The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives

Applied Mathematics and Computation 218 (2012) 8526–8536

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives q Xinguang Zhang a,⇑, Lishan Liu b,c, Yonghong Wu c a

School of Mathematical and Informational Sciences, Yantai University, Yantai, 264005 Shandong, People’s Republic of China School of Mathematical Sciences, Qufu Normal University, Qufu, 273165 Shandong, People’s Republic of China c Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia b

a r t i c l e

i n f o

Keywords: Fractional differential equation Positive solution Green function Eigenvalue problem

a b s t r a c t In this paper, we study the following singular eigenvalue problem for a higher order fractional differential equation

8 a l1 l2 ln1 xðtÞÞ; 0 < t < 1; > < D xðtÞ ¼ kf ðxðtÞ; D xðtÞ; D xðtÞ; . . . ; D p2 P l l l i > aj D xðnj Þ; 1 6 i 6 n  1; : xð0Þ ¼ 0; D xð0Þ ¼ 0; D xð1Þ ¼ j¼1

where n  3; n 2 N, n  1 < a 6 n; n  l  1 < a  ll < n  l, for l ¼ 1; 2; . . . ; n  2, and lP ln1 > 0; a  ln1  2; a  l > 1, aj 2 ½0; þ1Þ; 0 < n1 < n2 <    < np2 < 1, al1 < 1, Da is the standard Riemann–Liouville derivative, and 0 < p2 j¼1 aj nj n f : ð0; þ1Þ ! ½0; þ1Þ is continuous. Firstly, we give the Green function and its properties. Then we established an eigenvalue interval for the existence of positive solutions from Schauder’s fixed point theorem and the upper and lower solutions method. The interesting point of this paper is that f may be singular at xi ¼ 0; for i ¼ 1; 2; . . . ; n. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction The purpose of this paper is to establish the existence of positive solutions to the following higher order fractional differential equation

8 a l1 l2 ln1 xðtÞÞ; 0 < t < 1; > < D xðtÞ ¼ kf ðxðtÞ; D xðtÞ; D xðtÞ; . . . ; D p2 P l l > aj Dl xðnj Þ; 1 6 i 6 n  1; : xð0Þ ¼ 0; D i xð0Þ ¼ 0; D xð1Þ ¼

ð1:1Þ

j¼1

where n  3; n 2 N, n  1 < a 6 n; n  l  1 < a  ll < n  l, for l ¼ 1; 2; . . . ; n  2, and P al1 l  ln1 > 0; a  ln1  2; a  l > 1, aj 2 ½0; þ1Þ; 0 < n1 < n2 < . . . < np2 < 1, 0 < p2 < 1, Da is the standard j¼1 aj nj Riemann–Liouville derivative, and f : ð0; þ1Þn ! ½0; þ1Þ is continuous. Recently, fractional differential equations have been of great interest. It is caused both by the theory of fractional calculus itself and by the applications in various sciences such as physics, mechanics, chemistry, engineering, for details, see [1–22]

q The authors were supported financially by the National Natural Science Foundation of China (11071141) and the Natural Science Foundation of Shandong Province of China (ZR2010AM017). ⇑ Corresponding author. E-mail address: [email protected] (X. Zhang).

0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2012.02.014

X. Zhang et al. / Applied Mathematics and Computation 218 (2012) 8526–8536

8527

and the references therein. In [1], by using the fixed point index theory, Bai considered the following fractional differential equation with three-point boundary condition



Da uðtÞ þ f ðt; uðtÞÞ ¼ 0; uð0Þ ¼ 0; uð1Þ ¼ buðgÞ;

0 < t < 1;

ð1:2Þ

where 1 < a 6 2, 0 < bga1 < 1, 0 < g < 1, Da is the standard Riemann–Liouville derivative, f : ½0; 1  ½0; þ1Þ ! ½0; þ1Þ is continuous. In [1], Bai got the property of the Green function and the existence of positive solutions for (1.2) when f is sublinear. In [2], Zhang considered the following problem whose nonlinear term and boundary condition contain integer order derivatives of the unknown function

(

Da xðtÞ þ qðtÞf ðx; x0 ; . . . ; xðn2Þ Þ ¼ 0; 0

xð0Þ ¼ x ð0Þ ¼ . . . ¼ x

ðn2Þ

ð0Þ ¼ x

ðn2Þ

0 < t < 1; n  1 < a 6 n; ð1Þ ¼ 0;

ð1:3Þ

where Da is the standard Riemann–Liouville fractional derivative of order a, q may be singular at t ¼ 0, and f may be singular at x ¼ 0; xprime ¼ 0; . . . ; xðn2Þ ¼ 0: And then, Goodrich [3] was concerned with a partial extension of the problem (1.3)



Da xðtÞ ¼ f ðt; xðtÞÞ;

0 < t < 1; n  1 < a 6 n;

xðiÞ ð0Þ ¼ 0; 0 6 i 6 n  2; Da xð1Þ ¼ 0;

1 6 a 6 n  2;

ð1:4Þ

and the author derived the Green function for the problem (1.4) and showed that it satisfies certain properties. Then by using the cone theoretic techniques the author deduce a general existence theorem for this problem provided that f ðt; xÞ satisfies some growth conditions. Next, a significative work is also developed by Goodrich [4] to study a system of nonlinear differential equations of fractional order having the form of (1.4). Another fractional problem of nonlocal-type similar to (1.4) is treated by Wang et al. [5] through the cone theoretic techniques, where f ðt; xÞ can be singular at x ¼ 0. In recent work [15], M. Rehman and R. Khan investigated the multi-point boundary value problems for fractional differential equations of the form

8 a > < Dt yðtÞ ¼ f ðt; yðtÞ; > : yð0Þ ¼ 0;

Dbt yðtÞÞ; t 2 ð0; 1Þ; m2 P Dbt yð1Þ  fi Dbt yðni Þ ¼ y0 ;

