Some integral inequalities on time scales

Transcription

1 Al Mth Mech -Engl Ed (1:23 29 DOI /s c Editoril Committee of Al Mth Mech nd Sringer-Verlg 2008 Alied Mthemtics nd Mechnics (English Edition Some integrl ineulities on time scles Adnn Tun Servet Kutukcu (Dertment of Mthemtics Fcult of Science nd Arts Gzi Universit Beevler Ankr Turke Astrct In this rticle we stud the reverse Hölder te ineulit nd Hölder ineulit in two dimensionl cse on time scles We lso otin mn integrl ineulities using Hölder ineulities on time scles which give Hrd s ineulities s scil cses Ke words integrl ineulities Hölder s ineulities Hrd s ineulities time scles reverse ineulit Chinese Lirr Clssifiction O1757 O Mthemtics Suject Clssifiction 26D15 39A10 Introduction The theor of dnmic eutions on time scles (or mesure chins ws introduced Hilger [1] with the motivtion of roviding unified roch to continuous nd discrete nlsis The generlized derivtive or Hilger derivtive f Δ (t of function f : T R wheret is so-clled time scle (n ritrr closed nonemt suset of R ecomes the usul derivtive when T = R thtisf Δ (t =f (t On the other hnd if T = Z thenf Δ (t reduces to the usul forwrd difference tht is f Δ (t =Δf(t This theor not onl rought eutions leding to new lictions Also this theor llows one to get some insight into nd etter understnding of the sutle differences etween discrete nd continuous sstems [2 3] In this er we otin some generl results for estimting the integrl using Hölder ineulities nd the reverse Hölder te ineulit on time scles Now first we mention without roof severl fundmentl definitions nd results from the clculus on time scles in n excellent introductor text Bohner nd Peterson [3] 1 Generl definitions We ssume tht T =[ ] is n ritrr intervl on time scle For nottionl uroses the intersection of rel intervl [ ] withtimesclet is denoted [ ] T: [ ] T Definition 1 AtimescleT is nonemt closed suset of R We ssume throughout tht T hs the toolog tht is inherited from the stndrd toolog on R It lso ssumed throughout tht in T the intervl [ ] T mens the set {t T : s<t} for the oints <in T Received Se / Revised Nov Corresonding uthor Servet Kutukcu Doctor E-mil: skutukcu@hoocom

2 24 Adnn Tun nd Servet Kutukcu Since time scle m not e connected we need the following concet of jum oertors Definition 2 The mings σ ρ : T T defined σ(t =inf{s T : s>t} nd ρ(t = su {s T : s<t} re clled the jum oertors The jum oertors σ nd ρ llow the clssifiction of oints in T in the following w: Definition 3 A nonmximl element t T is sid to e right-dense if σ(t =t rightscttered if σ(t >tleft-dense if ρ(t =t left-scttered if ρ(t <t In the cse T = R wehveσ(t =t nd if T = hz h>0 then σ(t =t + h Definition 4 The ming μ : T R + defined μ(t =σ(t t is clled the grininess function The set T k is defined s follows: if T hs left-scttered mximum m then T k = T -{m} ; otherwise T k = T If T = R thenμ(t =0 nd when T = Z we hve μ(t =1 Definition 5 Let f : T R fis clled differentile t t T k with (delt derivtive f Δ (t R if given ε>0 there exists neighorhood U of t such tht for ll s U f σ (t f(s f Δ (t[σ(t s] ε σ(t s f σ = f σ If T = Z thenf Δ df (t (t = dt nd if T = R thenf Δ (t =f(t +1 f(t Some sic roerties of delt derivtives re the following [3] Theorem 1 Assume tht f : T R nd let t T k (i If f is differentile t t then f is continuous t t (ii If f is differentile t t nd t is right-scttered then f is differentile t t with f Δ (t = f σ (t f(t σ(t t (iii If f is differentile t t nd t is right-dense then f Δ f(t f(s (t = lim t s t s (iv If f is differentile t t then f σ (t =f(t+μ(tf Δ (t Exmle 1 (i If f : T R is defined f(t =α for ll t T whereα R is constnt then f Δ (t 0 (ii If f : T R is defined f(t =t for ll t T thenf Δ (t 1 Definition 6 The function f : T R is sid to e rd-continuous (denote f C rd (T R if t ll t T (i f is continuous t ever right-dense oint t T (ii lim s t f(s exists nd is finite t ever left-dense oint t T Definition 7 Let f C rd (T R Then g : T R is clled the ntiderivtive of f on T if it is differentile on T nd stisfies g Δ (t =f(t for n t T k In this cse we defined Theorem 2 t f(sδs = g(t g( t T If f is Δ-integrle on [ ] thensois f nd f(tδt f(t Δt

3 Some integrl ineulities on time scles 25 Theorem 3 (Hölder ineulit Let T For rd-continuous functions fg :[ ] R we hve ( ( f(tg(t Δt f(t Δt g(t Δt where >1 nd = /( 1 Our min results re given in the following theorems 2 Min result Lemm 1 For two ositive functions f nd g stisfing 0 <m f g set [ ] nd for >1 nd >1 with =1 we hve M< on the f(t Δt g(t Δt ( m M 1 f(tg(tδt Proof Since f g M g M 1 f therefore fg M 1 f 1+ 1 = M f nd so ( f(t Δt M 1 ( f(tg(tδt (1 On the other hnd since m f g f m 1 g hence f(tg(tδt m 1 g(t 1+ Δt = m 1 g(t Δt nd so ( f(tg(tδt ( m 1 g(t Δt Comining with (1 we hve the desired ineulit f(t Δt g(t Δt ( m M 1 f(tg(tδt Corollr 1 If we tke T = R it is cler tht we cn hve the sme Theorem 21 [5] Theorem 4 Let T For rd-continuous functions fg :[ ] [ ] R we hve ( f(x g(x Δx f(x Δx ( g(x Δx where >1 nd = /( 1

