Journal of Applied Mathematics and Physics, 2015, 3, 12071217 Published Online September 2015 in SciRes. http://www.scirp.org/journal/jamp http://dx.doi.org/10.4236/jamp.2015.39148 How to cite this paper: Bilgin, T. (2015) (f, p)Asymptotically Lacunary Equivalent Sequences with Respect to the Ideal I. Journal of Applied Mathematics and Physics, 3, 12071217. http://dx.doi.org/10.4236/jamp.2015.39148 (f, p)Asymptotically Lacunary Equivalent Sequences with Respect to the Ideal I Tunay Bilgin Department of Mathematics, Education Faculty, Yuzuncu Yil University, Van , Turk ey Email: tbilgin@yyu.edu.tr Received 6 Octo ber 20 14; accepted 27 September 2015; published 30 September 2015 Copyright © 2015 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativ ecommon s.org/l icenses/by/4. 0/ Abstract In this study, we define (f, p)Asymptotically Lacunary Equivalent Sequences with respect to the ideal I using a nontrivial ideal , a lacunary sequence , a strictly positive se quence , and a modulus function f, and obtain some revelent connections between these notions . Keywords Asymptotically Equivalenc e, Ideal Convergence , Lacunary Seque nce, Modulus Function, Statistically Limit 1. Introduction Let denote the spaces of all real sequences, bounded, and convergent sequences,respectively. Any subspace of s is called a sequence space. Following Freedman et al. [1], we call the sequence lacunary if it is an increasing sequence of integers such that as . The intervals determined by will be denoted by and . These notations will be used troughout the paper. The sequence space of lacunary stron gly co nvergent s equences was defined by Freedman et al. [1], a s follows: ( ) 1 :lim0for some r irr i iI Nx xshxss θ − ∈ = =∈−= ∑ . The notion of modulus function was introduced by Nakano [2]. We recall that a modulus f is a function fro m to such that 1) if and only if , 2) for , 3) f is increasing and 4) f is continuous from the right at 0. Hence f must be continuous everywhere on .
T. Bilgin Connor [3], Ko lk [4], Maddox [5], Öztürk a nd Bi lgin [6], Pehliva n and Fi sher [7], Ruck le [8] and others used a modulus function to construct sequence spaces. Marouf presented definitions for asymptotically equivalent sequences and asymptotic regular matrices in [9]. Patterson extended these concepts by presenting an asymptotically statistical equivalent analog of these defi nitions and natural regularity conditions for nonnegative summability matrices in [10]. Subsequently, many authors have shown their inter e st to solve differe nt p roblems arising in this area (see [11][13]). The concept of Iconvergence was introduced by Kostyrko et al. in a metric space [14]. Later it was further studie d b y Dass e t al. [15], D e ms [16], Savas and Gumus [17], Kumar and Sharma [18], Ku mar and Mursaleen [19] and many others. Recently, Bilgin [20] used modulus function to define some notions of asymptotically equivalent sequences and studied some of their connections. Kumar and Sharma extended these concepts by presenting a nontrivial ideal I This paper presents introduce some new notions, (f, p)asymptotically equivalent of multiple L, strong (f, p) asymptoticall y equivalent of multiple L, and strong (f, p)asympto tically lacunary equi valent of multip le L with respect to the ideal I which is a natural co mbinat ion of the d efinition for asymptoticall y equiva lent, a non trivial ideal I, Lacunary sequence, a strictly positive sequence , and Modulus function. In addition to these definitions, we obtain some revelent connections between these notions. 