论文部分内容阅读
Abstract： The notion of n-normed space was studied at the initial stage by Gahler （Gahler， 1965）， Gunawan （Gunawan， 2001） and many others. In this paper， we introduce some certain new generalized difference double sequence spaces via ideal convergence， double lacunary sequence and an Orlicz function in n-normed spaces and examine some properties of the resulting these spaces.
Key words： P-convergent； Double lacunary sequence； n-normed space； Orlicz function
By the convergence of a double sequence we mean the convergence on the Pringsheim sense that is， a double sequence x = （xk，l） has Pringsheim limit L （denoted by P？limx = L） provided that givenε> 0 there exists n∈N such that |xk，l？L| <εwhenever k，l > n [3]. We shall write more briefly as“P？convergent”.
The double sequence x = （xk，l） is bounded if there exists a positive number M such that |xk，l| < M for all k and l. Let l2∞the space of all bounded double such that
Recall in [5] that an Orlicz function M is continuous， convex， nondecreasing function define for x > 0 such that M（0） = 0 and M（x） > 0. If convexity of Orlicz function is replaced by M（x + y）≤M （x） + M （y） then this function is
called the modulus function and characterized by Ruckle [6]. An Orlicz function
M is said to satisfy？2？condition for all values u， if there exists K > 0 such that M（2u）≤KM（u）， u≥0. Lemma 1. Let M be an Orlicz function which satisfies？2？condition and let 0 <δ< 1. Then for each t≥δ， we have M（t） < Ktδ？1M （2） for some constant K > 0.
A double sequence space X is said to be solid or normal if （αk，lxk，l）∈X， and for all double sequencesα= （αk，l） of scalars with |αk，l|≤1 for all k，l∈N.
Let n∈N and X be a real vector space of dimension d， where n≤d. A real-valued function 3.，...，.3 on X satisfying the following four conditions：
（i） 3x1，x2，...，xn3 = 0 if and only if x1，x2，...，xnare linearly dependent，（ii） 3x1，x2，...，xn3 is invariant under permutation，（iii） 3αx1，x2，...，xn3 = |α|3x1，x2，...，xn3，α∈R，
（iv） 3x1+ x？1，x2，...，xn3≤3x1，x2，...，xn3+3x？1，x2，...，xn3 is called an n？norm on X， and the pair （X，3.，...，.3） is called an n？normed space [7，8]. Normed space
was studied by Mursaleen and Mohiuddine [9，10]， Mohiuddine and Lohani [11]，
Mohiuddine and Alghamdi [12] and many others from different aspects.
A trivial example of n？normed space is X = R equipped with the following Euclidean n？norm：
Let I2 be an ideal of 2N×N，θr，s be a double lacunary sequence， M be an Orlicz function， p = （pk，l） be a bounded double sequence of strictly positive real numbers and （X，3.，...，.3） be an n？normed space. Further w（n？X） denotes X？valued sequence space. Now， we define the following double generalized difference sequence spaces： Corollary 1. It can be noted that h = infn，m∈Nh（n，m）also gives a paranorm on the above sequence spaces. However if one consider the sequence space wθr，s[M，？m，p，3.，...，.3]∞
REFERENCES
[1] Kostyrko， P.， Salat， T.， & Wilczynski， W. （2000/2001）. I-convergence. Real Analysis Exchange， 26（2）， 669-686.
[2] Salat， T.， Tripathy， B. C.， & Ziman， M. （2005）. On I-convergencefield. Italian J. Pure and Appl. Math.， 17， 45-54.
[3] Pringsheim， A. （1900）.Zur theori der zweifach unendlichen zahlenfolgen. Math. Ann.， 53， 289-321.
[4] Sava？s， E.， & Patterson， R. F. （2011）. Some double lacunary sequence spaces defined by orlicz functions. Southeast Asian Bulletin of Mathematics， 35（1）， 103-110.
[5] Krasnoselski， M. A.， & Rutickii， Y. B. （1961）. Convex function and orlicz spaces. Nederland： Groningen.
[6] Ruckle， W. H. （1973）. FK-spaces in which the sequence of coordinate vectors is bounded. Canad. J. Math.， 25， 973-978.
[7] Gunawan， H. （2001）. On n-inner product， n-norms and the Cauchy-Schwarz inequality. Scientiae Mathematicae Japonicae Online， 5， 47-54.
[8] Gahler， S. （1965）. Lineare 2-normierte. Rume， Math. Nachr.， 28（164）， 1-43.
[9] Mursaleen， M.， & Mohiuddine， S. A. （2009）. On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space. J. Comput. Appl. Math.， 233， 142-149.
[10] Mursaleen， M.， & Mohiuddine， S. A. （2012）. On ideal convergence in probabilistic normed spaces. Math. Slovaca， 62（1）， 49-62.
[11] Mohiuddine， S. A.， & Danish Lohani， Q. M. （2009）. On generalized statistical convergence in intuitionistic fuzzy normed space. Chaos， Solitons Fractals， 42， 1731-1737.
[12] Mohiuddine， S. A.， & Alghamdi， M. A. （2012）. Statistical summability through a lacunary sequence in locally solid Riesz spaces. J. Inequal. Appl.， 2012， 225.
