ON NONNEGATIVE SOLUTIONS

OF NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEMS

FOR TWO-DIMENSIONAL DIFFERENTIAL SYSTEMS WITH ADVANCED ARGUMENTS

I. KIGURADZE^{∗} AND N. PARTSVANIA^{∗∗}

A. Razmadze Mathematical Institute of the Georgian Academy of Sciences Tbilisi, Georgia

1991 Mathematics Subject Classification. 34K10, 34K15

Key words and phrases. Nonlinear two-point boundary value problem, two-dimensional differential system with advanced arguments, nonnegative solution

§ 1. Statement of the Problem and Formulation of the Existence and Uniqueness Theorems

Consider the differential system

u′_{i}(t) = f_{i}(t, u_{1}(τ_{i1}(t)), u_{2}(τ_{i2}(t))) (i = 1, 2)  (1.1)

with the boundary conditions

φ(u_{1}(0), u_{2}(0)) = 0, u_{1}(t) = u_{1}(a), u_{2}(t) = 0 for t ≥ a,  (1.2)

where f_{i} : [0, a] × ℝ^{2} → ℝ (i = 1, 2) satisfy the local Carathéodory conditions, while φ : ℝ^{2} → ℝ and τ_{ik} : [0, a] → [0, +∞[ (i, k = 1, 2) are continuous functions. We are interested in the case, where

f_{i}(t, 0, 0) = 0, f_{i}(t, x, y) ≤ 0 for 0 ≤ t ≤ a, x ≥ 0, y ≥ 0 (i = 1, 2) (1.3)

and the function φ satisfies one of the following two conditions:

φ(0, 0) < 0, φ(x, y) > 0 for x > r, y ≥ 0  (1.4)

and

φ(0, 0) < 0, φ(x, y) > 0 for x ≥ 0, y ≥ 0, x + y > r,  (1.5)

∗ e-mail: kig@gmj.acnet.ge, http://www.rmi.acnet.ge/˜ kig

∗∗ e-mail: ninopa@rmi.acnet.ge, http://www.rmi.acnet.ge/˜ ninopa

where r is a positive constant.

Put

τ_{ik}^{0}(t) = \begin{cases}τ_{ik}(t) & for τ_{ik}(t) ≤ a \\ a & for τ_{ik}(t) > a \end{cases} (i, k = 1, 2).  (1.6)

Under a solution of the problem (1.1), (1.2) is understood an absolutely continuous vector function (u_{1}, u_{2}) : [0, a] → ℝ^{2} satisfying almost everywhere on [0, a] the differential system

u′_{i}(t) = f_{i}(t, u_{1}(τ_{i1}^{0}(t)), u_{2}(τ_{i2}^{0}(t))) (i = 1, 2)  (1.1′)

and also the boundary conditions

φ(u_{1}(0), u_{2}(0)) = 0, u_{2}(a) = 0.  (1.2′)

A solution (u_{1}, u_{2}) of the problem (1.1), (1.2) is called nonnegative if

u_{1}(t) ≥ 0, u_{2}(t) ≥ 0 for 0 ≤ t ≤ a.

If (1.3) holds, then it is obvious that each component of a nonnegative solution of the problem (1.1), (1.2) is a nonincreasing function.

For the case, where τ_{ik}(t) ≡ t (i, k = 1, 2), the boundary value problems of the type (1.1), (1.2) have been investigated by quite a number of authors (see, e.g., [4–8, 14–21] and the references therein). In this paper the case, where

τ_{ik}(t) ≥ t for 0 ≤ t ≤ a (i, k = 1, 2),  (1.7)

is considered and the optimal, in a certain sense, sufficient conditions are established for the existence and uniqueness of a solution of the problem (1.1), (1.2). Some of these results (see, e.g., Corollaries 1.1 and 1.4) are specific for advanced differential systems and have no analogues for the system

u′_{i}(t) = f_{i}(t, u_{1}(t), u_{2}(t)) (i = 1, 2).

Theorems 1.1–1.4 proven below and also their corollaries make the previous wellknown results [1–3, 9–12] on the solvability and unique solvability of the boundary value problems for the differential systems with deviated arguments more complete.

Along with (1.1) we will consider its perturbation

u′_{i}(t) = f_{i}(t, u_{1}(τ_{i1}(t) + ε), u_{2}(τ_{i2}(t) + ε)) (i = 1, 2),  (1.1_{ε})

where ε > 0. As it will be proved below^{1}, for every ε > 0 the problem (1.1_{ε}), (1.2) has at least one nonnegative solution provided the conditions (1.3), (1.4), and (1.7) are fulfilled.

1See Lemma 2.4.

Theorem 1.1 below reduces the question of the solvability of the problem (1.1), (1.2) to obtaining uniform a priori estimates of second components of solutions of the problem (1.1_{ε}), (1.2) with respect to the parameter ε. Such estimates can be derived in rather general situations and therefore from Theorem 1.1 the effective and optimal in a sense conditions are obtained for the solvability of the problem (1.1), (1.2) (see Corollaries 1.1–1.5 and Theorem 1.2).

Theorem 1.1. Let the conditions (1.3), (1.4), and (1.7) be fulfilled and let there exist positive numbers ε_{0} and ρ_{0} such that for any ε ∈]0, ε_{0}] the second component of an arbitrary nonnegative solution (u_{1}, u_{2}) of the problem (1.1_{ε}), (1.2) admits the estimate

u_{2}(0) ≤ ρ_{0}.  (1.8)

Then the problem (1.1), (1.2) has at least one nonnegative solution.

Corollary 1.1. Let the conditions (1.3), (1.4) be fulfilled and let

τ_{1i}(t) ≥ t (i = 1, 2), τ_{21}(t) ≥ t, τ_{22}(t) > t for 0 ≤ t ≤ a.  (1.9)

Then the problem (1.1), (1.2) has at least one nonnegative solution.

Remark 1.1. The condition (1.9) in Corollary 1.1 is essential and it cannot be replaced by the condition (1.7). To convince ourselves that this is so, consider the boundary value problem

u′_{1}(t) = 0, u′_{2}(t) = −(|u_{1}(t)| + |u_{2}(t)|)^{λ},  (1.10)
u_{1}(0) = 1, u_{2}(a) = 0,  (1.11)

where

λ ≥ \frac{1}{a} + 1.  (1.12)

It is seen that for that problem all the conditions of Corollary 1.1, except (1.9), are fulfilled. Instead of (1.9) there takes place the condition (1.7). Nevertheless, the problem (1.10), (1.11) has no solution. Indeed, should this problem have a solution (u_{1}, u_{2}), the function u_{2} would be positive on [0, a[ and

a = − ∫_{0}^{a} (1 + u_{2}(s))^{−λ} du_{2}(s) = \frac{1}{λ − 1} − \frac{1}{λ − 1} (1 + u_{2}(0))^{1−λ} < \frac{1}{λ − 1} .

