J. Mth. Anl. Appl. 337 2008) 36 48 www.elsevier.com/locte/jm Multiplictive clculus nd its pplictions Agmirz E. Bshirov,b, Emine Mısırlı Kurpınr c, Ali Özypıcı c, Deprtment o Mthemtics, Estern Mediterrnen University, Gzimgus, Mersin 10, Turkey b Institute o Cybernetics, Ntionl Acdemy o Sciences, F. Agyev St. 9, Az1141 Bku, Azerbijn c Deprtment o Mthemtics, Ege University, Izmir, Turkey Received 27 My 2006 Avilble online 3 April 2007 Submitted by Steven G. Krntz Abstrct Two opertions, dierentition nd integrtion, re bsic in clculus nd nlysis. In ct, they re the ininitesiml versions o the subtrction nd ddition opertions on numbers, respectively. In the period rom 1967 till 1970 Michel Grossmn nd Robert Ktz gve deinitions o new kind o derivtive nd integrl, moving the roles o subtrction nd ddition to division nd multipliction, nd thus estblished new clculus, clled multiplictive clculus. In the present pper our im is to bring up this clculus to the ttention o reserchers nd demonstrte its useulness. 2007 Elsevier Inc. All rights reserved. Keywords: Clculus; Derivtive; Integrl; Limit; Semigroup; Dierentil eqution; Clculus o vritions 1. Introduction Dierentil nd integrl clculus, the most pplicble mthemticl theory, ws creted independently by Isc Newton nd Gottried Wilhelm Leibnitz in the second hl o the 17th century. Lter Leonrd Euler redirect clculus by giving centrl plce to the concept o unction, nd thus ounded nlysis. Two opertions, dierentition nd integrtion, re bsic in clculus nd nlysis. In ct, they re the ininitesiml versions o the subtrction nd ddition opertions on numbers, respectively. * Corresponding uthor. E-mil ddresses: gmirz.bshirov@emu.edu.tr A.E. Bshirov), emine.kurpinr@ege.edu.tr E.M. Kurpınr), li.ozypici@emu.edu.tr A. Özypıcı). 0022-247X/$ see ront mtter 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jm.2007.03.081
A.E. Bshirov et l. / J. Mth. Anl. Appl. 337 2008) 36 48 37 In the period rom 1967 till 1970 Michel Grossmn nd Robert Ktz gve deinitions o new kind o derivtive nd integrl, moving the roles o subtrction nd ddition to division nd multipliction, nd thus estblished new clculus, clled multiplictive clculus. Sometimes, it is clled n lterntive or non-newtonin clculus s well. Unortuntely, multiplictive clculus is not so populr s the clculus o Newton nd Leibnitz lthough it perectly nswers to ll conditions expected rom theory tht cn be clled clculus. We, the uthors o this pper, think tht the gp is insuicient dvertising o multiplictive clculus. We cn ccount only two relted ppers [1,2]. Multiplictive clculus hs reltively restrictive re o pplictions thn the clculus o Newton nd Leibnitz. Indeed, it covers only positive unctions. Thereore, one cn sk whether it is resonble to develop new tool with restrictive scope while well-developed tool with wider scope hs lredy been creted. The nswer is similr to why do mthemticins use polr coordinte system while there is rectngulr coordinte system, well-describing the points on plne. We think tht multiplictive clculus cn especilly be useul s mthemticl tool or economics nd innce becuse o the interprettion given to multiplictive derivtive below in Section 2. In the present pper our im is to bring up multiplictive clculus to the ttention o reserchers in the brnch o nlysis nd demonstrte its useulness. Sections 2 nd 3 contin known results in the min. They re used in Section 4 to demonstrte some pplictions o multiplictive clculus. 