scholarly journals Continuity and differentiability properties of the Nemitskii operator in Hölder spaces

1988 ◽  
Vol 30 (1) ◽  
pp. 59-65 ◽  
Author(s):  
Rita Nugari
Keyword(s):  
Banach Space ◽  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Bounded Subset ◽  
Holder Space ◽  
Usual Norm ◽  
Image Position ◽  
Holder Spaces ◽  
Open Bounded Subset ◽  
Differentiability Properties

Let ℝn be the n-dimensional Euclidean space with the usual norm denoted by |·| In what follows 蒆 will denote an open bounded subset of ℝn, and its closure.For α ∊(0,1], is the space of all functions such that: is called the Holder space with exponent a and is a Banach space when endowed with the norm:where ‖u‖∞ is, as usual, defined by:

Some Imbedding Theorems for Sobolev Spaces

1971 ◽  
Vol 23 (3) ◽  
pp. 517-530 ◽  
Author(s):  
R. A. Adams ◽  
John Fournier
Keyword(s):  
Banach Space ◽  
Positive Integer ◽  
Euclidean Space ◽  
Sobolev Space ◽  
Dimensional Euclidean Space ◽  
Dense Subspace ◽  
Bounded Subset ◽  
Partial Derivatives ◽  
Image Position ◽  
Derivatives Of

We shall be concerned throughout this paper with the Sobolev space Wm,p(G) and the existence and compactness (or lack of it) of its imbeddings (i.e. continuous inclusions) into various LP spaces over G, where G is an open, not necessarily bounded subset of n-dimensional Euclidean space En. For each positive integer m and each real p ≧ 1 the space Wm,p(G) consists of all u in LP(G) whose distributional partial derivatives of all orders up to and including m are also in LP(G). With respect to the norm1.1Wm,p(G) is a Banach space. It has been shown by Meyers and Serrin [9] that the set of functions in Cm(G) which, together with their partial derivatives of orders up to and including m, are in LP(G) forms a dense subspace of Wm,p(G).

EXISTENCE OF A WEAK SOLUTION FOR A CLASS OF FRACTIONAL LAPLACIAN EQUATIONS

2016 ◽  
Vol 102 (3) ◽  
pp. 392-404
Author(s):  
V. RAGHAVENDRA ◽  
RASMITA KAR
Keyword(s):  
Weak Solution ◽  
Fractional Laplacian ◽  
Nonlocal Problem ◽  
Bounded Subset ◽  
Lipschitz Boundary ◽  
Integrodifferential Operator ◽  
Image Position ◽  
Laplacian Equations ◽  
Open Bounded Subset ◽  
Fractional Type

We study the existence of a weak solution of a nonlocal problem$$\begin{eqnarray}\displaystyle & \displaystyle -{\mathcal{L}}_{K}u-\unicode[STIX]{x1D707}ug_{1}+h(u)g_{2}=f\quad \text{in }\unicode[STIX]{x1D6FA}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle u=0\quad \text{in }\mathbb{R}^{n}\setminus \unicode[STIX]{x1D6FA}, & \displaystyle \nonumber\end{eqnarray}$$where${\mathcal{L}}_{k}$is a general nonlocal integrodifferential operator of fractional type,$\unicode[STIX]{x1D707}$is a real parameter and$\unicode[STIX]{x1D6FA}$is an open bounded subset of$\mathbb{R}^{n}$($n>2s$, where$s\in (0,1)$is fixed) with Lipschitz boundary$\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$. Here$f,g_{1},g_{2}:\unicode[STIX]{x1D6FA}\rightarrow \mathbb{R}$and$h:\mathbb{R}\rightarrow \mathbb{R}$are functions satisfying suitable hypotheses.

Properties of Harmonic Functions of Three Real Variables Given by Bergman-Whittaker Operators

1963 ◽  
Vol 15 ◽  
pp. 157-168 ◽  
Author(s):  
Josephine Mitchell
Keyword(s):  
Euclidean Space ◽  
Harmonic Functions ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Rectifiable Curve ◽  
Image Position

Let be a closed rectifiable curve, not going through the origin, which bounds a domain Ω in the complex ζ-plane. Let X = (x, y, z) be a point in three-dimensional euclidean space E3 and setThe Bergman-Whittaker operator defined by

On Asymptotic Centers and Fixed Points of Nonexpansive Mappings

1980 ◽  
Vol 32 (2) ◽  
pp. 421-430 ◽  
Author(s):  
Teck-Cheong Lim
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Nonempty Subset ◽  
Nonexpansive Mappings ◽  
Chebyshev Center ◽  
Bounded Subset ◽  
Uniformly Convex ◽  
Weakly Compact ◽  
Asymptotic Centers ◽  
Image Position

Let X be a Banach space and B a bounded subset of X. For each x ∈ X, define R(x) = sup{‖x – y‖ : y ∈ B}. If C is a nonempty subset of X, we call the number R = inƒ{R(x) : x ∈ C} the Chebyshev radius of B in C and the set the Chebyshev center of B in C. It is well known that if C is weakly compact and convex, then and if, in addition, X is uniformly convex, then the Chebyshev center is unique; see e.g., [9].Let {Bα : α ∈ ∧} be a decreasing net of bounded subsets of X. For each x ∈ X and each α ∈ ∧, define

Multiple points for symmetric Lévy processes

1978 ◽  
Vol 83 (1) ◽  
pp. 83-90 ◽  
Author(s):  
John Hawkes
Keyword(s):  
Markov Process ◽  
Euclidean Space ◽  
Lévy Process ◽  
Lévy Processes ◽  
Transition Function ◽  
Levy Processes ◽  
Levy Process ◽  
Dimensional Euclidean Space ◽  
Multiple Points ◽  
Image Position

Let Xt be a Lévy process in Rd, d-dimensional euclidean space. That is X is a Markov process whose transition function satisfies

scholarly journals Cores of Potential Operators for Processes With Stationary Independent Increments

1972 ◽  
Vol 48 ◽  
pp. 129-145
Author(s):  
Ken-Iti Sato
Keyword(s):  
Banach Space ◽  
Stochastic Process ◽  
Euclidean Space ◽  
Continuous Functions ◽  
Dimensional Euclidean Space ◽  
Semigroup Of Operators ◽  
Transition Semigroup ◽  
Potential Operators ◽  
Independent Increments

Let Xt(ω)) be a stochastic process with stationary independent increments on the N-dimensional Euclidean space RN, right continuous in t ≧ 0 and starting at the origin. Let C0(RN) be the Banach space of real-valued continuous functions on RN vanishing at infinity with norm . The process induces a transition semigroup of operators Tt on C0(RN) :Ttf(x) = Ef(x + Xt).

Symmetric Forms

1970 ◽  
Vol 13 (1) ◽  
pp. 83-87 ◽  
Author(s):  
K. V. Menon
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Image Position ◽  
Class Of Functions

Let Rm denote a m dimensional Euclidean space. When x ∊ Rm will write x = (x1, x2,..., xm). Let R+m ={x: x ∊ Rm, xi < 0 for all i} and R-m ={x: x ∊ Rm, xi < 0 for all i}. In this paper we consider a class of functions which consists of mappings, Er(K) and Hr(K) of Rm into R which are indexed by K ∊ R+m and K ∊ R-m respectively, and defined at any point α ∊ Rm by1.1

The number of polygons on a lattice

1961 ◽  
Vol 57 (3) ◽  
pp. 516-523 ◽  
Author(s):  
J. M. Hammersley
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
End Points ◽  
Unit Distance ◽  
Connective Constant ◽  
Image Position ◽  
Self Avoiding Walk ◽  
Self Avoiding Walks

In this paper an n-stepped self-avoiding walk is defined to be an ordered sequence of n + 1 mutually distinct points, each with (positive, negative, or zero) integer coordinates in d-dimensional Euclidean space (where d is fixed and d ≥ 2), such that any two successive points in the sequence are neighbours, i.e. are unit distance apart. If further the first and last points of such a sequence are neighbours, the sequence is called an (n + 1)-sided self-avoiding polygon. Clearly, under this definition a polygon must have an even number of sides. Let f(n) and g(n) denote the numbers of n-stepped self-avoiding walks and of n-sided self-avoiding polygons having a prescribed first point. In a previous paper (3), I proved that there exists a connective constant K such thatHere I shall prove the truth of the long-standing conjecture thatI shall also show that (2) is a particular case of an expression for the number of n-stepped self-avoiding walks with prescribed end-points, a distance o(n) apart, this being another old and popular conjecture.

Universal inequalities for eigenvalues of a system of elliptic equations

2009 ◽  
Vol 139 (2) ◽  
pp. 273-285 ◽  
Author(s):  
Qing-Ming Cheng ◽  
Hong-Cang Yang
Keyword(s):  
Euclidean Space ◽  
Eigenvalue Problem ◽  
Bounded Domain ◽  
Lower Order ◽  
Elliptic Equations ◽  
Elastic Vibration ◽  
Dimensional Euclidean Space ◽  
Explicit Method ◽  
Universal Inequalities ◽  
Image Position

Let D be a bounded domain in an n-dimensional Euclidean space ℝn. Assume thatare eigenvalues of an eigenvalue problem of a system of n elliptic equations:In particular, when n=3, the eigenvalue problem describes the behaviour of the elastic vibration. We obtain universal inequalities for eigenvalues of the above eigenvalue problem by making use of a direct and explicit method; our results are sharper than one of Hook. Furthermore, a universal inequality for lower-order eigenvalues of the above eigenvalue problem is also derived.

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