SOLVABILITY OF CERTAIN SEQUENCE SPACES EQUATIONS WITH OPERATORS

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1 Novi Sad J. Math. Vol. 44, No., 204, 9-20 SOLVABILITY OF CERTAIN SEQUENCE SPACES EQUATIONS WITH OPERATORS Bruno de Malafosse Abstract. In this paper we deal with special sequence space equations SSE with operators, which are determined by an identity whose each term is a sum or a sum of products of sets of the form χ a T and χ fx T where f map U + to itself and χ is any of the symbols s, s 0, or s c. Among other things under some conditions we solve SSE with operators χ a C λ D τ +χ x C µ D τ = χ b, and χ a C λ C µ+ χ x C λσ C µ = χ b where χ { s, s 0}, and χ a C λ D τ +s 0 x C µ D τ = χ b where χ is either of the symbols s, or s c and C ν D τ is a factorable matrix. AMS Mathematics Subject Classification 200: 40C05, 46A45 Key words and phrases: matrix transformations, BK space, multiplier of sets of sequences, sequence space inclusion equations, sequence space equations with operators. Introduction In [2] Wilansky introduced sets of the form a χ, where a = a n n is a sequence satisfying a n 0 for all n, and χ is any set of sequences. Recall that x = x n n belongs to a χ if a n x n n belongs to χ. In this way, for any strictly positive sequence a, are defined the sets s 0 a, s c a and s a by a χ where χ is either of the sets c 0, c, and l respectively. In [5, 8] the sum s a + s b and the product s a s b of the sets s a and s b were defined, and characterizations of matrix transformations mapping in the sets s a +s 0 b q and s a + s c b q were given, where is the operator of the first difference. In [6] de Malafosse and Malkowsky gave among other things properties of the matrix of weighted means considered as operator in the set s a. Characterizations of matrix transformations mapping in s 0 α λi h + s c β µi l with λ, µ, h, l C can be found in [9]. There are many other results using the sets s 0 a, s c a and s a, let us cite for instance applications to the following topics, σ core, [7], solvability of infinite tridiagonal systems, [6], measure of noncompactness, [8], Hardy theorem, [20] and statistical convergence, [9]. In this paper our aim is to solve special sequence spaces equations SSE, which are determined by an identity whose each term is a sum or a sum of products of sets of the form χ a T and χ fx T where f maps U + to itself, and χ is any of the symbols s, s 0, or s c, the sequence x is the unknown and T is LMAH Université du Havre, I.U.T Le Havre BP , Le Havre, France, bdemalaf@wanadoo.fr

2 0 Bruno de Malafosse a given triangle. The resolution of such SSE consists in determining the set of all sequences x satisfying the identity, see for instance [3,, 5, 2, 7, 4, 0]. This paper is organized as follows. In Section 2 we recall some results on the sum and the product of sets of the form χ a, where χ is either of the symbols s, or s 0. In Section 3 we solve the sequence spaces equations χ a C λ D τ + χ x C µ D τ = χ b and χ a N q + χx N p D q/p = χb for χ { s, s 0}, where N q is the operator of weighted means in some cases and we solve another type of SSE defined by χ a C λ D τ + s 0 x C µ D τ = s 0 b where χ is either of the symbols s, or s c. In Section 4 we deal with SSE with operators represented by products of triangles of the form χ a C λ C µ + χ x C λσ C µ = χ b where C ν D τ is a factorable matrix for χ { s, s 0}. 