GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 4, 1997, 303-312

A MULTIDIMENSIONAL SINGULAR BOUNDARY VALUE PROBLEM OF THE CAUCHY–NICOLETTI TYPE

J. DIBLÍK

Abstract. A two-point singular boundary value problem of the Cauchy–Nicoletti type is studied by introducing a two-point boundary value set and using the topological principle. The results on the existence of solutions whose graph lies in this set are proved. Applications and comparisons to the known results are given, too.

Introduction

Consider the system of ordinary differential equations

y′ = f(x, y),  (1)

where x ∈ I = (a, b), −∞ ≤ a < b ≤ ∞, y ∈ ℝ^{n} and n > 1.

We will study the following singular boundary value problem of the Cauchy–Nicoletti type:

y_{i}(a+) = A_{i} (i = 1, …, m), y_{k}(b−) = A_{k} (k = m + 1, …, n) (2)

where A_{i}, i = 1, …, n, are some constants and 1 ≤ m < n.

It is assumed that the vector-function f ∈ C(Ω, ℝ^{n}), where Ω is an open set such that Ω ∩ {(x^{∗}, y) : y ∈ ℝ^{n}} ≠ ∅ for each x^{∗} ∈ I and, moreover, f satisfies local Lipschitz condition in the variable y in Ω (f ∈ L_{loc}(Ω)). In this case the solutions of system (1) are uniquely determined by the initial data in Ω.

We define the solution of problem (1), (2) as a vector-function y = (y_{1}, …, y_{n}) ∈ C^{1}(I, ℝ^{n}) which satisfies system (1) on I, (x, y_{1}(x), …, y_{n}(x)) ⊂ Ω if x ∈ I and y_{i}(a+) = A_{i} (i = 1, …, m), y_{k}(b−) = A_{k} (k = m + 1, …, n).

303

1991 Mathematics Subject Classification. 34B10, 34B15.

Key words and phrases. Singular boundary value problem, nonlinear Cauchy–Nicoletti problem, topological principle.

1072-947X/97/0700-0303$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

In the paper certain sufficient conditions for the existence of solutions of problem (1), (2) will be given whose graph lies on the interval I in a two-point boundary value set Ω^{0} defined in the following way.

Definition 1. Let Ω^{0} ⊂ Ω and Ω^{0} ∩ {(x^{∗}, y) : y ∈ ℝ^{n}} ≠ ∅ for each x^{∗} ∈ I. We will call the set Ω^{0} a two-point boundary value set if each continuous curve l = {(x, y) : x ∈ I, y = y(x)} defined on I, for which the relation (x, y(x)) ⊂ \overline{Ω}^{0} holds on I, has the following limit values:

lim_{x→a+} y_{i}(x) = A_{i} (i = 1, …, m),  (3)
lim_{x→b−} y_{k}(x) = A_{k} (k = m + 1, …, n).  (4)

In the sequel Ω_{a,b}^{0} will denote such a type of set Ω^{0}.

Boundary value problems for systems of ordinary differential equations were considered by many authors (see [1]–[9], for example). Singular boundary value problems of such types were studied in [3]–[8], [10]–[16]. Our results are independent of the known ones. Some specific comparisons to the known results will be made in the paper. The main results are formulated as Theorems 2 and 3.

Main results

Let Ω^{0} ⊂ Ω be some open set with the boundary ∂Ω^{0}. According to Ważewski ([1], [17]), a point (x_{0}, y_{0}) ∈ ∂Ω^{0} ∩ Ω is a point of egress from Ω^{0} with respect to system (1) and the set Ω^{0} if, for the solution y = y(x) of the problem y(x_{0}) = y_{0}, there exists ε > 0 such that (x, y(x)) ∈ ∫ Ω^{0} if x ∈ [x_{0} − ε, x_{0}). A point of egress is a point of strict egress from Ω^{0} if, moreover, there exists ε_{1} > 0 such that (x, y(x)) ∉ \overline{Ω}^{0} if x ∈ (x_{0}, x_{0} + ε_{1}].

As usual, the set of all points of egress (strict egress) from Ω^{0} will be denoted by Ω_{e}^{0} (Ω_{se}^{0}).

Theorem 1 ([1], [17]). Let Ω^{0} ⊂ Ω be some open set such that Ω_{e}^{0} = Ω_{se}^{0}. Assume that S is a nonempty subset of Ω^{0} ∪Ω_{e}^{0} such that the set S ∩Ω_{e}^{0} is not a retract of S but is a retract of Ω_{e}^{0}.

