We obtain new result of the existence of positive solutions of a class of singular impulse periodic boundary value problem via a nonlinear alternative principle of Leray-Schauder. We do not require the monotonicity of functions in paper (Zhang and Wang, 2003). An example is also given to illustrate our result.

1. Introduction

Because of wide interests in physics and engineering, periodic boundary value problems have been investigated by many authors (see [1–19]). In most real problems, only the positive solution is significant.

In this paper, we consider the following periodic boundary value problem (PBVP in short) with impulse effects:

(1.1)

Here, J = [0,2π], 0 < t₁ < t₂ < ⋯<t_l < 2π, J′ = J∖{t₁, t₂, …, t_l}, M > 0, f ∈ C(J × R₊, R⁺), I_k ∈ C(R⁺, R), J_k ∈ C(R⁺, R⁺), R⁺ = [0, +∞), R₊ = (0, +∞) with −(1/m)J_k(u) < I_k(u) < (1/m)J_k(u), u ∈ R⁺,

, where

and

, i = 0,1, respectively, denote the right and left limit of u⁽ⁱ⁾(t) at t = t_k.

In [7], Liu applied Krasnoselskii′s and Leggett-Williams fixed-point theorem to establish the existence of at least one, two, or three positive solutions to the first-order periodic boundary value problems

(1.2)

Jiang [5] has applied Krasnoselskii′s fixed point theorem to establish the existence of positive solutions of problem

(1.3)

The work [5] proved that periodic boundary value problem (PBVP in short) (1.3) without singularity have at least one positive solutions provided f(t, x) is superlinear or sublinear at x = 0+ and x = +∞. In [14], Tian et al. researched PBVP (1.1) without singularity. They obtained the existence of multiple positive solutions of PBVP (1.1) by replacing the suplinear condition or sublinear condition of [4] with the following limit inequality condition:

(A₁)
(1.4)
(A₂)
(1.5)

Nieto [10] introduced the concept of a weak solution for a damped linear equation with Dirichlet boundary conditions and impulses. These results will allow us in the future to deal with the corresponding nonlinear problems and look for solutions as critical points of weakly lower semicontinuous functionals.

We note that the function f involved in above papers [5, 7, 10, 14] does not have singularity. Xiao et al. [16] investigate the multiple positive solutions of singular boundary value problem for second-order impulsive singular differential equations on the halfline, where the function f(t, u) is singular only at t = 0 and/or t = 1. Reference [19] studied PBVP (1.3), where the function f has singularity at x = 0. The authors present the existence of multiple positive solutions via the Krasnoselskii′s fixed point theorem under the following conditions.

There exist nonnegative valued ξ(x), η(x) ∈ C((0, ∞)) and P(t), Q(t) ∈ L¹[0,2π] such that

(1.6)

where η(x) is nonincreasing and ξ(x)/η(x) is nondecreasing on (0, ∞),

(1.7)
(1.8)
Here, δ_j, A_j, B_j are some constants.

In this paper, the nonlinear term f(t, u) is singular at u = 0, and positive solution of PBVP (1.1) is obtained by a nonlinear alternative principle of Leray-Schauder type in cone. We do not require the monotonicity of functions η, ξ/η used in [19]. An example is also given to illustrate our result.

This paper is organized as follows. In Section 1, we give a brief overview of recent results on impulsive and periodic boundary value problems. In Section 2, we present some preliminaries such as definitions and lemmas. In Section 3, the existence of one positive solution for PBVP (1.1) will be established by using a nonlinear alternative principle of Leray-Schauder type in cone. An example is given in Section 4.

2. Preliminaries

Consider the space PC[J, R] = {u : u is a map from J into R such that u(t) is continuous at t ≠ t_k, left continuous at t = t_k, and exists, for k = 1,2, …l.}. It is easy to say that PC[J, R] is a Banach space with the norm ∥u∥_pc = sup_t∈J |u(t)|. Let PC¹[J, R] = {u ∈ PC[J, R] : u′(t) exists at t ≠ t_k and is continuous at t ≠ t_k, and , exist and u^′(t) is left continuous at t = t_k, for k = 1,2, …l.} with the norm . Then, PC¹[J, R] is also a Banach space.

Lemma 2.1 (see [15].)u ∈ PC¹(J, R)∩C²(J′, R) is a solution of PBVP (1.1) if and only if u ∈ PC(J) is a fixed point of the following operator T:

(2.1)

where G(t, s) is the Green′s function to the following periodic boundary value problem:

(2.2)

here, Γ = 2m(e^2mπ − 1). It is clear that

(2.3)

Define

(2.4)

where

(2.5)

The following nonlinear alternative principle of Leray-Schauder type in cone is very important for us.

Lemma 2.2 (see [4].)Assume that Ω is a relatively open subset of a convex set K in a Banach space PC[J, R]. Let be a compact map with 0 ∈ Ω. Then, either

(i)
T has a fixed point in , or,
(ii)
there is a u ∈ ∂Ω and a λ < 1 such that u = λTu.