2. Sum and product of sequence spaces of the form χ a, where χ is either of the symbols s, s The sets χ a, where χ is either of the symbols s, s 0, or s c for a U + We write s, l, c and c 0 for the sets of all complex, bounded, convergent and convergent to naught sequences, respectively. For a given infinite matrix Λ = λ nm n,m we define the operators Λ n, for any integer n, by Λ n ξ = m= λ nmξ m, where ξ = ξ m m, and the series are assumed convergent for all n. So we are led to the study of the operator Λ defined by Λξ = Λ n ξ n mapping a sequence space into another sequence space. A Banach space E of complex sequences with the norm E is a BK space if each projection P n :E C defined by P n ξ = ξ n is continuous. A BK space E is said to have AK if every sequence ξ E has a unique representation ξ = n= ξ ne n where e n is the sequence with in the n-th position, and 0 otherwise. Let a be a nonzero sequence. Using Wilansky s notations we write /a E for the set of all ξ = ξ n n such that a n ξ n n E. Let U + be the set of all real sequences ξ with ξ n > 0 for all n. If ξ s we define D ξ as the diagonal matrix defined by [D ξ ] nn = ξ n for all n, we have D a E = /a E and it can be easily shown Λ D a E, D b F if and only if D /b ΛD a E, F where E, F s. Recall that for a U + we have s a = D a l, s 0 a = D a c 0 and s c a = D a c. Each of the previous sets is a BK space normed by ξ sa, where ξ sa = sup n ξ n /a n <. So we can define s a as the set of all sequences ξ such that ξ n /a n n l, s 0 a as the set of all sequences ξ such that ξ n /a n 0 n and s c a as the set of all sequences ξ such that ξ n /a n l n for some l C, cf. [3, 4]. If a = r n n, we write χ a = χ r where χ is any of the symbols s, s 0, or s c to simplify. When r =, we obtain s = l, s 0 = c 0 and s c = c. If we let e =,,..., then we have s e = s = l, s 0 e = s 0 = c 0 and s c e = s c = c. When Λ maps E into F we write Λ E, F, see [2]. So we have Λξ F for all ξ E, Λξ F means that for each n the series defined by Λ n ξ = m= λ nmξ m is convergent and Λ n ξ n F. The set S a of all infinite matrices Λ = λ nm n,m such that

3 Solvability of certain sequence spaces equations with operators Λ Sa = sup n a n m= λ nm a m < is a Banach algebra with identity normed by Λ Sa. Recall that if Λ s a, s a, then Λξ sa Λ Sa ξ sa for all ξ s a. It is well-known that S a = s 0 a, s a = s c a, s a = s a, s a Sum of sets of the form χ a where χ is either of the symbols s 0, or s. In this subsection we recall some properties of the sum E + F of sets of the form s 0 a, or s a. Let E, F s be two linear vector spaces. We write E + F for the set of all sequences ξ = ζ + ζ where ζ E and ζ F. In the next result we use the notation [max a, b] n = max a n, b n. We prove the following results. Proposition. Let a, b U + and assume χ is either of the symbols s 0, or s. Then we have i χ a χ b if and only if there is K > 0 such that a n Kb n for all n. ii χ a = χ b if and only if s a = s b, that is, there are K, K 2 > 0 such that K b n a n K 2 for all n. iii χ a + χ b = χ a+b = χ maxa,b. iv χ a + χ b = χ a if and only if b/a l. Proof. The case χ = s was shown in [5, Proposition, p. 244], and [8, Theorem 4, p. 293]. The case χ = s 0 can be shown similarly, since we have s a = s b if and only if s 0 a = s 0 b. Notice that χ a χ b is equivalent to a s b Solvability of the equation χ a + χ x = χ b where χ is either of the symbols s 0, or s. In the following we determine the set of all sequences x = x n n U + such that y n = b n O n if and only if there are u, v s such that y = u + v and u n = a n O and v n = x n O n for all y s. Similarly we determine the sequences x U + such that y n = b n o if and only if there are u, v s such that y = u + v and u n = a n o and v n = x n o n. Theorem 2. Let a, b U +, and consider the equation χ a + χ x = χ b where χ is either of the symbols s 0, or s and x = x n n U + is the unknown. Then i if a/b c 0, then equation holds if and only if there are K, K 2 > 0 depending on x, such that K b n x n K 2 b n for all n, that is s x = s b. ii If a/b, b/a l, then equation holds if and only if there is K > 0 depending on x such that 0 < x n Kb n for all n; that is, x s b. iii If a/b / l, then equation has no solution in U +.

