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In this paper, we for the first time prove local solvability and stability of the inverse Sturm‐Liouville problem with complex‐valued singular potential and with polynomials of the spectral parameter in the boundary conditions. The proof method is constructive. It is based on the reduction of the inverse problem to a linear equation in the Banach space of bounded infinite sequences. We prove that, under a small perturbation of the spectral data, the main equation of the inverse problem remains uniquely solvable. Furthermore, we derive new reconstruction formulas for obtaining the problem coefficients from the solution of the main equation and get stability estimates for the recovered coefficients. [ABSTRACT FROM AUTHOR]

Dynamical systems are one of the interesting concepts where iteration algorithms and chaos can be considered together. In iteration algorithms, one of the basic concepts of fixed point theory, it is well known that the behavior of the iteration mechanism is chaotic if the original transformation is taken as chaotic. One of the natural ways to transform a chaotic system into a dynamical system is through control mechanisms. In this paper, we first consider an iteration class defined on Banach spaces, which is prominent in the literature in terms of both speed and convergence rate. Then, we consider the transformation constituting the iteration class as chaotic and obtain the stability and unstability behaviors of the iterations according to the operator norm by using Gâteaux and Fréchet derivatives representing the direction‐dependent derivative. In this way, we aimed to obtain the chaos control intervals obtained with functions in real space with the help of operators in Banach spaces. Using the operator norm obtained with the Fréchet derivative, we derive interesting dynamical system intervals from a chaotic system with parameter variables of iteration algorithms. It is noteworthy that the parameter variables of the studied iteration classes are the same in some transformation classes. In addition, analytical proofs are followed by computer simulations of parameter‐dependent control intervals considering the logistic operator with a chaotic structure. At the same time, the periodic behavior of the iteration algorithms used in our study is illustrated by the Lyapunov exponent with parameter‐dependent control intervals according to operator norm. Finally, as a real‐life problem, it has been shown that the chaos of the logistic population growth model with dynamic features can be controlled by chaos control mechanisms established by fixed point iteration methods. [ABSTRACT FROM AUTHOR]

We present a complete characterization of the right-symmetric points in the one sum of two Banach spaces. We also obtain some basic properties of the left-symmetric (right-symmetric) points in the p sum, $ 1\leq p\not =2 \lt \infty $ 1 ≤ p ≠ 2 < ∞ ( $ 1 \lt p\not =2 \lt \infty $ 1 < p ≠ 2 < ∞), of two Banach spaces. Using these properties we (a) give examples of Banach spaces which do not have any non-zero left-symmetric points and (b) prove a complete characterization of those left-symmetric and right-symmetric points in the p sum, $ 1 \lt p\not =2 \lt \infty $ 1 < p ≠ 2 < ∞ , of two Banach spaces, whose components satisfy an additional norm assumption. We give examples of Banach spaces where all non-zero left-symmetric or right-symmetric points satisfy this additional norm assumption. We also present an alternative proof of the recently obtained characterization of the left-symmetric and the right-symmetric points in $ \ell _p^n $ ℓ p n , $ n\geq 3 $ n ≥ 3 , and $ \ell _p $ ℓ p , $ 1\leq p\not =2 \lt \infty $ 1 ≤ p ≠ 2 < ∞. [ABSTRACT FROM AUTHOR]

We introduce and study m-heavy subspaces of a Banach space and obtain a complete analytic characterization of the same. The importance of m-heavy subspaces in studying the geometry of a Banach space is illustrated by establishing its various applications, including a strengthening of the classical Birkhoff–James orthogonality, duality theory and linear Hahn–Banach extension operators. [ABSTRACT FROM AUTHOR]

We introduce the modified Picard–Ishikawa hybrid iterative scheme and establish some strong convergence results for the class of asymptotically generalized ϕ-pseudocontractive mappings in the intermediate sense in Banach spaces and approximate the fixed point of this class of mappings via the newly introduced iteration scheme. We construct some numerical examples to support our results. Furthermore, we apply the Picard–Ishikawa hybrid iteration scheme in solving the nonlinear Caputo type fractional differential equations. Our results generalize, extend and unify several existing results in literature. [ABSTRACT FROM AUTHOR]

