GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 4, 1995, 395-418

ON PROPER OSCILLATORY AND VANISHING AT INFINITY SOLUTIONS OF DIFFERENTIAL EQUATIONS

WITH A DEVIATING ARGUMENT

I. KIGURADZE AND D. CHICHUA

Abstract. Sufficient conditions are found for the existence of multiparametrical families of proper oscillatory and vanishing-at-infinity solutions of the differential equation

u^{(n)}(t) = g(t, u(τ_{0}(t)), …, u^{(m−1)}(τ_{m−1}(t))),

where n ≥ 4, m is the integer part of \frac{n}{2} , g : R_{+} × R^{m} → R is a function satisfying the local Carathéodory conditions, and τ_{i} : R_{+} → R (i = 0, …, m − 1) are measurable functions such that τ_{i}(t) → +∞ for t → +∞ (i = 0, …, m − 1).

Introduction

In this paper we consider the differential equation

u^{(n)}(t) = g(t, u(τ_{0}(t)), …, u^{(m−1)}(τ_{m−1}(t)))  (0.1)

and its particular cases

u^{(n)}(t) = p(t)|u(τ(t))|^{λ} sgn u(τ(t)),  (0.2)
u^{(n)}(t) = p(t)u(τ(t)),  (0.3)

u^{(n)}(t) = ∑_{i=0}^{m−1} p_{i}(t)u^{(i)}(τ_{i}(t)).  (0.4)

Throughout the paper it will be assumed that n ≥ 4, m is integer part of the number \frac{n}{2}, g : R_{+} ×R^{m} → R is a function satisfying the local Carathéodory conditions, p : R_{+} → R and p_{i} : R_{+} → R (i = 0, …, m − 1) are locally

1991 Mathematics Subject Classification. 34K15,34K10. Key words and phrases. Functional differential equation, proper solution, oscillatory

solution, vanishing at infinity solution.

395

1072-947X/95/0700-0395$07.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1995 Plenum Publishing Corporation

summable functions, while τ_{i} : R_{+} → R (i = 0, …, m − 1) and τ : R_{+} → R are measurable functions such that

lim_{t→+∞} τ_{i}(t) = +∞ (i = 0, …, m − 1)  (0.5)

and

lim_{t→+∞} τ(t) = +∞.  (0.6)

Let t_{0} ∈ R_{+}. A function u : [t_{0}, +∞[→ R is called a solution of equation (0.1) if it is locally absolutely continuous together with its derivatives up to order n − 1 inclusive and if there exists an m − 1 times continuously differentiable function \overline{u} : R → R such that \overline{u}(t) = u(t) for t ≥ t_{0} and the equality

u^{(n)}(t) = g(t, \overline{u}(τ_{0}(t)), …, \overline{u}^{(m−1)}(τ_{m−1}(t))).

is fulfilled almost everywhere on [t_{0}, +∞[.

A solution u of equation (0.1) determined on the interval [t_{0}, +∞[ is called proper if it is not identically zero in anyone of the neighborhoods of +∞ and is called vanishing-at-infinity if u(t) → 0 for t → +∞.

A proper solution is called oscillatory if it has a sequence of zeros converging to +∞, and nonoscillatory otherwise.

In the papers dealing with oscillatory properties of differential equations with deviating arguments it is always assumed a priori that the considered equation has proper solutions and sufficient conditions are established for these solutions to be oscillatory (see, for example, [1–9] and the references cited therein). However, the problem of the existence of proper solutions is far from being trivial and has not yet been investigated for a wide class of equations.^{1}

Therefore the question as to the existence of at least one oscillatory solution of such equations remains open. We do not know, for example, of a single result on the existence of oscillatory solutions of equations like (0.1), (0.2), or (0.3) when

τ_{i}(t) > t (i = 0, …, m − 1), τ(t) > t for t ≥ t_{0},  (0.7)

though such equations occur rather frequently in the oscillation theory. Further, it is not likewise clear for us whether (0.1), (0.2) or (0.3) have at least one proper solution vanishing-at-infinity. Hence this paper deals with these two open problems.

In §1 we prove, by means of the results of [10], theorems on the existence and uniqueness of two auxiliary boundary value problems with integral conditions for differential equations with a deviating argument. Using these theorems and the oscillation theorems from [11], in §§2 and 3 we establish sufficient conditions for equations (0.1)–(0.4) to have multiparametric families of proper oscillatory and vanishing-at-infinity solutions.^{2}

1Equations with a delay for which this problem is studied in [1] are an exception.

Throughout the paper the following notation will be used.

μ_{i}^{k} (i = 0, 1, …; k = 2i, 2i + 1, …) are the numbers given by the recurrent relations

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, …).

m is the integer part of \frac{n}{2}; m_{0} is the integer part of \frac{n}{4};

γ_{n} = ∑_{j=0}^{m_{0}−1} \frac{n!}{(2m − 2 − 4j)!} μ_{m−1−2j}^{n};

γ_{0n} = \frac{m − 1}{4}[\frac{(2m)!γ_{n}}{n!μ_{m}^{n}} + \frac{(m − 2)(4m^{2} − m − 3)}{3} + 4]^{m−1} − (−1)^{m} \frac{n!}{2}.

§ 1. Auxiliary Boundary Value Problems

For the differential equations

u^{(n)}(t) = h(t, u(t), u(τ_{0}(t)), …, u^{(m−1)}(τ_{m−1}(t))),  (1.1)
u^{(n)}(t) = ∑_{i=0}^{m−1} p_{i}(t)u^{(i)}(τ_{i}(t)) + q(t)  (1.1′)

we consider the boundary value problems

u^{(i)}(0) = c_{i} (i = 0, …, m − 1), ∫_{0}^{+∞} |u^{(m)}(t)|^{2}dt < +∞; (1.2)

\begin{matrix}u^{(i)}(0) = c_{i} (i = 0, …, m − 1), \\ ∫_{0}^{+∞} t^{2j}|u^{(j)}(t)|^{2}dt < +∞ (j = 0, …, m), \end{matrix}  (1.3)

where n ≥ 4, c_{i} ∈ R (i = 0, …, m − 1),

h : R_{+} × R^{m+1} → R satisfies the local
Carathéodory conditions,  (1.4)

p_{i} : R_{+} → R (i = 0, …, m − 1) and q : R_{+} → R are the locally summable functions, and τ_{i} : R_{+} → R_{+} (i = 0, …, m − 1) are measurable functions satisfying condition (0.5).

2When τ_{i}(t) ≡ t (i = 0, …, m − 1) and τ(t) ≡ t, sufficient conditions for equations (0.1)–(0.4) to have proper oscillatory and vanishing-at-infinity solutions are obtained in [11–15].

Alongside with the notation listed in the Introduction we shall need in this section the following notation as well:

τ_{0∗}(t) = min{t, τ_{0}(t)}, τ_{0}^{∗}(t) = max{t, τ_{0}(t)}.

L is the space of locally Lebesgue integrable functions v : R_{+} → R with a topology of convergence in the mean on each segment from R_{+}.

C^{n−1} is the topological space of (n − 1)-times continuously differentiable real functions given on R_{+}. By the convergence of the sequence (u_{k})_{k=1}^{+∞} of elements from this space we mean the uniform convergence of sequences (u_{k}^{(i)})_{k=1}^{+∞} (i = 0, …, n − 1) on each finite segment from R_{+}.

C_{0}^{n−1,m} = {u ∈ C^{n−1} : ∫_{0}^{+∞} |u^{(m)}(t)|^{2}dt < +∞};

C^{n−1,m} = {u ∈ C^{n−1} : ∫_{0}^{+∞} t^{2i}|u^{(i)}(t)|^{2}dt < +∞ (i = 0, …, m)};

∥u∥_{0,m} = [ ∑_{i=0}^{m−1} |u^{(i)}(0)|^{2} + ∫_{0}^{+∞} |u^{(m)}(t)|^{2}dt]^{\frac{1}{2}} ;

∥u∥_{m} = [ ∫_{0}^{+∞}(1 + t)^{2m}|u^{(m)}(t)|^{2}dt]^{\frac{1}{2}} .

Theorem 1.1. Let on R_{+} × R^{m+1} the conditions

|h(t, x, x_{0}, x_{1}, …, x_{m−1}) − h(t, x, x, 0, …, 0)| ≤
≤ a_{10}(t)|x − x_{0}|^{λ_{0}} + ∑_{i=1}^{m−1} a_{1i}(t)|x_{i}|^{λ_{i}},  (1.5)
(−1)^{n−m−1}h(t, x, x, 0, …, 0)x ≥ −a(t),  (1.6)

be fulfilled, where λ_{i} ∈ [0, 1] (i = 0, …, m − 1), a_{1i} : R_{+} → R_{+} (i = 0, …, m − 1), and a : R_{+} → R_{+} are measurable functions such that

∫_{0}^{+∞}(1 + t)^{n−m− \frac{1}{2}} [a_{10}(t)(1 + τ_{0}^{∗}(t))^{(m− \frac{3}{2} )λ_{0}}|τ_{0}(t) − t|^{λ_{0}} +

+ ∑_{i=1}^{m−1} a_{1i}(t)(1 + τ_{i}(t))^{(m−i− \frac{1}{2} )λ_{i}}]dt < μ_{m}^{n},  (1.7)

∫_{0}^{+∞}(1 + t)^{n−2m}a(t)dt < +∞.  (1.8)

Then problem (1.1), (1.2) has at least one solution.

Proof. Let r = ∑_{i=0}^{m−1} |c_{i}|. By (1.7) and (1.8) there is a positive number ε such that the functions

\begin{matrix}a_{1}(t) = (1 + ε)(1 + t)^{m− \frac{1}{2}} [a_{10}(t)(1 + τ_{0}^{∗}(t))^{(m− \frac{3}{2} )λ_{0}}|τ_{0}(t) − t|^{λ_{0}} + \\ + ∑_{i=1}^{m−1} a_{1i}(t)(1 + τ_{i}(t))^{(m−i− \frac{1}{2} )λ_{i}}], \\ a_{2}(t) = (1 + \frac{1}{ε})(1 + r)^{2}a_{1}(t) + a(t) \end{matrix}(1.9)

will satisfy the inequalities

∫_{0}^{+∞}(1+t)^{n−2m}a_{1}(t)dt<μ_{m}^{n}, ∫_{0}^{+∞}(1+t)^{n−2m}a_{2}(t)dt<+∞. (1.10)

For any u ∈ C^{n−1} we set

χ(u) = \begin{cases}1 & for ∑_{i=0}^{m−1} |u^{(i)}(0)| ≤ r \\ r + 1 − ∑_{i=0}^{m−1} |u^{(i)}(0)| & for r < ∑_{i=0}^{m−1} |u^{(i)}(0)| ≤ 1 + r \\ 0 & for ∑_{i=0}^{m−1} |u^{(i)}(0)| > 1 + r \end{cases} ,

f(u)(t) = χ(u)h(t, u(t), u(τ_{0}(t)), …, u^{(m−1)}(τ_{m−1}(t))).

The operator f : C^{n−1} → L is continuous on account of (1.4). On the other hand, it is obvious that problem (1.1),(1.2) is solvable if and only if the functionally differential equation

u^{(n)}(t) = f(u)(t)  (1.11)

has at least one solution satisfying the boundary conditions (1.2).

Using Theorem 1.1 from [10], we shall prove below that problem (1.11), (1.2) is solvable.

If u ∈ C^{n−1,m}, then by (1.5) and (1.6) we obtain

(−1)^{n−m−1}u(t)f(u)(t) =
= (−1)^{n−m−1}χ(u)[h(t, u(t), u(τ_{0}(t)), …, u^{(m−1)}(τ_{m−1}(t))) −
−h(t, u(t), u(t), 0, …, 0)]u(t) +
+(−1)^{n−m−1}χ(u)h(t, u(t), u(t), 0, …, 0)u(t) ≥
≥ −a_{10}(t)χ(u)|u(τ_{0}(t)) − u(t)|^{λ_{0}}|u(t)| −

400 I. KIGURADZE AND D. CHICHUA

− ∑_{i=1}^{m−1} a_{1i}(t)χ(u)|u^{(i)}(τ_{i}(t))|^{λ_{i}}|u(t)| − a(t),  (1.12)

|f(u)(t)| ≤ |h(t, u(t), u(t), 0, …, 0)| + χ(u)a_{10}(t)|u(τ_{0}(t)) − u(t)|^{λ_{0}} +

+χ(u) ∑_{i=1}^{m−1} a_{1i}(t)|u(τ_{i}(t))|^{λ_{i}} ≤

≤ b_{0}(t, |u(t)|) + χ(u) ∑_{i=0}^{m−1} a_{1i}(t)|u^{(i)}(τ_{i}(t))|,  (1.13)

where

b_{0}(t, x) = ∑_{i=0}^{m−1} a_{1i}(t) + a_{10}(t)x +

+ max {|h(t, s, s, 0, ⋯, 0)| : 0 ≤ s ≤ x}.  (1.14)

On the other hand, for an arbitrary i ∈ {0, …, m − 1} we have

|u^{(i)}(t)| = | ∑_{j=i}^{m−1} \frac{t^{j−i}}{(j − i)!}u^{(j)}(0) + \frac{1}{(m − i − 1)!} ∫_{0}^{t}(t − s)^{m−i−1}u^{(m)}(s)ds| ≤

≤ (1 + t)^{m−1−i} ∑_{j=0}^{m−1} |u^{(j)}(0)| +

+\frac{1}{(m − i − 1)!}( ∫_{0}^{t}(t − s)^{2m−2i−2}ds)^{\frac{1}{2}} ( ∫_{0}^{t} |u^{(m)}(s)|^{2}ds)^{\frac{1}{2}} ≤

≤ (1 + t)^{m−i− \frac{1}{2}} [ ∑_{j=0}^{m−1} |u^{(j)}(0)| + ( ∫_{0}^{+∞} |u^{(m)}(s)|^{2}ds)^{\frac{1}{2}} ] ≤

≤ (1 + t)^{m−i− \frac{1}{2}} [ ∑_{i=0}^{m−1} |u^{(i)}(0)| + ∥u∥_{0,m}] ≤

≤ (1 + t)^{m−i− \frac{1}{2}} [(1 + \frac{1}{ε})( ∑_{i=0}^{m−1} |u^{(i)}(0)|)^{2} + (1 + ε)∥u∥_{0,m}^{2}]^{\frac{1}{2}} . (1.15)

Therefore

χ(u)|u^{(i)}(τ_{i}(t))|^{λ_{i}}|u(t)| ≤

≤ (1 + τ_{i}(t))^{(m−i− \frac{1}{2} )λ_{i}}(1 + t)^{m− \frac{1}{2}} [(1 + \frac{1}{ε})(1 + r)^{2} +

+(1 + ε)∥u∥_{0,m}^{2}]^{\frac{1+λ_{i}}{2}} ≤ (1 + τ_{i}(t))^{(m−i− \frac{1}{2} )λ_{i}}(1 + t)^{m− \frac{1}{2}} ×

×[1 + (1 + \frac{1}{ε})(1 + r)^{2} + (1 + ε)∥u∥_{0,m}^{2}]  (1.16)

and

χ(u)|u(τ_{0}(t)) − u(t)|^{λ_{0}}|u(t)| = χ(u)| ∫_{t}^{τ_{0}(t)} u′(s)ds|^{λ_{0}}|u(t)| ≤
≤ (1 + τ_{0}^{∗}(t))^{(m− \frac{3}{2} )λ_{0}}|τ_{0}(t) − t|^{λ_{0}}(1 + t)^{m− \frac{1}{2}} ×
×[1 + (1 + \frac{1}{ε})(1 + r)^{2} + (1 + ε)∥u∥_{0,m}^{2}].  (1.17)

By (1.9) and (1.15)–(1.17) it follows from (1.12) and (1.13) that

(−1)^{n−m−1}u(t)f(u)(t) ≥ −a_{1}(t)∥u∥_{0,m}^{2} − a_{2}(t),
|f(u)(t)| ≤ b(t, |u(t)|, ∥u∥_{0,m}),  (1.18)

where

b(t, x, y) = b_{0}(t, x) +

+ ∑_{i=0}^{m−1} a_{1i}(t)(1 + τ_{i}(t))^{m−i− \frac{1}{2}} [(1 + \frac{1}{ε})(1 + r)^{2} + (1 + ε)y^{2}]^{\frac{1}{2}} .

Moreover, the functions a_{1} and a_{2} satisfy inequalities (1.10) and b the condition

lim_{\begin{matrix}t→0 \\ y→+∞ \end{matrix}} (y^{−2} ∫_{0}^{t} b(s, x, y)ds) = 0 for x ∈ R_{+}.  (1.19)

Thus all the conditions of Theorem 1.1 from [10] are fulfilled, thereby guaranteeing the solvability of problem (1.11),(1.2).

