Ulam stability of first-order nonlinear impulsive dynamic equations

This paper is devoted to the investigation of Ulam stability of first-order nonlinear impulsive dynamic equations on finite-time scale intervals. Our main objective is to formulate sufficient conditions under which the class of first-order nonlinear impulsive dynamic equations on time scales we consider exhibits Ulam stability. Our methods rely on the extended integral inequality on time scales for piecewise-continuous functions. We provide an example to support the validity of the results obtained.


Introduction
The theory of impulsive equations has been considered to be one of the significant research topics from both the theoretical as well as application points of view.Impulsive equations have been widely utilized for modeling the dynamics of processes in which discontinuous jumps occur unexpectedly during their evolution.Such types of equations have received appreciable consideration from researchers around the globe.Researchers have investigated impulsive differential equations for the past several years and the theory for these equations has been developed extensively, readers can refer to the popular books and interesting papers [1][2][3][4][5][6][7][8], and references therein.However, surprisingly, its discrete version, impulsive difference equations, has been less acknowledged and relatively few works are available for impulsive difference equations.
The topic of 'calculus and dynamic equations on time scales'emerged into the mathematics literature about 35 years ago.It provides a unified mathematical framework for difference equations and differential equations and found applicable in modeling the continuous-discrete hybrid time phenomena.This topic has now grown profoundly.In recent years, considerable progress has been made in the theory and applications of impulsive dynamic equations on time scales, we refer to [9][10][11][12][13], and references therein.In the study of dynamic equations time scales, the stability problem is substantial and of great interest for researchers working in the field.Although there are several types of stability, the Ulam stability is interesting and worthy of attention because it provides a relation between the exact solution and an approximate solution of the equation under consideration.To investigate the Ulam stability, there are several approaches available, one of which is by employing inequality.The main advantage of this approach is that it requires less restriction and is much simpler than any other approach.
Motivated by the work mentioned above, we assert that studying the Ulam stability of impulsive dynamic equations is worthwhile.In this paper, we introduce the Ulam stability for first-order nonlinear impulsive dynamic equations of the form where x is the delta derivative of x, p : T → R is positively regressive and rd-continuous, f : J × R → R is rd-continuous in the first variable and continuous in the second variable, each t i represents a priori known moments of impulse and satisfies t 0 < t i < t i+1 < T, {t i } i∈N ⊂ J, where N = {1, 2, . . ., m} ⊂ N, I i : R → R describes the discontinuity of x at each t i .
The rest of the paper is structured as follows.Section 2 covers some essential materials for the readership of this paper.An extended integral inequality on time scales is established in Sect.3. In Sect.4, we investigate Ulam stability of (1) on finite time scale intervals by means of the extended inequality.An illustrative example is given in Sect. 5. Finally, conclusions and future research directions are mentioned in Sect.6.

Preliminaries
In this section, we shall provide some essential materials from time scales calculus that are pertinent to the present paper.The reader can find more details on the topic in [50,51].A nonempty, closed subset of the real line R is a time scale T. We usually write Definition 2.1 A function f : T → R is said to be delta differentiable at t ∈ T κ if there exists f (t) ∈ R, a so-called delta derivative of f at t, with the following property: For any ε > 0 there is a neighborhood N of t such that is continuous at every rightdense point or maximal point in T and its left-sided limits exist at left-dense points in T. The symbol C rd (T, R) will be used for the set of all such functions.
If a function f : T × R → R is rd-continuous in the first variable and continuous in the second variable, then we write f ∈ C rd (T × R, R).
Note 2.1 The family, C rd (J, R), of all rd-continuous functions from J into R forms a Banach space coupled with the norm • defined as x := sup t∈J |x(t)|.

Definition 2.3 A function
The symbol R(T, R) will be used for the set of all rd-continuous regressive functions.
If 1 + μ(t)p(t) > 0 for all t ∈ T, then p is said to be positively regressive and R + (T, R) denotes the set of all rd-continuous positively regressive functions.

