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Search Results (6)

Search Parameters:
Keywords = Gateaux differential

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15 pages, 2229 KB  
Article
Convergence on Kirk Iteration of Cesàro Means for Asymptotically Nonexpansive Mappings
by Lale Cona and Deniz Şimşek
Symmetry 2025, 17(3), 393; https://doi.org/10.3390/sym17030393 - 5 Mar 2025
Viewed by 888
Abstract
This article addresses the convergence of iteration sequences in Cesàro means for asymptotically nonexpansive mappings. Specifically, this study explores the behavior of Kirk iteration in the Cesàro means in the context of uniformly convex and reflexive Banach spaces equipped with uniformly Gâteaux differentiable [...] Read more.
This article addresses the convergence of iteration sequences in Cesàro means for asymptotically nonexpansive mappings. Specifically, this study explores the behavior of Kirk iteration in the Cesàro means in the context of uniformly convex and reflexive Banach spaces equipped with uniformly Gâteaux differentiable norms. The focus is to determine the conditions under which the Kirk iteration sequence converges strongly or weakly to a fixed point. Finally, some examples are given in this article to demonstrate the advantages of the preferred iteration method and to verify the results obtained. Full article
(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points II)
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12 pages, 304 KB  
Article
Generalizations of Rolle’s Theorem
by Alberto Fiorenza and Renato Fiorenza
Mathematics 2024, 12(14), 2157; https://doi.org/10.3390/math12142157 - 10 Jul 2024
Viewed by 2747
Abstract
The classical Rolle’s theorem establishes the existence of (at least) one zero of the derivative of a continuous one-variable function on a compact interval in the real line, which attains the same value at the extremes, and it is differentiable in the interior [...] Read more.
The classical Rolle’s theorem establishes the existence of (at least) one zero of the derivative of a continuous one-variable function on a compact interval in the real line, which attains the same value at the extremes, and it is differentiable in the interior of the interval. In this paper, we generalize the statement in four ways. First, we provide a version for functions whose domain is in a locally convex topological Hausdorff vector space, which can possibly be infinite-dimensional. Then, we deal with the functions defined in a real interval: we consider the case of unbounded intervals, the case of functions endowed with a weak derivative, and, finally, we consider the case of distributions over an open interval in the real line. Full article
(This article belongs to the Special Issue Advanced Research in Functional Analysis and Operator Theory)
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23 pages, 357 KB  
Article
Existence and Properties of the Solution of Nonlinear Differential Equations with Impulses at Variable Times
by Huifu Xia, Yunfei Peng and Peng Zhang
Axioms 2024, 13(2), 126; https://doi.org/10.3390/axioms13020126 - 18 Feb 2024
Cited by 1 | Viewed by 1629
Abstract
In this paper, a class of nonlinear ordinary differential equations with impulses at variable times is considered. The existence and uniqueness of the solution are given. At the same time, modifying the classical definitions of continuous dependence and Gâteaux differentiability, some results on [...] Read more.
In this paper, a class of nonlinear ordinary differential equations with impulses at variable times is considered. The existence and uniqueness of the solution are given. At the same time, modifying the classical definitions of continuous dependence and Gâteaux differentiability, some results on the continuous dependence and Gâteaux differentiable of the solution relative to the initial value are also presented in a new topology sense. For the autonomous impulsive system, the periodicity of the solution is given. As an application, the properties of the solution for a type of controlled nonlinear ordinary differential equation with impulses at variable times is obtained. These results are a foundation to study optimal control problems of systems governed by differential equations with impulses at variable times. Full article
(This article belongs to the Special Issue Advances in Difference Equations)
11 pages, 241 KB  
Article
Relationship between Generalized Orthogonality and Gâteaux Derivative
by Peixuan Xu, Donghai Ji and Hongxu Zhang
Mathematics 2024, 12(3), 364; https://doi.org/10.3390/math12030364 - 23 Jan 2024
Viewed by 1068
Abstract
This paper investigates the relationship between generalized orthogonality and Gâteaux derivative of the norm in a normed linear space. It is shown that the Gâteaux derivative of x in the y direction is zero when the norm is Gâteaux differentiable in the y [...] Read more.
This paper investigates the relationship between generalized orthogonality and Gâteaux derivative of the norm in a normed linear space. It is shown that the Gâteaux derivative of x in the y direction is zero when the norm is Gâteaux differentiable in the y direction at x and x and y satisfy certain generalized orthogonality conditions. A case where x and y are approximately orthogonal is also analyzed and the value range of the Gâteaux derivative in this case is given. Moreover, two concepts are introduced: the angle between vectors in normed linear space and the ⊥Δ coordinate system in a smooth Minkowski plane. Relevant examples are given at the end of the paper. Full article
(This article belongs to the Section B: Geometry and Topology)
20 pages, 2155 KB  
Article
Generalized Inexact Newton-Landweber Iteration for Possibly Non-Smooth Inverse Problems in Banach Spaces
by Ruixue Gu, Hongsun Fu and Zhuoyue Wang
Mathematics 2023, 11(7), 1706; https://doi.org/10.3390/math11071706 - 3 Apr 2023
Viewed by 1589
Abstract
In this paper, we consider a generalized inexact Newton-Landweber iteration to solve nonlinear ill-posed inverse problems in Banach spaces, where the forward operator might not be Gâteaux differentiable. The method is designed with non-smooth convex penalty terms, including L1-like and total [...] Read more.
In this paper, we consider a generalized inexact Newton-Landweber iteration to solve nonlinear ill-posed inverse problems in Banach spaces, where the forward operator might not be Gâteaux differentiable. The method is designed with non-smooth convex penalty terms, including L1-like and total variation-like penalty functionals, to capture special features of solutions such as sparsity and piecewise constancy. Furthermore, the inaccurate inner solver is incorporated into the minimization problem in each iteration step. Under some assumptions, based on ε-subdifferential, we establish the convergence analysis of the proposed method. Finally, some numerical simulations are provided to illustrate the effectiveness of the method for solving both smooth and non-smooth nonlinear inverse problems. Full article
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13 pages, 401 KB  
Article
The Stability of the Aggregate Loss Distribution
by Riccardo Gatto
Risks 2018, 6(3), 91; https://doi.org/10.3390/risks6030091 - 5 Sep 2018
Cited by 2 | Viewed by 3194
Abstract
In this article we introduce the stability analysis of a compound sum: it consists of computing the standardized variation of the survival function of the sum resulting from an infinitesimal perturbation of the common distribution of the summands. Stability analysis is complementary to [...] Read more.
In this article we introduce the stability analysis of a compound sum: it consists of computing the standardized variation of the survival function of the sum resulting from an infinitesimal perturbation of the common distribution of the summands. Stability analysis is complementary to the classical sensitivity analysis, which consists of computing the derivative of an important indicator of the model, with respect to a model parameter. We obtain a computational formula for this stability from the saddlepoint approximation. We apply the formula to the compound Poisson insurer loss with gamma individual claim amounts and to the compound geometric loss with Weibull individual claim amounts. Full article
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