Solutions of the (n-1,1)-type multi-point boundary value problems for higher-order differential equations *.

Abstract

In this article, we study the higher-order differential equation 

[x.sup.(n)](t) = f(t, x(t), x'(t), ..., [x.sup.(n-1)](t)), 0 < t < 1, (**)

subject to one of following multi-point boundary value conditions

x(1) = [m.summation over (i=1)] [[alpha].sub.i]x([[xi].sub.i]), [x.sup.(i)](0) = 0 for i = 0, 1, ..., n - 2, (***)

and

[x.sup.(p)](1) = [m.summation over (i=1)] [[alpha].sub.i][x.sup.(p)][[xi].sub.i]), [x.sup.(i)](0) = 0 for i = 0, 1, ..., n - 2, (****)

We establish sufficient conditions for the existence of at least one solution of the BVP(**) and (***) and BVP(**) and (***) at resonance and another at non-resonance (Theorems 2.1, 2.2, 3.1, 3.2), respectively. Especially, we establish sufficient conditions for the existence of positive solutions of above BVPs (Corollaries 3.1 and 3.2). The emphasis in this paper is on the fact that f depends upon all higher-order derivatives.

Keywords and Phrases: Solvability, Resonance, Non-resonance, Multipoint boundary value problem, Higher-order differential equations.

1. Introduction

Eloe and Henderson [1] studied the following nonlinear (n-1,1) conjugate boundary value problems:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

where a and f are continuous and f(x) [greater than or equal to] 0 for all x [greater than or equal to] 0. They proved the existence of positive solutions under the assumptions that f is either sub-linear or super-linear.

Recently, Liu and Ge [2] considered the following higher-order differential equation:

[x.sup.(n)](t)+[lambda]a(t)f(x(t)) = 0, 0 < t < 1, (2)

subject to one of following boundary value conditions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2')

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2")

where a and f are continuous with f(x) [greater than or equal to] 0 for all x [greater than or equal to] 0 and f(0) > 0 and a may change sign, [lambda] > 0 is constant, [beta] [greater than or equal to] 0 and 0 < [eta] < 1 with [beta][[eta].sup.n-1] [not equal to] 1 for BVP(2) and (2") and [beta][[eta].sup.n-1] [not equal to] 1 for BVP(2) and (2"), respectively. They proved that under some assumptions, BVP(2) and (2'), BVP(2) and (2") have at least one positive solution for sufficiently small [lambda] > 0.

For the multi-point boundary value problems for the second order differential equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

where [[alpha].sub.i] [greater than or equal to] 0, 0 < [[xi].sub.1] < ... < [[xi].sub.m] < 1, f is nonnegative and continuous, there has been extensive studies for the existence of its positive solutions under the following assumption [[summation].sup.m.sub.i=1] [[alpha].sub.i][[xi].sub.i] < 1. We refer the readers to [3-5] and the references therein. We note that f only dependents on t and x. The existence results of the positive solutions of above BVPs have not been established when f dependents on both t, x and x'.

For the BVP

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

Feng and Webb [6,7] studied the solvability of above boundary value problem under the assumption [beta][eta] = 1. However, to the best of our knowledge, there is not many paper concerned with the existence of solutions of multi-point BVP for higher order differential equations, especially, concerned with the existence of solutions of multi-point boundary value problems for higher order differential equations at resonance.

In this paper, we are concerned with the following higher-order differential equation

[x.sup.(n)](t) = f(t, x(t), x'(t), ..., [x.sup.(n-1)](t)), 0 < t < 1, (5)

subject to one of the following boundary value conditions:

x(1) = [m.summation over (i=1)] [[alpha].sub.i]x([[xi].sub.i]), [x.sup.(i)](0) = 0 for i = 0, 1, ..., n - 2, (6)

and

[x.sup.(p)](1) = [m.summation over (i=1)] [[alpha].sub.i][x.sup.(p)][[xi].sub.i]), [x.sup.(i)](0) = 0 for i = 0, 1, ..., n - 2, (7)

where [[alpha].sub.i] [member of] R(i = 1, ..., m), n [greater than or equal to] 2 is an integer, and 0 < [[xi].sub.1] < [[xi].sub.2] < ... < [[xi].sub.m] < 1, 1 [less than or equal to] p [less than or equal to] n-1, an integer, are fixed, f is continuous. For BVP(5) and (6) and BVP(5) and (7), we shall consider cases resonance and non-resonance.

