Mem. Differential Equations Math. Phys. 5(1995), 131–132

Z. Sokhadze

ON A THEOREM OF MYSHKIS–TSALYUK

(Reported on June 19 and 26, 1995)

Consider the system of differential equations with delay

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

where f : [a, b] × R^{(m+1)n} → R^{n} is a vector function from the Caratheodory class and τ_{i} : [a, b] → [a, b] (i = 1, …, n) are measurable functions satisfying the inequalities

τ_{i}(t) ≤ t for a ≤ t ≤ b (i = 1, …, m).

Let b_{0} ∈]a, b]. A vector function x : [a, b_{0}] → R^{n} (a vector function x : [a, b_{0}[→ R^{n}) is said to be a solution of the system (1) on [a, b_{0}] (on [a, b_{0}[) if it is absolutely continuous on [a, b_{0}] (on every segment contained in [a, b_{0}[) and satisfies (1) a.e. on [a, b_{0}].

The solution x of the system (1) is said to be noncontinuable if one of the following conditions is fulfilled:

i) x is defined on the segment [a, b]; ii) x is defined on the segment [a, b_{0}[, where b_{0} ∈ [a, b[, and

lim_{t→b_{0}} sup ∥x(t)∥ = +∞.

A. D. Myshkis and Z. B. Tsalyuk [2] have proved a theorem on nonlocal continuability of solutions of the system (1), concerning the case where growth order of the vector function f with respect to the last mn arguments exceeds 1. Below we shall give a more general theorem of the same type.

The use will be made of the following notation: R is the set of real numbers; R_{+} = [0, +∞[; R^{l} is the space of vectors x = (x_{i})_{≡1}^{l} with the components x_{i} ∈ R (i = 1, …, l) and

the norm ∥x∥ = ∑_{i=1}^{l} |x_{i}|; R_{+}^{l} = {(x_{i})_{i=1}^{l} ∈ R^{l} : x_{i} ∈ R_{+}(i = 1, …, l)}; x · y is the scalar product of the vectors x and y ∈ R^{n}; if x = (x_{i})_{i=1}^{n}, then sgn(x) = (sgn(x_{i}))_{i=1}^{n}.

Theorem. Let for every s ∈]a, b] there exist a number δ_{s} ∈]0, s−a[, a vector c_{s} ∈ R^{n} and functions τ_{is} : [s − δ_{s}, s] → [s − δ_{s}, s] and φ_{s} : [s − δ_{s}, s] × R_{+}^{m+1} → R_{+} such that

τ_{i}(t) ≤ τ_{is}(t) for s − δ_{s} ≤ t ≤ δ_{s} (i = 1, …, m)

and the inequality

f(t, c_{s} + y_{0}, c_{s} + y_{1}, …, c_{s} + y_{m}) · sgn(y_{0}) ≤ ρ_{s}(t, ∥y_{0}∥, …, ∥y_{m}∥)

1991 Mathematics Subject Classification. 34K10. Key words and phrases: System of differential equations with delay, noncontinuable

solution, nonlocal continuability. is fulfilled on [s − δ_{s}, s] × R^{(m+1)n}. Moreover, let τ_{is} (i = 1, …, m) be measurable, φ be summable in the first argument, continuous and nondecreasing in the last m + 1 arguments and let for every ρ ∈ R_{+} the initial problem

\frac{du(t)}{dt} = φ_{s}(t, u(t), u(τ_{1s}(t)), …, u(τ_{ms}(t))); u(s − δ(s)) = ρ

have upper solution defined on [s − δ(s), s]. Then every noncontinuable solution of the system (1) is defined on [a, b].

Corollary 1. Let for every s ∈]a, b] there exist numbers m_{s} ∈ {0, …, m − 1}, δ_{s} ∈]0, s − a[, a vector c_{s} ∈ R^{n} and functions α_{s} : [s − δ(s), s] × R_{+} → R_{+} and φ_{s} : R_{+} →]0, +∞[ such that

τ_{k}(t) ≤ s − δ(s) for a ≤ t ≤ b (k = m_{s} + 1, …, m)

and the inequality

f(t, c_{s} + y_{0}, …, c(s) + y_{m}) · sgn(y_{0}) ≤ α_{s}(t, ∑_{k=m_{s}+1}^{m} ∥y_{k}∥)φ_{s}( ∑_{k=0}^{m_{s}} ∥y_{k}∥).

is fulfilled on the set [s − δ(s), s] × R^{(m+1)n}. Moreover, let α_{s} be summable in the first argument, continuous and nondecreasing in the second argument, φ_{s} be continuous, nondecreasing and

∫_{0}^{+∞}\frac{du}{φ_{s}(u)} = +∞.

Then every noncontinuable solution of the system (1) is defined on [a, b].

Corollary 2. Let for every s ∈]a, b] there exist numbers m_{s} ∈ {1, …, m − 1}, δ_{s} ∈]0, s−a[∩]0, 1[, λ_{ks} ∈ [1, +∞[, β_{s} ∈]−1, 0[ and a continuous function α_{s} : R_{+} → R_{+} such that the inequalities

τ_{k}(t) ≤ s − (s − t)^{\frac{1}{λks}} (k = 1, …, m_{s}), τ_{k}(t) ≤ s − δ(s) (k = m_{s} + 1, …, m)

and

f(t, c_{s} + y_{0}, …, c_{s} + y_{m}) · sgn(y_{0}) ≤

≤ α_{s}( ∑_{k=m_{s}+1}^{m} ∥y_{k}∥)(s − t)^{β_{s}}(1 + ∑_{k=1}^{m_{s}} ∥y_{k}∥^{λ_{ks}}) ln (2 + ∑_{k=1}^{m_{s}} ∥y_{k}∥).

are fulfilled on [s − δ(s), s] and on [s − δ(s), s] × R^{(m+1)n}, respectively. Then every noncontinuable solution of the system (1) is defined on [a, b].

Corollary 1 is an analogue of the wellknown A. Wintner’s theorem ([1], Ch. III, §3.5) for the system (1), while Corollary 2 is a generalization of the above-mentioned Myshkis-Tsalyuk theorem.

References

1. P. Hartman, Ordinary differential equations. John Wiley & Sons, New York-London-Sydney, 1964.

2. A. D. Myshkis, Z. B. Tsalyuk, On nonlocal continuability of solutions of differential systems with deviating argument. (Russian) Differentsial’nye Uravneniya 5(1969), No. 6, 1128–1130.

Author’s address:

Kutaisi A. Tsereteli State University 55, Queen Tamar St., Kutaisi 384000, Georgia