Some spaces of sequences of interval numbers defined by a modulus function

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1 Global Journal o athematical Analysis, c Science Publishing Corporation wwwsciencepubcocom/inexphp/gja oi: /gjmav2i12005 Research Paper Some spaces o sequences o interval numbers eine by a moulus unction Ayhan Esi, Sibel Yasemin Aiyaman University, Science an Art Faculty, Department o athematics, 02040, Aiyaman, Turey *Corresponing author aesi23@hotmailcom Copyright c 2014 Ayhan Esi, Sibel Yasemin This is an open access article istribute uner the Creative Commons Attribution License, which permits unrestricte use, istribution, an reprouction in any meium, provie the original wor is properly cite Abstract The main purpose o the present paper is to introuce c o, p, s, c, p, s l, p, s an l p, p, s o sequences o interval numbers eine by a moulus unction Furthermore, some inclusion theorems relate to these spaces are given Keywors: Complete space, interval number, moulus unction 1 Introuction Interval arithmetic was irst suggeste by Dwyer 8] in 1951 Development o interval arithmetic as a ormal system an evience o its value as a computational evice was provie by oore 11] in 1959 an oore an Yang 13] 1962 Furthermore, oore an others 9], 10], 11] an 14] have evelope applications to ierential equations Chiao in 6] introuce sequence o interval numbers an eine usual convergence o sequences o interval number Şengönül an Eryilmaz in 7] introuce an stuie boune an convergent sequence spaces o interval numbers an showe that these spaces are complete metric space Recently, Esi in 1], 2], 3], 4] an 5] eine an stuie ierent properties o interval numbers We enote the set o all real value close intervals by IR Any elements o IR is calle interval number an enote by A = x l, x r ] Let x l an x r be irst an last points o x interval number, respectively For A 1, A 2 IR, we have A 1 = A 2 x 1l =x 2l,x 1r =x 2r A 1 + A 2 = {x R : x 1l + x 2l x x 1r + x 2r },an i α 0, then αa = {x R : αx 1l x αx 1r } an i α < 0, then αa = {x R : αx 1r x αx 1l }, A 1 A 2 = {x R : min {x 1l x 2l, x 1l x 2r, x 1r x 2l, x 1r x 2r } x max {x 1l x 2l, x 1l x 2r, x 1r x 2l, x 1r x 2r }} The set o all interval numbers IR is a complete metric space eine by A 1, A 2 = max { x1l x 2l, x 1r x 2r } 12] In the special case A 1 = a, a] an A 2 = b, b], we obtain usual metric o R Let us eine transormation : N R by = A, A = A Then A = A is calle sequence o interval numbers The A is calle th term o sequence A = A w enotes the set o all interval numbers with real terms an the algebraic properties o w can be oun in 14] Now we give the einition o convergence o interval numbers:

2 12 Global Journal o athematical Analysis Deinition 11 6] A sequence A = A o interval numbers is sai to be convergent to the interval number xo i or each ε > 0 there exists a positive integer o such that A, A o < ε or all o an we enote it by A = A o Thus, A = A o A l = A ol an A r = A or We recall that moulus unction is a unction : 0, 0, such that a x = 0 i an only i x = 0, b x + y x + y or all x, y 0, c is increasing, is continuous rom the right at zero It ollows rom a an that must be continuous everywhere on 0, Let p = p be a boune sequence o strictly positive real numbers I H = sup p, then or any complex numbers a an b a + b p C a p + b p 1 where C = max1, 2 H 1 Deinition 12 A set o X sequences o interval numbers is sai to be soli or normal i B X whenever B, 0 A, 0 or all N, or some A X In this paper, we essentially eal with the metric spaces c o, p, s, c, p, s, l, p, s an l p, p, s o sequences o interval numbers eine by a moulus unction which are generalization o the metric spaces c o, c, l an l p o sequences o interval numbers We state an prove some topological an inclusion theorems relate to those sets 2 ain results Let be a moulus unction an s 0 be a real number an p = p be a sequence o strictly positive real numbers We introuce the sets o sequences o interval numbers as ollows: c o, p, s = {A = A : s A, 0 ] } p = 0, c, p, s = l, p, s = an l p, p, s = {A = A : s A, A o ] p = 0 }, { A = A : { A = A : sup s A, 0 ] } p < s A, 0 ] } p < Now, we may begin with the ollowing theorem Theorem 21 The sets c o, p, s, c, p, s, l, p, s an l p, p, s o sequences o interval numbers are close uner the coorinatewise aition an scalar multiplication Proo It is easy, so we omit the etail Theorem 22 The sets c o, p, s, c, p, s, l, p, s an l p, p, s o sequences o interval numbers are complete metric spaces with respect to the metrics A, B = sup s ] p A, A o

