Upper and Lower Solutions for 𝑚-Point Impulsive BVP with One-Dimensional 𝑝-Laplacian

Abstract

Upper and lower solutions theories are established for a kind of 𝑚-point impulsive boundary value problems with 𝑝-Laplacian. By using such techniques and the Schauder fixed point theorem, the existence of solutions and positive solutions is obtained. Nagumo conditions play an important role in the nonlinear term involved with the first-order derivatives.

1. Introduction

In this paper, we study the following 𝑚-point impulsive boundary value problem with one-dimensional 𝑝-Laplacian: 𝜙_𝑝𝑢^′(𝑡)^′+𝑞(𝑡)𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡)=0,𝑡∈𝐽^′,Δ𝑢|_{𝑡=𝑡𝑘}=𝐼_1𝑘𝑢𝑡_𝑘,𝑘=1,2,…,𝑛,Δ𝜙_𝑝𝑢^′_{𝑡=𝑡𝑘}=𝐼_2𝑘𝑢𝑡_𝑘,𝑢^′𝑡_𝑘𝑢(0)=,𝑘=1,2,…,𝑛,_{𝑚−2𝑖=1}𝑎_𝑖𝑢𝜂_𝑖,𝑢^′(1)=0, (1) where 𝜙_𝑝(𝑠)=|𝑠|^𝑝−2𝑠, 𝑝>1. 𝐽=[0,1], 𝑡_𝑘 (𝑘=1,2,…,𝑛, where 𝑛 is a fixed positive integer) are fixed points with 0<𝑡₁<𝑡₂<⋯<𝑡_𝑛<1. 𝐽^′=𝐽⧵{𝑡₁,𝑡₂,…,𝑡_𝑛}. 0<𝜂₁<𝜂₂<⋯<𝜂_𝑚−2<1, ∑^𝑚−2_𝑖=1𝑎_𝑖<1. Δ𝑢|_{𝑡=𝑡𝑘}=𝑢(𝑡⁺_𝑘)−𝑢(𝑡⁻_𝑘), 𝑘=1,2,…,𝑛, where 𝑢(𝑡⁺_𝑘) and 𝑢(𝑡⁻_𝑘) represent the right-hand limit and left-hand limit of 𝑢(𝑡) at 𝑡=𝑡_𝑘, respectively.

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. [1], Baĭnov and Simeonov [2], Samoilenko and Perestyuk [3], and the references therein. The theory of impulsive differential equations has become an important area of investigation in recent years and is much richer than the corresponding theory of differential equations (see for instance [4–9] and the references therein).

At the same time, it is well known that the method of lower and upper solutions is a powerful tool for proving the existence results for a large class of boundary value problems. For a few of such works, we refer the readers to [10–14].

In [15], Cabada and Pouso considered 𝜙_𝑝𝑢^′(𝑡)^′=𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡),fora.e[],𝑔𝑢(𝑎).𝑡∈𝐼=𝑎,𝑏,𝑢^′(𝑎),𝑢^′(𝑏)=0,𝑢(𝑏)=ℎ(𝑢(𝑎)), (2) by using the upper and lower method, the authors get the existence of solution to the above BVP.

In [16], Lü et al. studied 𝜙_𝑝𝑢^′(𝑡)^′=𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡),fora.e[],𝑢(0)(1).𝑡∈𝐼=0,1=0,𝑢=0, (3) by giving conditions on 𝑓 involving pairs of lower and upper solutions, they get the existence of at least three solutions to the above BVP.

In [12], Shen and Wang studied 𝑥^′′(𝑡)=𝑓𝑡,𝑥(𝑡),𝑥^′(𝑡),for𝑡∈𝐽,𝑡≠𝑡_𝑘,𝛿𝑥𝑡_𝑘=𝐼_𝑘𝑥𝑡_𝑘,𝑘=1,2,…,𝑝,𝛿𝑥^′𝑡_𝑘=𝐽_𝑘𝑥𝑡_𝑘,𝑥^′𝑡_𝑘𝑔𝑥(0),𝑘=1,2,…,𝑝,,𝑥^′(0)=0,ℎ𝑥(1),𝑥^′(1)=0, (4) they prove the existence of solutions to the problem under the assumption that there exist lower and upper solutions associated with the problem.

Motivated by the works mentioned above, in this paper, we considered BVP (1), the main tool is upper and lower method, and the Schäuder fixed point theorem. We not only get the existence of solutions, but also the existence of positive solutions.

The main structure of this paper is as follows. In Section 2, we give the preliminary and present some lemmas in order to prove our main results. Section 4 presents the main theorems of this paper, and at the end of Section 4, we give an example to illustrate our main results.

2. Preliminary

Define 𝑃𝐶[0,1]={𝑢∣𝑢∶[0,1]→𝑅 is continuous at 𝑡≠𝑡_𝑘, 𝑢(𝑡_𝑘)=𝑢(𝑡⁻_𝑘), 𝑢(𝑡⁺_𝑘) exists, left continuous}, and 𝑃𝐶¹[0,1]={𝑢∣𝑢∈𝑃𝐶[0,1], 𝑢^′∶[0,1]→𝑅 is continuous at 𝑡≠𝑡_𝑘, 𝑢^′(𝑡⁺_𝑘) and 𝑢^′(𝑡⁻_𝑘) exists, 𝑢^′(𝑡_𝑘)=𝑢^′(𝑡⁻_𝑘), 𝑘=1,2,…,𝑛}. Then, 𝑃𝐶¹[0,1] is a real Banach space with the norm ‖𝑢‖=max{‖𝑢‖_∞,‖𝑢^′‖_∞}, where ‖𝑢‖_∞=sup_0≤𝑡≤1|𝑢(𝑡)|.

By a solution of (1), we mean a function 𝑢∈𝐸 which satisfies (1), where 𝐸=𝑃𝐶[0,1]∩𝐶²(𝐽^′).

Definition 1. A function 𝛼∈𝐸, with 𝜙_𝑝(𝛼^′)∈𝑃𝐶¹(0,1), will be called a lower solution of (1) if 𝜙_𝑝𝛼^′(𝑡)^′+𝑞(𝑡)𝑓𝑡,𝛼(𝑡),𝛼^′(𝑡)≥0,𝑡∈𝐽^′,Δ𝛼|_{𝑡=𝑡𝑘}≤𝐼_1𝑘𝛼𝑡_𝑘,𝑘=1,2,…,𝑛,Δ𝜙_𝑝(𝛼^′)_{𝑡=𝑡𝑘}≤𝐼_2𝑘𝛼𝑡_𝑘,𝛼^′𝑡_𝑘𝛼(0)≤,𝑘=1,2,…,𝑛,_{𝑚−2𝑖=1}𝑎_𝑖𝛼𝜂_𝑖,𝛼^′(1)<0. (5) A function 𝛽∈𝐸, with 𝜙_𝑝(𝛽^′)∈𝑃𝐶¹(0,1), satisfying the reversed inequalities is an upper solution of problem (1).

Next, we define the Nagumo condition we are going to use. Note that the condition does not depend on the boundary data of the problem.

Definition 2. We say that 𝑓 satisfies a Nagumo condition relative to the pair 𝛼 and 𝛽, with 𝛼,𝛽∈𝐸, 𝛼≤𝛽 in [0,1], if there exists a function 𝜓∶𝐶([0,+∞),[0,+∞)) such that 𝑓𝑡,𝑥,𝑦≤𝜓𝑦,∀𝑡,𝑥,𝑦∈𝑃, (6) where 𝑃={(𝑡,𝑥,𝑦)∈[0,1]×[0,+∞)×𝑅; 𝛼(𝑡)≤𝑥(𝑡)≤𝛽(𝑡)}, and also that ^∞_{𝜙𝑝(𝜈)}𝑑𝑠𝜓𝜙_𝑝⁻¹(𝑠)>¹₀𝑞(𝑠)𝑑𝑠, (7) here, 𝜈=max_{𝑘=1,2,…,𝑛}{|(𝛽(𝑡_𝑘+1)−𝛼(𝑡_𝑘))/(𝑡_𝑘+1−𝑡_𝑘)|, |(𝛼(𝑡_𝑘+1)−𝛽(𝑡_𝑘))/(𝑡_𝑘+1−𝑡_𝑘)|}.

From Definition 2, we can find a real number 𝐿>𝜈>0 such that ^𝜙𝑝_𝜙^(𝐿)_𝑝(𝜈)𝑑𝑠𝜓𝜙_𝑝⁻¹(𝑠)>¹₀𝑞(𝑡)𝑑𝑡. (8)

In addition, we assume that the following three conditions hold:(H₁)𝑞∈𝐶(0,1) with 𝑞>0 on (0,1) and ∫¹₀𝑞(𝑠)𝑑𝑠<∞;(H₂)𝑓∈𝐶([0,1]×𝑅²,𝑅); (H₃)𝐼_1𝑘, 𝐼_2𝑘∈𝐶(𝑅,𝑅). 𝐼_2𝑘(𝑦,𝑧) is nondecreasing in 𝑧∈[−𝐿,𝐿] for all 1≤𝑘≤𝑛.

Firstly, we define 𝑃_𝛼,𝛽()=𝛽(𝑡)(𝑡)(𝑡),𝑥(𝑡)(𝑡)(𝑡)(𝑡)𝛽(𝑡)(𝑡)(𝑡).[),𝑡,𝑥,𝑥≥𝛽,𝛼≤𝑥≤𝛽,𝑥≤𝛼∀𝑥∈0,+∞ (9)

One can find the next result, with its proof, in [17].

Lemma 3. For each 𝑢∈𝐸, the next two properties hold:(i)d𝑃_𝛼,𝛽(𝑡,𝑢)/𝑑𝑡 exists for a.e. 𝑡∈𝐽^′,(ii)if 𝑢, 𝑢_𝑚∈𝐸 and 𝑢_𝑚→𝑢 in 𝐸, then 𝑑𝑃𝑑𝑡^𝛼,𝛽𝑡,𝑢_𝑚(𝑡)⟶𝑑𝑃𝑑𝑡^𝛼,𝛽(𝑡,𝑢(𝑡))𝑓𝑜𝑟𝑎.𝑒.𝑡∈𝐽^′. (10)

We consider the following modified problem: 𝜙_𝑝𝑢^′(𝑡)^′(𝑡)+𝑞(𝑡)𝑓^∗(𝑡),𝑑𝑡,𝑢𝑃𝑑𝑡^𝛼,𝛽(𝑡,𝑢)=0,𝑡∈𝐽^′,Δ𝑢|_{𝑡=𝑡𝑘}=𝐼_1𝑘𝑃_𝛼,𝛽(𝑡,𝑢),𝑘=1,2,…,𝑛,Δ𝜙_𝑝(𝑢^′)_{𝑡=𝑡𝑘}=𝐼_2𝑘𝑃_𝛼,𝛽(𝑡,𝑢),ℎ𝑡,𝑢^′𝑢(0)=,𝑘=1,2,…,𝑛,_{𝑚−2𝑖=1}𝑎_𝑖𝑢𝜂_𝑖,𝑢^′(1)=0, (11) with 𝑓^∗𝑡,𝑥,𝑦=𝑓𝑡,𝑃_𝛼,𝛽(𝑡,𝑥),ℎ𝑡,𝑦+tanh𝑥−𝑃_𝛼,𝛽(𝑡,𝑥), (12) where ℎ is defined by ℎ=(𝑡)𝑦(𝑡),𝑦(𝑡(𝑡)𝑡,𝑦𝐿,𝑦>𝐿,)≤𝐿,𝐿,𝑦<−𝐿.∀𝑦∈𝑅, (13)

Thus 𝑓^∗ is a continuous function on [0,1]×[0,∞)×𝑅 and satisfies 𝑓^∗𝑦𝑦𝑓𝑡,𝑥,𝑦≤𝜓+1,∀<𝐿,_∗𝑡,𝑥,𝑦≤𝑀,∀𝑡,𝑥,𝑦∈𝑃, (14) for some constant 𝑀. Moreover, we may choose 𝑀 so that ‖𝛼‖_∞,‖𝛽‖_∞<𝑀.

Lemma 4. Suppose that ∑^𝑚−2_𝑖=1𝑎_𝑖≠1, denote 𝑎=∑^𝑚−2_𝑖=1𝑎_𝑖, then the following boundary value problem: 𝜙_𝑝𝑢^′(𝑡)^′+𝑣(𝑡)=0,𝑡∈𝐽^′,Δ𝑢|_{𝑡=𝑡𝑘}=𝐼_1𝑘𝑢𝑡_𝑘,𝑘=1,2,…,𝑛,Δ𝜙_𝑝(𝑢^′)_{𝑡=𝑡𝑘}=𝐼_2𝑘𝑢𝑡_𝑘,𝑢^′𝑡_𝑘𝑢(0)=,𝑘=1,2,…,𝑛,_{𝑚−2𝑖=1}𝑎_𝑖𝑢𝜂_𝑖,𝑢^′(1)=0, (15) has a unique solution as follows: 𝑢(𝑡)=11−𝑎^𝑚−2_𝑖=1𝑎_{𝑖𝑡𝑘<𝜂𝑖}𝐼_1𝑘𝑢𝑡_𝑘+^𝜂𝑖₀𝜙_𝑝⁻¹¹_𝑠𝑣(𝜏)−𝑑𝜏_{𝑠<𝑡𝑘}𝐼_2𝑘𝑢𝑡_𝑘,𝑢^′𝑡_𝑘+𝑑𝑠_𝑡0𝜙_𝑝⁻¹¹_𝑠𝑣(𝜏)𝑑𝜏−_{𝑠<𝑡𝑘}𝐼_2𝑘𝑢𝑡_𝑘,𝑢^′𝑡_𝑘𝑑𝑠+_{𝑡𝑘<𝑡}𝐼_1𝑘𝑢𝑡_𝑘. (16)