2. Multiplictive derivtives For motivtion, ssume tht depositing $ in bnk ccount one gets $b ter one yer. Then the initil mount chnges b/ times. How mny times it chnges monthly? For this, ssume tht the chnge or month is p times. Then or one yer the totl mount becomes b = p 12.Now we cn compute p s p = b/) 1/12. Assuming tht deposits chnge dily, t ech hour, t ech minute, t ech second, etc. nd the unction, expressing its vlue t dierent time moments is, we ind the ormul ) 1 x+ h) h lim, 1) h 0 x) showing how mny times the mount x)chnges t the time moment x. Compring 1) with the deinition o the derivtive x+ h) x) x) = lim, 2) h 0 h we observe tht the dierence x+ h) x) in 2) is replced by the rtio x+ h)/ x) in 1) nd the division by h is replced by the rising to the reciprocl power 1/h. The limit 1) is clled the multiplictive derivtive or, briely, derivtive o t x nd it is denoted by x).i x) exists or ll x rom some open set A R, where R denotes the rel line, then the unction : A R is well deined. The unction itsel is clled the derivtive o : A R or which the symbol d /dt cn lso be used. The derivtive o is clled the second derivtive o nd it is denoted by. In similr wy the nth derivtive o cn be deined, which is denoted by n) or n = 0, 1,..., with 0) =. I is positive unction on A nd its derivtive t x exists, then one cn clculte
38 A.E. Bshirov et l. / J. Mth. Anl. Appl. 337 2008) 36 48 x+ h) x) = lim h 0 x) = lim h 0 1 + ) 1 h x+ h) x) x) ) x) x+h) x) x+h) x) h = e x) x) = e ln ) x), where ln )x) = ln x). I, dditionlly, the second derivtive o t x exists, then by n esy substitution we obtin x) = e ln ) x) = e ln ) x). Here ln ) x) exists becuse x) exists. Repeting this procedure n times, we conclude tht i is positive unction nd its nth derivtive t x exists, then n) x) exists nd n) x) = e ln )n) x), n= 0, 1,... 3) Note tht the ormul 3) includes the cse n = 0 s well becuse x)= e ln )x). Bsed on this, the unction : A R is sid to be dierentible t x or on A i it is positive nd dierentible, respectively, t x or on A. Cn we express n) in terms o n)? A ormul, similr to Newton s binomil ormul, cn be derived. For this, note tht by 3), we hve ln n) ) x) = ln ) n) x) = ln ) k)) n k) x) = ln k) ) n k) x). Hence, using x) = x) ln ) x), we clculte x) = x) ln ) x) + x) ln ) x) or, nd x) = x) ln ) x) + 2 x) ln ) x) + x) ln ) x) or. Repeting this procedure n times, we derive the ormul n 1 n) n 1)! x) = k!n k 1)! k) x) ln n k)) x). 4) k=0 For the constnt unction x)= c>0 on the intervl, b), where <b,wehve x) = e ln c) = e 0 = 1, x, b). Conversely, i x) = 1 or every x, b), then rom x) = e ln ) x) = 1, one cn esily deduce tht x)= const > 0, x, b). Thus, unction is positive constnt on n open intervl i nd only i its derivtive on this intervl is identiclly 1. Recll tht in similr condition, involving derivtive, the neutrl element 0 in ddition ppers insted o the neutrl element 1 o multipliction. 1 x)
A.E. Bshirov et l. / J. Mth. Anl. Appl. 337 2008) 36 48 39 Here re some rules o dierentition: ) c ) x) = x), b) g) x) = x)g x), c) /g) x) = x)/g x), d) h ) x) = x) hx) x) h x), e) h) x) = hx)) h x), where c is positive constnt, nd g re dierentible, h is dierentible nd in prt e) h is deined. For exmple, the rule b) cn be proved s ollows: g) x) = e ln g)) x) = e ln ) x)+ln g) x) = e ln ) x) e ln g) x) = x)g x). At the sme time, the respective rule or sum nd or dierence s well) is complicted x) gx) + g) x) = x) x)+gx) g x) x)+gx). Let us ormulte some useul results o dierentil clculus in terms o derivtive. They cn be proved by ppliction o the respective results o Newtonin clculus to the unction ln. Theorem 1 Multiplictive Men Vlue Theorem). I the unction is continuous on [,b] nd dierentible on, b), then there exists <c<bsuch tht b) ) = c) b. Corollry 1 Multiplictive Tests or Monotonicity). Let the unction :, b) R be dierentible. ) I x) > 1 or every x, b), then is strictly incresing. b) I x) < 1 or every x, b), then is strictly decresing. c) I x) 1 or every x, b), then is incresing. d) I x) 1 or every x, b), then is decresing. Corollry 2 Multiplictive Tests or Locl Extremum). Let the unction :, b) R be twice dierentible. ) I tkes its locl extremum t c, b), then c) = 1. b) I c) = 1 nd c) > 1, then hs locl minimum t c. c) I c) = 1 nd c) < 1, then hs locl mximum t c. Theorem 2 Multiplictive Tylor s Theorem or One Vrible). Let A be n open intervl nd let : A R be n + 1 times dierentible on A. Then or ny x,x + h A, there exists number θ 0, 1) such tht x+ h) = n m=0 m) x) ) hm m! n+1) x + θh) ) hn+1 n+1)!.
