Abstract
This paper establishes a criterion on integral -stability in terms of two measures for impulsive differential equations with “supremum” by using the cone-valued piecewise continuous Lyapunov functions, Razumikhin method, and comparative method. Meantime, an example is given to illustrate our result.
1. Introduction
In this paper, we discuss the integral -stability in terms of two measures for impulsive differential equations with “supremum”: and its perturbed impulsive differential equations with “supremum” where , , , , , , , , and . Let be -dimensional Euclidean space with norm , , and a sequence of fixed points in such that and . We denote by the solution of (1). In our further investigation we will assume that solution is defined on for any initial function .
The research on impulsive differential equations with “supremum” problem, Bainov et al. [1] justified the partial averaging for impulsive differential equations, He et al. [2] discussed the periodic boundary value problem for first order impulsive differential equations, Agarwal and Hristova [3] studied the strict stability in terms of two measures for impulsive differential equations, Stamova and Stamov [4] investigated the global stability of models based on impulsive differential equations and variable impulsive perturbations, and Hristova [5, 6] obtained the -stability in terms of two measures for impulsive differential equations.
In recent years, the integral stability theory has been rapid development (see [7–12]). For example, Soliman and Abdalla [10] introduced integral -stability of perturbed system of ordinary differential equations. Hristova [12] studied the integral stability in terms of two measures for impulsive differential equations with “supremum.” However, the corresponding theory of impulsive differential equations with “supremum” is still at an initial stage of its development, especially for integral -stability in terms of two measures. Motivated by the idea of [5, 6, 10, 12], in this work, by employing the cone-valued piecewise continuous Lyapunov functions, Razumikhin method, and comparative method, we extend the notions of -stability in terms of two measures to integral -stability in terms of two measures for impulsive differential equations with “supremum.”
2. Preliminaries
Denote by the set of all functions which are piecewise continuous in with points of discontinuity of the first kind at the points and which are continuous from the left at the points .
We denote by the set of all function which are continuously differentiable for .
Let . Denote by the dot product of both vectors and .
Let be a cone, and is adjoint cone.
We give the following notations for convenience:
Let , and . Define
Let and be constants, , . Define sets:
In our further investigations we use the following comparison scalar impulsive ordinary differential equation: the scalar impulsive ordinary differential equation: and its perturbed scalar impulsive ordinary differential equation: where , .
Assume that solutions of the scalar impulsive equations (7), (8), and (9) exist on for any initial values. Meanwhile, we give some definitions and lemmas. The details can be found in [5].
Definition 1 (see [5]). We say that function , belongs to the class if;for each and there exist the finite limits there exist constants , such that for any .
Definition 2 (see [5]). Let , be given. The function is said to be -strongly -decrescent if there exist a constant and a function such that implies that .
Let , and . We define a derivative of the function along the trajectory of solution of (1) as follows:
Similarly we define a derivative of the function along the trajectory of solution of the perturbed system (2) for , and as follows:
Definition 3 (see [5]). Let , be given. The function is -uniformly finer than if there exist a constant and a function , such that for any point the inequality holds.
Lemma 4 (see [5]). Let , be given, and is -uniformly finer than with a constant and a function . Then for any and inequality implies , where functions and are defined by (4), (5).
In our further investigations we use the following comparison result.
Lemma 5 (see [5]). Let the following conditions be fulfilled.The vector and function are such thatfor any number and any function such that for the inequality
holds, where ., , and , where functions .Function is a solution of (1) that is defined for , where .Function is the maximal solution of (7) with initial condition that is defined for .
Then the inequality implies the validity of the inequality for .
Definition 6. Let . System of impulsive differential equations with “supremum” (1) is said to be-equi-integral -stable if for every and for any there exists a positive function which is continuous in for each and such that for maximal solution of the perturbed system of impulsive differential equations with “supremum” (2) the inequality holds, provided that and for every , where is defined by (4) and ;-uniform-integrally -stable if is satisfied, where is independent on .
Remark 7. We note that in the case when and the -equi-integral (uniform-integral-stability reduces to equi-integral (uniform-integral-stability.
3. Main Result
Theorem 8. Let the following conditions be fulfilled.Functions ; is -uniformly finer than .There exists a function that is -strongly -decrescent andfor any number , , and any function , such that for and the inequality
holds, where is a constant., for , .For any number there exists a function such that for , where and .For any number , and any function , such that and for the inequality
holds. for , .Zero solution of the scalar impulsive differential equation (7) is equi-stable.Zero solution of the scalar impulsive differential equation (8) is uniform-integrally stable.
Then system of impulsive differential equations with “supremum” (1) is -uniform-integrally -stable.
Proof. Since function is -strongly -decrescent, there exist a constant and a function such that implies that
Since is -uniformly finer than , there exist a constant and a function such that implies that
where .
