Báo cáo hóa học: "DIFFERENTIAL INEQUALITIES METHOD TO nTh-ORDER BOUNDARY VALUE PROBLEMS"

DIFFERENTIAL INEQUALITIES METHOD TO nTh-ORDER BOUNDARY VALUE PROBLEMS

GUANGWA WANG, MINGRU ZHOU, AND LI SUN

Received 31 March 2005; Revised 18 October 2005; Accepted 7 December 2005

By the theory of differential inequality, bounding function method, and the theory of topological degree, this paper presents the existence criterions of solutions for the general nth-order differential equations under nonlinear boundary conditions, and extends many existing results.

Copyright © 2006 Guangwa Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

(cid:3) ,

From Nagumo [10], there have been many accomplishments on the study of the exis- tence of solutions for boundary value problems (BVPs) using the theory of differential inequality (cf. [1–9, 11–17]). However, for the nth-order nonlinear differential equations with the nonlinear boundary conditions, results are very few. The authors made some attempts to solve the nth-order Robin problem [14]. Now we are concerned with the nth-order nonlinear BVP:

(cid:2) t, y, y(cid:2),..., y(n−1) (cid:5)

(cid:4)

= 0,

y(n) = f

(cid:4)

= 0,

y(a), y(cid:2)(a),..., y(n−1)(a) Pi (1.1) i = 1,...,n − 1, (cid:5) y(b), y(cid:2)(b),..., y(n−1)(b) Pn

f (t,ξ0,ξ1,...,ξn−1) ∈ C(I × Rn, R), Pi(η0,η1,...,ηn−1) ∈ C(Rn, R), where t ∈ I = [a,b], Pn(ζ0,ζ1,...,ζn−1) ∈ C(Rn, R).

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 12040, Pages 1–12 DOI 10.1155/JIA/2006/12040

Our method is not only modifying the nonlinear function in the original equations, but also transforming the original nonlinear boundary conditions into some new bound- ary conditions which are easy to discuss. Thus, we get the new BVP which will be dis- cussed firstly, then the judgement of the existence of solutions for the original BVP will be attained naturally. This technique dealing with the nonlinear problem is simpler and

2 Differential inequalities method to nth-order BVPs

clearer compared with the method of shooting. However, it has scarcely been used in the available reference materials.

The paper is organized as follows. In Section 2, we give out some basic concepts and the preparative theorem. In Section 3, the main result is presented and proved. In Section 4, a more general BVP is studied. Finally, in Section 5, we use the results to solve an example which cannot be solved by [1–17].

2. Preparative theorem

2.1. Basic concepts. We first define a function

if x < r,

⎧ ⎪⎪⎪⎪⎨ r ⎪⎪⎪⎪⎩ s

(2.1) δ(r,x,s) ≡ x if r ≤ x ≤ s,

if s < x,

where r,x,s ∈ R, r ≤ s.

Definition 2.1. Assume that α(t),β(t) ∈ Cn(I, R). The pair of functions (α(t),β(t)) is called a bounding function pair (or simply, a bounding pair) of BVP (1.1) in case there exists N > 0 such that for all u(t) ∈ Cn(I, R):

(i) α( j)(t) ≤ β( j)(t), t ∈ I, j = 0,1,...,n − 2; (ii) α(n)(t) ≥ f (t,u(t),u(cid:2)(t),...,u(n−3)(t),α(n−2)(t),α(n−1)(t)), β(n)(t) ≤ f (t,u(t),u(cid:2)(t), ...,u(n−3)(t),β(n−2)(t),β(n−1)(t)), where u( j)(t) = δ(α( j)(t),u( j)(t),β( j)(t)), j = 0,1, ...,n − 3;

(iii) Pi(u(a),...,u(i−2)(a),α(i−1)(a),α(i)(a),u(i+1)(a),...,u(n−1)(a)) ≤ 0 ≤ Pi(u(a),..., u(i−2)(a),β(i−1)(a),β(i)(a),u(i+1)(a),...,u(n−1)(a)), Pn(u(b),...,u(n−3)(b),α(n−2)(b), α(n−1)(b)) ≤ 0 ≤ Pn(u(b),... ,u(n−3)(b),β(n−2)(b),β(n−1)(b)), where i = 1,2,..., n−1, u(n−2)(a) =δ(α(n−2)(a),u(n−2)(a),β(n−2)(a)), u(n−1)(a) =δ(−N,u(n−1)(a),N).

