Delay Differential Equations in Sequence Spaces

In this paper, we consider linear skew-product semiflows on bundles of Banach fibers over a locally compact metric space. Our aim is to give discrete-time theorems for the existence of global and pointwise continuous-time dichotomies with no invariant unstable manifolds. We involve here a concept of exponential dichotomy for skew-product semiflows weaker than the concept used by Sacker and Sell (J Differ Equ 15:429–458, 1974; 22:478–522, 1976) and Magalhaes (in: Chow, Hale (eds) Dynamics of infinite dimensional systems, NATO Advanced Science Institutes Series F: Computer and Systems Sciences, Springer, New York, vol 37, pp 161–168, 1987); our definition (of no past exponential dichotomy) follows roughly the definition given by Chow and Leiva (Proc Am Math Soc 124:1071–1081, 1996) in the sense that we allow the unstable subspace to have infinite dimension. The main improvement is that we go even more general and we do not assume a priori that the cocycle is invertible on the unstable space (actually we do not even assume that the unstable subspace is invariant under the cocycle). Roughly speaking, we prove that if the solution of the corresponding inhomogeneous variational difference equation belongs to any sequence space (on which the right shift is an isometry) for every inhomogeneity from the same class of sequence spaces, then the continuous-time solutions of the variational homogeneous differential equation will exhibit a (no past) exponential dichotomic behavior. This approach has many advantages among which we emphasize on the facts that the above condition is very general (since the class of sequence spaces that we use includes almost all the known sequence spaces, as the classical p-summable spaces, sequence Orlicz spaces, etc.). Since we use a discrete-time technique we are not forced to require any continuity or measurability hypotheses on the trajectories of the exponentially bounded cocycle. Also, it is worth to mention that from discrete-time conditions we get informations about the continuous-time behavior of the solutions of differential variational equations.

In this research, we introduce the regular Copper Lucas matrix operator, which is based on the Copper Lucas sequence. We investigate the sequence spaces c 0 (Γ) and c (Γ), as well as for 1 ≤ p ≤ ∞ , all of which are linked to the newly defined regular Copper Lucas matrix Γ. Our study explores various properties and inclusion relationships among these spaces, establishes a Schauder basis, and α ‐, β ‐, and γ ‐ duals. In addition, we characterize the connections between the newly defined matrix classes and classical sequence spaces. We also examine the compactness of matrix operators within these associated sequence spaces and provide results related to specific operator ideals. Finally, we investigate the geometric properties of the associated sequence spaces.

In this work, we introduce two sequence spaces c 0 λ ( G m ) and c λ ( G m ) generated by the composition of m th order generalized difference matrix and lambda matrix and define an isomorphism between new sequence spaces and classical sequence spaces. Afterward, we investigate inclusion relations and obtain the Schauder basis of those spaces. Furthermore, we determine their α-, β- and γ-duals. Lastly, we characterize some matrix classes related to those spaces. MSC:40C05, 40H05, 46B45.

We study geometric and isometric properties of two classes of sequence spaces. First, we consider the Banach sequence space X n X_n , the completion of c 00 c_{00} relatively to the norm \[ ‖ x ‖ n = sup { ∑ i ∈ F | x i | , with , F of cardinality at most n } . \|x\|_n = \sup \left \{\sum _{i \in F}|x_i|, \text { with }, F \text { of cardinality at most } n\right \}. \] For this space, we show that the isometry group of X n X_n consists of standard isometries. Second, we consider a Tsirelson-type space, T [ θ , A n ] T[\theta , A_n] ( 0 > θ > 1 0>\theta >1 ) associated with the regular family A n A_n , comprised of all subsets of N \mathbb {N} with at most n n elements. For this space, we characterize the extreme points and show that for θ ≤ 1 n \theta \leq \frac {1}{n} , T [ θ , A n ] T[\theta , A_n] supports standard isometries. We also derive the form for the surjective isometries of T [ θ , A 3 ] T[\theta , A_3] .

Motivated by $\Phi$-summability, novel matrix operators are produced by the aid of fascinating arithmetical functions. One of them is the Riesz-Jordan totient matrix operator, introduced very recently. The primary purpose of this study is to use this matrix to obtain new spaces as the matrix domain of it in some classical sequence spaces. After presenting basic topological properties and identifying dual spaces, the characterization of certain matrix transformations is studied, defined on the resulting spaces. In the final section, the compactness of the operators on the domain of Jordan totient type matrices in the space of null sequences is researched.

