Singular perturbations of third-order nonlinear differential equations with full nonlinear boundary conditions

Applied Mathematics and Computation 224 (2013) 88–95

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

Singular perturbations of third-order nonlinear differential equations with full nonlinear boundary conditions q Xiaojie Lin School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, China

a r t i c l e

i n f o

a b s t r a c t In this paper, we discuss singular perturbations of third-order nonlinear ordinary differential equations with full nonlinear boundary conditions. The emphasis here is that the nonlinear term depends on the first, second order derivatives and the boundary conditions are full nonlinear that is where the main novelty of this work lies. By applying the upper and lower solutions method, as well as analysis technique, the existence, uniqueness results for the singularly perturbed boundary value problem are established and asymptotic estimates of solutions is also obtained. Ó 2013 Elsevier Inc. All rights reserved.

Keywords: Singular perturbations Full nonlinear problems Asymptotic estimates Upper and lower solutions Leray–Schauder degree

1. Introduction Singularly perturbed boundary value problems (BVP, for short) arise very frequently in fluid mechanics and other branches of applied mathematics. These problems depend on a small positive parameter in such a way that the solution varies rapidly in some parts and varies slowly in some other parts. There are many results on the existence and asymptotic estimates of solutions for third order singularly perturbed boundary value problems [1,4–6,8–11] and therein. Many techniques arose in the studies of this kind of problems. For example, Howes [6] has considered problems of type

e2 y000 ¼ f ðyÞy0 þ gðx; yÞ; yðaÞ ¼ A;

yðbÞ ¼ B;

y0 ðbÞ ¼ C;

and discussed the existence and asymptotic estimates of the solutions by the method of descent. Zhao [10] has discussed a more general class of a third order singularly perturbed boundary value problems of the form

ey000 ¼ f ðx; y; y0 ; eÞ; y0 ð0Þ ¼ 0;

yð1Þ ¼ 0;

y0 ð1Þ ¼ 0;

and discussed the existence of solution and obtained asymptotic estimates using the theory of differential inequalities. Feckon [5] has studied high order problems and his approach was based on the nonlinear analysis involving fixed-point theory, Leray–Schauder theory. Valarmathi and Ramanujam [9] have considered singularly perturbed third-order ordinary differential equations of Convection–Diffusion type by using of an asymptotic numerical method. Du et al. [4] were concerned with the existence, uniqueness and asymptotic estimates of solutions of third order multi-point singularly perturbed boundary value problems were give by employing priori estimates, differential inequalities technique and Leray–Schauder degree theory. Lin [8] studied a two-point singularly perturbed boundary value problem to a class of nonlinear vector third order q This work is supported by the Natural Science Foundation of China (Grant No. 11071205 and 11101349), and partially supported by the Natural Science Foundation of Jiangsu Province (Grant No. BK2011042 and 11KJB110013) and PAPD of Jiangsu Higher Education Institutions. E-mail address: [email protected]

0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.08.048

X. Lin / Applied Mathematics and Computation 224 (2013) 88–95

89

integro-differential equation and proved the existence of the perturbed solution, also obtained the uniformly valid asymptotic expansion by employing the method of differential inequalities. However, the boundary conditions in the above-mentioned singularly perturbed boundary value problems for third-order nonlinear differential equation are all linear. Third order boundary value problems with nonlinear boundary conditions have been studied recently by several authors, see for example the papers [2,3,7]. Du et al. [2,3] have established some existence results for the third order nonlinear scalar and vector full boundary value problem, respectively. Lin et al. [7] discussed the existence and uniqueness results of the following nonlinear multi-point boundary value problem by using the upper and lower solutions method, as well as degree theory

x000 ðtÞ þ f ðt; xðtÞ; x0 ðtÞ; x00 ðtÞÞ ¼ 0;

0 < t < 1;

ð1Þ

xð0Þ ¼ 0; gðx0 ð0Þ; x00 ð0Þ; xðn1 Þ; xðn2 Þ; . . . ; xðnm2 ÞÞ ¼ A;

