Double sequences of interval numbers defined by Orlicz functions

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1 ACTA ET COENTATIONES UNIVERSITATIS TARTUENSIS DE ATHEATICA Volume 7, Numbe, June 203 Available online at Double sequences of inteval numbes efine by Olicz functions Ayhan Esi Abstact. We efine an stuy λ 2-convegence of ouble sequences of inteval numbes efine by Olicz function an λ 2-statistical convegence of ouble sequences of inteval numbes. We also establish some inclusion elations between them.. Intouction The iea of statistical convegence fo oinay sequences was intouce by Fast 7] in 95. Schoenbeg 7] stuie statistical convegence as a summability metho an liste some elementay popeties of statistical convegence. Both of these authos note that if a boune sequence is statistically convegent, then it is Cesào summable. Recently usaleen 5] efine an stuie λ-statistical convegence fo sequences as follows. Let λ = λ i be a non-eceasing sequence of positive numbes tening to infinity such that λ i+ λ i +, λ =. Then a sequence x = x k is sai to be λ-statistically convegent to a numbe L if fo evey ε > 0, lim {k I i λ i : x k, L ε } = 0, i whee I i = i λ i +, i]. Recall see 0] that an Olicz function is a continuous, convex, noneceasing function, efine fo u 0, such that 0 = 0 an u > 0 if u > 0. An Olicz function is sai to satisfy 2 -conition fo all values of u if thee exists a numbe K > 0 such that 2u Ku, u 0. Inteval aithmetic was fist suggeste by Dwye 2] in 95. Development of inteval aithmetic as a fomal system an evience of its value as a computational evice was povie by ooe 2] in 959 an ooe an Receive Febuay 7, athematics Subject Classification. 40A05, 40A35, 40C05, 46A45. Key wos an phases. Double sequence space, inteval numbes, Olicz function, statistical convegence. 57

2 58 AYHAN ESI Yang 3] in 962. Futhemoe, ooe an othes see 3], 8], 2] an ] have evelope applications to iffeential equations. Chiao ] intouce sequences of inteval numbes an efine the usual convegence of sequences of inteval numbes. Şengönül an Eyılmaz 8] intouce an stuie boune an convegent sequence spaces of inteval numbes an showe that these spaces ae complete metic spaces. Recently, Esi in 4] an 5] efine an stuie λ-statistical an lacunay statistical convegence of inteval numbes, espectively. We enote the set of all eal value close intevals by IR. Any element of IR is calle an inteval numbe an is enote by A = x l, x ]. Let x l an x be the smallest an the geatest points of an inteval numbe A, espectively. Fo inteval numbes A = x l, x ], A 2 = x 2l, x 2 ] an a numbe α R we have an A = A 2 x l = x 2l, x = x 2, A + A 2 = {x R : x l + x 2l x x + x 2 } αa = { {x R : αx l x αx } if α 0, {x R : αx x αx l } if α < 0. The set of all inteval numbes IR is a complete metic space with the istance A, A 2 = max { xl x 2l, x x 2 } see 4]. In the special case A = a, a] an A 2 = b, b] we obtain the usual metic of R. Now we give the efinition of a convegent sequence of inteval numbes see ]. Definition.. A sequence A k of inteval numbes is sai to be convegent to an inteval numbe A 0 if fo each ε > 0 thee exists a positive intege k 0 such that A k, A 0 < ε fo all k k0. We enote it by lim k A k = A 0. Thus, lim k A k = A 0 if an only if lim k x kl = x 0l an lim k x k = x 0. In this pape, we intouce an stuy the concepts of λ 2 -summable an statistically λ 2 -convegent ouble sequences of inteval numbes, an elations between them. 2. Definitions Recall that a ouble sequence a k,i of eal numbes is sai to be convegent in the Pingsheim sense to a numbe L if fo evey ε > 0 thee exists an inex n such that a k,i L < ε wheneve k, i > n see 6]. We tansfe

3 DOUBLE SEQUENCES OF INTERVAL NUBERS 59 this efinition to the ouble sequences of inteval numbes in the following way. Definition 2.. An inteval value ouble sequence A k,l is sai to be convegent in the Pingsheim sense to an inteval numbe A 0 if fo evey ε > 0 thee exists n N such that In this case we wite P -lim A k,l = A 0. A k,l, A 0 < ε fo k, l > n. We enote by c 2 the set of all convegent in the Pingsheim sense ouble sequences of inteval numbes. Definition 2.2. An inteval value ouble sequence A k,l is sai to be boune if thee exist an inteval numbe A 0 an a positive numbe B such that A k,l, 0 B fo all k, l N. We will enote the set of all boune ouble sequences of inteval numbe by the symbol l 2. It shoul be note that, similaly to the case of ouble numbe sequences, c 2 is not the subset of l 2. Let a ouble sequence λ 2 = of positive eal numbes ten to infinity an satisfy λ i+,j +, + +, Put λ i+,j + λ i+,j+, λ, =. I i,j = {k, l : i + k i, j + l j}. Definition 2.3. An inteval value ouble sequence A k,l is sai to be λ 2 -summable if thee exists an inteval numbe A 0 such that P - lim i,j = 0. Definition 2.4. Let be an Olicz function, let A k,l be an inteval value ouble sequence, an let p = p k,l be a ouble sequence of positive eal numbes. Let λ 2 = be the ouble sequence efine above. We

