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Journal articles on the topic 'Fréchet derivative operators'

Author: Grafiati

Published: 4 June 2021

Last updated: 13 February 2022

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1

CHUNG, B. K., K. G. JOO, and SOONKEON NAM. "HAMILTONIAN FORMULATION OF SL(3) Ur-KdV EQUATION." Modern Physics Letters A 08, no. 31 (1993): 2927–36. http://dx.doi.org/10.1142/s0217732393003342.

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We give a unified view of the relation between the SL(2) KdV, the mKdV, and the Ur-KdV equations through the Fréchet derivatives and their inverses. For this we introduce a new procedure of obtaining the Ur-KdV equation, where we require that it has no nonlocal operators. We extend this method to the SL(3) KdV equation, i.e. Boussinesq (Bsq) equation and obtain the Hamiltonian structure of Ur-Bsq .equation in a simple form. In particular, we explicitly construct the Hamiltonian operator of the Ur-Bsq system which defines the Poisson structure of the system, through the Fréchet derivative and i

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2

Mursaleen, M., S. A. Mohiuddine, Q. M. Danish Lohani, and M. Farhan Khan. "Nonlinear operators on fuzzy 2-normed spaces and Fréchet derivative." Journal of Intelligent & Fuzzy Systems 25, no. 4 (2013): 1043–51. http://dx.doi.org/10.3233/ifs-120709.

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3

Mursaleen, M., and S. A. Mohiuddine. "Nonlinear operators between intuitionistic fuzzy normed spaces and Fréchet derivative." Chaos, Solitons & Fractals 42, no. 2 (2009): 1010–15. http://dx.doi.org/10.1016/j.chaos.2009.02.041.

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4

Argyros, Ioannis K., and Santhosh George. "MODIFICATION OF THE KANTOROVICH-TYPE CONDITIONS FOR NEWTON'S METHOD INVOLVING TWICE FRECHET DIFFERENTIABLE OPERATORS." Asian-European Journal of Mathematics 06, no. 03 (2013): 1350026. http://dx.doi.org/10.1142/s1793557113500265.

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We expand the applicability of Newton's method for approximating a locally unique solution of a nonlinear equation in a Banach space setting. The nonlinear operator involved is twice Fréchet differentiable. We introduce more precise majorizing sequences than in earlier studied (see [Concerning the convergence and application of Newton's method under hypotheses on the first and second Fréchet derivative, Comm. Appl. Nonlinear Anal.11 (2004) 103–119; A new semilocal convergence theorem for Newton's method, J. Comp. Appl. Math.79 (1997) 131–145; A note of Kantorovich theorem for Newton iteration,

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5

Argyros, Gus I., Michael I. Argyros, Samundra Regmi, Ioannis K. Argyros, and Santhosh George. "On the Solution of Equations by Extended Discretization." Computation 8, no. 3 (2020): 69. http://dx.doi.org/10.3390/computation8030069.

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The method of discretization is used to solve nonlinear equations involving Banach space valued operators using Lipschitz or Hölder constants. But these constants cannot always be found. That is why we present results using ω− continuity conditions on the Fréchet derivative of the operator involved. This way, we extend the applicability of the discretization technique. It turns out that if we specialize ω− continuity our new results improve those in the literature too in the case of Lipschitz or Hölder continuity. Our analysis includes tighter upper error bounds on the distances involved.

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6

Singh, K., and R. K. Gupta. "On symmetries and invariant solutions of a coupled KdV system with variable coefficients." International Journal of Mathematics and Mathematical Sciences 2005, no. 23 (2005): 3711–25. http://dx.doi.org/10.1155/ijmms.2005.3711.

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We investigate symmetries and reductions of a coupledKdVsystem with variable coefficients. The infinitesimals of the group of transformations which leaves theKdVsystem invariant and the admissible forms of the coefficients are obtained using the generalized symmetry method based on the Fréchet derivative of the differential operators. An optimal system of conjugacy inequivalent subgroups is then identified with the adjoint action of the symmetry group. For each basic vector field in the optimal system, theKdVsystem is reduced to a system of ordinary differential equations, which is further stu

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7

Gilliam, D. S., T. Hohage, X. Ji, and F. Ruymgaart. "The Fréchet Derivative of an Analytic Function of a Bounded Operator with Some Applications." International Journal of Mathematics and Mathematical Sciences 2009 (2009): 1–17. http://dx.doi.org/10.1155/2009/239025.

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The main result in this paper is the determination of the Fréchet derivative of an analytic function of a bounded operator, tangentially to the space of all bounded operators. Some applied problems from statistics and numerical analysis are included as a motivation for this study. The perturbation operator (increment) is not of any special form and is not supposed to commute with the operator at which the derivative is evaluated. This generality is important for the applications. In the Hermitian case, moreover, some results on perturbation of an isolated eigenvalue, its eigenprojection, and i

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8

Argyros, Ioannis K., Ángel Alberto Magreñán, Lara Orcos, and Íñigo Sarría. "Unified Local Convergence for Newton’s Method and Uniqueness of the Solution of Equations under Generalized Conditions in a Banach Space." Mathematics 7, no. 5 (2019): 463. http://dx.doi.org/10.3390/math7050463.

