Solutions for 𝑚-Point BVP with Sign Changing Nonlinearity

Abstract

We study the existence of positive solutions for the following nonlinear 𝑚-point boundary value problem for an increasing homeomorphism and homomorphism with sign changing nonlinearity: {(𝜙(𝑢^′(𝑡)))^′+𝑎(𝑡)𝑓(𝑡,𝑢(𝑡))=0, 0<𝑡<1, 𝑢^′(0)=∑^𝑚−2_𝑖=1𝑎_𝑖𝑢^′(𝜉_𝑖), 𝑢(1)=∑^𝑘_𝑖=1𝑏_𝑖𝑢(𝜉_𝑖)−∑^𝑠_𝑖=𝑘+1𝑏_𝑖𝑢(𝜉_𝑖)−∑^𝑚−2_𝑖=𝑠+1𝑏_𝑖𝑢^′(𝜉_𝑖), where 𝜙∶𝑅→𝑅 is an increasing homeomorphism and homomorphism and 𝜙(0)=0. The nonlinear term 𝑓 may change sign. As an application, an example to demonstrate our results is given. The conclusions in this paper essentially extend and improve the known results.

1. Introduction

In this paper, we study the existence of positive solutions of the following nonlinear 𝑚-point boundary value problem with sign changing nonlinearity:(𝜙(𝑢^′(𝑡)))^′𝑢+𝑎(𝑡)𝑓(𝑡,𝑢(𝑡))=0,0<𝑡<1,_′(0)=∑^𝑚−2_𝑖=1𝑎_𝑖𝑢^′(𝜉_𝑖),𝑢(1)=∑^𝑘_𝑖=1𝑏_𝑖𝑢(𝜉_𝑖)−∑^𝑠_𝑖=𝑘+1𝑏_𝑖𝑢(𝜉_𝑖)−∑^𝑚−2_𝑖=𝑠+1𝑏_𝑖𝑢^′(𝜉_𝑖), (1.1) where 𝜙∶𝑅→𝑅 is an increasing homeomorphism and homomorphism and 𝜙(0)=0; 𝜉_𝑖∈(0,1) with 0<𝜉₁<𝜉₂<⋯<𝜉_𝑚−2<1 and 𝑎_𝑖,𝑏_𝑖,𝑎,𝑓 satisfy

(𝐻₁)𝑎_𝑖,𝑏_𝑖∈[0,+∞), 0<∑^𝑘_𝑖=1𝑏_𝑖−∑^𝑠_𝑖=𝑘+1𝑏_𝑖<1,0<∑^𝑚−2_𝑖=1𝑎_𝑖<1;(𝐻₂)𝑎(𝑡)∶(0,1)→[0,+∞) does not vanish identically on any subinterval of [0,1] and satisfies0<¹₀𝑎(𝑡)d𝑡<∞; (1.2)(𝐻₃)𝑓∈𝐶([0,1]×[0,+∞),(−∞,+∞)),𝑓(𝑡,0)≥0and𝑓(𝑡,0)≠0.

Definition 1.1. A projection 𝜙∶𝑅→𝑅 is called an increasing homeomorphism and homomorphism, if the following conditions are satisfied:
(i)if 𝑥≤𝑦, then 𝜙(𝑥)≤𝜙(𝑦), for all 𝑥,𝑦∈𝑅;(ii)𝜙 is a continuous bijection and its inverse mapping is also continuous;(iii)𝜙(𝑥𝑦)=𝜙(𝑥)𝜙(𝑦), for all 𝑥,𝑦∈𝑅.

The study of multipoint boundary value problems for linear second-order ordinary differential equations was initiated by Il'in and Moiseev [1, 2]. Motivated by the study of [1, 2], Gupta [3] studied certain three-point boundary value problems for nonlinear ordinary differential equations. Since then, more general nonlinear multipoint boundary value problems have been studied by several authors. We refer the reader to [4–12] for some references along this line. Multipoint boundary value problems describe many phenomena in the applied mathematical sciences. For example, the vibrations of a guy wire of a uniform cross-section and composed of 𝑁 parts of different densities can be set up as a multipoint boundary value problems (see Moshinsky [13]); many problems in the theory of elastic stability can be handled by the method of multipoint boundary value problems (see Timoshenko [14]).

In 2001, Ma [6] studied 𝑚-point boundary value problem (BVP):𝑢^′′(𝑡)+ℎ(𝑡)𝑓(𝑢)=0,0≤𝑡≤1,𝑢(0)=0,𝑢(1)=∑^𝑚−2_𝑖=1𝛼_𝑖𝑢^′(𝜉_𝑖), (1.3)where 𝛼_𝑖>0(𝑖=1,2,…,𝑚−2),∑^𝑚−2_𝑖=1𝛼_𝑖<1,0<𝜉₁<𝜉₂<⋯<𝜉_𝑚−2<1, and 𝑓∈𝐶([0,+∞),[0,+∞)), ℎ∈𝐶([0,1],[0,+∞)). Author established the existence of positive solutions under the condition that 𝑓 is either superlinear or sublinear.

