NON-NEWTONIAN IMPROPER INTEGRALS

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1 Journl of Science nd Arts Yer 8, No. (4), pp , 08 ORIGINAL PAPER NON-NEWTONIAN IMPROPER INTEGRALS MURAT ERDOGAN, CENAP DUYAR Mnuscript received:.09.07; Accepted pper: 4..07; Pulished online: Astrct: In this study, non-newtonin improper integrls were introduced nd their convergence conditions were investigted. Furthermore, some min theorems such s the second men vlue theorem nd intermedite vlue theorem were proved in the non- Newtonin sense to e given convergence tests. Keywords: Non-Newtonin improper integrls, Non-Newtonin clculus, Convergence tests.. INTRODUCTION The non-newtonin clculus provides wide diversity of mthemticl tools for use in engineering, mthemtics nd science. The notion of non-newtonin clculus ws firstly introduced nd worked y Grossmn nd Ktz. They pulished the ook out fundmentls of non-newtonin clculus nd which includes some specil clculuses such s geometric, hrmonic, igeometric etc. []. Non-Newtonin clculus ws used y Meginniss to crete theory of proility tht is dpted to humn ehvior nd decision mking []. Ryczuk nd Stopel used the igeometric clculus on frctls nd mteril science [3]. In study which is mde y Aniszewsk nd Ryczuk, the igeometric clculus ws used on multiplictive Lorenz system [4]. Uzer investigted multiplictive type comple clculus s lterntive to clssicl clculus [5]. In study which is mde y shirov nd Rız, differentition nlysed s comple multiplictive [6]. The non-newtonin clculus ws used in the study of iomedicl imge nlysis y Florck nd vn Assen [7]. Çkmk nd şr otined some results nd sequence spces with respect to non-newtonin clculus [8]. In study which is mde y Tekin nd şr, sequence spces were emined on non-newtonin comple field [9]. Duyr, Sğır nd Oğur got some sic topologic properties on non-newtonin rel line [0]. Duyr nd Erdoğn investigted non-newtonin rel numer series nd otined convergence tests for them []. As result of these studies, it hs risen the need of emintion the improper integrls on non-newtonin clculus. Hence, in this study, we introduce the non-newtonin improper integrls nd show some convergence tests for them. Ondokuz Myıs University, Grdute School of Sciences, Doctorl Student, Smsun, Turkey. E-mil: murt.erdogn@windowslive.com. Ondokuz Myıs University, Fculty of Arts nd Sciences, Deprtmnt of Mthemtics, Smsun, Turkey. E-mil: cenpd@omu.edu.tr. ISSN: Mthemtics Section

2 50 Non-Newtonin improper integrls Murt Erdogn nd Cenp Duyr. GENERAL INFORMATIONS.. -ARITHMETIC Definition : A genertor is one-to-one function whose domin is, the set of ll rel numers, nd whose rnge is suset of. The rnge of genertor is clled non- Newtonin rel line nd we denote it y ( N ). y rithmetic we men the rithmetic whose relm ( N ) nd whose opertions nd ordering reltion re defined s follows: α-ddition y z ( y) ( z) α-sutrction y z ( y) ( z) α-multipliction y z ( y) ( z) α-division y / z ( y)/ ( z) α-order y z y z ( y) ( z) ( y) ( z). In this cse, it is sid tht genertes rithmetic. For emple, the identity function I genertes the clssicl rithmetic nd the eponentil function ep genertes geometric rithmetic. Ech genertor genertes ectly one rithmetic nd, conversely, ech rithmetic is generted y ectly one genertor []. Definition : The positive numers re the numers in ( N ) such tht >0, similrly the negtive numers re the numers in ( N ) such tht <0. zero nd one numers re denoted y 0= 0 nd = respectively. integers re otined y successive ddition of to 0 nd successive sutrction of from 0. Hence integers re s follows:...,,, 0,,,... For ech integer n, we set sum of [, ]. n = n. If n is n positive integer, then it is n times Definition 3: solute vlue of numer ( ) is defined y if >0 = 0 if =0. 0 if <0 N This vlue is equivlent to the epression [, 8, 0]. Mthemtics Section

3 Non-Newtonin improper integrls Murt Erdogn nd Cenp Duyr 5 Definition 4: A closed intervl on ( N ) is represented y [, ]= ( N) : N = ( ) : =,, similrly n open intervl (, ) cn e represented. It is sid tht n intervl hs etent [, 0, ]. Definition 5: Let u n e n infinite sequence of the numers in ( N ). If ech open intervl contining n element u includes ll elements ecept for finite numers of elements of the sequence u n, then it is sid tht the sequence u n converges to u nd the element u is clled s limit of the sequence u n. This is denoted y limu n u. n This convergence ecomes the clssic convergence if = I. Clssic nd geometric convergence re equivlent in the sense tht positive numer sequence p n converges s geometric to positive numer p iff p converges s clssic to p [, 9, 5, ]. n Proposition : y y (non-newtonin tringle inequlity) nd y N y = y re hold for ny, ( ) [8]... *-CALCULUS Let nd e ritrry chosen genertors which imge the set to A nd respectively. *-Clculus is defined s n ordered pir of the rithmetics rithmetic, rithmetic nd the following nottions re used: rithmetic rithmetic Universe(Relm) A N N Summtion Sutrction Multipliction Division / or / or Ordering rithmetic is used on inputs nd rithmetic is used on outputs. In prticulr, the chnges of inputs nd outputs re mesured y rithmetic nd rithmetic, respectively. The opertors in *-clculus re pplied to functions whose inputs nd outputs elong to A nd, respectively. ISSN: Mthemtics Section

4 5 Non-Newtonin improper integrls Murt Erdogn nd Cenp Duyr If the genertors nd re chosen s one of I nd ep, the following specil clculuses re otined. Clculus Clssic I I Geometric I ep Angeometric ep I igeometric ep ep The isomorphism from rithmetic to rithmetic is the unique function (iot) tht possesses the following three properties:. is one to one,. is on A nd onto, 3. For ny numer u nd v in A, u v= u v, uv = u v, uv = u v, u v u v / = /, v 0 u< v u < v. It turns out tht = for every numer in A, nd tht n= n for every integer n. Any sttement in rithmetic cn esily trnsformed into sttement in rithmetic thnks to the isomorphism []. Definition 6: The *-limit t point A of function f is, if it eists, the unique f n for every infinite sequence n whose terms re distinct from nd which converges to. In this cse, numer in the set which is converged y outputs sequence is written[]. = lim f Definition 7: A function f is *-continuous t point A iff this point is n lim f = f []. input for f nd Definition 8: If the following *-limit eists, we denote it y *-derivtive of f t, nd sy tht f is *-differentile t : lim f f /. Df nd cll it the Mthemtics Section

5 Non-Newtonin improper integrls Murt Erdogn nd Cenp Duyr 53 If it eists, Df is necessrily in. The derivtive of f, denoted y D f A to the numer Df t, if it eists[]. M s r f, is the function tht ssigns ech numer t in Definition 9: The *-verge of *-continuous function f on [ rs,] is denoted y nd defined to e limit of the convergent sequence whose n th term is is the n fold prtition of [ rs,] []. verge of f,..., f n, where,..., n s Definition 0: The *-integrl of *-continuous function f on [ rs,], denoted y f d, is the numer r s r s r M f in []. The *-derivtive nd *-integrl re "inversely" relted in the sense indicted y the following two theorems. Theorem : (First fundmentl theorem of *-clculus) If f is *-continuous on [ rs,] nd g= f t d t for every [,] rs, then D g = f r on [ rs,] []. Theorem : (Second fundmentl theorem of *-clculus) If Dh s []. [,] rs, then D h d = h s h r r is *-continuous on. Let Remrk : Let = for given numer A f t = f t for function f whose inputs nd outputs re in A nd, respectively. Then the following reltions re true[]:. lim f nd lim f t coeist nd if they do eist t lim f = lim f t. t Furthermore, f is *-continuous t iff f is clssiclly continuous t.. The derivtives Df nd Df coeist nd if they do eist Df = Df. 3. If f is *-continuous on [ rs,], then s s s s M r f = Mr f nd f d = f d. r r ISSN: Mthemtics Section

