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Eigenvalue Assignment for Control of Time-Delay System
Published in Sambit Kumar Mishra, Zdzislaw Polkowski, Samarjeet Borah, Ritesh Dash, AI in Manufacturing and Green Technology, 2020
Jayanta Kumar Kar, Sovit Kumar Pradhan
where A and Ad are n × n matrices, and x(t) is a n × 1 state vector, and g(t) are specified preshape functions and an initial state defined in the Banach space, respectively. Informally, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit in the space. The existence and uniqueness of the solution for this system of linear DDE given in Equation (9.4) have been proved. A special case where the coefficient matrices A and Ad commute the solution of DDE is x(t)=Σk=−∞∞e(1hWk(Adhe−Ah)+A)tCkt
0 and Fixed Point Property
Published in Michael Ruzhansky, Hemen Dutta, Advanced Topics in Mathematical Analysis, 2019
Veysel Nezir, Nizami Mustafa, Hemen Dutta
It can be said that researches on fixed point theory started in 1912 by L.E.J. Brouwer’s Brouwer (1911) result: for n∈N, for C equal to the closed unit ball of Rn, every norm-to-norm continuous map f: C → C has a fixed point. His result was later extended to every compact convex subset of Rnand in 1930, Schauder Schauder (1930) extended to the same result to every Banach space. It was seen that the class of continuous maps was very large and fixed point theory was worked on smaller class of mappings. In 1922, the well known principle was introduced by Banach and he demonstrated so called Banach Contraction theorem Banach (1922): If (X, d) is a complete metric space, and f: X → X is a strict contraction, then f has a unique fixed point in X. Then, easily the following corollary was obtained in terms of Banach spaces: for a nonempty, closed, bounded, and convex subset C of a Banach space (X, ||⋅||), if T: C → C is a strict contraction for the metric d = d||⋅|| generated by the norm, then T has a fixed point.
Function Estimation
Published in M. Necati Özisik, Helcio R. B. Orlande, Inverse Heat Transfer, 2021
M. Necati Özisik, Helcio R. B. Orlande
A Hilbert space is a Banach space in which the norm is given by an inner (or scalar) product 〈.,.〉, that is, ‖u‖=〈u,u〉1/2
An open mapping theorem for nonlinear operator equations associated with elliptic complexes
Published in Applicable Analysis, 2023
In contrast to [6] we consider this problem in special Sobolev–Bochner type spaces, cf. [9] for the de Rham complex on the torus in the case where i = 1. Namely, using the standard tools, such as the interpolation Gagliardo–Nirenberg inequalities, Gronwall type lemmas and Faedo–Galerkin approximations, see, for instance, [1,2], we show that suitable linearizations of our problem have one and only one solution and nonlinear problem (4) has at most one solution in the constructed spaces. Applying the implicit function theorem for Banach spaces we prove that the image of is open in selected spaces (thus, obtaining the so-called ‘open mapping theorem’). However, we do not discuss here a much more delicate question on the existence of solution to nonlinear problem (4) aiming at the maximal generality of the nonlinear term for the open mapping theorem to be true. It is worth to note that even for the existence of weak ( distributional) solutions to (4) one has to impose rather restrictive but reasonable additional assumptions on the nonlinearity .
Convergence results of nonlinear problems based on Tikhonov regularization method
Published in Applicable Analysis, 2022
Considering ill-posed problems in Banach spaces is a topic of current research. Banach spaces often feature better mathematical properties in practical applications. Therefore, it is of high practical and theoretical interest to develop a convergence analysis for regularization methods. The rate of convergence of regularized solutions to an exact solution depend on the smoothness of all involved quantities. Typically the operator of the underlying equation has to be differentiable, the spaces should be smooth and the exact solution has to satisfy some abstract smoothness assumptions with respect to the operator. Among them the last type of smoothness is usually expressed in form of source conditions, which is important to attain the convergent rates. We refer e.g. to [1–3] for some convergent rate results and to [4,5] for an iterative method for solving ill-posed problems in Banach spaces. However, most of the current researches consider the exactly known forward operator.
Re-examination of Bregman functions and new properties of their divergences
Published in Optimization, 2019
Daniel Reem, Simeon Reich, Alvaro De Pierro
As a result of Proposition 5.6, it is of interest to provide some examples of Banach spaces which have the component-* property. Immediate examples are all finite-dimensional Banach spaces. Below we provide an infinite-dimensional example. More precisely, we claim that any Banach space which is isomorphic to for some has the component-* property. To see this, we first recall the well-known fact that if and are isomorphic Banach spaces, that is, there is a continuous and invertible linear operator , then is continuous too, as a consequence of the open mapping theorem, and their duals and are isomorphic too via the adjoint operator : see, for instance, [69, p.478–479].