THE FIBONACCI NUMBERS OF ASYMPTOTICALLY LACUNARY χ 2 OVER PROBABILISTIC p METRIC SPACES

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1 TWMS J. Pure Appl. Math. V.9, N., 208, pp THE FIBONACCI NUMBERS OF ASYMPTOTICALLY LACUNARY χ 2 OVER PROBABILISTIC p METRIC SPACES DEEPMALA, VANDANA 2, N. SUBRAMANIAN 3 AND LAKSHMI NARAYAN MISHRA 4 Abstract. In this paper we study the concept of almost asymptotically lacunary statistical χ 2 over probabilistic p metric spaces and discuss some general topological properties of above sequence spaces. Keywords: analytic sequence, double sequences, χ 2 space, Fibonacci number, Musielak-modulus function, probabilisitic p metric space, asymptotically equivalence, statistical convergent. AMS Subject Classification: 40A05, 40C05, 40D05.. Introduction In this paper we study the concept of almost asymptotically lacunary statistical χ 2 over probabilistic p metric spaces defined by Musielak. Since the study of convergence in PPspaces is fundamental to probabilistic functional analysis, we feel that the concept of almost asymptotically lacunary statistical χ 2 over probabilistic p-metric spaces defined by Musielak in a PP-space would provide a more general framework for the subject. Throughout w, χ and Λ denote the classes of all, gai and analytic scalar valued single sequences, respectively. We write w 2 for the set of all complex sequences (x mn ), where m, n N, the set of positive integers. Then, w 2 is a linear space under the coordinate wise addition and scalar multiplication. Some initial works on double sequence spaces is found in [4]. Later it was investigated by [3, 7 0, 2, 3, 20, 22 29, 3 43], We procure the following sets of double sequences: M u (t) := (x mn ) w 2 : sup m,n N x mn tmn <, C p (t) := (x mn ) w 2 : p x mn l t mn = for some l C, m,n C 0p (t) := (x mn ) w 2 : p x mn t mn = m,n L u (t) := (x mn ) w 2 : x mn t mn <, m= n= C bp (t) := C p (t) M u (t) and C 0bp (t) = C 0p (t) M u (t); PDPM Indian Institute of Information Technology, Design and Manufacturing, Jabalpur, India 2 Department of Management Studies, Indian Institute of Technology, Madras, India 3 Department of Mathematics, SASTRA University, India 4 School of Advanced Sciences, Vellore Institute of Technology, University, Vellore, India dmrai23@gmail.com, vdrai988@gmail.com, nsmaths@yahoo.com lakshminarayanmishra04@gmail.com Corresponding author Manuscript received January

2 DEEPMALA, et al: THE FIBONACCI NUMBERS OF ASYMPTOTICALLY LACUNARY where t = (t mn ) be the sequence of strictly positive reals t mn for all m, n N and p m,n denotes the it in the Pringsheim s sense. In the case t mn = for all m, n N; M u (t), C p (t), C 0p (t), L u (t), C bp (t) and C 0bp (t) reduce to the sets M u, C p, C 0p, L u, C bp and C 0bp, respectively. Now, we may summarize the knowledge given in some document related to the double sequence spaces. [6] have proved that M u (t) and C p (t), C bp (t) are complete paranormed spaces of double sequences and obtained the α, β, γ duals of the spaces M u (t) and C bp (t). Quite recently, [44] has essentially studied both the theory of topological double sequence spaces and the theory of summability of double sequences. ( [4]- [9]) and ( [3]- [42]) have independently introduced the statistical convergence and Cauchy for double sequences and established the relation between statistical convergent and strongly Cesàro summable double sequences. [] defined the spaces BS, BS (t), CS p, CS bp, CS r and BV of double sequences consisting of all