Then there is at least one point (x_{0}, y_{0}) ∈ S ∩ Ω^{0} such that the graph of the solution y(x) of the Cauchy problem y(x_{0}) = y_{0} lies in Ω^{0} on its right-hand maximal interval of existence.

In a further discussion we will suppose that all sets of the type Ω^{0} satisfy all the conditions of Definition 1, i.e., Ω^{0} = Ω_{a,b}^{0}.

Theorem 2. Let Ω^{0} = Ω_{a,b}^{0} and Ω_{e}^{0} = Ω_{se}^{0}. Assume that there are nonempty subsets S_{i} ⊂ {(x, y) ∈ Ω, x = x_{i}} ∩ (Ω^{0} ∪ Ω_{e}^{0}), i = 1, 2, …, where {x_{i}} is some decreasing sequence of real numbers with x_{i} ∈ (a, b) and lim_{i→∞} x_{i} = a such that S_{i} ∩ Ω_{e}^{0} is not a retract of S_{i} but is a retract of Ω_{e}^{0}.

A MULTIDIMENSIONAL SINGULAR BOUNDARY VALUE PROBLEM 305

Then there is at least one solution y = y(x) of problem (1), (2) such that its graph lies in \overline{Ω}^{0} on the interval (a, b).

Proof. Let the index i be fixed. Then, as follows from Theorem 1, there is at least one point (x_{i}, y^{i}) ∈ S_{i} such that the graph of the solution y^{i}(x) of the Cauchy problem y^{i}(x_{i}) = y^{i} for (1) lies in Ω^{0} on its right-hand maximal interval of existence, i.e., on the interval [x_{i}, b). Further, we denote by M_{i} the set of all initial points from the set Ω_{i} = \{(x, y) ∈ \overline{Ω}^{0}, x = x_{i}\} with the property that each point (x_{i}, y^{∗i}) ∈ Ω_{i} defines a solution y = y^{∗}(x) such that its graph lies in \overline{Ω}^{0} on [x_{i}, b). Obviously, M_{i} ≠ ∅. The set M_{i} is closed in Ω_{i} (including the case where M_{i} consists of one point only) since otherwise we get the contrary with continuous dependence of solutions on the initial data. Let χ{M_{i}, [x_{i}, b)} be the set of all solutions of (1) on [x_{i}, b) defined by the initial data from the set M_{i}. Then M′_{i} ⊂ M_{1} where M′_{i} ≡ χ{M_{i}, [x_{i}, b)} ∩ Ω_{1}, and if i > 2 then M′_{i} ⊂ M′_{i−1}. Then, as the sets M′_{i}, i = 1, 2 …, are compact, there is a nonzero set M_{0} = ∩_{i=1}^{∞}M′_{i}. If a point (x_{1}, y_{0}) ∈ M_{0} then for the corresponding solution y = y_{0}(x) we have (x, y_{0}(x)) ⊂ Ω^{0} on (a, b). As Ω^{0} = Ω_{a,b}^{0}, by (3) and (4) lim_{x→a+} y_{0i}(x) = A_{i}, i = 1, …, m, and lim_{x→b−} y_{0i}(x) = A_{i}, i = m + 1, …, n, i.e., the solution y_{0}(x) is a solution of problem (1), (2) with appropriate properties.

Now we will suppose that the open region Ω^{0} can be described by the functions n_{i} ∈ C^{1}(Ω), i = 1, …, l, and p_{j} ∈ C^{1}(Ω), j = 1, …, q, as follows:

Ω^{0} = {(x, y) ∈ Ω, x ∈ I, n_{i} < 0, i = 1, …, l, p_{j} < 0, j = 1, …, q}. (5)

For α ∈ {1, …, l} we denote

N_{α} = \{(x, y) ∈ \overline{Ω}^{0} ∩ Ω, n_{α} = 0, n_{i} ≤ 0, i = 1, …, l; i ≠ α,
p_{j} ≤ 0, j = 1, …, q}

and for β ∈ {1, …, q}

P_{β} = \{(x, y) ∈ \overline{Ω}^{0} ∩ Ω, p_{β} = 0, n_{i} ≤ 0, i = 1, …, l;
p_{j} ≤ 0, j = 1, …, q, j ≠ β}.

