MULTIPLE SOLUTIONS FOR NONLINEAR EIGENVALUE PROBLEM Текст научной статьи по специальности «Математика»

DIFFERENTIAL EQUATIONS AND

CONTROL PROCESSES N4, 2009 Electronic Journal, reg. N P2375 at 07.03.97 ISSN 1817-2172

http://www.neva.ru/journMl http://www.math.spbu.ru/diffjournal/ e-mail: jodiff@mail.ru

Multiple Solutions for Nonlinear Eigenvalue Problem 1

Victor F. Payne Senior Lecturer, Department of Mathematics, University of Ibadan, Ibadan, Nigeria. E-mail: vfolayapayne@yahoo.com Tel: +2348023902335

Abstract: We obtain a sequence of eigenvalues {Xjfor which there exist countably many positive solutions of a fourth-order, singular boundary value problem. These positive eigenfunctions correspond to the sequence of eigenvalues X > 1 (counting multiplicity). The methods involve application of the fixed point arguments of the Krasnosel'skii cone expansion-compression type and Holder's inequality.

Keywords: Singular boundary-value problem, Eigenvalue, Positive Solution, Holder's inequality, Multiple solution.

AMS(MOS) subject classification: 34B15, 34B16, 74K10. Introduction

1This work was done while the author was visiting Matematiska Institutionen of Stockholms Universitet, S-106 91, Sweden. He extends his thanks to all members of the Department for their warm hospitality during his stay.

In this work, we are concerned with the determination of a sequence of eigenvalues 0 < Ai < A2 < • • • < Aj_i < Aj • • • (counting multiplicity) for which the following singular boundary value problem which describes deformation of simply supported beam loaded uniformly in the transverse direction has countably many positive solutions:

w(4)(t) = Aa(i)/(i,w(i),w//(i)), t e (0,1) (1.1)

w(0) = w(1) = 0, w//(0) = w"(1) = 0 (1.2).

where

(Ei) f e C[(0,1) x (0, to) x (0, to), [0, to)] satisfies certain oscillatory growth condition

(E2) a(t) e L[(0,1)] for some p > 1 has countably many non-isolated singularities of the form 12 _ t|_e, e > 0 in [0, 2) which include the sequence {tj}TOLi such that 2 > ti > tj ^ to > 0, to e [0,2) with a(to) > 0 and lim a(t) < to V j = f, 2 • • •

t^-to

(E3) a(t) > c > 0 V t e [to, 1 _ to]

and A is a positive parameter whose values depend on tj (j = 1, 2, • • • ).

Singular boundary value problems describe many phenomena in applied sci-ences,particularly a vast class of elastic deflections. Equally numerous is the nature of techniques applied in the analysis of such problems by investigators. These techniques are dictated by the conditions imposed on the inhomogeneous term in the equation as well as the boundary conditions. The ancillary conditions in turn are prescribed by the mechanics of the problem. The literature reveals several forms of the equation studied. Amongst these is the existence and uniqueness result by Usmani (see [14] ) for the solution of the boundary value problem of the type:

w(4)(t) = /(t)«(t)+ s(t), t e (0,1) (1.3)

w(0) = wo, w(1) = w//(0) = wo, w//(0) = (1.4)

under the restriction sup |/(t)| < In his own case Afterbizadeh [1] proved an existence result under the sharp condition that f is a bounded function. The study has progressed to include the works of Del Pino and Manasevich [4], Yang [16], Hao [5], Ma and Wang [10], Agrawal and O'Regan [2] and other outstanding authors. Most of the contributions extended earlier works by im-

posing stricter conditions on the inhomogeneous term in the equations. Various methods employed include monotone [13], lower and upper solution techniques (11, [12]), operator approximation and degree theory ([6], [15]) and fixed point arguments of Leray-Schauder and Krasnosel'skii [8].

