This paper is concerned with dynamics in general ultrametric spaces. We prove the Fixed Point Theorem, the Stability Theorem, the Common Point Theorem, the Local Fixed Point Theorem, the Attractor Theorem and we derive conditions for a mapping to be surjective. We also discuss tightly continuous mappings and prove a theorem about the transfer of the property of principal completeness by a mappi...

In this paper, we study periodic linear systems on periodic time scales which include not only discrete and continuous dynamical systems but also systems with a mixture of discrete and continuous parts (e.g. hybrid dynamical systems). We develop a comprehensive Floquet theory including Lyapunov transformations and their various stability preserving properties, a unified Floquet theorem which es...

Abstract In this paper, we solve the Cauchy problem for a hyperbolic system of first-order PDEs defined on certain Banach space X . The has special semilinear structure because, one hand, evolution law can be expressed as sum linear unbounded operator and nonlinear Lipschitz function but, other perturbation takes values not in but larger Y which is related to order deal with situation use theor...

‎inspired by the work of suzuki in‎ ‎[t. suzuki‎, ‎a generalized banach contraction principle that characterizes metric completeness‎, proc‎. ‎amer‎. ‎math‎. ‎soc. ‎136 (2008)‎, ‎1861--1869]‎, ‎we prove a fixed point theorem for contractive mappings‎ ‎that generalizes a theorem of geraghty in [m.a‎. ‎geraghty‎, ‎on contractive mappings‎, ‎proc‎. ‎amer‎. ‎math‎. ‎soc., ‎40 (1973)‎, ‎604--608]‎an...

The Hahn-Banach theorem is one of the most fundamental theorems in the functional analysis theory. This theorem is well known in the case where the range space is the real number system as follows. Let p be a sublinear mapping from a vector space X into the real number system R, Y a subspace of X, and q a linear mapping from Y into R such that q ≤ p on Y. Then there exists a linear mapping g fr...

In this paper we are interested in spectral decomposition of an unbounded operator with discrete spectrum. We show that if A generates a polynomially bounded n-times integrated group whose spectrum set σ(A) = {iλk; k ∈ Z} is discrete and satisfies ∑ 1 |λk|δ k < ∞ (n and l nonnegative integers), then there exists projectors (Pk)k∈Z∗ such that ∑ Pkx = x (x ∈ D(A)), where δk = min ( |λk+1−λk| 2 , ...

Now we recall the statement of the central limit theorem (CLT) and give a proof in the case of IID (independent identically distributed) random variables. The weak law of large numbers says that if Xn is a sequence of IID random variables with E[Xn] = 0, then writing Sn = ∑n−1 k=0 Xk, the time averages 1 n Sn converge to 0 in probability, or equivalently (since the limit is a constant), in dist...

The continuity of the mapping which associates a J-spectral factor to a spectral density is analyzed. For the class of essentially bounded functions on the imaginary axis that are bounded away from zero it is well known that this mapping, even for the scalar case, is not continuous. In this paper three results concerning the continuity of the mapping which associates a J-spectral factor to a sp...