E L S E V I E R
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computers
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mathematics
with application,=Computers and Mathematics with Applications 49 (2005) 1-11
www.elsevier.com/loc a t e / c a m w a
Solvability of Multipoint Boundary Value
Problems at Resonance for
Higher-Order Ordinary Differential Equations
X I A O J I E L I N
D e p a r t m e n t o f M a t h e m a t i c s , X u z h o u N o r m a l U n i v e r s i t y X u z h o u , 221009, P.R. C h i n a
linxiaoj i e 1 9 7 3 ~ 1 6 3 , c o m
ZENGJI Du*
Department of Mathematics, Beijing Institute of Technology Beijing 100081, P.R. China
d u z e n g j i©163, com g e w © b i t , e d u . cn
W E I G A O
G E
Department of Mathematics, Beijing Institute of Technology Beijing 100081, P.R. China
gew0bit, edu. c n
(Received August 2002; accepted October 2003)
A b s t r a c t - - L e t f : [0, 1] x R n - - . R be a continuous function and e e LI[0, 1]. Let flj (1 _< j < rn - 2) E R, 0 < ~/1 < rl2 < "'" < ~/m--2 < 1 be given. This paper is concerned with t h e existence of solutions for the following nth-order multipoint boundary value problems at resonance case
x(")(t)=r(t,~(t),~'(t)
...
:~(n-~) (t)) + e (t), t e (0,1),m - - 2
(o) = ~' (o) . . . ~(,,-2) (o1 = o, ~ (i) = ~ ~j~ (,j), j=l and x (~) (t) = f ( t , x ( t ) , x ' (t) . . . x ( n - l ) (t)) q- e (t), t e (0, 1), rn-2 z (0) = x' (0) . . . x (n-2) (0) = 0, ~' (1) = ~ ~ / ( , j ) . j = l
Some existence results are obtained by using the coincidence degree theory of Mawhin. @ 2005 Elsevier Ltd. All rights reserved.
K e y w o r d s - - M u l t i p o i n t boundary value problem, Resonance, Coincidence degree theory.
Sponsored by t h e National Natural Science Foundation of China (10371006). *Author to whom all correspondence should be addressed.
T h e authors express their sincere gratitude to the referees for their valuable comments and helpful suggestions. 0898-1221/05/$ - see front m a t t e r @ 2005 Elsevier Ltd. All rights reserved. Typeset by fitA/~-TEX doi: 10.1016/j.camwa.2005.01.001
2 X. LIN et al.
1. I N T R O D U C T I O N
In this paper, we consider the following nth-order b o u n d a r y value p r o b l e m (BVP),
and x ( ~ ) ( t ) = f ( t , x ( t ) , x ' ( t ) , . . . , x ( ~ - i ) ( t ) ) + e ( t ) , t E ( 0 , 1 ) , r~z--2 x (0) = x ' (0) . . . x (n-2) (0) = O, x (1) = E fldx (~?J)' j = l x ( ~ ) ( t ) = f ( t , x ( t ) , x ' ( t ) , . . . , x ( ~ - t ) ( t ) ) + e ( t ) , t • ( 0 , 1 ) , rn--2 x (0) = x' (0) . . . x (n-2) (0) = 0, x ' (1)t = E fljx' (Vj). j = l
(i)
(2)(i)
(3)
W h e r e f : [0, 1] x R n --~ R is a continuous function, e E L 1[0, 1], ~j (j = 1 , . . . , r n - 2) e R a n d 0 < ~ l <U2 < ' ' ' < ~ ] m - 2 < l .We say t h a t B V P (1),(2) or B V P (1),(3) is a p r o b l e m at resonance, if the linear equation, x(") (t) = o, t • ( 0 , 1 ) ,
with the b o u n d a r y condition (2) or (3) has nontrivial solutions. Otherwise, we call t h e m a p r o b l e m at nonresonance.