The purpose of this paper is to establish sufficient conditions for the existence of solutions and positive solutions of BVP(5) and (6) and BVP(5) and (7). The results obtained improve the corresponding ones in [11] when n = 2 and are new when n > 2. The emphasis of this paper are as follows:

(i). The nonlinearity f depends on all derivatives, i.e. f depends on x, x', ..., [x.sup.(n-1)].

(ii). In case i.1(i=1,2), equation (5) with boundary value conditions (6) or (7) can not be transformed into an integral equation since the linear operator Lx(t) = [x.sup.(n)](t) defined in a Banach space is not invertible, which is called the resonance case.

(iii). In case i.2(i=1,2), we transform equation (5) with the corresponding boundary value conditions into the integral equations to establish the existence results for solutions.

This paper is organized as follows. In section 2, we establish existence results for solutions for Case i.1(i=1,2)(Theorems 2.1 and 2.2). And the existence results for solutions(Theorems 3.1 and 3.2) and positive solutions (Corollaries 3.1 and 3.2) for Case i.2(i=1,2) will be presented in section 3.

2. Existence Results for Cases i.1(i=1,2)

In this section, we establish sufficient conditions for the existence of at least one solution of BVP(5) and (6), BVP(5) and (7) at resonance case, i.e. [[summation].sup.m.sub.i=1] [[alpha].sub.i][[xi].sup.n-1.sub.i], and [[summation].sup.m.sub.i=1] [[alpha].sub.i][[xi].sup.n-1-p].sub.i] = 1, respectively. In these cases, the operator Lx(t) = [x.sup.(n)(t) is not invertible. For convenience, we need some notations and an abstract existence theorem, which can be see in [8,10-13] and are omitted.

We use the classical Banach space [C.sup.k][0, 1], let X = [C.sup.n-1][0, 1] and Y = [C.sup.0][0, 1]. Y is endowed with the norm [parallel]y[[parallel].sub.[infinity]] = [max.sub.t[member of][0,1]] [absolute value of y(t)], X is endowed with the norm [parallel]x[parallel] = max{[parallel]x[[parallel].sub.[infinity]], [parallel]x'[[parallel].sub.[infinity]], ..., [parallel][x.sup.(n-1)][[parallel].sub.[infinity]]}. Define the linear operator L and the nonlinear operator N by

L : X [intersection] domL [right arrow] Y, Lx(t) = [x.sup.(n)](t) for x [member of] X [intersection] domL, N : X [right arrow] Y Nx(t) = f(t, x(t), x'(t), ..., [x.sup.(n-1)](t)) for x [member of] X,

respectively. We first consider BVP(5) and (6) in Case 1.1. Let

domL = {x [member of] [C.sup.n][0, 1], [x.sup.(i)](0) = 0 i = 0, 1, ..., n - 2, x(1) = [m.summation over (i=1)] [[alpha].sub.i]x([[xi].sub.i])}

Lemma 2.1. The following results hold.

(i). KerL = {[ct.sup.n-1], t [member of] [0, 1], c [member of] R};

(ii). [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(iii). L is a Fredholm operator of index zero;

(iv). There is k [member of] {0, 1, ..., m} such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(v). There are projectors P : X [right arrow] X and Q : Y [right arrow] Y such that KerL = ImP and KerQ = ImL. Furthermore, let [OMEGA] [subset] X be an open bounded subset with [bar.[OMEGA]] [intersection] domL [not equal to] [empty set], then N is L-compact on [bar.[OMEGA]];

(vi). x(t) is a solution of BV P(5) and (6) if and only if x is a solution of the operator equation Lx = Nx in domL;

(vii). [paralell]x[paralell] = [paralell][x.sup.(n-1)][[paralell].sub.infinity] for all x [member of] domL.

Proof. It is simple and is omitted.

Theorem 2.1. Assume the following conditions hold.

([A.sub.1]). There exist functions [a.sub.i](i = 0, 1, ..., n - 1), b and r [member of] [L.sup.1][0, 1] and a constant [theta] [member of] [0, 1) such that for all [x.sub.i] [member of] R(i = 0, 1, ..., n - 1) and t [member of][0, 1], one of the following inequalities hold:

[absolute value of f(t, [x.sub.0], [x.sub.1], [x.sub.2], ..., [x.sub.n-1])] [less than or equal to] [n-1.summation over (i-0)] [a.sub.i](t) [absolute value of [x.sub.i]] + b(t) [[absolute value of [x.sub.n-1]].sup.theta]] + r(t), ([a.sub.n-1])

