Correspondence should be addressed to Yanmei Sun,

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1 Journal of Applied Mahemaics Volume 22, Aricle ID , 7 pages doi:.55/22/ Research Aricle Exisence of 2m Posiive Soluions for Surm-Liouville Boundary Value Problems wih Linear Funcional Boundary Condiions on he Half-Line Yanmei Sun and Zengqin Zhao 2 Deparmen of Mahemaics and Informaion Sciences, Weifang Universiy, Shandong, Weifang 266, China 2 Deparmen of Mahemaics, Qufu Normal Universiy, Shandong, Qufu 27365, China Correspondence should be addressed o Yanmei Sun, sunyanmei29@26.com Received January 22; Acceped 5 March 22 Academic Edior: Giuseppe Marino Copyrigh q 22 Y. Sun and Z. Zhao. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. By using he Legge-Williams fixed heorem, we esablish he exisence of muliple posiive soluions for second-order nonhomogeneous Surm-Liouville boundary value problems wih linear funcional boundary condiions. One explici example wih singulariy is presened o demonsrae he applicaion of our main resuls.. Inroducion In his paper, we consider he following Surm-Liouville boundary value problems on he half-line ( pu Φf (, u,u, <<, α u β lim pu Tu,. α 2 lim u β 2 lim pu Ku, where f : R R R R is a coninuous funcion, f / on any subinerval of R, here R, ; Φ : R R is a Lebesgue inegrable funcion and may be singular a

2 2 Journal of Applied Mahemaics ; p CR,R C R, ds/ps < ; α i,β i i, 2 wih α β 2 α 2 β α α 2 ds/ps; T, K are linear posiive funcionals on CR T, K which are called posiive if Tu, Ku foru CR. The heory of nonlocal boundary value problems for ordinary differenial equaions arises in differen areas of applied mahemaics and physics. There are many sudies for nonlocal, including hree-poin, m-poin, and inegral boundary value problems on finie inerval by applying differen mehods 3. I is well known ha boundary value problems on infinie inerval arise in he sudy of radial soluions of nonlinear ellipic equaions and models of gas pressure in a semi-infinie porous medium 4 6. Bu he heory of Surm- Liouville nonhomogeneous boundary value problems on infinie inerval is ye rare. The linear funcional boundary condiions cover some nonlocal hree-poin, m-poin, and inegral boundary condiions. In 7, Zhao and Li invesigaed some nonlinear singular differenial equaions wih linear funcional boundary condiions. However, he differenial equaions were defined only in a finie inerval. Recenly, Liu e al. 6 sudied muliple posiive soluions for Surm-Liouville boundary value problems on he half-line ( pu mf, u, <, α u β lim pu,.2 α 2 lim u β 2 lim pu. However, he auhors did no consider he case when Surm-Liouville boundary value problems are nonhomogeneous. Therefore BVP. is he direc exension of 7. Soiis worhwhile o invesigae BVP.. We denoe a β α ds ps, b β ds 2 α 2 ps,.3 u aby, u y 2, f (, u, u Ψ (, y,y 2, a lim a β, b lim b β ds 2 α 2 ps, a lim a β α ds ps, b lim b β 2..4 In his paper, we always assume ha he following condiions hold. H Ψ, y,y 2 qqy,y 2,q CR,R,Qy,y 2 CR R, R and Φsqsds <. H 2 For any consan τ,, < Taτ <, < Kbτ <and Tbτ Δ Kbτ Taτ Kaτ >..5