~ - - ~ - - 2 ~ ~ - - I
In the present work, if 2-.,j=1 PJ~j = 1, then, B V P (1),(2) is at resonance, since equation
x(n)(t) = 0 with b o u n d a r y condition (2) has nontrivial solutions x = ct n - l , c E R, t E [0, 1]. If
j = l j j -- I, then B V P (I),(3) is at resonance, since equation
x(n)(t)
= 0 with boundarycondition (3) has nontrivial solutions x -- et ~-1, a E _R, t E [0, I].
Feng [1,2], Prezeradzki and Stanezy [3], Liu [4], and G u p t a [5-7] studied the existence of solutions for s o m e second-order multipoint boundary value problems at resonance. G u p t a [8], M a [9], and Nagle and Pothoven [i0] studied the existence for s o m e third-order third-point boundary value problems at resonance. However, rare works are done for higher-order multipoint boundary value problems at resonance.
For the nonresonance case of higher-order boundary value problems, for instance, we refer to [11,12] and the references therein.
T h e purpose of our p a p e r is to discuss the existence of solutions for higher order multi- point B V P (1),(2) at resonance (i.e., V ' m - 2 / 3 @ - 1 = 1) and B V P (1),(3) at resonance (i.e., A ~ j = I J j ~ - 2 j = l ~ n - 2 J?~j = 1). Our m e t h o d is based on the coincidence degree t h e o r y of Mawhin [13,14].
2. M A I N R E S U L T S
For the convenience of the reader to u n d e r s t a n d the coincidence degree theory, we briefly recall some definitions [13-15].
DEFINITION 1. Let Y, Z be reaI Banach spaces, the linear operator L : d o m L C Y ~ Z is said to be a Fredholm map of index zero provided that KerL, the kernel of L, is of the same finite dimension as the Z / h n L , where I m L is the image of L.
Let L be a F r e d h o l m m a p of index zero, a n d P : Y > Y, Q : Z ~ Z be continuous
projectors, such that I m P = KerL, K e r Q = I m L and Y = K e r L ® K e r P , Z = I m L @ I m Q . We denote the inverse of the map
LldomnnKerP
: d o t a l M K e r P > I m L by K p , i.e., K p = ( L l d o m L n K e r P ) - 1 : I m L ) d o t a L • K e r P .Solvability of Multi-Point Boundary Value Problems 3 DEFINITION 2. Let L be a Fredholm m a p of index zero and f% be an open bounded subset of Y ,
such that d o m L N ~ ~ 0, the m a p N : Y ~ Z is said to be L - c o m p a c t on ~, if the m a p Q N ( ~ ) is bounded and K p ( I - Q ) N : ~ ~ Y is compact.
For more details, see [14,15].
The following theorem is the Mawhin theorem, which can be found in [13, Theorem IV.13]
or [14, Theorem 2.4].
T H E O R E M A. Let L be a Fredholm operator of index zero and let N be L-compact on ~. A s s u m e
that the following conditions are satisfied.
(i) L x # A N x , for every (x, A) E [(dotaL \ KerL) n @fi] x (0, 1). (ii) N x f~ ImL, for every x E KerL A 0f~.
(iii) deg(JQNIKern, t2 A KerL, 0) # 0, where J : ImQ ~ KerL is a linear isomorphism,
Q : Z ~ Z is a projection as above with ImL = KerQ.
Then, the equation L x = N x has at b a s t one solution in d o t a l n ~.
In the following, we shall use the classical spaces C[0, 1], C1[0, 1 ] , . . . , C "-1[0, 1] and L 1[0, 1]. For x • c n - l [ O , 1], we use the norm Ilxlloo = maxt~[0,1l [x(t)] and Ilxll = max{Hxlloo , Hx'lloo,..., ]lx(~-l)]loo}, and denote the norm in LI[0, 1] by I1" Hi. We will use the Sobolev space W~'I(0, 1), which may be defined by
(0,1) = { x : [0,1] --~ R I
wn,1 X, Fgl, . . . ~X ( n - l )
are absolutely continuous on [0, 1] with x (~) e L 1 [0, 1] } . Let Y -- Cn-l[O, 1], Z = LI[0, 1], L is the linear operator from dotaL C Y to Z with
d o m L =
z~W~'~(O, 1):z(O)=z'(O)
. . . x ( ~ - 2 / ( 0 ) = 0 , z ( 1 ) = 3 i x ( ~ J j = land L x = x (~), x • dotaL. We define N : Y --~ Z by setting
= s (t, ~ (t), ~' (t),.. ,~(~-1> (t)) + e (t),
t • (0, 1).
