SOLVABILITY OF HIGHER ORDER THREE-POINT ITERATIVE SYSTEMS Текст научной статьи по специальности «Математика»

ISSN 2074-1871 Уфимский математический журнал. Том 12. № 3 (2020). С. 109-124.

SOLVABILITY OF HIGHER ORDER THREE-POINT ITERATIVE SYSTEMS

K.R. PRASAD, M. RASHMITA, N. SREEDHAR

Abstract. In this paper, we consider an iterative system of nonlinear nth order differential equations:

yt\t) + *4Pi(t)fi(yi+i(t))=0, 1 < i < m, ym+i(t)= yi(t), t e [0,1], with three-point non-homogeneous boundary conditions

уг(0)= у' (0) = ... = yf-2)(0)=0,

«¿yin-2) (1) - Р%У(п-2)(л) = Vi, 1 < i < m,

where n ^ 3, ^ e (0,1) ^ e (0, ж) is a parameter, fi : R+ ^ R+ is continuous, Pi : [0,1] ^ R+ is continuous and pi does not vanish identically on any closed subinterval of [0,1] for 1 ^ i ^ m. We express the solution of the boundary value problem as a solution of an equivalent integral equation involving kernels and obtain bounds for these kernels. By an application of Guo-Krasnosel'skii fixed point theorem on a cone in a Banach space, we determine intervals of the eigenvalues X1,X2, ■ ■ ■ , Xm for which the boundary value problem possesses a positive solution. As applications, we provide examples demonstrating our results.

Keywords: boundary value problem, iterative system, kernel, three-point, eigenvalues, cone, positive solution. Mathematics Subject Classification: 334B18, 34A40, 34B15

1. Introduction

The existence of positive solutions for multi-point boundary value problems associated with ordinary differential equations are of a high interest and play a vital role in different areas of applied mathematics and physics. Multi-point boundary value problems appear in the mathematical modelling of deflection of a curve beam having a constant or varying cross section, three layer beam, electromagnetic waves and so on. For example, the vibration of a guy wire of a uniform cross-section and composed of different parts with different densities can be formulated as multi-point boundary value problems.

Due to the importance in both theory and applications, much attention is focussed on obtaining optimal eigenvalue intervals for the existence of positive solutions of the iterative systems of nonlinear multi-point boundary value problems by an application of Guo-Krasnosel'skii fixed point theorem. A few papers along these lines are Henderson and Ntouvas [5], Henderson, Ntouvas and Purnaras [6, 7] and Prasad, Sreedhar and Kumar [14]. In the past, the researchers have focussed and established the existence of positive solutions of the boundary value problems associated with homogeneous boundary conditions,

K.R. Prasad, M. Rashmita, N. Sreedhar, Solvability of Higher order three-point iterative

systems.

© Prasad K.R., Rashmita M., Sreedhar N. 2020.

Поступила 12 декабря 2019 г.

M. Rashmita is thankful to DST-INSPIRE, Government of India, New Delhi for awarding JRF.

see [3, 13, 2,12, 9, 18]. However, some works have been carried out in establishing the existence of positive solutions of the boundary value problems with non-homogeneous boundary conditions, see [15, 17, 11, 16, 10].

Motivated by the papers mentioned above, in this paper, we determine intervals of the eigenvalues ai,a2, ••• ,Am, which will give guarantee for the existence of positive solutions of the iterative system of nonlinear nth order differential equations

yit)it) + Alpl{t)fi{yi+i{t)) = 0, 1 ^i^m, ym+i(t) = Vl(t), te [0,1], (1.1)

satisfying three-point non-homogeneous boundary conditions

y i(0) = yi'(0) = ■ ■ ■ = y(ra-2)(0) = 0, aiy(n-2)(1) - fryt2fa) =/l, 1 ^ i ^ m, (1.2)

where n ^ 3, r] e (0,1) and /l e (0, ro) is a parameter for 1 ^ i ^ m. Our approach is based on application of Guo-Krasnosel'skii fixed point theorem on a cone in a Banach space.

Throughout the paper, we assume that the following conditions hold true: (B1) fi : R+ ^ R+ is continuous for 1 ^ i ^ m,

(B2) pl : [0,1] ^ R+ is continuous and pl does not vanish identically on any closed subinterval

of [0,1] for 1 ^ i ^ m, (B3) al and pl are constants such that al > 0 and pl e (0, ^) for 1 ^ i ^ m, (B4) each of

ZlG = lim and = lim

X X

for 1 ^ i ^ m exists as positive real number. The rest of the paper is organized as follows. In Section 2, we express the solution of the boundary value problem (1.1)—(1.2) as a solution of an equivalent integral equation involving kernels and find bounds for the these kernels. In Section 3, we establish the criteria determining the eigenvalues, for which the boundary value problems (1,1)-(1,2) has at least one positive solution in a cone; this is done by using the Guo-Krasnosel'skiis fixed point theorem. In Section 4, as an application, we provide some examples to illustrate our results.

2. Kernels and bounds

In this section, we express the solution of the boundary value problem (1,1)-(1,2) into an equivalent integral equation involving kernels by determining integral equation of yl for 1 ^ i ^ m and find bounds for the kernels, which will be needed to establish the main results.

Lemma 2.1. If h(t) e C([0,1],R+), then the boundary value problem

y(n\t) + h(t) = 0, 1 ^i^m, te [0,1], (2.1)

with (1.2) has a unique solution and is given by

i

//• tn-1 f ft- tn-1 *W=(n - ¡;(at- M +/ [GC■ ')+(n - ,ft) GC■ ')]hW^. (")

0

where

and

(\tn-1(1 -s)- (t -s )n-1 , 0 1,

G(t,s) = . 1 / J (2.3)

(n - 1)^ in-1(1 -s), 0 ^tO^ 1, 1 ;

fs(1 - n), 0 ^s^r?^ 1, G ( , ) =

^^ [V(1 - s), 0 ^ r] ^ s ^ 1. 1 ;

Proof. Let yi(t), 1 ^ i ^ m, be the solution of boundary value problem (2,1), (1.2), Then an equivalent integral equation of (2.1) is given by

t

yz(t) = Co + Cit + C2t2 + ■ ■ ■ + Cr-itn-1 - * f(t - s)n-1h(s)ds.

(n - 1)! J 0

Using the boundary conditions (1.2), we can determine Cj as

Cj = 0 as j = 0,1, 2, ■■■ , n - 2

and

1

C"-1 =(n - mc -, m + (n-wtk-ml11 - s)h(s) ds

0

v

1 (rj - s)h(s) ds.

(n - 1)\{ai - rißi)

0

Thus, the unique solution of boundary value problem (2.1), (1.2) is

1

»W = (n - IZ, - + / (G(i ■8) + (n - - rn Gl 8))h(,)da-

0

□

Lemma 2.2. Assume that the condition (B3) is satisfied. Then the kernels G(t, s) and G\(t, s) are satisfies the following inequalities:

(i) G(t, s) ^ 0 and Gi(t, s) ^ 0 for al11, s E [0,1],

(ii) G(t, s) ^ G(1, s) for al 11, s E [0,1],

(iii) G(t, s) ^ 4^-xG(1, s) for al 11 E I and s E [0,1], where I =

1 3

4 ' 4

Proof. We prove the inequality (z). For 0 ^ s ^ t ^ 1, then we have

G(t, s) = 1^Ty_(tn-1(1 - 8) - (t - sT-1) > (^rry (tn-1(1 - 8) - (t - StT-1)

fn— 1

-((1 - s) - (1 - sr1) ^ 0

(n - 1)! and

G1(t, s) = s(1 - t) ^ 0.

For 0 ^ t ^ s ^ 1, then we have

G(t ,* = >0

and

Gi(i, s) = t(1 - s) ^ 0. Now, we prove the inequality (iz). For 0 ^ s ^ t ^ 1, then we have 8 1 tn~2 WtG(t= (n^-2)T(< (1 -8) - « - s) ^ s ÔT-2)!(<1 - s) - (1 - 8) ' * a

For 0 ^ t ^ s ^ 1, then we have

8 tn~2

atG(t's)= (n-2)i(1 -s) s 0

Therefore, G(t, s) is increasing in t, which implies that G(t, s) ^ G(1, s).