(cid:10)(cid:2)

(cid:3)

(cid:12)

Definition 2.2. A continuous function f (t,ξ0,...,ξn−1) is said to satisfy a Nagumo condi- tion with respect to variable ξn−1 on the set

| t ∈ I;

(cid:11) (cid:11)ξ j

(cid:11) (cid:11) ≤ r j, j =0,1,...,n−2, r j is a positive constant; ξn−1 ∈ R (2.2)

(cid:11) (cid:3) (cid:11)

(cid:3)(cid:11) (cid:11) ≤ Φ

(cid:2) = t,ξ0,...,ξn−1

(cid:11) (cid:11) f

(cid:2)(cid:11) (cid:11)ξn−1

(cid:13)

,

= +∞.

(2.3)

in case there exists function Φ(t) ∈ C([0,+∞],(0,+∞)), such that (cid:2) t,ξ0,...,ξn−1 +∞ sds Φ(s)

2.2. The modified problem. Assume that there are two functions α(t), β(t) satisfying

(2.4) j = 0,1,...,n − 2. α( j)(t) ≤ β( j)(t),

Guangwa Wang et al. 3

(cid:3)

(cid:3)

(cid:2)

(cid:3) ,

We define function

(cid:2) t, y, y(cid:2),..., y(n−1)

≡ f

(cid:2) t, y, y(cid:2),..., y(n−1)

(2.5) f + h y(n−2)

(cid:14)

(cid:15)

(cid:11) (cid:11)

(cid:11) (cid:11),

where y( j)(t) = δ(α( j)(t), y( j)(t),β( j)(t)) ( j = 0,1,...,n − 2) and y(n−1)(t) = δ(−N, y(n−1) (t),N). N is a positive constant such that

(cid:11) (cid:11)α(n−1)(t)

(cid:11) (cid:11)β(n−1)(t)

N

, (2.6) , N > max t∈I 2M b − a (cid:13)

2M/(b−a)

(2.7) > 2M, sds Φ(s)

⎧

in which M > maxt∈I {|α(n−2)(t)|, |β(n−2)(t)|}. h(y(n−2)) is continuous, bounded, and

(cid:2)

(cid:3)

if y(n−2) < α(n−2),

⎪⎪⎪⎪⎨ < 0 = 0 ⎪⎪⎪⎪⎩ > 0

(2.8) y(n−2) h if α(n−2) ≤ y(n−2) ≤ β(n−2),

if y(n−2) > β(n−2).

(cid:2)

(cid:3)

≡

Such function h(·) is easy to obtain, for example, let

(cid:11) (cid:11) .

(2.9) h y(n−2) 1 + y(n−2) − y(n−2) (cid:11) (cid:11)y(n−2) − y(n−2)

(cid:4)

(cid:5)

In addition, we define

(cid:4)

(cid:5)

(cid:5) ,

≡ δ

y(t), y(cid:2)(t),..., y(n−1)(t) Pi

(cid:4) α(i−1)(t), y(i−1)(t) − Pi

(cid:4)

(cid:5)

y(t), y(cid:2)(t),..., y(n−1)(t) ,β(i−1)(t) i = 1,2,...,n − 1,

(cid:4)

(cid:5)

(cid:5)

≡ δ

y(t), y(cid:2)(t),..., y(n−1)(t) Pn

(cid:4) α(n−2)(t), y(n−2)(t) − Pn

y(t), y(cid:2)(t),... , y(n−1)(t) ,β(n−2)(t) . (2.10)

(cid:5)

Then we consider the following modified problem:

(cid:4) t, y, y(cid:2),..., y(n−1)

(cid:4)

, y(n) = f

(cid:5) , (cid:5) .