N x
Then, BVP (1),(2) can be written as L x = N x .
In order to apply Theorem A, in the following Lemma 1, we shall show t h a t L is a Fred- holm operator of index zero and construct a linear continuous projector operator Q satisfying Condition (iii) in Theorem A.
if v'm-2 ~ V . m - 2 /~ Tin--1 x-'~m--2 3 n
LEMMA 1. z_~j=l Pj ~--- 1, A.wj=l j j = 1,
Lj=I jr])
¢ I,then~
(i) KerL = {x E d o m L : x = ct ~-1, c • R, t • [0, 1]};
v , m - 2 / ~ . 1 fo "" f o ' Y ( T 1 ) d T l " ' ' d T n 0};
(ii) Imn
= { y • Z : Z z j = l 3 f £ j r . . ~.(iii) L : domL c Y ~ Z is a Fredholm operator o f index zero, and the linear continuous projector operator Q : Z ~ Z can be defined as
Q Y - - ra--2 E f l j "'" y(T1) d T l ' ' " dTn"
I -
E 9jr]~
J=~j:l
(iv) The linear operator K p : ImL ---* domL N K e r P can be written as
K p y = / t / ~ " . . . / ~ : y ( ~ - J d~h...dTn
4 X. LIN
st al.
PROOF.
(i) For Vx E K e r L , we have x(~) = 0. T h e n , we obtain
x (t) = ao + alt + a2t 2 + . . . + a~_lt n-l,
w h e r e
ai(i
= 0 , 1 , . . . , n - 1) E R. F r o m x(0) = x'(0) . . . x(~-2)(0) = 0, we r~--2have a0 = al . . .
an-2
= 0. Again, from x(1) = ~ j = l/3jx(~j),
one has a n - 1 = a ~ - i E T ~ 2 ~ j ~ - ' In view of E ~ - 2 ~j=l
J~j ~ - ' = 1, we have t h a t K e r L -- {x E d o m L : x =ct n-l, c E R, t C
[0, 1]}. (ii) We show t h a t{
m--2Jls. Jo
1 rn.}
I m L = y E Z : E ~j "" Y (~-1) d~-l--'d~'n = 0 . (4) j = l J Since t h e p r o b l e m x (~) = y (5)m--2
has a solution
x(t)
satisfied x(0) = x'(0) . . . x(n-2)(0) = 0, x(1) = E j = I /~jx(~j), if a n d only if/~j "' Y (~-1) dT, . . . d~-~ = 0. (6)
In fact, if (5) has solution
x(t)
satisfied x(0) = x'(0) . . . x(~-2)(0) = 0, x(1) = m--2E j = , ZT(~J), then, from (5), we h~ve
11 ' 11),X(n 1 ) t
n - '
fot fo r~
fO r2
x ( t ) = x ( O ) +
x'(O) t + . . . + ( n -
- (0)
+
.."
y(T1) dr,...d'r,~
~-(~1__i),x(n-l)(o)~'-l-r~Ot~O~'n...~or'y(Tl)dT1...dTn.
Thus, we haveL
-
i l l . .
x ( 1 ) - - ( n 1) ' x ( " ' ) ( 0 ) + j o Jo""~o r2y(7-')dT-''''d'r~'
Joz z
m - 2 I1)ix(.
l) m--2/~, n--1 'qj 'r~ "r2 j = l j = l m--2 ~ n--i According to ~ j = z J~/j = I, w e obtain J=,(n
(0) +
flj
"'"
Y (~'1) d71". d~',~.