ð1:5Þ

i¼1

P ab1 where 1 < a 6 2, 0 < b < 1, 0 < ni < 1, fi 2 ½0; þ1Þ with m2 < 1. By using the Schauder fixed point theorem and i¼1 fi ni the contraction mapping principle, the authors established the existence and uniqueness of nontrivial solutions for the BVP (1.5) provided that the nonlinear function f : ½0; 1  R  R is continuous and satisfies certain growth conditions. However when the f and boundary condition involve fractional derivatives of the unknown function and the nonlinear term possesses singularity at xi ¼ 0; for i ¼ 1; 2; . . . ; n, fewer results are established on fractional differential equations. Inspired by the above works, the aim of this paper is to establish the existence of positive solutions for the higher nonlocal fractional differential equations (1.1). The present paper has the following features. Firstly, the nonlinear term f involves fractional derivatives of unknown functions; Secondly, the BVP (1.1) possesses singularity, that is, f ðx1 ; x2 ; . . . ; xn Þ may be singular at xi ¼ 0; for i ¼ 1; 2; . . . ; n. In the end, the boundary conditions involving fractional derivatives of the unknown function are more general case, which include two-point,three point, multi-point and some nonlocal problems as special cases. Our technique is also rather different from the ones presented previously such as [1–7,11,13,15,18,22]. The rest of the paper is organized as follows. In Section 2, we present some preliminaries and lemmas on fractional calculus theory, and then give the associated Green function and develop some properties of the Green function, finally, establish a maximal principle. In Section 3, by constructing the upper and lower solutions, we establish an eigenvalue interval for the existence of positive solutions for the BVP (1.1) from Schauder’s fixed point theorem, we also give an example to demonstrate our main results. 2. Basic definitions and preliminaries We first wish to collect some basic lemmas that will be important to us in what follows. These and other related results and their proofs can be found, for example, in [3,4,9,10,18]. Definition 2.1 (see [3,4]). Let a > 0 with a 2 R: Suppose that x : ½a; 1Þ ! R. Then the ath Riemann–Liouville fractional integral is defined to be

Ia xðtÞ ¼

1 CðaÞ

Z

t

ðt  sÞa1 xðsÞ ds;

a

whenever the right-hand side is defined. Similarly, with a > 0 with a 2 R, we define the ath Riemann–Liouville fractional derivative to be

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X. Zhang et al. / Applied Mathematics and Computation 218 (2012) 8526–8536

 ðnÞ Z t 1 d ðt  sÞna1 xðsÞ ds; Cðn  aÞ dt a

Da xðtÞ ¼

where n 2 N is the unique positive integer satisfying n  1 6 a < n and t > a. Remark 2.1. If x; y : ð0; þ1Þ ! R with order a > 0, then

Da ðxðtÞ þ yðtÞÞ ¼ Da xðtÞ þ Da yðtÞ: Lemma 2.1 (see [10]). (1) If x 2 L1 ð0; 1Þ; q > r > 0; then

Iq Ir xðtÞ ¼ Iqþr xðtÞ; Dr Iq xðtÞ ¼ Iqr xðtÞ; Dr Ir xðtÞ ¼ xðtÞ: (2) If q > 0; m > 0; then

Dq t m1 ¼

CðmÞ mq1 : t Cðm  qÞ

Lemma 2.2 (see [8,18]). Assume that x 2 Cð0; 1Þ \ L1 ð0; 1Þ with a fractional derivative of order a > 0, then Ia Da xðtÞ ¼ xðtÞ þ c1 ta1 þ c2 t a2 þ    þ cn tan ; where ci 2 R; i ¼ 1; 2; . . . ; n ðn ¼ ½a þ 1Þ. Here Ia stands for the standard Riemann–Liouville fractional integral of order a > 0 and Da denotes the Riemann–Liouville fractional derivative as Definition 2.1. Let 8

k1 ðt; sÞ ¼

k2 ðt; sÞ ¼


aln1 1 ð1sÞal1 ðtsÞaln1 1

Cðaln1 Þ

; 0 6 s 6 t 6 1;

: ; Cðaln1 Þ 8 al1 ðtsÞal1 ðtð1sÞÞ < ; t aln1 1 ð1sÞal1

0 6 t 6 s 6 1; ð2:1Þ 0 6 s 6 t 6 1;

Cðaln1 Þ

: ðtð1sÞÞal1 ;

0 6 t 6 s 6 1;

Cðaln1 Þ

obviously for t; s 2 ½0; 1, we have

ki ðt; sÞ 6

ð1  sÞal1 ; i ¼ 1; 2: Cða  ln1 Þ

ð2:2Þ

Lemma 2.3.

Pp2 j¼1 aj k2 ðnj ; sÞ  ; GðsÞ ¼ P al1 Cða  ln1 Þ 1  p2 j¼1 aj nj

1þ M¼

1

Pp2 aj Pp2j¼1 al1 j¼1

aj nj

Cða  ln1 Þ

:

If h 2 L1 ½0; 1, then the boundary value problem

8 aln1 wðtÞ ¼ hðtÞ; > < D ll > : wð0Þ ¼ 0; D n1 wð1Þ ¼

p2 P

ð2:3Þ

aj Dlln1 wðnj Þ;

j¼1

has the unique solution

wðtÞ ¼

Z

1

Kðt; sÞhðsÞ ds; 0

where

Kðt; sÞ ¼ k1 ðt; sÞ þ

p2 X taln1 1 aj k2 ðnj ; sÞ; Pp2 al1 1  j¼1 aj nj j¼1

ð2:4Þ

is the Green function of the boundary value problem (2.3). Proof. By applying Lemma 2.2, we may reduce (2.3) to an equivalent integral equation

wðtÞ ¼ Ialn1 hðtÞ þ c1 t aln1 1 þ c2 taln1 2 ;

c1 ; c2 2 R:

ð2:5Þ

X. Zhang et al. / Applied Mathematics and Computation 218 (2012) 8526–8536

8529

From wð0Þ ¼ 0 and (2.5), we have c2 ¼ 0. Consequently the general solution of (2.3) is

wðtÞ ¼ Ialn1 hðtÞ þ c1 t aln1 1 :

ð2:6Þ

By (2.6) and Lemma 2.1, we have

Dlln1 wðtÞ ¼ Dlln1 Ialn1 hðtÞ þ c1 Dlln1 t aln1 1 ¼ Ial hðtÞ þ c1 ¼

Z

Cða  ln1 Þ al1 t Cða  lÞ

ðt  sÞal1 Cða  ln1 Þ al1 : hðsÞds þ c1 t Cða  lÞ Cða  lÞ

t

0

ð2:7Þ

So, from (2.7), we have

Dlln1 wð1Þ ¼ 

Z

1

0

D

lln1

wðnj Þ ¼ 

Z

nj

0

By Dlln1 wð1Þ ¼

R1 c1 ¼

0

Pp2 j¼1

ð1  sÞal1 Cða  ln1 Þ hðsÞds þ c1 ; Cð a  l Þ Cða  lÞ ðnj  sÞal1 Cða  ln1 Þ al1 ; hðsÞds þ c1 n Cða  lÞ Cða  lÞ j

ð2:8Þ for j ¼ 1; 2; . . . ; p  2:

aj Dlln1 wðnj Þ, combining with (2.8), we obtain

R nj P al1 ð1  sÞal1 hðsÞ ds  p2 hðsÞ ds j¼1 aj 0 ðnj  sÞ  : Pp2 al1  Cða  ln1 Þ 1  j¼1 aj nj

So, substituting c1 into (2.6), one has the unique solution of problem (2.3)

Z

ðt  sÞaln1 1 t aln1 1 hðsÞds þ Pp2 al1 Cða  ln1 Þ 1  j¼1 aj nj 0 (Z ) Z nj p2 al1 1 X ðnj  sÞal1 ð1  sÞ  aj hðsÞds  hðsÞds Cða  ln1 Þ 0 0 Cða  ln1 Þ j¼1 Z 1 Z t ðt  sÞaln1 1 ð1  sÞal1 t aln1 1 t aln1 1 hðsÞds þ hðsÞds þ ¼ Pp2 al1 Cða  ln1 Þ Cða  ln1 Þ 1  j¼1 aj n 0 0

wðtÞ ¼ 

t

j

p2 X

aj

¼

0

1

1

al1

ð1  sÞal1 nj

0

j¼1

Z

Z

aln1 1

hðsÞds 

Cða  ln1 Þ

1

t Pp2

p2 X

al1

j¼1 aj nj !

j¼1

Lemma 2.4. The function Kðt; sÞ has the following properties: (1) Kðt; sÞ > 0; for t; s 2 ð0; 1Þ; (2) t aln1 1 GðsÞ 6 Kðt; sÞ 6 Mð1  sÞal1 ; for t; s 2 ½0; 1, where

Pp2

aj k2 ðnj ; sÞ Pp2 al1 ; 1  j¼1 aj nj j¼1

1þ M¼

Pp2 aj Pp2j¼1 al1

1

j¼1

aj nj

Cða  ln1 Þ

:

Proof. It is obvious that (1) holds. In the following, we will prove (2). Firstly, by (2.4),

Kðt; sÞ P

0

nj

p2 X t aln1 1 aj k2 ðnj ; sÞ ¼ t aln1 1 GðsÞ: Pp2 al1 1  j¼1 aj nj j¼1

al1

ðnj  sÞ hðsÞds Cða  ln1 Þ

Z 1 p2 X t aln1 1 k1 ðt; sÞ þ aj k2 ðnj ; sÞ hðsÞds ¼ Kðt; sÞhðsÞds: Pp2 al1 1  j¼1 aj nj 0 j¼1

The proof is completed. h

GðsÞ ¼

aj

Z

ð2:9Þ

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X. Zhang et al. / Applied Mathematics and Computation 218 (2012) 8526–8536

On the other hand, it follows from (2.2) that

Pp2 al1 p2 X taln1 1 ð1  sÞal1 j¼1 aj ð1  sÞ   Kðt; sÞ ¼ k1 ðt; sÞ þ aj k2 ðnj ; sÞ 6 þ Pp2 al1 P Cða  ln1 Þ Cða  l Þ 1  p2 aj nal1 1  j¼1 aj nj j¼1 n1 j¼1 j ! Pp2 ð1  sÞal1 j¼1 aj : 6 1þ Pp2 al1 Cða  ln1 Þ 1  j¼1 aj n j

This completes the proof. h For the convenience of expression in the rest of the paper, we let l0 ¼ 0. Now let us consider the following modified problem of the BVP (1.1)

8 aln1 v ðtÞ ¼ kf ðIln1l0 v ðtÞ; Iln1 l1 v ðtÞ; . . . ; Iln1 ln2 v ðtÞ; v ðtÞÞ; > < D p2 P ll > aj Dlln1 v ðnj Þ: : v ð0Þ ¼ 0; D n1 v ð1Þ ¼

ð2:10Þ

j¼1

l

Lemma 2.5. Let xðtÞ ¼ I n1l0 v ðtÞ; v ðtÞ 2 C½0; 1. Then we can transform (1.1) into (2.10). Moreover, if solution of problem (2.10), then the function xðtÞ ¼ Iln1l0 v ðtÞ is a positive solution of problem (1.1).

v 2 Cð½0; 1; ½0; þ1ÞÞ is a

Proof. Substituting xðtÞ ¼ Iln1l0 v ðtÞ into (1.1), by the definition of Riemann–Liouville fractional derivative and Lemmas 2.1 and 2.2, we can obtain that n

n

n

d na d d xðtÞ ¼ n Ina Iln1l0 v ðtÞ ¼ n Inaþln1 v ðtÞ ¼ Daln1 v ðtÞ; nI dt dt dt Dl1 xðtÞ ¼ Dl1 Iln1l0 v ðtÞ ¼ Iln1 l1 v ðtÞ; Dl2 xðtÞ ¼ Dl2 Iln1l0 v ðtÞ ¼ Iln1 l2 v ðtÞ; ...

