Solutions for superlinear (n − 1, 1) conjugate boundary value problems

2001,21B(2):259-264

SOLUTIONS FOR SUPERLINEAR (n - 1,1) CONJUGATE BOUNDARY VALUE PROBLEMS

1

Lou Bendong ( ~*-,f- ) Department of Mathematics, Shandong University, Jinan 250100, China Abstract The author investigates the existence of positive and nontrivial solutions for superlinear (n - 1,1) conjugate boundary value problems by means of topological degree theory and cone theory. The main theorems improve some results published recently. Key words Boundary value problems, topological degree, cone, positive solutions, nontri vial solutions 1991 MR Subject Classification

1

34B15

Introduction In this paper, we are concerned with positive and nontrivial solutions for superlinear (n -

1,1) conjugate boundary value problem:

-u(n)(x):::: f(x,u(x)), { u(k)(O) :::: 0,

°

0< x < 1,

~ k ~ n - 2,

u(I):::: 0,

(1)

where f : [0,1] x R 1 -+ R 1 is continuous. In case n :::: 2, (1) describes various phenomena, such as nonlinear diffusion generated by nonlinear .sources, thermal ignition of gases, and concentration in chemical or biological problems, see, for example [1-5]. Also, [6] used the second order case in modeling the onedimensional Dirichlet problem. In case f(x, w) :::: a(x)f1(w), where a : [0,1] -+ [0,00) and tinuous, define fa::::

lim f1(w)/w, foo::::

w .... o+

lim

w .... +oo

f1 : [0,00)-+

[0,00) are conf1(w)/w. In [1,2,7], the authors investigated

the existence of solutions of (1) when 10 :::: 0, foo :::: +00 (i.e. f is superlinear) and when fa :::: +00, foo :::: (i.e, f is sublinear). Their main results were all followed from a theorem due to Krasnosel'skii (see [1,2,7]).

°

In this paper, we seek positive and nontrivial solutions of (1). Here we say 1 is superlinear if it satisfies conditions (Hd-(H 2 ) or (Fd-(F 3 ) in Section 3, which are more general than the cases discussed in [1,2,7]. 1 Receuved

December 7,1998; revised December 6,1999. The author is supported in part by NNSF of China

and Monbusho Scholarship of Japan. Current address: Graduate School of Mathematical Sciences, the University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153, Japan.

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In Section 2, we discuss some properties of the Green's function corresponding to BVP (1). In Section 3, we provide a new method to define an appropriate cone which the positive solutions may belong to. Instead of using Krasnosel'skii's theorem, we obtain the positive and nontrivial solutions of (1) by applying the properties of the fixed point index and the topological degree directly.

2 Some Lemmas Define an operator A by

=

Au(x)

1 1

G(x, y)f(y, u(y))dy,

(2)

where

G(x,y)

={

(I - y)n-1 xn-1 j(n - I)!,

o::; x

[(1- y)n-l xn-1 - (x - y)n-1]j(n - I)!,

O::;y::;x::;1.

::; y ::; 1,

(3)

Clearly,

G(x, y) > 0,

If 0 < z < 1, 0 < y < 1.

(4)

Set P = {u E C[O,l]1 u(x) 2: 0 for x E [0, I]}. Then P is a cone of C[O, 1] and by direct calculation we have Lemma 1 A maps C[O, 1] into cn[o, 1], and

(a) if u E C[O, 1] such that Au = u, then u is a solution of BVP (1). (b) if u E cn[o, 1] is a solution of BVP (1), then Au = u. Remark 1 It is easy to see that Lemma 1 remains valid when we replace C[O,l] and n[O,l] C by P and Cn[O, 1] n P, respectively. Lemma 2 Let G(x, y) be defined by (3). Then

1

1

3

4 ::; x ::; 4' z, y E [0,1].

(5)

113 If 4::; y::; 4' x,Z E [0,1].

(6)

G(x, y) 2: 4n- 1G(z, y),

If

G(x,y) 2: 4n _ 1G (x , z ),

Proof The proof of (5) can be found in Theorem 4 of [1]. We now prove (6). By the definition of G(x, y), it is easy to see that G(x, y) = G(l - y, 1 - x), Y z , Y E [0,1].