(2.11) i = 1,...,n − 1, y(i−1)(a) = Pi y(a), y(cid:2)(a),..., y(n−1)(a) (cid:4) y(b), y(cid:2)(b),..., y(n−1)(b) y(n−2)(b) = Pn

4 Differential inequalities method to nth-order BVPs

2.3. Preparative theorem.

Lemma 2.3. Assume that (A1) BVP (1.1) has a bounding pair (α(t),β(t)) on the interval I by Definition 2.1; (A2) the function f (t, y, y(cid:2),..., y(n−1)) in BVP (1.1) satisfies the Nagumo condition with respect to y(n−1)(t) by Definition 2.2. Then BVP (2.11) has a solution y(t) ∈ Cn(I, R) such that

(cid:11) (cid:11)y(n−1)(t)

(cid:11) (cid:11) ≤ N,

i = 0,1,...,n − 2, α(i)(t) ≤ y(i)(t) ≤ β(i)(t), (2.12) t ∈ I,

where N is the positive constant given in the definition of f .

The proof of Lemma 2.3 is a simple consequence of the following three propositions.

Proposition 2.4. The modified BVP (2.11) has a solution y(t) ∈ Cn(I, R).

(cid:4)

(cid:5)

Proof. Consider

≡ g(t),

(cid:4)

(cid:5)

t, y,..., y(n−1) y(n) = λ f

≡ gi(a), (cid:5)

≡ gn(b),

(2.13) i = 1,2,...,n − 1, y(i−1)(a) = λPi y(a),..., y(n−1)(a) (cid:4) y(b),..., y(n−1)(b) y(n−2)(b) = λPn

where λ ∈ [0,1]. From the representations of f , Pi, and Pn, we know that y(n)(t), y(i−1)(a) (i = 1,2,...,n − 1), and y(n−2)(b) all are bounded. Also, by the mean value theorem, we may ensure that y(n−1)(t),..., y(cid:2)(t), y(t) all are bounded functions in I. In fact, by the mean value theorem, there exists some ξ ∈ (a,b) satisfying

(2.14) y(n−2)(b) − y(n−2)(a) = y(n−1)(ξ)(b − a),

then y(n−1)(ξ) is bounded. From

(2.15) y(n−1)(t) − y(n−1)(ξ) = y(n)(η)(t − ξ) ∀t ∈ [a,b],

y(n−1)(t) is bounded. Thus, from

(2.16) η ∈ (a,b), i = 0,1,...,n − 2, y(i)(t) − y(i)(a) = y(i+1)(ζ)(t − a),

it is easy to see that y(n−2)(t),..., y(cid:2)(t), y(t) all are bounded in I.

(cid:13)

(cid:13)

(cid:13)

t

t1

tn−1

· · ·

Let Ω = {y(t) ∈ Cn(I, R) | (cid:9)y(i)(t)(cid:9) < K, for all t ∈ I, i = 0,1,...,n − 1,K is some sufficiently large positive constant}. Then Ω is a bounded open set. BVP (2.13) can be equivalently written as the following integral equation:

a

a

b

(2.17) y(t) = c1 + c2t + c3t2 + · · · + cntn−1 + g(s)dsdt1 · · · dtn−1 ≡ Tλ y,

Guangwa Wang et al. 5

where Tλ is an integral operator with a parameter λ and (c1,...,cn) is determined by the system of equations

c1 + c2a + c3a2 + · · · + cnan−1 = g1(a),

(2.18)

(cid:13)

(cid:13)

b

t1

c2 + c3 · 2a + · · · + cn(n − 1)an−2 = g2(a), ... cn−1(n − 2)(n − 3) · · ·3 + cn(n − 1)!a = gn−1(a),

a

b

g(s)dsdt1. cn−1(n − 2)(n − 3) · · ·3 + cn(n − 1)!b = gn(b) −

Let H(λ, y) = (I − Tλ)(y), then H : [0,1] × Ω → Rn is continuous, where I is identity mapping. Let hλ(y) = H(λ, y), then 0 /∈ hλ(∂Ω). In fact, for all y ∈ ∂Ω, (cid:9) y (cid:9)≥ K. Notic- ing that K is sufficiently large, we have

(cid:9) hλ(y) (cid:9) =(cid:9) y − Tλ y (cid:9)≥(cid:9) y (cid:9) − (cid:9) Tλ y (cid:9)≥ K − (cid:9) Tλ y (cid:9)> 0 ∀λ ∈ [0,1].

(2.19)

(cid:3)

(cid:3)

(cid:3)

(cid:2)

(cid:3)

Thus, 0 /∈ hλ(∂Ω). By the homotopy invariance theorem of topological degree, deg(hλ, Ω,0) is a constant, in particular, deg(h1,Ω,0) = deg(h0,Ω,0). Noticing that 0 ∈ Ω, by the normality of topological degree, we have

= deg

= deg

= deg

= 1.

