GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 2, 1995, 211-224

THE CAUCHY–NICOLETTI PROBLEM WITH POLES

T. WERNER

Abstract. The Cauchy–Nicoletti boundary value problem for a system of ordinary differential equations with pole-type singularities is investigated. The conditions of the existence, uniqueness, and nonuniqueness of a solution in the class of continuously differentiable functions are given. The classical Banach contraction principle is combined with a special transformation of the original problem.

Introduction

This paper deals with the Cauchy–Nicoletti problem for a system of differential equations with poles, i.e., with the problem

(t − a_{i})^{r_{i}}x′_{i} = ( ∑_{j=0}^{r_{i}−1} A_{i,j} · (t − a_{i})^{j}) x_{i} +
+ f_{i}(t, x_{1}, …, x_{n}) + g_{i}(t), t ∈ I_{i},  (1)
x_{i}(a_{i}) = 0,  i = 1, …, n,  (2)

where x_{i} are unknown vector variables, A_{i,j} are constant matrices, f_{i}, g_{i} are given vector functions, a_{i} are given real numbers, and I_{i} are intervals of real numbers, all specified below.

The systematic research of singular problems for ordinary defferential equations (ODE) with nonsummable right-hand side was started by Czeczik [1]. The first monograph on some classes of singular boundary value problems was written by Kiguradze [2]. There are very general results on the Cauchy and especially the Cauchy–Nicoletti problems in this monograph (see [2], Part II). Recently many different properties and applications of singular problems have been investigated. In [3] three boundary value problems arising in gas dynamics are investigated. In [4] the question of when solutions of singular equations are bounded in some general sense is discussed.

211 In [5] the existence theorems for nonlinear problems with a singularity “less” than pole of degree 1 but with a “great” nonlinearity are given. In [6] the existence, the uniqueness and also the absence of solutions of the two-point boundary value problem of second order with one pole of degree 2 is investigated. The investigation of the general local Cauchy initial value problem with poles at infinity was carried out by Konyuchova [7]. The boundary value problems with poles are investigated in different ways. For example, in [8] the topological method is used while in [9] formal fundamental solutions are calculated at poles using formal power series. This paper uses the Laurent expansion method similarly to [7,9]. The main result shows how the existence and uniqueness of the solution of the problem (1), (2) depends on the real parts of the eigenvalues of matrices A_{i,j} in (1) if suitable restrictions are imposed on functions f_{i} and g_{i}.

1991 Mathematics Subject Classification. 34B10, 34B15.

Key words and phrases. Differential inequalities, global structure of solutions, oscillatory solution.

1072-947X/95/0300-0211$07.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1995 Plenum Publishing Corporation

Notation. The letters ℝ, ℂ, ℕ denote the sets of real, complex, and natural numbers, respectively. Let 𝕂 ∈ {ℝ, ℂ} and N := {1, …, n}. C[X, Y] (resp. C^{1}[X, Y]) stand for spaces of continuous and continuously differentiable mappings from the normed space X into the normed space Y. The scalar product of two vectors u, v ∈ 𝕂^{m} will be denoted by

(u|v) = ∑_{i=1}^{m} u_{i}v̄_{i}.

For i ∈ N, m_{i} ∈ ℕ, and x_{i} ∈ 𝕂^{m_{i}} put m = ∑_{i∈N} m_{i} and x = (x_{1}, …, x_{n}) ∈ 𝕂^{m}. For real a_{1} < a_{2} < ⋯ < a_{n} we denote I_{i} = (a_{1}, a_{i}) ∪ (a_{i}, a_{n}), I = [a_{1}, a_{n}]. The symbols A_{i,j} will stand for constant (m_{i} × m_{i}) matrices, f_{i} ∈ C[I_{i} × 𝕂^{m}, 𝕂^{m_{i}}], and g_{i} ∈ C[I_{i}, 𝕂^{m_{i}}]. We will use the following norms:

∥x_{i}∥ := \sqrt{(x_{i}|x_{i})},

∥A∥ := \sqrt{∑_{i,j=1}^{m_{i}} |a_{i,j}|}

for any matrix A = (a_{i,j})_{i,j=1}^{m_{i}},

∥φ_{i}∥ := max_{t∈I} ∥φ_{i}(t)∥

for φ_{i} ∈ C[I, 𝕂^{m_{i}}] and

∥φ∥ := ∑_{i∈N} ∥φ_{i}∥

for φ = (φ_{1}, …, φ_{n}) ∈ C[I, 𝕂^{m}].

