Existence and Uniqueness of Positive Solutions to Nonlinear Second Order Impulsive Differential Equations with Concave or Convex Nonlinearities

Abstract

Using a new fixed point theorem of generalized concave operators, we present in this paper criteria which guarantee the existence and uniqueness of positive solutions to nonlinear two-point boundary value problems for second-order impulsive differential equations with concave or convex nonlinearities.

1. Introduction

In this paper, we study the existence and uniqueness of positive solutions to the following two-point boundary value problems for second-order impulsive differential equations: −𝑥^′′(𝑡)=𝑓(𝑡,𝑥(𝑡)),𝑡∈𝐽,𝑡≠𝑡_𝑘,𝑘=1,2,…,𝑚,Δ𝑥∣_{𝑡=𝑡𝑘}=𝐼_𝑘𝑥𝑡_𝑘,𝑘=1,2,…,𝑚,Δ𝑥^′∣_{𝑡=𝑡𝑘}=𝐼_𝑘𝑥𝑡_𝑘𝑥(0),𝑘=1,2,…,𝑚,=𝑥^′(0),𝑥(1)=𝑥^′(1), (1) where 𝑓∈𝐶[𝐽×𝐑,𝐑], 𝐽=[0,1], 0<𝑡₁<𝑡₂<⋯<𝑡_𝑘<⋯<𝑡_𝑚<1, Δ𝑥∣_{𝑡=𝑡𝑘}=𝑥(𝑡⁺_𝑘)−𝑥(𝑡⁻_𝑘), Δ𝑥^′∣_{𝑡=𝑡𝑘}=𝑥^′(𝑡⁺_𝑘)−𝑥^′(𝑡⁻_𝑘), 𝑥^′(𝑡⁺_𝑘), 𝑥^′(𝑡⁻_𝑘), 𝑥(𝑡⁺_𝑘), 𝑥(𝑡⁻_𝑘) denote the right limit (left limit) of 𝑥^′(𝑡) and 𝑥(𝑡) at 𝑡=𝑡_𝑘, respectively. 𝐼_𝑘,𝐼_𝑘∈𝐶[𝐑,𝐑], 𝑘=1,2,…,𝑚.

Impulsive differential equations have been studied extensively in recent years. Such equations arise in many applications such as spacecraft control, impact mechanics, chemical engineering, and inspection process in operations research. It is now recognized that the theory of impulsive differential equations is a natural framework for a mathematical modelling of many natural phenomena. There have appeared numerous papers on impulsive differential equations during the last ten years. Many of them are on boundary value problems, see [1–18], and it is interesting to note that some of them are about comparatively new applications like ecological competition, respiratory dynamics, and vaccination strategies, see [12, 19–25].

Second-order impulsive differential equations have been studied by many authors with much of the attention given to positive solutions. For a small sample of such work, we refer the reader to works by Feng and Xie [6], Hu et al. [8], Jankowski [10, 11], E. K. Lee and Y.-H. Lee [12], Lin and Jiang [13], Liu et al. [14], Agarwal and O’Regan [26], Wang et al. [27], Zhang [28], and Chu et al. [29]. The results of these papers are based on the Schauder fixed point theorem, Leggett-Williams theorem, fixed point index theorems in cones, Krasnoselski fixed point theorem, the method of upper-lower solutions, fixed point theorems in cones, and so on. But, in most of the existing works, in order to establish the existence of positive solutions, a key condition is the existence of upper-lower solutions. However, as we know, it is difficult to verify the existence of upper-lower solutions for concrete impulsive differential equations. In addition, few papers can be found in the literature on the existence and uniqueness of positive solutions for second-order impulsive differential equations. In this paper, we will study the problem (1) with concave or convex nonlinearities and not suppose the existence of upper-lower solutions and compactness condition. Different from the previously mentioned works, in this paper we will use a new fixed point theorem of generalized concave operators to show the existence and uniqueness of positive solutions for the problem (1).

For convenience, we list the following assumptions on the functions 𝑓(𝑡,𝑥),𝐼_𝑘(𝑥), and 𝐼_𝑘(𝑥):(𝐻₁)𝑓(𝑡,0)≤0, 𝑓(𝑡,1/2)<0, 𝑡∈[0,1], and 𝑓(𝑡,𝑥) is decreasing in 𝑥∈[0,∞) for each 𝑡∈[0,1],(𝐻₂)𝐼_𝑘(0)≤0, 𝐼_𝑘(0)≥0, and 𝐼_𝑘(𝑥) is decreasing, and 𝐼_𝑘(𝑥) is increasing in 𝑥∈[0,∞), 𝑘=1,2,…,𝑚, (𝐻₃) for any 𝜆∈(0,1), 𝑡∈[0,1], and 𝑥≥0, there exist 𝛼₁(𝜆),𝛼₂(𝜆),𝛼₃(𝜆)∈(𝜆,1) such that 𝑓(𝑡,𝜆𝑥)≤𝛼₁(𝜆)𝑓(),𝐼𝑡,𝑥_𝑘(𝜆𝑥)≤𝛼₂(𝜆)𝐼_𝑘(𝑥),𝐼_𝑘(𝜆𝑥)≥𝛼₃(𝜆)𝐼_𝑘(𝑥),𝑘=1,2,…,𝑚, (2) (𝐻₄)∑^𝑚_𝑘=1[−2𝐼_𝑘(3/2)+(1+𝑡_𝑘)𝐼_𝑘(3/2)]>0, (𝐻₁)^′𝑓(𝑡,3/2)<0, 𝑡∈[0,1], and 𝑓(𝑡,𝑥) is increasing in 𝑥∈[0,∞) for each 𝑡∈[0,1] and 𝑓(𝑡,𝑥)≤0 for [0,1]×[0,∞),(𝐻₂)^′  𝐼_𝑘(𝑥)≤0, 𝐼_𝑘(𝑥)≥0 for [0,∞), and 𝐼_𝑘(𝑥) is increasing, and 𝐼_𝑘(𝑥) is decreasing in 𝑥∈[0,∞), 𝑘=1,2,…,𝑚, (𝐻₃)^′ for any 𝜆∈(0,1), 𝑡∈[0,1], and 𝑥≥0, there exist 𝛽₁(𝜆),𝛽₂(𝜆),𝛽₃(𝜆)∈(0,1) such that 𝑓(𝑡,𝜆𝑥)≥𝜆^{−𝛽1(𝜆)}𝑓(),𝐼𝑡,𝑥_𝑘(𝜆𝑥)≥𝜆^{−𝛽2(𝜆)}𝐼_𝑘(𝑥),𝐼_𝑘(𝜆𝑥)≤𝜆^{−𝛽3(𝜆)}𝐼_𝑘(𝑥),𝑘=1,2,…,𝑚, (3)(𝐻₄)^′∑^𝑚_𝑘=1[−2𝐼_𝑘(1/2)+(1+𝑡_𝑘)𝐼_𝑘(1/2)]>0.

