GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 1, 1995, 21-36

ON THE PERIODIC BOUNDARY-VALUE PROBLEM FOR SYSTEMS OF SECOND-ORDER NONLINEAR ORDINARY

DIFFERENTIAL EQUATIONS

G. GAPRINDASHVILI

Abstract. The periodic boundary-value problem for systems of secondorder ordinary nonlinear differential equations is considered. Sufficient conditions for the existence and uniqueness of a solution are established.

§ 1. Statement of the Main Results

Consider the periodic boundary-value problem

x′′ = f(t, x, x′),  (1.1)
x(a) = x(b), x′(a) = x′(b),  (1.2)

where the vector-function f : [a, b] × ℝ^{2n} → ℝ^{n} (ℝ^{n} denotes the n-dimen-sional Euclidean space with the norm ∥ · ∥) satisfies the local Caratheodory conditions, i.e., f(·, x, y) : [a, b] → ℝ^{n} is measurable for each (x, y) ∈ ℝ^{2n}, f(t, ·) : ℝ^{2n} → ℝ^{n} is continuous for almost all t ∈ [a, b], and the function

f_{r}(·) = sup{∥f(·, x, y)∥ : ∥x∥ + ∥y∥ ≤ r}

is Lebesgue integrable on [a, b] for each positive r.

By a solution of the problem (1.1), (1.2) we mean a vector-function x : [a, b] → ℝ^{n} which has the absolutely continuous first derivative on [a, b] and satisfies the differential system (1.1) almost everywhere in [a, b], as well as the boundary conditions (1.2).

For the literature on (1.1),(1.2) we refer to [1,2] and the references cited therein. Note that [1] deals with the scalar variant of the boundary-value problem (1.1),(1.2) (i.e., when n = 1).

21

1991 Mathematics Subject Classification. 34B15, 34C25.

Key words and phrases. Second-order differential equation, periodic boundary-value problem, Nagumo pair.

1072-947X/95/0100-0021$07.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1995 Plenum Publishing Corporation

Below, the sufficient conditions for solvability and unique solvability of the boundary value problem (1.1), (1.2) are given. They supplement some of those mentioned above.

We use the following notation:

x · y is the inner product of vectors x, y ∈ ℝ^{n};

ℝ = ℝ^{1}, ℝ_{+} = [0, +∞[;

for each positive number r and vector y ∈ ℝ^{n}

η_{r}(y) = \begin{cases}0 & for ∥y∥ ≤ r, \\ \frac{y}{∥y∥} & for ∥y∥ > r; \end{cases}

U_{δ}(t_{0}) is the δ-neighborhood of t_{0} ∈ ℝ;

C̃^{1}([a, b]; S) (S ⊂ ℝ^{n}) is the set of vector-functions x : [a, b] → S which have an absolutely continuous first derivative on [a, b];

C(S_{1}; S_{2}) (S_{1} ⊂ ℝ,S_{2} ⊂ ℝ^{n}) is the set of continuous vector-functions x : S_{1} → S_{2};

L([a, b]; S) (S ⊂ ℝ) is the set of functions x : [a, b] → S which are Lebesgue integrable on [a, b].

Definition 1.1 (see [3, Definition 1.1] or [4, Definition 1.2]). Suppose that the functions φ :]a, b[→]0, ∞[ and z :]a, b[→ ℝ^{n} have a first derivative which is absolutely continuous on every segment contained in ]a, b[. A pair of functions (φ, z) is said to be a Nagumo pair of the differential system (1.1) if the condition

(x − z(t)) · (f(t, x, y) − z′′(t)) + ∥y − z′(t)∥^{2} − (φ′(t))^{2} ≥ φ(t)φ′′(t)

for a < t < b, ∥x − z(t)∥ = φ(t) and (x − z(t)) · (y − z′(t)) = φ(t)φ′(t)

is satisfied, the function ∥z′′(t)∥+φ(t) being essentially bounded from above on every segment contained in ]a, b[.

Remark 1.1. The Nagumo pair of differential system (1.1) serves as a vector analog for the upper and lower functions of the scalar equation (1.1), which were introduced by Nagumo [5] and which since then have been widely adopted in the theory of boundary-value problems (see [1] and the references cited therein; also [4, Remark 1.2], [3, Remark 1.1]). Namely, if n = 1 and σ_{1} and σ_{2} are, respectively, the upper and lower solutions of the differential equation (1.1), then the pair (φ, z) defined by

φ(t) = \frac{σ_{2}(t) − σ_{1}(t)}{2} and z(t) = \frac{σ_{2}(t) + σ_{1}(t)}{2}  (1.3)

is the Nagumo pair of (1.1) (and vice versa).

