ABEL-TYPE WEIGHTED MEANS TRANSFORMATIONS INTO

Let qk = ( k+α k ) for α > −1 and Qn = Σk=0qk. Suppose Aq = {ank}, where ank = qk/Qn for 0 ≤ k ≤ n and 0 otherwise. Aq is called the Abel-type weighted mean matrix. The purpose of this paper is to study these transformations as mappings into . A necessary and sufficient condition for Aq to be is proved. Also some other properties of the Aq matrix are investigated.


Introduction.
Throughout this paper, we assume that α > −1 and Q n is the partial sums of the sequence {q k }, where q k is as above. Let A q = {a nk }. Then the Abel-type weighted mean matrix, denoted by A q , is defined by for k > n. (1.1) The A q matrix is the weighted mean matrix that is associated with the Abel-type matrix introduced by M. Lemma in [5]. It is regular, indeed, totally regular.

Basic notation and definitions.
Let A = (a nk ) be an infinite matrix defining a sequence summability transformation given by a nk x k , (2.1) where (Ax) n denotes the nth term of the image sequence Ax. Let y be a complex number sequence. Throughout this paper, we use the following basic notation and definitions: (i) c = {The set of all convergent complex sequences}, (ii) = {y : ∞ k=0 |y k | < ∞}, (iii) P = {y : ∞ k=0 |y k | P < ∞}, (iv) (A) = {y : Ay ∈ }, (v) G = {y : y k = O(r k ) for some r ∈ (0, 1)}, (vi) G w = {y : y k = O(r k ) for some r ∈ (0,w), 0 < w < 1}. Definition 1. If X and Y are sets of complex number sequences, then the matrix A is called an X-Y matrix if the image Au of u under the transformation A is in Y , whenever u is in X.
3. Some basic facts. The following facts are used repeatedly.
(1) For any real number α > −1 and any nonnegative integer k, we have A k x k , (3.4) and suppose that Then it follows that

The main results
Proof. By the Knopp-Lorentz theorem [4], A q is an -matrix implies that and hence the assertion easily follows.
Proof. By Lemma 2, we have α > 0. To show that A q is an -matrix, we must show that the condition of the Knopp-Lorentz theorem [4] holds. Using (3.1), we have Hence, by the Knopp-Lorentz theorem [4], A q is an -matrix.
Theorem 1. The following statements are equivalent: Proof. The theorem easily follows by Lemmas 1, 2, and 3.

Remark 1.
In Theorem 1, we showed that A q is an -matrix if and only if 1/Q ∈ . But the converse is not true in general for any weighted mean matrix W p that corresponds to a sequence-to-sequence variant of the general J p power series method of summability [1]. To see this, let We show that 1/P ∈ but W p is not an -matrix. We have using integration by parts repeatedly. This yields and by the condensation test, it follows that 1/P ∈ .
Next, we show that W p is not an -matrix by showing that the condition of the Knopp-Lorentz theorem [4] fails to hold. Using (4.6), it follows that Thus, we have and hence W p is not an -matrix.
Corollary 3. Suppose that z k = k+β k and α < β; then A z is an -matrix whenever A q is an -matrix.
Proof. By the Knopp-Lorentz theorem [4], A α,t is an -matrix implies that This is equivalent to Now from (4.11), we can easily conclude that Hence, A α+1,t is an -matrix.
The next theorem compares the summability fields of the matrices A q and A α,t [5].
Theorem 2. If A α,t and A q are -matrices, then (A q ) ⊆ (A α,t ).
Proof. Let x ∈ (A q ). Then we show that x ∈ (A α,t ). Let y be the A q -transform of the sequence x. Then we have (4.13) Now since y n Q n is the partial sums of the sequence q x , using (3.6) it follows that q k x k t k n . (4.14) This yields and as a consequence we have (A α+1,t y) n = (A α,t x) n . By Lemma 4, A α,t is anmatrix implies that A α+1,t is also an -matrix, and from the assumption that x ∈ (A q ), it follows that y ∈ . Consequently, we have A α+1,t y ∈ and this is equivalent to A α,t x ∈ . Thus, x ∈ (A α,t ) and hence our assertion follows. Remark 2. Theorem 2 gives an important inclusion result in the -setting that parallels the famous inclusion result that exists between the power series method of summability and its corresponding weighted mean in the c-c setting [1].

Theorem 3. If B = A q is an -matrix, then B m is also an -matrix (for m a positive integer greater than 1.)
Proof. Let x ∈ . B is an -matrix implies that x ∈ (B). By Lemma 5, we have (B) ⊆ (B m ) and hence it follows that x ∈ (B m ). Hence, B m is an -matrix. In Corollary 1, we showed that A Q is an -matrix. Here, a question may be raised as to whether A Q maps P into for p > 1. But this is answered negatively by the following theorem.  and hence A Q does not map P into for p > 1 by [3,Thm. 2].
Our next theorem has the form of an extension mapping theorem. It indicates that a mapping of A q from G or G w into can be extended to a mapping of into .
Proof. Since A Q is an -matrix by Corollary 1, the assertion follows by Theorem 5. A q is a G-G matrix, then A q is an -matrix. (2) If A q is a G w -G w matrix, then A q is an -matrix.

Theorem 6. A q is a G-G matrix if and only if 1/Q ∈ G.
Proof. If A q is a G-G matrix, then the first column of A q is must in G. This gives 1/Q ∈ G since a n,0 = q 0 /Q n . Conversely, suppose 1/Q ∈ G. Then 1/Q n ≤ M 1 r n for M 1 > 0 and r ∈ (0, 1). Now let u ∈ G, say |u k | ≤ M 2 t k for some M 2 > 0 and t ∈ (0, 1). Let Y be the A q -transform of the sequence u. Then we have (4.20) Therefore, Y ∈ G and hence it follows that A q is a G-G matrix.
Proof. The proof follows easily using the same steps as in the proof of Theorem 6 by replacing G with G w . α+2 ∈ G, and hence by [5,Thm. 7], it follows that A α+1,t is a G-G matrix.
Proof. The assertion easily follows by replacing G with G w in the proof of Lemma 6. Theorem 8. If A α,t [5] and A q are G-G matrices, then the G(A α,t ) contains G(A q ).
Proof. The proof easily follows using the same techniques as in the proof of Theorem 3 by replacing with G and applying Lemma 6.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system. Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision. In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services. Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems. Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/ according to the following timetable: