Invenion Journl of Reerch Technology in Engineering & Mngemen (IJRTEM) ISSN: 2455-3689 www.ijrem.com Volume 2 Iue 9 ǁ Sepemer 28 ǁ PP 73-83 Poiive nd negive oluion of oundry vlue prolem for frcionl, -difference euion Yizhu Wng Chengmin Hou *,2,( Deprmen of Mhemic, Ynin Univeriy, Ynji, 332, P.R. Chin) ABSTRACT : In hi work, we udy oundry vlue prolem for frcionl, -difference euion. By uing he monoone ierive echniue nd lower-upper oluion mehod, we ge he exience of poiive or negive oluion under he nonliner erm i locl coninuiy nd locl monooniciy. The reul how h we cn conruc wo ierive euence for pproximing he oluion. KEYWORDS: frcionl, -difference euion; poiive nd negive oluion; lower nd upper oluion; ierive mehod; I. INTRODUCTION A unum clculu uiue he clicl derivive y difference operor, which llow one o del wih e of non-differenile funcion. There re mny differen ype of unum difference operor uch h- clculu, -clculu, Hhn clculu. Thee operor re lo found in mny pplicion of mhemicl re uch orhogonl polynomil, cominoric nd he clculu of vriion. Hhn [] inroduced hi difference operor D, f ( + ) f ( ) f ( ), ( ) + D, follow:, where f i rel funcion, nd (,) nd re rel fixed numer. Mlinowk nd Torre [2,3] inroduced he Hhn umum vriionl clculu, while Mlinowk nd Mrin [4] inveiged he generlized rnverliy condiion for he Hhn unum vriionl clculu. Recenly, Hmz e l. [5,6] udied he heory of liner Hhn difference euion, nd udied he exience nd uniuene reul of he iniil vlue prolem wih Hhn difference euion uing he mehod of ucceive pproximion. Moived y he foremenioned work, we conider he following nonliner oundry vlue prolem for frcionl, -difference euion: D, u( ) + f (, u( )), (, ), u D, u D, u, (.) where (,), 2 3, f :[, ] [, + ), i he frcionl, -derivive of he Riemnn - D, Liouville ype? II. BACKGROUND AND DEFINITIONS To how he min reul of hi work, we give in he following ome ic definiion, lemm nd heorem, which cn e found in [7]. Volume 2 Iue 9 www.ijrem.com 73
Poiive nd negive oluion of oundry vlue Definiion 2. [7] Le I e cloed inervl of uch h,, I. For f : I we define he, -inegrl of f from o y where f ( ) d : f ( ) d f ( ) d,,,, x f ( ) d : ( x( ) ) f ( x [ k] ), x I, k k, +, k wih k ( ) [ k], for k {}, provided h he erie converge x x. Lemm 2.2 [7] Aume f : I e coninuou. Define x F( x) : f ( Then ) d, F. i coninuou. Fuhermore, D, F( x) exi for every nd xid Converely,, F( x) f ( x). for ll, I. D F x d f f,, Lemm 2.3 [7] For, For ny poiive ineger k,, ( + ) [ ],, ().,,, ( k+ ) [ k]!. Definiion 2.4 [7] Le Riemnn-Liouville ype i given y nd f e funcion defined on [,]. The Hhn frcionl inegrion of ( I f )( ) f ( ), ( I f )( ) ( ) f d,, [, ]. Definiion 2.5 [7] The frcionl, -derivive of he Riemnn -Liouville ype i nd ( ),, where denoe he mlle ineger greer or eul o. Theorem 2.6 [7] Le ( N, N]. Then for ome conn c, i,2,, N, he following euliy hold: ( D f )( ) ( D I f )( ),,,,, ( I D f )( x) f ( x) + c ( x ) + c ( x ) + + c ( x ). ( ) ( 2) ( N ),, 2 N i Volume 2 Iue 9 www.ijrem.com 74
