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Marek T. Malinowski - One of the best experts on this subject based on the ideXlab platform.
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Symmetric Fuzzy Stochastic Differential Equations with Generalized Global Lipschitz Condition
Symmetry, 2020Co-Authors: Marek T. MalinowskiAbstract:The paper contains a discussion on solutions to symmetric type of fuzzy stochastic differential equations. The symmetric equations under study have drift and diffusion terms symmetrically on both sides of equations. We claim that such symmetric equations have unique solutions in the case that equations’ coefficients satisfy a certain generalized Lipschitz Condition. To show this, we prove that an approximation sequence converges to the solution. Then, a study on stability of solution is given. Some inferences for symmetric set-valued stochastic differential equations end the paper.
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Fuzzy and Set-Valued Stochastic Differential Equations With Local Lipschitz Condition
IEEE Transactions on Fuzzy Systems, 2015Co-Authors: Marek T. MalinowskiAbstract:We are concerned with the fuzzy stochastic differential equations driven by multidimensional Brownian motion viewed as a tool used to describe the behavior of dynamic systems operating in fuzzy environments with stochastic noises. Under the uniform Lipschitz Condition, we prove the local uniqueness theorem for the solutions of fuzzy stochastic differential equations. Next we show, assuming the Lipschitz Condition is satisfied only locally, that these equations have a unique solution. The fact that the solution is bounded is also proved. We conclude the paper with a number of corresponding results holding for the deterministic fuzzy differential equations and set-valued stochastic differential equations with local Lipschitz Condition.
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Random fuzzy differential equations under generalized Lipschitz Condition
Nonlinear Analysis: Real World Applications, 2012Co-Authors: Marek T. MalinowskiAbstract:Abstract We present the studies on two kinds of solutions to random fuzzy differential equations (RFDEs). The different types of solutions to RFDEs are generated by the usage of two different concepts of fuzzy derivative in the formulation of a differential problem. Under generalized Lipschitz Condition, the existence and uniqueness of both kinds of solutions to RFDEs are obtained. We show that solutions (of the same kind) are close to each other in the case when the data of the equation did not differ much. By an example, we present an application of each type of solutions in a population growth model which is subjected to two kinds of uncertainties: fuzziness and randomness.
Yan-mei Kang - One of the best experts on this subject based on the ideXlab platform.
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Exponential synchronization of delayed neutral-type neural networks with Lévy noise under non-Lipschitz Condition
Communications in Nonlinear Science and Numerical Simulation, 2018Co-Authors: Yan-mei KangAbstract:Abstract In this paper, the exponential synchronization of stochastic neutral-type neural networks with time-varying delay and Levy noise under non-Lipschitz Condition is investigated for the first time. Using the general Ito’s formula and the nonnegative semi-martingale convergence theorem, we derive general sufficient Conditions of two kinds of exponential synchronization for the drive system and the response system with adaptive control. Numerical examples are presented to verify the effectiveness of the proposed criteria.
Xiaofeng Wang - One of the best experts on this subject based on the ideXlab platform.
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New criteria on event-triggered cluster synchronization of neutral-type neural networks with Lévy noise and non-Lipschitz Condition
Neurocomputing, 2020Co-Authors: Yuqing Sun, Yihong Zhang, Wuneng Zhou, Xin Zhang, Xiaofeng WangAbstract:Abstract In this paper, two kinds of the exponential cluster synchronization of stochastic coupled neutral-type neural networks with Levy noise under non-Lipschitz Condition are investigated. The non-Lipschitz Condition has much weaker requirement than the usual Lipschitz Condition, so the neuron activation functions have a wider range of options. Using the general Ito’s formula and the nonnegative semi-martingale convergence theorem, the general sufficient Conditions of two kinds of exponential synchronization are derived for the systems with an event-triggered pinning controller. A numerical example is presented to verify the effectiveness of the proposed criteria.
Ioannis K Argyros - One of the best experts on this subject based on the ideXlab platform.
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Newton's method
A Contemporary Study of Iterative Methods, 2018Co-Authors: Á. Alberto Magreñán, Ioannis K ArgyrosAbstract:The center Lipschitz Condition is used in this chapter, together with the Lipschitz Condition, in order to obtain weaker convergence criteria to ensure the convergence pf Newton's method. Numerical examples and applications validating the theoretical results are also presented.
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A New Semi-local Convergence Analysis of the Secant Method
International Journal of Applied and Computational Mathematics, 2017Co-Authors: Ioannis K Argyros, Ekaterina NathansonAbstract:We provide a new semi-local convergence analysis for the secant method in a Banach space setting. Argyros and other authors have analyzed the method using a Lipschitz Condition and a simple center Lipschitz Condition. However, the secant method has two starting vectors \(u_0\), \(u_{-1}\), and it makes sense to analyze it using a mixed center-Lipschitz Condition based on both vectors. The direct analysis of the majorizing sequence employed in this paper can be used to obtain weaker convergence Conditions than those used in earlier studies. A numerical example is given to further justify the theoretical results.