ð2Þ

hðx0 ð1Þ; x00 ð1Þ; xðg1 Þ; xðg2 Þ; . . . ; xðgn2 ÞÞ ¼ B: By so far, very few results were established for third order nonlinear singularly perturbed boundary value problems with nonlinear boundary conditions. Motivated by the above works, the purpose of this article is to study the singular perturbations of the following third-order nonlinear ordinary differential equations

ex000 ðtÞ þ f ðt; xðtÞ; x0 ðtÞ; x00 ðtÞ; eÞ ¼ 0; 0 6 t 6 1; 0 < e  1;

ð3Þ

with full nonlinear boundary conditions

xð0; eÞ ¼ 0; gðx0 ð0; eÞ; x00 ð0; eÞ; xðn1 ; eÞ; xðn2 ; eÞ; . . . ; xðnm2 ; eÞÞ ¼ A;

ð4Þ

hðx0 ð1; eÞ; x00 ð1; eÞ; xðg1 ; eÞ; xðg2 ; eÞ; . . . ; xðgn2 ; eÞÞ ¼ B; where 0 < n1 < n2 <    < nm2 < 1, and 0 < g1 < g2 <    < gn2 < 1; A; B 2 R; f : ½0; 1  R3 ! R is continuous, g : Rm ! R; h : Rn ! R are continuous. In this paper, we discuss the existence, uniqueness and asymptotic estimates of solutions for the singularly perturbed boundary value problem (3) and (4). The emphases here is to construct appropriate upper and lower solutions BVP (3) and (4) not to assume that BVP (3) and (4) exist upper and lower solutions. The remaining part of this paper is organized as follows: in Section 2, we briefly present some definitions and lemmas which are important to obtain our main results. In Section 3, we obtain the existence and asymptotic estimates of solutions of singularly perturbed BVP (3) and (4) by constructing appropriate upper and lower solutions. We also establish the uniqueness result of BVP (3) and (4). 2. Preliminaries

Definition 1 [7]. Let

s : C½0; 1 ! Rm2 ; q : C½0; 1 ! Rn2 be defined by

sz ¼ ðzðn1 Þ; . . . ; zðnm2 ÞÞ; qz ¼ ðzðg1 Þ; . . . ; zðgn2 ÞÞ:

ð5Þ

Particularly, for constant M, we denote

sM ¼ ðM; M; . . . ; MÞ 2 Rm2 ; qM ¼ ðM; M; . . . ; MÞ 2 Rn2 : Definition 2 [7]. A function uðtÞ 2 C 3 ½0; 1 is called a lower solution of BVP (1) and (2), if

u000 ðtÞ þ f ðt; uðtÞ; u0 ðtÞ; u00 ðtÞÞ P 0;

0 6 t 6 1;

ð6Þ

and

uð0Þ ¼ 0; gðu0 ð0Þ; u00 ð0Þ; suÞ 6 A; 0

ð7Þ

00

hðu ð1Þ; u ð1Þ; quÞ 6 B: Similarly, a function v ðtÞ 2 C 3 ½0; 1 is called an upper solution of BVP (1) and (2), if

v 000 ðtÞ þ f ðt; v ðtÞ; v 0 ðtÞ; v 00 ðtÞÞ 6 0; and

0 6 t 6 1;

ð8Þ

90

X. Lin / Applied Mathematics and Computation 224 (2013) 88–95

v ð0Þ ¼ 0; gðv 0 ð0Þ; v 00 ð0Þ; sv Þ P A; hðv 0 ð1Þ; v 00 ð1Þ; qv Þ P B:

ð9Þ

Definition 3 ([7,11]). Let D be a subset of ½0; 1  R3 , we say that f ðt; x; y; zÞ satisfies Nagumo condition in D provided that f is continuous and given any a > 0, there exists a positive function U : ½0; 1Þ ! ½a; þ1Þ, for arbitrary ðt; x; y; zÞ 2 D, such that

jf ðt; x; y; zÞj 6 UðjzjÞ; and

Z 0

þ1

s

UðsÞ

ð10Þ

ds ¼ þ1:

ð11Þ

To study third-order singularly perturbed boundary value problem (3) and (4), we need the existence result of boundary value problem (1) and (2) and Lemma 1 [7]. Assume that (A1) BVP (1), (2) has lower and upper solutions uðtÞ; v ðtÞ, respectively, such that u0 ðtÞ 6 v 0 ðtÞ; t 2 ½0; 1; (A2) f ðt; x; y; zÞ is continuous and increasing with respect to x, for ðt; x; y; zÞ 2 ½0; 1  ½uðtÞ; v ðtÞ  R2 , and f ðt; x; y; zÞ satisfies Nagumo condition in ½0; 1  ½uðtÞ; v ðtÞ  ½u0 ðtÞ; v 0 ðtÞ  R; (A3) gðx1 ; x2 ; . . . ; xm Þ is continuous and decreasing with respect to x2 ; . . . ; xm ; hðy1 ; y2 ; . . . ; yn Þ is continuous and increasing in y2 and decreasing in y3 ; . . . ; yn . Then BVP (1), (2) has at least one solution xðtÞ 2 C 3 ½0; 1 such that

uðtÞ 6 xðtÞ 6 v ðtÞ; u0 ðtÞ 6 x0 ðtÞ 6 v 0 ðtÞ;

t 2 ½0; 1:

Similar the proof of Theorem 3 in [4], we have the following Lemma 2. Lemma 2. Assume that aðt; eÞ; bðt; eÞ; cðt; eÞ 2 Cð½0; 1  ½0; e0 Þ; cðt; eÞ P 0; ðt; eÞ 2 ½0; 1  ½0; e0 , constants p1 ; p2 ; q1 ; q2 satisfy q1 6 0; q2 P 0; r i 6 0 ði ¼ 1; . . . ; m  2Þ; dj 6 0 ðj ¼ 1; . . . ; n  2Þ, and there exists v ðt; eÞ 2 C 3 ð½0; 1  ½0; e0 Þ, such that v 0 ðt; eÞ > 0 and

ev 000 ðt; eÞ þ aðt; eÞv 00 ðt; eÞ þ bðt; eÞv 0 ðt; eÞ þ cðt; eÞv ðt; eÞ < 0; 0 6 t 6 1; 8 v ð0; eÞ P 0; > > > > m 2 > X > > < p1 v 0 ð0; eÞ þ q1 bv 00 ð0; eÞ þ r i v ðni ; eÞ > 0; i¼1

> > n2 > X > > 0 00 > p v ð1; e Þ þ q v ð1; e Þ þ dj v ðgj ; eÞ > 0: > 2 2 :

ð12Þ

ð13Þ

j¼1

Then the following singularly perturbed boundary value problem

ex000 ðt; eÞ þ aðt; eÞx00 ðt; eÞ þ bðt; eÞx0 ðt; eÞ þ cðt; eÞxðt; eÞ ¼ 0; 0 6 t 6 1; 8 xð0; eÞ ¼ 0; > > > > m2 > X > > < p1 x0 ð0; eÞ þ q1 x00 ð0; eÞ þ r i xðni ; eÞ ¼ 0; i¼1

> > n2 > X > > 0 00 > dj xðgj ; eÞ ¼ 0; > : p2 x ð1; eÞ þ q2 x ð1; eÞ þ j¼1

has only a zero solution.

3. Main results Theorem 1. Assume that

ð14Þ

ð15Þ

X. Lin / Applied Mathematics and Computation 224 (2013) 88–95

91

(H1) The reduced problem of BVP (3), (4) (i.e. f ðt; x; x0 ; x00 ; 0Þ ¼ 0; xð0Þ ¼ 0; gðx0 ð0Þ; x00 ð0Þ; xðn1 Þ; xðn2 Þ; . . . ; xðnm2 ÞÞ ¼ A) has a reduced solution x0 ðtÞ 2 C 3 ½0; 1; (H2) Let e0 be a constant, f ðt; x; y; z; eÞ is continuously differentiable and satisfies Nagumo condition on ½0; 1  R3  ½0; e0  and there exist some positive constants l; N; M, such that