Similarly to Theorem 1.1 we prove

Theorem 1.1′. Let on R_{+} × R^{m+1} the conditions

|h(t, x, x_{0}, x_{1}, ..., x_{m−1})−h(t, x, x, x_{1}, ..., x_{m−1})|≤a_{1}(t)|x−x_{0}|^{λ_{0}}, (1.5′)
(−1)^{n−m−1}h(t, x, x, x_{1}, …, x_{m−1})x ≥ −a(t),  (1.6′)

be fulfilled, where λ_{0} ∈ [0, 1], a_{1} : R_{+} → R_{+} and a : R_{+} → R_{+} are measurable functions such that

∫_{0}^{+∞}(1 + t)^{n−m− \frac{1}{2}} (1 + τ_{0}^{∗}(t))^{(m− \frac{3}{2} )λ_{0}} |τ_{0}(t) − t|^{λ_{0}}a_{1}(t)dt < μ_{m}^{n} (1.7′)

and inequality (1.8) is fulfilled. Besides, let for some t_{0} > 0 on the set [0, t_{0}] × R^{m+1} the inequality

|h(t, x, x, x_{1}, …, x_{m−1})| ≤ b_{0}(t, |x|) ∑_{i=1}^{m−1}(1 + x_{i}^{2})

hold, where b_{0} : [0, t_{0}] × R_{+} → R_{+} is the function summable with respect to the first argument and nondecreasing with respect to the second. Then problem (1.1), (1.2) has at least one solution.

Theorem 1.2. Let on R_{+} × R^{m} the conditions

|h(t, x, \overline{x}_{0}, ..., \overline{x}_{m−1})−h(t, x, x_{0}, ..., x_{m−1})|≤ ∑_{i=0}^{m−1} a_{1i}(t)|\overline{x}_{i}−x_{i}|, (1.20)
(−1)^{n−m−1}[h(t, \overline{x}, x_{0}, …, x_{m−1}) −
−h(t, x, x_{0}, …, x_{m−1})](\overline{x} − x) ≥ a_{00}(t)(\overline{x} − x)^{2},  (1.21)
(−1)^{n−m−1}[h(t, x, \overline{x}_{0}, x_{1}, …, x_{m−1}) −
−h(t, x, x_{0}, x_{1}, …, x_{m−1})](\overline{x}_{0} − x_{0}) ≥ a_{01}(t)(\overline{x}_{0} − x_{0})^{2}, (1.22)

be fulfilled, where a_{1i} : R_{+} → R_{+} (i = 0, ⋯, m − 1) and a_{0j} : R_{+} → R (j = 0, 1) are measurable functions satisfying inequality (1.7) for λ_{i} = 1 (i = 0, …, m − 1) and

a_{0}(t) = a_{00}(t) + a_{01}(t) ≥ 0 for t > 0.  (1.23)

Then problem (1.1), (1.2) has at most one solution. If, however, in addition to (1.7) and (1.20)–(1.23) we have the conditions

h^{2}(t, 0, ⋯, 0) ≤ l(t)a_{0}(t) for t > 0,
∫_{0}^{+∞}(1 + t)^{n−2m} l(t)dt < +∞ ,  (1.24)

then problem (1.1), (1.2) has one and only one solution.

Proof. First we shall prove the uniqueness of the solution. Let u and \overline{u} be two arbitrary solutions of problem (1.1), (1.2). It is assumed that v(t) = \overline{u}(t) − u(t),

Δ_{0}(t) = h(t, \overline{u}(t), \overline{u}(τ_{0}(t)), …, \overline{u}^{(m−1)}(τ_{m−1}(t))) −
−h(t, u(t), \overline{u}(τ_{0}(t)), …, \overline{u}^{(m−1)}(τ_{m−1}(t))),  (1.25)
Δ_{1}(t) = h(t, u(t), \overline{u}(τ_{0}(t)), \overline{u}′(τ_{1}(t)), …, \overline{u}^{(m−1)}(τ_{m−1}(t))) −
−h(t, u(t), u(τ_{0}(t)), \overline{u}′(τ_{1}(t)), …, \overline{u}^{(m−1)}(τ_{m−1}(t))),  (1.26)
Δ(t) = h(t, u(t), u(τ_{0}(t)), \overline{u}′(τ_{1}(t)), …, \overline{u}^{(m−1)}(τ_{m−1}(t))) −

ON PROPER OSCILLATING AND VANISHING 403

−h(t, u(t), u(τ_{0}(t)), u′(τ_{1}(t)), …, u^{(m−1)}(τ_{m−1}(t))),  (1.27)
l_{1}(t) = \begin{cases}\frac{Δ_{1}(t)}{v(τ_{0}(t))} & for v(τ_{0}(t)) ≠ 0 \\ 0 & for v(τ_{0}(t)) = 0 \end{cases} .  (1.28)

It is clear that

v^{(i)}(0) = 0 (i = 0, …, m − 1), v ∈ C_{0}^{n−1,m}.

Therefore

|v^{(i)}(t)| ≤ (1 + t)^{m−i− \frac{1}{2}} ∥v∥_{0,m} (i = 0, …, m − 1),
|v(τ_{0}(t)) − v(t)| ≤ (1 + τ_{0}^{∗}(t))^{m− \frac{3}{2}} |τ_{0}(t) − t|∥v∥_{0,m}.

On the other hand, on account of (1.20)–(1.23) and (1.25)–(1.28) we have

\begin{matrix}(−1)^{n−m−1}Δ_{0}(t)v(t) ≥ a_{00}(t)v^{2}(t), (−1)^{n−m−1}l_{1}(t) ≥ a_{01}(t), \\ |l_{1}(t)| ≤ a_{10}(t), |Δ(t)| ≤ ∑_{i=1}^{m−1} a_{1i}(t)|v^{(i)}(τ_{i}(t))| \end{matrix} (1.29)

and (−1)^{n−m−1}v(t)v^{(n)}(t) = (−1)^{n−m−1}Δ_{0}(t)v(t) + (−1)^{n−m−1}l_{1}(t)v^{2}(t) +

+(−1)^{n−m−1}l_{1}(t)[v(τ_{0}(t)) − v(t)]v(t) + +(−1)^{n−m−1}Δ(t)v(t) ≥ ≥ a_{0}(t)v^{2}(t) − |l_{1}(t)||v(τ_{0}(t)) − v(t)||v(t)| − |Δ(t)||v(t)| ≥

≥ −a_{10}(t)|v(τ_{0}(t)) − v(t)||v(t)| − ∑_{i=1}^{m−1} a_{1i}(t)|v^{(i)}(τ_{i}(t))||v(t)|.

Therefore

(−1)^{n−m}(1 + t)^{n−2m}v(t)v^{(n)}(t) ≤ (1 + t)^{n−2m}\overline{a}(t)∥v∥_{0,m}, (1.30)

where

\overline{a}(t)=(1+t)^{m− \frac{1}{2}} [a_{10}(t)(1+τ_{0}^{∗}(t))^{m− \frac{3}{2}} |τ_{0}(t)−t|+∑_{i=1}^{m−1} a_{1i}(t)(1+τ_{i}(t))^{m−i− \frac{1}{2}} ].

On integrating inequality (1.30) from 0 to t and applying Lemmas 4.1 and 4.4 from [11], we obtain

μ_{m}^{n} ∫_{0}^{t} |v^{(m)}(s)|^{2}ds ≤ w(t) + ∥v∥_{0,m}^{2} ∫_{0}^{t}(1 + s)^{n−2m}\overline{a}(s)ds, where

w(t) = (n − 2m) ∑_{i=0}^{n−m−1}(−1)^{n−m−i}(i + 1)v^{(i)}(t)v^{(n−2−i)}(t) −

−(1 + t)^{n−2m} ∑_{i=0}^{n−m−1}(−1)^{n−m−i}v^{(i)}(t)v^{(n−1−i)}(t);

moreover,

lim inf_{t→+∞} |w(t)| = 0.

It is therefore clear that

μ_{m}^{n}∥v∥_{0,m}^{2} ≤ ∥v∥_{0,m}^{2} ∫_{0}^{+∞}(1 + t)^{n−2m}\overline{a}(s)ds.

Hence by (1.7) we find that ∥v∥_{0,m} = 0. Thus problem (1.1), (1.2) has at most one solution.

To complete the proof of the theorem it remains to show that if in addition to (1.7) and (1.20)–(1.23) condition (1.24) is fulfilled, too, then problem (1.1), (1.2) is solvable.