But the latter inequality contradicts (1.12).

Corollary 1.2. Let the conditions (1.3), (1.4), and (1.7) be fulfilled and let there exist y_{0} > 0 such that

f_{2}(t, x, y) ≥ −h(t)ω(y) for 0 ≤ t ≤ a, 0 ≤ x ≤ r, y ≥ y_{0},  (1.13)

where h : [0, a] → [0, +∞[ is a summable function and ω : [y_{0}, +∞[ →]0, +∞[ is a nondecreasing continuous function satisfying the condition

∫_{y_{0}}^{+∞}\frac{dy}{ω(y)} = +∞.  (1.14)

Then the problem (1.1), (1.2) has at least one nonnegative solution.

Remark 1.2. As is shown above, if a = 1/ε and λ = 1+ε, then the problem (1.10), (1.11) has no solution. This fact shows that the condition (1.14) in Corollary 1.2 cannot be replaced by the condition

∫_{y_{0}}^{+∞}\frac{y^{ε} dy}{ω(y)} = +∞

no matter how small ε > 0 is.

Corollary 1.3. Let the conditions (1.3), (1.4), and (1.7) be fulfilled and let there exist numbers a_{i} ∈]0, a] (i = 1, 2) and y_{0} > 0 such that

τ_{12}(t) ≤ a_{2} for 0 ≤ t ≤ a_{1},  (1.15)
f_{1}(t, x, y) ≤ −δ(t, y) for 0 ≤ t ≤ a_{1}, 0 ≤ x ≤ r, y ≥ y_{0},  (1.16)

and

f_{2}(t, x, y) ≥ −[h(t) + |f_{1}(t, x, y)|]ω(y) for 0 ≤ t ≤ a_{2}, 0 ≤ x ≤ r, y ≥ y_{0}, (1.17)

where δ : [0, a_{1}]×[y_{0}, +∞[ → [0, +∞[ is a summable in the first and nondecreasing in the second argument function, while h : [0, a_{2}] → [0, +∞[ and ω : [y_{0}, +∞[ →]0, +∞[ are summable and nondecreasing continuous functions, respectively. Let, moreover, τ_{1i}(t) ≡ τ_{2i}(t) (i = 1, 2),

lim_{y→+_{∞}} ∫_{0}^{a_{1}}δ(t, y) dt > r,  (1.18)

and ω satisfy the condition (1.14). Then the problem (1.1), (1.2) has at least one nonnegative solution.

Remark 1.3. The condition (1.15) in Corollary 1.3 cannot be replaced by the condition

τ_{12}(t) ≤ a_{2} + ε for 0 ≤ t ≤ a_{1}  (1.19)

no matter how small ε > 0 is. As an example verifying this fact, consider the differential system

u′_{1}(t) = −u_{2}(a), u′_{2}(t) = −g(t)(|u_{1}(t)| + |u_{2}(t)|)^{1+ \frac{1}{ε}}  (1.20)

with the boundary conditions (1.11), where ε ∈]0, a[ and

g(t) = \begin{cases}0 & for 0 ≤ t ≤ a − ε \\ 1 & for a − ε < t ≤ a \end{cases} .

It is seen that for this problem the conditions (1.16)–(1.18) hold and instead of (1.15) the condition (1.19) is fulfilled, where τ_{12}(t) ≡ a, a_{1} = a, a_{2} = a − ε, δ(t, y) ≡ y, h(t) ≡ 0, ω(y) ≡ 1. But the problem (1.20), (1.11) has no solution. Indeed, if we assume that (1.20), (1.11) has a solution (u_{1}, u_{2}), then we will obtain the contradiction, i.e.,

ε = − ∫_{a−ε}^{a} (1 + u_{2}(s))^{− \frac{1}{ε} −1} du_{2}(s) = ε − ε(1 + u_{2}(a − ε))^{− \frac{1}{ε}} < ε.

In contrast to Corollary 1.3, Corollaries 1.4 and 1.5 below catch the effect of an advanced argument τ_{22}.

Corollary 1.4. Let the conditions (1.3), (1.4), and (1.7) be fulfilled and let for some a_{i} ∈]0, a] (i = 1, 2) and y_{0} > 0 the inequalities (1.15), (1.16), and

τ_{22}(t) ≥ a_{2} for 0 ≤ t ≤ a  (1.21)

hold, where δ : [0, a_{1}] × [y_{0}, +∞[ → [0, +∞[ is a summable in the first and nondecreasing in the second argument function satisfying the condition (1.18). Then the problem (1.1), (1.2) has at least one nonnegative solution.

Remark 1.4. It is obvious from the example (1.20), (1.11) that it is impossible in Corollary 1.4 to replace the condition (1.21) by the condition

τ_{22}(t) ≥ a_{2} − ε for 0 ≤ t ≤ a

no matter how small ε > 0 is.

Corollary 1.5. Let the conditions (1.3), (1.4), and (1.7) hold and let, moreover, for some a_{0} ∈]0, 1]∩]0, a^{1/α}] the inequalities

τ_{12}(t) ≤ t^{α} for 0 ≤ t ≤ a_{0},  (1.22)
f_{1}(t, x, y) ≤ −lt^{β}y^{λ_{1}} for 0 ≤ t ≤ a_{0}, 0 ≤ x ≤ r, y ≥ 0,  (1.23)

and

f_{2}(t, x, y) ≥ −h(t)(1 + y)^{1+λ_{2}} for 0 ≤ t ≤ a_{0}, 0 ≤ x ≤ r, y ≥ 0 (1.24)

be fulfilled, where 0 < α ≤ 1, β > −1, l > 0, λ_{1} > 0, λ_{2} ≥ 0 and h : [0, a_{0}] → [0, +∞[ is a measurable function satisfying the condition

∫_{0}^{a_{0}}[τ_{22}(t)]^{− \frac{(1+β)λ_{2}}{αλ_{1}}} h(t) dt < +∞.  (1.25)

Then the problem (1.1), (1.2) has at least one nonnegative solution.

