Mem. Differential Equations Math. Phys. 20(2000), 154–158

N. Partsvania

ON A BOUNDARY VALUE PROBLEM FOR THE TWO-DIMENSIONAL SYSTEM OF EVOLUTION FUNCTIONAL DIFFERENTIAL EQUATIONS

(Reported on December 20, 1999)

Suppose C([a, b]; ℝ^{2}) is the space of two-dimensional continuous vector functions (x_{1}, x_{2}) : [a, b] → ℝ^{2} with the norm

∥(x_{1}, x_{2})∥_{C} = max {|x_{1}(t)| + |x_{2}(t)| : a ≤ t ≤ b};

M([a, b]; ℝ_{+}^{2}) = {(x_{1}, x_{2}) ∈ C([a, b]; ℝ^{2}) : x_{1} and x_{2} are nonnegative nondecreasing

functions }; L([a, b]; ℝ) is the space of summable functions y : [a, b] → ℝ with the norm

∥y∥_{L} = ∫_{a}^{b} |y(t)| dt.

Consider the two-dimensional evolution differential system

\frac{du_{i}(t)}{dt} = f_{i}(u_{1}, u_{2})(t) (i = 1, 2)  (1)

with the boundary conditions

u_{1}(a) = φ_{1}(u_{2}(a)), φ_{2}(u_{1}(b), u_{2}(b)) = 0,  (2)

where f_{i} : M([a, b]; ℝ_{+}^{2}) → L([a, b]; ℝ) (i = 1, 2) are continuous operators, while φ_{1} : ℝ_{+} → ℝ and φ_{2} : ℝ_{+}^{2} → ℝ are continuous functions. We are interested in the case where

sup {|f_{i}(x_{1}, x_{2})(·)| : ∥(x_{1}, x_{2})∥_{C} ≤ ρ} ∈ L([a, b]; ℝ) for 0 < ρ < +∞,

and the function φ_{2} satisfies one of the following three conditions:

φ_{2}(0, 0) < 0, φ_{2}(x, y) > 0 for x ≥ 0, y ≥ 0, x + y > r;  (3)
φ_{2}(0, 0) < 0, φ_{2}(x, y) > 0 for x ≥ 0, y > r;  (4)
φ_{2}(0, 0) < 0, φ_{2}(x, y) > 0 for x > r, y ≥ 0,  (5)

where r is a positive number.

For the case f_{i}(u_{1}, u_{2})(t) ≡ f_{0i}(t, u_{1}(t), u_{2}(t)) (i = 1, 2), where f_{0i} : [a, b] × ℝ^{2} → ℝ (i = 1, 2) are functions satisfying the local Carathéodory conditions, boundary value problems of the type (1), (2) are investigated in full detail (see [1], [2], [4], [9]–[14], and the references therein). In the general case this problem have not been studied enough. The results below fill to some extent the existing gap.

2000 Mathematics Subject Classification. 34K10.

Key words and phrases. Boundary value problem, two-dimensional system of evolution functional differential equations, nonnegative nondecreasing solution, Volterra operator.

Let δ_{i} : [a, b] → ℝ (i = 1, 2) be continuous functions such that

0 ≤ δ_{i}(t) ≤ t − a for a ≤ t ≤ b (i = 1, 2).

f : M([a, b]; ℝ_{+}^{2}) → L([a, b]; ℝ) is called the (δ_{1}, δ_{2})-Volterra operator if for any t ∈]a, b] and for any vector functions (x_{1}, x_{2}) and (y_{1}, y_{2}) ∈ M([a, b]; ℝ_{+}^{2}) satisfying the equalities

x_{1}(s) = y_{1}(s) for 0 ≤ s ≤ t − δ_{1}(t), x_{2}(s) = y_{2}(s) for 0 ≤ s ≤ t − δ_{2}(t),

we have

f(x_{1}, x_{2})(s) = f(y_{1}, y_{2})(s) for almost all s ∈ [0, t].

f is called the Volterra operator if it is the (0, 0)-Volterra operator.

Unless the contrary is specified, throughout the paper we will assume that f_{1} and f_{2} are the Volterra operators.

Definition. A vector function (u_{1}, u_{2}) with the absolutely continuous components u_{i} : [a, b] → ℝ (i = 1, 2) is said to be a nonnegative nondecreasing solution of the problem (1), (2) if:

(i) (u_{1}, u_{2}) ∈ M([a, b]; ℝ_{+}^{2}) and almost everywhere on [a, b] the equalities (1) are fulfilled;

(ii) (u_{1}, u_{2}) satisfies the boundary conditions (2).

Theorem 1. Let

f_{i}(0, 0)(t) = 0, f_{i}(x_{1}, x_{2})(t) ≥ 0 (i = 1, 2)  (6)
for a ≤ t ≤ b, (x_{1}, x_{2}) ∈ M([a, b]; ℝ_{+}^{2}),
φ_{1}(0) = 0, φ_{1}(x) ≥ 0 for x ≥ 0,  (7)

and the condition (3) be fulfilled. Then the problem (1), (2) has at least one nonnegative nondecreasing solution.

Theorem 2. Let the conditions (4), (6), and (7) hold. Let, moreover, there exist a summable function h : [a, b] → ℝ_{+} and a positive constant ℓ such that

f_{1}(x_{1}, x_{2})(t) ≤ [h(t) + ℓf_{2}(x_{1}, x_{2})(t)](1 + x_{1}(t))  (8)
for a ≤ t ≤ b, (x_{1}, x_{2}) ∈ M([a, b]; ℝ_{+}^{2}), ∥x_{2}∥_{C} ≤ r.

