Some results on the fractional order Sturm-Liouville problems

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1 Ru e al. Advance in Difference Equaion (27) 27:32 DOI.86/ x R E S E A R C H Open Acce Some reul on he fracional order Surm-Liouville problem Yuanfang Ru, Fanglei Wang *, Tianqing An and Yukun An 2 * Correpondence: wang-fanglei@homail.com Deparmen of Mahemaic, College of Science, Hohai Univeriy, Nanjing, 298, China Full li of auhor informaion i available a he end of he aricle Abrac In hi work, we inroduce ome new reul on he Lyapunov inequaliy, uniquene and mulipliciy reul of nonrivial oluion of he nonlinear fracional Surm-Liouville problem { D q + (p()u ()) + ()f(u()), < q 2, (, ), αu() βp()u (), γ u() + δp()u (), where α, β, γ, δ are conan aifying βγ + αγ p(τ + αδ <+, p( )i ) poiive and coninuou on [,. In addiion, ome exience reul are given for he problem { D q + (p()u ()) + ()f(u(), λ), <q 2, (, ), αu() βp()u (), γ u() + δp()u (), where λ i a parameer. The proof i baed on he fixed poin heorem and he Leray-Schauder nonlinear alernaive for ingle-valued map. MSC: Primary 26A33; 34A8 Keyword: fracional differenial equaion; Surm-Liouville problem; Lyapunov inequaliy; fixed poin heorem Inroducion On he one hand, ince a Lyapunov-ype inequaliy ha found many applicaion in he udy of variou properie of oluion of differenial equaion, uch a ocillaion heory, diconjugacy and eigenvalue problem, here have been many exenion and generalizaion a well a improvemen in hi field, e.g., o nonlinear econd order equaion, o delay differenial equaion, o higher order differenial equaion, o difference equaion and o differenial and difference yem. We refer he reader o [ 4 (ineger order). Fracional differenial equaion have gained coniderable populariy and imporance due o heir numerou applicaion in many field of cience and engineering including phyic, populaion dynamic, chemical echnology, bioechnology, aerodynamic, elecrodynamic of complex medium, polymer rheology, conrol of dynamical yem. Wih he rapid developmen of he heory of fracional differenial equaion, here are many The Auhor() 27. Thi aricle i diribued under he erm of he Creaive Common Aribuion 4. Inernaional Licene (hp://creaivecommon.org/licene/by/4./), which permi unrericed ue, diribuion, and reproducion in any medium, provided you give appropriae credi o he original auhor() and he ource, provide a link o he Creaive Common licene, and indicae if change were made.

2 Ru e al. Advance in Difference Equaion (27) 27:32 Page 2 of 9 paper which are concerned wih he Lyapunov ype inequaliy for a cerain fracional order differenial equaion, ee [5 7 and he reference herein. Recenly, Ghanbari and Gholami [7 inroduced he Lyapunov ype inequaliy for a cerain fracional order Surm- Liouville problem in ene of Riemann-Liouville D α a +(p()u ()) + q()u(), <α 2, (a, b), b, u(a)u (a), u(b) like hi b b a a q() Ɣ(α) p(ω) d dω > 2(b a) α. On he oher hand, many auhor have udied he exience, uniquene and mulipliciy of oluion for nonlinear boundary value problem involving fracional differenial equaion, ee [8 9. Bu Lan and Lin [2 poined ou ha he coninuiy aumpion on nonlineariie ued previouly are no ufficien and obained ome new reul on he exience of muliple poiive oluion of yem of nonlinear Capuo fracional differenial equaion wih ome of general eparaed boundary condiion c D q z i ()f i (, z()), (, ), αz i () βz i (), γ z i() + δz i (), where z()(z (),...,z n ()), f i :[, R n + R + i coninuou on [, R n +, c D q i he Capuo differenial operaor of order q (, 2). The α, β, γ, δ are poiive real number. The relaion beween he linear Capuo fracional differenial equaion and he correponding linear Hammerein inegral equaion are udied, which how ha uiable Lipchiz ype condiion are needed when one udie he nonlinear Capuo fracional differenial equaion. Moivaed by hee excellen work, in hi paper we focu on he repreenaion of he Lyapunov ype inequaliy and he exience of oluion for a cerain fracional order Surm-Liouville problem D q +(p()u ()) + ()f (u()), < q 2, (, ), (.) αu() βp()u (), γ u() + δp()u (), where α, β, γ, δ are conan aifying βγ + αγ + αδ <+, p( ) iapo- iive coninuou funcion on [,, () :[, R i a nonrivial Lebegue inegrable funcion, f : R R i coninuou. In addiion, ome exience reul are given for he problem D q +(p()u ()) + ()f (u(), λ), <q 2, (, ), αu() βp()u (), γ u() + δp()u (), where λ iaparameer,f : R R + R i coninuou. For he Surm-Liouville problem, here are many lieraure work on he udie of he exience and behavior of o- (.2)

