Electronic Journal of Qualitative Theory of Differential Equations 2005, No. 7, 1-8; http://www.math.u-szeged.hu/ejqtde/

A NOTE ON THE THEOREM ON DIFFERENTIAL INEQUALITIES

H. ŠTĚPÁNKOVÁ

Abstract. It is proved that if a linear operator ℓ : C([a, b]; ℝ) → L([a, b]; ℝ) is nonpositive and for the initial value problem

u′′(t) = ℓ(u)(t) + q(t), u(a) = c_{1}, u′(a) = c_{2}

the theorem on differential inequalities is valid, then ℓ is an a−Volterra operator.

2000 Mathematics Subject Classification. 34K06, 34K12.

Key words and phrases: Second order linear functional differential equation, initial value problem, Volterra operator.

The following notation is used throughout the paper. ℕ is the set of natural numbers. ℝ is the set of real numbers, ℝ_{+} = [0, +∞[. C([a, b]; ℝ) is the Banach space of continuous functions u : [a, b] → ℝ

with the norm ∥u∥_{C} = max{|u(t)| : t ∈ [a, b]}.

C([a, b]; ℝ_{+}) = {u ∈ C([a, b]; ℝ) : u(t) ≥ 0 for t ∈ [a, b]}. C̃([a, b]; ℝ) is the set of absolutely continuous functions u : [a, b] → ℝ. C̃′([a, b]; ℝ) is the set of functions u ∈ C̃([a, b]; ℝ) such that u′ ∈ C̃([a, b]; ℝ). C̃′_{loc}([a, b[ ; ℝ) is the set of functions u ∈ C̃([a, b]; ℝ) such that u′ ∈

C̃([a, β]; ℝ) for every β ∈]a, b[ .

C̃′_{loc}(]a, b[ ; ℝ) is the set of functions u ∈ C̃([a, b]; ℝ) such that u′ ∈ C̃([α, β]; ℝ) for every [α, β] ⊂]a, b[ .

L([a, b]; ℝ) is the Banach space of Lebesgue integrable functions p : [a, b] →

ℝ with the norm ∥p∥_{L} = ∫_{a}^{b} |p(s)|ds.

L([a, b]; ℝ_{+}) = {p ∈ L([a, b]; ℝ) : p(t) ≥ 0 for almost all t ∈ [a, b]}. L_{ab} is the set of linear bounded operators ℓ : C([a, b]; ℝ) → L([a, b]; ℝ).

P_{ab} is the set of operators ℓ ∈ L_{ab} transforming the set C([a, b]; ℝ_{+}) into the set L([a, b]; ℝ_{+}).

We will say that ℓ ∈ L_{ab} is an a−Volterra operator if for arbitrary b_{0} ∈ ]a, b] and v ∈ C([a, b]; ℝ) satisfying the condition

v(t) = 0 for t ∈ [a, b_{0}]

we have

ℓ(v)(t) = 0 for almost all t ∈ [a, b_{0}].

We will say that an operator Ω : L([a, b]; ℝ) → C([a, b]; ℝ) is an a−Volterra operator, if for arbitrary b_{0} ∈]a, b] and q ∈ L([a, b]; ℝ) satisfying the condition

q(t) = 0 for almost all t ∈ [a, b_{0}]

we have

Ω(q)(t) = 0 for t ∈ [a, b_{0}].

In what follows, the equalities and inequalities with integrable functions are understood to hold almost everywhere.

Consider the problem on the existence and uniqueness of a solution of the equation

u′′(t) = ℓ(u)(t) + q(t)  (1)

satisfying the initial conditions

u(a) = c_{0}, u′(a) = c_{1},  (2)

where ℓ ∈ L_{ab}, q ∈ L([a, b]; ℝ) and c_{0}, c_{1} ∈ ℝ. By a solution of the equation (1) we understand a function u ∈ C̃′([a, b]; ℝ) satisfying this equation (almost everywhere) in [a, b].

Along with the problem (1), (2) consider the corresponding homogeneous problem

u′′(t) = ℓ(u)(t),  (1_{0})
u(a) = 0, u′(a) = 0.  (2_{0})

The following result is well-known from the general theory of boundary value problems for functional differential equations (see, e.g., [1,2,4,5,8]).

Theorem 1. The problem (1), (2) is uniquely solvable iff the corresponding homogeneous problem (1_{0}), (2_{0}) has only the trivial solution.

