On AP-Henstock Integrals of Interval-Valued. Functions and Fuzzy-Number-Valued Functions.

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Applied Mthemtis, 206, 7, 2285-2295 http://www.sirp.

1 Applied Mthemtis, 206, 7, ISSN Olie: ISSN Prit: O AP-Hestok Itegrls of Itervl-Vlued Futios d Fuzzy-Numer-Vlued Futios Muwy Elsheikh Hmid *, Alshikh Hmed Elmuiz, Mohmmed Eldirdiri Sheim 2 Shool of Mgemet, Ahfd Uiversity for Wome, Omdurm, Sud 2 Fulty of Egieerig, Uiversity of Khrtoum, Khrtoum, Sud How to ite this pper: Hmid, M.E., Elmuiz, A.H. d Sheim, M.E. (206) O AP-Hestok Itegrls of Itervl-Vlued Futios d Fuzzy-Numer-Vlued Futios. Applied Mthemtis, 7, Reeived: Otoer 8, 206 Aepted: Novemer 29, 206 Pulished: Deemer 2, 206 Copyright 206 y uthors d Sietifi Reserh Pulishig I. This work is liesed uder the Cretive Commos Attriutio Itertiol Liese (CC BY 4.0). Ope Aess Astrt I 2000, Wu d Gog [] itrodued the thought of the Hestok itegrls of itervl-vlued futios d fuzzy-umer-vlued futios d otied umer of their properties. The im of this pper is to itrodue the thought of the AP- Hestok itegrls of itervl-vlued futios d fuzzy-umer-vlued futios whih re extesios of [] d ivestigte umer of their properties. Keywords Fuzzy Numers, AP-Hestok Itegrls of Itervl-Vlued Futios, AP-Hestok Itegrls of Fuzzy-Numer-Vlued Futios. Itrodutio As it is well kow, the Hestok itegrl for rel futio ws first defied y Hestok [2] i 963. The Hestok itegrl is lot of powerful d esier th the Leesgue, Wieer d Rihrd Phillips Feym itegrls. Furthermore, it is lso equl to the Dejoy d the Perro itegrls [2] [3]. I 206, Hmid d Elmuiz [4] itrodued the oept of the Hestok-Stieltjes ( HS ) itegrls of itervl-vlued futios d fuzzy-umer-vlued futios d disussed umer of their properties. I this pper, we itrodue the oept of the AP-Hestok itegrls of itervl-vlued futios d fuzzy-umer-vlued futios d disuss some of their properties. The pper is orgized s follows. I Setio 2, we hve tedey to provide the prelimiry termiology used i this pper. Setio 3 is dedited to disussig the AP-Hestok itegrl of itervl-vlued futios. I Setio 4, we itrodue the AP- Hestok itegrl of fuzzy-umer-vlued futios. The lst setio provides olusios. DOI: /m Deemer 2, 206

2 2 Prelimiries Let E e mesurle set d let e rel umer. The desity of E t is defied y ( E( h, h) ) µ de = lim, h 0 2h (2.) provided the limit exists. The poit is lled poit of desity of E if de=. The set E. d E represets the set of ll poits x E suh tht x is poit of desity of A mesurle set Sx [, ] is lled pproximte eighorhood (r.p-d) of x [, ] if it otiig x s poit of desity. We hoose p-d Sx [, ] for eh x E [, ] d deote hoie o E y S = { Sx : x E}. A tgged itervl-poit pir ([ uv, ], ξ ) is sid to e S -fie if ξ [ uv, ] d uv, S ξ. A divisio P is fiite olletio of itervl-poit pirs ([ ui, vi] {[ u, v ]} = is i i i re o-overlppig suitervls of [, ] i i ; ) divisio of [, ] if [ u, v] = [, ] 2) S -fie divisio of [, ] i =, 2,,. { }, where P = {( ui, vi }. We sy tht [ ] if ξi [ ui, vi] d ([ i, i], i) u v ξ is S -fie for ll Defiitio 2.. [2] [3] A rel-vlued futio f: [, ] R is sid to e Hestok itegrle to A o [, ] if for every ε >0, there is futio δ >0 suh tht for y δ -fie divisio P = {[ u, v ]; ξ } = of [, ], we hve i i i i ( ξ )( ) f i vi ui A < ε, (2.2) where the sum is uderstood to e over P d we write ( ) ( )d [, ] f H. H f t t = A, d Defiitio 2.2. [5] A futio f: [, ] R is AP-Hestok itegrle if there exists rel umer A R suh tht for eh ε >0 there is hoie S suh tht for eh S -fie divisio P = ([ ui, vi] of [, ] f ( ξi)( vi ui) A < ε (2.3) { } A APH f. itegrl of f o [, ], d we write = ( ). A is lled AP-Hestok Theorem 2.. If f d g re AP-Hestok itegrle o [, ] d f lmost everywhere o [, ], the g ( APH ) f ( APH ) g. (2.4) 2286