6 54 Non-Newtonin improper integrls Murt Erdogn nd Cenp Duyr 4. Let A nd nd, respectively, then. If the genertors nd re clssiclly continuous t lim nd f lim f = iff f =. If nd re clssiclly continuous t is *-continuous t iff f is clssiclly continuous t., respectively, then f 3. RESULTS AND DISCUSSION Definition : Let function f : X ( N) ( N) e given. It is sid tht the left(right)-hnded *-limit of the function f t the point ( N) is the numer L ( N), when ny numer >0 = >0 such tht < f L for ll X is given if there eists t lest numer nd < < < < lim f = L lim f = L.. This *-limit is denoted y Definition : Let function f :(, ) ( N) ( N) f :(, ) ( N) ( N) e given. It is sid tht *-limit of the function f t the is the numer L ( N), when ny numer >0 = >0 such tht < for ll > < f L is given if there eists t lest numer. This *-limit is denoted y lim f = L lim f = L. Similr definitions cn e given for L = nd L =. Proposition : For, ( ) the followings re hold. N. If 0, then 0,.. Proof: One cn show this proposition esily y using definition nd properties of solute vlue. We omit the detils. Definition 3: Let the function f :[, ) ( N) ( N) e *-continuous on intervl [, ] for ech numer. The *-limit lim f d is clled improper Mthemtics Section

7 Non-Newtonin improper integrls Murt Erdogn nd Cenp Duyr 55 *-integrl of type of the function f on [, ] lim f d improper *-integrl eists nd equls to numer L ( ) f equls to one of nd nd is denoted y f N d. If, then it is sid tht the d is convergent(*-convergent). If this *-limit does not eist or, then it is sid tht the improper *-integrl f d divergent. Similrly, if the function f :(, ] ( N) ( N) is continuous on [, ] for ech numer, then the improper *-integrl of type of the function f on [, ] is defined y f d = lim f d. If the function f :(, ) ( N) is *-continuous on ech intervl [, ] with <, ( N), then the improper *-integrl of type of the function f on [, ] is defined y f d = lim f d for nd re independently from ech other. Emple : If the improper *-integrl of type is looked for geometric clculus, since = I = nd = e =lim = =lim =, =lim =,, then, =lim =0re otined. According to this, ( ) = nd ( N ) =(0, ). Let the function f :[, ) (0, ) e continuous on [, ] for ech, is. The improper geometric integrl of type of f on N is ln f ( ) d lim ln f ( ) d ln f ( ) d f( ) d lim f( ) d lim e e e. = nd For ngeometric clculus, since e = I =, ( N) =(0, ) nd ( N) =. Let the function f :[, ) (0, ) e continuous on [, ] for ech, is. The improper ngeometric integrl of type of f on ln f ( d ) lim f( d ) lim f( e) d. ln ISSN: Mthemtics Section

8 56 Non-Newtonin improper integrls Murt Erdogn nd Cenp Duyr For igeometric clculus, since = = e, ( N) = ( N) =(0, ). Let the function f :[, ) (0, ) (0, ) e continuous on [, ] for ech. The improper igeometric integrl of type of f on, is eistence of ln ln f ( e ) d ln f( ) d lim f( ) d lim e. f d f d lim f d for ll c ( N) c Remrk : Since eistence of = lim c lim f d nd is equivlent to, the integrl on the left is meningful ecept tht the two terms in the rightmost sum of the following equlities re infinite from the different signs, c = lim lim f d f d f d c = f d f d. Emple : We investigte convergence condition of improper integrls nd e d. c c e d, e d Solution : Since nd ln e d e d e d e e lim lim lim 0 ln e ln e d 0 e d lim e d lim e lim e 0 the improper geometric integrl divergent, ut the improper clssic integrl e d lim e d lim e e e, e d nd the improper igeometric integrl e d e d re is convergent. This shows us tht convergence condition of the improper integrl of type of function in clssicl clculus nd other clculuses cn e different from ech other. Mthemtics Section

9 Non-Newtonin improper integrls Murt Erdogn nd Cenp Duyr 57 p >: Since Emple 3: The improper *-integrl d= p lim d p d p = lim p is convergent for >0 nd d = lim d p = lim d p = lim d p = lim d p = lim d p p p = lim p p p = lim p p p = lim p p p = lim p p p = lim p ISSN: Mthemtics Section

10 58 Non-Newtonin improper integrls Murt Erdogn nd Cenp Duyr p p = lim p p = p for ech, then the improper *-integrl d p p >. is convergent for > 0 nd Definition 4: Let the function f :[, ) ( N) with, ( ) N e *-continuous on intervl [, ] for ech [, ). Let f e lso unounded on left d is clled improper *-integrl of type of the function f on [, ] nd is denoted y of the point lim f = or lim f =. The *-limit lim f f d. If lim f *-integrl one of nd d eists nd equls to numer L ( ) N, the improper f d is convergent (*-convergent). If this *-limit does not eist or equls to d is divergent. Similrly, if the function f :( A, ] ( N) with A, ( ) is *-continuous on, then the improper *-integrl f intervl [, ] for ech numer ( A, ] nd unounded on right of the point A, then the improper *-integrl of type of the function f on [ A, ] is defined s A A f d = lim f d. Emple 4: ( N) = nd ( N) = for geometric clculus. Let the function f :[, ) e continuous on, for ech, nd unounded on left of the point. Then the improper geometric integrl of type of the function f on [, ] is found tht N ln f ( ) d lim ln f ( ) d ln f ( ) d f d f d e e e ( ) lim ( ) lim. ( ) = nd ( ) = for ngeometric clculus. Let the function N N f :[, ),, nd unounded on left of the point. Then the improper ngeometric integrl of type of the function f on [, ] is found tht e continuous on for ech Mthemtics Section

11 Non-Newtonin improper integrls Murt Erdogn nd Cenp Duyr 59 ln f ( d ) lim f( d ) lim f( e) d. ln ( N) = ( N) = for igeometric clculus. Let the function f :[, ) e continuous on, for ech, nd unounded on left of the point. Then the improper igeometric integrl of type of the function f on [, ] is found tht ln ln f ( e ) d ln ln lim ln f ( e ) d ln f( ) d lim f( ) d lim e e. nd Emple 5: We investigte convergence condition of the improper integrls d. 0 0 d Solution : Since ln d.ln d = = = =.. = lim d lime lime e lim e nd 0 d = lim d = lim =, 0 0 the improper geometric integrl 0 d is convergent, ut the improper integrl d is 0 divergent. This shows tht convergence conditions of the improper integrl of type of function in clssicl clculus nd other clculuses cn e different from ech other. Definition 5: Let the function f :[, ) ( N) e *-continuous on ech [, ] [, ) where ( N) nd ( N) =[, ]. Let f e unounded on left of the point if ( ). If eists nd finite, then N lim f d f d = lim f d ISSN: Mthemtics Section

12 60 Non-Newtonin improper integrls Murt Erdogn nd Cenp Duyr is written nd it is sid tht the improper *-integrl Similr definition is lso given for ( A, ]. f d is convergent(*-convergent). Theorem 3: Let the functions f, g:[, ) ( N) with ( ) N nd ( N) e *-continuous on ech [, ] [, ), nd e unounded on left of the point if ( N). Let the improper *-integrls f d nd the improper integrl of the function f g on [, ] for ll, ( N ). g d re convergent. Then is = f g d f d g d Proof: Since the *-integrl is dditive nd homogeneous, it is written f g d= f d g d. If we tke *-limit s in this equlity, then lim f gd = lim fd gd nd thus = lim lim f g d f d g d = f d g d. Hence theorem is proved. Theorem 4: Let the functions f, g:[, ) ( N) with ( ) N nd ( N) e *-continuous on ech [, ] [, ), nd e unounded on left of the point if ( ). If c[, ), then the equlity N holds. c = f d f d f d c Mthemtics Section