3 Ru e al. Advance in Difference Equaion (27) 27:32 Page 3 of 9 luion o nonlinear Surm-Liouville equaion, for example, [2, 22 (ineger order) and [23, 24 (fracional order). The dicuion of hi manucrip i baed on he fixed poin heorem and he Leray- Schauder nonlinear alernaive for ingle-valued map. For convenience, we li he crucial lemma a follow. Lemma. ([25) Le ν be a poiive meaure and be a meaurable e wih ν( ). Le I be an inerval and uppoe ha u i a real funcion in L(dν) wih u() Iforall. If f i convex on I, hen ( ) f u() dν() f )u() dν(). (.3) If f i concave on I, hen inequaliy (.3) hold wih ubiued by. Lemma.2 ([26) Le E be a Banach pace, E be a cloed, convex ube of E, be an open ube of E, and. Suppoe ha T : E i compleely coninuou. Then eiher (i) T ha a fixed poin in, or (ii) here are u (he boundary of in E ) and λ (, ) wih u λtu. Lemma.3 ([26) Le E be a Banach pace and K EbeaconeinE. Aume ha, 2 are open ube of E wih, 2, and le T : K ( 2 \ ) Kbeacompleely coninuou operaor uch ha eiher (i) Tu u, u K and Tu u, u K 2 ; or (ii) Tu u, u K and Tu u, u K 2. Then T ha a fixed poin in K ( 2 \ ). Lemma.4 ([26) Le E be a Banach pace and K EbeaconeinE. Aume ha, 2 are open ube of E wih K, K 2 K. Le T : 2 K Kbea compleely coninuou operaor uch ha: (A) Tu u, u ( K), and (B) here exi e K \{} uch ha u Tu + μe, for u ( 2 K) and μ >. Then T ha a fixed poin in 2 K \ K. The ame concluion remain valid if (A) hold on ( 2 K) and (B) hold on ( K). 2 Preliminarie Definiion 2. ([26) For a funcion u given on he inerval [a,b, he Riemann-Liouville derivaive of fracional order q i defined a d n D q a +u() Ɣ(n q) d n where n [q+. a ( ) n q u() d,

4 Ru e al. Advance in Difference Equaion (27) 27:32 Page 4 of 9 Definiion 2.2 ([27) The Riemann-Liouville fracional inegral of order q for a funcion uidefineda I q a +u() provided ha uch inegral exi. a ( ) q u() d, q > Lemma 2.3 ([27) Le q >.Then n I q a +Dq a +u()u()+ c k q k, k n [q+. Lemma 2.4 Le h() AC[,. Then he fracional Surm-Liouville problem D q +(p()u ()) + h(), <q 2, (, ), αu() βp()u (), γ u() + δp()u () ha a unique oluion u() in he form where u() G(, ) G(, )h() d, βγ + αγ [β + α [β + α + αδ, [δ( )q + γ (τ ) q [δ( )q + γ [ H(, )α H(, ), ; (τ ) q Proof From Definiion 2., 2.2 and Lemma 2.3, i follow ha u () c p() p() c u()c 2 + Furhermore, we have ( ) q h() d, τ (τ ) q q(), ; (τ ) q δ + γ. d. u() c 2, u () c p(), c u() c 2 + u () c p() p() τ (τ ) q q() ( ) q q() d. d,

5 Ru e al. Advance in Difference Equaion (27) 27:32 Page 5 of 9 Combining he boundary condiion, we direcly ge c αγ τ (τ ) q h() d dω + αδ ( ) q h() d, c 2 βγ τ (τ ) q h() d + βδ ( ) q h() d. Finally, ubiuing c and c 2,weobain u() βγ τ (τ ) q h() d + βδ ( ) q h() d + αγ p(ω) τ (τ ) q h() ω (ω ) q h() d dω + αδ ( ) q h() d d βγ [ (τ ) q h() d + βδ αγ + [ For, G(, )h() d. (τ ) q ( ) q h() d [ (τ ) q h() d + αδ h() d ( ) q h() d u() βγ [ (τ ) q h() d + βδ + For, αγ [ β + α ( ) q h() d [ (τ ) q h() d + αδ [ δ( ) q + γ ( ) q h() d (τ ) q h() d. u() βγ [ (τ ) q h() d + βδ + [ ( ) q h() d ac [ (τ ) q h() d + αδ (τ ) q h() d ( ) q h() d

