New Results on the SSIE with an Operator of the form F ∆ ⊂ E + F (cid:48) x Involving the Spaces of Strongly Summable and Convergent Sequences using the Cesàro Method

: Given any sequence a = ( a n ) n ≥ 1 of positive real numbers and any set E of complex sequences, we can use E a to represent the set of all sequences y = ( y n ) n ≥ 1 such that y / a = ( y n / a n ) n ≥ 1 ∈ E . In this paper, we use the spaces w ∞ , w 0 and w of strongly bounded, summable to zero and summable sequences, which are the sets of all sequences y such that (cid:0) n − 1 ∑ nk = 1 | y k | (cid:1) n is bounded and tends to zero, and such that y − le ∈ w 0 , for some scalar l . These sets were used in the statistical convergence. Then we deal with the solvability of each of the SSIE F ∆ ⊂ E + F (cid:48) x , where E is a linear space of sequences, F = c 0 , c , (cid:96) ∞ , w 0 , w or w ∞ , and F (cid:48) = c 0 , c or (cid:96) ∞ . For instance, the solvability of the SSIE w ∆ ⊂ w 0 + s ( c ) x relies on determining the set of all sequences x = ( x n ) n ≥ 1 ∈ U + that satisfy the following statement. For every sequence y that satisﬁes the condition lim n → ∞ n − 1 ∑ nk = 1 | y k − y k − 1 − l | = 0, there are two sequences u and v , with y = u + v such that lim n → ∞ n − 1 ∑ nk = 1 | u k | = 0 and lim n → ∞ ( v n / x n ) = L for some scalars l and L .


Introduction
We write ω for the set of all complex sequences y = (y k ) k≥1 , ∞ , c and c 0 for the sets of all bounded, convergent and null sequences, respectively; and p = y ∈ ω : ∑ ∞ k=1 |y k | p < ∞ for 1 ≤ p < ∞. If y, z ∈ ω, then we write yz = (y n z n ) n≥1 . Let U = {y ∈ ω : y n = 0}; U + = {y ∈ ω : y n > 0}. We write z/u = (z n /u n ) n≥1 for all z ∈ ω and all u ∈ U, in particular, 1/u = e/u, where e is the sequence with e n = 1 for all n. Finally, if a ∈ U + and E is any subset of ω, then we put E a = (1/a) −1 * E = {y ∈ ω : y/a ∈ E}. Recall that the spaces w ∞ and w 0 of strongly bounded sequences that are summable to zero sequences using the Cesàro method, are the sets of all y such that n −1 ∑ n k=1 |y k | n is bounded and tends to zero. In this way, Hardy and Littlewood [1], defined the set w of strongly convergent sequences using the Cesàro method for real numbers as follows. A sequence y is said to be strongly Cesàro convergent to L, if y − Le ∈ w 0 . These spaces were studied by Maddox [2], Malkowsky and Rakočević [3] and Malkowsky and Başar in [4]. In [5], we gave some properties of well known operators defined by the sets W a = (1/a) −1 * w ∞ and W 0 a = (1/a) −1 * w 0 . In this paper, we deal with special sequence spaces inclusion equations (SSIE) (cf. [5,6]), which are determined by an inclusion, for which each term is a sum or a sum of products of sets of the form (E a ) T and E f (x) T where f maps U + to itself, E is any linear space of sequences and T is a triangle. In [5], we dealt with the class of SSIE of the form F ⊂ E a + F x , where F ∈ {c 0 , p , w 0 , w ∞ }, and E and F are any of the sets c 0 , c, s 1 , p , w 0 or w ∞ with p ≥ 1.
Then we stated some results on the solvability of the corresponding SSIE in the particular case of when a = (r n ) n , and we dealt with the case of when F = F . Then we dealt with the SSIE of the form F ⊂ E a + F x with e ∈ F, and we determined the solutions of these SSIE when a = (r n ) n≥1 , F is either c or s 1 , and E and F are any of the sets c 0 , c, s 1 , p , w 0 or w ∞ with p ≥ 1. Then we solved each of the SSIE c ⊂ D r * E ∆ + c x , with E ∈ {c 0 , c, s 1 }, and the SSIE s 1 ⊂ D r * (s 1 ) ∆ + s x . We also studied the SSIE c ⊂ D r * E C 1 + s (c) x with E ∈ {c, s 1 } and s 1 ⊂ D r * (s 1 ) C 1 + s x , where C 1 is the Cesàro operator defined by (C 1 ) n y = n −1 ∑ n k=1 y k for all y, and we dealt with the solvability of the SSE associated with the previous SSIE and In [6], we dealt with the solvability of the SSIE of the form ∞ ⊂ E + F x where E is a given linear space of sequences and F is either c 0 or ∞ . Then, for given linear space E of sequences, we solved each of the SSIE c 0 ⊂ E + s x and c ⊂ E + s (c) x and the SSE E + s In this paper, we use the difference sequence spaces (c 0 ) ∆ , c ∆ and ( ∞ ) ∆ introduced by Kizmaz (cf. [7]), and we deal with the solvability of each of the SSIE This paper is organized as follows. In Section 2, we recall some well known results on sequence spaces and matrix transformations. In Section 3, we recall some results on the multipliers of some sets. In Section 4, we recall some results used for the solvability of the SSIE. In Section 5, we deal with the solvability of the SSIE with an operator to solve each of the SSIE of the form In Section 6, we study each of the SSIE (w ∞ ) ∆ ⊂ E + F x , where F = c 0 , c or ∞ . Finally, in Section 7, we study the solvability of the SSIE F ∆ ⊂ E + F x where F = w 0 or w, and F = c 0 , c or ∞ .

