General Banach Spaces Research Articles

Index Concepts for Linear Differential-Algebraic Equations in Infinite Dimensions

Different index concepts for regular linear differential-algebraic equations are defined and compared in the general Banach space setting. For regular finite dimensional linear differential-algebraic equations, all these indices exist and are equivalent. For infinite dimensional systems, the situation is more complex. It is proven that although some indices imply others, in general they are not equivalent. The situation is illustrated with a number of examples.

Mordukhovich derivatives of the set-valued metric projection operator in general Banach spaces

ABSTRACT In this paper, we investigate the properties and the precise solutions of the Mordukhovich derivatives of the set-valued metric projection operator onto some closed balls in some general Banach spaces. In the Banach space c, we find the properties of Mordukhovich derivatives of the set-valued metric projection operator onto the closed subspace c 0. We show that the metric projection from C[0, 1] to polynomial with degree less than or equal to n is a single-valued mapping. We investigate its Mordukhovich derivatives and G a ˆ teaux directional derivatives.

Some refinements of the Fekete–Szegö inequalities for a class of biholomorphic mappings

Let B be the unit disk U in C or the unit ball of a general complex Banach space. In this paper, we obtain a refinement of the Fekete–Szegö inequality for the class of S k + 1 0 ( B ) of mappings f for which there exist g-Loewner chains f ( z , t ) , where g ( ξ ) = 1 + ξ 1 − ξ , ξ ∈ U , such that f ( z ) = f ( z , 0 ) and z = 0 is a zero of order k + 1 of e − t f ( z , t ) − z for each t ≥ 0 . The results presented generalize the classical Fekete–Szegö inequality and the results in Hamada et al. (Fekete–Szegö problem for univalent mappings in one and higher dimensions. J Math Anal Appl. 2022;516:126526).

Boundedness and compactness of Hausdorff operators on Fock spaces

We obtain a complete characterization of the bounded Hausdorff operators acting on a Fock space F α p F^p_\alpha and taking its values into a larger one F α q , 0 > p ≤ q ≤ ∞ F^q_\alpha ,\ 0 > p \leq q \leq \infty , as well as some necessary or sufficient conditions for a Hausdorff operator to transform a Fock space into a smaller one. Some results are written in the context of mixed norm Fock spaces. Also the compactness of Hausdorff operators on a Fock space is characterized. The compactness result for Hausdorff operators on the Fock space F α ∞ F^\infty _\alpha is extended to more general Banach spaces of entire functions with weighted sup norms defined in terms of a radial weight and conditions for the Hausdorff operators to become p p -summing are also included.

On the Controllability of Coupled Nonlocal Partial Integrodifferential Equations Using Fractional Power Operators

In this research paper, we investigate the controllability in the α-norm of a coupled system of integrodifferential equations with state-dependent nonlocal conditions in generalized Banach spaces. We establish sufficient conditions for the system’s controllability using resolvent operator theory introduced by Grimmer, fractional power operators, and fixed-point theorems associated with generalized measures of noncompactness for condensing operators in vector Banach spaces. Finally, we present an application example to validate the proposed methodology in this research.

Strong convergence theorem of a new modified Bregman extragradient method to solve fixed point problems and variational inequality problems in general reflexive Banach spaces

In this paper, we introduce a new modified self-adaptive extragradient method with the inertial technique for solving the variational inequality problem of a pseudomonotone mapping and the fixed point problem of a Bregman relatively nonexpansive operator in a general reflexive Banach space. We prove a strong convergence theorem of our algorithm under some suitable assumptions. Finally, some numerical experiments are provided to demonstrate the effectiveness of the suggested iterative method.

On Coarse Isometries and Linear Isometries between Banach Spaces

Let X,Y be two Banach spaces and f:X→Y be a standard coarse isometry. In this paper, we first show a sufficient and necessary condition for the coarse left-inverse operator of general Banach spaces to admit a linearly isometric right inverse. Furthermore, by using the well-known simultaneous extension operator, we obtain an asymptotical stability result when Y is a space of continuous functions. In addition, we also prove that every coarse left-inverse operator does admit a linear isometric right inverse without other assumptions when Y is a Lp(1<p<∞) space, or both X and Y are finite dimensional spaces of the same dimension. Making use of the results mentioned above, we generalize several results of isometric embeddings and give a stability result of coarse isometries between Banach spaces.

