Mem. Differential Equations Math. Phys. 10(1997), 115–118

A. Lomtatidze

ON A NONLOCAL BOUNDARY VALUE PROBLEM FOR SECOND ORDER NONLINEAR EQUATIONS

(Reported on March 11–18, 1996)

Below we will use the following notation: R is the set of real numbers. L([a, b]) is the set of the functions p : ]a, b[ → R which are Lebesgue integrable on

[a, b].

L_{loc}(]a, b[) is the set of the functions p : ]a, b[ → R which are Lebesgue integrable on [a + ε, b − ε] for arbitrarily small ε > 0.

K_{0}(]a, b[ ×R^{2}) is the set of the functions q : ]a, b[ ×R^{2} → R for which the mapping t ⟼ g(t, x_{1}(t), x_{2}(t)) is measurable for any continuous functions x_{i} : ]a, b[ → R (i = 1, 2).

σ : L_{loc}(]a, b[) → L_{loc}(]a, b[) is an operator defined by the equality

σ(p)(t) = exp [ ∫_{\frac{a+b}{2}}^{t} p(s)ds].

If σ(p) ∈ L([a, b]), α ∈ [a, b] and β ∈]α, b], then

σ_{α}(p)(t) = \frac{1}{σ(p)(t)} | ∫_{α}^{t} σ(p)(s)ds|,

σ_{αβ}(p)(t) = \frac{1}{σ(p)(t)} ∫_{α}^{t} σ(p)(s)ds · ∫_{t}^{β} σ(p)(s)ds.

u(s+) and u(s−) are the limits of the function u at the point s from the right and from the left, respectively.

If μ : [a, b] → R is a function of bounded variation, then by μ^{∗}(t) we denote the full variation of the function μ on the segment [a, t].

Under solution of the equation

u′′ = f(t, u, u′),  (1)

where f : ]a, b[ ×R^{2} → R satisfies the Carathéodory conditions on every compactum contained in ]a, b[ ×R^{2}, we understand a function u : ]a, b[ → R which is absolutely continuous along with its first derivative on every segment from ]a, b[ , and satisfies (1) a.e.

1991 Mathematics Subject Classification. 34K10. Key words and phrases. Second order equation, nonlocal boundary value problem,

existence and uniqueness.

In the present paper, we consider the problem of existence and uniqueness of solution of the equation (1) satisfying the boundary conditions

u(a+) = 0, u(b−) = ∫_{a}^{b} u(s)dμ(s),  (2)

where μ : [a, b] → R is a function of bounded variation.

Some criteria for unique solvability of the problem in a linear case are contained in [1, 2]. In the nonlinear case, a problem of the type (1), (2) has been considered in [3–5]. However, in those works μ is assumed to be a piecewise constant function (μ(t) = 0 for a ≤ t ≤ t_{0} and μ(t) = 1 for t_{0} < t ≤ b).

Theorems of existence and uniqueness of solution of the problem (1), (2) given in the present paper cover the case where μ is not, generaly speaking, piecewise constant, and f is not integrable in the first argument on the segment [a, b], having singularities at the points t = a and t = b.

Before passing to the formulation of basic results, let us introduce the following definitions.

Definition 1. We say that a vector-function (p_{1}, p_{2}) : ]a, b[ → R^{2} belongs to the class U_{μ}(]a, b[) if

σ(p_{2}), σ_{ab}(p_{2})p_{1} ∈ L(]a, b[)

and the solution u_{1} of the singular Cauchy problem

u′′ = p_{1}(t)u + p_{2}(t)u′; u(a+) = 0, lim_{t→a+} \frac{u′(t)}{σ(p_{2})(t)} = 1

satisfies the conditions

u_{1}(t) > 0 for a < t < b, u_{1}(b−) > ∫_{a}^{b} u_{1}(s)dμ^{∗}(s).

Definition 2. We say that a vector-function (p_{1}, p_{12}, p_{22}) : ]a, b[ → R^{3} belongs to the class V_{μ}(]a, b[) if

p_{12}(t) ≤ p_{22} for a < t < b,  (3)
p_{i2}, p_{1} ∈ L_{loc}(]a, b[), i = 1, 2,  (4)
σ(p_{i2}) ∈ L([a, b]), σ_{ab}(p_{i2})p_{1} ∈ L([a, b]), i = 1, 2,  (5)

and (p_{1}, p_{2}) ∈ U_{μ}(]a, b[) for any measurable function p_{2} : ]a, b[ → R satisfying

p_{12}(t) ≤ p_{2}(t) ≤ p_{22}(t) for a < t < b.

Theorem 1. On the set ]a, b[ ×R^{2}, let the inequalities

[f(t, x, y) − p_{1}(t)x − p_{2}(t, x, y)y] sgn x ≥ −p(t),
p_{12}(t) ≤ p_{2}(t, x, y) ≤ p_{22}(t)

be fulfilled, where p_{2} ∈ K_{0}(]a, b[ ×R^{2}) and (p_{1}, p_{2}, p_{22}) ∈ V_{μ}(]a, b[). Furthermore, let σ_{ab}(p_{i2})p ∈ L([a, b]) (i = 1, 2), and for some point t_{1} ∈]a, b[ let

|f(t, x, y) − p_{1}(t)x − p_{2}(t, x, y)y| ≤ p(t) for t_{1} < t < b, x ∈ R, y ∈ R.  (6)

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

Remark 1. Let μ be nondecreasing, the conditions (3)–(5) be fulfilled, and let (p_{1}, p_{12}, p_{22}) ∉ V_{μ}(]a, b[). Then there exists a function f satisfying the conditions of Theorem 1 for which the problem (1), (2) has no solution.

