Sharp Gronwall – Bellman type integral inequalities with delay

Various attempts have been made to give an upper bound for the solutions of the delayed version of the Gronwall–Bellman integral inequality, but the obtained estimations are not sharp. In this paper a new approach is presented to get sharp estimations for the nonnegative solutions of the considered delayed inequalities. The results are based on the idea of the generalized characteristic inequality. Our method gives sharp estimation, and therefore the results are more exact than the earlier ones.


Introduction
The Gronwall-Bellman integral inequality (see [5] and [8]) plays an important role in the qualitative theory of the solutions of differential and integral equations with and without delay.
It states that if a and x are nonnegative and continuous functions on the interval [t 0 , T[ (t 0 < T ≤ ∞) satisfying for some c ≥ 0, then This estimation is precise, since the function t → c exp t t 0 a(s)ds , t 0 ≤ t < T satisfies (1.1) with equality.
Corresponding author.Email: lhorvath@almos.uni-pannon.hu A number of generalizations of this inequality have been developed and studied, we refer to the classical books [3,7,16,17] and the literature cited these.
Various attempts have been made to give a sharp upper bound for the solutions of the following delayed version of (1.1) where a, x : [t 0 , T[ → R + are continuous, and α : [t 0 , T[ → [t 0 , T[ is a continuously differen- tiable and increasing function with α(t) ≤ t (t 0 ≤ t < T) (see [1,2,14,15,18,19]).The obtained estimation is which is not sharp in contrast with (1.2).
Essentially, there are two different methods to give upper bounds for the solutions of either (1.1) or (1.3): the first one is to obtain a differential inequality from the considered integral inequality (see [2,14,18,19]), while the second one is based on iterative techniques (see [4,11,12]).In case of applying iterative techniques, some standard integral inequalities are used, which can be found in a very general form in [13].
In this paper a new approach is presented to get sharp estimation for the nonnegative solutions of the delayed inequality where a : [t 0 , T[ → R + is locally integrable, τ : [t 0 , T[ → R + is a measurable function such that t 0 − r ≤ t − τ(t), t 0 ≤ t < T with some r ≥ 0, and x : [t 0 − r, T[ → R + is Borel measurable and locally bounded.By making a substitution, it can be shown that inequality (1.3) is a special case of (1.4).
Our treatment of the inequality (1.4) uses the following observation.Under suitable conditions, by introducing the function y : the integral inequality (1.4) can be transformed to the delayed differential inequality y (t) ≤ a(t)y(u − τ(u)), t 0 ≤ t < T. (1.5) Thus the nonnegative solutions of (1.4) can be estimated by the nonnegative solutions of the differential inequality (1.5) or the nonnegative solutions of the nonautonomous linear delay differential equation y (t) = a(t)y(u − τ(u)), t 0 ≤ t < T. (1.6) The results in this paper are based on the idea of the generalized characteristic inequality and the generalized characteristic equation which is obtained by looking for solutions of (1.6) in the form The generalized characteristic equation has been introduced for nonautonomous linear delay differential equations to obtain some powerful comparison results (see [10]).For a recent application we refer to [6].
We shall use the solutions γ : [t 0 − r, T[ → R + of (1.7) to estimate the solutions of (1.4).Our method gives sharp estimation for the solutions of (1.4), and therefore much better upper bounds can be obtained for the solutions of (1.3) than the earlier ones.