Remark 1.5. The condition (1.25) in Corollary 1.5 cannot be replaced by the condition

∫_{0}^{a_{0}}[τ_{22}(t)]^{− \frac{(1+β)λ_{2}}{αλ_{1}} +ε}h(t) dt < +∞  (1.26)

no matter how small ε > 0 is. As an example, consider the differential system

u′_{1}(t) = −u_{2}(t), u′_{2}(t) = −γt^{λ−1−δ}(|u_{1}(t)| + |u_{2}(t)|)^{1+λ}  (1.27)

with the boundary conditions (1.11), where 0 < δ < ε < λ,

γ > \frac{λ − δ}{λ} (\frac{λ}{δ})^{λ}η^{δ−λ}  (1.28)

and η = min{a, 1}. For the system (1.27) the conditions (1.22)–(1.24) hold and instead of (1.25) there takes place the condition (1.26), where τ_{12}(t) ≡ τ_{22}(t) ≡ t, a_{0} = η, α = 1, β = 0, l = 1, λ_{1} = 1, λ_{2} = λ, r = 1, and h(t) = 2^{1+λ}γt^{λ−1−δ}. Show that the problem (1.27), (1.11) has no solution. Assume the contrary that this problem has a solution (u_{1}, u_{2}). Then u_{1}(t) > u_{1}(η) > 0, u_{2}(t) > 0 for 0 ≤ t < η,

∫_{0}^{η} u_{2}(t) dt = 1 − u_{1}(η),

and

−(u_{1}(η) + u_{2}(t))^{−1−λ}u′_{2}(t) > γt^{λ−1−δ} for 0 < t < η.

The integration of the latter inequality from 0 to t yields

(u_{1}(η) + u_{2}(t))^{−λ} − (u_{1}(η) + u_{2}(0))^{−λ} > \frac{λγ}{λ − δ} t^{λ−δ} for 0 < t < η

and hence

u_{1}(η) + u_{2}(t) < (\frac{λ − δ}{λγ} )^{\frac{1}{λ}} t^{−1+ \frac{δ}{λ}} for 0 < t < η.

Integrating this inequality from 0 to η, we obtain

ηu_{1}(η) + 1 − u_{1}(η) < \frac{λ}{δ} (\frac{λ − δ}{λγ} )^{\frac{1}{λ}} η^{\frac{δ}{λ}} .

However, since 0 < u_{1}(η) < 1, we have ηu_{1}(η) + 1 − u_{1}(η) > η. Thus

\frac{λ}{δ} (\frac{λ − δ}{λγ} )^{\frac{1}{λ}} η^{\frac{δ}{λ}} > η,

which contradicts (1.28).

Theorem 1.2. If the conditions (1.3), (1.5), and (1.7) are fulfilled, then the problem (1.1), (1.2) has at least one nonnegative solution.

The uniqueness of a solution of the problem (1.1), (1.2) is closely connected with the uniqueness of a solution of the system (1.1) with the Cauchy conditions

u_{1}(t) = c, u_{2}(t) = 0 for t ≥ a.  (1.29)

The following theorem is valid.

Theorem 1.3. Let the condition (1.7) be fulfilled and for any c ∈ ℝ the Cauchy problem (1.1), (1.29) have no more than one solution. Let, moreover, the functions f_{i} (i = 1, 2) not increase in the last two arguments, while the function φ increase in the first argument and not decrease in the second argument. Then the problem (1.1), (1.2) has no more than one solution.

Theorem 1.4. Let (1.7) be fulfilled and for any c ∈ ℝ the Cauchy problem (1.1), (1.29) have no more than one solution. Let, moreover, the function f_{1} not increase in the last two arguments, f_{2} decrease in the second argument and not increase in the third argument, while the function φ be such that

φ(x, y) < φ(\overline{x}, \overline{y}) for x < \overline{x}, y < \overline{y}.  (1.30)

Then the problem (1.1), (1.2) has no more than one solution.

Remark 1.6. For the uniqueness of a solution of the problem (1.1), (1.29) it is sufficient that either the functions f_{i} (i = 1, 2) satisfy in the last two arguments the local Lipschitz condition or the functions τ_{ik} (i, k = 1, 2) satisfy the inequalities

τ_{ik}(t) > t for 0 ≤ t ≤ a (i, k = 1, 2).

As an example, consider the boundary value problem

u′_{i}(t) = − ∑_{k=1}^{2} p_{ik}(t)|u_{k}(τ_{ik}(t))|^{λ_{ik}} sgn(u_{k}(τ_{ik}(t))) (i = 1, 2),

u_{1}(0) + αu_{2}(0) = β, u_{1}(t) = u_{1}(a), u_{2}(t) = 0 for t ≥ a, where λ_{ik} > 0 (i, k = 1, 2), α ≥ 0, β > 0, p_{ik} : [0, a] → [0, +∞[ (i, k = 1, 2) are summable functions and τ_{ik} : [0, a] → [0, +∞[ (i, k = 1, 2) are continuous functions satisfying the inequalities (1.7). By virtue of Corollaries 1.1, 1.3 and Theorems 1.2, 1.3 this problem has the unique nonnegative solution if, besides the above given, one of the following three conditions is fulfilled:

(i) τ_{ik}(t) > t for 0 ≤ t ≤ a (i, k = 1, 2);

(ii) α > 0, λ_{ik} ≥ 1 (i, k = 1, 2);

(iii) α = 0, λ_{ik} ≥ 1 (i, k = 1, 2), λ_{22} ≤ 1 + λ_{12} and there exist a_{i} ∈]0, a] (i = 1, 2) such that τ_{12}(t) ≤ a_{2} for 0 ≤ t ≤ a_{1}, p_{12}(t) > 0 for 0 < t < a_{2}, and

vrai max \{\frac{p_{22}(t)}{p_{12}(t)} : 0 < t < a_{2}\} < +∞.

§ 2. Some Auxiliary Statements

2.1. Lemmas on properties of solutions of an auxiliary Cauchy problem.

Lemma 2.1. If the conditions (1.3) and

τ_{ik}(t) > t for 0 ≤ t ≤ a (i, k = 1, 2)  (2.1)

are fulfilled, then for any nonnegative c the problem (1.1), (1.29) has the unique solution (u_{1}(·, c), u_{2}(·, c)). Moreover, the functions (t, c) → u_{i}(t, c) (i = 1, 2) are continuous on the set [0, a] × [0, +∞[ and satisfy on the same set the inequalities

u_{1}(t, c) ≥ c, u_{2}(t, c) ≥ 0.  (2.2)

Proof. In view of (2.1) there exists a natural number m such that

τ_{ik}(t) − t > \frac{a}{m} for 0 ≤ t ≤ a (i, k = 1, 2).

Put

t_{j} = \frac{ja}{m} (j = 0, …, m).