By virtue of (1.21)–(1.24)

(−1)^{n−m−1}h(t, x, x, 0, …, 0)x =
= (−1)^{n−m−1}[h(t, x, x, 0, …, 0) − h(t, 0, x, …, 0)]x +

+(−1)^{n−m−1}[h(t, 0, x, …, 0)−h(t, 0, …, 0)]x+(−1)^{n−m−1}h(t, 0, …, 0)x≥

≥ a_{0}(t)x^{2} − l^{\frac{1}{2}} (t)a0^{\frac{1}{2}} (t)|x| ≥ −a(t),

where a(t) = \frac{1}{4}l(t) satisfies condition (1.8). Thus all the conditions of Theorem 1.1 are fulfilled, thereby guaranteeing the solvability of problem (1.1), (1.2).

When h(t, x, x_{0}, x_{1}, …, x_{m−1}) = ∑_{i=0}^{m−1} p_{i}(t)x_{i} + q(t) Theorem 1.2 implies

Corollary 1.1. Let (−1)^{n−m−1}p_{0}(t) ≥ 0 for t ∈ R_{+},

∫_{0}^{+∞}(1 + t)^{n−m− \frac{1}{2}} [|p_{0}(t)|(1 + τ_{0}^{∗}(t))^{m− \frac{3}{2}} |τ_{0}(t) − t| +

+ ∑_{i=1}^{m−1} |p_{i}(t)|(1 + τ_{i}(t))^{m−i− \frac{1}{2}} ]dt < μ_{m}^{n},

q^{2}(t) ≤ l(t)|p_{0}(t)| for t ∈ R_{+}, ∫_{0}^{+∞}(1 + t)^{n−2m}l(t)dt < +∞.

Then problem (1.1′), (1.2) has one and only one solution.

Theorem 1.3. Let on R_{+} × R^{m+1} condition (1.5) and

(−1)^{n−m−1}h(t, x, x, 0, …, 0)x ≥ γ(1 + t)^{−n}x^{2} − a_{2}(t)  (1.31)

be fulfilled, where λ_{i} ∈ [0, 1] (i = 0, …, m − 1), γ is a positive constant, a_{1i} : R_{+} → R_{+} (i = 0, …, m − 1), and a_{2} : R_{+} → R_{+} are measurable functions such that

δ = \frac{n!}{(2m)!}μ_{m}^{n} − ∫_{0}^{+∞}(1 + t)^{n− \frac{1}{2}} [a_{10}(t)(1 + τ_{0∗}(t))^{− \frac{3}{2} λ_{0}}|τ_{0}(t) − t|^{λ_{0}} +

+ ∑_{i=1}^{m−1} a_{1i}(t)(1 + τ_{i}(t))^{−(i+ \frac{1}{2} )λ_{i}}]dt > 0,  (1.32)

∫_{0}^{+∞}(1 + t)^{n}a_{2}(t)dt < +∞,  (1.33)
γ > \frac{m − 1}{4}γ_{n}[\frac{γ_{n}}{δ} + \frac{(m−2)(4m^{2}−m−3)}{3} +4]^{m−1}−(−1)^{m} \frac{n!}{2}. (1.34)

Then problem (1.1), (1.3) has at least one solution.

Proof. Problem (1.1), (1.3) is equivalent to problem (1.11), (1.3), where f(u)(t) = h(t, u(t), u(τ_{0}(t)), …, u^{(m−1)}(τ_{m−1}(t))). Using Theorem 1.3 from [10], we shall prove that problem (1.11), (1.2) is solvable. First of all we would like to note that the operator f : C^{n−1} → L is continuous on account of (1.4). On the other hand, for any u ∈ C^{n−1,m} inequalities (1.5) and (1.31) imply

(−1)^{n−m−1}u(t)f(u)(t) =
= (−1)^{n−m−1}[h(t, u(t), u(τ_{0}(t)), …, u^{(m−1)}(τ_{m−1}(t))) −

−h(t, u(t), u(t), 0, …, 0)]u(t) + (−1)^{n−m−1}h(t, u(t), u(t), 0, …, 0)u(t) ≥

≥ −a_{10}(t)|u(τ_{0}(t)) − u(t)|^{λ_{0}}|u(t)| − ∑_{i=1}^{m−1} a_{1i}(t)|u^{(i)}(τ_{i}(t))|^{λ_{i}}|u(t)| +

+γ(1 + t)^{−n}|u(t)|^{2} − a_{2}(t).  (1.35)

However, for any u ∈ C^{n−1,m} and i ∈ {0, …, m − 1} we have the representation

u^{(i)}(t) = \frac{1}{(m − 1 − i)!} ∫_{+∞}^{t}(t − s)^{m−1−i}u^{(m)}(s)ds,

Therefore

|u^{(i)}(t)| ≤ ∫_{t}^{+∞}(1 + s)^{m−1−i}|u^{(m)}(s)|ds ≤

≤ [ ∫_{t}^{+∞}(1 + s)^{−2−2i}ds]^{\frac{1}{2}} [ ∫_{t}^{+∞}(1 + s)^{2m}|u^{(m)}(s)|^{2}ds]^{\frac{1}{2}} ≤
≤ (1 + t)^{−i− \frac{1}{2}} ∥u∥_{m} (i = 0, …, m − 1) ,  (1.36)
|u(τ_{0}(t)) − u(t)|^{λ_{0}}|u(t)| = | ∫_{t}^{τ_{0}(t)} u′(s)ds|^{λ_{0}}|u(t)| ≤
≤ (1 + τ_{0∗}(t))^{− \frac{3}{2} λ_{0}}(1 + t)^{− \frac{1}{2}} |τ_{0}(t) − t|^{λ_{0}}∥u∥_{m}^{1+λ_{0}} ≤
≤ (1 + τ_{0∗}(t))^{− \frac{3}{2} λ_{0}}(1 + t)^{− \frac{1}{2}} |τ_{0}(t) − t|^{λ_{0}}(1 + ∥u∥_{m}^{2}).  (1.37)

On account of (1.36) and (1.37) inequality (1.35) implies

(−1)^{n−m−1}u(t)f(u)(t) ≥ γ(1 + t)^{−n}|u(t)|^{2} − a_{1}(t)∥u_{m}∥^{2} − ã_{2}(t),

where

a_{1}(t) = (1 + t)^{− \frac{1}{2}} [a_{10}(t)(1 + τ_{0∗}(t))^{− \frac{3}{2} λ_{0}}|τ_{0}(t) − t|^{λ_{0}} +
+ ∑_{i=1}^{m−1} a_{1i}(t)(1 + τ_{i}(t))^{−(i+ \frac{1}{2} )λ_{i}}], ã_{2}(t) = a_{1}(t) + a_{2}(t). (1.38)

Moreover, by virtue of (1.5), (1.14), and (1.36), inequality (1.18) holds, where

b(t, x, y) = b_{0}(t, x) + ∑_{i=0}^{m−1} a_{1i}(t)(1 + τ_{i}(t))^{−i− \frac{1}{2}} y  (1.39)

and b_{0} is the function given by equality (1.14). On the other hand, by (1.32), (1.38) it is obvious that

δ = \frac{n!}{(2m)!}μ_{m}^{n} − ∫_{0}^{+∞}(1 + t)^{n}a_{1}(t)dt > 0  (1.40)

and the function b satisfies condition (1.19).

Thus all the conditions of Theorem 1.3 from [10] are satisfied, thereby guaranteeing the solvability of problem (1.11), (1.3).

Similarly to Theorem 1.3 we prove

Theorem 1.3′. Let on R_{+} × R^{m+1} the conditions (1.5′) and

(−1)^{n−m−1}h(t, x, x, x_{1}, …, x_{m−1})x ≥ γ(1 + t)^{−n}x^{2} − a_{2}(t)

be fulfilled, where λ_{0} ∈ [0, 1], γ is a positive constant and a_{i} : R_{+} → R_{+} (i = 1, 2) are measurable functions such that

δ = \frac{n!}{(2m)!}μ_{m}^{n} − ∫_{0}^{+∞}(1 + t)^{n− \frac{1}{2}} (1 + τ_{0∗}(t))^{− \frac{3}{2} λ_{0}} |τ_{0}(t) − t|^{λ_{0}}a_{1}(t)dt > 0

and inequalities (1.33) and (1.34) are fulfilled. Moreover, let for some t_{0} > 0 on [0, t_{0}] × R^{m+1} the inequality

|h(t, x, x, x_{1}, …, x_{m−1})| ≤ b_{0}(t, |x|) ∑_{i=1}^{m−1}(1 + x_{i}^{2})

hold, where b_{0} : [0, t_{0}]×R_{+} → R_{+} is a function summable with respect to the first argument and nondecreasing with respect to the second. Then problem (1.1), (1.3) has at least one solution.

