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
(1.1)(Φp(x′))′+f(t,x,x′)=0,0<t<T,x(0)=x(η),x′(T)=0,
where Φp(s)=|s|p-2s, p>1, η∈(0,T) is a constant, T∈(0,+∞], and x′(T)=limt→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
(1.2)(Φp(x′))′=0,0<t<T,x(0)=x(η),x′(T)=0
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=(Φp(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, Φp(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
(1.3)(Φp(u′))′+f(t,u)=0,0<t<1,u(0)=0=G(u(η),u(1)),
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 Φp(s)=|s|p-2s, p>1. Then Φp satisfies the properties.

Φp is continuous, monotonically increasing, and invertible. Moreover Φp-1=Φq with q>1 a real number satisfying 1/p+1/q=1;

for any u,v⩾0,
(2.1)Φp(u+v)⩽Φp(u)+Φp(v),ifp<2,Φp(u+v)⩽2p-2(Φp(u)+Φp(v)),ifp⩾2.

Definition 2.2.

Let R2 be an 2-dimensional Euclidean space with an appropriate norm |·|. A function f:[0,T]×R2→R is called Φq-Carathéodory if and only if

for each x∈R2, t↦f(t,x) is measurable on [0,T];

for a.e. t∈[0,T], x↦f(t,x) is continuous on R2;

for each r>0, there exists a nonnegative function φr∈L1[0,T] with φr,q(t):=Φq(∫tTφr(τ)dτ)∈L1[0,T] such that
(2.2)|x|⩽rimplies|f(t,x)|φr(t),a.e.t∈[0,T].

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∩dom M→Z is called quasi-linear if and only if Im M is a closed subset of Z and Ker M is linearly homeomorphic to Rn, where n is an integer.

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

Definition 2.4.

A continuous operator Nλ:Ω¯→Z,λ∈[0,1] is said to be M-compact in Ω¯ if there is a vector subspace Z1⊂Z with dim Z1=dim Ker M and an operator R:Ω¯×[0,1]→X2 continuous and compact such that for λ∈[0,1],
(2.3)(I-Q)Nλ(Ω¯)⊂Im M⊂(I-Q)Z,(2.4)QNλx=0,λ∈(0,1)⟺QNx=0,∀x∈Ω,(2.5)R(·,0)isthezerooperator,R(·,λ)|Σλ=(I-P)|Σλ,(2.6)M[P+R(·,λ)]=(I-Q)Nλ,
where P, Q are projectors such that Im P=Ker M and Im Q=Z1, N=N1, Σλ={x∈Ω¯, Mx=Nλx}.

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 Nλ:Ω¯→Z, λ∈[0,1] is M-compact. In addition, if

Mx≠Nλx, for x∈
dom M∩∂Ω, λ∈(0,1),

QNx≠0, for x∈
Ker M∩∂Ω,

deg(JQN,Ω∩
Ker M,0)≠0,

where N=N1. Then the abstract equation Mx=Nx has at least one solution in
dom
M∩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∥∞=max0⩽t⩽T|x(t)| and Z=L1[0,T] with the usual Lebesgue norm denoted by ∥·∥z. Define the operator M by
(3.1)M:dom M∩X→Z,(Mx)(t)=(Φp(x′(t)))′,t∈[0,T],
where dom M={x∈C1[0,T],Φp(x′)∈C1[0,T],x(0)=x(η),x′(T)=0}. Then by direct calculations, one has
(3.2)Ker M={x∈dom M∩X:x(t)=c∈R,t∈[0,T]},Im M={y∈Z:∫0ηΦq(∫sTy(τ)dτ)ds=0}.

Obviously, Ker M≃R and Im M 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
(3.3)P:X→X,(Px)(t)=x(0),t∈[0,T],(3.4)Q:Z→Z,(Qy)(t)=1ρΦp(∫0ηΦq(∫sTy(τ)dτ)ds),t∈[0,T],
where ρ=((1/q)(Tq-(T-η)q))p-1. Define the operator Nλ:X→Z, λ∈[0,1] by
(3.5)(Nλx)(t)=-λf(t,x(t),x′(t)),t∈[0,T].

