A New Approach to Infinite Matrices of Interval Numbers

Transcription

1 Global Joural of Pure ad Applied Mathematics ISSN Volume 14, Number 3 (2018), pp Research Idia Publicatios A New Approach to Ifiite Matrices of Iterval Numbers Zarife Zararsız Nevşehir Hacı Betaş Veli Uiversity, Faculty of Art ad Sciece, Departmet of Mathematics, 50300, Nevşehir, Turey Abstract I this paper, Cesàro iterval ull, Cesàro iterval coverget ad Cesàro iterval bouded sequece spaces of iterval umbers are itroduced ad proved some iclusio relatios o them Additioally, a isomorphism is costructed o these iterval sequece spaces Furthermore, M(u), Y (u) ad O(u) symbols ad some properties of these otios are ivestigated Besides, defiitio of ifiite dimesioal Cesàro iterval matrix ad its left ad right parts are itroduced Moreover, a useful compariso is give betwee classical Cesàro matrix trasformatio ad Cesàro iterval matrix trasformatio Fially, completio of iterval metric spaces is give AMS subject classificatio: 03E72, 46A45, 40C05 Keywords: Iterval umber, iterval sequece space, iterval Cesàro matrix, isomorphism, completio of iterval metric spaces 1 Itroductio Iterval arithmetic was itroduced by Dwyer [7] I, [5], Chiao established sequece of iterval umbers ad gave the defiitio of usual covergece of sequece of iterval umbers Bouded ad coverget sequece spaces of iterval umbers are studied by Ṣegöül ad Eryılmaz I recet years, Esi [8] itroduced lacuary sequece spaces of iterval umbers Hase ad Smith [9] mae matrix calculatios by meas of iterval arithmetic, firstly After, may others such as Neumaier [11], Jauli et al [10] ad Roh [13], etc have wored o iterval matrices It is clear that matrices have a importat

2 486 Zarife Zararsız place i various fields such as mathematics, egieerig ad statistics Furthermore, itervals are geeralizatio of real umbers Therefore, arithmetic operatios o itervals are a geeralizatio of the arithmetic operatios defied o the real umber set I additio, itervals will be see as a bridge betwee fuzzy ad classical sets To build a ew sequece space by meas of the matrix domai of a certai limitatio method, may studies were employed by may authors, for example you ca see: Altay, Başar ad Mursalee [2], Başar ad Altay [3], [4], Ng ad Lee [12] ad Wag [15] The rest of our paper is orgaized, as follows: I Sectio 2, some basic defiitios ad theorems related with the iterval umbers are give Also, defiitios of iterval metric metric space, sequece of iterval umbers, iterval Cauchy sequece are give I Sectio 3, we have itroduced Cesàro iterval ull, Cesàro iterval coverget ad Cesàro iterval bouded sequece spaces of iterval umbers as the set of all sequeces such that C-trasforms of them are i the spaces c i, c0 i ad l i, respectively, by meas of Cesàro iterval matrix ad proved some iclusio relatios o these sequece spaces It is also established i Sectio 3 that the sequece spaces showed by E C 0, EC c ad EC b are liearly isomorphic to the spaces ci 0, ci ad l i, respectively Additioally, it is proved that the spaces E C 0, EC c ad EC b are complete metric spaces Furthermore, M(u), Y (u) ad O(u) symbols ad some properties are ivestigated Fially, i Sectio 3, iterval matrix otio ad algebraic structures o matrices are give Moreover, Cesàro iterval ad left, right Cesàro iterval matrices are itroduced ad a compariso is give betwee classical Cesàro matrix trasformatio ad Cesàro iterval matrix trasformatio I the fial Sectio 4, completio of iterval metric spaces is give 2 Prelimiaries, Bacgroud ad Notatio I this sectio, we recall some of the basic defiitios ad otios i the theory of iterval umbers ad sequece spaces such as otios of iterval metric space, algebraic operatios o E,(E, +) triple, iterval covergece ad iterval Cauchy sequece Let suppose that N, R ad E are the set of all o-egative itegers, all real umbers ad all bouded ad closed itervals o R, respectively Ay elemet of E is deoted by u ad called as iterval umber That is, E ={u = [u, u + ]:u, u + R, u u + } For u, v E, wehaveu = v u = v, u + = v + Additioally, we give the algebraic operatios additio, scalar multiplicatio ad multiplicatio as follows, respectively: + : E E E, +(u, v) = u + v = [u + v, u + + v + ], { [αu, αu + ], α 0 : R E E, αu = [αu +, αu ], α<0, : E E E, (u, v) = u v = [ mi R, max R], R ={u v, u v +, u + v, u + v + } We will use abbreviatio u 0 ɛ = [u 0 ɛ, u 0 + ɛ] = u 0 + ɛ[ 1, 1] where ɛ>0, i appropriate places