13 Non-Newtonin improper integrls Murt Erdogn nd Cenp Duyr 6 Proof: y the property of *-integrl, it cn e written c = f d f d f d c for ech [, ) nd c [, ). If we tke *-limit s in this equlity, then proof is completed. Theorem 5: If the function f :[, ) ( N) with ( ) N is *-nondecresing, then the *-limit lim f = L = sup f : [, ) eists. Further, if the function f is ounded from ove, then L ( ) L =. N, otherwise Proof: ) First, consider the function f is ounded from ove. It is ovious tht = L sup f : [, ) ( N ). We show tht lim f = L. Any numer >0 is given. y the property of supremum, following sttements hold: ) f L for ll [, ), )t lest [, ) eist such tht f > L. When ( ), if = >0 is tken, then N nmely, L < f f L< L, < f L for ll [, ) such tht < <, ecuse of the function f is *-nondecresing. This shows tht lim f = L. When =, if = [, ) is tken, nmely, L < f f L< L, < f L for ll >, ecuse of the function f is *-nondecresing. This shows tht lim f = L. ISSN: Mthemtics Section

14 6 Non-Newtonin improper integrls Murt Erdogn nd Cenp Duyr )Second, consider the function f is unounded from ove. We will show tht lim f =. Any numer >0 is given. There eists [, ) such tht f >. When ( N), if = >0 is tken, then one cn find tht for ll [, ) such tht < <, ecuse of the function f is lim f. When =, if = [, ) is f f > *-nondecresing. This shows tht = tken, then we hve f f > *-nondecresing. This shows tht f = for ll > ecuse of the function f is lim, hence proof is completed. Theorem 6: (Comprison test) Let the functions f, g:[, ) ( N) with ( N) nd ( N) e *-continuous on ech [, ] [, ), nd e unounded on left of the point if ( ) 0 f C g for ll [, ). If. Let N gd is convergent, then f f d C g d holds. Proof: Let the functions FG, :[, ) ( N) G= gd for ech [, ) nd *-integrls property, for ech [, ). Therefore d is convergent nd the inequlity e defined s =. According to 0 f Cg F f d nd for ll [, ) 0d f d C g d 0 f d C g d nd hence 0 F C G. 0< f Since for ll [, ), we hve s r = F s F r f d f d = s r r f d f d f d r Mthemtics Section

15 Non-Newtonin improper integrls Murt Erdogn nd Cenp Duyr 63 for ll r s < = s r s. Hence F s Fr 0d r f d, nmely, the function F is *-nondecresing. Similrly, the function G is lso *-nondecresing. y virtue of g d is convergent nd theorem 5, the function G is ounded from ove. In other words, there eists numer L 0 G L for ll [, ). Then the function F is ounded such tht from ove since F CG C L. So, y virtue of theorem 5, f inequlity F C G. convergent. Furthermore, d is f d C g d if we tke *-limit s of the Theorem 7: (Quotient test/*-limit comprison test) Let the functions f, g:[, ) ( N) 0 with ( N) nd ( N) e *-continuous on ech [, ] [, ), nd e unounded on left of the point if ( ). Let the *-limit eist. In this cse; f lim = L, 0 L g ) f d is convergent if ) f d is divergent if N g d is convergent when 0 L <, g d is divergent when 0<L, c) the improper *-integrls f d nd divergent when 0< L <. g d re oth convergent or oth f Proof: )Since lim = L g, for = there eists numer >0 such tht f L < for ll [, ) g with < <, for ll > if =. According to this, there eists numer c(, ), c > if =, such tht f L < < L g nd therefore f < L g for every [, c). In ISSN: Mthemtics Section

16 64 Non-Newtonin improper integrls Murt Erdogn nd Cenp Duyr ccordnce with theorem 4, gd is convergent since c g d is convergent. y f d is convergent. Agin y theorem 4, f virtue of theorem 6, d is lso c convergent. ) If 0< L <, for fied numer >0 such tht 0< <L there eists numer >0 f such tht L < for ll [, ) g with < <, for ll > if =. According to this, there eists numer c(, ), c > if =, such tht f L < < L nd hence g< f for ll [, c). y theorem 4, g L gd is divergent since c f d is divergent. Then, gin y theorem 4, f c If L = g d is divergent. In this cse, y virtue of theorem 6, d is lso divergent., for ny fied numer >0, there eists numer >0 such tht for every [, ) with < <, for ll > if = eists numer c(, ), c > if = [, c) < f g. According to this, there g < f for ll, such tht. y theorem 4, gd is divergent since cse, y virtue of theorem 6, c f d is lso divergent. c)the proof is ovious from () nd (). convergent. c g d is divergent. In this f d is divergent. Then, gin y theorem 4, Emple 6: Show tht the improper igeometric integrl ln. e e ln d is Solution 3: We know tht quotient test, we hve e e ln d is convergent from emple 3. Using the Mthemtics Section