Preliminaries and Notation
An FK space is a complete metric space, for which convergence implies coordinatewise convergence. A BK space is a Banach space of sequences that is an FK space. A BK space E is said to have AK if for every sequence y = (y k ) k≥1 ∈ E, then y = lim p→∞ ∑ p k=1 y k e (k) , where e (k) = (0, . . . , 1, . . . ), 1 being in the k − th position.
For a given infinite matrix A = (a nk ) n,k≥1 we define the operators A n = (a nk ) k≥1 for any integer n ≥ 1, by A n y = ∑ ∞ k=1 a nk y k , where y = (y k ) k≥1 , and the series are assumed to be convergent for all n. Hence, we are led to the study of the operator A defined by Ay = (A n y) n≥1 mapping between sequence spaces. When A maps E onto F, where E and F are subsets of ω, we write A ∈ (E, F) (cf. [2,[8][9][10]). It is well known that if E has AK, then the set B(E) of all bounded linear operators L mapping onto E, with norm L = sup y =0 ( L(y) E / y E ), satisfies the identity B(E) = (E, E). We denote by ω, c 0 , c and ∞ the sets of all sequences, and the sets of null, convergent and bounded sequences. For any subset F of ω, we write F A = {y ∈ ω : Ay ∈ F}, and for any subset E of ω we write AE = {y ∈ ω : there is x ∈ E such that y = Ax}. Then, for given sequence u ∈ ω we define the diagonal matrix D u by [D u ] nn = u n for all n. It is interesting to rewrite the set E u using a diagonal matrix. Let E be any subset of ω and u ∈ U + we have E u = D u * E = {y = (y n ) n≥1 ∈ ω : y/u ∈ E}. We use the sets s 0 a , s a and s a defined as follows (cf. [5], p. 160). For a given a ∈ U + we put D a * c 0 = s 0 a , D a * c = s (c) a and D a * ∞ = s a . We frequently write c a instead of s (c) a to simplify. Each of the spaces D a * E, where E ∈ {c 0 , c, ∞ } is a BK space normed by y s a = sup n≥1 (|y n |/a n ) and s 0 a has AK. If a = (R n ) n≥1 with R > 0, then we write s R , s 0 R and s a , respectively. We can also write D R for D (R n ) n≥1 . When R = 1, we obtain s 1 = ∞ , s 0 1 = c 0 and s (c) 1 = c. Recall that S 1 = (s 1 , s 1 ) is a Banach algebra and (c 0 , s 1 ) = (c, ∞ ) = (s 1 , s 1 ) = S 1 . We have A ∈ S 1 if and only if sup n (∑ ∞ k=1 |a nk |) < ∞. Recall the Schur's result (cf. [10], Theorem 1.17.8, p. 15) on the class (s 1 , c). We have A ∈ (s 1 , c) if and only if lim n→∞ a nk = l k for some scalar l k , k = 1, 2, . . . , and lim n→∞ ∑ ∞ k=1 |a nk | = ∑ ∞ k=1 |l k |, where the series ∑ ∞ k=1 |l k | is convergent.
We also use the following known lemmas, where the infinite matrix T is said to be a triangle, if T nk = 0 for k > n and T nn = 0 for all n. Lemma 1. Let T and T be any given triangles, and let E, F ⊂ ω. Then, for any given operator T represented by a triangle we have T ∈ (E T , F T ) if and only if T T T −1 ∈ (E, F).
By taking T = D 1/a and T = D b for a, b ∈ U + we obtain the next well-known result. Lemma 2. Let a, b ∈ U + , and let E, F ⊂ ω be any linear spaces. We have A ∈ (E a , F b ) if and only if D 1/b AD a ∈ (E, F).

