Some Remarks on Nonlinear Composition Operators in Spaces of Differentiable Functions

J. Appell; Z. Jesús; O. Mejía

Displaying similar documents to “Some Remarks on Nonlinear Composition Operators in Spaces of Differentiable Functions”

Blow-up of the solution to the initial-value problem in nonlinear three-dimensional hyperelasticity

J. A. Gawinecki, P. Kacprzyk (2008)

Applicationes Mathematicae

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We consider the initial value problem for the nonlinear partial differential equations describing the motion of an inhomogeneous and anisotropic hyperelastic medium. We assume that the stored energy function of the hyperelastic material is a function of the point x and the nonlinear Green-St. Venant strain tensor $e_{j k}$ . Moreover, we assume that the stored energy function is $C^{\infty}$ with respect to x and $e_{j k}$ . In our description we assume that Piola-Kirchhoff’s stress tensor $p_{j k}$ depends on the tensor...

Hypoellipticity and local solvability of anisotropic PDEs with Gevrey nonlinearity

Giuseppe De Donno, Alessandro Oliaro (2006)

Bollettino dell'Unione Matematica Italiana

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We propose a unified approach, based on methods from microlocal analysis, for characterizing the hypoellipticity and the local solvability in $C^{\infty}$ and Gevrey $G^{\lambda}$ classes of semilinear anisotropic partial differential operators with Gevrey nonlinear perturbations, in dimension $n\geq 3$ . The conditions for our results are imposed on the sign of the lower order terms of the linear part of the operator, see Theorem 1.1 and Theorem 1.3 below.

The restriction theorem for fully nonlinear subequations

F. Reese Harvey, H. Blaine Lawson (2014)

Annales de l’institut Fourier

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Let $X$ be a submanifold of a manifold $Z$ . We address the question: When do viscosity subsolutions of a fully nonlinear PDE on $Z$ , restrict to be viscosity subsolutions of the restricted subequation on $X$ ? This is not always true, and conditions are required. We first prove a basic result which, in theory, can be applied to any subequation. Then two definitive results are obtained. The first applies to any “geometrically defined” subequation, and the second to any subequation which can be...

On boundary value problems for systems of nonlinear generalized ordinary differential equations

Malkhaz Ashordia (2017)

Czechoslovak Mathematical Journal

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A general theorem (principle of a priori boundedness) on solvability of the boundary value problem $d x = d A (t) \cdot f (t, x), h (x) = 0$ is established, where $f : [a, b] \times ℝ^{n} \to ℝ^{n}$ is a vector-function belonging to the Carathéodory class corresponding to the matrix-function $A : [a, b] \to ℝ^{n \times n}$ with bounded total variation components, and $h : {BV}_{s} ([a, b], ℝ^{n}) \to ℝ^{n}$ is a continuous operator. Basing on the mentioned principle of a priori boundedness, effective criteria are obtained for the solvability of the system under the condition $x (t_{1} (x)) = ℬ (x) \cdot x (t_{2} (x)) + c_{0},$ where $t_{i} : {BV}_{s} ([a, b], ℝ^{n}) \to [a, b]$ $(i = 1, 2)$ and $ℬ : {BV}_{s} ([a, b], ℝ^{n}) \to ℝ^{n}$ are continuous...

A Note on $\mathcal{R}$ -Maps

Zbigniew Duszyński (2007)

Bollettino dell'Unione Matematica Italiana

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Richness of the class of $\mathcal{R}$ -maps [3] is investigated. Several consequences for the theory of quasi- $\mathcal{H}$ -closed spaces are indicated.

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Wadie Aziz, José A. Guerrero, L. Antonio Azócar, Nelson Merentes (2016)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this paper we study existence and uniqueness of solutions for the Hammerstein equation $u (x) = v (x) + λ \int_{I_{a}^{b}} K (x, y) f (y, u (y)) d y$ in the space of function of bounded total $ϕ$ -variation in the sense of Hardy-Vitali-Tonelli, where $λ \in ℝ$ , $K : I_{a}^{b} \times I_{a}^{b} ⟶ ℝ$ and $f : I_{a}^{b} \times ℝ ⟶ ℝ$ are suitable functions. The existence and uniqueness of solutions are proved by means of the Leray-Schauder nonlinear alternative and the Banach contraction mapping principle.

Entire solutions to a class of fully nonlinear elliptic equations

Ovidiu Savin (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We study nonlinear elliptic equations of the form $F (D^{2} u) = f (u)$ where the main assumption on $F$ and $f$ is that there exists a one dimensional solution which solves the equation in all the directions $ξ \in ℝ^{n}$ . We show that entire monotone solutions $u$ are one dimensional if their $0$ level set is assumed to be Lipschitz, flat or bounded from one side by a hyperplane.

