Landweber iteration for inverse problems using multiple repeated measurements data in Banach spaces

Abstract We show that every Banach space in which weakly compact sets are super weakly compact is automatically weakly sequentially complete answering a question from [11]. In the proof we show how to build a weakly compact set which is not super weakly compact from an arbitrary nontrivial weakly Cauchy sequence using the notion of a summing subsequence of Rosenthal or Singer.

We analyze a discrete-time, two-stage population model with migration between multiple spatial locations under periodic environmental conditions. The resulting system is formulated as a nonlinear matrix-vector recursion composed of irreducible linear parameter matrices and componentwise concave density-dependent functions (Beverton-Holt type). We establish sufficient conditions for the existence and global asymptotic stability of a nontrivial equilibrium in the interior of the positive cone. The analysis leverages the Cone Limit Set Trichotomy Theorem, together with a semigroup framework for monotone and strictly concave maps defined on an ordered Banach space.

Abstract This study investigates the existence and uniqueness of solutions for a coupled system of Langevin-type fractional differential equations featuring generalized Ψ-Caputo derivatives in arbitrary Banach spaces. While prior research has predominantly focused on finite-dimensional or specific function spaces, this work extends the framework to infinite-dimensional settings, offering a more versatile analytical approach. Uniqueness is established using Banach’s fixed-point theorem under Lipschitz-type conditions, while existence is proven via Monch’s fixed-point principle combined with the measure of noncompactness—a powerful tool for infinite-dimensional problems. Demonstrative examples, including cases in the space of null sequences, validate the theoretical framework. The findings enhance the study of fractional coupled systems, introducing a flexible and comprehensive approach that integrates diverse fractional operators and strengthens foundational results.

Abstract Given an open subset U of a complex Banach space E , a weight v on U , and a complex Banach space F , let $$\mathscr {H}_v^{\infty }(U,F)$$ H v ∞ ( U , F ) denote the Banach space of all weighted holomorphic mappings $$f:U\rightarrow F$$ f : U → F , under the weighted supremum norm $$\Vert f\Vert _v:=\sup \hspace{0.55542pt}\{v(x)\Vert f(x)\Vert \,{:}\, x\in U\}$$ ‖ f ‖ v : = sup { v ( x ) ‖ f ( x ) ‖ : x ∈ U } . In this paper, we introduce and study the class $$\Pi _p^{\mathscr {H}_v^{\infty }}(U,F)$$ Π p H v ∞ ( U , F ) of p -summing weighted holomorphic mappings. We prove that it is an injective Banach ideal of weighted holomorphic mappings. Variants for weighted holomorphic mappings of Pietsch Domination Theorem, Pietsch Factorization Theorem and Maurey Extrapolation Theorem are presented. We also identify the spaces of p -summing weighted holomorphic mappings from U into $$F^*$$ F ∗ under the norm $$\pi ^{\mathscr {H}_v^{\infty }}_p$$ π p H v ∞ with the duals of F -valued $$\mathscr {H}_v^{\infty }$$ H v ∞ -molecules on U under a suitable version $$d^{\mathscr {H}_v^{\infty }}_{p^*}$$ d p ∗ H v ∞ of the Chevet–Saphar tensor norms.

We develop new techniques for computing the metric entropy of ellipsoids -- with polynomially decaying semi-axes -- in Banach spaces. Besides leading to a unified and comprehensive framework, these tools deliver numerous novel results as well as substantial improvements and generalizations of classical results. Specifically, we characterize the constant in the leading term in the asymptotic expansion of the metric entropy of $p$-ellipsoids with respect to $q$-norm, for arbitrary $p,q \in [1, \infty]$, to date known only in the case $p=q=2$. Moreover, for $p=q=2$, we improve upon classical results by specifying the second-order term in the asymptotic expansion. In the case $p=q=\infty$, we obtain a complete, as opposed to asymptotic, characterization of metric entropy and explicitly construct optimal coverings. To the best of our knowledge, this is the first exact characterization of the metric entropy of an infinite-dimensional body. Application of our general results to function classes yields an improvement of the asymptotic expansion of the metric entropy of unit balls in Sobolev spaces and identifies the dependency of the metric entropy of unit balls in Besov spaces on the domain of the functions in the class. Sharp results on the metric entropy of function classes find application, e.g., in machine learning, where they allow to specify the minimum required size of deep neural networks for function approximation, nonparametric regression, and classification over these function classes.