3 Journal of Applied Mahemaics 3 Moivaed and inspired by 5 9, we are concerned wih he exisence of muliple posiive soluions for BVP. by applying Legge-Williams fixed heorem. The main new feaures presened in his paper are as follows. Firsly, Surm-Liouville nonhomogeneous boundary value problems wih linear funcional boundary condiions are seldom researched, i brings abou many difficulies when we imply he inegral equaions of BVP.. To solve he problem, we use a new mehod of undeermined coefficien o obain he inegral equaions of boundary value problems wih nonhomogeneous boundary condiions. Secondly, we discuss he exisence of riple posiive soluions and 2m posiive soluions of BVP.. Finally, he mehods used in his paper are differen from, 6, 7 and he resuls obained in his paper generalize and involve some resuls in 5. The res of paper is organized as follows. In Secion 2, we presen some preliminaries and lemmas. We sae and prove he main resuls in Secion 3. Finally, in Secion 4, one example wih a singular nonlineariy is presened o demonsrae he applicaion of Theorem Preliminary In order o discuss he main resuls, we need he following lemmas. Lemma 2.. Under he condiion ds/ps < and >, he boundary value problem ( pu y, <<, α u β lim pu Tu, 2. α 2 lim u β 2 lim pu Ku, has a unique soluion for any y L,. Moreover, his unique soluion can be expressed in he form u G, sysds A ( y a B ( y b, 2.2 where G, s, Ay, and By are defined by { abs, s<, G, s asb, s<<, 2.3 ( A ( y T Gτ, sysds Tbτ Δ (, K Gτ, sysds Kbτ a ( B ( y K Gτ, sysds Kbτ Δ (. b T Gτ, sysds Tbτ

4 4 Journal of Applied Mahemaics Proof. a and b in.3 are wo linear independen soluions of he equaion pu, so he general soluions for he equaion pu y can be expressed in he form u G, sysds Ca Db, 2.4 where C, D are undeermined consans. Through verifying direcly, when C and D saisfy a and b separaely, u in 2.4 is a soluion of BVP2.. Now we need o prove ha when u in 2.4 is a soluion of BVP2., C and D saisfy a and b separaely. Le u G, sysds CaDb be a soluion of BVP2., hen u asbysds absysds Ca Db, u b asysds a bsysds Cα p Dα 2 p ( α2 asysds α bsysds Cα Dα 2, p 2.5 ( pu α 2 ay α by α 2a α b y y. Tha is, pu y. By 2.4, we have u β u ( α p u β 2 u bsysds Cβ Dβ 2 Dα 2 bsysds Cα Dα 2, asysds Cβ Dβ 2 Cα ( α 2 p ds ps, ds ps, asysds Cα Dα 2, 2.6 hen D T C K ( ( Gτ, sysds CTaτ DTbτ, Gτ, sysds CKaτ DKbτ. 2.7 From 2.7,weobainhaC and D saisfy a and b separaely. The proof is compleed.

5 Journal of Applied Mahemaics 5 Remark 2.2. Assume ha H 2 holds. Then Ay <, By < for any y and any soluion u of BVP2. is nonnegaive. Lemma 2.3. From.3 and 2.3, i is easy o ge he following properies. G, s/ ab, a/ ab < /b /β 2,b/ab < /a /β. 2 Gs lim G, s β 2 /as <. 3 G, s Gs, s asbs/ <. Lemma 2.4. For any consan <a <b <, hereexiss <c <, such ha, for τ, s,, G, s/ ab c Gτ, s/ aτbτ, a/ ab c aτ/ aτbτ, b/ ab c bτ/ aτbτ, a,b. Proof. By.3, i is obvious ha a is increasing, and b is decreasing on, ; herefore, by 2.3, we have G, s ab abs ab aa β 2 ab ba, s, asb ab bb β ab ba, s. 2.8 We ake c min{aa β 2 / ab ba,bb β / ab ba }, hen <c < ; his is because ha aa β 2 ab ba ab ba ab ba <, bb β ab ba ab ba ab ba <. 2.9 By Lemma 2.3, we have Gτ, s/ aτbτ, hen G, s ab c c Gτ, s aτbτ, a ab aa ab ba aa β 2 ab ba c aτ β 2 b >c aτbτ. 2. Similarly, we can obain ha b/ ab c bτ/ aτbτ. The proof is compleed. In his paper, we use he space { E u C R : sup, u ab <, sup, u } < 2. wih he norm u max{ u, u }, where u sup, u / ab and u sup, u, hen E, u is a Banach space.