4 2 Bruno de Malafosse Proof. The case of equation where χ = s was shown in []. For equation with χ = s 0 it is enough to note that s a + s x = s b can be written in the form s a+x = s b which is turn in s 0 a+x = s 0 b and s0 a + s 0 x = s 0 b. This concludes the proof. In the following corollary we write cl u, u > 0, for the set of all sequences ξ such that Ku n ξ n K u n for all n and for some K, K > 0. This set is an equivalence class for the relation ξrξ if s ξ = s ξ with ξ = u n n. The following SSE is completely solved. Corollary 3. Let r, u > 0. The set Λ χ of all x U + that satisfy the equation 2 χ r + χ x = χ u where χ { s 0, s } is defined by Λ χ = cl u for r < u, s u U + for r = u, for r > u Product of sequence spaces of the form χ a for χ { s 0, s }. In this subsection we will deal with some properties of the product E F of particular subsets E and F of s. For any sequences ξ E and η F we put ξξ = ξ n ξ n. Most of the following results were shown in [5]. n For any sets of sequences E and F, we write E F for the set of all sequences ξξ such that ξ E and ξ F. We immediately have the following results where S χ, χ { s 0, s }, is constituted of all the sets of the form χ a with a U +. Proposition 4. The set S χ, where χ { s 0, s } with multiplication is a commutative group with χ as the unit element. Proof. First it can easily be seen that χ a χ b = χ ab. We deduce the map ψ : U + S χ defined by ψ a = χ a is a surjective homomorphism and since U + with the multiplication of sequences is a group it is the same for S χ. Then the unit element of S χ is ψ e = χ. Remark 5. Note that the inverse of χ a is χ /a with χ { s 0, s }. As a direct consequence of Proposition 4 we deduce the following corollary. Corollary 6. Let a, b, c U + and let χ { s 0, s }. Then i χ a χ b = χ ab. ii χ a χ b = χ a χ c if and only if χ b = χ c. iii The sequence x = x n n U + satisfies the equation χ a χ x = χ b if and only if K b n /a n x n K 2 b n /a n for all n and for some K, K 2 > 0 depending on x. Throughout this paper the unknown of each sequence spaces equation is a sequence x U +.

5 Solvability of certain sequence spaces equations with operators 3 3. The SSE with operators represented by factorable matrices In this section we deal with the resolution of SSE of the form χ a C λ D τ + χ x C µ D τ = χ b and χ a N q + χx N p D q/p = χb for χ { s, s 0} where N q is the operator of weighted means in some cases. Then we solve the SSE χ a C λ D τ + s 0 x C µ D τ = s 0 b, where χ is either of the symbols s, or sc. 3.. The operators C η, η and the sets Γ, Γ and Ĉ The infinite matrix T = t nm n,m is said to be a triangle if t nm = 0 for m > n and t nn 0 for all n. Now let U be the set of all sequences u n n s, with u n 0 for all n. The infinite matrix C η = c nm n,m, for η = η n n U, is defined by c nm = η n if m n, 0 otherwise. It can be shown that the