(cid:2) h1,Ω,0

(cid:2) h0,Ω,0

(cid:2) I − T0,Ω,0

deg (2.20) I,Ω,0

Hence, by the solvability theorem of topological degree, it is clear that there exists some (cid:2) y(t) satisfying (2.17), then this proposition is proved.

Proposition 2.5. Every solution y(t) of the modified BVP (2.11) satisfies

(2.21) α(i)(t) ≤ y(i)(t) ≤ β(i)(t), t ∈ I, i = 0,1,...,n − 2.

Proof. First, we show that

(2.22) α(n−2)(t) ≤ y(n−2)(t) ≤ β(n−2)(t), t ∈ I.

(cid:4)

(cid:5)

If α(n−2)(t) ≤ y(n−2)(t) is not true, then there exists some ξ ∈ [a,b], such that

= α(n−2)(ξ) − y(n−2)(ξ) > 0.

(2.23) α(n−2)(t) − y(n−2)(t) max t∈I

Then ξ (cid:11)= a,b by the boundary conditions of BVP (2.11). Thus

(2.24) α(n−1)(ξ) − y(n−1)(ξ) = 0,

(2.25) α(n)(ξ) − y(n)(ξ) ≤ 0.

6 Differential inequalities method to nth-order BVPs

(cid:5)

(cid:4)

However, on the other hand, from the definition of α(t) and that y(t) is a solution of (2.11), we have

(cid:5)

(cid:4)

(cid:5)

− h

− f (cid:4)

(cid:5)

= −h

α(n)(ξ) − y(n)(ξ) ≥ f ξ, y(ξ),..., y(n−3)(ξ),α(n−2)(ξ),α(n−1)(ξ) (cid:4) ξ, y(ξ),..., y(n−3)(ξ), y(n−2)(ξ), y(n−1)(ξ) y(n−2)(ξ)

y(n−2)(ξ) > 0. (2.26)

This contradicts (2.25). Hence,

(2.27) t ∈ I. α(n−2)(t) ≤ y(n−2)(t),

A similar proof shows that

(2.28) t ∈ I. y(n−2)(t) ≤ β(n−2)(t),

To sum up, (2.22) is true. From (2.22), the function y(n−3)(t) − α(n−3)(t) is increasing in I. Noticing

(2.29) α(n−3)(a) ≤ y(n−3)(a),

we know that α(n−3)(t) ≤ y(n−3)(t). A similar proof shows y(n−3)(t) ≤ β(n−3)(t). Using the same argument, it follows that α(i)(t) ≤ y(i)(t) ≤ β(i)(t), i = n − 4,n − 5,...,2,1. Thus, the (cid:2) proof of Proposition 2.5 is completed.

Proposition 2.6. For every solution y(t) of the modified BVP (2.11) holds

(cid:11) (cid:11)y(n−1)(t)

(cid:11) (cid:11) ≤ N,

(2.30) t ∈ I.

Proof. Suppose that there exists some τ ∈ [a,b] such that

(cid:11) (cid:11)y(n−1)(τ)

(cid:11) (cid:11) > N.

(2.31)

Without loss of generality, we assume that y(n−1)(τ) > N. There exists ξ ∈ (a,b), such that

≤ 2M b − a

(2.32) < N. y(n−1)(ξ) = y(n−2)(b) − y(n−2)(a) b − a

Hence, there exists some subinterval [c, d] (or [d, c]) ⊂ [a,b] such that

≤ y(n−1)(t) ≤ N, ∀t ∈ [c,d] (or [d,c]).

, y(n−1)(d) = N, y(n−1)(c) = 2M b − a (2.33)

2M b − a

Guangwa Wang et al. 7

(cid:13)

(cid:13)

d

d

≤

=

From condition (A2),

(cid:11) (cid:11) (cid:11)y(n−2)(d) − y(n−2)(c)

(cid:11) (cid:11) (cid:11) ≤ 2M.