3 A New Approach to Ifiite Matrices of Iterval Numbers 487 By w, we deote the set of all complex valued sequeces w is a liear space with the defied additio ad scalar multiplicatio Additioally, each liear subspace of w is called a sequece space We show l, c ad c 0 for the classical sequece spaces of all bouded, coverget ad ull sequeces, respectively We ca give the most geeral liear operator betwee two sequece spaces by meas of ifiite matrices For this reaso, matrix trasformatios have a importat place i sequece space studies For brevity i otatio, through all the text, we shall write sup, ad lim N, sup, ad lim istead of Let λ ad µ be two sequece spaces ad A = (a ) be a ifiite matrix of real or complex umbers a, where, N The, we ca say that A defies a matrix mappig from λ to µ, ad we deote it by writig A (λ : µ), if for every sequece x = (x )isiλ ad the sequece Ax ={(Ax) }, the A- trasform of x, isiµ, where rus from 0 to The domai λ A of a ifiite matrix A i a sequece space λ is defied by, =0 λ A ={x = (x ) w : Ax λ} (1) which is a sequece space If we tae λ = c, the c A is called covergece domai of A We write the limit of Ax as A lim x = lim a x, ad the A is called regular if lim x = lim x for every x c Also, a matrix A = (a ) is called triagle if a = 0 A for >ad a = 0 for all N Defiitio 21 Let (E, +) be Abel mooid ad the followig coditios are satisfied for all u, v E ad for all α, β R : 1 α(u + v) = αu + αv 2 α(βu) = (αβ)u 3 [1, 1]u = u 4 (α + β)u = αu + βu, α, β 0 Here, θ = [0, 0] ad [1, 1] are idetity elemets of E accordig to additio ad multiplicatio operatios, respectively I this case, we ca say that (E, +) triple is called almost liear space o R Defiitio 22 The sequece which terms cosists of compact subsets of R is called the sequece of iterval umbers Let us show the set of all sequece spaces of iterval umbers by w(e), for N as writte below: w(e) ={u = ([u, u+ ]) : f, g : N R, f () = u, g() = u+, u u+ } =0

4 488 Zarife Zararsız Let λ w(e) ad d : λ λ R be a metric For u w(e) we ca give followig defiitios by meas of [5] 1 A sequece u = ([u, u+ ]) w(e) is coverget to [u 0, u+ 0 ] accordig to metric d if ad oly if for every ɛ>0there exists a 0 N such that for every 0, d([u, u+ ], [u 0, u+ 0 ]) <ɛ It is show by lim [u, u+ ] = [u 0, u+ 0 ]or[u, u+ ] [u0, u+ 0 ], 2 A sequece u = ([u, u+ ]) of iterval umbers is said to be Cauchy sequece if for every ɛ>0there exist a 0 N such that d([u, u+ ], [u, u+ ]) <ɛfor all, 0 It is easy to see that every coverget sequece of iterval umbers is a Cauchy sequece Defiitio 23 Let X be a o-empty iterval set ad d : X X E + be iterval metric fuctio The couple (X, d) is called iterval metric space, if 1 d(u, v) = 0 if ad oly if u = v, 2 d(u, v) = d(v, u) = 0 for all u, v X, 3 d(u, z) d(u, v) + d(v, z) I the followig, some special sub-sets of E will be give [14]: c i ={u = ([u, u+ ]) : lim [u, u+ ] = [u 0, u+ 0 ]} (2) c0 i ={u = ([u, u+ ]) : lim [u, u+ ] = [0, 0]} (3) l i ={u = ([u, u+ ]) : sup M([u, u+ ]) < } (4) We called these subsets as iterval coverget, ull iterval coverget ad iterval bouded sequece spaces, respectively If we tae u = u+ for all N the (2), (3) ad (4) are correspod to coverget, ull ad bouded sequece spaces of real umbers, respectively 3 Cesàro Iterval Matrices ad sequece spaces I this sectio, matrices which cosist of itervals are itroduced ad some properties of these matrices are examied Defiitio 31 Let us suppose that N ={0, 1, 2,, }, N m ={0, 1, 2,, m} ad f : N N m E