matrix η = d nm n,m with η n if m = n, d nm = η n if m = n and n 2, 0 otherwise, is the inverse of C η, that is C η η ξ = η C η ξ for all ξ s. If η = e we get the well known operator of the first difference represented by e =. We then have ξ n = ξ n ξ n for all n, with the convention ξ 0 = 0. It is usually written Σ = C e. Note that = Σ and, Σ S R for any R >. Consider the sets { } Ĉ = ξ U + : [C ξ ξ] n = n ξ m = O, Γ = and { ξ U + : lim n ξn ξ n } <, Γ = ξ n m= { ξ U + : lim sup n ξn ξ n } < G = { ξ U + : there exist C > 0 and γ > such that ξ n Cγ n for all n }. By [4, Proposition 2., p. 786] and [6, Proposition 2.2 p. 88], we obtain the next lemma. Lemma 7. Γ Γ Ĉ G. We also need the following results. Lemma 8. [8, Proposition 9, p. 300] Let a, b U +. Then i the following statements are equivalent α χ a = χ b where χ is any of the symbols s, or s 0, β a Ĉ and s a = s b. ii a Γ if and only if s c a = s c a. rom the preceding results we deduce the following:

6 4 Bruno de Malafosse 3.2. Application to the equation χ a C λ D τ + χ x C µ D τ = χ b where x is the unknown Let a, b, λ, µ, τ U + and consider the equation 3 χ a C λ D τ + χ x C µ D τ = χ b, where χ = s 0, or s and x U + is the unknown. The operator represented by C λ D τ = D /λ ΣD τ is called a factorable matrix. For χ = s 0 solving the SSE 3 consists of determining all sequences x U + such that the condition y n /b n 0 n holds if and only if there are u, v s such that y = u + v and τ u τ n u n λ n a n We then have the following result. 0 and τ v τ n v n µ n x n 0 n for all y s. Theorem 9. Let a, b, λ, µ, τ U +. Then i If bτ / Ĉ, then equation 3 where x is the unknown has no solutions. ii If bτ Ĉ we then have a if aλ/bτ c 0, then equation 3 is equivalent to s x = s bτ/µ, that is K b n τ n /µ n x n K 2 b n τ n /µ n for all n and for some K, K 2 > 0. b if aλ/bτ, bτ/aλ l, then the solutions of 3 are all sequences that satisfy x s bτ/µ, that is, x n Kb n τ n /µ n for all n and for some K > 0. c If aλ/bτ / l, then 3 has no solution. Proof. We have [C λ D τ ] = D /τ λ and [C µ D τ ] = D /τ µ then χ a C λ D τ = [C λ D τ ] a χ a = D /τ λ χ a and χ x C µ D τ = D /τ µ χ x and equation 3 is equivalent to D /τ λ χ a + D /τ µ χ x = χ b, that is D /τ χ aλ + χ µx = χb. Since λ = D λ and µ = D µ we deduce 4 χ aλ + χ µx = χ b D/τ = χ bτ. Then 4 is equivalent to χ aλ+µx = χ bτ itself equivalent to χ aλ+µx = χ bτ and bτ Ĉ by Lemma 8. So if bτ / Ĉ equation 3 has no solution and if bτ Ĉ it is enough to apply Theorem to the equation χ aλ + χ µx = χ bτ. We can state the following corollaries. Corollary 0. Consider the equation 5 χ C λ D τ + χ x C µ D τ = χ with χ = s 0, or s. i If τ / Ĉ, then 5 has no solutions. ii If τ Ĉ, then a if λ s 0 τ, then 5 is equivalent to s x = s τ/µ ; b if λ s τ, τ s λ, then the solutions of SSE 5 are all sequences that satisfy x s τ/µ ; c if λ / s τ, then 5 has no solution.