Da xðtÞ ¼

ð2:11Þ

Dln2 xðtÞ ¼ Dln2 Iln1l0 v ðtÞ ¼ Iln1 ln2 v ðtÞ; Dln1 xðtÞ ¼ Dln1 Iln1l0 v ðtÞ ¼ v ðtÞ: we have Dln1 xð0Þ ¼ v ð0Þ ¼ 0, it follows from Dl xðtÞ ¼ Dl Iln1l0 v ðtÞ ¼ Dlln1 v ðtÞ that Dlln1 v ð1Þ ¼ lln1 a D v ðnj Þ. Hence, by xðtÞ ¼ Iln1l0 v ðtÞ; v 2 C½0; 1, we can transform (1.1) into (2.10). j¼1 j Now, let v 2 Cð½0; 1; ½0; þ1ÞÞ be a solution for problem (2.10). Then, from the definition of the Riemann–Liouville factional derivative and Lemmas 2.1 and (2.10) and (2.11), one has Also, Pp2

n

n

n

d na d d xðtÞ ¼  n Ina Iln1l0 v ðtÞ ¼  n Inaþln1 v ðtÞ ¼ Daln1 v ðtÞ nI dt dt dt ¼ kf ðIln1l0 v ðtÞ; Iln1 l1 v ðtÞ; . . . ; Iln1 ln2 v ðtÞ; v ðtÞÞ ¼ kf ðxðtÞ; Dl1 xðtÞ; Dl2 xðtÞ; . . . ; Dln1 xðtÞÞ;

Da xðtÞ ¼ 

Noting

Ia v ðtÞ ¼

1 CðaÞ

Z

t

ðt  sÞa1 v ðsÞds;

0

which implies that Ia v ð0Þ ¼ 0; from (2.11), for i ¼ 1; 2; . . . ; n  1; we have

xð0Þ ¼ 0;

Dli xð0Þ ¼ 0;

Dl xð1Þ ¼

p2 X

aj Dl xðnj Þ:

j¼1

Moreover, it follows from the monotonicity and property of Iln1l0 that

Iln1l0 v 2 Cð½0; 1; ½0; þ1ÞÞ: Consequently, xðtÞ ¼ Iln1l0 v ðtÞ is a positive solution of problem (1.1). h Definition 2.2. A continuous function wðtÞ is called a lower solution of the BVP (2.10), if it satisfies

8 aln1 wðtÞ 6 kf ðIln1l0 wðtÞ; Iln1 l1 wðtÞ; . . . ; Iln1 ln2 wðtÞ; wðtÞÞ; > < D p2 P ll > aj Dlln1 wðnj Þ: : wð0Þ P 0; D n1 wð1Þ P j¼1

0 < t < 1:

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X. Zhang et al. / Applied Mathematics and Computation 218 (2012) 8526–8536

Definition 2.3. A continuous function /ðtÞ is called a upper solution of the BVP (2.10), if it satisfies

8 aln1 /ðtÞ P kf ðIln1l0 /ðtÞ; Iln1 l1 /ðtÞ; . . . ; Iln1 ln2 /ðtÞ; /ðtÞÞ; > < D p2 P ll > aj Dlln1 /ðnj Þ: : /ð0Þ 6 0; D n1 /ð1Þ 6 j¼1

Now we make the following assumptions throughout this paper: ðH1 Þ f : ð0; þ1Þn ! ½0; þ1Þ is continuous and is non-increasing in xi > 0 for i ¼ 1; 2; . . . ; n; ðH2 Þ For all r 2 ð0; 1Þ, there exists constant  > 0 such that, for any ðx1 ; x2 ; . . . ; xn Þ 2 ð0; þ1Þn ,

f ðrx1 ; rx2 ; . . . ; rxn Þ 6 r  f ðx1 ; x2 ; . . . ; xn Þ: Remark 2.2. If ðH2 Þ holds, then for r  1 we have the following result: There exists constant ðx1 ; x2 ; . . . ; xn Þ 2 ð0; þ1Þn ,

>0

such that, for any

f ðrx1 ; rx2 ; . . . ; rxn Þ P r f ðx1 ; x2 ; . . . ; xn Þ: In fact, if r  1, then for any ðx1 ; x2 ; . . . ; xn Þ 2 ð0; þ1Þn , we have

    1 1 1 1 f r  x1 ; r  x2 ; . . . ; r  xn 6 f ðrx1 ; rx2 ; . . . ; rxn Þ; r r r r that is

f ðrx1 ; rx2 ; . . . ; rxn Þ P r f ðx1 ; x2 ; . . . ; xn Þ:

Lemma 2.6 (Maximal principle). If

Dlln1 v ð1Þ ¼

v ð0Þ ¼ 0;

v 2 Cð½0; 1; RÞ satisfies

p2 X

aj Dlln1 v ðnj Þ

j¼1

and Daln1 v ðtÞ  0 for any t 2 ½0; 1, then

v ðtÞ  0;

t 2 ½0; 1:

Proof. By Lemma 2.4, the conclusion is obvious, we omit the proof. h

3. Main results Let en1 ðtÞ ¼ t aln1 1 ; and for i ¼ 0; 1; 2; . . . ; n  2, define

ei ðtÞ ¼

Z 0

t

ðt  sÞln1 li 1 saln1 1 Cða  ln1 Þ ali 1 : ds ¼ t Cðln1  li Þ Cða  li Þ

Theorem 3.1. Suppose ðH1 Þ and ðH2 Þ hold, and the following condition is satisfied ðH3 Þ f ð1; 1; . . . ; 1Þ – 0, and

0<

Z

1

ð1  sÞal1 f ðe0 ðsÞ; e1 ðsÞ; . . . ; en2 ðsÞ; en1 ðsÞÞds < þ1:

0

Then there is a constant k > 0 such that for any k 2 ðk ; þ1Þ, the BVP (1.1) has at least one positive solution xðtÞ, and xðtÞ satisfies

e0 ðtÞ 6 xðtÞ 6 Ntln1 ; where

N¼

k ln1 M Cðln1 Þ

Z

1

ð1  sÞal1 f ðe0 ðsÞ; e1 ðsÞ; . . . ; en2 ðsÞ; en1 ðsÞÞds;

0

and M is defined by Lemma 2.4.