5 A New Approach to Ifiite Matrices of Iterval Numbers 489 (i, j) f (i, j) = a ij = a E,(1 i,1 j m) are give From here, we obtai iterval matrix A = [[aij, a+ ij ]] m that cosists of elemets a ij = [aij, a+ ij ] It is easy to see that if we tae a ij = a+ ij for 1 i ad 1 j m the [[aij, a+ ij ]] m is reduced to real matrices which has m dimesio We show the set of all iterval matrices with m dimesio as below: E m ={A : A is a iterval matrix with m dimesio} The matrix i the form showed as i the followig is called ifiite dimesioal iterval matrix [a11, a+ 11 ] [a 1m, a+ 1m ] [a21, a+ 21 ] [a 2m, a+ 2m ] A = [a1, a+ 1 ] [a m, a+ m ] Let A, B E m, λ R We give the algebraic structures of iterval matrices additio ad scalar multiplicatio as i the followig: A + B = [[a ij, a+ ij ]] m + [[b ij, b+ ij ]] m, { [[λa λu = ij, λa ij + ]] m, λ 0, [[λa ij +, λa ij ]] m, λ<0 Additioally, if we tae D E m r the multiplicatio of A D is defied as follows for 1 i,1 j r: where A D = [[a ij, a+ ij ]] m [[d ij, d+ ij ]] m r = [[c ij, c+ ij ]] r m cij = mi{ai d j, a i d+ j, a+ i d j, a+ i d+ j } =1 m c ij + = max{ai d j, a i d+ j, a+ i d j, a+ i d+ j } =1 Theorem 32 Let us suppose that A, B, C E m The, + operatio defied + : E m E m E m satisfies the followig properties: 1 A + B = B + A

6 490 Zarife Zararsız 2 (A + B) + C = A + (B + C) 3 A + θ = A Here θ = [[θ ij, θ + ij ]] = [[0, 0]] E m,1 i,1 j m Defiitio 33 A = [[aij, a+ ij ]] m be a iterval matrix of iterval umbers If there exists B = [[bij, b+ ij ]] m such that [bij, b+ ij ] [a ij, a+ ij ] for all i, j N, the we called B by sub-matrix of A ad showed by B A Because of the fact that every real umber is a iterval which has same first ad ed terms as metioed i Defiitio 33, it is clear that each fiite dimesioal matrix of real umbers is sub-matrix of a iterval matrix Theorem 34 A, B, C ad D be iterval matrices with type ad A = [a ij ] be real matrix with m type The, followig features are obtaied easily: 1 If C B ad D A the CD BA, 2 A(B + C) AB + AC, 3 A(A + B = AA + AB Iterval matrices of type 1 are called vectors of iterval umbers The iterval matrices of type m ca be expaded to ifiite dimesioal itervals of type as metioed i the followig defiitio Defiitio 35 Now, let us tae u = ([u, u+ ]) = ([u 1, u+ 1 ], [u 2, u+ 2 ],,[u, u+ ], ) be a sequece of iterval umbers If series i the form [a, a+ ][u, u+ ] are coverget for all N the Au = v is called as A trasformatio of iterval sequece u I additio this, if we choose a = a+ ad u = u+ for all, N, i this case the equatio [a 1 [a 11, a+ 11 ] [a 1m, a+ 1m ] [u [a 21, a+ 21 ] [a 2m, a+ 2m ] 11, u+ 11 ], a+ 1 ][u, u+ ] [u 21, u+ 21 ] [a 2, a+ 2 ][u, u+ ] [a 1, a+ 1 ] [a m, a+ m ] [u, u+ ] = [a, a+ ][u, u+ ] 1 is reduced to matrix trasformatios of sequeces i the classical sese 1