7 Solvability of certain sequence spaces equations with operators 5 Proof. It is enough to take a = b = e in Theorem 9. In the following remark where C λ = C n n is the Cesàro operator denoted by C, the SSE is completely solved. Remark. Consider the SSE 6 χ C D τ + χ x C D τ = χ with χ = s 0, or s. If τ / Ĉ then 6 has no solution. If τ Ĉ the solutions of 6 are all the sequences that satisfy s x = s τ n /n n. This means that there are K, K 2 > 0 such that K τ n /n x n K 2 τ n /n for all n. Indeed, we have λ n = n and since τ Ĉ implies that there is γ > such that τ n Kγ n for all n and for some K > 0, we deduce that n/τ n 0 n. So it is enough to apply Corollary 0 ii. To state the next result, consider the equation 7 χ a C λ + χ x C µ = χ b with χ = s 0, or s. Corollary 2. Let a, b, λ, µ U +. Then i If b / Ĉ, then equation 7 has no solution. ii If b Ĉ, then 3 cases are possible, a if aλ/b c 0 then the solutions x U + of equation 7 are all sequences that satisfy s x = s b/µ ; b if there are k, k 2 > 0 such that k a n λ n /b n k 2 for all n, then equation 7 is equivalent to x s b/µ ; c if aλ/b / l, then equation 7 has no solution. Proof. This result follows from Theorem 9 with τ = e. When a = e we obtain the next corollary where the SSE is totally solved. Corollary 3. The equation χ C + χ x C = χ b with χ = s 0, or s, has no solution if b / Ĉ and if b Ĉ the solutions are determined by K b n /n x n K 2 b n /n for all n and for some K, K 2 > 0. Proof. This result follows from Corollary 2 with a = e, λ n = µ n = n for all n. Indeed, the condition b Ĉ implies that there is γ > such that b n Kγ n for all n. Then we have a n λ n /b n Knγ n = o n. Now state the next result where we put λ 0 = n n. Here the SSE is also totally solved. Corollary 4. Let r, r 2 > 0 and consider the equation 8 χ r C D λ0 + χ x C D λ0 = χ r2 with χ = s 0, or s. i If r 2, then equation 8 has no solution. ii If r 2 >, then a if r < r 2, then equation 8 is equivalent to s x = s r2 ; b if r = r 2, then equation 8 is equivalent to x s r2 ; c if r > r 2, then equation 8 has no solution.

8 6 Bruno de Malafosse Proof. i If r 2, then we have r2 n n / Ĉ, since by Lemma 7 we have Ĉ G and by Corollary 2 equation 8 has no solutions. ii Case when r 2 >. a First we have n r n 2 lim n n r2 n = < r 2 and since Γ Ĉ we deduce nr2 n n Ĉ. So by Theorem 9 we have n a n λ n = nrn r b n τ n nr2 n = = o n r 2 and s x = s r2. The cases b and c can be shown similarly The SSE with operators of the weighted means In this subsection we use the operator of weighted means N q defined by the triangle [ ] N q nm = q m/q n for m n, where Q n = n m= q m, for all n, with q U +. Consider now the equation 9 χ a N q + χx N p D q/p = χb where χ = s 0, or s for p, q U +. The question in the case when χ = s is: what are the sequences x U + such that y n = O b n n if and only if there are u, v s such that y = u + v and q u q n u n Q n = O a n and q v q n v n P n = O x n n for all y? Since we have N q = D /Q ΣD q = C Q D q, it is enough to take Q = λ, µ = P and τ = q in Theorem 9. We then have Corollary 5. Let a, b, p, q U +. Then i If bq / Ĉ, then 9 has no solution; ii if bq Ĉ, then a aq/bq c 0 implies that SSE 9 is equivalent to s x = s bq/p. b If there are k, k 2 > 0 such that k a n Q n /b n q n k 2 for all n, the solutions of 9 are all the sequences that satisfy x s bq/p that is, x n Kb n q n /P n for all n. c If aq/bq / l, then 9 has no solution. This result leads to the following application. Example 6. Let R > 0 and let S be the set of all sequences x U + that satisfy the statement: y n /R n 0 n if and only if there are u, v such that y = u + v and 2 n n m= 2 m u m 0 and nx n n 2 m v m 0 n for all y s. m= It can be shown that the set S is empty if R < ; if R =, it is equal to s /nn and if R > it is determined by K 2R n /n x n K 2 2R n /n for all n. To end this section consider a new type of SSE using the sets s c a.

