The Ideal Convergence of Difference Strongly of

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1 International Journal o Mathematical Analyi Vol. 9, 205, no. 44, HIKARI Ltd, The Ideal Convergence o Dierence Strongly o χ 2 in p Metric Space Deined by Modulu C. Murugean Department o Mathematic Sathyabama Univerity Chennai-600 9, India N. Subramanian Department o Mathematic SASTRA Univerity Thanjavur-63 40, India Copyright c 205 C. Murugean and N. Subramanian. Thi article i ditributed under the Creative Common Attribution Licene, which permit unretricted ue, ditribution, and reproduction in any medium, provided the original work i properly cited. Abtract The aim o thi paper i to introduce and tudy a new concept o the χ 2 pace via ideal convergence o dierence deined by modulu. Some topological propertie o the reulting equence pace are alo examined. Mathematic Subject Claiication: 40A05, 40A99 Keyword: analytic equence, modulu unction, double equence, χ 2 pace, n-metric pace

2 290 C. Murugean and N. Subramanian. Introduction Throughout w, χ and Λ denote the clae o all, gai and analytic calar valued ingle equence, repectively. We write w 2 or the et o all complex equence x mn ), where m, n N, the et o poitive integer. Then, w 2 i a linear pace under the coordinate wie addition and calar multiplication. Some initial work on double equence pace i ound in. Later on it wa invetigated by 2,3,4,5,6,7,8,9,0,,2,3,4,5,6,7,8,9 and many other. We procure the ollowing et o double equence: M u t) := x mn ) w 2 : up m,n N x mn tmn <, C p t) := x mn ) w 2 : p lim m,n x mn l tmn = or ome l C, C 0p t) := x mn ) w 2 : p lim m,n x mn tmn =, L u t) := x mn ) w 2 : m= n= x mn tmn <, C bp t) := C p t) M u t) and C 0bp t) = C 0p t) M u t); where t = t mn ) be the equence o poitive real t mn or all m, n N and p lim m,n denote the limit in the Pringheim ene. In the cae t mn = or 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 et M u, C p, C 0p, L u, C bp and C 0bp, repectively. Now, we may ummarize the knowledge given in ome document related to the double equence pace. 20 have proved that M u t) and C p t), C bp t) are complete paranormed pace o double equence and obtained the α, β, γ dual o the pace M u t) and C bp t). Zelter 200) in her phd thei ha eentially tudied both the theory o topological double equence pace and the theory o ummability o double equence have independently introduced the tatitical convergence and Cauchy or double equence and etablihed the relation between tatitical convergent and trongly Ceàro ummable double equence. 28 have deined the pace BS, BS t), CS p, CS bp, CS r and BV o double equence coniting o all double erie whoe equence o partial um are in the pace M u, M u t), C p, C bp, C r and L u, repectively, and alo examined ome propertie o thoe equence pace and determined the α dual o the pace BS, BV, CS bp and the β ϑ) dual o the pace CS bp and CS r o double erie. 29 have introduced the Banach pace L q o double equence correponding to the well-known pace l q o ingle equence and examined ome propertie o the pace L q. Subramanian et al. 200) have tudied the pace χ 2 M p, q, u) o double equence and proved ome incluion relation. The cla o equence which are trongly Ceàro ummable with repect to a modulu wa introduced by 3 a an extenion o the deinition o trongly Ceàro ummable equence. Connor 989) urther extended thi deinition to a deinition o trong A ummability with repect to a modulu where A = a n,k ) i a nonnegative regular matrix and etablihed ome connection between trong A ummability, trong A ummability with repect to a modulu, and A tatitical convergence. In 33 the our dimenional matrix tranormation Ax) k,l = m= n= amn kl x mn wa tudied extenively by Robion and Hamilton.