Proof. 𝜙_𝑝𝑢^′(𝑡)^′=−𝑣(𝑡). (17) For 𝑡∈(𝑡_𝑛,1), integrating (17) from 𝑡 to 1, we get 𝑢^′(𝑡)=𝜙_𝑝⁻¹¹_𝑡𝑣(𝑠)𝑑𝑠. (18) For 𝑡∈(𝑡_𝑛−1,𝑡_𝑛], integrating (17) from 𝑡 to 𝑡_𝑛, we get 𝑢^′(𝑡)=𝜙_𝑝^−1𝑡𝑛_𝑡𝑣(𝑠)𝑑𝑠+𝜙_𝑝𝑢^′𝑡⁻_𝑛. (19) Combining (18), (19), and the boundary condition, for 𝑡∈(𝑡_𝑛−1,𝑡_𝑛], we have 𝑢^′(𝑡)=𝜙_𝑝^−1𝑡𝑛_𝑡𝑣(𝑠)𝑑𝑠+𝜙_𝑝𝑢^′𝑡⁺_𝑛−𝐼_2𝑛𝑢𝑡_𝑛,𝑢^′𝑡_𝑛=𝜙_𝑝^−1𝑡𝑛_𝑡𝑣(𝑠)𝑑𝑠+¹_𝑡𝑛𝑣(𝑠)𝑑𝑠−𝐼_2𝑛𝑢𝑡_𝑛,𝑢^′𝑡_𝑛=𝜙_𝑝⁻¹¹_𝑡𝑣(𝑠)𝑑𝑠−𝐼_2𝑛𝑢𝑡_𝑛,𝑢^′𝑡_𝑛. (20) For 𝑡∈(𝑡_𝑛−2,𝑡_𝑛−1], integrating (17) from 𝑡 to 𝑡_𝑛−1, and combining (20), we have 𝑢^′(𝑡)=𝜙_𝑝^{−1𝑡𝑛−1}_𝑡𝑣(𝑠)𝑑𝑠+𝜙_𝑝𝑢^′𝑡⁻_𝑛−1=𝜙_𝑝^{−1𝑡𝑛−1}_𝑡𝑣(𝑠)𝑑𝑠+𝜙_𝑝𝑢^′𝑡⁺_𝑛−1−𝐼_2(𝑛−1)𝑢𝑡_𝑛−1,𝑢^′𝑡_𝑛−1=𝜙_𝑝^{−1𝑡𝑛−1}_𝑡𝑣(𝑠)𝑑𝑠+¹_𝑡𝑛−1𝑣(𝑠)𝑑𝑠−𝐼_2𝑛𝑢𝑡_𝑛,𝑢^′𝑡_𝑛−𝐼_2(𝑛−1)𝑢𝑡_𝑛−1,𝑢^′𝑡_𝑛−1=𝜙_𝑝⁻¹¹_𝑡𝑣(𝑠)𝑑𝑠−𝐼_2𝑛𝑢𝑡_𝑛,𝑢^′𝑡_𝑛−𝐼_2(𝑛−1)𝑢𝑡_𝑛−1,𝑢^′𝑡_𝑛−1=𝜙_𝑝⁻¹¹_𝑡𝑣(𝑠)𝑑𝑠−_{𝑡<𝑡𝑘}𝐼_2𝑘𝑢𝑡_𝑘,𝑢^′𝑡_𝑘. (21)
By induction, for 𝑡∈(0,1), there holds 𝑢^′(𝑡)=𝜙_𝑝⁻¹¹_𝑡𝑣(𝑠)𝑑𝑠−_{𝑡<𝑡𝑘}𝐼_2𝑘𝑢𝑡_𝑘,𝑢^′𝑡_𝑘. (22) Integrating (22) from 0 to 𝑡₁, we have 𝑢𝑡₁=𝑢(0)+^𝑡1₀𝜙_𝑝⁻¹¹_𝑠𝑣(𝜏)𝑑𝜏−_{𝑠<𝑡𝑘}𝐼_2𝑘𝑢𝑡_𝑘,𝑢^′𝑡_𝑘𝑑𝑠. (23) For 𝑡∈(𝑡₁,𝑡₂], integrating (22) from 𝑡₁ to 𝑡, we have 𝑢(𝑡)=𝑢𝑡⁺₁+^𝑡_𝑡1𝜙_𝑝⁻¹¹_𝑠𝑣(𝜏)𝑑𝜏−_{𝑠<𝑡𝑘}𝐼_2𝑘𝑢𝑡_𝑘,𝑢^′𝑡_𝑘𝑑𝑠. (24) By the boundary condition Δ𝑢|_{𝑡=𝑡𝑘}=𝐼_1𝑘(𝑢(𝑡_𝑘)), (23), and (24), for 𝑡∈(𝑡₁,𝑡₂], we have 𝑢(𝑡)=𝑢(0)+𝐼₁₁𝑢𝑡₁+^𝑡₀𝜙_𝑝⁻¹¹_𝑠𝑣(𝜏)𝑑𝜏−_{𝑠<𝑡𝑘}𝐼_2𝑘𝑢𝑡_𝑘,𝑢^′𝑡_𝑘𝑑𝑠. (25) Repeating in the above manner, for 𝑡∈[0,1], we have 𝑢(𝑡)=𝑢(0)+_{𝑡𝑘<𝑡}𝐼_1𝑘𝑢𝑡_𝑘+^𝑡₀𝜙_𝑝⁻¹¹_𝑠𝑣(𝜏)𝑑𝜏−_{𝑠<𝑡𝑘}𝐼_2𝑘𝑢𝑡_𝑘,𝑢^′𝑡_𝑘𝑑𝑠. (26) By (26), we have 𝑢𝜂_𝑖=𝑢(0)+_{𝑡𝑘<𝜂𝑖}𝐼_1𝑘𝑢𝑡_𝑘+^𝜂𝑖₀𝜙_𝑝⁻¹¹_𝑠𝑣(𝜏)𝑑𝜏−_{𝑠<𝑡𝑘}𝐼_2𝑘𝑢𝑡_𝑘,𝑢^′𝑡_𝑘𝑑𝑠,𝑖=0,1,…,𝑚−2. (27) By the boundary condition 𝑢(0)=∑^𝑚−2_𝑖=1𝑎_𝑖𝑢(𝜂_𝑖), we obtain 𝑢(0)=11−𝑎^𝑚−2_𝑖=1𝑎_{𝑖𝑡𝑘<𝜂𝑖}𝐼_1𝑘𝑢𝑡_𝑘+^𝜂𝑖₀𝜙_𝑝⁻¹¹_𝑠𝑣(𝜏)𝑑𝜏−_{𝑠<𝑡𝑘}𝐼_2𝑘𝑢𝑡_𝑘,𝑢^′𝑡_𝑘𝑑𝑠. (28) Substituting (26) into (28), we have (16) which holds. The proof is complete.