Lemma 3.2.

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

Proof.

Choose Z1=Im Q and define the operator R:Ω¯×[0,1]→Ker P by
(3.6)R(x,λ)(t)=∫0tΦq(∫sTλ(f(τ,x(τ),x′(τ))-(Qf)(τ))dτ)ds,t∈[0,T].

Obviously, dim Z1=dim Ker M=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∈dom M∩Ω¯,
(3.7)M[Px+R(x,λ)](t)=(Φp[x(0)+∫0tΦq(∫sTλ(f(τ,x(τ),x′(τ))dτ-(Qf)(τ))dτ)ds]′)′=((I-Q)Nλx)(t),t∈[0,T].
So Nλ is M-compact in Ω¯.

Theorem 3.3.

Let f:[0,T]×R2→R be a Carathéodory function. Suppose that

there exist e(t)∈L1[0,T] and Carathéodory functions g1, g2 such that
(3.8)|f(t,u,v)|⩽g1(t,u)+g2(t,v)+e(t)fora.e.t∈[0,T]andall(u,v)∈R2,limx→∞∫0Tgi(τ,x)dτΦp(|x|)=ri∈[0,+∞),i=1,2;

there exists B1>0 such that for all tη∈[0,η] and x∈C1[0,T] with ∥x∥∞>B1,
(3.9)∫tηTf(τ,x(τ),x′(τ))dτ≠0;

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
(3.10)α1:=2q-2(Tp-1r1+r2)q-1<1,ifp<2,α2:=(2p-2Tp-1r1+r2)q-1<1,ifp⩾2.

Proof.

Let X, Z, M, Nλ, 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∈dom M:Mx=Nλx,λ∈(0,1)}. If x∈Ω1, then QNλx=0. Thus,
(3.11)Φp(∫0ηΦq(∫sTf(τ,x(τ),x′(τ))dτ)ds)=0.
The continuity of Φp and Φq together with condition (H2) implies that there exists ξ∈[0,T] such that |x(ξ)|⩽B1. So
(3.12)|x(t)|⩽|x(ξ)|+∫ξt|x′(s)|ds⩽B1+T∥x′∥∞,t∈[0,T].
Noting that Mx=Nλx, we have
(3.13)x′(t)=Φq(∫tTλf(τ,x(τ),x′(τ))dτ),x(t)=x(0)+∫0tΦq(∫sTλf(τ,x(τ),x′(τ))dτ)ds.

If p<2, choose ϵ>0 such that
(3.14)α1,ϵ:=2q-2(Tp-1(r1+ϵ)+(r2+ϵ))q-1<1.
For this ϵ>0, there exists δ>0 such that
(3.15)∫0Tgi(τ,x)dτ⩽(ri+ϵ)Φp(|x|)∀|x|>δ,i=1,2.
Set
(3.16)gi,δ=∫0T(max|x|⩽δgi(τ,x))dτ,i=1,2.
Noting (3.12)-(3.13), we have
(3.17)|x′(t)|=|Φq(∫tTλf(τ,x(τ),x′(τ))dτ)|⩽Φq(∫0T|f(τ,x(τ),x′(τ))|dτ)⩽Φq(∫0T(g1(τ,x)+g2(τ,x′)+e(τ))dτ)⩽Φq((r1+ϵ)Φp(|x|)+(r2+ϵ)Φp(|x′|)+g1,δ+g2,δ+∥e∥L1)⩽α1,ϵ∥x′∥∞+Bδ,
where Bδ=2q-2((r1+ϵ)B1p-1+g1,δ+g2,δ+∥e∥L1)q-1. So
(3.18)∥x′∥∞⩽Bδ1-α1,ϵ:=B′.
And then ∥x∥X⩽max{B1+TB′,B′}:=B.