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

Remark 1. The condition (8) in Theorem 2 cannot be replaced by the condition

f_{1}(x_{1}, x_{2})(t) ≤ [h(t) + ℓf_{2}(x_{1}, x_{2})(t)](1 + x_{1}(t))^{1+ε}
for a ≤ t ≤ b, (x_{1}, x_{2}) ∈ M([a, b]; ℝ_{+}^{2}), ∥x_{2}∥_{C} ≤ r

no matter how small ε > 0 would be. However, the condition (8) can be replaced by somewhat different type of condition. More precisely, the following theorem is valid.

Theorem 3. Let the conditions (4), (6), and (7) hold. Let, moreover, there exist a continuous function δ : [a, b] → ℝ such that

0 < δ(t) ≤ t − a for a < t ≤ b,  (9)

and f_{1} be the (δ, 0)-Volterra operator. Then the problem (1), (2) has at least one nonnegative nondecreasing solution.

Theorem 4. Let the conditions (5)–(7) be fulfilled. Let, moreover, there exist a summable in the first argument function g : [a, b] × ℝ_{+}^{2} → ℝ_{+}, a summable function h : [a, b] → ℝ_{+} and a positive constant ℓ such that

lim sup_{ρ→+∞} [φ_{1}(ρ) + ∫_{a}^{b} g(t, φ_{1}(ρ), ρ) dt] > r,  (10)

f_{1}(x_{1}, x_{2})(t) ≥ g(t, x_{1}(a), x_{2}(a)) for a ≤ t ≤ b,  (11)
(x_{1}, x_{2}) ∈ M([a, b]; ℝ_{+}^{2}), ∥x_{1}∥_{C} ≤ r,

and

f_{2}(x_{1}, x_{2})(t) ≤ [h(t) + ℓf_{2}(x_{1}, x_{2})(t)](1 + x_{2}(t)) for a ≤ t ≤ b,
(x_{1}, x_{2}) ∈ M([a, b]; ℝ_{+}^{2}), ∥x_{1}∥_{C} ≤ r.

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

Remark 2. The condition (10) in Theorem 4 cannot be replaced by the condition

lim sup_{ρ→+∞} [φ_{1}(ρ) + ∫_{a}^{b} g(t, φ_{1}(ρ), ρ) dt] ≥ r.  (10′)

Indeed, it is clear that the problem

u′_{1}(t) = 0, u′_{2}(t) = 0;

u_{1}(a) = r \frac{u_{2}(a)}{1 + u_{2}(a)} , u_{1}(b) = r

has no solution, although all the conditions of Theorem 4, except of (10), are fulfilled. Instead of (10) the condition (10′) holds.

Theorem 5. Let the conditions (5)–(7) be fulfilled. Let, moreover, there exist a summable in the first argument function g : [a, b] × ℝ_{+}^{2} → ℝ_{+} and a continuous function δ : [a, b] → ℝ_{+} such that the conditions (9)–(11) are fulfilled, and f_{2} be the (0, δ)-Volterra operator. Then the problem (1), (2) has at least one nonnegative nondecreasing solution.

As an example, consider the boundary value problem

\frac{du_{i}(t)}{dt} = f_{0i}(t, u_{1}(τ_{1}(t)), u_{2}(τ_{2}(t))) (i = 1, 2);  (12)
u_{1}(a) = αu_{2}(a), β_{1}u_{1}(b) + β_{2}u_{2}(b) = γ,  (13)

where f_{0i} : [a, b] × ℝ_{+}^{2} → ℝ (i = 1, 2) are functions satisfying the local Carathéodory conditions, τ_{i} : [a, b] → [a, b] (i = 1, 2) are measurable functions satisfying the inequalities

τ_{i}(t) ≤ t for a ≤ t ≤ b (i = 1, 2),

α ≥ 0, β_{1} ≥ 0, β_{2} ≥ 0, β_{1} + β_{2} > 0 and γ > 0.

From Theorem 1–5 we have

Corollary. Let

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

Then for the existence of at least one nonnegative nondecreasing solution of the problem (12), (13) it is sufficient one of the following five conditions to be fulfilled:

(i) β_{1} > 0, β_{2} > 0; (ii) β_{1} = 0, β_{2} = 1, and there exist a summable function h : [a, b] → ℝ_{+} and a

positive constant ℓ such that

f_{01}(t, x, y) ≤ [h(t) + ℓf_{02}(t, x, y)](1 + x) for a ≤ t ≤ b, x ≥ 0, 0 ≤ y ≤ γ;

(iii) β_{1} = 0, β_{2} = 1, and

ess inf {s − τ_{1}(s) : t ≤ s ≤ b} > 0 for a < t ≤ b;

(iv) α > 0, β_{1} = 1, β_{2} = 0, and there exist a summable function h : [a, b] → ℝ_{+} and a positive constant ℓ such that

f_{02}(t, x, y) ≤ [h(t) + ℓf_{01}(t, x, y)](1 + y) for a ≤ t ≤ b, 0 ≤ x ≤ γ, y ≥ 0;

(v) α > 0, β_{1} = 1, β_{2} = 0, and

ess inf {s − τ_{2}(s) : t ≤ s ≤ b} > 0 for a < t ≤ b.

The above-formulated theorems and their corollaries generalize some previous results from [3] and make the results from [5]–[8] more complete.

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Author’s address: A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia