Fredholm Type Integral Equations with Aleph-Function. and General Polynomials

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1 Iteto Mthetc Fou Vo. 8 3 o HIKI Ltd.-h.co Fedho Te Iteg uto th eh-fucto d Gee Poo u J K.J. o Ittute o Mgeet tude & eech Mu Id u5@g.co Kt e K.J. o Ittute o Mgeet tude & eech Mu Id dehuh_3@hoo.co Coght 3 u J d Kt e. Th oe cce tce dtuted ude the Cetve Coo ttuto Lcee hch et uetcted ue dtuto d eoducto edu ovded the og o oe cted. tct Th e devoted to the ueu ethod o ovg the oe-deo teg euto o Fedho te. The Me to techue o ovg gee edho te teg euto th the - ucto d geeed oo the ee codeed. ecg the coecet d vou ete the geeed oo d ucto ou theoe oud ed ed eve eut vovg e ee. Keod: Fedho te teg euto Me to techue & - ucto. Itoducto I the t eve e ge ue o Fedho te teg eutovovg vou oo o ec ucto the ee hve eetuded utho ot

2 99 u J d Kt e uch [ Hgg [ Love [3 d[ Ph d Kh [ vtv d [6 d othe. I the eet e e ot the oog oe deo Fedho teg euto. vovg the -ucto d geeed oo the ee c e oved eotg to the cto o Me to.. [ [ g The eh -ucto occug. toduced udd et. [ 9 hch deed cotou teg o Me e Te:. [ [ [ d Z Z L ω o o ω d.3 The tegto th γ γ eted o γ - to γ d uch tht the oe ued to e e o do ot cocde th the oe. The ete e o-egtve tege tg > o. The ete > d C. The et oduct.3 teeted ut. The etece codto o the deg teg. e gve eo: g < >.4 } g < < d.5 Whee

3 Fedho te teg euto The gee oo o ve gve vtv [4 deed d eeeted : [.8 [ Whee t otve tege d coecet [ e t cott e o coe. Let deote the ce o ucto hch e deed o [ d t γ L [ o o egtve tege. O. Fo coeodece to the ce o good ucto deed o the hoe e e - ee Lghth [8. The e Louve cto teg o ode µ deed µ µ µ } } d e µ > : ε.9 µ µ hee } udetood to e tht Φ oc tege outo o µ } g tht µ heeve ece e h te o cto teg oeto deed e..8 ove. W h } h } h. PLIMINY ULT µ the vee o the ctoteg oeto h µ µ o the e-louve d e h > : ε. We t ove the oog eut hch e eued ovg theoe eo.

4 99 u J d Kt e Le. Wth the et o ucet codto d.7 d et u uoe e > e e [ / > >. Let. [ [ [ [ [ [ - W Poo: To ove Le e t ue the deto o We cto teg gve. ee the -ucto d geeed oo the e chge the ode o uto d tegto hch uted ude the tted codto evute the t-teg d eteetg the eutg Me-e cotou teg te o the -ucto e e ve t the deed eut. Theoe - Ude the ucet codto d v o Le [ [ [ [ }.. Povded uthe d >.

5 Fedho te teg euto 993 Poo: Let η deote the t ee o the eto.. The ug Le d g. e hve η d.3. d.4 ug the veo o the ode o tegto to e ee ut the oo o Le. No deto.9.4 gve η }. d Whch the ecod ee o.. 3. OLUTION OF FHOLM TYP INTGL QUTION To ot the outo o edho te teg euto. e ue the Me To techue. Theoe. I ε } et > > g < >.7.6 } g gve d gve d < < e > e > d t otve tege d coecet e t cott e o coe the the outo o the teg euto 3. [ [ g gve

6 994 u J d Kt e 3.. d Z L ovded uthe tht e [ -/Φ} < e [ / < e / d Poo: O ecg - } 3. d g. e hve Mutg oth the ded o 3.3 h - d tegtg th eect to o to e hve 3.3 } [ [ [ [ g

7 Fedho te teg euto 995 } d g h h [ [ [ [ d 3.4 hee e hve ued the oute d uo covegece o the teg voved th ve to utg the veo o the ode o tegto. No evute the e teg 3.4 e chge o ve the eut c.. o ee [ 5 d [ 7 e. 3.4 educe to Z }

8 996 u J d Kt e 3.5 hee Φ gve 3.4. Z 3.6 } Ivetg 3.6 g the Me Iveo theoe e get 3.7. } d Z L Oetg uo oth de gve u

9 Fedho te teg euto } d Z L Whch ed 3.8. d Z L the outo o the teg euto ec Ce. I theoe e te d the e hve the ve teetg theoe.e. Theoe 3- uoe tht the codto coeodg to Theoe e ted. The / H

10 998 u J d Kt e [ [ [ [ g gve 4.. d Z L Povded teg et.. I e te d Theoe ee to coeod to eut gve vtv d [6 FNC [.. C. Ttchh Itoducto to the Theo o Foue Iteg ecod dto Ood Uvet Pe Lodo Cedo Pe Ood 948. [... Love hegeoetc teg euto Fcto Ccuu d t cto edted. o 7-88 ge-veg Ne Yo975. [3... Love oe teg euto hegeoetc ucto Poc. d-ugh Mth. oc.5o [4. H. M. vtv ute geetg ucto o the Kohu et o otogo oo uggeted the Lguee oo cc J.Mth

11 Fedho te teg euto 999 [5. H. M. vtv K. C. Gut d. P. Go The H ucto o oe d to ve th cto outh uhe Ne eh 98. [6. H. M. vtv d. K. O cet ethod o ovg c o teg euto o Fedho tej. ut. Mth. oc.ee [7. K.C. Gut d U.C. J Poc. Nt. cd. c. Id ect [8. M. J. Lghth toducto to Foue d Geeed ucto C. Mog. Mech..Mth.Cdge Uvet Pe Cdge Lodo d Ne Yo 958. [9.. N üdd u d TF Noeche. Oe oe: ho o out the eh ucto?. Fct. Cc... 4: [.. G. uch veo tegpoc. e. Mth. oc [. T. P. Hgg hegeoetc ucto toj. oc. Idut..Mth [. T.. Ph d N. K. Kh e c o hegeoetcteg euto Id J. ue. Mth eceved: 7 3

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