(cid:11) (cid:11) (cid:11) (cid:11) (cid:11)

(cid:11) (cid:11) (cid:11) (cid:3) ds (cid:11) (cid:11)

(cid:11) (cid:11) (cid:11) y(n−1)(s)ds (cid:11) (cid:11)

(cid:11) (cid:11) (cid:11) (cid:11) (cid:11)

c

c

(2.34) y(n−1)(s)y(n)(s) (cid:11) (cid:2)(cid:11) (cid:11) (cid:11)y(n−1)(s) Φ

(cid:13)

(cid:13)

(cid:13)

d

N

N

=

=

On the other hand, from (2.7) we know that

(cid:11) (cid:11) (cid:11) (cid:11) (cid:11)

(cid:11) (cid:11) (cid:11) (cid:3) ds (cid:11) (cid:11)

(cid:11) (cid:11) (cid:11) (cid:11) (cid:11)

(cid:11) (cid:11) (cid:11) (cid:11) (cid:11)

c

2M/(b−a)

2M/(b−a)

(2.35) > 2M. rdr Φ(r) rdr Φ(r) y(n−1)(s)y(n)(s) (cid:11) (cid:2)(cid:11) (cid:11) (cid:11)y(n−1)(s) Φ

(cid:2)

This inequality contradicts the above one and Proposition 2.6 holds.

3. Main theorem

Now, the main result of this paper is given in the following theorem.

Theorem 3.1. Assume that the conditions (A1), (A2) in Lemma 2.3 hold and added to (A3). The function Pi(η0,...,ηn−1) (i = 1,2,...,n) satisfies

(i) Pi(η0,...,ηn−1) is increasing in ηi−1 and decreasing in ηi, i = 1,2,...,n − 2; (ii) Pn−1(η0,...,ηn−1) is decreasing in ηn−1; (iii) Pn(η0,...,ηn−1) is increasing in ηn−1. Then BVP (1.1) has a solution y(t) ∈ Cn(I, R) such that

(cid:11) (cid:11)y(n−1)(t)

(cid:11) (cid:11) ≤ N,

α(i)(t) ≤ y(i)(t) ≤ β(i)(t), i = 0,1,...,n − 2, (3.1) t ∈ I,

where N is the positive constant given in the definition of f .

(cid:4)

(cid:5)

= 0,

Proof. From Lemma 2.3 and the definition of f , the solution y(t) of the modified BVP (2.11) satisfies (1.1). As soon as it is proved that y(t) satisfies the boundary conditions of (1.1) under condition (A3), we may say that y(t) is a solution of BVP (1.1). First, we prove

(3.2) y(a),..., y(n−1)(a) i = 1,2,...,n − 2. Pi

(cid:4)

(cid:5)

Case 1. Suppose that

≤ β(i−1)(a).

(3.3) y(a),..., y(n−1)(a) α(i−1)(a) ≤ y(i−1)(a) − Pi

(cid:5)

(cid:4)

(cid:4)

Then

(cid:5) .

= y(i−1)(a) − Pi

(3.4) y(a),..., y(n−1)(a) y(a),..., y(n−1)(a) y(i−1)(a) = Pi

(cid:4)

(cid:5)

Thus

= 0.

(3.5) y(a),..., y(n−1)(a) Pi

8 Differential inequalities method to nth-order BVPs

(cid:4)

(cid:5)

Case 2. Suppose that there exists some i ∈ {1,2,...,n − 2} such that

(3.6) y(a), y(cid:2)(a),..., y(n−1)(a) . α(i−1)(a) > y(i−1)(a) − Pi

(cid:4)

(cid:5)

Then

= α(i−1)(a).

(3.7) y(a), y(cid:2)(a),..., y(n−1)(a) y(i−1)(a) = Pi

(cid:5)

(cid:4)

Hence

(3.8) y(a), y(cid:2)(a),..., y(n−1)(a) > 0. Pi

(cid:4)

(cid:5)

From Propositions 2.5 and 2.6 and condition (A3),

(3.9) y(a),..., y(i−2)(a),α(i−1)(a),α(i)(a), y(i+1)(a),..., y(n−1)(a) > 0. Pi

(cid:5)

It is easy to see that the last inequality contradicts Definition 2.1(iii). Therefore, Case 2 is not true. Case 3. Suppose that there exists some i ∈ {1,2,...,n − 2} such that (cid:4) (3.10) y(a), y(cid:2)(a),..., y(n−1)(a) > β(i−1)(a). y(i−1)(a) − Pi

(cid:4)

(cid:5)

(cid:4)

(cid:5)

Then by the analogous analysis, we have

≤Pi

y(a),..., y(i−2)(a),β(i−1)(a),β(i)(a), y(i+1)(a),..., y(n−1)(a) y(a),..., y(n−1)(a) Pi <0. (3.11)

Obviously, the last inequality contradicts Definition 2.1(iii). Therefore, this case cannot hold.

