Positive and negative solutions of a boundary value problem for a

Size: px
Start display at page:

Download "Positive and negative solutions of a boundary value problem for a"

Transcription

1 Invenion Journl of Reerch Technology in Engineering & Mngemen (IJRTEM) ISSN: Volume 2 Iue 9 ǁ Sepemer 28 ǁ PP 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

2 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

3 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), c ( ) g d. ( 2) ( 2), ( ) u( ) ( ) g d + ( ) ( ), ( 2) ( ) ( ) g d ( 2), Volume 2 Iue

4 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

5 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

6 ( 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

7 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

8 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 8

9 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 8

10 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, (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), (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

11 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), Rerieved Sepemer 2, 28, from Volume 2 Iue

Hermite-Hadamard-Fejér type inequalities for convex functions via fractional integrals

Hermite-Hadamard-Fejér type inequalities for convex functions via fractional integrals Sud. Univ. Beş-Bolyi Mh. 6(5, No. 3, 355 366 Hermie-Hdmrd-Fejér ype inequliies for convex funcions vi frcionl inegrls İmd İşcn Asrc. In his pper, firsly we hve eslished Hermie Hdmrd-Fejér inequliy for

More information

GENERALIZATION OF SOME INEQUALITIES VIA RIEMANN-LIOUVILLE FRACTIONAL CALCULUS

GENERALIZATION OF SOME INEQUALITIES VIA RIEMANN-LIOUVILLE FRACTIONAL CALCULUS - TAMKANG JOURNAL OF MATHEMATICS Volume 5, Number, 7-5, June doi:5556/jkjm555 Avilble online hp://journlsmhkueduw/ - - - GENERALIZATION OF SOME INEQUALITIES VIA RIEMANN-LIOUVILLE FRACTIONAL CALCULUS MARCELA

More information

LAPLACE TRANSFORMS. 1. Basic transforms

LAPLACE TRANSFORMS. 1. Basic transforms LAPLACE TRANSFORMS. Bic rnform In hi coure, Lplce Trnform will be inroduced nd heir properie exmined; ble of common rnform will be buil up; nd rnform will be ued o olve ome dierenil equion by rnforming

More information

Contraction Mapping Principle Approach to Differential Equations

Contraction Mapping Principle Approach to Differential Equations epl Journl of Science echnology 0 (009) 49-53 Conrcion pping Principle pproch o Differenil Equions Bishnu P. Dhungn Deprmen of hemics, hendr Rn Cmpus ribhuvn Universiy, Khmu epl bsrc Using n eension of

More information

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SECOND-ORDER ITERATIVE BOUNDARY-VALUE PROBLEM

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SECOND-ORDER ITERATIVE BOUNDARY-VALUE PROBLEM Elecronic Journl of Differenil Equions, Vol. 208 (208), No. 50, pp. 6. ISSN: 072-669. URL: hp://ejde.mh.xse.edu or hp://ejde.mh.un.edu EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SECOND-ORDER ITERATIVE

More information

Integral Transform. Definitions. Function Space. Linear Mapping. Integral Transform

Integral Transform. Definitions. Function Space. Linear Mapping. Integral Transform Inegrl Trnsform Definiions Funcion Spce funcion spce A funcion spce is liner spce of funcions defined on he sme domins & rnges. Liner Mpping liner mpping Le VF, WF e liner spces over he field F. A mpping

More information

Research Article The General Solution of Differential Equations with Caputo-Hadamard Fractional Derivatives and Noninstantaneous Impulses

Research Article The General Solution of Differential Equations with Caputo-Hadamard Fractional Derivatives and Noninstantaneous Impulses Hindwi Advnce in Mhemicl Phyic Volume 207, Aricle ID 309473, pge hp://doi.org/0.55/207/309473 Reerch Aricle The Generl Soluion of Differenil Equion wih Cpuo-Hdmrd Frcionl Derivive nd Noninnneou Impule

More information

Exponential Decay for Nonlinear Damped Equation of Suspended String

Exponential Decay for Nonlinear Damped Equation of Suspended String 9 Inernionl Symoium on Comuing, Communicion, nd Conrol (ISCCC 9) Proc of CSIT vol () () IACSIT Pre, Singore Eonenil Decy for Nonliner Dmed Equion of Suended Sring Jiong Kemuwn Dermen of Mhemic, Fculy of

More information

On The Hermite- Hadamard-Fejér Type Integral Inequality for Convex Function

On The Hermite- Hadamard-Fejér Type Integral Inequality for Convex Function Turkish Journl o Anlysis nd Numer Theory, 4, Vol., No. 3, 85-89 Aville online h://us.scieu.com/jn//3/6 Science nd Educion Pulishing DOI:.69/jn--3-6 On The Hermie- Hdmrd-Fejér Tye Inegrl Ineuliy or Convex

More information

ON NEW INEQUALITIES OF SIMPSON S TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE CONVEX

ON NEW INEQUALITIES OF SIMPSON S TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE CONVEX Journl of Applied Mhemics, Sisics nd Informics JAMSI), 9 ), No. ON NEW INEQUALITIES OF SIMPSON S TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE CONVEX MEHMET ZEKI SARIKAYA, ERHAN. SET

More information

5.1-The Initial-Value Problems For Ordinary Differential Equations

5.1-The Initial-Value Problems For Ordinary Differential Equations 5.-The Iniil-Vlue Problems For Ordinry Differenil Equions Consider solving iniil-vlue problems for ordinry differenil equions: (*) y f, y, b, y. If we know he generl soluion y of he ordinry differenil

More information

FURTHER GENERALIZATIONS. QI Feng. The value of the integral of f(x) over [a; b] can be estimated in a variety ofways. b a. 2(M m)

FURTHER GENERALIZATIONS. QI Feng. The value of the integral of f(x) over [a; b] can be estimated in a variety ofways. b a. 2(M m) Univ. Beogrd. Pul. Elekroehn. Fk. Ser. M. 8 (997), 79{83 FUTHE GENEALIZATIONS OF INEQUALITIES FO AN INTEGAL QI Feng Using he Tylor's formul we prove wo inegrl inequliies, h generlize K. S. K. Iyengr's

More information

ON NEW INEQUALITIES OF SIMPSON S TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE CONVEX.