Recently there appears to be a fast growing interest in the investigation of existence result of countably many positive solutions of boundary value problems (see [5], [9], [13] and references therein). Therefore our motivation for this paper is in the direction of this trend. We obtain a sequence of eigenvalues for countably many positive solutions of (1.1)-(1.2) using Krasnosel'skii's cone type fixed point theorem and Holder's inequality. This work complements and extends known results. The outline of the paper continues in Section 2 with a presentation of preliminary notes involving the Green's function of (1.1)-(1.2). We also state a fixed point theorem due to Krasnosel'skii [7] which will be used to yield the multiple solutions of (1.1)-(1.2). We furnish an appropriate Banach space framework of cones in order to apply the fixed point arguments. Additional assumptions on f and standard results using Holder's inequality are also stated here. In the concluding Section, we state and prove the main result of the paper as well as furnish an example of functions a(t) that satisfies conditions (Ei) - (E3) 2. Preliminary notes

We begin this Section by defining a positive solution of (1.1) as a function

u(t) e C2[0,1] n C4(0,1) which satisfies (1.l)-(1.2) and u(t) > 0 for t e (0,1).

We assume additional conditions on a(t) and f as follows:

(E4) 3 a{(t) e C[(0,1), [0, to)] (i = 1, 2) and h(u) e C[[0, to), [0, to)] such that

ai(t)a(t) > maxie[0)i] fa(t) for fa(t) e p > 1 (i = 1, 2) with

fai(t)h(u) < f (t,u(t),u"(t)) < fa(t)h(u), u(t),u"(t) : [0,1) ^ [0, to)

The Green's function G(t, s) for

u№(t) = 0, t e (0,1) (2.1)

u(0) = u(1), u"(0) = u"(1) = 0 (2.2)

is

s(1 - t), 0 < s < t < 1 t(1 - s), 0 < t < s < 1

satisfying the following property:

G(t,s)

(E5) G(t,s) < G(s,s), 0 < s, t < 1

It is well known that the solutions of (1.1)-(1.2) are fixed points of the integral operator:

^(t) = L L G(t,s)^(s,r)a(r)f(r,w(r),w//(r))drds, (23)

t e [0,1]

Then the following statements are consequences of (^2) _ (^5) :

Ee)0 </ i G(s,s)G(s,r)^i(r)d<rds, / i G(s, s)G(s, r)^< to

Jo Jo Jo Jo

(E7) 3 to e (0,1) such that $(to) > 0 (f = 1, 2)

(Eg) minte[o,i] G(t,s) > to G(s,s) > to G(to, s) > 0, to e [0, 2].

It can than be seen that establishing the existence of positive solutions of (1.1)

- (1.2) is equivalent to proving the existence of fixed points of the operator

equation:

u(t) = ANu(t) (2.4)

We apply the following fixed point theorem to obtain positive solutions of (2.1)-(2.2) for certain values of A :

Theorem 1: (Krasnosel'skii) Let K be a cone in a Banach space E. Suppose are open subsets of E with 0 e e C and

N : K n —► K

is a completely continuous operator such that, either

(i) ||Nu|| < ||u||, u e K n and ||Nu|| > ||u||, u e K n or

(ii) ||Nu|| > ||u||, u e K n and ||Nu|| < ||u||, u e K n Then N has a fixed point in K n (^2)

Using (^8), we define a set of cones on the Banach space E = C2[0,1] endowed with the norm ||u|| = maxte[oi](|u(t)|, |u/(t)|, |u//(t)|). This is done by fixing to e [0, 2) and then defining Kto C E by

Kto = {u(t) e E : u(t) > 0, t e [0,1], min u(t) > to||u||}

te[o,i]

In the sequel, we claim in Lemmas 2 and 3 that the integral operator N is completely continuous and cone preserving i.e.

N : Kto Kto, to e [0, i)

is continuous and compact.

Lemma 2. The integral operator N : Kto —> Kto is continuous for every to e [0, 2).