Definition 2 ([1]). The open set Ω^{0} ⊂ Ω given by (5) is called an (n, p)- subset with respect to system (1) if for derivatives of the functions n_{α} (α = 1, …, l) and p_{β} (β = 1, …, q) along the trajectories of system (1)

dn_{α}(x, y)/dx < 0, for (x, y) ∈ N_{α},  (6)
dp_{β}(x, y)/dx > 0, for (x, y) ∈ P_{β}.  (7)

Theorem 3. Let f ∈ C(Ω, ℝ^{n}), f ∈ L_{loc}(Ω), Ω^{0} = Ω_{a,b}^{0} and Ω^{0} be an (n, p)-subset with respect to system (1). Let us assume that there are nonempty subsets S_{i} ⊂ {(x, y) ∈ Ω, x = x_{i}} ∩ (Ω^{0} ∪ Ω_{e}^{0}), i = 1, 2, …, where {x_{i}} is some decreasing sequence of numbers with x_{i} ∈ (a, b) and lim_{i→∞} x_{i} = a such that S_{i} ∩ Ω_{e}^{0} is not a retract of S_{i} but is a retract of Ω_{e}^{0}.

Then there is at least one solution y = y(x) of problem (1), (2) such that its graph lies in Ω^{0} on interval I, i.e., the inequalities

n_{i}(x, y(x)) < 0, i = 1, …, l,  (8)
p_{j}(x, y(x)) < 0, j = 1, …, q,  (9)

hold on interval I.

Proof. From the known result in [1] (Lemma 3.1, §3, Chapter X) it follows that Ω_{e}^{0} = Ω_{se}^{0} = ∩_{β=1}^{q}P_{β}\ ∩_{α=1}^{l} N_{α}. Then Theorem 3 is a consequence of Theorem 2 and that result. In this case (x, y(x)) ⊂ Ω^{0} on I (instead of (x, y(x)) ⊂ \overline{Ω}^{0} on I) because in view of (6), (7) \{(x, y) ∈ Ω, x ∈ I, y = y(x)} ∩ ∂Ω^{0} = ∅.

Applications

(A) Let system (1) be of the form

y′ = A(x)y + g(x, y),  (10)

where A = {a_{ij}}_{i,j=1,...,n}, a_{ij} ∈ C(I, ℝ), g ∈ C(Ω, ℝ^{n}) and g ∈ L_{loc}(Ω).

Let δ_{i}(x), i = 1, …, n be some functions continuously differentiable and positive on the interval I with the property

lim_{x→a+} δ_{i}(x) = 0 = lim_{x→b−} δ_{k}(x) (i = 1, …, m, k = m + 1, …, n). (11)

For some integers m_{1}, 0 ≤ m_{1} ≤ m and n_{1}, 0 ≤ n_{1} ≤ n − m and for (x, y) ∈ Ω we define the functions

N_{k}(x, y) ≡ N_{k}(x, y_{k}) ≡ (y_{k} − A_{k})^{2} − δ_{k}^{2}(x),  (12)

where k ∈ {1, …, m_{1}} ∪ {m + 1, …, m + n_{1}} and

P_{r}(x, y) ≡ P_{r}(x, y_{r}) ≡ (y_{r} − A_{r})^{2} − δ_{r}^{2}(x),  (13)

where r ∈ {m_{1} + 1, …, m} ∪ {m + n_{1} + 1, …, n}. If we put l = m_{1} + n_{1} and q = n − l then by formulas (12), (13) the functions n_{i} (i = 1, …, l) and p_{j} (j = 1, …, q) are defined as follows:

n_{i} ≡ \begin{cases}N_{i} & if i ∈ \{1, …, m_{1}\}, \\ N_{i−m_{1}+m} & if i ∈ \{m_{1} + 1, …, m_{1} + n_{1}\}, \end{cases}  (14)

p_{j} ≡ \begin{cases}P_{j+m_{1}} & if j ∈ \{1, …, m − m_{1}\}, \\ P_{j+m_{1}+n_{1}} & if j ∈ \{m − m_{1} + 1, …, n − m_{1} − n_{1}\}. \end{cases} (15)

In such a case the sets Ω^{0}, N_{α}, α ∈ {1, …, l}, and P_{β}, β ∈ {1, …, q}, have the following simpler form:

Ω^{0} = {x ∈ I, |y_{i} − A_{i}| < δ_{i}(x), i = 1, …, n},  (16)
N_{α} = {x ∈ I, |y_{α} − A_{α}| = δ_{α}(x), |y_{i} − A_{i}| < δ_{i}(x),
i = 1, …, n, i ≠ α},  (17)
P_{β} = {x ∈ I, |y_{β} − A_{β}| = δ_{β}(x), |y_{i} − A_{i}| < δ_{i}(x),
i = 1, …, n, i ≠ β},  (18)

In the proof of the next theorem we apply Theorem 3.

