In the present paper, we introduce some sequence spaces using ideal convergence and Musielak-Orlicz function ℳ=Mk. We also examine some topological properties of the resulting sequence spaces.

1. Introduction and Preliminaries

The notion of ideal convergence was first introduced by Kostyrko et al. [1] as a generalization of statistical convergence which was further studied in topological spaces by Das et al. see [2]. More applications of ideals can be seen in [2, 3]. We continue in this direction and introduce I-convergence of generalized sequences with respect to Musielak-Orlicz function.

A family ℐ⊂2X of subsets of a nonempty set X is said to be an ideal in X if

ϕ∈ℐ,

A,B∈ℐ imply A∪B∈ℐ,

A∈ℐ, B⊂A imply B∈ℐ,

while an admissible ideal ℐ of X further satisfies {x}∈ℐ for each x∈X; see [1]. A sequence (xn)n∈ℕ in X is said to be I-convergent to x∈X. If for each ϵ>0, the set A(ϵ)={n∈ℕ:∥xn-x∥≥ϵ} belongs to ℐ; see [1]. For more details about ideal convergent sequence spaces, see [4–10] and references therein.

Mursaleen and Noman [11] introduced the notion of λ-convergent and λ-bounded sequences as follows.

Let λ=(λk)k=1∞ be a strictly increasing sequence of positive real numbers tending to infinity; that is,
(1)0<λ0<λ1<⋯,λk⟶∞ask⟶∞.
The sequence x=(xk)∈w is λ-convergent to the number L, called the λ-limit of x, if Λm(x)→L, as m→∞, where
(2)Λm(x)=1λm∑k=1m(λk-λk-1)xk.

The sequence x=(xk)∈w is λ-bounded if supm|Λm(x)|<∞. It is well known [11] that if limmxm=a in the ordinary sense of convergence, then
(3)limm(1λm(∑k=1m(λk-λk-1)|xk-a|))=0.

This implies that
(4)limm|Λm(x)-a|=limm|1λm∑k=1m(λk-λk-1)(xk-a)|=0,
which yields that limmΛm(x)=a and hence x=(xk)∈w is λ-convergent to a.

Let X be a linear metric space. A function p:X→ℝ is called paranorm if

p(x)≥0, for all x∈X,

p(-x)=p(x), for all x∈X,

p(x+y)≤p(x)+p(y), for all x,y∈X,

if (λn) is a sequence of scalars with λn→λ as n→∞ and (xn) is a sequence of vectors with p(xn-x)→0 as n→∞, then p(λnxn-λx)→0 as n→∞.

A paranorm p for which p(x)=0 implies that x=0 is called total paranorm and the pair (X,p) is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (see [12, Theorem 10.4.2, P-183]). For more details about sequence spaces, see [13–15] and references therein.

An Orlicz function M is a function which is continuous, nondecreasing, and convex with M(0)=0, M(x)>0 for x>0 and M(x)→∞ as x→∞.

Lindenstrauss and Tzafriri [16] used the idea of Orlicz function to define the following sequence space. Let w be the space of all real or complex sequences x=(xk). Then,
(5)ℓM={x∈w:∑k=1∞M(|xk|ρ)<∞}
which is called an Orlicz sequence space. The space ℓM is a Banach space with the norm
(6)∥x∥=inf{ρ>0:∑k=1∞M(|xk|ρ)≤1}.

It is shown in [16] that every Orlicz sequence space ℓM contains a subspace isomorphic to ℓp(p≥1). The Δ2-condition is equivalent to M(Lx)≤kLM(x) for all values of x≥0 and for L>1.

A sequence ℳ=(Mk) of Orlicz function is called a Musielak-Orlicz function see; [17, 18]. A sequence 𝒩=(Nk) defined by
(7)Nk(v)=sup{|v|u-(Mk):u≥0},k=1,2,…,
is called the complementary function of a Musielak-Orlicz function ℳ. For a given Musielak-Orlicz function ℳ, the Musielak-Orlicz sequence space tℳ and its subspace hℳ are defined as follows:
(8)tℳ={x∈w:Iℳ(cx)<∞forsomec>0},hℳ={x∈w:Iℳ(cx)<∞forallc>0},
where Iℳ is a convex modular defined by
(9)Iℳ(x)=∑k=1∞Mk(xk),x=(xk)∈tℳ.

We consider tℳ equipped with the Luxemburg norm
(10)∥x∥=inf{k>0:Iℳ(xk)≤1},
or equipped with the Orlicz norm
(11)∥x∥0=inf{1k(1+Iℳ(kx)):k>0}.

Let ℳ=(Mk) be a Musielak-Orlicz function and let p=(pk) be a bounded sequence of positive real numbers. We define the following sequence spaces:
(12)cI(ℳ,Λ,p)={x=(xk)∈w:I-limkMk(|Λk(x)-L|ρ)pk=0,HhhHforsomeLandρ>0(|Λk(x)-L|ρ)pk},c0I(ℳ,Λ,p)={x=(xk)∈w:I-limkMk(|Λk(x)|ρ)pk=0,hhhhforsomeρ>0(|Λk(x)|ρ)pk},l∞(ℳ,Λ,p)={x=(xk)∈w:supkMk(|Λk(x)|ρ)pk<∞,hhhhhhhhhhhhforsomeρ>0(|Λk(x)|ρ)pk}.

We can write
(13)mI(ℳ,Λ,p)=cI(ℳ,Λ,p)∩l∞(ℳ,Λ,p),m0I(ℳ,Λ,p)=c0I(ℳ,Λ,p)∩l∞(ℳ,Λ,p).

If we take p=(pk)=1, for all k∈ℕ, we have
(14)cI(ℳ,Λ)={x=(xk)∈w:I-limkMk(|Λk(x)-L|ρ)=0,hhhh.hforsomeLandρ>0|Λk(x)-L|ρ},c0I(ℳ,Λ)={x=(xk)∈w:I-limkMk(|Λk(x)|ρ)=0,hhhhhhhhhh.forsomeρ>0(|Λk(x)|ρ)pk},l∞(ℳ,Λ)={x=(xk)∈w:supkMk(|Λk(x)|ρ)<∞,hhhhhhhhhh.forsomeρ>0(|Λk(x)|ρ)pk}.

The following inequality will be used throughout the paper. If 0≤pk≤suppk=H, D=max(1,2H-1), then
(15)|ak+bk|pk≤D{|ak|pk+|Bk|pk},
for all k, and ak,bk∈ℂ. Also |a|pk≤max(1,|a|H) for all a∈ℂ.

The main aim of this paper is to study some ideal convergent sequence spaces defined by a Musielak-Orlicz function ℳ=(Mk). We also make an effort to study some topological properties and prove some inclusion relations between these spaces.

2. Main Results Theorem 1.

Let ℳ=(Mk) be a Musielak-Orlicz function and let p=(pk) be a bounded sequence of positive real numbers. Then, the spaces cI(ℳ,Λ,p), c0I(ℳ,Λ,p), mI(ℳ,Λ,p), and m0I(ℳ,Λ,p) are linear.

Proof.

Let x,y∈cI(ℳ,Λ,p) and let α, β be scalars. Then, there exist positive numbers ρ1 and ρ2 such that
(16)I-limkMk(|Λk(x)-L1|ρ1)pk=0,forsomeL1∈ℂ,I-limkMk(|Λk(y)-L2|ρ2)pk=0,forsomeL2∈ℂ.

For a given ϵ>0, we have
(17)D1={k∈ℕ:Mk(|Λk(x)-L1|ρ1)pk},D2={k∈ℕ:Mk(|Λk(y)-L2|ρ2)pk}.

Let ρ3=max{2|α|ρ1,2|β|ρ2}. Since ℳ=(Mk) is nondecreasing convex function, so by using inequality (15), we have
(18)limkMk(|Λk((αx+βy)-(αL1+βL2))|ρ3)pk≤limkMk(|α||Λk(x)-L1|ρ3+|β||Λk(y)-L2|ρ3)pk≤limkMk(|Λk(x)-L1|ρ1)pk+limkMk(|Λk(y)-L2|ρ2)pk.

Now, by (17), we have
(19){k∈ℕ:limkMk(|Λk((αx+βy)-(αL1+βL2))|ρ3)pkh.>ϵ(|Λk((αx+βy)-(αL1+βL2))|ρ3)pk}⊂D1∪D2.

Therefore, αx+βy∈cI(ℳ,Λ,p). Hence cI(ℳ,Λ,p) is a linear space. Similarly, we can prove that c0I(ℳ,Λ,p), mI(ℳ,Λ,p), and m0I(ℳ,Λ,p) are linear spaces.

Theorem 2.

Let ℳ=(Mk) be a Musielak-Orlicz function. Then,
(20)c0I(ℳ,Λ,p)⊂cI(ℳ,Λ,p)⊂l∞(ℳ,Λ,p).

Proof.

Let x∈cI(ℳ,Λ,p). Then, there exist L∈ℂ and ρ>0 such that
(21)I-limkMk(|Λk(x)-L|ρ)pk=0.

We have
(22)Mk(|Λk(x)|2ρ)pk≤12Mk(|Λk(x)-L|ρ)pk+Mk12(|L|ρ)pk.

