Research Article Three-Point Boundary Value Problems of Nonlinear Second-Order q-difference Equations Involving Different Numbers of q

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1 Hindawi Publihing Corporaion Applied Mahemaic Volume 213, Aricle ID , 12 page hp://dx.doi.org/1.1155/213/ Reearch Aricle hree-poin Boundary Value Problem of Nonlinear Second-Order q-difference Equaion Involving Differen Number of q hanin Sihiwiraham, 1 Jeada ariboon, 1 and Soiri K. Nouya 2 1 Deparmen of Mahemaic, Faculy of Applied Science, King Mongku Univeriy of echnology Norh Bangkok, Bangkok, hailand 2 Deparmen of Mahemaic, Univeriy of Ioannina, Ioannina, Greece Correpondence hould be addreed o hanin Received 23 May 213; Acceped 7 Sepember 213 Academic Edior: Jin L. Kuang Copyrigh 213 hanin Sihiwiraham e al. hi i an open acce aricle diribued under he Creaive Common Aribuion Licene, which permi unrericed ue, diribuion, and reproducion in any medium, provided he original work i properly cied. We udy a new cla of hree-poin boundary value problem of nonlinear econd-order q-difference equaion. Our problem conain differen number of q in derivaive and inegral. By uing a variey of fixed poin heorem (uch a Banach conracion principle, Boyd and Wong fixed poin heorem for nonlinear conracion, Kranoelkii fixed poin heorem, and Leray-Schauder nonlinear alernaive) and Leray-Schauder degree heory, ome new exience and uniquene reul are obained. Illuraive example are alo preened. 1. Inroducion he q-difference calculu or quanum calculu i an old ubjec ha wa iniially developed by Jackon [1], Carmichael [2], Maon [3], and Adam [4], in he fir quarer of 2h cenury, ha been developed over he year, for inance, ee [5 14] and he reference herein. In fac, q-calculu ha a rich hiory, and he deail of i baic noion, reul, and mehod can be found in he ex [15]. In recen year, he opic ha araced he aenion of everal reearcher, and a variey of new reul can be found in he paper [16 28]and he reference cied herein. In [24], Ahmad e al. udied a boundary value problem of nonlinear q-difference equaion wih nonlocal boundary condiion given by D 2 q x () =f(, x ()), I1 q, α 1 x () 1 D q x () =γ 1 x( 1 ), (1) problem (1) i hown by mean of a variey of fixed poin heorem uch a Banach conracion principle, Kranoelkii fixed poin heorem, and Leray-Schauder nonlinear alernaive. Yu and Wang [28] conidered a boundary value problem wih he nonlinear econd-order q-difference equaion, D 2 q u () f(,u(),d qu ()) =, I 1 q, D q u () =, D q u (1) =αu(1), where f C(I 1 q R2, R) and α =i a fixed number. Exience and uniquene of he oluion are obained by mean of Banach conracion principle, Leray-Schauder nonlinear alernaive, and Leray-Schauder coninuaion heorem. Pongarm e al. [29] conidered equenial derivaive of nonlinear q-difference equaion wih hree-poin boundary condiion, (2) α 2 x (1) 2 D q x (1) =γ 2 x( 2 ), where f C(I 1 q R, R), I1 q = {qn : n N} {,1},and q (, 1) i a fixed conan. he exience of oluion for D q (D p λ)u() =f(, u ()), I q = [, ] I1 q, u () =, u() =α u () d r, (3)

2 2 Applied Mahemaic where <p,q,r<1, f C(I q R, R), <<,andλ, are given conan. Exience reul are proved baed on Banach conracion mapping principle, Kranoelkii fixed poin heorem, and Leray-Schauder degree heory. We noe ha in he above-menioned paper [24, 28]he q-number in he equaion and he boundary condiion are heame.afaraweknowhepaperbypongarmeal.[29] i he fir paper which ha differen value of he q-number in q-derivaive and q-inegral. In hi paper, we dicu he exience of oluion for he following nonlinear q-difference equaion wih hree-poin inegral boundary condiion D 2 q x () =f(, x ()), I q, αx () D r x()=, x () d p =, <<, where f C(I q R, R), I q =I1 q [,], I1 q ={qn :n N} {,1}, q (, 1) i a fixed conan, and I q \ {, } := (, ) q.alo,<p,q,r<1,andα, are given conan uch ha = α((/(1 p)) ). I i noeworhy ha, in he above problem (4), we have hree differen value of he q-number, in q-derivaive and he q-inegral. Moreover, we emphaize he fac ha, inead he value x() iuuallyuedinhelieraure,weuehe value of he funcion and i derivaive in an inermediae poin (, ). he aim of hi paper i o prove ome exience and uniquene reul for he boundary value problem (4). Our reul are baed on Banach conracion mapping principle, nonlinear conracion, Kranoelkii fixed poin heorem, Leray-Schauder nonlinear alernaive, and Leray-Schauder degree heory. he re of he paper i organized a follow. In Secion 2, we provide ome baic definiion, preliminarie fac, and a lemma, which are ued laer. he main reul are given in Secion 3. Inheend,Secion 4, ome reul illuraing he reul eablihed in hi paper are alo preened. 2. Preliminarie Le u recall ome baic concep of q-calculu [15, 18]. Definiion 1. For <q<1, one define he q-derivaive of a real valued funcion f a f () f(q) D q f () =, I 1 q (1 q) \ {}, D q f () = lim D q f (). he higher-order q-derivaive are given by D qf () =f(), Dn (4) (5) q f () =D qd n 1 q f (), n N. (6) For x one e J x ={xq n :n N {}} {}and define, he definie q-inegral of a funcion f:j x R by x I q f (x) = f () d q = n= provided ha he erie converge. For a, b J x,onee b f () d q =I q f (b) I q f (a) a x(1 q)q n f(xq n ) (7) =(1 q) q n [bf (bq n ) af(aq n )]. n= Noe ha for a, b J x,onehaa=xq n 1, b=xq n 2 for ome n 1,n 2 N; hu, he definie inegral b f()d a q i ju a finieum,onoqueionabouconvergenceiraied. One noe ha while if f i coninuou a x=,hen (8) D q I q f (x) =f(x), (9) I q D q f (x) =f(x) f(). (1) In q-calculu, he produc rule and inegraion by par formula are D q (gh) () =(D q g ())h() g(q)d q h (), x f () D q g () d q =[f() g ()] x x D q f () g(q)d q. Furher, revering he order of inegraion i given by f (r) d q rd q = (11) f (r) d q d q r. (12) qr In he limi q 1, he above reul correpond o heir counerpar in andard calculu. Lemma 2. Le <p,q,r<1and (, ) q.hen,forany y C(I q, R),heboundaryvalueproblem, D 2 q x () =y(), I q, (13) αx () D r x()=, x () d p =, (14)