Definitions and lemmas

Definition 1. We say that

i) the function φ = (φ_{1}, …, φ_{n}) ∈ C[I, 𝕂^{m}] is a solution of system (1) if φ_{i} ∈ C^{1}[I_{i}, 𝕂^{m_{i}}] and if (1) is satisfied for x_{i} = φ_{i}(t), i = 1, …, n; ii) the solution φ of system (1) is a solution of problem (1), (2) if

φ_{i}(a_{i}) = 0 (i = 1, …, n).

Consider the homogeneous part of system (1):

(t − a_{i})^{r_{i}}y′_{i} = ( ∑_{j=0}^{r_{i}−1} A_{i,j} · (t − a_{i})^{j}) y_{i}, t ∈ I_{i}, i = 1, …, n, (3)

which consists of n independent subsystems.

The following lemma on local transformation is an easy reformulation of the basic lemma in [10].

Lemma 1. There exist T_{i}, S_{i} ∈ ℝ, S_{1} = a_{1}, T_{1} > a_{1}, S_{n} < a_{n}, T_{n} = a_{n}, S_{i} < a_{i} < T_{i}, i = 2, …, n − 1, and transformations

z_{i} = P_{i}(t)y_{i}, i = 1, …, n, t ∈ (S_{i}, T_{i}) \ {a_{i}},  (4)

with continuously differentiable matrices P_{i}(t), which transform the subsystems of (3) into systems of the following special form:

(t − a_{i})^{r_{i}}z′_{i} = ( ∑_{j=0}^{r_{i}−1} B_{i,j} · (t − a_{i})^{j} + C_{i}(t)) z_{i}, t ∈ (S_{i}, T_{i}) \ {a_{i}}, (5)
i = 1, …, n,

where B_{i,j} are quasidiagonal constant matrices with non-zero blocks B_{i,j}^{1}, …, B_{i,j}^{k_{i}},

B_{i,0}^{k} = (\begin{matrix}λ_{i0}^{k} & γ_{i} & 0 & ⋯ & 0 & 0 \\ 0 & λ_{i0}^{k} & γ_{i} & ⋯ & 0 & 0 \\ · & · & · & ⋯ & · & · \\ 0 & 0 & 0 & ⋯ & λ_{i0}^{k} & γ_{i} \\ 0 & 0 & 0 & ⋯ & 0 & λ_{i0}^{k} \end{matrix}) , k = 1, …, k_{i},

γ_{i}, i = 1, …, n, are arbitrary small pozitive numbers,

rank B_{i,j}^{k} = rank B_{i,0}^{k} for j ≠ 0

and C_{i} (i = 1, …, n) are continuous and bounded remainder terms such that there exist limits

lim_{t→a_{i}} ∥\frac{C_{i}(t)}{(t − a_{i})^{r_{i}}} ∥ < +∞, i = 1, …, n.

Remark 1. As the proof of the basic lemma in [10] shows, the transformation matrix-functions P_{i}(t), i = 1, …, n, can be constructed in the polynomial form

P_{i}(t) = ∑_{j=0}^{r_{i}−1} P_{i,j} · (t − a_{i})^{j}

with the constant matrices P_{i,j}. It is the classical result that the matrices P_{i0} transforming A_{i0} to the modified Jordan matrices are regular (see, for example, [11]).

Definition 2. We say that system (1) satisfies the condition

CO1: if f_{i}(t, 0, …, 0) ≡ 0 on the interval I and if there exist functions μ_{i} ∈ C[I_{i}, ℝ^{+}] such that

∥f_{i}(t, x) − f_{i}(t, x̃)∥ ≦ μ_{i}(t)∥x − x̃∥, for each (t, x), (t, x̃) ∈ I_{i} × 𝕂^{m}

and

∫_{I_{i}} μ_{i}(t)|t − a_{i}|^{−r_{i}}dt = M_{i} < ∞, i = 1, …, n;

CO2: if

∫_{I_{i}} ∥g_{i}(t)∥ |t − a_{i}|^{−r_{i}}dt = G_{i} < ∞, i = 1, …, n.