40 A.E. Bshirov et l. / J. Mth. Anl. Appl. 337 2008) 36 48 The bove results cn be extended to unctions o severl vribles s well. For simplicity, consider the unction x,y) o two vribles deined on some open subset o R 2 = R R). We cn deine prtil derivtive o in x, considering y s ixed, which is denoted by x or / x. In similr wy, the prtil derivtive o in y cn be deined, which is denoted by y or / y. One cn go on nd deine higher-order prtil derivtives o or which the respective nottions re used. Two results, generlizing the property e) o dierentition nd Theorem 2, re s ollows. They cn lso be proved by ppliction o the respective results o Newtonin clculus to the unction ln. Theorem 3 Multiplictive Chin Rule). Let be unction o two vribles y nd z with continuous prtil derivtives. I y nd z re dierentible unctions on, b) such tht yx),zx)) is deined or every x, b), then d yx),zx)) dx = y ) y yx),zx) x) ) z z yx),zx) x). Theorem 4 Multiplictive Tylor s Theorem or Two Vribles). Let A be n open subset o R 2. Assume tht the unction : A R hs ll prtil derivtives o order n + 1 on A. Then or every x, y), x + h, y + k) A so tht the line segment connecting these two points belongs to A, there exists number θ 0, 1), such tht x+ h, y + k) = n 3. Multiplictive integrls m m=0 i=0 hi km i x, y) x i y m i i!m i)! m) n+1 n+1) i=0 hi kn+1 i x + θh,y + θk) x i y n+1 i i!n+1 i)!. Now let us deine n nlog o Riemnn integrl in multiplictive clculus. Let be positive bounded unction on [,b], where <<b<. Consider the prtition P ={x 0,...,x n } o [,b]. Tke the numbers ξ 1,...,ξ n ssocited with the prtition P. The irst step in the deinition o proper Riemnn integrl o on [,b] is the ormtion o the integrl sum n S,P) = ξ i )x i x i 1 ). i=1 To deine the multiplictive integrl o on [,b] we will replce the sum by product nd the product by rising to power n P,P) = ξ i ) x i x i 1 ). 5) i=1 The unction is sid to be integrble in the multiplictive sense or integrble i there exists number P hving the property: or every ε>0 there exists prtition P ε o [,b] such tht P,P) P <εor every reinement P o P ε independently on selection o the numbers ssocited with the prtition P. The symbol x) dx,
A.E. Bshirov et l. / J. Mth. Anl. Appl. 337 2008) 36 48 41 relecting the eture o the product in 5), is used or the number P nd it is clled the multiplictive integrl or integrl o on [,b]. It is resonble to let x) dx = 1 nd b x) dx = b x) dx ) 1. It is esily seen tht i is positive nd Riemnn integrble on [,b], then it is integrble on [,b] nd x) dx = e ln )x) dx. 6) Indeed, since the Riemnn integrl o ln on [,b] exists, the continuity o the exponentil unction nd P,P) = e n i=1 x i x i 1 )ln )ξ i ) Sln,P) = e imply the bove sttement. Conversely, one cn show tht i is Riemnn integrble on [,b], then x)dx= ln Some rules o integrtion re s ollows: e x) ) dx. 7) ) b) c) d) x) p ) b dx = b ) dx x)gx) = ) x) dx = gx) x) dx = c x) dx ) p, p R, x) dx x)dx, b gx)dx x) dx c gx) dx, x) dx, c b, where nd g re integrble on [,b]. For exmple, the rule b) ollows rom x)gx) ) dx = e ln )x)+ln g)x)) dx = x) dx gx) dx. Theorem 5 Fundmentl Theorem o Multiplictive Clculus). The ollowing sttements hold:
42 A.E. Bshirov et l. / J. Mth. Anl. Appl. 337 2008) 36 48 ) Let : [,b] R be dierentible nd let be integrble. Then x) dx = b) ). b) Let : [,b] R be integrble nd let Fx)= x t) dt, x b. I is continuous t c [,b], then F is dierentible t c nd F c) = c). Proo. Prt ) ollows rom x) dx = e ln ) x) ) dx = e ln ) x) dx = e ln )b) ln )) = b) ). For prt b), write Fx)= e x ln )t) dt, x b. Then F is dierentible t the point c o continuity o. Furthermore, F c) = e ln F) c) = e F c) Fc) completing the proo o prt b). Fc) ln )c) = e Fc) = e ln )c) = c) Theorem 6 Multiplictive Integrtion by Prts). Let : [,b] R be dierentible, let g : [,b] R be dierentible so the g is integrble. Then x) gx)) dx = b) gb) ) g) 1 x)g x) ). dx Proo. Use the property d) o dierentition nd Theorem 5). 4. Applictions 4.1. Support to Newtonin clculus Some cts o Newtonin clculus cn be proved esily through multiplictive clculus. For exmple, the unction { x)= e 1 x 2 i x 0, 0 ix = 0, clled Cuchy s unction, is ininitely mny times dierentible on R lthough it is not nlytic). The proo o this ct by tools o Newtonin clculus is time consuming nd, thereore, mny uthors void its proo. Using multiplictive clculus this cn be proved reltively esy. Indeed, veriiction o this sttement is esy t every x 0. It is more diicult to veriy it t x = 0.
A.E. Bshirov et l. / J. Mth. Anl. Appl. 337 2008) 36 48 43 Multiplictive clculus helps us to prove tht n) 0) = 0 or every n = 0, 1,... For this, let us consider x>0nd show tht n) x) = e 1)n+1 n+1)! x n+2, n= 0, 1,... 8) Theormul8)istrueorn = 0. Assume tht it is true or n nd clculte strightorwrdly the n + 1)st derivtive o, n+1) x) = lim e h 0 By the binomil ormul, Hence, 1) n+1 n+1)! x+h) n+2 1)n+1 n+1)! x n+2 ) 1h. ln n+1) x) = lim h 0 1) n+2 n + 1)! x + h) n+2 x n+2 ) hx n+2 x + h) n+2 n+1) x) = e 1)n+2 n+2)! x n+3. = lim h 0 1) n+2 n + 1)! n + 2)hx n+1 + +h n+2 ) hx n+2 x + h) n+2 = lim h 0 1) n+2 n + 1)! n + 2)x n+1 + +h n+1 ) x n+2 x + h) n+2 = 1)n+2 n + 2)! x n+1 x 2n+4 = 1)n+2 n + 2)! x n+3. By induction, 8) holds or every n = 0, 1,... Next, we use the ormul 4) nd obtin n 1 n) x) = k=0 1) n k+1 n 1)!n k + 1)! k) x) k!n k 1)! x n k+2, n= 1, 2,... Multiple ppliction o this ormul yields n) x) x N n = x) m=4 M n,m, n= 1, 2,..., xm where M n,m nd N n re integers nd N n 4. We need not the exct vlues o these integers since by multiple ppliction o L Hopitl s rule, x) lim x 0+ x m = lim x 0+ e 1 x 2 x m Thus, n) x) lim = 0, n= 0, 1,..., x 0+ x = lim 1/x m z m 2 = lim x 0+ e 1 x 2 z e z = 0. implying n+1) 0+) = 0 whenever n) 0+) = 0. Since 0) = 0, by induction, we conclude tht n) 0+) = 0 or every n = 0, 1,...Nowletx<0. Since is n even unction, we esily deduce n) 0 ) = 0 or every n = 0, 1,... Thus, n) 0) = 0 or every n = 0, 1,...