A modified subgradient extragradient algorithm for solving variational inequality problems in reflexive Banach spaces

We investigate the space of bounded linear operators on a Banach space equipped with a norm which is equivalent to the operator norm such that the subspace of compact operators is an M-ideal. In particular, we show that the space of compact operators on ℓ p ( 1 < p < ∞ ) equipped with the numerical radius norm is an M-ideal whenever the numerical index of ℓ p is not 0 (for all values of p in the complex case, for p ≠ 2 in the real case). On the other hand, we show that the space of compact operators on a Banach space containing an isomorphic copy of ℓ 1 whose numerical index is greater than 1/2 is not an M-ideal. We also study the proximinality, the existence of farthest points and the compact perturbation property for the numerical radius.

This paper investigates the existence of positive solutions for a specific category of p-Laplacian tempered fractional differential equations in which the nonlinear term f contains an integral operator θ. By employing fixed point theorems for sum operators in partially ordered Banach spaces, together with Krasnosel’skii fixed point theorem, the existence of positive solutions is established. Moreover, iterative sequences are constructed to approximate the unique positive solution of the problem. Finally, three examples are presented to illustrate the main results.

We consider a history-dependent hemivariational inequality in a reflexive Banach space X, stated on the interval of time [ 0 , T ] with T>0 and governed by a time-dependent set of constraints. We use arguments of pseudomonotonicity, Mosco convergence and fixed point in order to provide the existence of a unique solution u ∈ C ( [ 0 , T ] ; X ) of the inequality, together with a pointwise convergence result. Next, under additional assumptions, we provide necessary and sufficient conditions which guarantee the uniform convergence of a sequence of functions { u n } ⊂ C ( [ 0 , T ] ; X ) to the solution u. We then introduce and study two well-posedness concepts for the corresponding inequality. Our results give rise to various applications. To provide an example, we illustrate their use in the study of a mathematical model which describes the equilibrium of a viscoelastic rod in contact with a rigid-deformable obstacle, the so-called foundation.

We introduce the two-Arens-Eells space and the bi-Lipschitz tensor product for two pointed metric spaces, extending the classical Lipschitz tensor product. We show that the space of two-Lipschitz operators admits a canonical linearization through these constructions. We obtain isometric identifications between bi-Lipschitz tensor products and tensor products of Lipschitz-free Banach spaces. As an application, we characterize integral Lip-linear operators in terms of injective bi-Lipschitz tensor norms.

This study introduces a novel inertial-type iteration algorithm based on the Normal S iteration for the class of almost contraction mappings in Banach spaces. Traditional fixed point iterations often suffer from slow convergence and high computational cost; to address these limitations, the proposed framework incorporates an adaptive inertial-type parameter. We establish strong convergence of the algorithm and derive explicit a posteriori error estimates under weak contractive conditions. In addition, we demonstrate the asymptotic equivalence of the NS inertial-type trajectories with the classical Normal S iteration, provide a comprehensive weak w2−stability analysis, and obtain sharp upper bounds for the data dependence problem. The practical performance of the algorithm is evaluated in two distinct computational domains: image deblurring via wavelet-based ℓ1 regularization and the generation of complex fractal patterns, including Julia and Mandelbrot sets. Numerical results show that the proposed inertial-type iteration algorithm significantly outperforms existing methods—such as Picard, Mann, Ishikawa, and standard Normal S iterations—achieving faster convergence, higher PSNR values in image restoration, and more stable basins of attraction in fractal visualizations. These findings highlight the effectiveness and versatility of the NS inertial-type iteration algorithm approach for both theoretical analysis and real-world applications.

We consider Banach spaces ℓp(C),1≤p&lt;∞, where the index set C is the classical Cantor set and study various groups of symmetries of ℓp(C), associated with the binary representation of C. The main purpose of the paper is the investigation of polynomials on ℓp(C) that are symmetric (i.e., invariant) with respect to the constructed groups G. We are interested in finding systems of generators of algebras of G-symmetric polynomials for different groups G and we discuss possible applications of G-symmetric polynomials to highly composite physical systems. The generators are useful for descriptions of spectra of algebras of G-symmetric analytic functions on ℓp(C), and for the construction of some nontrivial complex homomorphisms of these algebras. Finally, we establish the topological transitivity and hypercyclicity of some shift-like operators on ℓp(C) and its subspaces, and translation operators on algebras of symmetric analytic functions on ℓp(C).

Abstract In this work, we consider a class of linear ill-posed problems with operators that map from the sequence space ℓ r ( r ⩾ 1 ) into a Banach space and in addition satisfy a conditional stability estimate in the scale of sequence spaces ℓ q , q ⩾ 0 . For the regularization of such problems in the presence of deterministic noise, we consider variational regularization with a penalty functional either of the form R p = ‖ ⋅ ‖ p p for some p &gt; 0 or in form of the counting measure R 0 = ‖ ⋅ ‖ 0 . The latter case guarantees sparsity of the corresponding regularized solutions. In this framework, we present first stability and then convergence rates for suitable a priori parameter choices. The results cover the oversmoothing situation, where the desired solution does not belong to the domain of definition of the considered penalty functional. The analysis of the oversmoothing case utilizes auxiliary elements that are defined by means of hard thresholding. Such technique can also be used for post processing to guarantee sparsity. Some numerical illustrations are included.

