Solvability of Three-Point Boundary Value Problems at Resonance with a 𝑝-Laplacian on Finite and Infinite Intervals

Abstract

Three-point boundary value problems of second-order differential equation with a p-Laplacian on finite and infinite intervals are investigated in this paper. By using a new continuation theorem, sufficient conditions are given, under the resonance conditions, to guarantee the existence of solutions to such boundary value problems with the nonlinear term involving in the first-order derivative explicitly.

1. Introduction

This paper deals with the three-point boundary value problem of differential equation with a 𝑝-Laplacian Φ_𝑝𝑥^′′+𝑓𝑡,𝑥,𝑥^′𝑥(0)𝜂=0,0<𝑡<𝑇,=𝑥,𝑥^′(𝑇)=0, (1.1) where Φ_𝑝(𝑠)=|𝑠|^𝑝−2𝑠, 𝑝>1, 𝜂∈(0,𝑇) is a constant, 𝑇∈(0,+∞], and 𝑥^′(𝑇)=lim_𝑡→𝑇⁻𝑥^′(𝑡).

Boundary value problems (BVPs) with a 𝑝-Laplacian have received much attention mainly due to their important applications in the study of non-Newtonian fluid theory, the turbulent flow of a gas in a porous medium, and so on [1–10]. Many works have been done to discuss the existence of solutions, positive solutions subject to Dirichlet, Sturm-Liouville, or nonlinear boundary value conditions.

In recent years, many authors discussed, solvability of boundary value problems at resonance, especially the multipoint case [3, 11–15]. A boundary value problem of differential equation is said to be at resonance if its corresponding homogeneous one has nontrivial solutions. For (1.1), it is easy to see that the following BVP Φ_𝑝𝑥^′′𝑥(0)𝜂=0,0<𝑡<𝑇,=𝑥,𝑥^′(𝑇)=0 (1.2) has solutions {𝑥∣𝑥=𝑎,𝑎∈𝑅}. When 𝑎≠0, they are nontrivial solutions. So, the problem in this paper is a BVP at resonance. In other words, the operator 𝐿 defined by 𝐿𝑥=(Φ_𝑝(𝑥^′))^′ is not invertible, even if the boundary value conditions are added.

For multi-point BVP at resonance without 𝑝-Laplacians, there have been many existence results available in the references [3, 11–15]. The methods mainly depend on the coincidence theory, especially Mawhin continuation theorem. At most linearly increasing condition is usually adopted to guarantee the existence of solutions, together with other suitable conditions imposed on the nonlinear term.

On the other hand, for BVP at resonance with a 𝑝-Laplacian, very little work has been done. In fact, when 𝑝≠2, Φ_𝑝(𝑥) is not linear with respect to 𝑥, so Mawhin continuation theorem is not valid for some boundary conditions. In 2004, Ge and Ren [3, 4] established a new continuation theorem to deal with the solvability of abstract equation 𝑀𝑥=𝑁𝑥, where 𝑀, 𝑁 are nonlinear maps; this theorem extends Mawhin continuation theorem. As an application, the authors discussed the following three-point BVP at resonance Φ_𝑝𝑢^′′()𝑢(0)𝑢𝜂(1,+𝑓𝑡,𝑢=0,0<𝑡<1,=0=𝐺,𝑢) (1.3) where 𝜂∈(0,1) is a constant and 𝐺 is a nonlinear operator. Through some special direct-sum-spaces, they proved that (1.3) has at least one solution under the following condition.

There exists a constant 𝐷>0 such that 𝑓(𝑡,𝐷)<0<𝑓(𝑡,−𝐷) for 𝑡∈[0,1] and 𝐺(𝑥,𝐷)<0<𝐺(𝑥,−𝐷) or 𝐺(𝑥,𝐷)>0>𝐺(𝑥,−𝐷) for |𝑥|⩽𝐷.

The above result naturally prompts one to ponder if it is possible to establish similar existence results for BVP at resonance with a 𝑝-Laplacian under at most linearly increasing condition and other suitable conditions imposed on the nonlinear term.

Motivated by the works mentioned above, we aim to study the existence of solutions for the three-point BVP (1.1). The methods used in this paper depend on the new Ge-Mawhin’s continuation theorem [3] and some inequality techniques. To generalize at most linearly increasing condition to BVP at resonance with a 𝑝-Laplacian, a small modification is added to the new Ge-Mawhin’s continuation theorem. What we obtained in this paper is applicable to BVP of differential equations with nonlinear term involving in the first-order derivative explicitly. Here we note that the techniques used in [3] are not applicable to such case. An existence result is also established for the BVP at resonance on a half-line, which is new for multi-point BVPs on infinite intervals [16, 17].

The paper is organized as follows. In Section 2, we present some preliminaries. In Section 3, we discuss the existence of solutions for BVP (1.1) when 𝑇 is a real constant, which we call the finite case. In Section 4, we establish an existence result for the bounded solutions to BVP (1.1) when 𝑇=+∞, which we call the infinite case. Some explicit examples are also given in the last section to illustrate our main results.

2. Preliminaries

For the convenience of the readers, we provide here some definitions and lemmas which are important in the proof of our main results. Ge-Mawhin’s continuation theorem and the modified one are also stated in this section.

