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1 International Journal of Mathematical Archive-8(5), 07, 5-55 Available online through wwwijmainfo ISSN GENERALIZED gic-rate SEQUENCE SPACES OF DIFFERENCE SEQUENCE OF MODAL INTERVAL NUMBERS DEFINED BY A MODULUS FUNCTION S ZION CHELLA RUTH*, G RAJANIRANJANA Assistant Professor, M Phil Scholar, & Department of Mathematics, Aditanar College of Arts and science, Tiruchendur, Tamilnadu India (Received On: ; Revised & Accepted On: ) ABSTRACT In this paper we introduce and study the concepts of generalized gic -Rate sequence spaces of difference sequence of modal interval numbers and prove some inclusion relations Keywords: FK-spaces, Modulus Function, Rate sequence space, Modal Interval Numbers, Difference Sequence Space, C -Summability Method Mathematics Subject Classification 000: 40C05, 40D5, 40G05, 4A05 4A0 INTRODUCTION Many mathematical structures have been constructed with real or comple numbers In recent years, these mathematical structures were replaced by fuzzy numbers or interval numbers and these mathematical structures have been very popular since 965 Interval arithmetic is a tool in numerical computing where the rules for the arithmetic of intervals are eplicity stated Interval arithmetic was first suggested by PSDwyer[3] in 95 Development of interval arithmetic as a formal system and evidence of its value as a computational device was provided by REMoore[3] in 96 Probably the most important paper for the development of interval arithmetic has been published by the Japanese scientist Sunaga [9] Recently, the sequence spaces of modal interval numbers gic ( m, f, p, q), gic cπ ( m, f, p, q) and gic (l ) π ( m, f, p, q) using a modulus function f and more general C -method in view of Armitage and Maddo [] Several properties of these spaces, and some inclusion relation have been eamined PRELIMINARIES Definition : A set consisting of a closed interval of real numbers such that a b is called an interval number A real interval can also be considered as a set We denote the set of all real valued closed intervals by I R Any elements of I R is called closed interval and denoted by That is = [ l, r ] = { R : l r} An interval number is a closed subset of real numbers Let l and r be respectively referred to as the infimum (lower bound) and supremum (upper bound) of theinterval number If = [0,0], then is said to be a zero interval It is denoted by 0 Chiao [] introduced sequence of interval numbers and defined usual convergence of sequences of interval number When >, ˆ is not an interval number But in modal analysis, [, ] is a valid interval ~ R is defined by a pair of real numbers Definition : A modal interval number = {[, ] :, } denoted by gi and = ma {, } If = [0,0], then is said to be a zero modal interval, and it is Corresponding Author: S Zion Chella Ruth* Assistant Professor, Department of Mathematics, Aditanar College of Arts and science, Tiruchendur, Tamilnadu India International Journal of Mathematical Archive- 8(5), May 07 5

2 S Zion Chella Ruth*, G Rajaniranjana / Generalized gic -Rate Sequence Spaces of Difference Sequence of Modal Interval Numbers Defined by a Modulus Function / IJMA- 8(5), May-07 Definition 3: For ~,, ~ gi ~ = [ a, a ] (Degenerate modal interval number) ~ ~ = if and only if = and = ~ ~ + = { R : + + )} ~ ~ = { R : min(,,, ) ma( ~ / ~ = ~, Definition 4: The distance between the two modal interval numbers is defined by { } d( ~ ~, ) = ma,, ~, ~ Clearly d is a metric on gi Definition 5: Let us define transformation : N gi, f ( ) =, then = ( ) is called sequence of th modal interval number ~ is called the term of sequence ~ = ( ~ ), ω (gi) denote the set of all sequence of modal interval number with real terms f ~,, )} Definition 6: Let = ( ) = ([, ]) ω( gi) If =, for all N, then the sequence ~ = ( ~ ) called degenerate sequence of modal interval number is Definition 7: A sequence = ( ) of modal interval number is said to be convergent to a modal interval number ~ 0 if for each ε > 0 there eists a positive integer 0 such that d( ~, ~ 0 ) < ε for all 0 and we denote it by lim ~ ~ = 0 Equivalently lim ~ ~ = 0 if and only if lim = 0 and lim = 0 Definition 8: A sequence of modal interval numbers ~ = ( ~ ) is said to be bounded if there eist a positive number A such that ~ d( ~, 0) A for all N Definition 9: A modulus f is a function from [0, ) to [0, ) such that i) f() = 0 if and only if = 