Theorem 4. Assume that:

(a) There are continuously differentiable and positive functions δ_{i}(x), i = 1, …, n, on the interval I with property (11).

(b) The inequality

∑_{j=1,j≠α^{0}}^{n} |a_{α^{0}j}(x)|δ_{j}(x) + ∑_{j=1}^{n} |a_{α^{0}j}(x)A_{j}| + |g_{α^{0}}(x, y)| <

< δ′_{α^{0}}(x) − a_{α^{0}α^{0}}(x)δ_{α^{0}}(x)  (19)

holds for each α^{0} ∈ {1, …, m_{1}}∪{m+1, …, m+n_{1}} and (x, y) ∈ N_{α}, where α = α^{0} if α^{0} ∈ {1, …, m_{1}} and α = α^{0}+m_{1}−m if α^{0} ∈ {m+1, …, m+n_{1}}.

(c) The inequality

∑_{j=1,j≠β^{0}}^{n} |a_{β^{0}j}(x)|δ_{j}(x) + ∑_{j=1}^{n} |a_{β^{0}j}(x)A_{j}| + |g_{β^{0}}(x, y)| <

< a_{β^{0}β^{0}}(x)δ_{β^{0}}(x) − δ′_{β^{0}}(x)  (20)

holds for each β^{0} ∈ {m_{1} + 1, …, m} ∪ {m + n_{1} + 1, …, n} and (x, y) ∈ P_{β}, where β = β^{0} − m_{1} if β^{0} ∈ {m_{1} + 1, …, m} and β = β^{0} − n_{1} − m_{1} if β^{0} ∈ {m + n_{1} + 1, …, n}.

Then there is at least one solution y = y(x) of problem (10), (2) such that for its components the inequalities

|y_{i}(x) − A_{i}| < δ_{i}(x), i = 1, …, n,  (21)

hold on the interval I.

Proof. First we prove that the set Ω^{0} described by (16) (where the functions n_{i} (i = 1, …, l), p_{j} (j = 1, …, q) are defined by formulas (14), (15)) satisfies the property Ω^{0} = Ω_{a,b}^{0} and generates some (n, p)-subset with respect to system (10). The property Ω^{0} = Ω_{a,b}^{0} is a consequence of formulas (11) and (16). Indeed, if l = {(x, y) : x ∈ I, y = y(x)} is a continuous curve for which the relation (x, y(x)) ⊂ \overline{Ω}^{0} holds on I, then from (11) and (16) it follows that

lim_{x→a+} y_{i}(x) = A_{i}, i ∈ {1, …, m}, lim_{x→b−} y_{k}(x) = A_{k} k ∈ {m + 1, …, n}.

Further we will compute the derivative of the function n_{α}, α ∈ {1, …, l}, along the trajectories of system (10) on the set N_{α}. In view of (17) and (19) we obtain

\frac{dn_{α}(x, y)}{dx} = 2(y_{α} − A_{α})y′_{α} − 2δ_{α}δ′_{α} = 2(y_{α} − A_{α})[ ∑_{j=1,j≠α}^{n} a_{αj}(y_{j} − A_{j}) +

+ ∑_{j=1}^{n} a_{αj}A_{j} + g_{α} + a_{αα}(y_{α} − A_{α})] − 2δ_{α}δ′_{α} <

< 2δ_{α}[a_{αα}δ_{α} − δ′_{α} + ∑_{j=1,j≠α}^{n} |a_{αj}|δ_{j} + ∑_{j=1}^{n} |a_{αj}A_{j}| + |g_{α}|] < 0.

By analogy we can compute that in view of (18) and (20) for the derivative of the function p_{β}, β ∈ {1, …, q}, along the trajectories of system (10) the inequality dp_{β}/dx > 0 holds on the set P_{β}. Inequalities (6) and (7) hold and, by Definition 2, the set Ω^{0} is an (n, p)-subset with respect to system (10).