17 Non-Newtonin improper integrls Murt Erdogn nd Cenp Duyr 65 ln. ln ln ln ln. ln ln e ln ln lim e = lime = lime = e. Hence the improper igeometric integrl ln. e e ln d is convergent. Theorem 8: Let the function f :[, ) ( N) with ( ) N nd ( N) e *-continuous on ech [, ] [, ), nd let the function f e unounded on left of the point if ( ). If the improper *-integrl N f d is convergent. f d is convergent, then Proof: Form the functions f, f :[, ) ( N) s f f f f f = nd f = for given function f. y theorem 6, the improper *-integrls re convergent since 0 f f nd other hnd, if we use the equlity f = f f f d nd f d 0 f f for ll [, ). On the nd refer to theorem 3, we otin f d = f d f d. Then f d is convergent. Definition 6: If the improper *-integrl improper *-integrl f d is convergent, then the f d is sid to e solute convergent( solute *-convergent). However, if the improper *-integrl f d is convergent ut the ISSN: Mthemtics Section

18 66 Non-Newtonin improper integrls Murt Erdogn nd Cenp Duyr improper *-integrl f d is divergent, then the improper *-integrl f d is sid to e conditionl convergent(conditionl *-convergent). In ccordnce with theorem 8, every solutely convergent integrl is lso convergent. Corollry : (Comprison test) Let the functions f :[, ) ( N) nd g:[, ) ( N) 0 with ( N) nd ( N) e *-continuous on ech [, ] [, ), nd let the functions f nd g e unounded on left of the point if ( N). Let f g for ll [, ) f d is convergent. e e cos(ln ) (ln ). If g d is convergent, then Emple 7 We investigte convergence condition of improper igeometric integrl d. Solution 4: y the emple 3, the improper igeometric integrl convergent. Since ln ln cos ln, we hve e e (ln ) d is cosln cosln ln ln ln ln e = e e = e for ll e, is convergent.. Thus, in view of corollry, the improper igeometric integrl e e cos(ln ) (ln ) d Theorem 9: (Cuchy criterion) For the function f : X ( N) ( N), lim f = A ( N) iff when ny numer >0 is given, there eists t lest numer >0 such tht f f " < for ll, " X with < nd " <. Proof: Suppose tht = ( ) lim f A N is eists. Then, when ny numer >0 is given, there eists t lest numer >0 such tht < f A nd Mthemtics Section

19 Non-Newtonin improper integrls Murt Erdogn nd Cenp Duyr 67 f " A < numers for ll, " nd ", we hve X with < nd " <. For the sme " = " f f f A A f f A f A <. Conversely suppose tht when ny numer >0 is given, there eists t lest numer >0 such tht f f " < for ll, " X with < nd " <. Let sequence n, which holds n X \ nd lim n = for ll n, e given. Since lim n =, there eist n such tht, n " X, < nd n " < for ll n n n n, n"> n. Then, y the hypothesis, f f for ll numers < n n" the sequence f n is Cuchy sequence in the spce ( ) complete, the sequence n lim f = A, y the definition of *-limit, we find tht lim = n n n n nd n ". Hence N. Since ( N ) is f converges to n element of this spce. If we set theorem. The proof of the following theorem cn esily seen from theorem 9. f A. This proves the Theorem 0: (Cuchy s test) Let the function f :[, ) ( N) with ( ) N nd ( ) e *-continuous on ech [, ] [, ), nd let the function f e N unounded on left of the point if ( ) N. Then, the improper *-integrl f d is convergent iff when every numer >0 is given, there eist one 0 0 = [, ) such tht for ll, [ 0, ). f d < Theorem Let function f :[, ] ( N) ( N) e *-continuous. At this sitution, the following sttements re hold: (i) The function f is ounded on [, ]. (ii) There eist numers, [, ] [, ]. m M such tht f = m f M = f m for ll M ISSN: Mthemtics Section

20 68 Non-Newtonin improper integrls Murt Erdogn nd Cenp Duyr (iii) f d f d. Proof: Let the function f e *-continuous on [, ]. (i) y remrk s first item, the function f = f is continuous on, since the function f is *-continuous on [, ]. Therefore the function f is ounded on,. Then, there eists numer C > 0 such tht f t C for ll, there eists one nd only one t, t, nd if we tke K = C, K >0 since C >0 t,. For ll [, ] such tht = f C for ll [, ] f C. This shows tht from f t C. We infer tht for ll t,, therefore we otin tht f K for ll [, ], nmely, the function f is ounded on [, ]. (ii) y remrk s first item, the function f = f is continuous on, since the function f is *-continuous on [, ]. Then there eist numers t,, m tm such tht f tm f t f tm for ll t,. For ll [, ], there eists one nd only one t, such tht = t, nd if t =, t = re tken, then, [, ]. Since m m M M f tm f t f tm for ll t, the definition of order, we hve for ll [, ]. f = f t f f t = f m m M M (iii) y proposition, 0 f f f, since f f Therefore, y *-integrls property nd homogeneity, 0 f d f d f d 0 f d f d f d 0 f d f d f d 0 f d f d f d m M, in view of for ll [, ]. Mthemtics Section

21 Non-Newtonin improper integrls Murt Erdogn nd Cenp Duyr 69 is written. Then, y virtue of lst inequlity nd first sttement of proposition, f d f d. This completes the proof. Theorem : (Intermedite vlue theorem) Let the function f : ( N) ( N) e *-continuous on [, ] f f. then there eist numer c[, ] such tht nd let f c = D for ll D [ f, f ] or D f f [, ]. Proof: y remrk s first item, the function f = f is continuous on, since the function f is *-continuous on [, ]. There eists t lest k, f k = f k = D since numer such tht or D f, f = f, f, D f f for ll D [ f, f ] or D [ f, f ]. If c= k is tken, then Therefore we otin f c = D since f k = D. c= k [, ]. Theorem 3: (Second men vlue theorem) Let the functions f, g:[, ] ( N) ( N) e *-continuous on [, ]. ) If the function g is *-nonincresing nd g 0 for ll [, ], then there eist point [, ] such tht f g d = g f d. ) If the function g is *-nondecresing nd 0 point [, ] such tht f g d = g f d. g for ll [, ], then there eist Proof: ) The n fold prtition,,..., n of [, ] is given. Then ISSN: Mthemtics Section

22 70 Non-Newtonin improper integrls Murt Erdogn nd Cenp Duyr f g d f g d n k = k = k = g f d g g f d n k n k k k k= k= k k () () = Sn Sn. () Now we show tht S n 0 s n. Since the function f is *-continuous on [, ], f is ounded, therefore there eist numer K >0 such tht f K for ll [, ]. Then n k () Sn = g gk f d k = k n k = k k n k = k k g g k f d n k ggk f d k = k n k K ggk d k = k K n k = sup g g y :, y[ k, k ] d k k k k = K sup g g y :, y[, ] nd *-continuity of the function g implies tht lims n n () =0. According to this, we hve Let F( )= f g d = g f d. n k lim k n k= k f t d t for ll [, ]. Since the function f is *-continuous, y the first fundmentl theorem of *-clculus, we write DF = f m F = min F : [, ] for ll [, ]. Now, let Mthemtics Section

23 Non-Newtonin improper integrls Murt Erdogn nd Cenp Duyr 7 M F = m F : [, ]. Since the function F is *-continuous, y the second fundmentl theorem of *-clculus, k f d F k F k k = is written. Therefore we otin tht n k () Sn = gk f d k = k = n k = g F F k k k n = F g F g F g g n k k k k = n = F g F g g. Since g nd k g k 0 n k k k k = g n 0 for k =,,..., n, y the lst equlity, n = m F g m F g m F g g n g If we tke limit of this inequlity s n, When =0 n k k k = F g F g g M F. n k k k k = lim m F g S () = f g d M F g. n n g, it is ovious tht desired eqution holds. When g >0 = f gd g is tken, y the lst inequlity, we hve mf MF *-continuous on [, ], there eist [, ] ) Proof is similr to ()., if. Since the function F is such tht = = f g d = g f d. F f d or ISSN: Mthemtics Section