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X. Zhang et al. / Applied Mathematics and Computation 218 (2012) 8526–8536

Proof. Let E ¼ C½0; 1, and denote a set P:

P ¼ fv ðtÞ 2 E : there exists a positive number lv such that v ðtÞ P lv en1 ðtÞ; t 2 ½0; 1g:

ð3:1Þ

Clearly, P is a nonempty set since en1 ðtÞ 2 P. Now define the operator T k in E:

ðT k v ÞðtÞ ¼ k

Z

1

Kðt; sÞf ðIln1 v ðsÞ; Iln1 l1 v ðsÞ; . . . ; Iln1 ln2 v ðsÞ; v ðsÞÞds:

ð3:2Þ

0

Obviously T k is well defined and T k ðPÞ P: To see this, for any . 2 P, by the definition of P, there exists a positive number l. 0 0 such that .ðtÞ P l. en1 ðtÞ for any t 2 ½0; 1, take l. ¼ minf12 ; l. g, then .ðtÞ P l. en1 ðtÞ for any t 2 ½0; 1 and 0 < l. < 1. It follows from Lemma 2.4 and ðH2 Þ ðH3 Þ that

ðT k .ÞðtÞ ¼ k

Z

1

0

Kðt; sÞf ðIln1 .ðsÞ; Iln1 l1 .ðsÞ; . . . ; Iln1 ln2 .ðsÞ; .ðsÞÞds

Z

1

6 kM 0 

ð1  sÞal1 f ðl. e0 ðsÞ; l. e1 ðsÞ; . . . ; l. en2 ðsÞ; l. en1 ðsÞÞds

Z

6 kMl.

1

ð1  sÞal1 f ðe0 ðsÞ; e1 ðsÞ; . . . ; en2 ðsÞ; en1 ðsÞÞds < þ1:

ð3:3Þ

0

Let B ¼ maxf2; maxt2½0;1 .ðtÞg, for i ¼ 0; 1; 2; . . . ; n  2,

Z

Iln1 li .ðtÞ ¼

t

0

ðt  sÞln1 li 1 .ðsÞ B ds 6 6 B; Cðln1  li Þ Cðln1  li Þ

then from ðH3 Þ, we have

ðT k .ÞðtÞ ¼ k

Z

1

Kðt; sÞf ðIln1 .ðsÞ; Iln1 l1 .ðsÞ; . . . ; Iln1 ln2 .ðsÞ; .ðsÞÞds

0

P ktaln1 1

Z

1

0

P kB taln1 1

GðsÞf ðB; B; . . . ; B; BÞds Z

1

GðsÞf ð1; 1; . . . ; 1; 1Þds

0

¼ kB f ð1; 1; . . . ; 1; 1Þ

Z

1

GðsÞdst

aln1 1

¼ kAB f ð1; 1; . . . ; 1; 1Þen1 ðtÞ;

ð3:4Þ

0

R1 where A ¼ 0 GðsÞds. (3.3) and (3.4) imply T k is well defined and T k ðPÞ P: Next we shall attempt to find the upper and lower solutions of the BVP (2.10). By ðH2 Þ and (3.2), we know the f is decreasing on xi for i ¼ 1; 2; . . . ; n. It follows

Z

1

Kðt; sÞf ðe0 ðsÞ; e1 ðsÞ; . . . ; en2 ðsÞ; en1 ðsÞÞds P t aln1 1

Z

0

that

1

GðsÞf ðe0 ðsÞ; e1 ðsÞ; . . . ; en2 ðsÞ; en1 ðsÞÞds;

8 t 2 ½0; 1;

0

Z

1

k1

Kðt; sÞf ðe0 ðsÞ; e1 ðsÞ; . . . ; en2 ðsÞ; en1 ðsÞÞds P t aln1 1 ;

8t 2 ½0; 1;

0

where

k1 ¼ R 1 0

1 GðsÞf ðe0 ðsÞ; e1 ðsÞ; . . . ; en2 ðsÞ; en1 ðsÞÞds

On the other hand, let bðtÞ ¼ for any k > k1 , by ðH3 Þ, we have

Z

1

R1 0

:

Kðt; sÞf ðe0 ðsÞ; e1 ðsÞ; . . . ; en2 ðsÞ; en1 ðsÞÞds; since f ðx1 ; x2 ; . . . ; xn1 ; xn Þ is decreasing in xi > 0,

Kðt; sÞf ðkIln1 bðsÞ; kIln1 l1 bðsÞ; . . . ; kIln1 ln2 bðsÞ; kbðsÞÞds

0

Z

1

6 0

6M

Kðt; sÞf ðk1 Iln1 bðsÞ; k1 Iln1 l1 bðsÞ; . . . ; k1 Iln1 ln2 bðsÞ; k1 bðsÞÞds 6

Z

Z 0

1

ð1  sÞal1 f ðe0 ðsÞ; e1 ðsÞ; . . . ; en2 ðsÞ; en1 ðsÞÞds < þ1:

0

Let C ¼ maxf2; maxt2½0;1 bðtÞg, for i ¼ 0; 1; 2; . . . ; n  2; we have

1

Kðt; sÞf ðe0 ðsÞ; e1 ðsÞ; . . . ; en2 ðsÞ; en1 ðsÞÞds

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X. Zhang et al. / Applied Mathematics and Computation 218 (2012) 8526–8536

C

Iln1 li bðtÞ 6

Cðln1  li Þ

6 C:

Now take

( k > max 1; k1 ;