ON NEW INEQUALITIES OF SIMPSON S TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE CONVEX. ON NEW INEQUALITIES OF SIMPSON S TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE CONVEX. MEHMET ZEKI SARIKAYA?, ERHAN. SET, AND M. EMIN OZDEMIR Asrc. In his noe, we oin new some ineuliies

More information

Application on Inner Product Space with. Fixed Point Theorem in Probabilistic

Application on Inner Product Space with. Fixed Point Theorem in Probabilistic Journl of Applied Mhemics & Bioinformics, vol.2, no.2, 2012, 1-10 ISSN: 1792-6602 prin, 1792-6939 online Scienpress Ld, 2012 Applicion on Inner Produc Spce wih Fixed Poin Theorem in Probbilisic Rjesh Shrivsv

More information

IX.1.1 The Laplace Transform Definition 700. IX.1.2 Properties 701. IX.1.3 Examples 702. IX.1.4 Solution of IVP for ODEs 704

IX.1.1 The Laplace Transform Definition 700. IX.1.2 Properties 701. IX.1.3 Examples 702. IX.1.4 Solution of IVP for ODEs 704 Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, 7 699 IX. THE APACE TRANSFORM IX.. The plce Trnform Definiion 7 IX.. Properie 7 IX..3 Emple 7 IX..4 Soluion of IVP for ODE 74 IX..5 Soluion

More information

Some New Dynamic Inequalities for First Order Linear Dynamic Equations on Time Scales

Some New Dynamic Inequalities for First Order Linear Dynamic Equations on Time Scales Applied Memicl Science, Vol. 1, 2007, no. 2, 69-76 Some New Dynmic Inequliie for Fir Order Liner Dynmic Equion on Time Scle B. İ. Yşr, A. Tun, M. T. Djerdi nd S. Küükçü Deprmen of Memic, Fculy of Science

More information

Citation Abstract and Applied Analysis, 2013, v. 2013, article no

Citation Abstract and Applied Analysis, 2013, v. 2013, article no Tile An Opil-Type Inequliy in Time Scle Auhor() Cheung, WS; Li, Q Ciion Arc nd Applied Anlyi, 13, v. 13, ricle no. 53483 Iued De 13 URL hp://hdl.hndle.ne/17/181673 Righ Thi work i licened under Creive

More information

A LIMIT-POINT CRITERION FOR A SECOND-ORDER LINEAR DIFFERENTIAL OPERATOR IAN KNOWLES

A LIMIT-POINT CRITERION FOR A SECOND-ORDER LINEAR DIFFERENTIAL OPERATOR IAN KNOWLES A LIMIT-POINT CRITERION FOR A SECOND-ORDER LINEAR DIFFERENTIAL OPERATOR j IAN KNOWLES 1. Inroducion Consider he forml differenil operor T defined by el, (1) where he funcion q{) is rel-vlued nd loclly

More information

Weighted Inequalities for Riemann-Stieltjes Integrals

Weighted Inequalities for Riemann-Stieltjes Integrals Aville hp://pvm.e/m Appl. Appl. Mh. ISSN: 93-9466 ol. Ie Decemer 06 pp. 856-874 Applicion n Applie Mhemic: An Inernionl Jornl AAM Weighe Ineqliie or Riemnn-Sielje Inegrl Hüeyin Bk n Mehme Zeki Sriky Deprmen

More information

Mathematics 805 Final Examination Answers

Mathematics 805 Final Examination Answers . 5 poins Se he Weiersrss M-es. Mhemics 85 Finl Eminion Answers Answer: Suppose h A R, nd f n : A R. Suppose furher h f n M n for ll A, nd h Mn converges. Then f n converges uniformly on A.. 5 poins Se

More information

New Inequalities in Fractional Integrals

New Inequalities in Fractional Integrals ISSN 1749-3889 (prin), 1749-3897 (online) Inernionl Journl of Nonliner Science Vol.9(21) No.4,pp.493-497 New Inequliies in Frcionl Inegrls Zoubir Dhmni Zoubir DAHMANI Lborory of Pure nd Applied Mhemics,

More information

Convergence of Singular Integral Operators in Weighted Lebesgue Spaces

Convergence of Singular Integral Operators in Weighted Lebesgue Spaces EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 2, 2017, 335-347 ISSN 1307-5543 www.ejpm.com Published by New York Business Globl Convergence of Singulr Inegrl Operors in Weighed Lebesgue

More information

IX.1.1 The Laplace Transform Definition 700. IX.1.2 Properties 701. IX.1.3 Examples 702. IX.1.4 Solution of IVP for ODEs 704

IX.1.1 The Laplace Transform Definition 700. IX.1.2 Properties 701. IX.1.3 Examples 702. IX.1.4 Solution of IVP for ODEs 704 Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 6, 8 699 IX. THE APACE TRANSFORM IX.. The plce Trnform Definiion 7 IX.. Properie 7 IX..3 Emple 7 IX..4 Soluion of IVP for ODE 74 IX..5 Soluion