Then

τ_{ik}(t) > t_{j} for t_{j−1} ≤ t ≤ t_{j} (i, k = 1, 2; j = 1, …, m).  (2.3)

Suppose that for some c≥0 the problem (1.1), (1.29) has a solution (u_{1}(·,c), u_{2}(·,c)). Then on account of

u_{i}(t, c) = u_{ij}(t, c) for t_{j} ≤ t ≤ t_{j+1} (i = 1, 2; j = m − 1, …, 0),

where

u_{1m}(t, c) = c, u_{2m}(t, c) = 0 for t ≥ a,  (2.4)

\begin{matrix}u_{ij}(t, c) = u_{i j+1}(t_{j+1}, c) − \\ − ∫_{t}^{t_{j+1}}f_{i}(s, u_{1 j+1}(τ_{i1}(s), c), u_{2 j+1}(τ_{i2}(s), c)) ds for t_{j} ≤ t ≤ t_{j+1}, \\ u_{ij}(t, c) = u_{i j+1}(t, c) for t > t_{j+1} (i = 1, 2; j = m − 1, …, 0), \end{matrix}  (2.5)

we conclude that if the problem (1.1), (1.29) has a solution, then this solution is unique since the functions u_{ij} (i = 1, 2; j = m, …, 0) are defined uniquely.

Let us now suppose that u_{ij} : [t_{j}, +∞[ ×[0, +∞[ → ℝ (i = 1, 2; j = m − 1, …, 0) are the functions given by (2.4) and (2.5). Then proceeding from the conditions (1.3) and (2.3) we prove by the induction that these functions are continuous,

u_{1j}(t, c) ≥ c, u_{2j}(t, c) ≥ 0 for t ≥ t_{j}, c ≥ 0 (j = m − 1, …, 0),

and for any j ∈ {0, …, m − 1} and c ≥ 0 the restriction of the vector function (u_{1j}(·, c), u_{2j}(·, c)) on [t_{j}, a] is a solution of the problem (1.1), (1.29) on [t_{j}, a]. Consequently, the vector function (u_{1}(·, c), u_{2}(·, c)) with the components

u_{i}(t, c) = u_{i0}(t, c) for 0 ≤ t ≤ a (i = 1, 2)

is the unique solution of the problem (1.1), (1.29). Moreover, u_{i} : [0, a]×[0, +∞[ → ℝ (i = 1, 2) are continuous and satisfy (2.2).

Lemma 2.2. Let (1.7) be fulfilled and the functions f_{i} (i = 1, 2) be nonincreasing in the last two arguments. Let, moreover, there exist c_{1} ≥ 0 and c_{2} > c_{1} such that for c ∈ {c_{1}, c_{2}} the problem (1.1), (1.29) has the unique solution (u_{1}(·, c), u_{2}(·, c)). Then

u_{1}(t, c_{2}) ≥ c_{2} − c_{1} + u_{1}(t, c_{1}), u_{2}(t, c_{2}) ≥ u_{2}(t, c_{1}) for 0 ≤ t ≤ a. (2.6)

Proof. For each j ∈ {1, 2} the vector function (u_{1}(·, c_{j}), u_{2}(·, c_{j})) is a solution of the differential system (1.1′) under the conditions

u_{1}(a, c_{j}) = c_{j}, u_{2}(a, c_{j}) = 0.

Using the transformation

s = a − t, v_{i}(s) = u_{i}(t) (i = 1, 2)

we can rewrite the system (1.1′) in the form

v′_{i}(s) = f̃_{i}(s, v_{1}(ζ_{i1}(s)), v_{2}(ζ_{i2}(s))) (i = 1, 2),  (2.7)

where

f̃_{i}(s, x, y) = −f_{i}(a − s, x, y), ζ_{ik}(s) = a − τ_{ik}^{0}(a − s) (i, k = 1, 2).

On the basis of (1.6) and (1.7) we have

0 ≤ ζ_{ik}(s) ≤ s for 0 ≤ s ≤ a (i, k = 1, 2).  (2.8)

According to one of the conditions of the lemma, for each j ∈ {1, 2} the system (2.7) under the initial conditions

v_{1}(0) = c_{j}, v_{2}(0) = 0

has the unique solution (v_{1}(·, c_{j}), v_{2}(·, c_{j})), and

v_{i}(s, c_{j}) = u_{i}(a − s, c_{j}) (i = 1, 2).

On the other hand,

c_{1} < c_{2},

the functions f̃_{i} (i = 1, 2) do not decrease in the last two arguments, and the functions ζ_{ik} (i, k = 1, 2) satisfy (2.8). By virtue of Corollary 1.9 from [13] the above conditions guarantee the validity of the inequalities

v_{i}(s, c_{1}) ≤ v_{i}(s, c_{2}) for 0 ≤ s ≤ a (i = 1, 2).

It is obvious that

v′_{i}(s, c_{1}) ≤ v′_{i}(s, c_{2}) for 0 ≤ s ≤ a (i = 1, 2).

Consequently,

u_{i}(t, c_{1}) ≤ u_{i}(t, c_{2}), u′_{i}(t, c_{1}) ≥ u′_{i}(t, c_{2}) for 0 ≤ t ≤ a (i = 1, 2).

Hence the inequalities (2.6) follow immediately.

For every natural m consider the Cauchy problem

u′_{i}(t) = f_{i}(t, u_{1}(τ_{i1m}(t)), u_{2}(τ_{i2m}(t))) (i = 1, 2),  (2.9)
u_{1}(t) = c_{m}, u_{2}(t) = 0 for t ≥ a,  (2.10)

where τ_{ikm} : [0, a] → [0, +∞[ (i, k = 1, 2) are continuous functions.

The following lemma holds.

Lemma 2.3. Let

lim_{m→+∞} τ_{ikm}(t) = τ_{ik}(t) uniformly on [0, a] (i, k = 1, 2),  (2.11)
lim_{m→+∞} c_{m} = c  (2.12)

and let there exist positive number ρ such that for every natural m the problem (2.9), (2.10) has a solution (u_{1m}, u_{2m}) satisfying the inequality

|u_{1m}(t)| + |u_{2m}(t)| ≤ ρ for 0 ≤ t ≤ a.  (2.13)

Then from the sequences (u_{im})_{m=1}^{+∞} (i = 1, 2) we can choose uniformly converging subsequences (u_{im_{j}})_{j=1}^{+∞} (i = 1, 2) such that the vector function (u_{1}, u_{2}), where

u_{i}(t) = lim_{j→+∞} u_{im_{j}}(t) for 0 ≤ t ≤ a (i = 1, 2),  (2.14)

is a solution of the problem (1.1), (1.29).