2. Definitions and Notations Now we recall some definitions of sequence spaces (see [2] [4][6] [15], and [20][25]). Definition 2 .1. A sequence is statisticall y convergent to L if for every , (denoted by ), where the vertical bars denote the cardinality of the Enclosed set. Definition 2.2. A sequence is strongly(Cesaro) summable to L if , (denoted by ). Definition 2 .3. Let f be any modulus; the sequence is strongl y (Cesaro) summable to L with respect to a modulus i f , (denot ed by ). Definition 2.4. Two nonnegative sequences and are said to be asymptotically equivalent if , (denoted by ). Definition 2 .5. T wo nonne gat ive se q uence s and are said to be asymptotica lly statistic al e quiva le nt of multiple L provided that for every , 1 lim :0, k nk x kn L ny ε ≤−≥ = (denoted by ) a nd si mply asymptotically statistical equ ivalent, if . Definition 2 .6. T wo nonnegative sequences and are said to be strong asymptotically equivalent of multiple L provided that (denoted by ) and simply strong asymptotically equivalent, if . Definition 2.7. Let be a lacunary sequence; the two nonnegative sequences and are said to be asymptotically lacunary statistical equivalent of multiple L provided that for every , 1 lim :0, k rr rk x kI L hy ε ∈−≥ = (denoted by ) and simply asymptotically lacunary statistical equi valent, if . Definition 2.8. Let be a lacunary sequence; the two nonnegative sequences and are said to be
T. Bilgin strong asymptotically lacunary equivalent of multiple L provided that (denoted by ) and simply strong asymptotically lacunary equivalent, if . Definition 2.9. Let f be any modulus; the two nonnegative sequences and are said to be f asymptotically equivalent o f multiple L provided that, (denoted by ) and s imply s t ro ng fasymptotically eq uiva lent, if . Definition 2.10. Let f b e an y modul us; the two nonne gative sequences and are said to be strong f asymptotically equivalent of multiple L provided that, 1 1 lim 0 nk nkk x fL ny = −= ∑ (denoted by ) and simply strong fasymptotic a lly equi valent , if . Definition 2.11. Let f be any modulus and be a lacunary sequence; the two nonnegative sequences and are said to be strong fasymptotically lacunary equivalent of multiple L provided that 1 lim 0 r k rkI rk x fL hy ∈ −= ∑ (denoted by ) and simply strong fasymptotically lacunary equivalent, if . For any nonempty set X, let denote the power set of X. Definition 2 .12. A family is said to be an ideal in X if 1) ; 2) imply and 3) imply . Definition 2 .13. A nonempty family is said to be a filter in X if 1) ; 2) imply and 3) imply . An ideal I is said to be nontrivial if and . A nontrivial ideal I is called admissible if it contains all the singleton sets. Moreover, if I is a nontrivial ideal on X, then is a filter on X and conversely. The filter is called the filter a ssociated with the ideal I. Definition 2 .14. Let be a nontrivial ideal in N and be a metric space. A sequence in X is said to b e Ico n ver ge nt to if for each , the set . In this case, we write Definition 2.15. A sequence of numbers is said to be Istatistical co nvergent or S(I)conver gent to L, if for every and , we have { } 1 ;: . k n NknxLI n εδ ∈≤−≥≥ ∈ In this case, we write or Definition 2.16 Let be a nontrivial ideal in N. The two nonnegative sequences and are said to be strongly asymptotically equivalent of multiple L with respect to the ideal I provided that for each 1 1 ;. nk kk x nNLI ny ε = ∈−≥ ∈ ∑ denoted by and simply strongly asymptotically equivalent with respect to the ide a l I, if . Definition 2.17. Let be a nontrivial ideal in N and be a lacunary sequence. The two nonne gative se quence s and are said to be asymptoticall y lacunary statistical equi valent of multiple L with respect to the id e a l I p r o vided that for each