According to Lemma 4, the inequality implies
Let be a fixed point. Choose a number such that .
According to condition of Theorem 8, there exists a function that is Lipshitz with a constant . Let be the Lipshitz constant of function .
Denote . Without loss of generality we assume .
Since the zero solution of the scalar impulsive differential equation (7) is equi-stable, there exists a function such that the inequality implies
where is a solution of (7).
Since the function there exists a , such that for the inequality
holds.
Since the zero solution of the scalar impulsive differential equation (8) is uniform-integrally stable, there exists a function , , such that for every solution of the perturbed impulsive equation (9) the inequality
holds, provided that
and for every ,
Since the function , and , we could choose a constant , such that
Since the function , and , we can find a , such that the inequalities
hold.
From (21) and (28) it follows that implies
that is, for .
Now let the initial functions be such that
and let the perturbed functions in impulsive equation with “supremum” (2) be such that
for every .
Let be a solution of (2), where the initial function and the perturbed functions satisfy (30) and (31); then
Suppose it is not true. There exists a point such that
Case 1. Let . Then from the continuity of the maximal solution at point follows that .
If we assume that then from the choice of and inequality (28) it follows that contradicts (33).
Therefore
Case 1.1. Let there exist a point , , such that and . Since and it follows that
Define a function for and let be the maximal solution of impulsive scalar differential equation (7) where . Let be the solution of the impulsive equations (1), . From conditions (i), (ii) of Theorem 8, according to Lemma 5, it follows that
From the choice of the point it follows that .
According to inequalities (19) and (23) we obtain
From inequalities (22) and (36) it follows that for , or
From inequality (28) and condition (iii) of Theorem 8, it follows that
Consider function that is defined in condition of Theorem 8, and define the function
the function satisfies the conditions of Lemma 5. Let point , and function be such that , , and for . Then using the Lipshitz conditions for functions and , and condition (iv) of Theorem 8, we obtain
where .
Let be such that . According to condition (v) of Theorem 8, we have
According to inequalities (41), (42) and Lemma 5, the inequality
holds.
Consider the scalar impulsive differential equation (9), where
According to above notations and inequality (31) for , we obtain
Let be the maximal solution of (9) through the point , where , and perturbations and are defined above and satisfy inequality (45).
Choose a point such that
Now define the continuous function :
and the sequence of numbers :
From (45), it follows that for every
Let be the maximal solution of the scalar impulsive differential equation (9) through the point , where perturbations of the right parts are defined above function and numbers . We note that
From inequalities (38) and (39), the definition of point , and inequality (49) follows the validity of (24) for the solution ; that is,
From inequalities (43) and (51), equality (50), the choice of point , and condition (iii) of Theorem 8, we obtain
The obtained contradiction proves the validity of the inequality (32) for .
Case 1.2. Let there exist a point such that , , and (35) is true.
We choose a number such that and . We repeat the proof of Case 1.1, where instead of we use and obtain a contradiction.
Case 2. Let there exist a natural number such that for and .
We repeat the proof of Case 1 as in this case we choose the constant , such that .
As in the proof of Case 1.1, we obtain the validity of inequalities (51) and (43). We apply conditions (iii) and (v) of Theorem 8 and obtain
and the obtained contradiction proves the validity of inequality (32) in this case. Inequality (32) proves -uniform-integral -stabilities of the considered system of the impulsive differential equations with “supremum.”
Next, we will provide an example which satisfies all the hypotheses of Theorem 8.
Example 9. Consider the system of impulsive differential equations with “supremum”
and its perturbed impulsive differential equations with “supremum”
where is enough small constant, . Without loss of generality we will assume further that .
Let .
Consider function , where is a cone.
Now, let us consider the vector . It is easy to check that the function is -strongly -decrescent with a function and the condition (iii) is satisfied for the function , where and .
Let be such that the inequality
or
then
Therefore if inequality (57) is satisfied then
or
Inequality (60) proves the validity of condition (i) of Theorem 8 for the function , where . Meanwhile, inequality (60) proves the validity of condition (iv) of Theorem 8 for the function , where .
From jump conditions (54) and the choice of vector and function we obtain the validity of conditions (ii) and (v) of Theorem 8 for the functions and , where and .
Consider following comparison scalar impulsive differential equation:
The solutions of the impulsive differential equation (61) and (62), correspondingly, are equi-stable and uniform-integrally stable. Thus, according to Theorem 8 the system of impulsive differential equations with “supremum” (54) is -uniform-integrally -stable.
4. Conclusion
This paper extends the notions of -stability in terms of two measures to integral -stability in terms of two measures for impulsive differential equations with “supremum” and establishes a criterion on integral -stability in terms of two measures for such system by using the cone-valued piecewise continuous Lyapunov functions, Razumikhin method and comparative method. Finally, an example is given to illustrate our result.
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).