Electronic Journal of Qualitative Theory of Differential Equations 2007, No. 23, 1-11; http://www.math.u-szeged.hu/ejqtde/

Sufficient condition for existence of solutions for higher-order

resonance boundary value problem

with one-dimensional p-Laplacian ^{1}

Liu Yang^{1} Chunfang Shen^{1} Xiping Liu^{2}

(1 Department of Mathematics, Hefei, Teacher’s College, Hefei Anhui Province, 236032, PR China 2 College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, PR China)

Abstract:By using coincidence degree theory of Mawhin, existence results for some higher order resonance multipoint boundary value problems with one dimensional p-Laplacian operator are obtained. Keywords: boundary value problem; one-dimensional p-Laplacian; resonance; coincidence degree.

1. Introduction

In this paper we consider higher-order multi-point boundary value problem with one-dimensional p-Laplacian

(φ_{p}(x^{(i)}(t)))^{(n−i)} = f(t, x(t), x′(t), ⋯, x^{(n−1)}(t)) + e(t), t ∈ (0, 1),  (1.1)

subject to one of the following boundary conditions:

x(1) = ∑_{j=1}^{m−2} α_{j}x(ξ_{j}), x′′(0) = ⋯ = x^{(i−1)}(0) = x^{(i+1)}(0) = ⋯ = x^{(n−1)}(0) = 0, x^{(i−1)}(1) = x^{(i−1)}(ξ), x^{(i)}(1) = x^{(i)}(η),

(1.2)

where p > 1 is a constant; φ_{p} : R → R, φ_{p}(u) = |u|^{p−2}u; f : [0, 1]×R^{n} → R is a continuous function and 1 ≤ i ≤ n−1 is a fixed integer, e(t) ∈ L^{1}[0, 1], α_{j}(1 ≤ j ≤ m − 2) ∈ R, η, ξ, ξ_{j} ∈ (0, 1), j = 1, ⋯, m − 2, 0 < ξ_{1} < ⋯ < ξ_{m−2} < 1.

We notice that the operator φ_{p}(u) = |u|^{p−2}u is called the (one-dimensional) p-Laplacian and it appears in many contexts. For example, it is used extensively in non-Newtonian fluids, in some reaction-diffusion problems, in flow through porous media, in nonlinear elasticity, glaceology and petroleum extraction.

The boundary value problem (1.1), (1.2) is said to be at resonance in the sense that the associate homogeneous problem

(φ_{p}(x^{(i)}(t)))^{(n−i)} = 0, 0 ≤ t ≤ 1

subject to boundary condition (1.2) has nontrival solutions.

The study on multi-point nonlocal boundary value problems for linear second-order ordinary differential equations was initiated by Il’in and Moiseev [1,2]. Since then some existence results have been obtained for general nonlinear boundary value problems by several authors. We refer the reader to some recent results, such as [3-7] at non-resonance and [8-12] at resonance. For resonance case, by using Leray-Schauder continuation theorem, nonlinear-alternative of Leray-Schauder and coincidence degree theorem, the main technique of these works is to convert the problem into the abstract form Lx = Nx, where L is a non-invertible linear operator. For problem (1.1) with some resonance conditions , if p = 2, some existence results are established by [10-12].

1 The work is sponsored by the Natural Science Foundation of Anhui Educational Department(Kj2007b055) and Youth Project Foundation of Anhui Educational Department(2007jqL101,2007jqL102)

But as far as we know, the existence results for high order resonance problems with p-Laplacian operator such as (1.1), (1.2) with p ≠ 2 have never been studied before. This is mainly due to the facts that in this situation, above methods are not applicable directly since the p-Laplacian operator (φ_{p}(x^{i}(t)))^{(n−i)} is not linear with respect to x. Inspired by [13,14], the goal of this paper is to fill the gap in this area. By using Mawhin continuation theorem the existence results for above problem are established.

2.Preliminaries

First we recall briefly some notations and an abstract existence results.

Let X,Y be real Banach spaces and let L : domL ⊂ X → Y be a Fredholm operator with index zero, here dom L denotes the domain of L. This means that ImL is closed in Y and dim KerL = dim(Y/ImL) < +∞. Consider the supplementary subspaces X_{1} and Y_{1} such that X = KerL ⊕ X_{1} and Y = ImL ⊕ Y_{1} and let P : X → KerL and Q : Y → Y_{1} be the natural projections. Clearly, KerL ∩ (domL ∩ X_{1}) = {0}, thus the restrictions L_{p} := L|_{domL∩X_{1}} is invertible. Denote by K the inverse of L_{p}.