Abstract In this work, we develop a new biological transmission model for Chagas disease. This model, set in two juxtaposed habitats with skew Brownian motion conditions at the interface, is composed of two reaction–diffusion equations and takes into account the sylvatic transmission. We write it as an abstract perturbed Cauchy problem using operator theory. Then, we show that the main operator, which models the dispersal process, generates an analytic semigroup in an adequate Banach space.

The aim of the present work is to give a mathematical underpinning for the use of quasi-probabilities and pseudo-metrics in infinite-dimensional Banach manifolds. The notion of a continuous binary structure is introduced. It is a triple consisting of a continuous symmetric bilinear form together with a pair of closed linear subspaces of a Banach space. Such binary structures are abundant in Hilbert spaces. In order to confirm their existence in arbitrary Banach spaces, the auxiliary notion is introduced of subspaces that are positive with respect to a given symmetric bilinear form. It is shown that any subspace which is maximally positive with respect to the bilinear form induces a continuous binary structure on the Banach space. The Wigner function of a system of quantum mechanical particles is treated as an example.

This paper proposes and studies a new class of nonlinear nonlocal problem driven by a tempered Caputo-type fractional derivative with respect to an arbitrary smooth kernel. The novelty lies in treating a single nonlocal integro-delay setting that simultaneously couples an arbitrary kernel, exponential tempering, a delayed state, a lower-order distributed fractional memory term, and multipoint nonlocal initial data, rather than introducing a new fractional operator. The resulting problem can be viewed as a rigorous well-posedness prototype for hereditary systems with delayed feedback, tempered memory, and nonlocal initialization. First, an equivalent Volterra integral equation is derived. Then, the existence and uniqueness of solutions are obtained by the Banach contraction principle in a suitable Banach space of continuous functions. Next, a Picard successive approximation procedure is introduced and shown to converge uniformly to the unique solution, together with an explicit a priori error estimate. Moreover, a continuous dependence result is proved with respect to perturbations in the initial constants, the multipoint coefficients, and the nonlinear term. Finally, the main results are illustrated with two examples enhanced by graphs of explicit Picard approximations and convergence tables.

This note focuses on contraction type mappings in 2-normed linear spaces with a graph. We extend Chatterjea, Kannan’s extensions and (N- S-G), (N-S-A-G) to build contractive mappings respectively for a single map and for two maps in 2-normed linear spaces which is complete (2-Banach space) via graph. The findings broaden, generalize, and augment previously established results in the literature. We outline the future scope of our determined outcomes at the end of this note.

This research paper demonstrates how to manage Caputo fractional neutral integro-differential equations which include both integral and nonlinear elements through a unified framework that models dynamic systems with memory-based dynamics. The research establishes sufficient conditions for controllability through fixed point theory in a Banach space framework which requires particular assumptions while the study focuses on the K1&lt;1 condition which leads to the existence of a controllable solution. The proposed criteria are demonstrated through a numerical example which tests the theoretical results. The real-world case study uses artificial neural network (ANN) technology to predict Litecoin prices through the application of the fractional controllability model which analyzes historical financial data. The hybrid framework enables precise forecasting of nonlinear time series because it combines fractional calculus mathematical principles with ANN learning abilities. The proposed method demonstrates its predictive efficiency. The method shows robust performance through experimental results using cross-validation and performance metrics. The proposed model demonstrates competitive performance while providing additional advantages such as incorporation of memory effects and theoretical controllability. The research establishes a novel connection between fractional dynamical systems and machine learning which serves as an essential tool for studying complicated systems in theoretical research and practical applications.

In this paper, we study iterative approximation of attractive points for finitely many families of generalized nonexpansive mappings in a uniformly convex Banach space. We introduce a new algorithm combining a viscosity step with an inertial extrapolation. Under suitable control of the inertial and viscosity parameters and standard conditions on the mappings, we prove that the generated sequence is bounded. We then show that every weak cluster point of the sequence is an attractive point common to all mapping families. The main result establishes that the sequence converges weakly to a unique such attractive point. This extends earlier results confined to two mappings by considering finite family. These findings confirm that the proposed viscosity-inertial iteration successfully approximates the common attractive point under the stated hypotheses. Overall, the work broadens convergence theory in Banach spaces by enabling new classes of algorithms for approximating solution points of generalized nonlinear problems.