double series whose sequence of partial sums are in the spaces M u, M u (t), C p, C bp, C r and L u, respectively, and also examined some properties of those sequence spaces and determined the α duals of the spaces BS, BV, CS bp and the β (ϑ) duals of the spaces CS bp and CS r of double series. [2] introduced the Banach space L q of double sequences corresponding to the well-known space l q of single sequences and examined some properties of the space L q. Recently [30] have studied the space χ 2 M (p, q, u) of double sequences and proved some inclusion relations. The class of sequences which are strongly Cesàro summable with respect to a modulus was introduced by Maddox [] as an extension of the definition of strongly Cesàro summable sequences. [5] further extended this definition to a definition of strong A summability with respect to a modulus where A = (a n,k ) is a nonnegative regular matrix and established some connections between strong A summability, strong A summability with respect to a modulus, and A statistical convergence. In [2] the four dimensional matrix transformation (Ax) = a mn x mn was studied extensively by Robison and Hamilton. m= n= We need the following inequality in the sequel of the paper. For a, b 0 and 0 < p <, we have The double series m,n= convergent, where s mn = (a + b) p a p + b p () x mn is called convergent if and only if the double sequence (s mn ) is m,n i,j= x ij (m, n N). A sequence x = (x mn ) is said to be double analytic if sup mn x mn /m+n <. The vector space of all double analytic sequences will be denoted by Λ 2. A sequence x = (x mn ) is called double gai sequence if ((m + n)! x mn ) /m+n 0 as m, n. The double gai sequences will be denoted by χ 2. Let ϕ = all finite sequences. Consider a double sequence x = (x ij ). The (m, n) th section x [m,n] of the sequence is defined by x [m,n] = x ij I ij for all m, n N ; where I ij denotes the double sequence whose only non i,j=0 m,n zero term is a (i+j)! in the (i, j) th place for each i, j N. An FK-space(or a metric space)x is said to have AK property if (I mn ) is a Schauder basis for X. Or equivalently x [m,n] x. An FDK-space is a double sequence space endowed with a complete metrizable; locally convex topology under which the coordinate mappings x = (x k ) (x mn )(m, n N) are also continuous. Let M and Φ be mutually complementary modulus functions. Then, we have (i) For all u, y 0, see [40] uy M (u) + Φ (y), (2)

3 96 TWMS J. PURE APPL. MATH., V.9, N., 208 (ii) For all u 0, uη (u) = M (u) + Φ (η (u)). (3) (iii) For all u 0, and 0 < λ <, M (λu) λm (u). (4) In [4] used the idea of Orlicz function to construct Orlicz sequence space ) l M = x w : M <, for some ρ > 0, The space l M with the norm x = inf k= ( xk ρ ρ > 0 : M k= ( ) xk ρ, becomes a Banach space which is called an Orlicz sequence space. For M (t) = t p ( p < ), the spaces l M coincide with the classical sequence space l p. A sequence f = (f mn ) of modulus function is called a Musielak-modulus function. A sequence g = (g mn ) defined by g mn (v) = sup v u f mn (u) : u 0, m, n =, 2, is called the complementary function of a Musielak-modulus function f. For a given Musielak modulus function f, the Musielak-modulus sequence space t f is defined by t f = x w 2 : I f ( x mn ) /m+n 0 as m, n, where I f need not be convex modular defined by I f (x) = f mn ( x mn ) (/m)+n, x = (x mn ) t f. m= n= We consider t f equipped with the Luxemburg metric ( ( ) ) d (x, y) = sup inf f xmn (/m)+n mn mn. mn m= n= The notion of difference sequence