9 Solvability of certain sequence spaces equations with operators On the SSE χ a C λ D τ + s 0 x C µ D τ = s 0 b where χ is either s, or s c Consider now another type of SSE with factorable matrices using the set s c a and that are totally solved. Here we determine the set of all the sequences x U + such that the condition y n /b n 0 n holds if and only if there are u, v s such that y = u + v and τ u τ n u n λ n a n l and τ v τ n v n µ n x n 0 n for all y s and for some scalar l. We state the next lemma, which is a direct consequence of [7, Theorem 4.4, p. 7]. Lemma 7. Let a, b U + and consider the SSE 0 χ a + s 0 x = s 0 b, where χ is either s, or s c. i if a/b c 0, then the solutions of 0 are all the sequences that satisfy s x = s b. ii if a/b / c 0, then 0 has no solution. From Lemma 7 and Theorem 9 we deduce the resolution of the SSE χ a C λ D τ + s 0 x C µ D τ = s 0 b where χ is either s, or s c. Theorem 8. Let a, b, λ, µ, τ U +. Then i if bτ / Ĉ, then SSE has no solution. ii If bτ Ĉ, then two cases are possible, a if aλ/bτ c 0, then the solutions of are all the sequences that satisfy s x = s bτ/µ ; b if aλ/bτ / c 0, then has no solution. Proof. Let χ be any of the symbols s, or s c. Show that if x satisfies, then χ aλ + s 0 µx = s 0 bτ and bτ Ĉ. Reasoning as in the proof of Theorem 9, we have that is equivalent to 2 χ aλ + s 0 µx = s 0 b D/τ = s 0 bτ, and since we have s 0 aλ χ aλ s aλ and s 0 µx s µx, we deduce s 0 aλ+µx = s 0 aλ + s 0 µx χ aλ + s 0 µx s aλ + s µx = s aλ+µx. Then s 0 aλ+µx s 0 bτ s aλ+µx. The first inclusion is equivalent to I s 0 aλ+µx, s0 bτ and to D /bτ D aλ+µx c 0, c 0. Since c 0, c 0 c 0, s = S, we deduce a n λ n + µ n x n b n τ n K for all n.

10 8 Bruno de Malafosse The second inclusion yields = Σ s 0 bτ, s aλ+µx, that is D/aλ+µx ΣD bτ c 0, l = S and b τ b n τ n a n λ n + µ n x n K for all n. We deduce b τ b n τ n b n τ n = b τ b n τ n a n λ n + µ n x n a n λ n + µ n x n b n τ n KK for all n. We conclude bτ Ĉ and by 2 and Lemma 8 we have χ aλ + s 0 µx = s 0 bτ. Conversely if χ aλ + s 0 µx = s 0 bτ and bτ Ĉ, then 2 and hold. We conclude the proof using Lemma 7. Remark 9. Note that the SSE in has solutions if and only if bτ Ĉ and aλ/bτ c On the equation χ a C λ C µ + χ x C λσ C µ = χ b In this section for a, b, λ, µ, σ U + we consider an equation that generalizes SSE 3 and defined for b Ĉ by 3 χ a C λ C µ + χ x C λσ C µ = χ b, where χ is any of the symbols s, or s 0. For χ = s 0 the resolution of equation 3 consists in determining the set of all x U + such that for every y s the condition y n /b n 0 n holds if and only if there are u, v s such that y = u + v and 4 n m n m u k 0 and v k 0 n. λ n a n λ n σ n x n m= µ m k= m= µ m k= To solve equation 3 we state the following proposition. Proposition 20. Assume that b Ĉ. Then i if b/µ / Ĉ, then equation 3 has no solution. ii Let b/µ Ĉ. Then a if aλµ/b c 0, then equation 3 holds if and only if s x = s b/λσµ ; b if aλµ/b, b/aλµ l, then equation 3 holds if and only if x s b/λσµ ; c if aλµ/b / l, then equation 3 has no solution. Proof. Equation 3 is equivalent to µ λ χ a + λσ χ x = χ b, that is 5 λ χ a + λσ χ x = χ b µ = D /µ χ b and since b Ĉ, we have D /µ χ b = D /µ χ b = χ b/µ. So equation 5 is equivalent to χ aλ + χ λσx = χ b/µ. Then by Lemma 8 equation 5 is equivalent to b/µ Ĉ and χ aλ + χ λσx = χ b/µ. We conclude by Theorem and Corollary 6 that if aλµ/b c 0 equation, χ aλ + χ λσx = χ b/µ is equivalent to s x = s b/λσµ. The cases b and c follow immediately from Theorem.