In [11], we considered the existence of positive solutions for the following nonlinear four-point singular boundary value problem with 𝑝-Laplacian:(𝜙_𝑝(𝑢^′(𝑡)))^′+𝑎(𝑡)𝑓(𝑢(𝑡))=0,0<𝑡<1,𝛼𝜙_𝑝(𝑢(0))−𝛽𝜙_𝑝(𝑢^′(𝜉))=0,𝛾𝜙_𝑝(𝑢(1))+𝛿𝜙_𝑝(𝑢^′(𝜂))=0, (1.4)where 𝜙_𝑝(𝑠)=|𝑠|^𝑝−2𝑠,𝑝>1,𝜙_𝑞=(𝜙_𝑝)⁻¹,1/𝑝+1/𝑞=1,𝛼>0,𝛽≥0,𝛾>0,𝛿≥0,𝜉,𝜂∈(0,1),𝜉<𝜂,𝑎∶(0,1)→[0,∞). By using the fixed-point theorem of cone, the existence of positive solution and many positive solutions for nonlinear singular boundary value problem 𝑝-Laplacian is obtained.

Recently, Ma et al. [5] used the monotone iterative technique in cones to prove the existence of at least one positive solution for 𝑚-point boundary value problem (BVP):(𝜙_𝑝(𝑢^′(𝑡)))^′𝑢+𝑎(𝑡)𝑓(𝑡,𝑢(𝑡))=0,0<𝑡<1,_′(0)=∑^𝑚−2_𝑖=1𝑎_𝑖𝑢^′(𝜉_𝑖),𝑢(1)=∑^𝑚−2_𝑖=1𝑏_𝑖𝑢(𝜉_𝑖), (1.5)where 0<∑^𝑚−2_𝑖=1𝑏_𝑖<1,0<∑^𝑚−2_𝑖=1𝑎_𝑖<1,0<𝜉₁<𝜉₂<⋯<𝜉_𝑚−2<1, 𝑎(𝑡)∈𝐿¹[0,1], 𝑓∈𝐶([0,1]×[0,+∞),[0,+∞)).

In [9], Wang and Hou investigated the following 𝑚-point BVP:(𝜙_𝑝(𝑢^′(𝑡)))^′𝜙+𝑓(𝑡,𝑢(𝑡))=0,𝑡∈(0,1),_𝑝(𝑢^′(0))=∑^𝑛−2_𝑖=1𝑎_𝑖𝜙_𝑝(𝑢^′(𝜉_𝑖)),𝑢(1)=∑^𝑛−2_𝑖=1𝑏_𝑖𝑢(𝜉_𝑖), (1.6)where 𝜙_𝑝(𝑢)=|𝑢|^𝑝−2𝑢, 𝑝>1, 𝜉_𝑖∈(0,1) with 0<𝜉₁<𝜉₂<⋯<𝜉_𝑛−2<1 and 𝑎_𝑖,𝑏_𝑖 satisfy 𝑎_𝑖,𝑏_𝑖∈[0,+∞), 0<∑^𝑛−2_𝑖=1𝑎_𝑖<1,0<∑^𝑛−2_𝑖=1𝑏_𝑖<1.

However, in all the above-mentioned paper, the authors discuss the boundary value problem (BVP) under the key conditions that the nonlinear term is positive continuous function. Motivated by the results mentioned above, in this paper we study the existence of positive solutions of 𝑚-point boundary value problem (1.1) for an increasing homeomorphism and homomorphism with sign changing nonlinearity. We generalize the results in [4–12].

By a positive solution of BVP (1.1), we understand a function 𝑢 which is positive on (0,1) and satisfies the differential equation as well as the boundary conditions in BVP (1.1).

2. The Preliminary Lemmas

In this section, we present some lemmas which are important to our main results.

Lemma 2.1. Let (𝐻₁) and (𝐻₂) hold. Then for 𝑢≥0∈𝐶[0,1], the problem(𝜙(𝑢^′(𝑡)))^′𝑢+𝑎(𝑡)𝑓(𝑡,𝑢(𝑡))=0,0<𝑡<1,_′(0)=∑^𝑚−2_𝑖=1𝑎_𝑖𝑢^′(𝜉_𝑖),𝑢(1)=∑^𝑘_𝑖=1𝑏_𝑖𝑢(𝜉_𝑖)−∑^𝑠_𝑖=𝑘+1𝑏_𝑖𝑢(𝜉_𝑖)−∑^𝑚−2_𝑖=𝑠+1𝑏_𝑖𝑢^′(𝜉_𝑖) (2.1)has a unique solution 𝑢(𝑡) if and only if 𝑢(𝑡) can be express as the following equation:𝑢(𝑡)=−¹_𝑡𝜔_𝑓(𝑠)d𝑠+𝐵, (2.2)where 𝐴,𝐵 satisfy𝜙⁻¹(𝐴)=^𝑚−2_𝑖=1𝑎_𝑖𝜙⁻¹(𝐴−^𝜉𝑖₀𝑎(𝑠)𝑓(𝑠,𝑢(𝑠))d𝑠),1𝐵=−1−∑^𝑘_𝑖=1𝑏_𝑖+∑^𝑠_𝑖=𝑘+1𝑏_𝑖[^𝑘_𝑖=1𝑏_𝑖¹_𝜉𝑖𝜔_𝑓(𝑠)d𝑠−^𝑠_𝑖=𝑘+1𝑏_𝑖¹_𝜉𝑖𝜔_𝑓(𝑠)d𝑠+^𝑚−2_𝑖=𝑠+1𝑏_𝑖𝜙⁻¹(𝐴−^𝜉𝑖₀𝑎(𝑠)𝑓(𝑠,𝑢(𝑠))d𝑠)], (2.3)(2.4)where𝜔_𝑓(𝑠)=𝜙⁻¹(−^𝑠₀𝑎(𝑟)𝑓(𝑟,𝑢(𝑟))d𝑟+𝐴). (2.5)Define 𝑙=𝜙(∑^𝑚−2_𝑖=1𝑎_𝑖)/(1−𝜙(∑^𝑚−2_𝑖=1𝑎_𝑖))∈(0,1), then there exists a unique 𝐴∈[−𝑙∫¹₀𝑎(𝑠)𝑓(𝑠,𝑢(𝑠))d𝑠,0] satisfying (2.3).