F r o m x(1) = E j = ~ ~ x ( ~ ) , w e have m - S" F
j = lff" F
Solvability of Multi-Point Boundary Value Problems 5 m - 2 again, f r o m ~-]d=l /3j = 1, o n e h a s m - 2 1 ~'~ T2
z J/ /0 I
j=l J O n t h e o t h e r h a n d , if (6) holds, s e t t i n g x (t) = ct n - 1 + . . . y ('rl) d ~ ' l . . , d'r,,, w h e r e c is a n a r b i t r a r y c o n s t a n t , t h e n x ( t ) is a s o l u t i o n of (5), a n d x (0) -- x ' (0) . . . x(n-2)(0) = 0, x(1) = Ej%~ ~ Zjx(v~). Hence, (4) is wlid.(iii) F o r y E Z , we t a k e t h e p r o j e c t o r Q y as ,--,m-2 5~1 N "'" Y ( n ) d T 1 . . , dTn. QY = 1 - ?-,j=l PJ~j = J L e t yl = y - Q y , we o b t a i n m - - 2 i r ~ T2 m -- 2 I r ~ ~-2 j = l J "= J
/?
rn--2 1 ~'n. - Qy ~ Zj .. d ~ . . . d ~ j = l J ~-- 0~t h e n , Yl E I m L . Hence, Z = I m L + R, since I m L N R = {0}, we have Z = I m L @ R, t h u s , d i m K e r L = d i m R = co d i m I m L = 1. Hence, L is a F r e d h o l m o p e r a t o r of i n d e x zero. (iv) T a k i n g P : Y ~ Y as follows, P x = x (~-~) (0) t n-~, t h e n , t h e g e n e r a l i z e d inverse K p : I m L ~ d o m L N K e r P of L c a n b e w r i t t e n as f t f ~ K p y = ." jo ~2 y ('el) dT1
f
dTn. J O J O . . . .I n fact, for y E I m L , we have
( L K p ) y (t) = [( K p y ) (t)] ('~) -- y ( t ) , a n d , for x E d o m L N K e r P , we k n o w ( K p L ) x (t) = ~o ~o " "" J o "~ x ( ~ ) (~-1) d~-i . . . d~-~ f 1 1 t n _ l --~ ~,(t) - - x ( O ) -- ~..x' (O) t . . . ( n -- 1)T x ( n - 1 ) (0) i n view of x E d o t a l ~ K e r P , x(0) = x ' ( 0 ) . . . x ( n - 2 ) ( 0 ) --- 0, a n d _Px -- 0, t h u s , ( K p L ) x (t) = x (t). T h i s shows t h a t K p = ( L l d o ~ L n K ~ p ) - 1 .
6 X. LINet al. (v) Prom the definition of Kp, we have
ItKpy[le~ <_ ~01 and from 1 • " ~ ly(71)l d r l ' " d r n = Ilytl~,
f t /'re_ I
(t) = jo ~f
(~py)'
JO dO
"'"
/0
( K p y ) (n-l) (t) = y (71) dT1, we obtainll( y)'ll -< HylI1,
then, IlKpyH < I[YlI1.This completes the proof of L e m m a 1.
y (r 0 d71.., drn_l,
• ., ( K P y ) <~<) ~ ~
Ilyll~,
THEOREM 1. Let f : [0, 1] x R ~ ~ R be a continuous function, assume that
(Hi) There exist functions al(t), a 2 ( t ) , . . . , an(t), b(t), r(t) E LI[0, 1], and constant 0 E [0, 1),
such that, for all (Xl, x 2 , . . . , Xn) E I I ~, t E [0, 1], satisfying one of the following inequali-
ties: If(t, xl,x2, if(t, xl,x2,
,)
• .,xn)l _<
a~(t) lx~+b(t) txn[°+r(t),
.,x~)l<_ a~(t) lx~ +b(t) lx,~-II°+r(t), [ f ( t , m ~ , x 2 ,.,xn)l
<_
a~(t)txd
÷b(t) lxll° +r(t).
(7)(8)
j=l
J
(H3) There exists a constant M* > O, such that, for c E R, if Icl > M*, then, either
m-2 f l ~Orn
j=lor
e/sem--2
I w nj=l
J
~0 "r2
..
If (71,cT?-l,c(n-1)7?%...,e.