44 A.E. Bshirov et l. / J. Mth. Anl. Appl. 337 2008) 36 48 4.2. Semigroups o liner opertors Consider the liner dierentil eqution y x) = x)yx), x > 0, y0) = y 0 > 0. We cn write it in the orm y x)/yx) = x) or e ln y) x) = e x), ssuming tht its solution is positive. Thus, y x) = e x), x >0, y0) = y 0, nd we cn express the solution o the bove dierentil eqution in terms o integrl s yx) = y 0 x 0 e t) ) dt, x 0. Thus, integrtion is closely relted to the representtion theory o liner dierentil equtions nd cn be used ter respective generliztion) or construction o semigroups o liner bounded opertors. 4.3. Multiplictive spces The concepts o derivtive nd integrl re bsed on the ordinry limit opertion. We cn deine multiplictive limit or limit s well. Given x R + = 0, ), we let the multiplictive bsolute vlue o x be number x such tht { x i x 1, x = 1/x i x<1. This llows to deine the multiplictive distnce d x, y) between x,y R + s d x, y) = x y. The ollowing properties o multiplictive distnce re obvious: ) x,y R +, d x, y) 1, b) d x, y) = 1 i nd only i x = y, c) x,y R +, d x, y) = d y, x), d) tringle inequlity) x,y,z R +, d x, z) d x, y)d y, z). On the bse o this, one cn deine multiplictive metric spces s lterntive to the ordinry metric spces. In prticulr, R + is multiplictive metric spce nd sequence {x n } in R +, converges to x R + in the multiplictive sense i or every ε>1, there exists N N such tht d x n,x)<εor every n>n. In ct, the convergence in R + in both multiplictive nd ordinry senses re equivlent. But they my be dierent in more generl cses. Another multiplictive metric spce cn be deined on the bse o the collection M + n o positive n n)-mtrices. An n n)-mtrix A is sid to be positive i x T Ax > 0 or every n-vector x, where x T is the trnspose o x. Iλ 1,...,λ n re eigenvlues o positive n n)-mtrix A, then they re positive numbers. One cn deine multiplictive norm o A s n A = λ i, i=1 nd the multiplictive distnce between A nd B s d A, B) = AB 1.
A.E. Bshirov et l. / J. Mth. Anl. Appl. 337 2008) 36 48 45 4.4. Multiplictive dierentil equtions It is nturl to cll the eqution y x) = x,yx) ), 9) contining the derivtive o y, smultiplictive dierentil eqution. This eqution hs sense i is positive unction deined on some subset G o R R +. To give theorem on existence nd uniqueness o its solution, stisying the condition yx 0 ) = y 0, 10) we need the nlog o the Lipschitz condition in multiplictive cse. Theorem 1 tells us tht such condition or should hve the orm x, y), x, z) G, x,y) x,z) L y z, 11) where L>1 is constnt. In prticulr, i hs prtil derivtive y in its second vrible nd y is bounded in the multiplictive sense, i.e., there exists M>1such tht x, y) G, y x, y) M, then x,y) stisies the multiplictive Lipschitz condition. Theorem 7 Multiplictive Dierentil Eqution). Let be continuous unction on the bounded open region G in R R + to, b), where 0 <<b<. Assume tht stisies the multiplictive nlog o the Lipschitz condition given in 11). Tkex 0,y 0 ) G. Then there exists ε>0 such tht Eq. 9) hs unique solution y : x 0 ε, x 0 + ε) R + stisying the condition 10). Proo. The multiplictive dierentil eqution 9) cn be trnsormed to the ollowing ordinry dierentil eqution y x) = yx)ln x,yx) ). 12) Let us show tht the unction Fx,y) = y ln x,y), x, y) G, stisies the ordinry Lipschitz condition in y, Fx,y) Fx,z) y ln x,y) y ln x,z) + y ln x,z) zln x,z) x,y) y ln x,z) + ln x,z) y z = y ln x,y) x,z) + ln x,z) y z y ln L + ln x,z) ) y z. Here ln is bounded since is bounded in the multiplictive sense. Also, y rnges in bounded set. Hence, F stisies the ordinry Lipschitz condition in y. We conclude tht Eq. 12) hs unique solution stisying the condition 10). This implies the conclusion o the theorem.
