STIELTJES DERIVATIVE METHOD FOR INTEGRAL INEQUALITIES WITH IMPULSES

Abstract

The purpose of this paper is to obtain some integral inequalities with impulses by using the method of Stieltjes derivatives, and we use our results in the study of Lyapunov stability of solutions of a certain nonlinear impulsive integro-differential equation.

1. INTRODUCTION

In this paper, we discuss various integral inequalities with impulses.

Differential equations with impulses arise in various real world phenomena in mathematical physics, mechanics, engineering, biology and so on. We refer to the monograph of Samoilenko and Perestyuk [6]. Also integral inequalities are very useful tools in global existence, uniqueness, stability and other properties of the solutions of various nonlinear di®erential equations, see,e.g., [5].

To obtain our results in the paper we need some preliminaries. Now we state them.

Assume that [a, b], [c, d]⊂ R are bounded intervals, where R is the set of all real numbers.

A function f : [a, b]→ R is called regulated on [a, b] if both

exist for every point s ∈ [a, b]. As a convention we define f(a-) = f(a) and f(b+) = f(b). Let G[a, b] be the set of all regulated functions on [a, b]. If we let for f ∈ G[a, b], ∥f∥ = sups∈[a, b] ｜f(s)｜, then (G[a, b], ∥ ᐧ ∥) becomes a Banach space. For regulated functions, see [1, 2].

For a closed interval I = [c, d], we define f(I) = f(d)- f(c). A function f : [a, b] → R is of bounded variation on [a, b] if

where the supremum is taken over all partitions

a = t0 < t1 < ᐧ ᐧ ᐧ < tn-1 < tn = b.

Let BV [a, b] be the set of all functions of bounded variation on [a, b]. We use the following notations for the convenience:

A tagged interval (τ, [c, d]) in [a, b] consists of an interval [c, d] ⊂ [a, b] and a point τ ∈ [c, d]. Let Ii = [ci, di] ⊂ [a, b]. A finite collection {(τi, [ci, di]) : i = 1,2, ..., m} of pairwise non-overlapping tagged intervals is called a tagged partition of [a, b] if =[a, b]. A positive function 𝛿 on [a, b] is called a gauge on [a, b].

Definition 1.1 ([4, 7]). Let 𝛿 be a gauge on [a, b]. A tagged partition P={(τi, [ti-1, ti]) : i = 1, 2,..., m} of [a, b] is said to be 𝛿-fine if for every i = 1, ...,m we have

τi∈ [ti-1, ti] ⊂ (τi -𝛿(τi), τi + 𝛿(τi)).

If moreover a 𝛿-fine partition P satisfies the implications

τi = ti-1⇒i = 1, τi = ti⇒i = m,

then it is called a 𝛿*-fine partition of [a, b].

The following lemma implies that for a gauge 𝛿 on [a, b] there exists a 𝛿*-fine partition of [a, b]. This also implies the existence of a 𝛿-fine partition of [a, b].

Lemma 1.2 ([4]). Let 𝛿 be a gauge on [a, b] and a dense subset ­ Ω ⊂(a, b) be given. Then there exists a 𝛿*-fine partition P = {(τi, [ti-1, ti]) : i = 1, 2,..., m} of [a, b] such that ti ∈Ω ­ for i = 1, ..., m-1.

We are now ready to give a formal definition of both types of the Kurzweil integral.

Definition 1.3 ([4, 7]). Assume that f, g : [a, b] → R are given. We say that fdg is Kurzweil integrable (or shortly, K-integrable) on [a, b] and v ∈ R is its integral if for every ε > 0 there exists a gauge 𝛿 on [a, b] such that for

we have

｜S(fdg, P) - v｜ ≤ ε,

provided P = {(τi, Ii):i=1,..., n} is a 𝛿-fine tagged partition of [a, b]. In this case we denote (or, shortly,)

If, in the above definition, 𝛿-fine is replaced by 𝛿*-fine, then we say that fdg is Kurzweil* integrable(or, shortly, K*-integrable) on [a, b] and we denote .

Remark 1.4. By the above definition it is obvious that K-integrability implies K*-integrability.

The integrals have the following properties. For the proofs, see, e.g., [7, 8].

Theorem 1.5. Assume that f, f1, f2, g : [a, b] → R and that f1dg and f2dg are integrable in the sense of Kurzweil or Kurzweil* on [a, b]. Let k1, k2 ∈ R. Then we have

If for c∈ [a, b], integrals, exist, then exists also and we have

For the integrability we have the following fundamental result.

Theorem 1.6. Assume that f ∈ G[a, b] and g ∈ BV [a, b]. Then fdg is K-integrable on [a, b].

Theorem 1.7. Assume that f, g : [a, b] → R and that fdg is K-integrable. If g is a regulated function on [a, b], then we have

 

2. THE STIELTJES DERIVATIVES

In this section we state the results in [3] that are essential to verify our main results.

Throughout this section, we assume that f ∈ G[a, b] and g is a nondecreasing function on [a, b].

A neighborhood of t ∈ [a, b] is an open interval containing t. We say that the function g is not locally constant at t ∈ (a, b) if there exists 𝜂 > 0 such that g is not constant on (t-ε, t+ε) for every ε < 𝜂. We also say that the function g is not locally constant at a and b, respectively if there exists 𝜂 > 0 such that g is not constant on [a, a +ε), (b- ε, b], respectively for every ε < 𝜂.