Taking supremum over k on both sides, we get x∈l∞(ℳ,Λ,p). The inclusion c0I(ℳ,Λ,p)⊂cI(ℳ,Λ,p) is obvious. Thus,
(23)c0I(ℳ,Λ,p)⊂cI(ℳ,Λ,p)⊂l∞(ℳ,Λ,p).

This completes the proof of the theorem.

Theorem 3.

Let ℳ=(Mk) be a Musielak-Orlicz function and let p=(pk) be a bounded sequence of positive real numbers. Then, l∞(ℳ,Λ,p) is a paranormed space with paranorm defined by
(24)g(x)=inf{ρ>0:supkMk(|Λk(x)|ρ)pk≤1}.

Proof.

It is clear that g(x)=g(-x). Since Mk(0)=0, we get g(0)=0. Let us take x,y∈l∞(ℳ,Λ,p). Let
(25)B(x)={ρ>0:supkMk(|Λk(x)|ρ)pk≤1},B(y)={ρ>0:supkMk(|Λk(y)|ρ)pk≤1}.

Let ρ1∈B(x) and ρ2∈B(y). If ρ=ρ1+ρ2, then we have
(26)supkMk(|Λk(x+y)|ρ)≤(ρ1ρ1+ρ2)supkMk(|Λk(x)|ρ1)+(ρ2ρ1+ρ2)supkMk(|Λk(y)|ρ2).

Thus, supkMk(|Λ(x+y)|/(ρ1+ρ2))pk≤1 and
(27)g(x+y)≤inf{(ρ1+ρ2)>0:ρ1∈B(x),ρ2∈B(y)}≤inf{ρ1>0:ρ1∈B(x)}+inf{ρ2>0:ρ2∈B(y)}=g(x)+g(y).

Let σs→σ, where σ,σs∈ℂ and let g(xs-x)→0 as s→∞. We have to show that g(σsxs-σx)→0 as s→∞. Let
(28)B(xs)={ρs>0:supkMk(|Λk(xs)|ρs)pk≤1},B(xs-x)={ρs′>0:supkMk(|Λk(xs-x)|ρs′)pk≤1}.

If ρs∈B(xs) and ρs′∈B(xs-x), then we observe that
(29)Mk(|Λk(σsxs-σx)|ρs|σs-σ|+ρs′|σ|)≤Mk(|Λk(σsxs-σxs)|ρs|σs-σ|+ρs′|σ|+|(σxs-σx)|ρs|σs-σ|+ρs′|σ|)≤|σs-σ|ρsρs|σs-σ|+ρs′|σ|Mk((|Λk(xs)|)ρs)+|σ|ρs′ρs|σs-σ|+ρs′|σ|Mk(|Λk(xs-x)|ρs′).

From the above inequality, it follows that
(30)Mk(|Λk(σsxs-σx)|ρs|σs-σ|+ρs′|σ|)pk≤1
and, consequently,
(31)g(σsxs-σx)≤inf{(ρs|σs-σ|+ρs′|σ|)>0:hhhhhhρs∈B(xs),ρs′∈B(xs-x)}≤(|σs-σ|)>0inf{ρ>0:ρs∈B(xs)}+(|σ|)>0inf{(ρs′)pn/H:ρs′∈B(xs-x)}⟶0ass⟶∞.

This completes the proof.

Theorem 4.

Let ℳ′=(Mk′) and ℳ′′=(Mk′′) be Musielak-Orlicz functions that satisfy the Δ2-condition. Then,

(i)Z(ℳ′′,Λ,p)⊆Z(ℳ′∘ℳ′′,Λ,p),

(ii)Z(ℳ′,Λ,p)∩Z(ℳ′′,Λ,p)⊆Z(ℳ′+ℳ′′,Λ,p) for Z=cI,c0I,mI,m0I.

Proof.

(i) Let x∈c0I(ℳ′′,Λ,p). Then, there exists ρ>0 such that
(32)I-limkℳk′′(|Λk(x)|ρ)pk=0.

Let ϵ>0 and choose δ with 0<δ<1 such that Mk′(t)<ϵ for 0≤t≤δ. Write yk=Mk′′(|Λk(x)|/ρ)pk and consider
(33)limk∈ℕ0≤yk≤δMk′(yk)=limk∈ℕyk≤δMk′(yk)+limk∈ℕyk>δMk′(yk).

Since ℳ=(Mk) satisfies Δ2-condition, we have
(34)limk∈ℕyk≤δMk′(yk)≤Mk′(2)limk∈ℕyk≤δ(yk).

For yk>δ, we have
(35)yk<ykδ<1+ykδ.

Since ℳ'=(Mk′) is nondecreasing and convex, it follows that
(36)Mk′(yk)<Mk′(1+ykδ)<12Mk′(2)+12Mk′(2yk)δ.