46 A.E. Bshirov et l. / J. Mth. Anl. Appl. 337 2008) 36 48 4.5. Multiplictive clculus o vritions Consider the problem o minimizing the unctionl Jy)= yx),y x) ) dx over ll continuously dierentible unctions yx) on [,b] with ixed end points y) = y 1 nd yb) = y 2. By the ormul 6), this problem is equivlent to the minimiztion o J 0 y) = ln Jy)= ln yx),y x) ) dx, or which the methods o clculus o vritions cn be used. Nevertheless, we re insisting to use multiplictive methods. At irst, let us consider the ollowing multiplictive nlog o the undmentl lemm o clculus o vritions. Lemm 1. I : [,b] R is positive continuous unction so tht x) hx) ) dx = 1 13) or every ininitely mny times dierentible unction h : [,b] R, then x)= 1 or every x b. Proo. From 1 = we obtin x) hx) ) dx = e hx) ln x)dx, hx) ln x)dx= 0. By the undmentl lemm o clculus o vritions, ln x)= 0orx)= 1 or every x b. Turning bck to the unctionl 13), ssume tht y,y ) hs the second-order continuous prtil derivtives in y nd y.lethx) be rbitrry continuously dierentible unction on [,b] stisying h) = hb) = 0 nd let ε R. Then Jy+ εh) 1 = yx) + εhx), y x) + εh x)) dx Jy) b yx),y x)) dx yx) + εhx), y x) + εh ) x)) dx = yx),y x))
A.E. Bshirov et l. / J. Mth. Anl. Appl. 337 2008) 36 48 47 = o ε) = o ε) b y yx),y x) ) εhx) y yx),y x) ) εh x)) dx y yx),y x) ) hx) y yx),y x) ) ) ε h x)) dx, where o ε) 1/ε 1iε 0. Rising both sides o the obtined inequlity to the power 1/ε nd consequently, moving ε to 0+ nd 0, one cn obtin y yx),y x) ) hx) y yx),y x) ) h x)) dx = 1 or y yx),y x) ) b hx)) dx y yx),y x) ) h x)) dx = 1. By multiplictive integrtion by prts ormul, y yx), y x)) hx) ) dx dx d y yx), y x)) hx) ) = 1 dx or y yx), y x)) d dx y yx), y x)) ) hx) ) dx = 1. By Lemm 1, y yx), y x)) d dx y yx), y x)) = 1 or y yx),y x) ) = d dx y yx),y x) ). 14) This is n nlog o the Euler Lgrnge eqution in the multiplictive cse. As n exmple, consider problem o minimizing Jy)= 1 0 y x) 2) dx over ll continuously dierentible unctions y on [0, 1] stisying y0) = λ nd y1) = μ.here y,y ) = y ) 2. To pply the ormul 14), we clculte y y, y ) = 1, y y, y ) = e 2 y nd d dx y y, y ) = e d dx 2 y ) = e 2y y ) 2.
48 A.E. Bshirov et l. / J. Mth. Anl. Appl. 337 2008) 36 48 By 14), we deduce 2y + y ) 2 = 0. This is Riccti eqution in y nd, hence, y x) = 2c 1 2 + c 1 x, implying yx) = 2c 1 ln2 + c 1 x) + c 2. 15) Here the constnts c 1 nd c 2 re determined rom the system o equtions { 2c1 ln 2 + c 2 = λ, 16) 2c 1 ln2 + c 1 ) + c 2 = μ. In prticulr, the cse λ = μ implies c 1 = 0 nd c 2 = λ, i.e., the unction t which 14) tkes its minimum is the constnt unction yx) = λ on [0, 1]. Otherwise, it is the unction 15) with c 1 nd c 2 deined rom 16). 5. Concluding remrk Assume ϕ is bijective unction o one vrible. Deine derivtive nd integrl o the unction by x) = ϕ ϕ 1 ) b x) nd x)d x = ϕ b ϕ 1 ) ) x) dx, where we ssume tht the rnge o is subset o the rnge o ϕ. On the bse o these deinitions one cn develop clculus. In ct, there re ininitely mny clculi. The multiplictive clculus is one o them. We reer to Grossmn nd Ktz [1] or other interesting clculi. Reerences [1] M. Grossmn, R. Ktz, Non-Newtonin Clculus, Lee Press, Pigeon Cove, MA, 1972. [2] D. Stnley, A multiplictive clculus, Primus IX 4) 1999) 310 326.