Lemma 2.1. Let Φ_𝑝(𝑠)=|𝑠|^𝑝−2𝑠, 𝑝>1. Then Φ_𝑝 satisfies the properties.(1)Φ_𝑝 is continuous, monotonically increasing, and invertible. Moreover Φ_𝑝⁻¹=Φ_𝑞 with 𝑞>1 a real number satisfying 1/𝑝+1/𝑞=1; (2)for any 𝑢,𝑣⩾0, Φ_𝑝(𝑢+𝑣)⩽Φ_𝑝(𝑢)+Φ_𝑝(𝑣)Φ,𝑖𝑓𝑝<2,_𝑝(𝑢+𝑣)⩽2^𝑝−2Φ_𝑝(𝑢)+Φ_𝑝(𝑣),𝑖𝑓𝑝⩾2. (2.1)

Definition 2.2. Let 𝑅² be an 2-dimensional Euclidean space with an appropriate norm |⋅|. A function 𝑓∶[0,𝑇]×𝑅²→𝑅 is called Φ_𝑞-Carathéodory if and only if (1)for each 𝑥∈𝑅², 𝑡↦𝑓(𝑡,𝑥) is measurable on [0,𝑇]; (2)for a.e. 𝑡∈[0,𝑇], 𝑥↦𝑓(𝑡,𝑥) is continuous on 𝑅²; (3)for each 𝑟>0, there exists a nonnegative function 𝜑_𝑟∈𝐿¹[0,𝑇] with 𝜑_𝑟,𝑞(𝑡)∶=Φ_𝑞(∫^𝑇_𝑡𝜑_𝑟(𝜏)𝑑𝜏)∈𝐿¹[0,𝑇] such that |𝑥|⩽𝑟implies𝑓(𝑡,𝑥)𝜑_𝑟(𝑡),a.e.𝑡∈[0,𝑇]. (2.2)

Next we state Ge-Mawhin’s continuation theorem [3, 4].

Definition 2.3. Let 𝑋, 𝑍 be two Banach spaces. A continuous opeartor 𝑀∶𝑋∩dom𝑀→𝑍 is called quasi-linear if and only if Im𝑀 is a closed subset of 𝑍 and Ker𝑀 is linearly homeomorphic to 𝑅^𝑛, where 𝑛 is an integer.

Let 𝑋₂ be the complement space of Ker𝑀 in 𝑋, that is, 𝑋=Ker𝑀⊕𝑋₂. Ω⊂𝑋 an open and bounded set with the origin 0∈Ω.

Definition 2.4. A continuous operator 𝑁_𝜆∶Ω→𝑍,𝜆∈[0,1] is said to be 𝑀-compact in Ω if there is a vector subspace 𝑍₁⊂𝑍 with dim𝑍₁=dimKer𝑀 and an operator 𝑅∶Ω×[0,1]→𝑋₂ continuous and compact such that for 𝜆∈[0,1], (𝐼−𝑄)𝑁_𝜆Ω⊂Im𝑀⊂(𝐼−𝑄)𝑍,𝑄𝑁_𝜆𝑥=0,𝜆∈(0,1)⟺𝑄𝑁𝑥=0,∀𝑥∈Ω,𝑅(⋅,0)isthezerooperator,𝑅(⋅,𝜆)|_Σ𝜆=(𝐼−𝑃)|_Σ𝜆,𝑀[𝑃+𝑅(⋅,𝜆)]=(𝐼−𝑄)𝑁_𝜆, (2.3)(2.4)(2.5)(2.6) where 𝑃, 𝑄 are projectors such that Im𝑃=Ker𝑀 and Im𝑄=𝑍₁, 𝑁=𝑁₁, Σ_𝜆={𝑥∈Ω, 𝑀𝑥=𝑁_𝜆𝑥}.

Theorem 2.5 (Ge-Mawhin’s continuation theorem). Let (𝑋,‖⋅‖_𝑋) and (𝑍,‖⋅‖_𝑍) be two Banach spaces, Ω⊂𝑋 an open and bounded set. Suppose 𝑀∶𝑋∩dom𝑀→𝑍 is a quasi-linear operator and 𝑁_𝜆∶Ω→𝑍, 𝜆∈[0,1] is 𝑀-compact. In addition, if (i)𝑀𝑥≠𝑁_𝜆𝑥, for 𝑥∈dom𝑀∩𝜕Ω, 𝜆∈(0,1), (ii)𝑄𝑁𝑥≠0, for 𝑥∈Ker𝑀∩𝜕Ω, (iii)deg(𝐽𝑄𝑁,Ω∩Ker𝑀,0)≠0, where 𝑁=𝑁₁. Then the abstract equation 𝑀𝑥=𝑁𝑥 has at least one solution in dom𝑀∩Ω.

According to the usual direct-sum spaces such as those in [3, 5, 7, 11–13], it is difficult (maybe impossible) to define the projector 𝑄 under the at most linearly increasing conditions. We have to weaken the conditions of Ge-Mawhin continuation theorem to resolve such problem.

Definition 2.6. Let 𝑌₁ be finite dimensional subspace of 𝑌. 𝑄∶𝑌→𝑌₁ is called a semiprojector if and only if 𝑄 is semilinear and idempotent, where 𝑄 is called semilinear provided 𝑄(𝜆𝑥)=𝜆𝑄(𝑥) for all 𝜆∈𝑅 and 𝑥∈𝑌.

Remark 2.7. Using similar arguments to those in [3], we can prove that when 𝑄 is a semiprojector, Ge-Mawhin’s continuation theorem still holds.

3. Existence Results for the Finite Case

Consider the Banach spaces 𝑋=𝐶¹[0,𝑇] endowed with the norm ‖𝑥‖_𝑋=max{‖𝑥‖_∞,‖𝑥′‖_∞}, where ‖𝑥‖_∞=max_{0⩽𝑡⩽𝑇}|𝑥(𝑡)| and 𝑍=𝐿¹[0,𝑇] with the usual Lebesgue norm denoted by ‖⋅‖_𝑧. Define the operator 𝑀 by 𝑀∶dom𝑀∩𝑋→𝑍,(𝑀𝑥)(𝑡)=Φ_𝑝(𝑥′(𝑡))^′,𝑡∈[0,𝑇], (3.1) where dom𝑀={𝑥∈𝐶¹[0,𝑇],Φ_𝑝(𝑥^′)∈𝐶¹[0,𝑇],𝑥(0)=𝑥(𝜂),𝑥^′(𝑇)=0}. Then by direct calculations, one has Ker𝑀={𝑥∈dom𝑀∩𝑋∶𝑥(𝑡)=𝑐∈𝑅,𝑡∈[0,𝑇]},Im𝑀=𝑦∈𝑍∶^𝜂₀Φ_𝑞^𝑇_𝑠𝑦(𝜏)𝑑𝜏𝑑𝑠=0. (3.2)

Obviously, Ker𝑀≃𝑅 and Im𝑀 is close. So the following result holds.

