China
Africa
Explore chapters and articles related to this topic
Locally Convex Topological Vector Spaces
Published in Kenneth Kuttler, Modern Analysis, 2017
In the case where X is normed linear space, this fixed point theorem is called the Schauder fixed point theorem. As an example of the usefulness of this fixed point theorem, consider the following application to the theory of ordinary differential equations. In the context of this theorem, X = C (0, T; ℝn), a Banach space with norm given by
setting and preliminary applications
Published in Mircea Sofonea, Stanisław Migórski, Variational-Hemivariational Inequalities with Applications, 2017
Mircea Sofonea, Stanisław Migórski
(The Alternate Schauder Fixed Point Theorem) Let K be a nonempty convex compact subset of a Banach space X and Λ:K → X be a continuous operator such that Λ(K) ⊂ K. Then Λ has at least one fixed point.
On null Controllability of fractional stochastic differential system with fractional Brownian motion
Published in Journal of Control and Decision, 2021
Next we prove that the family is an equicontinuous family of functions. Let and such that then From the above fact, we see that tends to zero independently of as Thus, is both equicontinuous and bounded. By the Arzela-Ascoli theorem is precompact in X. Hence is a completely continuous operator on X. From the Schauder fixed point theorem, has a fixed point in . Any fixed point of is a mild solution of (11) on Therefore the system (11) is exact null controllable on .
Optimal feedback control problems for a semi-linear neutral retarded integro-differential system
Published in International Journal of Control, 2023
Hai Huang, Tingting Feng, Xianlong Fu
We prove this theorem by using Schauder fixed point theorem. Put and define the set by is obviously a bounded, closed and convex subset of . We then define the operator P on the set as and Then by Theorem 2.1 (i), (iii), , (7) and Hölder inequality we can estimate, for any , Thus which shows that P maps into itself. To prove that P is a compact operator we first show that P is continuous on . Let with for some , then Hence, Since by carrying on the similar estimations as above we may apply the Lebesgue dominated convergence theorem to get that as , i.e. P is continuous.
Hierarchic control for a nonlinear parabolic equation in an unbounded domain
Published in Applicable Analysis, 2021
We now consider a nonlinear map such that, for every , where are solutions of (122)–(123). Proving that S has a fixed point will allows us to say that is solution of (115) and consequently, will be sufficient to finish the proof of Theorem 1.2. To this end, we use the Schauder fixed-point theorem.