Int. J. Open Problems Compt. Math., Vol. 2, No.2, June 2009 Lacunary Statistical Convergence on Probabilistic Normed Spaces Mohamad Rafi Segi Rahmat School of Applied Mathematics, The University of Nottingham Malaysia Campus, Jalan Broga, 43500 Semenyih,Selangor Darul Ehsan. Mohd.Rafi@nottingham.edu.my Abstract In this paper, we study the concepts of lacunary statistical convergent and lacunary statistical Cauchy sequences in probabilistic normed spaces and prove some basic properties. Keywords: lacunary sequence; lacunary statistical convergence; probabilistic norm; probabilistic normed spaces. Introduction Probabilistic normed (PN) spaces are real linear spaces in which the norm of each vector is an appropriate probability distribution function rather than a number. Such spaces were introduced by Šerstnev in 963 [5]. In [] Alsina, Schweizer and Sklar gave a new definition of PN spaces which includes Šerstnev s as a special case and leads naturally to the identification of the principle class of PN spaces, the Menger spaces. In [2], the continuity properties of probabilistic norms and the vector space operations (vector addition and scalar multiplication) are studied in details and it is shown that a PN space endowed with the strong topology turns out to be a topological vector space under certain conditions. A detailed history and the development of the subject up to 2006 can be found in [4]. Şençimen and Pehlivan [3] extended the results in paper [2] to a more general type of continuity, namely, the statistical continuity of probabilistic norms and vector space operations via the concepts of strong statistical convergence (see also [2, 8, 7]). Since the concept of Lacunary statistical convergence is a generalization of the concept of statistical convergence (see [4, 5, 0]), it seems reasonable to think if the concept of lacunary statistical convergence and lacunary statistical
286 Mohd. Rafi Cauchy sequences (see [3, 6, 9]) via the concepts of lacunary statistical strong convergence and lacunary statistical Cauchy can be extended to probabilistic normed spaces and in that case how the basic properties are effected. Since the study of convergence in PN spaces is fundamental to probabilistic functional analysis, we feel that the concept of lacunary statistical convergence and lacunary statistical Cauchy in a PN space would provide a more general framework for the subject. 2 Preinaries We recall some basic definition and results concerning PN spaces, see [, ]. A distance distribution function is a nondecreasing function F defined on R + = [0, + ], with F (0) = 0 and F (+ ) =, and is left-continuous on (0, + ). The set of all distance distribution functions will be denoted by +. The elements of + are partially order by the usual pointwise ordering of functions and has both a maximal element ε 0 and a minimal element ε : these are given, respectively, by ε 0 (x) = { 0, x 0,, x > 0. and ε (x) = { 0, x < +,, x = +. There is a natural topology on + that is induced by the modified Lévy metric d L (see, [], Sec. 4.2). Convergence witespect to the metric d L is equivalent to weak convergence of distribution functions, i.e., {F n } in + and F in +, the sequence {d L (F n, F )} converges to 0 if and only if the sequence {F n (x)} converges to F (x) at every point of continuity of the it function F. Moreover, the metric space ( +, d L ) is compact and complete. A triangle function is a binary operation τ on + that is commutative, associative, non-decreasing in each place, and has ε 0 as an identity element. Continuity of triangle function means uniform continuity witespect to the natural product topology on + +. Definition 2. A probabilistic normed space (briefly, a P N space) is a quadruple (V, η, τ, τ ) where V is a real linear space, τ and τ are continuous triangle functions, and, η be is a mapping from V into the space of distribution functions + such that - writing N p for η(p)-for all p, q in V, the following conditions hold: (N) N p = ε 0 if and only if p = θ, the null vector in V, (N2) N p = N p, (N3) N p+q τ(n p, N q ), (N4) N p τ (N αp, N ( α)p ), for all α in [0, ].