3 Proof. The proof is similr to the Theorem 3.6 i [3]. 3. The AP-Hestok Itegrl of Itervl-Vlued Futios I this setio, we shll give the defiitio of the AP-Hestok itegrls of itervl-vlued futios d disuss some of their properties. Defiitio 3.. [] Let IR = { I = I, I : I is the losed ouded itervl o the rel lie R}. For AB, IR, we defie A B iff A B d A B, A B = C iff C = A B d C A B A B= : A, B, where d =, d { } ( A B) mi { A B, A B, A B, A B } = (3.) ( A B) { A B A B A B A B } Defie d ( AB, ) mx ( A B, A B ) d B. = mx,,,. (3.2) = s the diste etwee itervls A Defiitio 3.2. [] Let [ ] every 0 we hve F:, IR e itervl-vlued futio. I0 IR, for { i, i, i} i ε > there is δ > 0 suh tht for y δ -fie divisio = [ ] the F( t ) is sid to e Hestok itegrle over [, ] ( IH ) F d t = I0. For revity, we write F IH [ ] P u v ξ = d F( ξi)( vi ui), I0 ε, (3.3) d write,. Defiitio 3.3. A itervl-vlued futio F: [, ] IR I I, if for every ε >0 there exists hoie S o [, ] is AP-Hestok itegr- suh tht le to 0 R { } d F( ξi)( vi ui), I0 ε, (3.4) wheever P = ([ ui, vi] is S -fie divisio of [, ] ( APIH ) F = I0 d F APIH [ ] Theorem 3.. If F APIH [, ] 2,., we write, the the itegrl vlue is uique. Proof. Let itegrl vlue is ot uique d let ( ) ( ) B = APIH F. Let 0 tht B = APIH F d ε > e give. The there exists hoie S o [, ] suh ε d F( ξi)( vi ui), B, (3.5)

4 { } ε d F( ξi)( vi ui), B2 (3.6) 2 wheever P = ([ ui, vi] is S -fie divisio of [, ] Whee it follows from the Trigle Iequlity tht:. ε ε d, d, d,. 2 2 ( B B2) = F( ξi)( vi ui) B F( ξi)( vi ui) B2 < = ε (3.7) Sie for ε > 0, there exists hoie S o [, ] Theorem 3.2. A itervl-vlued futio F APIH [, ], [, ] d F F APH Proof. Let F APIH [, ] s ove so B = B. 2 if d oly if ( APIH ) F = ( APH ) F,( APH ) F. (3.8), from Defiitio 3.3 there is uique itervl umer I = I, I with the property tht for y >0, suh tht { } ε there exists hoie S o [ ] d F( ξi)( vi ui), I0 ε, (3.9) wheever P = ([ ui, vi] is S -fie divisio of [, ] i, we hve. Sie v u 0 for i i d F( ξi)( vi ui), I0 ε = mx F( ξi)( vi ui) I0, F( ξi)( vi ui) I 0 < ε. = mx F ( ξi)( vi ui) I0, F ( ξi)( vi ui) I0 ε. (3.0) Hee F ( ξi)( vi ui) I0 < ε, F ( ξi)( vi ui) I0 < ε wheever = {([ i, i] ; i) } is S -fie divisio of [, ]. Thus F, F APH [, ] P u v ξ d ( APIH ) F = ( APH ) F,( APH ) F. (3.) Coversely, let F, F APH [, ] ε there exists hoie S o [, ]. The there exists H, H R with the property tht give >0 suh 2 tht F ( i)( vi ui) H, F ( i)( vi ui) H2 ξ < ε ξ < ε { } wheever P = ([ ui, vi] is S -fie divisio of [, ]. We defie = [, ], the if P = {[ ui, vi], ξ i} i = is S -fie divisio of [, ] I H H 0 2, we hve 2288