More information

LAPLACE TRANSFORM OVERCOMING PRINCIPLE DRAWBACKS IN APPLICATION OF THE VARIATIONAL ITERATION METHOD TO FRACTIONAL HEAT EQUATIONS

LAPLACE TRANSFORM OVERCOMING PRINCIPLE DRAWBACKS IN APPLICATION OF THE VARIATIONAL ITERATION METHOD TO FRACTIONAL HEAT EQUATIONS Wu, G.-.: Lplce Trnsform Overcoming Principle Drwbcks in Applicion... THERMAL SIENE: Yer 22, Vol. 6, No. 4, pp. 257-26 257 Open forum LAPLAE TRANSFORM OVEROMING PRINIPLE DRAWBAKS IN APPLIATION OF THE VARIATIONAL

More information

ENGR 1990 Engineering Mathematics The Integral of a Function as a Function

ENGR 1990 Engineering Mathematics The Integral of a Function as a Function ENGR 1990 Engineering Mhemics The Inegrl of Funcion s Funcion Previously, we lerned how o esime he inegrl of funcion f( ) over some inervl y dding he res of finie se of rpezoids h represen he re under

More information

Green s Functions and Comparison Theorems for Differential Equations on Measure Chains

Green s Functions and Comparison Theorems for Differential Equations on Measure Chains Green s Funcions nd Comprison Theorems for Differenil Equions on Mesure Chins Lynn Erbe nd Alln Peerson Deprmen of Mhemics nd Sisics, Universiy of Nebrsk-Lincoln Lincoln,NE 68588-0323 lerbe@@mh.unl.edu

More information

INTEGRALS. Exercise 1. Let f : [a, b] R be bounded, and let P and Q be partitions of [a, b]. Prove that if P Q then U(P ) U(Q) and L(P ) L(Q).

INTEGRALS. Exercise 1. Let f : [a, b] R be bounded, and let P and Q be partitions of [a, b]. Prove that if P Q then U(P ) U(Q) and L(P ) L(Q). INTEGRALS JOHN QUIGG Eercise. Le f : [, b] R be bounded, nd le P nd Q be priions of [, b]. Prove h if P Q hen U(P ) U(Q) nd L(P ) L(Q). Soluion: Le P = {,..., n }. Since Q is obined from P by dding finiely

More information

ON THE OSTROWSKI-GRÜSS TYPE INEQUALITY FOR TWICE DIFFERENTIABLE FUNCTIONS

ON THE OSTROWSKI-GRÜSS TYPE INEQUALITY FOR TWICE DIFFERENTIABLE FUNCTIONS Hceepe Journl of Mhemics nd Sisics Volume 45) 0), 65 655 ON THE OSTROWSKI-GRÜSS TYPE INEQUALITY FOR TWICE DIFFERENTIABLE FUNCTIONS M Emin Özdemir, Ahme Ock Akdemir nd Erhn Se Received 6:06:0 : Acceped

More information

( ) ( ) ( ) ( ) ( ) ( y )

( ) ( ) ( ) ( ) ( ) ( y ) 8. Lengh of Plne Curve The mos fmous heorem in ll of mhemics is he Pyhgoren Theorem. I s formulion s he disnce formul is used o find he lenghs of line segmens in he coordine plne. In his secion you ll

More information

Procedia Computer Science

Procedia Computer Science Procedi Compuer Science 00 (0) 000 000 Procedi Compuer Science www.elsevier.com/loce/procedi The Third Informion Sysems Inernionl Conference The Exisence of Polynomil Soluion of he Nonliner Dynmicl Sysems

More information

Applications of Prüfer Transformations in the Theory of Ordinary Differential Equations

Applications of Prüfer Transformations in the Theory of Ordinary Differential Equations Irih Mh. Soc. Bullein 63 (2009), 11 31 11 Applicion of Prüfer Trnformion in he Theory of Ordinry Differenil Equion GEORGE CHAILOS Abrc. Thi ricle i review ricle on he ue of Prüfer Trnformion echnique in

More information

0 for t < 0 1 for t > 0

0 for t < 0 1 for t > 0 8.0 Sep nd del funcions Auhor: Jeremy Orloff The uni Sep Funcion We define he uni sep funcion by u() = 0 for < 0 for > 0 I is clled he uni sep funcion becuse i kes uni sep = 0. I is someimes clled he Heviside

More information

graph of unit step function t

graph of unit step function t .5 Piecewie coninuou forcing funcion...e.g. urning he forcing on nd off. The following Lplce rnform meril i ueful in yem where we urn forcing funcion on nd off, nd when we hve righ hnd ide "forcing funcion"

More information

How to prove the Riemann Hypothesis

How to prove the Riemann Hypothesis Scholrs Journl of Phsics, Mhemics nd Sisics Sch. J. Phs. Mh. S. 5; (B:5-6 Scholrs Acdemic nd Scienific Publishers (SAS Publishers (An Inernionl Publisher for Acdemic nd Scienific Resources *Corresonding

More information

e t dt e t dt = lim e t dt T (1 e T ) = 1

e t dt e t dt = lim e t dt T (1 e T ) = 1 Improper Inegrls There re wo ypes of improper inegrls - hose wih infinie limis of inegrion, nd hose wih inegrnds h pproch some poin wihin he limis of inegrion. Firs we will consider inegrls wih infinie

More information

Research Article An Expansion Formula with Higher-Order Derivatives for Fractional Operators of Variable Order

Research Article An Expansion Formula with Higher-Order Derivatives for Fractional Operators of Variable Order Hindwi Pulishing Corporion The Scienific World Journl Volume 23, Aricle ID 95437, pges hp://dx.doi.org/.55/23/95437 Reserch Aricle An Expnsion Formul wih Higher-Order Derivives for Frcionl Operors of Vrile