3. Main Results

In this section, we establish the existence of positive solutions of PBVP (1.1).

Theorem 3.1. Assume that the following three hypothesis hold:

H₁ there exists nonnegative functions ξ(u), η(u), γ(u) ∈ C(0, +∞) and p(t), q(t) ∈ L¹([0,2π]) such that

(3.1)

(3.2)

H₂ there exists a positive number r > 0 such that

(3.3)

H₃ for the constant r in (H₂), there exists a function Φ_r > 0 such that

(3.4)

Then PBVP (1.1) has at least one positive periodic solution with 0 < ∥u∥<r, where

(3.5)

Proof. The existence of positive solutions is proved by using the Leray-Schauder alternative principle given in Lemma 2.2. We divide the rather long proof into six steps.

Step 1. From (3.3), we may choose n₀ ∈ {1,2, …} such that

(3.6)

Let N₀ = {n₀, n₀ + 1, …}. For n ∈ N₀. We consider the family of equations

(3.7)

where λ ∈ [0,1] and

(3.8)

For every λ and n ∈ N₀, define an operator as follows:

(3.9)

Then, we may verify that

(3.10)

To find a positive solution of (3.7) is equivalent to solve the following fixed point problem in PC[J, R]:

(3.11)

Let

(3.12)

then Ω is a relatively open subset of the convex set K.

Step 2. We claim that any fixed point u of (3.11) for any λ ∈ [0,1) must satisfies ∥u∥≠r.

Otherwise, we assume that u is a solution of (3.11) for some λ ∈ [0,1) such that ∥u∥ = r. Note that f_n(t, u) ≥ 0. u(t) ≥ 1/n for all t ∈ J and r ≥ u(t)≥(1/n) + σ∥u − 1/n∥. By the choice of n₀, 1/n ≤ 1/n₀ < r. Hence, for all t ∈ J, we get

(3.13)

From (3.2), we have

(3.14)

Consequently, for any fixed point u of (3.11), by (3.8), (3.13), and (3.14), we have

(3.15)

It follows from −(1/m)J_k(u) < I_k(u) < (1/m)J_k(u) that

(3.16)

So, we get from (3.1), (3.2), and (3.3) that

(3.17)

Therefore,

(3.18)

This is a contraction, and so the claim is proved.

Step 3. From the above claim and the Leray-Schauder alternative principle, we know that operator (3.9) (with λ = 1) has a fixed point denoted by u_n in . So, (3.7) (with λ = 1) has a positive solution u_n with

(3.19)

Step 4. We show that {u_n} have a uniform positive lower bound; that is, there exists a constant δ > 0, independent of n ∈ N₀, such that

(3.20)

In fact, from (3.4), (3.8), (3.16), and (3.19), we get

(3.21)

Step 5. We prove that

(3.22)

for some constant H > 0. Equations (3.19) and (3.20) tell us that δ ≤ u_n(t) ≤ r, so we may let

(3.23)

Then,

(3.24)

Step 6. Now, we pass the solution u_n of the truncation equation (3.7) (with λ = 1) to that of the original equation (1.1). The fact that ∥u_n∥<r and (3.22) show that is a bounded and equi-continuous family on [0,2π]. Then, the Arzela-Ascoli Theorem guarantees that has a subsequence , converging uniformly on [0,2π]. From the fact ∥u_n∥<r and (3.20), u satisfies δ ≤ u(t) ≤ r for all t ∈ J. Moreover, also satisfies the following integral equation:

(3.25)

Let j → +∞, and we get

(3.26)

where the uniform continuity of f(t, u) on J × [δ, r] is used. Therefore, u is a positive solution of PBVP (1.1). This ends the proof.

4. An Example

Consider the following impulsive PBVP:

(4.1)

where c_k > 0 are constants. Then, PBVP (4.1) has at least one positive solution u with 0 < ∥u∥<1.

To see this, we will apply Theorem 3.1.

Let

(4.2)

then f(t, u) has a repulsive singularity at u = 0

(4.3)

Denote

(4.4)

Then, it is easy to say that (3.1), (3.2), and (3.3) hold. From (3.5), we know

(4.5)

So, we may choose M large enough to guarantee that (3.3) holds. Then, the result follows from Theorem 3.1.

Remark 4.1. Functions ξ, η in example (4.1) do not have the monotonicity required as in [19]. So, the results of [19] cannot be applied to PBVP (4.1).

Acknowledgments

The authors are grateful to the anonymous referees for their helpful suggestions and comments. Zhaocai Hao acknowledges support from NSFC (10771117), Ph.D. Programs Foundation of Ministry of Education of China (20093705120002), NSF of Shandong Province of China (Y2008A24), China Postdoctoral Science Foundation (20090451290), ShanDong Province Postdoctoral Foundation (200801001), and Foundation of Qufu Normal University (BSQD07026).

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The Existence of Positive Solutions for Singular Impulse Periodic Boundary Value Problem

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. An Example

Acknowledgments

References

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The Existence of Positive Solutions for Singular Impulse Periodic Boundary Value Problem

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. An Example

Acknowledgments

References

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