1þ1 ) C : Af ð1; 1; . . . ; 1; 1Þ



By ðH2 Þ, for any t 2 ½0; 1, we have

k f ðk Iln1 bðsÞ; k Iln1 l1 bðsÞ; . . . ; k Iln1 ln2 bðsÞ; k bðsÞÞ P ðk Þþ1 f ðC; C; . . . ; C; CÞ P ðk Þþ1 C  f ð1; 1; . . . ; 1; 1Þ P A1 : Thus by Lemma 2.4, we obtain

k

Z

1

Kðt; sÞf ðk Iln1 bðsÞ; k Iln1 l1 bðsÞ; . . . ; k Iln1 ln2 bðsÞ; k bðsÞÞ P A1

0

Z

1

taln1 1 GðsÞds ¼ A1 t aln1 1

0

¼ t aln1 1 ; Let

/ðtÞ ¼ k

Z

Z

1

GðsÞds

0

8t 2 ½0; 1:

1

Kðt; sÞf ðe0 ðsÞ; e1 ðsÞ; . . . ; en2 ðsÞ; en1 ðsÞÞ; wðtÞ

0

¼ k

Z

1

Kðt; sÞf ðk Iln1 bðsÞ; k Iln1 l1 bðsÞ; . . . ; k Iln1 ln2 bðsÞ; k bðsÞÞds;

0

then by Lemma 2.3, for any t 2 ½0; 1, we have

8 R  1 aln1 1 ; > < /ðtÞ ¼ k 0 Kðt; sÞf ðe0 ðsÞ; e1 ðsÞ; . . . ; en2 ðsÞ; en1 ðsÞÞds P t p2 P ll > aj Dlln1 /ðnj Þ; : /ð0Þ ¼ 0; D n1 /ð1Þ ¼

ð3:5Þ

j¼1

8 R  1  ln1 bðsÞ; k Iln1 l1 bðsÞ; . . . ; k Iln1 ln2 bðsÞ; k bðsÞÞds P t aln1 1 ; > < wðtÞ ¼ k 0 Kðt; sÞf ðk I p2 P ll > aj Dlln1 wðnj Þ: : wð0Þ ¼ 0; D n1 wð1Þ ¼

t 2 ½0; 1; ð3:6Þ

j¼1

Obviously, /ðtÞ; wðtÞ 2 P. By (3.5) and (3.6), we have

taln1 1 6 wðtÞ ¼ ðT k /ÞðtÞ; which implies

wðtÞ ¼ ðT k /ÞðtÞ ¼ k 6 k

Z

Z

1

taln1 1 6 /ðtÞ;

8t 2 ½0; 1;

ð3:7Þ

Kðt; sÞf ðIln1 /ðsÞ; Iln1 l1 /ðsÞ; . . . ; Iln1 ln2 /ðsÞ; /ðsÞÞds

0 1

Kðt; sÞf ðe0 ðsÞ; e1 ðsÞ; . . . ; en2 ðsÞ; en1 ðsÞÞds ¼ /ðtÞ;

8t 2 ½0; 1:

ð3:8Þ

0

Thus, by (3.7) and (3.8), we have

Daln1 wðtÞ þ k f ðIln1l0 wðtÞ; Iln1 l1 wðtÞ; . . . ; Iln1 ln2 wðtÞ; wðtÞÞ P Daln1 ðT k /ÞðtÞ þ k f ðIln1l0 /ðtÞ; Iln1 l1 /ðtÞ; . . . ; Iln1 ln2 /ðtÞ; /ðtÞÞ ¼ k f ðIln1l0 /ðtÞ; Iln1 l1 /ðtÞ; . . . ; Iln1 ln2 /ðtÞ; /ðtÞÞ þ k f ðIln1l0 /ðtÞ; Iln1 l1 /ðtÞ; . . . ; Iln1 ln2 /ðtÞ; /ðtÞÞ ¼ 0;

ð3:9Þ

Daln1 /ðtÞ þ k f ðIln1l0 /ðtÞ; Iln1 l1 /ðtÞ; . . . ; Iln1 ln2 /ðtÞ; /ðtÞÞ ¼ k f ðe0 ðtÞ; e1 ðtÞ; . . . ; en2 ðtÞ; en1 ðtÞÞ þ k f ðIln1l0 /ðtÞ; Iln1 l1 /ðtÞ; . . . ; Iln1 ln2 /ðtÞ; /ðtÞÞ 6 k f ðe0 ðtÞ; e1 ðtÞ; . . . ; en2 ðtÞ; en1 ðtÞÞ þ k f ðe0 ðtÞ; e1 ðtÞ; . . . ; en2 ðtÞ; en1 ðtÞÞ ¼ 0:

ð3:10Þ

It is obvious that /; w satisfy the boundary conditions of the BVP (2.10). Thus it follows from (3.8)–(3.10) that wðtÞ; /ðtÞ are the upper and lower solutions of the BVP (2.10), and wðtÞ; /ðtÞ 2 P. Define the function F and the operator Ak in E by

8 l n1l0 wðtÞ; Iln1 l1 wðtÞ; . . . ; Iln1 ln2 wðtÞ; wðtÞÞ; u < wðtÞ; > < f ðI ln1l0 uðtÞ; Iln1 l1 uðtÞ; . . . ; Iln1 ln2 uðtÞ; uðtÞÞ; wðtÞ 6 u 6 /ðtÞ; FðuÞ ¼ f ðI > : n1l0 /ðtÞ; Iln1 l1 /ðtÞ; . . . ; Iln1 ln2 /ðtÞ; /ðtÞÞ; u > /ðtÞ f ðI

ð3:11Þ

8534

X. Zhang et al. / Applied Mathematics and Computation 218 (2012) 8526–8536

and

ðAk uÞðtÞ ¼ k

Z

1

8u 2 E:

Kðt; sÞFðuðsÞÞds; 0

Clearly, F : ½0; þ1Þ ! ½0; þ1Þ is continuous by (3.11). Consider the following boundary value problem

8 aln1 uðtÞ ¼ k FðuÞ; > < D

0 < t < 1; p2 P Dlln1 uð1Þ ¼ aj Dlln1 uðnj Þ:

> : uð0Þ ¼ 0;