More information

On Hadamard and Fejér-Hadamard inequalities for Caputo k-fractional derivatives

On Hadamard and Fejér-Hadamard inequalities for Caputo k-fractional derivatives In J Nonliner Anl Appl 9 8 No, 69-8 ISSN: 8-68 elecronic hp://dxdoiorg/75/ijn8745 On Hdmrd nd Fejér-Hdmrd inequliies for Cpuo -frcionl derivives Ghulm Frid, Anum Jved Deprmen of Mhemics, COMSATS Universiy

More information

Stability in Distribution for Backward Uncertain Differential Equation

Stability in Distribution for Backward Uncertain Differential Equation Sabiliy in Diribuion for Backward Uncerain Differenial Equaion Yuhong Sheng 1, Dan A. Ralecu 2 1. College of Mahemaical and Syem Science, Xinjiang Univeriy, Urumqi 8346, China heng-yh12@mail.inghua.edu.cn

More information

Refinements to Hadamard s Inequality for Log-Convex Functions

Refinements to Hadamard s Inequality for Log-Convex Functions Alied Mhemics 899-93 doi:436/m7 Pulished Online Jul (h://wwwscirporg/journl/m) Refinemens o Hdmrd s Ineuli for Log-Convex Funcions Asrc Wdllh T Sulimn Dermen of Comuer Engineering College of Engineering

More information

CSC 373: Algorithm Design and Analysis Lecture 9

CSC 373: Algorithm Design and Analysis Lecture 9 CSC 373: Algorihm Deign n Anlyi Leure 9 Alln Boroin Jnury 28, 2013 1 / 16 Leure 9: Announemen n Ouline Announemen Prolem e 1 ue hi Friy. Term Te 1 will e hel nex Mony, Fe in he uoril. Two nnounemen o follow

More information

How to Prove the Riemann Hypothesis Author: Fayez Fok Al Adeh.

How to Prove the Riemann Hypothesis Author: Fayez Fok Al Adeh. How o Prove he Riemnn Hohesis Auhor: Fez Fok Al Adeh. Presiden of he Srin Cosmologicl Socie P.O.Bo,387,Dmscus,Sri Tels:963--77679,735 Emil:hf@scs-ne.org Commens: 3 ges Subj-Clss: Funcionl nlsis, comle

More information

Analytic solution of linear fractional differential equation with Jumarie derivative in term of Mittag-Leffler function

Analytic solution of linear fractional differential equation with Jumarie derivative in term of Mittag-Leffler function Anlyic soluion of liner frcionl differenil equion wih Jumrie derivive in erm of Mig-Leffler funcion Um Ghosh (), Srijn Sengup (2), Susmi Srkr (2b), Shnnu Ds (3) (): Deprmen of Mhemics, Nbdwip Vidysgr College,

More information

4.8 Improper Integrals

4.8 Improper Integrals 4.8 Improper Inegrls Well you ve mde i hrough ll he inegrion echniques. Congrs! Unforunely for us, we sill need o cover one more inegrl. They re clled Improper Inegrls. A his poin, we ve only del wih inegrls

More information

Research Article Generalized Fractional Integral Inequalities for Continuous Random Variables

Research Article Generalized Fractional Integral Inequalities for Continuous Random Variables Journl of Proiliy nd Sisics Volume 2015, Aricle ID 958980, 7 pges hp://dx.doi.org/10.1155/2015/958980 Reserch Aricle Generlized Frcionl Inegrl Inequliies for Coninuous Rndom Vriles Adullh Akkur, Zeynep

More information

FM Applications of Integration 1.Centroid of Area

FM Applications of Integration 1.Centroid of Area FM Applicions of Inegrion.Cenroid of Are The cenroid of ody is is geomeric cenre. For n ojec mde of uniform meril, he cenroid coincides wih he poin which he ody cn e suppored in perfecly lnced se ie, is

More information

EECE 301 Signals & Systems Prof. Mark Fowler

EECE 301 Signals & Systems Prof. Mark Fowler EECE 31 Signal & Syem Prof. Mark Fowler Noe Se #27 C-T Syem: Laplace Tranform Power Tool for yem analyi Reading Aignmen: Secion 6.1 6.3 of Kamen and Heck 1/18 Coure Flow Diagram The arrow here how concepual

More information

FRACTIONAL-order differential equations (FDEs) are

FRACTIONAL-order differential equations (FDEs) are Proceedings of he Inernionl MuliConference of Engineers nd Compuer Scieniss 218 Vol I IMECS 218 Mrch 14-16 218 Hong Kong Comprison of Anlyicl nd Numericl Soluions of Frcionl-Order Bloch Equions using Relible

More information

Existence of positive solutions for fractional q-difference. equations involving integral boundary conditions with p- Laplacian operator

Existence of positive solutions for fractional q-difference. equations involving integral boundary conditions with p- Laplacian operator Invenion Journal of Researh Tehnology in Engineering & Managemen IJRTEM) ISSN: 455-689 www.ijrem.om Volume Issue 7 ǁ July 8 ǁ PP 5-5 Exisene of osiive soluions for fraional -differene euaions involving

More information

Approximation and numerical methods for Volterra and Fredholm integral equations for functions with values in L-spaces

Approximation and numerical methods for Volterra and Fredholm integral equations for functions with values in L-spaces Approximion nd numericl mehods for Volerr nd Fredholm inegrl equions for funcions wih vlues in L-spces Vir Bbenko Deprmen of Mhemics, The Universiy of Uh, Sl Lke Ciy, UT, 842, USA Absrc We consider Volerr

