Joural of Applied Mathematics & Bioiformatics, vol.5, o., 5, 85-97 ISSN: 79-66 (prit), 79-699 (olie) Sciepress Ltd, 5 Positive solutios of sigular (k,-k) cojugate eigevalue problem Shujie Tia ad Wei Gao Abstract Positive solutio of sigular oliear (k,-k) cojugate eigevalue problem is studied b emploig the positive propert of the Gree s fuctio, the fixed poit theorem of cocave fuctio ad Krasoselskii fixed poit theorem i coe. Mathematics Subject Classificatio: B8 Kewords: Eigevalue; Positive solutio; Fixed poit theorem Itroductio This paper deals with the followig sigular (k,-k) cojugate eigevalue problem k ( ) ( ) ( x) λhx ( ) f( ) x, = < < (.) School of Mathematics ad Statistics, Northeast Petroleum Uiversit, Daqig 68, Chia. Ε-mail: outset6@6.com School of Mathematics ad Statistics, Northeast Petroleum Uiversit, Daqig 68, Chia. Ε-mail: hit_gaowei@6.com Article Ifo: Received : March, 5. Revised : April, 5. Published olie : Jue, 5.
86 Positive solutios of sigular (k,-k) cojugate eigevalue problem () i ( j) () =, () = i k, j k, (.) Where k is a positive umber ad λ > is a parameter. If the cojugate eigevalue problem (.), (.) has a positive solutio x ( ) for a particular λ, the λ is called a eigevalue ad x ( ) a correspodig eigefuctio of (.), (.). Let Λ be the set of eigevalue of the problem (.), (.), i.e. Λ = { λ > ; (.), (.) has a positive solutio}. I recet ears, the cojugate eigevalue problem (.), (.) has bee studied extesivel, for special case of λ =, the existece results of positive solutio of the problem (.), (.) has bee established i [-6], ad as for twi positive solutios, several studies to the problem (.), (.) ca be foud i [7-9]. For the case of λ >, eigevalue itervals characterizatios of the problem (.), (.) has bee discussed i [] b usig Krasoselskii fixed poit theorem if f( ) is superliear or subliear. I this paper, b emploig propert of Gree fuctio, the fixed poit theorem of cocave fuctio ad Krasoselskii fixed poit theorem i coe, we give eigevalue characterizatios uder differet hpothesis coditio, ad we ma allow that f( x ) is sigular at x =,. B usig differet method from [] we establish ot ol existece of positive solutio but also multiplicit of positive solutios of the problem (.), (.). Our assumptios throughout are: (H ) hx ( ) is a oegative measurable fuctio defied i (,) ad do ot vaish ideticall o a subiterval i (,) ad k k k k ; < s ( s) hs ( )d s< s ( s) hs ( )ds< + (H ) f :[, +) [, +) is a odecreasig cotiuous fuctio ad f( ) > for > ; f( ) (H ) f = lim =, f f( ) = lim =.
Shujie Tia ad Wei Gao 87 B a positive solutio xof ( ) the problem (.), (.), we meas that x ( ) satisfies holds; (a) x k k ( ) C [,) C (,] C (,), f( ) > i (,) ad (.) (b) ( ) ( x) is locall absolutel cotiuous i (,) ad k ( ) ( ) ( x) λhx ( ) f( x ( )) = a.e. i (,). The mai results of this paper are as follows. Theorem Assume that (H ), (H ) ad (H ) hold. The there exist two positive umbers λ, λ with < λ λ < + such that (i). (.),(.) has o positive solutio for λ (, ) ; (ii). (.),(.) has at least oe positive solutio for λ ( λ, λ ]; (iii). (.),(.) has at least two positive solutios for λ ( λ, + ); (iv). (.),(.) has oegative solutio for λ =. λ λ Prelimiar Notes I this sectio, we provide some properties of the Gree s fuctio for the problem (.),(.) which are eeded later, ad state the fixed poit theorems required. As show i [6], the problem (.),(.) is equivalet to the itegral equatio where x () = λ Gxshs (,)() f( s ())d, s (.)
