Electronic Journal of Qualitative Theory of Differential Equations 2010, No. 14, 1-8; http://www.math.u-szeged.hu/ejqtde/

Existence of solutions for a class of fourth-order m-point boundary value problems^{∗}

Jian-Ping Sun, Qiu-Yan Ren

Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu, 730050, People’s Republic of China

E-mail: jpsun@lut.cn (Sun)

Abstract

Some existence criteria are established for a class of fourth-order m-point boundary value problem by using the upper and lower solution method and the Leray-Schauder continuation principle. Keywords: Fourth-order m-point boundary value problem; Upper and lower solution method; Leray-Schauder continuation principle; Nagumo condition 2000 AMS Subject Classification: 34B10, 34B15

1 Introduction

Boundary value problems (BVPs for short) of fourth-order differential equations have been used to describe a large number of physical, biological and chemical phenomena. For example, the deformations of an elastic beam in the equilibrium state can be described as some fourth-order BVP. Recently, fourth-order BVPs have received much attention. For instance, [3, 5, 6, 7] discussed some fourth-order two-point BVPs, while [1, 2, 4, 9] studied some fourth-order three-point or four-point BVPs. It is worth mentioning that Ma, Zhang and Fu [7] employed the upper and lower solution method to prove the existence of solutions for the BVP

\{ \begin{matrix}u^{(4)}(t) = f (t, u (t) , u′′ (t)) , t ∈ (0, 1), \\ u (0) = u′ (1) = u′′ (0) = u′′′ (1) = 0, \end{matrix}

and Bai [3] considered the existence of a solution for the BVP

\{ \begin{matrix}u^{(4)}(t) = f (t, u (t) , u′ (t) , u′′ (t) , u′′′ (t)) , t ∈ (0, 1), \\ u (0) = u′ (1) = u′′ (0) = u′′′ (1) = 0 \end{matrix}

by using the upper and lower solution method and Schauder′s fixed point theorem.

Although there are many works on fourth-order two-point, three-point or four-point BVPs, a little work has been done for more general fourth-order m-point BVPs [8]. Motivated greatly by the above-mentioned excellent works, in this paper, we will investigate the following fourth-order m-point BVP

∗Supported by the National Natural Science Foundation of China (10801068).

\{ \begin{matrix}u^{(4)}(t) + f(t, u (t) , u′ (t) , u′′ (t) , u′′′ (t)) = 0, t ∈ [0, 1] , \\ u (0) = ∑_{i=1}^{m−2} a_{i}u (η_{i}) , u′ (1) = 0, \\ u′′ (0) = ∑_{i=1}^{m−2} b_{i}u′′ (η_{i}) , u′′′ (1) = 0. \end{matrix}  (1.1)

Throughout this paper, we always assume that 0 < η_{1} < η_{2} < ⋯ < η_{m−2} < 1, a_{i} and b_{i} (i = 1, 2, ⋯, m − 2) are nonnegative constants and f : [0, 1] × R^{4} → R is continuous. Some existence criteria are established for the BVP (1.1) by using the upper and lower solution method and the Leray-Schauder continuation principle.

2 Preliminaries

Let E = C [0, 1] be equipped with the norm ∥v∥_{∞} = max_{t∈[0,1]} |v (t)| and

K = {v ∈ E| v (t) ≥ 0 for t ∈ [0, 1]} .

Then K is a cone in E and (E, K) is an ordered Banach space. For Banach space X = C^{1} [0, 1] , we use the norm ∥v∥ = max {∥v∥_{∞} , ∥v′∥_{∞}} .

Lemma 2.1 Let ∑_{i=1}^{m−2} a_{i} ≠ 1. Then for any h ∈ E, the second-order m-point BVP

\{ \begin{matrix}−u′′(t) = h (t) , t ∈ [0, 1] , \\ u (0) = ∑_{i=1}^{m−2} a_{i}u (η_{i}) , u′ (1) = 0 \end{matrix}  (2.1)

has a unique solution

u (t) = ∫_{0}^{1} G_{1} (t, s) h (s) ds,

where

G_{1} (t, s) = K (t, s) + \frac{1}{1 − ∑_{i=1}^{m−2} a_{i}} ∑_{i=1}^{m−2} a_{i}K (η_{i}, s) ,

here

K (t, s) = \{ \begin{matrix}s, 0 ≤ s ≤ t ≤ 1, \\ t, 0 ≤ t ≤ s ≤ 1 \end{matrix}

is Green’s function of the second-order two-point BVP

\{ \begin{matrix}−u′′(t) = 0, t ∈ [0, 1] , \\ u (0) = u′ (1) = 0. \end{matrix}

Proof. If u is a solution of the BVP (2.1), then we may suppose that

u (t) = ∫_{0}^{1} K(t, s)h(s)ds + At + B.