More information

Yan Sun * 1 Introduction

Yan Sun * 1 Introduction Sun Boundry Vlue Problems 22, 22:86 hp://www.boundryvlueproblems.com/conen/22//86 R E S E A R C H Open Access Posiive soluions of Surm-Liouville boundry vlue problems for singulr nonliner second-order

More information

Fractional Calculus. Connor Wiegand. 6 th June 2017

Fractional Calculus. Connor Wiegand. 6 th June 2017 Frcionl Clculus Connor Wiegnd 6 h June 217 Absrc This pper ims o give he reder comforble inroducion o Frcionl Clculus. Frcionl Derivives nd Inegrls re defined in muliple wys nd hen conneced o ech oher

More information

1. Introduction. 1 b b

1. Introduction. 1 b b Journl of Mhemicl Inequliies Volume, Number 3 (007), 45 436 SOME IMPROVEMENTS OF GRÜSS TYPE INEQUALITY N. ELEZOVIĆ, LJ. MARANGUNIĆ AND J. PEČARIĆ (communiced b A. Čižmešij) Absrc. In his pper some inequliies

More information

FUZZY n-inner PRODUCT SPACE

FUZZY n-inner PRODUCT SPACE Bull. Korean Mah. Soc. 43 (2007), No. 3, pp. 447 459 FUZZY n-inner PRODUCT SPACE Srinivaan Vijayabalaji and Naean Thillaigovindan Reprined from he Bullein of he Korean Mahemaical Sociey Vol. 43, No. 3,

More information

SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road QUESTION BANK (DESCRIPTIVE)

SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road QUESTION BANK (DESCRIPTIVE) QUESTION BANK 6 SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddhrh Ngr, Nrynvnm Rod 5758 QUESTION BANK (DESCRIPTIVE) Subjec wih Code :Engineering Mhemic-I (6HS6) Coure & Brnch: B.Tech Com o ll Yer & Sem:

More information

MTH 146 Class 11 Notes

MTH 146 Class 11 Notes 8.- Are of Surfce of Revoluion MTH 6 Clss Noes Suppose we wish o revolve curve C round n is nd find he surfce re of he resuling solid. Suppose f( ) is nonnegive funcion wih coninuous firs derivive on he

More information

Fractional operators with exponential kernels and a Lyapunov type inequality

Fractional operators with exponential kernels and a Lyapunov type inequality Abdeljwd Advnces in Difference Equions (2017) 2017:313 DOI 10.1186/s13662-017-1285-0 RESEARCH Open Access Frcionl operors wih exponenil kernels nd Lypunov ype inequliy Thbe Abdeljwd* * Correspondence: bdeljwd@psu.edu.s

More information

2k 1. . And when n is odd number, ) The conclusion is when n is even number, an. ( 1) ( 2 1) ( k 0,1,2 L )

2k 1. . And when n is odd number, ) The conclusion is when n is even number, an. ( 1) ( 2 1) ( k 0,1,2 L ) Scholrs Journl of Engineering d Technology SJET) Sch. J. Eng. Tech., ; A):8-6 Scholrs Acdemic d Scienific Publisher An Inernionl Publisher for Acdemic d Scienific Resources) www.sspublisher.com ISSN -X

More information

CALDERON S REPRODUCING FORMULA FOR DUNKL CONVOLUTION

CALDERON S REPRODUCING FORMULA FOR DUNKL CONVOLUTION Avilble online hp://scik.org Eng. Mh. Le. 15, 15:4 ISSN: 49-9337 CALDERON S REPRODUCING FORMULA FOR DUNKL CONVOLUTION PANDEY, C. P. 1, RAKESH MOHAN AND BHAIRAW NATH TRIPATHI 3 1 Deprmen o Mhemics, Ajy

More information

Numerical Approximations to Fractional Problems of the Calculus of Variations and Optimal Control

Numerical Approximations to Fractional Problems of the Calculus of Variations and Optimal Control Numericl Approximions o Frcionl Problems of he Clculus of Vriions nd Opiml Conrol Shkoor Pooseh, Ricrdo Almeid, Delfim F. M. Torres To cie his version: Shkoor Pooseh, Ricrdo Almeid, Delfim F. M. Torres.

More information

A new model for solving fuzzy linear fractional programming problem with ranking function

A new model for solving fuzzy linear fractional programming problem with ranking function J. ppl. Res. Ind. Eng. Vol. 4 No. 07 89 96 Journl of pplied Reserch on Indusril Engineering www.journl-prie.com new model for solving fuzzy liner frcionl progrmming prolem wih rning funcion Spn Kumr Ds

More information

Solving Evacuation Problems Efficiently. Earliest Arrival Flows with Multiple Sources

Solving Evacuation Problems Efficiently. Earliest Arrival Flows with Multiple Sources Solving Evcuion Prolem Efficienly Erlie Arrivl Flow wih Muliple Source Ndine Bumnn Univeriä Dormund, FB Mhemik 441 Dormund, Germny ndine.umnn@mh.uni-dormund.de Mrin Skuell Univeriä Dormund, FB Mhemik 441

More information

FIXED POINTS AND STABILITY IN NEUTRAL DIFFERENTIAL EQUATIONS WITH VARIABLE DELAYS

FIXED POINTS AND STABILITY IN NEUTRAL DIFFERENTIAL EQUATIONS WITH VARIABLE DELAYS PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 136, Number 3, March 28, Page 99 918 S 2-9939(7)989-2 Aricle elecronically publihed on November 3, 27 FIXED POINTS AND STABILITY IN NEUTRAL DIFFERENTIAL