Note also that the condition

x · f(t, x, y) + ∥y∥^{2} ≥ 0 for ∥x∥ = r_{0} and x · y = 0  (1.4)

(see [2, Theorem 3.1]) is necessary and sufficient for (φ, z) to be a Nagumo pair of (1.1), where z(t) ≡ 0 and φ(t) ≡ r_{0} > 0.

Definition 1.2. A Nagumo pair (φ, z) of the differential system (1.1) is said to be a Nagumo pair of the problem (1.1),(1.2) if φ ∈ C̃^{1}([a, b]; ℝ_{+}), z ∈ C̃^{1}([a, b]; ℝ^{n}) and the conditions

φ(a) = φ(b), z(a) = z(b)  (1.5_{1})

and

∥z′(a) − z′(b)∥ ≤ φ′(b) − φ′(a)  (1.5_{2})

are satisfied.

Remark 1.2. In the scalar case, (1.5_{1}) − (1.5_{2}) are equivalent to the conditions

σ_{i}(a) = σ_{i}(b), (−1)^{i}(σ′_{i}(a) − σ′_{i}(b)) ≤ 0 (i = 1, 2),

assuming that (1.3) is satisfied. See these conditions in [1, § 16].

Definition 1.3 (see [3, Definition 2.1] or [4, Definition 1.1]). Suppose that φ ∈ C([a, b]; ℝ_{+}) and z ∈ C([a, b]; ℝ^{n}). A vector-function f is said to have the property V ([a, b], φ, z) if there exist positive constants r and r_{1} such that if a ≤ t_{1} < t_{2} ≤ b, χ ∈ C(ℝ_{+}; [0, 1]) and x ∈ C̃^{1}([t_{1}, t_{2}]; ℝ^{n}) is an arbitrary solution of the differential system

x′′ = χ(∥x′∥)f(t, x, x′)  (1.6)

satisfying the inequalities

∥x(t) − z(t)∥ ≤ φ(t) for t_{1} ≤ t ≤ t_{2}  (1.7)

and

∥x′(t)∥ ≥ r for t_{1} ≤ t ≤ t_{2},  (1.8)

then x admits the estimate

∫_{t_{1}}^{t_{2}}∥x′(t)∥dt ≤ r_{1}.  (1.9)

Remark 1.3. It is clear that each scalar function has the property V ([a, b], φ, z) taking an arbitrary positive number for r and 2 max{φ(t) + ∥z(t)∥ : a ≤ t ≤ b} for r_{1}. The class of vector-functions with the property V ([a, b], φ, z) is introduced just to unify the approach to the problem (1.1),(1.2) in both the scalar and the vector cases. Some other boundaryvalue problems were also studied using this approach (see [3,4] and the references cited therein).

Effective sufficient conditions for a vector-function f to have the property V ([a, b], φ, z) are contained in [4, Propositions 1.1, 1.2] and [3, Proposition 2.1]. For example, if

\begin{matrix}(f(t, x, y) · y)(x · y) − (x · f(t, x, y))∥y∥^{2} ≤ l(t)∥y∥^{3} + k∥y∥^{4} \\ for a ≤ t ≤ b, ∥x − z(t)∥ ≤ φ(t) and ∥y∥ > ρ, \end{matrix} (1.10)

where l ∈L([a, b]; ℝ_{+}), k <1 and ρ>0, then f has the property V ([a, b], φ, z).

Theorem 1.1_{1}. Suppose that (φ, z) is a Nagumo pair of (1.1), (1.2), the vector-function f has the property V ([a, b], φ, z), and the inequality

f(t, x, y) · η_{ρ}(y) ≤ w(∥y∥)(l(t) + ∥y∥)  (1.11)

is satisfied on the set

{(t, x, y) : a < t < b, ∥x − z(t)∥ ≤ φ(t)},  (1.12)

where ρ > 0, l ∈ L([a, b]; ℝ_{+}), ω ∈ C(ℝ_{+}; ]0, +∞[), and

∫_{0}^{+∞}\frac{ds}{ω(s)} = +∞.  (1.13)

Then the boundary-value problem (1.1), (1.2) has at least one solution x ∈ C̃^{1}([a, b]; R^{n}) satisfying the estimate

∥x(t) − z(t)∥ ≤ φ(t) for a ≤ t ≤ b.  (1.14)

Theorem 1.1_{2}. The conclusion of Theorem 1.1_{1} remains valid if (1.11) is replaced by

f(t, x, y) · η_{ρ}(y) ≥ −ω(∥y∥)(l(t) + ∥y∥).  (1.15)

Theorem 1.2. Suppose that (φ, z) is a Nagumo pair of (1.1), (1.2), the vector-function f has the property V ([a, b], φ, z), the inequality (1.11) is satisfied on the set

{(t, x, y) : a_{0} ≤ t ≤ b, ∥x − z(t)∥ ≤ φ(t)},

and the inequality (1.15) on the set

{(t, x, y) : a < t < b_{0}, ∥x − z(t)∥ ≤ φ(t)},

where ρ > 0, a ≤ a_{0} < b_{0} ≤ b, l ∈ L([a, b]; ℝ_{+}), ω ∈ C(ℝ_{+}; ]0, +∞[), and (1.13) holds. Then the boundary-value problem (1.1), (1.2) has at least one solution x ∈ C̃^{1}([a, b]; ℝ^{n}) satisfying the estimate (1.14).

