Search Results (12)

In this paper, we introduce ${\omega }_n$-symmetric polynomials associated with the finite group ${\omega }_n,$ which consists of roots of unity, and groups of permutations acting on the Cartesian product of Banach spaces ${\ell }_1.$ These polynomials extend the classical notions of symmetric and supersymmetric polynomials on ${\ell }_1.$ We explore algebraic bases in the algebra of ${\omega }_n$-symmetric polynomials and derive corresponding generating functions. Building on this foundation, we construct rings of multisets (multinumbers), defined as equivalence classes on the underlying space under the action of ${\omega }_n$-symmetric polynomials, and investigate their fundamental properties. Furthermore, we examine the ring of integer multinumbers associated with the group ${\omega }_n,$ proving that it forms an integral domain when n is prime or $n=4.$ Full article

In this paper, the existence of optimal solutions for problems governed by differential equations involving feedback controls is established for when the problem must account for a Volterra-type distributed delay and is subject to the action of impulsive external forces. The problem is reformulated within the class of impulsive semilinear integro-differential inclusions in Banach spaces and is studied by using topological methods and multivalued analysis. The paper concludes with an application to a population dynamics model. Full article

Cauchy problems are considered for families of, generally speaking, non-Volterra functional differential equations of the second order. For each family considered, in terms of the parameters of this family, necessary and sufficient conditions for the unique solvability of the Cauchy problem for all equations of the family are obtained. Such necessary and sufficient conditions are obtained for the following four kinds of families: integral restrictions are imposed on positive and negative functional operators, namely, operator norms are specified; pointwise restrictions are imposed on positive and negative functional operators in the form of values of operators’ actions on the unit function; an integral constraint is imposed on a positive functional operator, a pointwise constraint is imposed on a negative functional operator; a pointwise constraint is imposed on a positive functional operator, an integral constraint is imposed on a negative functional operator. In all cases, effective conditions for the solvability of the Cauchy problem for all equations of the family are obtained, expressed through some inequalities regarding the parameters of the families. The set of parameters of families of equations for which Cauchy problems are uniquely solvable can be easily calculated approximately with any accuracy. The resulting solvability conditions improve the solvability conditions following from the Banach contraction principle. An example of the Cauchy problem for an equation with a coefficient changing sign is given. Taking into account various restrictions for the positive and negative parts of functional operators allows us to significantly improve the known solvability conditions. Full article

A class for systems of nonlinear second-order differential equations with periodic impulse action are considered. An urgent problem for this class of differential equations is the problem of the quantitative study (existence) in the case when the phase space of the equation is, in the general case, some Banach space. In this work, sufficient conditions for the existence of solutions for a system with parameters are obtained. The results are obtained by using fixed point theorems for operators on a cone. Our approach is based on Schaefer’s fixed point theorem more precisely. In addition, the existence of positive solutions is also investigated. Full article

The first goal of this paper is to find a representation of the Fock space on ${\mathbb{C}}^n$ in terms of the weighted Bergman spaces of the projective spaces ${\mathbb{CP}}^{n-1}$; i.e., every function in the Fock space can be written as a direct sum of elements in weighted Bergman space on ${\mathbb{CP}}^{n-1}$. Also, we study the $C^{*}$ algebras generated by Toeplitz operators where the symbols are taken from the following two families of functions: Firstly, the symbols depend on the moment map associated with the unit circle, and secondly, the symbols are invariant under the same action. Moreover, we analyze the commutative relations between these algebras, and we apply these results to find new commutative Banach algebras generated by Toeplitz operators on Fock space of ${\mathbb{C}}^n$. Full article

In this paper, a multivalued self-mapping is defined on the union of a finite number of subsets $p\left(\ge 2\right)$ of a metric space which is, in general, of a mixed cyclic and acyclic nature in the sense that it can perform some iterations within each of the subsets before executing a switching action to its right adjacent one when generating orbits. The self-mapping can have combinations of locally contractive, non-contractive/non-expansive and locally expansive properties for some of the switching between different pairs of adjacent subsets. The properties of the asymptotic boundedness of the distances associated with the elements of the orbits are achieved under certain conditions of the global dominance of the contractivity of groups of consecutive iterations of the self-mapping, with each of those groups being of non-necessarily fixed size. If the metric space is a uniformly convex Banach one and the subsets are closed and convex, then some particular results on the convergence of the sequences of iterates to the best proximity points of the adjacent subsets are obtained in the absence of eventual local expansivity for switches between all the pairs of adjacent subsets. An application of the stabilization of a discrete dynamic system subject to impulsive effects in its dynamics due to finite discontinuity jumps in its state is also discussed. Full article