Lemma 3.1. Let 𝑀 be defined as (3.1), then 𝑀 is a quasi-linear operator.

Set the projector 𝑃 and semiprojector 𝑄 by 𝑃∶𝑋→𝑋,(𝑃𝑥)(𝑡)=𝑥(0),𝑡∈[0,𝑇],(𝑡)=1𝑄∶𝑍→𝑍,𝑄𝑦𝜌Φ_𝑝^𝜂₀Φ_𝑞^𝑇_𝑠𝑦(𝜏)𝑑𝜏𝑑𝑠,𝑡∈[0,𝑇], (3.3)(3.4) where 𝜌=((1/𝑞)(𝑇^𝑞−(𝑇−𝜂)^𝑞))^𝑝−1. Define the operator 𝑁_𝜆∶𝑋→𝑍, 𝜆∈[0,1] by 𝑁_𝜆𝑥(𝑡)=−𝜆𝑓𝑡,𝑥(𝑡),𝑥′(𝑡),𝑡∈[0,𝑇]. (3.5)

Lemma 3.2. Let Ω⊂𝑋 be an open and bounded set. If 𝑓 is a Carathéodory function, 𝑁_𝜆 is 𝑀-compact in Ω.

Proof. Choose 𝑍₁=Im𝑄 and define the operator 𝑅∶Ω×[0,1]→Ker𝑃 by 𝑅(𝑥,𝜆)(𝑡)=^𝑡₀Φ_𝑞^𝑇_𝑠𝜆𝑓𝜏,𝑥(𝜏),𝑥′(𝜏)−𝑄𝑓(𝜏)𝑑𝜏𝑑𝑠,𝑡∈[0,𝑇]. (3.6)
Obviously, dim𝑍₁=dimKer𝑀=1. Since 𝑓 is a Carathéodory function, we can prove that 𝑅(⋅,𝜆) is continuous and compact for any 𝜆∈[0,1] by the standard theories.
It is easy to verify that (2.3)–(2.5) in Definition 2.3 hold. Besides, for any 𝑥∈dom𝑀∩Ω, 𝑀[𝑃𝑥+𝑅(𝑥,𝜆)](𝑡)=Φ_𝑝𝑥(0)+^𝑡₀Φ_𝑞^𝑇_𝑠𝜆𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)𝑑𝜏−𝑄𝑓(𝜏)𝑑𝜏𝑑𝑠^′′=(𝐼−𝑄)𝑁_𝜆𝑥(𝑡),𝑡∈[0,𝑇]. (3.7) So 𝑁_𝜆 is 𝑀-compact in Ω.

Theorem 3.3. Let 𝑓∶[0,𝑇]×𝑅²→𝑅 be a Carathéodory function. Suppose that (H1) there exist 𝑒(𝑡)∈𝐿¹[0,𝑇] and Carathéodory functions 𝑔₁, 𝑔₂ such that 𝑓(𝑡,𝑢,𝑣)⩽𝑔₁(𝑡,𝑢)+𝑔₂(𝑡,𝑣)+𝑒(𝑡)𝑓𝑜𝑟𝑎.𝑒.𝑡∈[0,𝑇]𝑎𝑛𝑑𝑎𝑙𝑙(𝑢,𝑣)∈𝑅²,lim_𝑥→∞∫^𝑇₀𝑔_𝑖(𝜏,𝑥)𝑑𝜏Φ_𝑝(|𝑥|)=𝑟_𝑖∈[0,+∞),𝑖=1,2; (3.8)(H2) there exists 𝐵₁>0 such that for all 𝑡_𝜂∈[0,𝜂] and 𝑥∈𝐶¹[0,𝑇] with ‖𝑥‖_∞>𝐵₁, ^𝑇_𝑡𝜂𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)𝑑𝜏≠0; (3.9)(H3) there exists 𝐵₂>0 such that for each 𝑡∈[0,𝑇] and 𝑢∈𝑅 with |𝑢|>𝐵₂ either 𝑢𝑓(𝑡,𝑢,0)⩽0 or 𝑢𝑓(𝑡,𝑢,0)⩾0. Then BVP (1.1) has at least one solution provided 𝛼₁∶=2^𝑞−2𝑇^𝑝−1𝑟₁+𝑟₂^𝑞−1𝛼<1,𝑖𝑓𝑝<2,₂∶=2^𝑝−2𝑇^𝑝−1𝑟₁+𝑟₂^𝑞−1<1,𝑖𝑓𝑝⩾2. (3.10)