Definition 3. We say that the ith subsystem of the transformed system (5)

(t − a_{i})^{r_{i}}z′_{i} = ( ∑_{j=0}^{r_{i}−1} B_{i,j} · (t − a_{i})^{j} + C_{i}(t)) z_{i}, t ∈ (S_{i}, T_{i}) \ {a_{i}}

satisfies the condition

CO3: if all eigenvalues of the matrix B_{i,0} have nonpositive real parts and those of them which lie on the imaginary axis of ℂ are simple, i.e.,

Re λ_{i0}^{k} ≦ 0,

and if Re λ_{i0}^{k} = 0 then mult λ_{i0}^{k} = 1, l = 1, …, k_{i}; CO4: if for each k such that λ_{i0}^{k} is simple and Re λ_{i0}^{k} = 0 there exists

j_{k} ∈ {1, …, r_{i} − 1} such that

Re λ_{i0}^{k} = Re B_{i,1}^{k} = ⋯ = Re B_{i,j_{k}−1}^{k} = 0, Re B_{i,j_{k}}^{k} > 0; (6)

CO5: if for each k such that λ_{i0}^{k} is simple and Re λ_{i0}^{k} = 0 there is no j_{k} ∈ {1, …, r_{i} − 1} with property (6); CO3’: if all eigenvalues of the matrix B_{i,0} have nonnegative real parts and

those which lie on the imaginary axis of ℂ are simple, i.e.,

Re λ_{i0}^{k} ≧ 0,

and if

Re λ_{i0}^{k} = 0 then mult λ_{i0}^{k} = 1, l = 1, …, k_{i};

CO4’: if for each k such that λ_{i0}^{k} is simple and Re λ_{i0}^{k} = 0 there exists j_{k} ∈ {1, …, r_{i} − 1} such that

Re λ_{i0}^{k} = Re B_{i,1}^{k} = ⋯ = Re B_{i,j_{k}−1}^{k} = 0, Re B_{i,j_{k}}^{k} < 0; (7)

CO5’: if for each k such that λ_{i0}^{k} is simple and Re λ_{i0}^{k} = 0 there is no j_{k} ∈ {1, …, r_{i} − 1} with property (7).

Definition 4. Let system (1) satisfy conditions CO1 and CO2. We say that the ith equation of (1)

(t − a_{i})^{r_{i}}x′_{i} = ( ∑_{j=0}^{r_{i}−1} A_{i,j} · (t − a_{i})^{j}) x_{i} + f_{i}(t, x) + g_{i}(t), t ∈ I_{i},

has:

a) property U_{R},
b) property \overline{U}_{R},
c) property U_{L},
d) property \overline{U}_{L},

if its transformed homogeneous part, i.e., the ith subsystem of (5) satisfies the conditions

a) CO3 ∧ CO5,

b) CO3’ ∧ CO4,

c) (CO3 for r_{i} odd ∨ CO3’ for r_{i} even) ∧ (CO5 with j_{k} odd ∨ CO5’

with j_{k} even),

d) (CO3’ for r_{i} odd ∨ CO3 for r_{i} even) ∧ (CO4 with j_{k} odd ∨ CO4’

with j_{k} even).

The following lemma summarizes some local topics of [7].

Lemma 2. Let the ith equation of (1) have property U_{R} for some fixed i ∈ {1, …, n − 1} [resp. have property U_{L} for some fixed i ∈ {2, …, n}]. Then for T_{i} (resp. S_{i}) sufficiently close to a_{i} (T_{i} > a_{i}, resp. S_{i} < a_{i}) there exist constants L_{i}^{+} (resp. L_{i}^{−}) such that for any fundamental matrix Ψ_{i} of the ith transformed subsystem of (5) we have

∥Ψ_{i}(t)Ψ_{i}^{−1}(s)∥ ≦ L_{i}^{+}, for a_{i} < s ≦ t ≦ T_{i},

resp.

∥Ψ_{i}(t)Ψ_{i}^{−1}(s)∥ ≦ L_{i}^{−}, for S_{i} ≦ t ≦ s < a_{i}.

Moreover, there are only trivial solutions of the ith subsystem of (5) on the intervals (a_{i}, T_{i}] (resp. [S_{i}, a_{i})) which vanish at the singular point a_{i}.