0, ii) f( + y) f() + f(y) for,y 0, iii) f is increasing, iv) f is continuous from the right at 0 It follows that f must be continuous everywhere on [0, ) Maddo [] used a modulus function to construct some sequence spaces Later on using a modulus different sequence spaces have been studied by Altın and et al [], et al [5], Nuray and Savas [5], Tripathy and Chandra [0] and many others Let π = (π n ) be a sequence of positive number ie, π n > 0, n N and X is FK- space We shall consider the sets of sequences of modal interval numbers = ( n ) X π (gi) = ω: (gi) n X(gI) π n The set X π (gi) may be considered as FK-space We shall call them as rate spaces of modal interval numbers (see, [7]) Let F be an infinite subset of N and F as the range of a strictly increasing sequence of positive integers, say F = {} n= The Cesaro submethod C is defined as (C ) n =, (n =,,, ) = where { } is a sequence of a modal interval numbers Therefore, the C -method yields a subsequence of the Cesaro method C and hence it is regular for any C is obtained by deleting a set of rows from Cesaro matri If = n is taen, then C = C is obtained On a range of sequences lim(c ) n = lim(c ) n, n n 07, IJMA All Rights Reserved 5

3 S Zion Chella Ruth*, G Rajaniranjana / Generalized gic -Rate Sequence Spaces of Difference Sequence of Modal Interval Numbers Defined by a Modulus Function / IJMA- 8(5), May-07 We will write C ~C The basic properties of C -method can be found in [] and [8] Let p = (p ) be a sequence of positive real numbers with G = sup p and D = ma(, G ) Then it is well nown that for all a, b R, the field of real numbers, for all N, a + b p D( a p + b p ) () Also for any real μ, μ p ma(, μ G ) () Let X(gI) be a sequence space of modal interval numbers Then X(gI) is called; i) Solid (or normal) if (α ) X(gI) whenever ( ) X(gI) for all sequences (α ) of scalars with α ii) Symmetric if ( ) X(gI) implies π() X(gI) where π is a permutation of N, iii) Sequence algebra if X(gI) is closed under multiplication 3 MAIN RESULTS Let f be a modulus function, X(gI) be a locally conve Hausdorff topological linear space whose topology is determined by a set Q of continuous seminorms q and p = (p ) be a sequence of positive real numbers Then defined the following sequence spaces of modal interval numbers gic ( m, f, p, q) = ω(gi): f q p m 0 as n π = gic cπ ( m, f, p, q) = ω(gi): f q m L π = p 0 as n for some L sup gic (l ) π ( m, f, p, q) = ω(gi): n f p m < π = as n where 0 =, m = m m + and m m = ( ) v m π π π π π + π v=0 v For f() = we shall write gic gic cπ ( m, f, p, q) and gic (l ) π ( m, p, q), gic cπ ( m, p, q)and gic (l ) π ( m, f, p, q) respectively 07, IJMA All Rights Reserved 53 +v π +v ( m, p, q) instead of gic ( m, f, p, q), Theorem 3: Let p = (p ) be bounded, then gic ( m, f, p, q), gic cπ ( m, f, p, q) and gic (l ) π ( m, f, p, q) are linear spaces of modal interval numbers Theorem 3: gic ( m, f, p, q)is a paranormed space of modal interval number (not totally paranormed), paranormed by g () = sup n f q p m π = where M = ma (, G = sup p ) Theorem 33: Let f, f, f are modulus function and 0 < h = inf p sup p = G then (i) gic ( m, f, p, q) gic ( m, f f, p, q) (ii) gic ( m, f, p, q) gic ( m, f, p, q) gic ( m, f + f, p, q) M Proof: (i) Let = gic π ( m, f, p, q) Let ε > 0 and choose δ with 0 < δ < such that f(t) < ε for 0 t δwrite y = f q m π and consider [f(y )] p = [f(y )] p + [f(y )] p = where the first summation is over y δ and the second over y > δ Since f is continuous, we get [f(y )] p < ma(ε h, ε G ) (3)

4 S Zion Chella Ruth*, G Rajaniranjana / Generalized gic -Rate Sequence Spaces of Difference Sequence of Modal Interval Numbers Defined by a Modulus Function / IJMA- 8(5), May-07 and for y > δ we use the fact that y < y δ < + y δ By the definition of f, we have y > δ, f(y ) f() + y δ f() y δ Hence [f(y )] p < ma, f() δ G [y ]p = 0 (4) By (3) and (4) we have gic (ii) Let = gic π ( m, f, p, q) gic ( m, f f, p, q) ( m, f, p, q) gic ( m, f, p, q) Then there eist f and f such that p f q m = 0 as n (5) π p f q m = 0 as n (6) π Then using (i) it can be shown that (f + f ) q m π = Therefore = gic π ( m, f + f, p, q) Hence, gic ( m, f, p, q) gic p = f q m π = 07, IJMA All Rights Reserved 54 p D f q m π = 0 as n ( m, f, p, q) gic ( m, f + f, p, q) The proof of the following result is a routine wor of the above theorem Corollary 34: Let f, f, f are modulus function then (i) gic cπ ( m, f, p, q) gic cπ ( m, f f, p, q), (ii) gic