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General solution to a higher-order linear difference equation and existence of bounded solutions

We present a closed-form formula for the general solution to the difference equation x n + k − q n x n = f n , n ∈ N 0 , $$x_{n+k}-q_{n}x_{n}=f_{n},\quad n\in \mathbb {N}_{0}, $$ where k ∈ N $k\in \mathbb {N}$ , ( q n ) n ∈ N 0 $(q_{n})_{n\in \mathbb {N}_{0}}$ , ( f n ) n ∈ N 0 ⊂ C $(f_{n})_{n\in \mathbb {N}_{0}}\subset \mathbb {C}$ , in the case q n = q $q_{n}=q$ , n ∈ N 0 $n\in...

General solution to a higher-order linear difference equation and existence of bounded solutions

We present a closed-form formula for the general solution to the difference equation x n + k − q n x n = f n , n ∈ N 0 , $$x_{n+k}-q_{n}x_{n}=f_{n},\quad n\in \mathbb {N}_{0}, $$ where k ∈ N $k\in \mathbb {N}$ , ( q n ) n ∈ N 0 $(q_{n})_{n\in \mathbb {N}_{0}}$ , ( f n ) n ∈ N 0 ⊂ C $(f_{n})_{n\in \mathbb {N}_{0}}\subset \mathbb {C}$ , in the case q n = q $q_{n}=q$ , n ∈ N 0 $n\in...

General solution to a higher-order linear difference equation and existence of bounded solutions

We present a closed-form formula for the general solution to the difference equation x n + k − q n x n = f n , n ∈ N 0 , $$x_{n+k}-q_{n}x_{n}=f_{n},\quad n\in \mathbb {N}_{0}, $$ where k ∈ N $k\in \mathbb {N}$ , ( q n ) n ∈ N 0 $(q_{n})_{n\in \mathbb {N}_{0}}$ , ( f n ) n ∈ N 0 ⊂ C $(f_{n})_{n\in \mathbb {N}_{0}}\subset \mathbb {C}$ , in the case q n = q $q_{n}=q$ , n ∈ N 0 $n\in...

General solution to a higher-order linear difference equation and existence of bounded solutions

We present a closed-form formula for the general solution to the difference equation x n + k − q n x n = f n , n ∈ N 0 , $$x_{n+k}-q_{n}x_{n}=f_{n},\quad n\in \mathbb {N}_{0}, $$ where k ∈ N $k\in \mathbb {N}$ , ( q n ) n ∈ N 0 $(q_{n})_{n\in \mathbb {N}_{0}}$ , ( f n ) n ∈ N 0 ⊂ C $(f_{n})_{n\in \mathbb {N}_{0}}\subset \mathbb {C}$ , in the case q n = q $q_{n}=q$ , n ∈ N 0 $n\in...

General solution to a higher-order linear difference equation and existence of bounded solutions

We present a closed-form formula for the general solution to the difference equation x n + k − q n x n = f n , n ∈ N 0 , $$x_{n+k}-q_{n}x_{n}=f_{n},\quad n\in \mathbb {N}_{0}, $$ where k ∈ N $k\in \mathbb {N}$ , ( q n ) n ∈ N 0 $(q_{n})_{n\in \mathbb {N}_{0}}$ , ( f n ) n ∈ N 0 ⊂ C $(f_{n})_{n\in \mathbb {N}_{0}}\subset \mathbb {C}$ , in the case q n = q $q_{n}=q$ , n ∈ N 0 $n\in...

Solvability of a class of product-type systems of difference equations

A solvable class of product-type systems of difference equations with two dependent variables on the complex domain is presented. The main results complement some recent ones in the literature, while their proofs contain some refined methodological details. We provide closed form formulas for general solutions to the system or give procedures for how to get them. MSC: 39A20, 39A45.

Solvability of boundary value problems for a class of partial difference equations on the combinatorial domain

By modifying our recent method of half-lines we show how the following boundary value problem for partial difference equations can be solved in closed form: d n , k = d n − 1 , k − 1 + f ( k ) d n − 1 , k , 1 ≤ k < n , d n , 0 = u n , d n , n = v n , n ∈ N , where ( u n ) n ∈ N and ( v n ) n ∈ N are given sequences of complex numbers, and f is a complex-valued function on N . MSC...