Lacunary Statistical Convergence on PN spaces 287 It follows from (N), (N2), (N3) that if F : S S + is defined via F(p, q) = F pq = N p q, () then (V, F, τ) is a PM space ([], Chap. 8). Furthermore, since τ is continuous, the system of neighborhoods {N p (t): p V, t > 0}, where N p (t) = {q V : d L (F pq, ε 0 ) < t} = {q V : F pq (t) > t} (2) determine a first-countable and Hausdorff topology on V, called the strong topology. Thus, the strong topology can be completely specified in terms of the convergence of sequences. Throughout this paper, V denotes a PN space endowed with the strong topology, written additively, which satisfies the first axiom of countability. A sequence {p n } in V converges strongly to a point p V, and we write n p n = p, if for any t > 0 there is a positive integer m such that p n N p (t) for all n m. Similarly, a sequence {p n } in V is a strong Cauchy sequence if for any t > 0 there is a positive integer i such that (p n, p m ) U(t) for all n, m i, where U(t) = {(p, q) V V : d L (F pq, ε 0 ) < t} for any t > 0 is called the strong vicinity(see []). Definition 2.2 [8] A sequence {p k } in V is statistically strong convergent to θ the null vector in V provided that for every t > 0 n n {k n: d L(N pk, ε 0 ) t} = 0. In this case we write S k p k = θ or p k θ(s). We shall use S to denote the set of all statistically strong convergent sequences in V. Of course, there is nothing special about θ as a it; if one wishes to consider the convergence of the sequence {p n } to the vector p, then it suffices to consider the sequence {p n p} and its convergence to θ. The statistical strong Cauchy sequence in PN space can be defined in a similar way as Definition 2.3 A sequence {p k } in V is a statistically strong Cauchy sequence if there there exists a positive integer m such that n n {k n: (p k p m ) / U(t)} = 0.
288 Mohd. Rafi 3 Lacunary statistical convergence and some basic properties By a lacunary sequence ϑ = {k r }; r = 0,, 2,, where k 0 = 0, we means an increasing sequence of nonnegative integers with = k r k r as r. The interval determined by ϑ will be denoted by I r = (k r, k r ] and q r = kr. A real number sequence {x k r k} is said to be lacunary statistically convergent (briefly S ϑ -convergent) to a R provided that for each t > 0 {k I r : x k a t} = 0. A sequence {x k } is a cauchy sequence if there exists a subsequence {x k (r)} of {x k } such that k (r) I r for eac, x k (r) = l and for every t > 0 {k I r : x k x k (r) t} = 0. Using these concepts, we extend the lacunary statistical convergence and lacunary statistical Cauchy to the setting of sequences in a PN space endowed with the strong topology as follows. Definition 3. Let ϑ be a lacunary sequence. A sequence {p k } in V is said to be lacunary statistically strong convergent (briefly, S ϑ -strong convergent) to θ in V if for each t > 0, or, equivalently, {k I r : d L (N pk, ε 0 ) t} = 0 (3) {k I r : p k / N θ (t)} = 0, (4) where N θ (t) = {p V : N p (t) > t} is the neighborhood of θ. In this case, we write S ϑ p k = p or p k θ(s ϑ ), and we will call θ, as the lacunary strong it of the sequence {p k }. We shall use S ϑ to denote the set of all lacunary strong convergent sequences from V. Of course, there is nothing special about θ as a it; if one wishes to consider the S ϑ -strong convergent of the sequence {p n } to the vector p, then it suffices to consider the sequence {p n p} and its S ϑ -strong convergent to θ. Theorem 3.2 Let ϑ be a lacunary sequence. If {p k } is a S ϑ -strong convergent sequence in V, then its it is unique.
Lacunary Statistical Convergence on PN spaces 289 Proof. Suppose the sequence {p k } is S ϑ -strong convergent to two distinct points p and q (say). Since p q, we have N p q ε 0, whence t = d L (N p q, ε 0 ) > 0. Set K = {k I r : p p k N θ (t/2)}, K 2 = {k I r : p k q N θ (t/2)}. K Then, clearly K 2 =, so K K 2 is a nonempty set. Let m K K 2, then d L (N p pm, ε 0 ) < t/2 and d L (N pm q, ε 0 ) < t/2. By uniform continuity of τ, we have d L (N p q, ε 0 ) d L (τ(n p pm, N pm q), ε 0 ) < t = d L (N p q, ε 0 ), a contradiction to the fact that K K 2 is a nonempty set. Therefore p = q and the proof is completed. Theorem 3.3 For any lacunary sequence ϑ, S ϑ S if sup r q r <. Proof. If sup r q r < then there exists a γ > 0 such that q r < γ for all r. Let S ϑ k p k = θ. We are going to prove that S k p k = θ. Set K r = {k I r : p k / N θ (t)}. Then, by definition, for a given t > 0, there exists r 0 N such that K r < t 2γ for all r r 0. Let M = max{k r : r r 0 } and let n N such that k r < n k r. Then we can write n {k n: p k / N θ (t)} = and the result follows immediately. {k k r : p k / N θ (t)} k r {K + K 2 + + K r0 + + K r } k r M r 0 + t k r 2γ q r M r 0 + t k r 2 Theorem 3.4 For any lacunary sequence ϑ, S S ϑ if sup r q r >. Proof. If sup r q r > then there exists a ξ > 0 and a positive integer r 0 such that q r + ξ for all r r 0. Hence for r r 0, k r = k r k r = ξ + ξ.
