VietnamJ Math, (2015)43'93-104 DOI 10.1007/S100I3-014-0072-4
Positive Solutions of a Fourtii-Order Differential Equation with Multipoint Boundary Conditions
Phan Dinh Phung
Received 21 February 2013 / Accepted 24 December 2013 / Published online: 5 June 2014
© Vietnam Academy of Science and Technology (VAST) and Springer Science+Busmess Media Singapore 2014
Abstract This paper is devoted to the study of the following fourth-order multipoint boundary value problem:
j x^'^Ht) - Xf(t,x(t),x'(t),x"(t)), 0 < I < 1, I x<2*+l>(0) - 0,^(2«(1) = YT^'-a,,x^'"'Kriki), (k - 0, 1).
We obtain some sufficient conditions for the existence of at least one or triple positive solutions by using the fixed point theory in cone.
Keywords Boundary value problem - Positive solution - Leggett-Williams fixed point theorem • Guo-Krasnosel'skii fixed point theorem
Mathematics Subject Classification (2010) 34B07 • 34B10 - 34B18 • 34B27
1 Introduction
The multipoint boundary value problems for ordinary differential equations arise m a variety of different areas of applied mathematics and physics There have been many papers dealing with the existence and multiplicity of positive solutions for high-order differential equations subjected to various boundary conditions. Let us quote here some basic works of If in and Moiseev [5], Leggett and Williams [9], Henderson and Kosmalov [4], Cabada et al. [1, 2], Webb and Infante [15], Pietramala [11], Kelevedjiev et al [6], etc
Liu et al. [8] considered the 2wi-th order multipoint boundary value problem x'-^''\t)==.f{l,x(t),x"(t) A ' ^ * " - ' " ( 0 ) , 0 < ( < 1,
^(2*+i)(oj^O, ; c < 2 * ' ( l ) = ' ^ a t , : . ( ^ ' ^ ' ( ^ , ) , K^O,l,. . , « - ! ,
P D, Phung (SJ)
Nguyen Tat Thanh University. 300A, Nguyen Tat Thanh Sfi-, Dishict 4, HoChiMinh City. Vieuiam e-mail: pdphung@ntt edu vn
a Sprii
94 P. D. Phung where aj,, > 0 for all /; = 0, 1 , , , . , « - I and i = 0 , 1 m — 2 and
Y^ak, < 1 V/t =
0,i,
..,n- i. (I) 1=1The authors established the existence of triple positive solutions using the Leggett-Williams fixed point theorem Assumption (I) plays an important role in obtaining positive solutions.
Recently, Eggensperger and Kosmatov [3] considered the three-point nonlinear boundary value problem
x^^Ht)^Xf(x(t)). 0 < ( < L (2)
;c<2*+"(0) = 0, ^'^^'(1) - atx^^''\r}k), k^0,\, (3) and obtained some results on the existence of positive solutions by using Ihe fixed point
index. Although the derivatives are not involved in the mhomogeneous term, the paper [3]
generalized the results of [8] (for n — 2 and m — 3) by considering the case 1 < a^ <
l/Vk, k = 0,l.
In [14], we extended problems (2)-(3) to the following m-point boundary value problem
x('^\t) = Xf(t,x(t)), 0 < r < l , (4)
x < ^ * + " ( 0 ) ^ 0 , x^^'H^Jj^a^x^^'Hr,,,), k = 0,l, (5)
<=i
widi die weaker assumptions related to the coefficients appearing in (5), namely m-2 ffl-2 m-2 m-1
X!«i''Ji' ^ 1 < I]«i'- J^"o>lo, <> < I]«o,--
The present paper is a continuation of the work in [14]. Here, we deal with Eq. (4), in which die inbomogeneous term depends on the derivatives of the first and the second orders ofx(t) and associated with the boundary condition (5).
The rest of this paper is organized as follows. In Section 2, we introduce some basic notions and state some needed resuhs. In Sections 3 and 4, we prove several sufficient conditions for the existence of at least one or triple positive solutions. Our main tools are die well-known fixed point theorems due to Guo-Krasnosel'skii or Leggett-Williams.
2 Preliminaries
Let X — C([0, I]) be tbe space of all continuous functions from [0, 1] into R endowed with tbe sup-norm, i.e.,
\\x\\ = sup lJ:(()| f o r ^ e C ( [ 0 , 1 ] ) .