Proof. The method of the proof is similar to [5, Lemma 2.1], we omit the details.

Lemma 2.2. Let (𝐻₁) and (𝐻₂) hold. If 𝑢∈𝐶⁺[0,1], the unique solution of the problem (2.1) satisfies𝑢(𝑡)≥0,𝑡∈[0,1]. (2.6)

Proof. According to Lemma 2.1, we first have−𝐴+^𝑠₀𝑎(𝑟)𝑓(𝑡,𝑢(𝑟))d𝑟≥0. (2.7)So1𝑢(1)=𝐵=−1−∑^𝑘_𝑖=1𝑏_𝑖+∑^𝑠_𝑖=𝑘+1𝑏_𝑖[^𝑘_𝑖=1𝑏_𝑖¹_𝜉𝑖𝜔_𝑓(𝑠)d𝑠−^𝑠_𝑖=𝑘+1𝑏_𝑖¹_𝜉𝑖𝜔_𝑓(𝑠)d𝑠+^𝑚−2_𝑖=𝑠+1𝑏_𝑖𝜙⁻¹(𝐴−^𝜉𝑖₀𝑎(𝑠)𝑓(𝑡,𝑢(𝑠))d=1𝑠)]1−∑^𝑘_𝑖=1𝑏_𝑖+∑^𝑠_𝑖=𝑘+1𝑏_𝑖[^𝑘_𝑖=1𝑏_𝑖¹_𝜉𝑖−𝜔_𝑓(𝑠)d𝑠−^𝑠_𝑖=𝑘+1𝑏_𝑖¹_𝜉𝑖−𝜔_𝑓(𝑠)d𝑠+^𝑚−2_𝑖=𝑠+1𝑏_𝑖𝜙⁻¹(−𝐴+^𝜉𝑖₀𝑎(𝑠)𝑓(𝑡,𝑢(𝑠))d≥1𝑠)]1−∑^𝑘_𝑖=1𝑏_𝑖+∑^𝑠_𝑖=𝑘+1𝑏_𝑖[^𝑘_𝑖=1𝑏_𝑖¹_𝜉𝑘−𝜔_𝑓(𝑠)d𝑠−^𝑠_𝑖=𝑘+1𝑏_𝑖¹_𝜉𝑘−𝜔_𝑓(𝑠)d=𝑠](∑^𝑘_𝑖=1𝑏_𝑖−∑^𝑠_𝑖=𝑘+1𝑏_𝑖)∫¹_𝜉𝑘−𝜔_𝑓(𝑠)d𝑠1−∑^𝑘_𝑖=1𝑏_𝑖+∑^𝑠_𝑖=𝑘+1𝑏_𝑖≥0. (2.8)If 𝑡∈[0,1), we have𝑢(𝑡)=𝐵−¹_𝑡𝜙⁻¹(𝐴−^𝑠₀𝑎(𝑟)𝑓(𝑟,𝑢(𝑟))d𝑟)d𝑠=𝑢(1)+¹_𝑡𝜙⁻¹(−𝐴+^𝑠₀𝑎(𝑟)𝑓(𝑟,𝑢(𝑟))d𝑟)d𝑠≥𝑢(1)≥0. (2.9)So 𝑢(𝑡)≥0,𝑡∈[0,1]. The proof of Lemma 2.2 is completed.

Lemma 2.3. Let (𝐻₁) and (𝐻₂) hold. If 𝑢∈𝐶⁺[0,1], the unique solution of the problem (2.1) satisfiesinf_𝑡∈[0,1]𝑢(𝑡)≥𝛾‖𝑢‖, (2.10)where 𝛾=(∑^𝑘_𝑖=1𝑏_𝑖−∑^𝑠_𝑖=𝑘+1𝑏_𝑖)(1−𝜉_𝑘)/(1−∑^𝑘_𝑖=1𝑏_𝑖𝜉_𝑘+∑^𝑠_𝑖=𝑘+1𝑏_𝑖𝜉_𝑘)∈(0,1),‖𝑢‖=max_𝑡∈[0,1]|𝑢(𝑡)|.