Corollary 1. Let the assumptions of Lemma 2 hold. Then there exists a constant K_{i}^{+} (resp. K_{i}^{−}) such that for any fundamental matrix Φ_{i} of the ith subsystem of (3) we have

∥Φ_{i}(t)Φ_{i}^{−1}(s)∥ ≦ K_{i}^{+}, for a_{i} < s ≦ t ≦ a_{n},

resp.

∥Φ_{i}(t)Φ_{i}^{−1}(s)∥ ≦ K_{i}^{−}, for a_{1} ≦ t ≦ s < a_{i}.

Moreover, there are only trivial solutions of the ith subsystem of (3) on the intervals (a_{i}, a_{n}) [resp. (a_{1}, a_{i})] which vanish at the singular point a_{i}.

Proof. Lemma 2 implies that

∥Φ_{i}(t)Φ_{i}^{−1}(s)∥ ≦ ∥P_{i}^{−1}(t)∥ · ∥P_{i}(s)∥L_{i}^{+}, for a_{i} < s ≦ t ≦ T_{i},

resp.

∥Φ_{i}(t)Φ_{i}^{−1}(s)∥ ≦ ∥P_{i}^{−1}(t)∥ · ∥P_{i}(s)∥L_{i}^{−}, for S_{i} ≦ t ≦ s < a_{i},

where the terms on the right-hand sides are bounded for T_{i} (resp. S_{i}) sufficiently close to a_{i}, because P_{i0} in Remark 1 is regular. For each fixed s the columns of Φ_{i}(t)Φ_{i}^{−1}(s) are the solutions of the ith subsystem of (3) with their norms bounded by sup_{t∈(a_{i},T_{i}]} ∥P_{i}^{−1}(t)∥ · ∥P_{i}(s)∥L_{i}^{+} =: Q_{i}^{+} (resp. sup_{t∈[S_{i},a_{i})} ∥P_{i}^{−1}(t)∥·∥P_{i}(s)∥L_{i}^{−} =: Q_{i}^{−}) at some point of the interval [T_{i}, a_{n})

(resp. (a_{1}, S_{i}]). Since ∥ ∑_{j=0}^{r_{i}−1} A_{i,j} · (t − a_{i})^{j−r_{i}}∥ is bounded on the interval

[T_{i}, a_{n}] (resp. [a_{1}, S_{i}]), we have

∥φ_{i}(t)∥ ≦ Q_{i}^{+}e^{A_{i}^{+}(a_{n}−T_{i})} resp. Q_{i}^{−}e^{A_{i}^{−}(S_{i}−a_{1})}

for any column φ_{i} of Φ_{i}(t)Φ_{i}^{−1}(s). Here

A_{i}^{+} denote sup ∥ ∑_{j=0}^{r_{i}−1} A_{i,j} · (t − a_{i})^{j−r_{i}}∥, t ∈ [T_{i}, a_{n}],

resp.

A_{i}^{−} denote sup ∥ ∑_{j=0}^{r_{i}−1} A_{i,j} · (t − a_{i})^{j−r_{i}}∥, t ∈ [a_{1}, S_{i}].

Consequently,

∥Φ_{i}(t)Φ_{i}^{−1}(s)∥ ≦\sqrt{m_{i}}Q_{i}^{+}e^{A_{i}^{+}(a_{n}−T_{i})} =: K_{i}^{+} (resp.\sqrt{m_{i}}Q_{i}^{−}e^{A_{i}^{−}(S_{i}−a_{1})} =: K_{i}^{−}) for a_{i} < s ≦ t ≦ a_{n} (resp. a_{1} ≦ t ≦ s < a_{i}). The nonexistence of any nontrivial solution vanishing at the point a_{i} directly follows, as in Lemma 2, from the regularity of P_{i0} and from the uniqueness of solutions of the ith subsystem of (3) on the intervals [T_{i}, a_{n}] (resp. [a_{1}, S_{i}]).