spaces (for single sequences) was introduced by [42] as follows Z ( ) = x = (x k ) w : ( x k ) Z, for Z = c, c 0 and l, where x k = x k x k+ for all k N. Here c, c 0 and l denote the classes of convergent, null and bounded scalar valued single sequences respectively. The difference sequence space bv p of the classical space l p is introduced and studied in the cases p and 0 < p < []. The spaces c ( ), c 0 ( ), l ( ) and bv p are Banach spaces normed by ( x = x + sup k x k and x bvp = ) /p x k p, ( p < ). k= Later on the notion was further investigated by many others. We now introduce the following difference double sequence spaces defined by Z ( ) = x = (x mn ) w 2 : ( x mn ) Z, where Z = Λ 2, χ 2 and x mn = (x mn x mn+ ) (x m+n x m+n+ ) = x mn x mn+ x m+n + x m+n+ for all m, n N. The generalized difference double notion has the following representation: m x mn = m x mn m x mn+ m x m+n + m x m+n+, and also this generalized difference double notion has the following binomial representation: m x mn = m m ( ( ) i+j m) ( ) m i j x m+i,n+j. i=0 j=0

4 DEEPMALA, et al: THE FIBONACCI NUMBERS OF ASYMPTOTICALLY LACUNARY Definition and preinaries Let n N and X be a real vector space of dimension w, where n w. A real valued function d p (x,..., x n ) = (d (x, 0),..., d n (x n, 0)) p on X satisfying the following four conditions: (i) (d (x, 0),..., d n (x n, 0)) p = 0 if and and only if d (x, 0),..., d n (x n, 0) are linearly dependent, (ii) (d (x, 0),..., d n (x n, 0)) p is invariant under permutation, (iii) (αd (x, 0),..., d n (x n, 0)) p = α (d (x, 0),..., d n (x n, 0)) p, α R (iv) d p ((x, y ), (x 2, y 2 ) (x n, y n )) = (d X (x, x 2, x n ) p + d Y (y, y 2, y n ) p ) /p for p < ; (or) (v) d ((x, y ), (x 2, y 2 ), (x n, y n )) := sup d X (x, x 2, x n ), d Y (y, y 2, y n ), for x, x 2, x n X, y, y 2, y n Y is called the p product metric of the Cartesian product of n metric spaces is the p norm of the n-vector of the norms of the n sub spaces. A trivial example of p product metric of n metric space is the p norm space is X = R equipped with the following Euclidean metric in the product space is the p norm: (d (x, 0),..., d n (x n, 0)) E = sup ( det(d mn (x mn, 0)) ) = d (x, 0) d 2 (x 2, 0)... d n (x n, 0) d 2 (x 2, 0) d 22 (x 22, 0)... d 2n (x n, 0). sup.. d n (x n, 0) d n2 (x n2, 0)... d nn (x nn, 0) where x i = (x i, x in ) R n for each i =, 2, n. If every Cauchy sequence in X converges to some L X, then X is said to be complete with respect to the p metric. Any complete p metric space is said to be p Banach metric space. ( ) Definition 2.. Let A = a mn denote a four dimensional summability method that maps the complex double sequences x into the double sequence Ax where the k, l th term of Ax is as follows: (Ax) = a mn x mn such transformation is said to be non-negative if a mn is m= n= non-negative. The notion of regularity for two dimensional matrix transformations was presented by Silverman and Toeplitz. Following Silverman and Toeplitz, Robison and Hamilton presented the following four dimensional analog of regularity for double sequences in which both added an adiditional assumption of boundedness. This assumption was made since a double sequence which is P convergent is not necessarily bounded. ( ) Let λ and µ be two sequence spaces and A = a mn be a four dimensional infinite matrix of ( ) real numbers, where m, n, k, l N. Then, we say A defines a matrix mapping from λ a mn into µ and we denoe it by writing A : λ µ if for every sequence x = (x mn ) λ the sequence Ax = (Ax), the A transform of x, is in µ. By (λ : µ), we denote the class of all matrices A such that A : λ µ. Thus A (λ : µ) if and only if the series converges for each k, l N. A sequence x is said to be A summable to α if Ax converges to α which is called as the A it of x. ( Lemma 2. (See []). Matrix A = a mn ) is regular if and only if the following three conditions hold: () There exists M > 0 such that for every k, l =, 2, the following inequality holds:

