Georgian Mathematical Journal 1(1994), No. 3, 303-314

ON SOME TWO-POINT BOUNDARY VALUE PROBLEMS

FOR TWO-DIMENSIONAL SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

A.LOMTATIDZE

Abstract. Sufficient conditions for the solvability of two-point boundary value problems for the system x′_{i} = f_{i}(t, x_{1}, x_{2}) (i = 1, 2) are given, where f_{1} and f_{2} : [a_{1}, a_{2}] × R^{2} → R are continuous functions.

1. Statement of the problems and formulation of the main results

Consider the system of ordinary differential equations

x′_{i} = f_{i}(t, x_{1}, x_{2}) (i = 1, 2)  (1.1)

with boundary conditions

λ_{i1}x_{1}(a_{i}) + λ_{i2}x_{2}(a_{i}) + g_{i}(x_{1}, x_{2}) = 0 (i = 1, 2)  (1.2)

or

λ_{i1}x_{1}(a_{i}) + λ_{i2}x_{2}(a_{i}) + h_{i}(x_{1}(a_{i}), x_{2}(a_{i})) = 0 (i = 1, 2), (1.3)

where −∞ < a_{1} < a_{2} < +∞, λ_{ij} ∈ R (i, j = 1, 2), the functions f_{i} : [a_{1}, a_{2}] × R^{2} → R, h_{i} : R^{2} → R (i = 1, 2) are continuous and g_{i} : C([a_{1}, a_{2}]; R^{2}) → R (i = 1, 2) are the continuous functionals.

The problems of the forms (1.1),(1.2) and (1.1),(1.3) have been studied earlier in [1-10]. In the present paper new criteria for solvability of these problems are established which have the nature of one-sided restrictions imposed on f_{1} and f_{2}.

We use the following notation:

R is the set of all real numbers; R_{+} = [0, +∞[,

D = [a_{1}, a_{2}] × R^{2},

D_{1} = [a_{1}, a_{2}] × (R\{0}) × R; D_{2} = [a_{1}, a_{2}] × R × (R\{0}),

C(A, B) is the set of continuous maps from A to B.

1991 Mathematics Subject Classification. 34B15.

A solution of the system (1.1) is sought in the class of continuously differentiable vector-functions (x_{1}, x_{2}) : [a_{1}, a_{2}] → R^{2}.

1.1. Problem (1.1), (1.2). We shall study the problem (1.1),(1.2) in the case when

(−1)^{i}σ · λ_{i1}λ_{i2} > 0 (i = 1, 2),

and the functionals g_{1} and g_{2} satisfy the inequality

|g_{1}(x, y)| + |g_{2}(x, y)| ≤ l,

on C([a_{1}, a_{2}]; R^{2}), where σ ∈ {−1, 1} and l ∈ R_{+}.

Theorem 1.1. Suppose that

σ[f_{1}(t, x, y) − p_{11}(t, x, y)x − p_{12}(t, x, y)y] sgn y ≥ −q_{0}
for (t, x, y) ∈ D_{2},  (1.4)
σ[f_{2}(t, x, y) − p_{21}(t, x, y)x − p_{22}(t, x, y)y] sgn x ≥ −q_{0}
for (t, x, y) ∈ D_{1},  (1.5)
σf_{1}(t, 0, y) sgn y ≥ 0 for a_{1} ≤ t ≤ a_{2}, |y| ≥ r_{0},  (1.6)
σf_{2}(t, x, 0) sgn x ≥ 0 for a_{1} ≤ t ≤ a_{2}, |x| ≥ r_{0},  (1.7)

where p_{11} : D_{2} → R, p_{22} : D_{1} → R, p_{12} and p_{21} : D → R are continuous bounded functions and q_{0}, r_{0} are positive constants. Then the problem (1.1.), (1.2) has at least one solution.

Corollary 1.1. Let the inequalities

σf_{1}(t, x, y) sgn y ≥ p_{0}(|y| − |x|) − q_{0},  (1.8)
σf_{2}(t, x, y) sgn x ≥ p_{0}(|x| − |y|) − q_{0},  (1.9)

hold on D, where p_{0} and q_{0} are positive constants. Then the problem (1.1), (1.2) is solvable.

Theorem 1.2. Suppose that

σ[f_{1}(t, x, y) − p_{11}(t, x, y)x] sgn y ≥ 0
for a_{1} ≤ t ≤ a_{2}, μxy > 0,  (1.10)
σ[f_{1}(t, x, y) − p_{11}(t, x, y)x − p_{12}(t, x, y)y] sgn y ≥ −q_{0}
for a_{1} ≤ t ≤ a_{2}, μxy < 0,  (1.11)
σ[f_{2}(t, x, y) − p_{22}(t, x, y)y] sgn x ≥ −q(x)
for a_{1} ≤ t ≤ a_{2}, μxy > 0,  (1.12)
σ[f_{2}(t, x, y) − p_{21}(t, x, y)x − p_{22}(t, x, y)y] sgn x ≥ −q_{0}
for a_{1} ≤ t ≤ a_{2}, μxy < 0  (1.13)

and the inequality (1.7) holds, where p_{11} : D_{2} → R, p_{22} : D_{1} → R, p_{12}, p_{21} : D → R are continuous bounded functions, q ∈ C(R; R_{+}), μ ∈ {−1, 1} and q_{0}, r_{0} are positive constants. Then the problem (1.1), (1.2) has at least one solution.

Theorem 1.3. Suppose that

σ[f_{1}(t, x, y) − p_{11}(t, x, y)x] sgn y ≥ −q_{0}
for a_{1} ≤ t ≤ a_{2}, μxy < 0,  (1.14)
σ[f_{2}(t, x, y) − p_{22}(t, x, y)y] sgn x ≥ −q(x)
for (t, x, y) ∈ D,  (1.15)

and the inequalities (1.7) and (1.10) hold, where p_{11} : D_{2} → R, p_{22} : D_{1} → R are continuous bounded functions, q ∈ C(R, R_{+}), μ ∈ {−1, 1} and r_{0}, q_{0} are positive constants. Then the problem (1.1), (1.2) has at least one solution.

1.2. The Problem (1.1), (1.3). We shall study the problem (1.1),(1.3) in the case when

(−1)^{i}σλ_{i1}λ_{i2} ≥ 0, |λ_{i1}| + |λ_{i2}| ≠ 0 (i = 1, 2)

and

sup{|h_{i}(x, y)| : (−1)^{i}σxy > 0} < +∞ (i = 1, 2),

where σ ∈ {−1, 1}.

Theorem 1.4. Let the inequalities (1.4)–(1.7) hold, where p_{11} : D_{2} → R, p_{22} : D_{1} → R, p_{12}, p_{21} : D → R are continuous bounded functions and q_{0}, r_{0} are positive constants. Then the problem (1.1), (1.3) has at least one solution.

Corollary 1.2. Let the inequalities (1.8) and (1.9) hold on D, where p_{0} and q_{0} are positive constants. Then the problem (1.1), (1.3) has one solution.

Consider as an example the boundary value problem

\begin{matrix}x′_{1} = p_{11}(t)x_{1} + p_{12}(t)x_{2} + g_{11}(t)x_{2}^{2k_{1}+1} + \\ +g_{12}(t)|x_{1}|^{n_{1}}x_{2}^{2m_{1}+1} + q_{1}(t), \\ x′_{2} = p_{21}(t)x_{1} + p_{22}(t)x_{2} + g_{21}(t)x_{1}^{2k_{2}+1} + \\ +g_{22}(t)|x_{2}|^{n_{2}}x_{1}^{2m_{2}+1} + q_{2}(t), \end{matrix}  (1.16)

x_{2}(a_{1}) = h_{1}(x_{1}(a_{1})), x_{2}(a_{2}) = h_{2}(x_{1}(a_{2})),  (1.17)

where n_{i}, k_{i}, m_{i} ∈ {1, 2, 3, … } (i = 1, 2), p_{ij}, g_{ij}, q_{i} ∈ C([a_{1}, a_{2}]; R) (i, j = 1, 2), h_{i} ∈ C(R; R) (i = 1, 2). It follows from Corollary 1.2 that if for some σ ∈ {−1, 1} and r ∈ R_{+} the inequalities

σg_{i1}(t) > 0, σg_{i2}(t) ≥ 0 for a_{1} ≤ t ≤ a_{2} (i = 1, 2)

and

(−1)^{i}h_{i}(x) sgn x ≤ 0 for |x| ≥ r (i = 1, 2)  (1.18)

hold, then the problem (1.16),(1.17) has at least one solution. Therefore, the problem (1.16),(1.17) is solvable in the resonance case, i.e. in the case when the corresponding homogeneous problem

\begin{matrix}x′_{1} = p_{11}(t)x_{1} + p_{12}(t)x_{2} \\ x′_{2} = p_{21}(t)x_{1} + p_{22}(t)x_{2} \end{matrix}, x_{2}(a_{1}) = 0, x_{2}(a_{2}) = 0

has a nontrivial solution.

Theorem 1.5. Let the inequalities (1.7), and (1.10)-(1.13) hold, where p_{11} : D_{2} → R, p_{22} : D_{1} → R, p_{12}, p_{21} : D → R are continuous bounded functions, q ∈ C(R, R_{+}), μ ∈ {−1, 1} and r_{0}, q_{0} are positive constants. Then the problem (1.1), (1.3) has at least one solution.

Theorem 1.6. Let the inequalities (1.7), (1.10), (1.14) and (1.15) hold, where p_{11} : D_{2} → R, p_{22} : D_{1} → R are continuous bounded functions, q ∈ C(R; R_{+}), μ ∈ {−1, 1} and r_{0}, q_{0} are positive constants. Then the problem (1.1), (1.3) has at least one solution.

Consider as an example the system

\begin{matrix}x′_{1} = p_{11}(t)x_{1} + p_{12}(t)x_{2} − σμ(|x_{1}x_{2}| − μx_{1}x_{2})^{n}x_{1}^{2k+1} + \\ +[\frac{|x_{1}x_{2}| − μx_{1}x_{2}}{|x_{1}x_{2}| + 1} ]^{m}q(t), \\ x′_{2} = p_{21}(t)x_{1} + p_{22}(t)x_{2} + f_{1}(x_{1}) · f_{2}(x_{2}), \end{matrix} (1.19)

where μ ∈ {−1, 1}, m, n, k ∈ {1, 2, 3, … }, p_{ij}, q ∈ C([a_{1}, a_{2}]; R) (i, j = 1, 2), f_{1} : R → R is a continuous function and f_{2} : R → R is a continuous bounded function. It follows from Theorem 1.6 that if for some σ ∈ {−1, 1} and r ∈ R_{+} the inequalities (1.18) and

σp_{12}(t) ≥ 0, σp_{21}(t) ≥ 0 for a_{1} ≤ t ≤ a_{2}

hold, then the problem (1.19),(1.17) is solvable.

2. Some Auxiliary Statements

In this section we shall give some lemmas on a priori estimates of the solutions of the system

x′_{i} = p_{i1}(t, x_{1}, x_{2})x_{1} + p_{i2}(t, x_{1}, x_{2})x_{2} + q_{i}(t, x_{1}, x_{2})
(i = 1, 2),  (2.1)

where q_{1} : D_{2} → R, q_{2} : D_{1} → R are continuous functions and p_{11} : D_{2} → R, p_{22} : D_{1} → R, p_{12}, p_{21} : D → R are continuous functions bounded by a positive number p_{0}.

Lemma 2.1. Suppose that

q_{1}(t, x, y) sgn y ≥ −q_{0}, q_{2}(t, x, y) sgn x ≥ −q_{0}
for (t, x, y) ∈ D,  (2.2)
q_{1}(t, 0, y) sgn y > −p_{12}(t, 0, y)|y|
for a_{1} ≤ t ≤ a_{2}, |x| ≥ r_{0},  (2.3)
q_{2}(t, x, 0) sgn x > −p_{21}(t, x, 0)|x|
for a_{1} ≤ t ≤ a_{2}, |y| ≥ r_{0},  (2.4)

where r_{0} and q_{0} are positive constants. Suppose, moreover, that an absolutely continuous vector-function (x_{1}, x_{2}) : [a_{1}, a_{2}] → R satisfies the system (2.1) almost everywhere and the conditions

either (−1)^{i}x_{1}(a_{i})x_{2}(a_{i}) ≤ 0
or |x_{1}(a_{i})| + |x_{2}(a_{i})| ≤ c (i = 1, 2).  (2.5)

Then the estimate

|x_{1}(t)| + |x_{2}(t)| ≤ (c + r_{0} + 2q_{0}(a_{2} − a_{1})) exp[4p_{0}(a_{2} − a_{1})]
for a_{1} ≤ t ≤ a_{2}  (2.6)

holds.

