We give necessary and sufficient conditions for uniform exponential dichotomy of evolution families in terms of the admissibility of the pair(Lp(ℝ,X),Lq(ℝ,X)). We show that the admissibility of the pair(Lp(ℝ,X),Lq(ℝ,X))is equivalent to the uniform exponential dichotomy of an evolution family if and only ifp≥q. As applications we obtain characterizations for uniform exponential dichotomy of semigroups.
This paper considers two trichotomy concepts in the context of abstract evolution operators. The first one extends the notion of exponential trichotomy in the sense of Elaydi-Hajek for differential equations to abstract evolution operators, and it is a natural extension of the generalized exponential dichotomy considered in the paper by Jiang (2006). The second concept is dual in a certain sense to the first one. We prove that these types of exponential trichotomy imply the existence of generalized exponential dichotomy on both half-lines. We emphasize that we do not assume the invertibility of the evolution operators on the whole spaceX(unlike the case of evolution operators generated by differential equations).
AbstractThe purpose of the present paper is to investigate the problem of nonuniform exponential stability of evolution families on the real line using the input-output technique known in the literature as the Perron method for the study of exponential stability. In this manuscript we describe an evolution family on the real line and we present sufficient conditions for the nonuniform exponential stability of an evolution family on the real line that does not have exponential growth.
The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.
Exponential dichotomy for evolution families on the real line