0<

@f ðt; x; y; z; eÞ 6 l; @x

@f ðt; x; y; z; eÞ 6 0; @z

j

@f ðt; x; y; z; eÞ 6 N < 0; @y

@f ðt; x; y; z; eÞj 6 M; @e

(H3) gðx1 ; x2 ; . . . ; xm Þ is continuous and decreasing with respect to x2 ; . . . ; xm ; hðy1 ; y2 ; . . . ; yn Þ is continuous and increasing in y2 and decreasing in y3 ; . . . ; yn . And there exist positive numbers M i ði ¼ 1; . . . ; 6Þ, such that x000 ð0Þ < M 1 ; x000 ð1Þ > M 2 , and

gðx00 ð0Þ; M1 ; sM 5 Þ 6 A 6 gðx00 ð0Þ; M 1 ; sM 3 Þ;

ð16Þ

hðx00 ð1Þ; M 2 ; qM 6 Þ 6 B 6 hðx00 ð1Þ; M 2 ; qM 4 Þ:

ð17Þ

Then for sufficiently small

e > 0, BVP (3), (4) has a solution xðt; eÞ such that

jxðt; eÞ  x0 ðtÞj 6 D1 ek1 t þ D2 ek2 ðt1Þ þ D3 e;

ð18Þ

3

where k1 ; k2 are two roots of equation ek  Nk þ l ¼ 0, such that

2

rffiffiffiffi N

e

rffiffiffiffi N ;

< k1 < 

e

1 2

rffiffiffiffi N

e

< k2 <

rffiffiffiffi N ;

ð19Þ

e

here D1 ; D2 ; D3 are three positive numbers.  Proof of Theorem 1. From condition ðH1Þ, there exists a constant M  > 0, such that jx000 0 ðtÞj 6 M . In the following, we assume 3 sufficiently small e > 0. Then the equation ek  Nk þ l ¼ 0 has three different real roots k1 ; k2 and k3 , since

!  2  3 2 1 l 1 N 1 l N3 < 0: þ  ¼ 2  4 e 27 e e 4 27e Furthermore, the estimates of k1 ; k2 are given in (19) and the estimate of k3 satisfies

l lþN < k3 < : N N

ð20Þ

Let 1

cðt; eÞ ¼ e2



  d1 k1 t d2 k2 ðt1Þ d3  k3 t þ e þ e 2e  1 ; k1 k2 k3

where

d1 ¼ 

M 1 þ jx000 ð0Þj þ 1 1 4

k1 e

;

d2 ¼

M 2 þ jx000 ð1Þj þ 1 1 2

k2 e

;

d3 ¼

k3 ð1 þ M þ M  Þ 1 e5 : l

From (20), one has 1





c0 ðt; eÞ ¼ e2 d1 ek1 t þ d2 ek2 ðt1Þ þ 2d3 ek3 t ; 1









c00 ðt; eÞ ¼ e2 d1 k1 ek1 t þ d2 k2 ek2 ðt1Þ þ 2d3 k3 ek3 t ; 1

c000 ðt; eÞ ¼ e2 d1 k21 ek1 t þ d2 k22 ek2 ðt1Þ þ 2d3 k23 ek3 t : It is clear that c0 ðt; eÞ > 0; c000 ðt; eÞ > 0; 0 6 t 6 1; e > 0, since d1 > 0; d2 > 0; d3 > 0. From (20), for sufficiently small e > 0, one has

  d1 d2 k2 d3 M 1 þ jx000 ð0Þj þ 1 1 M2 þ jx000 ð1Þj þ 1 1 þ M þ M þ ¼ þ e e4 þ þ e 1 k1 k2 k3 l k21 k22 e2 M1 þ jx000 ð0Þj þ 1 5 M2 þ jx000 ð1Þj þ 1 pffiffiNe 1 þ M þ M  1 > e4 þ ee e5 > 0: þ N N l 1

cð0; eÞ ¼ e2

Then cðt; eÞ ¼ cð0; eÞ þ

Rt

0

c0 ðs; eÞds > 0, for 0 6 t 6 1, since c0 ðs; eÞ > 0.

