HÖLDER SUMMABILITY METHOD OF FUZZY NUMBERS AND A TAUBERIAN THEOREM

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1 Iraia Joural of Fuzzy Systems Vol., No. 4, (204 pp HÖLDER SUMMABILITY METHOD OF FUZZY NUMBERS AND A TAUBERIAN THEOREM İ. C. ANAK Abstract. I this paper we establish a Tauberia coditio uder which covergece follows from Hölder summability of sequeces of fuzzy umbers.. Itroductio The cocept of the fuzzy set was itroduced by Zadeh [22]. Matloa [7] itroduced the cocepts of bouded ad coverget sequeces of fuzzy umbers ad showed that every coverget sequece is bouded. Nada [8] studied the spaces of bouded ad coverget sequeces of fuzzy umbers ad proved that every Cauchy sequece of fuzzy umbers is coverget. Furthermore, sequeces of fuzzy umbers have bee discussed i [5,, 3, 4] ad may others. Recetly differet classes of sequeces of fuzzy real umbers have bee ivestigated by Tripathy ad Baruah [3, 4], Tripathy ad Sarma [20, 2], Tripathy ad Borgohai [6, 7], Tripathy ad Dutta [8, 9], Tripathy et. al [5] ad may others. A umber of authors icludig Subrahmayam [0] ad Talo ad Caa [2] have established Tauberia theorems for some summability methods for sequeces of fuzzy umbers. Subrahmayam [0] itroduced Cesàro covergece of a sequece of fuzzy umbers ad obtaied a Tauberia theorem for Cesàro summability for the sequeces of fuzzy umbers. Tripathy ad Baruah [4] defied Nörlud ad Riesz meas of fuzzy real umbers ad obtaied a aalogue of a Tauberia theorem for Riesz summability method for the sequeces of fuzzy umbers. Talo ad Caa [2] obtaied ecessary ad sufficiet Tauberia coditios, uder which covergece follows from Cesàro covergece of sequeces of fuzzy umbers. Altı et. al [2] studied the cocept of statistical summability (C, for sequeces of fuzzy real umbers ad obtaied a Tauberia theorem. I this paper, we itroduce Hölder summability ad obtai a Tauberia coditio uder which covergece follows from Hölder summability of sequeces of fuzzy umbers. 2. Defiitos ad Notatios Let D deote the set of all closed ad bouded itervals A = [A, A] o the real lie R. Received: Jue 202; Revised: Jue 203; Accepted: Jue 204 Key words ad phrases: Hölder meas, Slow oscillatio, Tauberia theorems, Fuzzy umbers.

2 88 İ. C. aa For A, B D, we defie d(a, B = max( A, B, A, B where A = [A, A] ad B = [B, B]. It is ow that (D, d is a complete metric space. A fuzzy real umber u is a fuzzy set o R ad is a mappig u : R [0, ], associatig each real umber t with its grade of membership u(t. If there exists t 0 R, such that u(t 0 =, the the fuzzy real umber u is called ormal. A fuzzy real umber u is called covex if u(t u(s u(r = mi(u(s, u(r, where s < t < r. A fuzzy real umber u is said to be upper semi-cotiuous if for each ɛ > 0, u ([0, a + ɛ, for all a [0, ] is ope i the usual topology of R. Let E deote the set of all fuzzy umbers which are upper semi-cotiuous ad have a compact support. For 0 < α, the α-level set of a fuzzy umber u deoted by u α is defied by u α = {t R : u(t α}. u 0 is defied as the closure of the set {t R : u(t > 0}. We defie D : E E R + {0} by D(u, v = sup d(u α, v α. 0 α It is straightforward to see that the mappig D is a metric o E ad (E, D is a complete metric space. The followig properties [6] are eeded i the sequel. (i D(u + w, v + w = D(u, v for all u, v, w E (ii D(u, v = D(u, v for all u, v E, R (iii D(u + v, w + z D(u, w + D(v, z for all u, v, w, z E A sequece u = (u of fuzzy umbers is a fuctio u from the set N of all positive iteger ito E. A sequece (u of fuzzy umbers is said to be coverget to l, writte as lim u = l, if for every ɛ > 0 there exists a positive iteger N 0 such that D(u, l < ɛ for > N 0. A sequece (u is said to be slowly oscillatig if lim max D(u, u = 0, λ + < λ where λ deotes the iteger part of λ. The cocept of slow oscillatio was itroduced for the sequeces of real umbers by Staoević [9]. It is easy to see that every coverget sequece of fuzzy umbers is slowly oscillatig. Let (u be a sequece of fuzzy umbers. The arithmetic meas σ ( of (u are defied by σ ( = + =0 u for all N {0}. Repeatig this averagig process, we defie σ (m for all m N {0} by σ (0 = u ad σ (m+ = + =0 σ(m.