cπ ( m, f, p, q) gic cπ ( m, f, p, q) gic cπ ( m, f + f, p, q), ( m, f, p, q) gic (l ) π ( m, f f, p, q), (iv) gic (l ) π ( m, f, p, q) gic (l ) π ( m, f, p, q) gic (l ) π ( m, f + f, p, q) Proposition 35: gic cπ ( m, f, p, q) gic cπ ( m, f, p, q) Theorem 36: Let m, then the following inclusion are strict (i) gic ( m, f, q) gic ( m, f, q), (ii) gic cπ ( m, f, q) gic cπ ( m, f, q), ( m, f, q) gic (l ) π ( m, f, q) + f q m π p = p + D f q m π Eample 37: Let f() = and q() = Consider the sequences ( ) = ([0, m+α ]) and (π ) = ( α+ ), where = and m N, α R Then gic π ( m, f, q) but gic ( m, f, q), since m = 0, π m = [0, ( ) m (m )!] N π Theorem 38: For any two sequence p = (p ) and t = (t ) of positive real numbers and any two seminorms q, q we have (i) gic ( m, f, p, q ) gic ( m, f, p, q ), (ii) gic cπ ( m, f, p, q ) gic cπ ( m, f, p, q ), ( m, f, p, q ) gic (l ) π ( m, f, p, q ) = p

5 S Zion Chella Ruth*, G Rajaniranjana / Generalized gic -Rate Sequence Spaces of Difference Sequence of Modal Interval Numbers Defined by a Modulus Function / IJMA- 8(5), May-07 Proof: Since the zero element belongs to each of the above classes of sequences, thus the intersection is nonempty The following result is a consequence of Theorem 33 (i) and Corollary 34 (i) and (iii) Theorem 39: Let f be a modulus function Then (i) gic ( m, p, q) gic ( m, f, p, q), (ii) gic cπ ( m, p, q) gic cπ ( m, f, p, q), ( m, p, q) gic (l ) π ( m, f, p, q) Theorem 30: The sequence spaces gic ( m, f, p, q), gic cπ ( m, f, p, q) and gic (l ) π ( m, f, p, q) are neither solid nor symmetric, nor sequence algebras for m Proof: Let m =, p = for all N, f() = and q() = If the sequences ( ) = ([0, n+ ]) and (π ) = ( n ) are taen, then the sequence belongs to gic π (l ) π ( ) and gic cπ ( ) where n R Let α = ( ), then (α ) does not belong to gic (l ) π ( ) and gic cπ ( ) Hence gic cπ ( m, f, p, q) and gic (l ) π ( m, f, p, q) are not solid The other cases can be proved on considering similar eamples REFERENCES Altin, Y and Et, M Generalized difference sequence spaces defined by a modulus function in a locally conve space, Soochow J Math, 3() (005), Armitage DH and Maddo IJ A new type of Cesaro mean, Analysis 9 (989), Dwyer PS, Linear Comptation, New Yor, Wiley, (95) 4 Et, M; Altino, H and Altin, Y On some generalized sequence spaces Appl Math Comput 54() (004), Et, M Strongly almost summable difference sequences of order m defined by a modulus, Studia Sci Math Hungar 40(4) (003), Et, M and Cola, R On some generalized difference sequence spaces, Soochow JMath (4) (995), Et, M Generalized Cesaro difference sequence spaces of non-absolute type involving lacunary sequences, Appl Math Comput 9(7) (03), Gungor, M and Et, M r strongly almost summable sequences defined by Orlicz functions, Indian J Pure Appl Math 34(8) (003), Jurimae,E Matri mappings between rate-spaces and spaces with speed, Acta Comm UnivTartu 970(994), Kızmaz, H On certain sequence spaces, Canad Math Bull, 4() (98), Kuo-Ping Chiao, Fundamental properties of interval vector ma-norm, Tamsui Oford Journal of Mathematics, 8,9-33(00) Maddo I J Sequence spaces defined by a modulus, Math Proc Camb Philos Soc00 (986), REMoore and CT Yang, Interval Analysis I, LMSD-85875, Locheed Missiles and Space Company, (96) 4 Naano H Concave modulars, J Math Soc Japan 5 (953), Nuray, F and Savas, E Some new sequence spaces defined by a modulus function, Indian J Pure Appl Math 4() (993), Osiiewicz J A Equivalance Results for CesaroSubmethods, Analysis 0 (000), OzaltinB, DagadurI, Generalized C -Rate sequence spaces of difference sequence defined by a modulus function in a locally conve space, Math7-3, 05 8 Si, J Matri mappings for rate-space and $ \scr K$-multipliers in the theory of summability Tartu Riil Ul Toimetised 846 (989), Sunaga T, Theory of an interval algebra and its application to numeric analysis, RAAG Memories II, Ganutsu Punen Fueyu-ai, Toyo, (958) Tripathy BC and Chandra P On Some Generalized Difference Paranormed Sequence Spaces Associated with Multiplier Sequences Defined by Modulus Function, Anal Theory Appl 7() (0), -7 Wilansy A, Functional Analysis, Blaisdell Press, 964 Source of support: Nil, Conflict of interest: None Declared [Copy right 07 This is an Open Access article distributed under the terms of the International Journal of Mathematical Archive (IJMA), which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly cited] 07, IJMA All Rights Reserved 55