7 A New Approach to Ifiite Matrices of Iterval Numbers 491 Defiitio 36 A be a ifiite dimesioal iterval matrix i type The, orm of the matrix A is give as below: A = sup M([aij, a+ ij ]) i where M([a ij, a+ ij ]) = max{ a ij, a+ ij } j Defiitio 37 We give geeralized defiitio of Cesàro matrix by meas of iterval umbers for all, N Here, first of all we itroduce Cesàro iterval matrix as follows: [ C = (C ) = + 1, 1 ],, + 1 [0, 0], otherwise After, we divide Cesàro iterval matrix ito two parts amed left Cesàro iterval matrix ad right Cesàro iterval matrix We showed left ad right Cesàro iterval matrices by C ad C +, respectively, as give i the followig: C = (C ) = [, 0],, + 1 [0, 0], otherwise 1 C + = (C + ) = [0, ],, + 1 [0, 0], otherwise Norm of the matrices C, C + ad C is equal to 1 It meas that C = C + = C = 1 Defiitio 38 The matrix A = { [a, a + ],, [0, 0], otherwise is defied as the lower triagular iterval matrix of type Additioally, if there is o elemet o the pricipal diagoal, the A is called ormal iterval matrix A umber of rules ad properties ca be exteded to itervals i Summability Theory Let r = (r ) be a sequece of real umbers We say that [L, L + ] iterval umber is A iterval limit of the sequece (r ) if ad oly if the sequece (Ar) = [a, a+ ]r is coverget to [L, L + ] Iterval matrices ca be used to determie the rage of the limit, whe there is difficulty i the determiatio phase of the limit of a reel valued, bouded but ocoverget sequece

8 492 Zarife Zararsız Cesàro limit of the real valued but o-coverget sequece ( 1 ) is equal to 0 C, C + Cesàro limits of the sequece ( 1 ) is give as i the followig: [ lim (C r) = lim = 2, 1 ],, 2 ( + 1) [, ], otherwise [ lim (C + r) = lim 2, 1 ], if is eve, 2 ( + 1) [, ], if is odd [ The result of both limits is equal to 2, 1 ] Limits are coicidet If we calculate 2 the C- limit of the sequece ( 1) with a straightforward calculatio it ca be writte as a result that lim (Cr) = [ 1, 1] I spite of the fact that there are three itervals [ which iclude 0, Cesàro limit of the sequece ( 1), the iterval 2, 1 ] is arrower 2 tha [ 1, 1] Taig accout this example we give the followig theorem for explai our idea Theorem 39 If there exists both left ad right Cesàro limits of a sequece (r ), it meas that lim (C r) = [L 1, L+ 1 ] ad lim (C+ r) = [L 2, L+ 2 ] are preset the lim (C r) + lim (C + r) = lim (Cr) = [L 1 + L 2, L+ 1, L+ 2 ] Proof Whe matrix additio is cosidered proof is obtaied easily A geeralizatio of Theorem 39 will be give as below Theorem 310 Let A ad B be ifiite matrices of itervals I this case, 1 If r = (r ) is a sequece of real umbers the lim ((A + B)r) = lim (Ar) + lim (Br) 2 lim ((A + B)u) = lim (Au) + lim (Bu) where u is a sequece of iterval umbers Proof Proof is clear from ([11], Page 79 Propositio 312) Now, we give the theorem about regularity of iterval ifiite dimesioal matrices which is expressed ad proved by [6] [ ] Theorem 311 The matrix A = a, of iterval umbers is regular for every = 1, 2, if ad oly if it satisfies the followig statemets: =1 =1 a +

9 A New Approach to Ifiite Matrices of Iterval Numbers There exists M>0 such that [ 2 lim a, =1 =1 [ 3 lim a, =1 =1 a + a + ] ] a M ad a + M =1 =1 = [0, 0] for every = 1, 2, = [1, 1] From here, it is easy to see that iterval Cesàro matrix is regular 31 The Cesàro Iterval Sequece Spaces E C 0, EC c ad EC b I this sectio, we wish to itroduce the Cesàro iterval ull, Cesàro iterval coverget ad Cesàro iterval bouded sequece spaces of iterval umbers represeted by E C 0, EC c ad E C b, as the set of all sequeces such that C-trasforms of them are i the spaces ci 0, ci ad l i, respectively, that is ad E C 0 ={u = ([u, u+ ]) w(e) :Cu c i 0 } (5) E C c ={u = ([u, u+ ]) w(e) :Cu c i } (6) E C b ={u = ([u, u+ ]) w(e) :Cu l i } (7) It is easy to see that Cesàro iterval matrix is lower triagular ad regular iterval matrix of type Now, we defie the sequece of iterval umbers v = ([v, v+ ]) which will be frequetly used, as the C- trasform of a sequece of iterval umbers u = ([u, u+ ]) for, N, ie [ ] [v, v+ ] = + 1, 1 [uj + 1, u+ j ] (8) j=0 Theorem 312 The sequece spaces E C 0, EC c ad EC b are liearly isomorphic to the spaces c0 i, ci ad l i, respectively, ie EC 0 = c0 i, EC c = c i ad E C b = l i Proof We cosider oly the case E C b = l i sice others ca be doe i the same way First of all, we should show the existece of a liear bijectio betwee the spaces E C b ad l i Cosider the trasformatio defied T, with the otatio of (8) by T : EC b [ ] l i, u v = T u = + 1, 1 [uj + 1, u+ j ] The followig equatios hold for T : j=0