(~-l)!) +e (71)] dT~ .dT~ < O, (11)
~0 TM
. [f (Tl,er[~-l,c(n - 1)7~ -2 .... ,c. (n--1)!)÷e(T1)] dT1 .-dT~ > O. (12) (H2) There exists a constant M > O, such that, for x E dotaL, if [x(~-l)(t)l > M, for allt ~ [0, 1], then,
Solvability of Multi-Point Boundary Value Problems 7
Then,
for everye(t) e
LI[0, 1],the BVP (1),(2) with/--,j=l
J = 1, E y = l jr/j = 1, E i = I~j ~ ¢ 1, has at least one solution in
C~-1[0, 1],provided that
Ei~=l Iladll < 1/2.PROOF. We need to construct the set ft satisfying all the conditions in Theorem A, which is separated into the following four steps.
STEP 1. Let
t2L = {x C domL \ K e r L :
Lx = ANx,
for some A E [0, 1]}.Then, ~'~1
is bounded.Suppose that x E f~l,
Lx = ANx,
thus, A ¢ 0,N x E
ImL = KerQ, hence,m--2 f l~0T~ f0~'2
E /~J
'''
If ( TI'x(T1)'x, (T1)'''''x(n-1) (T1)) -~-e(T1)]
d T l ' ' ' d T n : O,j=l
J
thus, from (H2), there exists to e [0, 1], such that
Ix(~-l)(t0)l _< M.
In view of£°
x ("-1) (o) = x(~-1) (to) - x(") (t) dr,
then,
IIPxll = x(~-l) (0) _< M + x (") 1 = M + IIL~II~ _ M + fINally. (13)
Again for x E ftl, x E domL KerL, then
( I - P)x E
dotaL Cl KerP,LPx
= 0, thus from Lemma 1, we knowl[(I-P)xll = I I K p L ( I - P ) z l l < I l L ( I - P ) x l l l
= IILzI]I < ]lNxll I . (14) From (13) and (14), we have[fxtl <_ lIPxOl + ll(I- P)xll =
x(~-~)(O)+ N ( I - P)xlt < 211NxllI + M.
(15) If (7) holds, then from (15), we obtain)
'
,10,
where C1 =
II~lll + little +
M/2.
THus, from I1~11~ -< I[~ll
and (16), wehave
/[Xlloo_<l_2[lall[1 []ai[[ 1 x(i--1) oe +[[b[] 1 x (n-I) + C 1 • (17) From II='ll~ -< Ilzll, (16) and (17), one has
IIx'll~ _< IIxll
< 2
1 - 2 I/allI~ ~==
Ilx'll~ --< 1-- 2(llallll + I[a2111) ]la/lll x(i-1) + Ilblll x(n-') + C1 '
then,
z(n_m) ~ 2--<
n-1
1 - 2 E liad~
i=l
[ llanlll
x(n-1) oo-F]lblll x (n-l) o + C 1 ] , (18)8 X. LIN
et hi.
then,
_<
2_~1tb111
~(~_~) 0
+
3C(n-l)oo
1
-- 2 >_~lla~ll~
i = lSince 0 E [0, 1), from (19), there exist Mi > O, such t h a t
x ('~-1) ~ ~ M1, thus, from (20) and (18), there exist M2 > O, such t h a t
z (~-2) ~ <_ M2. Similarly, there exist M i > O, (i = 3, 4 , . . . , n), such t h a t
h e n c e , 2Ci
I- 2 ~
liudll
i = 1 x('~-i) oo <- M~, i = 3, 4 , . . . , n,=
IIz I L , . . .
_< max {M1, M2,..., M . }
Again, from (7), and (20)-(22), we have
x<~) 1 ~ Ilallll M~ + . . . + Ila~-1111 M2 + (11~1il + tlbll1) M1 -F IlriI~ + Ileile •
Then, we show t h a t
~ i
is bounded.If (8) or (9) holds, similar to the above argument, we can prove f~i is bounded too. STEP 2. T h e set f~2 = {x E K e r L : N x E ImL} is bounded.