Let {x_{i}} be some decreasing sequence of numbers with x_{i} ∈ I and lim_{i→∞} x_{i} = a. For each fixed i we denote S_{i} = (Ω^{0}∪Ω_{e}^{0})∩{(x_{i}, y) : y ∈ ℝ^{n}}, where

Ω_{e}^{0} = Ω_{se}^{0} = ⋃_{β=1}^{q} P_{β}\ ⋃_{α=1}^{l} N_{α}

(see the proof of Theorem 3). The set S_{i} ∩ Ω_{e}^{0} is a retract of the set Ω_{e}^{0} because the continuous mapping

Π : (x, y) ∈ Ω_{e}^{0} ↦ (x_{i}, y^{0}) ∈ S_{i} ∩ Ω_{e}^{0},

with

y_{j}^{0} = A_{j} − δ_{j}(x_{i}) + (y_{j} − A_{j} + δ_{j}(x))\frac{δ_{j}(x_{i})}{δ_{j}(x)} , j = 1, …, n,

is identical on S_{i} ∩ Ω_{e}^{0}. On the other hand, the set S_{i} ∩ Ω_{e}^{0} is not a retract of the set S_{i}. This follows from the fact that the set S̃_{i} ⊂ S_{i}, where

S̃_{i} = {(x, y) ∈ S_{i}, x = x_{i}, y_{j} = C_{j} ∈ (A_{j} − δ_{j}(x_{i}), A_{j} + δ_{j}(x_{i})),
C_{j} = const, j = 1, …, m_{1}; m + 1, …, m + n_{1}}

with the property that S̃_{i} ∩ Ω_{e}^{0} ⊂ S_{i} ∩ Ω_{e}^{0}, is not a retract of the set S̃_{i} ∩ Ω_{e}^{0} as the boundary of the sphere is not its retract ([18]). Consequently, all the assumptions of Theorem 3 are fulfilled and therefore Theorem 4 is valid. We obtain inequalities (21) from inequalities (8) and (9) or from (16).

Example 1. Let problem (10), (2) be of the form

y′_{1} = −4x^{−2}y_{1} + x^{5}(x − 1)^{−1}y_{2} + cos y_{2},
y′_{2} = (x − 1)^{4}x^{−1}y_{1} + 4(x − 1)^{−2}y_{2} + cos y_{1},
y_{1}(0+) = y_{2}(1−) = 0.

Then all the assumptions of Theorem 4 are fulfilled if we put n = 2, a_{11}(x) = −4x^{−2}, a_{12}(x) = x^{5}(x − 1)^{−1}, a_{21}(x) = (x − 1)^{4}x^{−1}, a_{22}(x) = 4(x − 1)^{−2}, g_{1}(x, y) = cos y_{2}, g_{2}(x, y) = cos y_{1}, m_{1} = 1, n_{1} = 0, m = 1, a = 0, b = 1, A_{1} = A_{2} = 0, δ_{1}(x) = x, δ_{2}(x) = 1 − x. Consequently, problem (10), (2) has at least one solution y = y(x) such that |y_{1}(x)| < x, |y_{2}(x)| < 1 − x on (0, 1).

Remark 1. [6] contains some theorems on the existence and uniqueness of solutions of singular Cauchy–Nicoletti problems for systems of ordinary differential equations. We note that these theorems are independent of the above-proved results. For example, if we apply Theorem 4.1 from [6, Chapter II, §4, pp. 37–38] to Example 1 then, in addition, the inequality

( − 4x^{−2}y_{1} + x^{5}(x − 1)^{−1}y_{2} + cos y_{2}) sign y_{1} ≤ −a(x)|y_{1}| + g(x, |y_{1}|, |y_{2}|)

must be valid on a set {(x, y) : 0 < x < 1, y ∈ ℝ^{2}}, where a(x) ≥ 0, a(x) ∈ L(0+, 1−) on (0, 1), and

sup {|g(x, |y_{1}|, |y_{2}|)| : |y_{1}| + |y_{2}| ≤ ρ} ∈ L(0, 1)  (22)

for each ρ ∈ (0, +∞). In our case a(x) ≡ −4x^{−2}, g(x, |y_{1}|, |y_{2}|) ≡ x^{5}(x − 1)^{−1}|y_{2}| + cos |y_{2}| and, consequently, relation (22) does not hold.

(B) Let system (23) be of the form

y′_{1} = f(x)y_{1} + F (x, y_{1}, y_{2}), y′_{2} = y_{1},  (23)

where f ∈ C(I, ℝ), F ∈ C(Ω, ℝ^{2}) ∩ L_{loc}(Ω). For system (23) we consider problem (2) if a = 0, b = T, 0 < T = const, m = 1, A_{1} = 0, A_{2} = −α, 0 ≤ α = const.