6 6 Journal of Applied Mahemaics Le { } u P u E : u, min a,b ab uτ c,τ R. 2.2 aτbτ Clearly P is a cone of E. Lemma 2.5 see. Le M C l R,R{x CR,R lim x exiss}, henm is precompac if he following condiions hold: a M is bounded in C l ; b he funcions belonging o M are locally equiconinuous on any inerval of R ; c he funcions from M are equiconvergen; ha is, given ε>, here corresponds Tε > such ha x x <εfor any Tε and x M. We shall consider nonnegaive coninuous and concave funcional α on P; ha is,α : P, is coninuous and saisfies α ( x y αx α ( y, x, y P,. 2.3 We denoe he se {x P a αx, x b}b >a> by Pα, a, b and P r {x P x <r}. 2.4 The key ool in our approach is he following Legge-Williams fixed poin heorem. Theorem 2.6 see. Le T : P c P c be compleely coninuous and α a nonnegaive coninuous concave funcional on P wih αx x for any x P c. Suppose ha here exis <a<b<d c such ha c {x Pα, b, d αx >b} / φ, and αtx >b,forx Pα, b, d; c 2 Tx <a,forx P a ; c 3 αtx >bfor x Pα, b, c wih Tx >d. Then T has a leas hree fixed poins x,x 2,x 3,wih x <a, b<αx 2, x 3 >a, αx 3 <b Exisence Resuls Define he operaor T : P P by Tu G, sφsf ( s, us,u s ds A ( Φf a B ( Φf b, <<. 3. Then u is a fixed poin of operaor T if and only if u is a soluion of BVP..

7 Journal of Applied Mahemaics 7 For convenience, we denoe δ, αx by <δ aa bb b ab ba Φsds, αu min a a,b u, u P. 3.2 ab Theorem 3.. Suppose ha H, H 2 hold, and assume here exis <r <b <l <r 2 wih l max{b /c, sup, b /c p}, such ha H 3 Qy,y 2 min{r 2 / Φsqsds AΦq/ β 2 BΦq/ β,r 2 / sup, /p Φsqsds AΦqα BΦqα 2 }, y r 2, y 2 r 2, H 4 Ψ, y,y 2 >b /δ, a,b,b y r 2, y 2 r 2, H 5 Qy,y 2 < min{r / Φsqsds AΦq/ β 2 BΦq/ β,r / sup, /p Φsqsds AΦqα BΦqα 2 }, y r, y 2 r. Then BVP. has a leas hree posiive soluions u, u 2, and u 3 wih u <r, b <αu 2, u 3 >r, αu 3 <b. 3.3 Proof. Firsly we prove ha T : P P is coninuous. We will show ha T : P P is well defined and TP P. For all u P, byh 2, Φ and f are nonnegaive funcions, and we have Tu. From H, H 2,weobain A ( Φf T Δ K ( ( Gτ, sφsf ( s, us,u s ds Gτ, sφsf ( s, us,u s ds max y, u, y 2 u Q ( y,y 2 T Δ K A ( Φq max Q ( y,y 2. y, u, y 2 u ( ( Tbτ Kbτ Gτ, sφsqsds Gτ, sφsqsds Tbτ Kbτ A In he same way, we have B ( Φf B ( Φq max ( y,y 2. y, u, y 2 u Q B

8 8 Journal of Applied Mahemaics By Lemma 2.3, A, B, andh, for all u P, we have Tu ab G, s ab Φsf( s, us,u s A ( Φf a ds ab B ( Φf b ab Φsf ( s, us,u s ds A( Φf B ( Φf β 2 β max Q ( ( y,y 2 Φsqsds A( Φq B ( Φq y, u, y 2 u β 2 β 3.4 <, Tu p sup, ( <. α 2 as Φsf ( s, us,u s ds α bs Φsf ( s, us,u s ds A ( Φf α B ( Φf α 2 p max y, u, y 2 u Q ( y,y 2 Φsqsds A ( Φq α B ( Φq α 2 Hence, T : P P is well defined. By 3., H, he Lebesgue dominaed convergence heorem and he coninuiy of p, for any u P,, 2 R, we have Tu Tu 2 α 2 a p p 2 Φsf ( s, us,u s ds 2 α 2a Φsf ( s, xs,x s ds p 2 α b p p 2 Φsf ( s, us,u s ds α 2 b Φsf ( s, xs,x s ds p 2 ( A ( Φf α B ( Φf α 2 p p 2, as Tha is, Tu C R ; herefore, Tu E.