(cid:4)

(cid:5)

(cid:4)

(cid:5)

= 0,

To sum up, (3.2) holds. A similar proof shows that

= 0.

(3.12) y(a), y(cid:2)(a),..., y(n−1)(a) y(b), y(cid:2)(b),..., y(n−1)(b) Pn−1 Pn

(cid:2)

The proof is completed.

4. A generalized problem

(cid:5)

(cid:4)

(cid:5)

Now, we consider the following boundary value problem with more general boundary conditions:

= 0,

(cid:4) t, y,..., y(n−1)

, (4.1) y(a),..., y(n−1)(a), y(b),..., y(n−1)(b) y(n) = f Pi

where t ∈ I, i = 1,2,...,n, f and Pi are continuous functions. Similarly to Definition 2.1, we give the following.

Guangwa Wang et al. 9

Definition 4.1. Assume α(t),β(t) ∈ Cn(I, R). The pair of functions (α(t),β(t)) is called a bounding function pair of BVP (4.1) in case that for all u(t) ∈ Cn(I, R)

(cid:5)

(cid:4) u(a),...,α(i−1)(a),α(i)(a),...,u(n−1)(a),u(b),...,u(n−1)(b)

(i) the same as Definition 2.1(i); (ii) the same as Definition 2.1(ii); (iii)(cid:2)

(cid:4)

(cid:5) ,

Pi

≤ 0 ≤ Pi

(cid:4)

(cid:5)

(cid:5)

(cid:4)

u(a),...,β(i−1)(a),β(i)(a),...,u(n−1)(a),u(b),...,u(n−1)(b) (4.2) u(a),...,u(n−1)(a),u(b),... ,u(n−3)(b),α(n−2)(b),α(n−1)(b) Pn

≤ 0 ≤ Pn

, u(a),...,u(n−1)(a),u(b),...,u(n−3)(b),β(n−2)(b),β(n−1)(b)

where i = 1,2,...,n − 1.

For BVP (4.1), we have the following existence theorem.

Theorem 4.2. Assume that (A1)(cid:2) BVP (4.1) has a bounding function pair (α(t),β(t)) in the interval I by Definition 4.1; (A2)(cid:2) the function f (t, y, y(cid:2),..., y(n−1)) in BVP (4.1) satisfies the Nagumo condition with respect to y(n−1)(t) by Definition 2.2; (A3)(cid:2) the function Pi(η0,...,ηn−1,ζ0,... ,ζn−1) (i = 1,2,...,n) satisfies (i) Pi(η0,...,ηn−1,ζ0,...,ζn−1) is increasing in ηi−1 and decreasing in ηi, i = 1,2,..., n − 2;

(ii) Pn−1(η0,...,ηn−1,ζ0,...,ζn−1) is decreasing in ηn−1; (iii) Pn(η0,...,ηn−1,ζ0,...,ζn−1) is increasing in ζn−1. Then BVP (4.1) has a solution y(t) ∈ Cn(I, R) such that

(cid:11) (cid:11)y(n−1)(t)

(cid:11) (cid:11) ≤ N,

i = 0,1,...,n − 2, α(i)(t) ≤ y(i)(t) ≤ β(i)(t), (4.3) t ∈ I,

where N is the positive constant given in the definition of f .

(cid:3) ,

(cid:2) t, y,..., y(n−1)

Proof. Consider the modified problem

y(n) = f y(i−1)(a) = Pi(a), y(n−2)(b) = Pn(b), (4.4) i = 1,2,...,n − 1.