Similarly, if p⩾2, we can obtain ∥x∥X⩽max{B1+TB~′,B~′}:=B~, where
(3.19)B~′=(2p-2(r1+ϵ)B1p-1+g1,δ+g2,δ+∥e∥L1)q-11-α2,ϵ,α2,ϵ=(2p-2Tp-1(r1+ϵ)+(r2+ϵ))q-1.
Above all, Ω1 is bounded.

Set Ω2,i:={x∈Ker M:(-1)iμx+(1-μ)JQNx=0,μ∈[0,1]}, i=1,2, where J:Im Q→Ker M is a homeomorphism defined by Ja=a for any a∈R. Next we show that Ω2,1 is bounded if the first part of condition (H3) holds. Let x∈Ω2,1, then x=a for some a∈R and
(3.20)μa=(1-μ)1ρΦp(∫0ηΦq(∫sTf(τ,a,0)dτ)ds).
If μ=0, we can obtain that |a|⩽B1. If μ≠0, then |a|⩽B2. Otherwise,
(3.21)μa2=a(1-μ)1ρΦp(∫0ηΦq(∫sTf(τ,a,0)dτ)ds)=(1-μ)1ρΦp(∫0ηΦq(∫sTaf(τ,a,0)dτ)ds)⩽0,
which is a contraction. So ∥x∥X=|a|⩽max{B1,B2} and Ω2,1 is bounded. Similarly, we can obtain that Ω2,2 is bounded if the other part of condition (H3) holds.

Let Ω={x∈X:∥x∥X<max{B(B~),B1,B2}+1}. Then Ω1∪Ω2,1(∪Ω2,2)⊂Ω. It is obvious that Mx≠Nλx for each (x,λ)∈(dom M∩∂Ω)×(0,1).

Take the homotopy Hi: (Ker M∩Ω¯)×[0,1]→X by
(3.22)Hi(x,μ)=(-1)iμx+(1-μ)JQNx,i=1or2.
Then for each x∈Ker M∩∂Ω and μ∈[0,1], Hi(x,μ)≠0, so by the degree theory
(3.23)deg={JQN,Ker M∩Ω,0}=deg{(-1)iI,Ker M∩Ω,0}≠0.
Applying Theorem 2.5 together with Remark 2.7, we obtain that Mx=Nx has a solution in dom M∩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

there exist nonnegative functions gi∈L1[0,T], i=0,1,2 such that
(3.24)|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.

Then BVP (1.1) has at least one solution provided
(3.25)2q-2(Tp-1∥g1∥L1+∥g2∥L1)q-1<1,ifp<2,(2p-2Tp-1∥g1∥L1+∥g2∥L1)q-1<1,ifp⩾2.

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

there exist B3, a>0, b,c⩾0 such that for all u∈R with |u|>B3, it holds that
(3.26)|f(t,u,v)|⩾a|u|-b|v|-c∀t∈[0,T]andallv∈R.

Then BVP (1.1) has at least one solution provided
(3.27)2q-2((ba+T)p-1r1+r2)q-1<1,ifp<2,(2p-2(ba+T)p-1r1+r2)q-1<1,ifp⩾2.Proof.

If x∈dom M such that Mx=Nλx for some λ∈(0,1), we have QNλx=0. The continuity of f and Φq imply that there exists ξ∈[0,T] such that f(ξ,x(ξ),x′(ξ))=0. From (H2^{'}), it holds
(3.28)|x(ξ)|⩽max{B3,ba∥x′∥∞+ca}.
Therefore,
(3.29)|x(t)|⩽|x(ξ)|+∫ξt|x′(s)|ds⩽(ba+T)∥x′∥∞+ca+B1,t∈[0,T].
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
(4.1)X={x∈C1[0,+∞),limt→+∞x(t)exists,limt→+∞x′(t)exists},Z={y∈L1[0,+∞),∫0+∞Φq(∫s+∞|y(τ)|dτ)ds<+∞},
with the norms ∥x∥X=max{∥x∥∞,∥x′∥∞} and ∥y∥Z=∥y∥L1, respectively, where ∥x∥∞=sup0⩽t<+∞|x(t)|. By the standard arguments, we can prove that (X,∥·∥X) and (Z,∥·∥Z) are both Banach spaces.