Proof. According to (2.13), for every m we have

|u′_{1m}(t)| + |u′_{2m}(t)| ≤ f^{∗}(t) for 0 ≤ t ≤ a,

where

f^{∗}(t) = max { ∑_{i=1}^{2} |f_{i}(t, x, y)| : |x| + |y| ≤ ρ}

and, as is evident, f^{∗} is summable on [0, a]. Therefore the sequences (u_{im})_{m=1}^{+∞} (i = 1, 2) are uniformly bounded and equicontinuous on [0, a]. By virtue of the Arzela– Ascoli lemma, these sequences contain subsequences (u_{im_{j}})_{j=1}^{+∞} (i = 1, 2) converging uniformly on [0, a]. Let u_{1} and u_{2} be functions given by (2.14). If in the equalities

u_{1m_{j}}(t) = c_{m_{j}} − ∫_{t}^{a} f_{1}(s, u_{1m_{j}}(τ_{11m_{j}}(s)), u_{2m_{j}}(τ_{12m_{j}}(s))) ds,

u_{2m_{j}}(t) = − ∫_{t}^{a} f_{2}(s, u_{1m_{j}}(τ_{21m_{j}}(s)), u_{2m_{j}}(τ_{22m_{j}}(s))) ds for 0 ≤ t ≤ a

we pass to the limit as j → +∞, then by virtue of (2.10)–(2.12) and the Lebesgue theorem concerning the passage to the limit under the integral sign we find that

u_{1}(t) = c − ∫_{t}^{a} f_{1}(s, u_{1}(τ_{11}(s)), u_{2}(τ_{12}(s))) ds,

u_{2}(t) = − ∫_{t}^{a} f_{2}(s, u_{1}(τ_{21}(s)), u_{2}(τ_{22}(s))) ds for 0 ≤ t ≤ a

and the vector function (u_{1}, u_{2}) satisfies (1.29). Consequently, (u_{1}, u_{2}) is a solution of the problem (1.1), (1.29).

2.2. Lemma on the solvability of the problem (1.1), (1.2).

Lemma 2.4. If the conditions (1.3), (1.4), and (2.1) are fulfilled, then the problem (1.1), (1.2) has at least one nonnegative solution.

Proof. By Lemma 2.1, for every nonnegative c the problem (1.1), (1.29) has the unique solution (u_{1}(·, c), u_{2}(·, c)), while the functions (t, c) → u_{i}(t, c) (i = 1, 2) are continuous on the set [0, a] × [0, +∞[ and satisfy on the same set the inequalities (2.2). Hence, according to (1.3), it becomes evident that

u_{i}(t, 0) = 0 for 0 ≤ t ≤ a (i = 1, 2)  (2.15)

and

u_{1}(0, c) ≥ c, u_{2}(0, c) ≥ 0 for c > 0.  (2.16)

Put

φ_{0}(c) = φ(u_{1}(0, c), u_{2}(0, c)).

It is clear that φ_{0} : [0, +∞[ → ℝ is a continuous function. On the other hand, in view of (1.4), (2.15), and (2.16) we have

φ_{0}(0) < 0, φ_{0}(r) ≥ 0.

Thus there exists c_{0} ∈]0, r] such that

φ_{0}(c_{0}) = 0.

Consequently,

φ(u_{1}(0, c_{0}), u_{2}(0, c_{0})) = 0

and thus the vector function (u_{1}(·, c_{0}), u_{2}(·, c_{0})) is a nonnegative solution of the problem (1.1), (1.2).

2.3. Lemmas on a priori estimates. First of all consider the system of differential inequalities

\begin{aligned}u′_{1}(t) & ≤ −δ(t, u_{2}(a_{0})), \\ u′_{2}(t) & ≥ −[h(t) + |u′_{1}(t)|]ω(u_{2}(t)) \end{aligned}  (2.17)

with the initial condition

u_{1}(0) ≤ r,  (2.18)

where δ : [0, a_{0}] × [0, +∞[ → [0, +∞[ is a continuous in the first and nondecreasing in the second argument function, h : [0, a_{0}] → [0, +∞[ is a summable function and ω : [0, +∞[ →]0, +∞[ is a nondecreasing continuous function.

A vector function (u_{1}, u_{2}) with the nonnegative components u_{i} : [0, a_{0}] → [0, +∞[ (i = 1, 2) is said to be a nonnegative solution of the problem (2.17), (2.18) if the functions u_{1} and u_{2} are absolutely continuous, the function u_{1} satisfies the inequality (2.18), and the system of differential inequalities (2.17) holds almost everywhere on [0, a_{0}].

Lemma 2.5. Let

lim_{y→+∞} ∫_{0}^{a_{0}}δ(s, y) ds > r  (2.19)

and

∫_{0}^{+∞}\frac{dy}{ω(y)} = +∞.  (2.20)

Then there exists a positive number ρ_{0} such that the second component of an arbitrary nonnegative solution (u_{1}, u_{2}) of the problem (2.17), (2.18) admits the estimate

u_{2}(t) ≤ ρ_{0} for 0 ≤ t ≤ a_{0}.  (2.21)

Proof. By virtue of (2.19) and (2.20) there exist positive numbers ρ_{0} and ρ_{1} such that

∫_{0}^{a_{0}}δ(s, y) ds > r for y > ρ_{1}  (2.22)

and

∫_{ρ_{1}}^{ρ_{0}} \frac{dy}{ω(y)} = r + ∫_{0}^{a_{0}}h(s) ds.  (2.23)

Let (u_{1}, u_{2}) be an arbitrary nonnegative solution of the problem (2.17), (2.18). Then

∫_{0}^{a_{0}}|u′_{1}(s)| ds = − ∫_{0}^{a_{0}}u′_{1}(s) ds = u_{1}(0) − u_{1}(a_{0}) ≤ r,

which, owing to (2.17), results in

r ≥ − ∫_{0}^{a_{0}}u′_{1}(s) ds ≥ ∫_{0}^{a_{0}}δ(s, u_{2}(a_{0})) ds,  (2.24)

∫_{u_{2}(a_{0})}^{u_{2}(t)} \frac{dy}{ω(y)} = − ∫_{t}^{a_{0}} \frac{u′_{2}(s) ds}{ω(u_{2}(s))} ≤ ∫_{0}^{a_{0}}h(s) ds + ∫_{0}^{a_{0}}|u′_{1}(s)| ds ≤

≤ r + ∫_{0}^{a_{0}}h(s) ds for 0 ≤ t ≤ a_{0}.  (2.25)

Taking into account (2.22), from (2.24) we get

u_{2}(a_{0}) ≤ ρ_{1}.