Proof. Clearly𝑢^′(𝑡)=𝜙⁻¹(𝐴−^𝑡₀𝑎(𝑠)𝑓(𝑠,𝑢(𝑠))d𝑠)=−𝜙⁻¹(−𝐴+^𝑡₀𝑎(𝑠)𝑓(𝑠,𝑢(𝑠))d𝑠)≤0. (2.11)This implies that‖𝑢‖=𝑢(0),min_𝑡∈[0,1]𝑢(𝑡)=𝑢(1). (2.12)It is easy to see that 𝑢^′(𝑡₂)≤𝑢^′(𝑡₁), for any 𝑡₁,𝑡₂∈[0,1] with 𝑡₁≤𝑡₂. Hence 𝑢^′(𝑡) is a decreasing function on [0,1]. This means that the graph of 𝑢(𝑡) is concave down on (0,1). So we have𝑢(𝜉_𝑘)−𝑢(1)𝜉_𝑘≥(1−𝜉_𝑘)𝑢(0). (2.13)Together with 𝑢(1)=∑^𝑘_𝑖=1𝑏_𝑖𝑢(𝜉_𝑖)−∑^𝑠_𝑖=𝑘+1𝑏_𝑖𝑢(𝜉_𝑖)−∑^𝑚−2_𝑖=𝑠+1𝑏_𝑖𝑢^′(𝜉_𝑖) and 𝑢^′(𝑡)≤0 on [0,1], we get∑𝑢(0)≤^𝑘𝑖=1𝑏_𝑖𝑢(𝜉_𝑘)−𝑢(1)∑^𝑘_𝑖=1𝑏_𝑖𝜉_𝑘−∑^𝑠_𝑖=𝑘+1𝑏_𝑖𝑢(𝜉_𝑘)+𝑢(1)∑^𝑠_𝑖=𝑘+1𝑏_𝑖𝜉_𝑘(∑^𝑘_𝑖=1𝑏_𝑖−∑^𝑠_𝑖=𝑘+1𝑏_𝑖)(1−𝜉_𝑘)≤∑^𝑘_𝑖=1𝑏_𝑖𝑢(𝜉_𝑖)−𝑢(1)∑^𝑘_𝑖=1𝑏_𝑖𝜉_𝑘−∑^𝑠_𝑖=𝑘+1𝑏_𝑖𝑢(𝜉_𝑖)+𝑢(1)∑^𝑠_𝑖=𝑘+1𝑏_𝑖𝜉_𝑘(∑^𝑘_𝑖=1𝑏_𝑖−∑^𝑠_𝑖=𝑘+1𝑏_𝑖)(1−𝜉_𝑘)≤𝑢(1)(1−∑^𝑘_𝑖=1𝑏_𝑖𝜉_𝑘+∑^𝑠_𝑖=𝑘+1𝑏_𝑖𝜉_𝑘)(∑^𝑘_𝑖=1𝑏_𝑖−∑^𝑠_𝑖=𝑘+1𝑏_𝑖)(1−𝜉_𝑘)=𝑢(1)𝛾. (2.14)The proof of Lemma 2.3 is completed.

Lemma 2.4 (see [8]). Let 𝐾 be a cone in a Banach space 𝑋. Let 𝐷 be an open bounded subset of 𝑋 with 𝐷_𝐾=𝐷∩𝐾≠𝜙 and 𝐷_𝐾≠𝐾. Assume that 𝐴∶𝐷_𝐾→𝐾 is a compact map such that 𝑥≠𝐴𝐾 for 𝑥∈𝜕𝐷_𝐾. Then the following results hold.
(1)If ‖𝐴𝑥‖≤‖𝑥‖, 𝑥∈𝜕𝐷_𝐾, then 𝑖(𝐴,𝐷_𝐾,𝐾)=1.(2)If there exists 𝑥₀∈𝐾⧵{𝜃} such that 𝑥≠𝐴𝑥+𝜆𝑥₀, for all 𝑥∈𝜕𝐷_𝐾 and all 𝑥>0, then 𝑖(𝐴,𝐷_𝐾,𝐾)=0.(3)Let 𝑈 be open in 𝑋 such that 𝑈⊂𝐷_𝐾. If 𝑖(𝐴,𝐷_𝐾,𝐾)=1 and 𝑖(𝐴,𝐷_𝐾,𝐾)=0, then 𝐴 has a fixed point in 𝐷_𝐾⧵𝑈_𝐾. The same results hold, if 𝑖(𝐴,𝐷_𝐾,𝐾)=0 and 𝑖(𝐴,𝐷_𝐾,𝐾)=1.

Let 𝐸=𝐶[0,1], then 𝐸 is Banach space, with respect to the norm ‖𝑢‖=sup_𝑡∈[0,1]|𝑢(𝑡)|. Denote𝐾={𝑢∣𝑢∈𝐶[0,1],𝑢(𝑡)≥0,inf_𝑡∈[0,1]𝑢(𝑡)≥𝛾‖𝑢‖}, (2.15)where 𝛾 is the same as in Lemma 2.3. It is obvious that 𝐾 is a cone in 𝐶[0,1].

We define 𝜑(𝑡)=min{𝑡,1−𝑡},𝑡∈(0,1) and𝐾_𝜌𝐾={𝑢(𝑡)∈𝐾∶‖𝑢‖<𝜌},_∗𝜌Ω={𝑢(𝑡)∈𝐾∶𝜌𝜑(𝑡)<𝑢(𝑡)<𝜌},_𝜌={𝑢(𝑡)∈𝐾∶min_{𝜉𝑚−2≤𝑡≤1}𝑢(𝑡)<𝛾𝜌}={𝑢(𝑡)∈𝐸∶𝑢≥0,𝛾‖𝑢‖≤min_{𝜉𝑚−2≤𝑡≤1}𝑢(𝑡)<𝛾𝜌}. (2.16)