Poiive nd negive oluion of oundry vlue Lemm 2.7 [8] Aume h X i Bnch pce nd K i norml cone in X, T :[ u, v] X i compleely coninuou increing operor which ifie u Tu, Tv Then v. T h miniml fixed poin u * * nd mximl fixed poin v wih u lim T n u, v u u v * v. * * lim T n v, n In ddiion, n where { Tu} n i n increing euence, { Tv n } n i decreing euence. III. EXISTENCE OF, -FRACTIONAL POSITIVE SOLUTIONS FOR PROBLEM Lemm 3. Aume g C[, ], hen he following oundry vlue prolem: D, u( ) + g( ), (, ), u D, u D, u, h uniue oluion * n where u( ) G(, ) g d,, ( ) ( ) ( 2) ( ) ( ) ( ), ( 2) ( ) G(, ), ( ) ( ) ( 2) ( ) ( ), ( 2). Proof In view of Theorem 2.6, Since we hve c c From he oundry condiion we ge Hence u( ) ( ) g d + c ( x ) + c ( x ), ( ) ( ) ( 2), 2 u D u, D, u, + c x ( 3), 3 2 3. c ( ) g d. ( 2) ( 2), ( ) u( ) ( ) g d + ( ) ( ), ( 2) ( ) ( ) g d ( 2), Volume 2 Iue 9 www.ijrem.com 75
Poiive nd negive oluion of oundry vlue ( ) ( ) ( 2) ( ) ( ) ( ) ( 2) g d, ( ) ( ) ( 2) + ( ) ( 2) g d, (, ). G g d, Lemm 3.2 The funcion G(, ) h he following properie: () () G(, ), G(, ) G(, ), ( ) G(, ) G(, ), ( ) ( 2) ( ),., ; Remrk 3.3 The funcion G(, ) h ome oher properie: () ( ) G(, ) ( ) ( ), ( ) ( 2) ( ) ( 2) ( ),. (2) According o he propery of eing non-decreing of funcion ( ) ( )( ) ( 2) i+ i+ 2 ( 2) i+ i ( ) ( )( ) on nd non-increing of ( ) 2 ( 2) on, We cn oin he following ineuliie: () For 2, we ge G(, ) G(, ) 2 ( ) ( ) ( ) ( ) ( ) ( ) 2 ( 2) ( 2) ( ) ( ) ( 2) ( 2) ( 2) ( ) ( ) ( ) ( 2) ( ) ( ) [( 2 ) ( ) ]. ( ) ( ) 2 () For, 2 we ge G(, ) G(, ) 2 Volume 2 Iue 9 www.ijrem.com 76
Poiive nd negive oluion of oundry vlue ( ) ( ) 2 ( 2) ( ) ( ) ( ( 2) 2 ) ( 2) ( ) ( ) ( ) ( ) ( 2) ( ) ( ) ( ) 2 2 (c) For ( ) ( ) ( 2) ( ) ( 2), we ge 2 ( ) ( ) ( ) [( 2 ) ( ) ( 2 ) + ]. G(, ) G(, ) 2 ( ) ( ) 2 ( 2) ( ) ( ) ( ( 2) 2 ) ( ) ( ) ( 2) ( ) ( ) ( ( 2) ) ( ) ( 2) ( ) ( ) ( ) ( 2) ( ) ( ) 2 + ( ) ( ) ( ) ( ) 2 ( ) ( ) ( ) ( ) ( 2 ) ( ) + ( 2 ) ( ). (3) G(, ), for (, ) (, ) (, ). Le X C[, ], he Bnch pce of ll coninuou funcion on [,], wih norm u mx{ u( ) : [, ]}. In our coniderion, we need he ndrd cone K X y K { u [, ]: u( ), }. I i cler h he cone K i norml. Theorem 3.4 Aume h ( F ) here exi rel numer d nd g L [, ], uch h ( i ) f :[, ] [, d] [, +) i coninuou, f (, u) g( ) for (, u) [, ] [, d] nd f (, u) f (, v) for, u v d; Volume 2 Iue 9 www.ijrem.com 77