A sharp Gronwall-Belmann type estimation for delay dependent linear integral inequalities
The set of nonnegative numbers and the set of nonnegative integers will be denoted by R + and N respectively.Throughout this paper measurable means Lebesgue measurable, while Borel measurability is always indicated.
(a) We say that f is locally integrable if it is integrable over [p, t] for every t ∈ [p, T[.(b) We say that f is locally bounded if it is bounded on [p, t] for every t ∈ [p, T[.We come now to one of the principal results of this paper.Theorem 2.2.Let t 0 ∈ R, t 0 < T ≤ ∞, c ≥ 0, and a : [t 0 , T[ → R + be locally integrable.Assume r ≥ 0, and τ : where the function γ : [t 0 − r, T[ → R + is locally integrable, and satisfies the characteristic inequality and ≥ 0, and a : [t 0 , T[ → R + be measurable.Assume r ≥ 0, and τ : [t 0 , T[ → R + is a measurable function such that is locally integrable, and (2.1) holds, then x is locally bounded on [t 0 , T[, since the function defined by the right hand side of (2.1) is continuous.This shows that the assumption of local boundedness on x is natural.
When a and τ are constant functions, we get the following corollary.
where the nonnegative number γ satisfies the inequality a ≤ γe γτ , and K := max ce γτ , sup Since a is locally integrable in Theorem 2.2, it is clear that the function where the function γ : [t 0 − r, T[ → R + is locally integrable and satisfies the inequality and Considerations similar to those involved in Corollary 2.5 give: under the conditions of Theorem 2.9 the function is locally integrable and satisfies the inequality (2.11), and thus we get the following Gronwall-Bellman type estimation for x.
Corollary 2.10.Let t 0 ∈ R, t 0 < T ≤ ∞, and c ≥ 0. Let a, b : [t 0 , T[ → R + be locally integrable.Assume r ≥ 0, and τ Borel measurable and locally bounded such that (2.10) holds, then where The next result gives a γ which provides a sharp estimation with respect to the nonnegative solutions of the inequality (2.10) (we need to slightly re-formulate Definition 2.6).
(a) There exists a unique locally integrable function γ : (2.12) with the initial condition is the unique solution of the integral equation Proof.
(a) By Theorem 2.7 (a), the initial value problem (2.12) and (2.13) has a unique solution.
(b) It follows from (a) by using Theorem 2.7 (b).
(c) We can follow the proof of Theorem 2.7 (c).
The proof is complete.

Applicability of the main results
First, we compare Theorem 2.2 to a frequently used result from [18].Another remarkable result in [14] is its special case.We need some notations.Theorem A (see [18]).
As we shall see in Remark 3.3, the above result is a consequence of the next theorem which is a far-reaching generalization of it, and which comes from Theorem 2.2.
where the function γ : [t 0 , T[ → R + is locally integrable, and satisfies the inequality Proof.
(a) By using a substitution, we get (b) It is worth to note that the proof of Theorem 3.2 (a) shows that inequality (3.1) can be transformed to an equivalent inequality having the form (2.10), but the converse is not true in general.
Remark 3.4.The extension of (3.1) have been studied in [19] under the conditions of Theorem A, and where c ∈ C([t 0 , T[, R + ) is positive and increasing.Like Theorem 3.2, it can be obtained an essential generalization of the main result of [19] from Theorem 2.12.
We illustrate by two examples that (3.3) can give much better explicit upper bound for the solutions of (3.1) than (3.2).
Example 3.5.(a) Consider the inequality where c ≥ 0 and Then by Theorem A, (3.9) Theorem 3.2 (a) gives by choosing which is not exponential estimation in contrast with (3.9).It can be checked easily that if (3.10) holds, then the inequality (3.4) is satisfied with equality, and hence Theorem 2.7 (b) and (c) show that (3.11) is the best upper bound for the solutions of (3.8).
From Theorem A, we have that Some easy calculations give that the function satisfies the inequality (3.4) which now has the form Therefore Theorem 3.2 (a) implies which is much better than (3.13).
Remark 3.6.We mention that Gronwall-Bellman type integral inequalities have been extended and studied in measure spaces in [12].Estimation for the solutions of (3.12) can be obtained from the results of [12] too: In spite of the very general settings in [12], the upper bound (3.15) is also sharper than the upper bound (3.13) coming from Theorem A for every t ≥ 0.
Next, we demonstrate the scope of the different estimations by applying them to the delay differential equation with the parameter q ∈ ]0, 1].We note that (3.16) is not a delay equation if q > 1.
We say that y ∈ It is an easy task to calculate that the function for any q ∈ ]0, 1].Thus it follows from Theorem 2.2 that which implies the result.
The proof is complete.
The previous result can be proved only partially by using other estimates.
It follows from this that every solution of (3.16) tends to zero at infinity if 1 2 < q ≤ 1.At the same time estimation (3.20) is useless if 0 < q ≤ 1 2 .(b) Theorem A cannot be applied for (3.18) directly: the inequality (3.18) is equivalent to where the range of the function α (c) It is not hard to check that the function and therefore Theorem 2.2 implies which shows that every solution of (3.16) tends to zero at infinity only if t is a solution of (3.16), (3.19) is the best estimation in the sense that it gives the best convergence rate for the solutions.
Finally, we consider the integral inequality Suppose that the latest inequality is an equality, that is The previous initial value problem can be solved easily: and hence This verifies again that estimation (3.25) is sharp.

