Banach Space Research Articles

A Novel Approach to Nonlinear Volterra-Fredholm Integral Equations Using Abaoub Shkheam Decomposition Method

In this study, we introduce a novel approach to the solution of a nonlinear Volterra -Fredholm integral equations by applying the Adomian decomposition method under the effect of the Abaoub- Shkheam transform. We demonstrate the existence and uniqueness of the solution in Banach space and illustrate this idea with an example.

Characterizing Nonuniform Hyperbolicity by Mather-Type Admissibility

AbstractWe consider linear cocycles acting on Banach spaces which satisfy the assumptions of the multiplicative ergodic theorem. A cocycle is nonuniformly hyperbolic if all Lyapunov exponents are non-zero, which is equivalent to the existence of a tempered exponential dichotomy. We provide an equivalent characterization of nonuniform hyperbolicity in terms of a Mather-type admissibility of a pair of weighted function spaces. As an application we give a short proof of the robustness of tempered exponential dichotomies under small linear perturbation.

Functional Inequalities in the Framework of Banach Spaces

AbstractThis paper aims to illustrate the usefulness of majorization theory and higher-order convexity in generalizing several classical inequalities as functional inequalities. This includes results due to Ball et al. (Invent Math 115:463-482, 1994), Hanner (Ark Mat 3:239-244, 1956), Popoviciu (Analele ştiinţifice Univ “Al. I. Cuza” Iasi, Secţia I-a Mat 11:155-164, 1965) and Zhang (Matrix Theory: Basic Results and Techniques, Springer, New York, 2011). Also, based on a substitute for the parallelogram law due to Clarkson, a quadrilateral inequality established by Schötz (in Quadruple inequalities: Between Cauchy-Schwarz and triangle. ArXiv preprint, arXiv:2307.01361 v.3, 2024) in the context of Hilbert spaces is extended to the general framework of Banach spaces.

On the large time behaviour of the solutions of an evolutionary-epidemic system with spatial dispersal.

This paper investigates some properties of the large time behaviour of the solutions of a spatially distributed system of equations modelling the evolutionary epidemiology of a plant-pathogen system. The model takes into account the phenotypic trait and the mutation of the pathogen, which is described by a non-local operator. We roughly speaking prove that the solutions separate the phenotype trait from the spatio-temporal evolution in the large time asymptotic. This feature is obtained by investigating the positive and bounded entire solutions of the problem, that are shown to exhibit such a separation of the variables property, by reformulating them as the positive solutions of suitable integral equations in some ordered Banach space. In addition, some numerical simulations are performed to support our theoretical results.

Gelfand–Phillips Type Properties of Locally Convex Spaces

We let 1≤p≤q≤∞. Being motivated by the classical notions of the Gelfand–Phillips property and the (coarse) Gelfand–Phillips property of order p of Banach spaces, we introduce and study different types of the Gelfand–Phillips property of order (p,q) (the GP(p,q) property) and the coarse Gelfand–Phillips property of order p in the realm of all locally convex spaces. We compare these classes and show that they are stable under taking direct product, direct sums and closed subspaces. It is shown that any locally convex space is a quotient space of a locally convex space with the GP(p,q) property. Characterizations of locally convex spaces with the introduced Gelfand–Phillips type properties are given.

Banach Spaces of Solenoidal Solutions of the Vector Helmholtz Equation in the Space and Its Associated Reproducing Kernel

Banach Spaces of Solenoidal Solutions of the Vector Helmholtz Equation in the Space and Its Associated Reproducing Kernel

Stability of Quadratic Mappings in 2-Banach Spaces and Related Topics

Functional analysis is an important branch of mathematics widely used to study the stability of different functional equations. This includes the stability of various quadratic functional equations, which typically involve a specific number of variables to reach results more easily in this field. One of the methods used to study the stability of functional equations is the direct method, which is known for its simplicity in proving the stability of this type of functional equation. In this research paper, we have successfully proven the Hyers-Ulam-Rassias stability of the quadratic functional equation in 2-Banach spaces. Specifically, we have shown that the equation: f(x+y+z)+f(x)+f(y)+f(z)=f(x+y)+f(y+z)+f(x+z) holds true within this context. Our approach involved using either the usual or the direct method to establish this stability. Furthermore, we have also demonstrated the generalized Hyers-Ulam stability of the quadratic functional equation in 2-Banach spaces using the usual process by considering various conditions. This has led to many exciting results and revealed many related applications.

Inertial Bilevel Variational Monotone Inclusion Problem in Banach Spaces

In this paper, we study accelerated Halpern-type iterative method for finding zero solution of sum of two monotone operators which solves variational inequality problem of inverse strongly monotone mapping in 2-uniformly convex and uniformly smooth real Banach spaces. The strong convergence of our proposed method is establish under some standard conditions imposed on parameters. Our theorems generalize many recently announced results in the literature.