For

i ::; y ::; ~,

T

n

== 1 - Y E [i,

G(x, y)

= Gn -

Therefore we have by (5)

y, 1- x)

1

= G(T, 1 -

2: 4n - 1 G(t, 1 - x), Thus for

i ::; y ::; ~, z = 1 -

Y

x)

1

3

4 ::; y ::; 4'

z , t E [0,1].

t E [0,1], we have

1 1 1

G(x, y) 2: 4n- 1G(t, 1 - x) = 4n- 1 G(x, 1- t) = 4n- 1G(x, z), (6) is proved. Define a linear integral operator L by

1 1

Lu(x) =

G(y, x)a(y)u(y)dy,

u E C[O, 1],

If z , z E [0,1].

Lou: SUPERLINEAR (n - 1,1) CONJUGATE BOUNDARY VALUE PROBLEMS

No.2

where G is given by (3) and a : [0,1]

261

[0,00) is continuous and does not vanish identically on

--+

any subinterval. It is easy to verify that L : C[O, 1] --+ C[O, 1] is a completely continuous positive operator with positive spectral radius: r(L). So Krein-Rutman theorem [8] implies that there exists p(x) E P, p(x) 1:- 0 such that

Lp(x)

=

1 1

G(y, x)a(y)p(y)dy

= r(L)p(x).

(7)

3 4

Lemma 3 There exists a b 2: 4 1 - n r(L )- 1J1 / 4 a(x)p(x)dx > 0 such that

p(y) 2: sci«, y),

V x, y E [0,1].

(8)

