(t, x_{1}, …, x_{m}) − ∑_{k=1}^{m} p_{k}(t, x_{1}, …, x_{m})x_{k}| ≤ q(t),  (6)

σ ∑_{k=1}^{m} p_{k}(t, x_{1}, …, x_{m}) ≥ α(t),  (7)
|p_{k}(t, x_{1}, …, x_{m})| ≤ β_{k}(t) (k = 1, …, m)  (8)

be fulfilled on the set [0, ω]×R^{m}, where p_{k} : [0, ω]×R^{m} → R (k = 1, …, m) are functions satisfying the local Carathéodory conditions, while q, α and β_{k} : [0, ω] → [0, +∞[ (k = 1, …, m) are summable functions. Let, moreover, α be different from zero on a set of positive measure and let there exist δ ∈]0, a[ such that on [0, ω] the inequality

∑_{i,k=1}^{m} β_{i}(t)| ∫_{t}^{τ_{0k}(t)}β_{k}(s)ds| ≤ δα(t)  (9)

is fulfilled. Then the equation (1) has at least one ω-periodic solution.

Proof. Due to the above remark, in order to prove the theorem it suffices to establish the solvability of the problem (4), (5). On the other hand, by Theorem 1.1 in [1] and the conditions (6)–(8), the problem (4), (5) is solvable if for any summable functions p_{0k} : [0, ω] → R (k = 1, …, m) satisfying

σ ∑_{k=1}^{m} p_{0k}(t) ≥ α(t),  (10)
|p_{0k}(t)| ≤ β_{k}(t) (k = 1, …, m),  (11)

the problem

\frac{dy(t)}{dt} = ∑_{k=1}^{m} p_{0k}(t)y(τ_{0k}(t)), y(0) = y(ω)  (12)

has only the trivial solution.

Let y be an arbitrary solution of the problem (12). Then almost everywhere on [0, ω] we have

\frac{dy(t)}{dt} = [ ∑_{k=1}^{m} p_{0k}(t)]y(t) + ∑_{k=1}^{m} p_{0k}(t) ∫_{t}^{τ_{0k}(t)}y′(s) ds =

= p(t)y(t) + q(t),  (13)

where

p(t) = ∑_{k=1}^{m} p_{0k}(t), q(t) = ∑_{i,k=1}^{m} p_{0k}(t) ∫_{t}^{τ_{0k}(t)}p_{0i}(s)y(τ_{0i}(s)) ds.

The functions α, β_{k} (k = 1, …, m), y, p and q will be assumed to be ω-periodically extended to the whole R. Then, because of (9)–(11), the following inequalities are fulfilled on R:

σp(t) ≥ α(t) ≥ 0, |q(t)| ≤ δσp(t)y_{0},  (14)

where

y_{0} = max {|y(t)| : t ∈ [0, ω]}.  (15)

Moreover,

σ ∫_{0}^{ω} p(t) dt ≥ ∫_{0}^{ω} α(t) dt > 0  (16)

since α is different from zero on a set of positive measure.

Owing to (13) and (16),

y(t) = [ exp ( − ∫_{0}^{ω} p(s) ds) − 1]^{−1} ∫_{t}^{t+ω}exp ( ∫_{s}^{t} p(ξ) dξ)q(s) ds,

whence, taking into account (14), (15) and ω-periodicity of p, we find

|y(t) ≤ δy_{0}| exp ( − ∫_{0}^{ω} p(s) ds) − 1|^{−1}σ ∫_{t}^{t+ω}exp ( ∫_{s}^{t} p(ξ) dξ)p(ξ) dξ = δy_{0}

and

y_{0} ≤ δy_{0}.

Therefore y_{0} = 0. Consequently, the problem (12) has only the trivial solution.

Theorem 2. Let the function f in the last m variables have partial derivatives which on [0, ω] × R^{m} satisfy both the local Carathéodory conditions and the inequalities

σ ∑_{k=1}^{m} \frac{∂f(t, x_{1}, …, x_{m})}{∂x_{k}} ≥ α(t),  (17)

|\frac{∂f(t, x_{1}, …, x_{m})}{∂x_{k}} | ≤ β_{k}(t) (k = 1, …, m),  (18)

where α and β_{k} : [0, ω] → [0, +∞[ (k = 1, …, m) are summable functions. Moreover, let α be different from zero on a set of positive measure and almost everywhere on [0, ω] the inequality (9) be fulfilled. Then the equation (1) has one and only one ω-periodic solution.

Proof. Suppose

f_{k}(t, x_{1}, …, x_{m}) = \frac{∂f(t, x_{1}, …, x_{m})}{∂x_{k}} (k = 1, …, m),

\overline{p}_{k}(t, x_{1}, …, x_{m}, y_{1}, …, y_{m}) = ∫_{0}^{1} f_{k}(t, sx_{1} + (1 − s)y_{1}, …, sx_{m} + (1 − s)y_{m}) ds,

p_{k}(t, x_{1}, …, x_{m}) = \overline{p}_{k}(t, x_{1}, …, x_{m}, 0, …, 0) (k = 1, …, m),
q(t) = |f(t, 0, …, 0)|.

Then, because of (17) and (18), conditions (6)–(8) are fulfilled. Therefore, by Theorem 1, the equation (1) has at least one ω-periodic solution.

To complete the proof of the theorem, it remains to show that the equation (1) has no more than one ω-periodic solution, i.e., the problem (4), (5) has no more than one solution. Let x and \overline{x} be two arbitrary solutions of this problem. Then the function y(t) = \overline{x}(t) − x(t) is a solution of the problem (12), where

p_{0k}(t) = \overline{p}_{k}(t, \overline{x}(τ_{01}(t)), …, \overline{x}(τ_{0m}(t)), x(τ_{01}(t)), …, x(τ_{0m}(t))) (k = 1, …, m).

Moreover, the functions p_{0k}(k = 1, …, m), as it follows from (17) and (18), satisfy the inequalities (10) and (11). However, as it is proved above, the conditions (9)–(11) and (16) guarantee for the problem (12) to have only the trivial solution. Consequently, \overline{x}(t) ≡ x(t).

Remark. This work was supported by Grant 619/1996 of the Development Fund of Czech Universities.

References

1. I. T. Kiguradze and B. Puža, Conti–Opial type theorems for nonlinear functional differential equations. (Russian) Differentsial’nye Uravneniya 33(1997), No. 2 (to appear).

Author’s Address:

A. Razmadze Mathematical Institute

Georgian Academy of Sciences

1, M. Aleksidze St., Tbilisi 380093

Georgia