( i 2) he following ineuliy hold: ( ) ( ) f, d d d. ( ) ( 2) ( 2), ( F 2) here exi c (, d) uch h Poiive nd negive oluion of oundry vlue ( ) ( ) G(, ), f c d ( 2), c. ( ) Then he prolem (.) h wo poiive oluion u, v D, where In ddiion, le ( ) ( ) D { u C[, ] c u( ) d, [, ]}. ( ) ( ) ( ) u ( ) c, ( ) ( 2) ( ) ( ) ( ) ( 2) ( 2) ( ) v d ( ) () ( 2) ( ) nd conruc he following euence: n+ (, ) (, ) n,, n+ n, u G f u d v G(, ) f (, v ) d, n,,2,, one h lim un u,lim vn v. n n Proof From he non-negivene nd coninuiy of G nd f, we cn define n operor T : C[, ] C[, ] y,. Tu( ) G(, ) f (, u) d, From Lemm 3., we cn ee h u i he oluion of prolem (.) if nd only if u i he fixed poin of T. We will how h T h fixed poin in he order inervl [ u, v ]. hve We need o how h T :[ u, v] C[, ] i compleely coninuou operor. For u [ u, v], we ( ) ( ) c u( ) d d, ( ) ( ) Since G(, ) i coninuou. So we only prove T i compc. Le M hen From he hypohei ( F) ( i2) nd g d,, M +. lemm 3.2, we ge ( ) ( ) ( 2) ( 2) Tu( ) mx G(, ) f (, u) d,. mx G (, ( )) f (, u ( )) d, M ( ) g d ( ). ( ) ( ), Volume 2 Iue 9 www.ijrem.com 78
Poiive nd negive oluion of oundry vlue Thi how h he e T ([ u, v ]) i uniform ounded in C[, ]. Afer h, for given, 2 [, ] wih, 2 nd u [ u, v], we oin Tu ( ) Tu ( ) G(, ) G(, ) f (, u) d 2 2, mx G(, ) G(, ) g d 2, M mx G(, ) G(, ). 2 In view of Remrk 3.3(2), one h Tu( ) Tu2( ),. 2 So we clim h he e T ([ u, v ]) i euiconinuou in C[, ]. By men of he Arzel-Acoli heorem, T :[ u, v ] C[, ] i compleely operor. By he hypohei ( F) ( i), T i n increing operor. From ( F),( F 2) nd Lemm 3.2, for ny [, ], one cn ee h Tu ( ) G(, ) f (, u ) d, ( ) ( ) G(, ), f c d ( 2), ( ) ( ) ( ) ( ) ( ) G(, ), ( 2) f c d ( 2), ( ) ( ) ( ) c u() ( ) ( ) ( 2) nd Tv ( ) G(, ) f (, v ) d, ( ) ( ) G(, ), f d d ( 2), ( ) ( ) ( ) ( ) ( ) ( 2) ( ), ( 2) f d d ( 2), ( ) ( ) d v ( ). ( ) ( ) ( 2) Hence, we ge Tu u, Tv v. We conruc he following euence: n+ (, ) (, ) n,, n+ n, u G f u d v G(, ) f (, v ) d, Volume 2 Iue 9 www.ijrem.com 79
Poiive nd negive oluion of oundry vlue n,, 2,. From he monooniciy of T, we hve un+ un, vn+ vn, n,, 2,. By uing Lemm 2.7, we know h he operor T h wo poiive oluion u, v C[, ] wih ( ) ( ) ( ) ( ) h i, c u ( ) v ( ) d,. In ddiion, lim un u,lim vn v. ( 2) ( 2) n n ( ) ( ) Theorem 3.5 Aume h u u v v * *, ( F 3) here exi rel numer d nd g L [, ], uch h ( i 3) f :[, ] [, d] i coninuou, f (, u) g( ) for (, u) [, ] [, d] nd f (, u) f (, v) for [, ], u v d; ( i 4) he following ineuliy hold: ( ) ( 2) ( ) mx{,,} ( 2), ( F 4) here exi c [, d] uch h ( ) f d d ( ) ( ) + G(, ) min{,,} f d d ( 2), d. ( ) ( ) ( ) G(, )mx{,,} f c d ( 2), ( ) ( ) ( ) ( 2) + ( ) min{,,} f c d ( 2), c. ( ) Then he prolem (.) h wo poiive oluion u, v D, where ( ) ( ) D { u C[, ] c u( ) d, [, ]}. ( ) ( ) ( ) ( ) ( 2) ( 2) In ddiion, le ( ) u ( ) c, ( ) ( 2) ( ) ( ) v d ( ) () ( 2) ( ) nd conruc he following euence: n+ (, ) (, ) n,, n+ n, u G f u d v G(, ) f (, v ) d, n,,2,, one h lim un u,lim vn v. n n Proof Conider he me operor T : C[, ] C[, ] defined in he proof of Theorem 3.4:, [, ]. Tu( ) G(, ) f (, u) d, Volume 2 Iue 9 www.ijrem.com 8