Lemma 5. If 𝑢 is a solution of BVP (11), 𝛼(𝑡) and 𝛽(𝑡) are lower and upper solutions of (1), respectively, 𝛼≤𝛽, and 𝐼_1𝑘𝛼𝑡_𝑘≤𝐼_1𝑘(𝑢)≤𝐼_1𝑘𝛽𝑡_𝑘𝑡,𝑘=1,…,𝑝,𝑓𝑜𝑟𝛼_𝑘≤𝑢≤𝛽𝑡_𝑘, (29) then, 𝛼(𝑡)≤𝑢(𝑡)≤𝛽(𝑡),𝑡∈[0,1]. (30)

Proof. Denote 𝑦(𝑡)=𝑢(𝑡)−𝛽(𝑡), we will only see that 𝑢(𝑡)≤𝛽(𝑡) for every 𝑡∈𝐽. An analogous reasoning shows that 𝑢(𝑡)≥𝛼(𝑡) for all 𝑡∈𝐽.
Otherwise, if 𝑢(𝑡)≤𝛽(𝑡), 𝑡∈[0,1] does not hold, then sup_0≤𝑡≤1(𝑢(𝑡)−𝛽(𝑡))>0. Since 𝑢^′(1)−𝛽^′(1)<0, there are three cases.
Case  1. Denote 𝑡₀=0, 𝑡_𝑛+1=1. There exists 𝑝∈{0,1,2,…,𝑛}, 𝑡₀∈(𝑡_𝑝,𝑡_𝑝+1) such that 𝑦(𝑡₀)=sup_0≤𝑡≤1𝑦(𝑡)>0. Then, sup_0≤𝑡≤1𝑦(𝑡)=max_0≤𝑡≤1𝑦(𝑡). Let 𝑡₁=max{𝑡∣𝑦(𝑡)≤0,𝑡∈[𝑡_𝑝,𝑡₀)}, 𝑡₂=min{𝑡∣𝑦(𝑡)≤0,𝑡∈(𝑡₀,𝑡_𝑝+1]}. If 𝑦(𝑡)>0 for all 𝑡∈[𝑡_𝑝,𝑡₀), let 𝑡₁=𝑡_𝑝, and if 𝑦(𝑡)>0 for all 𝑡∈(𝑡₀,𝑡_𝑝+1], let 𝑡₂=𝑡_𝑝+1. Thus, 𝜙_𝑝𝑢^′(𝑡)^′−𝜙_𝑝𝛽^′(𝑡)^′≥𝑓𝑡,𝛽(𝑡),𝛽^′(𝑡)+tanh𝑢(𝑡)−𝛽(𝑡)−𝑓𝑡,𝛽(𝑡),𝛽^′(𝑡)=tanh𝑢(𝑡)−𝛽(𝑡)>0,∀𝑡∈𝑡₁,𝑡₂. (31) As a consequence, 𝜙_𝑝(𝑢^′(𝑡))−𝜙_𝑝(𝛽^′(𝑡))>𝜙_𝑝(𝑢^′(𝑡₀))−𝜙_𝑝(𝛽^′(𝑡₀))=0 for all 𝑡∈(𝑡₀,𝑡₂) and since 𝜙_𝑝 is increasing and, in particular, one to one, we have 𝑢^′(𝑡)>𝛽^′(𝑡),∀𝑡∈𝑡₀,𝑡₂. (32) Thus, 𝑢(𝑡₂)−𝛽(𝑡₂)>𝑢(𝑡₀)−𝛽(𝑡₀)>0, which is a contradiction.
Case  2. According to Case  1, if sup_0≤𝑡≤1𝑦(𝑡)>0, then sup_0≤𝑡≤1𝑦(𝑡)=𝑦(𝑡⁻_𝑘)=𝑦(𝑡_𝑘) or sup_0≤𝑡≤1𝑦(𝑡)=𝑦(𝑡⁺_𝑘), 𝑘=0,1,2,…,𝑛. Since 𝑦^′(1)<0, 𝑦(1) cannot be sup_0≤𝑡≤1𝑦(𝑡).
Next, we claim that for 𝑘=0,1,2,…,𝑛−1, if sup_{𝑡∈[𝑡𝑘,𝑡𝑘+1]}𝑦(𝑡)>0, then sup_{𝑡∈[𝑡𝑘,𝑡𝑘+1]}𝑦(𝑡)=𝑦(𝑡⁺_𝑘). Otherwise, sup_{𝑡∈[𝑡𝑘,𝑡𝑘+1]}𝑦(𝑡)=𝑦(𝑡⁻_𝑘+1)=𝑦(𝑡_𝑘+1). Suppose that sup_{0≤𝑡≤𝑡1}𝑦(𝑡)=𝑦(𝑡⁻₁)=𝑦(𝑡₁)>0 and 𝑦^′(𝑡₁)=𝑦^′(𝑡⁻₁)≥0.
From the boundary condition and (29), we have 𝑢𝑡⁺₁−𝑢𝑡₁=𝑢𝑡⁺₁−𝑢𝑡⁻₁=𝐼_1𝑘𝑃_𝛼,𝛽(𝑡,𝑢)=𝐼_1𝑘𝛽𝑡₁=𝛽𝑡⁺₁−𝛽𝑡⁻₁=𝛽𝑡⁺₁−𝛽𝑡₁,𝜙_𝑝𝑢^′𝑡⁺₁−𝜙_𝑝𝑢^′𝑡₁=𝐼_2𝑘𝑃_𝛼,𝛽𝑡₁,𝑢,ℎ𝑡₁,𝑢^′=𝐼_2𝑘𝛽𝑡₁,𝛽^′𝑡₁≥𝜙_𝑝𝛽^′𝑡⁺₁−𝜙_𝑝𝛽^′𝑡₁. (33) Hence, 𝑦𝑡⁺₁=𝑢𝑡⁺₁−𝛽𝑡⁺₁=𝑢𝑡₁−𝛽𝑡₁𝑦>0,_′𝑡⁺₁=𝑢^′𝑡⁺₁−𝛽^′𝑡⁺₁≥𝑢^′𝑡₁−𝛽^′𝑡₁≥0. (34) Suppose that 𝑦^′(𝑡⁺₁)=0 and 𝑦 is nonincreasing on some interval (𝑡₁,𝑡₁+𝛾)⊂(𝑡₁,𝑡₂), where 𝛾>0 is sufficiently small such that 𝑦(𝑡)>0 on 𝑡∈(𝑡₁,𝑡₁+𝛾). For 𝑡∈(𝑡₁,𝑡₁+𝛾), 𝜙_𝑝𝑢^′(𝑡)^′−𝜙_𝑝𝛽^′(𝑡)^′(𝑡),𝑑≥𝑓𝑡,𝛽𝑃𝑑𝑡^𝛼,𝛽()𝑢(𝑡)(𝑡)(𝑡)𝑡,𝑢+tanh−𝛽−𝑓𝑡,𝛽,𝛽^′(𝑡𝑢(𝑡)(𝑡))=tanh−𝛽>0. (35) Note that 𝑓(𝑡,𝛽(𝑡),(𝑑/𝑑𝑡)𝑃_𝛼,𝛽(𝑡,𝑢))−𝑓(𝑡,𝛽(𝑡),𝛽^′(𝑡))→0 as 𝛾→0⁺. Hence, when 𝛾>0 is sufficiently small, [𝜙_𝑝(𝑢^′(𝑡))]^′−[𝜙_𝑝(𝛽^′(𝑡))]^′>0, 𝑡∈(𝑡₁,𝑡₁+𝛾), which contradicts the assumption of monotonicity of 𝑦. Thus, we obtain 0<𝑦𝑡⁺₁<𝑦𝑡₂=𝑦𝑡⁻₂,𝑦^′𝑡₂=𝑦^′𝑡⁻₂≥0. (36) We use the preceding procedure and deduce by induction that 𝑦𝑡_𝑘>0,𝑦^′𝑡_𝑘≥0,𝑘=1,2,…,𝑛+1, (37) which contradicts that 𝑦^′(1)<0. Essentially, by the same analysis, we can get that 𝑦(𝑡_𝑘)>0, 𝑘=2,3,…,𝑛 cannot hold.
Case  3. 𝑢(0)−𝛽(0)=sup_0≤𝑡≤1(𝑢(𝑡)−𝛽(𝑡))=max_0≤𝑡≤1(𝑢(𝑡)−𝛽(𝑡))>0. Easily, it holds that 𝑢^′(0⁺)−𝛽^′(0⁺)≤0. While by the boundary condition and ∑^𝑚−2_𝑖=1𝑎_𝑖<1, we have 𝑢(0)−𝛽(0)≤^𝑚−2_𝑖=1𝑎_𝑖𝑢𝜂_𝑖−𝛽𝜂_𝑖<max_{𝑖=1,2,…,𝑚−2}𝑢𝜂_𝑖−𝛽𝜂_𝑖, (38) which is also a contradiction.
Consequently, 𝑢(𝑡)≤𝛽(𝑡) holds for all 𝑡∈[0,1]. The proof is complete.

