Abstract:
C. Bu¸se and A. Zada in 2010 [10], proved a result for discrete systems as: The system Tα+1 = ATα is UES if and only if for each i ∈RN, b ∈CAn, A ∈ Mn(CA), α ∈ZN and n ∈ZP the solution of the Cauchy problem Tα+1 = ATα + eiiαb with T0 = 0 is bounded.
The above result is generalized in chapter 3 of this thesis by replacing A (square size matrix) on DS(1) and the forcing term eiiαb by general periodic sequences so the new statement become: The system Tα+1 = DS(1)Tα is UES if and only if for each ∂-periodic bounded sequence g(α) with g(0) = 0 then the corresponding solution of the Cauchy problem Tα+1 = DS(1)Tα+1 + g(α + 1), T0 = 0 (DS(1),0) is bounded. This generalization is not smooth, here we have also supposed that the operator eiiαP∂−1 ν=1 eiiνν(∂ −ν)x is not equal to zero for every non zero x in BS. This result is published in our research paper [38] with the link:
http://link.springer.com/article/10.1007/s12346-014-0124-x.
In the same chapter the above result is extended toward non-autonomous problems as: Let EF = {EF(α,ξ) : ξ,α ∈ ZN} be a discrete ∂-periodic evolution family on BS, i is any real number and g(α) ∈ W. If the sequence Tα = α P ξ=1 eiiξEF(α,ξ)g(ξ) is bounded then EF is UES. This result is published in our research paper [39] with the link: http://www.hindawi.com/journals/aaa/2014/784289/.
The results of the chapter 3 are also extended from periodic space to the space AAP0(ZN,BS), where AAP0(ZN,BS) is explained in chapter no 4. We have proved that the discrete semigroup DS = {DS(α) : α ∈ ZN} is UES if and only if for each g(α) ∈AAP0(ZN,BS) the solution of the Cauchy problem Tα+1 = DS(1)Tα + g(α + 1), T0 = 0 belongs to AAP0(ZN,BS). Our proof uses the approach of discrete evolution semigroups. This result is published in our research paper [42] with the link:
http://www.ems-ph.org/journals/showabstract.phpissn=0232-2064vol=34iss=4rank=6.
Further we have produced some more results related to the uniform exponen
tial stability of discrete semigroups and discrete evolution families toward self adjoint
discrete semigroups and ∂-periodic positive discrete evolution families. We have stud
ied the uniform exponential stability of self adjoint discrete semigroups over Hilbert
spaces. Which is extended toward self adjoint ∂-periodic discrete evolution families.
In the last the previous result is extended toward positive discrete ∂-periodic evo
lution families over Hilbert spaces. All these results are published in our research
paper [40] with the link: http://www.uab.ro/auajournal/upload/641146AUALatextemplate−23.pdf.