[absolute value of f(t, [x.sub.0], [x.sub.1], [x.sub.2], ..., [x.sub.n-1])] [less than or equal to] [n-1.summation over (i-0)] [a.sub.i](t) [absolute value of [x.sub.i]] + b(t) [[absolute value of [x.sub.n-2]].sup.theta]] + r(t), ([a.sub.n-2])

and

[absolute value of f(t, [x.sub.0], [x.sub.1], [x.sub.2], ..., [x.sub.n-1])] [less than or equal to] [n-1.summation over (i-0)] [a.sub.i](t) [absolute value of [x.sub.i]] + b(t) [[absolute value of [x.sub.0]].sup.theta]] + r(t); ([a.sub.0])

([A.sub.2]). Let [f.sub.x](s) = f(s, x(s), ..., [x.sup.(n-1)](s)). There is a constant M > 0 such that for any x [member of] domL/KerL, if [absolute value of [x.sup.(n-1)](t)] > M for all t [member of] [0, 1], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

([A.sub.3]). Let [f.sub.c](s) = f(s, [cs.sup.n-1], c(n-1)[s.sup.n-2], ..., c(n-1)!). There is a constant [M.sup.*] > 0 such that for any c [member of] R either

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] ([b.sub.1])

for all [absolute value of c] > [M.sup.*] or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] ([b.sub.2])

for all [absolute value of c] > [M.sup.*];

([A.sub.4]). [[summation].sup.n-1.sub.i=0] [parallel][a.sub.i][[parallel].sub.1] < 1.

Then BV P(5) and (6) has at least one solution.

Remark 1. Recently, Liu and Yu in [31] studied the solvability of BVP

[x.sup.n](t) = f(t, x(t), x'(t)) + e(t), t [member of] (0, 1),

x(0) = 0, x(1) = [[summation].sup.m.sub.i=1] [[beta].sub.i]x([[eta].sub.i]),

when [[summation].sup.m.sub.i=1][[beta].sub.i] = 1 = [[summation].sup.m.sub.i=1][[beta].sub.i][[eta].sub.i] and [[summation].sup.m.sub.i=1][[beta].sub.i]x([[eta].sup.2.sub.i] [not equal to] 1 (see Theorem 2.1 in [31]). He didn't establish the existence results for the case [[summation].sup.m.sub.i=1][[beta].sub.i] [not equal to] 1 or [[summation].sup.m.sub.i=1][[beta].sub.i]x([[eta].sup.2.sub.i] = 1. It is easy to see that the case n = 2 of Theorem 2.1 here improves Theorem 2.1 in [31] since the condition ([A.sub.4) is weaker than the corresponding one in Theorem 2.1(i.e. [paralell]a[[[paralell].sub.1] + [paralell]a[[[paralell].sub.1] < 1/2 in [31]) and the other conditions are same as those in [31]. The result here also generalizes and complements that in [31].

To prove Theorem 2.1, we need the following lemmas.

Lemma 2.2. Under the assumptions ([A.sub.1), ([A.sub.2]) and ([A.sub.4]) of Theorem 2.1, let

[[OMEGA].sub.1] = {x [member of] domL \ KerL, Lx = [lambda]Nx for some [lambda] [member of] (0, 1)},

then [[OMEGA].sub.1] is bounded.

Proof. For x [member of] [[OMEGA].sub.1], x [not member of] KerL, [lambda] [not equal to] 0 and Nx [member of] ImL, thus QNx = 0. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Hence by ([A.sub.2]), we know that there is [t.sub.0] [member of] [0, 1] such that [absolute value of x.sup.(n-1)]([t.sub.0])] [less than or equal to] M. Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Then by (vii) of Lemma 2.1 we get

[parallel]x[parallel] = [parallel][x.sup.(n-1)][[parallel].sub.[infinity]] [less than or equal to] M + [n-1.summation over (i=0)] [parallel][a.sub.i][[parallel].sub.1][parallel]x[parallel] + [parallel]b[[parallel].sub.1][parallel]x[[parallel].sup.[theta]] + [parallel]r[[parallel].sub.1].

So

(1 - [n-1.summation over (i=0)] [parallel][a.sub.i][[parallel].sub.1]) [parallel]x[parallel] [less than or equal to] [parallel] [parallel]b[[parallel].sub.1][parallel]x[[parallel].sup.[theta]] + [parallel]r[[parallel].sub.1].