4 26 Adnn Tun nd Servet Kutukcu Proof For nonnegtive rel numers α nd β the well known sic ineulit α 1 β 1 α + β (2 holds Now suose without loss of generlit tht ( ( f(x Δx Al (2 to g(x Δx 0 α(x = f(x f(τ nd β(x = 1τ 2 Δτ 1 Δτ 2 g(x g(τ 1τ 2 Δτ 1 Δτ 2 nd integrte the otined ineulit etween nd (this is ossile since ll occurring functions re rd-continuous to get = = 1 α(x 1 β(x Δx { { + 1 f(x g(x f(τ 1τ 2 Δτ 1 Δτ 2 g(τ Δx 1τ 2 Δτ 1 Δτ 2 { α(x + } f(x f(τ Δx 1τ 2 Δτ 1 Δτ 2 } g(x g(τ Δx 1τ 2 Δτ 1 Δτ 2 } β(x Δx = =1 This directl gives Hölder s ineulit Theorem 5 If =1with >1 nd K(x f(x g( ϕ(x ψ( e nonnegtive functions then the following ineulities re euivlent ϕ(x F (xf(x Δx ψ( G(g( (3 nd ( G( ψ( K(x f(xδx ϕ(x F (xf(x Δx (4 where F (x = K(x ψ( nd G(x = K(x ϕ(x Δx

5 Some integrl ineulities on time scles 27 Proof We strt with the following identit = Now if we l Hölder s ineulit we otin K(x f(x ϕ(x ψ( g(ψ( ϕ(x Δx ϕ(x F (xf(x Δx ψ( G(g( Let us show tht the ineulities (3 nd (4 re euivlent Suose tht the ineulit (3 is vlid If we ut ( 1 g( =G( ψ( K(x f(xδx tking into ccount = 1 nd using (3 we hve = ( G( ψ( K(x f(xδx = ϕ(x F (xf(x Δx ψ( G(g( ( ϕ(x F (xf(x Δx G( ψ( K(x f(xδx fromwherewehve(4 Now let s suose tht the ineulit (4 is vlid B ling Hölder s ineulit nd (4 we otin = ( ψ( 1 G( 1 K(x f(xδx ψ(g( g( ( ( G( ψ( K(x f(xδx ( ϕ(x F (xf(x Δx ψ( G(g( ψ( G(g( so we hve (3 While ineulit (3 is vlid the ineulit (4 holds too Theorem 6 If =1with >1 nd K(x f(x g( ϕ(x ψ( e nonnegtive functions nd F (x = K(x ψ( F 1 (x nd G(x = K(x ϕ(x Δx G 1 ( Then the

6 28 Adnn Tun nd Servet Kutukcu following ineulities re euivlent nd ϕ(x F 1 (xf(x Δx ψ( G 1 (g( ( G 1 ( ψ( K(x f(xδx ϕ(x F 1 (xf(x Δx (6 If we tke T = R then ineulities (4 nd (6 re Hrd te ineulities [5] If we ut { h( x K(x = 0 x > in Theorem 5 we otin following result Theorem 7 Let =1with >1 nd let h( f(x g( ϕ(x ψ( e nonnegtive functions Then the following ineulities re euivlent h(f(xg(δx ( ( ϕ(x f(x H( Δx nd x ( ( ( H( ϕ(x Δx ( ( ϕ(x f(x H( x (5 ( ψ( g( h( ϕ(x Δx (7 Δx f(xδx (8 where H( =h(ψ( If we ut { 0 x K(x = h( x > in Theorem 5 we otin following result Theorem 8 Let =1with >1 nd let h( f(x g( ϕ(x ψ( e nonnegtive functions Then the following ineulities hold euivlent h(f(xg(δx ( ( x ϕ(x f(x H( Δx ( ψ( g( h( ( ϕ(x Δx (9

7 Some integrl ineulities on time scles 29 nd ( ( H( ϕ(x Δx f(xδx ( ( x ϕ(x f(x H( Δx (10 If we tke T = R it is cler tht we cn hve the sme Theorem 1 Theorem 2 Theorem 3 nd Theorem 4 in [5] References [1] Hilger S Anlsis on mesure chins unified roch to continuous nd discrete clculus[j] Result Mth :18 56 [2] Agrwl R P Bohner M O Regn D Peterson A Dnmic eutions on time scle: A surve [J] J Comut Al Mth (12:1 26 [3] Bohner M Peterson A Dnmic eutions on time scle n introduction with lictions[m] Boston: Birkhuser 2001 [4] Sitoh S Tun V K Ymmoto M Reverse convolution ineulities nd lictions to inverse het source rolems[j] JIneinPurendAlMth (5 Article 80 [5] Krnıć M Pećrıć J Generl Hilert s n Hrd s ineulit[j] Mth Ine nd Al :29 51