Existence of Multiple Positive Solutions of a Type of Impulsive Functional Di ff erential Equations

In this paper, we consider the non-periodic boundary value problem for a type of first order impulsive functional differential equation in Banach spaces. The existence of pulse in differential equations makes them an important area of investigation. We make use of fixed point index theory on the cone to prove existence of positive solutions. The conditions for existence of two and three positive solutions are given.


Introduction and Preliminaries
In recent years, the theories of impulsive functional differential equations have been rapidly developed, and because such equations may exhibit several real world phenomena in physics, biology, engineering, and so forth (Bainov & Simeonov, 1993;Lakshmikantham, Bainov & Simeonov, 1989;Bainov & Hristova, 1993), they have received much attention (Ding, Mi & Han, 2005;Zhang & Liu, 2010),The periodic boundary value problem is an important research branch of the impulsive functional differential equations.Some conclusions have been made (Zhang, Li, Jiang & Wang, 2006;Zhimin & Weigao, 2002) about the existence of solutions and the multiplicity of positive solutions of the impulsive functional differential equations with periodic boundary value problems.Whereas the non periodic boundary value occurs more frequently in differential equations with pulse, researches are needed for the problem of existence of positive solutions and multiplicity of such equations.In this paper£we restrict our attention to the study of the following first order impulsive functional differential equations with non-periodic boundary value Where, f : J × C τ is continuous.C τ := {φ : [−τ, 0] → R ; φ(t) is continuous everywhere except a finite number of points t, φ(t + ), φ(t − ) exist, and The approaches used for the investigation of existence of positive solutions for differential equations with impulse are monotone iterative technique and upper and lower solution method (Zhimin & She, 2002;Juan & Rosana, 2006;Luo & Jing, 2008;He & He, 2004).Upper and lower solution method is often applied to discuss the minimal and maximal solutions of such equations, and monotone iterative technique is usually used to prove the existence of solution.Recently fixed point index theorem on cones in Banach space is introduced to investigate the multiplicity of solutions (Zhao, 2010).In (Zhao, 2010), Zhao studied the problem (1), the results are established using the fixed point index theorem on the cone, and they proved the existence of two solutions.
Motivated by the results mentioned above, in this paper, we give the conditions of the existence of two positive solutions and three positive solutions of equations (1) using fixed point index theory on the cone.

Preliminaries
Throughout the rest of this paper, we always assume that the points of impulse t k are right-dense for each k We define PC(J * ) = {u : J * → R is continuous everywhere except a finite number of points in [−τ, 0], for t Theorem 1.1 Suppose that E is a Banach space, K ∈ E is a cone in ,and r > 0, Ω r = {x ∈ K : ||x|| < r}.If S : Ω r → K is a complete continuous operator, and for ∀x ∈ ∂Ω r , S x x 1), if and only if u ∈ E 0 is a solution of the follow integral equation: Where For convenient, we always suppose that: Where Where δ = min{1, p} e M 2 T max{1, p} .
Where η and η k satisfy Then problem (1) has at least two positive solutions u 1 , u 2 , which satisfy 0 Proof: Following condition (H 4 ), and take N > (δmin t∈[0,T ] ∫ T 0 G(t, s)ds) −1 , then there exist constants r > 0 , R > 0 (r is sufficiently small, and R is sufficiently large, r < d < R), for ∀t ∈ [0, T ], we have: Therefore u t ∈ C * .According to condition (H 4 ) and inequality (6), we have Therefore ∀u ∈ C * .According to condition H 4 and inequality (7), we have Therefore when t ∈ J, and ∥S u∥ ≥ ∥u∥ [−τ,T ] = R holds. Let According to theorem 1.1, we have that S exits at least two fixed points u 1 , u 2 and It means that we have at least two positive solutions u 1 , u 2 for problem (1) and they satisfy inequality 0 Theorem 3.2 Suppose conditions (H 1 ), (H 2 ), (H 3 ) hold, together with the following conditions (H 6 ) and (H 7 ).(H 6 ) Then problem (1) has at least two positive solutions u 1 , u 2 , which satisfy Proof: Take ξ, ξ k > 0 sufficiently small such that max From condition (H 6 ), there exists constants r > 0, R > 0 (r is sufficiently small, and R is sufficiently large, r < d < R), for ∀t ∈ [0, T ], we have: Therefore we assert u t ∈ C * .According to condition (H 6 ) and inequality (8), we have For ∀u ∈ K ∩ ∂Ω R and ∀t ∈ J, we have hereby u t ∈ C * , according to condition (H 6 ) and inequality (9) , we have Consequently, for t ∈ J, we have Let Then we have We assert that S (u)(t) u(t).
Therefore from theorem 1.1, we have at least that two fixed points u 1 , u 2 for operator S .such and It means that problem (1) exist at least two positive solutions u 1 , u 2 , and they satisfy 0 Theorem 3.3 Suppose conditions (H 1 ), (H 2 ), (H 3 ) hold, and f (t, 0) 0, I k (0) 0, and with the following conditions (H 8 ), (H 9 ): Then problem (1) exits at least three positive solutions u 1 , u 2 , u 3 , and they satisfy 0 Proof: For three constants r < d < R, we define three open sets: For ∀u ∈ K ∩ ∂Ω r and ∀t ∈ J, we have For ∀u ∈ K ∩ ∂Ω R and ∀t ∈ J, we have According to condition (H 9 ) , for ∀u ∈ K ∩ ∂Ω d and ∀t ∈ [0, T ], we have Then we deduce from theorem 1.1 that S has at least three fixed points u 1 , u 2 , u 3 such that u According to lemma 3, we have u 1 0.