Since [theta] [member of] [0, 1), from the above inequality, there is [M.sub.1] > 0 such that [parallel]x[parallel] = [parallel][x.sup.(n-1)][[parallel].sub.[infinity]] [less than or equal to] [M.sub.1]. it follows that 1 is bounded. Similarly to above argument, we can prove that [[OMEGA].sub.1] is bounded if one of ([a.sub.i])(i = 0, 1, ..., n - 2) holds. This completes the proof.

Lemma 2.3. Under the assumption ([A.sub.3]) of Theorem 2.1, let [[OMEGA].sub.2] = {x [member of] KerL : Nx [member of] ImL}, then [[OMEGA].sub.2] is bounded.

Proof. For x [member of] [[OMEGA].sub.2], then x(t) = [ct.sup.n-1] for some c [member of] [0, 1]. Nx [member of] ImL implies QNx = 0. Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

From ([A.sub.3]), we get [absolute value of c] [less than or equal to] [M.sup.*]. So [[OMEGA].sub.2] is bounded. The proof is complete.

Lemma 2.4. Under the assumption ([A.sub.3]), if ([b.sub.1]) in ([A.sub.3]) holds, let

[[OMEGA].sub.3] = {x [member of] KerL : -[lambda][LAMBDA] x + (1 - [lambda])QNx = 0, [lambda] [member of] [0, 1]},

where [LAMBDA] is the isomorphism given by [LAMBDA]([ct.sup.n-1]) = [ct.sup.k] for all c [member of] R. If ([b.sub.2]) in ([A.sub.3]) holds, Let

[[OMEGA].sub.3] = {x [member of] KerL : [lambda][LAMBDA] x + (1 - [lambda])QNx = 0, [lambda] [member of] [0, 1]},

then [[OMEGA].sub.3] is bounded.

Proof. In fact, if ([b.sub.1]) holds, and x = [ct.sup.n-1] [member of] [[OMEGA].sub.3], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

if [lambda] = 1, then c = 0. Otherwise, if [absolute value of c] > [M.sup.*], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which contradicts [lambda][c.sup.2] [greater than or equal to] 0. So [absolute value of c] > [M.sup.*]. This shows that 3 is bounded. Similarly to above argument, we can prove that [[OMEGA].sub.3] is bounded if ([b.sub.2]) holds.

Proof of Theorem 2.1. Let [OMEGA] be a open bounded ball centered at zero of X such that [OMEGA] [contains] [[union].sup.3.sub.i=1][bar.[OMEGA]].sub.i]. By Lemma 2.1, L is a Fredholm operator of index zero and N is L-compact on [bar.[OMEGA]]. By Lemmas 2.2 and 2.3 and [OMEGA] [contains] [bar.[OMEGA]].sub.1] [union] [bar.[OMEGA]].sub.2], we have Lx [not equal to] [lambda]Nx for x [member of] (domL/KerL) [intersection] [differential][OMEGA] and [lambda] [member of] (0, 1) and Nx [not member of] ImL for x [member of] KerL [intersection] [differential][OMEGA].

Now, let H(x, [lambda]) = [+ or -] [lambda][LAMBDA] x + (1 - [lambda])QNx. According to Lemma 2.4 and [OMEGA] [contains] [bar.[OMEGA]].sub.3], we know H(x, [lambda]) [not equal to] 0 for x [member of] [differential][OMEGA] [intersection] KerL, thus by homotopy property of degree,

deg(QN|KerL, [OMEGA][intersection] KerL, 0) = deg(H(*, 0), [OMEGA] [intersection] KerL, 0) = deg(H(*, 1), [OMEGA][intersection] KerL, 0) = deg([LAMBDA], [OMEGA] [intersection] KerL, 0) [not equal to] 0.

Thus by Theorem GM, Lx = Nx has at least one solution in domL [intersection] [bar.[OMEGA]], which is a solution of BVP(5)-(6). The proof is complete.

Now, we consider BVP(5) and (7) in the Case 2.1, define

domL = {x [member of] [C.sup.n][0, 1], [x.sup.(i)](0) = 0, i = 0, 1, ..., n-2, [x.sup.(p)](1) = [m.summation over (i=1)] [[alpha].sub.i]x([[xi].sub.i])},

we have the following lemma.

Lemma 2.5. The following results hold.