4 60 AYHAN ESI efine ] ] pk,l V 2 λ 2,, p = Ak,l : P - lim =0, i,j } fo some > 0 an A 0 IR, ],, p = Ak,l, 0 ] pk,l Ak,l : P - lim =0, 0 i,j } fo some > 0 an ],, p whee 0 = 0, 0]. = Ak,l : sup i,j } fo some > 0, Ak,l, 0 ] pk,l <, If we consie vaious assignments of, λ 2 an p in Definition 2.4, then we obtain iffeent special sets of sequences. Fo example, if p k,l = fo all k, l ] N, then these sets euce ] to the sets enote, espectively, by,,, ]0 an,. In the special case = ij i, j N, we wite c 2, ] ] instea of,. Fo x = x we obtain ] V 2 λ 2, p = Ak,l : P - lim i,j } fo some A 0 IR, an similaly, ], p ]0 an, p. ] pk,l =0 3. ain theoems Theoem 3.. If 0 < p k,l < q k,l an ],, q. qk,l p k,l is boune, then ],, p

5 Poof. If we take DOUBLE SEQUENCES OF INTERVAL NUBERS 6 A k,l,a 0 ] pk,l = w k,l fo all k, l N, then using the same technique employe in the poof of Theoem 2.9 fom 9] we get the esult. Coollay 3.2. The following statements ae vali. ] ] i If 0 < inf k,l p k,l fo all k, l N, then,, p,. ] ii If p k,l sup k,l p k,l = H < fo all k, l N, then, ],, p. Poof. i follows fom Theoem 3. with q k,l = fo all k, l N an ii follows fom Theoem 3. with p k,l = fo all k, l N. The poof of the following esult is a outine wok, so we omit it. Poposition 3.3. Let be an Olicz function such that ] 2 -conition] is satisfie. Then we have, p ]0,, p ]0,, p,, p ] an, p ],, p. The following efinition was pesente by Esi 6] fo a single sequence of inteval numbes. A sequence of inteval numbes A k is sai to be statistically λ-convegent to an inteval numbe A 0 if fo evey ε > 0, { lim k In : } A k, A 0 ε = 0, n λ n whee the vetical bas inicate the numbe of elements in the enclose set. Now, we will give efinitions of statistical convegence an statistical λ 2 -convegence fo ouble sequences of inteval numbes. Definition 3.. A ouble sequence A k,l of inteval numbes is sai to be statistically convegent to an inteval numbe A 0 povie that fo each ε > 0, P - lim i,j { k, l N N; k i, l j : } A k,l, A 0 ε = 0. ij We enote the set of all statistically convegent ouble sequences of inteval numbes by s 2. Definition 3.2. A ouble sequence A k,l of inteval numbes is sai to be statistically λ 2 -convegent to an inteval numbe A 0 if fo each ε > 0, { P - lim k, l Ii,j : } A k,l, A 0 ε = 0. i,j We enote the set of all statistically λ 2 -convegent ouble sequences of inteval numbes by s 2 λ 2.

6 62 AYHAN ESI Theoem 3.4. Let be an Olicz ] function. sup k,l p k,l = H <, then,, p s 2 λ 2. If 0 < h inf k,l p k,l Poof. Let ] A k,l,, p. Then thee exists > 0 such that ] pk,l 0 in the Pingsheim sense if i, j. If ε > 0, then we obtain A k,l,a 0 ε A k,l,a 0 ε A k,l,a 0 ε ] pk,l ε ] pk,l ] pk,l { ε h ε } H min, { k, l Ii,j : { } A k,l, A 0 ε ε h ε } H min,. Hence A k,l s 2 λ2, which completes the poof. Theoem 3.5. Let be an Olicz function an ] let 0 < h inf k,l p k,l sup k,l p k,l = H <. Then s 2 λ 2 l 2,, p. Poof. Let A k,l s 2 λ2 l 2. Then thee is a constant N > 0 such that A k,l, A 0 N fo all k, l N. Given ε > 0, fo an abitaily fixe > 0

7 we have = + A k,l,a 0<ε DOUBLE SEQUENCES OF INTERVAL NUBERS 63 A k,l,a 0 ε A k,l,a 0<ε ] pk,l ε ] pk,l + ] pk,l ] pk,l A k,l,a 0 ε max { N h } N H, { ε h ε } H max, + { k, l Ii,j : { } A k,l, A 0 ε N h } N H max,. Hence ] A k,l,, p. This completes the poof. The following coollay follows iectly fom Theoems 3.4 an 3.5. Coollay 3.6. If 0 < h inf k,l p k,l sup k,l p k,l = H <, then s 2 λ 2 ] l 2 =,, p l 2. If we take = ij an p k,l = fo all k, l N in Theoems 3.4, 3.5 an Coollay 3.6, then we get Coollay 3.7. Let be an Olicz function. Then the following statements hol. i c 2, ] s 2. ii s 2 l 2 c 2, ]. iii c 2, ] l 2 = s 2 l 2. Acknowlegement The autho is extemely gateful to the efeee fo his/he many valuable comments an suggestions.

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