Georgian Mathematical Journal 1(1994), No. 5, 477-484

ON SOME PROPERTIES OF SOLUTIONS OF SECOND ORDER LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS

I. KIGURADZE

Abstract. The properties of solutions of the equation u′′(t) = p_{1}(t)u(τ_{1}(t)) + p_{2}(t)u′(τ_{2}(t)) are investigated where p_{i} : [a, +∞[→ R (i = 1, 2) are locally summable functions, τ_{1} : [a, +∞[→ R is a measurable function and τ_{2} : [a, +∞[→ R is a nondecreasing locally absolutely continuous one. Moreover, τ_{i}(t) ≥ t (i = 1, 2), p_{1}(t) ≥ 0, p_{2}^{2}(t) ≤ (4 − ε)τ′_{2}(t)p_{1}(t), ε = const > 0 and ∫_{a}^{+∞}(τ_{1}(t) − t)p_{1}(t)dt < +∞. In particular, it is proved that solutions whose derivatives are square integrable on [a, +∞[ form a one-dimensional linear space and for any such solution to vanish at infinity it is necessary and sufficient that ∫_{a}^{+∞} tp_{1}(t)dt = +∞.

Consider the differential equation

u′′(t) = p_{1}(t)u(τ_{1}(t)) + p_{2}(t)u′(τ_{2}(t)),  (1)

where p_{i} : [a, +∞[→ R (i = 1, 2) are locally summable functions, τ_{i} : [a, +∞[→ R (i = 1, 2) are measurable functions and

τ_{i}(t) ≥ t for t ≥ a (i = 1, 2).  (2)

We say that a solution u of the equation (1) is a Kneser-type solution if it satisfies the inequality u′(t)u(t) ≤ 0 for t ≥ a_{0} for some a_{0} ∈ [a, +∞[. A set of such solutions is denoted by K. By W we denote a space of solutions of (1) that satisfy ∫_{a}^{+∞} u′^{2}(t)dt < +∞. The results of [1, 2] imply that if p_{1}(t) ≥ 0 for t ≥ a and the condition

(i) τ_{i}(t) ≡ t, (i = 1, 2), ∫_{a}^{+∞}|p_{2}(t)|dt < +∞,

477 or

1991 Mathematics Subject Classification. 34K15.

1072-947X/1994/0900-0477$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1994 Plenum Publishing Corporation

(ii) p_{2}(t)≤0, for t≥0, ∫_{a}^{+∞}sp_{1}(s)ds<+∞, ∫_{a}^{+∞}\frac{s}{τ_{2}(s)}|p_{2}(s)|ds<+∞,

holds, then W ⊃ K and K is a one-dimensional linear space. The case when the conditions (i) and (ii) are violated, the matter of dimension of K and W and their interconnection has actually remained unstudied. An attempt is made in this note to fill up this gap to a certain extent.

Theorem 1. Let τ_{i}(t) ≥ t (i = 1, 2), p_{1}(t) ≥ 0 for t ≥ a,

∫_{a}^{+∞}[τ_{1}(t) − t]p_{1}(t)dt < +∞,  (3)

and let τ_{2} be a nondecreasing locally absolutely continuous function satisfying

p_{2}^{2}(t) ≤ (4 − ε)τ′_{2}(t)p_{1}(t) for t ≥ a ,  (4)

where ε = const > 0. Then

W ⊂ K, dim W = 1.  (5)

Before proceeding to the proof of the theorem we shall give two auxiliary statements.

Lemma 1. Let the conditions of Theorem 1 be fulfilled and let a_{0} ∈ [a, +∞[ be large enough for the equality

∫_{a_{0}}^{+∞}[τ_{1}(s) − s]p_{1}(s)ds ≤ 4δ^{2},  (6)

where δ = \frac{1}{4}[2 − (4 − ε)^{1/2}], to hold. Then any solution u of the equation (1) satisfies

δ ∫_{t}^{x} [u′^{2}(s) + p_{1}(s)u^{2}(s)]ds ≤ u′(x)u(x) − u′(t)u(t) +

+(1 − δ) ∫_{x}^{τ(x)}u′^{2}(s)ds for a_{0} ≤ t ≤ x < +∞ ,  (7)

SOLUTIONS OF FUNCTIONAL DIFFERENTIAL EQUATIONS 479

where τ(x) = ess sup_{a_{0}≤t≤x}[max_{1≤i≤2} τ_{i}(x)]. Moreover, if u ∈ W, then

u′(t)u(t) ≤ −δ ∫_{t}^{+∞}[u′^{2}(s) + p_{1}(s)u^{2}(s)]ds for t ≥ a_{0}  (8)

and

2δ ∫_{t}^{+∞}(s − t)[u′^{2}(s) + p_{1}(s)u^{2}(s)]ds ≤ u^{2}(t) for t ≥ a_{0}.  (9)

Proof. Let u be any solution of the equation (1). Then

−u′′(t)u(t) + p_{1}(t)u^{2}(t) = p_{1}(t)u(t) ∫_{τ_{1}(t)}^{t} u′(s)ds − p_{2}(t)u′(τ_{2}(t))u(t).

Integrating this equality from t to x, we obtain

u′(t)u(t) − u′(x)u(x) + ∫_{t}^{x} [u′^{2}(s) + p_{1}(s)u^{2}(s)]ds =

= ∫_{t}^{x} [p_{1}(s)u(s) ∫_{τ_{1}(s)}^{s} u′(y)dy]ds − ∫_{t}^{x} p_{2}(s)u′(τ_{2}(s))u(s)ds.

However, in view of (4) and (6),

∫_{t}^{x} [p_{1}(s)u(s) ∫_{τ_{1}(s)}^{s} u′(y)dy]ds ≤ δ ∫_{t}^{x} p_{1}(s)u^{2}(s)ds +

+\frac{1}{4δ} [ ∫_{t}^{x} [τ_{1}(s) − s]p_{1}(s)ds][ ∫_{t}^{τ(x)}u′^{2}(s)ds] ≤

≤ δ ∫_{t}^{x} p_{1}(s)u^{2}(s)ds + δ ∫_{t}^{τ(x)}u′^{2}(s)ds for a_{0} ≤ t ≤ x < +∞

and

−∫_{t}^{x}p_{2}(s)u′(τ_{2}(s))u(s)ds ≤

≤ 2(1 − 2δ) ∫_{t}^{x} [p_{1}(s)u^{2}(s)]^{1/2}[τ′_{2}(s)u′^{2}(τ_{2}(s))]^{1/2}ds ≤

≤ (1 − 2δ) ∫_{t}^{x} p_{1}(s)u^{2}(s)ds + (1 − 2δ) ∫_{t}^{x} τ′_{2}(s)u′^{2}(τ_{2}(s))ds ≤

≤ (1 − 2δ) ∫_{t}^{x} p_{1}(s)u^{2}(s)ds + (1 − 2δ) ∫_{t}^{τ(x)}u′^{2}(s)ds

for a_{0} ≤ t ≤ x < +∞.

Therefore

u′(t)u(t) − u′(x)u(x) + ∫_{t}^{x} [u′^{2}(s) + p_{1}(s)u^{2}(s)]ds ≤

≤(1 − δ) ∫_{t}^{x} [u′^{2}(s)+p_{1}(s)u^{2}(s)]ds+(1 − δ) ∫_{x}^{τ(x)}u′^{2}(s)ds

for a_{0} ≤ t ≤ x < +∞

and thus the inequality (7) holds.

Suppose now that u ∈ W. Then, as one can easily verify,

lim inf_{x→+∞} |u′(x)u(x)| = 0.

So (7) immediately implies (8). Integrating both sides of (8) from t to +∞, we obtain the estimate (9).

Lemma 2. Let the conditions of Lemma 1 be fulfilled and there exist b ∈]a_{0}, +∞[ such that

p_{i}(t) = 0 for t ≥ b (i = 1, 2).  (10)

Then for any c ∈ R there exists a unique solution of the equation (1) satisfying

u(a_{0}) = c, u′(t) = 0 for t ≥ b.  (11)

Proof. In view of (2) and (10), for any α ∈ R the equation (1) has a unique solution v(·; α) satisfying v(t; α) = α for b ≤ t < +∞. Moreover, v(t; α) = αv(t; 1). On the other hand, by Lemma 1 the function v(·; 1) : [a_{0}, +∞[→ R is non increasing and v(a_{0}; 1) ≥ 1. Therefore the function u(·) = \frac{c}{v(a_{0};1)}v(a_{0}; ·) is a unique solution of (1), (11).

Proof of Theorem 1. First of all we shall prove that for any c ∈ R the equation (1) has at least one solution satisfying

u(a_{0}) = c, ∫_{a_{0}}^{+∞}u′^{2}(s)ds < +∞.  (12)

For any natural k put

p_{ik}(t) = \begin{cases}p_{i}(t) & for a_{0} ≤ t ≤ a_{0} + k \\ 0 & for t > a_{0} + k \end{cases} (i = 1, 2).  (13)

According to Lemma 2, for any k the equation u′′(t) = p_{1k}(t)u(τ_{1}(t)) + p_{2k}(t)u′(τ_{2}(t)) has a unique solution u_{k} satisfying

u_{k}(a_{0}) = c, u′_{k}(t) = 0 for t ≥ a + k.  (14)

On the other hand, by Lemma 1

|u_{k}(t)| ≤ |c| for t ≥ a_{0}, 2δ ∫_{a_{0}}^{+∞}(s − a_{0})u′_{k}^{2}(s)ds ≤ c^{2}.  (15)

Taking (2) and (13)–(15) into account, it is easy to show that the sequences (u_{k})_{k=1}^{+∞} and (u′_{k})_{k=1}^{+∞} are uniformly bounded and equicontinuous on each closed subinterval of [a_{0}, +∞[. Therefore, by the Arzela-Ascoli lemma, we can choose a subsequence (u_{k_{m}})_{m=1}^{+∞} out of (u_{k})_{k=1}^{+∞}, which is uniformly convergent alongside with (u′_{k_{m}})_{m=1}^{+∞} on each closed subinterval of [a, +∞[. By (13)–(15) the function u(t) = lim_{m→+∞} u_{k_{m}}(t) for t ≥ a is a solution of the problem (1), (12).

We have thus proved that dim W ≥ 1. On the other hand, by Lemma 1 any solution u ∈ W satisfies (8) and is therefore a Kneser-type solution. To complete the proof it remains only to show that dim W ≤ 1, i.e., that the problem (1), (12) has at most one solution for any c ∈ R. Let u_{1} and u_{2} be two artbitrary solutions of this problem and u_{0}(t) = u_{2}(t) − u_{1}(t). Since u_{0} ∈ W and u_{0}(a_{0}) = 0, by Lemma 1

2 ∫_{a_{0}}^{+∞}(s − a_{0})u′_{0}^{2}(s)ds = 0 and u_{0}(t) = 0 for t ≥ a_{0},

i.e., u_{1}(t) ≡ u_{2}(t).

Remark 1. The condition (4) of Theorem 1 cannot be replaced by the condition

p_{2}^{2}(t) ≤ (4 + ε)τ′_{2}(t)p_{1}(t) for t ≥ a.  (16)

Indeed, consider the equation

u′′(t) = \frac{1}{(4 + ε)t^{2}} u(t) − \frac{1}{t}u′(t),  (17)

satisfying all conditions of Theorem 1 except (4), instead of which the condition (16) is fulfilled. On the other hand, the equation (17) has the solutions

u_{i}(t) = t^{λ_{i}} (i = 1, 2), where λ_{i} = (−1)^{i}(4 + ε)^{− \frac{1}{2}} (i = 1, 2). Clearly, u_{i} ∈ W (i = 1, 2). Therefore in our case instead of (5) we have K ⊂ W, dim W = 2.

Corollary 1. Let the conditions of Theorem 1 be fulfilled. Let, moreover,

p_{2}(t) ≤ 0 for t ≥ a.  (18)

Then

K = W, dim K = 1.  (19)

Proof. Let u ∈ K. Then by virtue of (18) and the non-negativity of p_{1} there exists t_{0} ∈ [a, +∞[ such that u(t)u′(t) ≤ 0, u′′(t)u(t) ≥ 0 for t ≥ t_{0}. Hence

∫_{t_{0}}^{+∞}u′^{2}(s)ds ≤ |u(t_{0})u′(t_{0})|.

Therefore u ∈ W. Thus we have proved that W ⊃ K. This fact, together with (5), implies (19).

A solution u of the equation (1) will be called vanishing at infinity if

lim_{t→+∞} u(t) = 0.  (20)

Theorem 2. Let the conditions of Theorem 1 be fulfilled. Then for any solution u ∈ W to vanish at infinity it is necessary and sufficient that

∫_{a}^{+∞}sp_{1}(s)ds = +∞.  (21)

Proof. Let u ∈ W. Then by Lemma 1 u^{2}(t) ≥ η for t ≥ a_{0}, where η = lim_{t→+∞} u^{2}(t), and ∫_{a_{0}}^{+∞}(s − a_{0})p_{1}(s)u^{2}(s)ds ≤ u^{2}(a_{0})/2δ. Hence it follows that (21) implies η = 0, i.e., u is a vanishing solution at infnity.

To complete the proof it is enough to establish that if

∫_{a}^{+∞}sp_{1}(s)ds < +∞,  (22)

then any nontrivial solution u ∈ W tends to a nonzero limit as t → +∞. Let us assume the contrary: the equation (1) has a nontrivial solution u ∈ W vanishing at infinity. Then by Lemma 1

u(t)u′(t) ≤ 0, ρ(t) ≤ η^{2}u^{2}(t) for t ≥ a_{0},  (23)

where

ρ(t) = ∫_{t}^{+∞}(s − t)[u′^{2}(s) + p_{1}(s)u^{2}(s)]ds, η = (2δ)^{− \frac{1}{2}} .

On the other hand, by (4), (20) and (22) we have

|u(t)| = | ∫_{t}^{+∞}(s − t)[p_{1}(s)u(τ_{1}(s)) + p_{2}(s)u′(τ_{2}(s))]ds| ≤

≤ [ ∫_{t}^{+∞}(s − t)p_{1}(s)ds]^{1/2}[ ∫_{t}^{+∞}(s − t)p_{1}(s)u^{2}(τ_{1}(s))ds]^{1/2} +

+2 ∫_{t}^{+∞}(s − t)[p_{1}(s)]^{1/2}[τ′_{2}(s)]^{1/2}|u′(τ_{2}(s)|ds ≤

≤ [ ∫_{t}^{+∞}(s − t)p_{1}(s)]^{1/2}[ ∫_{t}^{+∞}(s − t)p_{1}(s)u^{2}(τ_{1}(s))ds]^{1/2} +

+2[ ∫_{t}^{+∞}(s − t)p_{1}(s)ds]^{1/2}[ ∫_{t}^{+∞}(s − t)τ′_{2}(s)u′^{2}(τ_{2}(s))ds]^{1/2}

for t ≥ a_{0}.

Hence by (2) and (23) we find

|u(t)| ≤ [ ∫_{t}^{+∞}(s − t)p_{1}(s)ds|^{1/2}[ ∫_{t}^{+∞}(s − t)p_{1}(s)u^{2}(s)ds]^{1/2} +

+2[ ∫_{t}^{+∞}(s − t)p_{1}(s)ds]^{1/2}[ ∫_{t}^{+∞}(s − t)u′^{2}(s)ds]^{1/2} ≤

≤ 3η[ ∫_{t}^{+∞}(s − t)p_{1}(s)ds]^{1/2}|u(t)| for t ≥ a_{0}

and therefore u(t) = 0 for t ≥ a_{1}, where a_{1} is a sufficiently large number. By virtue of (2) the last equality implies u(t) = 0 for t ≥ a. But this is impossible, since by our assumption u is a nontrivial solution. The obtained contradiction proves the theorem.

References

1. I.T. Kiguradze, On the non-negative nonincreasing solutions of nonlinear second order differential equations. Ann. mat. pura ed appl. 81(1969), 169-192.

2. I.T. Kiguradze and D.I. Chichua, Kneser’s problem for functional differential equations. (Russian) Diffferentsial’nye Uravneniya 27(1992), No. 11, 1879-1892.

(Received 03.08.1993)

Author’s address:

A. Razmadze Mathematical Institute

Georgian Academy of Sciences

1, Z. Rukhadze St., Tbilisi 380093

Republic of Georgia