BSTRACT. In this paper, we determine the pα-duals of certain newly introduced difference se- quence spaces, namely ℓ∞(u, △m, q), c0(u, △m, q), and c(u, △m, q). Furthermore, we investigate v v v their topological properties and establish the conditions under which these spaces are perfect. 1. Introduction and preliminaries Assume that ω denotes the space of all sequences. Throughout this chapter, we adopt the notation q = (qk) to denote a sequence of positive real numbers. Some generalized sequence spaces, such as ℓ∞(q), c(q), and c0(q), have been studied by various authors [9–11]. Ahmad and Mursaleen [1] introduced the concept of some new generalized difference sequence spaces, defined as follows: △ℓ∞(q) = {x ∈ ω : △x ∈ ℓ∞(q)}. △c(q) = {x ∈ ω : △x ∈ c(q)}. △c0(q) = {x ∈ ω : △x ∈ c0(q)}. Where △xk = xk − xk+1 . After that, Et and Bas¸arer [8] defined these difference sequence spaces in a more general form by introducing the operator △mon these spaces.

In the present article, we define certain a set of new classes of sequence spaces by using Ces\`{a}ro mean and difference operator $\Delta^r\;\;r\in\mathbb{N}_0=\{0,1,2,3,\dots\}$. Also, we study the topological structures of the defined classes and determine their $\alpha-,\beta-$ and $\gamma-$ duals. Matrix transformations of given classes with their basic sequence spaces are characterized.

Genetic oscillatory networks can be mathematically modeled with delay differential equations (DDEs). Interpreting genetic networks with DDEs gives a more intuitive understanding from a biological standpoint. However, it presents a problem mathematically, for DDEs are by construction infinitely-dimensional and thus cannot be analyzed using methods common for systems of ordinary differential equations (ODEs). In our study, we address this problem by developing a method for reducing infinitely-dimensional DDEs to two- and three-dimensional systems of ODEs. We find that the three-dimensional reductions provide qualitative improvements over the two-dimensional reductions. We find that the reducibility of a DDE corresponds to its robustness. For non-robust DDEs that exhibit high-dimensional dynamics, we calculate analytic dimension lines to predict the dependence of the DDEs’ correlation dimension on parameters. From these lines, we deduce that the correlation dimension of non-robust DDEs grows linearly with the delay. On the other hand, for robust DDEs, we find that the period of oscillation grows linearly with delay. We find that DDEs with exclusively negative feedback are robust, whereas DDEs with feedback that changes its sign are not robust. We find that non-saturable degradation damps oscillations and narrows the range of parameter values for which oscillations exist. Finally, we deduce that natural genetic oscillators with highly-regular periods likely have solely negative feedback.

1Kernels of bounded operators on the classical transfinite Banach sequence spaces

In this note we study the lineability and algebrability problems for certain classes of sequence spaces and series. In particular, we extend some algebrability results to the context of p-adic fields. We also prove that the subset of nonconvergent sequences of ℓ∞(ℝ) whose set of accumulation points is not a Cantor-type set is strongly 𝔠-algebrable.

We know from the classical sequence spaces theory that there is a useful relationship between continuous and -duals of a scalar-valued FK-space originated by the AK-property. Our main interest in this work is to expose relationships between the operator space and and the generalized -duals of some -valued AK-space where and are Banach spaces and . Further, by these results, we obtain the generalized -duals of some vector-valued Orlicz sequence spaces.

In this paper we study the domain of generalized Riesz difference matrix RqΔ(α) of fractional order α in the classical sequence spaces c0 and c and introduced the sequence spaces r0q(Δ(α)) and rcq(Δ(α)). We obtain the α−, β− and γ−duals of these spaces and using Hausdorff measure of noncompactness, we characterize certain classes of compact operators on the space r0q(Δ(α)).

We find the best constants in inequalities relating the standard norm, the dual norm, and the norm∥x∥(p,s):=inf⁡{∑k∥x(k)∥p,s}, where the infimum is taken over all finite representationsx=∑kx(k)in the classical Lorentz sequence spaces. A crucial point in this analysis is the concept of level sequence, which we introduce and discuss. As an application, we derive the best constant in the triangle inequality for such spaces.

Generalised Morrey (function) spaces enjoyed some interest recently and found applications to PDE. Here we turn our attention to their discrete counterparts. We define generalised Morrey sequence spaces mφ,p=mφ,p(Zd). They are natural generalisations of the classical Morrey sequence spaces mu,p, 0<p≤u<∞, which were studied earlier. We consider some basic features of the spaces as well as embedding properties such as continuity, compactness and strict singularity.

Finite-dimensional approximations are developed for retarded delay differential equations (DDEs). The DDE system is equivalently posed as an initial-boundary value problem consisting of hyperbolic partial differential equations (PDEs). By exploiting the equivalence of partial derivatives in space and time, we develop a new PDE representation for the DDEs that is devoid of boundary conditions. The resulting boundary condition-free PDEs are discretized using the Galerkin method with Legendre polynomials as the basis functions, whereupon we obtain a system of ordinary differential equations (ODEs) that is a finite-dimensional approximation of the original DDE system. We present several numerical examples comparing the solution obtained using the approximate ODEs to the direct numerical simulation of the original non-linear DDEs. Stability charts developed using our method are compared to existing results for linear DDEs. The presented results clearly demonstrate that the equivalent boundary condition-free PDE formulation accurately captures the dynamic behaviour of the original DDE system and facilitates the application of control theory developed for systems governed by ODEs.