Mem. Differential Equations Math. Phys. 6(1995), 124–126

S. Mukhigulashvili

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

(Reported on November 13 and 20, 1995)

Let ℝ be the set of real numbers, ℝ_{0}^{+} = [0, +∞[ , ℝ^{+} = ]0, +∞[ , a, b ∈ ℝ^{+}, p ≥ 1. L_{p}([a, b]) is the space of functions f : ]a, b[ → ℝ such that |f(x)|^{p} is integrable on [a, b],

∥f∥_{L_{p}} = ∫_{a}^{b} |f(s)|^{p}ds.

_{C_{p}}([a, b]) is the space of functions u : [a, b] → ℝ such that u′ ∈ L_{p}([a, b]), ∥u∥_{C_{p}} =

|u(a)| + ∥u′∥_{L_{p}}.

C(I, ℝ) is the space of continuous functions u : I → ℝ, ∥u∥_{C} = sup{|u(t)| : t ∈ I}. _{C′_{p}}([a, b]) is the set of functions u ∈ _{C_{1}}([a, b]) such that u′ ∈ _{C_{p}}([a, b]). Consider the boundary value problem

u′′(t) = H(u, u′, u′′)(t), t ∈ [a, b]  (1)
u(a) = 0, u(b) = 0,  (2)

where H : C([a, b]) × C([a, b]) × L_{p}([a, b]) → L_{p}([a, b]) is a compact operator, i.e., H is continuous and H(B) is precompact for any bounded B ⊂ C([a, b])×C([a, b])×L_{p}([a, b]).

Under a solution of equation (1) we mean a function u ∈ _{C_{p}}([a, b]) satisfying a.e. equation (1).

Below two theorems on the solvability of the problem (1), (2) are given.

Theorem 1. Let the inequality

−g(t) ≤ H(x, x′, z)(t) · sign x(t), t ∈ [a, b], (x, z) ∈ _{C′_{p}}([a, b]) × L_{p}([a, b]) (3)

be fulfilled, where g ∈ L_{p}([a, b]). Moreover, let for any r > 0 there exist γ_{r}, α_{r} ∈ ℝ^{+} and f_{r} ∈ C(ℝ^{+}, ℝ^{+}) such that

∥H(x, x′, z)∥_{L_{p}} ≤ α_{r} · f_{r}(∥z∥_{L_{p}}) for ∥x′∥_{C} ≤ r, ∥z∥_{Lp} ≥ γ_{r}

and

lim inf_{ρ→+∞} \frac{ρ}{f_{r}(ρ)} > α_{r}.

Then the problem (1), (2) is solvable.

Theorem 2. Let the condition (3) be fulfilled. Moreover, let for any r ∈ ℝ^{+}, α ∈ ]0, (b − a)r[ and β ∈]0, α[ there exist γ_{r}, c_{r} ∈ ℝ^{+}, l_{r}, f_{r}, g_{β} ∈ C(ℝ_{0}^{+}, ℝ_{0}^{+}) and h_{β}(t) ∈ L_{p}([a, b]) such that

h_{β}(t) > 0 for t ∈ [a, b], l_{r}(0) = 0,
∥H(x, x′, z)∥_{L_{p}} ≤ l_{r}(∥x∥_{C} ) · f_{r}(∥z∥_{Lp} ) + c_{r} for ∥x∥_{C} < α,
∥x′∥_{C} ≤ r, ∥z∥_{Lp} ≥ γ_{r},

1991 Mathematics Subject Classification. 34K10.

Key words and phrases. Second order functional differential equation, two-point boundary value problem.

|H(x, x′, z)| ≥ h_{β}(t) · g_{β}(∥z∥_{Lp} ) for ∥x∥_{C} ≥ α, ∥x′∥_{C} ≤ r,
∥z∥_{Lp} ≥ γ_{r}, t ∈ {t ∈ [a, b] : |x(t)| ≥ β},

and

lim inf_{ρ→+∞} \frac{ρ}{f_{r}(ρ)} > 0, lim sup_{ρ→+∞} g_{β}(ρ) = +∞.

Then the problem (1), (2) is solvable.

Let us give some examples.

Let

G_{1} ∈ L_{p}([a, b] × [a, b]; ℝ^{+}), K(x, y)(t) · sign x(t) ≥ −g(t), t ∈ [a, b],

where

K : C([a, b]) × C([a, b]) → L_{p}([a, b]), q, g ∈ L_{p}([a, b]), k ∈ ℕ,  (4)
0 < G_{2}(t, s) ≤ g_{1}(t), (t, s) ∈ [a, b] × [a, b], g_{1} ∈ L_{p}([a, b]).  (5)

Consider the equation

u′′(t) = u^{2k+1}(t) ∫_{a}^{b} G_{1}(t, s)(1 + |u′(s)|^{α})[ ∫_{a}^{b} G_{2}(s, τ) · |u′′(τ)|^{p}dτ]^{μ}ds +
+ K(u, u′)(t) + q(t),  (6)

where α ∈ ℝ_{0}^{+}, p, λμ ≤ 1. Then according to Theorem 2, the problem (6), (2) is solvable. Analogously, the equations

u′′(t) = u^{2k+1}(t)(1 + |u′(t)|^{α})[ ∫_{a}^{b} G_{2}(t, s) · |u′′(s)|^{p}ds]^{∥u∥_{C} +ε} +

+ K(u, u′)(t) + q(t), for α ∈ ℝ_{0}^{+}, ε < \frac{1}{p}

and

u′′(t) = u^{2k+1}(t)∥u′∥_{C} [ ∫_{a}^{b} G_{2}(t, s) · |u′′(s)|^{∥u∥_{C} +ε}ds] + K(u, u′)(t) + q(t),

where

p ≥ (b − a) ∫_{a}^{b} |g(s)| + |q(s)|ds + ε, ε > 0

have solutions satisfying the boundary conditions (2).

Suppose now that the conditions (4) are fulfilled, and

0 ≤ G_{2}(t, s) ≤ g_{1}(t), (t, s) ∈ [a, b] × [a, b], g_{1} ∈ L_{p}([a, b]),
λμ < 1, λ ≤ p, β > 0, 0 < α < p, g_{0} ∈ L_{p}([a, b]).

Then by Theorem 1, the equations

u′′(t) = u^{2k+1}(t) ∫_{a}^{b} G_{1}(t, s) · |u′(s)|[ ∫_{a}^{b} G_{2}(s, τ) · |u(τ)|^{β} · |u′′(τ)|^{λ}dτ]^{μ}ds +

+ K(u, u′)(t) + q(t),

u′′(t) = u^{2k+1}(t) · |u′(t)| ln (1 + ∫_{a}^{b} G_{2}(t, r)|u(τ)|^{β} · |u′′(τ)|^{α}dτ) + K(u, u′)(t) + q(t)

have solutions satisfying the boundary conditions (2).

References

1. I. T. Kiguradze, Boundary value problems for systems of ordinary differential equations. (Russian) Current Problems in Mathematics. Newest Results, vol. 30 (Russian), 3–103, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vses. Inst. Nauchn. i Tekh. Inform., Moscow, 1987.

Author’s address:

A. Razmadze Mathematical Institute

Georgian Academy of Sciences

1, M. Aleksidze St., Tbilisi 380093

Georgia