5 d d F( ξi)( vi ui), I0 ε. (3.2) Hee F: [, ] IR is AP-Hestok itegrle o [, ]. Theorem 3.3. If F, G APIH [, ] d βγ, R. The βf γg APIH [, ] ( APIH ) ( βf γg) = β( APIH ) F γ ( APIH ) G. (3.3) Proof. If F, G APIH [, ], the F, F, G, G APH [, ] Hee βf γg βf γg βf γg βf γg APH [ ] () If β > 0 d γ > 0, the y Theorem 3.2.,,,,. ( APH ) ( βf γg) = ( APH ) ( βf γg ) (2) If β < 0 d γ < 0, the ( ) γ ( ) = β APH F APH G ( ) γ ( ) = β APIH F APIH G = β ( APIH ) F γ ( APIH ) G. ( APH ) ( βf γg) = ( APH ) ( βf γg ) ( ) γ ( ) = β APH F APH G ( ) γ ( ) = β APIH F APIH G = β (3) If β > 0 d γ < 0, (or β < 0 d γ < 0, ( APIH ) F γ ( APIH ) G. ), the ( APH ) ( βf γg) = ( APH ) ( βf γg ) Similrly, for four ses ove we hve ( ) γ ( ) = β APH F APH G = β ( ( APIH )) F γ ( APIH ) G = β( APIH ) F γ ( APIH ) G. 2289

6 ( APH ) ( βf γg) = β( APIH ) F γ ( APIH ) G. (3.4) Hee y Theorem 3.2 βf γg APIH [, ] d ( APIH ) ( βf γg) = β( APIH ) F γ ( APIH ) G. (3.5) Theorem 3.4. If F APIH [, ] d F APIH [, ], the F APIH [, ] d ( APIH ) F = ( APIH ) F ( APIH ) F. (3.6) Proof. If F APIH [, ] d F APIH [, ], the y Theorem 3.2, [, ] d F, F APH [, ]. Hee F, F APH [, ] F F APH ( ) = ( ) ( ) APH F APH F APH F = ( APIH ) F ( APIH ) F. d Similrly, ( APH ) F = ( APIH ) F ( APIH ) F. Hee y Theorem 3.2 [, ] F APIH d ( APIH ) F = ( APIH ) F ( APIH ) F. (3.7) Theorem 3.5. If F G erly everywhere o [, ] Proof. Let F G erly everywhere o [, ] [ ] F, F, G, G APH, d F G, F G d F, G APIH [, ], the ( APIH ) F ( APIH ) G. (3.8) d F, G APIH [, ] The erly everywhere o [, ] By Theorem 2. ( APH ) F ( APH ) G d ( APH ) F ( APH ) G. Hee y Theorem 3.2. Theorem 3.6. Let F, G APIH [, ] d d (, ) [, ]. The Proof. By defiitio of diste, ( APIH ) F ( APIH ) G, (3.9) d, d,. FG is Leesgue itegrle o ( APIH ) F ( APIH ) G ( L) ( F G) (3.20) 2290