5 98 TWMS J. PURE APPL. MATH., V.9, N., 208 m= n= (2) a mn M; amn (3) = 0 for every k, l =, 2, a mn =. m= n= Let (q mn ) be a sequence of positive numbers and Q = k m=0 n=0 Then, the matrix R q = (r mn)q of the Riesz mean is given by (r mn )q = qmn Q l q mn (k, l N). (5) if 0 m, n k, l 0 if (m, n) >. The fibonacci numbers are the sequence of numbers f mn (k, l, m, n N) defined by the linear recurrence equations f 00 = and f =, f mn = f m n + f m 2n 2 ; m, n 2. Fibonacci numbers have many interesting properties and applications in arts, sciences and architecture. Also, some basic properties of Fibonacci numbers are the following. m n f mn = f m+2n+2 ; m, n, k= l= m k= l= k= l= n fmn 2 = f mn f m+n+ ; m, n, f mn converges. In this paper, we define the fibonacci matrix F = (f mn) m,n=, which differs from existing Fibonacci matrix by using Fibonacci numbers f and introduce some new sequence spaces χ 2 and Λ 2. Now, we define the Fibonacci matrix F = (f mn) m,n=, by f (f mn) = f (k+2)(l+2) if 0 k m; 0 l n 0 if (m, n) > that is, It is obvious that the four dimensional infinite matrix F is triangular matrix. Also it follows from lemma 2.2 that the method F is regular. Definition 2.2. A function f : R R R + R + is called a distribution if it is non-decreasing and left continuous with inf t R R f (t) = 0 and sup f (t) =. We will denote the set of t R + + R all distribution functions by D. Definition 2.3. A triangular metric, briefly t-over probabilisitic p metric spaces, is a binary operation on [0, ] which continuous, commutative, associative, non-decreasing and has as neutral element, that is, it is the continous mapping : [0, ] [0, ] [0, ] [0, ] such that a, b, c [0, ] : () a =, (6)

6 DEEPMALA, et al: THE FIBONACCI NUMBERS OF ASYMPTOTICALLY LACUNARY (2) a b = b a (3) c d a b if c a and d b, (4) (a b) c = a (b c). Definition 2.4. A triple (X, P, ) is called a probabilistic p metric space or shortly P P space if X is a real vector space, P is a mapping from X X D D (for x X, the distribution function P (x) is denoted by P x and P x (t) is the value of P x at t R R) and is a t p metric satisfying the following conditions: (i) P x ( (d (x, 0),..., d n (x n, 0)) p ) = 0 if and and only if P x (d (x, 0),..., d n (x n, 0)) are linearly dependent, (ii) P x (t (d (x, 0),..., d n (x n, 0)) p ) = is invariant under permutation, (iii) P x ( (αd (x, 0),..., d n (x n, 0)) p ) = α P x ( (d (x, 0),..., d n (x n, 0)) p ), α R\ 0. (iv) P x (d p ((x, y ), (x 2, y 2 ) (x n, y n ))) = P x ( (d X (x, x 2, x n ) p ) + P x (d Y (y, y 2, y n ) p ) /p) for p < ; (or) (v) P x (d ((x, y ), (x 2, y 2 ), (x n, y n ))) := sup P x (d X (x, x 2, x n )), P x (d Y (y, y 2, y n )), for x, x 2, x n X, y, y 2, y n Y. Definition 2.5. A triple (X, P, ) be a P P space. Then a sequence x = (x mn ) is said to convergent 0 X with respect to the probabilistic p metric P if, for every ϵ > 0 and θ (0, ), there exists a positive integer m 0 n 0 such that P xmn 0 (ϵ) > θ whenever m, n m 0 n 0. It is denoted by P x = 0 or x mn P 0 as m, n. Definition 2.6. A triple (X, P, ) be a P P space. Then a sequence x = (x mn ) is called a Cauchy sequence with respect to the probabilistic p metric P if, for every ϵ > 0 and θ (0, ) there exists a positive integer m 0 n 0 such that P xmn x rs (ϵ) > θ for all m, n > m 0 n 0 and r, s > r 0 s 0. Definition 2.7. A triple (X, P, ) be a P P space. Then a sequence x = (x mn ) is said to analytic in X, if there is a u R such that P xmn (u) > θ, where θ (0, ). We denote by Λ 2P the space of all analytic sequences in P P space. Definition 2.8. Two non-negative sequences x = (x mn ) and y = ( ) are asymptotically equivalent 0 if x mn = 0 m,n and it is denoted by x 0. Definition 2.9. Let K be the subset of N, the set of natural numbers. Then the asymptotically density of K, denoted by δ (K), is defined as δ (K) = m, n k, l : m, n K, where the vertical bars denote the cardinality of the enclosed set. Definition 2.0. A number sequence x = (x mn ) is said to be statistically convergent to the number 0 if for each ϵ > 0, the set K (ϵ) = m k, n l : (m + n)! x mn 0 /m+n ϵ has asympototic density zero m k, n l : ((m + n)! x mn 0 ) /m+n ϵ = 0. In this case we write St x = 0. Definition 2.. The two non-negative double sequences x = (x mn ) and y = ( ) are said to be asymptotically double equivalent of multiple L provided that for every ϵ > 0,

7 00 TWMS J. PURE APPL. MATH., V.9, N., 208 (m, n) : m k, n l, x mn L ϵ = 0. and simply asymptotically double statistical equivalent if L =. Furthermore, let S L θ rs denote the set of all sequences x = (x mn ) and y = ( ) such that x is equivalentto y. Definition 2.2. Let θ rs = (m r, n s ) be a double lacunary sequence; the two double sequences x = (x mn ) and y = ( ) are said to be asymptotically double lacunary statistical equivalent of multiple L provided that for every ϵ > 0, r,s h r,s (m, n) I r,s : x mn L ϵ = 0 and simply asymptotically double lacunary statistical equivalent if L =.Furthermore, let S L θ rs denote the set of all sequences x = (x mn ) and y = ( ) such that x is equivalentto y. Definition 2.3. Let θ rs = (m r, n s ) be a double lacunary sequence; the two double sequences x = (x mn ) and y = ( ) are said to be strong asymptotically double lacunary equivalent of multiple L provided that r,s h r,s (m,n) I r,s x mn L = 0, that is x is equivalent to y and it is denoted by Nθ L rs and simply strong asymptotically double lacunary equivalent if L =. In addition, let Nθ L rs denote the set of all sequences x = (x mn ) and y = ( ) such that x is equivalent to y. Definition 2.4. The double sequence θ rs = (m r, n s ) is called double lacunary sequence if there exist two increasing of integers such that m = 0, h r = m r m r as r and n = 0, h s = n s n s as s. Notations: m r,s = m r m s, h r,s = h r hs and θ rs is determined by I rs = (m, n) : m r < m m r and n s < n n s, q r = m r m r, q s = n s n s and q rs = q r q s. Definition 2.5. Let M be an Musielak modulus function. The two non-negative double sequences x = (x mn ) and y = ( ) are said to be strong M asymptotically double equivalent of multiple 0 provided that [ ] χ 2F M, (d (x, 0), d (x 2, 0),, d (x n, 0)) p = F µ (x) = k l m= n= M f (k+2)(l+2) f mn k m= n= ( (m + n)! l M (/m)+n ) (d 0 ) )], (x, 0), d (x 2, 0),, d (x n, 0) p = 0 = ( (m + n)! x mn (/m)+n ) (d 0 ) )], (x, 0), d (x 2, 0),, d (x n, 0) p = 0, xmn f mn (k, l N), and [ ] Λ 2F M, (d (x, 0), d (x 2, 0),, d (x n, 0)) p = F η (x) k l ( = sup M f mn x mn (/m)+n ) 0 ) )], (d (x, 0), d (x 2, 0),, d (x n, 0) p < = sup f (k+2)(l+2) m= n= k l m= n= M (k, l N), it is denoted by f mn ( (m + n)! xmn (/m)+n ) (d 0 ) )], (x, 0), d (x 2, 0),, d (x n, 0) p <, ] 0 [ M and simply M asymptotically double. Definition 2.6. Let M be an Musielak modulus function and θ rs = (m r, n s ) be a double lacunary sequence; the two double sequences x = (x mn ) and y = ( ) are said to be strong M asymptotically double lacunary of multiple 0