Lemma 3. Let the ith equation of (1) has property U_{R} for some fixed i ∈ {1, …, n − 1} (resp. has property U_{L} for some fixed i ∈ {2, …, n}). Then for any (n − 1)-tuple of functions φ_{j} ∈ C[I, 𝕂^{m_{i}}], j = 1, …, n, j ≠ i, all solutions of the equation

(t − a_{i})^{r_{i}}x′_{i} = ( ∑_{j=0}^{r_{i}−1} A_{i,j} · (t − a_{i})^{j}) x_{i} +

+f_{i}(t, φ_{1}(t), …, φ_{i−1}(t), x_{i}, φ_{i+1}(t), …, φ_{n}(t)) + g_{i}(t), t ∈ I_{i}, (8)

satisfy the ith subcondition of (2), i.e., x_{i}(a_{i}) = 0.

Proof. Without loss of generality, we can consider only the case of property U_{R}. Let us consider the mentioned ith equation of (1) transformed by the the ith transformation from (4):

(t − a_{i})^{r_{i}}z′_{i} = ( ∑_{j=0}^{r_{i}−1} B_{i,j} · (t − a_{i})^{j}) z_{i} +

+f̃_{i}(t, φ_{1}(t), …, φ_{i−1}(t), z_{i}, φ_{i+1}(t), …, φ_{n}(t)) + g̃_{i}(t), t ∈ (a_{i}, T_{i}), (9)

where f̃_{i} contain also the remainder term C_{i}(t)z_{i} from (5):

f̃_{i}(t, φ_{1}(t), …, φ_{i−1}(t), z_{i}, φ_{i+1}(t), …, φ_{n}(t)) ≡
≡ P_{i}(t)f_{i}(t, φ_{1}(t), …, φ_{i−1}(t), P_{i}^{−1}z_{i}, φ_{i+1}(t), …, φ_{n}(t)) + C_{i}(t)z_{i}

and

g̃_{i}(t) ≡ P_{i}(t)g_{i}(t).

Due to the behavior of the ith transformation from (4), the continuity of f_{i}, g_{i} and conditions CO1 and CO2 are invariant with respect to this transformation. Let ψ_{i} be any solution of (9). If T_{i} is sufficiently close to a_{i}, then for the derivative of the norm of ψ_{i} we have

\frac{d}{dt}∥ψ_{i}(t)∥^{2} = 2 Re(ψ_{i}(t)| ( ∑_{j=0}^{r_{i}−1} B_{i,j}(t − a_{i})^{j−r_{i}} ) ψ_{i}(t) +

+f̃_{i}(t, φ_{1}(t), …, φ_{i−1}(t), ψ_{i}(t), φ_{i+1}(t), …, φ_{n}(t)) + g̃_{i}(t)) ≧

≧ ((2 Re λ − ϵ)(t − a_{i})^{L−r_{i}} − μ̃_{i}(t) ( ∑_{j∈N\{i}} ∥φ_{j}(t)∥)) · ∥ψ_{i}(t)∥^{2} −

−∥g̃_{i}(t)∥ · ∥ψ_{i}(t)∥,

where ϵ > 0 is arbitrarily small, the term μ̃_{i}(t) · (∑_{j∈N\{i}} ∥φ_{j}(t)∥) is bounded on the considered interval, L ≦ r_{i} − 1 is a greater value of j_{k}-s in the condition CO4 or L = 0 if no j_{k} exists, and Re λ is the smallest value of Re λ_{j_{k}}^{k} for j_{k} = L, i.e.,

Re λ = min{Re λ_{j_{k}}^{k}, j_{k} = L} > 0.

Consequently,

∥ψ_{i}(t)∥′ ≧ \frac{Re λ}{2}(t − a_{i})^{L−r_{i}}∥ψ_{i}(t)∥ − ∥g̃_{i}∥ ≧ \frac{Re λ}{2(t − a_{i})}∥ψ_{i}(t)∥ − ∥g̃_{i}∥

for T_{i} sufficiently close to a_{i}; hence

∥ψ_{i}(t)∥ ≦ ((\frac{∥ψ_{i}(T_{i})∥}{(T_{i} − a_{i})^{\frac{Re λ}{2}}} + \frac{(T_{i} − a_{i})^{1− \frac{Re λ}{2}}}{1 − \frac{Re λ}{2}} ∥g̃_{i}∥)(t − a_{i})^{\frac{Re λ}{2}} −

−\frac{∥g̃_{i}∥}{1 − \frac{Re λ}{2}} (t − a_{i})) ,

where the last term tends to zero as t → a_{i}+. This implies that each solution φ_{i} = P_{i}^{−1}ψ_{i} of (8) on (a_{i}, T_{i}) vanishes at the point a_{i}, too.