3 The ideal convergence o dierence trongly o χ We need the ollowing inequality in the equel o the paper. For a, b 0 and 0 < p <, we have.) a + b) p a p + b p The double erie m,n= x mn i called convergent i and only i the double equence mn ) i convergent, where mn = m,n i,j= x ijm, n N). A equence x = x mn )i aid to be double analytic i up mn x mn /m+n <. The vector pace o all double analytic equence will be denoted by Λ 2. A equence x = x mn ) i called double gai equence i m + n)! x mn ) /m+n 0 a m, n. The double gai equence will be denoted by χ 2. Let φ = all inite equence. Conider a double equence x = x ij ). The m, n) th ection x m,n o the equence i deined by x m,n = m,n i,j=0 x iji ij or all m, n N ; where I ij denote the double equence i+j)! in the i, j) th place or each i, j N. An FK-paceor a metric pace)x i aid to have AK property i I mn ) i a Schauder bai or X. Or equivalently x m,n x. An FDK-pace i a double equence pace endowed with a complete metrizable; locally convex topology under which the coordinate mapping x = x k ) x mn )m, n N) are alo continuou. An Orlicz unction i a unction : 0, ) 0, ) which i continuou, non-decreaing and convex with 0) = 0, x) > 0, or x > 0 and x) a x. I convexity o Orlicz unction i replaced by x + y) x)+ y), then thi unction i called modulu unction. An modulu unction i aid to atiy 2 condition or all value u, i there exit K > 0 uch that 2u) K u), u 0. Remark :An Modulu unction atiie the inequality λx) λx) or all λ with 0 < λ <... Lemma. Let be an modulu unction which atiie 2 condition and let 0 < δ <. Then or each t δ, we have t) < Kδ 2) or ome contant K > 0. Let M and Φ be mutually complementary modulu unction. Then, we have i) For all u, y 0,.2) uy M u) + Φ y), Y oung inequality)34 ii) For all u 0,.3) uη u) = M u) + Φ η u)).

4 292 C. Murugean and N. Subramanian iii) For all u 0, and 0 < λ <,.4) M λu) λm u). 35 ued the idea o Orlicz unction to contruct Orlicz equence pace l M = x w : ) k= M xk ρ <, or ome ρ > 0, The pace l M with the norm x = in ρ > 0 : ) k= M xk ρ, become a Banach pace which i called an Orlicz equence pace. For M t) = t p p < ), the pace l M coincide with the claical equence pace l p. A equence = mn ) o modulu unction i called a Muielak-modulu unction. A equence g = g mn ) deined by g mn v) = up v u mn u) : u 0, m, n =, 2, i called the complementary unction o a Muielak-modulu unction. For a given Muielak modulu unction, the Muielak-modulu equence pace t i deined by t = x w 3 : M x mnk ) /m+n+k 0 a m, n, k, where M i a convex modular deined by M x) = m= n= k= mnk x mnk ) /m+n+k, x = x mnk ) t. We conider t equipped with the Luxemburg metric pace, i.e)) Let X i, d i ), i I be a amily o metric pace uch that each two element o the amily are dijoint. Denote X : i I X i. I we deine d x, y) = d i x, y), i x, y X i + i x X i, y X j, i j then the pair X, d) i a Luxemburg metric pace. The notion o dierence equence pace or ingle equence) wa introduced by Kizmaz 36 a ollow Z ) = x = x k ) w : x k ) Z, or Z = c, c 0 and l, where x k = x k x k+ or all k N. Here c, c 0 and l denote the clae o convergent,null and bounded clar valued ingle equence repectively. The dierence equence pace bv p o the claical pace l p i introduced and tudied in the cae p by Başar and Altay and in the cae 0 < p <. The pace c ), c 0 ), l ) and bv p are Banach pace normed by x = x + up k x k and x bvp = k= x k p ) /p, p < ). Later on the notion wa urther invetigated by many other. We now introduce the ollowing dierence double equence pace deined by Z ) = x = x mn ) w 2 : x mn ) Z,

5 The ideal convergence o dierence trongly o χ 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+ or all m, n N. The generalized dierence double notion ha the ollowing repreentation: m x mn = m x mn m x mn+ m x m+n + m x m+n+, and alo thi generalized dierence double notion ha the ollowing binomial repreentation: m x mn = m m m) ) m i=0 j=0 )i+j i j x m+i,n+j. 2. Deinition and Preliminarie Let m X be a non empty et. A non-void cla I 2 mx power et, o m X) i called an ideal i I i additive i.e A, B I A B I) and hereditary i.e A I and B A B I). A non-empty amily o et F 2 mx i aid to be a ilter on m X i φ / F ; A, B F A B F and A F, A B B F. For each ideal I there i a ilter F I) given by F I) = K N : N \ K I. A non-trivial ideal I 2 mx i called admiible i and only i x : x m X I. A double equence pace E i aid to be olid or normal i α mn m x mn ) E, whenever m x mn ) E and or all double equence α = α mn ) o calar with α mn. or all m, n N. Let n N and X be a real vector pace o dimenion w, where n w. A real valued unction d p x,..., x n ) = d x, 0),..., d n x n, 0)) p on X atiying the ollowing our condition: i) d x, 0),..., d n x n, 0)) p = 0 i and and only i d x, 0),..., d n x n, 0) are linearly dependent, ii) d x, 0),..., d n x n, 0)) p i 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 or p < ; or) v) d x, y ), x 2, y 2 ), x n, y n )) := up d X x, x 2, x n ), d Y y, y 2, y n ), or x, x 2, x n X, y, y 2, y n Y i called the p product metric o the Carteian product o n metric pace i the p norm o the n-vector o the norm o the n ub pace. A trivial example o p product metric o n metric pace i the p norm pace i X = R equipped with the ollowing Euclidean metric in the product pace i the p norm: d x, 0),..., d n x n, 0)) E = up detd 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) up... d n x n, 0) d n2 x n2, 0)... d nn x nn, 0) where x i = x i, x in ) R n or each i =, 2, n. I every Cauchy equence in X converge to ome L X, then X i aid to be complete with repect to the p metric. Any complete p metric pace i aid to be p Banach metric pace.

6 294 C. Murugean and N. Subramanian 3. Main Reult In thi ection we introduce the notion o dierent type o I convergent double equence. Thi generalize and uniie dierent notion o convergence or χ 2. We hall denote the ideal o 2 N N by I 2. Let I 2 be an ideal o 2 N N, be an modulu unction, η = η mn ) be a double analytic equence o trictly poitive real number and m X, d x, 0),..., d n x n, 0)) p ) be an p product o n metric pace i the p norm o the n-vector o the norm o the n ubpace. Further χ 2 p m X) denote m X valued equence pace. Now, we deine the ollowing equence pace: χ 2I 2 d x, 0),..., d n x n, 0)) p η = x = m x mn ) χ 2 p m X) : ɛ > 0, r, ) N N : r r m= n= m x mn ) /m+n ηmn, d x, 0),..., d n x n, 0)) p ɛ I 2, or every d x, 0),..., d n x n, 0) m X. Λ 2I 2 d x, 0),..., d n x n, 0)) p η = x = x mn ) Λ 2 p m X) : K > 0, r, ) N N : r r m= n= m x mn /m+n ηmn, d x, 0),..., d n x n,0 )) p K I 2, or every d x, 0),..., d n x n, 0) m X. Λ 2 d x, 0),..., d n x n, 0)) p η = x = m x mn ) Λ 2 p m X) : K > 0, r, ) N N : r r m= n= every d x, 0),..., d n x n, 0) m X. I η = η mn = or all m, n N we obtain m x mn /m+n, d x, 0),..., d n x n, 0)) p ηmn K, or χ 2I 2 d x, 0),..., d n x n, 0)) p η = χ 2I 2 d x, 0),..., d n x n, 0)) p, Λ 2I 2 d x, 0),..., d n x n, 0)) p η = Λ 2I 2 d x, 0),..., d n x n, 0)) p, Λ 2 d x, 0),..., d n x n, 0)) p η = Λ 2 d x, 0),..., d n x n, 0)) p. The ollowing well-known inequality will be ued in thi tudy: 0 in mn η mn = H 0 η mn up mn = H <, D = max, 2 H ), then x mn + y mn ηmn D x mn ηmn + y mn ηmn or all m, n N and x mn, y mn C. Alo x mn ηmn/m+n max, x mn H/m+n) or all x mn C. 3.. Theorem. The clae o equence χ 2I 2 d x, 0),..., d n x n, 0)) p ηmn, Λ 2I 2 d x, 0),..., d n x n, 0)) p ηmn are linear pace over the complex ield C Proo: Now we etablih the reult or the cae χ 2I 2 d x, 0),..., d n x n, 0)) p ηmn and the other can be proved imilarly. Let x, y χ 2I 2 d x, 0),..., d n x n, 0)) p ηmn and α, β C. Then r, ) N N : r r m= n= I 2 and r, ) N N : r r m= n= m x mn ) /m+n, d x, 0),..., d n x n, 0)) p ηmn ɛ 2 m y mn ) /m+n, d x, 0),..., d n x n, 0)) p ηmn ɛ 2