Theorem 1.3. Suppose that (φ, z) is a Nagumo pair of (1.1), (1.2), the vector-function f has the property V ([a, b], φ, z), the inequality (1.11) is satisfied on the set

{(t, x, y) : t ∈]a_{1}, t_{0}[∪]b_{2}, b[, ∥x − z(t)∥ ≤ φ(t)},

and the inequality (1.15) on the set

{(t, x, y) : t ∈]a, a_{2}[∪]t_{0}, b_{1}[, ∥x − z(t)∥ ≤ φ(t)},

where ρ > 0, a < a_{1} < a_{2} < t_{0} < b_{2} < b_{1} < b, l ∈ L([a, b]; ℝ_{+}), ω ∈ C(ℝ_{+}; ]0, +∞[), and (1.13) holds. Then the boundary-value problem (1.1), (1.2) has at least one solution x ∈ C̃^{1}([a, b]; ℝ^{n}) satisfying the estimate (1.14).

Remark 1.4. Theorems 1.1–1.3 extend Theorem 3.1 [2] in the case of periodic boundary-value problem. As an example, define f_{i}(t, x, y) = −y_{i}∥y∥^{m}+ 1 − ∥x∥ and f = (f_{i})_{i=1}^{n}, where m is an arbitrary natural number. Let us verify the conditions of, e.g., Theorem 1.1_{1} assuming that z(t) ≡ 0, φ ≡ 1, ρ = 1, l(t) ≡ 1, and ω ≡ 1. First, according to (1.4) where r_{0} = 1, (φ, z) is the Nagumo pair of (1.1),(1.2). Further, according to (1.10) where k = 0, the vector-function f has the property V ([a, b], φ, z). Finally, the correctness of (1.11), as well as of (1.13), is evident. On the other hand, Theorem 3.1 [2] fails for this example when m > 2.

Theorem 1.2 can also be considered as a vector analog of Theorem 16.2 from [1].

Theorem 1.4. Suppose that for each positive r there exist l_{i}(t, r) ∈ L([a, b]; ℝ_{+}) (i = 1, 2) such that l_{1}(t, r) differs from zero on a subset of positive measure of the interval ]a, b[ and

[f(t, x_{1}, y_{1}) − f(t, x_{2}, y_{2})](x_{1} − x_{2}) ≥
≥ l_{1}(t, r)∥x_{1} − x_{2}∥^{2} − l_{2}(t, r)|(x_{1} − x_{2}) · (y_{1} − y_{2})|
for ∥x_{k}∥ ≤ r, ∥y_{k}∥ ≤ r (k = 1, 2).  (1.16)

Then the boundary-value problem (1.1), (1.2) has at most one solution in the class C̃^{1}([a, b]; ℝ^{n}).

Theorem 1.4 can be considered as a vector analog of Theorem 16.4 from [1].

§ 2. Some Auxiliary Results

Lemma 2.1. Suppose that a vector-function q : [a, b] × ℝ^{2n} → ℝ^{n} satisfies the local Caratheodory conditions and the inequality

∥q(t, x, y)∥ ≤ l(t)

holds on [a, b] × ℝ^{2n} where l ∈ L([a, b]; ℝ_{+}). Then the differential system

x′′ = x + q(t, x, y)

has at least one solution x ∈ C̃^{1}([a, b]; ℝ^{n}) satisfying the boundary conditions (1.2).

Proof. It is easy to verify that the differential system

x′′ = x

has no nontrivial solution satisfying the boundary conditions (1.2). Thus Lemma 2.1 immediately follows from Proposition 2.3 [1].

The next result deals with the solvability of an auxiliary differential system

x′′ = g(t, x, x′).  (2.1)

Lemma 2.2. Suppose that (φ, z) is a Nagumo pair of the boundary-value problem (2.1), (1.2) and on [a, b] × ℝ^{2n}

∥g(t, x, y)∥ ≤ h(t, x),  (2.2)

where the vector-functions g : [a, b] × ℝ^{2n} → ℝ^{n} and h : [a, b] × ℝ^{n} → ℝ^{n} satisfy the local Caratheodory conditions. Then the boundary-value problem (2.1), (1.2) has at least one solution x ∈ C̃^{1}([a, b]; ℝ^{n}) satisfying the estimate (1.14).