Proof. Let 𝑋, 𝑍, 𝑀, 𝑁_𝜆, 𝑃, and 𝑄 be defined as above. Then the solutions of BVPs (1.1) coincide with those of 𝑀𝑥=𝑁𝑥, where 𝑁=𝑁₁. So it is enough to prove that 𝑀𝑥=𝑁𝑥 has at least one solution.
Let Ω₁={𝑥∈dom𝑀∶𝑀𝑥=𝑁_𝜆𝑥,𝜆∈(0,1)}. If 𝑥∈Ω₁, then 𝑄𝑁_𝜆𝑥=0. Thus, Φ_𝑝^𝜂₀Φ_𝑞^𝑇_𝑠𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)𝑑𝜏𝑑𝑠=0. (3.11) The continuity of Φ_𝑝 and Φ_𝑞 together with condition (H2) implies that there exists 𝜉∈[0,𝑇] such that |𝑥(𝜉)|⩽𝐵₁. So |𝑥(𝑡)|⩽𝑥(𝜉)+^𝑡_𝜉𝑥^′(𝑠)𝑑𝑠⩽𝐵₁+𝑇𝑥′_∞,𝑡∈[0,𝑇]. (3.12) Noting that 𝑀𝑥=𝑁_𝜆𝑥, we have 𝑥^′(𝑡)=Φ_𝑞^𝑇_𝑡𝜆𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏),𝑥(𝑡)(0)+𝑑𝜏=𝑥^𝑡₀Φ_𝑞^𝑇_𝑠𝜆𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)𝑑𝜏𝑑𝑠. (3.13)
If 𝑝<2, choose 𝜖>0 such that 𝛼_1,𝜖∶=2^𝑞−2𝑇^𝑝−1𝑟₁+𝜖+𝑟₂+𝜖^𝑞−1<1. (3.14) For this 𝜖>0, there exists 𝛿>0 such that ^𝑇₀𝑔_𝑖(𝜏,𝑥)𝑑𝜏⩽𝑟_𝑖+𝜖Φ_𝑝(|𝑥|)∀|𝑥|>𝛿,𝑖=1,2. (3.15) Set 𝑔_𝑖,𝛿=^𝑇₀max_{|𝑥|⩽𝛿}𝑔_𝑖(𝜏,𝑥)𝑑𝜏,𝑖=1,2. (3.16) Noting (3.12)-(3.13), we have 𝑥^′(𝑡)=Φ_𝑞^𝑇_𝑡𝜆𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)𝑑𝜏⩽Φ_𝑞^𝑇₀𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)𝑑𝜏⩽Φ_𝑞^𝑇₀𝑔₁(𝜏,𝑥)+𝑔₂𝜏,𝑥^′+𝑒(𝜏)𝑑𝜏⩽Φ_𝑞𝑟₁+𝜖Φ_𝑝(|𝑥|)+𝑟₂+𝜖Φ_𝑝𝑥^′+𝑔_1,𝛿+𝑔_2,𝛿+‖𝑒‖_𝐿¹⩽𝛼_1,𝜖𝑥^′_∞+𝐵_𝛿, (3.17) where 𝐵_𝛿=2^𝑞−2((𝑟₁+𝜖)𝐵₁^𝑝−1+𝑔_1,𝛿+𝑔_2,𝛿+‖𝑒‖_𝐿¹)^𝑞−1. So 𝑥^′_∞⩽𝐵_𝛿1−𝛼_1,𝜖∶=𝐵^′. (3.18) And then ‖𝑥‖_𝑋⩽max{𝐵₁+𝑇𝐵^′,𝐵^′}∶=𝐵.
Similarly, if 𝑝⩾2, we can obtain ‖𝑥‖_𝑋⩽max{𝐵₁+𝑇𝐵^′,𝐵^′}∶=𝐵, where 𝐵^′=2^𝑝−2𝑟₁+𝜖𝐵₁^𝑝−1+𝑔_1,𝛿+𝑔_2,𝛿+‖𝑒‖_𝐿^1𝑞−11−𝛼_2,𝜖,𝛼_2,𝜖=2^𝑝−2𝑇^𝑝−1𝑟₁+𝜖+𝑟₂+𝜖^𝑞−1. (3.19) Above all, Ω₁ is bounded.
Set Ω_2,𝑖∶={𝑥∈Ker𝑀∶(−1)^𝑖𝜇𝑥+(1−𝜇)𝐽𝑄𝑁𝑥=0,𝜇∈[0,1]}, 𝑖=1,2, where 𝐽∶Im𝑄→Ker𝑀 is a homeomorphism defined by 𝐽𝑎=𝑎 for any 𝑎∈𝑅. Next we show that Ω_2,1 is bounded if the first part of condition (H3) holds. Let 𝑥∈Ω_2,1, then 𝑥=𝑎 for some 𝑎∈𝑅 and 1𝜇𝑎=1−𝜇𝜌Φ_𝑝^𝜂₀Φ_𝑞^𝑇_𝑠𝑓(𝜏,𝑎,0)𝑑𝜏𝑑𝑠. (3.20) If 𝜇=0, we can obtain that |𝑎|⩽𝐵₁. If 𝜇≠0, then |𝑎|⩽𝐵₂. Otherwise, 𝜇𝑎²1=𝑎1−𝜇𝜌Φ_𝑝^𝜂₀Φ_𝑞^𝑇_𝑠𝑓()=1𝜏,𝑎,0𝑑𝜏𝑑𝑠1−𝜇𝜌Φ_𝑝^𝜂₀Φ_𝑞^𝑇_𝑠𝑎𝑓(𝜏,𝑎,0)𝑑𝜏𝑑𝑠⩽0, (3.21) which is a contraction. So ‖𝑥‖_𝑋=|𝑎|⩽max{𝐵₁,𝐵₂} and Ω_2,1 is bounded. Similarly, we can obtain that Ω_2,2 is bounded if the other part of condition (H3) holds.
Let Ω={𝑥∈𝑋∶‖𝑥‖_𝑋<max{𝐵(𝐵),𝐵₁,𝐵₂}+1}. Then Ω₁∪Ω_2,1(∪Ω_2,2)⊂Ω. It is obvious that 𝑀𝑥≠𝑁_𝜆𝑥 for each (𝑥,𝜆)∈(dom𝑀∩𝜕Ω)×(0,1).
Take the homotopy 𝐻_𝑖∶ (Ker𝑀∩Ω)×[0,1]→𝑋 by 𝐻_𝑖𝑥,𝜇=(−1)^𝑖𝜇𝑥+1−𝜇𝐽𝑄𝑁𝑥,𝑖=1or2. (3.22) Then for each 𝑥∈Ker𝑀∩𝜕Ω and 𝜇∈[0,1], 𝐻_𝑖(𝑥,𝜇)≠0, so by the degree theory deg={𝐽𝑄𝑁,Ker𝑀∩Ω,0}=deg(−1)^𝑖𝐼,Ker𝑀∩Ω,0≠0. (3.23) Applying Theorem 2.5 together with Remark 2.7, we obtain that 𝑀𝑥=𝑁𝑥 has a solution in dom𝑀∩Ω. So (1.1) is solvable.