By the boundary conditions in (2.1), we know that

A = 0 and B = \frac{1}{1 − ∑_{i=1}^{m−2} a_{i}} ∑_{i=1}^{m−2} a_{i} ∫_{0}^{1} K(η_{i}, s)h (s) ds.

Therefore, the unique solution of the BVP (2.1)

u (t) = ∫_{0}^{1} K(t, s)h(s)ds + \frac{1}{1 − ∑_{i=1}^{m−2} a_{i}} ∑_{i=1}^{m−2} a_{i} ∫_{0}^{1} K(η_{i}, s)h (s) ds
= ∫_{0}^{1} G_{1} (t, s) h (s) ds.
□

In the remainder of this paper, we always assume that ∑_{i=1}^{m−2} a_{i} < 1 and ∑_{i=1}^{m−2} b_{i} < 1, which imply that G_{1} (t, s) and G_{2} (t, s) are nonnegative on [0, 1] × [0, 1], where

G_{2} (t, s) = K (t, s) + \frac{1}{1 − ∑_{i=1}^{m−2} b_{i}} ∑_{i=1}^{m−2} b_{i}K (η_{i}, s) .

Now, we define operators A and B : E → E as follows:

(Av) (t) = − ∫_{0}^{1} G_{1} (t, s) v (s) ds, t ∈ [0, 1]  (2.2)

and

(Bv) (t) = − ∫_{t}^{1} v (s) ds, t ∈ [0, 1] .  (2.3)

Remark 2.1 A and B are decreasing operators on E. Lemma 2.2 If the following BVP

\{ \begin{matrix}v′′(t) + f(t, (Av) (t) , (Bv) (t) , v (t) , v′ (t)) = 0, t ∈ [0, 1] , \\ v (0) = ∑_{i=1}^{m−2} b_{i}v (η_{i}) , v′ (1) = 0 \end{matrix}  (2.4)

has a solution, then does the BVP (1.1). Proof. Suppose that v is a solution of the BVP (2.4). Then it is easy to prove that u = Av is a solution of the BVP (1.1). □

EJQTDE, 2010 No. 14, p. 3 Definition 2.1 If α ∈ C^{2} [0, 1] satisfies

\{ \begin{matrix}α′′(t) + f(t, (Aα) (t) , (Bα) (t) , α (t) , α′ (t)) ≥ 0, t ∈ [0, 1] , \\ α (0) ≤ ∑_{i=1}^{m−2} b_{i}α (η_{i}) , α′ (1) ≤ 0, \end{matrix}  (2.5)

then α is called a lower solution of the BVP (2.4).

Definition 2.2 If β ∈ C^{2} [0, 1] satisfies

\{ \begin{matrix}β′′(t) + f(t, (Aβ) (t) , (Bβ) (t) , β (t) , β′ (t)) ≤ 0, t ∈ [0, 1] , \\ β (0) ≥ ∑_{i=1}^{m−2} b_{i}β (η_{i}) , β′ (1) ≥ 0, \end{matrix}  (2.6)

then β is called an upper solution of the BVP (2.4).

Remark 2.2 If the inequality in Definition (2.1)

α′′(t) + f(t, (Aα) (t) , (Bα) (t) , α (t) , α′ (t)) ≥ 0, t ∈ [0, 1]

is replaced by

α′′(t) + f(t, (Aα) (t) , (Bα) (t) , α (t) , α′ (t)) > 0, t ∈ [0, 1] ,

then α is called a strict lower solution of the BVP (2.4). Similarly, we can also give the definition of a strict upper solution for the BVP (2.4).

