Integral Stability in Terms of Two Measures for Nonlinear Differential Systems with “Maxima”

Abstract

This paper investigates relatively integral stability in terms of two measures for two differential systems with “maxima” by employing Lyapunov functions, Razumikhin method, and comparison principle. An example is given to illustrate our result.

1. Introduction

Recently, the interest in differential equations with “maxima” has increased exponentially. Such equations adequately model real world problems whose present state depends significantly on its maximum value on a past time interval. For example, many problems in the control theory correspond to the maximal deviation of the regulated quantity. Some qualitative properties of the solutions of ordinary differential equations with “maxima” can be found in [1–4] and references therein.

Integral stability for ordinary differential equations was introduced by Vrkoc [5]. The concept of integral stability occurs in connection with the stability under persistent perturbations when the perturbations are small enough everywhere except on a small interval. Recent developments in this field have been focused on various types of differential equations. In [6, 7], the integral stability and integral -stability properties of ordinary differential equations were discussed, respectively. Later, Hristova [8] discussed the integral stability in terms of two measures for impulsive differential equations with “supremum.” Moreover the same author in [9] discussed the integral stability in terms of two measures for impulsive functional differential equations. However, the integral stability in terms of two measures for two differential systems has not been obtained until now.

In this paper, we discuss the relatively integral stability in terms of two measures for two differential systems with “maxima.” Using Lyapunov functions, Razumikhin method, and comparison principle, sufficient conditions for uniform-relatively integral stability in terms of two measures are obtained.

2. Preliminaries

Firstly, we give the following sets for convenience:

Let , , and be constants, . Define the following sets:

We consider the following two differential systems with “maxima”: and the perturbed systems where , , with , is a given fixed number, , and ; denote the -dimensional Euclidean space with any convenient norm .

We denote by , the solutions of systems (3) satisfying the initial conditions , . Assume that solutions , are defined on for any initial functions .

In further investigations, we need the following comparison scalar ordinary differential equations: and its perturbed scalar ordinary differential equation where , , , and .

The following definitions will be needed in the sequel.

Definition 1. Letting , then (i) is finer than if there exits a and a function such that (ii) is uniformly finer than if there exists a and a function such that

Definition 2. The function belongs to class , if , , and is Lipschitz with respect to and .
Letting , we define a derivative of the function along the trajectory of systems (3) as follows: and a derivative of the function along the trajectory of systems (4) as follows:

Definition 3. Letting and , then is said to be(i)relatively -positive definite if there exists a and a function such that (ii)relatively -decrescent if there exists a and a function such that (iii)weak-relatively -decrescent if there exists a and a function such that
One will introduce relatively integral stability in terms of two measures for differential systems (3).

Definition 4. Letting , differential systems (3) are said to be uniform-relatively integrally stable in terms of measures , if for and any , there exists such that, for any initial functions and any perturbations , the inequality holds, provided that where , are the solutions of the initial value problem for perturbed differential systems with “maxima” (4).

3. Main Results

In further investigations, we need the following comparison result.

Lemma 5. Let the following conditions hold: , where , ;, andfor any number and any function such that for the inequality  holds, where ;, are the solutions of the initial value problem for differential systems with “maxima” (3); is the maximal solution of (6) with initial condition , which is defined for .
Then the inequality implies the validity of the inequality , .

Proof. Let and be such that Let be the maximal solution of the initial value problem Let .
Because of the fact that , it is enough to prove that for any the inequality holds. Then the inequality holds.
Assume that inequality (21) is not true; then there exists a point . Let . According to the assumption , we have where is a small enough number. From inequality (22) it follows that From on , it follows that the function is nondecreasing on .
Therefore for .
According to condition (i) of Lemma 5 and definition of function , we get that contradicts (23).
Therefore inequality (21) holds and the conclusion of Lemma 5 follows.

In the following results, we will obtain sufficient conditions for uniform-relatively integral stability in terms of two measures.

Theorem 6. Let the following conditions hold: , is uniformly finer than ;there exists , it is relatively -decrescent and(i)for any number and functions such that for and , the inequality  holds, where , is a constant;for any number , there exists such that(ii), where and ;(iii)for any number and functions such that and  for , the inequality  holds, where , ;the zero solution of differential equation (5) is equistable;the zero solution of differential equation (6) is uniform-integrally stable.
Then differential systems with “maxima” (3) are uniform-relatively integrally stable in terms of measures .

Proof. Since is relatively -decrescent, there exist and such that , the inequality holds.
Since is uniformly finer than , there exist and such that implies
Let be a number such that . According to condition , there exist with Lipschitz constant . Let be Lipschitz constant of the function .
Denote . Without loss of generality, we assume .
From condition , it follows that there exists a such that the inequality implies that where is a solution of (5) with the initial condition .
Since , there exists a , such that, for , the inequality holds.
From condition , it follows that there exist and such that, for every solution of perturbed equation (7) with the initial condition , the inequality holds, provided that and for every .
Since , , and , we choose , , such that
Since and , we can find such that the inequalities hold.
Now let the initial function and perturbation , of the right-hand side of differential systems (4) be such that and for every
We will prove that
From (28) and the choice of , it follows that implies that ; that is,
Suppose inequality (37) is not true. Therefore, there exists a point such that
From inequality (38) and , it follows the validity of the inclusions where .
Assume that ; then from the choice of and inequality (28) it follows , which contradicts (38). Therefore and there exists a point such that and for . Since and , it follows that
Let be the maximal solution of differential equation (5) with the initial condition , where . From condition of Theorem 6 and according to Lemma 5, we obtain
From inequality (30), we obtain where .
From inequalities (29), (42), and (43), we have or
From inequality (33) and condition of Theorem 6, it follows that
Consider defined by
From inequalities (45) and (46) it follows that
Let and be such that and , .
Using Lipschitz conditions for , and condition of Theorem 6, we obtain
Consider differential equation (7) where the perturbation on the right-hand side is given by
Let be the maximal solution of (7) with the initial condition , where . According to Lemma 5, the inequality holds, where is the interval of existence of .
Choose a point such that
Now define the continuous function by
From the choice of the perturbation , it follows that for every the inequality holds.
Let be the maximal solution of (7) with the initial condition , where the perturbation of the right-hand side is defined above function . Note that , .
From inequality (48) it follows that and therefore inequality (31) holds; that is,
From inequalities (52) and (56), the choice of the point , and condition , we obtain
The obtained contradiction proves the validity of inequality (37) for .
Inequality (37) proves uniform-relatively integral stability in terms of measures of the considered differential systems with “maxima.”

The following example is an application of Theorem 6.

Example 1. Consider the two differential systems with “maxima” and the perturbed systems
Let , , and , . Using the inequality , it is easy to check the validity of the conditions and (ii) of Theorem 6 for , .
Letting be such that , , , , then and
Letting , then .
Considering the comparison scalar differential system the solution is , , and we can prove that the solution is equistable; that is, the conditions and of Theorem 6 hold.
For , , the inequality holds. From (62), the inequality holds, where and its perturbed differential equation . We can prove that the differential equation is uniform-integrally stable. So the conditions and of Theorem 6 hold.
According to Theorem 6, differential systems with “maxima” (58) are uniform-relatively integrally stable in terms of two measures .

Conflict of Interests

The authors declare that they have no conflict of interests.

Authors’ Contribution

All authors completed the paper together. All authors read and approved the final paper.

Acknowledgments

The authors would like to thank the reviewers for their valuable suggestions and comments. This paper is supported by the National Natural Science Foundation of China (11271106) and the Natural Science Foundation of Hebei Province of China (A2013201232).