Since ℳ′=(Mk′) satisfies Δ2-condition, we have
(37)Mk′(yk)<12KykδMk′(2)+12KykδMk′(2)=KykδMk′(2).

Hence,
(38)limk∈ℕyk>δMk′(yk)≤max(1,Kδ-1Mk′(2))limk∈ℕyk≤δ(yk).
From (32), (34), and (38), we have x=(xk)∈c0I(ℳ′∘ℳ′′,Λ,p). Thus, c0I(ℳ′′,Λ,p)⊆c0I(ℳ′∘ℳ′′,Λ,p). Similarly, we can prove the other cases.

(ii) Let x∈c0I(ℳ′,Λ,p)∩c0I(ℳ′′,Λ,p). Then, there exists ρ>0 such that
(39)I-limkMk′(|Λk(x)|ρ)pk=0,I-limkMk′′(|Λk(x)|ρ)pk=0.

The rest of the proof follows from the following equality:
(40)limk∈ℕ(Mk′+Mk′′)(|Λk(x)|ρ)pk=limk∈ℕMk′(|Λk(x)|ρ)pk+limk∈ℕMk′′(|Λk(x)|ρ)pk.

Corollary 5.

Let ℳ=(Mk) be a Musielak-Orlicz function which satisfies Δ2-condition. Then, Z(p,Λ)⊆Z(ℳ,Λ,p) holds for Z=cI,c0I,mI, and m0I.

Proof.

The proof follows from Theorem 3 by putting Mk′′(x)=x and Mk′(x)=Mk(x)∀x∈[0,∞).

Theorem 6.

The spaces c0I(ℳ,Λ,p) and m0I(ℳ,Λ,p) are solid.

Proof.

We will prove for the space c0I(ℳ,Λ,pΛ). Let x∈c0I(ℳ,Λ,p). Then, there exists ρ>0 such that
(41)I-limkMk(|Λk(x)|ρ)pk=0.

Let (αk) be a sequence of scalars with |αk|≤1∀k∈ℕ. Then, the result follows from the following inequality:
(42)limkMk(|Λk(αx)|ρ)pk≤limkMk(|Λk(x)|ρ)pk
and this completes the proof. Similarly, we can prove for the space m0I(ℳ,Λ,p).

Corollary 7.

The spaces c0I(ℳ,Λ,p) and m0I(ℳ,Λ,p) are monotone.

Proof.

It is easy to prove, so we omit the details.

Theorem 8.

The spaces cI(ℳ,Λ,p) and c0I(ℳ,Λ,p) are sequence algebra.

Proof.

Let x,y∈c0I(ℳ,Λ,p). Then,
(43)I-limkMk(|Λk(x)|ρ1)pk=0,forsomeρ1>0,I-limkMk(|Λk(y)|ρ2)pk=0,forsomeρ2>0.

Let ρ=ρ1+ρ2. Then, we can show that
(44)I-limkMk(|Λk(x·y)|ρ)pk=0.

Thus, (x·y)∈c0I(ℳ,Λ,p). Hence, c0I(ℳ,Λ,p) is a sequence algebra. Similarly, we can prove that cI(ℳ,Λ,p) is a sequence algebra.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

KostyrkoP.ŠalátT.WilczyńskiW.I-convergenceDasP.KostyrkoP.WilczyńskiW.MalikP.I and I*-convergence of double sequencesDasP.MalikP.On the statistical and I variation of double sequencesKumarV.On I and I*-convergence of double sequencesMursaleenM.AlotaibiA.On I-convergence in random 2-normed spacesMursaleenM.MohiuddineS. A.EdelyO. H. H.On the ideal convergence of double sequences in intuitionistic fuzzy normed spacesMursaleenM.MohiuddineS. A.On ideal convergence of double sequences in probabilistic normed spacesMursaleenM.MohiuddineS. A.On ideal convergence in probabilistic normed spacesŞahinerA.GürdalM.SaltanS.GunawanH.Ideal convergence in 2-normed spacesTripathyB. C.HazarikaB.Some I-convergent sequence spaces defined by Orlicz functionsMursaleenM.NomanA. K.On some new sequence spaces of non-absolute type related to the spaces ℓp and ℓ∞ IWilanskyA.RajK.SharmaS. K.Some sequence spaces in 2-normed spaces defined by Musielak-Orlicz functionRajK.SharmaS. K.Some generalized difference double sequence spaces defined by a sequence of Orlicz-functionsRajK.SharmaS. K.Some multiplier sequence spaces defined by a Musielak-Orlicz function in n-normed spacesLindenstraussJ.TzafririL.On Orlicz sequence spacesMaligrandaL.MusielakJ.