Theorem 1.4. Let on R_{+} × R^{m+1} conditions (1.20) − (1.22) be fulfilled, where a_{1i} : R_{+} → R_{+} (i = 0, …, m − 1) and a_{0j} : R_{+} → R (j = 0, 1) are measurable functions, and there exists a positive number γ such that

a_{0}(t) = a_{00}(t) + a_{01}(t) > γ(1 + t)^{−n} for t ∈ R_{+}  (1.41)

and inequalities (1.32) and (1.34) hold for λ_{i} = 1 (i = 0, …, m − 1). Then problem (1.1), (1.3) has at least one solution. If in addition to (1.20)−(1.22), (1.32), (1.34), and (1.41) the condition

∫_{0}^{+∞}(1 + t)^{n} \frac{h^{2}(t, 0, …, 0)}{a_{0}(t)}dt < +∞  (1.42)

is fulfilled, too, then problem (1.1), (1.3) has one and only one solution.

Proof. As noted above, problem (1.1), (1.3) is equivalent to problem (1.11), (1.3), where f(u)(t) = h(t, u(t), u(τ_{0}(t)), …, u^{(m−1)}(τ_{m−1}(t))). Let us show that (1.11), (1.3) has at most one solution.

Let u and \overline{u} be arbitrary functions from C^{n−1,m} and v(t) = \overline{u}(t) − u(t). Then the representation

(−1)^{n−m−1}(\overline{u}(t) − u(t))(f(\overline{u})(t) − f(u)(t)) =
= (−1)^{n−m−1}l_{1}(t)v^{2}(t) + (−1)^{n−m−1}l_{1}(t)[v(τ_{0}(t)) − v(t)]v(t) +
+(−1)^{n−m−1}Δ_{0}(t)v(t) + (−1)^{n−m−1}Δ(t)v(t)  (1.43)

is valid, where Δ_{0}, Δ, and l_{1} are functions given by equalities (1.25)–(1.28). Inequalities (1.29) are fulfilled by (1.20)–(1.22). On the other hand,

|v^{(i)}(t)| ≤ (1 + t)^{−i− \frac{1}{2}} ∥v∥_{m} = (1 + t)^{−i− \frac{1}{2}} ∥\overline{u} − u∥_{m} (i = 0, …, m − 1),

|v(τ_{0}(t)) − v(t)| = | ∫_{t}^{τ_{0}(t)} v′(s)ds| ≤ (1 + τ_{0∗}(t))^{− \frac{3}{2}} |τ_{0}(t) − t|∥\overline{u} − u∥_{m}.

Therefore (1.41) and (1.43) imply

(−1)^{n−m−1}(\overline{u}(t) − u(t))(f(\overline{u})(t) − f(u)(t)) ≥
≥ γ(1 + t)^{−n}(\overline{u}(t) − u(t))^{2} − a_{1}(t)∥\overline{u} − u∥_{m}^{2},

where a_{1} is the function given by equality (1.38) for λ_{i} = 1 (i = 0, …, m−1) and satisfying condition (1.40). Therefore by Theorem 1.3 from [10] problem (1.1), (1.3) has at most one solution.

Now let condition (1.42) be fulfilled. Without loss of generality it can be assumed that the inequality (1 − ε)a_{0}(t) > γ(1 + t)^{−n}, where ε is a positive constant, holds instead of (1.41). Then (1.21) and (1.22) imply

(−1)^{n−m−1}h(t, x, x, 0, …, 0)x ≥ a_{0}(t)x^{2} − |h(t, 0, …, 0)||x| ≥

≥ γ(1 + t)^{−n}x^{2} + εa_{0}(t)x^{2} − 2ε^{\frac{1}{2}} a0^{\frac{1}{2}} (t)|x|a2^{\frac{1}{2}} (t) ≥ γ(1 + t)^{−n}x^{2} − a_{2}(t),

where a_{2}(t) = \frac{h^{2}(t,0,...,0)}{4εa_{0}(t)}. Moreover, since on account of (1.42) condition (1.33) is satisfied, by Theorem 1.3 problem (1.1), (1.3) is solvable.

The proven theorem immediately implies

Corollary 1.2. Let (−1)^{n−m−1}p_{0}(t) > γ(1 + t)^{−n} for t ∈ R_{+},

δ = \frac{n!}{(2m)!}μ_{m}^{n} − ∫_{0}^{+∞}(1 + t)^{n− \frac{1}{2}} [|p_{0}(t)|(1 + τ_{0∗}(t))^{− \frac{3}{2}} |τ_{0}(t) − t| +

+ ∑_{i=1}^{m−1} |p_{i}(t)|(1 + τ_{i}(t))^{−i− \frac{1}{2}} ]dt > 0,

∫_{0}^{+∞}(1 + t)^{n} \frac{q^{2}(t)}{|p_{0}(t)|}dt < +∞,

where γ is a positive constant satisfying inequality (1.34). Then problem (1.1′), (1.3) has one and only one solution.

§ 2. Oscillatory Solutions

2.1. Equations with Property O_{m}. We introduce

Definition 2.1. Equation (0.1) has property O_{m} if each proper solution u : [t_{0}, +∞[→ R of this equation, satisfying the condition

∫_{t_{0}}^{+∞} |u^{(m)}(t)|^{2}dt < +∞,  (2.1)

is oscillatory when m is even, and either oscilatory or satisfying, on some interval [t^{∗}, +∞[⊂ [t_{0}, +∞[, the inequalities

(−1)^{i}u^{(i)}(t)u(t) > 0 (i = 0, …, n − 1)  (2.2)

when m is odd.

Before we proceed to formulating the theorem on equation (0.1) having property O_{m} we shall give the following auxiliary statement.

Lemma 2.1. Let the function u : [t_{0}, +∞[→ R be locally absolutely continuous together with its derivatives up to order n − 1 inclusive and satisfy the inequalities

u(t) ≠ 0, mes {s ∈ [t, +∞[: u^{(n)}(s) ≠ 0} > 0 for t ≥ t_{0}, (2.3)
(−1)^{n−m−1}u^{(n)}(t)u(t) ≥ 0 for t ≥ t_{0}.  (2.4)

Then there are k ∈ {0, …, n} and t^{∗} ∈ [t_{0}, +∞[ such that k + m is odd and

\begin{matrix}u^{(i)}(t)u(t) > 0 (i = 0, …, k − 1), \\ (−1)^{i−k}u^{(i)}(t)u(t) > 0 (i = k, …, n − 1) for t ≥ t^{∗}. \end{matrix}  (2.5)

Moreover, if k = 0, then t^{∗} = t_{0} and therefore

(−1)^{i}u^{(i)}u(t) > 0 (i = 0, …, n − 1) for t ≥ t_{0}.  (2.6)

The above lemma immediately follows from Lemma 1.1 in the monograph [11].

For an arbitrary ε > 0 and an arbitrary positive λ ≠ 1 we set

D_{ε}(τ_{0}, …, τ_{m−1}) =

= \{(t, x_{0}, …, x_{m−1}) : t ≥ \frac{1}{ε}, |x_{i}| ≤ ε[τ_{i}(t)]^{m− \frac{1}{2} −i} (i = 0, …, m − 1)\},

σ(λ) = \begin{cases}n − m + (m − 1)λ & for 0 < λ < 1 \\ n − 1 & for λ > 1 and m is even \\ n + λ − 2 & for λ > 1 and m is odd \end{cases} .

Theorem 2.1. Let for some ε > 0

τ_{i}(t) ≥ t for t ≥ ε^{−1} (i = 0, …, m − 1)  (2.7)

and on the set D_{ε}(τ_{0}, …, τ_{m−1}) the inequality

(−1)^{n−m−1}g(t, x_{0}, …, x_{m−1}) sgn x_{0} ≥ p_{0}(t)|x_{0}|^{λ}  (2.8)

hold, where λ ≠ 1 is a positive constant and p_{0} : R_{+} → R_{+} is a locally summable function such that

∫_{0}^{+∞} t^{σ(λ)}p_{0}(t)dt = +∞.  (2.9)

Then equation (0.1) has property O_{m}.

Proof. Assume the contrary, i.e., that equation (0.1) has no property O_{m}. Then there is a proper nonoscillatory solution u : [t_{0}, +∞[→ R of this equation satisfying condition (2.1). Moreover, if m is odd, then on each interval [t^{∗}, +∞[⊂ [t_{0}, +∞[ at least one of inequalities (2.2) does not hold.