Definition 2.3 Assume that f ∈ C ([0, 1] × R^{4}, R) , α, β ∈ E and α(t) ≤ β(t) for t ∈ [0, 1]. We say that f satisfies Nagumo condition with respect to α and β provided that there exists a function h ∈ C ([0, +∞) , (0, +∞)) such that

|f(t, x_{1}, x_{2}, x_{3}, x_{4})| ≤ h (|x_{4}|) ,

for all (t, x_{1}, x_{2}, x_{3}, x_{4}) ∈ [0, 1] × [(Aβ) (t) , (Aα) (t)] × [(Bβ) (t) , (Bα) (t)] × [α(t), β(t)] × R, and

∫_{λ}^{+∞} \frac{s}{h (s)}ds > max_{t∈[0,1]} β (t) − min_{t∈[0,1]} α (t) ,  (2.7)

where λ = max {|β (1) − α (0)| , |β (0) − α (1)|} .

Lemma 2.3 Assume that α and β are, respectively, the lower and the upper solution of the BVP (2.4) with α (t) ≤ β (t) for t ∈ [0, 1] , and f satisfies the Nagumo condition with respect to α and β. Then there exists N > 0 (depending only on α and β) such that any solution ω of the BVP (2.4) lying in [α, β] satisfies

|ω′ (t)| ≤ N, t ∈ [0, 1].

Proof. It follows from the definition of λ and the mean-value theorem that there exists t_{0} ∈ (0, 1) such that

|ω′ (t_{0})| = |ω (1) − ω (0)| ≤ λ.  (2.8)

By (2.7), we know that there exists N > λ such that

∫_{λ}^{N} \frac{s}{h (s)}ds > max_{t∈[0,1]} β (t) − min_{t∈[0,1]} α (t) .  (2.9)

Now, we will prove that |ω′ (t)| ≤ N for any t ∈ [0, 1] . Suppose on the contrary that there exists t_{1} ∈ [0, 1] such that

|ω′ (t_{1})| > N.  (2.10)

In view of (2.8) and (2.10), we know that there exist t_{2}, t_{3} ∈ (0, 1) with t_{2} < t_{3} such that one of the following cases holds:

Case 1. λ < ω′ (t) < N for t ∈ (t_{2}, t_{3}), ω′ (t_{2}) = λ and ω′ (t_{3}) = N;

Case 2. λ < ω′ (t) < N for t ∈ (t_{2}, t_{3}), ω′ (t_{2}) = N and ω′ (t_{3}) = λ;

Case 3. −N < ω′ (t) < −λ for t ∈ (t_{2}, t_{3}), ω′ (t_{2}) = −N and ω′ (t_{3}) = −λ;

Case 4. −N < ω′ (t) < −λ for t ∈ (t_{2}, t_{3}), ω′ (t_{2}) = −λ and ω′ (t_{3}) = −N. Since the others is similar, we only consider Case 1. By the Nagumo condition, we have

|ω′′ (t)| · ω′ (t) = |f(t, (Aω) (t) , (Bω) (t) , ω (t) , ω′ (t))| · ω′ (t)
≤ h (|ω′ (t)|) · ω′ (t) , t ∈ [t_{2}, t_{3}] .

So,

\frac{|ω′′ (t)| · ω′ (t)}{h (ω′ (t))} ≤ ω′ (t) , t ∈ [t_{2}, t_{3}] ,

and so,

|∫_{t_{2}}^{t_{3}} \frac{ω′′ (t) · ω′ (t)}{h (ω′ (t))}dt| ≤ ∫_{t_{2}}^{t_{3}} |\frac{ω′′ (t) · ω′ (t)}{h (ω′ (t))}| dt ≤ ∫_{t_{2}}^{t_{3}} ω′ (t) dt,

which implies that

∫_{λ}^{N} \frac{s}{h (s)}ds ≤ ω (t_{3}) − ω (t_{2}) ≤ max_{t∈[0,1]} β (t) − min_{t∈[0,1]} α (t) ,

which contradicts with (2.9) and the proof is complete. □

3 Main result

Theorem 3.1 Assume that α and β are, respectively, the strict lower and the strict upper solution of the BVP (2.4) with α (t) ≤ β (t) for t ∈ [0, 1] , and f satisfies the Nagumo condition with respect to α and β. Then the BVP (2.4) has a solution v_{0} and

α (t) ≤ v_{0} (t) ≤ β (t) for t ∈ [0, 1] .

Proof. It follows from Lemma 2.3 that there exists N > 0 such that any solution ω of the BVP (2.4) lying in [α, β] satisfies

|ω′ (t)| ≤ N for t ∈ [0, 1] .