Lemma 6. If 𝑢 is a solution of (11), then −𝐿<𝑢^′(𝑡)<𝐿 for all 𝑡∈[0,1].

Proof. Let 𝑢∈𝐸 be a solution of (11). From Lemma 5, we have 𝛼(𝑡)≤𝑢(𝑡)≤𝛽(𝑡), and so 𝜙_𝑝𝑢^′(𝑡)^′+𝑞(𝑡)𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡)=0for𝑡∈𝐽^′. (39) By the mean value theorem, there exists 𝜉₀∈(𝑡_𝑘,𝑡_𝑘+1) with 𝑢^′(𝜉₀)=(𝑢(𝑡_𝑘+1)−𝑢(𝑡_𝑘))/(𝑡_𝑘+1−𝑡_𝑘), 𝑘=1,2,…,𝑛, and as a result, 𝛼𝑡−𝐿<−𝜈≤^𝑘+1−𝛽𝑡_𝑘𝑡_𝑘+1−𝑡_𝑘≤𝑢^′𝜉₀≤𝛽𝑡_𝑘+1−𝛼𝑡_𝑘𝑡_𝑘+1−𝑡_𝑘≤𝜈<𝐿. (40) Let 𝑣₀=|𝑢^′(𝜉₀)|. Suppose that there exists a point in the interval [0,1] for which 𝑢^′>𝐿 or 𝑢^′<−𝐿. From the continuity of 𝑢^′, we can choose 𝜉₁∈[0,1], such that one of the following situations holds: (i)𝑢^′(𝜉₀)=𝜈₀, 𝑢^′(𝜉₁)=𝐿 and 𝜈₀≤𝑢^′(𝑡)≤𝐿 for all 𝑡∈(𝜉₀,𝜉₁);(ii)𝑢^′(𝜉₁)=𝐿, 𝑢^′(𝜉₀)=𝜈₀ and 𝜈₀≤𝑢^′(𝑡)≤𝐿 for all 𝑡∈(𝜉₁,𝜉₀);(iii)𝑢^′(𝜉₀)=−𝜈₀, 𝑢^′(𝜉₁)=−𝐿 and −𝐿≤𝑢^′(𝑡)≤−𝜈₀ for all 𝑡∈(𝜉₀,𝜉₁);(iv)𝑢^′(𝜉₁)=−𝐿, 𝑢^′(𝜉₀)=−𝜈₀ and −𝐿≤𝑢^′(𝑡)≤−𝜈₀ for all 𝑡∈(𝜉₁,𝜉₀). Without loss of generality, suppose that −𝐿≤𝜈₀≤𝑢^′(𝑡)≤𝐿 for all 𝑡∈(𝜉₀,𝜉₁). Then, 𝜙_𝑝𝑢^′(𝑡)^′𝑛+𝑞(𝑡)𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡)=0for𝑡∈𝜉₀,𝜉₁, (41) and so 𝜙_𝑝𝑢^′(𝑡)^′=𝑞(𝑡)𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡)≤𝑞(𝑡)𝜓𝑢^′(𝑡),for𝑡∈𝜉₀,𝜉₁. (42) As a result, ^𝜙𝑝_𝜙^(𝐿)_𝑝(𝜈0)𝑑𝑠𝜓𝜙_𝑝⁻¹(𝑠=)^𝜉1𝜉₀𝜙_𝑝𝑢^′(𝑡)^′𝜓𝑢^′(𝑡)𝑑𝑡≤^𝜉1_𝜉0𝑞(𝑡)𝑑𝑡<¹₀𝑞(𝑡)𝑑𝑡. (43) Note also that 𝜙_𝑝⁻¹(𝑠)≥0 for 𝑠∈[𝜙_𝑝(𝜈₀),𝜙_𝑝(𝐿)], so we have 𝜈₀≤𝜈 and thus 𝜙_𝑝(𝜈₀)≤𝜙_𝑝(𝜈), which leads to the following: ^𝜙𝑝_𝜙^(𝐿)_𝑝𝜈0𝑑𝑢𝜓𝜙_𝑝⁻¹(𝑢)≥^𝜙𝑝_𝜙^(𝐿)_𝑝(𝜈)1𝜓𝜙_𝑝⁻¹(𝑠)>𝑑𝑠₁₀𝑞(𝑡)𝑑𝑡, (44) a contradiction. The proof is complete.