Proof Assume u e Kto and a fixed value of to e [0, |.)

Let al{s)a{s)f (s,u(s),u"(s)) > maxsG[o,i] A(s)f (s, u{s), u"(s)) > 0 (i = 1, 2) and G(t, s) > 0 V t,s e [0,1], then using (^6) we have, Nu(t) > 0 V t e [0,1], u e Kto

By (2.3) and (^8) and for u e Kto the following inequalities hold:

min XNu(t) > min X / G(t,s)G(s,r)fa(r)h(u)(r)drds (2.5)

te[o,1] te[o,1] Jo Jo

> min xf i G(t,s)G(s,r)p1(r)h(u)(r)drds te[o,1] Jo Jo

> X /o1 /o1 G(t,s)G(s,T)Pi(T)h(u)(r)drds

> Xto fd fd G(to, s)G(s, t)^1(t)h(u)(T)dTds

= XtoNu(to) V to e [0,1] to = to (2.6)

Therefore, min XNu(t) > Xto^Nu^. It follows that XNu : Kto —> Kto is te[o,1]

continuous and this concludes the proof of the Lemma. □

Lemma 3. The operator N : Kto —> Kto is compact for each to e [0,1) Proof: We prove this assertion by taking any family of bounded sets of E and show that N(Kto) is uniformly bounded and equicontinuous. Without loss of generality, we assume that R > 0 with ||u|| < R V u e Kj. C E. Now set m = maxtt€[o)fi] |h(u)| for each u e K. and we have

||Nu| < | i G(s,s)p2(s)h(u)(s))dsl< mi G(s, s)G(s,T)dT oo

This means N(Kto D N(K.o) is uniformly bounded. Next we have that V u e Kto,

|(Nu)/(t)| = | i i sG(s,r)a(r)f(r,u(r),u//(r)dTds oo

i f i

(1 _ s)G(s,r)a(r)f (r,u(r),u//(r)drds| (2.7)

to

< | / / sG(s,r)a(r)f(t,u(r),u//(r to

+ i i (1 _ s)G(s,T)a(T)f (t, u(t), u//(t))^tds| f f o o

< | / i sG(s,T(t)^(u(t to

+ / i (1 _ s)G(s,T)&(t)%(t))rfTrfs|

to

< max ^(u(t))| / / sG(s,T)^2(t «e[o,fl] Jo Jo

+ / i (1 _ s)G(s,T)&(t)^Tds| (2.8)

to

Now set

$(t)= i i sG(s,T)&(t)dTds + i i (1 _ s)G(s,T)&(t)dTds o o t o

Integrating the last expression and reversing the order of integration in the

resulting expression we obtain:

i |$(t)|dt < iff №(r )drdsdt

Jo Jo Js Jo

/>1 />s p1

+ / ds / / G(s,r)£2(r)drdt Jo Jo Jo

< 2 f s(1 - s) / G(s,r)&(r)drds

< 2/ G(s,s)G(s,r(r)drds< 00 (2.9)

o

From (2.6) and (2.7), we have 0 < / |(Nu/(t)| < 0. This shows that the last

o

integral is absolutely continuous and for 0 < i1 < t2 < l, u G ^ C E we have |N(u)(ii) - N(u)fo)| = if2 N(u/(t)dt| < f 2 |(N(u/(t)|dt.

Jtl Jtl

Thus N) D N(^o) is equicontinuous. An appeal to the Arzela-Ascoli theorem then results in N being compact as claimed. □.

In the main result, we will require the notion of the Holder's inequality so as to establish certain norm inequalities in our proof. The reader is referred to the details of the theory in Deimling [3]. Some useful properties of the Green's function in Lq[0, l] is stated hereunder.