2. Preliminaries

In this section, we state some definitions, notations, and known results. For convenience of readers, we suggest that one refers to [30] and references therein for details.

Suppose that 𝐸 is a real Banach space which is partially ordered by a cone 𝑃⊂𝐸. That is, 𝑥≤𝑦 if and only if 𝑦−𝑥∈𝑃. By 𝜃 we denote the zero element of 𝐸. Recall that a nonempty closed convex set 𝑃⊂𝐸 is called a cone if it satisfies (i) 𝑥∈𝑃,𝜆≥0⇒𝜆𝑥∈𝑃, (ii) 𝑥∈𝑃,−𝑥∈𝑃⇒𝑥=𝜃.

Moreover, 𝑃 is called normal if there exists a constant 𝑁>0 such that, for all 𝑥,𝑦∈𝐸,𝜃≤𝑥≤𝑦 implies ‖𝑥‖≤𝑁‖𝑦‖. In this case 𝑁 is called the normality constant of 𝑃. We say that an operator 𝐴∶𝐸→𝐸 is increasing (decreasing) if 𝑥≤𝑦 implies 𝐴𝑥≤𝐴𝑦(𝐴𝑥≥𝐴𝑦).

For all 𝑥,𝑦∈𝐸, the notation 𝑥∼𝑦 means that there exist 𝜆>0 and 𝜇>0 such that 𝜆𝑥≤𝑦≤𝜇𝑥. Clearly, ∼ is an equivalence relation. Given ℎ>𝜃(i.e.,ℎ≥𝜃 and ℎ≠𝜃), we denote by 𝑃_ℎ the set 𝑃_ℎ={𝑥∈𝐸∣𝑥∼ℎ}. Clearly, 𝑃_ℎ⊂𝑃 is convex and 𝜆𝑃_ℎ=𝑃_ℎ for all 𝜆>0.

We now present a fixed point theorem of generalized concave operators which will be used in the latter proof. See [30] for further information.

Theorem 1 (from [30, Lemma 2.1, and Theorem 2.1]). Let ℎ>𝜃, and let 𝑃 be a normal cone. Assume that (𝐷₁)𝐴∶𝑃→𝑃 is increasing and 𝐴ℎ∈𝑃_ℎ; (𝐷₂) for any 𝑥∈𝑃 and 𝑡∈(0,1), there exists 𝛼(𝑡)∈(𝑡,1) with respect to 𝑡 such that 𝐴(𝑡𝑥)≥𝛼(𝑡)𝐴𝑥. Then (i) there are 𝑢₀,𝑣₀∈𝑃_ℎ and 𝑟∈(0,1) such that 𝑟𝑣₀≤𝑢₀<𝑣₀,𝑢₀≤𝐴𝑢₀≤𝐴𝑣₀≤𝑣₀;  (ii) operator equation 𝑥=𝐴𝑥 has a unique solution in 𝑃_ℎ.

Remark 2. An operator 𝐴 is said to be generalized concave if 𝐴 satisfies condition (𝐷₂).
In what follows, for the sake of convenience, let PC[𝐽,𝐑]={𝑥∣𝑥∶𝐽→𝐑, 𝑥(𝑡) be continuous at 𝑡≠𝑡_𝑘 and left continuous at 𝑡=𝑡_𝑘, 𝑥(𝑡⁺_𝑘) exists, 𝑘=1,2,…,𝑚}, and let PC¹[𝐽,𝐑]={𝑥∈PC[𝐽,𝐑]∣𝑥^′(𝑡) be continuous at 𝑡≠𝑡_𝑘 and left continuous at 𝑡=𝑡_𝑘, 𝑥^′(𝑡⁺_𝑘) exists, 𝑘=1,2,…,𝑚}. Evidently, PC[𝐽,𝐑] is a Banach space with the norm ‖𝑥‖_PC=sup{|𝑥(𝑡)|∶𝑡∈𝐽}, and PC¹[𝐽,𝐑] is a Banach space with the norm ‖𝑥‖_PC¹=sup{‖𝑥‖_PC,‖𝑥^′‖_PC}. Let 𝐽^′=𝐽⧵{𝑡₁,𝑡₂,…,𝑡_𝑚}.

Definition 3. A function 𝑥∈PC¹[𝐽,𝐑]⋂𝐶²[𝐽^′,𝐑] is called a solution of the problem (1) if it satisfies problem (1).