Corollary 3.4. Let 𝑓∶[0,𝑇]×𝑅²→𝑅 be a Carathéodory function. Suppose that (H2), (H3) in Theorem 3.3 hold. Suppose further that(H1′) there exist nonnegative functions 𝑔_𝑖∈𝐿¹[0,𝑇], 𝑖=0,1,2 such that 𝑓(𝑡,𝑢,𝑣)⩽𝑔₁(𝑡)|𝑢|^𝑝−1+𝑔₂(𝑡)|𝑣|^𝑝−1+𝑔₀(𝑡)𝑓𝑜𝑟𝑎.𝑒.𝑡∈[0,𝑇]𝑎𝑛𝑑𝑎𝑙𝑙(𝑢,𝑣)∈𝑅². (3.24)
Then BVP (1.1) has at least one solution provided 2^𝑞−2𝑇^𝑝−1𝑔_1𝐿1+𝑔_2𝐿1^𝑞−12<1,𝑖𝑓𝑝<2,_𝑝−2𝑇^𝑝−1𝑔_1𝐿1+𝑔_2𝐿1^𝑞−1<1,𝑖𝑓𝑝⩾2. (3.25)

If 𝑓 is a continuous function, we can establish the following existence result.

Theorem 3.5. Let 𝑓∶[0,𝑇]×𝑅²→𝑅 be a continuous function. Suppose that (H1), (H3) in Theorem 3.3 hold. Suppose further that (H2′) there exist 𝐵₃, 𝑎>0, 𝑏,𝑐⩾0 such that for all 𝑢∈𝑅 with |𝑢|>𝐵₃, it holds that 𝑓(𝑡,𝑢,𝑣)⩾𝑎|𝑢|−𝑏|𝑣|−𝑐∀𝑡∈[0,𝑇]𝑎𝑛𝑑𝑎𝑙𝑙𝑣∈𝑅. (3.26)Then BVP (1.1) has at least one solution provided 2^𝑞−2𝑏𝑎+𝑇^𝑝−1𝑟₁+𝑟₂^𝑞−12<1,𝑖𝑓𝑝<2,_𝑝−2𝑏𝑎+𝑇^𝑝−1𝑟₁+𝑟₂^𝑞−1<1,𝑖𝑓𝑝⩾2. (3.27)

Proof. If 𝑥∈dom𝑀 such that 𝑀𝑥=𝑁_𝜆𝑥 for some 𝜆∈(0,1), we have 𝑄𝑁_𝜆𝑥=0. The continuity of 𝑓 and Φ_𝑞 imply that there exists 𝜉∈[0,𝑇] such that 𝑓(𝜉,𝑥(𝜉),𝑥^′(𝜉))=0. From (H2^'), it holds 𝑥(𝜉)⩽max𝐵₃,𝑏𝑎𝑥^′_∞+𝑐𝑎. (3.28) Therefore, |𝑥(𝑡)|⩽𝑥(𝜉)+^𝑡_𝜉𝑥^′(𝑠𝑏)𝑑𝑠⩽𝑎+𝑇𝑥^′_∞+𝑐𝑎+𝐵₁,𝑡∈[0,𝑇]. (3.29) With a similar way to those in Theorem 3.3, we can prove that (1.1) has at least one solution.

Corollary 3.6. Let 𝑓∶[0,𝑇]×𝑅²→𝑅 be a continuous function. Suppose that conditions in Corollary 3.4 hold except (H2) changed with (H2^'). Then BVP (1.1) is also solvable.

4. Existence Results for the Infinite Case

In this section, we consider the BVP (1.1) on a half line. Since the half line is noncompact, the discussions are more complicated than those on finite intervals.

Consider the spaces 𝑋 and 𝑍 defined by 𝑋=𝑥∈𝐶¹[0,+∞),lim_𝑡→+∞𝑥(𝑡)exists,lim_𝑡→+∞𝑥^′(𝑡)exists,𝑍=𝑦∈𝐿¹[0,+∞),₀^+∞Φ_𝑞𝑠^+∞𝑦(𝜏)𝑑𝜏𝑑𝑠<+∞, (4.1) with the norms ‖𝑥‖_𝑋=max{‖𝑥‖_∞,‖𝑥^′‖_∞} and ‖𝑦‖_𝑍=‖𝑦‖_𝐿¹, respectively, where ‖𝑥‖_∞=sup_{0⩽𝑡<+∞}|𝑥(𝑡)|. By the standard arguments, we can prove that (𝑋,‖⋅‖_𝑋) and (𝑍,‖⋅‖_𝑍) are both Banach spaces.

Let the operators 𝑀, 𝑁_𝜆, and 𝑃 be defined as (3.1), (3.3), and (3.5), respectively, expect 𝑇 replaced by +∞. Set 𝜔(𝑡)=((1−𝑒^{−(𝑞−1)𝜂})/(𝑞−1))^1−𝑝𝑒^−𝑡, 𝑡∈[0,+∞) and define the semiprojector 𝑄∶𝑌→𝑌 by 𝑄𝑦(𝑡)=𝑤(𝑡)Φ_𝑝^𝜂₀Φ_𝑞𝑠^+∞𝑦(𝜏)𝑑𝜏𝑑𝑠,𝑡∈[0,+∞). (4.2)

Similarly, we can show that 𝑀 is a quasi-linear operator. In order to prove that 𝑁_𝜆 is 𝑀-compact in Ω, the following criterion is needed.

Theorem 4.1 (see [16]). Let 𝑀⊂𝐶_∞={𝑥∈𝐶[0,+∞),lim_𝑡→+∞𝑥(𝑡)exists}. Then 𝑀 is relatively compact if the following conditions hold: (a)all functions from 𝑀 are uniformly bounded; (b)all functions from 𝑀 are equicontinuous on any compact interval of [0,+∞); (c)all functions from 𝑀 are equiconvergent at infinity, that is, for any given 𝜖>0, there exists a 𝑇=𝑇(𝜖)>0 such that |𝑓(𝑡)−𝑓(+∞)|<𝜖, for all 𝑡>𝑇, 𝑓∈𝑀.

Lemma 4.2. Let Ω⊂𝑋 an open and bounded set with 0∈Ω. If 𝑓 is a Φ_𝑞-Carathéodory function, 𝑁_𝜆 is 𝑀-compact in Ω.