T. Bilgin and , 1 ;: k r rk x rN kILI hy εγ ∈∈−≥≥ ∈ denoted by and simply asymptotically lacunary statistical equivalent with respect to the ideal I, if . Definition 2.18. Let be a nontrivial ideal in N and be a lacunary sequence. The two nonnegative seque nces and are said to be strongly asymptotically lacunary equivalent of multiple L with respect to the id e a l I provided that for 1 ; r k kI rk x rNL I hy ε ∈ ∈−≥ ∈ ∑ denoted by and simply strongly asymptotically lacunary equivalent with respect to the ideal I, if . Definition 2.19. Let be a nontrivial ideal in N and f be a modulus function. The two non negative sequences and are said to be fasymptotically equivalent ofmultiple L with respect to the ideal I provided that for each ;k k x k NfLI y ε ∈− ≥∈ denoted by and simply fasymptotically eq uivalent with resp e ct to the ideal I, i f . Definition 2.20. Let be a nontrivial ideal in N and f be a modulus function. The two non negative sequences and are said to be strongly fasymptotically equivalent of multiple L with respect to the ideal I provided that for each , 1 1 ;nk kk x nN fLI ny ε = ∈−≥∈ ∑ denoted by and simply strongly fasymptotically equivale nt with respect to the ide a l I, if . Definition 2.21. Let be a nontrivial ideal in N, f be a modulus function and be a lacunary sequence. The two nonnegative sequences and are said to be strongly fasymptotically lacunary e quiva l ent of mult iple L with respect to the ideal I provided that for each , 1 ; r k kI rk x rN fLI hy ε ∈ ∈− ≥∈ ∑ denoted by and simply strongly fasymptotically lacunary equivalent with respect to the ideal I, if . 3. Main Results We now con s ider our main results. We begin with the following definitions. Definition 3.1. Let be a nontrivial ideal in N, f be a modulus function, and be a sequence of positive real numbers. Two number sequences and are said to be strongly (f, p) asymptotically equivalent o f multiple L with respect to the ideal I provided that for each , 1 1 ; k p nk kk x nN fLI ny ε = ∈− ≥∈ ∑
T. Bilgin denoted by and simply strongly (f, p)asymptotically equi valent with respect to the ideal I, if . If we take for , we write instead of and simply strongly pasympto tically equivalent with respect to the ideal I, if . If we take for all , we write instead of Definition 3 .2. Let be a nontrivial ideal in N, f be a modulus function, be a lacunary sequence, and be a sequence of positive real numbers. Two number sequences and are said to be strongly (f, p)asymptotically lacunary equivalent of multiple L with respect to the ideal I provided that for each , 1 ; k r p k kI rk x rNf LI hy ε ∈ ∈− ≥∈ ∑ denoted by and simply strongly (f, p)asymptotically lacunary equivalent with respect to the ideal I, if . If we take for all , we write instead of Note that,we put , we write instead of . Hence is the same as the of Kumar and Sharma [15]. Also if we put for , we write instead of . Hence is the same as the of Savas and Gumus [25] We start this section with the following theorem to show that the relation between (f, p)asymptotically equivalence and strong pasymptotically equivalence with respect to the ideal I Theorem 3.1. Let be a nontrivial ideal in N, f be a modulus function, be a lacunary sequence and 0 infsup kk kkk h pppH< =≤≤=<∞ , then 1) if then , and 2) if , then ( ) ( ) () , . fp p Iw Iw x yxy ∼⇔ ∼ Proof. Part 1): Let and ε > 0. We choose such that for every u with . We can write 1 12 1 11 k kk p nk kk pp kk kk x fL ny xx fL fL ny ny = − =−+ − ∑ ∑∑ where the first summation is over and the second summation over By definition of f, we have { } ( ) ( ) { } 1 11 11 max,max 1, 21 kk pp nn H hH kk kk kk xx fLf L n yny εε δ − = = −≤ +− ∑∑
