Investigating Distributional Chaos for Operators on Fréchet Spaces

In this paper, various notions of chaos for continuous linear operators on Fréchet spaces are investigated. It is shown that an operator is Li–Yorke chaotic if and only if it is mean Li–Yorke chaotic in a sequence whose upper density equals one; that an operator is mean Li–Yorke chaotic if and only if it admits a mean Li–Yorke pair, if and only if it is distributionally chaotic of type 2, if and only if it has an absolutely mean irregular vector. As a consequence, mean Li–Yorke chaos is not conjugacy invariant for continuous self-maps acting on complete metric spaces. Moreover, the existence of invariant scrambled sets (with respect to certain Furstenberg families) of a class of weighted shift operators is proved.

Approximation of functions by Complex conformable derivative bases in Frechet spaces

In the present paper the representation, in different domains, of analytic functions by complex conformable fractional derivative bases (CCFDB) and complex conformable fractional integral bases (CCFIB) in Frechet space are investigated . Theorems are proved to show that such representation is possible in closed disks, open disks, open regions surrounding closed disks, at the origin and for all entire functions. Also, some results concerning the growth order and type of CCFDB and CCFIB are determined. Moreover the T-property of CCFDB and CCFIB are dis- cussed. The obtained results recover some known results when alpha = 1. Finally, some applications to the CCFDB and CCFIB of Bernoulli, Euler, Bessel and Chebyshev polynomials have been studied.

Localization of Fréchet Frames and Expansion of Generalized Functions

Matrix-type operators with the off-diagonal decay of polynomial or sub-exponential types are revisited with weaker assumptions concerning row or column estimates, still giving the continuity results for the frame type operators. Such results are extended from Banach to Fréchet spaces. Moreover, the localization of Fréchet frames is used for the frame expansions of tempered distributions and a class of Beurling ultradistributions.

Caputo-Fabrizio fractional differential equations in Frechet spaces

This paper deals with some existence and uniqueness of solutions for a class of functional Caputo-Fabrizio fractional differential equations. Some applications are made of a generalization of the classical Darbo fixed point theorem for Frechet spaces associate with the concept of measure of noncompactness. The last section illustrates our results with some examples.

On solvability of infinite system of integral equations of Volterra together with Hammerstein type in the Fréchet spaces

In this paper, we prove some fixed point theorems associated with Tychonoff fixed point theorem and measure of noncompactness in the Fr?chet spaces. Moreover, as an application of our results, we analyze the existence of solutions for infinite system of integral equations of Volterra together with Hammerstein type. Finally, we present an example to illustrate the effectiveness of our results.