Definition 1. We will say that an operator ℓ ∈ L_{ab} belongs to the set H̃_{ab}(a) if for every function u ∈ C̃′([a, b]; ℝ) satisfying

u′′(t) ≥ ℓ(u)(t) for t ∈ [a, b],  (3)

and (2_{0}), the inequality

u(t) ≥ 0 for t ∈ [a, b]  (4)

holds.

Remark 1. It follows from Definition 1 that if ℓ ∈ H̃_{ab}(a), then the homogeneous problem (1_{0}), (2_{0}) has only the trivial solution. Therefore, according to Theorem 1 the problem (1), (2) is uniquely solvable. Moreover, the inclusion ℓ ∈ H̃_{ab}(a) guarantees that if q ∈ L([a, b]; ℝ_{+}), then the unique solution of the problem (1), (2_{0}) is nonnegative.

Note also that ℓ ∈ H̃_{ab}(a) iff a certain theorem on differential inequalities hold. More precisely, whenever u, v ∈ C̃′([a, b]; ℝ) satisfy the inequalities

u′′(t) ≤ ℓ(u)(t) + q(t), v′′(t) ≥ ℓ(v)(t) + q(t) for t ∈ [a, b],
u(a) = v(a), u′(a) = v′(a),

then

u(t) ≤ v(t) for t ∈ [a, b].

In the paper [7], sufficient conditions are established guaranteeing the inclusion ℓ ∈ H̃_{ab}(a). In particular, in [7, Theorem 1.3] the following proposition is proved.

Proposition 1. Let −ℓ ∈ P_{ab} be an a−Volterra operator and let there exist a function γ ∈ C̃′_{loc}(]a, b[ ; ℝ) satisfying

γ′′(t) ≤ ℓ(γ)(t) for t ∈ [a, b],  (5)
γ(t) > 0 for t ∈]a, b[ ,  (6)
γ(a) + lim_{t→a+} γ′(t) ≠ 0.  (7)

Then ℓ ∈ H̃_{ab}(a).

Below we will prove (see Theorem 3) that in Proposition 1 the condition on ℓ to be a−Volterra operator is necessary. Analogous result for first order functional differential equations is proved in [3].

Before we formulate the main results, let us introduce the following definition.

Definition 2. Let the problem (1_{0}), (2_{0}) have only the trivial solution. Denote by Ω the operator, which assigns to every function q ∈ L([a, b]; ℝ) the solution of the problem (1), (2_{0}).

Remark 2. From Theorem 1 it follows that the operator Ω is well defined. It is also clear that Ω is a linear operator which maps the set L([a, b]; ℝ) into the set C([a, b]; ℝ).

Remark 3. It follows from [5, Theorem 1.4.1] that the operator Ω is continuous (bounded) (see also [1,4,6]).

Remark 4. It immediately follows from Definitions 1 and 2 that if ℓ ∈ H̃_{ab}(a), then the operator Ω is nonnegative, i.e., it transforms the set L([a, b]; ℝ_{+}) into the set C([a, b]; ℝ_{+}).

Theorem 2. Let −ℓ ∈ P_{ab} and ℓ ∈ H̃_{ab}(a). Then Ω is an a−Volterra operator.

Proof. Let t_{0} ∈]a, b[ and let the function q ∈ L([a, b]; ℝ) be such that

q(t) = 0 for t ∈ [a, t_{0}].  (8)

We will show that

Ω(q)(t) = 0 for t ∈ [a, t_{0}].  (9)

Denote by u the solution of the problem (1), (2_{0}) and by v the solution of the problem

v′′(t) = ℓ(v)(t) + |q(t)|,  (10)
v(a) = 0, v′(a) = 0.  (11)

According to Remark 1 (see also Remark 4) and the assumption −ℓ ∈ H̃_{ab}(a), we have

v(t) ≥ 0 for t ∈ [a, b],  (12)
u(t) ≤ v(t) for t ∈ [a, b].  (13)

Since −ℓ ∈ P_{ab}, it follows from (10) and (12) that

v′′(t) ≤ |q(t)| for t ∈ [a, t_{0}]

Hence, on account of (8), (11) and (12), we obtain

v(t) = 0 for t ∈ [a, t_{0}].  (14)

On the other hand, by virtue of (1), (10), (13), and the assumption −ℓ ∈ P_{ab}, we get

(u(t) − v(t))′′ = ℓ(u − v)(t) + q(t) − |q(t)| ≥ q(t) − |q(t)| for t ∈ [a, b].