On the Triangle C(λ) and on the Multipliers of Special Sets
In this section, we define the spaces of strongly bounded and summable sequences by the Cesàro method. Then we recall some results on the multipliers of sequence spaces involving the previous spaces.
3.1. On the Triangles C(λ) and ∆(λ) and the Sets w 0 , w and w ∞ For λ ∈ U, the infinite matrices C(λ) and ∆(λ) are triangles defined as follows. We have [C(λ)] nk = 1/λ n for k ≤ n; this triangle was used, for instance, in [5]; see also the Rhaly matrix studied by [11,12]). Then, the nonzero entries of ∆(λ) are determined by [∆(λ)] nn = λ n for all n, and [∆(λ)] n,n−1 = −λ n−1 for all n ≥ 2. It can be shown that the If λ = e we obtain the well known operator of the first difference represented by ∆(e) = ∆. We then have ∆ n y = y n − y n−1 for all n ≥ 1, with the convention y 0 = 0. We have Σ = C(e) and then, we may write C(λ) = D 1/λ Σ. Note that ∆ = Σ −1 . The Cesàro operator is defined by C 1 = C (n) n≥1 . In the following, we use the inverse of C 1 defined by We use the set of sequences that are a−strongly bounded and a−strongly convergent to zero, defined for a ∈ U + by W a = y ∈ ω : sup n n −1 ∑ n k=1 |y k |/a k < ∞ , and W 0 a = y ∈ ω : lim n→∞ n −1 ∑ n k=1 |y k |/a k = 0 (cf. [5], p. 202). For a = (r n ) n≥1 the sets W a and W 0 a are denoted by W r and W 0 r . For r = 1 we obtain the well-known spaces w ∞ and w 0 of strongly bounded and strongly null sequences by the Cesàro method (cf. [13]).

On the Multipliers of Some Sets
First, we need to recall some well known results. Let y and z be sequences, and let E and F be two subsets of ω. We then write M(E, F) = {y ∈ ω : yz ∈ F for all z ∈ E}; the set M(E, F) is called the multiplier space of E and F. We will use the next lemmas.
Lemma 4. Let a, b ∈ U + and let E and F be two subsets of ω.
From Lemma 2 we obtain the next result.

Lemma 6.
We have: To state results on the multipliers involving the set w, we need the next elementary lemmas.

On the Sequence Spaces Inclusions
In this section, we are interested in the study of the set of all positive sequences x that satisfy the inclusion F ⊂ E + F x where E , F and F are linear spaces of sequences. We may consider this problem as a perturbation problem. If we know the set M(F, F ), then the solutions of the elementary inclusion F x ⊃ F are determined by 1/x ∈ M(F, F ). Now, the question is: Let E be a linear space of sequences. What are the solutions of the perturbed inclusion F x + E ⊃ F? An additional question may be the following one: what are the conditions on E under which the solutions of the elementary and the perturbed inclusions are the same ?

Some Definitions and Results Used for the Solvability of Some SSIE
In the following, we use the notation I(E , F, F ) = {x ∈ U + : F ⊂ E + F x }, where E , F and F are linear spaces of sequences and a ∈ U + . We can state the next elementary results. Lemma 9. Let E , E 1 , F, F , F and F be linear spaces of sequences. Then we have: For any set χ of sequences we let χ = {x ∈ U + : 1/x ∈ χ}. Then we write Φ = {c 0 , c, ∞ , w 0 , w, w ∞ }. By c(1) we define the set of all sequences α ∈ U + that satisfy the condition lim n→∞ α n = 1. Then we consider the condition for any given linear space G of sequences. Notice that condition (1) is satisfied for all G ∈ Φ. Then we denote by U + 1 the set of all sequences α with 0 < α n ≤ 1 for all n. We consider the condition G ⊂ G 1/α for all α ∈ U + 1 .
for any given linear space G of sequences. To show some results on the SSIE, we introduce a linear space of sequences H which contains the spaces E and F and we will use the fact that if H satisfies the condition in (2) then we have H a + H b = H a+b for all a, b ∈ U + (cf. [5], Lemma 4.4, p. 162). Notice that c does not satisfy this condition, but each of the sets c 0 , ∞ , p , w 0 and w ∞ satisfies the condition in (2). Thus we have, for instance, s 0 a + s 0 b = s 0 a+b and W a + W b = W a+b .

Some Properties of the Set I(E , F, F )
We need the next lemma to involve the multiplier of F and F , which is an extension of Lemma 9.

On the Solvability of the SSIE with Operator of the form F
In this section, we determine multipliers involving some difference sequence spaces. Then we state a general result on the solvability of the SSIE with operator F ∆ ⊂ E + F x with e ∈ F. Then we apply these results to solve each of the SSIE c

On the Multipliers of the form M(X
In all that follows, for a ∈ U + , we use the triangle D a Σ, whose the nonzero entries are defined by (D a Σ) nk = a n for k ≤ n. We have (D a Σ) n y = a n ∑ n k=1 y k for all y ∈ ω and for all n. This triangle is also called the Rhally matrix (cf. [11,12]). We obtain some results on the multipliers involving the sets of the difference sequence spaces (c 0 ) ∆ , c ∆ and ( ∞ ) ∆ introduced by Kizmaz (cf. [7]; see also [14]), and stated in the next lemma. Proof. Part (i) follows from the proof of [5], Proposition 6.8, p. 289. (ii) We have a ∈ M(c ∆ , c 0 ) if and only if D a Σ ∈ (c, c 0 ) and by the characterization of (c, c 0 ) we have na n → 0 (n → ∞) and a ∈ s 0 (1/n) n≥1 . In the same way, we have a ∈ M(c ∆ , c) if and only if D a Σ ∈ (c, c), and by the characterization of (c, c) we obtain a ∈ s . For this, let a ∈ M(( ∞ ) ∆ , c). Then we have D a Σ ∈ ( ∞ , c) which implies D a Σ ∈ (c, c) and (na n ) n≥1 ∈ c. This implies lim n→∞ a n = 0, and by the Schur theorem we obtain lim n→∞ (|a n | ∑ n k=1 1) = 0 and a ∈ s 0 . Now, it can easily be seen that , and using Lemma 5, we obtain . Using (ii) and the inclusion ( ∞ ) ∆ ⊂ s (n) n≥1 , we can obtain and the identity M(( ∞ ) ∆ , ∞ ) = s (1/n) n≥1 holds. This completes the proof.

General Result on the Solvability of the SSIE with Operator F ∆ ⊂ E + F x with e ∈ F
In the following, we use the next result.
Proof. Let x ∈ I(E , F ∆ , F ). Then we have F ∆ ⊂ E + F x , and since e ∈ F, we have (n) n≥1 ∈ F ∆ , and there are α ∈ E and ϕ ∈ F such that n = α n + x n ϕ n for all n. Then we have n x n 1 − α n n = ϕ n for all n, and the condition E ⊂ s 0 implies lim n→∞ α n /n = 0. Since F satisfies the condition in (1), we obtain (n/x n ) n≥1 ∈ F and x ∈ F (1/n) n≥1 . Thus we have shown the inclusion This completes the proof.

Solvability of the SSIE c
As a direct consequence of Theorem 1 and Lemma 11, we obtain the following results on the sets of all positive sequences x = (x n ) n≥1 that satisfy each of the SSIE with operator be a linear space of sequences. We have Proof. The result follows from part (ii) of Lemma 11 and Theorem 1, where F = c, and F = c 0 , c and ∞ respectively.
We may state some immediate applications of Theorem 2.
Example 1. Using Lemma 10 and Theorem 2, it can easily be seen that the sets of the positive solutions x = (x n ) n≥1 of each of the SSIE with operator, c ∆ ⊂ ∞ + s (c) x , are determined by (n/x n ) n≥1 ∈ c. Then, the solutions of each of the SSIE c ∆ ⊂ (c 0 ) ∆ + s 0 x , c ∆ ⊂ ∞ + s 0 x and c ∆ ⊂ c + s 0 x are determined by n/x n → 0 (n → ∞). In a similar way, the solutions of each of the SSIE c ∆ ⊂ (c 0 ) ∆ + s x , c ∆ ⊂ ∞ + s x and c ∆ ⊂ c + s x are determined by (n/x n ) n≥1 ∈ ∞ . Example 2. It can easily be seen that w 0 ⊂ s 0 . This implies that the set of all sequences x = (x n ) n≥1 ∈ U + that satisfy the SSIE with operator c ∆ ⊂ w 0 + s 0 x is determined by n/x n → 0 (n → ∞).