' € 1 0 , 1 1
We also denote by K the following subset of C([0, I ])
\u e C((0, 1]) : u IS nonpositive and nonincreasing and — u(t) > Y\\U\\ Vr e [0, 1]), where
Q Springer
Positive Solutions of a Fourdi-Order Differential Equatio 95 It is clear that K is a
assumptions:
in X Throughout this paper, we shall use the following
>lk(m-2) < 1 and a^i (;' — 1, . ,m —2) are tbe positive constants (HI) 0 < ^ii < - •.
fork e { 0 , I),
(H2) x::ii^ «o, r,l<i< YT'^ m;
(H3) E r . f «i<'ji. < 1 < Er=?«i';
(H4) / : [0, 1] X (R+)3 ^ K+ is a continuous function.
Similarly to the method presented in [8, Theorem 2], the idea of our main results is based on die reduction order using an integral operator via the Oeen function. However, the method in [8] does not work for functions on the rigbt-hand side contaimng derivatives of odd order, meanwhile, our method will not be able to generalize to 2«-th-order differential equations First, we need some properties of a Green-type function that is an important tool in the statement of our main results.
Lemma 1 LetO < iji < ri2 < •• < J?m-2 < 1 and a, (i — 1,2 m-2) be the positive constants which satisfy the condition Y^'"Sj «, > 1. Suppose that
G(;-;ai,r],) : [0, 1] x [0, 1 ] - ^ R is the function defined by
G(t,s,a,,r)i) = 1 0 < J < r < 1, 0 < f <.^ < 1,
i-i:r=f "'•'?'+ (Er.
1 ) .0 < 5 < Jjl,
rik--\ <s<r)k.
where a — (1 - E I L : 'Xi) '• Then we have (al) G(-, •,€(,,rji) satisfies the following estimate
|G(r, j , a , , Tji)\ds <
Er.r
(a2) If we assume further that Y^Zi '^t^i < i-, then G(t,s, a,, i},) is negative and - y f • " ' ( ! - . ! ) <G(f,.v;a,,r?,) <-y"'•"'(\ - s) Vt,se [0,1], (7 where y""''' andyj" ' are two positive constants
^ Springe
Proof The esumation in (al) is straightforward, and we will omit it. Now, we prove (a2).
It follows from (6) that
l - s + E r . l ^ f . l s - * ) . O S ' i l l .
1-i + ES^".**-*). mss-sm.
G(t,s;cti,t!i) > s - l + a
1 -s, - l + a ( l - s )
>)»-2 < s < 1,
( l - i ) .
Git, s; at, tji) <
M ' -^ + X]"'^^''' ^'''
E,.i ". - 1
yz •
Er=f«,-i'
for all l,s € [0, 1]. These show that (7) is sabsfied with , \~-in-2
Tbe proof of this lemma is complete.
Now, in order to study the problem
ix'-'^Ht)^kf{t,x(t).x'(t),x"(t)). 0 < r < 1,
|^(2*+i)(oj ^ 0, :r<=*>(I) = E ; ' ; ? « * < ^ ' ' " ( ^ t , - ) , (k - 0,1), we consider first the auxiliary linear differential equation
- x " ( r ) = « ( / ) , 0 < ( < 1, together with the boundary conditions
m-2
^'(0) = 0, x{l) = J2ao,x(j,o,). (10)
Lemma 2 Let u € C([0, 1]) be a nonpositive and nonincreasing fiinction. Let (HI) and (H2) hold. Then there is a unique nonnegative solution of problems (9)-(IO) which is defined by
x(t)= G(t,s,ao,,mi)uU)ds •= Au(t), r e [0,1]. (11) Furthermore, we have the following estimate:
fl Springer
Positive Solutions of a Fourth-Order Differential Equation
Etr«0,(l-'?^,)
Proof The reader can see in [ 14). D Lemma 2 leads us to study the following second-order multipoint boundary value
problem:
\-u"(t) = Xf(l,Au(t),~j^u(s)ds,-u(t)y 0 < r < 1.
1 M ' ( 0 ) = 0 , «(l) = Er=f «n"('?ii)-
It is necessary to note that if problem (13) has a solution u E K, then, by Lemma 2, the original problem (8) owns a nonnegative solution satisfying (12), On die other hand, thanks to [14, Lemma 2.1], problem (13) can be ttansformed into die fixed point problem
M = Tu, where T is defined by
Tu(t) = XJ G(t, s; «!/, iiu)f (s, Au(.s), - f u(r)dr, -u(s)j ds. t e [0, 1]
The following lemma provides several important properties of the operator T.