Lemma 4. 𝑥∈𝑃𝐶¹[𝐽,𝐑]⋂𝐶²[𝐽^′,𝐑] is a solution of the problem (1) if and only if 𝑥∈𝑃𝐶¹[𝐽,𝐑] is the solution of the following integral equation: 𝑥(𝑡)=−¹₀𝐺(𝑡,𝑠)𝑓(𝑠,𝑥(𝑠))𝑑𝑠−(1+𝑡)^𝑚_𝑘=1𝐼_𝑘𝑥𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘, (4) where 𝐺()=𝑠()𝑡()𝑡,𝑠1+𝑡,0≤𝑡≤𝑠≤1,1+𝑠,0≤𝑠≤𝑡≤1. (5)

Proof. First suppose that 𝑥∈PC¹[𝐽,𝐑]⋂𝐶²[𝐽^′,𝐑] is a solution of the problem (1). It is easy to see by integration of (1) that 𝑥^′(𝑡)=𝑥^′(0)−^𝑡₀𝑓(𝑠,𝑥(𝑠))𝑑𝑠+_{0<𝑡𝑘<𝑡}𝑥^′𝑡⁺_𝑘−𝑥^′𝑡_𝑘=𝑥^′(0)−^𝑡₀𝑓(𝑠,𝑥(𝑠))𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘. (6) Integrate again, we can get 𝑥(𝑡)=𝑥(0)+𝑥^′(0)𝑡−^𝑡₀()𝑓((𝑠+𝑡−𝑠𝑠,𝑥))𝑑𝑠_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘𝑡−𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑥𝑡⁺_𝑘−𝑥𝑡_𝑘=𝑥(0)+𝑥^′(0)𝑡−^𝑡₀()𝑓((𝑠+𝑡−𝑠𝑠,𝑥))𝑑𝑠_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘𝑡−𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘. (7) Letting 𝑡=1 in (6) and (7), we find 𝑥^′(1)=𝑥^′(0)−¹₀𝑓(𝑠,𝑥(𝑠))𝑑𝑠+^𝑚_𝑘=1𝐼_𝑘𝑥𝑡_𝑘,𝑥(1)=𝑥(0)+𝑥^′(0)−¹₀()𝑓((𝑠+1−𝑠𝑠,𝑥))𝑑𝑠_𝑚𝑘=1𝐼_𝑘𝑥𝑡_𝑘1−𝑡_𝑘+^𝑚_𝑘=1𝐼_𝑘𝑥𝑡_𝑘. (8) From the boundary conditions 𝑥(0)=𝑥^′(0), and 𝑥(1)=𝑥^′(1), we have 𝑥(1)=𝑥(0)−¹₀𝑓(𝑠,𝑥(𝑠))𝑑𝑠+^𝑚_𝑘=1𝐼_𝑘𝑥𝑡_𝑘,𝑥(1)=2𝑥(0)−¹₀()𝑓((𝑠+1−𝑠𝑠,𝑥))𝑑𝑠_𝑚𝑘=1𝐼_𝑘𝑥𝑡_𝑘1−𝑡_𝑘+^𝑚_𝑘=1𝐼_𝑘𝑥𝑡_𝑘. (9) Then we obtain 𝑥(0)=−¹₀𝑠𝑓(𝑠,𝑥(𝑠))𝑑𝑠+^𝑚_𝑘=1𝐼_𝑘𝑥𝑡_𝑘−^𝑚_𝑘=1𝐼_𝑘𝑥𝑡_𝑘1−𝑡_𝑘−^𝑚_𝑘=1𝐼_𝑘𝑥𝑡_𝑘. (10) Substituting (10) into (7), we have 𝑥(𝑡)=−(1+𝑡)¹₀𝑠𝑓(𝑠,𝑥(𝑠))𝑑𝑠−^𝑡₀()𝑓((𝑠−()𝑡−𝑠𝑠,𝑥))𝑑𝑠1+𝑡^𝑚_𝑘=1𝐼_𝑘𝑥𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘=−¹₀𝐺(𝑡,𝑠)𝑓(𝑠,𝑥(𝑠))𝑑𝑠−(1+𝑡)^𝑚_𝑘=1𝐼_𝑘𝑥𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘. (11) Thus, the proof of sufficient is complete.
Conversely, if 𝑥 is a solution of (4). Then we can easily get Δ𝑥|_{𝑡=𝑡𝑘}=𝑥(𝑡⁺_𝑘)−𝑥(𝑡⁻_𝑘)=𝐼_𝑘(𝑥(𝑡_𝑘)). Direct differentiation of (4) implies that, for 𝑡≠𝑡_𝑘, 𝑥^′(𝑡)=−^𝑡₀()𝑓((𝑠()𝑓((𝑡−1+𝑠𝑠,𝑥))𝑑𝑠−𝑡1+𝑡𝑡,𝑥))_1𝑡((𝑠()𝑓((𝑡−𝑠𝑓𝑠,𝑥))𝑑𝑠+𝑡1+𝑡𝑡,𝑥))_𝑚𝑘=1𝐼_𝑘𝑥𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘+^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘. (12) Further 𝑥^′′(𝑡)=−𝑓(𝑡,𝑥(𝑡)),Δ𝑥^′_{𝑡=𝑡𝑘}=𝑥^′𝑡⁺_𝑘−𝑥^′𝑡⁻_𝑘=𝐼_𝑘𝑥𝑡_𝑘. (13) So 𝑥∈𝐶²[𝐽^′,𝐑] and it is easy to verify that 𝑥(0)=𝑥^′(0),𝑥(1)=𝑥^′(1), and the lemma is proved.

Define an operator 𝐴∶PC[𝐽,𝐑]→PC[𝐽,𝐑] by 𝐴𝑥(𝑡)=−¹₀𝐺(𝑡,𝑠)𝑓(𝑠,𝑥(𝑠))𝑑𝑠−(1+𝑡)^𝑚_𝑘=1𝐼_𝑘𝑥𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘. (14)

Lemma 5. 𝑥∈𝑃𝐶¹[𝐽,𝐑]⋂𝐶²[𝐽^′,𝐑] is a solution of problem (1) if and only if 𝑥∈𝑃𝐶¹[𝐽,𝐑] is a fixed point of the operator 𝐴.

3. Existence and Uniqueness of Positive Solutions for Problem (1)

In this section, we apply Theorem 1 to study the problem (1), and we obtain a new result on the existence and uniqueness of positive solutions. The method used in this paper is new to the literature and so is the existence and uniqueness result to the second-order impulsive differential equations. This is also the main motivation for the study of (1) in the present work.

Set 𝑃={𝑢∈PC[𝐽,𝐑]∣𝑢(𝑡)≥0,𝑡∈𝐽}, the standard cone. It is clear that 𝑃 is a normal cone in PC[𝐽,𝐑] and the normality constant is 1. Our main result is summarized in the following theorem.