11 Solvability of certain sequence spaces equations with operators 9 Example 2. The set of all x U + such that y n /2 n = O n holds if and only if there are u, v s such that y = u + v and 6 n n m u k m m= k= for all y is given by = O and x n n m v k = O n m n m= k= 7 K 2 n x n K 2 2 n for all n. Indeed, the previous statement is equivalent to the equation 8 l C 2 + sx C /n n C = s 2. We have b = 2 n n Ĉ, b/µ = 2 n /n n Ĉ and a n λ n µ n /b n = n 2 2 n 0 n. So we obtain 7. Furthermore for each x satisfying 7, we have l C 2 + sx C /n n C, s α = s2, s α for α U +. So A l C 2 + sx C /n n C, s α if and only if sup n α n m= a nm 2 m <. References [] Farés A., de Malafosse, B., Sequence spaces equations and application to matrix transformations Int. Math. Forum , [2] Maddox, I.J., Infinite matrices of operators. Berlin, Heidelberg and New York: Springer-Verlag, 980. [3] de Malafosse, B., Contribution à l étude des systèmes infinis. Thèse de Doctorat de 3è cycle, Univ. Paul Sabatier, Toulouse III, 980. [4] de Malafosse, B., On some BK space. Int. J. Math. Math. Sci , [5] de Malafosse, B., Sum and product of certain BK spaces and matrix transformations between these spaces. Acta Math. Hung. 04 3, 2004, [6] de Malafosse, B., The Banach algebra B X, where X is a BK space and applications. Mat. Vesnik, , [7] de Malafosse, B., Space S α,β and σ core. Studia Math , [8] de Malafosse, B., Sum of sequence spaces and matrix transformations. Acta Math. Hung , [9] de Malafosse, B., Sum of sequence spaces and matrix transformations mapping in s 0 α λi h + s c β µi l. Acta math. Hung., , [0] de Malafosse, B., Application of the infinite matrix theory to the solvability of certain sequence spaces equations with operators. Mat. Vesnik 54, 202,

12 20 Bruno de Malafosse [] de Malafosse, B., Applications of the summability theory to the solvability of certain sequence spaces equations with operators of the form B r, s. Commun. Math. Anal. 3, 202, [2] de Malafosse B., Solvability of certain sequence spaces inclusion equations with operators, Demonstratio Math. 46, 2 203, [3] de Malafosse, B., Solvability of sequence spaces equations using entire and analytic sequences and applications. J. Ind. Math. Soc. 8 N -2, 204, [4] de Malafosse, B., Malkowsky, E., On the solvability of certain SSIE with operators of the form B r, s. Math. J. Okayama. Univ , [5] de Malafosse, B., Malkowsky, E., On sequence spaces equations using spaces of strongly bounded and summable sequences by the Cesàro method. Antartica J. Math , [6] de Malafosse, B., Malkowsky, E., Matrix transformations in the sets χ N pn q where χ is in the form s ξ, or s ξ, or s c ξ. Filomat , [7] de Malafosse, B., Rakočević V., Matrix transformations and sequence spaces equations. Banach J. Math. Anal , -4. [8] de Malafosse, B., Rakočević, V., Applications of measure of noncompactness in operators on the spaces s α, s 0 α, s c α and lα. p J. Math. Anal. Appl , [9] de Malafosse, B., Rakočević, V., Matrix Transformations and Statistical convergence. Linear Algebra Appl , [20] de Malafosse, B., Rakočević V., A generalization of a Hardy theorem. Linear Algebra Appl , [2] Wilansky, A., Summability through Functional Analysis. North-Holland Mathematics Studies 85, 984. Received by the editors August 8, 20