9 Journal of Applied Mahemaics 9 By Lemma 2.4, we have min a,b Tu ab ( min a,b G, s ab Φsf( s, us,u s ds aa ( Φf ab ( bb Φf ab c ( Gτ, s aτbτ Φsf( s, us,u s ds aτa ( Φf ab ( bτb Φf ab 3.7 c Tuτ aτbτ, herefore T : P P. We show ha T : P P is coninuous. In fac suppose {u m } P, u P and u m u m, hen here exiss M>, such ha u m M. ByH, we have Φs ( f s, um s,u ms f ( s, u s,u s ds 2 Φsf ( s, us,u s ds 2 max Q( y,y 2 y,m, y 2 M <. Φsqsds 3.8 Therefore, by Lemma 2.3, he coninuiy of f and he Lebesgue dominaed convergence heorem imply ha Tu m Tu ab G, s ab Φs [ f ( s, u m s,u ms f ( s, u s,u s] ds Φs f ( s, um s,u ms f ( s, u s,u s ds, m,

10 Journal of Applied Mahemaics Tum Tu sup, p Φs f ( s, um s,u ms f ( s, u s,u s ds, m. 3.9 Thus, Tu m Tu m. Therefore T : P P is coninuous. Secondly we show ha T : P P is compac operaor. For any bounded se B P, here exiss a consan L> such ha u L, for all u B. ByLemma 2.3, A, B,andH, we have Tu Tu ab ab ( G, s a b ab Φsf( s, us,u s ds A ( Φf ( a ab B Φf b ab ( a b Φsf ( s, us,u s ds A( Φf B ( Φf β 2 β ( a b max Q( y,y 2 Φsqsds A( Φq B ( Φq y,l, y 2 L β 2 β <, Tu β 2 GsΦsf ( s, us,u s ds A ( Φf a B ( Φf b asφsf ( s, us,u s ds A ( Φf a B ( Φf b ( max Q( β2 a y,y 2 y,l, y 2 L Φsqsds A ( Φq a B ( Φq b <. 3. Therefore, Tu C l R,R. By 3.4 and 3.5, we have Tu sup, Tu ab max Q( y,y 2 y,l, y 2 L <, ( Φsqsds A( Φq B ( Φq β 2 β

11 Journal of Applied Mahemaics Tu max Tu, sup max Q ( ( y,y 2 Φsqsds A ( Φq α B ( Φq α 2 p, y,l, y 2 L <, 3. so TB is bounded. Given T>,, 2,T,byH and Lemma 2.3, we have G,s a b G 2,s a 2 b 2 Φsf( s, us,u s 2 max y,l, y 2 L Q ( y,y 2 Φsqs. 3.2 Therefore for any u B, by3., he Lebesgue dominaed convergence heorem and he coninuiy of G, s, a,andb, we have Tu a b Tx 2 a 2 b 2 G,s a b G 2,s a 2 b 2 Φsf ( s, us,u s ds A ( Φf a a b a 2 a 2 b 2 B ( Φf b a b b 2 a 2 b 2 3.3, as 2. By a similar proof as 3.6, weobain Tu Tu 2, as 2.Thus,TB is equiconinuous on,t. Since T>isarbirary, TB is locally equiconinuous on,. By Lemma 2.32, H 2 and he Lebesgue dominaed convergence heorem, we obain lim Tu ab a b β 2 asφsf ( s, xs,x s ds A ( Φf a B ( Φf b