Theorem 6. Assume that (H₁)–(H₄) hold. Then  (i) there exist 𝑢₀,𝑣₀∈𝑃_ℎ such that 𝑢₀(𝑡)≤−¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢₀(𝑠)𝑑𝑠−(1+𝑡)^𝑚_𝑘=1𝐼_𝑘𝑢₀𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑢₀𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝑢₀𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑢₀𝑡_𝑘𝑣,𝑡∈𝐽,₀(𝑡)≥−¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑣₀(𝑠)𝑑𝑠−(1+𝑡)^𝑚_𝑘=1𝐼_𝑘𝑣₀𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑣₀𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝑣₀𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑣₀𝑡_𝑘,𝑡∈𝐽, (15) (ii) the nonlinear impulsive problem (1) has a unique positive solution 𝑥^∗ in 𝑃_ℎ⋂𝑃𝐶¹[𝐽,𝐑], where ℎ(𝑡)=(1/2)(𝑡²+𝑡+1),𝑡∈[0,1].

Remark 7. It is easy to see that 1/2≤ℎ(𝑡)≤3/2,𝑡∈[0,1].

Proof of Theorem 6. Firstly, we show that 𝐴∶𝑃→𝑃 is increasing, generalized concave. For any 𝑥∈𝑃, 𝐴𝑥(𝑡)=−¹₀𝐺(𝑡,𝑠)𝑓(𝑠,𝑥(𝑠))𝑑𝑠−(1+𝑡)^𝑚_𝑘=1𝐼_𝑘𝑥𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘=−¹₀𝐺(𝑡,𝑠)𝑓(𝑠,𝑥(𝑠))𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘−(1+𝑡)_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘+_{𝑡≤𝑡𝑘<1}𝐼_𝑘𝑥𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘=−¹₀𝐺()𝑓((𝑠−𝑡,𝑠𝑠,𝑥))𝑑𝑠𝑡_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘+(1+𝑡)_{𝑡≤𝑡𝑘<1}𝐼_𝑘𝑥𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘. (16) From (𝐻₁)and(𝐻₂), we know that 𝑓(𝑡,𝑥(𝑠))≤0,𝐼_𝑘(𝑥(𝑡_𝑘))≤0, and 𝐼_𝑘(𝑥(𝑡_𝑘))≥0. So we have 𝐴𝑥(𝑡)≥0 for 𝑥∈𝑃. By Lemma 4, 𝐴∶𝑃→𝑃. It follows from (𝐻₁)and(𝐻₂) that 𝐴∶𝑃→𝑃 is increasing. Now we prove that 𝐴∶𝑃→𝑃 is generalized concave. Set 𝛼(𝑡)=min{𝛼₁(𝑡),𝛼₂(𝑡),𝛼₃(𝑡)},𝑡∈(0,1). Then 𝛼(𝑡)∈(𝑡,1). For any 𝑥∈𝑃 and 𝜆∈(0,1), from (𝐻₃) we have 𝐴(𝜆𝑥)(𝑡)=−¹₀𝐺()𝑓((𝑠−𝑡,𝑠𝑠,𝜆𝑥))𝑑𝑠𝑡_{0<𝑡𝑘<𝑡}𝐼_𝑘𝜆𝑥𝑡_𝑘+(1+𝑡)_{𝑡≤𝑡𝑘<1}𝐼_𝑘𝜆𝑥𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝜆𝑥𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝜆𝑥𝑡_𝑘≥𝛼₁(𝜆)−¹₀𝐺(𝑡,𝑠)𝑓(𝑠,𝑥(𝑠))𝑑𝑠+𝛼₂(𝜆)×−𝑡_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘−(1+𝑡)_{𝑡≤𝑡𝑘<1}𝐼_𝑘𝑥𝑡_𝑘+𝛼₃(𝜆)_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘≥𝛼(𝜆)−¹₀𝐺()𝑓((𝑠−𝑡,𝑠𝑠,𝑥))𝑑𝑠𝑡_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘+(1+𝑡)_{𝑡≤𝑡𝑘<1}𝐼_𝑘𝑥𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘=𝛼(𝜆)𝐴𝑥(𝑡). (17) That is, 𝐴(𝜆𝑥)≥𝛼(𝜆)𝐴𝑥,𝑥∈𝑃,𝜆∈(0,1).
Secondly, we prove that 𝐴ℎ∈𝑃_ℎ. Set 𝑟₁=min_𝑡∈[0,1]1−𝑓𝑡,2,𝑟₂=max_[]𝑡∈0,13−𝑓𝑡,2. (18) Then from (𝐻₁), we have 𝑟₂≥𝑟₁>0. Further, from (𝐻₁),(𝐻₂), and (𝐻₄), 𝐴ℎ(𝑡)=−¹₀𝐺(𝑡,𝑠)𝑓(𝑠,ℎ(𝑠))𝑑𝑠−(1+𝑡)^𝑚_𝑘=1𝐼_𝑘ℎ𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐼_𝑘ℎ𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘ℎ𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘ℎ𝑡_𝑘≥−¹₀𝐺()𝑓1𝑡,𝑠𝑠,2𝑑𝑠≥𝑟₁¹₀𝐺(𝑡,𝑠)𝑑𝑠=𝑟₁ℎ(𝑡),𝐴ℎ(𝑡)=−¹₀𝐺(𝑡,𝑠)𝑓(𝑠,ℎ(𝑠))𝑑𝑠−(1+𝑡)^𝑚_𝑘=1𝐼_𝑘ℎ𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐼_𝑘ℎ𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘ℎ𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘ℎ𝑡_𝑘≤−¹₀𝐺()𝑓3𝑡,𝑠𝑠,2𝑑𝑠−(1+𝑡)^𝑚_𝑘=1𝐼_𝑘ℎ𝑡_𝑘+^𝑚_𝑘=11−𝑡_𝑘𝐼_𝑘ℎ𝑡_𝑘+2^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘ℎ𝑡_𝑘≤𝑟₂ℎ(𝑡)+2−^𝑚_𝑘=1𝐼_𝑘32+^𝑚_𝑘=11+𝑡_𝑘𝐼_𝑘32=𝑟₂ℎ(𝑡)+^𝑚_𝑘=1−2𝐼_𝑘32+1+𝑡_𝑘𝐼_𝑘32≤𝑟₂ℎ(𝑡)+2^𝑚_𝑘=1−2𝐼_𝑘32+1+𝑡_𝑘𝐼_𝑘32(𝑡)=𝑟⋅ℎ₂+2^𝑚_𝑘=1−2𝐼_𝑘32+1+𝑡_𝑘𝐼_𝑘32ℎ(𝑡). (19) Hence, 𝑟₁ℎ≤𝐴ℎ≤𝑟₂+2^𝑚_𝑘=1−2𝐼_𝑘32+1+𝑡_𝑘𝐼_𝑘32ℎ. (20) That is, 𝐴ℎ∈𝑃_ℎ. Finally, an application of Theorem 1 implies that (i) there are 𝑢₀,𝑣₀∈𝑃_ℎ such that 𝑢₀≤𝐴𝑢₀,𝐴𝑣₀≤𝑣₀, (ii) operator equation 𝑥=𝐴𝑥 has a unique solution in 𝑃_ℎ. That is, 𝑢₀(𝑡)≤−¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢₀(𝑠)𝑑𝑠−(1+𝑡)^𝑚_𝑘=1𝐼_𝑘𝑢₀𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑢₀𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝑢₀𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑢₀𝑡_𝑘𝑣,𝑡∈𝐽,₀(𝑡)≥−¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑣₀(𝑠)−()𝑑𝑠1+𝑡^𝑚_𝑘=1𝐼_𝑘𝑣₀𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑣₀𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝑣₀𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑣₀𝑡_𝑘,𝑡∈𝐽, (21) and the problem (1) has a unique positive solution 𝑥^∗ in 𝑃_ℎ. Moreover, from Lemmas 4 and 5 we know that 𝑥^∗∈PC¹[𝐽,𝐑]. Evidently, 𝑥^∗ is a positive solution of the problem (1).