By condition (2.1) it can be assumed without loss of generality that t_{0} ≥ ε^{−1}, u(t) ≠ 0 and (t, u(τ_{0}(t)), …, u^{(m−1)}(τ_{m−1}(t))) ∈ D_{ε}(τ_{0}, …, τ_{m−1}) for t ≥ t_{0}. Then inequalities (2.3) and (2.4) are fulfilled on account of (2.7)–(2.9). By Lemma 2.1 there is t^{∗} ≥ t_{0} such that we have

u′(t)u(t) > 0 for t ≥ t^{∗},  (2.10)

but if m is odd, then

u′(t)u(t) > 0, u′′(t)u(t) > 0 for t ≥ t^{∗}.  (2.11)

Let g_{0}(t) = g(t, u(τ_{0}(t)), …, u^{(m−1)}(τ_{m−1}(t)))|u(t)|^{−λ} sgn u(t). Then

u^{(n)}(t) = g_{0}(t)|u(t)|^{λ} sgn u(t).  (2.12)

On the other hand, due to (2.8) and the fact that the function u has a constant sign we have

(−1)^{n−m−1}g_{0}(t) ≥ η(t)p_{0}(t) for t ≥ t_{0},  (2.13)

where η(t) = |u(τ_{0}(t))|^{λ}|u(t)|^{−λ}. Moreover, by (2.7) and (2.10) we have η(t) ≥ 1 for t ≥ t^{∗}. Therefore (2.9) and (2.13) imply

(−1)^{n−m−1}g_{0}(t)≥0 for t≥t_{0}, ∫_{0}^{+∞}t^{σ(λ)}|g_{0}(t)|dt=+∞. (2.14)

By virtue of condition (2.14) and Theorems 15.1, 15.2, and 15.4 from the monograph [11] we conclude that for the even m (odd m), equation (2.12) has no proper nonoscillatory solution satisfying condition (2.1) (conditions (2.1) and (2.11)). The obtained contradiction proves the theorem.

Quite similarly, using Theorems 1.6 and 1.7 from [11] we shall prove

Theorem 2.2. Let inequalities (2.7) be fulfilled for some ε > 0 and on the set D_{ε}(τ_{0}, …, τ_{m−1}) the condition

(−1)^{n−m−1}g(t, x_{0}, …, x_{m−1}) sgn x_{0} ≥ p_{0}(t)|x_{0}|  (2.15)

hold, where p_{0} : R_{+} → R_{+} is a locally summable function such that

lim sup_{t→+∞} (t ∫_{t}^{+∞} s^{n−2}p_{0}(s)ds) > (n − 1)! .  (2.16)

Then equation (0.1) has property O_{m}.

2.2. Theorem on the Existence of Proper Oscillatory Solutions of Equation (0.1).

Theorem 2.3. Let for some ε > 0

τ_{i}(t) ≥ t + Δ(t) for t ≥ ε^{−1} (i = 0, …, m − 1)  (2.17)

and on the D_{ε}(τ_{0}, …, τ_{m−1}) the conditions

(−1)^{n−m−1}g(t, x_{0}, …, x_{m−1})x_{0} ≥ 0,  (2.18)
|g(t, x, x_{1}, ..., x_{m−1})−g(t, x_{0}, x_{1}, ..., x_{m−1})|≤l(t)|x−x_{0}|^{λ_{0}} (2.19)

hold, where λ_{0} ∈ [0, 1], Δ : R_{+} →]0, +∞[ is a continuous function and l : R_{+} → R_{+} is a measurable function such that

∫_{ε^{−1}}^{+∞}(1+t)^{n−m− \frac{1}{2}} (1+τ_{0}(t))^{(m− \frac{3}{2} )λ_{0}}(τ_{0}(t)−t)^{λ_{0}}l(t)dt<+∞. (2.20)

Moreover, let equation (0.1) have property O_{m}. Then for the even m (odd m) this equation has an m-parametric ((m−1)-parametric) family of proper oscillatory solutions.

Proof. Choose t_{0} ≥ \frac{1}{ε} such that

∫_{t_{0}}^{+∞}(1+t)^{n−m− \frac{1}{2}} (1+τ_{0}(t))^{(m− \frac{3}{2} )λ_{0}}(τ_{0}(t)−t)^{λ_{0}}l(t)dt<μ_{m}^{n}. (2.21)

It can be assumed without loss of generality that τ_{i}(t) = t for 0 ≤ t ≤ t_{0} (i = 0, …, m − 1). We set

χ_{i}(t, x) = \begin{cases}x & for |x| ≤ ε[τ_{i}(t)]^{m− \frac{1}{2} −i} \\ ε[τ_{i}(t)]^{m− \frac{1}{2} −i} sgn x & for |x| > ε[τ_{i}(t)]^{m− \frac{1}{2} −i} \end{cases} ,

h(t, x, x_{0}, x_{1}, …, x_{m−1}) =
= \begin{cases}0 & for 0 ≤ t ≤ t_{0} \\ g(t, χ_{0}(t, x_{0}), …, χ_{m−1}(t, x_{m−1})) & for t ≥ t_{0} \end{cases}  (2.22)

and for any c_{0}, …, c_{m−1} which are not simultaneously equal to zero we consider problem (1.1), (1.2).

Due to (2.17)–(2.19), (2.21), and (2.22), conditions (1.5′), (1.6′), and (1.7′) are fulfulled with a_{1}(t) = 0 for a ≤ t ≤ t_{0}, a_{1}(t) = l(t) for t ≥ t_{0}, and a(t) = 0 for t ≥ 0.

By Theorem 1.1′, problem (1.1), (1.2) has a solution u. From (2.17), (2.18), and (2.22) it follows that u is a proper solution. On the other hand, by condition (2.1) there is t^{∗} ≥ t_{0} such that (t, u(τ_{0}(t)), …, u^{(m−1)}(τ_{m−1}(t))) ∈ D_{ε}(τ_{0}, …, τ_{m−1}) for t ≥ t^{∗}. Hence due to (2.22) it is obvious that u is a solution of equation (0.1) on [t^{∗}, +∞[.

However, by our assumption equation (0.1) has property O_{m}. Therefore, when m is even, u is the oscillatory solution, and when m is odd, it is either oscillatory or satisfies inequalities (2.2) on the interval [t^{∗}, +∞[. If u satisfies (2.2), then by (1.2), (2.18), and (2.22) we shall have

(−1)^{i}c_{i}c_{j} > 0 (i = 0, …, m − 1).  (2.23)

Thus if at least one of inequalities (2.23) is not fulfilled, say, c_{m−1} = 0, then u will be an oscillatory solution. We have thereby shown that when m is even (m is odd), to arbitrary numbers c_{0}, …, c_{m−2} (c_{0}, …, c_{m−1}), which are not simultaneously equal to zero, there corresponds at least one oscillatory solution of equation (0.1).

By Theorems 2.1 and 2.2, Theorem 2.3 gives rise to the following propositions.

Corollary 2.1. Let inequalities (2.17) be fulfilled for some ε > 0 and on the set D_{ε}(τ_{0}, …, τ_{m−1}) conditions (2.8) and (2.19) hold, where λ ≠ 1 is a positive constant λ_{0} ∈ [0, 1], Δ : R_{+} →]0, +∞[ is a continuous function, while p_{0} : R_{+} → R_{+} and l_{0} : R_{+} → R_{+} are locally summable functions satisfying conditions (2.9) and (2.20). Then for the even m (odd m) equation (0.1) has a m-parametric ((m − 1)-parametric) family of proper oscillatory solutions.

Corollary 2.2. Let inequalities (2.17) be fulfilled for some ε > 0 and on the set D_{ε}(τ_{0}, …, τ_{m−1}) conditions (2.15) and (2.19) hold, where λ_{0} ∈ [0, 1], Δ : R_{+} →]0, +∞[ is a continuous function, while p_{0} : R_{+} → R_{+} and l : R_{+} → R_{+} are locally integrable functions satisfying conditions (2.16) and (2.20). Then for even m (odd m) equation (0.1) has an m-parametric ((m − 1)-parametric) family of proper oscillatory solutions.

2.3. Sufficient Conditions for the Existence of Proper Oscillatory Solutions of Equations (0.2) and (0.3). Conditions 2.1 and 2.2 imply the following propositions.