Corollary 2. Let the assumptions of Lemma 3 hold. Then there exists a constant K_{i}^{+} > 0 (resp. K_{i}^{−} > 0) such that any fundamental matrix Φ_{i} of the ith subsystem of (3) satisfies

∥Φ_{i}(t)Φ_{i}^{−1}(s)∥ ≦ K_{i}^{+} resp. K_{i}^{−}

for

a_{i} < t ≦ s < a_{n} resp. a_{1} < s ≦ t < a_{i}.

Proof. Each column of Φ_{i}(t)Φ_{i}^{−1}(s) is a solution of the ith equation of (1) for f_{i} ≡ g_{i} ≡ 0 on some interval (a_{i}, T_{i}) (resp. (S_{i}, a_{i})). Such a solution vanishes at the point a_{i} and its extension on the interval (s, a_{n}) (resp.

(a_{1}, s)) is bounded because ∑_{j=0}^{r_{i}−1} A_{i,j} · (t − a_{i})^{j−r_{i}} is bounded there, too.

Thus ∥Φ_{i}(t)Φ_{i}^{−1}(s)∥ is bounded on the interval [a_{i}, a_{n}] (resp. [a_{1}, a_{i}]) (and vanishes at a_{i}), too.

Main results

Let us consider system (1). Denote by N_{L} and N_{R} the sets of all indices i for which the ith equation of (1) has property \overline{U}_{L} or \overline{U}_{R}, respectively, and by N_{L}^{0}, N_{R}^{0} the sets of all indices i for which the ith equation of (1) has property U_{L} or U_{R}, respectively. Then we have

Theorem 1. Let

card N_{L} + card N_{R} + card N_{L}^{0} + card N_{R}^{0} = 2(n − 1)

and let the inequality

∑_{i=1}^{n} K_{i}M_{i} < 1  (10)

hold.

Then there exists just a (∑_{i∈N_{L}} m_{i} + ∑_{i∈N_{R}} m_{i}) -parametric family of solutions of problem (1), (2).

Proof. Let φ = (φ_{1}, …, φ_{n}) be a solution of problem (1), (2). Then its ith component φ_{i} can be written in one of the following forms. If i ∈ N \ (N_{L} ∪ N_{R}), then

φ_{i}(t) = ∫_{a_{i}}^{t} Φ_{i}(t)Φ_{i}^{−1}(s)[f_{i}(s, φ(s)) + g_{i}(s)](s − a_{i})^{−r_{i}}ds, t ∈ I; (11)

if i ∈ N \ (N_{L}^{0} ∪ N_{R}^{0}), then

φ_{i}(t) = \begin{cases}Φ_{i}(t)Φ_{i}^{−1}(a_{1})φ_{i}(a_{1}) + ∫_{a_{1}}^{t} Φ_{i}(t)Φ_{i}^{−1}(s)[f_{i}(s, φ(s))+ \\ +g_{i}(s)](s − a_{i})^{−r_{i}}ds, t ∈ [a_{1}, a_{i}], \\ Φ_{i}(t)Φ_{i}^{−1}(a_{n})φ_{i}(a_{n}) + ∫_{a_{n}}^{t} Φ_{i}(t)Φ_{i}^{−1}(s)[f_{i}(s, φ(s))+ \\ +g_{i}(s)](s − a_{i})^{−r_{i}}ds, t ∈ [a_{i}, a_{n}]; \end{cases} (12)

if i ∈ N_{R}^{0} ∩ (N \ N_{L}^{0}), then

φ_{i}(t) = Φ_{i}(t)Φ_{i}^{−1}(a_{1})φ_{i}(a_{1}) +
+ ∫_{a_{1}}^{t} Φ_{i}(t)Φ_{i}^{−1}(s)[f_{i}(s, φ(s)) + g_{i}(s)](s − a_{i})^{−r_{i}}ds, t ∈ I,  (13)

and, finally, if i ∈ N_{L}^{0} ∩ (N \ N_{R}^{0}), then

φ_{i}(t) = Φ_{i}(t)Φ_{i}^{−1}(a_{n})φ_{i}(a_{n}) +
+ ∫_{a_{n}}^{t} Φ_{i}(t)Φ_{i}^{−1}(s)[f_{i}(s, φ(s)) + g_{i}(s)](s − a_{i})^{−r_{i}}ds, t ∈ I.  (14)

Here φ_{i}(a_{1}) ∈ 𝕂^{m_{i}}, i ∈ N_{L}, φ_{i}(a_{n}) ∈ 𝕂^{m_{i}}, i ∈ N_{R}, are arbitrary constants. Lemma 3 and Corollaries 1 and 2 ensure that the above integrations are correct. On the other hand, the solutions of the system of integral equations (11)–(14) are the solutions of problem (1), (2) which satisfy the boundary conditions

\begin{matrix}x_{i}(a_{1}) = φ_{i}(a_{1}), i ∈ N_{L}, \\ x_{i}(a_{n}) = φ_{i}(a_{n}), i ∈ N_{R}. \end{matrix}  (15)

Thus for any fixed values of φ_{i}(a_{1}) ∈ 𝕂^{m_{i}}, i ∈ N_{L}, φ_{i}(a_{n}) ∈ 𝕂^{m_{i}}, i ∈ N_{R}, problem (1), (2), (15) is equivalent to the system of integral equations (11)– (14). Define the integral operator F by means of the right sides of (11)–(14), which maps C[I, 𝕂^{m}] into itself and denote

I_{i}(ζ, ξ) := ∫_{ζ}^{ξ} Φ_{i}(t)Φ_{i}^{−1}(s)[f_{i}(s, φ(s)) + g_{i}(s)](s − a_{i})^{−r_{i}}ds.

Denote by B(c, R) a ball in the space C[I, 𝕂^{m}] with radius R and center at the fundamental solution of the homogeneous part of (1) c = (c_{1}, …, c_{n}), where

c_{i}(t) = \begin{cases}0, & t ∈ I, i ∈ N \ (N_{L} ∪ N_{R}), \\ Φ_{i}(t)Φ_{i}^{−1}(a_{1})φ_{i}(a_{1}), & t ∈ I, i ∈ N_{L} \ N_{R}, \\ Φ_{i}(t)Φ_{i}^{−1}(a_{1})φ_{i}(a_{1}), & t ∈ [a_{1}, a_{i}], i ∈ N_{L} ∩ N_{R}, \\ Φ_{i}(t)Φ_{i}^{−1}(a_{n})φ_{i}(a_{n}), & t ∈ [a_{i}, a_{n}], i ∈ N_{L} ∩ N_{R}, \\ Φ_{i}(t)Φ_{i}^{−1}(a_{n})φ_{i}(a_{n}), & t ∈ I, i ∈ N_{R} \ N_{L}. \end{cases}

For any φ ∈ B(c, R) we get

∥F φ − c∥ ≦ ∑_{i∈N\(N_{L}∪N_{R})} sup_{t∈I_{i}} ∥I_{i}(a_{i}, t)∥ +

+ ∑_{i∈(N_{L}∩N_{R})} max{ sup_{t∈(a_{1},a_{i})} ∥I_{i}(a_{1}, t)∥, sup_{t∈(a_{i},a_{n})} ∥I_{i}(a_{n}, t)∥} +

+ ∑_{i∈(N_{L}\N_{R})} sup_{t∈I_{i}} ∥I_{i}(a_{1}, t)∥ + ∑_{i∈(N_{R}\N_{L})} sup_{t∈I_{i}} ∥I_{i}(a_{n}, t)∥ ≦

≦ ∑_{i∈N} K_{i}(M_{i}∥φ∥ + G_{i}) ≦

≦ (∑_{i∈N} K_{i}M_{i}) ∥φ − c∥ + ∑_{i∈N} K_{i}(M_{i}∥c∥ + G_{i}).  (16)

Since ∑_{i∈N} K_{i}M_{i} < 1, we can select a radius R such that

(1 − ∑_{i∈N} K_{i}M_{i}) R > (∑_{i∈N} K_{i}(M_{i}∥c∥ + G_{i})) .

The last inequality implies that the operator F maps the ball B(c, R) into itself.

Similarly, we obtain the estimate

∥F φ − F φ̃∥ ≦ (∑_{i∈N} K_{i}M_{i}) ∥φ − φ̃∥ for any pair φ, φ̃ ∈ B(c, R); (17)

hence F is a contraction. The Banach theorem gives the existence of a unique solution of the system of integral equations (11)–(14) satisfying condition (15). This solution is simultaneously the solution of problem (1), (2) which satisfies the condition (15). The values φ_{i}(a_{1}), i ∈ N_{L}, φ_{i}(a_{n}), i ∈ N_{R} occurring in (15) can be selected arbitrarily and the total dimension of (15) is ∑_{i∈N_{L}} m_{i} + ∑_{i∈N_{R}} m_{i}.

Remark 2. Condition (10) is substantial in view of the following example.

Example. Consider a linear problem

x′_{1} = −\frac{x_{1}}{t} + 2(x_{1} − x_{2}), t ∈ (0, 1),  (18)
x′_{2} = −\frac{x_{2}}{t − 1} + 2(x_{1} − x_{2}), t ∈ (0, 1),  (19)
x_{1}(0) = x_{2}(1) = 0.  (20)

Equation (18) has property U_{R} at its singular point a_{1} = 0 and equation (19) has property U_{L} at the singular point a_{2} = 1. However, there exists a one-parametric system of solutions of problem (18), (19), (20)

x_{1} = ct, x_{2} = c(t − 1), c ∈ 𝕂

where c is arbitrary. This happens because condition (10) does not hold. In fact, we have

K_{1}M_{1} + K_{2}M_{2} ≧ 4 > 1

where

K_{1} ≧ sup_{0<s≤t≤1} \frac{s}{t} = 1,

K_{2} ≧ sup_{0≤t≤s<1} \frac{s − 1}{t − 1} = 1,

μ_{1} ≧ 2, μ_{2} ≧ 2

and so M_{1} ≧ ∫_{0}^{1} 2dt = 2, M_{2} ≧ 2.

The next Theorem 2 indicates the special case where condition (10) can be omitted.

Theorem 2. Let

card N_{L} = card N_{R}^{0} = n − 1,

or

card N_{L}^{0} = card N_{R} = n − 1.

Then there exists just a (∑_{i∈N_{L}} m_{i})-parametric [resp. a (∑_{i∈N_{R}} m_{i})-pa-rametric] family of solutions of problem (1), (2).

Proof. The system of integral equations in the proof of Theorem 1 reduces to (13) in the first case or to (14) in the second case. Let us consider the first case and define a new norm in the space C[I, 𝕂^{m}]:

∥φ∥_{p} := max_{t∈I} (∥φ(t)∥e^{−p(t−a_{1})}) .

The following estimates hold:

∥I_{i}(a_{1}, t)∥e^{−p(t−a_{1})} ≦

≦ K_{i} (M_{i} ∫_{a_{1}}^{t} ∥φ(s)∥e^{−p(s−a_{1})}e^{p(s−a_{1})}ds + G_{i}) e^{−p(t−a_{1})} ≦

≦ K_{i} (M_{i}∥φ∥_{p} ∫_{a_{1}}^{t} e^{p(s−a_{1})}ds + G_{i}) e^{−p(t−a_{1})} ≦

≦ K_{i} (M_{i} \frac{∥φ∥_{p}}{p} (e^{p(t−a_{1})} − 1) + G_{i}) e^{−p(t−a_{1})} ≦

≦ K_{i} (M_{i} \frac{∥φ∥_{p}}{p} + G_{i}) , i ∈ N.

So estimate (16) has the form

∥F φ − c∥_{p} ≦ ∑_{i∈N} K_{i} (M_{i} \frac{∥φ∥_{p}}{p} + G_{i}) ≦

≦ \frac{∑_{i∈N} K_{i}M_{i}}{p} ∥φ − c∥_{p} + ∑_{i∈N} K_{i} (M_{i} \frac{∥c∥_{p}}{p} + G_{i}) .

Similarly we obtain the modification of estimate (17)

∥F φ − F φ̃∥_{p} ≦ \frac{∑_{i∈N} K_{i}M_{i}}{p} ∥φ − φ̃∥_{p}.

When we select p such that

p > ∑_{i∈N} K_{i}M_{i},

the proof can be completed as that of Theorem 1.

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(Received 20.10.1993)

Author’s address: Department of Mathematics Czech Academy of Sciences Žižkova 22, 616 62 Brno Czech Republic