Let Ω be an open bounded subset of X with domL ∩ Ω ≠ ∅. A map N : \overline{Ω} → Y is said to be L-compact in \overline{Ω} if QN(\overline{Ω}) is bounded and the operator K(I − Q)N : \overline{Ω} → X is compact. We first give the famous Mawhin continuation theorem. Lemma 2.1(Mawhin [15]). Suppose that X and Y are Banach spaces, and L : domL ⊂ X → Y is a Fredholm operator with index zero. Furthermore, Ω ⊂ X is an open bounded set and N : \overline{Ω} → Y is L-compact on \overline{Ω}.If (1)Lx ≠ λNx, ∀x ∈ ∂Ω ∩ domL, λ ∈ (0, 1); (2)Nx ∉ ImL, ∀x ∈ ∂Ω ∩ KerL; (3)deg{JQN, Ω ∩ KerL, 0} ≠ 0, where J : KerL → ImQ is an isomorphism, then the equation Lx = Nx has a solution in \overline{Ω} ∩ domL.

3.Existence results for problem (1.1), (1.2)

In order to eliminate the dilemma that L isn’t linear for the case p ≠ 2, we set

\{ \begin{matrix}x_{1}(t) = x(t) \\ & (3.1) \\ x_{2}(t) = φ_{p}(x^{(i)}(t)) \end{matrix}

then problem (1.1), (1.2) is equivalent to system

\{ \begin{matrix}x_{1}^{(i)}(t) = φ_{q}(x_{2}) \\ & (3.2) \\ x_{2}^{(n−i)}(t) = f(t, x_{1}, ⋯, x_{1}^{(i−1)}, φ_{q}(x_{2}), ⋯, (φ_{q}(x_{2}))^{(n−i−1)}) + e(t) \end{matrix}

with boundary conditions

x_{1}(1) = ∑_{j=1}^{m−2} α_{j}x_{1}(ξ_{j}), x′′_{1}(0) = ⋯ = x_{1}^{(i−1)}(0) = x′_{2}(0) = ⋯ = x_{2}^{(n−i−1)}(0) = 0, x_{1}^{(i−1)}(1) = x_{1}^{(i−1)}(ξ), x_{2}(1) = x_{2}(η)

where φ_{q} is the inverse function of φ_{p}, φ_{q}(u) = |u|^{q−2}u, where 1/p + 1/q = 1.Clearly if x(t) = (x_{1}(t), x_{2}(t)) is a solution for system (3.2), then x_{1}(t) must be a solution for problem (1.1),(1.2).

Define

X = {u(t) = (u_{1}(t), u_{2}(t))| u_{1}(t) ∈ C^{i}[0, 1]u_{2}(t) ∈ C^{n−i}[0, 1]m} with the norm

∥u∥ = max{|u_{1}|_{∞}, |u′_{1}|_{∞}, ⋯, |u_{1}^{(i−1)}|_{∞}, |φ_{q}(u_{2})|_{∞}, ⋯, |φ_{q}(u_{2})^{(n−i−1)}|_{∞}},

Y = {v(t) = (v_{1}(t), v_{2}(t))| v_{i}(t) ∈ L^{1}[0, 1], i = 1, 2} with the norm ∥v∥ = max{|v_{1}|_{1}, |φ_{q}(v_{2})|_{1}}, where |u|_{∞} = max_{0≤t≤1} |u(t)|, |u|_{1} = ∫_{0}^{1} |u(t)|dt.Clearly X and Y are Banach spaces. We will use the Sobolev space W^{(i,n−i)}(0, 1) defined as W^{(i,n−i)}(0, 1) = {u = (u_{1}, u_{2}) : (0, 1) → R : u_{1}, u_{2} are absolutely continuous on [0,1] and u_{1}^{(i)}, u_{2}^{(n−i)} ∈ L^{1}[0, 1]}. Define L : dom L ⊂ X → Y by

Lx := (x_{1}^{(i)}(t), x_{2}^{(n−i)}(t))

where dom L = {x ∈ W^{(i,n−i)}(0, 1) : x_{1}(1) = ∑_{j=1}^{m−2} α_{j}x_{1}(ξ_{j}),

x′′_{1}(0) = ⋯ = x_{1}^{(i−1)}(0) = x′_{2}(0) = ⋯ = x_{2}^{(n−i−1)}(0) = 0, x_{1}^{(i−1)}(1) = x_{1}^{(i−1)}(ξ), x_{2}(1) = x_{2}(η)}

and N : X → Y by

Nx := (φ_{q}(x_{2}), f(t, x_{1}, ⋯, x_{1}^{(i−1)}, φ_{q}(x_{2}), ⋯, (φ_{q}(x_{2}))^{(n−i−1)}) + e(t)).

Then system (3.2) can be written as Lx = Nx, here L is a linear operator.

In this section we shall prove existence results for system (3.2) under the case ∑_{j=1}^{m−2} α_{j} = 1, ∑_{j=1}^{m−2} α_{j}ξ_{j} ≠ 1.

Lemma 3.1 If ∑_{j=1}^{m−2} α_{j} = 1, ∑_{j=1}^{m−2} α_{j}ξ_{j} ≠ 1, then

(1) ImL = {(y_{1}, y_{2}) ∈ Y : ∫_{ξ}^{1} y_{1}(t)dt = 0, ∫_{η}^{1} ∫_{0}^{s_{n−i}} ⋯ ∫_{0}^{s_{2}} y_{2}(s_{1})ds_{1} ⋯ ds_{n−i} = 0}.