Representative Functions, Variational Convergence and Almost Convexity

We develop a new epi-convergence based on the use of bounded convergent nets on the product topology of the strong topology on the primal space and weak star topology on the dual space of a general real Banach space. We study the propagation of the associated variational convergences through conjugation of convex functions defined on this product space. These results are then applied to the problem of construction of a bigger-conjugate representative function for the recession operator associated with a maximal monotone operator on this real Banach space. This is then used to study the relationship between the recession operator of a maximal monotone operator and the normal–cone operator associated with the closed, convex hull of the domain of that monotone operator. This allows us to show that the strong closure of the domain of any maximal monotone operator is convex in a general real Banach space.

Existence of solution sets for Φ-Laplacian for random impulsive differential equations

In this article, we provide results on the existence and compactness of the set of solutions for a class of initial-value problem representing a first-order random impulsive differential equation. The key tool to achieve our goal is based on an alternative nonlinear Leary-Schauder type theorem in the settings of generalized Banach spaces.

Investigation on continuous dependence and regularity solutions of functional integrodifferential equations

In this work, to show the existence and continuous dependence of integral solution for neutral integro-differential equation with finite delay situation. This result established by using integrated resolvent operator with support of Lipschitz continuous and uniqueness part using Banach contraction mapping. Also study the existence of strict solution on reflexive and general Banach spaces. In the last section, an example is provided related to this theory.

Differentiable Hartman-Grobman Theorem via modulus of continuity: A sharp result on linearization in general Banach space

Differentiable Hartman-Grobman Theorem via modulus of continuity: A sharp result on linearization in general Banach space

Local convergence study of tenth-order iterative method in Banach spaces with basin of attraction

<abstract><p>Many applications from computational mathematics can be identified for a system of non-linear equations in more generalized Banach spaces. Analytical methods do not exist for solving these type of equations, and so we solve these equations using iterative methods. We introduced a new numerical technique for finding the roots of non-linear equations in Banach space. The method is tenth-order and it is an extension of the fifth-order method which is developed by Arroyo et.al. <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>. We provided a convergence analysis to demonstrate that the method exhibits tenth-order convergence. Also, we discussed the local convergence properties of the suggested method which depends on the fundamental supposition that the first-order Fréchet derivative of the involved function $ \Upsilon $ satisfies the Lipschitz conditions. This new approach is not only an extension of prior research, but also establishes a theoretical concept of the radius of convergence. We validated the efficacy of our method through various numerical examples. Our method is comparable with the methods of Tao Y et al. <sup>[<xref ref-type="bibr" rid="b2">2</xref>]</sup>. We also compared it with higher-order iterative methods, and we observed that it either performs similarly or better for the numerical examples. We also gave the basin of attraction to demonstrate the behaviour in the complex plane.</p></abstract>

Investigation on integro-differential equations with fractional boundary conditions by Atangana-Baleanu-Caputo derivative.

We establish, the existence and uniqueness of solutions to a class of Atangana-Baleanu (AB) derivative-based nonlinear fractional integro-differential equations with fractional boundary conditions by using special type of operators over general Banach and Hilbert spaces with bounded approximation numbers. The Leray-Schauder alternative theorem guarantees the existence solution and the Banach contraction principle is used to derive uniqueness solutions. Furthermore, we present an implicit numerical scheme based on the trapezoidal method for obtaining the numerical approximation to the solution. To illustrate our analytical and numerical findings, an example is provided and concluded in the final section.

