Mem. Differential Equations Math. Phys. 20(2000), 129–132

L. Kokilashvili

ON A NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEM FOR HIGHER ORDER ORDINARY DIFFERENTIAL EQUATIONS

(Reported on September 27, 1999)

The present paper deals with the problem of existence of solution of the n-th order nonlinear nonautonomous ordinary differential equation

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

satisfying the nonlinear two-point boundary conditions

\begin{matrix}u^{(i)}(a) = φ_{1i}(u(a), u′(a), …, u^{(n−1)}(a)) (i = 0, …, n_{0} − 1), \\ u^{(j)}(b) = φ_{2j}(u(b), u′(b), …, u^{(n−1)}(b)) (j = 0, …, n − n_{0} − 1), \end{matrix}  (2)

where n ≥ 2, 0 < a < b < +∞, n_{0} is the entire part of \frac{n}{2} , each of the functions p_{k} : [a, b] → ℝ for k ∈ {1, …, n−1} is absolutely continuous together with its derivatives up to the order k − 1 inclusive, the function f : [a, b] × ℝ^{n} → ℝ satisfies the local Carathéodory conditions and the functions φ_{1i} : ℝ^{n} → ℝ (i = 0, …, n_{0} − 1) and φ_{2j} : ℝ^{n} → ℝ are continuous and satisfy the conditions

\begin{matrix}∑_{i=0}^{n_{0}−1} |φ_{1i}(x_{0}, x_{1}, …, x_{n−1})| ≤ c_{1}(1 + ∑_{k=n_{0}}^{n−1} |x_{k}|)^{−ϑ_{1}}, \\ ∑_{j=0}^{n−n_{0}−1} |φ_{2j}(x_{0}, x_{1}, …, x_{n−1})| ≤ c_{2}(1 + ∑_{k=n−n_{0}}^{n−1} |x_{k}|)^{−ϑ_{2}} \end{matrix}  (3)

on ℝ^{n}, where c_{i} ≥ 0 and ϑ_{i} ∈ [0, 1] (i = 1, 2).

The above-given theorems on the existence and uniqueness of a solution of the problem (1), (2) supplement the results of the works [1–3, 6].

Everywhere in the sequel we will assume that ℝ =] − ∞, +∞[, ℝ_{+} = [0, +∞[ and ℝ^{n} is an n dimensional real Euclidean space.

Following [4], by μ_{i}^{k}(k = 1, 2, …; k = 2i, 2i + 1, …) we denote real constants defined by the recurrence relation

μ_{0}^{i+1} = 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, …).

2000 Mathematics Subject Classification. 34B15.

Key words and phrases. Higher order ordinary differential equation, linear two-point boundary value problem, solvability, uniqueness.

Below in Theorems 1 and 2 it will be assumed that the function f satisfies the inequalities

(−1)^{n−n_{0}−1}f(t, x_{0}, x_{1}, …, x_{n−1})s_{0} ≥ − ∑_{i=0}^{n_{0}−1} α_{i}(t)|x_{i}|, −α(t),  (4)
|f(t, x_{0}, x_{1}, …, x_{n−1})| ≤ h(t, |x_{0}|, |x_{1}|, …, |x_{n_{0}−1}|)  (5)

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

lim sup\begin{matrix}t→a_{k} \\ ρ→+∞ \end{matrix} \frac{1}{ρ^{2}} ( ∫_{a_{k}}^{t} h(τ, ρ_{0}, ρ, …, ρ) dτ)^{1−λ_{k}} < +∞ (k = 1, 2),  (6)

where λ_{k} ∈ [0, 1], a_{1} = a and a_{2} = b.

Theorem 1. Let 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−1} \frac{iγ_{i}}{n_{0}} η^{i−n_{0}}δ  (7)

and the inequalities

\begin{matrix}∑_{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(t)]^{(k)} − t^{n−2n_{0}} ∑_{i=0}^{n_{0}−1} α_{i}(t) ≥ \\ ≥ ∑_{i=1}^{n_{0}−1} \frac{(n_{0} − i)γ_{i}}{n_{0}} η^{i} + δ \end{matrix}  (8)

hold on [a, b]. Then the problem (1), (2) is solvable.

In the case where n is odd, we complement Theorem 1 with

Theorem 2. Let n = 2n_{0} + 1 and there exist a constant δ > 0 such that the inequalities

\begin{matrix}p_{n−1}(t) ≤ −δ, ∑_{k=2i}^{n−1}(−1)^{n−n_{0}+k−i}μ_{i}^{k}[p_{k}(t)]^{(k−2i)} ≤ 0 (i = 1, …, n_{0} − 1), \\ ∑_{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 \end{matrix} (9)

hold on [a, b]. Then the problem (1), (2) is solvable.

In the special case where

f(t, x_{0}, x_{1}, …, x_{n−1}) ≡ f(t, x_{0}, x_{1}, …, x_{n_{0}−1}),
φ_{1i}(x_{0}, x_{1}, …, x_{n−1}) ≡ 0 (i = 0, …, n_{0} − 1),
φ_{2j}(x_{0}, x_{1}, …, x_{n−1}) ≡ 0 (j = 0, …, n − n_{0} − 1),

i.e., when the problem (1), (2) is of the form

u^{(n)} + ∑_{k=1}^{n−1} p_{k}(t)u^{(k)} = f(t, u, u′, …, u^{(n_{0}−1)}),  (1_{0})
u^{(i)}(a) = 0 (i = 0, …, n_{0} − 1), u^{(j)}(b) = 0 (j = 0, …, n − n_{0} − 1),  (2_{0})

we have the following results on the unique solvability of the problem (1_{0}), (2_{0}).

Theorem 3. Let the inequality

(−1)^{n−n_{0}−1}[f(t, x_{0}, x_{1}, …, x_{n_{0}−1}) − f(t, y_{0}, y_{1}, …, y_{n_{0}−1})] ≥ −

− ∑_{i=0}^{n_{0}−1} α_{i}(t)|x_{i} − y_{i}|  (10)

hold on [a, b]×ℝ^{n_{0}}, where the functions α_{0}[a, b] → ℝ and α_{i} : [a, b] → ℝ_{+} (i = 1, …, n_{0}− 1) are summable. Moreover, assume that the condition

∫_{a}^{b} |f(t, 0, 0, …, 0)|^{2}dt < +∞  (11)

is fulfilled and there exist constants γ_{i} ≥ 0 (i = 1, …, n_{0} − 1), η > 0 and δ > 0 satisfying (7) and such that the inequalities (8) hold on [a, b]. Then the problem (1_{0}), (2_{0}) is uniquely solvable.

Theorem 4. Let n = 2n_{0}+1, the inequality (10) hold on [a, b]×ℝ^{n_{0}} and the condition (11) be fulfilled. Moreover, assume that there exists a constant δ > 0 such that the inequalities (3) are satisfied on [a, b]. Then the problem (1_{0}), (2_{0}) is uniquely solvable.

For proving the above-formulated theorems the use was made of the method of a priori estimates and the principle of a priori boundedness proven in [5].

References

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3. G. T. Gegelia, On bounded and periodic solutions of nonlinear ordinary differential equations. (Russian) Differentsial’nye Uravneniya 22(1986), No. 3, 390–396.

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5. I. Kiguradze and B. Půža, On the solvability of boundary value problems for systems of nonlinear differential equations with deviating arguments. Mem. Differential Equations Math. Phys. 10(1997), 157–161.

6. I. Kiguradze and G. Tskhovrebadze, On the two-point boundary value problems for systems of higher order ordinary differential equations with singularities. Georgian Math. J. I. 1(1994), No. 1, 31–45.

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