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We first give a sufficient condition for positivity of the solution semigroup of linear functional difference equations. Then, we obtain a Perron–Frobenius theorem for positive linear functional difference equations. Next, we offer a new explicit criterion for exponential stability of a wide class of positive equations. Finally, we study stability radii of positive linear functional difference equations. It is proved that complex, real and positive stability radius of positive equations under structured perturbations (or affine perturbations) coincide and can be computed by explicit formulae. [ABSTRACT FROM AUTHOR]

We study stability radii of higher order linear difference systems under multi-perturbations. A formula for complex stability radius of higher order linear difference systems under multi-perturbations is given. Then, for the class of positive systems, we prove that the complex stability radius and real stability radius of the system under multi-perturbations coincide and they are computed via a simple formula. These are extensions of corresponding results of Hinrichsen and Son, Hinrichsen et al., Ngoc and Son, and Pappas and Hinrichsen. An example is given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

We study robustness of 𝒟-stability of linear difference equations under multiperturbation and affine perturbation of coefficient matrices via the concept of 𝒟-stability radius. Some explicit formulae are derived for these 𝒟-stability radii. The obtained results include the corresponding ones established earlier in Hinrichsen and Son and Ngoc and Son as particular cases. [ABSTRACT FROM AUTHOR]

We give a new characterization of exponential dichotomy of linear difference equations in terms of -admissibility. The obtained result is a discrete version of the recent new result of Preda et al. [( L p , L q )-admissibility and exponential dichotomy of evolution processes on the half-plane, Integral Equations Operator Theory , 49 (2004), 405–418] on exponential dichotomy of evolution processes on the half-plane. Then we derive a new characterization of exponential stability of linear difference equations. This extends the recent new result of Pituk M. [A criterion for the exponential stability of linear difference equations, Applied Mathematics Letters , 17 (2004), 779–783] and of Aulbach and Nguyen Van Minh. [The concept of spectral dichotomy for linear difference equations II, Journal of Difference Equations and Applications , 2 (1996), 251–262]. [ABSTRACT FROM AUTHOR]

This article is concerned with the Massera criterion for periodic mild solutions of the equations of the form where A generates an analytic semigroup on a Banach space X , B is an L ( X )-valued function of bounded variation, and f is a continuous 1-periodic function. The obtained results extend recent ones [13 , 17] on the subject. [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]

We give a short summary of recent development of spectral analysis of autonomous or periodic linear functional differential equations with infinite delay on the abstract phase space with uniformly fading memory property. The considered functional differential equation is transformed into an operator equation on function spaces. The admissibility theory of function spaces is reduced to the problem of the solvability of this operator equation. The spectra of bounded solutions are studied from the decomposition technique of solutions. The method of commuting operators is effectively used by combining the general result about the distribution of the spectrum of the generator of solution semigroup of autonomous equation. [ABSTRACT FROM AUTHOR]

We deal with linear inhomogeneous differential equations of the form dx/dt = Ax(t) + f(t) in a Banach space X, where A is the generator of a C[SUB0]semigroup on X and f is a periodic function. In this paper, we present a Massera type theorem, a method to show the existence of bounded solutions and their structure of them. As results, we obtain criteria for the existence of quasi-periodic, periodic, asymptotically periodic solutions. [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.