12 2 Journal of Applied Mahemaics max y,l, y 2 LQ ( ( y,y 2 ( β 2 a β β 2 Φsqsds A ( Φq a B ( Φq b <, Tu ab Tu a b asb ab a b Φsf( s, xs,x s ds as ( b β2φsf s, xs,x s ds a b a as bs ab Φsf( s, xs,x s ds asbs ab a b Φsf( s, xs,x s ds as ( bs β2φsf s, xs,x s ds A ( Φf a a a b ab B ( Φf b b ab [ A ( Φf a B ( Φf b ] ab a b max Q ( y,y 2 y,m, y 2 M { ba ab a b Φsqsds a b β2φsqsds b a as Φsqsds a b β β 2 a b ab a b Φsqsds a a b ( bs β2φsqsds a a A Φq ab B ( Φq b b ab ( A ( Φq a B ( Φq b } ab a b, as. 3.4

13 Journal of Applied Mahemaics 3 By 3.5, we know ha lim Tu <, hen Tu Tu p α 2 as Φsf ( s, us,u s ds p p A( Φf α p p B( Φf α 2 p α 2 as Φsf ( s, us,u s ds p max Q ( y,y 2 y,l, y 2 L p α bs Φsf ( s, us,u s ds α 2 as Φsf ( s, us,u s ds p A( Φf α p B( Φf α 2 { α 2 as Φsqsds p as. α bs Φsqsds p α 2 as Φsqsds ( A ( Φq α B ( Φq α 2 p } p, 3.5 Therefore, TB is equiconvergen a.bylemma 2.5, TB is compleely coninuous. Finally we will show ha all condiions of Theorem 2.6 hold. From he definiion of α, we can ge αu u for all u P. For all u P r2, we have u r 2 ; herefore y r 2, y 2 r 2.By3.4, 3.5,andH 3, we have Tu ab max Q ( ( y,y 2 Φsqsds A( Φq B ( Φq y,r 2, y 2 r 2 β 2 β Tu r 2, sup, ( r 2, max Q ( y,y 2 p y,r 2, y 2 r 2 Φsqsds A ( Φq α B ( Φq α ha is, Tu r 2 for u P r2.thust : P r2 P r2. Similarly for any u P r, we have Tu < r, which means ha condiion c 2 of Theorem 2.6 holds.

14 4 Journal of Applied Mahemaics In order o apply condiion c of Theorem 2.6, we choose u b ab/ c, R, hen u l ; his is because u sup u, u b c l, b a b ab sup c, sup, b p c l, 3.7 and αu min a,b u/ ab b /c > b, which means ha {u Pα, b,l αu > b } / φ. For all u Pα, b,l, we have αu b and u l,hus b u/ ab l, u l,hais,b y l, y 2 l.byh 4, we can ge αtu Tu min ab ( a a,b a min a,b asb Φsf ( s, us,u s ds b ab asb Φsf ( s, us,u s b ds abs Φsf ( s, us,u s ds abs Φsf ( s, us,u s ds > > aa bb b ab ba Φsf ( s, us,u s ds a aa bb ab ba b. b a Φsds b δ 3.8 Consequenly condiion c of Theorem 2.6 holds. We will prove ha condiion c 3 of Theorem 2.6 holds. If u Pα, b,r 2,and Tu > l,byh 4, we have αtu min a,b Tu ab > aa bb b ab ba Φsds b a δ b. 3.9 Therefore, condiion c 3 of Theorem 2.6 is saisfied. Then we can complee he proof of his heorem by Legge-Williams fixed poin heorem. Theorem 3.2. Suppose ha H, H 2 hold, and assume here exis <r <b <l <r 2 <b 2 < l 2 <r 3 < <r m wih l i max{b i /c, sup, b i /c p}, such ha

15 Journal of Applied Mahemaics 5 H 6 Qy,y 2 < min{r i / Φsqsds AΦq/ β 2 BΦq/ β,r i / sup, /p Φsqsds AΦqα BΦqα 2 }, y r i, y 2 r i, i m, H 7 Ψ, y,y 2 >b i /δ, a,b, b i y r i, y 2 r i, i m. Then BVP. has a leas 2m posiive soluions. Proof. When m, i follows from H 6 ha T has a leas one posiive soluion by he Schauder fixed poin heorem. When m 2, i is clear ha Theorem 3. holds. Then we can obain hree posiive soluions. In his way, we can finish he proof by he mehod of inducion. 4. Example Consider he following singular Surm-Liouville singular boundary value problems for second-order differenial equaion on he half-line ( 2 u e f (, u,u, <<, u lim pu m 2 i ( 3 i uξ i, <ξ i <, lim u lim pu 3 e s susds, 4. where f (, u,u Ψ ( y 4 y2, y, 55, y,y 2 y2, y, p 2,α α 2 β β 2, a 2 /, b /, Φ e / which is singular a, 3, Tu m 2 i /3i uξ i, Ku /3e s susds. Se q and Q ( y 4 y2, y, 55 y,y 2 y 2, y, hen Φsqsds < 3, a, a 2, b 2, b, /2 < Taτ <, /2 < Tbτ <, Kaτ Kbτ, Δ > 3,AΦq < 26/9, BΦq < 2/9.