Corollary 2.3. Let for some t_{0} > 0 the inequalities

τ(t) ≥ t + Δ(t), (−1)^{n−m−1}p(t) ≥ 0 for t ≥ t_{0}  (2.24)

hold, where Δ : [t_{0}, +∞[→]0, +∞[ is a continuous function. Moreover, let

∫_{t_{0}}^{+∞} t^{σ(λ)}|p(t)|dt = +∞

and

∫_{t_{0}}^{+∞}(1 + t)^{n−m− \frac{1}{2}} (1 + τ(t))^{(m− \frac{1}{2} )λ−λ_{0}}(τ_{0}(t) − t)^{λ_{0}}|p(t)|dt < +∞,

where λ_{0} = λ for 0 < λ < 1 and λ_{0} = 1 for λ > 1. Then for even m (odd m) equation (0.2) has an m-parametric ((m − 1)-parametric) family of proper oscillatory solutions.

Corollary 2.4. Let for some t_{0} > 0 inequalities (2.24) hold, where Δ : [t_{0}, +∞[→]0, +∞[ is a continuous function. Moreover, let

lim sup_{t→+∞} (t ∫_{t}^{+∞} s^{n−2}|p(s)|ds) > (n − 1)!

and

∫_{t_{0}}^{+∞}(1 + t)^{n−m− \frac{1}{2}} (1 + τ(t))^{m− \frac{3}{2}} (τ(t) − t)|p(t)|dt < +∞.

Then for even m (odd m) equation (0.3) has an m-parametric ((m − 1)- parametric) family of proper oscillatory solutions.

§ 3. Vanishing-at-Infinity Solutions

3.1. Existence Theorem for Equation (0.1). For any s ∈ R and ε > 0 we set

[s]_{+} = \frac{1}{2}(|s| + s),

D_{ε}^{∗}(τ_{1}, …, τ_{m−1}) = \{(t, x_{0}, x_{1}, …, x_{m−1}) : t ≥ \frac{1}{ε},

|x_{0}| ≤ εt^{− \frac{1}{2}} , |x_{i}| ≤ [τ_{i}(t)]^{−i− \frac{1}{2}} (i = 0, …, m − 1)\}.

Theorem 3.1. Let for some ε > 0

τ_{i}(t) ≥ t + Δ(t) for t ≥ ε^{−1} (i = 0, …, m − 1)  (3.1)

and on the set D_{ε}^{∗}(τ_{1}, …, τ_{m−1}) the inequalities

(−1)^{n−m−1}g(t, x_{0}, …, x_{m−1})x_{0} ≥ [γ(1 + t)^{−n}x_{0}^{2} − l_{0}(t)]_{+}, (3.2)
|g(t, x, x_{1}, …, x_{m−1}) − g(t, x_{0}, x_{1}, …, x_{m−1})| ≤ l(t)|x − x_{0}|^{λ_{0}} (3.3)

hold, where

γ > γ_{0n},  (3.4)

λ_{0} ∈ [0, 1], Δ : R_{+} →]0, +∞[ is a continuous function, and l : R_{+} → R_{+} are measurable functions such that

∫_{ε^{−1}}^{+∞}t^{n}l_{0}(t)dt<+∞, ∫_{ε^{−1}}^{+∞}t^{n− \frac{1}{2} − \frac{3}{2} λ_{0}}(τ_{0}(t)−t)^{λ_{0}}l(t)dt<+∞. (3.5)

Then for even m (odd m) equation (0.1) has an m-parametric ((m − 1)- parametric) family of vanishing-at-infinity proper oscillatory solutions.

Proof. By the definition of γ_{0n} and condition (3.5) there is t_{0} > \frac{1}{ε} such that

δ = \frac{n!}{(2m)!}μ_{m}^{n} − ∫_{t_{0}}^{+∞}(1 + t)^{n− \frac{1}{2} − \frac{3}{2} λ_{0}}(τ_{0}(t) − t)^{λ_{0}}l(t)dt > 0 (3.6)

and inequality (1.34) is fulfilled. It can be assumed without loss of generality that τ_{i}(t) = t for 0 ≤ t ≤ t_{0} (i = 0, …, m − 1).

Let

χ_{0}(t, x) = \begin{cases}x & for |x| ≤ εt^{− \frac{1}{2}} \\ εt^{− \frac{1}{2}} sgn x & for |x| > εt^{− \frac{1}{2}} \end{cases} ,

χ(t, x) = \begin{cases}1 & for x = 0 \\ \frac{χ_{0}(t,x)}{x} & for x ≠ 0 \end{cases} .

If i ∈ {1, …, m − 1}, then

χ_{i}(t, x) = \begin{cases}x & for |x| ≤ ε[τ_{i}(t)]^{− \frac{1}{2} −i} \\ ε[τ_{i}(t)]^{− \frac{1}{2} −i} sgn x & for |x| > ε[τ_{i}(t)]^{− \frac{1}{2} −i} \end{cases} .

We set

h(t, x, x_{0}, …, x_{m−1}) = γ(1 + t)^{−n}x for 0 ≤ t ≤ t_{0},  (3.7)
h(t, x, x_{0}, …, x_{m−1}) = γ(1 + t)^{−n}x +
+χ(t, x)[g(t, χ_{0}(t, x_{1}), …, χ_{m−1}(t, x_{m−1}))−γ(1+t)^{−n}χ_{0}(t, x)] (3.8)
for t > t_{0}.

Let c_{0}, …, c_{m−1} be arbitrary numbers which are not simultaneously equal to zero. Moreover, if m is odd, then c_{m−1} = 0. We shall consider problem (1.1), (1.3).

By virtue of (3.1)–(3.8) all the conditions of Theorem 1.3′ are fulfilled, where a_{i}(t) = 0 for 0 ≤ t ≤ t_{0} (i = 1, 2), a_{1}(t) = l(t) and a_{2}(t) = l_{0}(t) for t ≥ t_{0}, b_{0}(t, x) ≡ γ(1 + t)^{−n}x. Therefore problem (1.1), (1.3) has a solution u which due to (3.1), (3.2), (3.7), and (3.8) is proper and satisfies the inequalities

\begin{matrix}(−1)^{n−m−1}u^{(n)}(t)u(τ_{0}(t)) ≥ 0, \\ mes \{s ∈ [t, +∞[: u^{(n)}(s) ≠ 0\} > 0 for t ≥ 0. \end{matrix}  (3.9)

On the other hand, by Lemma 4.5 from [11],

lim_{t→+∞} (t^{t+ \frac{1}{2}} u^{(i)}(t)) = 0 (i = 0, …, m − 1).  (3.10)

By Lemma 2.1 it follows from (3.9) and (3.10) that for even m the solution u is oscillatory and for odd m it is either oscillatory or satisfies the inequalities

(−1)^{i}u^{(i)}(t)u(t) > 0 for t ≥ 0 (i = 0, …, n − 1).

The latter assertion, however, can be discarded because when m is odd, then u^{(m−1)}(0) = c_{m−1} = 0. Therefore u is an oscillatory solution for odd m as well.

By (3.10) there is t^{∗} > t_{0} such that (t, u(τ_{0}(t)), …, u^{(m−1)}(τ_{m−1}(t))) ∈ D_{ε}^{∗}(τ_{1}, …, τ_{m−1}) for t > t^{∗}. Hence on account of (3.8) it is clear that u is a solution of equation (0.1) on [t^{∗}, +∞[. We have thereby shown that when m is even (m is odd), to arbitrary numbers c_{0}, …, c_{m−1} (c_{0}, …, c_{m−2}) which are not simultaneously zero, there corresponds at least one vanishing-at-infinity proper oscillatory solution of equation (0.1).

3.2. Corollaries for Equation (0.2).

Corollary 3.1. Let λ > 1 and the conditions

τ(t) ≥ t + Δ(t), (−1)^{n−m−1}p(t) > 0,  (3.11)

\begin{matrix}∫_{t_{0}}^{+∞} |t^{n}p(t)|^{− \frac{2}{λ−1}} dt < +∞, \\ ∫_{t_{0}}^{+∞} t^{n− \frac{3+λ}{2}} (τ(t) − t)|p(t)|dt < +∞ \end{matrix}  (3.12)

be fulfilled for some t_{0} > 0 and a continuous function Δ : [t_{0}, +∞[→]0, +∞[. Then for the even m (odd m) equation (0.2) has the m-parametric ((m−1)- parametric) family of proper oscillatory solutions.

Proof. Let γ be an arbitrary positive number satisfying inequality (3.4). Then by the Young inequality we obtain

|p(t)||x_{0}|^{λ+1} ≥ γx_{0}^{2} − l_{0}(t) for t ≥ t_{0},  (3.13)

where

l_{0}(t) = γ^{\frac{λ+1}{λ−1}} (1 + t)^{− \frac{n(n+1)}{λ−1}} |p(t)|^{− \frac{2}{λ−1}} .

We set ε = \frac{1}{t_{0}} , τ_{0}(t) = τ(t), τ_{i}(t) = t + Δ(t) (i = 1, …, m − 1), and g(t, x_{0}, …, x_{m−1}) = p(t)|x_{0}|^{λ} sgn x_{0}. By (3.11)–(3.13) inequalities (3.1) are now fulfilled and on the set D_{ε}^{∗}(τ_{1}, …, τ_{m−1}) conditions (3.2) and (3.3) hold, where λ_{0} = 1 and l(t) = λt^{− \frac{λ−1}{2}} |p(t)|. Moreover, l_{0} and l satisfy conditions (3.5). Thus all the conditions of Theorem 3.1 are fulfilled.

The propositions below are proved quite similarly.

Corollary 3.2. Let 0 < λ < 1 and the conditions

τ(t) ≥ t + Δ(t), (−1)^{n−m−1}t^{n+ \frac{1−λ}{2}} p(t) ≥ η for t ≥ t_{0},
∫_{t_{0}}^{+∞} t^{n− \frac{1+3λ}{2}} |τ(t) − t|^{λ}|p(t)|dt < +∞

hold for some t_{0} > 0, η > 0 and a continuous function Δ : [t_{0}, +∞[→ ]0, +∞[. Then for even m (odd m) equation (0.2) has an m-parametric ((m − 1)-parametric) family of vanishing-at-infinity proper oscillatory solutions.

3.3. Biernacki’s Problem for Equations (0.3) and (0.4). ByZ^{(n)}(p; τ) and Z^{(n)}(p_{0}, …, p_{m−1}; τ_{0}, …, τ_{m−1}) we denote respectively the spaces of vanishing-at-infinity solutions of equations (0.3) and (0.4), and by dim Z we denote the dimension of the space Z. For the case τ(t) ≡ t we set Z^{(n)}(p) = Z^{(n)}(p; τ). M. Biernacki [12] showed that if p is continuously differentiable and p(t) ↓ −∞ for t → +∞, then dim Z^{(4)}(p) ≥ 1, and he put forward the hypothesis that the inequality dim Z^{(4)}(p) ≥ 2 holds under the same restrictions on p. This hypothesis was later substantiated by M. Švec [13]. More exactly, he proved a more general proposition: if p is continuous and for some t_{0} > 0 and η > 0 satisfies the inequality p(t) ≤ −η for t ≥ t_{0}, then dim Z^{(4)}(p) ≥ 2. The question about dimension of the space of vanishing-at-infnity solutions of linear homogeneous differential equations of an arbitrary order was initially treated in [14]. ^{3} In particular, it is shown there that if p is locally summable and (−1)^{n−m−1}t^{n}p(t) → +∞ for t → +∞, then dim Z^{(n)}(p) ≥ m. The problem of dimensions of the spaces Z^{(n)}(p; τ) and Z^{(n)}(p_{0}, …, p_{m−1}; τ_{0}, …, τ_{m−1}) has never been studied for the cases τ(t) ≢ 0 and τ_{i}(t) ≢ t (i = 0, …, m − 1).

Theorem 3.2. If

lim inf_{t→+∞} [(−1)^{n−m−1}t^{n}p_{0}(t)]>γ_{0n}, ∫_{0}^{+∞}t^{n− \frac{1}{2}} p̃(t)dt<+∞, (3.14)

where p̃(t) = (1 + τ_{0∗}(t))^{− \frac{3}{2}} |τ_{0}(t) − t||p_{0}(t)| + ∑_{i=1}^{m−1}(1 + |τ_{i}(t)|)^{−i− \frac{1}{2}} |p_{i}(t)| and τ_{0∗}(t) = min{t, |τ_{0}(t)|}, then

dim Z^{(n)}(p_{0}, …, p_{m−1}; τ_{0}, …, τ_{m−1}) ≥ m.  (3.15)

Proof. By (0.5) and (3.14) there are positive numbers t_{0} and γ such that τ_{i}(t) > 0 (i = 0, …, m − 1), (−1)^{n−m−1}p_{0}(t) > γ(1 + t)^{−n} for t ≥ t_{0},

δ = \frac{n!}{(2m)!}μ_{m}^{n} − ∫_{t_{0}}^{+∞} t^{n− \frac{1}{2}} p̃(t)dt > 0

3See also §§4 and 5 of [11] where a detailed account of the results connected with this problem is given.

and inequality (1.34) holds. It can be assumed without loss of generality that p_{0}(t) = 2γ(1 + t)^{−n}, p_{i}(t) = 0 (i = 1, …, m − 1), and τ_{i}(t) = t (i = 0, …, m − 1) for 0 ≤ t ≤ t_{0}. Now, obviously, all the conditions of Corollary 1.2 will be fulfilled. Therefore problem (0.2), (1.3) has one and only one solution for any c_{0}, …, c_{m−1}. However, as mentioned above, this solution is vanishing at infinity, and therefore inequality (3.15) is valid.

The theorem proved for equation (0.3) gives rise to

Corollary 3.3. If

lim inf_{t→+∞} [(−1)^{n−m−1}t^{n}p(t)] > γ_{0n}, ∫_{0}^{+∞} t^{n− \frac{1}{2}} p̃(t)dt < +∞,

where p̃(t) = (1 + τ_{∗}(t))^{− \frac{3}{2}} |τ(t) − t||p(t)| and τ_{∗}(t) = min\{t, |τ(t)|\}, then

dim Z^{(n)}(p; τ) ≥ m.

References

1. R. G. Koplatadze and T. A. Chanturia, On oscillatory properties of differential equations with a deviating argument. (Russian) Tbilisi Univ. Press, Tbilisi, 1977.

2. Christos G. Philos, An oscillatory and asymptotic classification of the solutions of differential equations with deviating arguments. Atti. Acad. Naz. Lincei. Rend. Cl. Sci. fis. mat. e natur. 63(1977), No. 3–4, 195–203.

3. V. N. Shevelo, Oscillation of solutions of differential equations with a deviating argument. (Russian) Naukova Dumka, Kiev, 1978.

4. Yu. I. Domshlak, A comparison method by Shturm for investigation of behavior of solutions of differential-operator equations. (Russian) Elm, Baku, 1986.

5. U. Kitamura, Oscillation of functional differential equations with general deviating arguments. Hiroshima Math. J. 15(1985), 445–491.

6. M. E. Drakhlin, On oscillation properties of some functional differential equations. (Russian) Differentsial’nye Uravneniya 22(1986), No. 3, 396–402.

7. J. Jaroš and T. Kusano, Oscillation theory of higher order linear functional differential equations of neutral type. Hiroshima Math. J. 18(1988), 509–531.

8. R. G. Koplatadze, On differential equations with deviating arguments having properties A and B. (Russian) Differentsial’nye Uravneniya 25(1989), No. 11, 1897–1909.

9. R. G. Koplatadze, On monotone and oscillatory solutions of nth order differential equations with deviating arguments. (Russian) Mathematica Bohemica 116(1991), No. 3, 296–308.

10. I. Kiguradze and D. Chichua, On some boundary value problems with integral conditions for functional differential equations. Georgian Math. J. 2(1995), No. 2, 165–188.

11. I. T. Kiguradze and T. A. Chanturia, Asymptotic properties of solutions of nonautonomous ordinary differential equations. Kluwer Academic Publishers, Dodrecht, Boston, London, 1993.

12. M. Biernacki, Sur l’équation differentielle y′′ + A(x)y = 0. Prace Ann. Univ. M. Curie–Sklodiwska 6(1953), 65–78.

13. M. Švec, Sur le comportement asymptotique des intégrales de l’équation differentielle y^{(u)} + Q(x)y = 0. Czechosl. Math. J. 8(1958), No. 2, 450–462.

14. I. T. Kiguradze, On vanishing-at-infinity solutions of ordinary differential equations. Czechosl. Math. J. 33(1983), No. 4, 613–646.

15. M. Bartušek, Asymptotic properties of oscillatory solutions of differential equations of the nth order. Masaryk University, Brno, 1992.

(Received 08.12.1993)

Authors’ addresses:

I. Kiguradze A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, Rukhadze St., Tbilisi 380093 Republic of Georgia

D. Chichua I. Vekua Institute of Applied Mathematics Tbilisi State University 2, University St., Tbilisi 380043 Republic of Georgia