Let the operators M, Nλ, 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
(4.2)(Qy)(t)=w(t)Φp(∫0ηΦq(∫s+∞y(τ)dτ)ds),t∈[0,+∞).

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

Theorem 4.1 (see [<xref ref-type="bibr" rid="B1">16</xref>]).

Let M⊂C∞={x∈C[0,+∞),
lim
t→+∞x(t)exists}. Then M is relatively compact if the following conditions hold:

all functions from M are uniformly bounded;

all functions from M are equicontinuous on any compact interval of [0,+∞);

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 Φq-Carathéodory function, Nλ is M-compact in Ω¯.

Proof.

Let Z1=Im Q and define the operator R:Ω¯×[0,1]→Ker P by
(4.3)R(x,λ)(t)=∫0tΦq(∫s+∞λ(f(τ,x(τ),x′(τ))-(Qf)(τ))dτ)ds,t∈[0,+∞).
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 Φq-Carathéodory function, there exists φr∈L1[0,+∞) with φr,q∈L1[0,+∞) such that
(4.4)|f(t,x(t),x′(t))⩽|φr(t),a.e.t∈[0,+∞).
Therefore
(4.5)|R(x,λ)(t)|=|∫0tΦq(∫s+∞λ(f(τ,x(τ),x′(τ))-(Qf)(τ))dτ)ds|⩽∫0+∞Φq(∫s+∞(φr(τ)+Υrω(τ))dτ)ds<+∞,∀t∈[0,+∞),
where Υr=Φp(∫0ηΦq(∫s+∞φr(τ)dτ)ds). Meanwhile, for any t1,t2∈[0,+∞), we have
(4.6)|R(x,λ)(t1)-R(x,λ)(t2)|⩽∫t1t2Φq(∫s+∞λ|f(τ,x(τ),x′(τ))-(Qf)(τ)|dτ)ds⩽∫t1t2Φq(∫s+∞(φr(τ)+Υrω(τ))dτ)ds→0,ast1→t2,(4.7)|∫t1t2λ(f(τ,x(τ),x′(τ))-(Qf)(τ))dτ|⩽∫t1t2(φr(τ)+Υrω(τ))dτ→0,ast1→t2.
The continuity of Φq concludes that
(4.8)|R(x,λ)′(t1)-R(x,λ)′(t2)|→0,ast1→t2.
It is easy to see that limt→+∞R(x,λ)(t) exists and limt→+∞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], xn→x in Ω as n→+∞. In fact,
(4.9)|∫0+∞(f(τ,xn,xn′)-f(τ,x,x′))dτ|⩽2∥φr∥L1,|∫0t[Φq(∫s+∞f(τ,xn,xn′)dτ)-Φq(∫s+∞f(τ,x,x′)dτ)]ds|⩽2∥φr,q∥L1.
So by Lebesgue Dominated Convergence theorem and the continuity of Φq, we can obtain
(4.10)∥R(xn,λ)-R(x,λ)∥X→0,asn→+∞.

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
(4.11)∥R(x,λ)∥X=max{∥R(x,λ)∥∞,∥R′(x,λ)∥∞}⩽max{∫0+∞Φq(∫s+∞(φr0(τ)+Υr0ω(τ))dτ)ds,Φq(∫0+∞(φr0(τ)+Υω(τ))dτ)},|R(x,λ)(t)-R(x,λ)(+∞)|=|∫t+∞Φq(∫s+∞λ(f(τ,x(τ),x′(τ))-(Qf)(τ))dτ)ds|⩽∫t+∞Φq(∫s+∞(φr0(τ)+Υr0ω(τ))dτ)ds→0,uniformlyast→+∞,|R(x,λ)′(t)-R(x,λ)′(+∞)|=|Φq(∫t+∞λ(f(τ,x(τ),x′(τ))-(Qf)(τ))dτ)|⩽Φq(∫t+∞(φr0(τ)+Υω(τ))dτ)→0,uniformlyast→+∞.
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 Φq-Carathéodory function. Suppose that