T. Bilgin Thus, { } ( ) ( ) { } 1 11 max , 11 ;; . max1,21 kk pphH nn kk H kk kk xx nN fLnNL n ynyf ε εε ε δ − = = − ∈−≥⊆∈−≥ ∑∑ Since , it follows the later set, and hence, the first set in above expression belongs to I. This prove s that Part 2): If , then for all . Let , clearly { } 1 1 =1 1 11 min , k kk p nk kk pp nn hH kk kk kk x fL ny xx LL n yny β ββ = = − ≥ −≥− ∑ ∑∑ it follows that for each , we have { } 11 11 ;;min , k k p p nn hH kk kk kk xx nNLnN fL nyn y εε ββ = = ∈− ≥⊆∈−≥ ∑∑ Since , it follows that the later set belong s to I, and therefore, the theorem is proved. Theorem 3.2. Let be a nontrivial ideal in N, f be a modulus function, be a lacunary sequence and 0 infsup kk kkk h pppH< =≤≤=<∞ , then 1) if then , and 2) if , then () ( ) , . fp p IN IN x yxy θ θ ∼⇔ ∼ Proof. The proof of Theorem 3.2 is very similar to the Theorem 3.1. Then, we omit it. The next theorem shows the relationship between the strong (f, p)asymptotically equivalence and the strong (f, p)asymptotically lacunary equivalence with respect to the ideal I. Theorem 3.3. Let be a nontrivial ideal in N, f be a modulus function, be a lacunary sequence and be a sequence of positive real numbers. Then 1) if then implies ; 2) if then implies ; 3) if 1 liminflimsup rr rr qq< ≤<∞ , then ( ) () ( ) , ,fp fp IN Iw x yxy θ ∼⇔∼ . Proof. Part (i): If then there exi st s such that for every r. Now suppose that and . Let 1 ; k r p k kI rk x A rNfL hy ε ∈ =∈ −< ∑
T. Bilgin Hence, for all we have 1. k j p k jkI jk x H fL hy ε ∈ = −< ∑ Let n be any integer with . Now write ( ) 1 11 11 11 11 11 1 1 11 11 sup sup k kk r m k m p nk kk pp kr kk km kI rk rk p rr mm kmmj jA m kIm rmk r r jA r x fL ny xx fL fL ky ky kkx f LkkH khy k kH k = ==∈ −− −−∈ =∈= −− ∈ − − ≤ −=− − =−= − = ∑ ∑ ∑∑ ∑∑ ∑  sup jr j jA q HK εε ∈ = <= it follows that for any , ( )  1 1 ; k p nk kk x n NfLFI ny ε = ∈− <∈ ∑ which yields that . Because for any set , . Part (ii): Let and . There exists such that for all . We have, for sufficientl y lar ge r, that and . Let and define the set 1 1 ;. k r p kk rk rk x AkNf L ky ε = =∈ −< ∑ We have , which is the f ilte r of the ideal I, For each , we have 1 1 11 1 11 1 1 1 11 k kk rr r kk rr r ppp kk kkk kI kk r krkrk pp kk kk rr kk rr krrk k r k rr xxx fLfL fL h yhyhy xx kk fL fL khyhky k kh − − ∈== − = = − = −= −− − = −−− ≤ ∑ ∑∑ ∑∑ ∑  1 k p k k x fL y δεε δ + −<= it follows that for any , ( )  1 ; k r p k kI rk x r NfLFI hy ε ∈ ∈− <∈ ∑ which yiel ds that Part (iii): This immediatel y follows from (i) and (ii). Now we give relation between asymptotically statistical equivalence and strong (f, p) asymptotically equi valence with respect to the ideal I. Also we give relation between asymptotically lacunary statistical equivalence and strong (f, p)asymptotically lacunary equivalence with respect to the ideal I. Theorem 3.4. Let be a nontrivial ideal in , be a modulus function, be a lacunary se quence and 0 infsup kk kkk h pppH< =≤≤=<∞ , then
T. Bilgin 1) if then , 2) if f is bounded then . Proof. Part 1): Suppose , and let and denote the sum over with then we can write ( )( ) { } 11 11 1 min ,: kk pp nhH kk k kkkk xx x fLfLff knL nynyn y εε ε = −≥ −≥≤−≥ ∑∑ Consequently, for any , we have ( )( ) { } 1 1 ;: 1 ;min ,. k k k p nhH k kk x nN knL ny x nNfLf fI ny εγ γ εε = ∈≤−≥ ≥ ⊆∈− ≥∈ ∑ Therefore we have 2) Suppose f is bounded and . Since f is bo unded, there exists an integer T such t hat for all . Moreover, for , We split the s um for into sums over and . Then { } ( )( ) { } , 1 11 max ,:max, k p nhH hH kk kkk xx fLTTknLf f nyn y ε εε = −≤≤−≥ + ∑ Consequen tly, we have ( )( ) { } { } 1 , 1 ; max , 1 ;: . max , k p nk kk hH k hH k x nN fL ny ff x nN knLI ny TT ε ε εε ε = ∈ −≥ − ⊆ ∈≤−≥≥∈ ∑ Therefore we have Theorem 3.5. Let be a nontrivial ideal in N, f be a modulus function, be a lacunary sequence and 0 infsup kk kkk h pppH< =≤≤=<∞ , then 1) if then , 2) if f is bounded then ( ) ( ) ,fp IN IS xyx y θθ ∼ ⇔∼ . Proof. Part 1): Take and let denote the sum over with . Then ( )( ) { } 1 11 1 min ,:, kk r pp hH kk k r kI r krkrk xx x fLfLffkI L h yhyhy εε ε ∈ −≥ −≥∈−≥ ∑∑