Proof. Let t_{0} ∈]a_{1}, a_{2}[. Suppose that (−1)^{k}x_{1}(t_{0})x_{2}(t_{0}) > 0, where k ∈ {1, 2}. Then either

(−1)^{k}x_{1}(t)x_{2}(t) > 0 for min{t_{0}, a_{k}} ≤ t ≤ max{t_{0}, a_{k}},
|x_{1}(a_{k})| + |x_{2}(a_{k})| ≤ c

or t_{1} ∈ [min{t_{0}, a_{k}}, max{t_{0}, a_{k}}] can be found such that

(−1)^{k}x_{1}(t)x_{2}(t) > 0 for min{t_{0}, t_{1}} < t < max{t_{0}, t_{1}},
x_{1}(t_{1})x_{2}(t_{1}) = 0.  (2.7)

In the case when (2.7) holds, the inequality

x′_{1}(t_{1})x_{2}(t_{1}) + x′_{2}(t_{1})x_{1}(t_{1}) ≤ 0

together with (2.3) and (2.4) implies

|x_{1}(t_{1})| + |x_{2}(t_{1})| ≤ r_{0}.

Therefore, if (−1)^{k}x_{1}(t_{0})x_{2}(t_{0}) > 0, then t_{1} ∈ [min{t_{0}, a_{k}}, max{t_{0}, a_{k}}] can be found such that

(−1)^{k}x_{1}(t)x_{2}(t) > 0 for min{t_{0}, t_{1}} < t < max{t_{0}, t_{1}},
|x_{1}(t_{1})| + |x_{2}(t_{1})| ≤ c + r_{0}.  (2.8)

Integrating the sum x′_{1}(t) + (−1)^{k}x′_{2}(t) from t_{1} to t and taking into consideration (2.1),(2.2) and (2.8) we easily see that

|x_{1}(t)|+|x_{2}(t)|≤c+r_{0}+2q_{0}(a_{2}−a_{1})−(−1)^{k}4p_{0}∫_{t_{1}}^{t}[|x_{1}(τ)|+|x_{2}(τ)|]dτ

for min{t_{0}, t_{1}} < t < max{t_{0}, t_{1}}.

Applying the Gronwall-Bellman lemma, we obtain that the estimate (2.6) holds for t = t_{0}.

Suppose now that x_{1}(t_{0})x_{2}(t_{0}) = 0. Then either a sequence (t_{n})_{n=1}^{+∞}, t_{n} ∈]a_{1}, a_{2}[ n ∈ {1, 2, 3, … } can be found such that

lim_{n→+∞} t_{n} = t_{0}, x_{1}(t_{n})x_{2}(t_{n}) ≠ 0 n ∈ {1, 2, … }  (2.9)

or for some ε ∈]0, min(b − t_{0}, t_{0} − a)[

x_{1}(t)x_{2}(t) = 0 for t_{0} − ε < t < t_{0} + ε.  (2.10)

If (2.9) is true, then as it already was shown above, the estimate (2.6) holds for t = t_{n} n ∈ {1, 2, … }. Hence (2.6) holds for t = t_{0} also. And if (2.10) is true, then according to (2.3) and (2.4) we obtain from the equality

x′_{1}(t_{0})x_{2}(t_{0}) + x′_{2}(t_{0})x_{1}(t_{0}) = 0

that |x_{1}(t_{0})| + |x_{2}(t_{0})| ≤ r_{0}.

Lemma 2.2. Suppose that

q_{1}(t, x, y) sgn y ≥ −p_{12}(t, x, y)|y|
for a_{1} < t < a_{2}, xy > 0,  (2.11)
q_{1}(t, x, y) sgn y ≥ −q_{0} for a_{1} < t < a_{2}, xy < 0,  (2.12)
q_{1}(a_{1}, 0, y) sgn y > p_{12}(a_{1}, 0, y)|y| for |y| ≥ r_{0},  (2.13)
q_{2}(t, x, y) sgn x ≥ −q(x) − p_{21}(t, x, y)|x|
for a_{1} < t < a_{2}, xy > 0,  (2.14)
q_{2}(t, x, y) sgn x ≥ −q_{0} for a_{1} < t < a_{2}, xy < 0  (2.15)

and (2.4) holds, where q ∈ C(R; R_{+}) and r_{0} and q_{0} are positive constants. Then for any absolutely continuous vector-function (x_{1}, x_{2}) : [a_{1}, a_{2}] → R satisfying the system (2.1) and the conditions (2.5), the estimate

|x_{1}(t)| + |x_{2}(t)| ≤ 2(c + r_{0} + 2q_{0}(a_{2} − a_{1}) + max{q(x) : |x| ≤
≤ (c + r_{0}) exp[p_{0}(a_{2} − a_{1})]}) exp[4p_{0}(a_{2} − a_{1})]
for a_{1} ≤ t ≤ a_{2}  (2.16)

holds.

Proof. Let t_{0} ∈]a_{1}, a_{2}[. Suppose first x_{1}(t_{0})x_{2}(t_{0}) > 0. Then either

x_{1}(t)x_{2}(t) > 0 for t_{0} < t ≤ a_{2} |x_{1}(a_{2})| + |x_{2}(a_{2})| ≤ c

or t_{1} ∈]t_{0}, a_{2}[ can be found such that

x_{1}(t)x_{2}(t) > 0 for t_{0} < t < t_{1}, x_{1}(t_{1})x_{2}(t_{1}) = 0.  (2.17)

If (2.17) is true, then according to (2.11) from (2.1) we have

|x_{1}(t)|′ ≥ −p_{0}|x_{1}(t)| for t_{0} < t < t_{1}.  (2.18)

Therefore if x_{1}(t_{1}) = 0, then x_{1}(t) ≡ 0 for t_{0} < t < t_{1} which contradicts (2.17). So x_{2}(t_{1}) = 0, and hence x′_{2}(t_{1}) sgn x_{1}(t_{1}) ≤ 0. From this according to (2.4) we see that |x_{1}(t_{1})| ≤ r_{0}.

Thus if x_{1}(t_{0})x_{2}(t_{0}) > 0, then t_{1} ∈]t_{0}, a_{2}] can be found such that

x_{1}(t)x_{2}(t) > 0 for t_{0} < t < t_{1}, |x_{1}(t_{1})| + |x_{2}(t_{1})| ≤ c + r_{0}.

By virtue of the above-said and from (2.18) we easily find that

|x_{1}(t)| ≤ (c + r_{0}) exp[p_{0}(a_{2} − a_{1})] for t_{0} ≤ t ≤ t_{1}.  (2.19)

According to (2.11),(2.14) and (2.19) the second of the equalities (2.1) implies

|x′_{2}(t)| ≥ −p_{0}|x_{2}(t)| − q(x_{1}(t)) for t_{0} < t < t_{1}

and

|x_{2}(t)| ≤ (c + r_{0} + max{q(x) : |x| ≤ (c + r_{0})} exp[p_{0}(a_{2} − a_{1})]) ×
× exp[p_{0}(a_{2} − a_{1})] for t_{0} ≤ t ≤ t_{1}.

Therefore the estimate (2.16) holds for t = t_{0}.

Suppose now that x_{1}(t_{0})x_{2}(t_{0}) < 0. Then either

x_{1}(t)x_{2}(t) ≤ 0, x_{2}(t) ≠ 0 for a_{1} < t < t_{0}, |x_{1}(a_{1})| + |x_{2}(a_{1})| ≤ c,

or

x_{1}(t)x_{2}(t) ≤ 0, x_{2}(t) ≠ 0 for a_{1} < t < t_{0}, x_{1}(a_{1}) = 0, (2.20)

or t_{1} ∈ [a_{1}, t_{0}[ can be found such that

x_{1}(t)x_{2}(t) ≤ 0, x_{2}(t) ≠ 0 for t_{1} < t < t_{0}, x_{2}(t_{1}) = 0. (2.21)

If (2.20) ((2.21)) is true, then according to (2.13) ((2.4)) we obtain from the inequality x′_{1}(a_{1}) sgn x_{2}(a_{1}) ≤ 0 (x′_{2}(t_{1}) sgn x_{1}(t_{1}) ≤ 0) that |x_{2}(a_{1})| ≤ r_{0} (|x_{1}(t_{1})| ≤ r_{0}).