Theorem 7 (Schäuder fixed point theorem). Let 𝑃 be a convex subset of a normed Linear space 𝐸. Each continuous, compact map 𝑇∶𝑃→𝑃 has a fixed point.

3. The Main Results

Theorem 8. Suppose that conditions (𝐻₁)–(𝐻₃) hold. Then, BVP (1) has at least one solution 𝑢∈𝐸∩𝐶²(𝐽^′) such that 𝛼(𝑡)≤𝑢(𝑡)≤𝛽(𝑡),−𝐿<𝑢^′(𝑡)<𝐿,𝑡∈[0,1]. (45)

Proof. Solving (11) is equivalent to finding a 𝑢∈𝐸 which satisfies 𝑢(𝑡)=11−𝑎^𝑚−2_𝑖=1𝑎_{𝑖𝑡𝑘<𝜂𝑖}𝐼^∗_1𝑘𝑢𝑡_𝑘+^𝜂𝑖₀𝜙_𝑝⁻¹¹_𝑠𝐹^∗_𝑢(𝜏)𝑑𝜏−_{𝑠<𝑡𝑘}𝐼^∗_2𝑘𝑢𝑡_𝑘,𝑢^′𝑡_𝑘+𝑑𝑠_𝑡0𝜙_𝑝⁻¹¹_𝑠𝐹^∗_𝑢𝑑𝜏𝑑𝑠−_{𝑠<𝑡𝑘}𝐼^∗_2𝑘𝑢𝑡_𝑘,𝑢^′𝑡_𝑘+_{𝑡𝑘<𝑡}𝐼^∗_1𝑘𝑢𝑡_𝑘, (46) where 𝐹^∗_𝑢(𝑡)=𝑞(𝑡)𝑓^∗(𝑡,𝑢(𝑡),𝑢^′(𝑡)), 𝐼^∗_1𝑘(𝑢(𝑡_𝑘))=𝐼_1𝑘(𝑃_𝛼,𝛽(𝑡,𝑢)), 𝐼^∗_2𝑘(𝑢(𝑡_𝑘),𝑢^′(𝑡_𝑘))=𝐼_2𝑘(𝑃_𝛼,𝛽(𝑡,𝑢),ℎ(𝑡,𝑢^′)).
Now, define the following operator 𝑇∶𝐸→𝐸 by (𝑡)=1𝑇𝑢)(1−𝑎^𝑚−2_𝑖=1𝑎_{𝑖𝑡𝑘<𝜂𝑖}𝐼^∗_1𝑘𝑢𝑡_𝑘+^𝜂𝑖₀𝜙_𝑝⁻¹¹_𝑠𝐹^∗_𝑢(𝜏)𝑑𝜏−_{𝑠<𝑡𝑘}𝐼^∗_2𝑘𝑢𝑡_𝑘,𝑢^′𝑡_𝑘+𝑑𝑠_𝑡0𝜙_𝑝⁻¹¹_𝑠𝐹^∗_𝑢(𝜏)𝑑𝜏−_{𝑠<𝑡𝑘}𝐼^∗_2𝑘𝑢𝑡_𝑘,𝑢^′𝑡_𝑘+𝑑𝑠_{𝑡𝑘<𝑡}𝐼^∗_1𝑘𝑢𝑡_𝑘. (47) It is obvious that 𝑇∶𝐸→𝐸 is completely continuous.
By the Schäuder fixed point Theorem 7, we can easily obtain that 𝑇 has a fixed point 𝑢∈𝐸, which is a solution of BVP (11). And by Lemmas 5 and 6, we know that 𝛼(𝑡)≤𝑢(𝑡)≤𝛽(𝑡), −𝐿<𝑢^′(𝑡)<𝐿, then BVP (11) becomes BVP (1), therefore 𝑢(𝑡) is a solution of BVP (1). The proof is complete.

Theorem 9. Suppose that conditions (𝐻₁)–(𝐻₃) hold. Assume that there exist two lower solutions 𝛼₁ and 𝛼₂ and two upper solutions 𝛽₁ and 𝛽₂ for problem (1), satisfying the following:(i)𝛼₁≤𝛼₂≤𝛽₂; (ii)𝛼₁≤𝛽₁≤𝛽₂; (iii)𝛼₂≰𝛽₁, which means that there exists 𝑡∈[0,1] such that 𝛼₂(𝑡)>𝛽₁(𝑡);(iv)if 𝑢 is a solution of (1) with 𝑢≥𝛼₂, then 𝑢>𝛼₂ on (0,1);(v)if 𝑢 is a solution of (1) with 𝑢≤𝛽₁, then 𝑢<𝛽₁ on (0,1). If 𝑓 satisfies the Nagumo condition with respect to 𝛼₁, 𝛽₂, then problem (1) has at least three solutions 𝑢₁, 𝑢₂, and 𝑢₃ satisfying 𝛼₁≤𝑢₁≤𝛽₁,𝛼₂≤𝑢₂≤𝛽₂,𝑢₃≰𝛽₁,𝑢₃≱𝛼₂. (48)