(2) L : domL ⊂ X → Y is a Fredholm operator with index zero,

(3) Define projector operator P : X → KerL as

P x = (x_{1}(0), x_{2}(0)),

then the generalized inverse of operator L, K_{P} : ImL → domL ∩ KerP can be written as K_{P} (y) =

(−\frac{∑_{j=1}^{m−2} α_{j}}{1 − ∑_{j=1}^{m−2} α_{j}ξ_{j}} ∫_{ξ_{j}}^{1} ∫_{0}^{s_{i}} ⋯ ∫_{0}^{s_{2}} y_{1}(s_{1})ds_{1} ⋯ ds_{i}+∫_{0}^{t} ⋯ ∫_{0}^{s_{2}} y_{1}(s_{1})ds_{1} ⋯ ds_{i}, ∫_{0}^{t} ∫_{0}^{s_{n−i}} ⋯ ∫_{0}^{s_{2}} y_{2}(s_{1})ds_{1} ⋯ ds_{n−i})

satisfying ∥K_{P} (y(t))∥ ≤ △∥y∥, where △ = 1 + \frac{∑_{j=1}^{m−2} |α_{j}|(1 − ξ_{j})}{|1 − ∑_{j=1}^{m−2} α_{j}ξ_{j}|} is a constant.

Proof: (1):First we show

ImL = {(y_{1}, y_{2}) ∈ Y : ∫_{ξ}^{1} y_{1}(t)dt = 0, ∫_{η}^{1} ∫_{0}^{s_{n−i}} ⋯ ∫_{0}^{s_{2}} y_{2}(s_{1})ds_{1} ⋯ ds_{n−i} = 0}.

First suppose y(t) = (y_{1}(t), y_{2}(t)) ∈ ImL, then there exists x(t) = (x_{1}(t), x_{2}(t)) ∈ domL such that Lx = y. That is

x_{1}(t) = ∫_{0}^{t} ∫_{0}^{s_{i}} ⋯ ∫_{0}^{s_{2}} y_{1}(s_{1})ds_{1} ⋯ ds_{i} + a_{i−1}t^{i−1} + ⋯ + a_{1}t + a_{0}
x_{2}(t) = ∫_{0}^{t} ∫_{0}^{s_{n−i}} ⋯ ∫_{0}^{s_{2}} y_{2}(s_{1})ds_{1} ⋯ ds_{n−i} + b_{n−i−1}t^{n−i−1} + ⋯ + b_{1}t + b_{0}

Then boundary condition

x_{1}(1) = ∑_{j=1}^{m−2} α_{j}x_{1}(ξ_{j}), x′′_{1}(0) = ⋯ = x_{1}^{(i−1)}(0) = x′_{2}(0) = ⋯ = x_{2}^{(n−i−1)}(0) = 0, x_{1}^{(i−1)}(1) = x_{1}^{(i−1)}(ξ), x_{2}(1) = x_{2}(η)

imply that

∫_{ξ}^{1} y_{1}(t)dt = 0, ∫_{η}^{1} ∫_{0}^{s_{n−i}} ⋯ ∫_{0}^{s_{2}} y_{2}(s_{1})ds_{1} ⋯ ds_{n−i} = 0.

Next we suppose y(t) ∈ {(y_{1}, y_{2}) ∈ Y : ∫_{ξ}^{1} y_{1}(t)dt = 0, ∫_{η}^{1} ∫_{0}^{s_{n−i}} ⋯ ∫_{0}^{s_{2}} y_{2}(s_{1})ds_{1} ⋯ ds_{n−i} = 0}. Let x(t) = (x_{1}(t), x_{2}(t)), where

x_{1}(t) = −\frac{∑_{j=1}^{m−2} α_{j}}{1 − ∑_{j=1}^{m−2} α_{j}ξ_{j}} ∫_{ξ_{j}}^{1} ∫_{0}^{s_{i}} ⋯ ∫_{0}^{s_{2}} y_{1}(s_{1})ds_{1} ⋯ ds_{i} + ∫_{0}^{t} ⋯ ∫_{0}^{s_{2}} y_{1}(s_{1})ds_{1} ⋯ ds_{i},

x_{2}(t) = ∫_{0}^{t} ∫_{0}^{s_{n−i}} ⋯ ∫_{0}^{s_{2}} y_{2}(s_{1})ds_{1} ⋯ ds_{n−i}

then Lx = (x_{1}^{(i)}(t), x_{2}^{(n−i)}(t)) = (y_{1}(t), y_{2}(t)). Furthermore consider

∫_{ξ}^{1} y_{1}(t)dt = 0, ∫_{η}^{1} ∫_{0}^{s_{n−i}} ⋯ ∫_{0}^{s_{2}} y_{2}(s_{1})ds_{1} ⋯ ds_{n−i} = 0,

by a simple computation,

x_{1}(1) = ∑_{j=1}^{m−2} α_{j}x_{1}(ξ_{j}), x′′_{1}(0) = ⋯ = x_{1}^{(i−1)}(0) = x′_{2}(0) = ⋯ = x_{2}^{(n−i−1)}(0) = 0, x_{1}^{(i−1)}(1) = x_{1}^{(i−1)}(ξ), x_{2}(1) = x_{2}(η)

Then x(t) ∈ domL, thus y(t) ∈ ImL. Sum up all above we obtain that

ImL = {(y_{1}, y_{2}) ∈ Y : ∫_{ξ}^{1} y_{1}(t)dt = 0, ∫_{η}^{1} ∫_{0}^{s_{n−i}} ⋯ ∫_{0}^{s_{2}} y_{2}(s_{1})ds_{1} ⋯ ds_{n−i} = 0}.

(2):Following we claim that L is a Fredholm operator with index zero. It’s easy to see that KerL = (a, b), a, b ∈ R. Suppose y(t) ∈ Y, define the projector operator Q as

Q(y) = (Q(y_{1}(t)), Q(y_{2}(t))) = (\frac{∫_{ξ}^{1} y_{1}(t)dt}{1 − ξ}, \frac{(n − 1)!}{1 − η^{n−i}} ∫_{η}^{1} ∫_{0}^{s_{n−i}} ⋯ ∫_{0}^{s_{2}} y_{2}(s_{1})ds_{1} ⋯ ds_{n−i}).

Let y^{∗} = y(t) − Q(y(t)) = (y_{1} − Q(y_{1}), y_{2} − Q(y_{2})), it’s easy to see that y^{∗} ∈ ImL. Hence Y = ImL + KerL, furthermore considering ImL ∩ KerL = {0}, we have Y = ImL ⊕ KerL. Thus

dim KerL = co dim ImL,

which means L is a Fredholm operator with index zero. (3):Define the projector operator P : X → KerL as

P x = (x_{1}(0), x_{2}(0)),

for y(t) ∈ ImL, we have

(LK_{P})(y(t)) = y(t),

and for x(t) ∈ domL ∩ KerP, following facts

−\frac{∑_{j=1}^{m−2} α_{j}}{1 − ∑_{j=1}^{m−2} α_{j}ξ_{j}} ∫_{ξ_{j}}^{1} ∫_{0}^{s_{i}} ⋯ ∫_{0}^{s_{2}} x_{1}^{(i)}(s_{1})ds_{1} ⋯ ds_{i} + ∫_{0}^{t} ⋯ ∫_{0}^{s_{2}} x_{1}^{(i)}(s_{1})ds_{1} ⋯ ds_{i} = x_{1}(t) − x_{1}(0) = x_{1}(t),

∫_{0}^{t} ∫_{0}^{s_{n−i}} ⋯ ∫_{0}^{s_{2}} x_{2}^{(n−i)}(s_{1})ds_{1}ds_{n−i} = x_{2}(t) − x_{2}(0) = x_{2}(t). show that K_{P} = (L_{domL∩KerP})^{−1}. Furthermore from the definition of the norms in the X, Y, we have

∥K_{P} (y(t))∥ ≤ △∥y∥.

The above arguments complete the proof of Lemma 3.1. Theorem 3.1: Let f : [0, 1] × R^{n} → R be a continuous function. Assume there exists m_{1} ∈ 1, 2, ⋯, m − 3 such that α_{j} > 0 for 1 ≤ j ≤ m_{1} and α_{j} < 0 for m_{1} + 1 ≤ j ≤ m − 2, furthermore following conditions are satisfied: (C_{1}) There exist functions a_{k}(t) ∈ L^{1}[0, 1], k = 1, 2, ⋯, n and constant θ ∈ [0, 1) such that for all (x_{1}, x_{2}, ⋯, x_{n}) ∈ R^{n}, t ∈ [0, 1],one of following conditions is satisfied:

|f(t, x_{1}, x_{2}, ⋯, x_{n}) + e(t)| ≤ (∑_{k=1}^{n} a_{k}(t)|x_{k}| + b(t)|x_{n}|^{θ} + r(t))^{p−1},  (3.3)
|f(t, x_{1}, x_{2}, ⋯, x_{n}) + e(t)| ≤ (∑_{k=1}^{n} a_{k}(t)|x_{k}| + b(t)|x_{n−1}|^{θ} + r(t))^{p−1},  (3.4)
· · · · · ·
|f(t, x_{1}, x_{2}, ⋯, x_{n}) + e(t)| ≤ (∑_{k=1}^{n} a_{k}(t)|x_{k}| + b(t)|x_{1}|^{θ} + r(t))^{p−1},  (3.5)

(C_{2}) There exists a constant M > 0 such that for x ∈ dom L, if |x_{1}(t)| > M,

