, R we have. x x. ) 1 x. R and is a positive bounded. det. International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:10 No:06 11

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raioal Joral of asic & ppli Scics JS-JENS Vol: No:6 So Dirichl ors a Pso Diffrial Opraors wih Coiioall Epoial Cov cio aa. M. Kail Dpar of Mahaics; acl of Scic; Ki laziz Uivrsi Jah Sai raia Eail: fkail@ka..sa srac-- his papr w prs a cocp of coiioall poial cov fcios a w prov ha r sial asspios i is possil o fi a Dirichl fors cosrc a class of pso iffrial opraors wih sols fi i rs of coiioall poial cov fcios. Tr-- Pso iffrial opraors Dirichl forscoiioall poial cov fcio.. - CONDTONLLY EXPONENTL CONVEX UNCTONS. Dfiiio.. ral val fcio : is sai o coiioall poial cov if for a a c c W hav k k c ck k. Thor.. [8] Thor 3.7. coios fcio : is coiioall poial cov o if a ol if i ca rprs i h for: ; C p ; \{}. whr C is a cosa : is a coios aiv qaraic for o asr o iql ri h fcio. a is a posiiv o \.Morovr h ripl ; C is La.. L coios coiioall poial cov fcios. Th i ; ii C for so cosa C iii C C is a coios coiioall poial cov fcio. or h proof s [3]. l iv is a coios coiioall cov fcio a h iqali l l l ;. hols for all or h proof of iv w hav h followi: La.. L : a coios coiioall poial cov fcio a is sqar roo h for w hav.3 fiiio of coiioall poial cov fcio w hav - hrfor i a ii. Or Ths w fi Which prov h la. L s iroc so soolv spacs rla o a coios coiioall poial cov fcio :. w fi 76-88 JS-JENS Dcr JENS

raioal Joral of asic & ppli Scics JS-JENS Vol: No:6 L : l.4 Whr o l w hav h or l s or w oai h sal sool v spacs. or l or ails s [45] wih his or h spac is a ilr spac a C l l a for all l Whr l D l D l Usi La. w ca cosrc h chai L l.5 is a s sspac of Proposiio.. Sppos a ar wo coios coiioall poial cov fcio a ha.6 ols for all. Th i follows ha C l.7 l or all. pariclar l is coiosl io h l l spac. Sic h fcio is a coios coiioall cov fcio h for a ] i follows ha l C Dfiiio...8 ] Dirichl for o aiv iliar for D L wih oai D iplis a is a clos sric o D Th pair D is call Dirichl spac [9]. CLSS O CLOSED LNE OMS L N ; N for sch ha : a coios coiioall poial cov fcio cosh \. whr is a posiiv fii sric asr o \. W o h ia of r h appi T : ; is o. L L ip of. l w o ; k k k a w iif wih a sspac ; of. L : coios coiioall poial cov fcio i his cas / L : / ;.. is a Dirichl spac. or / / / h fcio. or C w fi L Whr. ˆ Whr ˆ w osrv 76-88 JS-JENS Dcr JENS

L Sic raioal Joral of asic & ppli Scics JS-JENS Vol: No:6 3 is ip of sric iliar for fi o w ca associa wih C / / L Whr / has h sol a /. Proposiio.. or all C w hav C / /. Thrfor has a coios sio o / Sic C. / ; is s i i is sffici o prov. for all s fcios. or C i follows ha / ; / C / / Sic / C / for all C L i follows ha ca rar as a coios iliar for o /. Proposiio.. L as i Proposiio. a sppos ha hr iss a cosa sch ha for all.3 / Th w hav for all.4 /.5 / or w hav a / / / /.6 Which ivs.4 No ha.5 is a Cari-iqali [9] a w.6 a.3 / ro proposiio. a. i follows ha wih oai / D is a clos sric iliar for L o. Thor.. / L as i Proposiio. or w hav J Whr.7 J.8 Usi h oaios iroc aov. w wri or C fi.9 or or rall for L w 76-88 JS-JENS Dcr JENS

raioal Joral of asic & ppli Scics JS-JENS Vol: No:6 4 76-88 JS-JENS Dcr JENS / / / / Whr or L f l z z f f follows ha cosh ; ; Which fiall ivs.. hrfor h hor is prov. Now i is asil s ha Thor.. Sppos ha saisfis h asspios of Proposiio. a Thor.. Th ; / is a Dirichl spac. f h w oai h rsl ro [67]. EEENCES []. S. Ok El-a ``Coiioall poial cov fcio o Locall Copac rops Gar Uiv. Sci 5.3 9933-6 [].S. Ok El-a a M. S. El. Shazli`` Characrizaio of covolio Si-rops Proc. Pakisa ca Sci. 43 987 49-59.. [3] C. r a G. ors Poial hor o locall copac alia rop. rli ilr Nw Yark Sprir Vrla 975 [4] E. Popsc o o llr si rops Poial alsis 47-9. [5] E. Popsc O h Sols associa wih a sirop of opraor. v. oai Mah. Prs ppl. 46 77-84.. [6].. li `` Dirichl fors ra Coiioall poial Cov fcios ll. ac. Sc. ssi Uiv 33-c 4-8.

raioal Joral of asic & ppli Scics JS-JENS Vol: No:6 5 [7].. li Pso Diffrial opraor wih coiioall poial Cov fcio a llr sirops. M. S. E. Vol 4 No 3 3 3-43. [8] M. S. Elshazl Ph. D. Thsis l-zhar Uivrsi Cairo. Ep 99. [9] N. Jaco G â ri iqali for crai aisoropic Pso iffrial opraors wih o-sooh sols Osaka J. Mah. T.6 989 p.857-879. [] N. Jaco Pso-iffrial opraors a Markov procsss vol.: orir aalsis a si rops. prial Coll Prss Loo. [] N. Jaco Pso-iffrial opraors a Markov procsss vol.3: Markov procsss a applicaios. prial Coll Prss Loo 5. 76-88 JS-JENS Dcr JENS

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