On the basis of the last inequality and (2.23), from (2.25) we find the estimate (2.21).

Finally, on the segment [0, a_{0}] consider the system of differential inequalities

\begin{matrix}u′_{1}(t) ≤ −lt^{β}u_{2}^{λ_{1}}(t^{α}), \\ −h(t)(1 + u_{2}(τ(t)))^{1+λ_{2}} ≤ u′_{2}(t) ≤ 0, \end{matrix}  (2.26)

where

a_{0} ∈]0, 1],  (2.27)

0 < α ≤ 1, β > −1, l > 0, λ_{1} > 0, λ_{2} ≥ 0, while h : [0, a_{0}] → [0, +∞[ and τ : [0, a_{0}] → [0, a_{0}] are measurable functions.

Lemma 2.6. Let τ(t) ≥ t for 0 ≤ t ≤ a_{0} and

∫_{0}^{a_{0}}[τ(t)]^{− \frac{(1+β)λ_{2}}{αλ_{1}}} h(t) dt < +∞.  (2.28)

Then there exists a positive number ρ_{0} such that the second component of an arbitrary nonnegative solution (u_{1}, u_{2}) of the problem (2.26), (2.18) admits the estimate

u_{2}(0) < ρ_{0}.

Proof. Put

h_{0}(t) = (1 + [\frac{r(1 + β)}{l}]^{\frac{1}{λ_{1}}} [τ(t)]^{− \frac{1+β}{αλ_{1}}} )^{λ_{2}}h(t)

and

ρ_{0} = (1 + [\frac{r(1 + β)}{l}]^{\frac{1}{λ_{1}}} a_{0}^{− \frac{1+β}{αλ_{1}}} ) exp ( ∫_{0}^{a_{0}}h_{0}(s) ds).

Then by (2.28), ρ_{0} < +∞.

Let (u_{1}, u_{2}) be an arbitrary nonnegative solution of the problem (2.26), (2.18). Then

r ≥ u_{1}(t) − ∫_{0}^{t} u′_{1}(s) ds ≥ l ∫_{0}^{t} s^{β}u_{2}^{λ_{1}}(s^{α}) ds ≥

≥ u_{2}^{λ_{1}}(t^{α})l ∫_{0}^{t} s^{β} ds = \frac{l}{1 + β} t^{1+β}u_{2}^{λ_{1}}(t^{α}) for 0 < t ≤ a_{0}.

Thus

u_{2}(t^{α}) ≤ (\frac{r(1 + β)}{l})^{\frac{1}{λ_{1}}} t^{− \frac{1+β}{λ_{1}}} for 0 < t ≤ a_{0},

whence by virtue of (2.27) we get

u_{2}(t) ≤ (\frac{r(1 + β)}{l})^{\frac{1}{λ_{1}}} t^{− \frac{1+β}{αλ_{1}}} for 0 < t ≤ a_{0}.

According to the latter estimate and the inequalities (2.26), we have

u_{2}(a_{0}) ≤ (\frac{r(1 + β)}{l})^{\frac{1}{λ_{1}}} a_{0}^{− \frac{1+β}{αλ_{1}}} ,
u_{2}(τ(t)) ≤ u_{2}(t),

and

(1 + u_{2}(t))′ ≥ −h(t)[1 + u_{2}(τ(t))]^{λ_{2}}[1 + u_{2}(τ(t))] ≥
≥ −h_{0}(t)(1 + u_{2}(t)) for 0 < t ≤ a_{0}.

Thus

1 + u_{2}(t) ≤ (1 + u_{2}(a_{0})) exp ( ∫_{t}^{a_{0}}h_{0}(s) ds) ≤ ρ_{0} for 0 ≤ t ≤ a_{0}.

§ 3. Proofs of the Existence and Uniqueness Theorems

Proof of Theorem 1.1. Let

τ_{ikm}(t) = τ_{ik}(t) + \frac{ε_{0}}{m} (i, k = 1, 2; m = 1, 2, …).

Then by virtue of Lemma 2.4, for every natural m the system (2.9) has a nonnegative solution (u_{1m}, u_{2m}) satisfying the boundary conditions

φ(u_{1m}(0), u_{2m}(0)) = 0, u_{1m}(t) = u_{1m}(a), u_{2m}(t) = 0 for t ≥ a. (3.1)

By the condition of the theorem,

u_{2m}(0) ≤ ρ_{0} (m = 1, 2, …).  (3.2)

On the other hand, taking into account (1.4) and (3.1), we find

u_{1m}(0) ≤ r (m = 1, 2, …).  (3.3)

In view of (1.3) and the fact that u_{1m} and u_{2m} are nonnegative, we can conclude that these functions are nonincreasing. Thus it becomes clear from (3.2) and (3.3) that for every natural m the estimate (2.13), where ρ = r + ρ_{0}, is valid. It is also obvious that (u_{1m}, u_{2m}) is a solution of the problem (2.9), (2.10), where c_{m} = u_{1m}(a). Without loss of generality it can be assumed that the sequence (c_{m})_{m=1}^{+∞} is convergent. Denote by c the limit of that sequence.

According to Lemma 2.3, we can choose from (u_{im})_{m=1}^{+∞} (i = 1, 2) uniformly converging subsequences (u_{im_{j}})_{j=1}^{+∞} (i = 1, 2), and the vector function (u_{1}, u_{2}) whose components are given by (2.14) is a solution of the problem (1.1), (1.29). On the other hand, if in the equality

φ(u_{1m_{j}}(0), u_{2m_{j}}(0)) = 0

we pass to the limit as j → +∞, then, taking into account the fact that φ is continuous, we obtain

φ(u_{1}(0), u_{2}(0)) = 0.

Therefore (u_{1}, u_{2}) is a nonnegative solution of the problem (1.1), (1.2).

Proof of Corollary 1.1. Choose a natural number m so that

τ_{22}(t) > t + \frac{a}{m} for 0 ≤ t ≤ a.

Put

t_{j} = \frac{ja}{m} , τ(t) = t_{j+1} for t_{j} < t ≤ t_{j+1} (j = 0, …, m − 1).  (3.4)

Then

τ_{22}(t) > τ(t) for 0 ≤ t ≤ a.  (3.5)

Introduce the function

h(t, y) = max {|f_{2}(t, x, z)| : 0 ≤ x ≤ r, 0 ≤ z ≤ y}  (3.6)

and the numbers

ρ_{m} = 0, ρ_{j} = ρ_{j+1} + ∫_{t_{j}}^{t_{j+1}}h(s, ρ_{j+1}) ds (j = m − 1, …, 0).  (3.7)

Let ε ∈]0, 1] be an arbitrarily fixed number and (u_{1}, u_{2}) be a nonnegative solution of the problem (1.1_{ε}), (1.2). By Theorem 1.1, to prove Corollary 1.1 it is sufficient to show that u_{2} admits the estimate (1.8).