Theorem 8. Assume that (H₁)′–(H₄)′ hold. Then  (i) there exist 𝑢₀,𝑣₀∈𝑃_ℎ such that 𝑢₀(𝑡)≤−¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢₀(𝑠)𝑑𝑠−(1+𝑡)^𝑚_𝑘=1𝐼_𝑘𝑢₀𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑢₀𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝑢₀𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑢₀𝑡_𝑘𝑣,𝑡∈𝐽,₀(𝑡)≥−¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑣₀(𝑠)𝑑𝑠−(1+𝑡)^𝑚_𝑘=1𝐼_𝑘𝑣₀𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑣₀𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝑣₀𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑣₀𝑡_𝑘,𝑡∈𝐽, (22)where 𝑢₀(𝑡)=−¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢₀(𝑠)𝑑𝑠−(1+𝑡)^𝑚_𝑘=1𝐼_𝑘𝑢₀𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑢₀𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝑢₀𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑢₀𝑡_𝑘,𝑡∈𝐽,𝑣₀(𝑡)=−¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑣₀(𝑠)𝑑𝑠−(1+𝑡)^𝑚_𝑘=1𝐼_𝑘𝑣₀𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑣₀𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝑣₀𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑣₀𝑡_𝑘,𝑡∈𝐽, (23) (ii) the nonlinear impulsive problem (1) has a unique positive solution 𝑥^∗ in 𝑃_ℎ⋂𝑃𝐶¹[𝐽,𝐑], where ℎ(𝑡)=(1/2)(𝑡²+𝑡+1), 𝑡∈[0,1].