Thus if x_{1}(t_{0})x_{2}(t_{0}) < 0, then t_{1} ∈ [a_{1}, t_{0}[ can be found such that

x_{1}(t)x_{2}(t) ≤ 0, x_{2}(t) ≠ 0 for t_{1} < t < t_{0},
|x_{1}(t_{1})| + |x_{2}(t_{1})| ≤ c + r_{0}.  (2.22)

Integrating the difference of the equalities (2.1) from t_{1} to t, taking into consideration (2.12),(2.15) and applying the Gronwall-Bellman lemma, we see that the estimate (2.16) holds for t = t_{0}.

Consider, at least, the case when x_{1}(t_{0})x_{2}(t_{0}) = 0. Then either a sequence (t_{n})_{n=1}^{+∞}, t_{n} ∈]a_{1}, a_{2}[, n ∈ {1, 2, 3, … } can be found such that (2.9) holds or for some ε ∈]0, min(b − t_{0}, t_{0} − a)[ (2.10) is valid. Suppose that (2.10) is true. Then either x_{1}(t_{0}) = x_{2}(t_{0}) = 0 or

x_{1}(t_{0}) ≠ 0, x_{2}(t_{0}) = 0  (2.23)

or

x_{2}(t_{0}) ≠ 0, x_{1}(t_{0}) = 0.  (2.24)

Let (2.23) be fulfilled. Then ε_{1} ∈]0, ε[ can be found such that

x_{1}(t) ≠ 0, x_{2}(t) = 0 for t_{0} − ε_{1} < t < t_{0}.

According to (2.4) from the equality x′_{2}(t_{0}) sgn x_{1}(t_{0}) = 0 we have that |x_{1}(t_{0})| ≤ r_{0}. Therefore, the estimate (2.16) is true for t = t_{0}.

Let (2.24) be fulfilled. Then ε_{1} ∈]0, ε] can be found such that

x_{2}(t) ≠ 0, x_{1}(t) = 0 for t_{0} − ε_{1} < t < t_{0}.

Put

α = inf{τ ∈]a_{1}, t_{0}[: x_{1}(t) ≡ 0, x_{2}(t) ≠ 0 for τ < t < t_{0}}.

If α = a_{1}, then according to (2.13) from the equality x′_{1}(a_{1}) sgn x_{2}(a_{1}) = 0 we find that |x_{2}(a_{1})| ≤ r_{0}. And if α > a_{1}, then either

x_{1}(α) = x_{2}(α) = 0

or ε_{0} ∈]0, α − a_{1}[ can be found such that

x_{1}(t)x_{2}(t) < 0 for α − ε_{0} ≤ t < α.

Since x_{1}(α−ε_{0})x_{2}(α−ε_{0}) < 0, as it was already shown above, t_{1} ∈ [a_{1}, α−ε_{0}[ can be found such that (2.22) holds.

Thus, if (2.24) is valid, then t_{1} ∈ [a_{1}, t_{0}[ can be found such that (2.22) is true.

Integrating the difference of the equalities (2.1) from t_{1} to t, taking into consideration (2.12),(2.15) and applying the Gronwall-Bellman lemma, we see that the estimate (2.16) is true for t = t_{0}.

The proof of the following lemma is quite analogous.

Lemma 2.3. Suppose that

q_{1}(t, x, y) sgn y ≥ −p_{12}(t, x, y)|y| − q_{0} for a_{1} < t < a_{2}, xy < 0, q_{2}(t, x, y) sgn x ≥ −q(x) − p_{21}(t, x, y)|x| for (t, x, y) ∈ D,

and the conditions (2.4), (2.11) and (2.13) hold, where q ∈ C(R; R_{+}) and r_{0}, q_{0} are positive constants. Then for any absolutely continuous vectorfunction (x_{1}, x_{2}) : [a_{1}, a_{2}] → R satisfying the system (2.1) and conditions (2.5), the estimate (2.16) holds.