(cid:4)

(cid:5)

The modified function f (t, y,..., y(n−1)) is defined as BVP (2.11), and

(cid:4)

y(t),..., y(n−1)(t), y(b + a − t),..., y(n−1)(b + a − t)

≡ δ

(cid:5)

(4.5)

, y(t),..., y(n−1)(t), (cid:5) ,β(i−1)(t) Pi(t) ≡ Pi (cid:4) α(i−1)(t), y(i−1)(t) − Pi y(b + a − t),..., y(n−1)(b + a − t)

10 Differential inequalities method to nth-order BVPs

(cid:5)

(cid:4)

where i = 1,2,...,n − 1, (cid:4) y(b + a − t),..., y(n−1)(b + a − t), y(t),..., y(n−1)(t)

≡ δ

(cid:5)

(cid:5)

(4.6) y(b + a − t),..., y(n−1)(b + a − t), Pn(t) ≡ Pn (cid:4) α(n−2)(t), y(n−2)(t) − Pn

y(t),..., y(n−1)(t) ,β(n−2)(t) .

Using the same argument as the proof of Lemma 2.3, it follows that under the conditions (A1)(cid:2) and (A2)(cid:2), BVP (4.4) has a solution y(t) satisfying the two inequalities in the con- clusions of Lemma 2.3. Furthermore, in an analogous way to the proof of Theorem 3.1, it follows that the solution y(t) of BVP (4.4) is a solution of BVP (4.1). Consequently, the (cid:2) proof of Theorem 4.2 is completed. The details of the proof will be omitted.

5. An example

In this section, we study an example by making use of Theorems 3.1 and 4.2.

(cid:3)

(cid:3)

(cid:3)2(cid:2)

(cid:3) ,

(cid:2) 1 + t2

Example 5.1. Consider the 4th-order nonlinear boundary value problem

(cid:2) 1 + (y(cid:2))2

(cid:2) t + t2

(cid:2)

y(cid:2) + sin(y(cid:2)(cid:2)) + 1 + (y(cid:2)(cid:2)(cid:2))2 y(iv) = (t − y)2 − t 112 sin2

(cid:3)3 − y(cid:2)(cid:2)(1) +

(cid:2)

y(cid:2)(1) y(2) = A,

(5.1) y(cid:2)(2) k 6 (cid:3)2 = B, y(cid:2)(cid:2)(1) + k 8

y(cid:2)(cid:2)(2) = C,

(cid:3)3 = D,

4y(1) − 1 8 5y(cid:2)(1) − 1 2 y(1) + 2y(cid:2)(cid:2)(1) − y(cid:2)(cid:2)(cid:2)(1) − k 2 (cid:2) (cid:3)2 + 4 (cid:2) y(cid:2)(cid:2)(2) k y(1) − y(cid:2)(2) − 4 y(cid:2)(cid:2)(cid:2)(2)

(cid:3)

(cid:2)

(cid:3)

(cid:2)

(cid:3)2(cid:2)

=

(cid:3) ,

(cid:3)2 − t

where t ∈ [1,2], k is a constant. Let

(cid:2) t + t2

(cid:2) t,ξ0,ξ1,ξ2,ξ3

(cid:2) t − ξ0

(cid:3) ξ1 +

(cid:3)

(cid:2) η0,η1,η2,η3,ζ0,ζ1,ζ2,ζ3

1 + t2 f sinξ2 + 1 + ξ2 3 1 + ξ2 1

− η2 +

(cid:3)

P1 ζ0 − A, η3 1 k 6

− B,

(cid:2) η0,η1,η2,η3,ζ0,ζ1,ζ2,ζ3

(cid:3)

(5.2) P2 η2 + ζ 2 1

(cid:2) η0,η1,η2,η3,ζ0,ζ1,ζ2,ζ3 (cid:3)

− D.

ζ2 − C,

= kη0 − ζ1 − 4ζ 2

P3 (cid:2) η0,η1,η2,η3,ζ0,ζ1,ζ2,ζ3 P4 112 sin2 = 4η0 − 1 8 = 5η1 − 1 k 2 8 = η0 + 2η2 − η3 − k 2 2 + 4ζ 3 3

Guangwa Wang et al. 11

Let

(5.3) β(t) = t. α(t) = −t2,

Then, for the case of k = 0, A ∈ [−1,31/8], B ∈ [−9,5], C ∈ [−3, −1], D ∈ [−12, −1], and the case of k = 1, A ∈ [−2/3,77/24], B ∈ [−7,5], C ∈ [−2, −1], D ∈ [−11, −2], it is easy to prove that (α(t),β(t)) is a bounding pair of BVP (5.1) and all assumptions of Theorems 3.1 and 4.2 are fulfilled, respectively. Hence, for any of the two cases, BVP (5.1) has at least one solution y(t) satisfying

−t2 ≤ y(t) ≤ t, −2t ≤ y(cid:2)(t) ≤ 1, −2 ≤ y(cid:2)(cid:2)(t) ≤ 0,

(5.4) t ∈ [1,2].