Lemma 4 Given q > 0, (q-1 + p-1 = l) with G(t,.) G Lq, then

max ||G(i,.)||q = i ^9 (2.10)

ie[o,1]1 v ;||q 4V1 + q/

and

max ||G(t,.)||o = 7 (2.11)

iG[o,1] 4

Proof Assume that to G [0, 2). This yields

f t0(G(t, s))qds = (tq(1 - t)q) - (*o)(q+1)((1 - t)q + tq)) (2.12)

which on setting to = 0, yields

ji1 G(t, s))qds = (j^) tq(1 - t)q (2.13)

from whence (2.10) follows. Letting q —> to in (2.10) yields (2.11) □

3. Main Result

In this section, we state and prove our main result as the following:

Proposition 1. Let ai(t)a(t) = fa(t) (i = 1, 2) satisfy (^2) — (E4) and let the sequence {(to)jbe defined by tj > (to)j > tj+1,j = 1, 2, • • •. Furthermore, suppose the sequences {£j}jXL1, {Vj}j=L1 are such that £j > Dllj > Vj > (to)jVj >^jj = 1, 2 • • • where

f 4 , 16 "I D = max{-4} - t0 E [0 1] ^

and f (t,u,u") < h(u) satisfies the following oscillatory growth condition for each j E N:

16

(Eg) f (t,u,u") < h(u) < M£j V u E [0- , ] where M < ^^^ and

(E10) DVj < f (t,u,u") < h(u) V u E [(to)jVj,Vj] *

Then the boundary value problem (1.1) - (1.2) has infinitely many solutions

{uj}j<L1. Moreover, Vj < ||uj || < £j for every j = 1, 2, • • • and

_1_ X

p 1—(H r 1 < X

/ / G(t, s)G(s,r)&(t)(h — 6)(u(r))drds J(t 0)j Jo

< -p-x-1-E A. (3-2)

/ / G(s, s)G(s, t)Pi(t)(h — 6)(u(r))drds oo

where A = {A : 0 < A1 < A2 < • • • < Xj < • • • } for each tj. Proof: The proof of the proposition is in two parts :

(I) existence of countably many positive solutions {ujand

(II) existence result for infinitely many corresponding eigenvalues {Aj}jl1 Part I

Define the sequences of open subsets of E by

Q1:j = {u E E : ||u|| < £j} ^2,j = {u E E : ||u|| < Vj}

Suppose {(to) ■ }0L1 is as in the hypothesis and 1 > tj > (to) ■ > tj+1 > to j j 2 V j G N. Next define the cone Xj by

X = {u(t) G E : u(t) > 0, t G [0,1), min , ,u(t) > (to)j||u||} Keep j fixed and let u G Xj R . For s G [(to)j, 1 - (to)j]

^ = ||u| > u(s) > min u(s) > (to)j||u| = (to)j^ (3.3)

tG[(io)J- ,1-(io)J-]

Using (^ 1o), we have

||ANu|| = A max / / G(t,s)G(s,r)a^(r)a(r)/(r,u(r),u//(r))drds ^G[o,1] Jo Jo

r MM, /• 1

> A max / G(t,s)G(s,r)#(r)^(u(r))drds (3.4)

tG[o,1] J(to), ./o

f MM, f1

> A max / / G(t,s)G(s,r)drds^2(r

tG[o,1] J(to), ./o

f MM, f1

> A / max ^2(rdr / G(t, s)G(s, s)ds

./(to),- tG[o'1] ./o

f1 f 1-(M,

= A / max dr / G(t, s)G(s, s)ds ./o tGM J(to),

r MM, Z-1

> A max ^2(r/ G(s,t)dr / toG(to,s)ds

tG[o'1] ./(to), ./o

r MM, r 1

> AD^jC1 max G(t,s)G(i°,s)dW todr (3.5)

tG[o'1] ./(to), ./o

Using (E4), (2.12) with q = 1, we have

f 1-(to)j f1

A||Nu|| = A max / G(t,s)toG(to,s)ds D^3cxdT tEt0'1^(to)J- Jo

= A max]1[t(1 — t) — ((to)j)2][to(1 — to) — ((*,),.)2]Dvd £ todT = A ¿(i — ((to)j)2) DvjC1 to (3.6)