6 Ru e al. Advance in Difference Equaion (27) 27:32 Page 6 of 9 α [ β + α [ δ + γ [ δ( ) q + γ (τ ) q h() d. (τ ) q Lemma 2.5 Aume ha α, β, γ, δ >,and p( ) :[, (, + ). The Green funcion G(, ) aifie he following properie: (i) G(, ) for, ; (ii) For,, here exi C() >uch ha G(, ) aifie he inequaliie C()G(, ) G(, ) and min C()< [θ, θ ( for θ, ). 2 (iii) ThemaximumvalueeimaeofG(, ) where G max G(, ), { max max [, G(, ), max [, G( (), )}, [ αδ( ) q + αγ (τ ) q () + q. Proof (i) On he one hand, ince α, β, γ, δ >,andβγ + αγ + αδ >,iiclear ha G(, ) for. On he oher hand, for, we can verify he following inequaliie: ( ) q (τ ) q αδ αδ, αγ αγ (τ ) q (τ ) q Then we ge G(, ) for. (ii) For, G(, ) Then i i eay o obain [ α δ( ) q + γ p() ( ) q ( ) q. (τ ) q. (2.) G(, ) G(, ) for.

7 Ru e al. Advance in Difference Equaion (27) 27:32 Page 7 of 9 For, Le G(, ) αγ { βγ αγ p() ( )q p() + αδ( )q p() ( ) q ( )q αδ p() p() } + αγ (τ ) q p() + αγ p() ( ) q p() [ ( ) q + αδ( ) q + αγ F() ( ) q + αδ( ) q + αγ (τ ) q. (τ ) q (τ ) q. (2.2) I i clear ha F () (q )( ) q 2 <, which implie ha F( )idecreaingon (,. Since F()>andF() <, here exi unique () (,)uchhaf( ),namely, [ αδ( ) q + αγ (τ ) q () + q. From he above dicuion, we ge he concluion G(, ) G(, ),, for [,, and G(, ) G(, ) G(, ), for [,, andg(, ) G(, ) G(, ). Furhermore, we obain he eimae G(, ) G ( (), ), for. For, G(, ) G(, ) β + α β + α For, β + α β + α C (). G(, ) G(, ) [β + α [β + α [β + α [δ( )q + γ (τ ) q H(, ) [δ( )q + γ (τ ) q [δ( )q + γ (τ ) q H(, ) [β + α [δ + γ

8 Ru e al. Advance in Difference Equaion (27) 27:32 Page 8 of 9 [β + α βδ( )q + βγ αδ( )q [δ( )q + γ ( ) q α[δ + γ [β + α [β + α ( ) q [β + α + αγ [δ + γ βδ( )q + γ ( ) q [β + αδ( )q [ [β + α [β + α [δ + γ βδ( )q + γ ( ) q [β + [β + α [β + α αδ( ) q [β + α [δ + γ βδ( ) q [δ + γ [δ + γ + αδ( ) q [δ + γ ( ) q + α [δ + γ + α C 2(). + αγ [δ + γ α α ( ) q ( ) q Chooing C()min{C (), C 2 ()},wegec()g(, ) G(, ). 3 Exience reul I Theorem 3. (Lyapunov ype inequaliy) Aume ha α, β, γ, δ >,p( ):[, (, + ), and le ():[, R be a nonrivial Lebegue inegrable funcion. Then, for any nonrivial oluion of he fracional Surm-Liouville problem D q +(p()u ()) + ()u(), <q 2, (, ), αu() βp()u (), γ u() + δp()u (), he following o-called Lyapunov ype inequaliy will be aified: () d > G, where G idefined in(iii) of Lemma 2.5. Proof From Lemma 2.4 and he riangular inequaliy, we ge u() G(, ) ()u() d G(, ) ()u() d.