In this work, we shifted a recent multiplicity result by B. Ricceri from a Hilbert space to a Banach space setting by making use of a duality mapping relative to some increasing function. Using the min–max arguments, we provide conditions for an action functional to have at least two global minima. Full article

In this paper, we study a semilinear integro-differential inclusion in Banach spaces, under the action of infinitely many impulses. We provide the existence of mild solutions on a half-line by means of the so-called extension-with-memory technique, which consists of breaking down the problem in an iterate sequence of non-impulsive Cauchy problems, each of them originated by a solution of the previous one. The key that allows us to employ this method is the definition of suitable auxiliary set-valued functions that imitate the original set-valued nonlinearity at any step of the problem’s iteration. As an example of application, we deduce the controllability of a population dynamics process with distributed delay and impulses. That is, we ensure the existence of a pair trajectory-control, meaning a possible evolution of a population and of a feedback control for a system that undergoes sudden changes caused by external forces and depends on its past with fading memory. Full article

Let $A,X,Y$ be Banach spaces and $A\times X\to Y$, $(a,x)\mapsto ax$ be a continuous bilinear function, called a Banach action. We say that this action preserves unconditional convergence if for every bounded sequence ${\left(a_n\right)}_{n\in \omega }$ in A and unconditionally convergent series ${\sum }_{n\in \omega }{x}_n$ in X, the series ${\sum }_{n\in \omega }{a}_n{x}_n$ is unconditionally convergent in Y. We prove that a Banach action $A\times X\to Y$ preserves unconditional convergence if and only if for any linear functional $y^{*}\in Y^{*}$ the operator $D_{y^{*}}:X\to A^{*}$, $D_{y^{*}}\left(x\right)\left(a\right)=y^{*}\left(ax\right)$ is absolutely summing. Combining this characterization with the famous Grothendieck theorem on the absolute summability of operators from ${\ell }_1$ to ${\ell }_2$, we prove that a Banach action $A\times X\to Y$ preserves unconditional convergence if A is a Hilbert space possessing an orthonormal basis ${\left(e_n\right)}_{n\in \omega }$ such that for every $x\in X$, the series ${\sum }_{n\in \omega }{e}_{n}x$ is weakly absolutely convergent. Applying known results of Garling on the absolute summability of diagonal operators between sequence spaces, we prove that for (finite or infinite) numbers $p,q,r\in [1,\infty ]$ with $\frac{1}{r}\le \frac{1}{p}+\frac{1}{q}$, the coordinatewise multiplication ${\ell }_p\times {\ell }_q\to {\ell }_r$ preserves unconditional convergence if and only if one of the following conditions holds: (i) $p\le 2$ and $q\le r$, (ii) $2<p<q\le r$, (iii) $2<p=q<r$, (iv) $r=\infty$, (v) $2\le q<p\le r$, (vi) $q<2<p$ and $\frac{1}{p}+\frac{1}{q}\ge \frac{1}{r}+\frac{1}{2}$. Full article

In this paper, we investigate various classes of multi-dimensional Doss $\rho$-almost periodic type functions of the form $F:\mathsf{\Lambda }\times X\to Y,$ where $n\in \mathbb{N},$$\varnothing \ne \mathsf{\Lambda }\subseteq {\mathbb{R}}^n,$ X and Y are complex Banach spaces, and $\rho$ is a binary relation on $Y.$ We work in the general setting of Lebesgue spaces with variable exponents. The main structural properties of multi-dimensional Doss $\rho$-almost periodic type functions, like the translation invariance, the convolution invariance and the invariance under the actions of convolution products, are clarified. We examine connections of Doss $\rho$-almost periodic type functions with $(\omega ,c)$-periodic functions and Weyl-$\rho$-almost periodic type functions in the multi-dimensional setting. Certain applications of our results to the abstract Volterra integro-differential equations and the partial differential equations are given. Full article

In this paper, we introduce and analyze several different notions of almost periodic type functions and uniformly recurrent type functions in Lebesgue spaces with variable exponent $L^{p\left(x\right)}$ . We primarily consider the Stepanov and Weyl classes of generalized almost periodic type functions and generalized uniformly recurrent type functions. We also investigate the invariance of generalized almost periodicity and generalized uniform recurrence with variable exponents under the actions of convolution products, providing also some illustrative applications to the abstract fractional differential inclusions in Banach spaces. Full article

In this paper, we introduce and analyze Stepanov uniformly recurrent functions, Doss uniformly recurrent functions and Doss almost-periodic functions in Lebesgue spaces with variable exponents. We investigate the invariance of these types of generalized almost-periodicity in Lebesgue spaces with variable exponents under the actions of convolution products, providing also some illustrative applications to the abstract semilinear integro-differential inclusions in Banach spaces. Full article