More information

5. Network flow. Network flow. Maximum flow problem. Ford-Fulkerson algorithm. Min-cost flow. Network flow 5-1

5. Network flow. Network flow. Maximum flow problem. Ford-Fulkerson algorithm. Min-cost flow. Network flow 5-1 Nework flow -. Nework flow Nework flow Mximum flow prolem Ford-Fulkeron lgorihm Min-co flow Nework flow Nework N i e of direced grph G = (V ; E) ource 2 V which h only ougoing edge ink (or deinion) 2 V

More information

Research Article New General Integral Inequalities for Lipschitzian Functions via Hadamard Fractional Integrals

Research Article New General Integral Inequalities for Lipschitzian Functions via Hadamard Fractional Integrals Hindwi Pulishing orporion Inernionl Journl of Anlysis, Aricle ID 35394, 8 pges hp://d.doi.org/0.55/04/35394 Reserch Aricle New Generl Inegrl Inequliies for Lipschizin Funcions vi Hdmrd Frcionl Inegrls

More information

AN EIGENVALUE PROBLEM FOR LINEAR HAMILTONIAN DYNAMIC SYSTEMS

AN EIGENVALUE PROBLEM FOR LINEAR HAMILTONIAN DYNAMIC SYSTEMS AN EIGENVALUE PROBLEM FOR LINEAR HAMILTONIAN DYNAMIC SYSTEMS Mrin Bohner Deprmen of Mhemics nd Sisics, Universiy of Missouri-Roll 115 Roll Building, Roll, MO 65409-0020, USA E-mil: ohner@umr.edu Romn Hilscher

More information

NEW FRACTIONAL DERIVATIVES WITH NON-LOCAL AND NON-SINGULAR KERNEL Theory and Application to Heat Transfer Model

NEW FRACTIONAL DERIVATIVES WITH NON-LOCAL AND NON-SINGULAR KERNEL Theory and Application to Heat Transfer Model Angn, A., e l.: New Frcionl Derivives wih Non-Locl nd THERMAL SCIENCE, Yer 216, Vol. 2, No. 2, pp. 763-769 763 NEW FRACTIONAL DERIVATIVES WITH NON-LOCAL AND NON-SINGULAR KERNEL Theory nd Applicion o He

More information

Introduction to SLE Lecture Notes

Introduction to SLE Lecture Notes Inroducion o SLE Lecure Noe May 13, 16 - The goal of hi ecion i o find a ufficien condiion of λ for he hull K o be generaed by a imple cure. I urn ou if λ 1 < 4 hen K i generaed by a imple curve. We will

More information

REAL ANALYSIS I HOMEWORK 3. Chapter 1

REAL ANALYSIS I HOMEWORK 3. Chapter 1 REAL ANALYSIS I HOMEWORK 3 CİHAN BAHRAN The quesions re from Sein nd Shkrchi s e. Chper 1 18. Prove he following sserion: Every mesurble funcion is he limi.e. of sequence of coninuous funcions. We firs

More information

Laplace Transform. Inverse Laplace Transform. e st f(t)dt. (2)

Laplace Transform. Inverse Laplace Transform. e st f(t)dt. (2) Laplace Tranform Maoud Malek The Laplace ranform i an inegral ranform named in honor of mahemaician and aronomer Pierre-Simon Laplace, who ued he ranform in hi work on probabiliy heory. I i a powerful

More information

NECESSARY AND SUFFICIENT CONDITIONS FOR LATENT SEPARABILITY

NECESSARY AND SUFFICIENT CONDITIONS FOR LATENT SEPARABILITY NECESSARY AND SUFFICIENT CONDITIONS FOR LATENT SEPARABILITY Ian Crawford THE INSTITUTE FOR FISCAL STUDIES DEPARTMENT OF ECONOMICS, UCL cemmap working paper CWP02/04 Neceary and Sufficien Condiion for Laen

More information

Fractional Ornstein-Uhlenbeck Bridge

Fractional Ornstein-Uhlenbeck Bridge WDS'1 Proceeding of Conribued Paper, Par I, 21 26, 21. ISBN 978-8-7378-139-2 MATFYZPRESS Fracional Ornein-Uhlenbeck Bridge J. Janák Charle Univeriy, Faculy of Mahemaic and Phyic, Prague, Czech Republic.

More information

Network Flows: Introduction & Maximum Flow

Network Flows: Introduction & Maximum Flow CSC 373 - lgorihm Deign, nalyi, and Complexiy Summer 2016 Lalla Mouaadid Nework Flow: Inroducion & Maximum Flow We now urn our aenion o anoher powerful algorihmic echnique: Local Search. In a local earch

More information

Introduction to Congestion Games

Introduction to Congestion Games Algorihmic Game Theory, Summer 2017 Inroducion o Congeion Game Lecure 1 (5 page) Inrucor: Thoma Keelheim In hi lecure, we ge o know congeion game, which will be our running example for many concep in game

More information

LAGRANGIAN AND HAMILTONIAN MECHANICS WITH FRACTIONAL DERIVATIVES

LAGRANGIAN AND HAMILTONIAN MECHANICS WITH FRACTIONAL DERIVATIVES LAGRANGIAN AND HAMILTONIAN MEHANIS WITH FRATIONAL DERIVATIVES EMIL POPESU 2,1 1 Asronomicl Insiue of Romnin Acdemy Sr uiul de Argin 5, 40557 Buchres, Romni 2 Technicl Universiy of ivil Engineering, Bd

