Mem. Differential Equations Math. Phys. 13(1998), 145–147

L. Kokilashvili

ON A CERTAIN BOUNDARY VALUE PROBLEM FOR NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS

(Reported on November 24, 1997)

Consider the n-th order ordinary differential equation

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

on the interval [a, +∞[, where a > 0, n ≥ 2, each of the functions p_{k} : [a, +∞[ → ℝ for k ∈ {1, …, n−1} is locally absolutely continuous together with its derivatives up to order k − 1 inclusively (i.e., the functions p_{k}^{(i)} (i = 0, …, k − 1) are absolutely continuous on any finite segment contained in [a, +∞[), and the function f : [a, +∞[ ×ℝ^{n} → ℝ satisfies the local Carathéodory conditions.

Let n_{0} be an integer part of the number \frac{n}{2} , and let c_{i} (i = 0, …, n_{0} − 1) be arbitrary real numbers. Consider the problem on existence of a solution u : [a, +∞[ → ℝ of the equation (1), satisfying the conditions

u^{(i)}(a) = c_{i} (i = 0, …, n_{0} − 1), ∫_{a}^{+∞}[u^{(j)}(t)]^{2} dt < +∞ (j = 0, …, n_{0}).  (2)

In the case where p_{k}(t) ≡ 0 (k = 1, …, n − 1), the problems of the type (1), (2) have been investigated by I. Kiguradze [1]. The theorems given below complement the results of this work.

Let μ_{i}^{k} (i = 0, 1, …, n − n_{0} − 1; k = 2i, 2i + 1, …, n − 1) be real numbers given 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, …, n−n_{0} −1; k = 2i+3, …, n−1),

and ℝ_{+} = [0, +∞[ .

Theorem 1. Let the inequalities

|f(t, x_{0}, x_{1}, …, x_{n−1})| ≤ φ(t, |x_{0}|, |x_{1}|, …, |x_{n_{0}−1}|),  (3)

(−1)^{n−n_{0}−1}f(t, x_{0}, x_{1}, …, x_{n−1}) sgn x_{0} ≥ − ∑_{i=0}^{n_{0}−1} α_{i}(t)|x_{i}| + α(t),

1991 Mathematics Subject Classification. 34B15.

Key words and phrases. Nonlinear ordinary differential equation, boundary value broblem. hold on [a, +∞[ ×ℝ^{n}, where the functions α_{0} : [a, +∞[ → ℝ, α_{i} : [a, +∞[ → ℝ_{+} (i = 1, …, n_{0} − 1) are locally summable, α : [a, +∞[ → ℝ_{+} is measurable and

∫_{a}^{+∞}t^{2n−4n_{0}}α^{2}(t) dt < +∞,

while the function φ : [a, +∞[ ×ℝ_{+}^{n_{0}} → ℝ_{+} is locally summable with respect to the first argument, non-decreasing with respect to the last n_{0} arguments and for any ρ_{0} ∈]0, +∞[ satisfies the condition

lim_{\begin{matrix}t→a \\ ρ→+∞ \end{matrix}} \frac{1}{ρ^{2}} ∫_{a}^{t} φ(τ, ρ_{0}, ρ, …, ρ) dτ = 0.  (4)

Further, suppose that there exist constants γ_{i} ≥ 0 (i = 1, …, n_{0} − 1), η > 0 and δ > 0 such that

μ_{n_{0}}^{n} − ∑_{i=1}^{n_{0}−1} \frac{iγ_{i}}{n_{0}} η^{i−n_{0}} ≥ δ

and the inequalities

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

∑_{k=1}^{n−1}(−1)^{n−n_{0}+k}μ_{0}^{k}[t^{n−2n_{0}}p_{k}(t)]^{(k)} − t^{n−2n_{0}}α_{0}(t) ≥

≥ ∑_{i=1}^{n_{0}−1} t^{n−2n_{0}}α_{i}(t) + ∑_{i=1}^{n_{0}−1} \frac{(n_{0} − i)γ_{i}}{n_{0}} η^{i} + δ

hold on [a, +∞[ . Then there exists at least one solution of the problem (1), (2).

Corollary 1. Let the inequalities (4) and

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

hold on [a, +∞[ ×ℝ^{n}, where λ > 1, the function φ be taken as it were in Theorem 1 and γ : [a, +∞[ →]0, +∞[ is a measurable function satisfying the condition

∫_{a}^{+∞}t^{n−2n_{0}}[γ(t)]^{− \frac{2}{λ−1}} dt < +∞.

Further, suppose that there exists a constant r ∈]0, +∞[ such that the inequalities

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

∑_{k=1}^{n−1}(−1)^{n−n_{0}+k}μ_{0}^{k}[t^{n−2n_{0}}p_{k}(t)]^{(k)} > −r

hold on [a, +∞[ . Then the problem (1), (2) is solvable.

Corollary 2. Let all the conditions of Corollary 1, except (4) be fulfilled. Then the problem (1), (2) has an n_{0}-parametric family of solutions satisfying the conditions

∫_{a}^{+∞}[u^{(j)}(t)]^{2} dt < ∞ (i = 0, …, n_{0}).

Remark 1. In the case where n = 2n_{0}, p_{n−2}(t) ≡ 1 and p_{k}(t) ≡ 0 (k ≠ n − 2; k = 1, …, n − 1), from Corollary 1 it follows Theorem 1.2 of the paper [2].

References

1. I. T. Kiguradze, On a boundary value problem with a condition at infinity for higher order ordinary differential equations. (Russian) In: Trudy Vsesojuznogo Simpoziuma po Diff. Uravneniyam v Chastnykh Proizvodnykh, Tbilisi University Press, Tbilisi, 1986, 91–105.

2. L. Kokilashvili, On the existence of proper solutions of high order nonlinear differential equations. Fasc. Math., 1996, No. 26, 37–60.

Author’s Address: A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia