Mem. Differential Equations Math. Phys. 10(1997), 150–152

A. Lomtatidze and S. Mukhigulashvili

ON A TWO-POINT BOUNDARY VALUE PROBLEM FOR SECOND ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS, II

(Reported on October 21–28, 1996)

In the present note, we continue the investigation of the question of solvability of the boundary value problem

u′′(t) = F (u)(t),  (1)
u(a) = 0, u(b) = 0,  (2)

which was begun in [1]. Notation introduced there remains valid. Moreover, we assume that:

L_{2}([a, b]) is the space of quadratically summable functions p : ]a, b[ → R with the norm ∥p∥_{L_{2}} = ∫_{a}^{b} p^{2}(s) ds;

σ_{ab} : L([a, b]) → L([a, b]) is an operator defined by

σ_{ab}(p)(t) = \frac{1}{σ(p)(t)} ∫_{a}^{t} σ(p)(s) ds · ∫_{t}^{b} σ(p)(s) ds.

Theorem 1. Let on the set C′_{0}([a, b]) the inequalities

∫_{a}^{b} \frac{[v(s)(F (v)(s) − g(s)v′(s)]_{−}}{σ(g)(s)} ds ≤ α_{0}(∥\frac{v′}{\sqrt{σ(g)}} ∥_{L_{2}}^{2}) +

+ α_{1}(∥v∥_{C′}) · β_{1}(∥\frac{v′}{\sqrt{σ(g)}} ∥_{L_{2}}^{2γ}),

∥[(F (v) − gv′) sgn v]_{−}∥_{L} ≤ α_{2}(∥v∥_{C′}) + β_{2}(∥v′∥_{L_{2}}^{2δ})

be fulfilled, where g ∈ L([a, b]), γ + δ = 1, and continuous, nondecreasing functions α_{i}, β_{j}: R_{+} → R_{+}, i = \overline{0, 2}, j = 1, 2, satisfy the conditions

lim sup_{x→+∞} \frac{1}{x} α_{0}(x) < 1,

lim_{x→+∞} \frac{1}{x} α_{i}(x) = 0, β_{i}(x) = O(x), i = 1, 2.

Then the problem (1), (2) has at least one solution.

Corollary 1. Let on the set C′_{0}([a, b]) the inequality

[F (v)(t) − p(t)v(t) − g(t)v′(t) − l(v)(t)] sgn v(t) ≥ −q(t, ∥v∥_{C′})  (3)

1991 Mathematics Subject Classification. 34K10.

Key words and phrases. Functional differential equation, boundary value problem. be fulfilled, where l ∈ L([a, b]), p, g ∈ L([a, b]), q ∈ K_{1}([a, b] × R, R_{+}) is nondecreasing in the second argument, and

lim_{x→+∞} \frac{1}{x} ∫_{a}^{b} q(s, x) ds = 0.  (4)

Moreover, let

\frac{1}{∫_{a}^{b} σ(g)(s) ds} · ∫_{a}^{b} σ_{ab}(g)(s)[p(s)]_{−} ds + \frac{1}{2} ∥_{l}∥ < 1,

where _{l}: L([a, b]) → L([a, b]) is the operator defined by the equality

_{l}(v)(t) = \frac{1}{\sqrt{σ(g)(t)}} \sqrt{σ_{ab}(g)(t)} l(v)(t).

Then the problem (1), (2) has at least one solution.

Mention two corollaries of Theorem 1 for the equation

u′′(t) = p(t)u(t) + g(t)u′(t) + h(t)u(τ(t)) + G(u)(t),  (5)

where p, g, h ∈ L([a, b]), τ ∈ M([a, b], [a, b]) and G ∈ K([a, b]).

Corollary 2. Let on the set C′_{0}([a, b]) the inequality

G(v)(t) sgn v(t) ≥ −q(t, ∥v∥_{C′})  (6)

be fulfilled, where q ∈ K_{1}([a, b] × R, R_{+}) is nondecreasing in the second argument and satisfies (4). Let, moreover,

\frac{1}{∫_{a}^{b} σ(g)(s) ds} · ∫_{a}^{b} σ_{ab}(g)(s)[p(s) + h(s)]_{−} ds +

+ \frac{1}{\sqrt{∫_{a}^{b} σ(g)(s) ds}} · ∫_{a}^{b} \sqrt{σ_{ab}(g)(s)} \sqrt{\frac{| ∫_{s}^{τ(s)} σ(g)(ξ) dξ|}{σ(g)(s)}} |h(s)| ds < 1.  (7)

Then the problem (5), (2) has at least one solution.

Remark 1. Note that, unlike Corollaries 1–3 in [1], the restriction imposed on the sign of the function h is not required here.

In the case where

h(t) ≥ 0 for a < t < b,  (8)

the condition (7) can be somewhat improved. More exactly, we have

Corollary 3. Let on the set C′_{0}([a, b]) the inequality (6) be fulfilled, where q ∈ K_{1}([a, b] × R, R_{+}) is nondecreasing in the second argument and satisfies (4). Let, moreover, the inequality (8) holds, and

\frac{1}{∫_{a}^{b} σ(g)(s) ds} · ∫_{a}^{b} σ_{ab}(g)(s)[p(s)]_{−} ds + \frac{1}{4} ∫_{a}^{b} \frac{h(s)}{σ(g)(s)} | ∫_{s}^{τ(s)}σ(g)(ξ) dξ| ds < 1.

Then the problem (5), (2) has at least one solution.

Theorem 2. Let on the set C′_{0}([a, b]) the inequality (3) be fulfilled, where l ∈ L([a, b]) is a positive operator, p, g ∈ L([a, b]), and q ∈ K([a, b] × R, R_{+}) is nondecreasing in the second argument and satisfies (4). Let, moreover,

∫_{a}^{b} σ_{ab}(g)(s)[p(s)]_{−} ds < ∫_{a}^{b} σ(g)(s) ds

and

∫_{a}^{b} \frac{l(1)(s)}{σ(g)(s)} ds < \frac{16}{∫_{a}^{b} σ(g)(s) ds} (1 − \frac{1}{∫_{a}^{b} σ(g)(s) ds} · ∫_{a}^{b} σ_{ab}(g)(s)[p(s)]_{−} ds).

Then the problem (1), (2) has at least one solution.

References

1. A. Lomtatidze and S. Mukhigulashvili, On a two-point boundary value problem for second order ordinary differential equations I. Mem. Differential Equations Math. Phys. 10(1997), 125–128.

2. I. T. Kiguradze and B. L. Shekhter, Singular boundary value problems for second-order differential equations. In: “Current Problems in Mathematics: Newest Results”, vol. 30, pp. 105–201, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuzn. Inst. Nauchn. i Tekhn. Inform., Moscow, 1987.

Authors’ addresses:

A. Lomtatidze N. Muskhelishvili Institute of Computational Mathematics Georgian Academy of Sciences 8, Akuri St., Tbilisi 380093 Georgia

S. Mukhigulashvili A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 390093 Georgia