Spectral Criteria of Abstract Sequences; Application to Difference Equations |
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Authors: | Alaa E Hamza |
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Institution: | Department of Mathematics, Faculty of Sciences , Cairo University , Giza, Egypt |
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Abstract: | We suppose that M is a closed subspace of l ∞(J, X), the space of all bounded sequences {x(n)} n?J ? X, where J ? {Z+,Z} and X is a complex Banach space. We define the M-spectrum σM (u) of a sequence u ? l ∞(J,X). Certain conditions will be supposed on both M and σM (u) to insure the existence of u ? M. We prove that if u is ergodic, such that σM (u,) is at most countable and, for every λ ? σM (u), the sequence e?iλnu(n) is ergodic, then u ? M. We apply this result to the operator difference equationu(n + 1) = Au(n) + ψ(n), n ? J,and to the infinite order difference equation Σ r k=1 ak (u(n + k) ? u(n)) + Σ s ? Z?(n ? s)u(s) = h(n), n?J, where ψ?l ∞(Z,X) such that ψ| J ? M, A is the generator of a C 0-semigroup of linear bounded operators {T(t)} t>0 on X, h ? M, ? ? l 1(Z) and ak ?C. Certain conditions will be imposed to guarantee the existence of solutions in the class M. |
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Keywords: | 35Q58 37K40 |
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