88 Positive solutios of sigular (k,-k) cojugate eigevalue problem Gxs (, ) s( x) k k t t+ x s t s x ( k )!( k )! = x( s) k k t t+ s x t x s ( k )!( k )! ( ) d,, ( ) d,. Moreover, the followig results have bee offered b Kog ad Wag [6]. Lemma. For a xs, [,], we have (.) α() xgs () Gxs (,) β() xgs (), (.) where g() s, s x, Gxs (, ) s x g() s, x s. s (.) x α( x) = k ( x) k k k k k x ( x) s ( s), β ( x) =, gx ( ) =. mi{ k, k} ( k )!( k )! Let K be a coe i Baach space E. We sa that a mapψ is a oegative cotiuous cocave fuctio o K, if it satisfied: Ψ : K [, +) is cotiuous ad Ψ ( α x+ ( α) ) αψ ( x)+ ( α) Ψ ( ) for all x, Kad α. Theorem [] Le K be a coe i Baach space E. For give R >, defie R { ; } K = u K u < R. Assume that T : KR KR is a completel cotiuous operator ad Ψ is a oegative cotiuous cocave fuctio o Κ such thatψ ( ) for all KR. Suppose that there exist < a< b Rsuch that (A) { K( Ψ, ab, ); Ψ ( )> a} φ, ad T a K( Ψ, ab, ) { K; Ψ a; b} Ψ ( )> for all K( Ψ, ab, ) where = ( ) ; (B) T < r for all Kr ;
Shujie Tia ad Wei Gao 89 (C) Ψ ( T)> a for all K( Ψ, ar, ) with T > b, the T has at least three fixed poits, ad i { Ψ, Ψ } K( ar, ); ( )> a ad K \( K( Ψ, ar, ) K). R K satisfig K r, r R Theorem [] Let E be a Baach space, ad K E a coe i E. Assume Ω, Ω are ope subset of E with Ω, Ω Ω, ad let Φ: K Ω ( \ Ω) Kbe a completel cotiuous operator such that either (I) Φu u for u K Ω, ad Φu u for u K Ω ; or (II) Φu u for u K Ω, ad Φu u for u K Ω, the Φ has a fixed poit i K Ω ( \ Ω ). Mai Results α Letα = mi α( x), β = max β( x) adγ =. Defie the coe i Baach space x [,] β C [,] give as We defie the operator T : P P= { x ( ) C[,]; x ( ) }, K= { x ( ) C[,]; mi x ( ) γ }. Pb x, ( T)( x) : = λ G( x, s) h( s) f ( ( s))ds. (.) Lemma. Suppose that (H )-(H ) hold. The T : P P cotiuous mappig adt(k) K. Moreover, for K we have k- -k- - ( T)( x) C [,) C (,] C (,), is a completel
9 Positive solutios of sigular (k,-k) cojugate eigevalue problem T x = λh x f x x, k ( ) ( ) ( ) ( ) ( ) ( ( )) a.e. (,) () ( ) ( ) i j T (),( T) (), i k, j k Proof We ol prove T( K) be foud i [6]. = =. For K, b emploig (.) we have this implies T( K) K. The proof of the remaider of Lemma. ca mi( T)() x λ mi α() s g()() s h s f ( ())d s s λα max β x [,] = γ T, Gxshs (,)() f( s ())ds K.The proof is complete. It follows from the lemma. that we kow that T( K) K of T is a solutio of the problem (.),(.) ad vice versa. K ad fixed poit i Lemma. Suppose that (H ), (H ) ad (H ) hold. The there exists < λ < + such that the problem (.), (.) has at least two positive solutios for λ ( λ, + ). f( ) Proof It follows from lim = that there exists R > such that f( ) ε for + all R, where ε satisfies ελβ gshs ()()ds<. Let M = max f( ), R the f( ) M + ε for all. B (H ) we get lim = lim = +, f( ) + f( ) thus, there exists < b < + such that b f b = mi, ad hece there f ( ) ( ) exists < λ < + such that have ab λ = [ gshs ()()d] s f( b). Clearl, for all we
Shujie Tia ad Wei Gao 9 b f( ). (.) f( b) We shall ow show that the coditios of Theorem are satisfied. Choose R max{ b, Mλβ( ελβ gshs ()()d) s gshs ()()d} s = +. For KR this shows T( K ) K K. R R R Let : K [, ) Ψ + be defied b x [,] ( T )( x ) λ max G ( x, s ) h ( s ) f ( ( s ))d s x [,] λ max β () x gshs ()()( M+ εs ())d s λβ( M+ ε ) gshs ()()ds = λβ( M+ ε R) gshs ()()ds< R, Ψ ( ) = mi x ( ), we have clearl,ψ is a oegative cotiuous cocave fuctio o K such that Ψ ( ) for all K. It is oted that x ( ) = ( γb+ b) { K( Ψ, γbb, ); Ψ ( ) γb} φ, let K( Ψ, γbb, ), the mi x ( ) = Ψ ( ) γbad b. Usig this together with (.), for λ > λ we get Ψ ( T)( x) = mi ( T)( x) α λ mi () x gshsf ()() ( s ())ds λαb f( b) gshss ()()()ds
