Function Theorem Research Articles

Decorated Discrete Conformal Maps and Convex Polyhedral Cusps

Abstract We discuss a notion of discrete conformal equivalence for decorated piecewise Euclidean surfaces (PE-surface), that is, PE-surfaces with a choice of circle about each vertex. It is closely related to inversive distance and hyperideal circle patterns. Under the assumption that the circles are non-intersecting, we prove the corresponding discrete uniformization theorem. The uniformization theorem for discrete conformal maps corresponds to the special case that all circles degenerate to points. Our proof relies on an intimate relationship between decorated PE-surfaces, canonical tessellations of hyperbolic surfaces and convex hyperbolic polyhedra. It is based on a concave variational principle, which also provides a method for the computation of decorated discrete conformal maps.

A note on the mapping theorem of essential pseudospectra on a Banach space

UDC 517.98 The main goal of the paper is to determine some basic properties of the essential pseudospectrum of a bounded linear operator A defined on a Banach space X . We also prove two different versions of the essential pseudospectral mapping theorem.

Browder operators on quaternionic Hilbert space

In this paper, we study the notions of Browder operator and Browder S-spectrum of bounded right linear operator defined over the right quaternionic Hilbert space. Some properties of Browder operator and stability of the ascent, descent and Browder S-spectrum have been investigated in the right quaternionic setting. We also characterize the property of invariant Browder operators and study the spectral mapping theorem of Browder S-spectrum for self-adjoint operators in quaternionic setting. This investigation concludes by exploring the Browder S-spectrum of the sum of two bounded linear operators.

Milnor fibration theorem for differentiable maps

In Cisneros-Molina et al. (São Paulo J Math Sci, 2023. https://doi.org/10.1007/s40863-023-00370-y) it was proved the existence of fibrations à la Milnor (in the tube and in the sphere) for real analytic maps f:(Rn,0)→(Rk,0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:({\\mathbb {R}}^n,0) \\rightarrow ({\\mathbb {R}}^k,0)$$\\end{document}, where n≥k≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge k\\ge 2$$\\end{document}, with non-isolated critical values. In the present article we extend the existence of the fibrations given in Cisneros-Molina et al. (São Paulo J Math Sci, 2023. https://doi.org/10.1007/s40863-023-00370-y) to differentiable maps of class Cℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{\\ell }$$\\end{document}, ℓ≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell \\ge 2$$\\end{document}, with possibly non-isolated critical value. This is done using a version of Ehresmann fibration theorem for differentiable maps of class Cℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{\\ell }$$\\end{document} between smooth manifolds, which is a generalization of the proof given by Wolf (Michigan Math J 11:65–70, 1964) of Ehresmann fibration theorem. We also present a detailed example of a non-analytic map which has the aforementioned fibrations.

The Gevrey class implicit mapping theorem with application to UQ of semilinear elliptic PDEs

This paper is concerned with a regularity analysis of parametric operator equations with a perspective on uncertainty quantification. We study the regularity of mappings between Banach spaces near branches of isolated solutions that are implicitly defined by a residual equation. Under [Formula: see text]-Gevrey assumptions on the residual equation, we establish [Formula: see text]-Gevrey bounds on the Fréchet derivatives of the locally defined data-to-solution mapping. This abstract framework is illustrated in a proof of regularity bounds for a semilinear elliptic partial differential equation with parametric and random field input.

Fixed point theorems for continuous single-valued and upper semicontinuous set-valued mappings in $p$-vector and locally $p$-convex spaces

The goal of this paper is to establish a general fixed point theorem for compact single-valued continuous mappings in Hausdorff $p$-vector spaces, and a fixed point theorem for upper semicontinuous set-valued mappings in locally $p$-convex spaces for $p\in (0, 1]$. These results not only provide a solution to Schauder conjecture in the affirmative under the setting of $p$-vector spaces for compact single-valued continuous mappings, but also show the existence of fixed points for upper semicontinuous set-valued mappings defined on $s$-convex subsets in Hausdorff locally $p$-convex spaces, which would be fundamental for nonlinear functional analysis, where $s, p \in (0, 1]$.

