Chet Spaces Research Articles

Study on the controllability of delayed evolution inclusions involving fractional derivatives

<abstract><p>This paper dealt with the infinite controllability of delayed evolution inclusions with $ \alpha $-order fractional derivatives in Fr$ \acute{e} $chet spaces, where $ \alpha\in (1, 2) $. The controllability conclusion was acquired without any compactness for the nonlinear term, the cosine family, and the sine family. The investigation was based on a nonlinear alternative and the cosine family theory. An application of our findings was provided.</p></abstract>

Existence Results for Fractional Integral Equations in Frechet Spaces

The objective of this paper is to present results on the existence of solutions for a class of fractional integral equations in Fr\'{e}chet spaces of Banach space-valued functions on the unbounded interval. Our main tool is the technique of measures of noncompactness and fixed points theorems.

On existence and controllability results for some cylindrical stochastic integro-differential equations in Fr\'{e}chet spaces

On existence and controllability results for some cylindrical stochastic integro-differential equations in Fr\'{e}chet spaces

A Global Diffeomorphism Theorem for Fréchet Spaces

We give sufficient conditions for a $ C^1_c $-local diffeomorphism between Fr\'{e}chet spaces to be a global one. We extend the Clarke's theory of generalized gradients to the more general setting of Fr\'{e}chet spaces. As a consequence, we define the Chang Palais-Smale condition for Lipschitz functions and show that a function which is bounded below and satisfies the Chang Palais-Smale condition at all levels is coercive. We prove a version of the mountain pass theorem for Lipschitz maps in the Fr\'{e}chet setting and show that along with the Chang Palais-Smale condition we can obtain a global diffeomorphism theorem.

Measure of noncompactness and fractional integro-differential equations with state-dependent nonlocal conditions in Fréchet spaces

This paper deals with the existence of mild solutions for non-linear fractional integrodifferential equations with state-dependent nonlocal conditions. The technique used is a generalization of the classical Darbo fixed point theorem for Frechet spaces associated with the concept of measures ′ of noncompactness. An application of the main result has been included.

Integrated Semigroups on Fréchet Spaces II: F_sBSV ([a,b],F)-space, FsLip_0,w([0,+∞),F)-space, Isometric Theorem and Application

In this article we will study the Riemann Stieltjes Laplace integral of vectorial functions in Fréchet spaces. Particularly we will prove a isometric theorem and a generation theorem for integrated semigroups on Fréchet spaces.

Integrated Semigroups on Fréchet Spaces I: Bochner Integral, Laplace Integral and Real Representation Theorems

In this articles we will study the integration of the vectorial functions in Fréchet spaces. Particularly we will introduce and we will study a new functional space and we will prove some theorems of representation.

Integrated Semigroups on Fréchet Spaces I: Bochner Integral, Laplace Integral and Real Representation Theorems

In this articles we will study the integration of the vectorial functions in Fréchet spaces. Particularly we will introduce and we will study a new functional space and we will prove some theorems of representation.

A Generalized Palais-Smale Condition in the Fr\'{e}chet space setting

The Palais-Smale condition was introduced by Palais and Smale in the mid-sixties and applied to an extension of Morse theory to infinite dimensional Hilbert spaces. Later this condition was extended by Palais for the more general case of real functions over Banach-Finsler manifolds in order to obtain Lusternik-Schnirelman theory in this setting. Despite the importance of Fr\'{e}chet spaces, critical point theories have not been developed yet in these spaces.Our aim in this paper is to extend the Palais-Smale condition to the cases of $C^1$-functionals on Fr\'{e}chet spaces and Fr\'{e}chet-Finsler manifolds of class $C^1$. The difficulty in the Fr\'{e}chet setting is the lack of a general solvability theory for differential equations. This restricts us to adapt the deformation results (which are essential tools to locate critical points) as they appear as solutions of Cauchy problems. However, Ekeland proved the result, today is known as Ekleand’s variational principle, concerning the existence of almost-minimums for a wide class of real functions on complete metric spaces. This principle can be used to obtain minimizing Palais-Smale sequences. We use this principle along with the introduced conditions to obtain some customary results concerning the existence of minima in the Fr\'{e}chet setting.Recently it has been developed the projective limit techniques to overcome problems (such as solvability theory for differential equations) with Fr\'{e}chet spaces. The idea of this approach is to represent a Fr\'{e}chet space as the projective limit of Banach spaces. This approach provides solutions for a wide class of differential equations and every Fr\'{e}chet space and therefore can be used to obtain deformation results. This method would be the proper framework for further development of critical point theory in the Fr\'{e}chet setting.