Hence in view of (8) and (14) we get

u′′(t) ≥ v′′(t) = 0 for t ∈ [a, t_{0}].

The latter inequality, together with (13), (14) and (2_{0}), implies

u(t) = 0 for t ∈ [a, t_{0}].

Consequently (since u(t) = Ω(q)(t) for t ∈ [a, b]), the equality (9) is fulfilled. □

Theorem 3. Let −ℓ ∈ P_{ab} and ℓ ∈ H̃_{ab}(a). Then ℓ is an a−Volterra operator.

Proof. Assume the contrary, let ℓ be not an a−Volterra operator. Then there exist v_{0} ∈ C([a, b]; ℝ) and b_{0} ∈]a, b[ such that

v_{0}(t) = 0 for t ∈ [a, b_{0}]

and

mes{t ∈ [a, b_{0}] : ℓ(v_{0})(t) ≠ 0} > 0.

Without loss of generality we can assume that

mes{t ∈ [a, b_{0}] : ℓ(v_{0})(t) < 0} > 0.  (15)

First we will show that

Ω(ℓ(|v_{0}|))(t) = 0 for t ∈ [a, b_{0}],  (16)

where Ω is the operator introduced in Definition 2.

Choose a sequence of functions v_{k} ∈ C̃′([a, b]; ℝ), k ∈ ℕ, such that

lim_{k→+∞} ∥v_{k} − |v_{0}|∥_{C} = 0  (17)

and

v_{k}(t) = 0 for t ∈ [a, b_{0}], k ∈ ℕ.  (18)

According to Remark 3 and (17), we get

lim_{k→+∞} ∥Ω(ℓ(v_{k})) − Ω(ℓ(|v_{0}|))∥_{C} = 0.  (19)

It is clear that

v′′_{k}(t) = ℓ(v_{k})(t) + q_{k}(t) for t ∈ [a, b], k ∈ ℕ,  (20)

where

q_{k}(t) =^{def} v′′_{k}(t) − ℓ(v_{k})(t) for t ∈ [a, b], k ∈ ℕ.  (21)

Consequently,

v_{k}(t) = Ω(q_{k})(t) for t ∈ [a, b], k ∈ ℕ.  (22)

It follows from (20)–(22) that

v_{k}(t) = Ω(v′′_{k})(t) − Ω(ℓ(v_{k}))(t) for t ∈ [a, b], k ∈ ℕ.  (23)

Hence, taking into account the fact that Ω is an a−Volterra operator (see Theorem 2) and the condition (18), we obtain

Ω(ℓ(v_{k}))(t) = −v_{k}(t) = 0 for t ∈ [a, b_{0}], k ∈ ℕ.

Thus, in view of (19), we get the equality (16).

Let u be a solution of the problem (1), (2_{0}), where

q(t) = \begin{cases}−ℓ(|v_{0}|)(t) & for t ∈ [a, b_{0}[ \\ 0 & for t ∈ [b_{0}, b] \end{cases} .  (24)

It is evident that

u(t) = Ω(q)(t) for t ∈ [a, b]  (25)

and

q(t) ≥ 0 for t ∈ [a, b].  (26)

Moreover, on account of the assumption −ℓ ∈ P_{ab}, the inequality

ℓ(|v_{0}|)(t) ≤ ℓ(v_{0})(t) for t ∈ [a, b]

holds. Consequently, due to (15) and (24)

mes{t ∈ [a, b_{0}] : q(t) > 0} > 0.  (27)

According to Theorem 2, Ω is an a−Volterra operator. Hence by virtue of (16) and (24) we get from (25) that

u(t) = 0 for t ∈ [a, b_{0}].  (28)

On the other hand, the inequality (26) and the assumption ℓ ∈ H̃_{ab}(a) imply

u(t) ≥ 0 for t ∈ [a, b].  (29)

In view of (29) and the assumption −ℓ ∈ P_{ab}, it follows from (1) that

u′′(t) ≤ q(t) for t ∈ [a, b].

Hence, on account of (24), we obtain

u′′(t) ≤ 0 for t ∈ [b_{0}, b].

The latter inequality, together with (28) and (29), yields

u(t) = 0 for t ∈ [a, b].

Therefore, it follows from (1) that q ≡ 0, which contradicts (27).

□

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(Received February 7, 2005)

Author’s address:

Hana Štěpánková Faculty of Education, University of South Bohemia, Jeronýmova 10, 371 15 České Budějovice, Czech Republic E-mail: stepanh@pf.jcu.cz