3. Proof of the Main Results

We shall carry out the proof only in the case σ = μ = 1, since the general case by the change of variables

\overline{x}_{1}(t) = −σμx_{1}(σμt + \frac{1 − σμ}{2}),

\overline{x}_{2}(t) = −σx_{2}(σμt + \frac{1 − σμ}{2})

can be reduced to this one. Proof of Theorem 1.1. Assume first that instead of (1.6) and (1.7) the conditions

f_{1}(t, 0, y) sgn y > 0 for a_{1} ≤ t ≤ a_{2}, |y| ≥ r_{0},
f_{2}(t, x, 0) sgn x > 0 for a_{1} ≤ t ≤ a_{2}, |x| ≥ r_{0},

are fulfilled.

Put

\begin{matrix}η = ∑_{i,j=1}^{2} |λ_{ij}|^{−1}, p_{0} = sup\{|p_{ij}(t, x, y)| : i, j = 1, 2, (t, x, y) ∈ D\}, \\ r_{1} = 1 + (ηl + r_{0} + 2q_{0}(a_{2} − a_{1})) exp[4p_{0}(a_{2} − a_{1})], \end{matrix}(3.1)

χ(τ) = \begin{cases}1 & for 0 ≤ τ ≤ r_{1} \\ 2 − \frac{τ}{r_{1}} & for r_{1} < τ < 2r_{1} \\ 0 & for τ ≥ 2r_{1} \end{cases} ,  (3.2)

q_{i}(t, x, y) = χ(|x| + |y|)[f_{i}(t, x, y) − p_{i1}(t, x, y)x − p_{i2}(t, x, y)y]

(i = 1, 2) for (t, x, y) ∈ D,  (3.3)
q̃_{1}(t, x, y) = χ(|x| + |y|)[f_{1}(t, x, y) − y] for (t, x, y) ∈ D,
q̃_{2}(t, x, y) = χ(|x| + |y|)[f_{2}(t, x, y) − x] for (t, x, y) ∈ D,
g̃_{i}(x, y) = χ(∥x∥_{C} + ∥y∥_{C})g_{i}(x, y) (i = 1, 2)
for x, y ∈ C([a_{1}, a_{2}]; R),  (3.4)

where ∥p∥_{C} = max{|p(t)| : t ∈ [a_{1}, a_{2}]}, and consider the boundary value problem

\begin{matrix}x′_{1} = x_{2} + q̃_{1}(t, x_{1}, x_{2}), \\ x′_{2} = x_{1} + q̃_{2}(t, x_{1}, x_{2}), \end{matrix}  (3.5)
λ_{i1}x_{1}(a_{i}) + λ_{i2}x_{2}(a_{i}) + g̃_{i}(x_{1}, x_{2}) = 0 (i = 1, 2).  (3.6)

According to Theorem 2.1 from [2], the problem (3.5),(3.6) has at least one solution (x_{1}, x_{2}). It is easy to see that (x_{1}, x_{2}) is a solution of the system

x′_{i} = p̃_{i1}(t, x_{1}, x_{2})x_{1} + p̃_{i2}(t, x_{1}, x_{2})x_{2} + q_{i}(t, x_{1}, x_{2}) (i = 1, 2), (3.7)

where

p̃_{ij}(t, x, y) = 1 + (p_{ij}(t, x, y) − 1)χ(|x| + |y|) (i, j = 1, 2, i ≠ j),
p̃_{ii}(t, x, y) = p_{ii}(t, x, y)χ(|x| + |y|) (i = 1, 2).  (3.8)

In view of (3.2),(3.4) and (3.6) we have

∑_{i=1}^{2} |λ_{i1}x_{1}(a_{i})| + |λ_{i2}x_{2}(a_{i})| ≤ l.

from which we get that the solution (x_{1}, x_{2}) of the system (3.7) satisfies the conditions

either x_{1}(a_{1})x_{2}(a_{1}) > 0 or |x_{1}(a_{1})| + |x_{2}(a_{1})| ≤ ηl

and

either x_{1}(a_{2})x_{2}(a_{2}) < 0 or |x_{1}(a_{2})| + |x_{2}(a_{2})| ≤ ηl.

According to Lemma 2.1 and (3.1) we have

|x_{1}(t)| + |x_{2}(t)| ≤ r_{1} for a_{1} ≤ t ≤ a_{2}.  (3.9)

This estimate together with (3.2)-(3.4) and (3.6)-(3.8) implies that (x_{1}, x_{2}) is a solution of the problem (3.1),(3.2). Moreover, (3.9) holds.