7 The ideal convergence o dierence trongly o χ I 2. Since d x, 0),..., d n x n, 0)) p be an p product o n metric pace i the p norm o the n-vector o the norm o the n ubpace and i an modulu unction, the ollowing inequality hold: r r m= n= α m x mn+β m y mn /m+n ηmn, d α /m+n + β /m+n x, 0),..., d n x n, 0)) p D r r m= n= D α /m+n α /m+n + β /m+n β /m+n α /m+n + β /m+n m x mn ) /m+n ηmn, d x, 0),..., d n x n, 0)) p + 2 ) ηmn y mn /m+n, d x, 0),..., d n x n, 0)) p r r m= n= D r r m= n= m x mn ) /m+n ηmn, d x, 0),..., d n x n, 0)) p + D r r m= n= m y mn ) /m+n ηmn, d x, 0),..., d n x n, 0)) p. From the above inequality we get r, ) N N : r r m= n= α m x mn+β m y mn ) /m+n ηmn, d α /m+n + β /m+n x, 0),..., d n x n, 0)) p ɛ m x mn ) /m+n ηmn, d x, 0),..., d n x n, 0)) p ɛ 2 r, ) N N : D r r m= n= I 2 r, ) N N : D r r m= I 2. Thi complete the proo. n= m y mn /m+n ηmn, d x, 0),..., d n x n, 0)) p ɛ Theorem. The cla o equence χ 2I 2 d x, 0),..., d n x n, 0)) p η i a paranormed pace with repect to the paranorm deined by g r x) = in up r r r m= n= m x mn ) /m+n, d x ),..., d n x n )) p ηmn ) H, or every d x, 0),..., d n x n, 0) X. Proo: g r θ) = 0 and g r x) = g r x) are eay to prove, o we omit them. Let u take x, y χ 2I 2 d x, 0),..., d n x n, 0)) p ηmn. Let g r x) = in up r r r m= n= m x mn ) /m+n ηmn, d x, 0),..., d n x n, 0)) p, x X and g r y) = in up r r r m= n= m y mn ) /m+n ηmn, d x, 0),..., d n x n, 0)) p, x X. Then we have up r r r m= n= m x mn + m y mn ) /m+n ηmn, d x, 0),..., d n x n, 0)) p up r r r m= n= up r r r m= n= Thu up r r r m= n= m x mn ) /m+n ηmn, d x, 0),..., d n x n, 0)) p + m y mn ) /m+n ηmn, d x, 0),..., d n x n, 0)) p. m x mn + m y mn ) /m+n, d x, 0),..., d n x n, 0)) p ηmn and g r x + y) = g r x) + g r y). Now, let λ u mn λ, where λ u mn, λ C and g r m x u mn m x mn ) 0 a u. We have to prove that g r λ mn m x u mn λ m x mn ) 0 a u. Let g r x u ) =