We proceed to proving inequality (in). Hence, as0 1 and t G I, we have

G(t, S) = (tn~'(1 - 8) - (t - s)n~ ')

tn~1 1 > (¡¡-i)! ((1 -') - (1 - 1) > inrG(1-')•

As 0 ^ t ^ s ^ 1 and t G I we then get

n 1 1 G(t, s) = 1)! (1 - S) ^ ii^IG(1,

□

We note that an m-tuple (yi(t), y2(t), • • • , ym(t)) is a solution of the boundary value problem (1,1)-(1,2) if and only if yi(t) satisfies the following equations

yw

+ X

and

(n - 1)!(a, - №) i

/3itn~1

G(t, s) + --—-w\G'(V, s) Pi(s) fi( Уi+l(s))ds, 1 ^i^m te [0,1],

(n - 1)!(Q!j - T] Pi)

ym+i(t) = yi(t), te [0,1],

so that, in particular,

Vl(t) = (n - Wai - ,Pi) + ■XlJ (G(t, Sl) + (n - mai - VPi)Gl(v, Sl))P1(Sl)

0

y 1

H (n - M + X*I (G<S 1' ^ + (n - 1)£- VP,) Gi("'S2))^2<^2)

V n

i ( /mSm_i i f . . pmSm_ i \

• • •f-'I(n - 1)! (a„, - r,0m) + (G(S-S™> + (n - 1)!(am - vSm)Gl("'

0

Pm(Sm)fm(yi(sm))dsm) ■■■ds2 ds 1

^ •••ds

The following Guo-Krasnosel'skii fixed point theorem is a fundamental tool to establish our main results.

Theorem 2.1. [1, 4, 8] Let B be a Banach space, P C B be a cone and suppose that Q', Q2 are open subsets of B with 0 e Q' and Q' C Q2. Suppose further that T : P if (Q2 \ Q') ^ P is completely continuous operator such that either

(i) \\Ty II ^ ||y||, yeP n dQi, and \\Ty |y, yeP n dÜ2, or

(ii) \\Ty II ^ \\y\\, yEP nô Q^a'nd \\Ty II ^ \\y\\, yEP n ô Q2. Then T has a fixed point in P n (Q2 \

3. Positive solutions in a cone

In this section, we establish a criteria to determine the eigenvalues, for which the iterative system of three-point non-homogeneous boundary value problem (1.1)—(1.2) has at least one positive solution in a cone.

Let B = {x : x E C[0,1]} be the Banach space equipped with the norm

||x|| = max \x(t)\. te[o,1]

Define a cone P C B by

P = S^x eB : x(t) ^ 0 on te [0,min x(t) ^ Hx^wJ .

We define an operator T : P ^ B for y1 e P by

i

Tyi(t) = (n - mai - Vßi) +XlI (G(t, Sl) + (n - 1ßti- nß,)Gl(V, Sl))Pl(Sl)

0

, i

h (n - ma. -.M+x*i "*'+(n - - ^ g «>■

0

s 2) ■ ■ ■fm-i ( (n - > ,ßm)+AmI i'Sm)

(3.1)

lm 'lymj

0

m) ) Pm( Sm^ fm(yi ( Sm))ds m I ■ ds 2 m - 1 Ißm) J J I

+ 7-HmSm l-~a T Gl (V, Sm) ) Pm( Sm) fm(Vl( Sm))ds m ) ■ ■ ■ ds 2] dSi.

(n - 1 )\(am - Tjßm) Lemma 3.1. The operator T : P ^ B is a self map on P.

Proof From the positivitv of the kernels G(t, s) and G\(t, s) in Lemma 2.2 that for yx e P, Tyi(t) ^ 0 on t e [0,1]. Also, for y\ e P, and by Lemma 2.2, we have

i

Tv.№=,_ "T'.», + W (GC1) + (nnßEnS)Gs'0

(n - 1)!(a - rißt) i M (n - 1)!(a, - vß,)

0

1

w?"i x I (r„ > , ß&TTi

^ 4(n - W» -, M +X,2 I{G(S ^

0

0

ßmSm-1

(n - 1)!(a, - riß,)

n— l

Gl (r], S2) P2( S2) ■ ■ ■ fm-1 -,-7T"m_m--¿-z + A m G( m 1, m)

) \(n - 1)!(am - vßm) J

0

+ 7Z-ißmSm l O ) G\(V, Sm) ) Pm( Sm) fm(yi( Sm))ds m) ■ ■ ■ ds 2) dsX

m VPm)

(n - 1)!(am - Vßm)

Pm( Sm) fm(yi( Sm))ds ^ ■ ■ ■ ds 2 ^

«(n -1)!/::, +x'/ +(n - 1)!pa. - „&) g'("- 0

0

(1

(n - 1/)Ka2- „&) + X2j (G(S" S2) + (n - ,/2)

n

0

g ^,s2))p2(s 2) • • •fm- 1 ( (n - i)uOz- v/m)+XmS {g(s -',sm)