Definition 2.1. If g is not locally constant at t ∈ (a, b), we define

provided that the limit exists. If g is not locally constant at t = a and t = b respectively, we define

respectively. Sometimes we use instead of

If both f and g are constant on some neighborhood of t, we define = 0.

Remark 2.2. It is obvious that if g is not continuous at t then exists. Thus if does not exist then g is continuous at t. is called the Stieltjes derivative.

K*-integrals recover Stieltjes derivatives.

Theorem 2.3. Assume that if g is constant on some neighborhood of t then f is also constant there. Suppose that exists at every t∈ [a, b] -{c1, c2, ...}, where f is continuous at every t∈ {c1, c2, ...}. Then we have

 

3. MAIN RESULTS

In this section we will state and prove our results.

Let

0 < t1 < t2 < ᐧ ᐧ ᐧ

and let 0 < a < 1: Two sorts of Heaviside functions Ha, : [0, 1] → {0, 1}are defined respectively by

Using the Heaviside functions Ha, a we define functions 𝜙, 𝜓 : [0, 1] → [0,∞) by

respectively.

It is obvious that the functions 𝜙, 𝜓 are strictly increasing and of bounded variation on [0, 1].

From now on, we assume that c ≥ 0 and that all the functions u, f, g, gi, i = 1, . . . , n are nonnegative functions defined on [0, 1] that are regulated on [0, 1] and continuous at every t ≠ tk, k = where = 1, . . . , m.

Lemma 3.1. Assume that f ’(t) exists for t ≠= tk, k = . Then we have

(b) If a left-continuous function f is positive, nondecreasing, and differentiable at t ≠= tk, k = , then

(c)

Proof. (a) By definition, for tk < t < tk+1 and for sufficiently small 𝛿 and 𝜂 we have

so we have

And

This implies

Similarly we can verify

And

This completes the proof for (a).

(b) By (a) if t ≠= tk then it is obvious that

and by the Mean Value Theorem and since f is nondecreasing and left-continuous we have

(c) By Theorem 1.7, we have for tk < t

Through the same process, we can obtain that By the same method we can easily verify that

Since for tk> t, Htk (s) = 0 = (s) for every s ∈ [0, t] we have

Using the above results and the definition and properties of K-integral we get

Considering

we get

The proof is complete. □

Now we define functions A, Bi : [0, 1] → [0,∞) as follows:

The following theorem is a Gronwall-Bellman type integral inequality with impulses.

Theorem 3.2 ([6]). Let ak≥ 0, k = , If

then we have

Proof. Define a function z(t) by the right side of (3.1), then we observe that z(0) = c, u(t)≤z(t) and for t ≠ tk, k = , we have by Lemma 3.1

So, we have

By Lemma 3.1 this implies

By setting t = s in (3.2) and integrating it with respect to 𝜙 from 0 to t then by Theorem 2.3 and Lemma 3.1 we get

Since z(0) = c we get

This completes the proof. □

A generalization of Theorem 3.2 is the following result.

Theorem 3.3. Let 0 < m1

If

then

where

provided that M(t) + N(t) < 1.

Proof. Inequality (3.3) is written as:

By applying Theorem 3.2, we get

Then for every mj , j = , we have

Multiplying the last inequality by a negative term -mngj(t), we have

By summing the inequality for j = , we obtain

This implies that for t ≠= tk, k =

And by Lemma 3.1 we have

By the Mean Value Theorem we get for some

So we conclude that

By (3.6) and (3.7) we obtain

Integrating from 0 to t with respect to 𝜙 we get by Theorem 2.3

This implies that

So inequality (3.5) becomes

The proof is complete. □

From now on a function :[0, 1]→ [0,∞) is defined by

Theorem 3.4. Let 1 < p and let ak, bk ≥ 0, k =. If

and 1-M(t)- N(t) > 0, where

then we have

where

Proof. Define a function z(t) by the right side of (3.8), then we observe that z(0) = c, u(t) ≤ z(t) and for t ≠= tk, k = we have

and

This implies that

Define a function v(t) by

and then for t ≠ tk, k = , by Lemma 3.1 and (3.9)

and

and

Thus we have

This implies that

Since z(t) is left-continuous we have

So we have

Thus

By Theorem 3.3 we have

And

v(tk) ≤ akv(tk-) + bkvp(tk-)≤ c[akW(tk) + bkcp-1Wp(tk)].

Thus we get

So we have by Theorem 2.3,

This completes the proof. □

 

4. AN EXAMPLE

There are many applications of the inequalities obtained in Section 3. Here we shall give an example which is su±cient to show the usefulness of our results.

Consider the following impulsive integro-differential equation

where 0< t1 < ⋯ < tk < ⋯ < tm < 1, where a function F : [0, 1] × R2 → R is continuous on [0, 1] × R2 and satisfies

｜F(s, x, y)｜ ≤ f(s)(｜x｜ + ｜y｜)

for some continuous function f : [0, 1] → [0,∞), and a function G : [0, 1]×R → R is continuous on [0, 1] × R and satisfies

｜G(s, x)｜ ≤ g(s)｜x｜p

for some continuous function g : [0, 1] → [0,∞) and p > 1. Then we have

Assume that F(t, 0, 0) = 0 and that the function Ik : R→R is continuous and ｜Ik(x)｜ ≤ ak｜x｜ + bk｜x｜p, ak, bk ≥ 0, k = .

Let q = p- 1 and suppose that M(t) + N(t) < 1,where

Then we have

Applying Theorem 3.4 to the above inequality, we get

Since the function K(t) is bounded on [0, 1], the above inequality implies that the zero solution of equation (4.1) is Lyapunov stable. □