Approximate solutions of impulsive integro-differential equations

In this paper, we consider impulsive integro-differential equations in Banach space and we establish the bound on the difference between two approximate solutions. We also discuss nearness and convergence of solutions of the problem under consideration. The impulsive integral inequality of Grownwall type is used to obtain results.


Introduction
In the recent decade, the study of impulsive integro-differential equations has become an important thrust area for many researchers across the world. Many real-life phenomena and processes are subject to short-term perturbations whose duration is negligible as compared to the whole phenomena. These problems mostly arise in medicine, economics, biological sciences, engineering, etc. Such type of problems can be modelled with impulsive integro-differential equations. Thus, many researchers [2,4,5,[8][9][10]13] have opted for this research area and contributed to the development of the theory of impulsive differential equations. More information related to this can be found in monographs of Bainov and Simeonov [2] and Lkashmikantham et al. [8].
As it is difficult to provide explicit solutions to most of the physical problems, the method of approximate solution is the best analytic tool for such situation which provides the required information of solutions without finding the explicit solution. Tidke and Dhakne [14,15], Pachpatte [11], Pachpatte [12], Kucche et al. [7] used this technique to study the qualitative properties of solutions of different initial value problems. Since there is less information available in the literature about the approximate solutions of impulsive integro-differential equations, we apply this technique for the following impulsive integro-differential equation of the type: where A is the infinitesimal generator of strongly continuous semigroup of bounded linear operators {T (t)} t≥0 and I k (k = 1, 2, . . . , m) are the linear operators acting in a Banach space X . Let k be a real-valued continuous function on [0, T ] × [0, T ] and the functions and f and h are given functions satisfying some assumptions. The impulsive moments τ k are such that 0 , where x(τ k + 0) and x(τ k − 0) are, respectively, the right and the left limits of x at τ k . In [6], Kendre and Dhakne studied the existence, uniqueness, continuation and continuous dependence of solutions of IVP: using theory of analytic semigroups and fractional power of operators. The problem of existence, uniqueness and other basic properties of IVP (1)-(3) and their special forms have been studied by several authors using different methods such as Banach fixed point theorem, semigroup approach, progressive contractions, etc. See [3,10,13]. Our aim is to find the bound on the difference between two approximate solutions, nearness, convergence and continuous dependence of solutions on parameters of mild solutions of IVP (1)-(3).
The paper is organised as follows: Sect. 2 consists of preliminaries and hypotheses. In Sect. 3, we establish the bound on the difference between two approximate solutions, nearness and convergence properties of solutions and, finally, we give continuous dependence of solutions on parameters and functions involved therein.

Preliminaries and hypotheses
Let X be a Banach space with the norm · . Let PC([0, T ], X ) = {x : [0, T ] → X |x(t) be piecewise continuous at t = τ k , left continuous at t = τ k , that is, x(τ k − h) = x(τ k ) and the right limit , X ) satisfying the equations: is said to be the mild solution of the initial value problem (1)-(3).
be the function such that x i (t) exists for each t ∈ [0, T ] and satisfies the inequality: for a given constant ε i ≥ 0, where it is considered that the initial and impulsive conditions, are satisfied. Then, x i (t) are called i -approximate solutions to the IVP (1)-(3).

Lemma 2.3 [1]
Assume the following inequality holds for t ≥ t 0 : where, u, a ∈ PC([t 0 , ∞), R + ), a is nondecreasing, b(t, s) and k(t, s, τ ) are continuous and non negative functions for t, s, τ ≥ t 0 and are nondecreasing with respect to t, β k (t)(k ∈ N) are nondecreasing for t ≥ t 0 . Then, for t ≥ t 0 , the following inequality holds: Now, we introduce the following hypotheses: for every t ∈ [0, T ], ψ ∈ X and x ∈ X . (H 2 ) Let I k : X → X be functions such that there exist positive constants L k satisfying In this paper, we consider that there exists a constant K 0 > 0 such that where δ is a nonnegative constant, then the following inequality holds: Proof Let x i (i = 1, 2) be approximate solutions of Eq. (1) with Conditions (5) and (6). Then we get Taking t = ξ in (7) and integrating with respect to ξ from 0 to t, we obtain Using the inequalities Using hypotheses (H 1 ) and (H 2 ), we get Let u(t) = x 1 (t) − x 2 (t) . Then, we have Applying Lemma 2.3, we get Remark The inequality obtained in (8) establishes the bound on the difference between the two approximate solutions of Eqs. (1)-(3). If x 1 (t) is a solution of Eq. (1) with x(t) = x 1 0 , then we have ε 1 = 0 and, from (8), we see that x 2 (t) → x 1 (t) as ε 2 → 0 and δ → 0. Moreover, if we put ε 1 = ε 2 = 0 and x 1 0 = x 2 0 in (8), then the uniqueness of the solutions of (1)-(3) is established.
Consider the impulsive IVP (1)-(3), along with the following initial value problem: where k, h are as given in (1)

Theorem 3.2 Suppose that the functions f, k, h in (1)-(3) satisfy the hypotheses (H 1 ) and (H 2 )
and there exist nonnegative constants 3 , δ k such that Let x(t) and y(t) be, respectively, solutions of the initial value problem (1)-(3) and (9)-(11) on [0,T]. Then, the following inequality holds: Proof Using the facts that x(t) and y(t) are, respectively, the solutions of the initial value problem (1)-(2) and (9)-(11) and hypotheses (H 1 ) and (H 2 ), we get Let Now, applying the inequality given in Lemma 2.3, we get Consequently, which implies that This completes the proof.
Remark If f is nearer tof , x 0 to y 0 , then the corresponding solutions of the initial value problem (1)-(3) and (9)-(11) are nearer to each other, and it also depends on the functions continuously involved therein. Thus, the above inequality gives the relation between the solutions of IVP (1)- (3) and (9)- (11).
Consider the initial value problem (1)-(3) with the initial value problem: where k, h are as given in (1), and f n : [0, T ] × X × X → X is a sequence in X.
As an immediate consequence of the above theorem, we have the following corollary: Suppose that the functions f, k, h in (1)-(3) satisfy the hypotheses (H 1 ) and (H 2 ) and there exist nonnegative constants n , δ n , δ kn such that with n → 0, δ n → 0, δ kn → 0 as n → ∞. If x(t) and y n (t), n = 1, 2, . . . are, respectively, solutions of the initial value problems Remark The result obtained in this corollary provides sufficient conditions to ensure that the solutions of the initial value problem (17)-(19) will converge to solutions of the initial value problem (1)-(3).
If x(t) and y(t) are solutions of Eqs. (23)-(25) and (26)-(28), then Proof It is an easy consequence of our main result, so we have omitted the proof.