Lemma 3 Lei (HI )-(H4) hold. Then we have (bl) the operator T : K -* K is completely continuous, (b2) min,e|o,i] |7-M(/)| > / | | r M | | VM € K.
Proof The proof of this lemma is straightforward. D To finish this section, we state two fixed-point theorems due to Guo-Krasnosel'skii and
Leggett-Williams that will be used in the next sections.
Theorem I (Guo-Krasnosel'skii, [7]) Let X bea Banach space and let K 1:1 X bea cone in X. Assume that J^j and Qi are two bounded open subsets of X withQ e S2i andQ] C fii Let
T : Kn(a2\^\) -^ K be a completely continuous operator such that either
(i) \\Tx\\< \\x\\.xE Kr\a^\.and\\Tx\\> \\x\\,x € Kr\d^2,or (ii) \\Tx\\ > lijrll.j; e Kr\B^\,and\\Tx\\ < \\x\\.x e KndQ2- Then T has a fixed point in K fl (£22\£2i).
Theorem 2 (Leggett-Williams. [9]) Let K bea cone in a Banach space X. K^ ^ [x € K :
\lx\\ < c] and a bea nonnegative continuous concave fimctional on K with a(x) < ||A:|j/or all X e Kr- Assume that
S(a. b,d) = [xeK :b < a(x), \\x\\ < d).
•^ Springer
98 P D. Phung
and T . Kc —* Kc is a completely continuous map such that there exist the constants Q<a<b<d<c satisfying the conditions
(I) [xe S(a,b,d):a(x) > b] ^?ianda(Tx) >bforx€S(a,b,d);
(u) \\Tx\\<afor]\x\\<a:
(iii) a(Tx) > b forx € S(a, b, c) with \\Tx\\ > d Then T has at least three fixed points x\, X2, and Arj with
ll-iill Sa, b < a(x2), a < \\xi\\ with aix^) < b.
3 One Positive Solution
In this section, we shall study the existence of one positive solution to problem (8) by using Theorem I. First, we introduce the notations:
f , - F F fi'-x<y.l) ^ ,. ^ , fii^x,y,z) / o . = h m i n f inf , /oo : = h m i n f inf — • ,
(r,y.!)->-(0+,0-i-,0+]rs[0,1] z : ^ ™ {i.joOeLO.ilxtK-^]^ 2 f(t X y z) f(t :c V z) f := h m s u p s u p • , / ™ : = l i n i s u p sup '•—'—^—,
(i,y.;)^(0+,0+,0+)/e|0,l| Z ; - , o o (f,j,j>)6[0,|]!<(E+)2 Z
Theorem 3 Let (H1)-(H4) hold. Assume furthermore that there exist two positive numbers R] and Ri such that
MR\ NRj Al - ~ A 7 ' where
Al ^mir>{f(t.x,y,z) : il,x,y,z) £ [ 0 , 1 ] x [O.pRy] x [0, Ri] x [yRu Ri]], A2 = max(/((, X, y, z) : (t, x, y, z) e [0. 1] x [0, pRz] x (0, R2] x [yRj, Ri]).
Then problem (8) has at least one positive solution for every MR\ NR2
- r — < ^ < — ^ (14)
A | A 2
for/ — 1,2. Without loss of generality, we can assume that R] < R2. Then Q.\ and ^2 a two open-bounded subsets of C([0, I ]) and
0 e £2] and £2| c ^2- Let /. satisfy (14). If « e A: n 3f2i, then we have
j/ff, = y\\u\\ < -u(i) = \u(t)\ < {lull = Ri V( e [0, 1], 0 < -f^u(s)ds < tR\ < Ri Vr e [0. 1],
fl Spnnger
Positive Solutions of a Fourth-Order Differeniial EquaUon it follows from ( U ) that
0 < Au(t) = I G(l,s;aoi,r}o,)u(s)ds Jo
< f \G(t,s:ao,,r]Q,)\\u(s)\ds Jo
< pRi Vr € [0,1].
