The concept of convergence of a sequence of real numbers has been extended to statistical convergence independently by Fast  and Schoenberg . There has been an effort to introduce several generalizations and variants of statistical convergence in different spaces. One such very important generalization of this notion was introduced by Kostyrko et al.  by using an ideal I of subsets of the set of natural numbers, which they called I-convergence. After that the idea of I-convergence for double sequence was introduced by Das et al.  in 2008.
Throughout a double sequence is defined by and we denote showing the space of all real or complex double sequences.
Let be a nonempty set then a family is said to be an idealin if , is additive, i.e., for all and is hereditary, i.e., for all . A nonempty family of sets is said to be a filteron if for all implies and for all with implies . An ideal is said to be nontrivialif , this non trivial ideal is said to be admissible if and is said to be maximalif there cannot exist any nontrivial ideal containing as a subset. For each ideal there is a filter called as filter associate with ideal , that is
A double sequence is said to be -convergent[5, 6, 7, 8] to a number if for every , we have In this case, we write A double sequence is said to be -Cauchyif for every there exists numbers such that
A continuous linear functional on is said to be an invariant mean[9, 10] or -mean if and only if:
1. where the sequence has for all k,
2. where ,
3. for all ,
where be an injective mapping of the set of the positive integers into itself having no finite orbits.
If , write , so we have
where denote the mth-iterate of at k. In this case is the translation mapping, that is, mean is called a Banach limit  and , the set of bounded sequences of all whose invariant means are equal, is the set of almost convergent sequences. The special case of (3) in which was given by Lorentz  and the general result can be proved in a similar way. It is familiar that a Banach limit extends the limit functional on in the sense that
Definition 1.1A sequence is of -bounded variationif and only if:
(i) converges uniformly in k,
(ii) which must exist, should take the same value for all k.
We denote by , the space of all sequences of -bounded variation:
is a Banach space normed by
A function is said to be an Orlicz function[13, 14] if it satisfies the following conditions:
(i) M is continuous, convex and non-decreasing,
(ii) and as
Remark 1.1If the convexity of an Orlicz function is replaced by , then this function is called Modulus function[15, 16, 17]. If is an Orlicz function, then for all with An Orlicz function is said to satisfy -condition for all values of if there exists a constant such that for all values of .
Definition 1.2A double sequence space is said to be:
[i] solidor normalif implies that for all sequence of scalars with for all .
[ii] symmetricif whenever , where is a permutation on .
[iii] sequence algebraif whenever
[iv] convergence freeif whenever and implies , for all .
Definition 1.3Let and be a double sequence space. A -step space of is a sequence space
A canonical preimage of a sequence is a sequence defined as follows:
A sequence space is said to be monotoneif it contains the canonical preimages of all its step spaces.
The following subspaces where is a sequence of positive real numbers. These subspaces were first introduced and discussed by Maddox . The following inequalities will be used throughout the section. Let be a double sequence of positive real numbers . For any complex with , we have
Let , then for the sequences and in the complex plane, we have
2. Bounded variation sequence spaces defined by Orlicz function
In this section, we define and study the concepts of -convergence for double sequences defined by Orlicz function and present some basic results on the above definitions [8, 20].
Now, we read some theorems based on these sequence spaces. These theorems are of general importance as indispensable tools in various theoretical and practical problems.
Theorem 2.1Let be two Orlicz functions with condition, then
Proof.(a) Let be an arbitrary element, so there exists such that
Let and choose with such that for .
Write . Consider,
Now, since is an Orlicz function so we have . Therefore, we have
For , we have . Now, since is non-decreasing and convex, it follows that,
Since satisfies the -condition, so we have
This implies that,
Hence, we have
Therefore from (12) and (16), we have
This implies that . Hence for The other cases can be proved in similar way.
(b) Let . Let be given. Then there exist such that
from Eqs. (17) and (18), we get
so we have
Hence, For the inclusion are similar.
Proof.For this let , for all . Let us suppose that . Then for any given , we have
Now let be such that . Consider for ,
This implies that , which shows that .
Hence, we have
Using the definition of convergence free sequence space, let us give another theorem which will be of particular importance in our future work:
Theorem 2.2The spaces and are not convergence free.
Example 2.1To show this let and , for all . Now consider the double sequence which defined as follows:
Then we have belong to both and but does not belong to and . Hence, the spaces and are not convergence free.
To gain a good understanding of these double sequence spaces and related concepts, let us finally look at this theorem on inclusions:
Theorem 2.3Let M be an Orlicz function. Then
Proof.For this let us consider It is obvious that it must belong to Now consider
Now taking the limit on both sides we get
Hence Now it remains to show that
For this let us consider this implies that there exist s.t
Now taking the supremum on both sides, we get
3. Paranorm bounded variation sequence spaces
In this section we study double sequence spaces by using the double sequences of strictly positive real numbers with the help of space and an Orlicz function M. We study some of its properties and prove some inclusion relations related to these new spaces. For m, n , we have
We also denote
We can now state and proof the theorems based on these double sequence spaces which are as follows:
Theorem 3.1Let then the classes of double sequence and are paranormed spaces, paranormed by
Proof.It is clear that if and only if
Let . Now for , we denote
Let us take . Then by using the convexity of M, we have
which in terms give us
Let be a double sequence of scalars with and such that
in the sense that
Then, since the inequality
holds by subadditivity of g, the sequence is bounded.
as . That implies that the scalar multiplication is continuous. Hence is a paranormed space. For another space , the result is similar.