0

(n - 1)!(am - r]/m)

m) I Pm( sm) fm(yi( sm)) dsm I • • •d S 2 m - i l/m) J J I

+ 7-f,ZSm '—o \ Gi(v, sm) ) pm( sm) fm(yi( sm)) dsm) • • • ds2] dsi,

so that

1

»^«'l < (n - 1)Kai -,,ft) + X'J (G(1- S') + (n - 1)^ -,,ft)^, S'))

0

(1

(n - ma* - ^ + X2I (G(S1 ■s,) + (n - ot- r/h)

n

0

Gi(V, S2?)Ms2) • • • U-1 ((n - ^ vpm) + X™j (G(s-', O

0

^jpm(Sm) fm(yi(sm)) ds^ • • • ds

+ 7-fX^ 1—o~\ Gi(v, sm.) ) pm( Sm) fm(yi( sm)) dsm) • • • ds 2 ) ds'•

(n - 1)!( am - r]pm)

Further, if yi e P, we have from Lemma 2,2 and the above inequalities that

!i

(n - ill- VPi) +XlJ {G(t, Sl) + (n - 1^!1(£:ni - ,Pi)Gl(V, Sl))

0

(i

(n - 17: - VB2) +X2j(G(S 82) + (n - 1^2 - „P,) G: ("' s2))

0

u 1 1

P2(S2) • • • fm~' ((n - ) + *»)+

0

--p™Sm —— G'(r], smM pm( sm) fm( yi( sm))dsm] •••ds2 Idsi

(n - 1)!(am - r]pm) J J

411 ( (n - 1)!Ua1i - vPi) +XlS (G(1, Sl) + (n - D!^ - „Pi)Gl(V, Sl))

0

1

^ i) 4 (n - „p.)+(g(s ^+(n - !£-, m g:«>- ^

0

P2(S2) ' ' ' fm~^ (u - l)!^- r]ßm) +Xmj {G(S-^ +

0

(u - 1)\(am - r]ßm) 1

ßmSm 1 G (V, Srn^ Pm(sm) frn(yi(Sm))ds^j ■■■ds^jds^j

m) I ymy^m) J m\yiv am J J^^m,

m ' Ißm)

4nT WTV

Therefore, T : P ^ P and this completes the proof, □

T

completely continuous,

T P

result, we define positive numbers F', F2, F3 and F4 by the formulae

F' = (i^/K s) + (n - ^ -,P)Gi("• s))p-(s)<is) }, <3'2>

F = 2(/a/ (G(1s)+(n - m* -,,Pi)Gi(",>))»(')d>) }, <3-3>

F3=ms. {(43^ / (g(1- s)+(n - 1)!Pa, - „a)Gi("- s))p-(s) <is) } <3'4)

F = iSS.^'"» / (G(1-s)+(n - ^ -,,P,)Gi("■ s))p'(s)ds) }• (3'5)

Theorem 3.1. Assume that the conditions (B1)-(B4) are satisfied. Then, for each X:,X2, • • • ,Xm satisfying either

Fi <Xi < F,, Fi <X2 < F,, • • • , Fi < Xm < F,, (3.6)

or

F3 < Xi < F4, F3 < X, < F4, • • • , F3 < Xm < F4, (3.7)

there exists an m-tuple (yi, y2, • • • , ym) satisfying (1.1)—(1.2) su,ch that yi(t) > 0 on (0,1) and Ui G (0, <x>) is sufficiently small for 1 S i S m.

Proof Let Xi, 1 S i S m, be given as in (3.6). Let e > 0 be chosen such that

fioo £ f

^nax K 1 _ I

| \ 42ra_2 J

sei

G(1, s) + 7--wrGi(r], s) pi(s)ds 1 S ^ min{Ai, A2, ■ ■ ■ , \m]

(u - 1)!(«i - vßi)

y1}

and

max{ Ai, A2, ■ ■ ■ , Am}

< i!SiS, {1 (( f" +f) / (G<1's) + (U - me -, ,ßi) G1("' '>)*«d)

Now, we seek a fixed point of the completely continuous operator T : P ^ P defined by (3,1), By the definition of fi0, 1 ^ z ^ m, there exists an H > 0 such that, for each 1 ^ i ^ m, the inequality

fi(x) ^ (ho + t)x, 0 <x ^ H1,

holds true.