So, by using the definition of operator T. we obtain tbe following estimate:
|rM(f)| = X I \G(t,s,au,>lu)\fUAu(s),-l' u(r)dT,~u(s)\ds
> A A , / \G(i,s;au,rii,)\ds Jo
> kAiy^'-"'' f'(l-s)ds>Ri Jo
for all r e [0, I ], This implies
\\Tu\\ > \\u\\ Vw e KndSli. (15) On the odier hand, if w £ ^ D 9 £22, then by arguing sinularly, we have
YR2 < - « ( ' ) < «^, 0 < - / u{s)ds <R2, 0 < Au(t) < pRj Jo
foralKE [0,1]. Thus,
\Tu(l)l <xf \G(t.s:au,7ju)\f(s.Au(s),-ru(r)dT,-u(s)^ds
< AAsyj"'""" [ (I - s-)ds < R2 = \\u\\
Jo for all ( e [0, I], So we get
IITMII < ||»{| VuEKndQi- (16) Combining (15) and (16) and using the second part of Theorem 1, we conclude that T has
at least a fixed point M e S" n f22\n|. The proof of the theorem is complete. • Corollary 1 Let (H1)-{H4) hold. Then the boundary value problem (8) has at least one positive solution for every
\yfo f^l
iffo, f°° e (0, oo) satisfy y'ft, > / ° ° , or
If fee, f € (0, OO) satisfy f" < y^f^.
Proof Case 1 / o , / ' ^ e (0, oo) satisfy/^/o > / ° ° . L e t l € ( ^ . - p r ) - T h e n , thereexisis e > 0 such that
M N
• * * • " = 7 ; ^ < " )
yC/o-t) f^ + z'
a Spn,
100 p. D. Phung Since /o e (0, oo), there exists S | > 0 such that
/ ( M . y. z) i (/o - e)z for all t s [0, 1], ix, y, z) e ([0, R i ] ) ^ This implies
mm [fil.x, y,z):{t,x,y,z)<^ [0, IJ x [0, pR,] X [0, Ri] X [ / R , , R,]) > (/o - e)y«|
which means that
Rl 1
M^Vif^> ™
On tbe other hand, since f^ s (0, co), there is fi > 0 such that
sup fU,x,y,z) ^ yco _|_ ^ ^gj (',.t.j)e[0,1]x(K+)2 2
foraUz > -R. If we choose fi2^max{fli-M,)^-'fi|, then (19) holds for aUz e [YRI^RII This implies
/ ( f , x , y,z) < ( / ° ° + e ) z Vr e [0, 1], (x, y, z) e [0, pRj] x [0, ffa] x [yR2, R2I Hence, we get
^2 1
Combining (17), (18), and (20) and applying Theorem 3, we deduce that problem (8) has at least one positive solution.
Case 2. foo, J E > 0 such that
y(f^-s) / » + £- Since / " € (0, CO), there exists R2 > 0 such that
fit.x.y.z.}<(f'' + z)z for all I E [0, 1], ix,y,z}^ ([0, Rz])^ This implies
A 2 < ( / ° + e)R2 or | 1 > _ L _ ^ A2 / » + B Since / ^ e (0, 00), there exists R > Rz such that
f(t,x,y,z)^(fco-s)z
for all I e [0.1], (J:, j ) € (R+)^ and z > R. Select Ri = max{)/-' R, S2 - H I - Then we have
Rl 1 Al - 7 ( / = o - e ) '
Therefore, by applying Theorem 3, we conclude that problem (8) has at least one positive
solution. • Theorem 4 Let (H1)-(H4) hold. Assume that one of the followtng cotiditiott.s holds
/o, /°° e (0, (X>), yVo > / ~ attd i. s (~, — ) , Vy/o / ~ / / » , / " s ( 0 , c » ) f'<y^f^ and i E f J ^ , 4 ; )
r/ie« ;/ie set of positive solutions of problem (8) is compact in C([0, 1])^ Springer
Positive Solutions of a Fourth-Order Differential Equa
Proof Since the operator A, defined by (11), is continuous, it is sufficient to prove that tbe solutions set
S={u e C ( [ 0 , l ] ) : M - r « }
of problem (13) is compact in C([0, 1]). We can obtain this fact by using the similar
arguments as in [14, Theorem 4.1], D Example I L&i a, b.c, andi/ be positive numbers such that 6c > 32(a -i- c)d. We consider
the boundary value problem
x^^\t) = (t^~ -H 1) h ' ' ' ' ' Z : i V ' ' + . » s m V ( . ( 0 ) ^ + (x-(0)^ I [a(x"it))'-+t [ cx"(l) + Vd
x'(0)=jr'3>(0) = 0, x(\)^^x{^\, / ' ( l ) ^ l . v " Q V In this case, we have
j/l"'""" = 1. y^"'""" = 4 , y - 1 / 4 , M = 2 and N = 1/2.