ð3:12Þ

j¼1

Obviously, a fixed point of the operator Ak is a solution of the BVP (3.12). For all u 2 E, it follows from Lemma 2.4 and (3.11) and wðtÞ P t aln1 1 that

ðAk uÞðtÞ 6 k M

Z

1

sð1  sÞal1 FðuðsÞÞds 6 k M

Z

0

6 k M

Z

1

ð1  sÞal1 f ðIln1l0 wðsÞ; Iln1 l1 wðsÞ; . . . ; Iln1 ln2 wðsÞ; wðsÞÞds

0 1

ð1  sÞal1 f ðe0 ðsÞ; e1 ðsÞ; . . . ; en2 ðsÞ; en1 ðsÞÞds < þ1:

0

So Ak is bounded. It is easy to see Ak : E ! E is continuous from the continuity of FðuÞ and Kðt; sÞ. On the other hand, let X E be bounded, by using the fact that the function Kðt; sÞ is uniformly continuous on [0, 1], it is easy to see Ak ðXÞ is equicontinuous. Thus by the means of the Arzela–Ascoli theorem, we have Ak : E ! E is completely continuous. Thus, by using the Schauder fixed point theorem, Ak has at least one fixed point w such that w ¼ Ak w. Now we prove

wðtÞ 6 wðtÞ 6 /ðtÞ;

t 2 ½0; 1:

Let zðtÞ ¼ /ðtÞ  wðtÞ; t 2 ½0; 1. By /ðtÞ is the upper solution of the BVP (2.10) and w is a fixed point of Ak , we know

Dlln1 wð1Þ ¼

wð0Þ ¼ 0;

p2 X

aj Dlln1 wðnj Þ:

ð3:13Þ

j¼1

From the definition of F, (3.7) and (3.8), we obtain

f ðIn1l0 /ðtÞ; Iln1 l1 /ðtÞ; . . . ; Iln1 ln2 /ðtÞ; /ðtÞÞ 6 FðuðtÞÞ 6 f ðIln1l0 wðtÞ; Iln1 l1 wðtÞ; . . . ; Iln1 ln2 wðtÞ; wðtÞÞ 6 f ðe0 ðtÞ; e1 ðtÞ; . . . ; en2 ðtÞ; en1 ðtÞÞ; 8u 2 E; 8t 2 ½0; 1:

ð3:14Þ

Thus (3.5) and (3.14) imply

Daln1 zðtÞ ¼ Daln1 /ðtÞ  Daln1 wðtÞ ¼ k f ðe0 ðtÞ; e1 ðtÞ; . . . ; en2 ðtÞ; en1 ðtÞÞ þ k FðwðtÞÞ 6 0:

ð3:15Þ

aln1

It follows from (3.15) that D zðtÞ  0, by Lemma 2.6, we have zðtÞ P 0 which implies wðtÞ 6 /ðtÞ on ½0; 1: By the same way, it is easy to prove wðtÞ P wðtÞ on ½0; 1: So we obtain

wðtÞ 6 wðtÞ 6 /ðtÞ;

t 2 ½0; 1: ln1 l0

Consequently, FðwðtÞÞ ¼ f ðI BVP (2.10). Finally, by (3.16), we know

wðtÞ; I

ð3:16Þ ln1 l1

wðtÞ; . . . ; I

ln1 ln2

wðtÞ; wðtÞÞ; t 2 ½0; 1: Then wðtÞ is a positive solution of the

wðtÞ P wðtÞ P taln1 1 ¼ en1 ðtÞ:

ð3:17Þ

On the other hand, it follows from (3.17) and Lemma 2.4 that

wðtÞ ¼ k

Z

1

Kðt; sÞf ðIln1 l0 wðsÞ; Iln1 l1 wðsÞ; . . . ; Iln1 ln2 wðsÞ; wðsÞÞds

0

6 k

Z

1

Kðt; sÞf ðe0 ðsÞ; e1 ðsÞ; . . . ; en2 ðsÞ; en1 ðsÞÞds

0

6 k M

Z

1

ð1  sÞal1 f ðe0 ðsÞ; e1 ðsÞ; . . . ; en2 ðsÞ; en1 ðsÞÞds ¼ N0 ;

0

where

N 0 ¼ k M

Z

1 0

ð1  sÞal1 f ðe0 ðsÞ; e1 ðsÞ; . . . ; en2 ðsÞ; en1 ðsÞÞds:

ð3:18Þ

X. Zhang et al. / Applied Mathematics and Computation 218 (2012) 8526–8536

8535

(3.17) and (3.18) yield

taln1 1 6 wðtÞ 6 N0 : Consequently,

Z

e0 ðtÞ 6 xðtÞ ¼ Iln1 wðtÞ 6 N 0

ðt  sÞln1 1 l N0 ln1 ¼ Ntln1 ; ds ¼ n1 t Cðln1 Þ Cðln1 Þ

t

0

where N ¼

ln1 N0 Cðln1 Þ :

h

If we adopt a stronger condition instead of ðH3 Þ, then we have the following nice result. Corollary 3.1. Suppose that conditions ðH1 ÞðH2 Þ hold. Furthermore, suppose f satisfies the following condition: ðH03 Þ f ð1; 1; . . . ; 1Þ – 0, and

0<

Z

1

f ðe0 ðsÞ; e1 ðsÞ; . . . ; en2 ðsÞ; en1 ðsÞÞds < þ1:

0

Then there is a constant k > 0 such that for any k 2 ðk ; þ1Þ, the BVP (1.1) has at least one positive solution xðtÞ, and xðtÞ satisfies

e0 ðtÞ 6 xðtÞ 6 Ntln1 ; where

N¼

k ln1 M Cðln1 Þ

Z

1

f ðe0 ðsÞ; e1 ðsÞ; . . . ; en2 ðsÞ; en1 ðsÞÞds:

0

An example consider the following problem

8   1  1  > < D52 xðtÞ þ k x29 ðtÞ þ D98 xðtÞ 2 þ D54 xðtÞ 8 ¼ 0; > :