there exist functions g0,g1,g2∈L1[0,+∞) such that
(4.12)|f(t,u,v)|⩽g1(t)|u|p-1+g2(t)|v|p-1+g0(t)fora.e.t∈[0,+∞)andall(u,v)∈R2,∥gi,q∥L1:=∫0+∞Φq(∫s+∞|gi(τ)|dτ)ds<+∞,i=0,1,2,∥g1∥1:=∫0+∞tp-1|g1(τ)|dτ<+∞;

there exists γ>0 such that for all ζ satisfying
(4.13)f(ζ,u,v)=0,f(t,u,v)≠0,t∈[0,ζ),(u,v)∈R2,
it holds ζ⩽γ;

there exist B4, a>0, b,c⩾0 such that for all u∈R with |u|>B4, it holds
(4.14)|f(t,u,v)|⩾a|u|-b|v|-c∀t∈[0,γ],v∈R;

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
(4.15)max{2q-2∥g1,q∥L1,β1}<1,ifp<2,max{∥g1,q∥L1,β2}<1,ifp⩾2,

where
(4.16)β1:=2q-2((ba+γ)p-1∥g1∥L1+∥g1∥1+∥g2∥L1)q-1,β2:=(22(p-2)(ba+γ)p-1∥g1∥L1+22(q-2)∥g1∥1+∥g2∥L1)q-1.Proof.

Let X, Z, M, Nλ, P, and Q be defined as above. Let Ω1={x∈dom M:Mx=Nλx, λ∈(0,1)}. We will prove that Ω1 is bounded. In fact, for any x∈Ω1, QNλx=0, that is,
(4.17)ω(t)Φp(∫0ηΦq(∫s+∞λf(τ,x(τ),x′(τ))dτ)ds)=0.
The continuity of Φp and Φq together with conditions (H5) and (H6) implies that there exists ξ⩽γ such that
(4.18)|x(ξ)|⩽max{B4,ba∥x′∥∞+ca}.
So, we have
(4.19)|x(t)|⩽|x(ξ)|+|∫ξtx′(s)ds|⩽max{B4,ba∥x′∥∞+ca}+(t+γ)∥x′∥∞,t∈[0,+∞).

If p<2, it holds
(4.20)|x(t)|p-1⩽((ba+γ)p-1+tp-1)∥x′∥∞p-1+(ca+B4)p-1,t∈[0,+∞).
Therefore
(4.21)|x′(t)|=|Φq(∫t+∞λf(τ,x(τ),x′(τ))dτ)|⩽Φq(∫0+∞(g1(τ)|x(τ)|p-1+g2(τ)|x′(τ)|p-1+g0(τ))dτ)⩽β1∥x′∥∞+2q-2((c/a+B4)p-1∥g1∥L1+∥g0∥L1)q-1,t∈[0,+∞)
concludes that
(4.22)∥x′∥∞⩽2q-2((c/a+B4)p-1∥g1∥L1+∥g0∥L1)q-11-β1:=C.
Meanwhile
(4.23)|x(t)|=|x(0)+∫0tΦq(∫s+∞λf(τ,x(τ),x′(τ))dτ)ds|⩽|x(0)|+∫0+∞Φq(∫s+∞(g1|x|p-1+g2|x′|p-1+g0)dτ)ds⩽2q-2∥g1,q∥L1∥x∥∞+C0
implies that
(4.24)∥x∥∞⩽C01-2q-2∥g1,q∥L1,
where C0=(b/a+γ+22(q-2)∥g2,q∥L1)C+B4+c/a+22(q-2)∥g0,q∥L1.

