β – Hyers – Ulam – Rassias Stability of Semilinear Nonautonomous Impulsive System

Xiaoming Wang 1,†, Muhammad Arif 2,† and Akbar Zada 2,* 1 School of Mathematics & Computer Science, Shangrao Normal University, Shangrao 334001, China; wxmsuda03@163.com 2 Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan; arifjanmath@gmail.com * Correspondence: zadababo@yahoo.com or akbarzada@uop.edu.pk; Tel.: +92-345-9515060 † These authors contributed equally to this work.


Introduction
Differential equations are the key tools for modeling the physical problems in nature.To understand the sudden changes in physical problems, differential equations are the best option for use.Examples of these sudden changes are Plague deforestation, volcano eruption and rivers overflow [1].Physical problems which have rapid changes are blood flows, biological systems such as heart beats, theoretical physics, engineering, control theory, population dynamics, mechanical systems with impact, pharmacokinetics, biotechnology processes, mathematical economy, chemistry, medicine and many more.These problems can be modeled by systems of differential equations with impulses.One can obtain the impulsive conditions by taking the short-term perturbation parameters and the initial value problem.For the details of the impulsive differential equations see the results by Ahmad et al. [2], Bainov et al. [3], Benchohra et al. [4], Berger et al. [5], Bianca et al. [6], Gala et al. [7], Hernandez et al. [8], Pierri et al. [9], Samoilenko et al. [10,11], Tang et al. [12] and Wang et al. [13,14].
Ulam stability problem was put forward for the first time at Wisconsin University in 1940.The problem was to discuss the relationship between approximate solution of homomorphism from a group H 1 to a metric group H 2 [15].Considering H 1 and H 2 as Banach spaces, Hyers solved the above problem with the help of direct method [16].The extension of the famous work of Hyers and Ulam can be seen in Aoki [17] and Rassias [18] work.In this work they found the bound for the norm of difference, Cauchy difference, f (t + s) − f (t) − f (s).Answers to this problem, its inductions and attractions for different categories of equations, is a vast region of research and has well elaborated of what is now called Ulam's type stability.
Recently, Yu et al. [36] studied β-Hyers-Ulam stability of the system Motivated from the above work, we investigate the β-Hyers-Ulam-Rassias stability of the system: where 0 In this article, we present four different types of β-Ulam type stability for the system of semilinear nonautonomous impulsive differential equations.Our main objective of this work is to discuss the uniqueness of solution for the given system and analyze the β-Hyers-Ulam-Rassias stability of semilinear nonautonomous system (2) with the help of evolution family.Evolution family has its great importance in every field of research.Different researchers are working to discuss stability analysis of different systems using evolution family.For more details of evolution family we prefer [20,28,[37][38][39][40][41][42][43][44].
Definition 4 ([45]).The semilinear nonautonomous system of differential equations with impulses gives the solution in the form where Q(t, s) = Υ(t)Υ −1 (s) and is known as evolution family and Υ(t) is the fundamental matrix of Θ (t) = H(t)Θ(t) + B(t)u(t).
Definition 5.If Υ(t) is the fundamental matrix of The above system is exponentially bounded if we can find some constants M > 0 and κ < 0 such that Choose > 0, ψ ≥ 0 and ϕ from P C(I, S).Take the inequality With the help of inequality (4) we will define β-Hyers-Ulam-Rassias stability for the system (2).

Definition 6.
(2) is said to be β-Hyers-Ulam-Rassias stable with respect to (ψ β , ϕ β ) if ∃ positive K f ,M,ϕ,β such that for any > 0 and for any solution Θ ∈ PC(I , S) C(I , S) of ( 4) ∃ a solution y of (2) in PC(I , S) satisfying Remark 1.It is direct consequence of inequality (4) that a function y ∈ P C(I , S) C(I , S) is the solution for the inequality (4) if and only if we can find h ∈ C(I ), ψ ≥ 0 and a sequence h k , k ∈ M satisfying On the basis of Remak 1 we can say that the solution of the system For the inequality (4) we obtain Now we state an important lemma known as Gr önwall lemma, which is used in our main result.
Definition 8.The function f from X to X , has a unique fixed point if it is a contraction, where (X , d) is complete metric space.
To discuss β-Hyers-Ulam-Rassias stability of the given system, we need some assumptions which can be used later on.The assumptions are: Now we are able to prove that the nonautonomous differential system (2) has only one solution.
Now for any Θ, Θ ∈ P C(I, S) we have Then, F is contractive with respect to ||.|| P C .By using contraction mapping theorem, which shows that the mapping F has a unique fixed point which is the solution of the system (2).

β-Hyers-Ulam-Rassias Stability on a Compact Interval
To discuss β-Hyers-Ulam-Rassias stability of system (2) on a compact interval, we need to introduce other conditions along with [A 1 ], [A 3 ] and [A 4 ], which can be used to prove our required results.The assumptions are given as follows: [A * 2 ] : f : I × S → S which satisfies Caratheodory conditions and ∃ function L f ∈ C(I, S) so that for every t ∈ I and Θ, Θ ∈ S.
By considering the inequality (4) and above assumptions, we present our first result as follows.

Proof of Theorem 2. Unique solution of the impulsive Cauchy problem
can be written as . . .
Let y be the solution of the inequality (4).Then for every t ∈ (t k , t k+1 ], we can obtain that, Therefore for every t ∈ (t k , t k+1 ], we get Thus, , where x, y, z ≥ 0, and γ > 1.

β-Hyers-Ulam-Rassias Stability on an Unbounded Interval
Here we study β-Hyers-Ulam-Rassias stability on an unbounded interval.For the desired proof we need the following assumptions which can be used in our later work.
[A 0 ]: The operators family {Q(t, s) : t ≥ s ≥ 0} is exponentially stable, that is we can find M ≥ 1 and for every t ∈ R + and Θ, Θ ∈ S. Also we assume that for every t ∈ R + and Θ, Θ ∈ S. Furthermore, we assume that [A 8 ]: A function ϕ ∈ P C(R + , S) and a constant η ϕ > 0 so that t 0 e κ(t−s)+3 By considering the inequality (4) and above assumptions we state our second result as follows.

β-Hyers-Ulam-Rassias Stability with Infinite Impulses
Now to discuss β-Hyers-Ulam-Rassias stability for the system (2) with infinite impulses, that is when M = N.For this case inequality (4) will become where ϕ(.) has the same definition and ψ := {ψ k } k∈N is a nonconstant sequence of nonnegative entries ψ k ≥ 0, for each k ∈ N. Then definition (6) can be written as We call it as extended β-Hyers-Ulam-Rassias stability.To prove β-Hyers-Ulam-Rassias stability with infinite impulses, we consider: for every t ∈ R + and Θ, Θ ∈ S.
[A 11 ] : I k : S → S and there exists a constant L I k > 0 so that and Proof of Theorem 4. Consider Θ is the mild solution of the semilinear nonautonomous impulsive differential system: Let y be the solution of the inequality (10).To prove the required result we follow the method of Theorem 3, for any t ∈ (t k , t k+1 ], we obtain that Thus, The proof is complete.

Conclusions
In the last few decades, many mathematicians showed their interests in the qualitative theory of impulsive differential equations.In particular, to discuss β-Hyers-Ulam-Rassias stability of differential equations, different types of conditions were used in the form of integral inequalities.For the case of semilinear nonautonomous differential system a strong Lipschitz condition of functions were common among them and mostly results were obtained via Grönwall integral inequality.In this article, we present β-Hyers-Ulam-Rassias stability of the semilinear nonautonomous impulsive differential system with the help of evolution family and Grönwall integral inequality.