9

5

11

xð0Þ ¼ D8 xð0Þ ¼ D4 xð0Þ ¼ 0;

D 8 xð1Þ ¼

0 < t < 1;

ð3:19Þ

pffiffi 11  2 D 8 12 : 2

Proof. Let 2

1

1

f ðx1 ; x2 ; x3 Þ ¼ x1 9 þ x2 2 þ x3 8 ; then ðH1 Þ holds, and for all r 2 ð0; 1Þ and for any ðx1 ; x2 ; x3 Þ 2 ð0; þ1Þ3 , 2

2

1

1

1

1

1

f ðrx1 ; rx2 ; . . . ; rxn Þ ¼ r 9 x1 9 þ r 2 x2 2 þ r8 x3 8 6 r2 f ðx1 ; x2 ; x3 Þ; which implies that ðH2 Þ also holds. 1 On the other hand, by direct calculation, we have f ð1; 1; 1Þ ¼ 3 – 0, e2 ðtÞ ¼ t4 ; and then

e0 ðtÞ ¼

Z

t

0

e1 ðtÞ ¼

Z 0

t

1 1

ðt  sÞ4 s4 1 ds P 5 Cð54Þ Cð4Þ 7 1

ðt  sÞ8 s4 ds P Cð18Þ

Z

t

0

Z

t

1

ðt  sÞ4 sds ¼

0

Z

1 Cð54Þ

7

ðt  sÞ8 s 1 ds ¼ 1 Cð18Þ Cð8Þ

t

9

1

s4 ðt  sÞds ¼

0

Z

t

16 9 t4 t4 P ; 45Cð54Þ 3Cð54Þ 9

7

s8 ðt  sÞds ¼

0

9

64t 8 t8 P 1 ; 9Cð18Þ Cð8Þ

thus

0<

Z

1

1 8

ð1  sÞ f ðe0 ðsÞ; e1 ðsÞ; e2 ðsÞÞds 6

Z

0

Z 6 0

0 1

2 4

1 3Cð54Þ

!29 1

1

s2 ð1  sÞ8 þ

1

1

2

!12

Cð18Þ

9

s4 ð1  sÞ 4 3Cð54Þ 1 8

9

1

!29

1

3 !12 9  1 18 s8 þ þ s4 5ds Cð18Þ 3 1

s16 ð1  sÞ8 þ s32 ð1  sÞ8 5ds < þ1:

Hence ðH3 Þ holds. Then by Theorem 3.1 there is a constant k > 0 such that for any k 2 ðk ; þ1Þ the BVP (3.19) has at least one positive solution xðtÞ , and there exists a constant N > 0 such that 1

5

t4 6 xðtÞ 6 Nt4 : 

8536

X. Zhang et al. / Applied Mathematics and Computation 218 (2012) 8526–8536

Acknowledgment The authors thank the (anonymous) referees for their careful reading of the manuscript and suggestions for its improvement, and especially thank to the referee who found an error in the proof of Lemma 2.3. References [1] Z. Bai, On positive solutions of a nonlocal fractional boundary value problem, Nonlinear Anal. 72 (2010) 916–924. [2] S. Zhang, Positive solutions to singular boundary value problem for nonlinear fractional differential equation, Comput. Math. Appl. 59 (2010) 1300– 1309. [3] C.S. Goodrich, Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett. 23 (2010) 1050–1055. [4] C.S. Goodrich, Existence of a positive solution to systems of differential equations of fractional order, Comput. Math. Appl. 62 (2011) 1251–1268. [5] Y. Wang, L. Liu, Y. Wu, Positive solutions for a nonlocal fractional differential equation, Nonlinear Anal. 74 (2011) 3599–3605. [6] B. Ahmad, A. Alsaedi, Existence and uniqueness of solutions for coupled systems of higher-order nonlinear fractional differential equations, Fixed Point Theory Appl. 2010 (2010) (Article ID 364560, 17 pages). [7] B. Ahmad, J. Nieto, Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations, Abstr. Appl. Anal. 2009 (2009) (Article ID 494720, 9 pages). [8] K. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. [9] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, London, Toronto, 1999. [10] A. Kilbas, H. Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier B.V., Netherlands, 2006. [11] M. Feng, X. Zhang, W. Ge, New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions, Bound. Value Probl. 2011 (2011) (Article ID 720702, 20 pages). [12] M. El-Shahed, J. Nieto, Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order, Comput. Math. Appl. 59 (2010) 3438–3443. [13] B. Ahmad, J. Nieto, Existence results for higher order fractional differential inclusions with nonlocal boundary conditions, Nonlinear Stud. 17 (2010) 131–138. [14] A. Hussein Salem, On the fractional order m-point boundary value problem in reflexive Banach spaces and weak topologies, J. Comput. Appl. Math. 224 (2009) 565–572. [15] M. Rehman, R. Khan, Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations, Appl. Math. Lett. 23 (2010) 1038–1044. [16] V. Lakshmikantham, A. Vatsala, General uniqueness and monotone iterative technique for fractional differential equations, Appl. Math. Lett. 21 (2008) 828–834. [17] T. Bhaskar, V. Lakshmikantham, S. Leela, Fractional differential equations with a Krasnoselskii–Krein type condition, Nonlinear Anal. 3 (2009) 734–737. [18] Z. Bai, H. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. 311 (2005) 495–505. [19] X. Zhang, Y. Han, Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations, Appl. Math. Lett. 25 (2012) 555–560. [20] B. Ahmad, A. Alsaedi, Existence of solutions for anti-periodic boundary value problems of nonlinear impulsive functional integro-differential equations of mixed type, Nonlinear Anal. Hybrid Syst. 3 (2009) 501–509. [21] M. Benchohraa, S. Hamania, S. Ntouyas, Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Anal. 71 (2009) 2391–2396. [22] X. Zhang, L. Liu, Y. Wu, Multiple positive solutions of a singular fractional differential equation with negatively perturbed term, Math. Comput. Modelling 55 (2012) 1263–1274.