290 Mohd. Rafi Let S k p k = θ. Then for every t > 0 and for every r r 0, we have k r {k k r : p k / N θ (t)} k r {k I r : p k / N θ (t)} Therefore S ϑ k p k = θ. ξ + ξ {k I r : p k / N θ (t)}. Corollary 3.5 Let ϑ be a lacunary sequence, then S = S ϑ if < inf r q r sup r <. Proof. By combining the Theorem 3.3 and Theorem 3.4. Definition 3.6 Let ϑ be a lacunary sequence. A sequence {p k } in V is said to be S ϑ -strong Cauchy sequence if there exists a subsequence {p k (r)} of {p k } such that k (r) I r for eac, p k (r) = p and for every t > 0 {k I r : p k p k (r) / N θ (t)} = 0, (5) Theorem 3.7 The sequence {p k } in V is S ϑ -strong convergent if and only if it is S ϑ -strong Cauchy sequence in V. Proof. Let S ϑ k p k = θ and write K n = {k N: p k N θ (/n)}, K for each n N. Then, obviously K n+ K n for each n and n I r =. This implies that there exists m such that r m and K I r > 0, i.e., K I r. We next choose m 2 m such that r m 2 implies that K 2 I r. Thus for eac satisfying m r m 2, we choose k (r) I r such that k (r) I r K, i.e., p k (r) N θ (). In general we choose m n+ > m n such that r m n+ implies that k (r) I r K n, i.e., p k (r) N θ (/n). Thus k (r) I r, for eac and p k (r) N θ (/n) implies that r p k (r) = θ. Furthermore, for t > 0 and the uniform continuity of τ implies that {k I r : d L (N pk p k (r), ε 0 ) t} {k I r : d L (τ(n pk, N pk (r) ), ε 0 ) t} {k I r : d L (N pk, ε 0 ) t/2} {k I r : d L (N pk (r), ε 0 ) t/2}. The above inclusion implies {k I r : p k p k (r) / N θ (t)} {k I r : p k / N θ (t/2)} + {k I r : p k (r) / N θ (t/2)}.
Lacunary Statistical Convergence on PN spaces 29 Since S ϑ k p k = θ and r p k (r) = θ, it follows that {p k } is a S ϑ -strong Cauchy sequence. Conversely, suppose that {p k } is a S ϑ -strong Cauchy sequence. For every t > 0 and the uniform continuity of τ, we have {k I r : d L (N pk, ε 0 ) t} {k I r : d L (τ(n pk p k (r), N pk (r) ), ε 0 ) t} {k I r : d L (N pk p k (r), ε 0 ) t/2} {k I r : d L (N pk (r), ε 0 ) t/2}. The above inclusion implies {k I r : p k / N θ (t)} {k I r : p k p k (r) / N θ (t/2)} + {k I r : p k (r) / N θ (t/2)} for which it follows that S ϑ k p k = θ. Corollary 3.8 If {p k } in V is a S ϑ -strong convergent sequence, then {p k } has a strong convergent subsequence. Proof. The proof is an immediate consequence of Theorem 3.4. 4 Conclusion We study the concepts of lacunary statistical convergent and lacunary statistical Cauchy sequences in probabilistic normed spaces and proved several important properties of sequences in probabilistic normed spaces. 5 Open Problem It can be easily proved that if a sequence {p n } in V is a strong convergent sequence in V then it is a S ϑ -strong convergent sequence in V. But the converse is not necessarily true. Find a suitable condition(s) so that the converse of the above proposition valid. ACKNOWLEDGEMENTS. The author would like to thank the referee for giving useful comments and suggestions for improving the paper. References [] C. Alsina, B. Schweizer, and A. Sklar, On the definition of a probabilistic normed space, Aequationes Math. 46(993), 9-98.
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