9 Ru e al. Advance in Difference Equaion (27) 27:32 Page 9 of 9 Le E denoe he Banach pace C[, wih he norm defined by u max [, u(). Via ome imple compuaion, we can obain namely, u() max [, G(, ) ()u() d u() max G(, ) () d [, [ () u() max G(, ) d, [, () d > G. Theorem 3.2 (Generalized Lyapunov ype inequaliy) Aume ha α, β, γ, δ >,p( ) : [, (, + ), and le () :[, R be a nonrivial Lebegue inegrable funcion, f (u) i a poiive funcion on R. Then, for any nonrivial oluion of he fracional Surm- Liouville problem (.), hefollowingo-calledlyapunovypeinequaliywillbeaified: () u d > G max u [u,u f (u), where u min [, u(), u max [, u(). Proof From he imilar proof of Theorem 3.,wege u() G(, ) () f ( u() ) d. Since f i coninuouand concave, hen uing Jenen inequaliy (.3), we obain u() max G(, ) () f ( u() ) d [, [ () max G(, ) ( ) f u() d [, G () L () f ( u() ) d () L G max f (u) () u [u,u L, namely, () u d > G max u [u,u f (u).

10 Ru e al. Advance in Difference Equaion (27) 27:32 Page of 9 For convenience, we give ome noaion: [ ϖ max G ( (), ) () d + [, ς min C() [θ, θ G(, ) () d. G(, ) () d ; Theorem 3.3 Le ():[, R + be a nonrivial Lebegue inegrable funcion and f : R R be a coninuou funcion aifying he Lipchiz condiion f (x) f (y) L x y, x, y R, L >. Then problem (.) ha a unique oluion if Lϖ <. Proof By Lemma 2.4, he oluion of problem (.) i equivalen o a fixed poin of he operaor T : E E defined by T(u()) G(, ) ()f (u()) d. Le up [, f () ν. Now we how ha T : B r B r,whereb r {u C[, : u < r} wih r > νϖ Lϖ.Foru B r,oneha f (u) f (u) f() + f () L u + ν Lr + ν.furhermore, we have T(u)() G(, ) ()f ( u() ) d G(, ) ()f ( u() ) d + G(, ) ()f ( u() ) d G ( (), ) ()f ( u() ) d + G(, ) ()f ( u() ) d [ G ( (), ) () d + G(, ) () d (Lr + ν) max [, (Lr + ν)ϖ r, which yield T : B r B r. For any x, y E,wehave T(x) T(y) G(, ) ()f ( x() ) d { up [, + G(, ) ()f ( y() ) d G ( (), ) () ( ) ( ) f x() f y() d G(, ) () } ( ) ( ) f x() f y() d [ L max G ( (), ) () d + [, Lϖ x y. G(, ) () d x y Since Lϖ <, from he Banach conracion mapping principle i follow ha here exi a unique fixed poin for he operaor T which correpond o he unique oluion for problem (.). Thi complee he proof.

11 Ru e al. Advance in Difference Equaion (27) 27:32 Page of 9 Theorem 3.4 Le ():[, R + be a nonrivial Lebegue inegrable funcion and f : R R be a coninuou funcion aifying he following: (F) There exi a poiive conan K uch ha f (u) K for u R. Then problem (.) ha a lea one oluion. Proof Fir, ince he funcion p : [, (, + ) i coninuou, we ge p min [, p()>.furher,from(2.)and(2.2), we ge he following eimae repecively: for, < G(, ) α p α p() [ δ + γ for, G(, ) p() p [ δ( ) q + γ ; (τ ) q [ ( ) q + αδ( ) q + αγ [ + αδ + αγ ; (τ ) q which implie ha G(,) i bounded for,, namely, here exi S >uch ha G(,) S. Combining wih f (, u) K for [,, R, we obain (Tu) () G(, ) ()f ( u() ) d SK () L. Hence, for any, 2 [,, we have (Tu)( 2 ) (Tu)( ) 2 (Tu) () d SK () L 2. Thi mean ha T i equiconinuou on [,. Thu, by he Arzelà-Acoli heorem, he operaor T i compleely coninuou. Finally, le B r {u E : u < r} wih r Kϖ +.Ifu i a oluion for he given problem, hen, for λ (, ), we obain u λ Tu() λ G(, ) ()f ( u() ) d λ G(, ) ()f ( u() ) d + G(, ) ()f ( u() ) d < max [, K max [, Kϖ, G ( (), ) () ( ) f u() d + G(, ) ( ) ()f u() d [ G ( (), ) () d + G(, ) () d