Proof. Put

σ(s, t) = \begin{cases}0 & for s ≤ 0 and τ ∈ ℝ, \\ τ & for |τ| < s, \\ s sign τ & for |τ| ≥ s > 0, \end{cases}

γ(t, x) = \begin{cases}1 & for ∥x − z(t)∥ ≤ φ(t), \\ \frac{φ(t)}{∥x−z(t)∥} & for ∥x − z(t)∥ > φ(t), \end{cases}

σ_{1}(t, x, y) = σ(∥x − z(t)∥ − φ(t), φ′(t)∥x − z(t)∥ − (x − z(t)) · (y − z′(t))),

ỹ(t, x, y) = \begin{cases}y & for ∥x − z(t)∥ ≤ φ(t), \\ y + \frac{σ_{1}(t,x,y)}{∥x−z(t)∥^{2}} (x − z(t)) & for ∥x − z(t)∥ > φ(t), \end{cases}

g_{1}(t, x, y) = x − z(t) + γ(t, x)[f(t, z(t) + γ(t, x)(x − z(t)),

ỹ(t, x, y)) − x + z(t)] − (γ(t, x) − 1)z′′(t),

σ_{2}(t, x, y) = σ[∥x − z(t)∥ − φ(t), ∥ỹ(t, x, y) − z′(t)∥^{2} −

−∥y − z′(t)∥^{2} − (φ′(t))^{2} + (\frac{(x − z(t)) · (y − z′(t))}{∥x − z(t)∥})^{2}],

g_{2}(t, x, y) = \begin{cases}0 for ∥x − z(t)∥ ≤ φ(t), \\ \frac{σ_{2}(t,x,y)+(∥x−z(t)∥−φ(t))(φ′′(t)+1)}{∥x−z(t)∥^{2}}(x − z(t)) \\ for ∥x − z(t)∥ > φ(t) \end{cases}

and

g̃(t, x, y) = g_{1}(t, x, y) + g_{2}(t, x, y).

By the condition (2.2) on [a, b] × ℝ^{2n} we have

∥g̃(t, x, y) − x∥ ≤ h^{∗}(t) + |φ′′(t)| + ∥z′′(t)∥ + 2 + ∥z(t)∥ + φ(t)

where

h^{∗}(t) = sup{h(t, x) : ∥x − z(t)∥ ≤ φ(t)}.

Therefore according to Lemma 2.1 the differential system

x′′ = g̃(t, x, x′)

has at least one solution x ∈ C̃^{1}([a, b]; ℝ^{n}) satisfying the boundary conditions (1.2). Due to the definition of g̃ it remains to show that x admits the estimate (1.14). Assume, on the contrary, that (1.14) is violated. Then there exists t_{0} ∈ [a, b] where the function

u(t) = ∥x(t) − z(t)∥ − φ(t)

reaches its positive maximum on [a, b]. Assume first that t_{0} ∈]a, b[. Then by the Fermat theorem we have

u′(t_{0}) = \frac{(x(t_{0}) − z(t_{0})) · (x′(t_{0}) − z′(t_{0}))}{∥x(t_{0}) − z(t_{0})∥} − φ′(t_{0}) = 0.

Hence

lim_{t→t_{0}} ỹ(t, x(t), x′(t)) = x′(t_{0})

and for a certain positive δ the inequalities

|φ′(t)∥x(t) − z(t)∥ − (x(t) − z(t)) · (x′(t) − z′(t))| <
< ∥x(t) − z(t)∥ − φ(t)  (2.3)

and

|∥ỹ(t, x(t), x′(t)) − z′(t)∥^{2} − ∥x′(t) − z′(t)∥^{2} − (φ′(t))^{2} +

+(\frac{(x(t) − z(t)) · (x′(t) − z′(t))}{∥x(t) − z(t)∥})^{2}| < ∥x(t) − z(t)∥

hold in U_{δ}(t_{0}). Therefore due to the definition of g̃ we obtain

u′′(t) = \frac{(x(t) − z(t)) · (x′′(t) − z′′(t)) + ∥x′(t) − z′(t)∥^{2}}{∥x(t) − z(t)∥} −

−\frac{[\frac{(x(t)−z(t))·(x′(t)−z′(t))}{∥x(t)−z(t)∥}]2}{∥x(t) − z(t)∥} − φ′′(t) =

= \frac{(x(t) − z(t)) · (g_{1}(t, x(t), x′(t)) − z′′(t))}{∥x(t) − z(t)∥} +

+\frac{∥ỹ(t, x(t), x′(t)) − z′(t)∥^{2} − φ(t)φ′′(t) − (φ′(t))^{2}}{∥x(t) − z(t)∥} +

+\frac{∥x(t) − z(t)∥ − φ(t)}{∥x(t) − z(t)∥} for t ∈ U_{δ}(t_{0}).