Proof. Let 𝑍₁=Im𝑄 and define the operator 𝑅∶Ω×[0,1]→Ker𝑃 by 𝑅(𝑥,𝜆)(𝑡)=^𝑡₀Φ_𝑞𝑠^+∞𝜆𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)−𝑄𝑓(𝜏)𝑑𝜏𝑑𝑠,𝑡∈[0,+∞). (4.3) We just prove that 𝑅(⋅,𝜆)∶Ω×[0,1]→𝑋 is what we need. The others are similar and are omitted here.
Firstly, we show that 𝑅 is well defined. Let 𝑥∈Ω, 𝜆∈[0,1]. Because Ω is bounded, there exists 𝑟>0 such that for any 𝑥∈Ω, ‖𝑥‖_𝑋⩽𝑟. Noting that 𝑓 is a Φ_𝑞-Carathéodory function, there exists 𝜑_𝑟∈𝐿¹[0,+∞) with 𝜑_𝑟,𝑞∈𝐿¹[0,+∞) such that 𝑓𝑡,𝑥(𝑡),𝑥^′(𝑡)⩽𝜑_𝑟(𝑡),a.e.𝑡∈[0,+∞). (4.4) Therefore |𝑅(𝑥,𝜆)(𝑡)|=^𝑡₀Φ_𝑞𝑠^+∞𝜆𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)−𝜏)⩽𝑄𝑓(𝑑𝜏𝑑𝑠_0+∞Φ_𝑞𝑠^+∞𝜑_𝑟(𝜏)+Υ_𝑟𝜔(𝜏)𝑑𝜏𝑑𝑠<+∞,∀𝑡∈[0,+∞), (4.5) where Υ_𝑟=Φ_𝑝(∫^𝜂₀Φ_𝑞(∫_𝑠^+∞𝜑_𝑟(𝜏)𝑑𝜏)𝑑𝑠). Meanwhile, for any 𝑡₁,𝑡₂∈[0,+∞), we have 𝑅(𝑥,𝜆)𝑡₁−𝑅(𝑥,𝜆)𝑡₂⩽^𝑡2_𝑡1Φ_𝑞𝑠^+∞𝜆𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)−𝜏⩽𝑄𝑓()𝑑𝜏𝑑𝑠_𝑡2𝑡1Φ_𝑞𝑠^+∞𝜑_𝑟(𝜏)+Υ_𝑟𝜔(𝜏)𝑑𝜏𝑑𝑠→0,as𝑡₁→𝑡₂,_𝑡2𝑡1𝜆𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)−𝑄𝑓(𝜏)𝑑𝜏⩽^𝑡2_𝑡1𝜑_𝑟(𝜏)+Υ_𝑟𝜔(𝜏)𝑑𝜏→0,as𝑡₁→𝑡₂. (4.6)(4.7) The continuity of Φ_𝑞 concludes that 𝑅(𝑥,𝜆)^′𝑡₁−𝑅(𝑥,𝜆)^′𝑡₂→0,as𝑡₁→𝑡₂. (4.8) It is easy to see that lim_𝑡→+∞𝑅(𝑥,𝜆)(𝑡) exists and lim_𝑡→+∞𝑅(𝑥,𝜆)′(𝑡)=0. So 𝑅(𝑥,𝜆)∈𝑋.
Next, we verify that 𝑅(⋅,𝜆) is continuous. Obviously 𝑅(𝑥,𝜆) is continuous in 𝜆 for any 𝑥∈Ω. Let 𝜆∈[0,1], 𝑥_𝑛→𝑥 in Ω as 𝑛→+∞. In fact, ₀^+∞𝑓𝜏,𝑥_𝑛,𝑥^′_𝑛−𝑓𝜏,𝑥,𝑥^′𝑑𝜏⩽2𝜑_𝑟𝐿1,_𝑡0Φ_𝑞𝑠^+∞𝑓𝜏,𝑥_𝑛,𝑥^′_𝑛𝑑𝜏−Φ_𝑞𝑠^+∞𝑓𝜏,𝑥,𝑥^′𝑑𝜏𝑑𝑠⩽2𝜑_{𝑟,𝑞𝐿1}. (4.9) So by Lebesgue Dominated Convergence theorem and the continuity of Φ_𝑞, we can obtain 𝑅𝑥_𝑛,𝜆−𝑅(𝑥,𝜆)_𝑋→0,as𝑛→+∞. (4.10)
Finally, 𝑅(⋅,𝜆) is compact for any 𝜆∈[0,1]. Let 𝑈⊂𝑋 be a bounded set and 𝜆∈[0,1], then there exists 𝑟₀>0 such that ‖𝑥‖_𝑋⩽𝑟₀ for any 𝑥∈𝑈. Thus we have ‖𝑅(𝑥,𝜆)‖_𝑋=max‖𝑅(𝑥,𝜆)‖_∞,𝑅^′(𝑥,𝜆)_∞⩽max₀^+∞Φ_𝑞𝑠^+∞𝜑_𝑟0(𝜏)+Υ_𝑟0𝜔(𝜏)Φ𝑑𝜏𝑑𝑠,_𝑞0+∞𝜑_𝑟0(𝜏)(𝜏),|𝑅(𝑡)(=+Υ𝜔𝑑𝜏𝑥,𝜆)(−𝑅𝑥,𝜆)(+∞)|_𝑡^+∞Φ_𝑞𝑠^+∞𝜆𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)−(𝜏)⩽𝑄𝑓𝑑𝜏𝑑𝑠_𝑡+∞Φ_𝑞𝑠^+∞𝜑_𝑟0(𝜏)+Υ_𝑟0𝜔(𝜏)𝑑𝜏𝑑𝑠→0,uniformlyas𝑅()𝑡→+∞,𝑥,𝜆^′(𝑡)−𝑅(𝑥,𝜆)^′(+∞)=Φ_𝑞𝑡^+∞𝜆𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)−𝑄𝑓(𝜏)𝑑𝜏⩽Φ_𝑞𝑡^+∞𝜑_𝑟0(𝜏)+Υ𝜔(𝜏)𝑑𝜏→0,uniformlyas𝑡→+∞. (4.11) Those mean that 𝑅(⋅,𝜆) is uniformly bounded and equiconvergent at infinity. Similarly to the proof of (4.3) and (4.6), we can show that 𝑅(⋅,𝜆) is equicontinuous. Through Lemma 4.2, 𝑅(⋅,𝜆)𝑈 is relatively compact. The proof is complete.