∫_{ξ}^{1} φ_{q}(∫_{σ}^{t} ∫_{0}^{s_{n−i}} ⋯ ∫_{0}^{s_{2}}(f(s_{1}, x_{1}, ⋯, x_{n}) + e(s_{1}))ds_{1} ⋯ ds_{n−i})dt ≠ 0;  (3.6)

for all x_{2}, ⋯, x_{n} ∈ R^{n−1}, σ ∈ (0, 1), t ∈ (0, 1)\{σ}. (C_{3}) There exist M^{∗} > 0 such that for any c_{1} ∈ R, if |c_{1}| > M^{∗}, for all c_{2} ∈ R, then either

c_{2} × ∫_{η}^{1} ∫_{0}^{s_{n−i}} ⋯ ∫_{0}^{s_{2}}(f(s_{1}, c_{1}, 0, ⋯, 0, φ_{q}(c_{2}), 0, ⋯, 0) + e(s_{1}))ds_{1} ⋯ ds_{n−i} < 0  (3.7)

or else

c_{2} × ∫_{η}^{1} ∫_{0}^{s_{n−i}} ⋯ ∫_{0}^{s_{2}}(f(s_{1}, c_{1}, 0, ⋯, 0, φ_{q}(c_{2}), 0, ⋯, 0) + e(s_{1}))ds_{1} ⋯ ds_{n−i} > 0  (3.8)

Then for each e ∈ L^{1}[0, 1], the resonance problem (1.1), (1.2) with ∑_{j=1}^{m−2} α_{j} = 1, ∑_{j=1}^{m−2} α_{j}ξ_{j} ≠ 1 has at least one solution in C^{n−1}[0, 1] provided that

∑_{k=1}^{n} |a_{k}|_{1} < \frac{1}{1 + △}.

Proof : We divide the proof into the following steps. Step 1.Let

Ω_{1} = {x ∈ dom L \ KerL : Lx = λNx} for some λ ∈ [0, 1].

Then Ω_{1} is bounded.

Suppose that x ∈ Ω_{1}, Lx = λNx, thus λ ≠ 0, Nx ∈ ImL = KerQ, hence

∫_{ξ}^{1} φ_{q}(x_{2}(t))dt = 0, ∫_{η}^{1} ∫_{0}^{s_{n−i}} ⋯ ∫_{0}^{s_{2}}(f(s_{1}, x_{1}, ⋯, x_{1}^{(i−1)}, φ_{q}(x_{2}), ⋯, (φ_{q}(x_{2}))^{(n−i−1)}) + e(s_{1}))ds_{1} ⋯ ds_{n−i} = 0.

For x_{1}^{(i−1)}(1) = x_{1}^{(i−1)}(ξ), there exist σ_{1} ∈ (ξ, 1) such that x_{1}^{(i)}(σ_{1}) = 0. Integrate both sides of (1.1), we have

x_{2}(t) = ∫_{σ_{1}}^{t} ∫_{0}^{s_{n−i}} ⋯ ∫_{0}^{s_{2}}(f(s_{1}, x_{1}, ⋯, x_{1}^{(i−1)}, φ_{q}(x_{2}), ⋯, (φ_{q}(x_{2}))^{(n−i−1)}) + e(s_{1}))ds_{1} ⋯ ds_{n−i} = 0,  (3.9)

Thus

∫_{ξ}^{1} φ_{q}(∫_{σ_{1}}^{t} ∫_{0}^{s_{n−i}} ⋯ ∫_{0}^{s_{2}}(f(s_{1}, x_{1}, ⋯, x_{1}^{(i−1)}, φ_{q}(x_{2}), ⋯, (φ_{q}(x_{2}))^{(n−i−1)}) + e(s_{1}))ds_{1} ⋯ ds_{n−i})dt = 0. (3.10)

Then (3.10) together with condition (C_{2}) imply that there exists t_{0} ∈ [0, 1] such that |x_{1}(t_{0})| < M. In view of x_{1}(t) = x_{1}(t_{0}) + ∫_{t_{0}}^{t} x′_{1}(s)ds, we obtain that

|x(t)| < M + |x′_{1}|_{1}.  (3.11)

Furthermore, for α_{j} > 0, 1 ≤ j ≤ m_{1} and α_{j} < 0, m_{1} + 1 ≤ j ≤ m − 2 and x_{1}(1) = ∑_{j=1}^{m−2} α_{j}x_{1}(ξ_{j}), we have

x_{1}(1) − ∑_{j=m_{1}+1}^{m−2} α_{j}x_{1}(ξ_{j}) = ∑_{j=1}^{m_{1}} α_{j}x_{1}(ξ_{j}),

then there exists t_{1} ∈ [ξ_{m_{1}+1}, 1], t_{2} ∈ [0, ξ_{m_{1}}] such that

x_{1}(t_{1}) = \frac{x_{1}(1) − ∑_{j=m_{1}+1}^{m−2} α_{j}x_{1}(ξ_{j})}{1 − ∑_{m_{1}+1}^{m−2} α_{j}}, x_{1}(t_{2}) = \frac{∑_{j=1}^{m_{1}} α_{j}x_{1}(ξ_{j})}{∑_{j=1}^{m_{1}} α_{j}},