Example 3.
The set of all positive sequences that satisfy the SSIE c ∆ ⊂ c C 1 + s 0 x is determined by I c C 1 , c ∆ , c 0 = s 0 . Then, the set of all positive sequences that satisfy the SSIE c ∆ ⊂

Solvability of the SSIE of the Form
In this part, Theorem 1 cannot be applied since e / ∈ c 0 . Thus, we need to use some results stated in Section 4.
be a linear space of sequences, and let F = c 0 , c or ∞ . Then, the set of all the solutions of the SSIE (c 0 ) ∆ ⊂ E + F x is determined by Then we have (c 0 ) ∆ ⊂ E + F x , and since F ⊂ s 1 and s 1 satisfies the condition in (2), we obtain E + F x ⊂ s θ + s x = s θ+x and (c 0 ) ∆ ⊂ s θ+x . Then we have D 1/(θ+x) Σ ∈ (c 0 , s 1 ), and by the characterization of (c 0 , s 1 ) we have n/(θ n + x n ) = O(1) (n → ∞). Using the inclusion E ⊂ s θ with θ ∈ s 0 The converse follows from Theorem 1 and part (i) of Lemma 11, where M((c 0 ) ∆ , s 1 ) = s (1/n) n≥1 .
We consider another example, where bv p = p ∆ with p > 1 is the set of p−bounded variations (cf. [14]).

Solvability of the SSIE of the Form bv ∞ ⊂ E + F x
In this part, we use the notation bv ∞ for the difference sequence space ( ∞ ) ∆ (cf. [14]) and we study each of the SSIE bv ∞ ⊂ E + F x , where F ∈ {c 0 , c, ∞ }.
be a linear space of sequences. Then, the sets of all positive sequences x are determined by Proof. First, we show the identities I(E , bv ∞ , ∞ ) = s (1/n) n≥1 and I(E , bv ∞ , c 0 ) = s 0 , F = ∞ and F = ∞ and c 0 , respectively, we obtain . We have s 0 Then we have x n n + x n = 1 n x n + 1 for all n, and as we have just seen, we have lim n→∞ n/x n = l for some scalar l and lim n→∞ 1 n x n + 1 Thus, we have shown the inclusion s . These statements imply the we obtain (1/(n + x n )) n≥1 ∈ s 0 . Then we have n/(n + x n ) → 0 (n → ∞) and (n/x n ) n≥1 ∈ c 0 , and we have shown the inclusion , by part (i) of Lemma 10, we conclude I(E , ( ∞ ) ∆ , c) = s 0 . This completes the proof.
We obtain the following result, where bs = ( ∞ ) Σ is the set of all bounded series.

Example 6. The solutions of each of the SSIE bv
x and bv ∞ ⊂ bs + s (c) x are determined by I( ∞ , bv ∞ , c) = I(bs, bv ∞ , c) = s 0 By using similar arguments as in Example 5, we obtain the following result.
x . For instance, the solvability of the SSIE (w ∞ ) ∆ ⊂ s 0 (n) n≥1 + s x consists of determining the set of all positive sequences x = (x n ) n≥1 that satisfy the next statement. For every y such that n −1 ∑ n k=1 |y k − y k−1 | = O(1) there are two sequences u and v with y = u + v where lim n→∞ u n /n = 0 and v n /x n = O(1) (n → ∞).

Determination of the Sets
We state the next Lemma.
6.2. Application to the Solvability of the SSIE of the form (w ∞ ) ∆ ⊂ E + F x .
In the following theorem, we solve each of the SSIE (w ∞ ) ∆ ⊂ E + F x , where F ∈ {c 0 , c, ∞ }.
Funding: This research received no external funding.

Conflicts of Interest:
The authors declare no conflict of interest.