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On the Generalized Method of Lines Applied to the Time-Independent Incompressible Navier-Stokes System
Published in Fabio Silva Botelho, Functional Analysis, Calculus of Variations and Numerical Methods for Models in Physics and Engineering, 2020
In the first part of this article, we obtain a linear system whose the solution solves the time-independent incompressible Navier-Stokes system for the special case in which the external forces vector is a gradient. In a second step we develop approximate solutions, also for the time independent incompressible Navier-Stokes system, through the generalized method of lines. We recall that for such a method, the domain of the partial differential equation in question is discretized in lines and the concerning solution is written on these lines as functions of the boundary conditions and boundary shape Finally, we emphasize these last main results are established through applications of the Banach fixed point theorem.
Algorithmic Optimization
Published in Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene, Modern Engineering Mathematics, 2017
Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene
Convergence analysis of iterative methods for both linear and nonlinear systems is based on the following important from mathematical analysis, known as the Banach Fixed Point Theorem or Contraction Mapping Theorem for a proof of this theorem we refer to [18] or [19].
Fixed Point Theory
Published in Eberhard Malkowsky, Vladimir Rakočević, Advanced Functional Analysis, 2019
Eberhard Malkowsky, Vladimir Rakočević
In this section we are going to study the famous Banach fixed point theorem, also commonly referred to as the Banach contraction principle. This principle of 1922 marks the beginning of fixed point theory in metric spaces.
A contribution to best proximity point theory and an application to partial differential equation
Published in Optimization, 2023
Sakan Termkaew, Parin Chaipunya, Dhananjay Gopal, Poom Kumam
Banach fixed point theorem (generalized fixed point theorem) is the most widely used analytical tool in solving various non-linear problems, such as integral equation, differential equation, functional equation etc. Since the solution of such equations can be found as the fixed point of corresponding self operator equation. Thus, it is desirable to have appropriate theory when the corresponding operator become non-self operator. This exactly evolute the concept of best proximity point. Let be two non-empty subsets of a metric space . A best proximity point of a non-self mapping is a point satisfying the equality , where . In the case, when V = U, we get i.e. becomes fixed point of T. Thus, every best proximity point is a natural generalization of fixed point. Hence to this reason many authors have studied on this topic (see for instance [1–15]).
Approximate controllability for semilinear second-order stochastic evolution systems with infinite delay
Published in International Journal of Control, 2020
This paper is organised as follows. The notations and terminologies concerning cosine operators and phase space for infinite delay used in the paper are presented in Section 2. In Section 3, we first discuss the fundamental solution of the corresponding deterministic linear system. Then, we apply it to define the mild solutions of System (1). After that, we also state two lemmas to be used in the proofs of the main results. In Section 4, we start by showing the existence and uniqueness of mild solution of System (1) making use of the Banach fixed point theorem. Then, we investigate its approximate controllability on and obtain the main result of this paper. Finally, in Section 5, an example is given to show the applications of the obtained results.