We lo how h T h fixed poin in he order inervl [ u, v ]. Poiive nd negive oluion of oundry vlue Similr o he proof of Theorem 3.4, T : C[, ] C[, ] i compleely coninuou operor. From he hypohei ( F3) ( i3), T i n increing operor. Furher, y uing he condiion ( F3),( F 4), Remrk 3.3 nd Lemm 3.2, for ny [, ], one oin Tu ( ) G(, ) f (, u ) d, ( ) ( ) G(, ) mx{,,} f c d ( 2), ( ) ( ) ( ) + G(, )min{,,} f c d ( 2), ( ) ( ) ( ) ( ) ( ) G(, ) mx{,,} ( 2) f c d ( 2), ( ) ( ) ( ) ( ) ( 2) + ( ) min{,,} f c d ( 2), ( ) ( ) c u() ( ) ( ) ( 2) nd Tv ( ) G(, ) f (, v ) d, ( ) ( ) G(, )mx{,,} f d d ( 2), ( ) ( ) ( ) + G(, )min{,,} f d d ( 2), ( ) ( ) d v ( ). ( ) ( ) ( 2) Hence, we ge Tu u, Tv v. We conruc he following euence: n+ (, ) (, ) n,, n+ n, u G f u d v G(, ) f (, v ) d, n,, 2,. According o he monooniciy of T, we ge u u, v v, n,, 2,. n+ n n+ n Volume 2 Iue 9 www.ijrem.com 8
Poiive nd negive oluion of oundry vlue By uing Lemm 2.7, we know h he operor T h wo poiive oluion u u v v * *, h i, ( ) ( ) c u ( ) v ( ) d d, ( ) ( ) ( ) ( ) ( 2) ( 2) u, v C[, ] wih. In ddiion, lim un u,lim vn v. n n By uing he me proof Theorem 3.5, we cn eily oin he following concluion. Theorem 3.6 Aume h ( F 5) here exi rel numer c nd g L [, ], uch h ( i 5) f :[, ] [ c, ] i coninuou, f (, u) g( ) for (, u) [, ] [ c, ] nd f (, u) f (, v) for [, ], c u v. In ddiion, here exi d ( c, ) uch h ( F ) ( i ) nd ( F 4) in Theorem 3.5 re lo ified. Then 3 4 he prolem (.) h wo negive oluion u, v D, where ( ) ( ) D { u C[, ] c u( ) d, [, ]} ( ) ( ) ( ) ( ) ( 2) ( 2) Le ( ) u ( ) c, ( ) ( 2) ( ) ( ) v d ( ) () ( 2) ( ) nd we conruc he following euence: n+ (, ) (, ) n,, n+ n, u G f u d v G(, ) f (, v ) d, n,,2,, we cn oin lim un u,lim vn v. n n REFERENCES [] Hhn,W: Üer Orhogonlpolynome, die -Differenzenlgleichungen genügen. Mh. Nchr. 2, 4-34(949) [2] Mlinowk, AB, Torre, DFM: The Hhn unum vriionl clculu. J. Opim. Theory Appl. 47, 49-442(2) [3] Mlinowk, AB, Torre, DFM: Qunum Vriionl Clculu. Springer Berlin (24) [4] Mlinowk, AB, Mrin, N: Generlized rnverliy condiion for he Hhn unum vriionl clculu. Opimizion 62(3),323-344(23) [5] Hmz, AE, Ahmed, SM: Theory of liner Hhn difference euion. J. Adv. Mh. 4(2), 44-46(23) [6] Hmz, AE, Ahmed, SM: Exience nd uniuene of oluion of Hhn difference euion. Adv. Differ. Eu. 23, 36(23) [7] Yizhu Wng, Yiding Liu, Chengmin Hou: New concep of frcionl Hhn, -derivive of Riemnn-Liouville ype nd Cpuo ype nd pplicion. Adv. Differ. Eu. 292(28) Volume 2 Iue 9 www.ijrem.com 82
Poiive nd negive oluion of oundry vlue [8] Guo, D, Lkhmiknhm, V: Nonliner Prolem in Arc Cone. Acdemic Pre, New York(988) Yizhu Wng, Poiive nd negive oluion of oundry vlue prolem for frcionl - difference euion. Invenion Journl of Reerch Technology in Engineering & Mngemen (IJRTEM),2(9), 73-82. Rerieved Sepemer 2, 28, from www.ijrem.com. Volume 2 Iue 9 www.ijrem.com 83