GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 1, 1995, 93-98

EXISTENCE OF CONJUGATE POINTS FOR SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS

A. LOMTATIDZE

Abstract. The sufficient conditions are established under which the second-order linear differential equation is conjugate.

1. Consider the differential equation

u′′ = p(t)u,  (1)

where −∞ < a < b < +∞ and the function p :]a, b[→ ℝ is Lebesgue integrable on each compact subset of ]a, b[.

Definition 1. A function p :]a, b[→ ℝ belongs to the class 𝕆(]a, b[) if the condition

∫_{a}^{b}(s − a)(b − s)|p(s)|ds < +∞  (2)

is satisfied and the solution of equation (1), satisfying the initial conditions

u(a+) = 0, u′(a+) = 1,  (3)

has at least one zero on ]a, b[.

It is known [1] that if condition (2) is satisfied then any solution u of (1) has the finite left- and right-hand side limits u(a+) and u(b−) and problem (1),(3) is uniquely solvable.

Definition 2. A function p :]a, b[→ ℝ belongs to the class 𝕆′(]a, b[) if

∫_{a}^{b}(s − a)|p(s)|ds < +∞  (4)

and the derivative of the solution of problem (1),(3) has at least one zero on ]a, b].

93

1991 Mathematics Subject Classification. 34C10.

Key words and phrases. Second-order linear differential equation, conjugate point.

1072-947X/95/0100-0093$07.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1995 Plenum Publishing Corporation

While investigating two-point singular boundary-value problems there arises a question as to an effective description of the classes 𝕆(]a, b[) and 𝕆′(]a, b[) (see, for example, [2]). Earlier attempts in this direction were undertaken in [3-7]. The statements given below complete the results of these papers.

2. On the Class 𝕆(]a, b[). In this section the function p :]a, b[→ ℝ is assumed to satisfy (2).

Theorem 1. Let there exist a point t_{0} ∈]a, b[ and absolutely continuous functions f_{1} : [a, t_{0}] → [0, +∞[ and f_{2} : [t_{0}, b] → [0, +∞[ such that f_{1}(a) = f_{2}(b) = 0, f_{1}(t) > 0 for a < t ≤ t_{0}, f_{2}(t) > 0 for t_{0} ≤ t < b,

g_{1}(t_{0}) = ∫_{a}^{t_{0}} \frac{[f_{1} ′(s)]^{2}}{f_{1}(s)} ds < +∞, g_{2}(t_{0}) = ∫_{t_{0}}^{b} \frac{[f_{2} ′(s)]^{2}}{f_{2}(s)} ds < +∞ (5)

and

f_{2}(t_{0}) ∫_{a}^{t_{0}} f_{1}(s)p(s)ds + f_{1}(t_{0}) ∫_{t_{0}}^{b} f_{2}(s)p(s)ds ≤
≤ −\frac{1}{4}(g_{1}(t_{0})f_{2}(t_{0}) + g_{2}(t_{0})f_{1}(t_{0})).  (6)

Then p ∈ 𝕆(]a, b[).

Corollary 1. Let there exist t_{0} ∈]a, b[ and α ∈]1, +∞[ such that

(b − t_{0})^{α} ∫_{a}^{t_{0}}(s − a)^{α}p(s)ds + (t_{0} − a)^{α} ∫_{t_{0}}^{b}(b − s)^{α}p(s)ds ≤

≤ −\frac{α^{2}}{4(α − 1)}[(t_{0} − a)(b − t_{0})]^{α−1}.

Then p ∈ 𝕆(]a, b[).

When t_{0} = \frac{a+b}{2} and α = 2 this proposition implies the result of R. Putnam ([4], p. 177).

Corollary 2. Let one of the following three conditions hold:

∫_{a}^{b}[(s − a)(b − s)]^{\frac{3}{2}} p(s)ds ≤ −\frac{9π(b − a)^{2}}{32} ,

∫_{a}^{b}[(s − a)(b − s)]^{n}p(s)ds ≤ −\frac{n!^{2}}{2(n − 1)(2n − 1)!}(b − a)^{2n−1},

n ≥ 2, n ∈ ℕ,

∫_{a}^{b} sin^{2} (\frac{π(s − a)}{b − a})p(s)ds ≤ −\frac{π^{2}}{2(b − a)}.

Then p ∈ 𝕆(]a, b[).

EXISTENCE OF CONJUGATE POINTS 95

Corollary 3. Let the inequality (b − t_{0}) ∫_{a}^{t_{0}}(s − a)^{\frac{3}{2}} (b − s)^{\frac{1}{2}} p(s)ds + (t_{0} − a) ∫_{t_{0}}^{b}(s − a)^{\frac{1}{2}} (b − s)^{\frac{3}{2}} p(s)ds ≤

≤ −\frac{5}{4}[(t_{0} − a)(b − t_{0})]^{\frac{1}{2}} (b − a)

hold for some t_{0} ∈]a, b[. Then p ∈ 𝕆(]a, b[).

Theorem 2. Let the inequality

(b − t) ∫_{a}^{t}(s − a)^{\frac{3}{2}} (b − s)^{\frac{1}{2}} [p(s)]_{−}ds + +(t − a) ∫_{t}^{b}(s − a)^{\frac{1}{2}} (b − s)^{\frac{3}{2}} [p(s)]_{−}ds < [(t − a)(b − t)]^{\frac{1}{2}} (b − a)

for a < t < b  (7)

hold, where [p(t)]_{−} = \frac{|p(t)|−p(t)}{2}. Then p ∉ 𝕆(]a, b[).

3. On the Class 𝕆′(]a, b[). In this section the function p :]a, b[→ ℝ is assumed to satisfy (4).

Theorem 3. Let there exist t_{0} ∈]a, b[ and absolutely continuous functions f_{1} : [a, t_{0}] → [0, +∞[ and f_{2} : [t_{0}, b] → [0, +∞[ such that f_{1}(a) = 0, f_{1}(t) > 0 for a < t < t_{0}, f_{2}(t) > 0 for t_{0} < t < b and conditions (5) and (6) are satisfied. Then p ∈ 𝕆′(]a, b[).

Corollary 4. Let there exist t_{0} ∈]a, b[ and α ∈]1, +∞[ such that

∫_{a}^{t_{0}}(s − a)^{α}p(s)ds + (t_{0} − a)^{α} ∫_{t_{0}}^{b} p(s)ds ≤ −\frac{α^{2}}{4(α − 1)}(t_{0} − a)^{α−1}.

Then p ∈ 𝕆′(]a, b[).

Corollary 5. Let the inequality

∫_{a}^{t_{0}}(s − a)^{\frac{3}{2}} p(s)ds + (t_{0} − a) ∫_{t_{0}}^{b}(s − a)^{\frac{1}{2}} p(s)ds ≤

≤ −\frac{5}{4}(t_{0} − a)^{\frac{1}{2}} + \frac{t_{0} − a}{8(b − a)^{\frac{1}{2}}}

hold for some t_{0} ∈]a, b[. Then p ∈ 𝕆′(]a, b[).

4. Proof of the Main results.

Proof of Theorem 1 (Theorem 3). Admit on the contrary that p ∉ 𝕆(]a, b[) (p ∉ 𝕆′(]a, b[)). Then equation (1) has the solution u satisfying

u(a+) = 0, u′(a+) = 1, u(t) > 0 for a < t < b.
( u(a+) = 1, u′(b−) = 0, u(t) > 0 for a ≤ t ≤ b ).

Denote

ρ(t) = \frac{u′(t)}{u(t)} for a < t < b.

It is clear that

ρ′(t) = p(t) − ρ^{2}(t) for a < t < b.  (8)

Multiplying both sides of this equality by f_{1} and integrating from a + ε to t_{0} where ε ∈]0, t_{0} − a[, we have

− ∫_{a+ε}^{t_{0}} f_{1}(s)p(s)ds + f_{1}(t_{0})ρ(t_{0}) − f_{1}(a + ε)ρ(a + ε) =

= ∫_{a+ε}^{t_{0}} [f_{1} ′(s)ρ(s) − f_{1}(s)ρ^{2}(s)]ds < \frac{1}{4} ∫_{a+ε}^{t_{0}} \frac{[f_{1} ′(s)]^{2}}{f_{1}(s)} ds < \frac{1}{4}g_{1}(t_{0}).

From (5) it easily follows that

lim_{t→a+} f_{1} ′(t) = 0.

Therefore

lim_{t→a+} f_{1}(t)ρ(t) = 0.

In view of this the last inequality can be rewritten as

− ∫_{a}^{t_{0}} f_{1}(s)p(s)ds + f_{1}(t_{0})ρ(t_{0}) < \frac{1}{4}g_{1}(t_{0}).  (9)

Now multiplying both sides of (8) by f_{2} and integrating from t_{0} to b − ε where ε ∈]0, b − t_{0}[, we have

− ∫_{t_{0}}^{b−ε} f_{2}(s)p(s)ds − f_{2}(t_{0})ρ(t_{0}) + f_{2}(b − ε)ρ(b − ε) =

= ∫_{t_{0}}^{b−ε} [f_{2} ′(s)ρ(s) − f_{2}(s)ρ^{2}(s)]ds < \frac{1}{4} ∫_{t_{0}}^{b−ε} \frac{[f_{2} ′(s)]^{2}}{f_{2}(s)} ds < \frac{1}{4}g_{2}(t_{0}).

Taking into account

lim_{t→b−} f_{2} ′(t) = 0 and lim_{t→b−}(b − t)u′(t) = 0,

EXISTENCE OF CONJUGATE POINTS 97

from the last inequality we obtain

− ∫_{t_{0}}^{b} f_{2}(s)p(s)ds − f_{2}(t_{0})ρ(t_{0}) < \frac{1}{4}g_{2}(t_{0}).  (10)

From (9) and (10) we have

−[f_{2}(t_{0}) ∫_{a}^{t_{0}} f_{1}(s)p(s)ds + f_{1}(t_{0}) ∫_{t_{0}}^{b} f_{2}(s)p(s)ds] <

< \frac{1}{4}[f_{2}(t_{0})g_{1}(t_{0}) + f_{1}(t_{0})g_{2}(t_{0})],

which contradicts (6). Proof of Theorem 2. Admit on the contrary that p ∈ 𝕆(]a, b[). Then equation (1) has the solution u satisfying

u(a+) = u(b_{1}) = 0, u(t) > 0 for a < t < b_{1} ≤ b.

According to the Green formula

u(t) =

= −\frac{1}{b_{1} − a}[(b_{1} − t) ∫_{a}^{t}(s − a)p(s)u(s)ds + (t − a) ∫_{t}^{b_{1}}(b_{1} − s)p(s)u(s)ds]

for a ≤ t ≤ b_{1}.

Hence we easily obtain

u(t) ≤ \frac{1}{b_{1} − a}[(b_{1} − t) ∫_{a}^{t}(s − a)[p(s)]_{−}u(s)ds +

+(t − a) ∫_{t}^{b_{1}}(b_{1} − s)[p(s)]_{−}u(s)ds] for a ≤ t ≤ b_{1},

i.e.,

v(t) ≤

≤ \frac{1}{[(t − a)(b − t)]^{\frac{1}{2}}} [\frac{b_{1} − t}{b_{1} − a} ∫_{a}^{t}(s − a)^{\frac{3}{2}} (b − s)^{\frac{1}{2}} [p(s)]_{−}v(s)ds +

+\frac{t − a}{b_{1} − a} ∫_{t}^{b}(s − a)^{\frac{1}{2}} (b_{1} − s)(b − s)^{\frac{1}{2}} [p(s)]_{−}v(s)ds] <

< λ\frac{1}{[(t − a)(b − t)]^{\frac{1}{2}}} [\frac{b − t}{b − a} ∫_{a}^{t}(s − a)^{\frac{3}{2}} (b − s)^{\frac{1}{2}} [p(s)]_{−})ds +

+\frac{t − a}{b − a} ∫_{t}^{b}(s − a)^{\frac{1}{2}} (b − s)^{\frac{3}{2}} [p(s)]_{−}ds] for a < t < b_{1},

where

v(t) = \frac{u(t)}{[(t − a)(b − t)]^{\frac{1}{2}}} for a < t < b_{1}

and

λ = sup{v(t) t ∈]a, b_{1}[}.

Taking into account (7), we obtain the contradiction λ < λ.

Corollaries 1-5 are obtained from Theorems 1 and 3 by an appropriate choice of functions f_{1} and f_{2}.

References

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2. A. G. Lomtatidze, On positive solutions of boundary-value problems for second-order ordinary differential equations with singularities. (Russian) Differentsial’nye Uravneniya 23(1987), 1685-1692.

3. F. Hartman, Ordinary differential equations. Wiley, New York/London/Sydney, 1964.

4. M. A. Krasnosel’ski, Vector fields on the plane. (Russian) Moscow, Nauka, 1963.

5. N. l. Korshikova, On zeros of solutions of high-order linear equations. (Russian) Differential equations and their applications. (Russian) 143-148, Moscow University Press, Moscow, 1984.

6. A. G. Lomtatidze, On oscillatory properties of solutions of secondorder linear differential equations. (Russian) Sem. I. Vekua Inst. Appl. Math. Tbiliss. St. Univ. Reports. 19(1985), 39-53.

7. O. Došlý, The multiplicity criteria for zero points of second-order differential equations. Math. Slovaca 42(1992), No. 2, 181-193.

(Received 01.12.1993)

Author’s address:

N.Muskhelishvili Institute of Computational Mathematics

Georgian Academy of Sciences

8, Akuri St., Tbilisi 380093

Republic of Georgia