3 Hölder Summability Method of Fuzzy Numbers ad a Tauberia Theorem 89 A sequece (u of fuzzy umbers is said to be summable by the Hölder s method (H, m to l, writte as lim u = l (H, m, if D(σ (m, l 0 as. For ay m N {0}, the idetity where v (m = σ (m + + σ (m+ =0 = v (m, ( v (m, m (u u, m = 0, =0 will be used i the sequel. The idetity ( for m = 0 is ow as the Kroecer idetity. By the expressio of σ (m+ i the form we may rewrite ( as σ (m σ (m+ = u 0 + = v (m + = = v (m v (m, (2 + u 0. The Hölder s method (H, is the same as the Cesàro summability method. Sice Cesàro summability method is regular ([0], the Hölder s method (H, m is also regular. Subrahmayam [0] proved that every coverget sequece of fuzzy umbers is Cesàro summable ad costructed a sequece of fuzzy umbers showig that the coverse of this implicatio does ot hold. The coverse may be true uder some additioal coditios which are called Tauberia coditios. A theorem which states that covergece of a sequece of fuzzy umbers follows from a summability method ad some Tauberia coditio(s is called a Tauberia theorem. 3. Prelimiary Results We eed the followig Lemmas for the proof of our result. Lemma 3.. If (u is slowly oscillatig, the D(v (0, 0 = O( for all N {0}, where v (0 = + =0 (u u. Proof. Defie ρ(, λ = max < λ D(u, u for λ >. Represet the fiite sum = (u u as the series =0 2 + (u u. Assume that 2 w(λ := ρ(, λ 0, λ +. If w(λ exists for λ >, the w(2 exists. By summatio by parts, we have ( 2 2 D (u u, 0 D(u, u 2 + 2D(u 2, u = = ( 2 ρ(, 2 + ρ(, 2 = 2ρ(, 2.

4 90 İ. C. aa For 2 = [ ] ad 2 2 2, we obtai ( D (u u, 0 C 2 = O(, 0 for some positive costat C. By the last iequality, we have D(v (0, 0 = O( for all N {0}. The proof of the Lemma 3. was give by Szász [] i his lecture otes for the sequeces of real umbers. For the completeess of this paper we give the proof of the Lemma 3. for the sequeces of fuzzy umbers by the mior chages i the proof of Szász. Lemma 3.2. If (u is slowly oscillatig, the the sequeces (σ ( ad (v (0 are slowly oscillatig. Proof. By Lemma 3., we have D(v (0, 0 C for all N {0} ad some positive costat C. The for all <, we have D(σ (, σ( = D =+ v, 0 =+ =0 D(v, 0 C =+ Taig max of both sides of the iequality over, we obtai ( max D(σ (, λ + < λ σ( C log. + ( + C log. + Taig of both sides of the iequality above as, we have max D(σ ( < λ, σ( C log λ. Taig the limit of both sides as λ +, we obtai lim max D(σ ( < λ which shows that (σ ( is slowly oscillatig. It follows from the iequality D(v (0, v(0 = D(u σ (, u σ (, σ( = 0, D(u, u + D(σ (, σ( that (v (0 is slowly oscillatig. 4. The Mai Result We prove that a sequece (u of fuzzy umbers summable by the Hölder s method (H, m is coverget if it is slowly oscillatig. Theorem 4.. If lim u = l (H, m for some positive iteger m ad (u is slowly oscillatig, the lim u = l.