More information

On Tempered and Substantial Fractional Calculus

On Tempered and Substantial Fractional Calculus On Tempered nd Subsnil Frcionl Clculus Jiniong Co,2, Chngpin Li nd YngQun Chen 2, Absrc In his pper, we discuss he differences beween he empered frcionl clculus nd subsnil frcionl operors in nomlous diffusion

More information

Honours Introductory Maths Course 2011 Integration, Differential and Difference Equations

Honours Introductory Maths Course 2011 Integration, Differential and Difference Equations Honours Inroducory Mhs Course 0 Inegrion, Differenil nd Difference Equions Reding: Ching Chper 4 Noe: These noes do no fully cover he meril in Ching, u re men o supplemen your reding in Ching. Thus fr

More information

, the. L and the L. x x. max. i n. It is easy to show that these two norms satisfy the following relation: x x n x = (17.3) max

, the. L and the L. x x. max. i n. It is easy to show that these two norms satisfy the following relation: x x n x = (17.3) max ecure 8 7. Sabiliy Analyi For an n dimenional vecor R n, he and he vecor norm are defined a: = T = i n i (7.) I i eay o how ha hee wo norm aify he following relaion: n (7.) If a vecor i ime-dependen, hen

More information

SLOW INCREASING FUNCTIONS AND THEIR APPLICATIONS TO SOME PROBLEMS IN NUMBER THEORY

SLOW INCREASING FUNCTIONS AND THEIR APPLICATIONS TO SOME PROBLEMS IN NUMBER THEORY VOL. 8, NO. 7, JULY 03 ISSN 89-6608 ARPN Jourl of Egieerig d Applied Sciece 006-03 Ai Reerch Publihig Nework (ARPN). All righ reerved. www.rpjourl.com SLOW INCREASING FUNCTIONS AND THEIR APPLICATIONS TO

More information

FLAT CYCLOTOMIC POLYNOMIALS OF ORDER FOUR AND HIGHER

FLAT CYCLOTOMIC POLYNOMIALS OF ORDER FOUR AND HIGHER #A30 INTEGERS 10 (010), 357-363 FLAT CYCLOTOMIC POLYNOMIALS OF ORDER FOUR AND HIGHER Nahan Kaplan Deparmen of Mahemaic, Harvard Univeriy, Cambridge, MA nkaplan@mah.harvard.edu Received: 7/15/09, Revied:

More information

Syntactic Complexity of Suffix-Free Languages. Marek Szykuła

Syntactic Complexity of Suffix-Free Languages. Marek Szykuła Inroducion Upper Bound on Synacic Complexiy of Suffix-Free Language Univeriy of Wrocław, Poland Join work wih Januz Brzozowki Univeriy of Waerloo, Canada DCFS, 25.06.2015 Abrac Inroducion Sae and ynacic

More information

The Concepts and Applications of Fractional Order Differential Calculus in Modelling of Viscoelastic Systems: A primer

The Concepts and Applications of Fractional Order Differential Calculus in Modelling of Viscoelastic Systems: A primer The Concep nd Applicion of Frcionl Order Differenil Clculu in Modelling of Vicoelic Syem: A primer Mohmmd Amirin Mlob, Youef Jmli,2* Biomhemic Lborory, Deprmen of Applied Mhemic, Trbi Modre Univeriy, Irn

More information

Example on p. 157

Example on p. 157 Example 2.5.3. Le where BV [, 1] = Example 2.5.3. on p. 157 { g : [, 1] C g() =, g() = g( + ) [, 1), var (g) = sup g( j+1 ) g( j ) he supremum is aken over all he pariions of [, 1] (1) : = < 1 < < n =

More information

f t f a f x dx By Lin McMullin f x dx= f b f a. 2

f t f a f x dx By Lin McMullin f x dx= f b f a. 2 Accumulion: Thoughs On () By Lin McMullin f f f d = + The gols of he AP* Clculus progrm include he semen, Sudens should undersnd he definie inegrl s he ne ccumulion of chnge. 1 The Topicl Ouline includes

More information

Variational Iteration Method for Solving System of Fractional Order Ordinary Differential Equations

Variational Iteration Method for Solving System of Fractional Order Ordinary Differential Equations IOSR Journal of Mahemaics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 1, Issue 6 Ver. II (Nov - Dec. 214), PP 48-54 Variaional Ieraion Mehod for Solving Sysem of Fracional Order Ordinary Differenial

More information

Asymptotic relationship between trajectories of nominal and uncertain nonlinear systems on time scales

Asymptotic relationship between trajectories of nominal and uncertain nonlinear systems on time scales Asympoic relionship beween rjecories of nominl nd uncerin nonliner sysems on ime scles Fim Zohr Tousser 1,2, Michel Defoor 1, Boudekhil Chfi 2 nd Mohmed Djemï 1 Absrc This pper sudies he relionship beween

More information

1 Motivation and Basic Definitions

1 Motivation and Basic Definitions CSCE : Deign and Analyi of Algorihm Noe on Max Flow Fall 20 (Baed on he preenaion in Chaper 26 of Inroducion o Algorihm, 3rd Ed. by Cormen, Leieron, Rive and Sein.) Moivaion and Baic Definiion Conider

More information

Bipartite Matching. Matching. Bipartite Matching. Maxflow Formulation

Bipartite Matching. Matching. Bipartite Matching. Maxflow Formulation Mching Inpu: undireced grph G = (V, E). Biprie Mching Inpu: undireced, biprie grph G = (, E).. Mching Ern Myr, Hrld äcke Biprie Mching Inpu: undireced, biprie grph G = (, E). Mflow Formulion Inpu: undireced,

More information

Analysis of Boundedness for Unknown Functions by a Delay Integral Inequality on Time Scales