T. Bilgin and ( )( ) { } 1 ;: 1 ;min ,. k r k r rk p hH k kI rk x rN kIL hy x rNfLf fI hy εγ γ εε ∈ ∈∈−≥ ≥ ⊆∈−≥∈ ∑ But then, by definition of an ideal, later set belongs to I, and therefore Part 2): Suppose that f is bounded and . Since f is bounded, there exists an integer T such that for all . We see that { } ( )( ) { } , 11 max ,:max,, k r p hH hH kk r kI rkr k xx fLTTkILff hyh y ε εε ∈ −≤∈−≥ + ∑ so we have ( )() {} {} , 1 ; max , 1 ;: max , k r p k kI rk hH k rhH rk x rNfL hy ff x rN kILI hy TT ε ε εε ε ∈ ∈ −≥ − ⊆ ∈∈−≥≥∈ ∑ Therefore we have Let for all k, for all k and . Then it follows following T heorem. Theorem 3.6. Let be a nontrivial ideal in N, f be a modulus function, and be a lacunary sequence, then implies , Proof. Let It follo ws from Holder’s inequality 11 rr pt pt kk kI kI rk rk xx fL fL hy hy ∈∈ −≤ − ∑∑ and 11 ;; rr pt tp kk kI kI rk rk xx rNf LrNf LI hyhy εε ∈∈ ∈−≥⊆∈−≥∈ ∑∑ . Thus we have We now consider that and are not constant sequences. Theorem 3.7. Let be a nontrivial ideal in N, f be a modulus function, be a lacunary sequence, for all k and be bounde d, then implies Proof. Let . k t k kk x zfL y = − and , so that : We define the sequences and as follows: For ; let and and for ; let and . T hen we have ; Now it follows that and . There fore
T. Bilgin ( ) 1 111 k kk r rrr kkkk k kIkIkIkI r rrr zuvz v h hhh λ λλλ ∈ ∈∈∈ =+≤+ ∑ ∑∑∑ Now for each r; 1 1 11 11 1 11111 rrr rr kk kk kI kIkIkIkI r rrrrr vv vv h hhhhh λλ λλ λ λλ λλ λ − − −− ∈∈∈ ∈∈ =≤< ∑∑∑ ∑∑ and so 1 111 11 , 1,1 11 1 ,12, 1 k k r rrr rr rr r p kk kk kIkI kI kI rk rrr kkkk kI kI rr kk kkk kI kIkI rrr x fLzzv hy hhh zzzz hh zzzzz hh h λ λ λλ ∈ ∈∈∈ ∈∈ ∈∈ ∈ −=≤ + ≥≥ = ≤ +< < ∑ ∑∑∑ ∑∑ ∑∑ ∑ If 1 k r p k kI rk x fL hy ε ∈ −≥ ∑ then 1 1,1 1, <1 2 k r k r t kk kI rk t kk kI rk x fL z hy x fL z hy λ ε ε ∈ ∈ −≥ ≥ −≥ ∑ ∑ Hence 1 11 ;;min ,. 2 kk rr pt kk kI kI rk rk xx rNf LrNf LI hy hy λ ε εε ∈∈ ∈−≥⊆∈−≥∈ ∑∑ Thus we ha ve Acknowledgements The wor k is s uppo rte d b y the P res ide ncy o f scientific Research P rojects of Y uz unc u Yil Univ ers ity ( No. KONGRE 2014/94) and is presented at 3rd International Eurasian Conference on Mathematical Sciences and Applications (IECMSA2014). References [1] Freedman, A.R, Sember, J.J. and Raphel, M. (1978) Some CesaroType Summability Spaces. Proceedings London Mathematical Society, 37, 508520. http://dx.doi.org/10.1112/plms/s337.3.508 [2] Nakano, H. (1953) Concave Modulars. Journal of the Mathematical Society of Japan, 5, 2949. http://dx.doi.org/10.2969/jmsj/00510029 [3] Connor, J.S. (1989) On Strong Matrix Summability with Respect to a Modulus and St atistical Con vergence. Canadian Mathematical Bulletin, 32, 194198. http://dx.doi.org/10.4153/CMB19890293 [4] Kolk, E. (1993) On Strong Boundedness and Summability with Respect to a Sequence Moduli. Tartu Ülikooli Toime tised, 960, 4150. [5] Maddox, I.J. (1986) Sequence Spaces Defin ed b y a Modulus. Mathematical Proceedings of the Cambridge Philosoph ical Society, 100, 161166. http://dx.doi.org/10.1017/S0305004100065968 [6] Öztürk, E. and Bilgin, T. (1994) Strongly Summable S equence S paces Defined b y a Modulus. Indian Journal of Pure and Applied Mathematics, 25, 621625.
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