Showing papers on "Banach space published in 2023"

Hardy spaces associated with ball quasi‐Banach function spaces on spaces of homogeneous type: Characterizations of maximal functions, decompositions, and dual spaces

Abstract: Let ( X , ρ , μ ) $({\mathcal {X}},\rho ,\mu )$ be a space of homogeneous type in the sense of Coifman and Weiss, and let Y ( X ) $Y({\mathcal {X}})$ be a ball quasi-Banach function space on X ${\mathcal {X}}$ , which supports both a Fefferman–Stein vector-valued maximal inequality and the boundedness of the powered Hardy–Littlewood maximal operator on its associate space. The authors first introduce the Hardy space H Y ∗ ( X ) $H_{Y}^*({\mathcal {X}})$ , associated with Y ( X ) $Y({\mathcal {X}})$ , via the grand maximal function and then establish its various real-variable characterizations, respectively, in terms of radial or nontangential maximal functions, atoms or finite atoms, and molecules. As an application, the authors give the dual space of H Y ∗ ( X ) $H_{Y}^*({\mathcal {X}})$ , which proves to be a ball Campanato-type function space associated with Y ( X ) $Y({\mathcal {X}})$ . All these results have a wide range of generality and, particularly, even when they are applied to variable Hardy spaces, the obtained results are also new. The major novelties of this paper exist in that, to escape both the reverse doubling condition of μ and the triangle inequality of ρ, the authors cleverly construct admissible sequences of balls and fully use the geometrical properties of X ${\mathcal {X}}$ expressed by dyadic reference points or dyadic cubes and, to overcome the difficulty caused by the lack of the good dense subset of H Y ∗ ( X ) $H_{Y}^*({\mathcal {X}})$ , the authors further prove that Y ( X ) $Y({\mathcal {X}})$ can be embedded into the weighted Lebesgue space with certain special weight and then can fully use the known results of the weighted Lebesgue space.

Solvability and pullback attractor for a class of differential hemivariational inequalities with its applications

Abstract: In this paper, we consider an abstract system which consists of a nonlinear differential inclusion and a parabolic hemivariational inequality (DPHVI) in Banach spaces. The objective of this paper is four fold. The first target is to deal with the existence of solutions and the properties which involve the boundedness and continuous dependence results of the solution set to parabolic hemivariational inequality. The second aim is to investigate the existence of mild solutions to DPHVI by means of a fixed point technique. The third one is to study the existence of a pullback attractor for the multivalued processes governed by DPHVI. Finally, the fourth goal is to demonstrate a concrete application of our main results arising from the dynamic thermoviscoelasticity problems.

New mixed finite element methods for the coupled Stokes and Poisson–Nernst–Planck equations in Banach spaces

Abstract: In this paper we employ a Banach spaces-based framework to introduce and analyze new mixed finite element methods for the numerical solution of the coupled Stokes and Poisson–Nernst–Planck equations, which is a nonlinear model describing the dynamics of electrically charged incompressible fluids. The pressure of the fluid is eliminated from the system (though computed afterwards via a postprocessing formula) thanks to the incompressibility condition and the incorporation of the fluid pseudostress as an auxiliary unknown. In turn, besides the electrostatic potential and the concentration of ionized particles, we use the electric field (rescaled gradient of the potential) and total ionic fluxes as new unknowns. The resulting fully mixed variational formulation in Banach spaces can be written as a coupled system consisting of two saddle-point problems, each one with nonlinear source terms depending on the remaining unknowns, and a perturbed saddlepoint problem with linear source terms, which is in turn additionally perturbed by a bilinear form. The well-posedness of the continuous formulation is a consequence of a fixed-point strategy in combination with the Banach theorem, the Babuˇska–Brezzi theory, the solvability of abstract perturbed saddle-point problems, and the Banach–Neˇcas–Babuˇska theorem. For this we also employ smallness assumptions on the data. An analogous approach, but using now both the Brouwer and Banach theorems, and invoking suitable stability conditions on arbitrary finite element subspaces, is employed to conclude the existence and uniqueness of solution for the associated Galerkin scheme. A priori error estimates are derived, and examples of discrete spaces that fit the theory, include, e.g., Raviart–Thomas elements of order k along with piecewise polynomials of degree ďk. Finally, rates of convergence are specified and several numerical experiments confirm the theoretical error bounds. These tests also illustrate the balance-preserving properties and applicability of the proposed family of methods.

On $ \mathcal{A B C} $ coupled Langevin fractional differential equations constrained by Perov's fixed point in generalized Banach spaces

Abstract: Nonlinear differential equations are widely used in everyday scientific and engineering dynamics. Problems involving differential equations of fractional order with initial and phase changes are often employed. Using a novel norm that is comfortable for fractional and non-singular differential equations containing Atangana-Baleanu-Caputo fractional derivatives, we examined a new class of initial values issues in this study. The Perov fixed point theorems that are utilized in generalized Banach spaces form the foundation for the new findings. Examples of the numerical analysis are provided in order to safeguard and effectively present the key findings.