Let X,Y$X, Y$ be two Banach spaces, F:X→Y$F:X\rightarrow Y$ be an ε$\varepsilon$‐norm‐additive map for some ε⩾0$\varepsilon \geqslant 0$, that is, |∥F(x)+F(y)∥−∥x+y∥|⩽ε,for allx,y∈X.$$\begin{equation*} \big\vert \Vert F(x)+F(y)\Vert -\Vert x+y\Vert \big\vert \leqslant \varepsilon, {\text{\,for all\,}} x,y\in X. \end{equation*}$$In this paper, we prove that if F$F$ is surjective with F(0)=0$F(0)=0$, then there exists a linear surjective isometry U:X→Y$U:X\rightarrow Y$ such that ∥F(x)−U(x)∥⩽32ε,for allx∈X.$$\begin{equation*} \Vert F(x)-U(x)\Vert \leqslant \frac{3}{2}\varepsilon, {\text{\,for all\,}} x\in X. \end{equation*}$$The estimate 32ε$\frac{3}{2}\varepsilon$ is sharp. We also approximate standard surjective ε$\varepsilon$‐norm‐additive maps between the positive cones of continuous function spaces by linear isometries within a sharp approximation error 32ε$\frac{3}{2}\varepsilon$. [ABSTRACT FROM AUTHOR]

This paper studies the basic topological properties of weighted composition operators on the weighted sequence spaces , with , given by a weight sequence of positive reals such as boundedness, compactness, compactness of differences of two operators, formulas for their essential norms, and description of closed range operators. Previously these properties were studied by Luan and Khoi in the case of Hilbert space . Their methods can be also applied with some minor modifications to the case of Banach spaces with . They based essentially on using the dual spaces of continuous linear functionals and, consequently, cannot be applied to the quasi-Banach case . Moreover, some of them do not work even in . Motivated by these reasons, we develop a more universal approach that allows studying the whole scale of spaces . To this end, we establish the necessary and sufficient conditions for a linear operator to be compact on an abstract quasi-Banach sequence space. These conditions are new even in the case of Banach spaces. Moreover, we introduce the new characteristic, the -essential norm of a continuous linear operator on a quasi-Banach space . This characteristic measures the distance in the operator metric, between and the set of all -compact operators on . Here an operator is -compact on if is compact and coordinatewise continuous on . We show that for with the essential and -essential norms of a weighted composition operator coincide, whereas for we do not know whether the same is true or not. Our main results for weighted composition operators in are as follows: We provide criteria for an operator to be bounded, compact, or closed range, and completely describe the pairs of operators with compact difference; as well as some exact formula for the -essential norm. Some key aspects of our approach can be used for other operators and scales of spaces. [ABSTRACT FROM AUTHOR]

We obtain different results about the mean ergodicity of weighted composition operators when acting on the spaces H(B)$H(B)$, Hb(B)$H_b(B)$, and H∞(B)$H^\infty (B)$, where B$B$ is the open unit ball of a Banach space, as well as about the compactness and the mean ergodicity of the multiplication operator. This study relates these properties of the operators with properties of the symbol and the weight defining such operators. [ABSTRACT FROM AUTHOR]

We study families of analytic semigroups, acting on a Banach space, and depending on a parameter, and give sufficient conditions for existence of uniform with respect to the parameter norm bounds using spectral properties of the respective semigroup generators. In particular, we use estimates of the resolvent operators of the generators along vertical segments to estimate the growth/decay rate of the norm for the family of analytic semigroups. These results are applied to prove the Lyapunov linear stability of planar traveling waves of systems of reaction–diffusion equations, and the bidomain equation, important in electrophysiology. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the extrapolation theory of Rubio de Francia on general function spaces. We present endpoint extrapolation results including A1$A_1$, Ap$A_p$, and A∞$A_\infty$ extrapolation in the context of Banach function spaces, and also on modular spaces. We also include several applications that can be easily obtained using extrapolation: local decay estimates for various operators, Coifman–Fefferman inequalities that can be used to show some known sharp A1$A_1$ inequalities, Muckenhoupt–Wheeden and Sawyer's conjectures are also presented for many operators, which go beyond Calderón–Zygmund operators. Finally, we obtain two‐weight inequalities for Littlewood–Paley operators and Fourier integral operators on weighted Banach function spaces. [ABSTRACT FROM AUTHOR]