8 296 C. Murugean and N. Subramanian up r r r m= n= m x u mn ) /m+n ηmn, d x, 0),..., d n x n, 0)) p, x X and g r x u x) = up r r r m= n= m x u mn m x mn ) /m+n ηmn, d x, 0),..., d n x n, 0)) p, orall x X. We oberve that ) λu mn m x u mn λ m x mn ) /m+n, d λ u mn λ /m+n + λ /m+n x, 0),..., d n x n, 0)) p ) λu mn xu mn λ m x u mn )/m+n, d λ u mn λ /m+n + λ /m+n x, 0),..., d n x n, 0)) p + ) λ m x u mn λ m x mn ) /m+n, d λ u mn λ /m+n + λ /m+n x, 0),..., d n x n, 0)) p ) λ u mn λ λ u mn λ + λ m x u mn ) /m+n, d x, 0),..., d n x n, 0)) p + ) λ λ u mn λ + λ m x u mn m x mn ) /m+n, d x, 0),..., d n x n, 0)) p. From thi inequality, it ollow that λu mn m x u mn λ 2 x mn ) /m+n λ u mn λ /m+n + λ /m+n, d x, 0),..., d n x n, 0)) p ηmn and conequently g r λ u mn m x u mn λ m x mn ) λ u mn λ ) ηmn H in gr m x u mn) + λ ) ηmn H in gr m x u mn x) max λ, λ ) ηmn H g r m x u mn m x mn ). Hence by our aumption the right hand ide tend to zero a u, m and n. Thi complete the proo Theorem. i) I 0 < in mn η mn = H 0 η mn <, then χ 2I 2 d x, 0),..., d n x n, 0)) p η χ 2I 2 d x, 0),..., d n x n, 0)) p. ii) I η mn up mn η mn = H <, then χ 2I 2 d x, 0),..., d n x n, 0)) p χ 2I 2 d x, 0),..., d n x n, 0)) p η. iii) I 0 < η mn < µ mn < and µmn η mn i double analytic, then χ 2I 2 d x, 0),..., d n x n, 0)) p η χ 2I 2 d x, 0),..., d n x n, 0)) p µ. Proo: The proo can be etablihed uing tandard technique. The ollowing reult i well known Lemma. I a equence pace E i olid, then it i monotone Theorem. The cla o equence χ 2I 2 d x, 0),..., d n x n, 0)) p η i not olid and hence not monotone Proo: It i routine veriication. Thereore we omit the proo Theorem. Let, and 2 be modulu unction. Then we have i) χ 2I 2 d x, 0),..., d n x n, 0)) p η χ 2I 2 d x, 0),..., d n x n, 0)) p η ii)χ 2I 2 d x, 0),..., d n x n, 0)) p η χ 2I 2 2 d x, 0),..., d n x n, 0)) p η

9 The ideal convergence o dierence trongly o χ χ 2I d x, 0),..., d n x n, 0)) p η Proo: i) Let in mn η mn = H 0. For given ɛ > 0, we irt chooe ɛ 0 > 0 uch that max ɛ H 0, ɛh 0 0 ɛ. Now uing the continuity o, chooe 0 < δ < uch that 0 < t < δ implie t) < ɛ 0. Let m x χ 2I 2 d x, 0),..., d n x n, 0)) p η We oberve that A δ) = r, ) N N : r r m= n= m x mn ) /m+n ηmn, d x, 0),..., d n x n, 0)) p δ H I 2. Thu i r, ) / A δ) then r r m= n= r m= n= m x mn ) /m+n ηmn, d x, 0),..., d n x n, 0)) p < δ H m x mn ) /m+n ηmn, d x, 0),..., d n x n, 0)) p < rδ H, m x mn ) /m+n, d x, 0),..., d n x n, 0)) p ηmn < δ H, or all m, n =, 2,. m x mn ) /m+n, d x, 0),..., d n x n, 0)) p ) < δ, or all m, n =, 2,. Hence rom above inequality and uing continuity o, )) we mut have m x mn ) /m+n, d x, 0),..., d n x n, 0)) p < ɛ 0, or all m, n =, 2,. n= m x mn ) /m+n ηmn, d x, 0),..., d n x n, 0)) p < r max r m= r ɛ r r m= n= m x mn ) /m+n ηmn, d x, 0),..., d n x n, 0)) p < ɛ. ɛ H 0, ɛh 0 0 Hence we have r, ) N N : r r m= n= m x mn ) /m+n ηmn, d x, 0),..., d n x n, 0)) p ɛ A δ) I 2. ii) Let x χ 2I 2 d x, 0),..., d n x n, 0)) p η χ 2I 2 2 d x, 0),..., d n x n, 0)) p η. Then the act that r r m= n= + 2 ) m x mn ) /m+n ηmn, d x, 0),..., d n x n, 0)) p D r r m= n= m x mn ) /m+n ηmn, d x, 0),..., d n x n, 0)) p + 2 m x mn ) /m+n ηmn, d x, 0),..., d n x n, 0)) p. Thi complete the D r r m= n= proo Theorem. The cla o equence Λ 2I 2 d x, 0),..., d n x n, 0)) p η i a equence algebra Proo: Let m x mn ), m y mn ) Λ 2I 2 d x, 0),..., d n x n, 0)) p η and 0 < ɛ <. Then the reult ollow rom the ollowing incluion relation: r, ) N N : r r m= n= m x mn m y mn /m+n ηmn, d x, 0),..., d n x n, 0)) p < ɛ I 2 r, ) N N : r r m= n= m x mn /m+n ηmn, d x, 0),..., d n x n, 0)) p < ɛ I 2 r, ) N N : r r m= n= m y mn /m+n ηmn, d x, 0),..., d n x n, 0)) p < ɛ I 2. Similarly we can prove the reult or other cae. < <