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Under the hypotheses that a function and its Fréchet derivative satisfy some generalized Newton–Mysovskii conditions, precise estimates on the radii of the convergence balls of Newton’s method, and of the uniqueness ball for the solution of the equations, are given for Banach space-valued operators. Some of the existing results are improved with the advantages of larger convergence region, tighter error estimates on the distances involved, and at-least-as-precise information on the location of the solution. These advantages are obtained using the same functions and Lipschitz constants as in ea

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9

Argyros, Ioannis K., Yeol Cho, and Hongmin Ren. "Convergence of Halley’s method for operators with the bounded second Fréchet-derivative in Banach spaces." Journal of Inequalities and Applications 2013, no. 1 (2013): 260. http://dx.doi.org/10.1186/1029-242x-2013-260.

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10

Singh, Sukhjit, Eulalia Martínez, Abhimanyu Kumar, and D. K. Gupta. "Domain of Existence and Uniqueness for Nonlinear Hammerstein Integral Equations." Mathematics 8, no. 3 (2020): 384. http://dx.doi.org/10.3390/math8030384.

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In this work, we performed an study about the domain of existence and uniqueness for an efficient fifth order iterative method for solving nonlinear problems treated in their infinite dimensional form. The hypotheses for the operator and starting guess are weaker than in the previous studies. We assume omega continuity condition on second order Fréchet derivative. This fact it is motivated by showing different problems where the nonlinear operators that define the equation do not verify Lipschitz and Hölder condition; however, these operators verify the omega condition established. Then, the s

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11

Montagu, E. L., and John Norbury. "Solving nonlinear non-local problems using positive square-root operators." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, no. 2239 (2020): 20190817. http://dx.doi.org/10.1098/rspa.2019.0817.

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A non-constructive existence theory for certain operator equations L u = D u , using the substitution u = B 1 2 ξ with B = L −1 , is developed, where L is a linear operator (in a suitable Banach space) and D is a homogeneous nonlinear operator such that Dλu = λ α D u for all λ ≥ 0 and some α ∈ R , α ≠ ~1. This theory is based on the positive-operator approach of Krasnosel’skii. The method has the advantage of being able to tackle the nonlinear right-hand side D in cases where conventional operator techniques fail. By placing the requirement that the operator B must have a positive square root,

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12

Derbazi, Choukri, Zidane Baitiche, Mouffak Benchohra та G. N’Guérékata. "Existence, Uniqueness, and Mittag–Leffler–Ulam Stability Results for Cauchy Problem Involving ψ -Caputo Derivative in Banach and Fréchet Spaces". International Journal of Differential Equations 2020 (13 жовтня 2020): 1–16. http://dx.doi.org/10.1155/2020/6383916.

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Our aim in this paper is to investigate the existence, uniqueness, and Mittag–Leffler–Ulam stability results for a Cauchy problem involving ψ -Caputo fractional derivative with positive constant coefficient in Banach and Fréchet Spaces. The techniques used are a variety of tools for functional analysis. More specifically, we apply Weissinger’s fixed point theorem and Banach contraction principle with respect to the Chebyshev and Bielecki norms to obtain the uniqueness of solution on bounded and unbounded domains in a Banach space. However, a new fixed point theorem with respect to Meir–Keeler

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13

Mahale, Pallavi. "Simplified Iterated Lavrentiev Regularization for Nonlinear Ill-Posed Monotone Operator Equations." Computational Methods in Applied Mathematics 17, no. 2 (2017): 269–85. http://dx.doi.org/10.1515/cmam-2016-0044.

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AbstractMahale and Nair [12] considered an iterated form of Lavrentiev regularization for obtaining stable approximate solutions for ill-posed nonlinear equations of the form ${F(x)=y}$, where ${F:D(F)\subseteq X\to X}$ is a nonlinear monotone operator and X is a Hilbert space. They considered an a posteriori strategy to find a stopping index which not only led to the convergence of the method, but also gave an order optimal error estimate under a general source condition. However, the iterations defined in [12] require calculation of Fréchet derivatives at each iteration. In this paper, we co

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14

Sano, Takashi. "Fréchet derivatives for operator monotone functions." Linear Algebra and its Applications 456 (September 2014): 88–92. http://dx.doi.org/10.1016/j.laa.2013.05.018.

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15

Le Merdy, Christian, and Anna Skripka. "HIGHER ORDER DIFFERENTIABILITY OF OPERATOR FUNCTIONS IN SCHATTEN NORMS." Journal of the Institute of Mathematics of Jussieu 19, no. 6 (2019): 1993–2016. http://dx.doi.org/10.1017/s1474748019000033.