10 494 Zarife Zararsız 1 T (u + v ) = T u + T v 2 T (αu ) = [ ] + 1, j=0 [ ] + 1, j=0 where u, v E C b Thus, T is liear [αu j, αu+ j ] = α [ + 1, 1 [αu + j, αu j ] = α [ + 1, Let v l i ad defie the sequece u = ([u, u+ ]) by From here, we ca write sup N d(cu, θ) = sup d ] [u j + 1, u+ j ] = αt u, for α 0 j=0 ] [u j, u+ j ] = αt u, for α<0 j=0 [u, u+ ] = [v, v+ [ ] ] [v, v+ [ ] +1, 1 +1, 1 ],( N) = sup d ( [ + 1, [ + 1, = sup d([v, v+ ], θ) < ] ) [u, u+ ], θ =0 ] =0 [v, v+ [ ] ] [v, v+ [ ] +1, 1 +1, 1 ], θ which obtai from the above equatio is u E C b Furthermore, [ ] u E C = sup d b + 1, 1 [v, v+ [ ] ] [v, v+ [ ] + 1 =0 +1, 1 +1, 1 ], θ = sup d([v, v+ ], θ) = v l i < Namely, T is orm preservig As a result, the spaces E C b ad li are liearly isomorphic It is clear that if the spaces E C b ad li are replaced by the spaces EC 0 ad ci 0, EC c ad ci, we obtai the fact that E C 0 = c0 i ad EC c = c i This completes the proof Let us defie the sequece c ={c () } N of the space E C c as below: { c () [( 1) = ( + 1), ( 1) ( + 1)], 1 [0, 0], 0 1or>

11 A New Approach to Ifiite Matrices of Iterval Numbers 495 for every fixed N The we ca say that the sequece c is a basis for the space E C c ad every u E C c has a uique represetatio of the form u = µ c where µ = (Cu) for all N Theorem 313 The sets (E C c, d), (EC 0, d) ad (EC b, d) are complete metric spaces with the metric defied by sup max { c u =1 =1 c v, c + u+ =1 =1 c + v+ } (9) Proof It was see that i Theorem 312, the sequece spaces of iterval umbers (E C c, d), (EC 0, d) ad (EC b, d) are liearly isomorphic to the spaces ci, c0 i ad li, respectively Additioally, sice the Cesàro iterval matrix is ormal (see, [16]) ad c i, c0 i ad li are complete ormed sequece spaces, it is clear that the sequece spaces (E C c, d), (EC 0, d) ad (EC b, d) are complete metric spaces with the metric defied i (9) Theorem 314 The iclusios E C 0 EC c EC b strictly hold Proof It is clear that E C 0 EC c Additioally, if we tae the sequece (( 1) ) = ([( 1),( 1) ]) the C(( 1) ) E C c but C(( 1) ) / E C 0 Now, we show EC c EC b Let u E C c the Cu c i l i Namely, u E C b It meas that EC c E C b Furthermore, if we tae ito accout the sequece ( ) ( ) C ( 1) E Cb + 1 but C ( 1) + 1 the theorem ( ( 1) + 1 ) it is easy to see that / E C c This step completes the proof of Theorem 315 The iclusios c i 0 EC 0, ci E C c ad li EC b strictly hold Proof Because of the fact that C is regular iterval matrix, iclusios c0 i EC 0 ad c i E C c are clear Besides, let us tae the sequece u = ([u, u+ ]) = ([(),() ]), the for every N we ca write that u / c i but u E C c Hece, iclusio ci E C c strictly holds Let us show that u = ([u, u+ ]) li From here, we ca say that there is a elemet of R i the form M>0such that sup{ u, u+ } <Mie u <Mad

