Frechet Space Research Articles

Using Frechet space to solve non linear functional integral equations

In this paper we have used Frechet space to study a unique solution on a semi-infinite integral equation by applying a nonlinear alternative of the Leray-Schauder type for contraction maps.

Statistical Bochner Integral on Frechet Space

Probability theory including the Bochner integral is a very important part of modern mathematical concepts including the modern theory of probabilities, especially in the concept of mathematical expectation and dispersion. In this, study a statistical approach form of Bochner’s theory is given and extended, but some fundamental properties of statistical integral were previously studied in the Banach case. Our approach formulates an extended integration concept of Bochner. By using the statistical convergence on general locally convex space it is possible to obtain very similar results referring to the Frechet space type. From our results, some interesting comparable outputs to Banach space are carried out. At the end of our research, it is conducted that if a function “f” is Bochner integrable in the classic report then it is statistically Bochner integrable, but conversely, this is not true. Hence, the value of the extension of Bochner integration is a need and is the focus of our work. This extension is given by modifying the model published by Schvabik and Guoju. Mathematically it is substantiated that on the space of Frechet types the space of functions of statistical Bochner integrable is a Frechet space.

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.

Darboux problem for fractional partial hyperbolic di erential inclusions on unbounded domains with delay

"In this paper we investigate the existence of solutions of initial value problems (IVP for short), for partial hyperbolic functional and neutral differential inclusions of fractional order involving Caputo fractional derivative with finite delay by using the nonlinear alternative of Frigon type for multivalued admissible contraction in Frechet spaces."

Approximate Identities in Algebras of Compact Operators on Frechet Spaces

Bounded approximate identity(bai) is a key concept in the theory of amenability of algebras. In this paper, we show that algebra of compact operators on Frechet space X has both the right and left locally bounded approximate identities. Sufficient conditions for the existence of these identities are established based on the geometry properties of the Frechet space X and its dual space X' respectively.

On the space of Schwartz operators in the symmetric Fock space and its dual

A long-standing problem that arises when constructing the mathematical apparatus of quantum mechanics is the need to work with unbounded operators. Since the space of nuclear operators is preconjugate for the algebra of all bounded operators, we can consider the states of a quantum system as nuclear operators, and consider bounded operators as observables. In this case, taking the trace for the product of a nuclear operator (a quantum state) and a bounded operator (a quantum observable) gives the average value of a quantum observable in a fixed state. The existence of such an average for unbounded operators is not guaranteed. If we want to define a space of observables that includes such naturally occurring unbounded operators as the position and momentum operators, for which average values are always determined, we should consider a space of states smaller than all nuclear operators. Recently, this approach has been mathematically correct implemented in the Hilbert space H = L^2(R^N). The so-called space of Schwarz operators, equipped with a system of semi-norms and being a Frechet space, was chosen as the space of states. Schwarz operators appeared to be integral operators whose kernels are functions belonging to the usual Schwarz space. The dual space to the space of Schwarz operators should be considered as the space of quantum observables and it really includes such standard observables as polynomials from the products of the position and momentum operators. In the present work we transfer this approach to the symmetric Fock space H = F(H) over an infinite dimensional separable Hilbert space H. We introduce the space of Schwartz operators in F(H) over an infinite dimensional separable space H and investigate which of the standard operators of quantum white noise belong to the space dual to the space of Schwartz operators.

Delay Differential Equations with Differentiable Solution Operators on Open Domains in C((-∞, 0], Rn) and Processes for Volterra Integro-Differential Equations

For autonomous delay differential equations x'(t)=f(xt){x'(t)=f(x_t)} we construct a continuous semiflow of continuously differentiable solution operators x0xt{x_0 \to x_t}, t0{t \le 0}, on open subsets of the Frechet space C((-,0],Rn){C((-\infty, 0], R^n)}. For nonautonomous equations this yields a continuous process of differentiable solution operators. As an application, we obtain processes which incorporate all solutions of Volterra integro-differential equations x'(t)=∫0tk(t,s)h(x(s))ds{x'(t)={\int_0}^t k(t,s) h(x(s)) ds}.