Dominated and absolutely summing operators on the space Crc(X,E)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\,C_{rc}(X,E)$$\\end{document} of vector-valued continuous functions

Let X be a completely regular Hausdorff space and E and F be Banach spaces. Let Crc(X,E)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{rc}(X,E)$$\\end{document} denote the Banach space of all continuous functions f:X→E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:X\\rightarrow E$$\\end{document} such that f(X) is a relatively compact set in E, and βσ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta _\\sigma $$\\end{document} be the strict topology on Crc(X,E)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{rc}(X,E)$$\\end{document}. We characterize dominated and absolutely summing operators T:Crc(X,E)→F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T:C_{rc}(X,E)\\rightarrow F$$\\end{document} in terms of their representing operator-valued Baire measures. It is shown that every absolutely summing (βσ,‖·‖F)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\beta _\\sigma ,\\Vert \\cdot \\Vert _F)$$\\end{document}-continuous operator T:Crc(X,E)→F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T:C_{rc}(X,E)\\rightarrow F$$\\end{document} is dominated. Moreover, we obtain that every dominated operator T:Crc(X,E)→F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T:C_{rc}(X,E)\\rightarrow F$$\\end{document} is absolutely summing if and only if every bounded linear operator U:E→F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U:E\\rightarrow F$$\\end{document} is absolutely summing.

Fibers and Gleason parts for the maximal ideal space of Au(Bℓp)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {A}}_u(B_{\\ell _p})$$\\end{document}

In the early nineties, R. M. Aron, B. Cole, T. W. Gamelin and W. B. Johnson initiated the study of the maximal ideal space (spectrum) of Banach algebras of holomorphic functions defined on the open unit ball of an infinite dimensional complex Banach space. Within this framework, we investigate the fibers and Gleason parts of the spectrum of the algebra of holomorphic and uniformly continuous functions on the unit ball of ℓp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _p$$\\end{document} (1≤p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le p<\\infty $$\\end{document}). We show that the inherent geometry of these spaces provides a fundamental ingredient for our results. We prove that whenever p∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\in {\\mathbb {N}}$$\\end{document} (p≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\ge 2$$\\end{document}), the fiber of every z∈Bℓp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z\\in B_{\\ell _p}$$\\end{document} contains a set of cardinal 2c\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2^{{\\mathfrak {c}}}$$\\end{document} such that any two elements of this set belong to different Gleason parts. For the case p=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=1$$\\end{document}, we complete the known description of the fibers, showing that, for each z∈B¯ℓ1′′\\Sℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z\\in \\overline{B}_{\\ell _1''}{\\setminus } S_{\\ell _1}$$\\end{document}, the fiber over z is not a singleton. Also, we establish that different fibers over elements in Sℓ1′′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_{\\ell _1''}$$\\end{document} cannot share Gleason parts.

The Lomonosov type theorems and the invariant subspace problem for non-archimedean Banach spaces

In this paper, we study the existence of invariant (and even hyperinvariant) subspaces of bounded operators on a non-archimedean Banach space E=(E,‖.‖) over a valued field K equipped with a non-trivial non-archimedean valuation |.|. Specifically, we consider compact operators and operators that commute with a compact operator. First we show that if E has a base, then any compact operator T such that limn⁡‖Tn‖1n>0 has a finite-dimensional hyperinvariant subspace. Next we show that if K is locally compact, then every compact operator T on E has a hyperinvariant subspace. Afterward, assuming that K is spherically complete or E is of countable type, we provide a necessary condition for a bounded operator on E to have a hyperinvariant subspace. We demonstrate that the classical Lomonosov Invariant Subspace theorem does not hold in the case where K is non-spherically complete. Finally, we prove Lomonosov type theorem for spectral quasinilpotent operators, when K is locally compact.

The Riemann-Liouville fractional integral in Bochner-Lebesgue spaces III

In this manuscript, we examine the continuity properties of the Riemann-Liouville fractional integral for order α=1/p, where p>1, mapping from Lp(t0,t1;X) to the Banach space BMO(t0,t1;X)∩K(p−1)/p(t0,t1;X). This improvement, refines a result by Hardy-Littlewood. To achieve this, we study properties between spaces BMO(t0,t1;X) and K(p−1)/p(t0,t1;X). Additionally, we obtained the boundedness of the fractional integral of order α≥1 from L1(t0,t1;X) into the Riemann-Liouville fractional Sobolev space WRLs,p(t0,t1;X).