8 DEEPMALA, et al: THE FIBONACCI NUMBERS OF ASYMPTOTICALLY LACUNARY... 0 ] [χ 2F M, (d (x, 0), d (x 2, 0),, d (x n, 0)) p = F µ (x) = r,s h r,s M f mn m I r,s n I r,s = r,s h r,s f (k+2)(l+2) m I r,s n I r,s (k, l N), and ( (m + n)! M x mn f mn (/m)+n ) (d 0 ) )], (x, 0), d (x 2, 0),, d (x n, 0) p = 0. ( x mn ] [Λ 2F M, (d (x, 0), d (x 2, 0),, d (x n, 0)) p = F η (x) = sup r,s h r,s m I r,s n I r,s f (k+2)(l+2) M f mn m I r,s n I r,s ( x mn M f mn ( x mn (/m)+n ) (d 0 ) )], (x, 0), d (x 2, 0),, d (x n, 0) p = 0, (/m)+n ) (d 0 ) )], (x, 0), d (x 2, 0),, d (x n, 0) p < = sup h r,s r,s (/m)+n ) (d 0 ) )], (x, 0), d (x 2, 0),, d (x n, 0) p <, [ M] 0 (k, l N), provided that is denoted by N θ r,s and simply strong M asymptotically double lacunary. Consider the metric space ] [Λ 2F M, (d (x, 0), d (x 2, 0),, d (x n, 0)) p with the metric d (x, y) = sup M (Fη (x) F η (y)) : m, n =, 2, 3,. (7) Consider the metric space [χ 2F M, (d (x, 0), d (x 2, 0),, d (x n, 0)) p ] d (x, y) = sup with the metric M (Fµ (x) F µ (y)) : m, n =, 2, 3,. (8) 3. Almost asymptotically lacunary convergence of P P spaces The idea of statistical convergence was first introduced by Steinhaus in 95 and then studied by various authors. In this paper has studied the concept of statistical convergence in probabilistic p metric spaces. Definition 3.. A triple (X, P, ) be a P P space. Then a sequence x = (x mn ) is said to statistically convergent to 0 with respect to the probabilistic p metric P provided that for every ϵ > 0 and γ (0, ) ) δ (m, n N : P ((m+n)! xmn ) /m+n (ϵ) γ = 0 or equivalently In this case we write St P x = 0. m k, n l : P ((m+n)! xmn ) /m+n (ϵ) γ = 0 Definition 3.2. A triple (X, P, ) be a P P space. The two non-negative sequences x = (x mn ) and y = ( ) are said to be almost asymptotically statistical equivalent of multiple 0 in P P space X if for every ϵ > 0 and γ (0, ).

9 02 TWMS J. PURE APPL. MATH., V.9, N., 208 or equivalently δ ( m, n N : P( m k, n l : P( In this case we write x Ŝ(P P ) y. (m+n)! xmn ymn (m+n)! xmn ymn ) /m+n 0 (ϵ) γ ) = 0 ) /m+n 0 (ϵ) γ = 0. Definition 3.3. A triple (X, P, ) be a P P space and θ = (m r n s ) be a lacunary sequence. The two non-negative sequences x = (x mn ) and y = ( ) are said to be a almost asymptotically lacunary statistical equivalent of multiple 0 in P P space X if for every ϵ > 0 and γ (0, ) or equivalently δ θ (m, n I r,s : P( rs h rs m, n I rs : P( (m+n)! x mn ymn (m+n)! xmn ymn ) /m+n 0 (ϵ) γ ) = 0 (9) ) /m+n 0 (ϵ) γ = 0. In this case we write x Ŝθ(P P ) y. Lemma 3.. A triple (X, P, ) be a P P space. Then for every ϵ > 0 and γ (0, ), the following statements are equivalent: () rs h rs m, n I rs : P( ) (m+n)! /m+n 0 (ϵ) γ = 0, (2) δ θ (m, n I r,s : P( (3) δ θ (m, n I r,s : P( (4) rs h rs m, n I rs : P( (m+n)! x mn ymn (m+n)! x mn ymn xmn ymn (m+n)! x mn ymn ) /m+n 0 (ϵ) γ ) = 0, ) /m+n 0 (ϵ) γ ) =, ) /m+n 0 (ϵ) γ =. 4. Main results Theorem 4.. A triple (X, P, ) be a P P space. If two sequences x = (x mn ) and y = ( ) are almost asympototically lacunary statistical equivalent of multiple 0 with respect to the probabilistic p metric P, then 0 is unique sequence. S 0 θ (P P ) Proof. Assume that x y. For a given λ > 0 choose γ (0, ) such that ( γ) > λ. Then, for any ϵ > 0, define the following set: K = m, n I r,s : P( ) /m+n 0 (ϵ) γ. Then, clearly (m+n)! x mn ymn rs K 0 h rs =,

10 DEEPMALA, et al: THE FIBONACCI NUMBERS OF ASYMPTOTICALLY LACUNARY S 0 θ (P P ) so K is non-empty set,since x y, δ θ (K) = 0 for all ϵ > 0, which implies δ theta (N K) =. If m, n N K, then we have P 0 (ϵ) = P( since λ is arbitrary, we get P 0 (ϵ) =. This completes the proof. ) (m+n)! x mn /m+n 0 ymn (ϵ) > ( γ) λ Theorem 4.2. A triple (X, P, ) be a P P space. For any lacunary sequence θ = (m r n s ), Ŝθ (P P ) Ŝ (P P ) if sup q rs <. rs Proof. If sup q rs <. then there exists a B > 0 such that q rs < B for all r, s. Let rs x Ŝθ(P P ) y and ϵ > 0. Now we have to prove Ŝ (P P ). Set K rs = m, n I r,s : P( ) /m+n 0 (ϵ) > γ. (m+n)! xmn ymn Then by definition, for given ϵ > 0, there exists r 0 s 0 N such that K rs h rs < ϵ 2B for all r > r 0 and s > s 0. Let M = max K rs : r r 0, s s 0 and let uv be any positive integer with m r < u m r and n s < v n s. Then uv m u, n v : P( (m+n)! x mn ) /m+n 0 (ϵ) > γ ymn m r n s m m r, n n s : P( ) /m+n 0 (ϵ) > γ = (m+n)! m r n s K + K rs M m r n s r 0 s 0 + ϵ 2B q rs This completes the proof. xmn ymn M m r n s r 0 s 0 + ϵ 2. Theorem 4.3. A triple (X, P, ) be a P P space. For any lacunary sequence θ = (m r n s ), Ŝ (P P ) Ŝ θ (P P ) if inf rs q rs >. Proof. If inf rs q rs >, then there exists a β > 0 such that q rs > + β for sufficiently large rs, which implies S 0 (pp) h rs K rs β +β. Let x y, then for every ϵ > 0 and for sufficiently large r, s, we have m r n s m m r, n n s : P( (m+n)! x mn ) /m+n 0 (ϵ) > γ ymn m rn s m, n I rs : P( (m+n)! xmn ) /m+n 0 (ϵ) > γ ymn β +β h rs m, n I rs : P( ) /m+n 0 (ϵ) > γ. (m+n)! Therefore x S 0 θ (pp) y. x mn ymn

11 04 TWMS J. PURE APPL. MATH., V.9, N., 208 This completes the proof. Corollary 4.. A triple (X, P, ) be a P P space. For any lacunary sequence θ = (m r n s ), with < inf q rs sup q rs <, then Ŝ (P P ) = rs Ŝθ (P P ). rs Proof. The result clearly follows from Theorem 4.2 and Theorem Competing interests The authors declare that there is no conflict of interests regarding the publication of this research paper. 6. Conclusion This paper contains generalized results for the concept of almost asymptotically lacunary statistical χ 2 over probabilistic p metric spaces with some general topological properties. Researchers can extend the results for more general cases. References [] Altay, J., Başar, F., (2005), Some new spaces of double sequences, Journal of Mathematical Analysis and Applications, 309(), pp [2] Başar, F., Sever, Y., (2009), The space L p of double sequences, Mathematical Journal of Okayama University, 5, pp [3] Basarir, M., Solancan, O., (999), On some double sequence spaces, Journal of Indian Academy Mathematics, 2(2), pp [4] Bromwich, T.J.I A., (965), An Introduction to the Theory of Infinite Series, Macmillan and Co.Ltd., New York. [5] Cannor, J., (989), On strong matrix summability with respect to a modulus and statistical convergence, Canadian Mathematical Bulletin, 32(2), pp [6] Gökhan, A., Çolak, R., (2005), Double sequence spaces l 2, ibid., 60(), pp [7] Hardy, G.H., (97), On the convergence of certain multiple series, Proceedings of the Cambridge Philosophical Society, 9, pp [8] Kamthan, P.K., Gupta, M., (98), Sequence Spaces and Series, Lecture Notes, Pure and Applied Mathematics, 65 Marcel Dekker, In c., New York. [9] Kizmaz, H., (98), On certain sequence spaces, Canadian