24 7 Non-Newtonin improper integrls Murt Erdogn nd Cenp Duyr Theorem 4: (Generlized second men vlue theorem) Let the functions f, g:[, ] ( N) e *-continuous nd let the function g e *-nondecresing or *-nonincresing. Then, there eists point [, ] such tht f g d = g f d g f d. Proof: Suppose tht the function g is *-nondecresing. The function G= g g is continuous nd *-nonincresing on [, ] since the function g is *-continuous nd *-nondecresing on [, ]. Further, G 0 for ll [, ]. Then, y the second men vlue theorem, there eists point [, ] such tht Agin, since nd f G d = G f d. = f G d f g g d = g f d f g d () G f d = g f d g f d, () y the equlity of right sides of equlities () nd (), = g f d f g d g f d g f d is written. Hence, we see tht the equlity = f g d g f d g f d holds for point [, ]. When the function g is *-nonincresing, since the function G= g g is *-continuous nd *-nondecresing on [, ], nd since G 0 for ll [, ], the proof is completed similr to ove. Mthemtics Section

25 Non-Newtonin improper integrls Murt Erdogn nd Cenp Duyr 73 Theorem 5: (Ael s-dirichlet s test) Let the functions f, g:[, ) ( N) with ( ) nd ( ) e *-continuous on ech [, ] [, ), nd e unounded N N on left of the point if ( ) N. In this cse, ) if the improper *-integrl f d is convergent nd if the function g is *-nonincresing(or *-nondecresing) nd ounded, then the improper *-integrl f g d is convergent(ael s test). ) if the function = F f d with [, ) is ounded on [, ), nd if the function g is *-nonincresing or *-nondecresing which holds lim g=0, then the improper *-integrl f g d is convergent(dirichlet s test). Proof: y virtue of generlized men vlue theorem, there eist t lest [, ] etween numers nd such tht for ll, [, ). = f g d g f d g f d ) Since g is ounded nd f f g d is convergent. ) Since the function = f d nd f d is convergent, y Cuchy s test, F f d is ounded on [, ) nd the *-integrls d re ounded lso. Thus, y Cuchy s test, f gd is convergent since f d nd f since lim g=0. cos e d re ounded nd Emple 8: We investigte convergence condition of the improper geometric integrl d. ISSN: Mthemtics Section

26 74 Non-Newtonin improper integrls Murt Erdogn nd Cenp Duyr g cos ln e cos Solution 5: The equlity e = e = e. For ll,, sinsin We hve F function F is ounded. f = e nd holds. Now we tke cos cos d cos sinsin F = e d = e = e. = e since sin sin for ll,. Therefore, the lim g = lime ==0 holds nd the function f is = = geometric nonincresing since f e e f for numers,, which re <. Then, y Dirichlet s test, the improper geometric integrl convergent. cos e d is REFERENCES [] Grossmn, M., Ktz, R., Non-Newtonin Clculus, Lee Press, Msschusetts, 97. [] Meginniss, J.R., Americn Sttisticl Assoition: Proceedings of the usiness nd Economic Sttistics Section, 405, 980. [3] Ryczuk, M., Stoppel, S., Springer, 03(), 7, 000. [4] Aniszewsk, D., Ryczuk, M., Chos, Solitons nd Frctls, 5(), 79, 005. [5] Uzer, A., Computers nd Mthemtics with Applictions, 60, 75, 00. [6] shirov, A.E., Riz M., TWMS Journl of Applied nd Engineering Mthemtics, (), 75, 0. [7] Florck, L., vn Assen, H., Journl of Mthemticl Imging nd Vision, 4(), 64, 0. [8] Çkmk, A.F., şr, F., Journl of Inequlities nd Applictions, 8(),, 0. [9] Tekin, S., şr, F., Astrct nd Apllied Anlysis, 03,, 03. [0] Duyr, C., Sgir,., Ogur, O., ritish Journl of Mthemtics & Computer Science, 9(4), 300, 05. [] Duyr, C., Erdogn, M., IOSR Journl of Mthemtics, (6)(IV), 34, Mthemtics Section