Analysis of Boundedness for Unknown Functions by a Delay Integral Inequality on Time Scales Inernaional Conference on Image, Viion and Comuing (ICIVC ) IPCSIT vol. 5 () () IACSIT Pre, Singaore DOI:.7763/IPCSIT..V5.45 Anali of Boundedne for Unknown Funcion b a Dela Inegral Ineuali on Time Scale

More information

Flow Networks Alon Efrat Slides courtesy of Charles Leiserson with small changes by Carola Wenk. Flow networks. Flow networks CS 445

Flow Networks Alon Efrat Slides courtesy of Charles Leiserson with small changes by Carola Wenk. Flow networks. Flow networks CS 445 CS 445 Flow Nework lon Efr Slide corey of Chrle Leieron wih mll chnge by Crol Wenk Flow nework Definiion. flow nework i direced grph G = (V, E) wih wo diingihed erice: orce nd ink. Ech edge (, ) E h nonnegie

More information

Chapter Direct Method of Interpolation

Chapter Direct Method of Interpolation Chper 5. Direc Mehod of Inerpolion Afer reding his chper, you should be ble o:. pply he direc mehod of inerpolion,. sole problems using he direc mehod of inerpolion, nd. use he direc mehod inerpolns o

More information

Main Reference: Sections in CLRS.

Main Reference: Sections in CLRS. Maximum Flow Reied 09/09/200 Main Reference: Secion 26.-26. in CLRS. Inroducion Definiion Muli-Source Muli-Sink The Ford-Fulkeron Mehod Reidual Nework Augmening Pah The Max-Flow Min-Cu Theorem The Edmond-Karp

More information

HUI-HSIUNG KUO, ANUWAT SAE-TANG, AND BENEDYKT SZOZDA

HUI-HSIUNG KUO, ANUWAT SAE-TANG, AND BENEDYKT SZOZDA Communicions on Sochsic Anlysis Vol 6, No 4 2012 603-614 Serils Publicions wwwserilspublicionscom THE ITÔ FORMULA FOR A NEW STOCHASTIC INTEGRAL HUI-HSIUNG KUO, ANUWAT SAE-TANG, AND BENEDYKT SZOZDA Absrc

More information

Supplement for Stochastic Convex Optimization: Faster Local Growth Implies Faster Global Convergence

Supplement for Stochastic Convex Optimization: Faster Local Growth Implies Faster Global Convergence Supplemen for Sochasic Convex Opimizaion: Faser Local Growh Implies Faser Global Convergence Yi Xu Qihang Lin ianbao Yang Proof of heorem heorem Suppose Assumpion holds and F (w) obeys he LGC (6) Given

More information

T-Rough Fuzzy Subgroups of Groups

T-Rough Fuzzy Subgroups of Groups Journal of mahemaic and compuer cience 12 (2014), 186-195 T-Rough Fuzzy Subgroup of Group Ehagh Hoeinpour Deparmen of Mahemaic, Sari Branch, Ilamic Azad Univeriy, Sari, Iran. hoienpor_a51@yahoo.com Aricle

More information

FRACTIONAL EULER-LAGRANGE EQUATION OF CALDIROLA-KANAI OSCILLATOR

FRACTIONAL EULER-LAGRANGE EQUATION OF CALDIROLA-KANAI OSCILLATOR Romnin Repors in Physics, Vol. 64, Supplemen, P. 7 77, Dediced o Professor Ion-Ioviz Popescu s 8 h Anniversry FRACTIONAL EULER-LAGRANGE EQUATION OF CALDIROLA-KANAI OSCILLATOR D. BALEANU,,3, J. H. ASAD

More information

Journal of Mathematical Analysis and Applications. Two normality criteria and the converse of the Bloch principle

Journal of Mathematical Analysis and Applications. Two normality criteria and the converse of the Bloch principle J. Mh. Anl. Appl. 353 009) 43 48 Conens liss vilble ScienceDirec Journl of Mhemicl Anlysis nd Applicions www.elsevier.com/loce/jm Two normliy crieri nd he converse of he Bloch principle K.S. Chrk, J. Rieppo

More information

Explicit form of global solution to stochastic logistic differential equation and related topics

Explicit form of global solution to stochastic logistic differential equation and related topics SAISICS, OPIMIZAION AND INFOMAION COMPUING Sa., Opim. Inf. Compu., Vol. 5, March 17, pp 58 64. Publihed online in Inernaional Academic Pre (www.iapre.org) Explici form of global oluion o ochaic logiic

More information

TEACHING STUDENTS TO PROVE BY USING ONLINE HOMEWORK Buma Abramovitz 1, Miryam Berezina 1, Abraham Berman 2

TEACHING STUDENTS TO PROVE BY USING ONLINE HOMEWORK Buma Abramovitz 1, Miryam Berezina 1, Abraham Berman 2 TEACHING STUDENTS TO PROVE BY USING ONLINE HOMEWORK Bum Armoviz 1, Mirym Berezin 1, Arhm Bermn 1 Deprmen of Mhemics, ORT Brude College, Krmiel, Isrel Deprmen of Mhemics, Technion IIT, Hf, Isrel Asrc We

More information

RESPONSE UNDER A GENERAL PERIODIC FORCE. When the external force F(t) is periodic with periodτ = 2π

RESPONSE UNDER A GENERAL PERIODIC FORCE. When the external force F(t) is periodic with periodτ = 2π RESPONSE UNDER A GENERAL PERIODIC FORCE When he exernl force F() is periodic wih periodτ / ω,i cn be expnded in Fourier series F( ) o α ω α b ω () where τ F( ) ω d, τ,,,... () nd b τ F( ) ω d, τ,,... (3)

More information