A New Structure of Random Approach Normed Space via Banach Space

The goal of this research is to define the convergent sequence in A-random approach space and sequentially convergent are discussed and the cluster point, open and closed ball and linear transformation. We are going to explain a new structure of Random approach normed space via Banach space in and discussed all the relations between metric space in this research.

Controllability of some nonlocal‐impulsive Volterra evolution systems via measures of noncompactness

This study investigates the controllability of a Volterra evolution equation with impulsive terms and nonlocal initial conditions. With the aid of the resolvent operator generated by the linear part of the equation, mild solutions can be defined. Notably, the resolvent operator lacks compactness and equicontinuity. Additionally, the compactness of the impulsive and nonlocal functions is not required. Sufficient conditions for controllability are obtained through measures of noncompactness in Banach spaces. Functional differential equations and hyperbolic partial differential equations can be solved with these results. An example is given to illustrate the validity of the presented results.

An Algorithm for the Solution of Integro-Fractional Differential Equations with a Generalized Symmetric Singular Kernel

This work studies an integro-fractional differential equation (I-FrDE) with a generalized symmetric singular kernel. The scientific approach in this study was to transform the integro-differential equation (I-DE) into a mixed integral equation (MIE) with an Able kernel in fractional time and a generalized symmetric singular kernel in position. Additionally, the authors first set conditions on the singular kernels, whether related to time or position, and then transform the integral equation into an integral operator. Secondly, the solution is unique, which is proven by means of fixed-point theorems. In combination with the solution rules, the convergence of the solution is studied, and the error equation resulting from the solution is a stable error-integral influencer equation. Next, to solve this MIE, the authors apply a special technique to separate the variables and produce an integral equation in position with coefficients, in the form of an integral operator in time. As the most effective technique for resolving singular integral equations, the Toeplitz matrix method (TMM) is utilized to convert the integral equation into an algebraic system for the purpose of solving the position problem. The existence of a solution to the linear algebraic system in Banach space is then demonstrated. Lastly, certain applications where the functions of the generalized symmetric kernel are cubic or exponential and it assumes the logarithmic, Cauchy, or Carleman form are discussed. In each case, Maple 18 is also used to compute the error estimate.

Unconditional Cesàro convergence of sequences of super-reflexive valued random variables

We establish that if X X is a super-reflexive Banach space, every L X 1 L_{X}^{1} -bounded sequence has a subsequence for which any subsequences Cesàro converges unconditionally a.e. in X X to the same limit, generalizing a similar result for a Hilbert space.

Disjoint p$p$‐convergent operators and their adjoints

AbstractFirst, we give conditions on a Banach lattice so that an operator from to any Banach space is disjoint ‐convergent if and only if is almost Dunford–Pettis. Then, we study when adjoints of positive operators between Banach lattices are disjoint ‐convergent. For instance, we prove that the following conditions are equivalent for all Banach lattices and : (i) a positive operator is almost weak ‐convergent if and only if is disjoint ‐convergent; (ii) has order continuous norm or has the positive Schur property of order . Very recent results are improved, examples are given and applications of the main results are provided.

Properties of generalized polyhedral convex multifunctions

This paper presents a study of generalized polyhedral convexity under basic operations on multifunctions. We address the preservation of generalized polyhedral convexity under sums and compositions of multifunctions, the domains and ranges of generalized polyhedral convex multifunctions, and the direct and inverse images of sets under such mappings. Then we explore the class of optimal value functions defined by a generalized polyhedral convex objective function and a generalized polyhedral convex constrained mapping. The new results provide a framework for representing the relative interior of the graph of a generalized polyhedral convex multifunction in terms of the relative interiors of its domain and mapping values in locally convex topological vector spaces. Among the new results in this paper is a significant extension of a result by Bonnans and Shapiro on the domain of generalized polyhedral convex multifunctions from Banach spaces to locally convex topological vector spaces.

On the Implementation of Iterative Methods Without Inverse Updating for Solving Equations in Banach Spaces

The implementation of iterative methods using inverses to solve equations is a computationally expensive or impossible task in general. This is the case, since the analytical form of the inverse is difficult to find in practice. That is why, we replace the inverse by a sum of linear operators which is well defined. The convergence of the resulting hybrid methods is studied based on majorizing sequences under generalized continuity assumptions on the operators involved and in the setting of a Banach space. It is demonstrated by numerical experimentations that the convergence order as well as the number of iterations required to obtain a predetermined error tolerance when comparing the original to the hybrid method is essentially the same.

The solution of split common fixed point problems for enriched asymptotically nonexpansive mappings in Banach spaces

The solution of split common fixed point problems for enriched asymptotically nonexpansive mappings in Banach spaces