5 Hölder Summability Method of Fuzzy Numbers ad a Tauberia Theorem 9 Proof. Let lim u = l (H, m. The, we have lim u = l (H, m+ ad lim v (m = 0. By defiitio, v (m = σ ( (v (m. Sice (u is slowly oscillatig, (v (m is slowly oscillatig by Lemmas 3. ad 3.2. We ow prove that v (m 0 as. By the triagle iequality, we have D(v (m, 0 D(v (m, τ (m + D(τ (m, v (m λ =+ v(m. + D(v (m, 0, (3 where τ (m = λ For the first term o the right-had side of the iequality (3, we have D(v (m (, τ (m D λ λ ( = λ D λ λ λ =+ =+ =+ v (m, λ v (m, λ =+ D(v (m, v (m max D(v m, v m. < λ λ =+ v (m v (m For the secod term o the right-had side of the iequality (3, we have D(τ (m, v (m = D λ v (m, v (m λ + =+ =0 = D λ v (m + v (m, λ =+ =0 = D λ v (m λ +, v (m λ (λ =0 ( + =0 ( λ + = D λ vm λ, λ + λ vm = λ + λ D(vm λ, v(m. Sice λ + λ < 2λ λ for all λ > ad sufficietly large, ( + + λ =0 v (m D(τ (m, v (m 2λ λ D(vm λ, v (m. (5 From the calculatios for the first ad secod terms of the right-had side of the iequality (3, we have D(v (m, 0 max D(v m, v m + 2λ < λ λ D(vm λ, v (m + D(v (m, 0. (6 Note that sice lim v (m = 0, the secod ad third terms o the right-had side of the iequality of (6 vaishes as. Taig of both sides of the iequality (6 as, we coclude that D(v (m (4, 0 max D(v m, v m. < λ (7

6 92 İ. C. aa Taig the limit of both sides of the iequality (7 as λ +, we obtai lim D(v(m, 0 = 0. (8 It follows from the iequality D(σ (m, l = D(σ (m + v (m, l D(σ (m, l + D(v (m, 0 that lim σ (m = l. Cotiuig i this way, we obtai that lim v (0 = 0 ad lim σ ( = l. By the iequality D(u, l = D(σ ( + v (0, l D(σ (, l + D(v (0, 0 we have lim u = l. Corollary 4.2. If lim u = l (H, m ad D(u, u = O( for all N, the lim u = l. Proof. Assume that D(u, u C for all N {0} ad some positive costat C. The for all <, we have ( + D(u, u D(u, u C C log. + (9 =+ It follows from the iequality (9 that lim which shows that (u is slowly oscillatig. =+ max D(u, u = 0, < λ Sice every coverget sequece of fuzzy umbers is slowly oscillatig, we have the followig immediate corollary. Corollary 4.3. If lim u = l (H, ad (v (0 is slowly oscillatig, the lim u = l. Acowledgemet. The author would lie to tha the aoymous referees for their valuable commets ad suggestios to improve the quality ad readability of the paper. Refereces [] Y. Alti, M. Et ad M. Basarir, O some geeralized differece sequeces of fuzzy umbers, Kuwait J. Sci. Eg., 34(A (2007, 4. [2] Y. Alti, M. Mursalee ad H. Altio, Statistical summability (C, for sequeces of fuzzy real umbers ad a Tauberia theorem, J. Itell. Fuzzy Syst., 2(6 (200, [3] H. Altio, Y. Alti ad M. Isi, Statistical covergece ad strog p-cesàro summability of order β i sequeces of fuzzy umbers, Iraia Joural of Fuzzy Systems, 9(2 202, [4] H. Altio, R. Cola ad Y. Alti, O the class of λ-statistically coverget differece sequeces of fuzzy umbers, Soft Comput., 6(6 (202, [5] H. Altio, R. Cola ad M. Et, λ-differece sequece spaces of fuzzy umbers, Fuzzy Sets ad Syst., 60(2 (2009, [6] D. Dubois ad H. Prade, Fuzzy umbers: a overview, aalysis of fuzzy iformatio, Mathematical Logic, CRC Press, Boca, FL, (987, [7] M. Matloa, Sequeces of fuzzy umbers, BUSEFAL, 28 (986, [8] S. Nada, O sequeces of fuzzy umbers, Fuzzy Sets ad Systems, 33( (989,

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