Let x E f~2, x E K e r L -- {x E d o m L : x = c t n - l , c E R , t E [0, 1]}, and Q N x = 0, thus,
m-2~ [i
~J
£
[T~...IT2 If (TI'CT?--I'c(n--1) T~--2'''''C'(n--1)!)-Fe(T1)] dTl'"dTnO,
f
f
J0 Jo
(19)
(20)
(21)
(22)
Let n~ C 2 - m - 2 _ ~ n j=l f~3 = {x e K e r L : - A x + (1 - A) J Q N x = 0, A e [0,1]}, m - 2 1f
T,~c2E jJ
Jo
j=l [f('ri, c.rr -i, c (n-l)~.~-2,..., c. ( n - 1)!
• /0 T2
+ e (~'z)] d r 1 . . , dT~ < O,(23)
whereso the set ~12 is bounded.
STEP 3. If the first part of (Ha) holds, t h a t is, there exists M* > 0, such that, for any c E R, if lel > M*, then,
[Ix[I = m a x { l l x l l o ~ , . . . , x (n-l) ~ } = x (~-1) ~o -< M,
From (He), there exists to C [0,1], such t h a t Ix(n-1)(to)[ ~ M, i.e., I(n - 1)[c I _< M. Then, we have Hx(~-l)lt~ = Ix(~-l)l = [(n - 1)!c I <_ M, thus,
Solvability of Multi-Point Boundary Value Problems 9
here, Y : ImQ ----* KerL is the linear isomorphism given by J(e) = ct n - l , Y c • R, t • [0, 1]. Then, ~3 is bounded.
Since, for x = cot "-1 • ~3, then, for t E [0, 1], we obtain
where
or equivalently,
m - -
2
1Tn
7-2
/~cotn-1:C37~n-1
E~j/ ~0 ""/
j = l J
If (~'1, co~'~ -1, co ( n - 1 ) ~ . ? - 2 , . . . , co' ( n - 1)!) + e (~-1)] d~-i.., d~-~,
c3 = (1 - )~) c2 - ( 1 - A ) . n! m - 2 1 _ E fl J ~ n j = l m - 2 r l [f (71, co~-~ -1, co (n - 1) ~_~-2..., co. (n - 1)!) + e (71)] d T l . . , d'rn.
If A = l, then, co = 0. Otherwise, if Ic01 > M*, in view of (23), one has
l/0 /o
m - - 2 1 r n . = c o
j = l J
If (T,, co~-[ ~-1, co (n - 1) "r~-2,..., Co. (n - 1)!) + e (T1)] d~h.., d~-n < 0,
which contradicts ~,co ~ _> 0. Thus, r~3 c {= • K e r L : Ilzll -< (n - 1 ) ~ M * } is bounded.
STEP 4. If the second part of
(H3)
holds, t h a t is, there exists M* > 0, such that, for any c • R, if [cl > M*, then,m - 2 1 ~-n T2
c'C2 E ~j a~
f / ""~0 If
( T I ' C T ~ - I ' c ( ? ~ - - 1 ) 7 - ~ - 2 ' ' ' ' ' c ' ( n - - 1 ) ' ) - ~ - e ( T 1 ) ]dTl...dTn ~0.
j = l JLet Qa = {x • KerL : ),x + (1 - A) J Q N x = 0, A • [0, 1]}, here, J as in Step 3. Similar to the argument in Step 3, we can verify 123 is bounded.
Now, we shall prove t h a t all the conditions of Theorem A are satisfied. 3
Let t2 be a bounded open subset of Y, such t h a t Ui=l £1i c ~2. By the Ascoli-Arzela theorem, we can show t h a t K p ( I - Q ) N : ~ ~ Y is compact, thus, N is L-compact on ~. Then, by the
above argument, we have
(i) Lx ¢ ANx, for every (x, ~) e [(domL\Kern) rh 0t]] x (0, 1); (ii) N x ~ ImL, for every x • KerL r~ 0t];
(iii) If the first part of (Ha) holds, then, we let
H (x, A) : - l x + (1 - ,k) 7 Q N x .
According to the above argument, we know H(x, A) ~ 0, for x C KerL Q 0~, by the homotopy property of degree, we get
deg ( J Q N
IKerL, ~
FI KerL, 0) = deg (H (., 0), fl rh KerL, 0) = deg ( H (., 1), ~ A KerL, 0) = deg ( - l , g~ r7 KerL, 0).10 X. LIN et al.