We shall see about the separability of these new defined double sequence spaces in the next theorem.
Theorem 3.2The spaces and are not separable.
Example 3.1By counter example, we prove the above result for the space .
Let A be an infinite subset of increasing natural numbers, i.e., such that .
Since A is infinite, so is uncountable. Consider the class of open balls
Let be an open cover of containing .
Since is uncountable, so cannot be reduced to a countable subcover for . Thus is not separable.
We shall now introduce a theorem which improves our work.
Theorem 3.3Let and be two double sequences of positive real numbers. Then if and only if , where such that .
Proof.Let and Then, there exists such that for sufficiently large
Since For a given there exist such that
Let Then for all sufficiently large
Therefore, . The converse part of the result follows obviously.
Remark 3.1Let and be two double sequences of positive real numbers. Then if and only if and if and only if and , where such that .
Theorem 3.4The set is closed subspace of .
Proof.Let be a Cauchy double sequence in such that . We show that Since, , then there exists , and such that
We need to show that
(1) converges to a.
(2) If , then .
Since be a Cauchy double sequence in then for a given there exists such that
For a given , we have
Then . Let
where then . We choose , then for each , we have
Then is a Cauchy double sequence in . So, there exists a scalar such that
(2) For the next step, let be given. Then, we show that if
then Since , then there exists such that,
where implies . The number can be so chosen that together with (25), we have
such that Since .
Then we have a subset such that , where
Therefore, for , we have
Hence the result follows.
Since the inclusions and are strict so in view of Theorem (3.3), we have the following result.
The above theorem is interesting and itself will have various applications in our future work.
4. Bounded variation sequence spaces defined by modulus function
In this section, we study some new double sequence spaces of invariant means defined by ideal and modulus function. Furthermore, we also study several properties relevant to topological structures and inclusion relations between these spaces. The following classes of double sequence spaces are as follows:
We also denote
We shall now consider important theorems of these double sequence spaces by using modulus function.
Theorem 4.1For any modulus function, the classes of double sequence, andare linear spaces.
Proof.Suppose and be any two arbitrary elements. Let are scalars. Now, since . Then this implies that there exists some positive numbers and such that the sets
be such that Since is a modulus function, we have
This implies that Thus As and are two arbitrary element then for all and for all scalars . Hence is linear space. The proof for other spaces will follow similarly.▪
We may go a step further and define another theorem on ideal convergence which basically depends upon the set in the filter associated with the same ideal.
Theorem 4.2A sequence -convergent if and only if for every , there exists such that
Proof.Let . Suppose . Then, the set
Fix Then we have
which holds for all
Conversely, suppose that
Then, being a modulus function and by using basic triangular inequality, we have
Then, the set
If we fix then, we have as well as .
Hence . This implies that
This shows that
where the denotes the length of interval . In this way, by induction we get the sequence of closed intervals
with the property that for and for .
Then there exists a where such that
showing that is -convergent. Hence the result holds.
As the reader knows about solid and monotone sequence space now turn to theorem on solid and monotone double sequence spaces of invariant mean defined by ideal and modulus function.
Theorem 4.3For any modulus function , the spaces and are solid and monotone.
Proof.We consider and for the proof shall be similar.
Let be an arbitrary element, then the set
Let be a sequence of scalars with for all
Now, since is a modulus function. Then the result follows from (2.18) and the inequality
Thus we have Hence is solid. Therefore is monotone. Since every solid sequence space is monotone.
Remark 4.1The space and are neither solid nor monotone in general.
Example 4.1Here we give counter example for establishment of this result. Let and . Let us consider and , for all and . Consider, the K-step space of defined as follows:
Let and be such that
Consider the sequence defined by for all .
Then, and , but -step space preimage does not belong to and . Thus, and are not monotone and hence they are not solid.
After discussing about solid and monotone sequence space now we come to the concept of sequence algebra which will help to understand our further work.
Theorem 4.4For any modulus function , the spaces and are sequence algebra.
Proof.Let be any two arbitrary elements.
Then, the sets
Thus, we have Hence is sequence algebra. And for the result can be proved similarly.
Remark 4.2If is not maximal and then the spaces and are not symmetric.
Example 4.2Let be an infinite set and for all and If
Then, it is clearly seen that
Let be such that Let be a bijective maps (as all four sets are infinite). Then, the mapping defined by
is a permutation on
But and hence showing that and are not symmetric double sequence spaces.
In this chapter, we study different forms of double sequence spaces of invariant means with the help of ideal, operators and some functions such as Orlicz function and modulus function. The chapter shows the potential of the new theoretical tools to deal with the convergence problems of sequences in sigma bounded variation, occurring in many branches of science, engineering and applied mathematics.
The authors express their gratitude to our Chairman Prof. Mohammad Imdad for his advices and continuous support. We are grateful to the learned referees for careful reading of our manuscript and considering the same for publication.
Conflict of interest
The authors declare that they have no competing interests.