Theorem 4.3. Let 𝑓∶[0,+∞)×𝑅²→𝑅 be a continuous and Φ_𝑞-Carathéodory function. Suppose that (H4) there exist functions 𝑔₀,𝑔₁,𝑔₂∈𝐿¹[0,+∞) such that 𝑓(𝑡,𝑢,𝑣)⩽𝑔₁(𝑡)|𝑢|^𝑝−1+𝑔₂(𝑡)|𝑣|^𝑝−1+𝑔₀(𝑡)𝑓𝑜𝑟𝑎.𝑒.𝑡∈[0,+∞)𝑎𝑛𝑑𝑎𝑙𝑙(𝑢,𝑣)∈𝑅²,𝑔_{𝑖,𝑞𝐿1}∶=₀^+∞Φ_𝑞𝑠^+∞𝑔_𝑖(𝜏𝑔)𝑑𝜏𝑑𝑠<+∞,𝑖=0,1,2,₁₁∶=₀^+∞𝑡^𝑝−1𝑔₁(𝜏)𝑑𝜏<+∞; (4.12)(H5) there exists 𝛾>0 such that for all 𝜁 satisfying 𝑓(𝜁,𝑢,𝑣)=0,𝑓(𝑡,𝑢,𝑣)≠0,𝑡∈0,𝜁),(𝑢,𝑣)∈𝑅², (4.13) it holds 𝜁⩽𝛾; (H6) there exist 𝐵₄, 𝑎>0, 𝑏,𝑐⩾0 such that for all 𝑢∈𝑅 with |𝑢|>𝐵₄, it holds 𝑓(𝑡,𝑢,𝑣)⩾𝑎|𝑢|−𝑏|𝑣|−𝑐∀𝑡∈0,𝛾,𝑣∈𝑅; (4.14)(H7) there exists 𝐵₅>0 such that for all 𝑡∈[0,+∞) and 𝑢∈𝑅 with |𝑢|>𝐵₅ either 𝑢𝑓(𝑡,𝑢,0)⩽0 or 𝑢𝑓(𝑡,𝑢,0)⩾0. Then BVP (1.1) has at least one solution provided max2^𝑞−2𝑔_1,𝑞𝐿1,𝛽₁𝑔<1,𝑖𝑓𝑝<2,max_1,𝑞𝐿1,𝛽₂<1,𝑖𝑓𝑝⩾2, (4.15)where 𝛽₁∶=2^𝑞−2𝑏𝑎+𝛾^𝑝−1𝑔_1𝐿1+𝑔₁₁+𝑔_2𝐿1^𝑞−1,𝛽₂∶=2^2(𝑝−2)𝑏𝑎+𝛾^𝑝−1𝑔_1𝐿1+2^2(𝑞−2)𝑔₁₁+𝑔_2𝐿1^𝑞−1. (4.16)

Proof. Let 𝑋, 𝑍, 𝑀, 𝑁_𝜆, 𝑃, and 𝑄 be defined as above. Let Ω₁={𝑥∈dom𝑀∶𝑀𝑥=𝑁_𝜆𝑥, 𝜆∈(0,1)}. We will prove that Ω₁ is bounded. In fact, for any 𝑥∈Ω₁, 𝑄𝑁_𝜆𝑥=0, that is, 𝜔(𝑡)Φ_𝑝^𝜂₀Φ_𝑞𝑠^+∞𝜆𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)𝑑𝜏𝑑𝑠=0. (4.17) The continuity of Φ_𝑝 and Φ_𝑞 together with conditions (H5) and (H6) implies that there exists 𝜉⩽𝛾 such that 𝑥(𝜉)⩽max𝐵₄,𝑏𝑎𝑥′_∞+𝑐𝑎. (4.18) So, we have |𝑥(𝑡)|⩽𝑥(𝜉)+^𝑡_𝜉𝑥^′(𝑠)𝑑𝑠⩽max𝐵₄,𝑏𝑎𝑥′_∞+𝑐𝑎+𝑡+𝛾𝑥^′_∞,𝑡∈[0,+∞). (4.19)
If 𝑝<2, it holds |𝑥(𝑡)|^𝑝−1⩽𝑏𝑎+𝛾^𝑝−1+𝑡^𝑝−1𝑥^′_∞^𝑝−1+𝑐𝑎+𝐵₄^𝑝−1,𝑡∈[0,+∞). (4.20) Therefore 𝑥^′(𝑡)=Φ_𝑞𝑡^+∞𝜆𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)𝑑𝜏⩽Φ_𝑞0^+∞𝑔₁(𝜏)|𝑥(𝜏)|^𝑝−1+𝑔₂(𝜏)𝑥^′(𝜏)^𝑝−1+𝑔₀(𝜏)𝑑𝜏⩽𝛽₁𝑥^′_∞+2^𝑞−2𝑐/𝑎+𝐵₄^𝑝−1𝑔_1𝐿1+𝑔_0𝐿1^𝑞−1,𝑡∈[0,+∞) (4.21) concludes that 𝑥^′_∞⩽2^𝑞−2𝑐/𝑎+𝐵₄^𝑝−1𝑔_1𝐿1+𝑔_0𝐿1^𝑞−11−𝛽₁∶=𝐶. (4.22) Meanwhile |𝑥(𝑡)|=𝑥(0)+^𝑡₀Φ_𝑞𝑠^+∞𝜆𝑓𝜏,𝑥(𝜏),𝑥^′(𝜏)⩽|𝑥(0+𝑑𝜏𝑑𝑠)|₀^+∞Φ_𝑞𝑠^+∞𝑔₁|𝑥|^𝑝−1+𝑔₂𝑥^′𝑝−1+𝑔₀𝑑𝜏𝑑𝑠⩽2^𝑞−2𝑔_1,𝑞𝐿1‖𝑥‖_∞+𝐶₀ (4.23) implies that ‖𝑥‖_∞⩽𝐶₀1−2^𝑞−2𝑔_1,𝑞𝐿1, (4.24) where 𝐶₀=(𝑏/𝑎+𝛾+2^2(𝑞−2)‖𝑔_2,𝑞‖_𝐿¹)𝐶+𝐵₄+𝑐/𝑎+2^2(𝑞−2)‖𝑔_0,𝑞‖_𝐿¹.
If 𝑝⩾2, we can prove that 𝑥^′_∞⩽2^𝑝−2𝐵₄+𝑐/𝑎^𝑝−1𝑔_1𝐿1+𝑔_0𝐿1^𝑞−11−𝛽₂‖𝑥‖∶=𝐶,_∞⩽𝑏/𝑎+𝛾+𝑔_2,𝑞𝐿1𝐶+𝐵₄+𝑐/𝑎+𝑔_0,𝑞𝐿11−𝑔_1,𝑞𝐿1. (4.25)
So Ω₁ is bounded. With the similar arguments to those in Theorem 3.3, we can complete the proof.