We denote C = max {N, max_{t∈[0,1]} |α′ (t)| , max_{t∈[0,1]} |β′ (t)|} and define the auxiliary functions f_{1}, f_{2}, f_{3}

and F : [0, 1] × R^{4} → R as follows:

f_{1}(t, x_{1}, x_{2}, x_{3}, x_{4}) = \{ \begin{matrix}f(t, x_{1}, x_{2}, x_{3}, C), x_{4} > C, \\ f(t, x_{1}, x_{2}, x_{3}, x_{4}), − C ≤ x_{4} ≤ C, \\ f(t, x_{1}, x_{2}, x_{3}, −C), x_{4} < −C; \end{matrix}

f_{2}(t, x_{1}, x_{2}, x_{3}, x_{4}) = \{ \begin{matrix}f_{1}(t, (Aα) (t) , x_{2}, x_{3}, x_{4}), x_{1} > (Aα) (t) , \\ f_{1}(t, x_{1}, x_{2}, x_{3}, x_{4}), (Aβ) (t) ≤ x_{1} ≤ (Aα) (t) , \\ f_{1}(t, (Aβ) (t) , x_{2}, x_{3}, x_{4}), x_{1} < (Aβ) (t) ; \end{matrix}

f_{3}(t, x_{1}, x_{2}, x_{3}, x_{4}) = \{ \begin{matrix}f_{2}(t, x_{1}, (Bα) (t) , x_{3}, x_{4}), x_{2} > (Bα) (t) , \\ f_{2}(t, x_{1}, x_{2}, x_{3}, x_{4}), (Bβ) (t) ≤ x_{2} ≤ (Bα) (t) , \\ f_{2}(t, x_{1}, (Bβ) (t) , x_{3}, x_{4}), x_{2} < (Bβ) (t) \end{matrix}

and

F (t, x_{1}, x_{2}, x_{3}, x_{4}) = \{ \begin{matrix}f_{3}(t, x_{1}, x_{2}, β (t) , x_{4}), x_{3} > β (t) , \\ f_{3}(t, x_{1}, x_{2}, x_{3}, x_{4}), α (t) ≤ x_{3} ≤ β (t) , \\ f_{3}(t, x_{1}, x_{2}, α (t) , x_{4}), x_{3} < α (t) . \end{matrix}

Consider the following auxiliary BVP

\{ \begin{matrix}v′′(t) + F (t, (Av) (t) , (Bv) (t) , v (t) , v′ (t)) = 0, t ∈ [0, 1] , \\ v (0) = ∑_{i=1}^{m−2} b_{i}v (η_{i}) , v′ (1) = 0. \end{matrix}  (3.1)

If we define an operator T : X → X by

(T v) (t) = ∫_{0}^{1} G_{2} (t, s) F (s, (Av) (s) , (Bv) (s) , v (s) , v′ (s))ds, t ∈ [0, 1] ,

then it is obvious that fixed points of T are solutions of the BVP (3.1). Now, we will apply the Leray-Schauder continuation principle to prove that the operator T has a fixed point. Since it is easy to verify that T : X → X is completely continuous by using the Arzela-Ascoli theorem, we only need to prove that the set of all possible solutions of the homotopy group problem v = λT v is a priori bounded in X by a constant independent of λ ∈ (0, 1) . Denote

α_{m} = min_{t∈[0,1]} α (t) , β_{M} = max_{t∈[0,1]} β (t) ,

(Aβ)_{m} = min_{t∈[0,1]} (Aβ) (t) , (Aα)_{M} = max_{t∈[0,1]} (Aα) (t) ,

(Bβ)_{m} = min_{t∈[0,1]} (Bβ) (t) , (Bα)_{M} = max_{t∈[0,1]} (Bα) (t) ,

L = sup {|f(t, x_{1}, x_{2}, x_{3}, x_{4})| : (t, x_{1}, x_{2}, x_{3}, x_{4}) ∈ [0, 1] × [(Aβ)_{m} , (Aα)_{M}]
× [(Bβ)_{m} , (Bα)_{M}] × [α_{m}, β_{M}] × [−C, C]} .

