Solvability for Second-order Nonlocal Boundary Value Problems with a P-laplacian at Resonance on a Half-line *

This paper investigates the solvability of the second-order boundary value problems with the one-dimensional p-Laplacian at resonance on a half-line    (c(t)φ p (x (t))) = f (t, x(t), x (t)), 0 < t < ∞, x(0) = n i=1 µ i x(ξ i), lim t→+∞ c(t)φ p (x (t)) = 0 and    (c(t)φ p (x (t))) + g(t)h(t, x(t), x (t)) = 0, 0 < t < ∞, x(0) = ∞ 0 g(s)x(s)ds, lim t→+∞ c(t)φ p (x (t)) = 0 with multi-point and integral boundary conditions, respectively, where φ p (s) = |s| p−2 s, p > 1. The arguments are based upon an extension of Mawhin's continuation theorem due to Ge. And examples are given to illustrate our results.


INTRODUCTION
In this paper, we consider the second-order boundary value problems with a p-Laplacian on a half line (c(t)φ p (x (t))) = f (t, x(t), x (t)), 0 < t < ∞, (1.1) and (c(t)φ p (x (t))) + g(t)h(t, x(t), x (t)) = 0, 0 < t < ∞, (1.4) Throughout this paper, we assume c(s) )ds = 0. Due to the conditions (1.3) and (1.6), the differential operator d dt (cφ p ( d dt •)) in (1.1) and (1.4) is not invertible under the boundary conditions (1.2) and (1.5), respectively.In the literature, boundary value problems of this type are referred to problems at resonance.The theory of boundary value problems (in short: BVPs) with multi-point and integral boundary conditions arises in a variety of different areas of applied mathematics and physics.For example, bridges of small size are often designed with two supported points, which leads to a standard two-point boundary condition and bridges of large size are sometimes contrived with multi-point supports, which corresponds to a multi-point boundary condition [1].Heat conduction, chemical engineering, underground water flow, thermo-elasticity and plasma physics can be reduced to the nonlocal problems with integral boundary conditions [2,3].The study of multi-point BVPs for linear second-order ordinary differential equations was initiated by Il'in and Moiseev [4] in 1987.
Second-order BVPs on infinite intervals arising from the study of radially symmetric solutions of nonlinear elliptic equation and models of gas pressure in a semi-infinite porous medium, have received much attention.For an extensive collection of results on BVPs on unbounded domains, we refer the readers to a monograph by Agarwal and O'Regan [16].Other recent results and methods for BVPs on a half-line can be found in [14,15] and the references therein.
From the existed results, we can see a fact: for the resonance case, only BVPs with linear differential operator on half-line were considered.The BVPs with multi-point and integral boundary conditions on a half-line have not investigated till now.Although some authors (see [5,9,10,12,17 Motivated by the above works, we intend to discuss the BVPs (1.1)-(1.2) and (1.4)-(1.5)at resonance on a half-line.Due to the fact that the classical Mawhin's continuation theorem can't be directly used to discuss the BVP with nonlinear differential operator, in this paper, we investigate the BVPs (1.1)-(1.2) and (1.4)-(1.5)by applying an extension of Mawhin's continuation theorem due to Ge [5].Furthermore, examples are given to illustrate the results.

PRELIMINARIES
For the convenience of readers, we present here some definitions and lemmas.Definition 2.1.We say that a mapping f : [0, ∞)×R 2 → R satisfies the Carathéodory conditions, if the following two conditions are satisfied: In addition, f is called a L 1 -Carathéodory function if (B1), (B2) and (B3) hold, f is called a g-Carathéodory function if (B1), (B2) and (B4) are satisfied.
A continuous operator M : X∩ domM → Z is said to be quasi-linear if Definition 2.3 [6] .Let X be a Banach spaces and X 1 ⊂ X a subspace.The operator P : X → X 1 is said to be a projector provided P 2 = P , P (λ Let X 1 = ker M and X 2 be the complement space of X 1 in X, then X = X 1 ⊕ X 2 .On the other hand, suppose Z 1 is a subspace of Z and Z 2 is the complement of Z 1 in Z, then Z = Z 1 ⊕ Z 2 .
Let P : X → X 1 be a projector and Q : Z → Z 1 be a semi-projector, and Ω ⊂ X an open and bounded set with the origin θ ∈ Ω, where θ is the origin of a linear space.Suppose Definition 2.4 [5] .N λ is said to be M-compact in Ω if there is a vector subspace Z 1 of Z with dimZ 1 = dimX 1 and an operator R : Ω × [0, 1] → X 2 being continuous and compact such that for QN λ x = 0, λ ∈ (0, 1) ⇐⇒ QNx = 0, (2.2) ) (2.4) Theorem 2.1 [5] .Let X and Z be two Banach spaces with norms || • || X and || • || Z , respectively, and Ω ⊂ X an open and bounded set.Suppose M : X ∩ domM → Z is a quasi-linear operator and Then the abstract equation Mx = Nx has at least one solution in Ω.

RELATED LEMMAS
Let AC[0, ∞) denote the space of absolutely continuous functions on the interval [0, ∞).In this paper, we work in the following spaces and where C is a constant.In view of (1.2) and (1.3), we have Conversely, if (3.3) holds for y ∈ Y , we take x ∈ dom M 1 as given by (3.2), then (c(t)φ p (x (t))) = 2) is satisfied.Hence, we have Similarly, we can calculate that and prove that Hence, M 2 is also a quasi-linear operator.
In order to apply Theorem 2.1, we have to prove that R is completely continuous, and then to prove that N is M-compact.Because the Arzelà-Ascoli theorem fails to the noncompact interval case, we will use the following criterion.
Lemma 3.2 [14] .Let X be the space of all bounded continuous vector-valued functions on [0, ∞) and S ⊂ X.Then S is relatively compact if the following conditions hold: (F1) S is bounded in X; (F2) all functions from S are equicontinuous on any compact subinterval of [0, ∞); (F3) all functions from S are equiconvergent at infinity, that is, for any given ε > 0, there Proof.We recall the condition (A2) and define the continuous operator where a semi-projector and dimX 1 =1=dimY 1 .Moreover, (3.4) and (3.6) imply that ImM 1 =ker Q 1 .

It is easy to see
where X 2 is the complement space of We first assert that R 1 is relatively compact for any λ ∈ [0, 1].In fact, since U ⊂ X is a bounded set, there exists r > 0 such that U ⊂ {x ∈ X : ||x|| X ≤ r}.Because the function f is for x ∈ U. Then for any x ∈ U , λ ∈ [0, 1], we have From (A1), we can see that φ q ( 1 c ) is bounded.Hence, In additional, we claim that R 1 (•, λ)U is equiconvergent at infinity.In fact, which implies (2.3).For any x ∈ U, we have where Ω ⊂ X is an open and bounded subset with θ ∈ Ω.
Then the BVP (1.1)-(1.2) has at least one solution provided EJQTDE, 2009 No. 19, p. 9 Before the proof of the main result, we first prove two lemmas.
is uniformly bounded.Meanwhile, for any t 1 , t 2 ∈ [0, T ] with T a positive constant, as in the proof of Lemma 3.3, we can also show that R 2 (•, λ)Ω is equicontinuous on [0,T] and equiconvergent at infinity.Thus, Lemma 3.2 yields that R 2 (•, λ)Ω is relatively compact.Since f is a g-Carathéodory function, the continuity of R 1 on Ω follows from the Lebesgue dominated convergence theorem.