Acknowledgments

Sincere thanks to the anonymous Referee for his/her careful reading and the Editors for their careful and patient work. This work is supported by the NSF (04XLB03,05XLA02, and 05XLB02) of Xuzhou Normal University.

[1] A. Cabada, M. do R. Grossinho, and F. Minh ´os, On the solvability of some discontinuous third or- der nonlinear differential equations with two point boundary conditions, Journal of Mathematical Analysis and Applications 285 (2003), no. 1, 174–190.

[2] K. W. Chang and F. A. Howes, Nonlinear Singular Perturbation Phenomena: Theory and Applica-

tions, Applied Mathematical Sciences, vol. 56, Springer, New York, 1984.

[3] Z. Du, W. Ge, and X. Lin, Existence of solutions for a class of third-order nonlinear boundary value problems, Journal of Mathematical Analysis and Applications 294 (2004), no. 1, 104–112. [4] J. Ehme, P. W. Eloe, and J. Henderson, Upper and lower solution methods for fully nonlinear

boundary value problems, Journal of Differential Equations 180 (2002), no. 1, 51–64.

[5] L. H. Erbe, Existence of solutions to boundary value problems for second order differential equations,

Nonlinear Analysis 6 (1982), no. 11, 1155–1162.

[6] Ch. Fabry and P. Habets, Upper and lower solutions for second-order boundary value problems

with nonlinear boundary conditions, Nonlinear Analysis 10 (1986), no. 10, 985–1007.

[7] F. A. Howes, Differential inequalities and applications to nonlinear singular perturbation problems,

Journal of Differential Equations 20 (1976), no. 1, 133–149.

[8] W. G. Kelley, Some existence theorems for nth-order boundary value problems, Journal of Differ-

ential Equations 18 (1975), no. 1, 158–169.

[9] Z. Lin and M. Zhou, Perturbation Methods in Applied Mathematics, Jiangsu Education Press,

Nanjing, 1995.

[10] M. Nagumo, ¨Uber die Differentialgleichung y(cid:2)(cid:2) = f (x, y, y(cid:2)), Proceedings of the Physico-

Mathematical Society of Japan 19 (1937), 861–866.

[11] M. A. O’Donnell, Semi-linear systems of boundary value problems, SIAM Journal on Mathemati-

cal Analysis 15 (1984), no. 2, 316–332.

[12] M. H. Pei, Nonlinear two-point boundary value problems for nth-order nonlinear differential equa-

tions, Acta Mathematica Sinica 43 (2000), no. 5, 921–930.

[13] F. Sadyrbaev, Nonlinear fourth-order two-point boundary value problems, The Rocky Mountain

Journal of Mathematics 25 (1995), no. 2, 757–781.

References

[14] G. Wang, M. Zhou, and L. Sun, Existence of solutions of two-point boundary value problems for the systems of nth-order differential equations, Journal of Nanjing University. Mathematical Bi- quarterly 19 (2002), no. 1, 68–79 (Chinese).

[15] X. Yang, The method of lower and upper solutions for systems of boundary value problems, Applied

Mathematics and Computation 144 (2003), no. 1, 169–172.

, Upper and lower solutions for periodic problems, Applied Mathematics and Computation

[16]

12 Differential inequalities method to nth-order BVPs

137 (2003), no. 2-3, 413–422.

[17] X. Zhang, An existence theorem and estimation of the solution of the Robin problem for a class of nth order equations, Acta Mathematicae Applicatae Sinica 14 (1991), no. 3, 397–403 (Chinese).

Guangwa Wang: Department of Mathematics, Xuzhou Normal University, Xuzhou 221116, China E-mail address: wgw7653@xznu.edu.cn

Mingru Zhou: Department of Mathematics, Xuzhou Normal University, Xuzhou 221116, China E-mail address: zhoumr@xznu.edu.cn

Li Sun: Department of Mathematics, Xuzhou Normal University, Xuzhou 221116, China E-mail address: slwgw-7653@xznu.edu.cn