9 Positive solutios of sigular (k,-k) cojugate eigevalue problem λαγ b f( b) λγb = > γ b. λ gshs ()()ds Hece, coditio (A) of Theorem is satisfied. f( ) Blim =, there exists < r < γ bsuch that f( ) < ε for all r, where ε satisfies ελβ gshs ()()ds<. For Kr we have, T λβ gshsf ()() ( s ())d s ελβ gshs ()()ds< r this implies that coditio (B) of Theorem is satisfied. Fiall, for K( Ψ, γbb, ) with T > b, we obtai Ψ ( T)( x) = mi ( T)( x) λ mi α() x gshsf ()() ( s ())ds αλ max β x [,] Gxshsf (,)() ( s ())ds = γ T > γb. Therefore, the coditio (C) of Theorem is also satisfied. Cosequetl, a applicatio of Theorem shows that the problem (.), (.) has at least three solutios,, KR. Further, K, { K( Ψ,γbR, ); Ψ ( )> γb} ad K \( K( Ψ,γbR, ) K). This r shows that ( x) ad ( ) x are two positive solutio of the problem (.), (.), ad R r ( ) x is a oegative solutio of (.), (.). Lemma. Suppose that (H ), (H ) ad (H ) hold. If λ is sufficietl small,
Shujie Tia ad Wei Gao 9 the λ Λ. Proof If λ Λ, the the problem (.),(.) has a positive solutio λ ( x) K ad it satisfies (.). We ote that (H ) implies the existece of a costatη > such that f( ) η for all. B emploig (.) we have λ λβ gshsf ()() ( λ())d s s λβη gshs ()()ds, λ this meas λβη gshs ()()ds, which cotradicts with λ sufficietl small. Lemma. Suppose that (H ), (H ) ad (H ) hold. The there exists a λ > such that[ λ, +) Λ. Proof Let K { K, } for K K = <, choose ad λ [ λ, ) (H ) we have - = ( f( ))( gshs ( ) ( )d s), λ α γ +, the mi x ( ) γ = γ, b usig (H ) ad x, α x, T λ mi () x g()() s h s f ( ())d s s λα γ = =, f() gshs ()()ds i.e. T for K K. It follows from (H ) that there exist R > such that f( ) where ε satisfiesελ β gshs s < ()()d. Let R= R + λβ max f ( ) gshs ()()d s, R ε for all R,
9 Positive solutios of sigular (k,-k) cojugate eigevalue problem ad KR { K, R} = <,the for K KR, we have T λβ g()() s h s f ( ())d s s λβ[ gshs ()() f( s ())d s+ gshs ()() f( s ())d] s ( s) R R ( s) R λβ(max f( ) + εr) gshs ()()ds R < λβmax f( ) gshs ()()ds+ R R = R=, i.e. T for K KR. I view of the Theorem, we kow that T has a fixed poit xi ( ) K ( K \ K). That is to sa, the itegral equatio (.) has at least R oe positive solutio x ( ), ad hece x ( ) is a positive solutio of (k,-k) cojugate eigevalue problem (.),(.). Lemma.5 Suppose that (H ), (H ) ad (H ) hold. The the problem (.),(.) has a oegative solutio for λ =. λ Proof Without loss of geeralit, let { λ } = be a mootoe decreasig sequece, lim λ =, ad{ λ ( x)} = be correspodig positive solutio sequece λ where λ Λ. We claim that{ λ ( x)} = is uiforml bouded. If it is ot true, the lim λ f( ) = +. It follows from lim = ; that there exists M > such + that f( ) M + ε for all, whereε satisfiesελβ gshs ()()ds<. So we get λ λβ ()() ( ())d gshsf λ s s
Shujie Tia ad Wei Gao 95 λβ ( M+ ε λ ) gshs ()()d. s (.) Let i (.) to ield lim λ λβ gshs ()()ds < + ελβ gshs ()()ds which is a cotradictio. Thus, there exists a umber L with < L < + such, that λ Lfor all. It follows from (H ) ad (.) that we have Gxs (, ) λ λ ( ) ( ( ))d hs f s s λ x ( ) ( s ( s) hs ( )d s+ s ( s) hs ( )d s) ( k )!( k )! λ f L x k k k k x λ f( L) k k s ( s) hs ( )d s: Q, = ( k )!( k )! this show that { λ ( x)} = is equicotiuous. Ascoli-Arzela theorem claims that { λ ( x)} = has a uiforml coverget subsequece, deoted agai b { λ ( x)} =, ad { λ ( x)} = coverges to () x uiforml o [,]. Isertig ( x) ito (.) ad lettig, usig the Lebesgue domiated λ covergece theorem, we obtai () x = λ Gxshsf (,)() ( ())d s s, thus, ( x) is a oegative solutio of (.),(.). Remark It is possible that ( x) =. Let λ = if Λ, the from Lemma. ad lemma. we kow λ > ad < λ λ < +. So (k,-k) cojugate eigevalue problem
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