In this article, we address a problem posed by Bayart regarding the existence of an infinite‐dimensional closed vector subspace (excluding the null operator) within the set of supercyclic operators on Banach spaces. We fully resolve this problem by establishing the existence of the closed subspace. Furthermore, we prove that the set of supercyclic operators on ℓ1$\ell _1$ contains, up to the null operator, an isometric copy of ℓ1$\ell _1$. [ABSTRACT FROM AUTHOR]

We devise an iterative algorithm incorporating inertial techniques to approximate the shared solution of a generalized equilibrium problem, a fixed point problem for a finite family of relatively nonexpansive multivalued mappings, and a variational inequality problem. Our discussion encompasses the strong convergence of the proposed algorithm and highlights specific outcomes derived from our theorem. Additionally, we provide a computational analysis to underscore the significance of our findings and draw comparisons. The results presented in this paper serve to extend and unify numerous previously established outcomes in this particular research domain. [ABSTRACT FROM AUTHOR]

Let 퐷 be the nonsplit quaternion algebra over Q p . We prove that a class of admissible unitary Banach space representations of D × of global origin are topologically of finite length. [ABSTRACT FROM AUTHOR]

The concepts of various kinds of asymptotically Toeplitz operators have been originally defined and studied for the Hardy–Hilbert space. The aim of this article is to extend these definitions to the case of operators acting on Banach spaces of holomorphic functions, including Hardy spaces, Hardy–Lorentz spaces, and Hardy–Orlicz spaces. We give a function-theoretic characterization of uniformly asymptotically Toeplitzness and mean weakly asymptotically Toeplitzness for weighted composition operators. We also investigate strong and weak asymptotic Toeplitzness of weighted composition operators. [ABSTRACT FROM AUTHOR]

In this article, we consider two parametric classes of derivative free iterative methods of high order of convergence in Banach spaces. We study local and semilocal convergence results in Banach spaces for both families of iterative processes. We carry out a numerical application to solve a nonlinear and nondifferentiable Fredholm integral equation in the Banach space of continuous functions with the maximum norm. [ABSTRACT FROM AUTHOR]

In this paper, we present a new characterization for finding a solution of common fixed points problem involving multivalued and single-valued mappings in Banach spaces by applying the properties of composition without commuting assumption. We equally obtain a convergence result to a solution of system of inclusion problems in Banach spaces. Our results unify, generalize and complement various known comparable results from the current literature. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the strong convergence of the iteration scheme given by Akutsah et al., for mean nonexpansive mappings in uniformly convex Banach space endowed with directed graph. Also, we show fastness of Akutsah et al. iteration scheme with some well-known iteration schemes with the help of numerical examples. We, also present an application of fixed point theory in solution of integral equation. [ABSTRACT FROM AUTHOR]

We consider a general class of convex optimization problems in which one seeks to minimize a strongly convex function over a closed and convex set which is by itself an optimal set of another convex problem in Banach space. Regularized forward–backward splitting method is applied to find the minimum like-norm solution of the minimization problem under investigation. We also introduce a gradient-based method, called the minimal like-norm gradient method, for solving this class of problems and establish the convergence of the sequence generated by the algorithm as well as a rate of convergence of the sequence of function values. [ABSTRACT FROM AUTHOR]

