Topological structure of solution sets to differential problems in Fréchet spaces (original) (raw)

Topological structure of solution sets: current results

2000

We shall present current results concerning Browder–Guptatype theorems, Banach principle for multivalued mappings and the inverse limit method including its applications to ordinary differential equations and differential inclusions. Consequently, Aronszajn’s-type topological characterization of the set of solutions for differential equations and inclusions is considered. Note that some new results of the above mentioned type will be discussed. AMS Subject Classification. 59B99, 54C60, 47H10, 54H25, 55M20, 34A60, 46A04, 55M25

Topological structure of solution sets for impulsive differential inclusions in Fr�chet spaces

Fuel and Energy Abstracts, 2011

In this paper, we consider the existence of solutions as well as the topological and geometric structure of solution sets for first-order impulsive differential inclusions in some Fréchet spaces. Both the initial and terminal problems are considered. Using ingredients from topology and homology, the topological structures of solution sets (closedness and compactness) as well as some geometric properties (contractibility, acyclicity, ARAR and RδRδ) are investigated. Some of our existence results are obtained via the method of taking the inverse system limit on noncompact intervals.

Fixed point theorems for maps on cones in Frechet spaces via the projective limit approach

2016

We present fixed point results for admissibly compact maps on cones in Fr´echet spaces. We first extend the Krasnosel’ski˘i fixed point theorem with order type cone-compression and cone-expansion conditions. Then, we extend the monotone iterative method to this context. Finally, we present fixed point results under a combination of the assumptions of the previous results. More precisely, we combine a cone-compressing or cone-extending condition only on one side of the boundary of an annulus with an assumption on the existence of an upper fixed point. In addition, we show that the usual monotonicity condition can be weaken

Ekeland's inverse function theorem in graded Fréchet spaces revisited for multifunctions

Journal of Mathematical Analysis and Applications

In this paper, we present some inverse function theorems and implicit function theorem for set-valued mappings between Fréchet spaces. The proof relies on Lebesgue's Dominated Convergence Theorem and on Ekeland's variational principle. An application to the existence of solutions of differential equations in Fréchet spaces with non-smooth data is given.

On the structure of the set of solutions of equations involving AAA-proper mappings

Transactions of the American Mathematical Society, 1974

Let X and Y be Banach spaces having complete projection schemes (say, for example, they have Schauder bases). We consider various properties of mappings T : D ⊂ X → Y T:D \subset X \to Y which are either Approximation-proper (A-proper) or the uniform limit of such mappings. In §1 general properties, including those of the generalized topological degree, of such mappings are discussed. In §2 we give sufficient conditions in order that the solutions of an equation involving a nonlinear mapping be a continuum. The conditions amount to requiring that the generalized topological degree not vanish, and that the mapping involved be the uniform limit of well structured mappings. We devote §3 to proving a result connecting the topological degree of an A-proper Fréchet differentiable mapping to the degree of its derivative. Finally, in §4, various Lipschitz-like conditions are discussed in an A-proper framework, and constructive fixed point and surjectivity results are obtained.

On the structure of attainable sets for generalized differential equations and control systems

Journal of Differential Equations, 1970

Section 1 deals with the study of properties of the set of solutions of (1) k E R(t), x(O) = 0, where the set valued map R is measurable with nonempty compact subsets (of a ball of finite radius in E") as values. This is equivalent to the study of solutions of a linear control system. If n/r, CLm[O, T] denotes the set of all measurable selections of R, and for T E n/r, , (Yr)(t) = si r(7) d7, then 4(MR) C C[O, T] is the space of all solutions of (1). One type of typical "cost functional" for an associated optimization problem is a continuous mapF: C[O, T]-+ El. An extension of Aumanns theorem is used, together with the Stone-Weierstrass theorem, to show that the set of F: CIO, T]-f E' such that F(Y(MR) is compact is dense in the space of all continuous maps from C[O, T]-+ E' with the uniform topology. The implications to optimal control problems are evident. Section 2 deals with the nonlinear problem (2) % E R(x), x(O) = 0, where R has values as in Section 1. Using the machinery of Section 1, the existence of solutions of Eq. (2) when R is Lipschitzian (a result of Filippov, 1966) is shown to be a trivial consequence of a fixed point theorem for contracting set valued mappings. If R is continuous and convex valued, the fact that the set of solutions in CIO, T] is compact and has compact fixed time cross section (Filippov-Roxin theorem) is also an immediate consequence of, now, the Bohnenblust-Karlin fixed point theorem for set valued maps. The remainder of Section 2 gives an example in which the set of points attainable by solutions of an equation of the form (2), at some time T > 0, is actually open! In fact, in this example R has the control representation R(x) = {f (x, u) : u E U} with U compact and f smooth. To construct this, every point of the boundary of the attainable set of the convexified problem is attainable only as a limit of "chattering solutions" of the original system. This is quite difficult to accomplish (in fact many people conjectured it was not possible).

On weighted inductive limits of spaces of Fréchet-valued continuous functions

Journal of the Australian Mathematical Society, 1991

In this article we continue the study of weighted inductive limits of spaces of Frechet-valued continuous functions, concentrating on the problem of projective descriptions and the barrelledness of the corresponding "projective hull." Our study is related to the work of Vogt on the study of pairs (E, F) of Frechet spaces such that every continuous linear mapping from E into F is bounded and on the study of the functor Ext 1 (E, F) for pairs (E, F) of Frechet spaces.

An inverse function theorem in Fréchet spaces

Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2011

I present an inverse function theorem for differentiable maps between Fréchet spaces which contains the classical theorem of Nash and Moser as a particular case. In contrast to the latter, the proof does not rely on the Newton iteration procedure, but on Lebesgue's dominated convergence theorem and Ekeland's variational principle. As a consequence, the assumptions are substantially weakened: the map F to be inverted is not required to be C 2 , or even C 1 , or even Fréchet-differentiable. 2 k := p1+...+pn≤p Ω ∂ p1+...+pn x ∂ p1 ω 1 ...∂ pn ω n 2 dω Date: October 14, 2010; to appear, Annales de l'Institut Henri Poincaré, Analyse Non Linéaire. Key words and phrases. Inverse function theorem, implicit function theorem, Fréchet space, Nash-Moser theorem. The author thanks Eric Séré, Louis Nirenberg, Massimiliano Berti and Philippe Bolle, who were the first to see this proof. Particular thanks go to Philippe Bolle for spotting a significant mistake in an earlier version, and to Eric Séré, whose careful reading led to several improvements and clarifications. Their contribution and friendship is gratefully acknowledged.

On ω-limit sets of ordinary differential equations in Banach spaces

Journal of Mathematical Analysis and Applications, 2010

We classify ω-limit sets of autonomous ordinary differential equations x = f (x), x(0) = x 0 , where f is Lipschitz, in infinite dimensional Banach spaces as being of three types I-III. Let S ⊂ X be a Polish subset of a Banach space X. S is of type I if there exists a Lipschitz function f and a solution x such that S = Ω(x) and x ∩ S = ∅. S is of type II if it has non-empty interior and there exists a Lipschitz function f and a solution x such that S = Ω(x). S is of type III if it has empty interior and x ⊂ S for every solution x (of an equation where f is Lipschitz) such that S = Ω(x). Our main results are the following: S is a type I set if and only if there exists an open and connected set U ⊂ X such that S ⊂ ∂U . Suppose that there exists an open separable and connected set U ⊂ X such that S = U . Then S is a type II set. Every separable Banach space with a Schauder basis contains a type III set. Moreover in all these results we show that in addition f may be chosen C k -smooth whenever the underlying Banach space is C k -smooth.