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Convergence of Halley's method under centered Lipschitz Condition on the second Fréchet derivative
2013Co-Authors: Ioannis K Argyros, Hongmin RenAbstract:We present a semi-local as well as a local convergence analysis of Halley's method for approximating a locally unique solution of a nonlinear equation in a Banach space setting. We assume that the second Frechet-derivative satisfies a centered Lipschitz Condition. Numerical examples are used to show that the new convergence criteria are satisfied but earlier ones are not satisfied.
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on the semi local convergence of halley s method under a center Lipschitz Condition on the second frechet derivative
Applied Mathematics and Computation, 2012Co-Authors: Hongmin Ren, Ioannis K ArgyrosAbstract:Abstract We expand the applicability of Halley’s method for solving nonlinear equations in a Banach space setting. We assume the existence of the center-Lipschitz Condition on the second Frechet-derivative of the operator involved instead of Lipschitz Condition used extensively in the literature [1] , [2] , [4] , [5] . The center-Lipschitz Condition is satisfied in many interesting cases, where the Lipschitz Condition is not satisfied [3] , [4] , [6] , [7] , [13] . We show that the semi-local convergence theorem established in [X.B. Xu, Y.H. Ling, Semilocal convergence for Halley’s method under weak Lipschitz Condition, Appl. Math. Comput. 215 (2009) 3057–3067] is not true. A new semi-local convergence theorem is established for Halley’s method under the same Condition. Our results are illustrated using a nonlinear Hammerstein integral equation of the second kind where our convergence criteria are satisfied but convergence criteria in earlier studies such as [1] , [2] are not satisfied.
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On the semi-local convergence of Halley’s method under a center-Lipschitz Condition on the second Fréchet derivative
Applied Mathematics and Computation, 2012Co-Authors: Hongmin Ren, Ioannis K ArgyrosAbstract:Abstract We expand the applicability of Halley’s method for solving nonlinear equations in a Banach space setting. We assume the existence of the center-Lipschitz Condition on the second Frechet-derivative of the operator involved instead of Lipschitz Condition used extensively in the literature [1] , [2] , [4] , [5] . The center-Lipschitz Condition is satisfied in many interesting cases, where the Lipschitz Condition is not satisfied [3] , [4] , [6] , [7] , [13] . We show that the semi-local convergence theorem established in [X.B. Xu, Y.H. Ling, Semilocal convergence for Halley’s method under weak Lipschitz Condition, Appl. Math. Comput. 215 (2009) 3057–3067] is not true. A new semi-local convergence theorem is established for Halley’s method under the same Condition. Our results are illustrated using a nonlinear Hammerstein integral equation of the second kind where our convergence criteria are satisfied but convergence criteria in earlier studies such as [1] , [2] are not satisfied.
Á. Alberto Magreñán - One of the best experts on this subject based on the ideXlab platform.
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starting points for newton s method under a center Lipschitz Condition for the second derivative
Journal of Computational and Applied Mathematics, 2018Co-Authors: José Antonio Ezquerro, M A Hernandezveron, Á. Alberto MagreñánAbstract:Abstract We analyze the semilocal convergence of Newton’s method under a center Lipschitz Condition for the second derivative of the operator involved different from that used by other authors until now. In particular, we propose to center the Lipschitz Condition for the second derivative in a different point from that where Newton’s method starts. This allows us to obtain different starting points for Newton’s method and modify the domain of starting points.
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Starting points for Newton’s method under a center Lipschitz Condition for the second derivative
Journal of Computational and Applied Mathematics, 2018Co-Authors: José Antonio Ezquerro, Miguel Ángel Hernández-verón, Á. Alberto MagreñánAbstract:Abstract We analyze the semilocal convergence of Newton’s method under a center Lipschitz Condition for the second derivative of the operator involved different from that used by other authors until now. In particular, we propose to center the Lipschitz Condition for the second derivative in a different point from that where Newton’s method starts. This allows us to obtain different starting points for Newton’s method and modify the domain of starting points.
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Newton's method
A Contemporary Study of Iterative Methods, 2018Co-Authors: Á. Alberto Magreñán, Ioannis K ArgyrosAbstract:The center Lipschitz Condition is used in this chapter, together with the Lipschitz Condition, in order to obtain weaker convergence criteria to ensure the convergence pf Newton's method. Numerical examples and applications validating the theoretical results are also presented.