Let ^i, 1 ^ i ^ m, be such that

0<ft ^ (n- 1)!c-,w,

Let y1 E P with ||y1\\ = H1. By Lemma 2.2 and the choice of e, for 0 ^ sm-1 ^ 1 we have

^msm-1 , a f (rU \ 1 0msm-1 n ,

+ Km I Sm-1, Sm) + --—---, Sm)

)

Pm(Sm) fm(V1(Sm)) dsm 1

(n - 1)!(tm - V0m) J V (n - 1)!(tm - VPm)

0

^T-T\TTm-+ ^mf (G(1, Sm) + ~f-7Tr/m--^-G 1(7], SmU

(n - 1)!(am - VPm) J \ (n - 1)!(Cm - VPm) J

Pm(Sm)( f m0 + V1(sm) dSm

1

+ I ( G(1, Sm)+ ( 1)!rm a ) G^, Sm) I Pm (Sm)dSm( fm0 + e) || V1 J \ (n 1)!(cm 1]Pm) J

0

H + Kn J [G(l, S.m) + jn-Y^m _ ri/3m)

« Hi + Hi = H,

2 2 1

In the same way, it follows from Lemma 2,2 and the choice of e that, for 0 ^ sm-2 ^ 1,

_^m-1 sm-2__+ K if G(s s ) +__0m-1 $m-2_

(n - 1)!(Cm-1 - V0m-1) ^ J V ^2, ^1 (n - 1)!(tm-1 - VPm-1)

0

G1(v , ^m—1) ) Pm—1(Sm—1) fm— 1) 1)!(t ) + Km J ^G(Sm—1, Sm)

m m 0

^Pm-1( s m-1) fm-1 ^

m ) m( ) m( 1( m )) mm m - 110m) J I

+ 7-^T^ 1-o 7 G1(V, Sm) ) Pm( Sm) fm(iJ1( Sm))dSm ) dSm-1

(n - 1)!(Cm - VPm)

^7-^r/^-5--T + if ( G(1, 1) + ^1

(n - 1)!(am-1 - V0m-1) J V ' (n - 1)!(tm-1 - VPm-1)

0

G1(l], Sm-1) Pm-1(Sm- 1)dsm-1( fm-1,0 + t)H1

iHi + H = Hv

2 2

Continuing with this bootstrapping argument, we have, for 0 ^ t ^ 1,

1

tn~1 ^ ] (m ^ 01 1

tn~ 1 \

G(t ■ >J + (n - »¡fr - „H)G1(rt'1)

K (U - ffC „ft) + A2 / (G<» 1-) + (U - jgl „ft) Gi(". ^ 2)

\ n

fm— 1

t

„n— 1

ßmSm_ i

ßmSm_i

+ Am I G(1, Sm) +

(u - 1)!(am - r]ßm)

0

Gi(rj, pm( sm) fm( yi( sm))ds ^ ■■■ds ^ dsi ^ Hi,

Tyi(t) ^ Hi.

(u - 1)!(am - Tjßm)

so that for 0 ^ t ^ 1,

Hence, \\Ty1\\ ^ H1 = ||y1\\. If we let

fii = {x e B : \\x\\ < Hi},

then

\\Tyi\ ^ \\yi\\ for yi e P n <9Qi. (3.8)

By the definitions of fix>, 1 ^ i ^ m, there exists H2 ^ 0, such that, for each 1 ^ i ^ m,

fi(x) ^ (fi^ - e)x, x ^ H2.

Let

H2 = max^H, 4n_ 'H,} • We choose j/: e P and W ^W = H2. Then

min y1 (t) ^

1

i\\

tel ' ' 4n_1 By Lemma 2.2 and the choice of e, for 1 ^ s m_ 1 ^ |, we have:

n— 1

ßm^m-1

ß „n-1

(u - 1)!(am - Tjßm) + AJ v( ^ 1, Sm ) + (u - 1)!(«rn - Vßm)

0

'mPm( Sm) fm ( j/i ( Sm)) d Sm

Gi(rj, Sm)

)

ß Hn~1

^ Am J ( G(Sm—1, 0

1

m^m-1

^ ^A™ I (G(1, ^

sei

> A™J (G(1, ^

e

^ \\yi\\ = H2.