3 Global Journal o athematical Analysis 13 an { p A, B = s ] A, B p } 1 respectively, where A = A an B = B are elements o the sets co, p, s, c o, p, s, l, p, s an l p, p, s an = max 1, sup p = H Proo We consier only the space c o, p, s, since the proo is similar or the spaces c, p, s, l, p, s an l p, p, s One can easily establish that eines a metric on c o, p, s which is a routine veriication, so we omit it It remains to prove the completeness o the space c o, p, s Let A i be any Cauchy sequence in the space c o, p, s, where A i = A i o, A i 1, A i 2, Then, or a given ε > 0 there exists a positive integer n o ε such that A i, A j = sup s A i ], A j p < ε 2 or all i, j > n o ε We obtain or each ixe N rom 2 that s A i ], A j p < ε or all i, j > n o ε 3 means that i,j s i,j A i ], A j p = 0 Since s 0 or all N an is continuous, we have rom 4 that ] A i, A j = Thereore, since is a moulus unction one can erive by 5 that A i, A j = 0 6 i,j which means that is a Cauchy sequence in IR or every ixe N Since IR is complete, it converges, say A i A i A as i Using these ininitely many its, we eine the interval sequence A = Ao, A 1, A 2, Let us pass to it irstly as j an nextly taing supremum over N in 3 we obtain A i, A ε Since s A i c o, p, s or each i N, there exists o N such that A i p, 0] < ε or every o ε an or each ixe i N Thereore, since s A, 0 ] p C s A i, A ] p + C s ] A i p, 0 hol by triangle inequality or all i, N, where C = max1, 2 H 1 Now or all o ε, we have s A, 0 ] p 2ε This shows that A co, p, s Since complete A i was an arbitrary Cauchy sequence, the the space c o, p, s is Theorem 23 The spaces c o, p, s, l, p, s an l p, p, s are soli

4 14 Global Journal o athematical Analysis Proo Let X, p, s enotes the anyone o the spaces c o, p, s, l, p, s an l p, p, s Suppose that B, 0 A, 0 7 hols or some A X, p, s Since the moulus unction is increasing, one can easily see by 7 that s B, 0 ] p s A, 0 ] p, sup s B, 0 ] p sup s A, 0 ] p an s B, 0 ]p s A, 0 ]p The above inequalities yiel the esire that B X, p, s Theorem 24 Let in p = h > 0 Then a A c implies A c, p, s, b A c p, s implies A c, p, s, c β = t t t > 0 implies c p, s = c, p, s Proo a Suppose that A c Then A, A o = 0 As is moulus unction, then A, A o = A, A o ] = 0 = 0 As in p = h > 0, then A, A o ] h = 0 So, or 0 < ε < 1, o such that or all > o A, A o ] h < ε < 1, an as p h or all, A, A o ] p A, A o ] h < ε < 1, then we obtain ] p A, A o = 0 As s is boune, we can write s ] p A, A o = 0 Thereore A c, p, s b Let A c p, s, then s A, A o p = 0 Let ε > 0 an choose δ with 0 < δ < 1, such that t < ε or 0 t δ Now we write I ı = { N : A, A o δ } an I 2 = { N : A, A o > δ } For A, A o > δ A, A o < A, A o δ 1 < 1 + A, A o ] where I 2 an t ] enotes the integer o t By using properties o moulus unction, or A, A o > δ, we have A, A o ] < 1 + A, A o ] A, A o δ 1

5 Global Journal o athematical Analysis 15 For A, A o δ, A, A o ] < ε, where I1 Hence s ] p A, A o = s ] p I1 A, A o + s ] p I2 A, A o s ε H δ 1] H s A, A o ] p 0 as Then we obtain A c, p, s c In b, it was shown that c p, s c, p, s We must show that c, p, s c p, s For any moulus unction, the existence o positive it given by β in aox16, Proposition 1] Now, β > 0 an let A c, p, s As β > 0 or every t > 0, we write t βt From this inequality, it is easy seen that A c p, s Theorem 25 Let an g be two moulus unctions an s, s 1, s 2 0 Then a c, p, s c g, p, s c + g, p, s, b s 1 s 2 implies c, p, s 1 c, p, s 2 Proo a Let A c, p, s c g, p, s From 1, we have + g A, A o ] p = A, A o + g A, A o ] p C A, A o ] p + C g A, A o ] p As s is boune, we can write s + g A, A o ] p C s A, A o ] p + C s g A, A o ] p Hence we obtain A c + g, p, s b Let s 1 s 2 Then s 2 s 1 or all N Hence s 2 A, A o ] p s 1 A, A o ] p This inequality implies that c, p, s 1 c, p, s 2 Theorem 26 Let be a moulus unction, then a l l, p, s, b I is boune then l, p, s = w Proo a Let A l Then there exists a positive integer such that A, 0 Since is boune then A, 0 ] is also boune Hence s A, 0 ] p s 1] p s 1] H < Thereore A l, p, s b I is boune, then or any A w, s A, 0 ] p s L p s L H < Hence l, p, s = w Reerences 1] A Esi, Strongly almost λ-convergence an statistically almost λ-convergence o interval numbers, Scientia agna, , ] A Esi, Lacunary sequence spaces o interval numbers, Thai Journal o athematics, , ] A Esi, λ-sequence spaces o interval numbers, Appl ath In Sci, , ] A Esi, A new class o interval numbers, Journal o Qaqaz University, athematics an Computer Sciences, 2012,

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