Lemma 2.5 (see [13]). Ω_𝜌 defined above has the following properties:
(a)𝐾_𝛾𝜌⊂Ω_𝜌⊂𝐾_𝜌;(b)Ω_𝜌 is open relative to 𝐾;(c)𝑋∈𝜕Ω_𝜌 if and only if min_{𝜉𝑚−2≤𝑡≤1𝑐}𝑥(𝑡)=𝛾𝜌;(d) If 𝑥∈𝜕Ω_𝜌, then 𝛾𝜌≤𝑥(𝑡)≤𝜌 for 𝑡∈[𝜉_𝑚−2,1]. Now, for the convenience, one introduces the following notations:𝑓^𝜌_𝛾𝜌=min{min_{𝜉𝑚−2≤𝑡≤1}𝑓(𝑡,𝑢)𝑓𝜙(𝜌)∶𝑢∈[𝛾𝜌,𝜌]},_𝜌0=max{max_0≤𝑡≤1𝑓(𝑡,𝑢)𝑓𝜙(𝜌)∶𝑢∈[0,𝜌]},_{𝜌𝜑(𝑡)𝜌}=max{max_0≤𝑡≤1𝑓(𝑡,𝑢)𝑓𝜙(𝜌)∶𝑢∈[𝜑(𝑡)𝜌,𝜌]},_𝛼=lim_𝑢→𝛼supmax_0≤𝑡≤1𝑓(𝑡,𝑢),𝑓𝜙(𝑢)_𝛼=lim_𝑢→𝛼infmin_{𝜉𝑚−2≤𝑡≤1}𝑓(𝑡,𝑢)𝜙(𝑢),(𝛼∶=∞𝑜𝑟0⁺∑),𝑚={(1+^𝑠_𝑖=𝑘+1𝑏_𝑖+∑^𝑚−2_𝑖=𝑠+1𝑏_𝑖)𝜙⁻¹((𝑙+1)∫¹₀𝑎(𝑠)d𝑠)1−∑^𝑘_𝑖=1𝑏_𝑖+∑^𝑠_𝑖=𝑘+1𝑏_𝑖}⁻¹,∑𝑀={^𝑘𝑖=1𝑏_𝑖−∑^𝑠_𝑖=𝑘+1𝑏_𝑖1−∑^𝑘_𝑖=1𝑏_𝑖+∑^𝑠_𝑖=𝑘+1𝑏_𝑖^1𝜉_𝑘𝜙⁻¹(^𝑠₀𝑎(𝑟)d𝑟)d𝑠}⁻¹. (2.17)

3. The Main Result

In the rest of the section, we also assume the following conditions.

(𝐴₁)There exist 𝜌₁,𝜌₂∈(0,+∞) with 𝜌₁<𝛾𝜌₂ such that(1)𝑓(𝑡,𝑢)>0,𝑡∈[0,1],𝑢∈[𝜌₁𝜑(𝑡),+∞),(2)𝑓^𝜌1_{𝜑(𝑡)𝜌1}≤𝜙(𝑚),𝑓^𝜌2_𝛾𝜌2≥𝜙(𝑀𝛾). (3.1)(𝐴₂)There exist 𝜌₁,𝜌₂∈(0,+∞) with 𝜌₁<𝜌₂ such that(3)𝑓(𝑡,𝑢)>0,𝑡∈[0,1],𝑢∈[min{𝛾𝜌₁,𝜌₂𝜑(𝑡)},+∞),(4)𝑓^𝜌1_𝛾𝜌1≥𝜙(𝑀𝛾),𝑓^𝜌2_{𝜑(𝑡)𝜌2}≤𝜙(𝑚). (3.2)(𝐴₃)There exist 𝜌₁,𝜌₂,𝜌₃∈(0,+∞) with 𝜌₁<𝛾𝜌₂ and 𝜌₂<𝜌₃ such that(1)𝑓(𝑡,𝑢)>0,𝑡∈[0,1],𝑢∈[𝜌₁𝜑(𝑡),+∞),(2)𝑓^𝜌1_{𝜑(𝑡)𝜌1}≤𝜙(𝑚),𝑓^𝜌2_𝛾𝜌2≥𝜙(𝑀𝛾),𝑓^𝜌3_{𝜑(𝑡)𝜌3}≤𝜙(𝑚). (3.3)(𝐴₄)There exist 𝜌₁,𝜌₂,𝜌₃∈(0,+∞) with 𝜌₁<𝜌₂<𝛾𝜌₃ such that(3)𝑓(𝑡,𝑢)>0,𝑡∈[0,1],𝑢∈[min{𝛾𝜌₁,𝜌₂𝜑(𝑡)},+∞),(4)𝑓^𝜌1_𝛾𝜌1≥𝜙(𝑀𝛾),𝑓^𝜌2_{𝜑(𝑡)𝜌2}≤𝜙(𝑚),𝑓^𝜌3_𝛾𝜌3≥𝜙(𝑀𝛾). (3.4)(𝐴₅)There exist 𝜌^′,𝜌∈(0,+∞) with 𝜌^′<𝛾𝜌 such that(1)𝑓(𝑡,𝑢)>0,𝑡∈[0,1],𝑢∈[𝜌^′𝜑(𝑡),+∞),(2)𝑓^𝜌′_{𝜑(𝑡)𝜌′}≤𝜙(𝑚),𝑓^𝜌_𝛾𝜌≥𝜙(𝑀𝛾),0≤𝑓^∞<𝜙(𝑚). (3.5)(𝐴₆)There exist 𝜌^′,𝜌∈(0,+∞) with 𝜌^′<𝜌 such that(3)𝑓(𝑡,𝑢)>0,𝑡∈[0,1],𝑢∈[min{𝛾𝜌^′,𝜌𝜑(𝑡)},+∞),(4)𝑓^𝜌′_𝛾𝜌′≥𝜙(𝑀𝛾),𝑓^𝜌_{𝜑(𝑡)𝜌}≤𝜙(𝑚),𝜙(𝑀)<𝑓_∞≤∞. (3.6)

Our main results are the following theorems.