It is easy to see that the conditions (H1)-(H4) hold with
f(t.x.y.z)^(t- + \) \zsin-y/x^-\-y^ + '^~^^~lj' , Moreover, we have
r / ( f ' ^ - v . ^ ) ,- '^z^+bz b fa = lim mf — lim -^ — - .
(.r,>.:)-*(0+,0+,0+)i6[0,l] Z z^0+ CZ^ + dz d hm sup
^~"^(i.^.y)€[0.\]>:
azr -\-bz\ I a\
: lim \\ + —. f - 2 ( 1 - 1 - - } .
; ^ ^ I c7} + dz\ V cl Since be > 32(a -|- c)d, we deduce that y-fo > / ' " • So, by Corollary 1, we conclude that for each
die solution sel of our problem is nonempty and compact in C([0, 1]).
4 TViple Positive Solutions
In this section, we will further discuss the existence criteria for at least three positive solu- tions of problem (8) by using Leggett-Williams fixed point theorem (Theorem 2). For diis.
we assume that X = I and define a nonnegative continuous concave functional or on ^ by
ll is clear that a(w) < ||»ll Vu e A!' Now, we state the mam results mthis section.
Theorem 5 Let (H1 )-(H4) hold Suppose that there exist positive constants a. b, and c with a < b < - < c and f satisfies the following growth conditions:
(Al) / ( ; , ^ , y , z ) < A ' c / o r ( ( , . r . y . z ) e [ 0 . I J x [0, pc] x [0. c] xiyc.c];
£ Springer
102 PD. Phung (A2) f(t,x,y,z) < / V a / o r ( / , j c , y , z ) e [ 0 , l ] x [ 0 , p a ] x [0,a] x [ya,a].- (A3) f(l,x,y,z) > Mb for(t,x,y,z) e [0,1] x [0,/J^] x [0, ^ ] x [fc, ^ ] . Then the boundary value problem (8) has at least three positive solutions J: i, J:2, andx^ witk
\[x'{\\ < a. b < min \x2(t)], a < \\x';^\\ and min 1x3(1)1 < b.
Proof We need to show^ that all the conditions of Theorem 2 hold with respect to the operator T. First, if M e K^, then we have
y c < -u(t) < c, 0 < - / u(s)ds <c, 0 < (AM)(r) < pc Jo
for all ( e [0,1]. So by assumption (Al), we obtain
|7'«(f)l - \[ G(t.s;au,y}u)f(s,(Au)(s),-l u(T)dT,-(s)\ds\
< j \G(t, s; au, vu)\f (s,(Au)(s),- I u(T)dr,-u(s)\ds
< Nc \G(t,s\au,i]]i)\ds <c Jo
for a l l ; 6 [0, 1]. Hence, | | r « | | < c which means diat T :'KC ^ ^K^. By using the similar arguments and assumption (A2), we get the condition (li) of Theorem 2.
Next, we will check condition (i). Let e > 0 be such that / < y -I- e < I and /b b \ b
I'd) = -{ t - . / e [0.1].
\y y-\-eJ y + s It is easy to see that
a(u) ^ min | H ( / ) | - min (-u(t)) = > b, and ||u|| = - . This shows that
\u esia,b,-\ :a(u) > i U i 0.
On the other hand, for u e S(a, b, ~), we have
b < -~u(t) < - , 0 < - / u(s)ds < - , 0 < Au(t) < p- Y Jo Y - • ' - "^ y for all f e [0, 1] By assumption (A3), we deduce that
/ ( ' - » « ( ' ) . -j^ u(s)ds. -u(t)') > Mb for all r G [0, 1]. Hence,
a(Tu) - mm \Tu{t)\
(e[0,l]
> ^b inin / \G(t.s;au,r,i,)\ds>b.
'e[0,J| Jo
© Springer
Positive Solutions of a Fourth-Order Differential Equa Finally, ifM e S(a,b,c) and \\Tu\\ > ^, th en we have
a(Tu) = mm \Tu(t)\ >Y\\TU\\ > b.