7 d ( APIH ) F, ( APIH ) G = mx ( APIH ) F ( APIH ) G, ( APIH ) F ( APIH ) G = mx, ( APH ) ( F G ) ( APH ) ( F G ) mx ( L) F G,( L) F G ( L) mx ( F G, F G ) ( L) ( FG) d,. = (3.2) 4. The AP-Hestok Itegrl of Fuzzy-Numer-Vlued Futios This setio itrodues the oept of the AP-Hestok itegrl of fuzzy-umervlued futios d ivestigtes some of their properties. Defiitio 4.. [6] [7] [8] Let A F( R) e fuzzy suset o R. If for y [ 0, ], A = A, A d A φ, where A = { t: At ( ) }, the A is lled fuzzy umer. If A is ovex, orml, upper semi-otiuous d hs the ompt support, we sy tht A is ompt fuzzy umer. Let R deote the set of ll fuzzy umers. Defiitio 4.2. [6] Let AB, R, we defie A B iff A B for ll ( 0, ], A B = C iff A B C 0,, A B = D iff A B = D for y ( 0, ]. C For AB, R, = for y ( ] ( ) ( ) D AB, = sup d A, B is lled the diste etwee A d [ 0,] B. Lemm 4.. [9] If mppig H :[ 0, ] I R, H( ) = [ m, ], stisfies m, m, A : = H( ) R (4.) ( 0,] d where =. = ( ) A = H, (4.2) ( ) Defiitio 4.3. [] Let F :[, ] R. If the itervl-vlued futio F = F, F is Hestok itegrle o [, ] for y ( ] sy tht F is Hestok itegrle o [, ] 0,, the we d the itegrl vlue is defied y 229

8 ( ) ( ) ( 0,] ( ) ( ) FH F t d t: = IH F t dt = ( H) F d, t ( H) F d t. ( 0,] For revity, we write F FH [, ]. Defiitio 4.4. Let F :[, ] R. If the itervl-vlued futio F = F, F ) is AP-Hestok itegrle o [, ] for y ( ] the F is lled AP-Hestok itegrle o [, ] d the itegrl vlue is defied y ( ) ( ) ( 0,] ( ) ( ) APFH F t d t : = APIH F t dt 0,, = ( APH ) F dt, ( APH ) F d t. ( 0,] We write F APFH [, ]. Theorem 4.. F APFH [, ], the ( ) ( )d where =. ( ) Proof. Let ( ] APFH F t t R d ( APFH ) F dt = ( APIH ) F d, t (4.3) = H : 0, I R, e defied y Sie ( ) d ( ) 0 <, we hve F F, F F, o [ ] H ( ) = ( APH ) F d, t ( APH ) F d t. F t F t re iresig d deresig o respetively, therefore, whe 2 From Theorem 3.5 we hve 2 2 ( APH ) F d, t ( APH ) F dt ( APH ) F d, t ( APH ) F d t. (4.4) 2 2 From Theorem 3.2 d Lemm 4. we hve APFH F t d t : = APH F d, t APH F dt R (4.5) (0,] ( ) ( ) ( ) ( ) APFH F t dt APIH F t d, t where = d for ll ( 0, ], ( ) ( ) = ( ) ( ) =. ( ) d Theorem 4.2. If F, G APFH [, ] d, R.,. βγ The βf γg APFH [, ] ( ) ( β γ ) = β( ) ( ) γ ( ) ( ) APFH F G dt APFH F t dt APFH G t d. t (4.6) Proof. If F, G APFH [, ], the the itervl-vlued futio 2292

9 = ( ) ( ) d G G, G F () t F t, F t = re AP-Hestok itegrle o APFH F t dt = ( APIH ) F dt ( 0,] d = ( ) ( 0,] ( ) d. From Theorem 3.3 we hve F γg APIH [, ] d [, ] for y ( 0,] d ( ) ( ) ( ) ( ) β APFH G t t APIH G t t ( APIH ) ( β F γ G) dt = β ( APIH ) Fdt γ ( APIH ) Gdt for y ( 0,] Hee βf γg APFH [, ] d ( ) ( ) ( 0,] ( ) ( ) APFH βf γg dt = APIH βf γg dt ( ) d ( ) ( 0, ] (0,] ( ) ( ) γ ( ) ( ). = β( APIH ) Fdt γ ( APIH ) Gdt ( 0,] = β APIH F t γ APIH G dt = β APFH F t dt APFH G t d. t F APFH, d Theorem 4.3. If F APFH [, ] d F APFH [, ], the [ ] ( ) ( ) = ( ) ( ) ( ) ( ) APFH F t dt APFH F t dt APFH F t d. t (4.7) Proof. If F APFH [, ] d F APFH [, ] F = F, F is AP-Hestok itegrle o [, ] d [, ] ( 0,] d ( APFH ) F dt = ( APIH ) F dt d ( ) ( ), the the itervl-vlued futio for y ( 0,] d = ( ) ( 0,] ( ) d. From Theorem 3.4 we hve APFH F t t APIH F t t APIH [, ] d ( APIH ) F d t ( APIH ) F d t ( APIH ) F d = t for y F d ( 0,]. Hee F APFH [, ] ( ) ( ) ( 0,] ( ) ( ) APFH F t dt = APIH F t dt = ( APIH ) F dt ( APIH ) F dt ( 0,] ( ) ( ) d ( ) ( ) = APIH F t t APIH F t dt ( 0, ] (0,] ( ) ( ) ( ) ( ) = APFH F t dt APFH F t d. t 2293

