Approximation of a bounded solution of one difference equation with unbounded operator coefficient by solutions of the corresponding boundary-value problems

2000 ◽  
Vol 52 (4) ◽  
pp. 628-632
Author(s):  
M. F. Gorodnii ◽  
V. M. Romanenko
Keyword(s):  
Difference Equation ◽  
Boundary Value Problems ◽  
Unbounded Operator ◽  
Bounded Solution ◽  
Boundary Value ◽  
Operator Coefficient ◽  
Unbounded Operator Coefficient

scholarly journals Bounded solutions of a difference equation with finite number of jumps of operator coefficient

2020 ◽  
Vol 12 (1) ◽  
pp. 165-172
Author(s):  
A. Chaikovs'kyi ◽  
O. Lagoda
Keyword(s):  
Difference Equation ◽  
Finite Number ◽  
Existence And Uniqueness ◽  
Bounded Solution ◽  
Sufficient Conditions ◽  
Operator Coefficient ◽  
Bounded Solutions ◽  
Piecewise Constant ◽  
Variable Operator ◽  
Necessary And Sufficient

We study the problem of existence of a unique bounded solution of a difference equation with variable operator coefficient in a Banach space. There is well known theory of such equations with constant coefficient. In that case the problem is solved in terms of spectrum of the operator coefficient. For the case of variable operator coefficient correspondent conditions are known too. But it is too hard to check the conditions for particular equations. So, it is very important to give an answer for the problem for those particular cases of variable coefficient, when correspondent conditions are easy to check. One of such cases is the case of piecewise constant operator coefficient. There are well known sufficient conditions of existence and uniqueness of bounded solution for the case of one jump. In this work, we generalize these results for the case of finite number of jumps of operator coefficient. Moreover, under additional assumption we obtained necessary and sufficient conditions of existence and uniqueness of bounded solution.

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