Theorem 3.1. Assume that (𝐻₁),(𝐻₂),(𝐻₃),(𝐴₃) hold. Then BVP (1.1) has at least three positive solutions.

Theorem 3.2. Assume that (𝐻₁),(𝐻₂),(𝐻₃),(𝐴₄) hold. Then BVP (1.1) has at least two positive solutions.

Theorem 3.3. Assume that (𝐻₁),(𝐻₂),(𝐻₃) hold and also assume that (𝐴₁) or (𝐴₂) hold. Then BVP (1.1) has at least a positive solution.

Theorem 3.4. Assume that (𝐻₁),(𝐻₂),(𝐻₃) hold and also assume that (𝐴₅) or (𝐴₆) hold. Then BVP (1.1) has at least two positive solutions.

Proof of Theorem 3.1. Without loss of generality, we suppose that (𝐴₃) hold. Denote𝑓^∗(𝑡,𝑢)=𝑓(𝑡,𝑢),𝑢≥𝜌₁𝜑(𝑡),𝑓(𝑡,𝜌₁𝜑(𝑡)),0≤𝑢<𝜌₁𝜑(𝑡), (3.7)it is easy to check that 𝑓^∗(𝑡,𝑢)∈𝐶([0,1]×[0,+∞),(0,+∞)).
Now define an operator 𝑇∶𝐾→𝐶[0,1] by setting1(𝑇𝑢)(𝑡)=−1−∑^𝑘_𝑖=1𝑏_𝑖+∑^𝑠_𝑖=𝑘+1𝑏_𝑖[^𝑘_𝑖=1𝑏_𝑖¹_𝜉𝑖𝜔(𝑠)d𝑠−^𝑠_𝑖=𝑘+1𝑏_𝑖¹_𝜉𝑖𝜔(𝑠)d𝑠+^𝑚−2_𝑖=𝑠+1𝑏_𝑖𝜙⁻¹(𝐴−^𝜉𝑖₀𝑎(𝑠)𝑓^∗(𝑠,𝑢(𝑠))d𝑠)]−¹_𝑡𝜔(𝑠)d𝑠, (3.8)where𝜔(𝑠)=𝜙⁻¹(−^𝑠₀𝑎(𝑟)𝑓^∗(𝜏,𝑢(𝑟))d𝑟+𝐴). (3.9)By Lemma 2.3, we have 𝑇(𝐾)⊂𝐾. So by applying Arzela-Ascoli's theorem, we can obtain that 𝑇(𝐾) is relatively compact. In view of Lebesgue's dominated convergence theorem, it is easy to prove that 𝑇 is continuous. Hence, 𝑇∶𝐾→𝐾 is completely continuous.
Now, we consider the following modified BVP (1.1):(𝜙(𝑢^′))^′+𝑎(𝑡)𝑓^∗𝑢(𝑡,𝑢(𝑡))=0,0<𝑡<1,_′(0)=∑^𝑚−2_𝑖=1𝑎_𝑖𝑢^′(𝜉_𝑖),𝑢(1)=∑^𝑘_𝑖=1𝑏_𝑖𝑢(𝜉_𝑖)−∑^𝑠_𝑖=𝑘+1𝑏_𝑖𝑢(𝜉_𝑖)−∑^𝑚−2_𝑖=𝑠+1𝑏_𝑖𝑢^′(𝜉_𝑖). (3.10)Obviously, BVP (3.10) has a solution 𝑢(𝑡) if and only if 𝑢 is a fixed point of the operator 𝑇. From the condition (𝐴₃)(2), we have𝑓^∗𝜌1_{𝜑(𝑡)𝜌1}≤𝜙(𝑚),𝑓^∗𝜌2_𝛾𝜌2≥𝜙(𝑀𝛾),𝑓^∗𝜌3_{𝜑(𝑡)𝜌3}≤𝜙(𝑚). (3.11)
Next, we will show that 𝑖(𝑇,𝐾^∗_𝜌1,𝐾)=1.