Remark 2. The condition (6) can be replaced by the condition

|f(t, x, y) − p_{1}(t)x − _{p_{2}}(t, x, y)y| ≤ p(t) for t_{1} < t < b, x ∈ R, y ∈ R,

where _{p_{2}} ∈ K_{0}(]a, b[ ×R^{2}) and p_{12}(t) ≤ _{p_{2}}(t) ≤ p_{22}(t, x, y) for (t, x, y) ∈]t_{1}, b[ ×R^{2}.

As an example, let us consider the problem (1), (2) in the case where μ increases, μ(b) − μ(a) < 1, and

f(t, x, y) = p_{0}(t) + p_{1}(t)x + p_{2}(t)y + p_{3}(t)x^{2n+1}|y|^{k}, p_{i} ∈ L_{loc}(]a, b[) i = \overline{0, 3},

where n and k are positive integers. Assume that λ > 0, 0 ≤ δ < 1, p_{1}(t) ≥ 0, p_{3}(t) ≥ 0, |p_{2}(t)| ≤ λ + \frac{δ}{t−a} + \frac{δ}{b−t} for a < t < b,

∫_{a}^{b} (s − a)(b − s)|p_{i}(s)|ds < +∞, i = 0, 1,

and p_{3}(t) ≡ 0 in a neighborhood of the point b. Taking into account Theorem 1.2 in [2], we obtain from Theorem 1 that in this case the problem (1), (2) has at least one solution. As it is seen from the example, the function f may have nonintegrable singularities for t = a and t = b.

Corollary 1. On the set ]a, b[ ×R^{2}, let the inequality

f(t, x, y) sgn x ≥ p_{1}(t)|x| − p_{2}(t)|y| − p(t)  (7)

be fulfilled, where (p_{1}, −p_{2}, p_{2}) ∈ V_{μ}(]a, b[) and

σ_{ab}((−1)^{i}p_{2})p ∈ L([a, b]), i = 1, 2.

Furthermore, let for some point t_{1} ∈]a, b[

|f(t, x, y) − p_{1}(t)x − p_{2}(t)y| ≤ p(t) for t_{1} < t < b, x ∈ R, y ∈ R.  (8)

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

Corolary 2. Let there exist numbers λ_{i} ∈ [0, 1[ , l_{i} ∈ [0, +∞[ , γ_{i} ∈ [0, +∞[ , i = 1, 2, c ∈]a, b[ and the function p : ]a, b[ →]0, +∞[ such that

∫_{0}^{+∞}\frac{ds}{l_{1} + l_{2}s + s^{2}} ≥ \frac{(c − a)^{1−λ_{1}}}{1 − λ_{1}} , ∫_{γ_{1}}^{γ_{2}} \frac{ds}{l_{1} + l_{2}s + s^{2}} ≥ \frac{(b − c)^{1−λ_{2}}}{1 − λ_{2}} ,

∫_{γ_{1}}^{γ_{2}} \frac{sds}{l_{1} + l_{2}s + s^{2}} < − ln(μ^{∗}(b) − μ^{∗}(a)),

let the function t ⟼ (t − a)(b − t)p(t) be summable on [a, b], and on the set ]a, b[ ×R^{2} let the inequality (7) be fulfilled, where

p_{1}(t) = \begin{cases}−l_{1}(t − a)^{−2λ_{1}} & for a < t ≤ c \\ −l_{1}(b − t)^{−2λ_{2}} & for c < t < b \end{cases} ,

p_{2}(t) = \begin{cases}l_{2}(t − a)^{−λ_{1}} + λ_{1}(t − a)^{−1} & for a < t ≤ c \\ l_{2}(b − t)^{−λ_{2}} + λ_{2}(b − t)^{−1} & for c < t < b \end{cases} .

Moreover, let (8) be fulfilled for some point t_{1} ∈]a, b[ . Then the problem (1), (2) has at least one solution.

References

1. A. G. Lomtatidze, Nonlocal boundary value problem for linear ordinary differential equations of second order. (Russian) Differentsial’nye Uravneniya 31(1995), No. 3, 446–455.

2. A. Lomtatidze, On a nonlocal boundary value problem for second order linear ordinary differential equations. J. Math. Anal. Appl. 193(1995), 889–908.

3. A. G. Lomtatidze, On a singular three-point boundary value problem. (Russian) Trudy Inst. Prikl. Mat. I. N. Vekua 17(1986), 122–133.

4. A. G. Lomtatidze, On a boundary value problem for second order linear ordinary differential equations with singularities. (Russian) Differentsial’nye Uravneniya 22(1986), No. 3, 416–426.

5. A. G. Lomtatidze, To the question on solvability of singular boundary value problems for second order ordinary differential equations. (Russian) Trudy Inst. Prikl. Mat. I. N. Vekua 22(1987), 135–149.

Author’s address:

N. Muskhelishvili Institute of Computational Mathematics

Georgian Academy of Sciences

8, Akuri St., Tbilisi 380093

Georgia