On periodic solutions of a class of k-dimensional systems of max-type difference equations

Some sufficient conditions are given such that the following system of difference equations: x ( 1 ) ( n + 1 ) = max 1 ≤ j ≤ l 1 { f 1 j ( n , x ( 1 ) ( n ) , … , x ( k ) ( n ) ) } , x ( 2 ) ( n + 1 ) = max 1 ≤ j ≤ l 2 { f 2 j ( n , x ( 1 ) ( n ) , … , x ( k ) ( n ) ) } , ⋮ x ( k ) ( n + 1 ) = max 1 ≤ j ≤ l k { f k j ( n , x ( 1 ) ( n ) , … , x ( k ) ( n ) ) } , n ∈ N 0 , where k...

Two-dimensional product-type system of difference equations solvable in closed form

A solvable two-dimensional product-type system of difference equations of interest is presented. Closed form formulas for its general solution are given. MSC: 39A10, 39A20.

Boundedness and compactness of a new product-type operator from a general space to Bloch-type spaces

We characterize the boundedness and compactness of a product-type operator, which, among others, includes all the products of the single composition, multiplication, and differentiation operators, from a general space to Bloch-type spaces. We also give some upper and lower bounds for the norm of the operator. MSC: 47B38, 46E15.

Generalized weighted composition operators from α-Bloch spaces into weighted-type spaces

Some criteria for the boundedness, as well as for the compactness, of the generalized weighted composition operator D φ , u n from α-Bloch spaces into weighted-type spaces are given. Estimates for the norm and the essential norm of the operator are also given. Our results extend and complement some results in the literature. MSC: 47B33, 30H30.

Weighted composition operators from weighted Bergman spaces with Békollé weights to Bloch-type spaces

Let σ be a Békollé weight function and ν be a weight function. In this paper, we characterize the boundedness and compactness of weighted composition operators acting from Bergman-type spaces A p ( σ ) to Bloch-type spaces B ν and B ν , 0 , considerably extending some results in the literature. MSC: 47B33, 46E10, 30D55.

Boundedness character of a fourth-order system of difference equations

The boundedness character of positive solutions of the following system of difference equations: x n + 1 = A + y n p x n − 3 r , y n + 1 = A + x n p y n − 3 r , n ∈ N 0 , when min { A , r } > 0 and p ≥ 0 , is studied. MSC: 39A10, 39A20.

On a product-type system of difference equations of second order solvable in closed form

It is shown that the following system of difference equations z n + 1 = z n a w n − 1 b , w n + 1 = w n c z n − 1 d , n ∈ N 0 , where a , b , c , d ∈ Z , z − 1 , z 0 , w − 1 , w 0 ∈ C , is solvable in closed form. MSC: 39A10, 39A20.

On a close to symmetric system of difference equations of second order

Closed form formulas of the solutions to the following system of difference equations: x n = y n − 1 y n − 2 x n − 1 ( a n + b n y n − 1 y n − 2 ) , y n = x n − 1 x n − 2 y n − 1 ( α n + β n x n − 1 x n − 2 ) , n ∈ N 0 , where a n , b n , α n , β n , n ∈ N 0 , and initial values x − i , y − i , i ∈ { 1 , 2 } are real numbers, are found. The domain of undefinable solutions to the...

Some new characterizations of the Bloch space

We obtain some new characterizations for the Bloch space on the open unit disk in the complex plane ℂ and the open unit ball of Cn.MSC: 32A18.

On the Periodicity of Some Classes of Systems of Nonlinear Difference Equations

: Josef Diblik Copyright © 2014 Stevo Stević et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and

On a Class of Solvable Difference Equations

, P.O. Box 42641, Jeddah 21551, Saudi Arabia Received 25 September 2013; Accepted 22 October 2013 Academic Editor: Josef Diblík Copyright © 2013 Stevo Stević et al. This is an open access article

On a Class of Solvable Difference Equations

, P.O. Box 42641, Jeddah 21551, Saudi Arabia Received 25 September 2013; Accepted 22 October 2013 Academic Editor: Josef Diblík Copyright © 2013 Stevo Stević et al. This is an open access article