Almost Periodic Solutions of Differential Equations in Banach Spaces
CRC Press, Oct 25, 2001 - Mathematics - 264 pages
This monograph presents recent developments in spectral conditions for the existence of periodic and almost periodic solutions of inhomogenous equations in Banach Spaces. Many of the results represent significant advances in this area. In particular, the authors systematically present a new approach based on the so-called evolution semigroups with an original decomposition technique. The book also extends classical techniques, such as fixed points and stability methods, to abstract functional differential equations with applications to partial functional differential equations. Almost Periodic Solutions of Differential Equations in Banach Spaces will appeal to anyone working in mathematical analysis.
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admissibility AP(X assume assumption asymptotically almost periodic Banach space BC-TS bounded linear operator bounded set bounded solution bounded uniformly continuous boundedness BUC(R BUC(R,X closed linear operator closed subspace commuting operators compact Co-semigroup compactly condition H1 consider continuous function converges Corollary countable deﬁned deﬁnition denote equicontinuous evolution equations evolution semigroup exists a unique exponential dichotomy fading memory space ﬁnite ﬁrst ﬁxed point function space functional differential equations Ha(w Hence homogeneous equation implies integral invariant subspace Lemma Lipschitz continuous monodromy operator Moreover nonlinear null solution periodic function periodic mild solution periodic solutions positive constants Proof properties Proposition prove quasi-process R+ X R+ real line relatively compact satisﬁes satisfy condition solution u(t sp(f spectrum Stability strongly continuous strongly continuous evolutionary strongly continuous semigroup sufﬁciently Suppose trigonometric polynomials uniform fading memory unique almost periodic unique solution uniquely solvable