In this paper, we present a new analytical scheme to generate atomic solutions of partial differential equations. The theory of tensor product in Banach spaces coupled with some features of atomics operators are utilized to attain our results. Some demonstrative examples are given for completeness. [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes a mixed variational formulation for the Navier-Stokes equations with variable viscosity that depends nonlinearly on the velocity gradient. Differently from previous works in which augmented terms are added to the formulation, here we employ a technique that had been previously applied to the stationary Boussinesq problem and the Navier-Stokes equations with constant viscosity. Firstly, a modified pseudostress tensor is introduced involving the diffusive and convective terms and the pressure. Secondly, by using the incompressibility condition, the pressure is eliminated, and the gradient of velocity is incorporated as an auxiliary unknown to handle the aforementioned nonlinearity. As a consequence, a Banach spaces-based formulation is obtained, which can be written as a perturbed twofold saddle point operator equation. We address the continuous and discrete solvabilities of this problem by linearizing the perturbation and employing a fixed-point approach along with a particular case of a known abstract theory. Given an integer ℓ ≥ 0 , feasible choices of finite element subspaces include discontinuous piecewise polynomials of degree ≤ ℓ for each entry of the velocity gradient, Raviart-Thomas spaces of order ℓ for the pseudostress, and discontinuous piecewise polynomials of degree ≤ ℓ for the velocity as well. Finally, optimal a priori error estimates are derived, and several numerical results confirming in general the theoretical rates of convergence, and illustrating the good performance of the scheme, are reported. [ABSTRACT FROM AUTHOR]

The examination of nonlinear equations is essential in diverse domains such as science, business, and engineering because of the widespread occurrence of nonlinear phenomena. The primary obstacle in computational science is to create numerical algorithms that are both computationally efficient and possess a high convergence rate. This work tackles these problems by presenting a three-step nonlinear, time-efficient numerical method for solving nonlinear models. The selected method exhibits a convergence of sixth order and an efficiency index of 1.4310, with the added benefit of only requiring five function evaluations per iteration. This study diverges from earlier research by emphasizing the use of first-order derivatives instead of higher-order derivatives. As a result, the method becomes more versatile and applicable. It is possible to estimate solutions that are locally different in Banach spaces by looking at convergence on both local and semi-local levels. The stability and performance of the scheme are also checked by using polynomiography to do a visual analysis and compare it to other schemes in terms of convergence, speed, and CPU time. [ABSTRACT FROM AUTHOR]

Let $ (X, T) $ (X , T) be a linear dynamical system defined on a Banach space X and $ {L_T}:{K_{{X^ * }}} \to {K_{{X^ * }}} $ L T : K X ∗ → K X ∗ be defined by $ {L_T}S = TS $ L T S = TS , where $ {K_{{X^ * }}} = \overline {span\{ \langle \cdot,{y^ * } \rangle x:{y^ * } \in {X^ * },x \in X\} } $ K X ∗ = span { 〈 ⋅ , y ∗ 〉 x : y ∗ ∈ X ∗ , x ∈ X } ¯ . This paper first obtains the equivalence of the (syndetically) weak multi-sensitive compactness between $ (X, T) $ (X , T) and $ ({K_{{X^ * }}},{L_T}) $ (K X ∗ , L T). Then, it is shown that (1) there exists a non-mean-sensitive linear dynamical system $ (X ,T) $ (X , T) satisfying that $ ({K_{{X^ * }}},{L_T}) $ (K X ∗ , L T) is mean sensitive; (2) There exists a non-topologically-transitive and weakly multi-sensitive compact system, which is not syndetically weakly multi-sensitive compact. Besides, it is proved that every weakly mixing linear dynamical system defined on a separable Banach space is weakly multi-sensitive compact. [ABSTRACT FROM AUTHOR]

Let X be a completely regular Hausdorff space, let E and F denote Banach spaces. Let C_b(X,E) denote the space of E-valued bounded continuous functions on X and let \beta be the strict topology on this space. We establish the relationship between nuclear operators T:C_b(X,E)\rightarrow F between the locally convex space (C_b(X,E),\beta) and the Banach space F and their representing operator-valued Borel measures. [ABSTRACT FROM AUTHOR]

The purpose in this article is to discuss under what conditions a continuous pseudocontractive mapping, in particular a nonexpansive map, has a fixed point when its domain is a closed, convex and unbounded subset of a Banach space. In fact, we will give a characterization of this property. The Halpern iterative algorithm for nonexpansive mappings is also investigated. [ABSTRACT FROM AUTHOR]

Let BX be the unit ball of a complex Banach space X, which may be infinite dimensional. The authors characterize the boundedness of the product of the radial derivative operator and the weighted composition operator from the natural Bloch space (or the little Bloch-type space) into the weighted-type space (or the little weighted-type space) on BX. [ABSTRACT FROM AUTHOR]

The paper investigated the exact controllability of delayed fractional evolution systems of order α ∈ (1, 2) in abstract spaces. At first, the exact controllability result is obtained when the nonlinear term f is locally Lipschitz continuous. Then, the certain compactness conditions and the measure of noncompactness conditions were applied to demonstrate the exact controllability of the concerned problem. The discussion was based on the fixed point theorems and the cosine family theory. [ABSTRACT FROM AUTHOR]

The aim of this paper is to provide new results on the Wolf and Schechter essential pseudospectra of bounded linear operators on a Banach space. More precisely, we characterize the Wolf and Schechter essential pseudospectra by using the notion of Fredholm perturbation. Also, we state the condition under which the Wolf (respectively, Schechter) essential pseudospectrum of two different bounded linear operators coincides. Furthermore, we give some characterizations of the Wolf and Schechter essential pseudospectra of 3 × 3 upper triangular block operator matrices. [ABSTRACT FROM AUTHOR]

In the theory of Banach spaces, the normalized duality mapping assumes a pivotal role. The analytic depiction of this mapping holds paramount significance in the associated analysis. Given that Bochner spaces serve as foundational underpinnings in stochastic variational analysis and stochastic optimizations, delving into the analytic representations of the normalized duality mapping becomes imperative, especially in uniformly convex and uniformly smooth Bochner spaces. The study of the analytic representations of normalized duality mapping contributes to our understanding of various geometric properties inherent in Bochner spaces. Leveraging the analytic representation of the normalized duality mapping, we establish and substantiate certain non-convex properties linked to this mapping in uniformly convex and uniformly smooth Bochner spaces. [ABSTRACT FROM AUTHOR]

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Let 퐷 be the nonsplit quaternion algebra over Q p . We prove that a class of admissible unitary Banach space representations of D × of global origin are topologically of finite length. [ABSTRACT FROM AUTHOR]

Linear coefficient inverse problems for equations with the Dzhrbashyan–Nersesyan fractional derivative in Banach spaces are studied in the cases of a bounded and of a sectorial operator at the unknown function. The equation is endowed by the Dzhrbashyan– Nersesyan initial value conditions and the overdetermination condition, the form of which is different for the cases of a constant and variable unknown coefficient in the equation. In the case of a constant unknown coefficient we obtain a criterium for the well-posedness of the inverse problem in terms of the characteristic function of the inverse problem. In the case of a variable unknown coefficient, we establish the unique solvability of the problem. are proved. Some applications of the obtained results are discussed. [ABSTRACT FROM AUTHOR]

A dynamic direct and inverse problem for a one-dimensional system of Hopf type equations is considered. The solvability of the problem under consideration is established in the scale of Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the local and semilocal convergence of an iterative method for solving nonlinear systems of equations. We first establish the conditions under which these methods converge locally to the solution. Then, we extend our analysis to examine the semilocal convergence of these methods, considering their behavior when starting from initial guesses that are not necessarily close to the solution. Iterative approaches for solving nonlinear systems of equations must take into account the radius of convergence, computable upper error bounds, and the uniqueness of solutions. These points have not been addressed in earlier studies. Moreover, we provide numerical examples to demonstrate the theoretical findings and compare the performance of these methods under different circumstances. Finally, we conclude that our examination offers a significant understanding of the convergence characteristics of previous iterative techniques for solving nonlinear equation systems. [ABSTRACT FROM AUTHOR]

We explore an oscillator with nonlinear functions and equilibrium lines that displays chaos. The equilibrium stability and complexity of the oscillator have been analysed and investigated. The presence of multiple equilibrium lines sets it apart from previously reported oscillators. The synchronization of the oscillator is considered as an application for secure communications. An observer is designed by considering a transmitted signal as a state, in other words, by injecting a linear function satisfying Lipschitz's condition to the proposed oscillator. Moreover, the adaptive control of the new oscillator is obtained. [ABSTRACT FROM AUTHOR]

In uniformly convex Banach spaces, we study within this research Fibonacci–Ishikawa iteration for monotone asymptotically nonexpansive mappings. In addition to demonstrating strong convergence, we establish weak convergence result of the Fibonacci–Ishikawa sequence that generalizes many results in the literature. If the norm of the space is monotone, our consequent result demonstrates the convergence type to the weak limit of the sequence of minimizing sequence of a function. One of our results characterizes a family of Banach spaces that meet the weak Opial condition. Finally, using our iterative procedure, we approximate the solution of the Caputo-type nonlinear fractional differential equation. [ABSTRACT FROM AUTHOR]

Predicting complex dynamics in physical applications governed by partial differential equations in real-time is nearly impossible with traditional numerical simulations due to high computational cost. Neural operators offer a solution by approximating mappings between infinite-dimensional Banach spaces, yet their performance degrades with system size and complexity. We propose an approach for learning neural operators in latent spaces, facilitating real-time predictions for highly nonlinear and multiscale systems on high-dimensional domains. Our method utilizes the deep operator network architecture on a low-dimensional latent space to efficiently approximate underlying operators. Demonstrations on material fracture, fluid flow prediction, and climate modeling highlight superior prediction accuracy and computational efficiency compared to existing methods. Notably, our approach enables approximating large-scale atmospheric flows with millions of degrees, enhancing weather and climate forecasts. Here we show that the proposed approach enables real-time predictions that can facilitate decision-making for a wide range of applications in science and engineering. Real-time prediction of dynamics for complex physical systems governed by partial differential equations is challenging and computationally expensive. The authors propose a framework for learning neural operators in latent spaces that allows real-time predictions of high-dimensional nonlinear systems. [ABSTRACT FROM AUTHOR]

In this paper, we provide some appropriate conditions for the existence of solutions for a perturbed fractional neutral integro-differential system under the deformable derivative in a Banach space. Using the Banach contraction principle and Krasnoselskii's fixed point theorem, we establish some new existence theorems. Moreover, we provide two numerical examples to demonstrate the applicability of the theoretical results [ABSTRACT FROM AUTHOR]

We investigate the roughness of nonuniform exponential dichotomies in Banach spaces subject to a new class of small linear time variable perturbations that satisfy an integral inequality which can benefit from a smallness integrability condition. We establish the continuous dependence of constants in terms of a dichotomy notion. Our proofs introduce a new development based on integral inequalities. Notably, we do not require the notion of admissibility for bounded nonlinear perturbations. Furthermore, we derive related roughness results for nonuniform exponential contractions and expansions. Our results are also new to uniform exponential dichotomy. We construct the evolution operator and projections directly, without the need for admissibility. [ABSTRACT FROM AUTHOR]

The study of the zero existence problem for a nonnegative set-valued functional on a metric space is continued. The zero existence problem for a functional related by a certain -continuity condition to a parametric family of -search functionals on an open subset of a metric space is examined. A theorem containing several sufficient conditions for this functional to have zeros is proved. As corollaries of this result, theorems on the existence of coincidence and fixed points are also proved for set-valued mappings related by the -continuity condition to families of set-valued mappings with the property that the existence of coincidence and fixed points in an open subset of a metric space is preserved under parameter variation. For uniformly convex metric spaces, analogs of M. Edelstein's 1972 asymptotic center theorem and M. Frigon's 1996 fixed point theorem for nonexpansive mappings of Banach spaces are obtained and compared with the main results of the paper. [ABSTRACT FROM AUTHOR]

In this paper we give growth estimates for ‖ T n ‖ for n → ∞ in the case T is a strongly Kreiss bounded operator on a UMD Banach space X. In several special cases we provide explicit growth rates. This includes known cases such as Hilbert and L p -spaces, but also intermediate UMD spaces such as non-commutative L p -spaces and variable Lebesgue spaces. [ABSTRACT FROM AUTHOR]

In this article, we establish a version of Banach’s contraction principle in ordered Banach spaces. This version is adapted to prove existence and uniqueness results for an integral equation or a boundary value problem depending on the derivative. [ABSTRACT FROM AUTHOR]

A method without memory as well as a method with memory are developed free of derivatives for solving Banach space valued equations. Their convergence order was established in the scalar case using Taylor expansions and hypotheses on higher order derivatives not appearing on these methods. But this way their applicability is limited. That is why in this paper their ball convergence analysis is provided using only the divided differences of order one that actually appear on these methods. Moreover, we provide computable error distances and uniqueness of the solution results not given before. Our technique is very general, so it can be used to extend the applicability of other methods along the same lines. This way their applicability is expanded. Numerical experiments testing the convergence criteria complete this paper. [ABSTRACT FROM AUTHOR]

This work is devoted to establish the existence of multiple positive radial solutions to the following equations associated with certain boundary conditions. The Six Functionals Fixed Point Theorem serves as the basis for the technique. To illustrate the key findings, an application is given. [ABSTRACT FROM AUTHOR]

The main goal of this research article is to investigate the stability of generalized norm-additive functional equations. This study demonstrates that these equations are Hyers-Ulam stable for surjective functions from an arbitrary group G to a real Banach space B using the large perturbation method. Furthermore, hyperstability results are investigated for a generalized Cauchy equation. [ABSTRACT FROM AUTHOR]

We develop a new Kantorovich-like convergence analysis of Newton-type methods to solve nonsingular and singular nonlinear equations in Banach spaces. The outer or generalized inverses are exchanged by a finite sum of linear operators making the implementation of these methods easier than in earlier studies. The analysis uses relaxed generalized continuity of the derivatives of operators involved required to control the derivative and on real majorizing sequences. The same approach can also be implemented on other iterative methods with inverses. The examples complement the theory by verifying the convergence conditions and demonstrating the performance of the methods. [ABSTRACT FROM AUTHOR]

Dynamical systems subject to perturbations that decay over time are relevant in describing many physical models, e.g. when considering the effect of a laser pulse on a molecule, in epidemiological studies, and celestial mechanics. For this purpose, we consider time-dependent Hamiltonian vector fields that are the sum of two components. The first has an invariant torus supporting quasiperiodic solutions, and the second decays as time tends to infinity. The time decay is modelled by functions satisfying suitable conditions verified by a proper polynomial decay in time. We prove the existence of orbits converging as time tends to infinity to the quasiperiodic solutions associated with the unperturbed system. The proof of this result relies on a new strategy based on a refined analysis of the Banach spaces and the functionals involved in the resolution of suitable nonlinear invariant equations. This result is proved for finite differentiable and real-analytic Hamiltonians. Analogous statements for time-dependent vector fields on the torus are also obtained as a corollary. These results extend a previous work of Canadell and de la Llave, where only exponential decay in time is considered. The relaxation of the decay in time makes the results in the present paper suited for applications in many physical problems, such as celestial dynamics. [ABSTRACT FROM AUTHOR]

We present a result concerning the essential norm of linear operators mapping some general weighted‐type spaces of holomorphic functions on the open unit ball in ℂn$$ {\mathrm{\mathbb{C}}}&amp;#x0005E;n $$ to Banach spaces. As some applications, we compare essential norms of a product‐type operator of integral type and a weighted composition operator between some of the general weighted‐type spaces. [ABSTRACT FROM AUTHOR]

We give an example of a pair of real Banach spaces such that they are neither linearly isomorphic nor isomorphic with respect to the structure of Birkhoff–James orthogonality, but have mutually homeomorphic geometric structure spaces. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate a class of nonconvex-constrained differential hemivariational inequalities consisting of nonlinear evolution equations and evolutionary hemivariational inequalities. The admissible set of constraints is closed and star-shaped with respect to a certain ball in a reflexive Banach space. We construct an auxiliary inclusion problem and obtain the existence results by applying a surjectivity theorem for multivalued pseudomonotone operators and the properties of Clarke subgradient operator. Moreover, the existence of a solution to the original problem is established by hemivariational inequality approach and a penalization method in which a small parameter does not have to tend to zero. Finally, an application of the main results is provided. [ABSTRACT FROM AUTHOR]

In this paper, an inertial subgradient extragradient algorithm is proposed to solve the pseudomonotone variational inequality problems in Banach space. This iterative scheme employs a new line-search rule. Strong convergence theorems for the proposed algorithms are established under the assumptions that the operators are non-Lipschitz continuous. Furthermore, several numerical experiments are given to show that our method has better convergence performance than the known ones in the literatures. [ABSTRACT FROM AUTHOR]