thus in view of ∑_{j=1}^{m−2} α_{j} = 1,we obtain that x_{1}(t_{1}) = x_{1}(t_{2}), and t_{1} ≠ t_{2}. This implies that there exists t_{3} ∈ (t_{1}, t_{2})

such that x′_{1}(t_{3}) = 0. Then from x′_{1}(t) = x′_{1}(t_{3}) + ∫_{t_{3}}^{t} x′′_{1}(s)ds, we obtain

|x′_{1}| ≤ |x′′_{1}|_{1}.  (3.12)

Consider the boundary condition x′′_{1}(0) = x′′′_{1}(0) = ⋯ = x_{1}^{(i−1)}(0) = x′_{2}(0) = ⋯ = x_{2}^{(n−i−1)}(0) = 0 together with x_{2}(σ_{1}) = 0, it’s easy to get

|x′′_{1}|_{∞} ≤ |x′′′_{1}|_{∞} ≤ ⋯ |x_{1}^{(i)}|_{∞} = |φ_{q}(x_{2})|_{∞} ⋯ ≤ |(φ_{q}(x_{2})^{(n−i−1)})|_{∞}.  (3.13)

Consider (3.11),(3.12),(3.13) we have ∥P x∥ ≤ max{|x_{1}(0)|, |φ_{q}(x_{2}(0))|}

≤ max{M + |φ_{q}(x_{2})|_{1}, |φ_{q}(f(t, x_{1}, ⋯, x_{1}^{(i−1)}, φ_{q}(x_{2}), ⋯, (φ_{q}(x_{2}))^{(n−i−1)} + e(t))|_{1}}  (3.14)

Again for x ∈ Ω_{1}, x ∈ domL \ KerL, then (I − P)x ∈ domL ∩ KerP, LP x = 0, thus from Lemma 3.1, we have

∥(I − P)x∥ = ∥K_{P} L(I − P)x∥ ≤ △∥L(I − P)x∥_{1} = △∥Lx∥ ≤ △∥Nx∥

≤ △ max{|φ_{q}(x_{2})|_{1}, |φ_{q}(f(t, x_{1}, ⋯, x_{1}^{(i−1)}, φ_{q}(x_{2}), ⋯, (φ_{q}(x_{2}))^{(n−i−1)}) + e(t))|_{1}}.  (3.15)

From (3.14),(3.15) we have

∥x∥ ≤ ∥P x∥ + ∥(I − P)x∥ ≤ M + (1 + △)|φ_{q}(f(t, x_{1}, ⋯, x_{1}^{(i−1)}, φ_{q}(x_{2}), ⋯, (φ_{q}(x_{2}))^{(n−i−1)}) + e(t))|_{1}.  (3.16)

If assumption (3.3) holds,we obtain that

∥x∥ ≤ M + (1 + △)|φ_{q}(f(t, x_{1}, ⋯, x_{1}^{(i−1)}, φ_{q}(x_{2}), ⋯, (φ_{q}(x_{2}))^{(n−i−1)}) + e(t))|_{1}
≤ (1 + △)(|a|_{1}|x_{1}|_{∞} + ⋯ + |a_{i}|_{1}|x_{1}^{(i−1)}|_{∞} + |a_{i+1}|_{1}|φ_{q}(x_{2})|_{∞} + ⋯
+|a_{n}|_{1}|(φ_{q}(x_{2}))^{(n−i−1)}|_{∞} + |b|_{1}|(φ_{q}(x_{2}))^{(n−i−1)}|_{∞}^{θ} + C)

where C = |r|_{1} + |e|_{1} + \frac{M}{1 + △}.

From |x_{1}|_{∞} ≤ ∥x∥,we obtain

|x_{1}|_{∞} ≤ \frac{1 + △}{1 − (1 + △)|a_{1}|_{1}} [|a_{2}|_{1}|x′_{1}|_{∞}+⋯+|a_{i+1}|_{1}|φ_{q}(x_{2})|_{∞}+⋯+|a_{n}|_{1}|(φ_{q}(x_{2}))^{(n−i−1)}|_{∞}+|b|_{1}|(φ_{q}(x_{2}))^{(n−i−1)}|_{∞}^{θ}+C]

From |x′_{1}| ≤ ∥x∥,we obtain

|x′_{1}|_{∞} ≤ \frac{1 + △}{1 − (1 + △)(|a_{1}|_{1} + |a_{2}|_{1})}[|a_{3}|_{1}|x′′_{1}|_{∞} + ⋯

+|a_{i+1}|_{1}|φ_{q}(x_{2})|_{∞} + ⋯ + |a_{n}|_{1}|(φ_{q}(x_{2}))^{(n−i−1)}|_{∞} + |b|_{1}|(φ_{q}(x_{2}))^{(n−i−1)}|_{∞}^{θ} + C].