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We establish the following results on higher order ${\mathcal{S}}^{p}$-differentiability, $1<p<\infty$, of the operator function arising from a continuous scalar function $f$ and self-adjoint operators defined on a fixed separable Hilbert space:(i)$f$ is $n$ times continuously Fréchet ${\mathcal{S}}^{p}$-differentiable at every bounded self-adjoint operator if and only if $f\in C^{n}(\mathbb{R})$;(ii)if $f^{\prime },\ldots ,f^{(n-1)}\in C_{b}(\mathbb{R})$ and $f^{(n)}\in C_{0}(\mathbb{R})$, then $f$ is $n$ times continuously Fréchet ${\mathcal{S}}^{p}$-differentiable at every self-adjoin

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16

Argyros, Ioannis K., and Á. Alberto Magreñán. "Extending the convergence domain of Newton’s method for twice Fréchet differentiable operators." Analysis and Applications 14, no. 02 (2016): 303–19. http://dx.doi.org/10.1142/s0219530515500013.

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We present a semi-local convergence analysis of Newton’s method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Using center-Lipschitz condition on the first and the second Fréchet derivatives, we provide under the same computational cost a new and more precise convergence analysis than in earlier studies by Huang [A note of Kantorovich theorem for Newton iteration, J. Comput. Appl. Math. 47 (1993) 211–217] and Gutiérrez [A new semilocal convergence theorem for Newton’s method, J. Comput. Appl. Math. 79 (1997) 131–145]. Numerical examples wh

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17

MAAS, JAN, and JAN VAN NEERVEN. "ON THE DOMAIN OF NONSYMMETRIC ORNSTEIN–UHLENBECK OPERATORS IN BANACH SPACES." Infinite Dimensional Analysis, Quantum Probability and Related Topics 11, no. 04 (2008): 603–26. http://dx.doi.org/10.1142/s0219025708003245.

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We consider the linear stochastic Cauchy problem [Formula: see text] where A generates a C0-semigroup on a Banach space E, WH is a cylindrical Brownian motion over a Hilbert space H, and B: H → E is a bounded operator. Assuming the existence of a unique minimal invariant measure μ∞, let Lp denote the realization of the Ornstein–Uhlenbeck operator associated with this problem in Lp (E, μ∞). Under suitable assumptions concerning the invariance of the range of B under the semigroup generated by A, we prove the following domain inclusions, valid for 1 < p ≤ 2: [Formula: see text] Here [Formula:

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18

Finster, Felix, and Magdalena Lottner. "Banach manifold structure and infinite-dimensional analysis for causal fermion systems." Annals of Global Analysis and Geometry 60, no. 2 (2021): 313–54. http://dx.doi.org/10.1007/s10455-021-09775-4.

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AbstractA mathematical framework is developed for the analysis of causal fermion systems in the infinite-dimensional setting. It is shown that the regular spacetime point operators form a Banach manifold endowed with a canonical Fréchet-smooth Riemannian metric. The so-called expedient differential calculus is introduced with the purpose of treating derivatives of functions on Banach spaces which are differentiable only in certain directions. A chain rule is proven for Hölder continuous functions which are differentiable on expedient subspaces. These results are made applicable to causal fermi

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19

DE BEER, RICHARD J. "TAUBERIAN THEOREMS AND SPECTRAL THEORY IN TOPOLOGICAL VECTOR SPACES." Glasgow Mathematical Journal 55, no. 3 (2013): 511–32. http://dx.doi.org/10.1017/s0017089512000699.

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AbstractWe investigate the spectral theory of integrable actions of a locally compact abelian group on a locally convex vector space. We start with an analysis of various spectral subspaces induced by the action of the group. This is applied to analyse the spectral theory of operators on the space, generated by measures on the group. We apply these results to derive general Tauberian theorems that apply to arbitrary locally compact abelian groups acting on a large class of locally convex vector spaces, which includes Fréchet spaces. We show how these theorems simplify the derivation of Mean Er

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20

Badriev, I. B., V. Ju Bujanov, M. V. Makarov, and N. V. Kalacheva. "Gâteaux and Fréchet derivatives of the operator of geometrically nonlinear bending problem of sandwich plate." Journal of Physics: Conference Series 1158 (February 2019): 022015. http://dx.doi.org/10.1088/1742-6596/1158/2/022015.

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21

Dragomir, S. Silvestru. "Reverse Jensen Integral Inequalities for Operator Convex Functions in Terms of Fréchet Derivative." Bulletin of the Iranian Mathematical Society, January 11, 2021. http://dx.doi.org/10.1007/s41980-020-00482-7.

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22

Djordjević, Bogdan D. "Singular Lyapunov operator equations: applications to $$C^*-$$algebras, Fréchet derivatives and abstract Cauchy problems." Analysis and Mathematical Physics 11, no. 4 (2021). http://dx.doi.org/10.1007/s13324-021-00596-z.

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