Proof. From the proof of Theorem 6, for any 𝑥∈𝑃, 𝐴𝑥(𝑡)=−¹₀𝐺()𝑓((𝑠−𝑡,𝑠𝑠,𝑥))𝑑𝑠𝑡_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘+(1+𝑡)_{𝑡≤𝑡𝑘<1}𝐼_𝑘𝑥𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘. (24) From (𝐻₁)^′and(𝐻₂)^′, we know that 𝐴𝑥(𝑡)≥0, 𝑡∈[0,1]. By Lemma 4, 𝐴∶𝑃→𝑃. It follows from (𝐻₁)^′and(𝐻₂)^′ that 𝐴∶𝑃→𝑃 is decreasing. Set 𝛽(𝑡)=max{𝛽₁(𝑡),𝛽₂(𝑡),𝛽₃(𝑡)}, 𝑡∈(0,1). Then 𝛽(𝑡)∈(0,1). For any 𝑥∈𝑃 and 𝜆∈(0,1), from (𝐻₃)^′ we have 𝐴(𝜆𝑥)(𝑡)=−¹₀𝐺()𝑓((𝑠−𝑡,𝑠𝑠,𝜆𝑥))𝑑𝑠𝑡_{0<𝑡𝑘<𝑡}𝐼_𝑘𝜆𝑥𝑡_𝑘+(1+𝑡)_{𝑡≤𝑡𝑘<1}𝐼_𝑘𝜆𝑥𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝜆𝑥𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝜆𝑥𝑡_𝑘≤𝜆^{−𝛽1(𝜆)}−¹₀𝐺(𝑡,𝑠)𝑓(𝑠,𝑥(𝑠))𝑑𝑠+𝜆^{−𝛽2(𝜆)}×−𝑡_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘−(1+𝑡)_{𝑡≤𝑡𝑘<1}𝐼_𝑘𝑥𝑡_𝑘+𝜆^{−𝛽3(𝜆)}_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘≤𝜆^{−𝛽(𝜆)}−¹₀𝐺()𝑓((𝑠−𝑡𝑡,𝑠𝑠,𝑥))𝑑𝑠_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘+(1+𝑡)_{𝑡≤𝑡𝑘<1}𝐼_𝑘𝑥𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘=𝜆^{−𝛽(𝜆)}𝐴𝑥(𝑡). (25) That is, 𝐴(𝜆𝑥)≤𝜆^{−𝛽(𝜆)}𝐴𝑥,𝑥∈𝑃,𝜆∈(0,1). Further, for 𝜆∈(0,1) and 𝑥∈𝑃, 1𝐴𝑥=𝐴𝜆⋅𝜆𝑥≤𝜆^{−𝛽(𝜆)}𝐴1𝜆𝑥. (26) So we obtain 𝐴((1/𝜆)𝑥)≥𝜆^𝛽(𝜆)𝐴𝑥,𝑥∈𝑃,𝜆∈(0,1). Consequently, 𝐴²∶𝑃→𝑃 is increasing, and, for 𝑥∈𝑃,𝜆∈(0,1), 𝐴²(𝜆𝑥)=𝐴(𝐴(𝜆𝑥))≥𝐴𝜆^{−𝛽(𝜆)}1𝐴𝑥=𝐴𝜆^𝛽(𝜆)≥𝜆𝐴𝑥_{𝛽(𝜆)𝛽(𝜆𝛽(𝜆))}𝐴²𝑥≥𝜆^𝛽(𝜆)𝐴²𝑥. (27)
Let 𝛼(𝑡)=𝑡^𝛽(𝑡),𝑡∈(0,1). Then 𝛼(𝑡)∈(𝑡,1) and 𝐴²(𝜆𝑥)≥𝛼(𝜆)𝐴²𝑥,𝑥∈𝑃,𝜆∈(0,1). So the operator 𝐴²∶𝑃→𝑃 is generalized concave. Next we prove that 𝐴²ℎ∈𝑃_ℎ. Set 𝑟₁=min_𝑡∈[0,1]3−𝑓𝑡,2,𝑟₂=max_[]𝑡∈0,11−𝑓𝑡,2. (28) Then from (𝐻₁)^′, we have 𝑟₂≥𝑟₁>0. Further, from (𝐻₁)^′,(𝐻₂)^′, and (𝐻₄)^′, 𝐴ℎ(𝑡)=−¹₀𝐺()𝑓((𝑠−()𝑡,𝑠𝑠,ℎ))𝑑𝑠1+𝑡^𝑚_𝑘=1𝐼_𝑘ℎ𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐼_𝑘ℎ𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘ℎ𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘ℎ𝑡_𝑘≥−¹₀𝐺()𝑓3𝑡,𝑠𝑠,2𝑑𝑠≥𝑟₁¹₀𝐺(𝑡,𝑠)𝑑𝑠=𝑟₁ℎ(𝑡),𝐴ℎ(𝑡)=−¹₀𝐺()𝑓((𝑠−()𝑡,𝑠𝑠,ℎ))𝑑𝑠1+𝑡^𝑚_𝑘=1𝐼_𝑘ℎ𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐼_𝑘ℎ𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘ℎ𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘ℎ𝑡_𝑘≤−¹₀𝐺()𝑓1𝑡,𝑠𝑠,2𝑑𝑠−(1+𝑡)^𝑚_𝑘=1𝐼_𝑘ℎ𝑡_𝑘+^𝑚_𝑘=11−𝑡_𝑘𝐼_𝑘ℎ𝑡_𝑘+2^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘ℎ𝑡_𝑘≤𝑟₂ℎ(𝑡)+2−^𝑚_𝑘=1𝐼_𝑘12+^𝑚_𝑘=11+𝑡_𝑘𝐼_𝑘12=𝑟₂ℎ(𝑡)+^𝑚_𝑘=1−2𝐼_𝑘12+1+𝑡_𝑘𝐼_𝑘12≤𝑟₂ℎ(𝑡)+2^𝑚_𝑘=1−2𝐼_𝑘12+1+𝑡_𝑘𝐼_𝑘12(𝑡)=𝑟⋅ℎ₂+2^𝑚_𝑘=1−2𝐼_𝑘12+1+𝑡_𝑘𝐼_𝑘12ℎ(𝑡). (29) Hence, 𝑟₁ℎ≤𝐴ℎ≤𝑟₂+2^𝑚_𝑘=1−2𝐼_𝑘12+1+𝑡_𝑘𝐼_k12ℎ. (30) We can choose a sufficiently small number 𝑟₀∈(0,1) such that 𝑟₀≤𝑟₁<𝑟₂+2^𝑚_𝑘=1−2𝐼_𝑘12+1+𝑡_𝑘𝐼_𝑘12≤1𝑟₀. (31) Then we get 𝑟₀ℎ≤𝐴ℎ≤(1/𝑟₀)ℎ. Further, 𝐴²ℎ=𝐴(𝐴ℎ)≤𝐴𝑟₀ℎ≤𝑟₀^{−𝛽(𝑟0)}1𝐴ℎ≤𝑟₀^{1+𝛽(𝑟0)}𝐴ℎ,₂()1ℎ=𝐴𝐴ℎ≥𝐴𝑟₀ℎ≥𝑟₀^𝛽(𝑟0)𝐴ℎ≥𝑟₀^{1+𝛽(𝑟0)}ℎ. (32) That is, 𝐴²ℎ∈𝑃_ℎ. Finally, an application of Theorem 1 implies that (i) there are 𝑢₀,𝑣₀∈𝑃_ℎ such that 𝑢₀≤𝐴²𝑢₀,𝐴²𝑣₀≤𝑣₀, (ii) operator equation 𝑥=𝐴²𝑥 has a unique solution in 𝑃_ℎ. Let 𝑢₀=𝐴𝑢₀,𝑣₀=𝐴𝑣₀. Then, 𝑢₀≤𝐴𝑢₀,𝐴𝑣₀≤𝑣₀. That is, 𝑢₀(𝑡)≤−¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢₀(𝑠)−()𝑑𝑠1+𝑡^𝑚_𝑘=1𝐼_𝑘𝑢₀𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑢₀𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝑢₀𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑢₀𝑡_𝑘𝑣,𝑡∈𝐽,₀(𝑡)≥−¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑣₀(𝑠)−()𝑑𝑠1+𝑡^𝑚_𝑘=1𝐼_𝑘𝑣₀𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑣₀𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝑣₀𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑣₀𝑡_𝑘,𝑡∈𝐽. (33) Next we show that 𝑥^∗ is the unique fixed point of 𝐴 in 𝑃_ℎ. In view of 𝐴²(𝐴𝑥^∗)=𝐴(𝐴²𝑥^∗)=𝐴𝑥^∗, and by the uniqueness of solutions for the operator equation 𝑥=𝐴²𝑥, we have 𝐴𝑥^∗=𝑥^∗. Suppose that 𝑦^∗ is another fixed point of 𝐴 in 𝑃_ℎ. Then 𝐴²𝑦^∗=𝐴(𝐴(𝑦^∗))=𝐴𝑦^∗=𝑦^∗. Hence, by the uniqueness of solutions for the operator equation 𝑥=𝐴²𝑥, we obtain 𝑥^∗=𝑦^∗. So the problem (1) has a unique positive solution 𝑥^∗ in 𝑃_ℎ. Moreover, from Lemmas 4 and 5 we know that 𝑥^∗∈PC¹[𝐽,𝐑]. Evidently, 𝑥^∗ is a positive solution of the problem (1).