Proof In fact, by (5) and (7), it is easy to prove that p(y) 2:

~~~;

it

a(x)p(x)dx· G(z,y),

(9)

V y,z E [0,1].

4

Remark 2 By (7) and (8) we have

r(L)p(x)

1 fl = 10r G(y, x)a(y)p(y)dy:s "610 p(x)a(y)p(y)dY,

1

i.e.

1

p(y)a(y)dy 2: 8r(L).

(10)

Lemma 4 ([8]) Let K be a cone of a real Banach space E and B : K --+ K a completely continuous operator. Assume that B is order-preserving and positively homogeneous of degree 1 and that there exist v E K\ {O}, A > 1 such that Bv 2: AV. Then r( B) > 1.

3

Main Theorems In what follows, denote by

< l}(l > 0).

11·11 the maximum norm of C[O, 1], denote U, = {u

E C[O,

1J1llull

In this section, we shall prove the existence of positive solutions and nontrivial solutions for BVP (1). We first list some conditions for convenience: (Ho) f: [0,1] x [0,+00) --+ [0,+00) is continuous. (HI) There exist a(x), b(x) E P, a(x) does not vanish identically on any subinterval, such that f(x, w) 2: a(x)w - b(x), V x E [0,1], w 2: O. (H 2 ) There exist c(x) E P and Ro > 0 such that f(x,w) :s c(x)w, V x E [0,1], O:S w:s Ro. (FI) There exist a(x), b(x) E P, a(x) does not vanish identically on any subinterval, such

that f(x, w) 2: a(x)w - b(x), V x E [0,1], wE R 1. (F 2 ) There exist c(x) E P and Ro > 0 such that If(x, w)1 :s c(x)lwl, V x E [0,1], Iwl :s Ro· (F 3 ) There exist g(x), h(x) E P such that f(x, w) 2: g(x)w.- h(x), V x E [0,1], wE R 1. Let L be the integral operator defined as above with a(x) given by (HI) or (F 1 ) . Define u, : C[O, 1] --+ C[O, 1] by

Lou(x)

=

1 1

G(x, y)c(y)u(y)dy,

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Vol.21 Ser.B

where c(x) is given by (H 2 ) or (F 2 ) . Denote by r(L) and r(L o) the spectral radii of Land Lo, respectively.

Theorem 1 Let (H o), (Ht} and (H 2 ) be satisfied and r(L o) ::; 1 < r(L). Then BVP (1) has at least one positive solution. Proof Set

Q

= {u E

pill

p(x)a(x)u(x)dx 2: or(L)u(y), 'if y E [O,I]},

(11)

where a(x) is given by (HI)' p(x) and r(L) are given by (7) and 0 is given by (8). By (10), we know that Q f:- 0, Q f:- {8} and Q is also a cone in e[O, 1]. We now prove that operator A defined by (2) maps Pinto Q: A(P) C Q. In fact, for u E P, we have by (7) and (8)

1 1

p(x)a(x)Au(x)dx

1 =1

=

1

1 1 1

p(x)a(x)dx

1

G(x,y)f(y,u(y))dy

1

f(y,u(y))dy

= r(L)

1 1

G(x,y)p(x)a(x)dx

1

p(y)f(y,u(y))dy

1

2: or(L)

G(x, y)f(y, u(y))dy

= or(L)Au(x),

'if x E [0,1].

Choose Uo E Q\{8} and R> R o such that

r R> or(L)(r(L) _ 1) Jo Jo p(x)a(x)G(x, y)b(y)dxdy. 1

1

We assert that

(

f:- Auo, 'if u E P, Ilull = R, A 2: o. U1 E P, IIu111 = R and A 2: 0 such that U1 - AU1 u - Au

In fact, if there exist account of uo, AU1 E Q, and therefore by (7) we have

1 1 -1 1 1 -1 1 = 1

(12)

(13)

= AUo, then U1 E Q on

1

02: -A

p(x)a(x)uo(x)dx

1

=

2:

p(x)a(x)AU1(X)dx

1

1

p(x)a(x)u1(x)dx

1

p(x)a(x)dx 1

G(x, y)a(y)u1(y)dy

-1

1

p(x)a(x)u1(x)dx

1

G(x, y)p(x)a(x)b(y)dxdy 1

(r(L) - 1)

p(y)a(y)u1(y)dy - {3

2: (r(L) - l)or(L)U1(X) - {3, 1 1

'if x E [0,1],

where {3 = fo fo G(x, y)p(x )a(x )b(y)dxdy 2: O. Thus {3 2: (r(Lf-l )or(L )U1 (x) ~ 0, 'if x E [0,1]. Since IIu111 = R, we have {3 ~ (r(L)-I)or(L)·R, in contradiction with (12). Consequently (13)

No.2

Lou: SUPERLINEAR (n - 1,1) CONJUGATE BOUNDARY VALUE PROBLEMS

263

holds, i.e. the direction Uo is left out by I-A. Since A : P n UR ~ P is completely continuous, it follows from the property ofthe fixed point index (see e.g.[10, Lemma 4.2 in Chapter 3] and its Corollary or [11, Corollary 2.3.1]) that

i(A,pnUR,p)=O.

(14)

On the other hand, we have

Au of j.LU,

If J], ~ 1,u E P, Ilull

= R o.

(15)

In fact, if Uz E P, Iluzll = Ro and J],1 ~ 1 (without loss of generality, assume that J],1 that Auz = J],1UZ, then by (Hz) we have

J],1UZ(X) =

1 1

1 1

G(x,y)f(y,uz(y))dy:S

> 1) such

G(x,y)c(y)uz(y)dy = Louz(x).