Definition 2.4
For p ∈ R(T, R), the exponential function e p (t, s) on the time scale T is defined as For p, q ∈ R(T, R), we define the following.
p ⊕ q := p + q + μpq, p := -p 1 + μp , p q := p ⊕ ( q).We let ) is a solution of (1) without any impulse, if and only if Let C(J, R) be the Banach space of all continuous functions x with domain J and taking values in R with the norm x := sup t∈J |x(t)|.We write J 0 := [t 0 , t 1 ] and for each i ∈ N , It can be readily seen that the set PC is a Banach space coupled with the norm x PC := max i∈N { x i }, where x i = sup t∈J i |x(t)|, and the set PC 1 (J, R) is also a Banach space coupled with the norm x PC 1 := max{ x PC , x PC }.
Definition 2.5 A function x ∈ PC 1 is said to be a solution of the IDP (1), if x satisfies the dynamic equation Now, we introduce stability definitions that will be used in this paper.
Definition 2. 6 The IDP ( 1) is Hyers-Ulam stable if there exists a constant K f ,N > 0 with the following property: For any ε > 0, if y ∈ PC 1 (J, R) is such that then there exists x ∈ PC 1 (J, R) satisfying ( 1) such that The constant K f ,N > 0 is known as the HUS constant.
Definition 2.9 The IDP ( 1) is generalized Hyers-Ulam-Rassias stable with respect to (φ, ψ) if there exists K f ,N ,φ > 0 with the following property: For any nondecreasing φ ∈ PC 1 (J, R + ) and ψ ≥ 0, if y ∈ PC 1 (J, R) is such that then there exists x ∈ PC 1 (J, R) satisfying (1) such that The constant K f ,N ,φ > 0 is known as the GHURS constant.
Remark 2.2 A function y ∈ PC 1 (J, R) satisfies ( 7) if and only if there exists a function g ∈ PC 1 (J, R) and a sequence {g i } i∈N (which depends on y) with the following properties: Similar arguments hold for the inequalities ( 5) and (9).and c, b

Extended integral inequality on time scales
In this section, we establish the timescale analog of the integral inequality for the piecewise-continuous functions studied by Samoilenko and Perestyuk [6].Theorem 3.1 Let t 0 , t ∈ T, t ≥ t 0 , and the following inequality hold: That is, This completes the proof.

Main results
In this section, we investigate the Ulam stability for first-order nonlinear impulsive dynamic equations on time scales (1), by means of the impulsive integral inequality given in Theorem 3.1.For this, below we list some essential conditions: Also, set L * f := sup t∈J L f (t).(C 3 ) For I : R → R, there exists a constant L I i > 0 such that (C 4 ) For a nondecreasing function φ ∈ PC(J, R), there exists a constant L φ such that Theorem 4.1 Consider the IDP (1).Under the conditions (C 1 )-(C 4 ), the following assertions hold: , then the IDP (1) has a unique solution x ∈ PC 1 (J, R) satisfying initial condition x(t 0 ) = A for any initial value A ∈ R. (ii) The IDP (1) is Hyers-Ulam-Rassias stable with respect to (φ, ψ) and the HURS constant is Proof (i) First, fix A ∈ R and define the mapping F : PC 1 (J, R) → PC 1 (J, R) by In view of Remark 2.1, it is clear that the fixed points of F are the solutions of (1).Hence, we show that F has a fixed point and for this we use the contraction mapping principle.
For any x, y ∈ PC 1 (J, R), we can write Thus, for all x, y ∈ PC 1 (J, R) Since (E p L * f (Tt 0 )+ m i=1 |e p (T, t i )|L I i ) < 1, the above inequality implies that the mapping F is contraction on PC 1 (J, R).Therefore, F has a unique fixed point x * ∈ PC 1 (J, R), which is the unique solution of the IDP (1) satisfying x * (t 0 ) = A.

Illustrative example
In this section, we give an illustrative example to show the validity of our result obtained in Sect. 5 concerning the Ulam stability of IDP (1) in the finite time scale domain.

Conclusion
We have investigated the Ulam stability for first-order nonlinear impulsive dynamic equations on bounded timescale intervals.We have achieved our results by effectively employing an extended integral inequality on time scales.The Ulam stability is an essential and reliable tool in solving the considered problem approximately, when it is not easy to find the exact solution.Also, impulsive dynamic equations are committed to model the continuous-discrete hybrid phenomena that unexpectedly undergo some discontinuous jumps during their evolution.Thus, we are confident that the results of the present paper are valuable and find their place in approximation theory, control theory, optimization, and other related fields.We believe that the results obtained in this paper can be easily extended to systems of impulsive dynamic equations and to Banach spaces.Further, it would be an exciting topic to study impulsive problems (1) with suitable delays.