16 6 Journal of Applied Mahemaics Choose r /3, b 7/5, r 2 9. When a, b 2, by he definiion of δ, we may choose δ 8/5. By direc calculaions, we imply ha min min { { r Φsqsds A ( Φq / β 2 B ( Φq / β, r ( ( sup, /p Φsqsds A ( Φq α B ( Φq α 2 r 2 Φsqsds A ( Φq / β 2 B ( Φq / β, r 2 ( ( sup, /p Φsqsds A ( Φq α B ( Φq α 2 > 3r 55, > 3r 2 55, 4.4 Q ( y,y 2 ( < 55 3r 55, for y 3, y 2 3, Q ( 9 y,y 2 55 < 3 9 3r , for y 9, Ψ (, y,y 2 > 7/5 8/5 b δ, for, 2, 7/5 y 9, y2 9, y2 9. Therefore, he condiions H H 5 hold. Applying Theorem 3. we conclude ha BVP4. has a leas hree posiive soluions. Acknowledgmens The research was suppored by he Naional Naural Science Foundaion of China 876 and he Naural Science Foundaion of Shandong Province of China ZR2AM5. References J. Mao, Z. Zhao, and N. Xu, The exisence and uniqueness of posiive soluions for inegral boundary value problems, Bullein of he Malaysian Mahemaical Sciences Sociey. Second Series, vol. 34, no., pp , 2. 2 A. Boucherif, Second-order boundary value problems wih inegral boundary condiions, Nonlinear Analysis A, vol. 7, no., pp , X. P. Liu, M. Jia, and W. G. Ge, Exisence of monoone posiive soluions o a ype of hree-poin boundary value problem, Aca Mahemaicae Applicaae Sinica, vol. 3, no., pp. 9, X. Ni and W. Ge, Exisence of muliple posiive soluions for boundary value problems on he halfline, Journal of Sysems Science and Mahemaical Sciences, vol. 26, no., pp. 3 2, M. H. Xing, K. M. Zhang, and H. L. Gao, Exisence of muliple posiive soluions for general Surm- Liouville boundary value problems on he half-line, Aca Mahemaica Scienia A, vol. 29, no. 4, pp , L. Liu, Z. Wang, and Y. Wu, Muliple posiive soluions of he singular boundary value problems for second-order differenial equaions on he half-line, Nonlinear Analysis A, vol.7,no.7-8,pp , 29.

17 Journal of Applied Mahemaics 7 7 Z. Q. Zhao and F. S. Li, Exisence and uniqueness of posiive soluions for some singular boundary value problems wih linear funcional boundary condiions, Aca Mahemaica Sinica, vol. 27, no., pp , 2. 8 X. Zhang, L. Liu, and Y. Wu, Exisence of posiive soluions for second-order semiposione differenial equaions on he half-line, Applied Mahemaics and Compuaion, vol. 85, no., pp , H. Lian, H. Pang, and W. Ge, Triple posiive soluions for boundary value problems on infinie inervals, Nonlinear Analysis A, vol. 67, no. 7, pp , 27. C. Corduneanu, Inegral Equaions and Sabiliy of Feedback Sysems, Academic Press, New York, NY, USA, 973. R. W. Legge and L. R. Williams, Muliple posiive fixed poins of nonlinear operaors on ordered Banach spaces, Indiana Universiy Mahemaics Journal, vol. 28, no. 4, pp , 979.

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