92

X. Lin / Applied Mathematics and Computation 224 (2013) 88–95

Similarly, from the expression of c00 ðt; eÞ, we obtain 1





1

c00 ð0; eÞ ¼ e2 d1 k1 þ d2 k2 ek2 þ 2d3 k3 > ðM1 þ jx000 ð0Þj þ 1Þe4 þ ðM2 þ jx000 ð1Þj þ 1Þe

pffiffiN e

þ

2lð1 þ M þ M Þ

Rt Then c00 ðt; eÞ ¼ c00 ð0; eÞ þ 0 c000 ðs; eÞds > 0, for 0 6 t 6 1 since c000 ðs; eÞ > 0. Define functions v ðt; eÞ; uðt; eÞ be as

v ðt; eÞ ¼ x0 ðtÞ þ cðt; eÞ;

N2

1

e5 > 0:

uðt; eÞ ¼ x0 ðtÞ  cðt; eÞ:

Then for ðt; eÞ 2 ½0; 1  ½0; e0 , one has

uðt; eÞ < v ðt; eÞ; u0 ðt; eÞ < v 0 ðt; eÞ; u00 ðt; eÞ < v 00 ðt; eÞ; uð0; eÞ 6 0 6 v ð0; eÞ and

ev 000 ðt; eÞ þ f ðt; v ðt; eÞ; v 0 ðt; eÞ; v 00 ðt; eÞ; eÞ ¼ ev 000 ðt; eÞ þ f ðt; v ðt; eÞ; v 0 ðt; eÞ; v 00 ðt; eÞ; eÞ  f ðt; v ðt; eÞ; v 0 ðt; eÞ; x000 ðtÞ; eÞ þ f ðt; v ðt; eÞ; v 0 ðt; eÞ; x000 ðt; eÞ; eÞ  f ðt; v ðt; eÞ; x00 ðtÞ; x000 ðtÞ; eÞ þ f ðt; v ðt; eÞ; x00 ðtÞ; x000 ðtÞ; eÞ  f ðt; x0 ðtÞ; x00 ðtÞ; x000 ðtÞ; eÞ þ f ðt; x0 ðtÞ; x00 ðtÞ; x000 ðtÞ; eÞ  f ðt; x0 ðtÞ; x00 ðtÞ; x000 ðtÞ; 0Þ þ f ðt; x0 ðtÞ; x00 ðtÞ; x000 ðtÞ; 0Þ Z

1

@f ðt; v ðt; eÞ; v 0 ðt; eÞ; x000 ðtÞ þ hc00 ðt; eÞ; eÞdh  c00 ðt; eÞ 00 @x 0 Z 1 Z 1 @f @f þ ðt; v ðt; eÞ; x00 ðtÞ þ hc0 ðt; eÞ; x000 ðtÞ; eÞdh  c0 ðt; eÞ þ ðt; x0 ðtÞ 0 0 @x 0 @x Z 1 @f ðt; x0 ðtÞ; x00 ðtÞ; x000 ðtÞ; heÞdh  e þ hcðt; eÞ; x00 ðtÞ; x000 ðtÞ; eÞdh  cðt; eÞ þ 0 @e

¼ ev 000 ðt; eÞ þ

000 0 6 eðx000 0 ðtÞ þ c ðt; eÞÞ  N c ðt; eÞ þ lcðt; eÞ þ M e 1

¼ eðM þ M  Þ þ þ

e2 d1 k1

1

ek1 t ðek31  Nk1 þ lÞ þ

e2 d2 k2

ek2 ðt1Þ ðek32  Nk2 þ lÞ

2d3 k3 t 3 ld3 e ðek3  Nk3 þ lÞ  k3 k3

¼ eðM þ M  Þ 

ld3 1 4 ¼ e5 ½ð1 þ M þ M  Þ  ðM þ M  Þe5  < 0; k3

i.e.

ev 000 ðt; eÞ þ f ðt; v ðt; eÞ; v 0 ðt; eÞ; v 00 ðt; eÞ; eÞ 6 0:

v 0 ðt; eÞ, we obtain v 0 ð0; eÞ ¼ x00 ð0Þ þ c0 ð0; eÞ P x00 ð0Þ:

From the expression of

Similarly, v 0 ð1; eÞ P x00 ð1Þ, and there exists e1 > 0, for 0 < e 6 e1 , one has v 00 ð0; eÞ 6 M 1 . Again there exists 0 < e 6 e2 , we have v 00 ð1; eÞ P M 2 . There exists e3i > 0, when 0 < e 6 e3i (i ¼ 1; 2; . . . ; m  2), the following is hold

e2 > 0, when

   d1 k1 ni d2 k1 ðni 1Þ d3  k3 ni þ e þ e 2e  1 k1 k2 k3 5 1 1

pffiffiN

1pffiffiN e4 e2 1 þ M þ M  lþNni 2e N  1 e5 M 1 þ jx000 ð0Þj þ 1 e2 e ni þ M 2 þ jx000 ð1Þj þ 1 e2 e ðni 1Þ þ 6 x0 ðni Þ  4N 4N l 6 x0 ðni Þ þ 1 6 jx0 ðni Þj þ 1 :¼ m1i ; i ¼ 1; 2; . . . ; m  2: 1 2

v ðni ; eÞ ¼ x0 ðni Þ þ e

Similarly there exist

e4j > 0, when 0 < e 6 e4j (j ¼ 1; 2; . . . ; n  2), the following inequalities are hold

v ðgj ; eÞ 6 jx0 ðgj Þj þ 1 :¼ m2j ;

j ¼ 1; 2; . . . ; n  2:

Let

M3 ¼ max fm1i g; i¼1;...;m2

M 4 ¼ max fm2j g; j¼1;...;n2

e0 ¼ minfe1 ; e2 ; min fe3i g; min fe4j gg: i¼1;...;m2

j¼1;...;n2

For 0 < e 6 e0 , from condition ðH3Þ, we have

X. Lin / Applied Mathematics and Computation 224 (2013) 88–95

93

gðv 0 ð0; eÞ; v 00 ð0; eÞ; v ðn1 ; eÞ; . . . ; v ðnm2 ; eÞÞ P gðx00 ð0Þ; M1 ; M 3 ; . . . ; M 3 Þ ¼ gðx00 ð0Þ; M 1 ; sM3 Þ P A; hðv 0 ð1; eÞ; v 00 ð1; eÞ; v ðg1 ; eÞ; . . . ; v ðgm2 ; eÞÞ P hðx00 ð1Þ; M2 ; M 4 ; . . . ; M 4 Þ ¼ gðx00 ð1Þ; M 2 ; qM 4 Þ P B: Thus we show v ðt; eÞ is an upper solution of BVP (3) and (4). Similar to the above argument, we can prove uðt; eÞ is a lower solution of BVP (3) and (4). Therefore the conditions of Lemma 1 are all satisfied, hence BVP (3), (4) has a solution xðt; eÞ satisfying

uðt; eÞ 6 xðt; eÞ 6 v ðt; eÞ;

0 6 t 6 1;

and the inequality (18) holds on ½0; 1  ½0; e0 . h Theorem 2. Assume (H1), (H2) and (H3) in Theorem 1 hold, and the following inequalities hold

!

N  lþN 2e N  1 > 0; l

ð21Þ

!   n2 X q l N N  lþN 2 p2 þ 2 e l þ dj 2e N  1 > 0; N l j¼1

ð22Þ

m2 X

p1 þ

ri

i¼1

where constants p1 ; p2 ; q1 ; q2 , and r i ði ¼ 1; . . . ; m  2Þ; dj ðj ¼ 1; . . . ; n  2Þ are expressed in Lemma 2 and l; N are given in Theorem 1. Then BVP (3), (4) has a unique solution. Proof of Theorem 2. From Theorem 1, we show that BVP (3), (4) has a solution. Now we only need to show BVP (3), (4) has at most one solution. If the assertion is not true, then BVP (3), (4) has two difficult solutions x1 ðt; eÞ; x2 ðt; eÞ. Let zðt; eÞ ¼ x2 ðt; eÞ  x1 ðt; eÞ, then it is easy to show that zðt; eÞ is a solution of the following boundary value problem:

ex000 ðt; eÞ þ aðt; eÞx00 ðt; eÞ þ bðt; eÞx0 ðt; eÞ þ cðt; eÞxðt; eÞ ¼ 0; 0 6 t 6 1; 8 xð0; eÞ ¼ 0; > > > > m2 > X > > < p1 x0 ð0; eÞ þ q1 x00 ð0; eÞ þ r i xðni ; eÞ ¼ 0; i¼1