By virtue of (1.3), the functions u_{1} and u_{2} are nonincreasing. From this fact, on account of (1.4) and (3.5) it follows that

u_{1}(t) ≤ u_{1}(0) ≤ r for 0 ≤ t ≤ a  (3.8)

and

u_{2}(τ_{22}(t) + ε) ≤ u_{2}(τ(t)) for 0 ≤ t ≤ a.  (3.9)

In view of (3.6), (3.8), and (3.9), from (1.1_{ε}) we get

u′_{2}(t) ≥ −h(t, u_{2}(τ(t))) for 0 ≤ t ≤ a,

whence with regard for (3.4) and since u_{2}(t_{m}) = 0, we find that

u_{2}(t) ≤ ρ_{j} for t_{j} ≤ t ≤ a (j = m − 1, …, 0).

Therefore the estimate (1.8) is valid.

Proof of Corollary 1.2. Without loss of generality it can be assumed below that

h(t) > \frac{1}{ω(y_{0})} max \{|f_{2}(t, x, y)| : 0 ≤ x ≤ r, 0 ≤ y ≤ y_{0}\}.  (3.10)

If now we suppose

ω(y) = ω(y_{0}) for 0 ≤ y ≤ y_{0},

then owing to (1.13), we will have

f_{2}(t, x, y) ≥ −h(t)ω(y) for 0 ≤ t ≤ a, 0 ≤ x ≤ r, y ≥ 0.  (3.11)

On the other hand, by virtue of (1.14) there exists ρ_{0} > 0 such that

∫_{0}^{ρ_{0}} \frac{dy}{ω(y)} = ∫_{0}^{a} h(s) ds.  (3.12)

Let (u_{1}, u_{2}) be a nonnegative solution of the problem (1.1_{ε}), (1.2) for some ε ∈]0, 1]. By Theorem 1.1, to prove Corollary 1.2 it suffices to show that u_{2} admits the estimate (1.8).

By (1.3), the functions u_{1} and u_{2} are nonincreasing. Hence, taking into account (1.4) and (1.7), we get (3.8) and

u_{2}(τ_{22}(t) + ε) ≤ u_{2}(t) for 0 ≤ t ≤ a.  (3.13)

According to (3.8), (3.11), and (3.13), for almost all t ∈ [0, a] we have

−u′_{2}(t) ≤ h(t)ω(u_{2}(t))

as long as ω is a nondecreasing function. Moreover, u_{2}(a) = 0. Thus

∫_{0}^{u_{2}(0)}\frac{dy}{ω(y)} = − ∫_{0}^{a} \frac{u′_{2}(s)}{ω(u_{2}(s))} ds ≤ ∫_{0}^{a} h(s) ds.

Hence by virtue of (3.12) we obtain the estimate (1.8).

Proof of Corollary 1.3. Without loss of generality we assume that the function h satisfies the inequality (3.10) on [0, a_{2}]. If we now suppose a_{0} = a_{2},

ω(y) = ω(y_{0}) for 0 ≤ y ≤ y_{0}, δ(t, y) = 0 for 0 ≤ t ≤ a_{1}, 0 ≤ y ≤ y_{0}

and

δ(t, y) = 0 for a_{1} < t ≤ a_{0}, y ≥ 0,

then in view of (1.15)–(1.17) we get

τ_{12}(t) ≤ a_{0} for 0 ≤ t ≤ a_{1},
f_{1}(t, x, y) ≤ −δ(t, y) for 0 ≤ t ≤ a_{0}, 0 ≤ x ≤ r, y ≥ 0,

f_{2}(t, x, y) ≥ −[h(t) + |f_{1}(t, x, y)|]ω(y) for 0 ≤ t ≤ a_{0}, 0 ≤ x ≤ r, y ≥ 0.

(3.14)

On the other hand, by (1.18) and (1.14) the functions δ and ω satisfy the conditions (2.19) and (2.20), respectively.

Let ρ_{0} be the positive constant appearing in Lemma 2.5. Put

ψ(y) = \begin{cases}1 & for |y| ≤ ρ_{0} \\ 2 − \frac{|y|}{ρ_{0}} & for ρ_{0} < |y| < 2ρ_{0} \\ 0 & for |y| ≥ 2ρ_{0} \end{cases} ,  (3.15)

f̃_{2}(t, x, y) = ψ(y)f_{2}(t, x, y)  (3.16)

and consider the differential system

u′_{1}(t) = f_{1}(t, u_{1}(τ_{11}(t)), u_{2}(τ_{12}(t))), u′_{2}(t) = f̃_{2}(t, u_{1}(τ_{21}(t)), u_{2}(τ_{22}(t))). (3.17)

In view of (3.15) and (3.16)

f̃_{2}(t, x, y) ≥ −h^{∗}(t) for 0 ≤ t ≤ a, 0 ≤ x ≤ r, y ≥ 0,  (3.18)

where

h^{∗}(t) = max {|f_{2}(t, x, y)| : 0 ≤ x ≤ r, 0 ≤ y ≤ 2ρ_{0}}  (3.19)

and h^{∗} is summable on [0, a]. By virtue of Corollary 1.2 the condition (3.18) guarantees the existence of a nonnegative solution (u_{1}, u_{2}) of the problem (3.17), (1.2).

By the conditions (1.3) and (1.4), the functions u_{1} and u_{2} are nonincreasing and u_{1} satisfies the inequalities (3.8). If along with this fact we take into account the conditions (1.7), (3.14)–(3.16), then we will see that the restriction of (u_{1}, u_{2}) on [0, a_{0}] is a solution of the problem (2.17), (2.18). Hence if we take into consideration how ρ_{0} is, then we will get the estimate (2.21). Consequently,

u_{2}(τ_{22}(t)) ≤ ρ_{0} for 0 ≤ t ≤ a.  (3.20)

According to this estimate, from (3.15)–(3.17) follows that (u_{1}, u_{2}) is a solution of the system (1.1).