· · · · · ·

|(φ_{q}(x_{2}))^{(n−i−1)}|_{∞} ≤ \frac{1 + △}{1 − (1 + △) ∑_{k=1}^{n−1} |a_{k}|_{1}} [|a_{n}|_{1}|(φ_{q}(x_{2}))^{(n−i−1)}|_{∞} + |b|_{1}|(φ_{q}(x_{2}))^{(n−i−1)}|_{∞}^{θ} + C],

then

|(φ_{q}(x_{2}))^{(n−i−1)}|_{∞} ≤ \frac{(1 + △)|b|_{1}}{1 − (1 + △) ∑_{k=1}^{n−1} |a_{k}|_{1}} |(φ_{q}(x_{2}))^{(n−i−1)}|_{∞}^{θ} + \frac{2C}{1 − 2 ∑_{k=1}^{n−1} |a_{k}|_{1}} .

Consider θ ∈ [0, 1) together with ∑_{k=1}^{n} |a_{k}|_{1} < \frac{1}{1 + △}, we claim that there exists constant M_{1} > 0 such that

|(φ_{q}(x_{2}))^{(n−i−1)}|_{∞} ≤ M_{1}  (3.17)

Then there exist constants M_{k} > 0, k = 2, ⋯, i, M_{j} > 0, j = i + 1, ⋯, n such that

|x_{1}^{(k)}|_{∞} < M_{k}, |(φ_{q}(x_{2}))^{(n−j)}|_{∞} < M_{j},

thus there exists N > 0 such that ∥x∥ < N,therefor we show that Ω_{1} is bounded. Step 2.The set Ω_{2} = {x ∈ KerL : Nx ∈ ImL} is bounded.

The fact x ∈ Ω_{2} implies that x = (c_{1}, c_{2}) and

N(x) = (φ_{q}(c_{2}), f(t, c_{1}, 0, ⋯, 0, φ_{q}(c_{2}), ⋯, 0) + e(t))

From QNx = 0,we have

∫_{ξ}^{1} φ_{q}(c_{2})dt = 0, ∫_{η}^{1} ∫_{0}^{s_{n−i}} ⋯ ∫_{0}^{s_{2}}(f(s_{1}, c_{1}, 0, ⋯, 0, φ_{q}(c_{2}), ⋯, 0) + e(s_{1}))ds_{1} ⋯ ds_{n−i} = 0,

which implies c_{2} = 0 and

∫_{η}^{1} ∫_{0}^{s_{n−i}} ⋯ ∫_{0}^{s_{2}}(f(s_{1}, c_{1}, 0, ⋯, 0) + e(s_{1}))ds_{1} ⋯ ds_{n−i} = 0.

Consider condition (C3), we obtain that |c_{1}| ≤ M^{∗}, then the set Ω_{2} is bounded. Step 3. If the first part of condition (C3) is satisfied,there exists M^{∗} > 0 such that for any c ∈ R, if c_{1} > M^{∗}, then

c_{2} ∫_{η}^{1} ∫_{0}^{s_{n−i}} ⋯ ∫_{0}^{s_{2}}(f(s_{1}, c_{1}, 0, ⋯, 0, φ_{q}(c_{2}), ⋯, 0) + e(s_{1}))ds_{1} ⋯ ds_{n−i} < 0.

Let Ω_{3} = {x ∈ KerL : −λx + (1 − λ)JQNx = 0, λ ∈ [0, 1],here J : ImQ → KerL is the linear isomorphism given by J(c_{1}, c_{2}) = (c_{1}, c_{2}),we obtain

λc_{1} = (1 − λ)φ_{q}(c_{2})

λc_{2} = (1 − λ)\frac{(n − i)!}{1 − η^{n−i}} ∫_{η}^{1} ∫_{0}^{s_{n−i}} ⋯ ∫_{0}^{s_{2}}(f(s_{1}, c_{1}, 0, ⋯, φ_{q}(c_{2}), 0, ⋯, 0) + e(s_{1}))ds_{1} ⋯ ds_{n−i}.

If λ = 1,it’s easy to see c_{1} = c_{2} = 0.If λ = 0, φ_{q}(c_{2}) = 0 implies c_{2} = 0, then

∫_{η}^{1} ∫_{0}^{s_{n−i}} ⋯ ∫_{0}^{s_{2}}(f(s_{1}, c_{1}, 0, ⋯, 0) + e(s_{1}))ds_{1} ⋯ ds_{n−i} = 0.