English

The space $${\cal H}{\cal K}$$ of Henstock-Kurzweil integrable functions on [a, b] is the uncountable union of Frechet spaces $${\cal H}{\cal K}$$ (X). In this paper, on each Frechet space $${\cal H}{\cal K}$$ (X), an F-norm is defined for a continuous linear operator. Hence, many important results in functional analysis, like the Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem, hold for the $${\cal H}{\cal K}$$ (X) space. It is known that every control-convergent sequence in the $${\cal H}{\cal K}$$ space always belongs to a $${\cal H}{\cal K}$$ (X) space for some X. We illustrate how to apply results for Frechet spaces $${\cal H}{\cal K}$$ (X) to control-convergent sequences in the $${\cal H}{\cal K}$$ space. Examples of compact linear operators are given. Existence of solutions to linear and Hammerstein integral equations is proved.

Operators between different weighted Fréchet and (LB)-spaces of analytic functions

We study some classical operators defined on the weighted Bergman Frechet space $A^p_{\alpha+}$ (resp. weighted Bergman (LB)-space $A^p_{\alpha-}$) arising as the projective limit (resp. inductive limit) of the standard weighted Bergman spaces into the growth Frechet space $H^\infty_{\alpha+}$ (resp. growth (LB)-space $H^\infty_{\alpha-}$), which is the projective limit (resp. inductive limit) of the growth Banach spaces. We show that, for an analytic self map $\varphi$ of the unit disc $\mathbb{D}$, the continuities of the weighted composition operator $W_{g,\varphi}$, the Volterra integral operator $T_g$, and the pointwise multiplication operator $M_g$ defined via the identical symbol function are characterized by the same condition determined by the symbol's state of belonging to a Bloch-type space. These results have consequences related to the invertibility of $W_{g,\varphi}$ acting on a weighted Bergman Frechet or (LB)-space. Some results concerning eigenvalues of such composition operators $C_\varphi$ are presented.

The coreflective hull of $$\mathcal {U}$$-Fréchet spaces

It is known that the category of all sequential spaces is a coreflective subcategory of the category of all topological spaces, and the coreflective hull of the category of all Frechet spaces is the category of all sequential spaces. In this paper, we generalize these results to $$\mathcal {U}$$ -sequential spaces and $$\mathcal {U}$$ -Frechet spaces where $$\mathcal {U}$$ is a free ultrafilter on $$\mathbb {N}$$ . Also we prove that the category of $$\mathcal {U}$$ -sequential spaces is a bi-coreflective subcategory of the category of all topological spaces where $$\mathcal {U}$$ is a free filter. As a consequence of these results, we get that some bi-coreflective subcategories of Top can be constructed by using free filters or free ultrafilters on $$\mathbb {N}$$ .

On Fermat-Torricelli Problem in Frechet Spaces

We study the Fermat-Torricelli problem (FTP) for Frechet space X, where X is considered as an inverse limit of projective system of Banach spaces. The FTP is defined by using fixed countable collection of continuous seminorms that defines the topology of X as gauges. For a finite set A in X consisting of n distinct and fixed points, the set of minimizers for the sum of distances from the points in A to a variable point is considered. In particular, for the case of collinear points in X, we prove the existence of the set of minimizers for FTP in X and for the case of non collinear points, existence and uniqueness of the set of minimizers are shown for reflexive space X as a result of strict convexity of the space.

Measure of Noncompactness and Fractional Differential Equations in Frechet Spaces

In this paper, the existence of solutions for an initial value problem of a fractional differential equation is obtained by means of Monch's fixed point theorem and the technique of measures of noncompactness. AMS (MOS) Subject Classification. 26A33, 34B10, 34G20

Stochastic calculus on Fréchet spaces

We develop a stochastic calculus on Frechet spaces for a particular family of stochastic processes. The formulae obtained is applied to the case of tame Frechet spaces and tame maps. In particular, an Ito formula for tame maps is proved.