Furthermore, by (2.3) for each t ∈ U_{δ}(t_{0}) we have

∥γ(t, x(t))(x(t) − z(t))∥ = φ(t)

and

γ(t, x(t))(x(t) − z(t)) · (ỹ(t, x(t), x′(t)) − z′(t)) = φ(t)φ′(t).

Taking into account the last three equalities and Definition 1.1, it can be shown that u′′(t) is positive for each t ∈ U_{δ}(t_{0}). But this is impossible, since t_{0} ∈]a, b[ and t_{0} is a point of maximum for u. Thus t_{0} ∉]a, b[. In view of (1.2) and (1.5_{1}) both a and b are the points of maxima for the function u. Therefore u′(a) ≤ 0 and u′(b) ≥ 0. Assuming u′(a) = 0 or u′(b) = 0, an argument similar to the one carried out above leads us to a contradiction. Thus

u′(a) = \frac{(x(a) − z(a)) · (x′(a) − z′(a))}{∥x(a) − z(a)∥} − φ′(a) < 0

and

u′(b) = \frac{(x(b) − z(b)) · (x′(b) − z′(b))}{∥x(b) − z(b)∥} − φ′(b) > 0.

But since x′(a) = x′(b) and

\frac{x(a) − z(a)}{∥x(a) − z(a)∥} = \frac{x(b) − z(b)}{∥x(b) − z(b)∥},

the last two inequalities yield

\frac{x(a) − z(a)}{∥x(a) − z(a)∥} · (z′(a) − z′(b)) > φ′(b) − φ′(a),

which contradicts (1.5_{2}). Therefore the estimate (1.14) is proved.

Lemma 2.3. Suppose that ρ, r_{1}, and ρ′ are positive constants, ω ∈ C(ℝ_{+}; ]0, +∞[), l ∈ L([t_{1}, t_{2}]; ℝ_{+}) and

∫_{ρ}^{ρ′} \frac{ds}{ω(s)} > r_{1} + ∫_{t_{1}}^{t_{2}} l(t)dt.  (2.4)

Then an arbitrary x ∈ C̃^{1}([t_{1}, t_{2}]; ℝ^{n}), satisfying (1.9) and the inequalities

∥x′(t_{i})∥ ≤ ρ  (2.5_{i})

and

(−1)^{i−1}x′′(t) · η_{ρ}(x′(t)) ≤ ω(∥x′(t)∥)(l(t) + ∥x′(t)∥)
for t_{1} ≤ t ≤ t_{2}  (2.6_{i})

with i ∈ {1, 2}, admits the estimate

∥x′(t)∥ ≤ ρ′ for t_{1} ≤ t ≤ t_{2}.  (2.7)

Proof. Assume for definiteness that i = 1. Admit to the contrary that (2.7) is violated, i.e., there exists t^{∗} ∈]t_{1}, t_{2}] such that

∥x′(t^{∗})∥ > ρ′.  (2.8)

By (2.5_{1}) there exists t_{∗} ∈ [t_{1}, t^{∗}[ such that

∥x′(t)∥ = ρ and ∥x′(t)∥ > ρ′ for t_{∗} ≤ t ≤ t^{∗}.

Hence, taking into account the definition of the function η_{r}, from (2.6_{1}) we get

∥x′(t)∥′ ≤ ω(∥x′(t)∥)(l(t) + ∥x′(t)∥) for t_{∗} ≤ t ≤ t^{∗}.

Dividing this inequality by ω(∥x′(t)∥), integrating from t_{∗} to t^{∗}, and using (1.9) and (2.5_{1}), we obtain

∫_{ρ}^{∥x′(t^{∗})∥}\frac{ds}{ω(s)} ≤ r_{1} + ∫_{t_{1}}^{t_{2}} l(t)dt,

which, on account of (2.8), contradicts (2.4).

Definition 2.1. Suppose that r and r_{1} are positive constants. A vectorfunction x : [α, β] → ℝ^{n} is said to belong to the set W_{n}([α, β], r, r_{1}) if (1.8) implies the estimate (1.9) for arbitrary t_{1} ∈ [α, β] and t_{2} ∈]t_{1}, β].