10 298 C. Murugean and N. Subramanian Reerence T.J.I A. Bromwich, An Introduction to the Theory o Ininite Serie, Macmillan and Co.Ltd., New York, G.H. Hardy, On the convergence o certain multiple erie, Proceeding o the Cambridge Philoophical Society, 9 97), F. Moricz, Extention o the pace c and c 0 rom ingle to double equence, Acta Mathematica Hungariga, 57 99), no. -2, F. Moricz and B.E. Rhoade, Almot convergence o double equence and trong regularity o ummability matrice, Mathematical Proceeding o the Cambridge Philoophical Society, ), M. Baarir and O. Solancan, On ome double equence pace, Journal o Indian Academy Mathematic, 2 999), no. 2, B.C. Tripathy and S. Mahanta, On a cla o vector-valued equence aociated with multiplier equence, Acta Mathematica Applicata Sinica Engl. Ser.), ), no. 3, B.C. Tripathy and A.J. Dutta, On uzzy real-valued double equence pace 2 l p F, Mathematical and Computer Modelling, ), no. 9-0, B.C. Tripathy and B. Sarma, Statitically convergent dierence double equence pace, Acta Mathematica Sinica, ), no. 5, B. Tripathy and B. Sarma, Vector valued double equence pace deined by Orlicz unction, Mathematica Slovaca, ), no. 6, B.C. Tripathy and A.J. Dutta, Bounded variation double equence pace o uzzy real number, Computer and Mathematic with Application, ), no. 2, B.C. Tripathy and B. Sarma On I convergent double equence pace o uzzy number, Kyungpook Math. Journal, ), no. 2, B.C. Tripathy and P. Chandra, On ome generalized dierence paranormed equence pace aociated with multiplier equence deined by modulu unction, Analyi in Theory and Application, 27 20), no., B.C. Tripathy and A.J. Dutta, Lacunary bounded variation equence o uzzy real number, Journal in Intelligent and Fuzzy Sytem, ), no., B.C. Tripathy, On tatitically convergent double equence, Tamkang Journal o Mathematic, ), no. 3, B.C. Tripathy and M. Sen, Characterization o ome matrix clae involving paranormed equence pace, Tamkang Journal o Mathematic, ), no. 2, B.C. Tripathy and B. Hazarika, I convergent equence pace aociated with multiplier equence, Mathematical Inequalitie and Application, 2008), no. 3, B.C. Tripathy and H. Dutta, On ome new paranormed dierence equence pace deined by Orlicz unction, Kyungpook Mathematical Journal, ), no.,

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12 2200 C. Murugean and N. Subramanian 36 B.C. Tripathy, M. Sen and S. Nath, I-convergence in probabilitic n-normed pace, Sot Computing, 6 202), B.C. Tripathy, B. Hazarika and B. Choudhary, Lacunary I-convergent equence, Kyungpook Mathematical Journal, ), no. 4, Received: Augut 0, 205; Publihed: September, 205

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

TWMS J. Pure Appl. Math. V.9, N., 208, pp.94-07 THE FIBONACCI NUMBERS OF ASYMPTOTICALLY LACUNARY χ 2 OVER PROBABILISTIC p METRIC SPACES DEEPMALA, VANDANA 2, N. SUBRAMANIAN 3 AND LAKSHMI NARAYAN MISHRA