/e[0,il
Applying Theorem 2, we deduce T has three fixed points u\,U2, andM3 such that
||Mil|<fl, b<a(u2), a < llMall with a(u^) < b.
Therefore, Theorem 5 is proved completely
Example 2 Let B : [0, 1] -^ M"*" be a continuous fiinction satisfying the condition
^ < e ( f ) < ^ V f e [ 0 , I ] . and (p : (K+)^ -^ M+ defined by
zlX fl iTTx-s I 1 1 /irv\ n 'P(x,y,z) •• V 6 0 / V 2 0 /
0 < s < 1, 5 6 S / 3 - 5 5 / 3 , 1 < s < 5 / 4 , 5. 5 / 4 < s < 5 , 1/z -I- 24/5, ; > 5.
We consider the followmg boundary value problem
x^'^Ht) - d(t)^(x(t).x'(t), x"(t)), t e (0, I), x'(O)=x"'(O) = 0,
x ( l ) - 3 / 2 . * : ( 3 / 4 ) ,
; t " ( l ) - 4 / 3 x " ( l / 2 )
In diis case, we have m — 3, aoi — 3/2, ;)oi - 3/4, a n = 4 / 3 , f/n — 1/2. So we deduce that
y = 1/4, M = 2, TV = 1/2, p ^ i .
If we take a — 1, fo — 5/4, and c = 10, then we have a < b < b/y < c On die odier hand, we can show that conditions (A1)-(A3) of Theorem 5 hold. Thus, we conclude that our problem has at least three positive solutions X], X2, and x^ such that
\\xi\\ < I; 5/4 < min|A:2l; 1 < 11x3 11 and min[;>:3l < 5/4.
Acknowledgements The authors wish to express their sincere thanks to the referees for their valuable suggestions and remarks leading to improvement of the original manuscript
1 Cabada. A. Cid, J.A„ Infante, G • N ew cnteria for the existence of noQ-tnvial fixed pomts m cones.
Fixed Point Theory Appl 2013, 125(2013)
2 Cabada, A„ Pouso, R L , Minhos, P.M. Extremal solutions to fourth-order functional boundary value problems including multipoint conditions Nonlinear Anal.: Real World Appl. 10, 2157-2170 (2009) 3. Eggesperger, M , Kosmatov, N 'Posmvesolutionsof a fourth-order multi-point boundary value problem
Commun. Math. Anal 6. 22-30 (2009)
4 Henderson, J , Kosmalov, N ; The existence and mutiplicity of constant sign solutions to a three-point boundary value problem Commun Appl. Nonhnear Anal 14(3), 63-78 (2007)
5, H'in, VA , Moiseev, E,I Nonlocal boundary-value problem of the first kind for a Sturm-Liouville operator Differ Equations 23, 979-987 (1987) TransI Differ Uravn, 23,1422-1431 (1987) 6, Kelevedjiev, PS„ Palamides, P.K,, Popivanov, N I. Another understanding of fourth-order four-point
boundary-value problems, Electi-on J Differ Eqns 2008(47), 1-13 (2008)
^ Sprii
104 P. D. Phung
7. KrasnoseTskii, M A.: Positive Solution of Operator Equations. Noordhoff, Groningen (1964) 8. Liu, X--J., Jiang, W.-H,, Guo, Y.-R: Multi-poml boundary value problems for higher order differeniial
equations. Appl. Math E-notes4,106-113(2004)
9. Leggett, R.W., Wilhams, L R Muiuple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Matii. J. 28, 673-688 (1979)
10. Ma, R. Existence results of a m-point boundary value problem al resonance. J. Math. Anal Appl, 294, 147-157(2004)
11. Pietramala. P. A note on a beam equation with nonlinear boundary conditions. Bound, Value Probl, 2011(376782), 14(2011)
12. Sun, Y.: Positive solutions of nonlinear second-order m-point boundary value problem. Nonlinear Anal.
61, 1283-1294(2005)
13 Tniong, L . X , Ngoc, L.TR, Long, N,T., Positive solutions for an m-poinl boundary value problem.
Electron J Differ. Eqns 2008(111), 1-11 (2008)
14^. Tniong, L X , Phung, P.D • Existence of positive solutions for a multi-point four-order boundary-value problem Electi-on LDiffer Eqns 2011(119), 1-10(2011)
15- Webb, J.R L., Infante, G Nonlocal boundary value problems of arbitrary order J Lond Math. Soc 79, 238-258(2009)
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