Fuzzy multiplier, sum and intersection rules in non-Lipschitzian settings: Decoupling approach revisited

We revisit the decoupling approach widely used (often intuitively) in nonlinear analysis and optimization and initially formalized about a quarter of a century ago by Borwein & Zhu, Borwein & Ioffe and Lassonde. It allows one to streamline proofs of necessary optimality conditions and calculus relations, unify and simplify the respective statements, clarify and in many cases weaken the assumptions. In this paper we study weaker concepts of quasiuniform infimum, quasiuniform lower semicontinuity and quasiuniform minimum, putting them into the context of the general theory developed by the aforementioned authors. Along the way, we unify the terminology and notation and fill in some gaps in the general theory. We establish rather general primal and dual necessary conditions characterizing quasiuniform ε-minima of the sum of two functions. The obtained fuzzy multiplier rules are formulated in general Banach spaces in terms of Clarke subdifferentials and in Asplund spaces in terms of Fréchet subdifferentials. The mentioned fuzzy multiplier rules naturally lead to certain fuzzy subdifferential calculus results. An application from sparse optimal control illustrates applicability of the obtained findings.

Characterizations of the metric and generalized metric projections on subspaces of Banach spaces

The variational principles of the metric projection onto closed and convex sets are commonly used as the basis for characterizations of metric projections onto subspaces of Hilbert spaces. However, it is important to realize that the variational principles do not hold as necessary and sufficient conditions in general Banach spaces to characterize the metric projection and its related extensions. This technical handicap prevents the study of metric projection onto subspaces of general Banach spaces from being fully explored. Motivated by this existing lacuna in the literature, we investigate the metric projection and the generalized metric projection onto subspaces of general Banach spaces in this study. Focusing on concrete Banach spaces, we show that even though there are no general variational principles in such spaces for the projection onto subspaces, specific elements satisfy a variational characterization. We provide concrete examples to illustrate various notions of projections in general Banach spaces.

Inertial method for solving split equality Hammerstein-type equation problems in reflexive Banach spaces

The purpose of this paper is to introduce an inertial scheme for solving Hammerstein-type equation problems in general reflexive real Banach spaces, where the underlying mappings are generally continuous monotone, not just uniformly continuous monotone. A strong convergence theorem is proved under some mild conditions, and finally numerical examples are provided to demonstrate the applicability of the algorithm.

Some noncompact types of fixed point results in the generalized Banach spaces with respect to the G–weak topology contexts and applications

This research deals with Krasnoselskii’s fixed point theorem where the entries operators do not need to be G-weakly compact and contraction. These results were obtained by using the so-called generalized measure of weak noncompactness and some user-friendly lemmas. Moreover, these gained fixed point results are applied to study the existence of solutions of a coupled system for integral equations in the generalized Banach space mathcal{{ C }} ( [0,1], E_{1} )times mathcal{{ C }} ( [0,1], E_{2} ) .

On Lagrange Multiplier Theorems for Non-smooth Optimization for a Large Class of Variational Models in Banach Spaces

This article develops optimality conditions for a large class of non-smooth variational models. The main results are based on standard tools of functional analysis and calculus of variations. Firstly we address a model with equality constraints and, in a second step, a more general model with equality and inequality constraints, always in a general Banach space context. We highlight some novelties are introduced related to the proof procedures developed in this text, which are in general softer than those concerning the present literature.

Results on existence of solutions in nonlocal partial functional integrodifferential equations with finite delay in nondense domain

In this work, to show the existence and uniqueness solution of functional integrodifferential equation with nonlocal condition and finite delay function. This article contains continuously dependence of integral solution and existence of strict solution established by using integrated resolvent operator theory with support of Lipschitz continuous. In this article the operator (T(t))t⩾0 is assumed the locally Lipschitz continuous integrated resolvent operator and proved the existence of strict solution on reflexive and general Banach spaces. Finally an example is given related to this work.

A Mathematical Theoretical Study of a Coupled Fully Hybrid (k, Φ)-Fractional Order System of BVPs in Generalized Banach Spaces

In this paper, we study a coupled fully hybrid system of (k,Φ)–Hilfer fractional differential equations equipped with non-symmetric (k,Φ)–Riemann-Liouville (RL) integral conditions. To prove the existence and uniqueness results, we use the Krasnoselskii and Perov fixed-point theorems with Lipschitzian matrix in the context of a generalized Banach space (GBS). Moreover, the Ulam–Hyers (UH) stability of the solutions is discussed by using the Urs’s method. Finally, an illustrated example is given to confirm the validity of our results.