Remark 9. Here, we provide an alternative approach to study the same type of problems under different conditions. Our result can guarantee the existence of a unique positive solution without supposing the existence of upper-lower solutions. The method used in this paper is relatively new to the literature and so is the existence and uniqueness result to the impulsive differential equations.
In the following we consider two special cases of the problem (1): −𝑥^′′(𝑡)=𝑓(𝑡,𝑥(𝑡)),𝑡≠𝑡_𝑘|,𝑘=1,2,…,𝑚,Δ𝑥_{𝑡=𝑡𝑘}=𝐼_𝑘𝑥𝑡_𝑘𝑥(0),𝑘=1,2,…,𝑚,=𝑥^′(0),𝑥(1)=𝑥^′(1),−𝑥^′′(𝑡)=𝑓(𝑡,𝑥(𝑡)),𝑡≠𝑡_𝑘,𝑘=1,2,…,𝑚,Δ𝑥^′_{𝑡=𝑡𝑘}=𝐼_𝑘𝑥𝑡_𝑘𝑥(0),𝑘=1,2,…,𝑚,=𝑥^′(0),𝑥(1)=𝑥^′(1), (34)(35) where 𝑓∈𝐶[𝐽×𝐑,𝐑], 𝐼_𝑘, 𝐼_𝑘∈𝐶[𝐑,𝐑], 𝑘=1,2,…,𝑚.
From Theorems 6 and 8, we have the following conclusions.

Corollary 10. Assume that (𝐻₁) holds and(𝐺₁)𝐼_𝑘(0)≤0 and 𝐼_𝑘(𝑥) is decreasing in 𝑥∈[0,∞),𝑘=1,2,…,𝑚 with ^𝑚_𝑘=1𝐼_𝑘32<0, (36)(𝐺₂) for any 𝜆∈(0,1),𝑡∈[0,1] and 𝑥≥0, there exist 𝛼₁(𝜆),𝛼₂(𝜆)∈(𝜆,1) such that 𝑓(𝑡,𝜆𝑥)≤𝛼₁(𝜆)𝑓(),𝐼𝑡,𝑥_𝑘(𝜆𝑥)≤𝛼₂(𝜆)𝐼_𝑘(𝑥),𝑘=1,2,…,𝑚. (37)Then  (i) there exist 𝑢₀,𝑣₀∈𝑃_ℎ such that 𝑢₀(𝑡)≤−¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢₀(𝑠)−()𝑑𝑠1+𝑡^𝑚_𝑘=1𝐼_𝑘𝑢₀𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑢₀𝑡_𝑘𝑣,𝑡∈𝐽,₀(𝑡)≥−¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑣₀(s)−()𝑑𝑠1+𝑡^𝑚_𝑘=1𝐼_𝑘𝑣₀𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑣₀𝑡_𝑘,𝑡∈𝐽, (38) (ii) the nonlinear impulsive problem (34) has a unique positive solution 𝑥^∗ in 𝑃_ℎ, where ℎ(𝑡)=(1/2)(𝑡²+𝑡+1), 𝑡∈[0,1], and 𝐺(𝑡,𝑠) is given as in Lemma 4.

Corollary 11. Assume that (𝐻₁)^′ hold and(𝐺₁)^′ 𝐼_𝑘(𝑥)≤0 for [0,∞) and 𝐼_𝑘(𝑥) is increasing in 𝑥∈[0,∞),𝑘=1,2,…,𝑚 with ^𝑚_𝑘=1𝐼_𝑘12<0, (39)(𝐺₂)^′ for any 𝜆∈(0,1),𝑡∈[0,1] and 𝑥≥0, there exist 𝛽₁(𝜆),𝛽₂(𝜆)∈(0,1) such that 𝑓(𝑡,𝜆𝑥)≥𝜆^{−𝛽1(𝜆)}𝑓(),𝐼𝑡,𝑥_𝑘(𝜆𝑥)≥𝜆^{−𝛽2(𝜆)}𝐼_𝑘(𝑥),𝑘=1,2,…,𝑚. (40)
Then  (i) there exist 𝑢₀,𝑣₀∈𝑃_ℎ such that 𝑢₀(𝑡)≤−¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢₀(𝑠)𝑑𝑠−(1+𝑡)^𝑚_𝑘=1𝐼_𝑘𝑢₀𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑢₀𝑡_𝑘𝑣,𝑡∈𝐽,₀(𝑡)≥−¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑣₀(𝑠)−()𝑑𝑠1+𝑡^𝑚_𝑘=1𝐼_𝑘𝑣₀𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑣₀𝑡_𝑘,𝑡∈𝐽, (41)where 𝑢₀(𝑡)=−¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢₀(𝑠)−()𝑑𝑠1+𝑡^𝑚_𝑘=1𝐼_𝑘𝑢₀𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑢₀𝑡_𝑘,𝑡∈𝐽,𝑣₀(𝑡)=−¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑣₀(𝑠)−()𝑑𝑠1+𝑡^𝑚_𝑘=1𝐼_𝑘𝑣₀𝑡_𝑘+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑣₀𝑡_𝑘,𝑡∈𝐽, (42) (ii) the nonlinear impulsive problem (34) has a unique positive solution 𝑥^∗ in 𝑃_ℎ, where ℎ(𝑡)=(1/2)(𝑡²+𝑡+1) and  𝑡∈[0,1] and 𝐺(𝑡,𝑠) is given as in Lemma 4.