12 Ru e al. Advance in Difference Equaion (27) 27:32 Page 2 of 9 which yield a conradicion. Therefore, by Lemma.2, he operaor T ha a fixed poin in E. Theorem 3.5 Le () :[, R + be a nonrivial Lebegue inegrable funcion and f : R + R + be a coninuou funcion aifying (F). In addiion, he following aumpion hold: (F) There exi a poiive conan r uch ha f (u) ς r for u [, r. Then problem (.) ha a lea one oluion. Proof Define a cone P of he Banach pace E a P {u E : u }. Fromheproofof Theorem 3.4,weknowhaT : P P i compleely coninuou. Se P ri {u P : u < r i }. For u P r,oneha u r.for [θ, θ, we have T ( u() ) G(, ) ()f ( u() ) d C()G(, ) ()f ( u() ) d min C() [θ, θ min C() [θ, θ > r u. G(, ) ()f ( u() ) d G(, ) () d r Chooing r 2 > Kϖ.Then,foru P r2,wehave ( ) T u() G(, ) ()f ( u() ) d G(, ) ()f ( u() ) d + G(, ) ()f ( u() ) d [ max G ( (), ) () d + G(, ) () d K [, < r 2 u. Then, by Lemma.3,problem(.) ha a lea one poiive oluion u()belongingoe uch ha r u r 2. Theorem 3.6 Le ():[, R + be a nonrivial Lebegue inegrable funcion, f : R R be a coninuou funcion and aify he following aumpion: (F2) There exi a nondecreaing funcion ϕ : R + R + uch ha f (u) ϕ ( u ), u R; (F3) There exi a conan R >uch ha Then problem (.) ha a lea one oluion. R ϖϕ(r) >.

13 Ru e al. Advance in Difference Equaion (27) 27:32 Page 3 of 9 Proof From he proof of Theorem 3.4,weknowhaT i compleely coninuou. Now we how ha (ii) of Lemma.2 doe no hold. If u i a oluion of (.), hen, for λ (, ), we obain u λ ( ) T u() λ G(, ) ()f ( u() ) d λ G(, ) ()f ( u() ) d + G(, ) ()f ( u() ) d [ < max G ( (), ) () ( ) f u() d + G(, ) () ( ) f u() d [, [ max G ( (), ) () d + G(, ) () d ϕ ( u ) [, ϖϕ ( u ). Le B R {u E : u < R}. From he above inequaliy and (F3), i yield a conradicion. Therefore, by Lemma.2, he operaor T ha a fixed poin in B R. Theorem 3.7 Le ():[, R + be a nonrivial Lebegue inegrable funcion and f : [, R + R + be a coninuou funcion. Suppoe ha (F2) and (F3) hold. In addiion, he following aumpion hold: (F4) There exi a poiive conan r wih r < R and a funcion ψ : R + R + aifying f (u) ψ ( u ), for u [, ςr, ψ(ςr) r. If ς <,hen (.) ha a lea one poiive oluion u(). Proof Le B r {u E : u < r}. Par (I). For any u (B R P), from (F3) and (F4) i follow ha ( ) T u() G(, ) ()f ( u() ) d G(, ) ()f ( u() ) d + G(, ) ()f ( u() ) d < max [, max [, ϖ ϕ(r) R u, G ( (), ) ()f ( u() ) d + G(, ) ()f ( u() ) d [ G ( (), ) () d + G (, () ) d ϕ ( u ) which implie ha (A) of Lemma.4 hold. Now we prove ha u T(u)+μ for u (B ςr P) andμ >.Onheconrary,ifhere exi u (B ςr P)andμ >uchhau T(u )+μ,hen,for [θ, θ, one ha