Direct trajectory optimization of macro-micro robotic system using a Gauss pseudospectral framework

Trajectory planning is a crucial aspect of macro-micro robotic systems (MMRSs), especially when the system has high degrees of freedom (DOFs). In the field of robotic polishing, the MMRS is usually composed of an industrial robot and an end-effector, which is responsible for polishing force control. Therefore, the compliance of the macro-robot can be minimized by trajectory planning to reduce its impact on the micro-robot. This study proposes a trajectory planning strategy based on Gauss pseudospectral method for a 9-DOF MMRS. Different from traditional sequential solution strategies, it can be used to obtain an approximate global optimal trajectory. Firstly, the velocity-level kinematics model of MMRS is built, which comprehensively considers the workpiece placement pose and task redundancy. Secondly, an optimal control model for trajectory planning is developed through an effective variable allocation. On the premise of considering traditional trajectory smoothness constraints, a constraint on manipulability is additionally analyzed to avoid reaching a singular configuration during compliance optimization. Thirdly, a Gauss pseudospectral framework based on the optimal control model is proposed, and the costate mapping theorem is proved. The latter provides a theoretical basis for the efficiency and accuracy of the proposed method. Finally, comparison results demonstrate the effectiveness of the proposed method.

The Robin problems for the coupled system of reaction–diffusion equations

This article investigates the local well-posedness of Turing-type reaction–diffusion equations with Robin boundary conditions in the Sobolev space. Utilizing the Hadamard norm, we derive estimates for Fokas unified transform solutions for linear initial-boundary value problems subject to external forces. Subsequently, we demonstrate that the iteration map, defined by the unified transform solutions and incorporating nonlinearity instead of external forces, acts as a contraction map within an appropriate solution space. Our conclusive result is established through the application of the contraction mapping theorem.

Fixed points of Suzuki-generalized nonexpansive mappings in CATp(0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$CAT_p(0)$$\\end{document} metric spaces

In this work, we obtain fixed point theorems and convergence theorems for Suzuki-generalized nonexpansive mappings in complete CATp(0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$CAT_p(0)$$\\end{document} metric spaces for 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}. Our results extend and improve many results in the literature.

Results in Cone Metric Spaces and Related Fixed Point Theorems for Contractive Type Mappings With Application

In this article, we prove some new fixed point theorems for contractive type mappings in the setting of complete cone metric spaces and provide examples to illustrate the concepts and results developed in the article. We consider some consequences of our results to establish fixed point theorems in the context of cone metric spaces. As an application of our results, periodic point results for the contractive type mappings are proved.

Fixed point theorems for fuzzy contractions mappings in a dislocated b-metric spaces with applications

The purpose of this article, is to establish some fixed point results for fuzzy mappings in a complete dislocated b-metric space. Examples are provided to support results and concepts presented herein. As an application of our results, multivalued mappings and fuzzy mappings are studied.

Fixed Point Theorems for Non-self Mappings Using Disconnected Graphs

Objectives: To prove the fixed point theorems for non-self mappings using disconnected graphs. Method: Graph theoretical approach is adopted to prove the fixed point theorems for non-self mappings. In all the previous works, connected graphs were used for establishing the results, but it is demonstrated in this work that disconnected graphs are best suited, and this new approach simplifies the proofs to a greater extent. Findings: The fixed point theorems by Banach, Kannan, Chatterjea, and Bianchini are proved using the new methodology. Novelty: An important part of the results concerning fixed point theorems is proving the iterated sequence to be a Cauchy sequence, and this is amalgamated with the edge sequence of the disconnected graph. Subject Classification: 54H25, 47H10 Keywords: Non-self mapping, Iterated sequence, Disconnected graph, Edge sequence, Fixed point

Generalized fixed points for fuzzy and nonfuzzy mappings in strong b-metric spaces

The main purpose of this research article is to generalize Kannan-type fixed-point (FP) theorems for single-valued mappings and Chatterjea-type FP result for fuzzy mappings (FMs) in the context of complete strong b-metric spaces (MSs). Moreover, fuzzy FPs are established for Suzuki-type fuzzy contraction in the setting of complete strong b-MSs. The conclusions are supported by nontrivial examples to enhance the validity of the results obtained in this study. In addition, previous findings have been made as corollaries from the relevant literature. The numerous implications that this technique has across the literature improve and integrate our findings. Applications of some of the results obtained are also incorporated.