(i). KerL = {[ct.sup.n-1], t [member of] [0, 1], c [member of] R};

(ii). [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(iii). L is a Fredholm operator of index zero;

(iv). There is k [member of] {0, 1, ..., m} such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(v). There are projectors P : X [right arrow] X and Q : Y [right arrow] Y such that KerL = ImP and KerQ = ImL. Furthermore, let [OMEGA] [subset] X be an open bounded subset with [bar.[OMEGA]] [intersection] domL [not equal to] [empty set], then N is L-compact on [bar.[OMEGA]];

(vi). x(t) is a solution of BV P(5) and (7) if and only if x is a solution of the operator equation Lx = Nx in domL;

(vii). [parallel]x[parallel] = [parallel][x.sup.(n-1)][[parallel].sub.1] for all x [member of] domL.

Proof. The proof is similar to that of Lemma 2.1 and is omitted.

Theorem 2.2. Assume ([A.sub.1]) of Theorem 2.1 holds and the following conditions hold.

([A.sub.5]). There is M > 0 such that for any x [member of] domL/KerL, if [absolute value of x[(n-1)(t)] > M for all t [member of] [0, 1], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

([A.sub.6). There is [M.sup.*] > 0 such that for any c [member of] R either

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for all [absolute value of c] > [M.sup.*] or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for all [absolute value of c] > [M.sup.*].

([A.sub.7]). [[summation].sup.n-1.sub.i=0] [parallel][a.sub.i][[parallel].sub.1] < 1.

Then BV P(5) and (7) has at least one solution.

Remark 2. Recently, Liu and Yu in [31] studied the solvability of BVP

x"(t) = f(t, x(t), x'(t)) + e(t), t [member of] (0, 1), x(0) = 0, x'(1) = [[summation].sup.m.sub.i=1] [[beta].sub.i]x'([[eta].sub.i]),

when [[summation].sup.m.sub.i=1] [[beta].sub.i] and [[summation].sup.m.sub.i=1] [[beta].sub.i][[eta].sub.i] [not equal to] 1 (see Theorem 3.6 in [31]). He didn't establish the existence results for the case [[summation].sup.m.sub.i=1] [[beta].sub.i][[eta].sub.i] = 1. It is easy to see that the case n = 2 of Theorem 2.2 here improves Theorem 3.6 in [31] since the condition ([A.sub.7]) is weaker than the corresponding one in Theorem 3.6(i.e. [parallel]a[[parallel].sub.1] + [parallel]b[[parallel].sub.1] < 1/2 in [31]) and the other conditions are same as those in [31]. The result here also generalizes and complements that in [31]. Proof of Theorem 2.2. The proof is similar to that of Theorem 2.1 and is omitted.

3. Existence Results for Cases i.2(i=1,2)

In this section, we obtain sufficient conditions for the existence of at least one solution of BVP(5) and (6), BVP(5) and (7) at non-resonance case, i.e. [[summation].sup.n.sub.i=1][[alpha].sub.i][[xi].sup.n-1.sub.i] [not equal to] 1 and [[summation].sup.m.sub.i=1][[alpha].sub.i][[xi].sup.n-1-p.sub.i] [not equal to] 1, respectively. The method employed is based upon Scheaffer's theorem, see, for example [9], we omit its details.

Combining the differential equation (5) with the boundary value conditions (6), if x(t) is a solution of BVP(5) and (6), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Define the Banach space

X = {x [member of] [C.sup.n-1][0, 1] : [x.sup.(i)](0) = 0 for i = 0, 1, ..., n - 2 }, whose norm is [parallel]x[parallel] = max{[parallel]x[[parallel].sub.[infinity]], ..., [parallel][x.sup.(n-1)][[parallel].sub.[infinity]]}, where [parallel]x[[parallel].sub.[infinity]] = [max.sub.t[member of][0,1]] [absolute value of x(t)]. It is easy to show that [parallel]x[parallel] = [parallel][x.sup.(n-1)][[parallel].sub.[infinity]]. Define the nonlinear operator T : X [right arrow] X by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for x [member of] X. It is easy to check that T is continuous and compact on each bounded subset of X.

Theorem 3.1. Assume ([A.sub.1) of Theorem 2.1 holds and that

([B.sub.1]). [[summation].sup.n-1.sub.i=0] [parallel][a.sub.i][[parallel].sub.1] < 1/[sigma], where [sigma] = 1 + [[summation].sup.m.sub.i=1] [absolute value of [[alpha].sub.i]] [[xi].sup.n-1.sub.i] + [absolute value of 1 - [[summation].sup.m.sub.i=1] [[alpha].sub.i] [[xi].sup.n-1.sub.i]]/[absolute value of 1 - [[summation].sup.m.sub.i=1] [[alpha].sub.i] [[xi].sup.n-1.sub.i]].