Corollary 12. Assume that (𝐻₁) holds and(𝐺₃)𝐼_𝑘(0)≥0 and 𝐼_𝑘(𝑥) is increasing in 𝑥∈[0,∞),𝑘=1,2,…,𝑚 with ^𝑚_𝑘=11+𝑡_𝑘𝐼_𝑘32>0, (43)(𝐺₄) for any 𝜆∈(0,1),𝑡∈[0,1] and 𝑥≥0, there exist 𝛼₁(𝜆),𝛼₂(𝜆)∈(𝜆,1) such that 𝑓(𝑡,𝜆𝑥)≤𝛼₁(𝜆)𝑓(𝑡,𝑥),𝐼_𝑘(𝜆𝑥)≥𝛼₂(𝜆)𝐼_𝑘(𝑥),𝑘=1,2,…,𝑚. (44)Then  (i) there exist 𝑢₀,𝑣₀∈𝑃̆_ℎ such that 𝑢₀(𝑡)≤−¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢₀(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝑢₀𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑢₀𝑡_𝑘𝑣,𝑡∈𝐽,₀(𝑡)≥−¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑣₀(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_k𝑣₀𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑣₀𝑡_𝑘,𝑡∈𝐽, (45) (ii) the nonlinear impulsive problem (35) has a unique positive solution 𝑥^∗ in 𝑃̆_ℎ⋂𝑃𝐶¹[𝐽,𝐑], where ℎ(𝑡)=(1/2)(𝑡²+𝑡+1), 𝑡∈[0,1], 𝑃̆={𝑥∈𝐶[𝐽,𝐑]∣𝑥(𝑡)≥0,𝑡∈𝐽}, and 𝑃𝐶¹[𝐽,𝐑]={𝑥∈𝐶[𝐽,𝐑]∣𝑥^′(𝑡) is continuous at 𝑡≠𝑡_𝑘 and left continuous at 𝑡=𝑡_𝑘, 𝑥^′(𝑡⁺_𝑘) exists, 𝑘=1,2,…,𝑚}.

Corollary 13. Assume that (𝐻₁)^′ hold and (𝐺₃)^′𝐼_𝑘(𝑥)≥0 for [0,∞) and 𝐼_𝑘(𝑥) is decreasing in 𝑥∈[0,∞), 𝑘=1,2,…,𝑚 with ^𝑚_𝑘=11+𝑡_𝑘𝐼_𝑘12>0, (46) (𝐺₄)^′ for any 𝜆∈(0,1), 𝑡∈[0,1], and 𝑥≥0, there exist 𝛽₁(𝜆),𝛽₂(𝜆)∈(0,1) such that 𝑓(𝑡,𝜆𝑥)≥𝜆^{−𝛽1(𝜆)}𝑓(𝑡,𝑥),𝐼_𝑘(𝜆𝑥)≤𝜆^{−𝛽2(𝜆)}𝐼_𝑘(𝑥),𝑘=1,2,…,𝑚. (47)
Then  (i) there exist 𝑢₀,𝑣₀∈𝑃̆_ℎ such that 𝑢₀(𝑡)≤−¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢₀(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝑢₀𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑢₀𝑡_𝑘𝑣,𝑡∈𝐽,₀(𝑡)≥−¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑣₀(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝑣₀𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑣₀𝑡_𝑘,𝑡∈𝐽, (48)where 𝑢₀(𝑡)=−¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢₀(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝑢₀𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑢₀𝑡_𝑘,𝑡∈𝐽,𝑣₀(𝑡)=−¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑣₀(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝑡−𝑡_𝑘𝐼_𝑘𝑣₀𝑡_𝑘+(1+𝑡)^𝑚_𝑘=1𝑡_𝑘𝐼_𝑘𝑣₀𝑡_𝑘,𝑡∈𝐽, (49) (ii) the nonlinear impulsive problem (35) has a unique positive solution 𝑥^∗ in 𝑃̆_ℎ⋂PC¹[𝐽,𝐑], where ℎ(𝑡)=(1/2)(𝑡²+𝑡+1), 𝑡∈[0,1], 𝑃̆={𝑥∈𝐶[𝐽,𝐑]∣𝑥(𝑡)≥0,𝑡∈𝐽}, and PC¹[𝐽,𝐑]={𝑥∈𝐶[𝐽,𝐑]∣𝑥^′(𝑡) is continuous at 𝑡≠𝑡_𝑘 and left continuous at 𝑡=𝑡_𝑘,𝑥^′(𝑡⁺_𝑘) exists, 𝑘=1,2,…,𝑚}.

4. An Example

To illustrate how our main results can be used in practice we present an example.

Example 1. Consider the following boundary value problem: −𝑥^′′(𝑡)=−√1𝑡𝑥+4,𝑡∈𝐽,𝑡≠2,Δ𝑥|_𝑡=1/2=−𝑥^1/312,Δ𝑥^′_𝑡=1/2=𝑥^1/412,𝑥(0)=𝑥^′(0),𝑥(1)=𝑥^′(1). (50)

Conclusion. BVP (50) has a unique positive solution in 𝑃_ℎ, where ℎ(𝑡)=(1/2)(𝑡²+𝑡+1), 𝑡∈[0,1].

Proof. BVP (50) can be regarded as a BVP of the form (1), where 𝑡₁=(1/2), 𝑓(𝑡,𝑥)=−√𝑡𝑥+4, 𝐼₁(𝑥)=−𝑥^1/3, and 𝐼₁(𝑥)=𝑥^1/4. It is not difficult to see that the conditions (𝐻₁),(𝐻₂), and (𝐻₄) hold. In addition, let 𝛼₁(𝜆)=𝜆^1/2, 𝛼₂(𝜆)=𝜆^1/3, and 𝛼₃(𝜆)=𝜆^1/4. Then, the condition (𝐻₃) of Theorem 6 holds. Hence, by Theorem 6, the conclusion follows, and the proof is complete.

Remark 14. Example 1 implies that there is a large number of functions that satisfy the conditions of Theorem 6. In addition, the conditions of Theorem 6 are also easy to check.

Acknowledgments

The research was supported by the Fund of National Natural Science of China (61250011, 11201272) and the Science Foundation of Shanxi Province (2012011004-4, 2010021002-1).