integro-differential equations with a

This paper is concerned with a class of boundary value problems for the nonlinear impulsive functional integro-differential equations with a parameter by establishing new comparison principles and using the method of upper and lower solutions together with monotone iterative technique. Sufficient conditions are established for the existence of extremal system of solutions for the given problem. Finally, we give an example that illustrates our results.


Introduction
Impulsive differential equations have become more important in recent years in some mathematical models of real processes and phenomena studied in physics, chemical technology, biotechnology and economics.There has been a significant development in impulse theory ( [1] [2]).
The differential equations with parameters play important roles and tools not only in mathematics but also in physics, population dynamics, control systems, dynamical systems and engineering to create the mathematical modelling of many physical phenomena.It is more accurate than the average differential equations to describe the objective world.And the existence of solutions for the BVPS of these equations have been studied by many authors( [11]- [13]).
Especially, there is an increasing interest in the study of nonlinear mixed integro-differential equations with deviating arguments and multipiont BVPS( [4]- [10]) for impulsive differential equations.And theorems about existence, uniqueness of differential and impulsive functional differential abstract evolution Cauchy problem with nonlocal conditions have been studied by Byszewski and Lakshmikantham [21], by G.Infants [22], by Chang et al. [20] [25], by Anguraj et al. [19], and by Akca et al. [24] and the references therein.
In this paper, we are concerned with the following BVPS for the nonlinear mixed impulsive functional integro-differential equations with a parameter: (ii) If (iii) If λ 2 = 0, a i = k = 0, (i = 1, 2, • • • p) then the Eq.(1.1) reduces to the integral boundary value problem which has been studied in [23].The article is organized as follow.In section 2, we establish new comparison principles.In section 3, by using of the monotone iterative technique and the method of upper and lower solutions, we obtain the existence result for the extremal solutions of BVPS(1.1).In section 4, we give an example that illustrates our results.EJQTDE, 2013 No. 4, p. 2 Let P C(J) = {x : J → R; x(t) is continuous everywhere except for some t k at which x(t + k ) and m}, P C(J) and P C 1 (J) are Banach spaces with the norms x P C = sup{|x(t)| : t ∈ J} and For conveniences, we set Lemma 2.1 Assume that (H 1 )(H 2 ) hold and q ∈ P C 1 (J) such that where the operator H is defined as T 0 h(t, s)q(δ(s))ds.
Proof :Let p(t) = q(t)e R t 0 M(s)ds .Obviously p(t) and q(t) have the same sign on J.In view of (2.2), we where (H * p)(t) = N * (t)p(α(t)) + K * (t) Suppose, to the contrary, that p(t) > 0 for some t ∈ J.
By (H 1 ) we get that C ≤ 0 which is a contradiction.
By the two inequalities above, we obtain Therefore, we get that (µ * + m k=1 L k ) > r * , which is in contradiction to (H 2 ).Hence p(t) ≤ 0, q(t) ≤ 0. We complete the proof.
Lemma 2.2Assume that (H 1 ), (H 2 ) and Then the linear problem has a unique solution x ∈ P C 1 (J, E) and it is represented by: where (2.6) EJQTDE, 2013 No. 4, p. 5 Proof : First, differentiating (2.5), we have It is easy to check that u(0) = ru(T ) + d.Hence, we know that (2.5) is a solution of (2.4).
Next we show that the solution of (2.4) is unique.Let u 1 , u 2 are the solutions of (2.4) and set In view of Lemma 2.1, we get p ≤ 0 which implies u 1 ≤ u 2 .Similarly, we have The proof is complete. where T 0 h(t, s)ds]dt, then Eq.(2.5) has a unique solution u in P C(J).
Proof : Define an operator F by it is easy to see that max{G(t, s), (t, s) Now, for x, y ∈ P C(J), we have Consequently, the Banach fixed point theorem implies that F has a unique fixed point u in P C(J), and the lemma is proved.

Finally, we assert
By Lemma 2.1, we have p(t) ≤ 0 for all t ∈ J, that is u n+1 (t) ≤ u(t).
The proof is almost similar to theorem 3.1, so we omit it.
then the Eq.(1.1) can be regarded as the nonlocal Cauchy problem.