Then BV P(5) and (6) has at least one solution.

Proof. For [mu] [member of] [0, 1], consider the equation

x = [mu]Tx. (8)

If x [member of] X, then we have

[absolute value of f(t, x(t), x'(t), ..., [x.sup.(n-1)](t))] [less than or equal to] a(t) + [n-1.summation over (i=0)] [a.sub.i](t) [absolute value of [x.sub.i]] + b(t) [[absolute value of [x.sup.(i)](t)].sup.theta]] + r(t).

Thus if x(t) is a solution of (9), then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Thus we get by using ([B.sub.1])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Then (1 - [sigma] [n-1.summation over (i=0)] [parallel][a.sub.i][[parallel].sub.1])[parallel]x[parallel] [less than or equal to] [parallel]b[[parallel].sub.1][parallel]x[[parallel].sup.[theta]] + [parallel]r[[parallel].sub.1].

It follows from ([B.sub.2]) that there is M > 0 sufficiently large and independent on [mu] such that [parallel]x[parallel] = [parallel][x.sup.(n-1)][[parallel].sub.[infinity] [less than or equal to] M. On the other hand, T is continuous and compact on each bounded subset of X. Therefore, by Schaeffer's theorem [9], we obtain the existence of at least one fixed point for the operator T, which is a solution of BVP(5)-(6). The proof is complete.

Remark 3. When [[alpha].sub.i] [less than or equal to] 0 for all i = 1, ..., m and [[summation].sup.m.sub.i=1] [[alpha].sub.i][[xi].sup.n-1.sub.i] < 1, it is easy to check that if x(t) is solution of BVP(5) and (6), i.e. x is a solution of the integral equation (9), then x(t) > 0 for all t [member of] (0, 1)(similar to Lemma 2 in [2]). Hence we have the following corollary, whose proof is omitted.

Corollary 3.1. Suppose that [[alpha].sub.i] [greater than or equal to] 0 for all i = 1, ..., m and [[summation].sup.m.sub.i=1] [[alpha].sub.i][[xi].sup.n-1.sub.i] < 1 and all conditions of Theorem 3.1 hold. Then BV P(5) and (6) has at least one positive solution.

Similarly, if x(t) is a solution of BVP(5) and (7), then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)

Define the operator T : X [right arrow] X by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for x [member of] X. Similarly we have the following theorem.

Theorem 3.2. Assume ([A.sub.1) of Theorem 2.1 holds and that

([B.sub.2]). [[summation].sup.n-1.sub.i-0] [parallel][a.sub.i][[parallel].sub.1] < 1/[sigma], where [sigma] = 1 + [[summation].sup.m.sub.i=1] [absolute value of [[alpha].sub.i]] + [absolute value of 1 - [[summation].sup.m.sub.i=1] [[alpha].sub.i] [[xi].sup.n-1-p.sub.i]]/[absolute value of 1 - [[summation].sup.m.sub.i=1] [[alpha].sub.i] [[xi].sup.n-1.sub.i]].

Then BV P(5) and (7) has at least one solution.

Proof. The proof is similar to that of Theorem 3.1 and is omitted.

Remark 4. When [[alpha].sub.i] [greater than or equal to] 0 for all i = 1, ..., m and [[summation].sup.m.sub.i=1] [[alpha].sub.i] [[xi].sup.n-1-p.sub.i]] < 1, it is easy to check that if x(t) is solution of BVP(5) and (7), i.e. x is a solution of the corresponding integral equation, then x(t) > 0 for all t [member of] (0, 1)(similar to Lemma 2 in [2]). Hence we have the following corollary, whose proof is omitted.

Corollary 3.2. Suppose that [[alpha].sub.i] [greater than or equal to] 0 for all i = 1, ..., m and [[summation].sup.m.sub.i=1] [[alpha].sub.i] [[xi].sup.n-1-p.sub.i]] < 1 and all conditions of Theorem 3.2 hold. Then BV P(5) and (7) has at least one positive solution.

Received December 1, 2003, Accepted February 18, 2004.

* 2000 Mathematics Subject Classification. Primary 34B10, 34B15; Secondary 35B18.

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Yuji Liu ([dagger])

Department of Mathematics, Beijing Institute of Technology 

Beijing 100081, People's Republic of China

and

Weigao Ge

Department of Mathematics, Hunan Institute of Technology

Yueyang, Hunan 414000, People's Republic of China

([dagger]) E-mail:[e-mail]