[PDF] Hardy-type inequalities quantum calculus





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Hardy-type inequalities quantum calculus

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Hardy-type inequalities quantum calculusDepartment of Engineering Sciences and Mathematics

ISSN 1402-1544

ISBN 978-91-7790-240-9

(print)

ISBN 978-91-7790-241-6 (pdf)

Luleå University of

Technology DOCTORAL THESIS

Serikbol Shaimardan

Hardy-type inequalities quantum calculus

Serikbol Shaimardan

Hardy-type inequalities quantum calculusSerikbol Shaimardan

Luleå University of Technology

Department of Engeneering Sciences and Mathematics Printed by Luleå University of Technology, Graphic Production 2018

ISSN 1402-1544

ISBN 978-91-7790-240-9 (print)

ISBN 978-91-7790-241-6 (pdf)

Luleå

www.ltu.se2010 Mathematics Subject Classication. 35B27, 76D08

Key words and phrases

. Inequalities, Hardy-type inequalities,

Riemann-Liouville operator, Integral operator,

q -analysis, q -analog, weights, h -calculus, h-integral, discrete fractional calculus

Abstract

q h q n2 N q q prq q I Ifx x Z x xsfssds; q iii ivABSTRACT 0 @1 Z 0 u rx0@x Z 0 t ∞x xΓtftdt1Ar dx1A1 r C0 @1 Z 0 f pxdx1A1 p ;8f; ;1 q q < pr <1 p q 1Z 00@ x∞xZ 0 tftdt1Ap dxppΓΓ p1Z 0 f pxdx; fp0jfxΓfyjp j xΓyj∞+pdxdy1Ap C0@1 Z 0 j f

0xjpx(∞)pdx1Ap

<< < p< 1 h C?1 p-1 p p )1p h

Preface

Some newHar dy-type

inequalitiesfor Riemann-Liouvil lefractionalq-integralop- erator

Some newHar dy-typeinequalitiesinq-

analysis

A Hardy-typeinequalityfor thefractional

integraloper atorinq-analysis

Hardy-type

inequalitiesin fractional h-discretec alculus

Fractionalorder Hardy-typeine qualityin

fractionalh-discretec alculus

Remark

v

Acknowledgment

vii

Introduction

John vonNeumann

Calculus orinβnitesimal calculushas afascinating history. In the 17thcen tury,I.NewtonandG. Leibnizindep enden tlyin ven ted calculus basedon theco nceptof limit(butelements ofit hav eal- ready appearedinancien tGreece). Theusualmeaningof limit im- plies thatspace andtime arecon tinuous, andw ehavemaintained that allnatural processes happencontin uouslyonsmooth curves and surfaces.Ho wever,theatomictheoryinp hysics andc hemistry in the19th century pavedthat thenatureprocessofdividing it intoev ersmallerpartswill terminatein anindivisible oran atom, a partwhic h,lackingprop erpartsitself,ca nnotbefurther divided. In aw ord,continua aredivisiblewithoutlimitorinβnitely divisi- ble. Thisb ecomestheoriginof developing anothert ype ofcalculus based onβnite diαerence principle,orcalculuswithou tlimit which is quantumcalculus(the calculusof βnited iαerencesw asdev eloped at thesame time). In mathematics,the quantum calculusisequivalen tto usual inβnitesimal calculuswithout theconcept oflimits orthe inv esti- gation ofcal culuswithoutlimits(quan tumis fromthe Latinword "quantus"and literallyit meansho wm uch, inSwedish"Kv ant"). It hast womajorbranches, q-calculus andthe h-calculus. Andb oth of themw ereworked outbyP.Cheung andV. Kac[36]inthe early twentiethcentury.

2INTRODUCTION

h

One ofthe popular quantumcalculus ish-calculus.

This calculusis thestudy ofthe deβnitions,prop erties,and ap- plications ofrelated concepts,th efractional calculusanddiscrete fractional calculus.How ever,theinvestigationforfractional calcu- lus wasstudied alreadybyG. Leibnizafter thatG.L'Hospitalin

1695 askedhim:" whatw ouldb ethe one-halfderivative ofx?" (see

[74]). In1772, J.L.Lagrange intro ducedthe diαerentialoperators of integerorderand wrote(see [72]): d m dxmd ndxny/dn+mdxn+my: In1819, S.F.Lacro ixdeveloped amoremathematicalgeneraliz- ing froma caseof integer order[73]. Namelyhepresen tedthe nth derivativeinthefollo wingform: D nxm/dn dxn(xm)/m!(n+m)!xmn; withy/xmandm;n2Zsuchthatmn. Replacingthe factorial symbolby theGammafunction,he develop edt heform ulafor the fractional derivativeofap ow erfunc tion: D x/(+ 1) (Γ+1)x whereandare fractionaln umbers.Thenhegavethe example that thederiv ativeoforder

2fory/xisasfollows:

d 1 2x dx12/2p xp: Note thatthisinterestingresult ofS.F. Lacroixis thesame as the Riemann-Liouvilledeβnition ofa fractionald erivativ e. However,thisabo ve authorsdidnotdeβnederivatives ofarbi- trary orderand theyga ve noapplicationsorexamples.Th eβrst application waspresented byN.H.Ab el[2]in1823. Heapplied the fractional calculusinthe solutionof anin tegralequation. Abel's so- lution wassoelegan tthat itattractedthe attention ofJ. Liouville. In 1832he took theβrststepto solve diαerential equationsin volv- ing fractionalop erators(see[75]).Moreo ver, hega vehisdeβnition

INTRODUCTION3

D

αxa-αa

(ax-a-α ;a> ; D

αfx

(x Z c x-t

1ftdt x;a> :

c x N N

αfx ∞X

k/0-N k fxk;

4INTRODUCTION

t )t (t-; t n )nΓ1Y |/0t-j t (t-n t -j j h h q-calculus: q q qqΓ111Q k/01 1 qk+1jqj< p n pn n q q q 211X
n /0aqnbqn qqncqnzn;jzj<; q a qn8 :;n ; n 1Q m /0-aqm;n?N;

INTRODUCTION5