According to definition of degree on a space which is isomorphic to R l, l < e% and fl N K e r L = {ct ~ - I : Icl < d}.
W e have
deg ( - I , ~t N K e r L , 0) = deg ( - J - 1 I J , j - 1 (~-~ (-] K e r L ) , J - l { 0 } ) = d e g ( - I , ( - d , d ) , 0 ) = - 1 # O.
If the second p a r t of condition (3) of T h e o r e m 2.2 holds, let H ( x , A ) = Ax + (1 - A) J Q N x . Similar to the above argument, we have
deg ( J Q N ]KerL, ~ N KerL, 0) = deg (I, f~ n K e r L , 0) = 1, since 0 E f~ C~ KerL.
Then, we have
deg ( J Q N I K e r L , f~ • KerL, 0) 7~ 0.
T h e n by, T h e o r e m A, L x = N x has at least one solution in d o t a L n ~, so t h a t the B V P (1),(2) has at least one solution in C n - l [ 0 , 1]. T h e proof is completed. |
Now, we discuss the existence of solution of B V P (1),(3).
Let Y = C n-1 [0, 1], Z = L 1 [0, 1], the m a p N and the linear o p e r a t o r L are the s a m e as above, and let
d o m L = z E W n'l (0, 1): X (0) : X' (0) . . . X (n-2) (0) = 0, X' (1) = ~ j X ' (~lj) • j = l
B y using the same m e t h o d as in the proof of L e m m a 1 and T h e o r e m 1, we can show the following L e m m a 2 and T h e o r e m 2. v--~ra-2 ~ n--1 m--2 rn--2 ~ . 7 7 n - 2 = 1, 2.-,j=l PJ?]j
5~ 1, then,
LEMMA 2.If~j=l
/~j = 1, E j = I 3 j (i) K e r L = {x C d o m L : x = c t ~ - l , c C R , t C [0,1]}; rn--2 (ii) I m L = {y E Z : ~ j = l flJ f£1j ~,_~ .fo
• fo y(,1)d l
. . .0};
(iii) L : dotaL C Y ~ Z is a Fredholm o p e r a t o r of index zero, and the linear continuous p r o j e c t o r operator Q : Z ---* Z can be defined as
m - 2 1 T n - - I I T 2
~'~ ~ . ~ n - - 1 j = l J 1 - - a - ~ 3 j
j = l
(iv) the linear operator K p : I m L ~ d o m L A K e r P can be written by
/?
K p y . . . . y ('rl) d ' r l . . , d~%; (v)
HKpyll <_
Ilyll~, for all y e I m L .Solvability of Multi-Point Boundary Value Problems 11 THEOREM 2. Let f : [0, 1] x R ~ > R be a continuous function, assume that (H1) in Theorem 1 is satisfied, and
(H4) There exists a constant M > O, such that, for any x E d o m L , if [x(n-1)(t)] > M , for all
t E [0, 1],
then,
~j "'" [f (TI,X(T1),X/ ( T 1 ) , . . . , X ( n - l ) (T1)) "~-e(T1) ] dT 1 . . . d T n _ 1 ~= O.
j = l 4
(Hh) There
exists a constant M* > O, such that, for c E R, if[c[ >
M*, then,either
c'rn--2j=l E ~j J~Jfl[f~-i , , d O " fO TM [f (T1, c~-f-l,c (n - 1) T ? - 2 , . . . , C. ( n -- 1)!) + e (T1)] dTl ... dTn-1 < O, or else
m--2
r I [ . r ~ - i I f ( ~ - l , C r ? - l , c ( n - 1 ) T r - 2 , . . . ,C. ( n -- 1)[) + e ( r l ) ] d~'1...dTn-X > O. r~m--2 ,1 n - 2 Then, for everye(t) e LI[0,1],
the BVP (1),(3) with ~j~__12flj = 1, 2_.j=1 PJ~/j = 1,~-~m-2
j = l~
jr/j n - 1~ 1,
has at least one solution in Ca-l[0, 1], provided t h a tY~=I [[ai[[1 < 1/2.
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