(u - 1)!(am - rjßm)

ßm

(u - 1)!(am - tj ßm)

ßm

(u - 1)!(am - Tj ßm,)

Gi(v , Sm) ) Pm( Sm) fm(yi( Sm)) dsm Gi(rj, pm(sm)(fmix - e)yi(sm) dsr, Gi(rq, sm) )pm(sm) dsm(fmix - e)\\yi\\

In the same way, it follows from Lemma 2.2 and the choice of e, that for ' S sm-' S 4, we have

n— 1

ßm-1 $ m-2

(u - 1)!(am_ 1 - Tjßm-1)

-il G( Sm-2, S m— 1

ß

C n-1 'm-1 2

(u - 1)!(am_1 - r]ßm-i)

1

1

1

1

m

G\(rj, Sm-1) Pm-1( Sm-1) fm-17-7TT7-m-Z.-T + Km G( m 1 , m)

(n - 1)!(am-1 - Tj Pm-1) J V

0

Pm-1(Sm- 0 fm-1 ^

) m( m) m( 1( m)) m

+ 7-1 ^ 1 0-r G1 (rj, Sm) ) Pm( Sm) fm(ij1( Sm))dSm \dSm-1

(n - 1)!(am-1 - V Pm-1)

> ijzT Xm-1 f (G(1, Sm-O + 7-UUPm 1-o-7 G1(V, Sm- 1U

4 1 J \ (n - 1)!(am-1 - V Pm-1) J

sei

Pm-1(Sm- 1)dSm-1(fm-- t)H2

^ Aln-2 Km-1 i \G(1, Sm—1) + 7-x,/ Pm 1-n-7 G1('1, Sm- 1)J

42n 2 J \ (n - 1)!(am-1 - Tj Pm— 1) J

sei

Pm-1(Sm- 1)dSm-1(fm-- t)H2 ^ H2.

Proceeding as above, we get:

1

»1 tn~1 ^ ](r^u \ P1 tn~1

^P -j-n— 1 \

G(t,S1) + (n - ma - VP1)G1(r],S1))P1(s 1)

(n - 1)!(a - VP1) VV (n - 1)!(c - VP1)

0

(1

(n - mk- vk) + X2IiG(s + (n -„02)Gl<"-S2 0P2<S2)

n

0

n— 1 1 / 0 jn— 1

f i ^msm-1 . a f (\ 1 PmSm-1

' " " Jm-1 7 7T77 n 7 + Km G(Sm-1, Sm) +

(n - 1)!(Cm-1 - VPm-1) J V ' (n - 1)!(tm - VPm)

0

G1('ll, Sm)j Pm( Sm) fm( V1( Sm))ds ^ •••d S^ dS1 ^ H2,

so that, for 0 ^ t ^ 1,

TV1(t) >H2 = ||yM

Hence, HTy^ ^ ||y1^. If we let

^2 = {x EB : ||x|| < H2},

then

HTy^ > ||yj, for V1 EP n d^2. (3.9)

Applying Theorem 2,1 to (3,8) and (3,9), we obtain that T has a fixed point y1 E P n (&2 \ ^). Since ym+1 = y1, we obtain a positive solution (y1, y2, ■ ■ ■ , ym) of (1.1)—(1.2) given iterativelv by

U(t) = tz-^t—,.— + I ( G(t, s) + 1(V, s)jPl(s)h(yt+1(s))ds,

(n - 1)\(at - vP,) +Xlj {G(t, S)+ (n - 1)!(at - vP,)

= m, m - 1 , ■ ■ ■ , 1

Let \i, 1 ^ i ^ m, ^e given as in (3,7) and let e > 0 be chosen such that

{ [^ / (G(1, + (n - mai - VP) G1(V, P*(S) ^ > ^ min{K1,K2, ■ ■ ,Km}

and

max{Ai, A2, ■ ■ ■ , Am}

S iSin {2 (/ (G(1,s ) + (n - - nfr)Gl(v, s))Pl(s) ds(^ +e)

We seek fixed point of the completely continuous operator T : P ^ P defined in (3,1), By the definition of fi0,1 S i S m, there exists an H3 > 0 such that, for each 1 S i S m,

fi(x) ^ (fio - t)x, 0 < x S S3.

Also, it follows from the definition of /¿0 that /¿0 = 0,1 S i S m, and so there exists 0 < lm < Im-' < • • • < I2 < H3 such that

Aifi(x) ^ —-—-r-, x e [0, h],

2/ G(1, sm) + (n -iy.(ai-Vßi)G1(r?, Sm) )Pi(s) ds

(u — 1)!(ai — rißi) L_ 1 0 < ßi <---for 3 ^i^m,

and

H3

X, f,(x) S—Q-3-r-, x e [0, /2],

2/ G(1, Sm) + (n

(n - 1)!(a2 - VP2)H3

0 <u, <-2-•

Choose y' e P with W^W = lm. Then we have

UmSm-1 , A f (¡^i \ 1 PmSm-i

+ Xm I G(Sm-1, Sm) +

(u - 1)!(am - 7]ßm) J v ' (u - 1)!(am - rjßm)

0

)

Gi(v , S m

) Pm( Sm) fm( Ui( Sm))ds m

^T--«"T + A™[ (G(1, ^ + 7-J*-, Sm)l

(u - 1)!(;m - vßm) J \ (U - 1)!(am - r]ßm) J

m 'lymj J \ V'1, -V'V^m, 'lymj

0

Pm( S m) fm( Vi( S m)) d Sm J ^G(1, Sm) + (n-iyjan-vßm)G1('ll, Im-iPm (Sm)dsr

< t m-1 . ^_/_ . ,

^ —2—1 1-^ 1 m-i.

