The method of upper and lower solutions to impulsive differential equations with integral boundary conditions

This paper considers a second-order impulsive differential equation with integral boundary conditions. Some sufficient conditions for the existence of solutions are proposed by using the method of upper and lower solutions and Leray-Schauder degree theory.


Introduction
The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments. Processes with such a character arise naturally and often, especially in phenomena studied in physics, chemical technology, population dynamics, biotechnology, and economics. For an introduction of the basic theory of impulsive differential equations in R n , see Lakshmikantham et al. [], Bainov and Simeonov [], Samoilenko and Perestyuk [], and the references therein. The theory of impulsive differential equations has become an important area of investigation in recent years, and it is much richer than the corresponding theory of differential equations (see, for instance, [-] and the references therein).
On the other hand, the theory of boundary value problems with integral boundary conditions for ordinary differential equations arises in different areas of applied mathematics and physics. For example, heat conduction, chemical engineering, underground water flow, thermo-elasticity, and plasma physics can be reduced to nonlocal problems with integral boundary conditions. For boundary value problems with integral boundary conditions and comments on their importance, we refer the reader to the papers in Gallardo [-]  , deal with impulsive differential equations with integral boundary conditions. However, most of these results are obtained by using the fixed point theorem in cones. It is well known that the method of upper and lower solutions is a powerful tool for proving the existence results for a large class of boundary value problems, see [-]. ©2014 Pang et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. http://www.advancesindifferenceequations.com/content/2014/1/183 In [], Shen and Wang applied the method of upper and lower solutions to solve impulsive differential equations with nonlinear boundary conditions as follows: Motivated by the works mentioned above, in this paper, we shall employ the method of upper and lower solutions together with Leray-Schauder degree theory to study the existence of a solution of the impulsive BVP ) and x(tk ) represent the right and left limits of x(t) at t = t k , respectively, x (t k ) has a similar meaning for x (t).

Preliminaries
) and x (ti ) exist, and x (ti ) = x (t i ), i = , , . . . , p}. Note that PC(J) and PC  (J) are Banach spaces with the respective norms is called a solution of (.) if it satisfies the differential equation ) and the boundary con- Definition . The function α ∈ E is said to be a lower solution for boundary value problem (.) if The function β ∈ E is said to be an upper solution for boundary value problem (.) if We say that f satisfies the Nagumo condition relative to α, β if for there exists a constant D such that In addition, we assume that the following conditions hold: (H  ) f satisfies the Nagumo condition relative to α, β; We consider the modified problem where One can find the next result, with its proof, in [].
Lemma . For each x ∈ E, the following two properties hold: Lemma . For any v(t) ∈ C(J), the following boundary value problem has a unique solution as follows: for Proof Denote y(t) = x(t)β(t), we will only see that x(t) ≤ β(t) for every t ∈ J. An analogous reasoning shows that , does not hold, then sup ≤t≤ (x(t)β(t)) > , there are three cases. Case . Suppose that max ≤t≤ y(t) = sup ≤t≤ (x(t)β(t)) = y(), or max ≤t≤ y(t) = y(), we only see that max ≤t≤ y(t) = y(). Easily, it holds that y() > . From Definition . and (H  ), we have which is a contradiction.
Case . Suppose that there exist k ∈ {, , . . . , p} and τ ∈ (t k , t k + ) such that sup t∈(t k ,t k +] Then y (τ ) =  and y (τ ) ≤ . On the other hand, which is a contradiction. Hence the function y cannot have any positive maximum interior to the interval (t k , t k + ) for k = , , . . . , p.

Lemma . -D ≤ x (t) ≤ D on J, where x(t) is the solution of (.).
Proof Here we only show . . , p}, and as a result, Therefore, there exist r  , r  ∈ (t k , t k+ ) such that x (r  ) = ω, x (r  ) = D and either or ω ≤ x (t) ≤ D, t ∈ (r  , r  ). http://www.advancesindifferenceequations.com/content/2014/1/183 We only consider the first case, since the other case can be handled similarly. It follows from the assumption that This implies that This, obviously, contradicts the choice of D. The proof is complete.
Lemma . (Schauder's fixed point theorem) Let K be a convex subset of a normed linear space E. Each continuous, compact map L : K → K has a fixed point.

Existence results
Theorem . Suppose that conditions (H  )-(H  ) hold. Then BVP (.) has at least one solution x ∈ E ∩ C  (J * ) such that Proof Solving (.) is equivalent to finding x ∈ E which satisfies It is obvious that T : E → E is completely continuous. By the Schauder fixed point theorem, we can easily obtain that T has a fixed point x ∈ E, which is a solution of BVP (.). And by Lemma . and Lemma ., we know that ( ). If f satisfies the Nagumo condition with respect to α  , β  , then problem (.) has at least three solutions x  , x  and x  satisfying Proof We consider the following modified problem: x(t k ) =Ī k x(t k ) , k = , , . . . , p,  Since the functions m α  ,β  and n are continuous and bounded, we obtain that there exists M F >  such that [-d,d] I k (x) + max x∈ [-d,d],y∈ [-D,D] J k (x, y) , It is standard that T : E → E is completely continuous. It is immediate from the argument above that T(¯ ) ⊂ .