If p⩾2, we can prove that
(4.25)∥x′∥∞⩽(2p-2(B4+c/a)p-1∥g1∥L1+∥g0∥L1)q-11-β2:=C~,∥x∥∞⩽(b/a+γ+∥g2,q∥L1)C~+B4+c/a+∥g0,q∥L11-∥g1,q∥L1.

So Ω1 is bounded. With the similar arguments to those in Theorem 3.3, we can complete the proof.

5. ExamplesExample 5.1.

Consider the three-point BVPs for second-order differential equations
(5.1)(x′(t)|x′(t)|)′=a2(t)x′(t)+a1(t)x2(t)sgnx(t)+a0(t),0<t<1,x(0)=x(η),x′(1)=0,
where ai(t)∈C1[0,1], i=0,1,2 with a1=min|a1(t)|>0.

Take
(5.2)f(t,u,v)=a1(t)u2sgnu+a2(t)v+a0(t),g1(t,u)=|a1(t)|u2,g2(t,v)=|a2(t)||v|,
and e(t)=|a0(t)|. Then, we have
(5.3)|f(t,u,v)|⩽g1(t,u)+g2(t,v)+e(t),for(t,u,v)∈[0,1]×R2max0⩽t⩽1g1(t,x)|x|=∥a1∥L1∈[0,+∞),max0⩽t⩽1g1(t,x)|x|=0,|f(t,u,v)|⩾a1|u|-∥a2∥∞|v|-∥a0∥∞,for (t,|u|,v)∈[0,T]×[1,+∞)×R,uf(t,u,0)=a1(t)|u|3+a0(t)u⩾0,for(t,|u|)∈[0,1]×[∥a0∥∞a1,+∞).
By using Theorem 3.5, we can concluded that BVP (5.1) has at least one solution if
(5.4)(∥a2∥∞a1+1)2∥a1∥∞<12.

Example 5.2.

Consider the three-point BVPs for second-order differential equations on a half line
(5.5)x′′(t)+e-αtp(t)x(t)+q(t)=0,0<t<+∞,x(0)=x(η),limt→+∞x′(t)=0,
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, ∥g1,q∥L1=∥g1∥1=1/α2 and ∥g0,q∥L1⩽∥g0∥L1⩽∥q∥∞. Furthermore,
(5.6)|f(t,u)|⩽|g1(t)||u|+|g0(t)|,|f(t,u)|⩾12e-α|u|-∥q∥∞.
If there exists ξ∈[0,+∞) such that f(ξ,u)=0, then ξ⩽1. Otherwise
(5.7)uf(ξ,u)=e-αξp(ξ)u2⩾12e-αξu2>0,∀u∈R∖{0}
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.

CabadaA.PousoR. L.Existence results for the problem (φ(u'))'=f(t,u,u') with nonlinear boundary conditionsGarcía-HuidobroM.ManásevichR.YanP.ZhangM.A p-Laplacian problem with a multi-point boundary conditionGeW.RenJ.An extension of Mawhin's continuation theorem and its application to boundary value problems with a p-LaplacianGeW.JiangD.Upper and lower solutions method and a singular superlinear boundary value problem for the one-dimensional p-LaplacianLüH.ZhongC.A note on singular nonlinear boundary value problems for the one-dimensional p-LaplacianMawhinJ.Some boundary value problems for Hartman-type perturbations of the ordinary vector p-Laplaciandel PinoM.DrábekP.ManásevichR.The Fredholm alternative at the first eigenvalue for the one-dimensional p-LaplacianO'ReganD.Some general existence principles and results for (ϕ(y′))′=qf(t,y,y′), 0<t<1ZhangM.Nonuniform nonresonance at the first eigenvalue of the p-LaplacianDuZ.LinX.GeW.On a third-order multi-point boundary value problem at resonanceFengW.WebbJ. R. L.Solvability of three point boundary value problems at resonanceGuptaC. P.A second order m-point boundary value problem at resonanceLiuB.Solvability of multi-point boundary value problem at resonance. IVMaR.AgarwalR. P.O'ReganD.LianH.GeW.Solvability for second-order three-point boundary value problems on a half-line