[13] Esi， A. （2012）. Strongly summable double sequence spaces in n-normed space defined by ideal convergence and an orlicz function. Kyungpook Math. J.， 52， 137-147.
Key words： P-convergent； Double lacunary sequence； n-normed space； Orlicz function
By the convergence of a double sequence we mean the convergence on the Pringsheim sense that is， a double sequence x = （xk，l） has Pringsheim limit L （denoted by P？limx = L） provided that givenε> 0 there exists n∈N such that |xk，l？L| <εwhenever k，l > n [3]. We shall write more briefly as“P？convergent”.
The double sequence x = （xk，l） is bounded if there exists a positive number M such that |xk，l| < M for all k and l. Let l2∞the space of all bounded double such that
Recall in [5] that an Orlicz function M is continuous， convex， nondecreasing function define for x > 0 such that M（0） = 0 and M（x） > 0. If convexity of Orlicz function is replaced by M（x + y）≤M （x） + M （y） then this function is
called the modulus function and characterized by Ruckle [6]. An Orlicz function
M is said to satisfy？2？condition for all values u， if there exists K > 0 such that M（2u）≤KM（u）， u≥0. Lemma 1. Let M be an Orlicz function which satisfies？2？condition and let 0 <δ< 1. Then for each t≥δ， we have M（t） < Ktδ？1M （2） for some constant K > 0.
A double sequence space X is said to be solid or normal if （αk，lxk，l）∈X， and for all double sequencesα= （αk，l） of scalars with |αk，l|≤1 for all k，l∈N.
Let n∈N and X be a real vector space of dimension d， where n≤d. A real-valued function 3.，...，.3 on X satisfying the following four conditions：
（i） 3x1，x2，...，xn3 = 0 if and only if x1，x2，...，xnare linearly dependent，（ii） 3x1，x2，...，xn3 is invariant under permutation，（iii） 3αx1，x2，...，xn3 = |α|3x1，x2，...，xn3，α∈R，
（iv） 3x1+ x？1，x2，...，xn3≤3x1，x2，...，xn3+3x？1，x2，...，xn3 is called an n？norm on X， and the pair （X，3.，...，.3） is called an n？normed space [7，8]. Normed space
was studied by Mursaleen and Mohiuddine [9，10]， Mohiuddine and Lohani [11]，
Mohiuddine and Alghamdi [12] and many others from different aspects.
A trivial example of n？normed space is X = R equipped with the following Euclidean n？norm：
Let I2 be an ideal of 2N×N，θr，s be a double lacunary sequence， M be an Orlicz function， p = （pk，l） be a bounded double sequence of strictly positive real numbers and （X，3.，...，.3） be an n？normed space. Further w（n？X） denotes X？valued sequence space. Now， we define the following double generalized difference sequence spaces： Corollary 1. It can be noted that h = infn，m∈Nh（n，m）also gives a paranorm on the above sequence spaces. However if one consider the sequence space wθr，s[M，？m，p，3.，...，.3]∞
REFERENCES
[1] Kostyrko， P.， Salat， T.， & Wilczynski， W. （2000/2001）. I-convergence. Real Analysis Exchange， 26（2）， 669-686.
[2] Salat， T.， Tripathy， B. C.， & Ziman， M. （2005）. On I-convergencefield. Italian J. Pure and Appl. Math.， 17， 45-54.
[3] Pringsheim， A. （1900）.Zur theori der zweifach unendlichen zahlenfolgen. Math. Ann.， 53， 289-321.
[4] Sava？s， E.， & Patterson， R. F. （2011）. Some double lacunary sequence spaces defined by orlicz functions. Southeast Asian Bulletin of Mathematics， 35（1）， 103-110.
[5] Krasnoselski， M. A.， & Rutickii， Y. B. （1961）. Convex function and orlicz spaces. Nederland： Groningen.
[6] Ruckle， W. H. （1973）. FK-spaces in which the sequence of coordinate vectors is bounded. Canad. J. Math.， 25， 973-978.
[7] Gunawan， H. （2001）. On n-inner product， n-norms and the Cauchy-Schwarz inequality. Scientiae Mathematicae Japonicae Online， 5， 47-54.
[8] Gahler， S. （1965）. Lineare 2-normierte. Rume， Math. Nachr.， 28（164）， 1-43.
[9] Mursaleen， M.， & Mohiuddine， S. A. （2009）. On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space. J. Comput. Appl. Math.， 233， 142-149.
[10] Mursaleen， M.， & Mohiuddine， S. A. （2012）. On ideal convergence in probabilistic normed spaces. Math. Slovaca， 62（1）， 49-62.
[11] Mohiuddine， S. A.， & Danish Lohani， Q. M. （2009）. On generalized statistical convergence in intuitionistic fuzzy normed space. Chaos， Solitons Fractals， 42， 1731-1737.
[12] Mohiuddine， S. A.， & Alghamdi， M. A. （2012）. Statistical summability through a lacunary sequence in locally solid Riesz spaces. J. Inequal. Appl.， 2012， 225.
[13] Esi， A. （2012）. Strongly summable double sequence spaces in n-normed space defined by ideal convergence and an orlicz function. Kyungpook Math. J.， 52， 137-147.