So by Lemma 4 we have r(L o) > 1, in contradiction with r(L o) :S 1. Hence (15) holds, and therefore by the homotopy invariance of the fixed point index,

i(A, P n URo , P)

= 1.

(16)

By (14), (16) and the additivity of the fixed point index we get

i(A,pn(UR\URo)'P)

= -1,

which implies that A has at least one fixed point in P n (UR\U Ro)' that is, by Lemma 1 and Remark 1, BVP (1) has at least one positive solution. This completes the proof. Theorem 2 Let (Fd-(F 3 ) be satisfied, r(L o) :S 1 < r(L). Suppose that a(x) and g(x) satisfy

1 !4

a(x)dx > 2r(L)· 43 n -

4

3

1 1

g(x)dx.

(17)

Then BVP (1) has at least onenontrivial solution. Remark 3 (i) Under the conditions of Theorem 2, the conclusion of Theorem 1 and its proof are not valid since f does not satisfy (H o), i.e. A is not a cone-mapping. (ii) For n 2, Lou[9] discussed the nontrivial solutions of BVP (1) in case g == 0, and in such a case, (17) can be omitted. Proof of Theorem 2 By (F 3 ) , f(x,w) - g(x)w + h(x) ~ 0, If z E [0,1], wE R i . So replacing f(y, u(y)) by f(y, u(y)) - g(y)u(y) + h(y) in (2), we get by (12) that

=

1 1

G(x, y)[f(y, u(y)) - g(y)u(y)

+ h(y)]dy

E Q,

Ifu E e[O, 1].

By (7) and (6), we have for ~:S x :S ~,

r(L)p(x)

=

1 1

G(y,x)a(y)p(y)dy

~ 41- n

1 1

G(y, z)a(y)p(y)dy = 41 -

nr(L)p(z),

If z E [0,1].

264 i.e. p(x)

(17),

ACTA MATHEMATICA SCIENTIA

2: 4 I -

21

1

np(z),

"Ix E

[i,

Vo1.21 Ser.B

n z E [0,1]. Therefore, for any given Xo E [i, ~], we have by

1 1 1t I t 1

p(x)g(x)dx :::; 2· 4n -

1

I

g(x)dx· p(xo)

< r(L)4 2n -

a(x)dx· p(xo) :::; r(L)4 n -

2

I

4

Hence

.!!. 4

a == 4 1 - n r(L )- I 1

a(x)p(x)dx -

4

21

1

p(x)g(x)dx

Let Q be defined by (11). Choose Uo E Q\ {B} and R> 1

a(x)p(x)dx.

4

r t'

= const. > O.

Ro such 'that 2

r

R> ar(L)(r(L) _ 1) Jo Jo p(x)a(x)G(x, y)b(y)dxdy + ~ Jo p(y)h(y)dy. In a similar way as proving (14) and (16), one can get deg(I - A, UR, 0)

= 0,

deg(I - A, URo'O)

= 1,

which implies by the additivity of Leray-Schauder degree that deg(I - A, UR\UR o, 0)

= -1.

This means that A has at least one fixed point in UR \ UR o ' i.e. BVP (1) has at least one nontrivial solution. This completes the proof of Theorem 2. Remark 4 One can also discuss the existence of positive and nontrivial solutions of (1) in case f is sublinear and for (k, n - k) conjugate boundary value problems in a similar way. References 1 Eloe P W, Henderson J. Positive solutions for (n - 1,1) conjugate boundary value problems. Nonlinear Analysis TMA, 1997, 28: 1669-1680 2 Agarwal R P, Henderson J. Superlinear and sublinear focal boundary value problems. Appl Anal, 1996, 60: 189-200 3 De Figueiredo D G, Lions P L, Nussbaum R D. A priori estimates and existence of positive solutions of semilinear elliptic equations. J Math pures appl, 1982, 61: 41-63 4 Guo Dajun, Lakshmikantham V. Multiple solutions of two-point boundary value problems od ordinary differential equations in Banach spaces. J Math Anal Appl, 1988, 129: 211-222 5 Sun Jingxian. Nontrivial solutions of superlinear Hammerstein integral equations and applications. Acta Math Sinica, 1985, 28: 347-359 6 Fink A M, Gatica J A, Hernandez G E. Eigenvalues of generalized Gel'fand models. Nonlinear Analysis TMA, 1993, 20: 1453-1468 7 Erbe L H, Wang H. On the existence of positive solutions of ordinary differential equations. Proc Amer Math Soc, 1994, 120: 743-748 8 Nussbaum R D. Eigenvectors of nonlinear positive operators and the linear Krein-Rutman theorem. In: Fixed Point Theory LNM 886. New York, Berlin: Springer-Verlag, 1980. 309-330 9 Lou Bendong. Solutions of superlinear Sturm-Liouville problems in Banach spaces. J Math Anal Appl, 1996, 201: 169-179 10 Guo Dajun, Nonlinear Functional Analysis. Jinan: Shandong Sci Tech Publishing House, 1985 11 Guo Dajun, Lakshmikantham V. Nonlinear Problems in Abstract Cones. Boston, New York: Academic Press, Inc, 1988