Lemma 2.4_{1}. Suppose that r and r_{1} are positive constants, l ∈ L([t_{1}, t_{2}]; ℝ_{+}), ω ∈ C(ℝ_{+}; ]0, +∞[), and (1.13) holds. Then there exists a positive constant r′ such that if δ ∈]0, \frac{b−a}{4}[ and an arbitrary x ∈ C̃^{1}([a, b]; ℝ^{n}) ∩W_{n}([a+δ, b−δ], r, r_{1}) satisfies the boundary conditions (1.2), the inequality

∥x′′(t)∥ ≤ l(t) for t ∈]a, a + δ[∪]b − δ, b[,  (2.9)

and furthermore the inequality

x′′(t) · η_{r}(x′(t)) ≤ ω(∥x′(t)∥)(l(t) + ∥x′(t)∥)  (2.10)

on the set [a + δ, b − δ], then x admits an estimate

∥x′(t)∥ ≤ r′ for a ≤ t ≤ b.  (2.11)

Proof. Due to Definition 2.1, without loss of generality we may assume that r(b − a) > 2r_{1}. In view of (1.13) there exist positive numbers r^{∗} and r′ such that

∫_{r}^{r^{∗}} \frac{ds}{ω(s)} > r_{1} + ∫_{a}^{b} l(t)dt  (2.12)

and

∫_{μ}^{r′} \frac{ds}{ω(s)} > r_{1} + ∫_{a}^{b} l(t)dt  (2.13)

where

μ = r^{∗} + ∫_{a}^{b} l(t)dt.

Let us show that r′ is a suitable constant.

First note that for a certain t_{0} ∈[\frac{3a+b}{4}, \frac{a+3b}{4}] we have

∥x′(t_{0})∥ ≤ r.  (2.14)

Indeed, assuming the contrary implies that

∥x′(t)∥ > r for \frac{3a + b}{4} ≤ t ≤ \frac{a + 3b}{4}.

Therefore, taking into account the inequality r(b − a) ≥ 2r_{1}, we obtain

∫_{\frac{3a+b}{4}}^{\frac{a+3b}{4}}∥x′(t)∥dt > r_{1}.

But this is impossible, since x ∈ W_{n}([a + δ, b − δ], r, r_{1}). Thus (2.14) is proved.

Now let us show that

∥x′(t)∥ ≤ r^{∗} for t_{0} ≤ t ≤ b − δ.  (2.15)

Suppose to the contrary that for arbitrary t_{2} ∈]t_{0}, b − δ] we have

∥x′(t_{2})∥ > r^{∗}.  (2.16)

Then by (2.14) there exists t_{1} ∈ [t_{0}, t_{2}[ such that

∥x′(t_{1})∥ = r and ∥x′(t)∥ > r for t_{1} < t ≤ t_{2}.

Hence, taking into account x ∈ W_{n}([a + δ, b − δ], r, r_{1}), we get the estimate (1.9). Assuming i = 1, ρ = r, and ρ′ = r^{∗}, it is easy to verify that x satisfies the other conditions of Lemma 2.4_{1} too. Therefore x admits the estimate (2.7), which contradicts (2.16). Thus (2.15) is proved. In view of (1.2) and (2.9) it implies that

∥x′(t)∥ ≤ μ for t ∈ [a, a + δ] ∪ [b − δ, b].  (2.17)

In particular, ∥x′(a + δ)∥ ≤ μ. Applying Lemma 2.3 where ρ = μ, ρ′ = r′ and i = 1, an argument similar to the one carried out above yields the estimate

∥x′(t)∥ ≤ r′ for a + δ ≤ t ≤ t_{0}.  (2.18)

Finally, from (2.15), (2.17), and (2.18) we obtain the estimate (2.11).

In a similar manner we can prove

Lemma 2.4_{2}. Suppose that the conditions of Lemma 2.4_{1} are satisfied, except that the inequality (2.10) is replaced by

x′′(t) · η_{r}(x′(t)) ≥ −ω(∥x′(t)∥)(l(t) + ∥x′(t)∥).  (2.19)

Then x admits the estimate (2.11).

Lemma 2.5. Suppose that r and r_{1} are positive constants, a ≤ a_{0} < b_{0} ≤ b, l ∈ L([t_{1}, t_{2}]; ℝ_{+}), ω ∈ C(ℝ_{+}; ]0, +∞[), and (1.13) holds. Then there exists a positive constant r′ such that if 0 < δ < min{a_{0} − a, b − b_{0}} and an arbitrary x ∈ C̃^{1}([a, b]; ℝ^{n}) ∩ W_{n}([a + δ, b − δ], r, r_{1}) satisfies the conditions (2.9), the inequality (2.10) on the set ]a_{0}, b−δ[, and, furthermore, the inequality (2.19) on the set ]a + δ, b_{0}[, then x admits the estimate (2.11).

The proof of Lemma 2.5 is similar to the one carried out above for Lemma 2.4_{1}, so we shall note only the main points.

Due to Definition 2.1, without loss of generality we may assume that r(b_{0} − a_{0}) ≥ r_{1}. In view of (1.13) there exists a positive number r^{∗} such that (2.12) holds. The constant

r′ = r^{∗} + ∫_{a}^{b} l(t)dt

is just the suitable one.