Isomorphic properties in spaces of compact operators

Ioana Ghenciu (2023)

Commentationes Mathematicae Universitatis Carolinae

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We introduce the definition of $p$ -limited completely continuous operators, $1 \leq p < \infty$ . The question of whether a space of operators has the property that every $p$ -limited subset is relative compact when the dual of the domain and the codomain have this property is studied using $p$ -limited completely continuous evaluation operators.

A Note on Sectorial and R-Sectorial Operators

Alberto Venni (2008)

Bollettino dell'Unione Matematica Italiana

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The following results are proved: (i) if $\alpha$ , $\beta\in\mathbb{R}^{+}$ and $A$ is a sectorial operator, then the set $\{t^{\alpha}A^{\beta}(t+A);t>0\}$ is bounded; (ii) the same set of operators is R-bounded if $A$ is R-sectorial.

Unconditional uniqueness of higher order nonlinear Schrödinger equations

Friedrich Klaus, Peer Kunstmann, Nikolaos Pattakos (2021)

Czechoslovak Mathematical Journal

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We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation with the initial data $u_{0} \in X$ , where $X \in {M_{2, q}^{s} (ℝ), H^{σ} (𝕋), H^{s_{1}} (ℝ) + H^{s_{2}} (𝕋)}$ and $q \in [1, 2]$ , $s \geq 0$ , or $σ \geq 0$ , or $s_{2} \geq s_{1} \geq 0$ . Moreover, if $M_{2, q}^{s} (ℝ) ↪ L^{3} (ℝ)$ , or if $σ \geq \frac{1}{6}$ , or if $s_{1} \geq \frac{1}{6}$ and $s_{2} > \frac{1}{2}$ we show that the Cauchy problem is unconditionally wellposed in $X$ . Similar results hold true for all higher order nonlinear Schrödinger equations and mixed order NLS due to a factorization property of the corresponding phase factors. For the proof we employ...

On the Existence of Solutions for Abstract Nonlinear Operator Equations

Marek Galewski (2007)

Bollettino dell'Unione Matematica Italiana

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We provide a duality theory and existence results for a operator equation $\nabla T(x)=\nabla N(x)$ where $T$ is not necessarily a monotone operator. We use the abstract version of the so called dual variational method. The solution is obtained as a limit of a minimizng sequence whose existence and convergence is proved.

Nonlinear $*$ -Lie higher derivations of standard operator algebras

Mohammad Ashraf, Shakir Ali, Bilal Ahmad Wani (2018)

Communications in Mathematics

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Let $ℋ$ be an infinite-dimensional complex Hilbert space and $𝔄$ be a standard operator algebra on $ℋ$ which is closed under the adjoint operation. It is shown that every nonlinear $*$ -Lie higher derivation $𝒟 = {δ_{n}}_{n \in ℕ}$ of $𝔄$ is automatically an additive higher derivation on $𝔄$ . Moreover, $𝒟 = {δ_{n}}_{n \in ℕ}$ is an inner $*$ -higher derivation.

Order bounded composition operators on the Hardy spaces and the Nevanlinna class

Nizar Jaoua (1999)

Studia Mathematica

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We study the order boundedness of composition operators induced by holomorphic self-maps of the open unit disc D. We consider these operators first on the Hardy spaces $H^{p}$ 0 < p < ∞ and then on the Nevanlinna class N. Given a non-negative increasing function h on [0,∞[, a composition operator is said to be X,Lh-order bounded (we write (X,Lh)-ob) with $X = H^{p}$ or X = N if its composition with the map f ↦ f*, where f* denotes the radial limit of f, is order bounded from X into $L_{h}$ . We give...

Spaces of compact operators on $C (2^{} \times [0, α])$ spaces

Elói Medina Galego (2011)

Colloquium Mathematicae

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We classify, up to isomorphism, the spaces of compact operators (E,F), where E and F are the Banach spaces of all continuous functions defined on the compact spaces $2^{} \times [0, α]$ , the topological products of Cantor cubes $2^{}$ and intervals of ordinal numbers [0,α].

On a Kirchhoff-Carrier equation with nonlinear terms containing a finite number of unknown values

Nguyen Vu Dzung, Le Thi Phuong Ngoc, Nguyen Huu Nhan, Nguyen Thanh Long (2024)

Mathematica Bohemica

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We consider problem (P) of Kirchhoff-Carrier type with nonlinear terms containing a finite number of unknown values $u (η_{1}, t), \dots, u (η_{q}, t)$ with $0 \leq η_{1} < η_{2} < \dots < η_{q} < 1 .$ By applying the linearization method together with the Faedo-Galerkin method and the weak compact method, we first prove the existence and uniqueness of a local weak solution of problem (P). Next, we consider a specific case $(P_{q})$ of (P) in which the nonlinear term contains the sum $S_{q} [u^{2}] (t) = q^{- 1} \sum_{i = 1}^{q} u^{2} (\frac{(i - 1)}{q}, t)$ . Under suitable conditions, we prove that the solution of $(P_{q})$ converges to the solution...

Inverse results for generalized Favard-Kantorovich and Favard-Durrmeyer operators in weighted function spaces

Nowak Grzegorz (2006)

Bollettino dell'Unione Matematica Italiana

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We consider the Kantorovich and the Durrmeyer type modifications of the generalized Favard operators and we prove inverse approximation theorems for functions $f$ such that $w_{2m}f\in L^{p}(R)$ , where $1\leq p\leq\infty$ and $w_{2m}(x)=(1+x^{2m})^{-1}$ , $m\in N_{0}$ .

Analytical theory of nonlinear oscillations $X$ : some classes of equations $\ddot{x}+g(x)=0$ with no finite Fourier series solution

Chike Obi (1979)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Si dimostra che l'equazione considerata nel titolo ammette soluzioni periodiche rappresentabili da una serie di Fourier con un numero finito di termini solo se $g(x)$ è lineare.

Breathers for nonlinear wave equations

Michael W. Smiley (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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The semilinear differential equation (1), (2), (3), in $\mathbb{R}\times\Omega$ with $\Omega\in\mathbb{R}^{N}$ , (a nonlinear wave equation) is studied. In particular for $\Omega=\mathbb{R}^{3}$ , the existence is shown of a weak solution $u(t,x)$ , periodic with period $T$ , non-constant with respect to $t$ , and radially symmetric in the spatial variables, that is of the form $u(t,x)=\nu(t,|x|)$ . The proof is based on a distributional interpretation for a linear equation corresponding to the given problem, on the Paley-Wiener criterion for the Laplace Transform, and on the alternative...

Real $C^{k}$ Koebe principle

Weixiao Shen, Michael Todd (2005)

Fundamenta Mathematicae

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We prove a $C^{k}$ version of the real Koebe principle for interval (or circle) maps with non-flat critical points.

On the number of positive solutions of singularly perturbed 1D nonlinear Schrödinger equations

Patricio Felmer, Salomé Martínez, Kazunaga Tanaka (2006)

Journal of the European Mathematical Society

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We study singularly perturbed 1D nonlinear Schrödinger equations (1.1). When $V (x)$ has multiple critical points, (1.1) has a wide variety of positive solutions for small $ε$ and the number of positive solutions increases to $\infty$ as $ε \to 0$ . We give an estimate of the number of positive solutions whose growth order depends on the number of local maxima of $V (x)$ . Envelope functions or equivalently adiabatic profiles of high frequency solutions play an important role in the proof.

On hyponormal operators in Krein spaces

Kevin Esmeral, Osmin Ferrer, Jorge Jalk, Boris Lora Castro (2019)

Archivum Mathematicum

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In this paper the hyponormal operators on Krein spaces are introduced. We state conditions for the hyponormality of bounded operators focusing, in particular, on those operators $T$ for which there exists a fundamental decomposition $𝕂 = 𝕂^{+} \oplus 𝕂^{-}$ of the Krein space $𝕂$ with $𝕂^{+}$ and $𝕂^{-}$ invariant under $T$ .

On the rate of convergence of the Bézier-type operators

Grażyna Anioł (2006)

Bollettino dell'Unione Matematica Italiana

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For bounded functions $f$ on an interval $I$ , in particular, for functions of bounded p-th power variation on $I$ there is estimated the rate of pointwise convergence of the Bezier-type modification of the discrete Feller operators. In the main theorem the Chanturiya modulus of variation is used.

Nonlinear Elliptic Equations with Lower Order Terms and Symmetrization Methods

Angelo Alvino, Anna Mercaldo (2008)

Bollettino dell'Unione Matematica Italiana

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We consider the homogeneous Dirichlet problem for nonlinear elliptic equations as $-\operatorname{div}a(x,\nabla u)=b(x,\nabla u)+\mu$ where $\mu$ is a measure with bounded total variation. We fix structural conditions on functions $a$ , $b$ which ensure existence of solutions. Moreover, if $\mu$ is an $L^{1}$ function, we prove a uniqueness result under more restrictive hypotheses on the operator.

Partial Boundary Regularity of Solutions of Nonlinear Superelliptic Systems

Christoph Hamburger (2007)

Bollettino dell'Unione Matematica Italiana

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We prove global partial regularity of weaksolutions of the Dirichlet problem for the nonlinear superelliptic system $\operatorname{div}A(x,u,Du)+B(x,u,DU)=0$ , under natural polynomial growth of the coefficient functions $A$ and $B$ . We employ the indirect method of the bilinear form and do not use a Caccioppoli or a reverse Hölder inequality.