Proof of Corollary 1.4. By virtue of (1.18) we can find ρ_{1} ≥ y_{0} so that

∫_{0}^{a_{1}}δ(s, ρ_{1}) ds > r.  (3.21)

Put

h(t) = max {|f_{2}(t, x, y)| : 0 ≤ x ≤ r, 0 ≤ y ≤ ρ_{1}}  (3.22)

and

ρ_{0} = ρ_{1} + ∫_{0}^{a_{2}}h(s) ds.  (3.23)

Let ψ, f̃_{2} and h^{∗} be the functions defined by (3.15), (3.16) and (3.19), respectively. Then f̃_{2} satisfies the condition (3.18). By this condition and Corollary 1.2 the problem (3.17), (1.2) has a nonnegative solution (u_{1}, u_{2}). By virtue of (1.3) and (1.4) the functions u_{1} and u_{2} do not increase and u_{1} admits the estimate (3.8).

Let us now show that

u_{2}(a_{2}) < ρ_{1}.  (3.24)

Assume the contrary that u_{2}(a_{2}) ≥ ρ_{1}. Then in view of (1.15) and (1.16) we get

u_{2}(τ_{12}(t)) ≥ u_{2}(a_{2}) ≥ ρ_{1} for 0 ≤ t ≤ a_{1}

and

−u′_{1}(t) ≥ δ(t, ρ_{1}) for 0 ≤ t ≤ a_{1}.

Integrating the latter inequality from 0 to a_{1} and taking into account (3.8), we obtain

r ≥ ∫_{0}^{a_{1}}δ(s, ρ_{1}) ds.

But this contradicts (3.21). The contradiction obtained proves that the estimate (3.24) is valid. Hence by virtue of (1.21) we get

u_{2}(τ_{22}(t)) < ρ_{1} for 0 ≤ t ≤ a.

According to this estimate and the conditions (3.8), (3.22), we have

|u′_{2}(t)| ≤ h(t) for 0 ≤ t ≤ a.

If along with the above inequality we take into account (3.23) and (3.24), then we will get

u_{2}(0) ≤ u_{2}(a_{2}) + ∫_{0}^{a_{2}}h(s) ds < ρ_{0}.

Consequently, the estimate (3.20) is valid. In view of this estimate, from (3.15)–(3.17) follows that (u_{1}, u_{2}) is a solution of the system (1.1).

Proof of Corollary 1.5. Introduce the function

τ(t) = min {a_{0}, τ_{22}(t)}.

Then by virtue of (1.25), the condition (2.28) is fulfilled.

Let ρ_{0} be the positive constant appearing in Lemma 2.6 and let ψ and f̃_{2} be the functions defined by (3.15) and (3.16). By the conditions (1.3), (1.4) and Corollary 1.2, the problem (3.17), (1.2) has a nonnegative solution (u_{1}, u_{2}), the functions u_{1} and u_{2} do not increase and u_{1} admits the estimate (3.8). If along with this fact we take into account the conditions (1.7), (1.22)–(1.24), then it will become evident that the restriction of (u_{1}, u_{2}) on [0, a_{0}] is a solution of the problem (2.26), (2.18). Hence if we take into consideration how ρ_{0} is, then we obtain the estimate (3.20). According to this estimate, from (3.15)–(3.17) follows that (u_{1}, u_{2}) is a solution of the system (1.1).

Proof of Theorem 1.2. First of all note that (1.4) follows from (1.5).

Let us now suppose that (u_{1}, u_{2}) is an arbitrary nonnegative solution of the problem (1.1_{ε}), (1.2) for some ε ∈]0, 1]. Then according to (1.5) we have

u_{1}(0) + u_{2}(0) ≤ r.

Therefore the estimate (1.8), where ρ_{0} = r, is valid.

From the above reasoning it is clear that all the conditions of Theorem 1.1 are fulfilled, which guarantees the existence of at least one nonnegative solution of the problem (1.1), (1.2).

Proof of Theorem 1.3. Assume the contrary that the problem (1.1), (1.2) has two different solutions (u_{1}, u_{2}) and (\overline{u}_{1}, \overline{u}_{2}). Suppose

u_{1}(a) = c_{0}, \overline{u}_{1}(a) = \overline{c}_{0}.  (3.25)

Then (u_{1}, u_{2}) is a solution of the problem (1.1), (1.29) with c = c_{0}, while (\overline{u}_{1}, \overline{u}_{2}) is a solution of the same problem with c = \overline{c}_{0}. Thus \overline{c}_{0} ≠ c_{0}, since according to one of the conditions of the theorem the problem (1.1), (1.29) has no more than one solution for an arbitrarily fixed c.

Without loss of generality it can be assumed that

\overline{c}_{0} > c_{0}.

Then by virtue of Lemma 2.2 we have

\overline{u}_{1}(t) ≥ \overline{c}_{0} − c_{0} + u_{1}(t) > u_{1}(t), \overline{u}_{2}(t) ≥ u_{2}(t) for 0 ≤ t ≤ a.  (3.26)

Consequently,

\overline{u}_{1}(0) > u_{1}(0), \overline{u}_{2}(0) ≥ u_{2}(0)

and

φ(\overline{u}_{1}(0), \overline{u}_{2}(0)) > φ(u_{1}(0), u_{2}(0)),  (3.27)

since φ is an increasing in the first argument and nondecreasing in the second argument function. But the latter inequality contradicts the equalities

φ(\overline{u}_{1}(0), \overline{u}_{2}(0)) = 0, φ(u_{1}(0), u_{2}(0)) = 0.  (3.28)

The contradiction obtained proves the validity of the theorem.

Proof of Theorem 1.4. Assume the contrary that the problem (1.1), (1.2) has two different solutions (u_{1}, u_{2}) and (\overline{u}_{1}, \overline{u}_{2}). Then, as is shown when proving Theorem 1.3, \overline{c}_{0} ≠ c_{0}, where c_{0} and \overline{c}_{0} are the numbers defined by (3.25). For the sake of definiteness we assume that \overline{c}_{0} > c_{0}. Then by virtue of Lemma 2.2 the inequalities (3.26) are fulfilled. Thus

\overline{u}_{2}′(t) = f_{2}(t, \overline{u}_{1}(τ_{21}(t)), \overline{u}_{2}(τ_{22}(t))) <
< f_{2}(t, u_{1}(τ_{21}(t)), u_{2}(τ_{22}(t))) = u′_{2}(t) for 0 ≤ t ≤ a,  (3.29)

since f_{2} is a function decreasing in the second and nonincreasing in the third argument.

From (3.26) and (3.29) we have

\overline{u}_{1}(0) > u_{1}(0), \overline{u}_{2}(0) > u_{2}(0).

Hence by virtue of (1.30) we obtain the inequality (3.27), which contradicts (3.28). The contradiction obtained proves the validity of the theorem.

Acknowledgement

This work was supported by INTAS Grant No. 96–1060.

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