scholarly journals Periodic Problems of Difference Equations and Ergodic Theory

2011 ◽  
Vol 2011 ◽  
pp. 1-12
Author(s):  
B. A. Biletskyi ◽  
A. A. Boichuk ◽  
A. A. Pokutnyi
Keyword(s):  
Difference Equations ◽  
Periodic Boundary Condition ◽  
Generalized Inverse ◽  
Bounded Operator ◽  
Inverse Operator ◽  
Sufficient Conditions ◽  
Linear Bounded Operator ◽  
Periodic Problems ◽  
The Family ◽  
Necessary And Sufficient

The necessary and sufficient conditions for solvability of the family of difference equations with periodic boundary condition were obtained using the notion of relative spectrum of linear bounded operator in the Banach space and the ergodic theorem. It is shown that when the condition of existence is satisfied, then such periodic solutions are built using the formula for the generalized inverse operator to the linear limited one.

Using binary operations to construct a transitive set of block transformations

2020 ◽  
Vol 30 (6) ◽  
pp. 375-389
Author(s):  
Igor V. Cherednik
Keyword(s):  
Binary Operation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Binary Operations ◽  
The Family ◽  
Transitive Set ◽  
Necessary And Sufficient

AbstractWe study the set of transformations {ΣF : F∈ 𝓑∗(Ω)} implemented by a network Σ with a single binary operation F, where 𝓑∗(Ω) is the set of all binary operations on Ω that are invertible as function of the second variable. We state a criterion of bijectivity of all transformations from the family {ΣF : F∈ 𝓑∗(Ω)} in terms of the structure of the network Σ, identify necessary and sufficient conditions of transitivity of the set of transformations {ΣF : F∈ 𝓑∗(Ω)}, and propose an efficient way of verifying these conditions. We also describe an algorithm for construction of networks Σ with transitive sets of transformations {ΣF : F∈ 𝓑∗(Ω)}.

On the Necessary and Sufficient Conditions for the Stability of Linear Generalized Ordinary Differential, Linear Impulsive and Linear Difference Systems

2009 ◽  
Vol 16 (4) ◽  
pp. 597-616
Author(s):  
Shota Akhalaia ◽  
Malkhaz Ashordia ◽  
Nestan Kekelia
Keyword(s):  
Linear System ◽  
Difference Equations ◽  
Closed Interval ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Difference Systems ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Generalized Ordinary Differential Equations ◽  
Lyapunov Sense

Abstract Necessary and sufficient conditions are established for the stability in the Lyapunov sense of solutions of a linear system of generalized ordinary differential equations 𝑑𝑥(𝑡) = 𝑑𝐴(𝑡) · 𝑥(𝑡) + 𝑑𝑓(𝑡), where and are, respectively, matrix- and vector-functions with bounded total variation components on every closed interval from . The results are realized for the linear systems of impulsive, ordinary differential and difference equations.

scholarly journals Asymptotic iteration method for solving Hahn difference equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Lucas MacQuarrie ◽  
Nasser Saad ◽  
Md. Shafiqul Islam
Keyword(s):  
Difference Equations ◽  
Iteration Method ◽  
Difference Operator ◽  
Sufficient Conditions ◽  
Asymptotic Iteration Method ◽  
Order Linear ◽  
Polynomial Solutions ◽  
Hahn Difference Equations ◽  
Iteration Methods ◽  
Necessary And Sufficient

AbstractHahn’s difference operator $D_{q;w}f(x) =({f(qx+w)-f(x)})/({(q-1)x+w})$ D q ; w f ( x ) = ( f ( q x + w ) − f ( x ) ) / ( ( q − 1 ) x + w ) , $q\in (0,1)$ q ∈ ( 0 , 1 ) , $w>0$ w > 0 , $x\neq w/(1-q)$ x ≠ w / ( 1 − q ) is used to unify the recently established difference and q-asymptotic iteration methods (DAIM, qAIM). The technique is applied to solve the second-order linear Hahn difference equations. The necessary and sufficient conditions for polynomial solutions are derived and examined for the $(q;w)$ ( q ; w ) -hypergeometric equation.

scholarly journals Existence of a Period-Two Solution in Linearizable Difference Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
E. J. Janowski ◽  
M. R. S. Kulenović
Keyword(s):  
Difference Equation ◽  
Linear Equation ◽  
Difference Equations ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Minimal Period ◽  
Real Numbers ◽  
Existence And Nonexistence ◽  
The Difference ◽  
Necessary And Sufficient

Consider the difference equationxn+1=f(xn,…,xn−k),n=0,1,…,wherek∈{1,2,…}and the initial conditions are real numbers. We investigate the existence and nonexistence of the minimal period-two solution of this equation when it can be rewritten as the nonautonomous linear equationxn+l=∑i=1−lkgixn−i,n=0,1,…,wherel,k∈{1,2,…}and the functionsgi:ℝk+l→ℝ. We give some necessary and sufficient conditions for the equation to have a minimal period-two solution whenl=1.

Generalized WG inverse

2020 ◽  
pp. 2150072
Author(s):  
Sanzhang Xu ◽  
Hongxing Wang ◽  
Jianlong Chen ◽  
Xiaofeng Chen ◽  
Tiwei Zhao
Keyword(s):  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Algebraic Form ◽  
Necessary And Sufficient

In this paper, we introduce a generalized inverse of a matrix, namely, the generalized WG inverse, which a generalization of the WG inverse. Several necessary and sufficient conditions such that a matrix to be generalized WG invertible are obtained. Moreover, the formulae of generalized WG inverse of a matrix are given. Finally, we give a algebraic form of the generalized WG inverse.

scholarly journals Gaps in Discrete Random Samples

2009 ◽  
Vol 46 (04) ◽  
pp. 1038-1051 ◽  
Author(s):  
Rudolf Grübel ◽  
Paweł Hitczenko
Keyword(s):  
Analysis Of Algorithms ◽  
Sufficient Conditions ◽  
Borderline Case ◽  
Convergence In Probability ◽  
Enumerative Combinatorics ◽  
Random Samples ◽  
The Family ◽  
Heavy Tailed ◽  
Convergence Almost Surely ◽  
Necessary And Sufficient

Let (X i ) i∈ℕ be a sequence of independent and identically distributed random variables with values in the set ℕ0 of nonnegative integers. Motivated by applications in enumerative combinatorics and analysis of algorithms we investigate the number of gaps and the length of the longest gap in the set {X 1,…,X n } of the first n values. We obtain necessary and sufficient conditions in terms of the tail sequence (q k ) k∈ℕ0 , q k =P(X 1≥ k), for the gaps to vanish asymptotically as n→∞: these are ∑ k=0 ∞ q k+1/q k <∞ and limk→∞ q k+1/q k =0 for convergence almost surely and convergence in probability, respectively. We further show that the length of the longest gap tends to ∞ in probability if q k+1/q k → 1. For the family of geometric distributions, which can be regarded as the borderline case between the light-tailed and the heavy-tailed situations and which is also of particular interest in applications, we study the distribution of the length of the longest gap, using a construction based on the Sukhatme–Rényi representation of exponential order statistics to resolve the asymptotic distributional periodicities.

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