3 Applied Mahemaic 3 i equivalen o he inegral equaion (1 p) x () = ( q) y () d q Ω where [α ( q)y() d q y () d q (1 r) ( q) y () d q ] 1p Ω (α ( ) ) ( qv)y(v) d q Vd p, (15) Ω=(α)(1p) α=. (16) Proof. aking double q-inegral for (13), we have x () = y (V) d q V d q c 1 c 2. (17) By changing he order of q-inegraion, we have x () = In paricular, for =, we ge y (V) d q d q V c 1 c 2 qv = ( q) y () d q c 1 c 2. (18) x()= ( q) y () d q c 1 c 2. (19) aking r-derivaive for (18), for =,weobain D r x () =D r [ ( q) y () d q c 1 c 2 ] r 1 = (1 r) [ ( q) y () d q (r q) y () d q ]c 1 r = y () d q r For =,wehave D r x () = lim D r x () q (1 r) y () d qc 1. (1 q) = lim q n (1 q n1 ) 1 r =c 1. n= [h(q n ) r 2 h(rq n )] c 1 (2) (21) herefore, D r x()= y () d q q (1 r) y () d qc 1. (22) Now, uing he fir condiion of (14)wih(19), (22), we have (α ) c 1 αc 2 = α ( q) y () d q y () d q (1 r) ( q)y() d q. aking he p-inegral for (18)from o,weobain x () d p = ( qv) y (V) d q V d p (23) 2 1p c 1 c 2. (24) Subiuing =in (24) and uing he econd condiion of (14), we ge 2 1p c 1 c 2 = ( qv)y(v) d q V d p. (25) Solving he yem of linear equaion (23) and(25) for he unknown conan c 1 and c 2,wehave c 1 = 1p Ω α(1p) Ω [α ( q) y () d q y () d q (1 r) ( q) y () d q ] c 2 = (α)(1p) Ω Ω [α ( qv)y(v) d q V d p, ( qv)y(v) d q V d p ( q)y() d q y () d q (1 r) ( q) y () d q ], (26) where Ω i defined by (16). Subiuing he value of c 1 and c 2 in (18), we obain (15). hi complee he proof. Le C = C(I q, R) denoe he Banach pace of all he coninuou funcion from I q o R endowed wih he norm

4 4 Applied Mahemaic defined by x = up{ x(), I q }. Define an operaor A:C C by (Ax)() = ( q) f (, x ()) d q (1p) Ω [α ( q) f (, x ()) d q f (, x ()) d q (1 r) ( q)f(, x ()) d q ] (1p)(α( ) ) Ω ( qv)f(v,x(v)) d q V d p. (27) Oberve ha he problem (4) ha oluion if and only if he operaor A ha fixed poin. For he ake of convenience, we e a conan Λ a Λ= 2 1q p 2 [ α 1q 3. Main Reul (1q(2r)) ] 1q (1 p) ( α ( ) )2. (1q)(1pp 2 ) (28) Now, we are in he poiion o eablih he main reul. Our fir reul i baed on Banach fixed poin heorem. heorem 3. Aume ha f:i q R R i a coninuou funcion aifying he condiion (H 1 ) f(, x) f(, y) L x y,forall I q and x, y R, (H 2 ) LΛ < 1, where L i a Lipchiz conan, and Λ i defined by (28). hen, he boundary value problem (4) ha a unique oluion. Proof. We ranform he boundary value problem (4) ino a fixed poin problem x=ax,wherea:c C i defined by (27). Aume ha up I q f(, ) = M,andchooeaconan R aifying R MΛ 1 LΛ. (29) Now, we will how ha AB R B R,whereB R ={x C : x R}.Foranyx B R,wehave Ax = up I q up I q up I q ( q) f (, x ()) d q (1p) Ω [α ( q) f (, x ()) d q f (, x ()) d q (1 r) ( q) f (, x ()) d q ] (1p)(α( ) ) Ω ( qv)f(v,x(v)) d q V d p { ( q) ( f (, x ()) f(, ) f (, ) )d q (1 p) [ α ( q) ( f (, x ()) f(, ) f (, ) )d q ( f (, x ()) f(, ) f (, ) )d q (1 r) ( q)( f (, x ()) f(, ) f (, ) )d q] (1 p) α( ) ( qv) ( f (V,x()) f(v,) f (V,) )d qv d p } { ( q) (LR M) d q (1 p)