But (to)J <t1 < 2, and as such (4 — ((to))j)2) > (4 — (t^2) > 0.

f 4 1

Since we have D > < -——:--—— >, it therefore follows that

"I Mo( 4 — (to2)/'

A||Nu|||| > 1 C1to( 1 — (t1)^ DVj > Vj = ||u|| (3.7)

The next step proceeds as follows:

Let u E Xj R 6Q1;j then u(s) < ||u|| = £j V s E [0,1]. Using (^g), we have that

A||Nu|| = A max / / G(t,s)G(s,T)ai(T)a(T)f (t,u(t),u"(t))dTds tE[o,1]Jo Jo

f 1 — (to), f1

< A max / / G(t,s)G(s,T)(31(t)h(u(T))dTds

tE[°,1]J(t o))3- ./o

f 1 — (to), f 1

< A max G(t,s)G(s,T)ds ^(t)dTM£3

tE[o'1] AH- ^

/• 1—(to), r 1

< A M£j max / G(t,s)G(to,s)ds A(t)todT (3.8)

tE[o'1^(to)J. ./o

There are two cases to consider: Case 1, p =1

Let p = 1, so that q = . Using Lemma 4, we have

A||Nu|| < A max ||G(t,.)||q||G(to,.)||q||^1|q||A||PM£jto (3.9)

tE[o,1]

Using (2.10) and (^9), we obtain

2

A||Nu|| < A^ (^J ' II^iMp^Cî < ^^M«,-i0 < A«, (3.10)

Case 2, p =1.

By (2.11) and (E9), we obtain

A||Nu|| = < A max i'(A)*(r)(A)*(t)^t||G(i, .)|U|G(i0, OIUI M«, ie [0,1^0 1°

< AIIAIU"%C, < AC, (M < fHi;)

(3.11)

In either case and because ||u|| = C, V u e X, R , we have

A||Nu|| < ||u||. (3.12)

But 0 e ^2,, C C ^i,,. Therefore by (3.7), (3.12) and Theorem 1 N has a fixed point u, e X, R (Ù1,,\^2,,) such that < ||u,|| < C,. But j e N is arbitrary thus completing the proof of the first part of the Proposition.

Part II.

The arguments in the second part run as follows: Let Aj be given as in (3.2) and choose 5 > 0 such that

1

r 1-(to), C1 - Aj

/ / G(t,s)G(s,r)ft(r)(fo - 5)drds

./(to), ./o

1

i f G(s,s)G(s,r)&(r)(fo + 5)drds oo

Let N be the cone preserving, continuous and compact operator defined in (2.3). Assume 3 > 0 such that fo(u) — (fo + 5)u whenever fo(u) G [0, to) for u G [0, ^1]. We then obtain for u G Xto, ||u|| = tj G [0,1],

ANu(t) — A i i G(s,s)G(s,r)^(r)fo(u(r))drds oo

— A / i G(s,s)G(s,r)&(r)(fo + 5)u(r)drds (3.13)

oo

— a( i G(s,s)G(s,r)A(r)(fo + 5)drds||u|| — ||u||

oo

Define E1 c E = {u G E : ||u|| — F1}. We have A||Nu|| — ||u|| V u G XtoR5E1 It remains to consider the case when 3 > 0 such that fo(u) > (fo -5)u, (fo is bounded), whenever fo G [0, to), for u G [0,#2). We have that

u E Kto, ||u|| < H2 V tj E [(to)j, 1 — (to)j] which leads to

ANu(t) < A i i G(t,s)G(s,T)A(t)h(u(T))dTds oo

r 1—(to), r 1

< A / / G(t,s)G(s,T)A(t)h(u(T))dTds

./(to )j Jo

f 1—(to), r 1

< A / / G(t,s)G(s,T)A(t)(h — 6)u(t)dTds

./(to )j Jo

r 1—(to), r 1

< a/ / G(s,s)G(s,T)fa(T)(h — 6)dT ||u|| < ||u||.(3.14)