10 Theorem 4.4. If F G, ], APFH [, ], the F G ( ) ( ) ( ) ( ) d APFH F t dt APFH G t d. t (4.8) Proof. If F G erly everywhere o [, ] d F, G APFH [, ] ( ) ( ) erly everywhere o [, ] for y ( 0,] d F d re AP-Hestok itegrle o [, ] for y ( 0,] d F t G t G ( ) ( ) d = ( ) ( ) d d ( 0,] APFH F t t APIH F t t ( ) ( ) d = ( ) ( ) d. From Theorem 3.5 we hve ( 0,] APFH G t t APIH G t t ( APIH ) F dt ( APIH ) G dt for y ( 0,] 5. Colusio ( ) ( ) ( 0,] ( 0,]. Hee ( ) ( ) APFH F t dt = APIH F t dt = ( ) ( ) APIH G t dt ( ) ( ) APFH G t d. t, the I this pper, we hve tedey to itrodue the oept of the AP-Hestok itegrls of itervl-vlued futios d fuzzy umer-vlued futios d ivestigte some properties of those itegrls. Referees [] Wu, C.X. d Gog, Z. (2000) O Hestok Itegrls of Itervl-Vlued Futios d Fuzzy-Vlued Futios. Fuzzy Sets d Systems, 5, [2] Hestok, R. (963) Theory of Itegrtio. Butterworth, Lodo. [3] Peg-Yee, L. (989) Lzhou Letures o Hestok Itegrtio. World Sietifi, Sigpore. [4] Hmid, M.E. d Elmuiz, E.H. (206) O Hestok-Stieljes Itegrls of Itervl-Vlued Futios d Fuzzy-Vlued Futios. Jourl of Applied Mthemtis d Physis, 4, [5] Prk, J.M., Prk, C.G., Kim, J.B., Lee, D.H. d Lee, W.Y. (2004) The Itegrls of s-perro, sp-perro d p-mshe. Czehoslovk Mthemtil Jourl, 54, [6] Nd, S. (989) O Itegrtio of Fuzzy Mppigs. Fuzzy Sets d Systems, 32, [7] Wu, C.X. d M, M. (99) Emeddig Prolem of Fuzzy Numer Spes: Prt I. Fuzzy 2294

11 Sets d Systems, 44, [8] Wu, C.X. d M, M. (992) Emeddig Prolem of Fuzzy Numer Spes: Prt II. Fuzzy Sets d Systems, 45, [9] Cheg, Z.L. d Demou, W. (983) Extesio of the Itegrl of Itervl-Vlued Futio d the Itegrl of Fuzzy-Vlued Futio. Fuzzy Mth, 3, Sumit or reommed ext musript to SCIRP d we will provide est servie for you: Aeptig pre-sumissio iquiries through Emil, Feook, LikedI, Twitter, et. A wide seletio of jourls (ilusive of 9 sujets, more th 200 jourls) Providig 24-hour high-qulity servie User-friedly olie sumissio system Fir d swift peer-review system Effiiet typesettig d proofredig proedure Disply of the result of dowlods d visits, s well s the umer of ited rtiles Mximum dissemitio of your reserh work Sumit your musript t: Or ott m@sirp.org 2295

Convergence rates of approximate sums of Riemann integrals

Convergence rates of approximate sums of Riemann integrals Covergece rtes of pproximte sums of Riem itegrls Hiroyuki Tski Grdute School of Pure d Applied Sciece, Uiversity of Tsuku Tsuku Irki 5-857 Jp tski@mth.tsuku.c.jp Keywords : covergece rte; Riem sum; Riem