Considering condition C3,|c_{1}| < M^{∗}. For λ ≠ 0, λ ≠ 1, if |c_{1}| ≥ M^{∗},we obtain that

λc_{2}^{2} = c_{2}(1 − λ)\frac{(n − i)!}{1 − η^{n−i}} ∫_{η}^{1} ∫_{0}^{s_{n−i}} ⋯ ∫_{0}^{s_{2}}(f(s_{1}, c_{1}, 0, ⋯, φ_{q}(c_{2}), 0, ⋯, 0) + e(s_{1}))ds_{1} ⋯ ds_{n−i} < 0,

which contradicts toλ c_{2}^{2} ≥ 0.Thus |c_{1}| < M^{∗}.From λc_{1} = (1 − λ)φ_{q}(c_{2}) and λ ≠ 0, λ ≠ 1, |c_{2}| < (\frac{λ}{1 − λ}M^{∗})^{p−1}. Thus the set Ω_{3} is bounded. Step 4.If the second part of condition (C3) is satisfied, similar with above argument, the set Ω_{4} = {x ∈ KerL : λx + (1 − λ)JQNx = 0, λ ∈ [0, 1]} is bounded too. Now we show all the conditions of Lemma 2.1 are satisfied. Let Ω be a bounded open set of Y such that ⋃_{i=1}^{3} \overline{Ω_{i}} ⊂ Ω. By the Ascoli-Arezela theorem, we can show that K_{P} (I − Q)N : \overline{Ω} → Y is compact, thus N is L-compact on \overline{Ω}. Then by the above arguments, we have

(i) Lx ≠ Nx, for every (x, λ) ∈ [(domL\KerL) ⋂ ∂Ω] × (0, 1); (ii) Nx ≠ ImL, for every x ∈ KerL ⋂ ∂Ω; (iii)If the first part of condition (C3) holds, we let

H(x, λ) = −λx + (1 − λ)JQNx.

According to the above argument, we know that H(x, λ) ≠ 0, for x ∈ KerL ⋂ ∂Ω, by the homotopy property of degree, we get

deg(JQN|_{KerL}, Ω ∩ KerL, 0) =deg(H(x, 0), Ω ∩ KerL, 0)
=deg(H(x, 1), Ω ∩ KerL, 0)
=deg(−I, Ω ∩ KerL, 0) ≠ 0.

If the second part of condition C3 holds, we let

H(x, λ) = λx + (1 − λ)JQNx,

Similar to argument above, we have deg(JQN|_{KerL}, Ω ∩ KerL, 0)=deg(I, Ω ∩ KerL, 0) ≠ 0. Then by Lemma 2.1, Lx = Nx has at least one solution in domL ∩ \overline{Ω}, so that problem (1.1), (1.2) has at least one solution in C^{n−1}[0, 1]. The proof of Theorem 3.1 is completed.

References

1 V.A.Il’in and E.I.Moiseev, Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator. Differential Equation, 1987, 23(8):979-987.

2 V.A.Il’in and E.I.Moiseev, Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects. Differential Equation, 1987, 23(7):803-810

3 R.Y.Ma, Positive solutions for a nonlinear three-point boundary value problem, Electronic Journal of Differential Equations. 34(1999), 1-8.

4 B.Liu, Positive solutions of a nonlinear four-point boundary value problems, Appl.Math.Comput, 155(2004)179-203.

5 R.Y.Ma, N.Cataneda, Existence of solution for nonlinear m-point boundary value problem, J.Math.Anal.Appl, 256(2001)556-567.

6 Yanping Guo and Weigao Ge, Positive solutions for three-point boundary-value problems with dependence on the first order derivative, J.Math.Anal.Appl, 290(2004):291-301.

7 Z.B.Bai, W.G.Ge and Y.F.Wang, Multiplicity results for some second-order four-point boundary-value problems, Nonlinear Analysis. 60(2004), 491-500.

8 Feng W and Webb J R L, Solvability of three-point boundary value problems at resonance. Nonlinear Analysis, 30(6), 1997:3227-3238.

9 B.Liu, Solvability of multi-point boundary value problem at resonance(II), Appl.Math.Comput, 136(2003), 353-377.

10 S.P.Lu and W.G.Ge, On the existence of m-point boundary value problem at resonance for higher order differential equation, J.Math.Anal.Appl, 287(2003), 522-539.

11 X.J.Lin, Z.J.Du and W.G.Ge, Solvability of multipoint boundary value problems at resonance for higher order ordinary differential equations, Comput.Math.Appl, 49(2005), 1-11.

12 Z.J.Du, X.J.Lin and W.G.Ge, Some higher order multi-point boundary value problems at resonance, J.Math. Anal.Appl, 177(2005), 55-65.

13 W. Ge and J. Ren, An extension of Mawhin’s continuation theorem and its application to boundary value problems with a p-Laplacian. Nonlinear Analysis TMA 58 (2004), no. 3-4, 477-488.

14 W.S. Cheung and J. Ren, Periodic solutions for p-Laplacian differential equations with multiple deviating arguments, Nonlinear Analysis TMA, 62(2005), no.4, 727-742.

15 J.Mawhin, Topological degree methods in nonlinear boundary value problems, in: NSFCBMS Regional Conference Series in Math, American Mathematics Society, Providence, RI 1979.

(Received April 27, 2007)