./(to )j Jo

Define E2 c E = {u E E : ||u|| < H2}, hence by the above inequality, we obtain, V tj E [(to)j, 1 — (to)j] u E Kto R 6^2 and A||Nu|| > ||u|| from which the result follows by (3.13) and (3.14). □

Example : As our example, we define a 1-parameter class of functions a(t; e) : [0,1] ^ M+ by

to ]

a(t; e) = ^^ j, j~ where we have set Tj = (to)j j=1 1 j 1

and

1 1

32 + j+32

tj = 7^ + , j > 1

1 .11.1 1 To = 1 and Tj = 2(tj + tj+1) = 32 + 2( JT32 + j +12 + 32), j > 1

We have that

90 1

a(t : e) > a(1; e) = (-)e > 0 tj | 9

Now if 0 < e < 1, we show that

a(t) E L1(0,1) (and so aia(t) E L1(0,1) (i = 1, 2) as follows:

Consider

to „ 1 r 1 TO

y f f,, rj-1] dt = y P' |t - tj

^Jo |t - tj|« £I, 1 j 1

TO /»t - i*T' '

= '[/' (tj - t)-edt + '- (t - tj)-edt]

j=1T' Jt' 1 [(J1)1-+(s)1-e]

1 - eLV22a 90 11 TO 1 1

+ — R^V [(—___1_u-£

+ 1 - e(2) j +32 j + 12 + 32

+ (j+22 - j+32 )1 £]

<

2 2e 1

+ 2(1 — e) S

1 - e 2(1 - e) j=£L(j + 32)1-

11

+ (j+22)'-e )]

<

<

(j+12+32)1-e (j+22)1-

2 2e ^ 1 1 +1-^ ^ (j + 32)1- (j + 22)1-

2 2£ y. 1

1 - e + 1 - e ^ j 2 (1-^)

j=2

We have that

TO

2e ^ 1 3n , 1 _ 1

1-/ —V converges since -(1 - e) > 1 for e < -

1 — e —' j 2(1-e) 2 3

j=2

and it therefore follows that

TO r 1 r 1

y n[T, - Tj-1] dt

f^Jo |i - ij

TO r1

V^1 I K[rj - r?'_ 1] converges. Thus j —^—^p— dt

j=1

/> 1 TO

V- Mrj - rj-1] dt Z^ |t_t.|e dt

ro -=1 |t - tj 1

a(t)dt,

o

1

showing that a(t; e) G L1[0,1]. By induction on p G N, it can be shown that a(t; e) G L^[0,1] for e G (0, 3p). In this general framework, it is also necessary

to relax condition (S3) to read 0 < c < a(t) on A (of positive measure) C [0,1] such that 1 E A. We next define

f (t,u,u") < h(u) = <

M, u>^1

Dm +

(u — Vj), m < u < &,j E N

MCj+1 + DV—Sr1, <u < Tj Vj ,j E N

Tj Vj 1

0, u = 0

1

1

-, j E IN and

where = 1, = 3 + (j + 1)2, V> 32 + (32 + j2 + 1) since the conditions ^j > Dvj > TjVj > Cj+1 j > 1 on the sequences {ij}j=1 and {vj}to=1 ^ Cj i 0, and Vj i 0, we thus have that for A > 1, the solutions uj of (1.1) - (1.2) satisfy ||uj|| i 0 and (3.2) holds for each tj

4. Acknowledgement.

The author appreciates immensely the useful comments and suggestions of Professor A. Szulkin of Matematiska institutionen, Stockholm University. He is also grateful to him for his help in facilitating his stay in Stockholm. The author also immensely appreciates the useful corrections of the referees.

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