© Hindawi Publishing Corp. INCLUSION RESULTS FOR CONVOLUTION SUBMETHODS

If B is a summability matrix, then the submethod Bλ is the matrix obtained by deleting a set of rows from the matrix B. Comparisons between Euler-Knopp submethods and the Borel summability method are made. Also, an equivalence result for convolution submethods is established. This result will necessarily apply to the submethods of the Euler-Knopp, Taylor, Meyer-König, and Borel matrix summability methods.


Introduction and notation.
Let E be an infinite subset of N∪{0} and consider E as the range of a strictly increasing sequence of nonnegative integers, say E := {λ(n)} ∞ n=0 .If B := (b n,k ) is a summability matrix, then the submethod B λ is the matrix whose nkth entry is B λ [n, k] := b λ(n),k .Thus, for a given sequence x, the B λ -transform of x is the sequence B λ x with (1.1) Since B λ is a row submatrix of B, it is regular (i.e., limit preserving) whenever B is regular.Row submatrices have appeared throughout the literature [5,6,8,12], but they were first studied as a class unto themselves by Goffman and Petersen [7], and later by Steele [14].The class of Cesàro submethods has been studied by Armitage and Maddox [1] and Osikiewicz [11].
Let A and B be two summability matrices.If every sequence which is A-summable is also B-summable to the same limit, then B includes A, denoted by A ⊆ B. Also, B is called a triangle if b n,k = 0 for all k > n and b n,n ≠ 0 for all n.The following lemma extends [1,Theorem 1].
(2) If B is a triangle and B λ ⊆ B ρ , then F \ E is finite.
(3) If B is a triangle, then B λ is equivalent to B ρ if and only if the symmetric difference E F is finite.
In particular, B ⊆ B λ for any λ.
Proof.Assume F \ E is finite and let x be a sequence that is B λ -summable to L. Then there exists an N such that {ρ(n Now assume B is a triangle, and hence invertible, and j=0 with ρ(n(j)) < ρ(n(j + 1)).Consider the sequence y defined by and let x be the sequence B −1 y.Then, for every n, Hence, lim n (B λ x) n = 0.However, for every j, Thus x is not B ρ -summable.Therefore B ρ does not include B λ , which completes the contrapositive of assertion (2).Lastly, assertion (3) follows from ( 1) and ( 2) since To show the reason for the necessity of B being a triangle in assertion (2) of Lemma 1.1, consider the matrix B whose nkth entry is 1, if n odd and n = k. (1.5) Then if λ(n) := 2n and ρ(n

Inclusion results for Euler-Knopp submethods.
For r ∈ C \ {0, 1}, the Euler-Knopp method of order r is given by the matrix E r whose nkth entry is (2.1) For the case r = 1, E 1 is the identity matrix, and E 0 is the matrix whose nkth entry is It is well known that E r is regular if and only if 0 < r ≤ 1 (see [4]).
Let E := {λ(n)} be an infinite subset of N∪{0} and r ∈ C\{0, 1}.The submethod E r ,λ is the matrix whose nkth entry is ( Proof.If S is finite, then by Lemma 2.1, E r and E r ,λ are equivalent.But the Borel summability method includes E r for r > 0 (see [4]).Hence, it also includes E r ,λ .If S is infinite, then it may be written as a strictly increasing sequence of nonnegative integers, say consider the sequence y defined by and let x be the sequence Hence, x is E r ,λ -summable to 0. Now observe that for a given n, Thus, for n = ρ(m), we have and it follows that lim sup for all nonzero t and hence x is not Borel summable.

Theorem 2.3.
There exists a sequence which is Borel summable but not E r ,λ -summable for any λ and r > 0.
Proof.Let r > 0 and consider the sequence x defined by Then it can be shown that (E r ,λ x) n = (−1) λ(n) λ(n).Hence x is not E r ,λ -summable for any λ.However, and hence x is Borel summable to 0.
Lemma 3.1.The convolution method C * is regular if and only if p 0 < 1.
Proof.Note that for n = 0, the result holds.So assume the result holds for some integer n > 0. Then By induction, the result follows.
Lemma 3.3.Let C * be the convolution method formed from the sequences p and q and D * := (d n,k ) the convolution method formed from the sequences p and q := {1, 0, 0,...}.Then for nonnegative integers n, k, and j, (3.5) The proof of this lemma is a straightforward induction argument left to the reader.
Lemma 3.4.Let C * be the convolution method formed from the sequences p and q.

and
Proof.Let D * := (d n,k ) be the convolution method formed from the sequences p and q := {1, 0, 0,...}.We first prove that the result holds for D * . Let (3.7) The first and the third terms on the right-hand side of the inequality are bounded by a result of Bikjalis and Jasjunas [2].For the middle term, the mean value theorem yields where ξ n,k ∈ (x n,k ,x n,k+1 ) and K > 0 is some constant.Thus, the result holds for the convolution method D * .Then, by Lemma 3.3, (3.9)

Equivalence results for convolution submethods.
Let E := {λ(n)} be an infinite subset of N ∪{0}.The convolution submethod C * λ is the matrix whose nkth entry is The following theorem compares C * λ with C * for bounded sequences.
λ and C * are equivalent for all sequences.So assume S is infinite.Then there exists an N such that for n ≥ N, ρ(n) > λ(0).Since E and S are disjoint, for n ≥ N, there exists an integer m such that λ(m) < ρ(n) < λ(m + 1).We write ρ(n) := λ(m) + j, where 0 By Lemma 3.3, this becomes ( By Lemma 3.4, there exists an M > 0 such that Then, by Lemma 3.2, Thus, Therefore, the sequence C * x may be partitioned into two disjoint subsequences, namely (C * λ x) n = (C * x) λ(n) and (C * ρ x) n = (C * x) ρ(n) , each having the common limit L. Thus, x must be C * -summable to L, and hence C * and C * λ are equivalent for bounded sequences.
The following theorem is a well-known result due to Meyer-König (see [10,Theorem 25]).
Since the Euler-Knopp methods of order 0 < r < 1, Taylor methods of order 0 < r < 1, Meyer-König methods of order 0 < r < 1, and the Borel matrix method all have generating sequences satisfying the conditions in Theorem 4.2, the following corollary is immediate.Corollary 4.4.Let E := {λ(n)} be an infinite subset of N ∪ {0} and 0 < r < 1.If λ satisfies condition (4.6), then E r ,λ , E r , T r ,λ , T r , S r ,λ , S r , B * λ , B * , and the Borel method are all equivalent for bounded sequences.
The next theorem presents an equivalence relationship between the C * λ submethods.
Proof.Let x be a bounded sequence and consider the sequences Then, as in the proof of Theorem 4.2, we have Thus, C * λ and C * ρ are equivalent for bounded sequences.

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: The method E r ,λ ⊆ E r ,ρ if and only if F \ E is finite.(2)The method E r ,λ is equivalent to E r ,ρ if and only if the symmetric difference E F is finite.