Theorem 5. Let there exist a positive function h ∈ C^{1}(I_{1}, ℝ^{+}), I_{1} = (0, T) and a negative function ω ∈ C^{1}(I_{1}, ℝ^{−}) such that h(0+) = 0, ω(x) < −α on I_{1}, ω(T −) = −α, h(x) < ω′(x) on I_{1} and on the set

D = {(x, y_{2}) : x ∈ I_{1}, ω(x) < y_{2} < −α}

the following inequalities hold:

f(x)h(x) − h′(x) + F (x, h(x), y_{2}) < 0 < F (x, 0, y_{2}).  (24)

Then there is at least one solution y = y(x) of problem (23), (25) where

y_{1}(0+) = 0, y_{2}(T −) = −α  (25)

such that the inequalities 0 < y_{1}(x) < h(x), ω(x) < y_{2}(x) < −α hold on I_{1}.

Proof. Let n_{1} ≡ y_{1}(y_{1} −h(x)) and p_{1} ≡ (y_{2} −ω(x))(y_{2} +α). Then the set Ω^{0} defined by (5) satisfies the condition Ω^{0} = Ω_{0,T}^{0} . Compute the derivatives along the trajectories of system (23). We obtain

\frac{dn_{1}(x, y)}{dx} = [f(x)y_{1} + F (x, y_{1}, y_{2})](y_{1} − h(x)) +
+ y_{1}[f(x)y_{1} + F (x, y_{1}, y_{2}) − h′(x)].

For the value y_{1} we have y_{1} = h(x) or y_{1} = 0 on the set N_{1}(x, y). Then from (24) it follows that dn_{1}(x, y)/dx < 0. Analogously, dp_{1}(x, y)/dx > 0 on the set P_{1}(x, y). Consequently, the set Ω^{0} is an (n, p)-subset.

The property that for some decreasing sequence of numbers {x_{i}} with x_{i} ∈ I_{1} and lim_{i→∞} x_{i} = 0 there is a set S_{i} with the properties described in Theorem 3 can be verified in a similar fashion as in the corresponding part of the proof of Theorem 4. Now all the assumptions of Theorem 3 are fulfilled and therefore Theorem 5 holds.

Example 2. In system (23) let us put f(x) = −Lx^{−m}, where 0 < L = const and 0 < m = const. Let h(x) = εx^{p} where εT^{p} < α, 0 < ε = const, p is an even positive number, ω(x) = [−α−(x−T)^{p}] exp(T −x), F (x, 0, y_{2}) > 0, and F (x, h(x), y_{2}) < εx^{p}(Lx^{−m} + px^{−1}) on D. Then all the assumptions of Theorem 5 are valid and its conclusion is true.

Remark 2. Some classes of singular problems were studied in [14], [15]. For example, in [15] the problem

y′_{1} = −(N − 1)y_{1}/x + F_{1}(y_{2}, x), y′_{2} = y_{1},  (26)
y_{1}(0+) = 0, y_{2}(T −) = −α ≤ 0,  (27)

where 2 ≤ N, N is an integer, x ∈ I_{1} and F_{1} ∈ C^{1}(ℝ^{−} × I_{1}, ℝ^{+}), is considered in connection with the study of increasing negative radial solutions of semilinear elliptic equations. In particular, this work contains the following result:

Let 0 ≤ d ≤ l ≤ NT^{−1}, 0 < K < sT^{−1}, and 0 < F_{1}(y_{2}, x) < (N − lx)K exp(−lx) hold for some constants d, l, K, and s if ψ(x) ≡ −α − s(T − x) exp(−dx) < y_{2} < −α and x ∈ I_{1}. Then problem (26), (27) has at least one solution y = y(x) which satisfies the inequalities 0 < y_{1}(x) < φ(x) ≡ Kx exp(−lx) and ψ(x) < y_{2}(x) < −α on I_{1}.

We note that problem (23), (25) is more general that the one given above. If we put f(x) = −(N − 1)x^{−1} and F (x, y_{1}, y_{2}) ≡ F_{1}(y_{2}, x) then from Theorem 5 (if h ≡ φ and ω ≡ ψ) it follows that there is at least one solution of problem (26), (27) with the mentioned properties. Moreover, as Example 2 shows, we may obtain more precise estimations of this solution if the functions h and ω are chosen in a proper way.

Acknowledgements

The author was supported by grant 201/93/0452 of the Czech Grant Agency (Prague).

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(Received 15.02.1995)

Author’s address:

Department of Mathematics

Faculty of Electrical Engineering

and Computer Science

Technical University (VUT)

Technická 8, 616 00 Brno

Czech Republic