> > n2 > X > > 0 00 > dj xðgj ; eÞ ¼ 0; > : p2 x ð1; eÞ þ q2 x ð1; eÞ þ j¼1

where

aðt; eÞ ¼

Z

1

@f t; x1 ðt; eÞ; x01 ðt; eÞ; x001 ðt; eÞ þ hz00 ðt; eÞ; e dh; 00 @x

1

@f t; x1 ðt; eÞ; x01 ðt; eÞ þ hz0 ðt; eÞ; x001 ðt; eÞ; e dh; @x0

1

@f t; x1 ðt; eÞ þ hzðt; eÞ; x01 ðt; eÞ; x001 ðt; eÞ; e dh: @x

0

bðt; eÞ ¼

Z 0

cðt; eÞ ¼

Z 0

1 ¼ p

Z

1

@g 0 x ð0; eÞ þ hx00 ð0; eÞ; x001 ð0; eÞ; sx1 ðt; eÞ dh; @z1 1

1

@g 0 x ð0; eÞ; x001 ð0; eÞ þ hx000 ð0; eÞ; sx1 ðt; eÞ dh; @z2 0

1

@h 0 x1 ð1; eÞ þ hx00 ð1; eÞ; x001 ð1; eÞ; qx1 ðt; eÞ dh; @z1

1

@h 0 x ð1; eÞ; x001 ð1; eÞ þ hx000 ð1; eÞ; qx1 ðt; eÞ dh; @z2 1

0

1 ¼ q

Z 0

2 ¼ p

Z 0

2 ¼ q

Z 0

ð23Þ

ð24Þ

94

X. Lin / Applied Mathematics and Computation 224 (2013) 88–95

ri ¼

Z

1

0

dj ¼

Z 0

1

@g 0 x1 ð0; eÞ; x001 ð0; eÞ; sx1 ðt; eÞ þ hx0 ðni ; eÞ dh; @ziþ2

i ¼ 1; 2; . . . ; m  2;

@h  0 x1 ð1; eÞ; x001 ð1; eÞ; qx1 ðt; eÞ þ hx0 ðgj ; eÞ dh; @zjþ2

In the expressions of constants ri ;  dj , the arguments ðj ¼ 1; 2; . . . ; n  2Þ are defined as

j ¼ 1; 2; . . . ; n  2:

sx1 ðt; eÞ þ hx0 ðni ; eÞ ði ¼ 1; 2; . . . ; m  2Þ, qx1 ðt; eÞ þ hx0 ðgj ; eÞ

sx1 ðt; eÞ þ hx0 ðni ; eÞ :¼ ðx1 ðn1 ; eÞ; . . . ; x1 ðni ; eÞ þ hx0 ðni ; eÞ; . . . ; x1 ðnm2 ; eÞÞ; qx1 ðt; eÞ þ hx0 ðgj ; eÞ :¼ ðx1 ðg1 ; eÞ; . . . ; x1 ðgj ; eÞ þ hx0 ðgj ; eÞ; . . . ; x1 ðgn2 ; eÞÞ: From condition ðH2Þ; ðH3Þ in Theorem 1, we obtain that aðt; eÞ; bðt; eÞ; cðt; eÞ 2 Cð½0; 1  ½0; e0 Þ, and aðt; eÞ 6 0, 1 6 0; q 2 P 0, ri 6 0 q dj 6 bðt; eÞ 6 N < 0; 0 6 cðt; eÞ 6 l, for ðt; eÞ 2 ½0; 1  ½0; e0 , and ði ¼ 1; 2; . . . ; m  2Þ;   eÞ; cðt; eÞ; q ðt; eÞ; bðt; 1 ; q 2 ; r i ði ¼ 1; 2; . . . ; m  2Þ,  0; ðj ¼ 1; 2;    ; n  2Þ. That is a dj ðj ¼ 1; 2;    ; n  2Þ satisfy the Eq. (14) and boundary conditions (15). Define

wðt; eÞ ¼

2ek3 t  1 2k3 ek1 t  : k3 k21

It is obvious that wðt; eÞ > 0, w0 ðt; eÞ > 0, w00 ðt; eÞ P 0, and

 eÞw0 ðt; eÞ þ cðt; eÞwðt; eÞ 6 ew000 ðt; eÞ  Nw0 ðt; eÞ þ lwðt; eÞ ew000 ðt; eÞ þ aðt; eÞw00 ðt; eÞ þ bðt; ¼