2 I G(1, Sm) + (n-1)!(am-?7ßm) G1(^, Sm) Pm(Sm

Continuing as above, we get:

ß sn-i f f ß sn-i \

21 + av [G(s i,s2)+(u - D^ - ^2)Gi(v,s2))p2 (s 2) 0

(u - 1)!(;2 - ^2)

, 1

/2 (U-Ißö) + (G<S2, ") + (U - 1g- „ß.) g<"'

0

)

Then,

P:i(s3) ■ ■ ■ fm(y1(Sm))dSm ■ ■ ■ ds3 ds2 ^ H3.

1

T*W = (n - Wai-,,A) I '")+ (n - mci-,M Gl(h S1))

0

1

P1(s 1) Ml-^ ^ +kJ( G( s 1, 82) + P2Sr 1

(n - 1)!(a2 - VP2) J V ' (n - 1)!(a2 - VP2)

(

G1(r], S2)^j P2 («2 ) ■ ■ ■ fm (v( Sm))dSm ■ ■ ■ ds^ ds 1

> iZ10;--^ A! / (Gll, S1)+{n _ ^ _ ^ GM S^ i)d S1

sei

So, WTyH > ||y1l We let

a1 = {x EB : ||x|| < lm),

then

HTV1H > ||yH for V1 EP n dQ1. (3.10)

Since each fi™ is assumed to be a positive real number, it follows that fi, 1 ^ i ^ m, is unbounded at <x. For each 1 ^ i ^ m, let

f*(x) = sup s).

Then, it is straightforward that, for each 1 ^ i ^ m, f*(x) is a non-decreasing real-valued function, fi ^ f* and

lim =

x^<x x

By the definition of f^, 1 ^ i ^ m, there exists H4 such that, for each 1 ^ i ^ m,

fi(x) ^ (fi™ + t)x, x ^ HA. This implies that there exists H4 > max{2H3, H4j such that, for each 1 ^ i ^ m,

H(x) ^ f*(H4), 0 < x ^ H4.

Let ji, 1 ^ i ^ m, satisfy

(n - 1)!(ai - Tj Pi)H4

0 < J <-2-.

We choose y1 E P with ||y1H = H4. Then, using the above bootstrapping argument, we obtain

1

« = (n-lJO-P) + A1I (G«'S1) + (n - m«'- ,81)^

0

( 1

(n - mL - V P2) +A2J {G(S 1 S2) + (n - №2 - VP2)

0

Gi(v, p2( s2) • • • fm( yi( sm))dsm •••dsz) dsi

ßi

(n - 1)!(ai - vßi)

+ Xi I (o(t, si) +

ßi

(n - 1)!(ai - vßi)

Gi('q, Si)

)

ß2Snr i

Pi( S i) A*^

Gi(ï], S2)^j P2 ( S2) • • • fm( yi( sm))ds m •••ds 2 I ds i

M

ni

(n - l)!(a2 - Vß2) {G(sU S2) + (n - 1)!(CX2 - vß2)

0

2

^H + Xi S (G(1, si) + 0

ßi

(n - 1)!(ai - T]ßi)

Gi(rj, si)^jpi(si)fi*(H4) dsi

^H + A1 (/i~ + e)H I ( G(1, s-,) +

ß

(n - 1)!(ai - vßi)

M 1))

G ( , ) ( )

^ H + H = H4. 2 2 4

Hence, \\TyJ ^ \\yi\\. So, if we set

Ü2 = {x EB : \\a;\\ < H4},

then

\\TVl\\ ^ |M|, for yi EP ndQ2. (3.11)

Applying Theorem 2.1 to (3.10) and (3.11), we obtain that T has a fixed point y\ E Pfl(Q2\^). In view of the identity ym+i = y\ this yields that the m-tuple (y 1, y2, • • • , ym) satisfies boundary value problem (1,1)-(1,2) for the values Ai, 1 ^ i ^ m. The proof is complete. □

4. Examples

Here we consider two examples demonstrating our results.