12 496 Zarife Zararsız u + <Mfor all N That is, we ca write the followigs for each N: [ ] C(u ) = c u, c + u+ =1 =1 { } = max c u, c + u+ max = M =1 =1 { } c M, c + M =1 =1 Cosequetly, we coclude the iclusio l i EC b Some properties of the sets E C c, EC 0 ad EC b, defied above, are listed below Their proof ca be structured as if it were i real-valued sequeces Propositio 316 Let us suppose that α R, lim [u, u+ ] = [u 0, u+ 0 ] ad lim [v, v+ ] = [v0, v+ 0 ] The we ca write the followigs: 1 If equatio lim [u, u+ ] = [u 0, u+ 0 ] hold, the [u 0, u+ 0 ] is oe ad oly 2 lim ([u, u+ ] + [v, v+ ]) = lim [u, u+ ] + lim [v, v+ ] = [u 0, u+ 0 ] + [v 0, v+ 0 ] 3 lim α[u, u+ ] = α lim [u, u+ ] = α[u 0, u+ 0 ] [u 4 lim, u+ ] [v, v+ ] = lim [u, u+ ] lim [ 1 v + N,0 / v ), 1 v ] = [u0, u+ 0 ][ 1 v 0 +, 1 v 0 ], (For every 5 Let us suppose that a = mi{lim u v, lim u v+, lim u+ v, lim u+ v+ } ad b = max{lim u v, lim u v+, lim u+ v, lim u+ v+ } The it is clear that lim ([u, u+ ][v, v+ ]) = [a, b] 6 E C 0 EC c EC B hold Defiitio 317 A sequece space W of iterval umbers is called solid if y = ([y, y+ ]) W wheever y x for all N ad x = ([x, x+ ]) W Theorem 318 The spaces E C 0, EC c ad EC b are solid

13 A New Approach to Ifiite Matrices of Iterval Numbers 497 Proof Let y x for all N ad for some x E C 0 From here, it is easy to see that d(y, θ) d(x, θ) Namely, we have y x ady+ x + So, y E C 0 Cosequetly, E C 0 is solid The others ca be proved i similar way 32 M(u), Y(u), O(u) Symbols ad Some Properties Defiitio 319 Let us tae u = [u, u + ] E Now, we give the followig defiitios with [1] The real umbers B(u) = u + u, M(u) = max{ u, u + } ad O(u) = 1 2 (u+ u ) are called size of u, orm of u ad middle poit of u, respectively Additioally, by cosiderig the set B(u) it is clear that the set of real umbers will be writte by R ={u = [u, u + ]:B(u) = 0} Defiitio 320 Let u, v E Ifu v ad u + v +, the u is said to be less tha v This ca be showed by u v It is obvious that this sort of order ca ot compare ay elemet of E For istace, we ca ot compare the iterval umbers u = [0, 8] ad v = [ 1, 9] Shortly, if u v or v u, it is ot possible to determie which of u ad v iterval umbers is smaller, by usig Defiitio 320 A secod way to compare itervals is give as follows Let S ={u i = [ui, u+ i ]: i N} E be give For each i N, legth of u i is equal to B(u i ) = u i + ui I this case, if B(v i ) B(u i ) for v i E for all i N the S is said to be bouded below I the same way, if B(u i ) B(w i ) for w i E for all i N the S is said to be bouded above Defiitio 321 Let [ui, u+ i ] E,1 i,( N) ad B(u i) = u i + case, if max B(u i ) = M for i =, the max u i = u i i u i I this I Defiitio 321, legth of iterval is cosidered i order to choose the largest size of itervals istead of ed poits of itervals With this sort of order, the problems that occur i the order give by usig the ed poits of itervals, will be removed Additioally, we have the possibility to compare each elemet of E 4 Completio of Iterval Metric Spaces Defiitio 41 Let (X, d 1 ) ad (Y, d 2 ) be iterval metric spaces A mappig T of X to Y is called iterval isometry if for all u, v X, d 2 (Tx, Ty) = d 1 (u, v) I additio this, X is cosidered as iterval-isometric to the space Y, if there exists a bijective iterval isometry betwee the spaces X ad Y The, X ad Y are said to be iterval-isometric spaces Theorem 42 Let ( X, d) be a complete iterval metric space Every iterval metric space (X, d) has a completio X which has a subspace Z dese i X ad there exists a isometry betwee X ad X

