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In this paper we study the asymptotic behavior of solutions of fractional differential equations of the form D^{\alpha }_Cu(t)=Au(t)+f(t), u(0)=x, 0<\alpha \le 1, (*) where D^{\alpha }_Cu(t) is the derivative of the function u in the Caputo's sense, A is a linear operator in a Banach space \mathbb {X} that may be unbounded and f satisfies the property that \lim _{t\to \infty } (f(t+1)-f(t))=0 which we will call asymptotic 1-periodicity. By using the spectral theory of functions on the half line we derive analogs of Katznelson-Tzafriri and Massera Theorems. Namely, we give sufficient conditions in terms of spectral properties of the operator A for all asymptotic mild solutions of Eq. (*) to be asymptotic 1-periodic, or there exists an asymptotic mild solution that is asymptotic 1-periodic. [ABSTRACT FROM AUTHOR]

We consider a Katznelson-Tzafriri type theorem for linear difference equations of the form x(n+1)=Tx(n)+y(n)\ (*), where T is a bounded operator in a Banach space \mathbb {X} and \{y(n)\}_{n=1}^\infty is a bounded sequence that is asymptotically constant, that is, \lim _{n\to \infty } [y(n+1)-y(n)]=0. A sequence \{u(n)\}_{n=1}^\infty is said to be an asymptotic solution to (*) if u(n+1)=Tu(n)+y(n)+\epsilon (n), where \lim _{n\to \infty }\epsilon (n)=0. We show that if 1 is either not in \sigma (T)\cap \{z\in \mathbb {C}:\ |z|=1\}, or is its isolated element, then if (*) has a bounded asymptotic solution, it has an asymptotic solution that is asymptotically constant. Furthermore, if \sigma (T)\cap \{z\in \mathbb {C}:\ |z|=1\}\subset \{ 1\}, then every asymptotic solution of (*) is asymptotic constant. An application to the evolution periodic equations is given. [ABSTRACT FROM AUTHOR]

We consider the asymptotic behaviour of solutions of the difference equations of the formin a Banach space X, wheren = 0,1,2,…;A, B(n) are linear bounded operators in X. Our method of study is based on the concept of spectrum of a unilateral sequence. The obtained results on asymptotic stability and almost periodicity are stated in terms of spectral properties of the equation and its solutions. To this end, a relation between theZ-transform and spectrum of a unilateral sequence is established. The main results extend previous results. [ABSTRACT FROM PUBLISHER]

This paper is a survey of the (new) concept of uniform spectrum of bounded functions. We review its properties and relationship with classical concepts of spectra of bounded functions, and some applications to differential equations in Banach spaces. [ABSTRACT FROM AUTHOR]

We extend the method of sums of commuting operators to the study of the existence and uniqueness of bounded solutions of Volterra equations of the form u(t) = Au(t) + &integ;0∞ dB(τ)u(t - τ) +f(t) with bounded f in the infinite dimensional case. The main results are necessary and sufficient conditions for the above equations to have a unique bounded solution with spectrum not intersecting the spectrum of the equation under consideration. Applications are made to illustrate the main results. [ABSTRACT FROM AUTHOR]

Presents the periodic solutions of differential equations and functional differential equations in Banach spaces. Assumption that Banach spaces generate a strongly continuous periodic evolutionary process; Spectral properties of either the operator-coefficients in the case of autonomous equations or the monodromy operators in the case of periodic equations; Spectral theory of functions.

We study the well-posedness of evolution equation of the form u ′ (t) = A u (t) + C u (t) , t ≥ 0 , where A generates a C 0 -semigroup (T A (t)) t ≥ 0 with ‖ T A (t) ‖ ≤ M e ω t and C is a (possibly unbounded) linear operator in a Banach space X. We prove that if C satisfies D (A) ⊂ D (C) , ‖ C R (μ , A) ‖ ≤ K / (μ − ω) for each μ > ω , then, the equation is well-posed, i.e., A + C generates a C 0 -semigroup (T A + C (t)) t ≥ 0 satisfying ‖ T A + C (t) ‖ ≤ M e (ω + M K) t. Our approach is to use the Hille-Yosida's theorem. We consider the space of such operators C in X with norm ‖ C ‖ A : = 1 M sup μ > ω ⁡ ‖ (μ − ω) C R (μ , A) ‖ < ∞ , in which we show that the exponential dichotomy of (T A (t)) t ≥ 0 persists under small perturbation C. The obtained results seem to be new. [ABSTRACT FROM AUTHOR]

We present an analog of Massera Theorem for asymptotic periodic solutions of linear equations x ′ (t) = A (t) x (t) + f (t) , t ≥ 0 (⁎) , where the family of linear operators A (t) generates a 1-periodic process (U (t , s)) t ≥ s ≥ 0 in a Banach space X and f is asymptotic 1-periodic in the sense that lim t → ∞ ⁡ (f (t + 1) − f (t)) = 0. The main result says that if 1 is isolated in σ (U (1 , 0)) on the unit circle Γ, then (⁎) has an asymptotic 1-periodic mild solution if and only if it has an asymptotic mild solution that is bounded and uniformly continuous with precompact range. If 1 ∉ σ (U (1 , 0)) ∩ Γ , such an asymptotic 1-periodic mild solution always exists and unique within a function g (t) with lim t → ∞ ⁡ g (t) = 0. Our study relies on a spectral theory of functions on the half line and the evolution semigroups associated with linear equations. The obtained results seem to be new, even in the finite dimensional case. [ABSTRACT FROM AUTHOR]

For A(t) and f(t,x,y) T-periodic in t, we consider the following evolution equation with infinite delay in a general Banach space X: (0.1)u′(t)+A(t)u(t)=ft,u(t),ut, t>0, u(s)=φ(s), s&les;0, where the resolvent of the unbounded operator A(t) is compact, and ut(s)=u(t+s), s&les;0. By utilizing a recent asymptotic fixed point theorem of Hale and Lunel (1993) for condensing operators to a phase space Cg, we prove that if solutions of Eq. (0.1) are ultimate bounded, then Eq. (0.1) has a T-periodic solution. This extends and improves the study of deriving periodic solutions from boundedness and ultimate boundedness of solutions to infinite delay evolution equations in general Banach spaces; it also improves a corresponding result in J. Math. Anal. Appl. 247 (2000) 627–644 where the local strict boundedness is used. [Copyright &y& Elsevier]

For linear functional differential equations with infinite delay in a Banach space, a variation-of-constants formula is established in the phase space. As an application one applies it to study the admissibility of some spaces of functions whose spectra are contained in a closed subset of the real line. [Copyright &y& Elsevier]