First, taking into account the conditions x ∈ W_{n}([a + δ, b − δ], r, r_{1}), we conclude that for a certain t_{0} ∈ [a_{0}, b_{0}] (2.14) is satisfied. Further, applying Lemma 2.3 (i = 1) and the inequality (2.10), we get the estimate (2.15). Finally, from (2.9) and (2.15) it follows that we have ∥x′(t)∥ ≤ r′ on the set [b − δ, b]. Thus the last estimate holds on [t_{0}, b]. Analogously, applying Lemma 2.3 (i = 2) and the inequality (2.19), it can be proved on [a, t_{0}].

In a similar manner we can prove

Lemma 2.6. Suppose that r and r_{1} are positive constants, a < a_{1} < a_{2} < t_{0} < b_{2} < b_{1} < b, l ∈ L([t_{1}, t_{2}]; ℝ_{+}), ω ∈ C(ℝ_{+}; ]0, +∞[), and (1.13) holds. Then there exists a positive constant r′ such that if 0 < δ < min{a_{1}−a, b−b_{1}} and an arbitrary x ∈ C̃^{1}([a, b]; ℝ^{n})∩W_{n}([a+δ, b−δ], r, r_{1}) satisfies the conditions (2.9), the inequality (2.10) on the set ]a_{1}, t_{0}[∪]b_{2}, b[, and furthermore the inequality (2.19) on the set ]a, a_{2}[∪]t_{0}, b_{1}[, then x admits the estimate (2.11).

§ 3. Proof of the Main Results

Proof of Theorem 1.1_{1}. Without loss of generality we can assume that

l(t) ≥ ∥z′′(t)∥ + |φ′′(t)| for a < t < b.

Put

a_{k} = a + \frac{b − a}{4k}, b_{k} = b − \frac{b − a}{4k}

and

χ_{ρ}(s) = \begin{cases}0 & for s ≤ ρ, \\ \frac{2ρ−s}{ρ} & for ρ < s < 2ρ, \\ 0 & for 2ρ ≤ s. \end{cases}

By Definition 1.1 there exists a sequence (ρ_{k})_{k=1}^{+∞} such that lim_{k→+∞} ρ_{k} =

+∞ and for each k ∈ {1, 2, … }

(x − z(t)) · (χ_{ρ_{k}}(∥y∥)f(t, x, y) − z′′(t)) + ∥y − z′(t)∥^{2} ≥ φ(t)φ′′(t)

for a_{k} < t < b_{k}, ∥x − z(t)∥ = φ(t) and (x − z(t)) · (y − z′(t)) = φ(t)φ′(t).

Put

h(t, x, y) = \begin{cases}z′′(t) + \frac{|φ′′(t)|}{φ(t)}(x − z(t)) & for ∥x − z(t)∥ ≤ φ(t), \\ z′′(t) + \frac{|φ′′(t)|}{φ(t)}(x − z(t)) & for ∥x − z(t)∥ > φ(t) \end{cases}

and

f_{k}(t, x, y) = \begin{cases}h(t, x, y) & for t ∉ [a_{k}, b_{k}], \\ χ_{ρ_{k}} (∥y∥)f(t, x, y) & for t ∈ [a_{k}, b_{k}] \end{cases}

(k = 1, 2….). It is easy to verify that for each k ∈ {1, 2, … } the vectorfunction g(t, x, y) = f_{k}(t, x, y) satisfies all the conditions of Lemma 2.2. Therefore the differential system

x′′ = f_{k}(t, x, x′)  (3.1_{k})

has at least one solution x_{k} ∈ C̃^{1}([a, b]; ℝ^{n}) satisfying the boundary conditions (1.2) and the estimate

∥x_{k}(t) − z(t)∥ ≤ φ(t) for a ≤ t ≤ b (k = 1, 2, …).  (3.2_{k})

Choose the positive constants r and r_{1} according to Definition 1.3 and the constant r′ according to Lemma 2.4_{1}, assuming without loss of generality that r ≥ ρ. Then by Lemma 2.4_{1} we obtain

∥x′_{k}(t)∥ ≤ r′ for a ≤ t ≤ b (k = 1, 2, …).  (3.3_{k})

In view of (3.2_{k}) and (3.3_{k}) the sequences (x_{k})_{k=1}^{+∞} and (x′_{k})_{k=1}^{+∞} are uniformly bounded and equicontinuous on [a, b]. So due to the well-known Arzela–Ascoli theorem there exists a sequence (k_{j})_{j=1}^{+∞} such that (x_{k_{j}})_{j=1}^{+∞} and (x′_{k_{j}})_{j=1}^{+∞} uniformly converge on [a, b]. Put

x(t) = lim_{j→∞} x_{k_{j}}(t) for a ≤ t ≤ b.

Due to the definition of the functions f_{k}(k = 1, 2 …) x belongs to the set C̃^{1}([a, b]; ℝ^{n}) and is a solution of (1.1),(1.2).

The proof of Theorems 1.1_{2}, 1.2, and 1.3 is similar to the one carried out above for Theorem 1.1_{1}. The only difference is that one has to apply Lemmas 2.4_{2},2.5, and 2.6 respectively instead of Lemma 2.4_{1}.

Remark 3.1. Theorem 1.1_{1} (Theorem 1.1_{2}) can be strengthened by a slight complication of Lemma 2.4_{1} (Lemma 2.4_{2}). Namely, we can assume that the vector-function f has the property V ([α, β], φ, z) for each segment [α, β] contained in the interval ]a, b] (in the interval [a, b[). In that case the vector-function f may fail to have the property V ([a, b], φ, z).

Proof of Theorem 1.4. Assume to the contrary that x_{i} ∈ C̃^{1}([a, b]; ℝ^{n}) (i = 1, 2) are solutions of the boundary-value problem (1.1),(1.2) and x_{1}(t) ≢ x_{2}(t). Put

x(t) = x_{1}(t) − x_{2}(t) for a ≤ t ≤ b,
u(t) = ∥x(t)∥

and

r = max{∑_{i=1}^{2} ∥x_{i}(t)∥ + ∥x′_{i}(t)∥ : a ≤ t ≤ b}.

Choose the functions l_{i}(t, r)(i = 1, 2) according to the condition of Theorem 1.4. First prove that u′(t) ≢ 0. Indeed, assuming the contrary we have

0 = u′′(t) = \frac{x′′(t) · x(t) + ∥x′(t)∥^{2}}{∥x(t)∥} − \frac{(x(t) · x′(t))^{2}}{∥x(t)∥^{3}} ≥

≥ \frac{x′′(t) · x(t)}{∥x(t)∥} for a < t < b.

Hence by (1.16) we obtain

l_{1}(t, r) ≤ 0 for a < t < b.

But this is impossible, since l_{1}(t, r) is nonnegative and differs from zero on a subset of positive measure of the interval ]a, b[. Thus u′(t) ≢ 0. Therefore, there exists t_{0} ∈]a, b[ such that either

u(t_{0}) > 0, u′(t_{0}) > 0  (3.4)

or

u(t_{0}) > 0, u′(t_{0}) < 0.  (3.5)

Without loss of generality assume that (3.4) holds. Then on [t_{0}, b]

u(t) > 0, u′(t) > 0.  (3.6)

Indeed, if this is not so, then there exists t_{1} ∈]t_{0}, b] such that u′(t_{1}) = 0 and (3.6) holds on [t_{0}, t_{1}[. Applying (1.16) once more we obtain

u′′(t) ≥ \frac{x′′(t) · x(t)}{∥x(t)∥} ≥ l_{1}(t, r)u(t) − l_{2}(t, r)|u′(t)| ≥

≥ −l_{2}(t, r)|u′(t)| for t_{0} < t < t_{1}.

According to the Gronwall–Bellman lemma (see e.g. [6]) the last inequality yields

u′(t_{1}) ≥ u′(t_{0}) exp[− ∫_{t_{0}}^{t_{1}} l_{2}(t, r)dt] > 0.

The obtained contradiction shows that (3.6) holds on [t_{0}, b]. Hence, taking into account the equalities

u(a) = u(b), u′(a) = u′(b)  (3.7)

which follow from the boundary conditions (1.2),we get

u(a) > 0, u′(a) > 0.

Repeating the argument that was carried out above we can show the validity of (3.6) on [a, b], but this contradicts (3.7).

References

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2. L. H. Erbe and P. K. Palamides, Boundary-value problems for secondorder differential equations. J. Math. Anal. Appl. 127(1987), 80-92.

3. G. D. Gaprindashvili, On Solvability of the Dirichlet boundary-value problem for systems of ordinary nonlinear differential equations with singularities. (Russian) Differential’nye Uravneniya 27(1991), No. 9, 1521-1525.

4. G. D. Gaprindashvili, On certain boundary-value problems for systems of second-order nonlinear ordinary differential equations. (Russian) Proc. I.Vekua Inst. Appl. Math. Tbilis. St. Univ. 31(1988), 23-52.

5. M. Nagumo, Über die Differentialgleichung y′′ = f(x, y, y′). Proc. Phys.-Math. Soc. Japan 19(1937), 861-866.

6. Ph. Hartman, Ordinary differential equations. John Wiley & Sons, New York, 1964.

(Received 14.09.1993)

Author’s address: Dept. of Applied Mathematics Georgian Technical University 77, Kostava St., Tbilisi 380075 Republic of Georgia