A generalized Halpern-type forward-backward splitting algorithm for solving variational inclusion problems

Abstract: In this paper, we investigate the problem of finding a zero of sum of two accretive operators in the setting of uniformly convex and $ q $-uniformly smooth real Banach spaces ($ q > 1 $). We incorporate the inertial and relaxation parameters in a Halpern-type forward-backward splitting algorithm to accelerate the convergence of its sequence to a zero of sum of two accretive operators. Furthermore, we prove strong convergence of the sequence generated by our proposed iterative algorithm. Finally, we provide a numerical example in the setting of the classical Banach space $ l_4(\mathbb{R}) $ to study the effect of the relaxation and inertial parameters in our proposed algorithm.

On symplectic Banach spaces

Abstract: Abstract We extend and generalize the result of Kalton and Swanson ( $$Z_2$$ Z 2 is a symplectic Banach space with no Lagrangian subspace) by showing that all higher order Rochgberg spaces $${\mathfrak {R}}^{(n)}$$ R ( n ) are symplectic Banach spaces with no Lagrangian subspaces. The nontrivial symplectic structure on Rochberg spaces of even order is the one induced by the natural duality; while the nontrivial symplectic structure on Rochberg spaces of odd order requires perturbation with a complex structure. We will also study symplectic structures on general Banach spaces and, motivated by the unexpected appearance of complex structures, we introduce and study almost symplectic structures.

Bounded solutions and Hyers–Ulam stability of quasilinear dynamic equations on time scales

Abstract: We consider the quasilinear dynamic equation in a Banach space on unbounded above and below time scales T with rd-continuous, regressive right-hand side.We define the corresponding Green-type map. Using the integral functional technique, we find a new simpler, but at the same time, more general sufficient condition for the existence of a bounded solution on the time scales expressed in terms of integrals of the Green-type map. We construct previously unknown linear scalar differential equation, which does not possess exponentially dichotomy, but for which the integral of the corresponding Green-type map is uniformly bounded. The existence of such example allows, on the one hand, to obtain the new sufficient condition for the existence of bounded solution and, on the other hand, to prove Hyers–Ulam stability for a much broader class of linear dynamic equations even in the classical case.

Policy Iteration Method for Time-Dependent Mean Field Games Systems with Non-separable Hamiltonians

Abstract: We introduce two algorithms based on a policy iteration method to numerically solve time-dependent Mean Field Game systems of partial differential equations with non-separable Hamiltonians. We prove the convergence of such algorithms in sufficiently small time intervals with Banach fixed point method. Moreover, we prove that the convergence rates are linear. We illustrate our theoretical results by numerical examples, and we discuss the performance of the proposed algorithms.

Banach-valued Bloch-type functions on the unit ball of a Hilbert space and weak spaces of Bloch-type

Abstract: In this article, we study the space $\mathcal B_\mu(B_X,Y)$ of $Y$-valued Bloch-type functions on the unit ball $B_X$ of an infinite dimensional Hilbert space $X$ with $\mu$ is a normal weight on $B_X$ and $Y$ is a Banach space. We also investigate the characterizations of the space $\mathcal{WB}_\mu(B_X)$ of $Y$-valued, locally bounded, weakly holomorphic functions associated with the Bloch-type space $\mathcal B_\mu(B_X)$ of scalar-valued functions in the sense that $f\in \mathcal{WB}_\mu(B_X)$ if $w\circ f \in \mathcal B_\mu(B_X)$ for every $w \in \mathcal W,$ a separating subspace of the dual $Y'$ of $Y.$

Almost Periodic Solutions of Abstract Impulsive Volterra Integro-Differential Inclusions

Abstract: In this paper, we introduce and systematically analyze the classes of (pre-)(B,ρ,(tk))-piecewise continuous almost periodic functions and (pre-)(B,ρ,(tk))-piecewise continuous uniformly recurrent functions with values in complex Banach spaces. We weaken substantially, or remove completely, the assumption that the sequence (tk) of possible first kind discontinuities of the function under consideration is a Wexler sequence (in order to achieve these aims, we use certain results about Stepanov almost periodic type functions). We provide many applications in the analysis of the existence and uniqueness of almost periodic type solutions for various classes of the abstract impulsive Volterra integro-differential inclusions.

Study of Nonlocal Multiorder Implicit Differential Equation Involving Hilfer Fractional Derivative on Unbounded Domains

Abstract: This paper aims to study the existence and uniqueness of the solution for nonlocal multiorder implicit differential equation involving Hilfer fractional derivative on unbounded domains a , ∞ , a ≥ 0 , in an applicable Banach space by utilizing the Banach contraction principle. Furthermore, we discuss various types of stability such as Ulam–Hyers–Rassias (UHR), Ulam–Hyers (UH), and semi-Ulam–Hyers–Rassias (sUHR) for nonlocal boundary value problem. Absolutely, our results cover that several outcomes have existed in the literature.

Inverse problem for the Atangana–Baleanu fractional differential equation

Abstract: Abstract In this manuscript, we examine a fractional inverse problem of order 0 < ρ < 1 {0<\rho<1} in a Banach space, including the Atangana–Baleanu fractional derivative in the Caputo sense. We use an overdetermined condition on a mild solution to identify the parameter. The major strategies for determining the outcome are a direct approach using the Volterra integral equation for sufficiently regular data. For less regular data, an optimal control approach uses Euler–Lagrange (EL) equations for the fractional order control problem (FOCP) and a numerical approach for solving FOCP. At last, a numerical example is provided in the support of our results.

On variational methods for interval-valued functions with some applications

Abstract: In this paper, we introduce the concept of sequential weak lower semicontinuity of interval-valued functions defined on a Banach space. We focus on the direct methods of the calculus of variations for interval-valued functions and extend some well-known results of F.E. Browder from scalar results to interval-valued cases. Moreover, we obtain some new results on the Palais–Smale condition and the coercivity for interval-valued functions. Finally, we apply the obtained results to interval-valued integral functions, to interval-valued optimal control problems, and to variational problems for interval-valued functions.

Vector‐valued singular integral operators with rough kernels

Abstract: In this paper, we establish a weak‐type (1,1) boundedness criterion for vector‐valued singular integral operators with rough kernels. As applications, we obtain weak‐type (1,1) bounds for the convolution singular integral operator taking value in the Banach space Y with a rough kernel, the maximal operator taking vector value in Y with a rough kernel and several square functions with rough kernels. Here, Y=[H,X]θ$Y=[H,X]_\theta$ is a complex interpolation space between a Hilbert space H and a UMD space X.

Existence and stability analysis for Caputo generalized hybrid Langevin differential systems involving three-point boundary conditions

Abstract: Abstract This research inscription gets to grips with two novel varieties of boundary value problems. One of them is a hybrid Langevin fractional differential equation, whilst the other is a coupled system of hybrid Langevin differential equation encapsuling a collective fractional derivative known as the ψ -Caputo fractional operator. Such operators are generated by iterating a local integral of a function with respect to another increasing positive function Ψ. The existence of the solutions of the aforehand equations is tackled by using the Dhage fixed point theorem, whereas their uniqueness is handled using the Banach fixed point theorem. On the top of this, the stability within the scope of Ulam–Hyers of solutions to these systems are also considered. Two pertinent examples are presented to corroborate the reported results.

Convergence Analysis for Yosida Variational Inclusion Problem with Its Corresponding Yosida Resolvent Equation Problem through Inertial Extrapolation Scheme

Abstract: In this paper, we study a Yosida variational inclusion problem with its corresponding Yosida resolvent equation problem. We mention some schemes to solve both the problems, but we focus our study on discussing convergence criteria for the Yosida variational inclusion problem in real Banach space and for the Yosida resolvent equation problem in q-uniformly smooth Banach space. For faster convergence, we apply an inertial extrapolation scheme for both the problems. An example is provided.

Metrical almost periodicity, metrical approximations of functions and applications

Abstract: In this paper, we analyze Levitan and Bebutov metrical approximations of functions $F :\Lambda \times X \rightarrow Y$ by trigonometric polynomials and $\rho$-periodic type functions, where $\emptyset eq \Lambda \subseteq {\mathbb R}^{n}$, $X$ and $Y$ are complex Banach spaces, and $\rho$ is a general binary relation on $Y$. We also analyze various classes of multidimensional Levitan almost periodic functions in general metric and multidimensional Bebutov uniformly recurrent functions in general metric. We provide several applications of our theoretical results to the abstract Volterra integro-differential equations and the partial differential equations.

The Baire theorem, an analogue of the Banach fixed point theorem and attractors in T1 compact spaces

Abstract: We prove that if X is a T1 second countable compact space, then X is a Baire space if and only if every nonempty open subset of X contains a closed subset with nonempty interior. We also prove an analogue of Banach's fixed point theorem for all T1 compact spaces. Applying the analogue of Banach's fixed point theorem we prove the existence of unique attractors for so called contractive iterated function systems whose Hutchinson operators are closed in compact T1 spaces.

Observability for Non-autonomous Systems

Abstract: We study non-autonomous observation systems , where is a strongly measurable family of closed operators on a Banach space and is a family of bounded observation operators from to a Banach space . Based on an abstract uncertainty principle and a dissipation estimate, we prove that the observation system satisfies a final-state observability estimate in for measurable subsets . We present applications of the above result to families of uniformly strongly elliptic differential operators as well as non-autonomous Ornstein–Uhlenbeck operators on with observation operators . In the setting of non-autonomous strongly elliptic operators, we derive necessary and sufficient geometric conditions on the family of sets such that the corresponding observation system satisfies a final-state observability estimate.