Mathematical Bulletin, 24(2), pp [0] Lindenstrauss, J., Tzafriri, L., (97), On Orlicz sequence spaces, Israel Journal of Mathematics, 0, pp [] Maddox, I.J., (986), Sequence spaces defined by a modulus, Mathematical Proceedings of the Cambridge Philosophical Society, 00(), pp [2] Moricz, F., (99), Extentions of the spaces c and c 0 from single to double sequences, Acta Mathematica Hungariga, 57(-2), pp [3] Moricz, F., Rhoades, B.E., (988), Almost convergence of double sequences and strong regularity of summability matrices, Mathematical Proceedings of the Cambridge Philosophical Society, 04, pp [4] Mursaleen, M., Edely, O.H.H., (2003), Statistical convergence of double sequences, Journal of Mathematical Analysis and Applications, 288(), pp [5] Mursaleen, M., (2004), Almost strongly regular matrices and a core theorem for double sequences, Journal of Mathematical Analysis and Applications, 293(2), pp [6] Mursaleen, M., Edely, O.H.H., (2004), Almost convergence and a core theorem for double sequences, Journal of Mathematical Analysis and Applications, 293(2), pp [7] Mursaleen, M., Sharma, S.K., Kilicman, A., (203), Sequence spaces defined by Musielak-Orlicz function over n-normed space, Abstract and Applied Analysis, Article ID , pages 0.

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13 06 TWMS J. PURE APPL. MATH., V.9, N., 208 [44] Zeltser, M., (200), Investigation of Double Sequence Spaces by Soft and Hard Analitical Methods, Dissertationes Mathematicae Universitatis Tartuensis, 25, Tartu University Press, Univ. of Tartu, Faculty of Mathematics and Computer Science, Tartu. Deepmala - is working currently as Assistant Professor at Mathematics Discipline, PDPM Indian Institute of Information Technology, Design Manufacturing, Jabalpur, Madhya Pradesh, India. Her research interests are in the areas of pure and applied mathematics including optimization, mathematical programming, fixed point theory and applications, integral equations, operator theory, approximation theory, air craft modeling, dynamic programming etc. Vandana - is a Post Doctoral Researcher at Department of Management Studies, Indian Institute of Technology, Madras, Chennai , Tamil Nadu. Her research interests are in the areas of applied mathematics including optimization, mathematical programming, inventory control, supply chain management, operations research, operations management, approximation theory, summability through functional analysis etc. N. Subramanian - received the Ph.D. degree in Mathematics for Alagappa University at Karaikudi, Tamil Nadu, India and also getting Doctor of Science (D.Sc.) degree in Mathematics for Berhampur University, Berhampur, Odissa, India. His research interests are in the areas of summability through functional analysis of applied mathematics and pure mathematics.

14 DEEPMALA, et al: THE FIBONACCI NUMBERS OF ASYMPTOTICALLY LACUNARY Lakshmi Narayan Mishra - is working as Assistant Professor in the School of Advanced Sciences, Vellore Institute of Technology (VIT) University, Vellore, Tamil Nadu, India. He completed his Ph.D. programme from National Institute of Technology, Silchar, Assam, India. His research interests are in the areas of pure and applied mathematics including special functions, non-linear analysis, fractional integral and differential equations, measure of non-compactness, local global attractivity, approximation theory, Fourier approximation, fixed point theory and applications, q-series and q-polynomials, signal analysis and image processing etc.