5 Applied Mahemaic 5 [ α ( q) (LR M) d q (LR M) d q (1 r) ( q)(lr M) d q ] (1 p) α( ) ( qv) (LR M) d q V d p } (LR M) { 2 1q p α 2 [ 1q (1q(2r)) ] 1q (1 p) ( α ( ) )2 } (1 q) (1 p p 2 ) = (LR M) Λ R. (3) herefore, AB R B R. Nex, we will how ha A i a conracion. For any x, y C and for each I q,wehave Ax Ay = up (Ax)() (Ay)() Iq up I q ( q) f (, x ()) f(,y()) d q (1 p) Ω [α ( q) f (, x ()) f (, y ()) d q f (, x ()) f (, x 2 ()) d q (1 r) (1 p) (α ( ) ) Ω up I q ( qv) ( q) f (, x () f (, y ()) d q] f (V,x(V)) f(v,y(v)) d qv d p {L x y ( q) d q L x y (1p) [ α ( q)d q d q (1 r) ( q) d q ] (1 p) ( α ( ) ) L x y ( qv)d q V d p } L x y { 2 1q p α 2 [ 1q (1q(2r)) ] 1q (1 p) ( α ( ) )2 } (1q)(1pp 2 ) =LΛ x y. (31) Since LΛ < 1, A i a conracion. hu, he concluion of he heorem follow by Banach conracion mapping principle. hi complee he proof. Nex, we can ill deduce he exience and uniquene of a oluion o he boundary value problem (4). We will ue nonlinear conracion o accomplih hi. Definiion 4. Le E be a Banach pace and le F:E E be a mapping. F i aid o be a nonlinear conracion if here exi a coninuou nondecreaing funcion Ψ:R R

6 6 Applied Mahemaic uch ha Ψ() = and Ψ(ρ) < ρ for all ρ > wih he following propery: Fx Fy Ψ( x y ), x,y E. (32) Lemma 5 (Boyd and Wong [3]). Le E be a Banach pace and le F:E Ebe a nonlinear conracion. hen, F ha a unique fixed poin in E. heorem 6. Suppoe ha (H 3 ) here exi a coninuou funcion h:i q R uch ha hu, Ax () Ay() ( q) h () x () y() G x () y() (1p) [ α ( q) h() d q x () y() G x () y() d q h () x () y() G x () y() d q f (, x) f(,y) h() for all I q and x, y,where G= ( q) h () d q p [ α x y G x y ( q)h() d q h () d q (1 r) ( q) h () d q ] (1 p) ( α ( ) ) ( qv)h(v) d q Vd p (33) (34) (1 r) ( q) h () (1 p) α( ) ( qv) h(v) x () y() G x () y() x (V) y(v) G x (V) y(v) { ( q) h () d q p [ α ( q)h() d q (1 r) ( q) h () d q ] d q V d p d q ] h () d q and Ω i defined in (16). hen, he boundary value problem (4) ha a unique oluion. Proof. Le he operaor A:C C be defined a (27). We define a coninuou nondecreaing funcion Ψ:R R by Ψ(ρ)= Gρ, ρ, (35) Gρ uch ha Ψ() = and Ψ(ρ) < ρ,forallρ>. Le x, y C. hen, we ge f (, x ()) f(,y()) h () G Ψ( x y ). (36) (1 p) ( α ( ) ) ( qv)h(v) d q V d p } x y G x y = G x y G x y, I q. (37) hi implie ha Ax Ay Ψ( x y ). Hence,A i a nonlinear conracion. herefore, by Lemma 5, he operaor A ha a unique fixed poin in C,whichiauniqueoluionof problem (4). he hird reul i baed on he following Kranoelkii fixed poin heorem [31].

7 Applied Mahemaic 7 heorem 7. Le K be a bounded cloed convex and nonempy ube of a Banach pace X.LeA, B be operaor uch ha: (i) Ax By K whenever x, y K, (ii) A i compac and coninuou, (iii) B i a conracion mapping. hen, here exi z Kuch ha z=azbz. heorem 8. Aume ha (H 1 ) and (H 2 ) hold. In addiion one uppoe ha: (H 4 ) f(, x) μ(), forall(, x) I q R, wihμ L 1 (I q, R ). For x, y B R,bycompuingdirecly,wehave F 1xF 2 y μ ( q) d q μ [ α (1 p) ( q) d q d q (1 r) ( q) d q ] μ (1 p) α( ) ) ( qv)d q V d p μ Λ R. (41) If Λ<1, (38) where Λ i given by (28), hen he boundary value problem (4) ha a lea one oluion on I q. Proof. Seing max I q μ() = μ and chooing a conan R μ Λ, (39) we conider B R ={x C : x R}. In view of Lemma 2, we define he operaor F 1 and F 2 on he ball B R a (F 1 x) () = ( q) f (, x ()) d q, (F 2 x) () = (1p) Ω [α ( q)f(, x ()) d q f (, x ()) d q (1 r) (1 p) (α ( ) ) Ω ( q) f (, x ()) d q ] ( qv)f(v,x(v)) d q V d p. (4) herefore, F 1 x F 2 y B R. Condiion (38) implie ha F 2 i a conracion mapping. Nex, we will how ha F 1 i compac and coninuou. Coninuiy of f coupled wih he aumpion (H 3 ) implie ha he operaor F 1 i coninuou and uniformly bounded on B R. We define up (,x) I q B R f(, x) = f max <.For 1, 2 I q wih 1 2 and x B R,wehave F 1x( 2 ) F 1 x( 1 ) = 2 ( 2 q)f(, x ()) d q = 1 1 ( 1 q)f(, x ()) d q ( 2 1 )f(, x ()) d q 2 ( 2 q)f(, x ()) d q (12q 1q )f max. (42) Acually, a 2 1,herigh-handideofheabove inequaliy end o be zero. So, F 1 i relaively compac on B R. Hence, by he Arzelá-Acoli heorem, F 1 i compac on B R. herefore, all he aumpion of heorem 7 are aified, and he concluionof heorem 7 implie ha he boundary value problem (4) ha a lea one oluion on I q.hicomplee he proof. A he fourh reul, we prove he exience of oluion of (4) by uing Leray-Schauder nonlinear alernaive. heorem 9 (Nonlinear Alernaive for Single Valued Map [32]). Le E be a Banach pace, C a cloed convex ube of E, U an open ube of C, and U.SuppoehaF:U C i a coninuou, compac (ha i, F(U) i a relaively compac ube of C)map.hen,eiher

8 8 Applied Mahemaic (i) F ha a fixed poin in U or (ii) here i a u U(he boundary of U in C) andλ (, 1) wih u = λf(u). heorem 1. Aume ha: (H 5 ) here exi a coninuou nondecreaing funcion ψ : [, ) (, ) and a funcion z L 1 (I q, R ) uch ha f (, u) z() ψ ( u ), for each (, u) I q R; (43) (H 6 ) here exi a conan M>uch ha M >1. (44) ψ (M) z L 1Λ hen, he boundary value problem (4) ha a lea one oluion on I q. Proof. We will how ha A map bounded e (ball) ino bounded e in C. Forapoiivenumberρ, leb ρ ={x C(I q, R) : x ρ}be a bounded ball in C(I q, R).hen,for I q,wehave (Ax)() ( q) f (, x ()) d q (1 p) [ α ( q) f (, x ()) d q f (, x ()) d q (1 r) ( q) f (, x ()) d q] (1 p) α( ) ( qv) f (V,x(V)) d qv d p ψ( x ) ( q) z () d q ψ ( x ) (1 p) [ α ( q)z() d q z () d q (1 r) ( q) z () d q ] ψ ( x ) (1 p) α( ) ( qv)z() d q V d p ψ( x ) z L 1 ( q) d q ψ ( x ) z L 1 (1 p) [ α ( q)d q d q (1 r) ( q)d q ] ψ ( x ) z L 1 (1 p) α( ) ( qv)d q Vd p ψ ( x ) z L 12 1q ψ ( x ) z L 1p α 2 [ 1q (1q(2r)) ] 1q ψ ( x ) z L 1 (1 p) ( α ( ) )2 (1q)(1pp 2 ) =ψ( x ) z L 1Λ. Conequenly, (45) Ax ψ( x ) z L 1Λ. (46) Nex,wewillhowhaA map bounded e ino equiconinuou e of C(I q, R). Le 1, 2 I q wih 1 2 and x B ρ.hen,wehave (Ax) ( 2) (Ax) ( 1 ) 2 ( 2 q) f (, x ()) d q 1 ( 1 q) f (, x ()) d q (1 p) 2 1 [ α ( q) f (, x ()) d q