Proof. We consider the following modified problem: 𝜙_𝑝𝑢^′(𝑡)^′+𝑞(𝑡)𝑓^∗(𝑡),𝑑𝑡,𝑢𝑃𝑑𝑡^𝛼1,𝛽2(𝑡,𝑢)=0,𝑡∈𝐽^′,Δ𝑢|_{𝑡=𝑡𝑘}=𝐼_1𝑘𝑢𝑡_𝑘,𝑘=1,2,…,𝑛,Δ𝜙_𝑝(𝑢^′)_{𝑡=𝑡𝑘}=𝐼_2𝑘𝑢𝑡_𝑘,𝑢^′𝑡_𝑘𝑢(0)=,𝑘=1,2,…,𝑛,_{𝑚−2𝑖=1}𝑎_𝑖𝑢𝜂_𝑖,𝑢^′(1)=0. (49) Now, define the following operator 𝑇∶𝐸→𝐸 by (𝑡)=1𝑇𝑢1−𝑎^𝑚−2_𝑖=1𝑎_{𝑖𝑡𝑘<𝜂𝑖}𝐼_1𝑘𝑢𝑡_𝑘+^𝜂𝑖₀𝜙_𝑝⁻¹¹_𝑠𝐹_𝑢(𝜏)𝑑𝜏−_{𝑠<𝑡𝑘}𝐼_2𝑘𝑢𝑡_𝑘,𝑢^′𝑡_𝑘+𝑑𝑠_𝑡0𝜙_𝑝⁻¹¹_𝑠𝐹_𝑢(𝜏)𝑑𝜏−_{𝑠<𝑡𝑘}𝐼_2𝑘𝑢𝑡_𝑘,𝑢^′𝑡_𝑘+𝑑𝑠_{𝑡𝑘<𝑡}𝐼_1𝑘𝑢𝑡_𝑘, (50) where 𝐹_𝑢=𝑞(𝑡)𝑓^∗(𝑡,𝑢(𝑡),(𝑑/𝑑𝑡)𝑃_𝛼1,𝛽2(𝑡,𝑢)), 𝐼_1𝑘(𝑢(𝑡_𝑘))=𝐼_1𝑘(𝑃_𝛼1,𝛽2(𝑡_𝑘,𝑢)), and 𝐼_2𝑘=𝐼_2𝑘(𝑃_𝛼1,𝛽2(𝑡,𝑢),ℎ(𝑡,𝑢^′)).
It is standard that 𝑇∶𝐸→𝐸 is completely continuous.
Let 𝑁>(4/(1−𝑎))((𝑀𝑄)^1/(𝑝−1)+𝑛𝐼), where 𝑀=max{𝑓(𝑡,𝑥,𝑦)∣(𝑡,𝑥,𝑦)∈[0,1]×[𝛼₁,𝛽₂]×[−𝐿,𝐿]}, 𝑄=∫¹₀𝑞(𝑡)𝑑𝑡, 𝐼=max{max_𝑡𝑘<1(𝐼_1𝑘(𝑃_𝛼1,𝛽2(𝑡_𝑘,𝑢)), 𝐼_2𝑘(𝑃_𝛼,𝛽(𝑡,𝑢),ℎ(𝑡,𝑢^′))}.
Let Ω={𝑢∈𝐸∶‖𝑢‖<𝑁}.
It is immediate from the argument above that 𝑇(Ω)⊂Ω. Thus, deg𝐼−𝑇,Ω,0=1. (51) Let Ω_𝛼2=𝑢∈Ω∶𝑢>𝛼₂(,Ωon0,1)_𝛽1=𝑢∈Ω∶𝑢<𝛽₁on(0,1). (52) Since 𝛼₂≰𝛽₁, 𝛼₂>−𝑀, 𝛽₁<𝑀 (i.e., choose 𝑀 such that ‖𝛼₂‖_∞, ‖𝛽₁‖_∞<𝑀). It follows that Ω^𝛽1≠∅≠Ω_𝛼2, Ω^𝛽1∩Ω_𝛼2=∅, and Ω⧵(Ω^𝛽1∪Ω_𝛼2)≠∅.
By assumptions (iv) and (v), there are no solutions in 𝜕Ω^𝛽1∪𝜕Ω_𝛼2. Thus, deg𝐼−𝑇,Ω,0=deg𝐼−𝑇,Ω⧵Ω^𝛽1∪Ω_𝛼2,0+deg𝐼−𝑇,Ω^𝛽1,0+deg𝐼−𝑇,Ω_𝛼2,0. (53) We show that deg(𝐼−𝑇,Ω^𝛽1,0)=deg(𝐼−𝑇,Ω_𝛼2,0)=1, then deg𝐼−𝑇,Ω⧵Ω^𝛽1∪Ω_𝛼2,0=−1, (54) and there are solutions in Ω^𝛽1∪Ω_𝛼2, Ω^𝛽1, Ω_𝛼2 as required.
We now show deg(𝐼−𝑇,Ω_𝛼2,0)=1. The proof that deg(𝐼−𝑇,Ω^𝛽1,0)=1 is similar and hence omitted. We define 𝐼−𝑊, the extension to Ω of the restriction of 𝐼−𝑇 to Ω_𝛼2 as follows. Let 𝑤𝑡,𝑥,𝑦=𝑓𝑡,𝑃_𝛼2,𝛽2(𝑡,𝑥),ℎ𝑡,𝑦+tanh𝑥−𝑃_𝛼2,𝛽2(𝑡,𝑥), (55) where 𝑃_𝛼2,𝛽2(𝑡,𝑥) (in 𝑃_𝛼2,𝛽2(𝑡,𝑥) replace 𝛼₁ by 𝛼₂) and ℎ are previously defined. Thus, 𝑤 is a continuous function on [0,1]×𝑅² and satisfies 𝑤𝑦+𝜋𝑡,𝑥,𝑦≤𝜓2,for𝑦𝑤<𝐿,𝑡,𝑥,𝑦≤𝑀₁,for𝑡,𝑥,𝑦∈[0,1]×𝑅², (56) for some constants 𝑀₁. Moreover, we may choose 𝑀₁ so that ‖𝛼₂‖_∞, ‖𝛽₂‖_∞<𝑀₁.
Consider the following problem: 𝜙_𝑝𝑢^′(𝑡)^′(𝑡)𝑤(𝑡),𝑑+𝑞𝑡,𝑢𝑃𝑑𝑡^𝛼2,𝛽2(𝑡,𝑢)=0,𝑡∈𝐽^′,Δ𝑢|_{𝑡=𝑡𝑘}=𝐼̃_1𝑘𝑢𝑡_𝑘,𝑘=1,2,…,𝑛.Δ𝜙_𝑝(𝑢^′)_{𝑡=𝑡𝑘}=𝐼̃_2𝑘𝑢𝑡_𝑘,𝑢^′𝑡_𝑘𝑢(0)=,𝑘=1,2,…,𝑛._{𝑚−2𝑖=1}𝑎_𝑖𝑢𝜂_𝑖,𝑢^′(1)=0. (57) Now, define the following operator: (𝑡)=1𝑊𝑢)(1−𝑎^𝑚−2_𝑖=1𝑎_𝑖^𝜂𝑖₀𝜙_𝑝⁻¹¹_𝑠𝑞(𝜏)𝑊^∗_𝑢(𝜏)𝑑𝜏𝑑𝑠+_{𝑡𝑘<𝜂𝑖}𝐼_1𝑘𝑢𝑡_𝑘+^𝑡₀𝜙_𝑝⁻¹¹_𝑠𝑞(𝜏)𝑊^∗_𝑢(𝜏)𝑑𝜏−_{𝑠<𝑡𝑘}𝐼̃_2𝑘𝑢𝑡_𝑘,𝑢^′𝑡_𝑘𝑑𝑠+_{𝑡𝑘<𝑡}𝐼̃_1𝑘𝑢𝑡_𝑘, (58) where 𝑊^∗_𝑢(𝑡)=𝑞(𝑡)𝑤(𝑡,𝑢(𝑡),(𝑑/𝑑𝑡)𝑃_𝛼2,𝛽2(𝑡,𝑢)), 𝐼̃_1𝑘(𝑢(𝑡_𝑘))=𝐼_1𝑘(𝑃_𝛼2,𝛽2(𝑡_𝑘,𝑢)), and 𝐼̃_2𝑘=𝐼_2𝑘(𝑃_𝛼2,𝛽2(𝑡,𝑢),ℎ(𝑡,𝑢^′)). Again, it is easy to check (from a previous argument and (v)) that 𝑢 is a solution of (57) if 𝑢∈Ω_𝛼2 and 𝑊𝑢=𝑢 (note that 𝑊∶𝐸→𝐸 is compact).
Thus, deg(𝐼−𝑇,Ω⧵Ω_𝛼2,0)=0. Moreover, it is easy to see that 𝑊(Ω)⊂Ω. By assumptions (iv) and (v), there are no solutions in 𝜕Ω_𝛼2∩𝜕Ω^𝛽1. Thus, deg𝐼−𝑇,Ω_𝛼2,0=deg𝐼−𝑊,Ω⧵Ω_𝛼2,0+deg𝐼−𝑊,Ω_𝛼2(),0=deg𝐼−𝑊,Ω,0=1. (59) Thus there are three solutions, as required. The proof is complete.