Fractional order differential inclusions on an unbounded domain with infinite delay

We provide sufficient conditions for the existence of solutions to initial value problems, for partial hyperbolic differential inclusions of fractional order involving Caputo fractional derivative with infinite delay by applying the nonlinear alternative of Frigon type for multivalued admissible contraction in Frechet spaces.

Operators acting in sequence spaces generated by Dual Banach spaces of discrete Cesàro spaces

The dual spaces $d(p), 1 \lt p \lt \infty ,$ of the discrete Cesaro (Banach) spaces $\mathrm{ces} (q)$, $1 \lt q \lt \infty ,$ were studied by G. Bennett, A. Jagers and others. These (reflexive) dual Banach spaces induce the non-normable Frechet spaces $d (p+) := \bigcap_{r \gt p} d (r), $ for $1 \leq p \lt \infty ,$ and the (LB)-spaces $d (p-) := \bigcup_{ 1 \lt r \lt p } d (r), $ for $ 1 \lt p \leq \infty ,$ recently introduced and investigated in [11]. Here a detailed study is made of various aspects, such as the spectrum, continuity, compactness, mean ergodicity and supercyclicity of the Cesaro operator, multiplication operators and inclusion operators when they act on (and between) such spaces.

Meromorphic Extensions of $$(\cdot , W)$$-Meromorphic Functions

In this paper we establish some classes of subspaces W of the dual $$F'$$ of a locally convex space F such that every F-valued (F, W)-meromorphic function (with/without local boundedness) on a domain D in $$\mathbb {C}^n,$$ in the sense $$u \circ f$$ is meromorphic for all $$u \in W,$$ is meromorphic. Further, combining those results with studing on (BB)-Zorn property we give conditions for Frechet spaces E, F and subspaces W of $$F'$$ under which (F, W)-meromorphic functions can be meromorphically extended to a domain D of E from a subset $$D \cap E_B$$ where $$E_B$$ is the linear hull of some balanced convex compact subset B of E. Using these results we get the answers of the following questions: (1) When does the domain of meromorphy of a $$(\cdot , W)$$ -meromorphic function on a Riemann domain D over a Frechet space coincide with the envelope of holomorphy of D? (2) When will $$(\cdot , W)$$ -meromorphic functions be able to extend meromorphically through an analytic subset of codimension $$\ge 2$$ of a domain in a Frechet space?

On the absence of stability of bases in some Fréchet spaces

We show that, for each compact subset of the real line of infinite cardinality with an isolated point, the space of Whitney jets on the set does not possess a basis consisting only of polynomials. On the other hand, polynomials are dense in any Whitney space. Thus, there are no general results about stability of bases in Frechet spaces.

Semigroups on time scales and applications to abstract Cauchy problems

In this paper, we introduce the definition of a $C_0$-semigroup on a time scale, which unifies the continuous, discrete and other cases which lie between them. Also, it extends the classical theory of operator semigroups to the quantum case. We study the relationship between the semigroup and its infinitesimal generator. We apply our theory to study the homogeneous and non homogeneous abstract Cauchy problem in Banach and Frechet spaces.

Disjoint Distributional Chaos in Fréchet Spaces

We introduce two different notions of disjoint distributional chaos for sequences of continuous linear operators in Frechet spaces. We focus special attention to the analysis of some specific classes of linear continuous operators having a certain disjoint distributionally chaotic behaviour, providing also a great number of illustrative examples and applications of our abstract theoretical results.

Formal Functional Calculus for Weakly Locally Nilpotent Operators in Fréchet Spaces

We generalize the classical Riesz–Dunford holomorphic functional calculus with weakly locally nilpotent operators in a Frechet space to the case of the algebra of all formal power series. We obtain a counterpart of the spectral mapping theorem and formulas similar to the Taylor and Riesz–Dunford formulas. We show that the weak local nilpotency of an operator is necessary for constructing the formal functional calculus. Bibliography: 12 titles.