5. Examples

Example 5.1. Consider the three-point BVPs for second-order differential equations 𝑥^′(𝑡)𝑥^′(𝑡)^′=𝑎₂(𝑡)𝑥^′(𝑡)+𝑎₁(𝑡)𝑥²(𝑡)sgn𝑥(𝑡)+𝑎₀(𝑡)𝑥(0)𝜂,0<𝑡<1,=𝑥,𝑥^′(1)=0, (5.1) where 𝑎_𝑖(𝑡)∈𝐶¹[0,1], 𝑖=0,1,2 with 𝑎₁=min|𝑎₁(𝑡)|>0.
Take 𝑓(𝑡,𝑢,𝑣)=𝑎₁(𝑡)𝑢²sgn𝑢+𝑎₂(𝑡)𝑣+𝑎₀(𝑡),𝑔₁(𝑡,𝑢)=𝑎₁(𝑡)𝑢²,𝑔₂(𝑡,𝑣)=𝑎₂(𝑡)|𝑣|, (5.2) and 𝑒(𝑡)=|𝑎₀(𝑡)|. Then, we have 𝑓(𝑡,𝑢,𝑣)⩽𝑔₁(𝑡,𝑢)+𝑔₂(𝑡,𝑣)+𝑒(𝑡),for(𝑡,𝑢,𝑣)∈[0,1]×𝑅²max_0⩽𝑡⩽1𝑔₁(𝑡,𝑥)|𝑥|=𝑎_1𝐿1∈[0,+∞),max_0⩽𝑡⩽1𝑔₁(𝑡,𝑥)|𝑥|𝑓(=0,𝑡,𝑢,𝑣)⩾𝑎₁|𝑢|−𝑎_2∞|𝑣|−𝑎_0∞,for(|𝑢|)∈[]×[)()𝑡,,𝑣0,𝑇1,+∞×𝑅,𝑢𝑓𝑡,𝑢,0=𝑎₁(𝑡)|𝑢|³+𝑎₀(𝑡)𝑢⩾0,for(𝑡,|𝑢|)∈[0,1]×𝑎_0∞𝑎₁,+∞. (5.3) By using Theorem 3.5, we can concluded that BVP (5.1) has at least one solution if ‖𝑎₂‖_∞𝑎₁+1²𝑎_1∞<12. (5.4)

Example 5.2. Consider the three-point BVPs for second-order differential equations on a half line 𝑥^′′(𝑡)+𝑒^−𝛼𝑡𝑝(𝑡)𝑥(𝑡)(𝑡)𝑥(0)𝜂+𝑞=0,0<𝑡<+∞,=𝑥,lim_𝑡→+∞𝑥^′(𝑡)=0, (5.5) where 𝛼>(1+√5)/2,𝑝(𝑡)=max{sin𝛽𝑡,1/2} and 𝑞(𝑡) continuous on [0,+∞) with 𝑞(𝑡)>0(or 𝑞(𝑡)<0) on [0,1) and 𝑞≡0 on [1,+∞).
Denote 𝑓(𝑡,𝑢)=𝑒^−𝛼𝑡𝑝(𝑡)𝑢+𝑞(𝑡). Set 𝑔₁(𝑡)=𝑒^−𝛼𝑡, 𝑔₀(𝑡)=𝑞(𝑡). By direct calculations, we obtain that ‖𝑔₁‖_𝐿¹=1/𝛼, ‖𝑔_1,𝑞‖_𝐿¹=‖𝑔₁‖₁=1/𝛼² and ‖𝑔_0,𝑞‖_𝐿¹⩽‖𝑔₀‖_𝐿¹⩽‖𝑞‖_∞. Furthermore, 𝑓(𝑡,𝑢)⩽𝑔₁(𝑡)|𝑢|+𝑔₀(𝑡,𝑓(⩾1)𝑡,𝑢)2𝑒^−𝛼|𝑢|−𝑞_∞. (5.6) If there exists 𝜉∈[0,+∞) such that 𝑓(𝜉,𝑢)=0, then 𝜉⩽1. Otherwise 𝑢𝑓(𝜉,𝑢)=𝑒^−𝛼𝜉𝑝(𝜉)𝑢²⩾12𝑒^−𝛼𝜉𝑢²>0,∀𝑢∈𝑅⧵{0} (5.7) which is a contraction.
Obviously max{1/𝛼,1/𝛼+1/𝛼²}<1. Meanwhile, it is easy to verify that condition (H7) holds. So Theorem 4.3 guarantees that (5.5) has at least one solution.

Acknowledgments

The paper is supported by the National Natural Science Foundation of China (no. 11101385, 11226133) and by the Fundamental Research Funds for the Central Universities.