The Borsuk-Ulam Theorem for n-valued maps between surfaces

The Borsuk-Ulam Theorem for n-valued maps between surfaces

Landau-Bloch type theorem for elliptic and quasiregular harmonic mappings

In this paper, we establish a coefficient bound for quasiregular and elliptic harmonic mappings and using these bounds, we establish Landau-Bloch type theorem for (K,K′)-elliptic and K-quasiregular harmonic mappings in the plane. Furthermore, we prove the coefficient estimates for K-quasiconformal harmonic self maps defined on the unit disc D.

On semi-continuity and continuity of the smallest and largest minimizing point of real convex functions with applications in probability and statistics

We prove that the smallest minimizer σ(f)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma (f)$$\\end{document} of a real convex function f is less than or equal to a real point x if and only if the right derivative of f at x is non-negative. Similarly, the largest minimizer τ(f)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au (f)$$\\end{document} is greater than or equal to x if and only if the left derivative of f at x is non-positive. From this simple result we deduce measurability and semi-continuity of the functionals σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document} and τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}. Furthermore, if f has a unique minimizing point, so that σ(f)=τ(f)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma (f)=\ au (f)$$\\end{document}, then the functional is continuous at f. With these analytical preparations we can apply Continuous Mapping Theorems to obtain several Argmin theorems for convex stochastic processes. The novelty here are statements about classical distributional convergence and almost sure convergence, if the limit process does not have a unique minimum point. This is possible by replacing the natural topology on R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}$$\\end{document} with the order topologies. Another new feature is that not only sequences but more generally nets of convex stochastic processes are allowed.

Stability in nonlinear integro-differential equations with functional delay on Time scale

The main purpose of this work is to show the stability of the zero solution for a non-linear neutral differential equation on space called the Time scale. In order to achieve our purposes, we assume a time scale , that is unbounded above and below, and we employ the contractive mapping theorem by converting the neutral differential equation into an equivalent integral equation, which allows us to reach the desired result of the asymptotic stability of the zero solution on Time scale provided that

A New faster Iteration Process to fixed points of mean-non expansive Mappings in Banach Space

Let X be a Banach space and F a subset of X. In this paper, we introduce a new iterative scheme to approximate fixed point of mean non-expansive mappings. We first prove that proposed iteration process is faster than all of Mann, Ishikawa, Picard, Agarwal, Noor and Thakur processes for contractive mappings. We also show that some weak and strong convergence theorems for mean non-expansive mappings. Using example presented in mean non-expansive mappings in Banach space. We compare the convergence behavior of the new iterative process with other iterative processes. Keywords. Mean non-expansive mappings, Iterative Process, Uniformly Convex Banach Space, Convergence Theorem.

Value iteration for LQR control of unknown stochastic-parameter linear systems

This paper focuses on the linear-quadratic optimal control problem for unknown stochastic-parameter linear systems using reinforcement learning methods. Based on the second moments of random system matrices, a model-based value iteration algorithm is proposed to solve the problem and is proved to be convergent by using the contraction mapping theorem. For the case without knowing any information about the random system matrices, a normalized model-free value iteration algorithm is presented to learn the optimal control law by estimating the data for the next time. The collected data are normalized first in our model-free algorithm to reduce the error of the least squares method. It is proved that our algorithm can obtain an approximate optimal solution. Our algorithm is applicable for any distribution of the random system matrices and does not require an initial mean-square stabilizing control policy. Finally, an example illustrates that our algorithm converges to an approximate optimal control policy and that the normalization step can significantly reduce convergence errors.

Results in Cone Metric Spaces and Related Fixed Point Theorems for Contractive Type Mappings

The purpose of this article, is to establish some fixed point results for contractive type mappings in cone metric spaces. Examples are provided to support results and concepts presented herein. As an application of our results, we deduce other established fixed point theorems in cone metric spaces.