9 Applied Mahemaic 9 f (, x ()) d q (1 r) ( q) f (, x ()) d q] (1 p) α 2 1 ( qv) f (V,x(V)) d qv d p z () ψ(ρ)d q 2 ( 2 q)z() ψ(ρ)d q 1 (1 p) 2 1 [ α ( q)z() ψ(ρ)d q z () ψ(ρ)d q (1 r) ( q)z() ψ(ρ)d q ] (1 p) α 2 1 ( qv)z() ψ(ρ)d q V d p. (47) A 2 1, he righ-hand ide of he above inequaliy end o zero independenly of x B ρ.aa aifie he above aumpion; herefore, i follow by he Arzelá-Acoli heorem ha A : C(I q, R) C(I q, R) i compleely coninuou. Le x be a oluion. hen, for I q and following he imilar compuaion a in he fir ep, we have x () ψ( x ) z L 1Λ. (48) Conequenly, we have x 1. (49) ψ ( x ) z L 1Λ In view of (H 5 ), here exi M uch ha x =M.Leue U={x C(I q, R) : x <M}. (5) Noe ha he operaor A : U C(I q, R) i coninuou and compleely coninuou. From he choice of U, herei no x Uuch ha x=λaxfor ome λ (, 1). Conequenly, by he nonlinear alernaive of Leray-Schauder ype (heorem 9), we deduce ha A ha a fixed poin x Uwhich iaoluionofheproblem(4). hi complee he proof. Finally, we prove ha problem (4) ha a lea one oluion on I q by uing Leray-Schauder degree heory. heorem 11. Le f:i q R R be a coninuou funcion. Aume ha: (H 7 ) here exi conan κ<λ 1,whereΛ i given by (28) and N>uch ha f(, x) κ x N for all I q, x C. hen, he boundary value problem (4) ha a lea one oluion. Proof. Le u define an operaor A : C C a (27). We wih o prove ha here exi a lea one oluion x C of he fixed poin equaion x=ax. (51) We define a ball B R C,wihaconanradiuR>given by B R ={x C : max x () <R}. (52) Iq hen, i i ufficien o how ha A:B R C(I q ) aifie Now, we e x =λax, x B R, λ [, 1]. (53) H (λ, x) =λax, x C, λ [, 1]. (54) hen, by he Arzelá-Acoli heorem, we conclude ha a coninuou map h λ defined by h λ (x) = x H(λ, x) = x λax i compleely coninuou. If (53) hold, hen he following Leray-Schauder degree are well defined. From he homoopy invariance of opological degree, i follow ha deg (h λ,b R,)=deg (I λa, B R,)=deg (h 1,B R,) = deg (h,b R,)=deg (I, B R,)=1=, B R, (55) where I denoe he uni operaor. By he nonzero propery of Leray-Schauder degree, h 1 (x) = x Ax = for a lea one x B R. Le u aume ha x=λaxfor ome λ [,1]. hen, for all I q,weobain x () = λ (Ax)() ( q) f (, x ()) d q (1p) [ α ( q) f (, x ()) d q

