Mem. Differential Equations Math. Phys. 21(2000), 131-133

L. Kokilashvili

ON THE EXISTENCE OF PROPER AND VANISHING AT INFINITY

SOLUTIONS OF ODD ORDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS

(Reported on February 21, 2000)

Consider a nonlinear ordinary differential equation

u^{(n)} + ∑_{k=1}^{n−1} p_{k}(t)u^{(k)} = f(t, u, u′, …, u^{(n−1)})  (1)

on [a, +∞[, where n ≥ 2, 0 < a < +∞ each of the functions p_{k} : [a, +∞[→ ℝ, k ∈ {1 < ⋯ < n − 1} are locally absolutely continuous together with its derivatives up to order k − 1, inclusive, and the function f : [a, +∞[×ℝ^{n} → ℝ is locally summable in the first argument and satisfies the local Lipschitz condition in the last n variables.

As is well-known, even in the case where p_{k}(t) ≡ 0, the general theory of the Cauchy problem does not answer the question on the existence of a global nontrivial solution of the equation (1) if the increase order with respect to at least one phase variable is greater than 1.

In 1986, I. Kiguradze [1] investigated a certain boundary value problem for (1) and obtained sufficient conditions for the existence of proper solutions in the above-mentioned case. The same question was considered by the author in [2] for equation (1). In the present paper we complement the results of [2], in the case, where n is odd.

Throughout this work, the use will be made of the following notation:

ℝ is the set of real numbers;

ℝ_{+} = [0, +∞[;

ℝ^{n} is the n-dimensional real Euclidean space;

μ^{k}(k = 1, 2, …; k = 2i, 2i + 1, …) are the real constants defined by the recurrence relation

μ_{0}^{i+1} = \frac{1}{2} ; μ_{i}^{2i} = 1; μ_{i+1}^{k} = μ_{i+1}^{k−1} + μ_{i}^{k−2} (i = 0, 1, …; k = 2i + 3, …).

The solution u of the equation (1) is said to be proper, if it is defined on [t_{0}, +∞[⊂ [a, +∞[ and does not equal identically to zero in any neighborhood of +∞.

We say that the proper solution u of the equation (1) vanishes at infinity, if

lim_{t→+∞} u(t) = 0.

Let n_{0} be the entire part of the number \frac{n}{2} and φ_{i} : ℝ^{n} → ℝ be continuous functions satisfying

0 < ∑_{i=0}^{n_{0}−1} |φ_{i}(x_{0}, x_{1}, …, x_{n−1})| ≤ c(1 + ∑_{k=n_{0}}^{n−1} |x_{k}|)^{−λ}  (2)

2000 Mathematics Subject Classification. 34B40 Key words and phrases. Odd order nonlinear ordinary differential equation, vanishing

at infinity solution, existence. on ℝ^{n}, where c ≥ 0 and λ ∈ [0, 1].

Below, unless otherwise specified, the function f is assumed to satisfy the conditions

\begin{matrix}(−1)^{n_{0}}f(t, x_{0}, x_{1}, …, x_{n−1}) sgn x_{0} ≥ − ∑_{i=0}^{n_{0}−1} α_{i}(t)|x_{0}|, \\ |f(t, x_{0}, x_{1}, …, x_{n−1})| ≤ h(t, |x_{0}|, |x_{1}|, …, |x_{n_{0}−1}|) \end{matrix}  (3)

on [a, +∞[×ℝ^{n}, where the functions α_{i} : [a, +∞[→ ℝ (i = 0, …, n_{0} − 1) are locally summable, while the function h : [a, +∞[→ ℝ_{+}^{n_{0}} → ℝ_{+} is locally summable in the first argument, nondecreasing in the last n_{0} arguments and for any ρ_{0} > 0 satisfies the condition

lim sup\begin{matrix}t→a_{k} \\ ρ→+∞ \end{matrix} \frac{1}{ρ^{2}} ( ∫_{a}^{t} h(τ, ρ_{0}, ρ, …, ρ) dτ)^{1−λ} < +∞.  (4)

Theorem 1. Let there exist the constant δ > 0 such that the inequalities

∑_{k=2i}^{n−1}(−1)^{n_{0}+k−i}μ_{i}^{k}[p_{k}(t)]^{(k−2i)} ≤ 0 (i = 1, …, n_{0} − 1),

p_{n−1}(t) ≤ −δ and ∑_{k=1}^{n−1}(−1)^{n_{0}+k−1}μ_{0}^{k}[p_{k}(t)]^{(k)} − ∑_{i=0}^{n_{0}−1} α_{i}(t) ≥ δ

hold on [a, +∞[. Then there exists a proper, vanishing at infinity, solution u of the equation (1) satisfying the initial conditions

u^{(i)}(a) = φ_{i}(u(a), u′(a), …, u^{(n−1)}(a)) (i = 0, …, n_{0} − 1).  (5)

The theorem below deals with the case, where h(x_{0}, x_{1}, …, x_{n−1}) ≡ h_{0}(t, x_{0}x), α_{i}(t) ≡ 0 (i = 1, …, n_{0} − 1) λ = 0 i.e., when the conditions (2) and (3) are of the form

(−1)^{n_{0}}f(t, x_{0}, …, x_{n−1}) sgn x_{0} ≥ −α_{0}(t)|x_{0}|,

0 < ∑_{i=0}^{n_{0}−1} |φ_{i}(x_{0}, …, x_{n−1})| ≤ c.

The condition (4) in this case is fulfilled automatically.

Theorem 2. Let there exist a continuous solution δ : [a, +∞[→ [0, +∞[ such that δ(a) > 0 and the inequalities

∑_{k=2i}^{n−1}(−1)^{n_{0}+k−i}μ_{i}^{k}p_{k}^{(k−2i)}(t) ≤ 0 (i = 1, …, n_{0} − 1),

p_{n−1}(t) ≤ 0, ∑_{k=1}^{n−1}(−1)^{n_{0}+k−1}μ_{0}^{k}p_{k}^{(k)}(t) ≥ α_{0}(t) + δ(t)

hold on [a, +∞[. Then there exists at least one proper solution of the equation (1) satisfying the initial conditions (5).

In a special case where p_{k}(t) ≡ 0 for k ∈ {1, …, n − 3, n − 1} and p_{n−2}(t) ≡ 1, Theorem 2 implies one result from [3].

References

1. I. Kiguradze, On a boundary value problem with a condition at infinity for higher order differential equations. (Russian) Trudy Vsesojuznogo Simpoziuma po Diff. Uravnenijam v Chastnykh Proizvodnykh, Tbilis. Univ. Press, 1986, 91–105.

2. L. Kokilashvili, On the existence of proper solutions of higher order nonlinear ordinary differential equations. Fasc. Math. 26(1996), 37–60.

3. L. Kokilashvili, On a certain boundary value problem for nonlinear ordinary differential equations. Mem. Differential Equations Math. Phys. 13(1998), 145–147.

Author’s address: Faculty of Mechanics and Mathematics I. Javakhishvili Tbilisi State University 2, University St., Tbilisi 380043 Georgia