14 Ru e al. Advance in Difference Equaion (27) 27:32 Page 4 of 9 min [θ, θ C() >. Furhermore, from (F5) i follow ha u ()T ( u () ) + μ G(, ) ()f ( u () ) d + μ C()G(, ) ()f ( u () ) d + μ min C() [θ, θ min C() [θ, θ ςr + μ. G(, ) ()f ( u () ) d + μ G(, ) ()ψ(ςr) d + μ Furhermore, we ge ςr > min u () ςr + μ > ςr, [θ, θ which yield a conradicion. So (B) of Lemma.4 hold. Therefore, Lemma.3 guaranee ha T ha a lea one fixed poin. Theorem 3.8 Le ():[, R + be a nonrivial Lebegue inegrable funcion and f : R + R + be a coninuou funcion aifying (F). In addiion, he following aumpion hold: (F5) lim u + f (u) u ; (F6) There exi R >uch ha min u [ϑr,r f (u)>σr, where [ <ϑ η [ <η max [ σ min C() [θ, θ min C() [θ, θ <, G( (), ), G(, ) θ θ G(, ) () d. Then problem (.) ha a lea wo oluion. Proof FromLemma 2.5, we can derive he following inequaliie: ( ) T u() G(, ) ()f ( u() ) d G(, ) ()f ( u() ) d + G(, ) ()f ( u() ) d [ max G ( (), ) ()f ( u() ) d + G(, ) ()f ( u() ) d [,

15 Ru e al. Advance in Difference Equaion (27) 27:32 Page 5 of 9 and T ( u() ) [ max [, max G( (), ) G(, ) ()f ( u() ) d + G(, ) [ G(, ) ()f ( u() ) d G( (), ) G(, ) G(, ) ()f ( u() ) d C()G(, ) ()f ( u() ) d. Combining he wo inequaliie, we have G(, ) ()f ( u() ) d T ( u() ) C()η T ( u() ). Define a ubcone P of he Banach pace E a P {u E : u C()η u() }.Fromheandard proce, we know ha T : P P i compleely coninuou. Se P r {u P : u < r}. Since lim u + f (u),hereexiɛ >andr >uchhaf (u)<ɛu,for u r,where u ɛ aifie ɛϖ <.Foru P r,wehave ( ) T u() G(, ) ()f ( u() ) d G(, ) ()f ( u() ) d + G(, ) ()f ( u() ) d [ ɛ max G ( (), ) () d + G(, ) () d u [, < u. In a imilar way, we chooe R > Kϖ.Then,foru P R,wehave ( ) T u() G(, ) ()f ( u() ) d G(, ) ()f ( u() ) d + [ max G ( (), ) () d + [, < R u. G(, ) ()f ( u() ) d G(, ) () d K For any u P R,chooing (θ, θ), i i eay o verify ha u( ) [ϑr, R. Furhermore, we have T ( u ( )) G (, ) ()f ( u() ) d C ( ) θ G(, ) ()f ( u() ) d θ

16 Ru e al. Advance in Difference Equaion (27) 27:32 Page 6 of 9 C ( ) θ G(, ) () min f ( u() ) d θ u [ϑr,r [ θ min C() G(, ) ()σ Rd [θ, θ θ R u. Then by Lemma.3,problem(.) ha a lea wo poiive oluion r u () Rand R u 2 () R. Example Le u conider he problem D q +(p()u ()) + () arcan u, <q 2, (, ), αu() βp()u (), γ u() + δp()u (). Since f (u) arcan u < π, hi problem ha a oluion by Theorem 3.4.If ()aifie [ ϖ max G ( (), ) () d + [, I i eay o ge ha f (u)(arcan u) +u 2 L. G(, ) () d <. Therefore, hi problem ha a unique oluion by Theorem 3.3. Example 2 Le u conider he problem D q +(p()u ()) + ()e u, <q 2, (, ), αu() βp()u (), γ u() + δp()u (). Since f (u)e u, we can chooe r ϖ +.Theniiclearha f (u) <ϖ r for u [, r, which implie ha (F) hold. Finally, for any r >,wehavef (u) e r for u [, r. Since lim r + e r +, hereexir ς r 2 < r uch ha f (u) ς r 2 for u [, r 2, which implie ha (F) hold. Therefore, hi problem ha a unique oluion by Theorem 3.5. Example 3 Le u conider he problem D q +(p()u ()) + ()e u2 (arcan u + in u +2), <q 2, (, ), αu() βp()u (), γ u() + δp()u (). I i clear ha f (u) e u2 (arcan u 5 + in u 3 +2) u 5 + u 3 +2ϕ( u ), u R. R Then (F2) hold. Furhermore, for ufficienly large R >, heinequaliy ϖϕ(r) >obviouly hold, namely, (F3) hold. Then hi problem ha a lea one oluion by Theorem 3.6.