14 498 Zarife Zararsız Proof Let (X, d) be a iterval metric space ad u = ([u, u+ ]) ad v = ([v, v+ ]) are Cauchy sequeces i X Defie a relatio betwee u ad v as i the followig: u v lim d(u, v ) = θ This clearly defies a equivalece relatio I fact: 1 lim d(u, u ) = max{ u u, u+ u+ } = 0 u u, 2 u v lim d(u, v ) = 0 = lim d(v, u ) v u, 3 Sice lim d(u, v ) lim d(u, z ) + lim d(z, v ), u z ad z v meas that u v Let X be the set of all equivalece classes of iterval Cauchy sequeces It meas that Now, we defie d : X X E + by X ={ũ : u is a Cauchy sequece i X} d(ũ, ṽ) = lim d(u, v ) (10) for ũ, ṽ X Let u 1 ad v 1 be two Cauchy sequece of X such that u u 1 ad v v 1 The lim d(u, u 1 ) = lim d(v, v 1 ) = θ ad By taig ito cosideratio the triagle iequality, we coclude From here, we ca write, d(u, v ) d(u, u 1 ) + d(u 1, v 1 ) + d(v 1, v ) d(u 1, v 1 ) d(u 1, u ) + d(u, v ) + d(v, v 1 ) d(u, v ) d(u 1, v 1 ) d(u, u 1 ) + d(v, v 1 ) 0 Because of the fact that (d(u, v )) ad (d(u 1, v 1 )) are coverget d is well-defied Now, we prove that d i (10) is a iterval metric o X I fact, 1 d(ũ, ũ) = lim d(u, u ) = 0, 2 d(ũ, ṽ) = lim d(u, v ) = lim d(v, u ) = d(ṽ, ũ), 3 d(ũ, ṽ) = lim d(u, v ) lim d(u, z ) + lim d(z, v ) = d(ũ, z) + d( z, ṽ)

15 A New Approach to Ifiite Matrices of Iterval Numbers 499 Cosequetly, d is a metric o X With each u X, we costruct the class ũ = ([u, u + ], [u, u + ], ) X, the equivalece classes of the costat sequece ([u, u + ], [u, u + ], ) This deduce a mappig T : X X with T (u) =ũ The, for ay u, v X d(t (u), T (v)) = d(ũ, ṽ) = lim d(u, v) = d(u, v) It meas that T is a isometry from X ito X Now, let us show the deseess of Z X Supposig that ũ X, ε>0 ad u is a member of ũ Because of the fact that u is iterval Cauchy sequece there exists a 0 N such that for ay m, N, d(u m, u ) < ε 2 Cosider the costat Cauchy sequece un = (u N, u N, ) ad ũ N be its equivalece class Sice, d(ũ, ũ N ) = lim d(u, u N ) = lim d(u, u N ) ε 2 <ε Hece, ũ N Z Thus, Z is dese i X The completeess of X is show as below: Let ũ be ay Cauchy sequece i X Sice Z is dese X, there exists z Z for every ũ as follows: d(ũ, z ) < 1 (11) From here, we have by the triagle iequality d( z m, z ) d( z m, ũ m ) + d(ũ m, ũ ) + d(ũ, z ) < 1 m + d(ũ m, ũ ) + 1 <ε Sice ũm is Cauchy, z m is also Cauchy ad T : X Z is isometric, z m Z, the the sequece z m, where z m = T z m, is Cauchy i X Now, let us show that ũ is the limit of ũ By usig (11) we obtai d(ũ, ũ) d(ũ, z ) + d( z, ũ) < 1 + d( z, ũ) (12) Because of the fact that z m ũ ad (z, z, ) z Z, (12) tur ito d(ũ, ũ) < 1 + lim m d(z, z m ) <ε Hece, arbitrary Cauchy sequece ũ X has the limit ũ X From here, X is complete Refereces [1] G Alefeld ad J Herzberger, Itroductio to Iterval Computatios, 1-352, Academic Press, USA (1983) [2] B Altay, F Başar ad M Mursalee, O the Euler sequece spaces which iclude the spaces l p ad l, Iform Sci, 176, o:10, (2006)

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