On sharpness of the classical estimation in the delayed case
Consider the inequality under the conditions in Theorem 2.2.It follows from Corollary 2.5 that the estimation holds for every nonnegative solution x of (4.The next proposition shows that the estimation (4.2) is sharp under some integral condition.Proposition 4.1.Let t 0 ∈ R, t 0 < T ≤ ∞, and a : [t 0 , T[ → R + be locally integrable.Assume r ≥ 0, and τ : [t 0 , T[ → R + is a measurable function such that The function a provides a sharp estimation with respect to the nonnegative solutions of the inequality (4.1) if and only if ) Proof.(a) Assume (4.3) holds.By Theorem 2.7 (c), is a solution of (2.1), and hence (2.5) is true with this solution.
Conversely, assume the existence of a solution x 0 : [t 0 − r, T[ → R + of (4.1) such that (2.5) holds.By Theorem 2.2, The proof is complete.

Monotone dependence of the estimation with respect to the delay
As an illustration, consider the delay differential equation where a > 0 is fixed and τ ≥ 0 is a parameter.The characteristic equation of (5.1) is The unique solution of (5.2) is denoted by γ(τ).It is easy to check that γ : R of the initial value problems These simple observations are generalized in this subsection. Let It is well known that the initial value problem is equivalent to the integral equation with the same initial condition.Consequently, we consider integral equations first.
is the unique solution (see Theorem 2.7 (c)) of the integral equation Proof.
(a) Since γ1 is nonnegative, (5.5) yields that (b) It is an immediate consequence of (a).
The proof is complete.
We arrive now at an application of the foregoing results to differential equations.
with the initial condition Since γ 1 satisfies the inequality (5.7), Theorem 2.7 (b) and (5.8) yield According to the equivalence of (5.
Consider the unstable type delay differential equation (5.10) The interesting meaning of the above theorem is that the positive solutions of (5.10) growth faster at infinity if τ is replaced by a smaller delay.which contradicts y(t 1 ) = L 1 .Hence Since L 1 > L is arbitrary, (6.2) shows that From what we have proved already follows that (ii) Now assume x is Borel measurable and locally bounded.
Introduce the function z Then (2.1) and the definition of z imply that x(t) ≤ z(t) (t 0 − r ≤ t < T), and therefore Since z is continuous on [t 0 , T[, it follows from the first part of the proof that where The proof is complete.
It can be proved by induction easily that 0 ≤ γ n (t) ≤ a(t) (t 0 − r ≤ t < T) and γ n is locally integrable for all n ∈ N.

We show by another induction argument on k that for every
By using (6.3) and (6.4), it is easy to check that Then and thus and therefore We have proved that (6.6) implies In the next step we show that the sequence (γ n ) ∞ n=0 is convergent and the limit is a solution of the integral equation (2.6) with the initial condition (2.7).
Due to (6.5), the sequences (γ 2k ) It can be proved by induction that 0 ≤ λ n (t) ≤ a(t) (t 0 − r ≤ t < T) for all n ≥ 1 and λ n is locally integrable for all n ∈ N.
As in (a), another induction argument on k gives that for every k ≥ 1 where (γ n ) is the sequence defined by (6.3) and (6.4).

Definition 3 . 1 .
Let p ∈ R and p < T ≤ ∞.(c) The set of all continuous and nonnegative functions on [p, T[ will be denoted by C([p, T[, R + ).
(d) The set of all continuously differentiable functions from [p, T[ into [p, T[ will be denoted by C 1 ([p, T[, [p, T[)
1) (x : [t 0 − r, T[ → R + is Borel measurable and locally bounded), but Theorem 2.7 ensures that (2.2) is more exact than (4.2).This means (see Remark 2.8) that there exist a function γ :[t 0 − r, T[ → R + satisfying (2.3) which provides a sharp estimation with respect to the nonnegative solutions of the inequality (4.1) in the sense of Definition 2.6, but a does not provide a sharp estimation with respect to the nonnegative solutions of the inequality (4.1) in general.Thus a natural question is: for which classes of inequalities (4.1) is the estimation (4.2) sharp?
is nonnegative, and the measurability of τ yields that it is also locally integrable.This and(2.11)showthatTheorem2.2 can be applied to (6.12), and thereforey(t) ≤ K exp Proof of Theorem 2.12.Extend the function c to [t 0 − r, T[ such that c(t) = c(0) if t 0 − r ≤ t < t 0 .Since c is positive and increasing, and the other functions are nonnegative, we have from (2.14) that n→∞ λ n ≤ γ. a(u)y(u − τ(u))du, t 0 ≤ t < T,and therefore Theorem 2.9 can be applied.