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C 1 -smoothness of Nemytskii operators on Sobolev-type spaces of periodic functions

Irina Kmit (2011)

Commentationes Mathematicae Universitatis Carolinae

We consider a class of Nemytskii superposition operators that covers the nonlinear part of traveling wave models from laser dynamics, population dynamics, and chemical kinetics. Our main result is the C 1 -continuity property of these operators over Sobolev-type spaces of periodic functions.

Calcul fonctionnel dans certains espaces de Besov

G. Bourdaud, D. Kateb (1990)

Annales de l'institut Fourier

On montre que les fonctions qui opèrent, par composition a gauche, sur l’espace de Besov d’exposant s , avec 0 < s < 1 / q , dans l’espace euclidien de dimension n , sont précisément les fonctions lipschitziennes.

Caristi's fixed point theorem and its equivalences in fuzzy metric spaces

Naser Abbasi, Hamid Mottaghi Golshan (2016)

Kybernetika

In this article, we extend Caristi's fixed point theorem, Ekeland's variational principle and Takahashi's maximization theorem to fuzzy metric spaces in the sense of George and Veeramani [A. George , P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems. 64 (1994) 395-399]. Further, a direct simple proof of the equivalences among these theorems is provided.

Caristi's fixed point theorem in probabilistic metric spaces

Kianoush Fathi Vajargah, Hamid Mottaghi Golshan, Abbas Arjomand Far (2021)

Kybernetika

In this work, we define a partial order on probabilistic metric spaces and establish some new Caristi's fixed point theorems and Ekeland's variational principle for the class of (right) continuous and Archimedean t-norms. As an application, a partial answer to Kirk's problem in metric spaces is given.

Cauchy problems in weighted Lebesgue spaces

Jan W. Cholewa, Tomasz Dłotko (2004)

Czechoslovak Mathematical Journal

Global solvability and asymptotics of semilinear parabolic Cauchy problems in ℝ n are considered. Following the approach of A. Mielke [15] these problems are investigated in weighted Sobolev spaces. The paper provides also a theory of second order elliptic operators in such spaces considered over ℝ n , n ∈ ℕ . In particular, the generation of analytic semigroups and the embeddings for the domains of fractional powers of elliptic operators are discussed.

Characterization of Globally Lipschitz Nemytskiĭ Operators Between Spaces of Set-Valued Functions of Bounded φ-Variation in the Sense of Riesz

N. Merentes, J. L. Sánchez Hernández (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

Let (X,∥·∥) and (Y,∥·∥) be two normed spaces and K be a convex cone in X. Let CC(Y) be the family of all non-empty convex compact subsets of Y. We consider the Nemytskiĭ operators, i.e. the composition operators defined by (Nu)(t) = H(t,u(t)), where H is a given set-valued function. It is shown that if the operator N maps the space R V φ ₁ ( [ a , b ] ; K ) into R W φ ₂ ( [ a , b ] ; C C ( Y ) ) (both are spaces of functions of bounded φ-variation in the sense of Riesz), and if it is globally Lipschitz, then it has to be of the form H(t,u(t)) = A(t)u(t)...

Characterizations of the Solution Sets of Generalized Convex Minimization Problems

Ivanov, Vsevolod (2003)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 90C26, 90C20, 49J52, 47H05, 47J20.In this paper we obtain some simple characterizations of the solution sets of a pseudoconvex program and a variational inequality. Similar characterizations of the solution set of a quasiconvex quadratic program are derived. Applications of these characterizations are given.

Characterizations of weakly compact sets and new fixed point free maps in c₀

P. N. Dowling, C. J. Lennard, B. Turett (2003)

Studia Mathematica

We give a basic sequence characterization of relative weak compactness in c₀ and we construct new examples of closed, bounded, convex subsets of c₀ failing the fixed point property for nonexpansive self-maps. Combining these results, we derive the following characterization of weak compactness for closed, bounded, convex subsets C of c₀: such a C is weakly compact if and only if all of its closed, convex, nonempty subsets have the fixed point property for nonexpansive mappings.

Classification of homotopy classes of equivariant gradient maps

E. N. Dancer, K. Gęba, S. M. Rybicki (2005)

Fundamenta Mathematicae

Let V be an orthogonal representation of a compact Lie group G and let S(V),D(V) be the unit sphere and disc of V, respectively. If F: V → ℝ is a G-invariant C¹-map then the G-equivariant gradient C⁰-map ∇F: V → V is said to be admissible provided that ( ∇ F ) - 1 ( 0 ) ∩ S ( V ) = ∅ . We classify the homotopy classes of admissible G-equivariant gradient maps ∇F: (D(V),S(V)) → (V,V∖0).

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