In fact, by 𝑓^∗𝜌1_{𝜑(𝑡)𝜌1}≤𝜙(𝑚), for ∀𝑢∈𝜕𝐾^∗_𝜌1, we have1(𝑇𝑢)(𝑡)=−1−∑^𝑘_𝑖=1𝑏_𝑖+∑^𝑠_𝑖=𝑘+1𝑏_𝑖×(^𝑘_𝑖=1𝑏_𝑖¹_𝜉𝑖𝜔(𝑠)d𝑠−^𝑠_𝑖=𝑘+1𝑏_𝑖¹_𝜉𝑖𝜔(𝑠)d𝑠+^𝑚−2_𝑖=𝑠+1𝑏_𝑖𝜙⁻¹(𝐴−^𝜉𝑖₀𝑎(𝑠)𝑓^∗(𝑠,𝑢(𝑠))d−𝑠))_1𝑡𝜔(𝑠)d𝑠≤11−∑^𝑘_𝑖=1𝑏_𝑖+∑^𝑠_𝑖=𝑘+1𝑏_𝑖(^𝑘_𝑖=1𝑏_𝑖¹₀𝜙⁻¹((𝑙+1)¹₀𝑎(𝑟)𝑓^∗(𝑟,𝑢(𝑟))d𝑟)d𝑠+^𝑚−2_𝑖=𝑠+1𝑏_𝑖𝜙⁻¹((𝑙+1)¹₀𝑎(𝑠)𝑓^∗(𝑠,𝑢(𝑠))d+𝑠))₁₀𝜙⁻¹((𝑙+1)¹₀𝑎(𝑟)𝑓^∗(𝑟,𝑢(𝑟))d𝑟)d𝑠≤(1+∑^𝑠_𝑖=𝑘+1𝑏_𝑖+∑^𝑚−2_𝑖=𝑠+1𝑏_𝑖)𝜙⁻¹((𝑙+1)∫¹₀𝑎(𝑠)d𝑠)1−∑^𝑘_𝑖=1𝑏_𝑖+∑^𝑠_𝑖=𝑘+1𝑏_𝑖𝜙⁻¹(𝜙(𝜌₁)𝜙(𝑚))=𝜌₁=‖𝑢‖. (3.12)This implies that ‖𝑇𝑢‖≤‖𝑢‖ for 𝑢∈𝜕𝐾^∗_𝜌. By Lemma 2.4(1), we have𝑖(𝑇,𝐾^∗_𝜌1,𝐾)=1. (3.13)
Furthermore, we will show that 𝑖(𝑇,𝐾_𝜌2,𝐾)=1.
Let 𝑒(𝑡)≡1, for 𝑡∈[0,1], then 𝑒∈𝜕𝐾₁. We claim that𝑢≠𝑇𝑢+𝜆𝑒,𝑢∈𝜕Ω_𝜌2,𝜆>0. (3.14)
In fact, if not, there exist 𝑢₀∈𝜕Ω₂ and 𝜆₀>0 such that 𝑢₀=𝑇𝑢₀+𝜆₀𝑒.
By (𝐴₃) and Lemma 2.1, we have for 𝑡∈[0,1],−^𝑠₀𝑎(𝜏)𝑓^∗(𝜏,𝑢(𝜏))𝑑𝜏+𝐴≤−𝜙(𝜌₂)𝜙(𝑀𝛾)(^𝑠₀𝑎(𝜏)𝑑𝜏), (3.15)so that−𝜔(𝑠)=𝜙⁻¹(−^𝑠₀𝑎(𝜏)𝑓^∗(𝜏,𝑢(𝜏))𝑑𝜏+𝐴)≥𝜌₂𝑀𝛾𝜙⁻¹[^𝑠₀𝑎(𝜏)𝑑𝜏]. (3.16)Then, we have that𝑢₀(𝑡)=𝑇𝑢₀(𝑡)+𝜆₀≥1𝑒(𝑡)1−∑^𝑘_𝑖=1𝑏_𝑖+∑^𝑠_𝑖=𝑘+1𝑏_𝑖^𝑘_𝑖=1(𝑏_𝑖¹_𝜉𝑘(−𝜔(𝑠))d𝑠−^𝑠_𝑖=𝑘+1𝑏_𝑖¹_𝜉𝑘(−𝜔(𝑠))d𝑠)+𝜆₀≥∑^𝑘_𝑖=1𝑏_𝑖−∑^𝑠_𝑖=𝑘+1𝑏_𝑖1−∑^𝑘_𝑖=1𝑏_𝑖+∑^𝑠_𝑖=𝑘+1𝑏_𝑖𝜌₂𝑀𝛾¹_𝜉𝑘𝜙⁻¹(^𝑠₀𝑎(𝑟)d𝑟)d𝑠+𝜆₀=𝛾𝜌₂+𝜆₀. (3.17)This implies that 𝛾𝜌₂≥𝛾𝜌₂+𝜆₀, this is a contradiction. Hence, by Lemma 2.4(2), it follows that𝑖(𝑇,Ω_𝜌2,𝐾)=0. (3.18)
Finally, similar to the proof of 𝑖(𝑇,𝐾^∗_𝜌1,𝐾)=1, we can show that 𝑖(𝑇,𝐾^∗_𝜌3,𝐾)=1.
By Lemma 2.5(a) and 𝜌₁<𝛾𝜌₂ and 𝜌₂<𝜌₃, we have 𝐾_𝜌1⊂𝐾_𝛾𝜌2⊂Ω_𝜌2⊂𝐾_𝜌2⊂𝐾_𝜌3. It follows from Lemma 2.4(3) that 𝑇 has three positive fixed points 𝑢₁,𝑢₂,𝑢₃ in 𝐾^∗_𝜌1,Ω_𝜌2⧵𝐾^∗_𝜌1,𝐾^∗_𝜌3, respectively. Therefore, BVP (3.10) has three positive solutions 𝑢₁,𝑢₂,𝑢₃ in 𝐾^∗_𝜌1,Ω_𝜌2⧵𝐾^∗_𝜌1,𝐾^∗_𝜌3, respectively.
Then, BVP (3.10) has three positive solutions 𝑢₁,𝑢₂,𝑢₃∈[𝜌₁𝜑(𝑡),∞), which means that 𝑢₁,𝑢₂,𝑢₃ are also the positive solutions of BVP (1.1).

Proof of Theorem 3.2. The proof of Theorem 3.2 is similar to that of Theorem 3.1, and so we omit it here. The proof of Theorem 3.2 is completed.

Proof of Theorem 3.3. Theorem 3.3 is corollary of Theorem 3.1. The proof of Theorem 3.3 is completed.

Proof of Theorem 3.4. We show that condition (𝐴₅) implies condition (𝐴₃). Let 𝑘∈(𝑓^∞,𝜙(𝑚)), then there exists 𝑟>𝜌 such that max_𝑡∈[0,1]𝑓(𝑡,𝑢)≤𝑘𝜙(𝑢),𝑢∈[𝑟,∞) since 0≤𝑓^∞<𝜙(𝑚). Denote𝛽=max{max_𝑡∈[0,1]𝑓(𝑡,𝑢)∶𝜌^′𝜑(𝑡)≤𝑢≤𝑟},𝜌₃>max{𝜙⁻¹(𝛽𝜙(𝑚)−𝑘),𝜌}. (3.19)Then we havemax_𝑡∈[0,1]𝑓(𝑡,𝑢)≤𝑘𝜙(𝑢)+𝛽≤𝑘𝜙(𝜌₃)+𝛽≤𝜙(𝑚)𝜙(𝜌₃),𝑢∈[𝜌^′𝜑(𝑡),∞). (3.20)This implies that 𝑓^𝜌3_{𝜑(𝑡)𝜌3}≤𝜙(𝑚) and (𝐴₃) holds. Similarly condition (𝐴₆) implies condition (𝐴₄).
By an argument similar to that Theorem 3.1, we can obtain the result of Theorem 3.4. The proof of Theorem 3.4 is completed.

4. Examples

Example 4.1. Consider the following five-point boundary value problem with 𝑝-Laplacian:(𝜙(𝑢^′))^′𝑢+𝑓(𝑡,𝑢)=0,0<𝑡<1,_′1(0)=𝑢128^′(141)+𝑢256^′(121)+𝑢64^′(341),𝑢(1)=81𝑢(41)−164𝑢(2), (4.1)where 𝑎₁=1/128,𝑎₂=1/256,𝑎₃=1/64,𝑏₁=1/8,𝑏₂=1/64,𝑏₃=0,𝜉₁=1/4,𝜉₂=1,/2,𝜉₃=3/4:𝜙(𝑢)=−𝑢²𝑢,𝑢≤0,₂1,𝑢>0,𝑓(𝑡,𝑢)=5(1+𝑡)(𝑢(𝑡)−𝜑(𝑡)2)³⁰1,(𝑡,𝑢)∈[0,1]×(0,2],5(1+𝑡)(2−𝜑(𝑡)2)³⁰,(𝑡,𝑢)∈[0,1]×(2,+∞). (4.2)
It is easy to check that 𝑓∶[0,1]×[0,+∞)→[0,+∞) is continuous. It follows from a direct calculation that𝑚={(1+∑^𝑠_𝑖=𝑘+1𝑏_𝑖+∑^𝑚−2_𝑖=𝑠+1𝑏_𝑖)𝜙⁻¹((𝑙+1)∫¹₀𝑎(𝑠)d𝑠)1−∑^𝑘_𝑖=1𝑏_𝑖+∑^𝑠_𝑖=𝑘+1𝑏_𝑖}⁻¹=0.96,𝛾=(∑^𝑘_𝑖=1𝑏_𝑖−∑^𝑠_𝑖=𝑘+1𝑏_𝑖)(1−𝜉_𝑘)1−∑^𝑘_𝑖=1𝑏_𝑖𝜉_𝑘+∑^𝑠_𝑖=𝑘+1𝑏_𝑖𝜉_𝑘=21,∑250𝑀={^𝑘𝑖=1𝑏_𝑖−∑^𝑠_𝑖=𝑘+1𝑏_𝑖1−∑^𝑘_𝑖=1𝑏_𝑖+∑^𝑠_𝑖=𝑘+1𝑏_𝑖∫¹_𝜉𝑘𝜙⁻¹(∫^𝑠₀𝑎(𝑟)d𝑟)d𝑠}⁻¹=0.76. (4.3)
Choose 𝜌₁=1,𝜌₂=250, it is easy to check that 𝜌₁<𝛾𝜌₂ and𝑓𝑓(𝑡,𝑢)>0,𝑡∈[0,1],𝑢∈[𝜑(𝑡),+∞),_{𝜌1𝜌1𝜑(𝑡)}=max{max_0≤𝑡≤1(1/5)(1+𝑡)(𝑢(𝑡)−𝜑(𝑡)/2)³⁰1²}=(1/5)(1+1)1³⁰1²=25<𝜙(𝑚)=𝑚²=0.92,𝑡∈[0,1],𝑢∈[𝜑(𝑡)𝜌₁,𝜌₁𝑓],_{𝜌2𝛾𝜌2}=min{min_{3/4≤𝑡≤1}(1/5)(1+𝑡)(2−𝜑(𝑡)/2)³⁰250²}=(1/5)(1+3/4)(2−1/2)³⁰250²=1.0742>𝜙(𝑀𝛾)=(𝑀𝛾)²3=0.004,𝑡∈[4,1],𝑢∈[𝛾𝜌₂,𝜌₂]. (4.4) It follows that 𝑓 satisfies the condition (𝐴₁) of Theorem 3.3, then problems (1.1) have at least two positive solutions.

Remark 4.2. Let 𝜑(𝑢)=𝑢, the problem is second-order 𝑚-point boundary value problem.

Remark 4.3. Let 𝜙_𝑝(𝑠)=|𝑠|^𝑝−2𝑠,𝑝>1, the problem is boundary value problem with 𝑝-Laplacian operators.

Hence our results generalize boundary value problem with 𝑝-Laplacian operators.

Acknowledgment

This research is supported by the Doctor of Scientific Startup Foundation of Shandong University of Finance of China (08BSJJ32).