Consider now the case when (1.6) and (1.7) are fulfilled. According to what has been proved above, for any natural n the system of the differential equations

x′_{1} = f_{1}(t, x_{1}, x_{2}) + \frac{x_{2}}{n(1 + |x_{2}|)}

x′_{2} = f_{2}(t, x_{1}, x_{2}) + \frac{x_{1}}{n(1 + |x_{1}|)}

has the solution (x_{1n}, x_{2n}) satisfying the boundary conditions (1.2) and the inequality

|x_{1n}(t)| + |x_{2n}(t)| ≤ r_{1} for a_{1} ≤ t ≤ a_{2}.

It is clear that the sequences of functions (x_{in})_{n=1}^{+∞} (i = 1, 2) are uniformly bounded and equicontinuous on [a_{1}, a_{2}]. Therefore, without loss of generality, we can assume that they are uniformly convergent. Putting

x_{i}(t) = lim_{n→+∞} x_{in}(t) for a_{1} ≤ t ≤ a_{2} (i = 1, 2)

it is easy to see that (x_{1}, x_{2}) is a solution of the problem (1.1),(1.2). ■

The proofs of the other theorems are quite analogous to the one of Theorem 1.1. The difference is that instead of Lemma 2.1 one has to apply Lemma 2.2 in proving Theorems 1.2, 1.5 and Lemma 2.3 in proving Theorems 1.3, 1.6.

Applying Theorems 1.1 and 1.4 in the case when

p_{11}(t, x, y) = \begin{cases}p_{11}(t) sgn(xy) & for x ≠ 0 \\ p_{11}(t) & for x = 0 \end{cases}

p_{22}(t, x, y) = \begin{cases}p_{22}(t) sgn(xy) & for y ≠ 0 \\ p_{22}(t) & for y = 0 \end{cases}

one can easily be convinced in the validity of Corollaries 1.1 and 1.2.

References

1. I.T. Kiguradze, On a boundary value problem for a system of two differential equations. (Russian) Trudy Tbiliss. Univ. Mat. Mekh. Astron. 1(137)A(1971), 77-87.

2. I.T. Kiguradze, The boundary value problems for systems of ordinary differential equations. (Russian) Current problems in mathematics. Newest results, v. 30 (Russian), 3-103, Itogi nauki i tekhniki, Akad. Nauk SSSR, Vsesoyuzn. Inst. Nauchn. i Tekhn. Inform., Moscow, 1987.

3. N.P. Lezhava, On the solvability of a nonlinear problem for a system of two differential equations. (Russian) Bull. Acad. Sci. Georgian SSR 68(1972), No. 3, 545-547.

4. G.N. Mil’shtein, On a boundary value problem for a system of two differential equations. (Russian) Differentsial’nie Uravneniya 1(1961), No. 12, 1628-1639.

5. A.I. Perov, On a boundary value problem for a system of two differential equations. (Russian) Dokl. Akad. Nauk SSSR 144(1962), No. 3, 493-496.

6. B.L. Shekhter, On one boundary value problem for two-dimensional discontinuous differential systems. (Russian) Some problems of ordinary differential equations theory (Russian), Proceedings of I.N. Vekua Institute of Applied Mathematics v. 8, 79-161. Tbilisi University Press, Tbilisi, 1980.

7. B.L. Shekhter, On singular boundary value problems for two-dimen-sional differential systems. Arch. Math. 19(1983), No. 1, 19-41.

8. N.I. Vasil’ev, Some boundary value problems for a system of two first order differential equations, I. (Russian) Latv. Mat. Ezhegodnik 5(1969), 11-24.

9. N.I. Vasil’ev, Some boundary value problems for a system of two first order differential equations, II. (Russian) Latv. Mat. Ezhegodnik 6(1969), 31-39.

10. P. Waltman, Existence and uniqueness of solutions of boundary value problems for two-dimensional systems of nonlinear differential equations. Trans. Amer. Math. Soc. 153(1971), No. 1, 223-234.

(Received 21.10.1992)

Author’s address:

N. Muskhelishvili Institute of

Computational Mathematics

Georgian Academy of Sciences

8, Akuri St., 380093 Tbilisi

Republic of Georgia