2 k3 t 3 2k3 l l e ðek3  Nk3 þ lÞ  2 ek1 t ðek31  Nk1 þ lÞ  ¼  k3 k3 k3 k1

< 0: For 0 < e 6 e0 , from (21) we have

  k3 w0 ðt; eÞ ¼ 2 ek3 t  ek1 t ; k1

w00 ðt; eÞ ¼ 2k3 ðek3 t  ek1 t Þ:

One has

p1 w0 ð0; eÞ þ q1 w00 ð0; eÞ þ

m2 X

m2 X k3 2ek3 ni  1 2k3 ek1 ni Þþ ri  k3 k1 k21 i¼1 ! m2 X N  lþN ri 2e N  1 > 0; P p1 þ l i¼1

ri wðni ; eÞ ¼ 2p1 ð1 

i¼1

p2 w0 ð1; eÞ þ q2 w00 ð0; eÞ þ

! P p1 þ

m2 X N  lþNni ri 2e N  1 l i¼1

  n2 n2 X q l N X N  lþNgj dj wðgj ; eÞ P 2 p2 þ 2 e l þ dj 2e N  1 N l j¼1 j¼1 !   n2 X q2 l N N  lþN l P 2 p2 þ e þ dj 2e N  1 > 0: N l j¼1

Then wðt; eÞ satisfies the condition in Lemma 2, hence the BVP (23) and (24) has only a zero solution, which contradicts x1 ðt; eÞ – x2 ðt; eÞ. Then BVP (3) and (4) has a unique solution. h Remark 1. If we take m ¼ n, and take the nonlinear boundary functions g; h are the following linear functions as:

gðx1 ; x2 ; . . . ; xn Þ ¼ ax1  bx2 þ

n X

ai xi ;

i¼3

hðy1 ; y2 ; . . . ; yn Þ ¼ cy1 þ dy2 þ

n X b j xj ; j¼3

we find that BVP (3) and (4) become the singularly perturbed boundary value problem (1.1) and (1.2) in [4] that is be a particular case of our BVP (3) and (4). Theorem 2 and Theorem 4 in [4] are the particular case of our Theorem 1 and 2.

X. Lin / Applied Mathematics and Computation 224 (2013) 88–95

95

4. An example

Example 1. Consider the following singularly perturbed boundary value problem

ex000  x00  2x0 þ 3x þ 2te ¼ 0; 0 6 t 6 1; 0 < e  1;

ð25Þ

8 xð0Þ ¼ 0; > > < 0 1 ðx00 ð0ÞÞ3  14 3 xð13Þ ¼ 1; x ð0Þ  512 e3  e > > : 0 2 3 00 ðx ð1ÞÞ þ 2ðx ð1ÞÞ  12 xð12Þ ¼ 1;

ð26Þ

where m ¼ n ¼ 3 and n ¼ 13 ; g ¼ 12 and

f ðt; x; y; z; eÞ ¼ z  2y þ 3x þ 2t e; gðx1 ; x2 ; x3 Þ ¼ x1 

1 4 ðx2 Þ3  1 x3 ; 512 e3  3 e

1 hðy1 ; y2 ; y3 ; y4 Þ ¼ y21 þ 2ðy2 Þ3  y3 : 2 We show the reduced problem (25), (26) has the following reduced solution

x0 ðtÞ ¼ et  e3t ;

0 6 t 6 1:

Then the condition ðH1Þ hold. 1 2 We can choose a constant e0 > 0 and l ¼ 3; N ¼ 2; M ¼ 3; M1 ¼ 8; M 2 ¼ 1; M 3 ¼ M 5 ¼ e3  3e ; M4 ¼ M6 ¼ 2ðe þ 3e3 Þ , 1 1 p1 ¼ 10; p2 ¼ 6; r ¼  13 ; d ¼ 10, satisfying the conditions ðH2Þ and ðH3Þ hold the inequalities (21) and (22) hold. Then conditions in Theorem 2 are all satisfied, which implies for sufficiently small e > 0, BVP (25), (26) has a unique solution xðt; eÞ. Acknowledgment The author wish to express her thanks to the referees for their very valuable comments and suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

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