10 1 Applied Mahemaic f (, x ()) d q (1 r) ( q) f (, x ()) d q] (1 p) α( ) y (V,x(V)) d qv d p ( qv) (κ x N) ( q) d q (κ x N) [ α (1p) ( q)d q d q (1 r) ( q) d q ] (κ x N) (1 p) α( ) ( qv)d q Vd p (κ x N) { 2 1q p 2 [ α 1q (1q(2r)) ] 1q (1 p) ( α ( ) )2 } (1q)(1pp 2 ) = (κ x N) Λ. (56) aking norm up I q x() = x and olving i for x, hi yield x NΛ 1 κλ. (57) Le R = (NΛ/(1 κλ)) 1, hen(53) hold. hi complee he proof 4. Example In hi ecion, we illurae our main reul wih ome example. Le u conider he following boundary value problem of nonlinear econd-order q-difference equaion wih hree-poin boundary condiion D 2 1/2x () =f(, x ()), I1/2 1/2 =I1 1/2 [,1 2 ], 2 3 x(1 8 ) 1 3 D 3/4x( 1 (58) 1/2 8 )=, x () d 1/4 =. Here,wehaveq=1/2, p=1/4, r=3/4, =1/2, α=2/3, = 1/3,and=1/8.Wefindha Λ= 2 1q p 2 [ α 1q (1q(2r)) 1q (1 p) ( α ( ) )2 (1q)(1pp 2 ) = [ ] ] (59) (a) Le f:i 1/2 1/2 R R be a coninuou funcion given by f (, x) = 2 e in 1e x () co2 1 x (). (6) Since, f(, x) f(, y) (1/2) x y,hen(h 1 ) i aified wih L=1/2.Wecanfindha LΛ < 1. (61) Hence, by heorem 3,problem(58)wihf(, x) given by (6) ha a unique oluion on I 1/2 1/2. (b) If f:i 1/2 1/2 R R i a coninuou funcion given by f (, x) = Chooing h() = 1,wefindha (1) x. (62) 1 x G= ( q) h () d q p [ α ( q) h () d q (1) d q (1 r) ( q) (1) d q ] (1 p) ( α ( ) ) ( qv) (V 1) d q Vd p (63)

11 Applied Mahemaic 11 Here, f (, x) f(,y) (1) x y x y. (64) herefore, by heorem 6, he problem (58) wih f(, x) given by (62)haauniqueoluiononI 1/2 1/2. (c) Conider a coninuou funcion f:i 1/2 1/2 R R given by f (, x) = in 2x We can how ha 3 e x2 2. (65) f (, x) = in 2x 3 e x2 2 2 x 3 2, (66) wih κ=2< , (67) Λ and N=3/2.By heorem 11, he problem(58) wih he f(, x) given by (65) ha a lea one oluion on I 1/2 1/2. Acknowledgmen hi reearch of. Sihiwiraham and J. ariboon i uppored by King Mongku Univeriy of echnology Norh Bangkok, hailand. Soiri K. Nouya i a Member of Nonlinear Analyi and Applied Mahemaic (NAAM), Reearch Group a King Abdulaziz Univeriy, Jeddah, Saudi Arabia. Reference [1] F. H. Jackon, On q-difference equaion, American Mahemaic,vol.32,no.4,pp ,191. [2] R. D. Carmichael, he general heory of linear q-difference equaion, American Mahemaic,vol.34,no.2,pp , [3]. E. Maon, On properie of he oluion of linear q- difference equaion wih enire funcion coefficien, American Mahemaic,vol.37,no.4,pp ,1915. [4] C. R. Adam, On he linear ordinary q-difference equaion, American Mahemaic II,vol.3,pp ,1929. [5]W.J.rjizinky, Analyicheoryoflinearq-difference equaion, Aca Mahemaica,vol.61,no.1,pp.1 38,1933. [6]. Ern, A new noaion for q-calculu and a new q-aylor formula, U.U.D.M. Repor 1999:25, Deparmen of Mahemaic, Uppala Univeriy, Uppala, Sweden, [7] R. J. Finkelein, q-field heory, Leer in Mahemaical Phyic,vol.34,no.2,pp ,1995. [8] R. J. Finkelein, q-deformaion of he Lorenz group, Journal of Mahemaical Phyic,vol.37,no.2,pp ,1996. [9] R. Floreanini and L. Vine, Auomorphim of he q-ocillaor algebra and baic orhogonal polynomial, Phyic Leer A, vol. 18, no. 6, pp , [1] R. Floreanini and L. Vine, Symmerie of he q-difference hea equaion, Leer in Mahemaical Phyic,vol.32,no.1,pp.37 44, [11] R. Floreanini and L. Vine, q-gamma and q-bea funcion in quanum algebra repreenaion heory, Compuaional and Applied Mahemaic,vol.68,no.1-2,pp.57 68,1996. [12] P. G. O. Freund and