Example 1. Consider the iterative system of third order three-point non-homogeneous boundary value problem

y'" + Ai/i(V2(t)) = 0, tE [0,1],

yZ + A2Î2(ys(t)) = 0, tE [0,1], y" + A3f3(yi(t)) = 0, tE [0,1],

(4.1)

Vl(0) = 0, y[ (0) = 0, V2(0) = 0, y2 (0) = 0,

y3(0) = 0, y'3 (0) = 0,

2y 1 (1) - Vi Q 3y2(1) - (2

3y'3(1) - 2y'3 (2

ßi, ß2,

ß3,

(4.2)

where

fi(y2) = y2(476.5 - 468.7e-V2)(210 - 202.7e-3y2),

1

1

f2(y3) = y3(872.5 - 867.2e—5y3)(162 - 149.5e—2y3), h(y1) = y'(374.6 - 366.5e-3yi)(250 - 238.5e-2yi).

and

Pi(t) = P,(t) = P3(t) = 1. The kernels G(t, s^d G'(i, s) are

g W t2(1 - s) - (t - s)2, 0 OSK 1,

(, =2) t2(1 - s), 0 SiOS 1,

and

(s 1

0 <»< 1«i,

^ 0 S 1 2 ' 2

By direct calculation, we find that

/10 =56.94, /20 = 66.25, /30 = 93.15, /1» =100065, /2» = 141345, /3» = 93650,

0.75 X -1

Fi = max ^ ( y (G(1, s) + 3Gi(1, s)^(100065)j

0.25

(0.15 x —1

256 /(g(1, s) + 2Gi(2, s))dS(141345)j ,

0.25

/ 0.75 \ —M

[2k j (g(1, s) + (0.5)Gi (1, sfjds(93650) j I

0.25

0.0031692, 0.0032499, 0.0149956

0.25

max 0.0031692, 0.0032499, 0.0149956 0.0149956

F2 = min {0.019158895, 0.022641543, 0.0368069256} = 0.019158895.

Applying Theorem 3,1, we get an optimal eigenvalue interval 0.0149956 < Xi < 0.019158895, = 1, 2, 3

by choosing u:, u, and /3 are sufficiently small.

Example 2. Here we consider the iterative system of third order three-point non-homogeneous boundary value problem

y'" + Xi/i(y2(t)) = 0, t e [0,1],

y'2' + X,/2(y3(t)) = 0, t e [0,1], (4.3)

1/3" + Xsf3(yi(t)) = 0, t e [0,1],

2/l(0) = 0, 3/1(0) = 0, 2y'i(1) - 3yl(2)=Ui,

3/2(0) = 0, y,(0) = 0, 3y2(1) - 4^2)=U2, (44)

2/3(0) = 0, 2/3(0) = 0, 3y3(1) - 2^2)=U3,

where

/1(2/2) = 2/2(11 + 1001 e—2y2 )(11 + 1101 e—4y2), /2(2/3) = 2/3(21 + 1011 e—3yi )(21 + 1111 e—5y3), /3(3/1) = yi(31 + 1021 e—4yi )(31 + 1121 e—6yi),

and

Pi(t) = P,(t) = j?3(i) = 1.

By direct calculation, we find that

/10 = 1125344, /20 = 1168224, /30 = 1211904, /1» = 121, /2» = 441, /3» = 961,

F3 = max {0.00028178, 0.0003932, 0.00115891} = 0.00115891,

F4 = min {0.009015,0.0034013, 0.0035676} = 0.0034013.

Applying Theorem 3,1, we get an optimal eigenvalue interval 0.00115891 < Xi < 0.0034013, i = 1, 2, 3 for which boundary value problem (4,3)-(4,4) has at least one positive solution once u^d /3 are sufficiently small.

Acknowledgements The authors thank the referees for their valuable suggestions and comments,

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Kapula Eajendra Prasad Department of Applied Mathematics, College of Science and Technology, Andhra University, Visakhapatnam, 530 003, India E-mail: rajendra92@rediffmail. com

Mahantv Rashmita Department of Applied Mathematics, College of Science and Technology, Andhra University, Visakhapatnam, 530 003, India E-mail: rashmita.mahantySgmail. com

Sreedhar Namburi, Department of Mathematics, Institute of Science, GITAM Deemed to be University), Visakhapatnam, 530 045, India E-mail: sreedharnamburil3@gmail. com