Evolution equations in Fréchet spaces

This paper deals with the existence of mild solutions for a class of evolution equations. The technique used is a generalization of the classical Darbo fixed point theorem for Fr\'{e}chet spaces associated with the concept of measure of noncompactness.

Infinite systems of integral equations in Fr\'{e}chet spaces using the notion of L-functions

Infinite systems of integral equations in Fr\'{e}chet spaces using the notion of L-functions

On operators commuting with a Pommiez type operator in weighted spaces of entire functions

A description is presented for continuous linear operators defined on a countable inductive limit of weighted Fréchet spaces of entire functions and commuting with a Pommiez type operator.

Hereditarily hypercyclic subspaces

We say that a sequence of operators $(T_n)$ possesses hereditarily hypercyclic subspaces along a sequence $(n_k)$ if for any subsequence $(m_k)\subset(n_k)$, the sequence $(T_{m_k})$ possesses a hypercyclic subspace. While so far no characterization of the existence of hypercyclic subspaces in the case of Fr\'{e}chet spaces is known, we succeed to obtain a characterization of sequences $(T_n)$ possessing hereditarily hypercyclic subspaces along $(n_k)$, under the assumption that the sequence $(T_n)$ satisfies the Hypercyclicity Criterion along $(n_k)$. We also obtain a characterization of operators possessing a hypercyclic subspace under the assumption that $T$ satisfies the Frequent Hypercyclicity Criterion.

Hahn spaces in Fréchet spaces and applications to real sequence spaces

In a joint paper with Grahame Bennett and Toivo Leiger (cf. [3]) the second author introduced for real sequence spaces the Hahn property and the notion of Hahn spaces. The aim of this paper is to extend these notions and a series of results in [3] to subspaces of Fréchet spaces by replacing the space $\omega$ of all real sequences and the set $\chi$ of all sequences of 0's and 1's by any Fréchet space $H$ and a suitable subset $\chi$ of $H$, respectively. Applications of the general considerations to the original case of Hahn spaces of real sequences are also a main subject matter of the paper.

Resolutions of the identity in Fr�chet spaces

It is a classical result that every Bade σ-complete Boolean algebra of (selfadjoint) projections in a separable Hilbert space coincides with the projections forming the resolution of the identity of some bounded selfadjoint operator. This result is extended to the setting of separable Frechet spaces. Namely, every Bade σ-complete Boolean algebra of projections in such a space coincides with the resolution of the identity of some (continuous) scalar-type spectral operator having spectrum a compact subset ofℝ.

Weakly sequentially complete Fr�chet spaces of integrable functions

For a Frechet space E let \(L^1(\mu , E)\) be the space of all E-valued integrable functions with respect to a probability measure \(\mu \) and, when E is a real Frechet space, let \(L^1 ({\cal G}, E)\) be the space of all real-valued integrable functions with respect to an E-valued vector measure \({\cal G}\). We prove that if E is weakly sequentially complete then both \(L^1(\mu , E)\) and \(L^1({\cal G}, E)\) are weakly sequentially complete. We also prove that if E is a Frechet lattice, then E is weakly sequentially complete if and only if E does not contain any (lattice) copy of c0. By combining these main theorems, we also give some results about the existence of copies of c0 in \(L^1(\mu , E)\) and \(L^1({\cal G}, E)\).

Bounded sets in (LF)-spaces

The behaviour of bounded sets is important in the theory of countable inductive limits of Fréchet spaces, the (LF)-spaces, and its applications. An (LF)-space is called regular if every bounded set is contained and bounded in one of the steps. In the present paper necessary conditions and sufficient conditions are given for the regularity of an (LF)-space. The conditions are expressed in terms of the behaviour of the neighbourhoods of the steps. It is proved that the conditions are equivalent for (LF)-spaces of sequences or of continuous functions.

Ordinary differential equations with a continuous right-hand side in Fr�chet spaces

Ordinary differential equations with a continuous right-hand side in Fr�chet spaces

Peano's theorem fails for infinite-dimensional Fr�chet spaces

Peano's theorem fails for infinite-dimensional Fr�chet spaces

Some results on Fr�chet spaces with the density condition

Some results on Fr�chet spaces with the density condition