17 Ru e al. Advance in Difference Equaion (27) 27:32 Page 7 of 9 For u R +,incef(u) e u2 (arcan u 5 + in u 3 +2) e u2 e u 2 ψ( u ), we have f (u) ψ( u ) foru [, ςr, for any r >. Via ome imple compuaion, we ge lim r + ψ(ςr) r +. Then here exi ufficienly mall r >uchhaψ(ςr) r. From he above dicuion, we have ha (F4) hold. Therefore, hi problem ha a lea one poiive oluion u()forς <bytheorem3.7. Example 4 Le u conider he problem D q +(p()u 2σ ()) + () u 2 e u, <q 2, (, ), (2ϑ) 2 e 2ϑ αu() βp()u (), γ u() + δp()u (). 2σ + Since f (u) u 2 e u, via ome imple compuaion, we can verify ha (F) and (F5) (2ϑ) 2 e 2ϑ hold. In addiion, ince f 2σ + (u) e u (2u u 2 2σ + ) e u u(2 u), i i clear ha (2ϑ) 2 e 2ϑ (2ϑ) 2 e 2ϑ f (u)>foru (, 2); f (u)<foru (2, + ). Le R 2, hen for any u [2ϑ,2,wehave 2σ + min u [2ϑ,2 f (u) (2ϑ) 2 e 2ϑ >2σ. Therefore, hi problem ha a lea wo poiive (2ϑ) 2 e 2ϑ oluion u()bytheorem Exience reul II Theorem 4. Le ():[, R + be a nonrivial Lebegue inegrable funcion and f : R [,+ ) R be a coninuou funcion aifying he following: (H) There exi a poiive conan K uch ha f (u, λ) K for u R, λ R +. Then problem (.2) ha a lea one oluion. Thi reul can be direcly derived from he proof of Theorem 3.4. Now define a cone P of he Banach pace E a P {u E : u }. LeP ri {u P : u < r i }.DefineT by T ( u() ) G(, ) ()f ( u(), λ ) d. From he proof of Theorem 3.4,weknowhaT : P P i compleely coninuou. Theorem 4.2 Le ():[, R + be a nonrivial Lebegue inegrable funcion and f be a nonnegaive coninuou funcion aifying (H). If f (, ) >, hen here exi λ >uch ha problem (.2) ha a lea one oluion for λ < λ. Proof Since f (u, λ) i coninuou and f (, ) >, for any given ɛ > (ufficienly mall), here exi δ >uchhaf (u, λ) >f (, ) ɛ if u < δ, λ < δ. Chooingr < min{δ, ς(f (, ) ɛ)} and λ δ.then,foranyu P r and [θ, θ, we have T ( u() ) G(, ) ()f ( u(), λ ) d C()G(, ) ()f ( u(), λ ) d min C() [θ, θ G(, ) ()f ( u(), λ ) d

18 Ru e al. Advance in Difference Equaion (27) 27:32 Page 8 of 9 min C() [θ, θ > r u. G(, ) () d (f (, ) ɛ ) Chooing r 2 > Kϖ.Then,foru P r2,wehave ( ) T u() G(, ) ()f ( u() ) d max [, max [, < r 2 u. G ( (), ) ()f ( u() ) d + G(, ) ()f ( u() ) d [ G ( (), ) () d + G(, ) () d K Then, by Lemma.3,problem(.2) ha a lea one poiive oluion u()belongingoe uch ha r 2 u r. Corollary 4.3 Le () :[, R + be a nonrivial Lebegue inegrable funcion and f be a nonnegaive coninuou funcion aifying (H). If lim u + f (u, λ) f (, ) >, hen problem (.2) ha a lea one oluion for any λ. Example 5 Le u conider he problem D q +(p()u ()) + ()(arcan u 2 + e λ ), <q 2, (, ), αu() βp()u (), γ u() + δp()u (). I i clear ha (H) hold and f (, ) >. Then here exi λ > uch ha hi problem ha a lea one oluion for λ < λ. Example 6 Le u conider he problem D q +(p()u ()) + ()e λu, <q 2, (, ), αu() βp()u (), γ u() + δp()u (). I i clear ha (H) hold and lim u + f (u, λ)f (, ) >. Then hi problem ha a lea one oluion for any λ >. 5 Concluion Inhimanucrip,heauhorproveomenewexiencereulawellauniquene and mulipliciy reul on fracional boundary value problem. Acknowledgemen The auhor would like o hank he referee for he helpful uggeion. The econd auhor i uppored by NNSF of China (No. 565), he Fundamenal Reearch Fund for he Cenral Univeriie (25B944). Compeing inere The auhor declare ha hey have no compeing inere.

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