A. V. Zabrodin, he pecral problem for he q-knizhnik-zamolodchikov equaion and coninuou q- Jacobi polynomial, Communicaion in Mahemaical Phyic, vol.173,no.1,pp.17 42,1995. [13] G. Gaper and M. Rahman, Baic Hypergeomeric Serie,Cambridge Univeriy Pre, Cambridge, UK, 199. [14] G. N. Han and J. Zeng, On a q-equence ha generalize he median Genocchi number, Annale de Science Mahémaique du Québec,vol.23,pp.63 72,1999. [15] V.KacandP.Cheung,Quanum Calculu,Springer,NewYork, NY, USA, 22. [16] G. Bangerezako, Variaional q-calculu, Mahemaical Analyi and Applicaion,vol.289,no.2,pp ,24. [17] A. Dobrogowka and A. Odzijewicz, Second order q-difference equaion olvable by facorizaion mehod, Compuaional and Applied Mahemaic, vol.193,no.1,pp , 26. [18] G. Gaper and M. Rahman, Some yem of mulivariable orhogonal q-racah polynomial, Ramanujan Journal, vol.13, no. 1 3, pp , 27. [19] M. E. H. Imail and P. Simeonov, q-difference operaor for orhogonal polynomial, Compuaional and Applied Mahemaic,vol.233,no.3,pp ,29. [2] M. Bohner and G. S. Gueinov, he h-laplace and q-laplace ranform, Mahemaical Analyi and Applicaion, vol. 365, no. 1, pp , 21. [21] M. El-Shahed and H. A. Haan, Poiive oluion of q- difference equaion, Proceeding of he American Mahemaical Sociey,vol.138,no.5,pp ,21. [22] B. Ahmad, Boundary-value problem for nonlinear hirdorder q-difference equaion, Elecronic Differenial Equaion,vol.94,pp.1 7,211. [23] B. Ahmad, A. Alaedi, and S. K. Nouya, A udy of econd-order q-difference equaion wih boundary condiion, Advance in Difference Equaion,vol.212,p.35,212. [24] B. Ahmad, S. K. Nouya, and I. K. Purnara, Exience reul for nonlinear q-difference equaion wih nonlocal boundary condiion, Communicaion on Applied Nonlinear Analyi, vol. 19, pp , 212. [25] B. Ahmad and J. J. Nieo, On nonlocal boundary value problem of nonlinear q-difference equaion, Advance in Difference Equaion,vol.212,p.81,212. [26] B. Ahmad and S. K. Nouya, Boundary value problem for q- difference incluion, Abrac and Applied Analyi, vol. 211, Aricle ID 29286, 15 page, 211. [27] W. Liu and H. Zhou, Exience oluion for boundary value problem of nonlinear fracional q-difference equaion, Advance in Difference Equaion, vol. 213, 113 page, 213. [28] C. Yu and J. Wang, Exience of oluion for nonlinear econdorder q-difference equaion wih fir-order q-derivaive, Advance in Difference Equaion,vol.213,p.124,213. [29] N. Pongarm, S. Aawaamri, and J. ariboon, Sequenial derivaive of nonlinear q-difference equaion wih hree-poin q- inegral boundary condiion, Applied Mahemaic, vol.213,aricleid65169,9page,213.

12 12 Applied Mahemaic [3]D.W.BoydandJ.S.W.Wong, Onnonlinearconracion, Proceeding of he American Mahemaical Sociey, vol.2,pp , [31] M. A. Kranoelkii, wo remark on he mehod of ucceive approximaion, Upekhi Maemaichekikh Nauk, vol. 1, pp , [32] A. Grana and J. Dugundji, Fixed Poin heory, Springer, New York, NY, USA, 23.

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