Let v = λT v. Then we have

|v (t)| = |λ (T v) (t)|
= λ |∫_{0}^{1} G_{2} (t, s) F (s, (Av) (s) , (Bv) (s) , v (s) , v′ (s))ds|

≤ ∫_{0}^{1} (K (t, s) + \frac{1}{1 − ∑_{i=1}^{m−2} b_{i}} ∑_{i=1}^{m−2} b_{i}K (η_{i}, s)) |F (s, (Av) (s) , (Bv) (s) , v (s) , v′ (s))| ds

≤ \frac{L}{1 − ∑_{i=1}^{m−2} b_{i}} =: R, t ∈ [0, 1]

and

|v′ (t)| = |λ (T v)′ (t)|
= λ |∫_{0}^{1} \frac{∂G_{2} (t, s)}{∂t}F (s, (Av) (s) , (Bv) (s) , v (s) , v′ (s))ds|
≤ ∫_{t}^{1} |F (s, (Av) (s) , (Bv) (s) , v (s) , v′ (s))| ds
≤ L ≤ R, t ∈ [0, 1] ,

which imply that

∥v∥ = max {∥v∥_{∞} , ∥v′∥_{∞}} ≤ R.

It is now immediate from the Leray-Schauder continuation principle that the operator T has a fixed point v_{0}, which solves the BVP (3.1).

Now, let us prove that v_{0} is a solution of the BVP (2.4). Therefor, we only need to verify that α (t) ≤ v_{0} (t) ≤ β (t) and |v′_{0} (t)| ≤ C for t ∈ [0, 1] .

First, we will verify that v_{0} (t) ≤ β (t) for t ∈ [0, 1] . Suppose on the contrary that there exists t_{0} ∈ [0, 1] such that

v_{0} (t_{0}) − β (t_{0}) = max_{t∈[0,1]} {v_{0} (t) − β (t)} > 0.

We consider the following three cases:

Case 1: If t_{0} ∈ (0, 1) , then v_{0} (t_{0}) > β (t_{0}) , v′_{0} (t_{0}) = β′ (t_{0}) and v′′_{0} (t_{0}) ≤ β′′ (t_{0}) . Since β is a strict upper solution of the BVP (2.4), one has

v′′_{0} (t_{0}) = −F (t_{0}, (Av_{0}) (t_{0}) , (Bv_{0}) (t_{0}) , v_{0} (t_{0}) , v′_{0} (t_{0}))
= −f_{3}(t_{0}, (Av_{0}) (t_{0}) , (Bv_{0}) (t_{0}) , β (t_{0}) , β′ (t_{0}))
= −f_{2}(t_{0}, (Av_{0}) (t_{0}) , (Bβ) (t_{0}) , β (t_{0}) , β′ (t_{0}))
= −f_{1}(t_{0}, (Aβ) (t_{0}) , (Bβ) (t_{0}) , β (t_{0}) , β′ (t_{0}))
= −f(t_{0}, (Aβ) (t_{0}) , (Bβ) (t_{0}) , β (t_{0}) , β′ (t_{0}))
> β′′ (t_{0}) ,

which is a contradiction.

Case 2: If t_{0} = 0, then v_{0} (0) > β (0) . On the other hand, v_{0} (0) = ∑_{i=1}^{m−2} b_{i}v_{0} (η_{i}) ≤ ∑_{i=1}^{m−2} b_{i}β (η_{i}) ≤

β (0) . This is a contradiction.

Case 3: If t_{0} = 1, then v_{0} (1) − β (1) = max_{t∈[0,1]} {v_{0} (t) − β (t)} > 0, which shows that v′_{0} (1) ≥ β′ (1) .

On the other hand, v′_{0} (1) = 0 ≤ β′ (1) . Consequently, v′_{0} (1) = β′ (1) , and so, v′′_{0} (1) ≤ β′′ (1) . With the similar arguments as in Case 1, we can obtain a contradiction also.

Thus, v_{0} (t) ≤ β (t) for t ∈ [0, 1] . Similarly, we can prove that α (t) ≤ v_{0} (t) for t ∈ [0, 1] .

Next, we will show that |v′_{0} (t)| ≤ C for t ∈ [0, 1] . In fact, since f satisfies the Nagumo condition with respect to α and β, with the similar arguments as in Lemma 2.3, we can obtain that

|v′_{0} (t)| ≤ N ≤ C for t ∈ [0, 1] .

Therefore, v_{0} is a solution of the BVP (2.4) and α (t) ≤ v_{0} (t) ≤ β (t) for t ∈ [0, 1] . □

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(Received October 8, 2009)