A slight modification of the argument in Theorem 9 yields the next result.

Theorem 10. Suppose that conditions (𝐻₁)–(𝐻₃) hold. Assume that there exist two lower solutions 𝛼₁ and 𝛼₂ and two upper solutions 𝛽₁ and 𝛽₂ for problem (1), satisfying(i)𝛼₁≤𝛼₂≤𝛽₂; (ii)𝛼₁≤𝛽₁≤𝛽₂; (iii)𝛼₂≰𝛽₁; (iv)there exists 0<𝜀<min_𝑡∈[0,1]{𝛼₂(𝑡)−𝛼₁(𝑡),𝛽₂(𝑡)−𝛽₁(𝑡)} such that all 𝜀∈(0,𝜀], the function 𝛼₂−𝜀, and 𝛽₁+𝜀 are, respectively, lower and upper solution of (1);(v)𝛼₂−𝜀≰𝛽₁+𝜀. If 𝑓 satisfies the Nagumo condition with respect to 𝛼₁, 𝛽₂, then problem (1) has at least three solutions 𝑢₁, 𝑢₂, and 𝑢₃ satisfying 𝛼₁≤𝑢₁≤𝛽₁,𝛼₂≤𝑢₂≤𝛽₂,𝑢₃≰𝛽₁,𝑢₃≱𝛼₂. (60)

Proof. In the proof of Theorem 9, define Ω_𝛼2=𝑢∈Ω∶𝑢>𝛼₂−(,Ω𝜀on0,1)_𝛽1=𝑢∈Ω∶𝑢<𝛽₁+𝜀on(0,1), (61) where Ω is defined in Theorem 9.

Theorem 11. Let 𝑓∈𝐶([0,1]×(0,+∞)×(0,+∞),[0,+∞)) and 𝑞 satisfies (𝐻₁). Furthermore, the following conditions hold.(i)BVP (1) has a pair of positive upper and lower solutions 𝛽, 𝛼∈𝑋 satisfying 𝛼(𝑡)≤𝛽(𝑡),𝑡∈[0,1]. (62)(ii)𝑓 satisfies a Nagumo condition relative to the pair 𝛼 and 𝛽.
Then, BVP (1) has at least one positive solution such that 𝛼(𝑡)≤𝑢(𝑡)≤𝛽(𝑡),𝑡∈[0,1]. (63)

Proof. It can be proved easily, we omit it here.

4. Examples

Example 1. Consider the following BVP: 𝜙_𝑝𝑢^′(𝑡)^′+𝑞(𝑡)𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡)=0,𝑡∈𝐽^′,Δ𝑢|_{𝑡=𝑡𝑘}=𝑢𝑡_𝑘14,𝑘=1,2,…,𝑛,Δ𝜙_𝑝(𝑢^′)_{𝑡=𝑡𝑘}=−24𝑢^′𝑡_𝑘+𝑢𝑡_𝑘𝑢(0)=1,𝑘=1,2,…,𝑛,5𝑢13+16𝑢23,𝑢^′(1)=0, (64) where 𝑞(𝑡)=1, 𝑓(𝑡,𝑢(𝑡),𝑢^′(𝑡))=4𝑡²+sin²𝑢(𝑡)+20𝑢^′(𝑡)³. Let 𝑝=3; then, 𝑞=3/2, which means that 𝜙_𝑝⁻¹=𝜙_𝑞.

It is clear that 𝛼(𝑡)=−𝑡²1+4,𝑡∈0,2,−1𝑡−2²1+3,𝑡∈2,34,−3𝑡−4²3+2,𝑡∈4,𝛽(𝑡)=−3,1𝑡−4²+89116,𝑡∈0,2,−(𝑡−1)²+2541,𝑡∈2,34,−(𝑡−2)²3+8,𝑡∈4,1 (65) are lower and upper solutions of BVP (64), respectively. The figures of 𝛼(𝑡) and 𝛽(𝑡) are as shown in Figure 1.

Figure 1 

Clearly, 𝛼(𝑡)≤𝛽(𝑡). After substituting 𝛼(𝑡), 𝛽(𝑡) into (𝜙_𝑝(𝑢^′(𝑡)))^′+𝑞(𝑡)𝑓(𝑡,𝑢(𝑡),𝑢^′(𝑡))=0, 𝑡∈𝐽^′, denote upper=(𝜙_𝑝(𝛽^′(𝑡)))^′+𝑞(𝑡)𝑓(𝑡,𝛽(𝑡),𝛽^′(𝑡)) and lower=(𝜙_𝑝(𝛼^′(𝑡)))^′+𝑞(𝑡)𝑓(𝑡,𝛼(𝑡),𝛼^′(𝑡)), we can get Figure 2. From Figure 2, we can see that when 𝑡∈𝐽^′, (𝜙_𝑝(𝛽^′(𝑡)))^′+𝑞(𝑡)𝑓(𝑡,𝛽(𝑡),𝛽^′(𝑡))≤0 and (𝜙_𝑝(𝛽^′(𝑡)))^′+𝑞(𝑡)𝑓(𝑡,𝛽(𝑡),𝛽^′(𝑡))≥0. Other inequalities can be verified easily. So, according to Definition 1, we can get that 𝛽(𝑡) and 𝛼(𝑡) are upper and lower solutions of BVP (64), respectively.

Figure 2 

Let 𝜓(𝑦)=20𝑦³+5, and it is clear that 𝑓 satisfies Nagumo condition relative to 𝛼 and 𝛽. Obviously, all the conditions of Theorem 11 hold. Hence, (64) has at least one positive solution 𝛼(𝑡)≤𝑢(𝑡)≤𝛽(𝑡).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by the National Natural Science Foundation (no. 11226133), the Fundamental Research Funds for the Central Universities (no. 2652012141), and the Young Talents Programme of Beijing.