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    Massera-Type Results For CN -Almost Automorphic Solutions Of Some Differential Equations

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    Nelson_morgan_0755M_10052.pdf (220.5Kb)
    Permanent Link
    http://hdl.handle.net/11603/10406
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    • MSU Student Collection
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    Author/Creator
    Nelson, Valerie N.
    Date
    2009
    Type of Work
    Text
    theses
    Department
    Mathematics
    Program
    Master of Arts
    Rights
    This item is made available by Morgan State University for personal, educational, and research purposes in accordance with Title 17 of the U.S. Copyright Law. Other uses may require permission from the copyright owner.
    Subjects
    Mathematics
    Differential equations
    Abstract
    We are concerned with the existence of C(n)-almost automorphic solutions of the equation (1) x'(t) = A(t)x(t) + f(t) where A(t) is a τ-periodic operator and f is C(n)-almost automorphic. In 1950, Massera proved that if the equation (2) x'(t)=Ax(t) + f(t) where f(t) is periodic and A is a constant n x n matrix, has a bounded solution on R+, then it has a periodic solution on all of R. This result has since been generalized for many different cases, including when f(t) is almost periodic in an infinite dimensional space. In this thesis, we apply Floquet's Theory and some recent results on C(n)-almost automorphic functions to prove some Massera-type results for the nonautonomous differential equation (3) x'(t)=A(t)x(t)+f(t) in Ck. We show that if A(t) is periodic, f(t) is C(n)-almost automorphic, and Equation (3) has a bounded solution on R+, then it has a C(n+1)-almost automorphic solution on the whole real line. Moreover, every bounded solution of Equation (3) is C(n+1)-almost automorphic. We published these results in CUBO A Mathematical Journal in 2008. Secondly, we study the convolution operator of f with h, denoted (f*h) defined by (f*h)(t) := integral of f(σ)h(t-σ)dσ for all t in R where h is in L1(R) and explore the existence of C(n)-almost automorphic solutions of the convolution equation (4) u(t)=f(t) + integral of a(t-s)u(s)ds, for all t in R, where f: R /&rarr ; R is C(n)-almost automorphic and a is Lebesgue measurable. These generalize some recent results.


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    Growing the Future, Leading the World!


    If you wish to submit a copyright complaint or withdrawal request, please email mdsoar-help@umd.edu.