The Multidirectional Mean Value Theorem in Banach Spaces

1997 ◽  
Vol 40 (1) ◽  
pp. 88-102 ◽  
Author(s):  
M. L. Radulescu ◽  
F. H. Clarke
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Continuous Functions ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Locally Lipschitz Functions ◽  
Bump Function ◽  
Lipschitz Continuous ◽  
Lower Semi ◽  
Locally Lipschitz

AbstractRecently, F. H. Clarke and Y. Ledyaev established a multidirectional mean value theorem applicable to lower semi-continuous functions on Hilbert spaces, a result which turns out to be useful in many applications. We develop a variant of the result applicable to locally Lipschitz functions on certain Banach spaces, namely those that admit a C1-Lipschitz continuous bump function.

scholarly journals Locally Lipschitz functions are generically pseudo-regular on separable Banach spaces

1993 ◽  
Vol 47 (2) ◽  
pp. 205-212 ◽  
Author(s):  
J.R. Giles ◽  
Scott Sciffer
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Lipschitz Function ◽  
Separable Banach Space ◽  
Locally Lipschitz Function ◽  
Locally Lipschitz Functions ◽  
Dini Derivative ◽  
Locally Lipschitz ◽  
Set Of Points ◽  
Clarke Derivative

For a locally Lipschitz function on a separable Banach space the set of points of Gâteaux differentiability is dense but not necessarily residual. However, the set of points where the upper Dini derivative and the Clarke derivative agree is residual. It follows immediately that the set of points of intermediate differentiability is also residual and the set of points where the function is Gâteaux but not strictly differentiable is of the first category.

scholarly journals The subdifferential of the sum of two functions in Banach spaces II. Second order case

1995 ◽  
Vol 51 (2) ◽  
pp. 235-248 ◽  
Author(s):  
Robert Deville ◽  
El Mahjoub El Haddad
Keyword(s):  
Hilbert Spaces ◽  
Type Theorem ◽  
Continuous Functions ◽  
Second Order ◽  
Infinite Dimensional ◽  
Jacobi Operators ◽  
Uniqueness Results ◽  
Lower Semi ◽  
Order Case ◽  
Jacobi Equations

We prove a formula for the second order subdifferential of the sum of two lower semi continuous functions in finite dimensions. This formula yields an Alexandrov type theorem for continuous functions. We derive from this uniqueness results of viscosity solutions of second order Hamilton-Jacobi equations and singlevaluedness of the associated Hamilton-Jacobi operators. We also provide conterexamples in infinite dimensional Hilbert spaces.

The Rank Theorem for Locally Lipschitz Continuous Functions

1988 ◽  
Vol 31 (2) ◽  
pp. 217-226 ◽  
Author(s):  
G. J. Butler ◽  
J. G. Timourian ◽  
C. Viger
Keyword(s):  
Continuous Functions ◽  
Constant Rank ◽  
Lipschitz Continuous Functions ◽  
Generalized Derivative ◽  
Lipschitz Continuous ◽  
Locally Lipschitz ◽  
Rank Theorem ◽  
Locally Lipschitz Continuous

AbstractThe Rank Theorem is proved for locally Lipschitz continuous functions f:Rn → Rp with generalized derivative of constant rank.

scholarly journals 2-SYMMETRIC CRITICAL POINT THEOREMS FOR NON-DIFFERENTIABLE FUNCTIONS

2008 ◽  
Vol 50 (3) ◽  
pp. 447-466 ◽  
Author(s):  
PASQUALE CANDITO ◽  
ROBERTO LIVREA ◽  
DUMITRU MOTREANU
Keyword(s):  
Critical Point ◽  
Critical Values ◽  
Lipschitz Continuous ◽  
Unbounded Sequence ◽  
Differentiable Functions ◽  
Lower Semi ◽  
Locally Lipschitz ◽  
Critical Point Theorems ◽  
Locally Lipschitz Continuous

AbstractIn this paper, some min–max theorems for even andC1functionals established by Ghoussoub are extended to the case of functionals that are the sum of a locally Lipschitz continuous, even term and a convex, proper, lower semi-continuous, even function. A class of non-smooth functionals admitting an unbounded sequence of critical values is also pointed out.

scholarly journals Calculus for the intermediate point associated with a mean value theorem of the integral calculus

2020 ◽  
Vol 28 (1) ◽  
pp. 59-66
Author(s):  
Emilia-Loredana Pop ◽  
Dorel Duca ◽  
Augusta Raţiu
Keyword(s):  
Continuous Functions ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Integral Calculus ◽  
Intermediate Point

AbstractIf f, g: [a, b] → 𝕉 are two continuous functions, then there exists a point c ∈ (a, b) such that\int_a^c {f\left(x \right)} dx + \left({c - a} \right)g\left(c \right) = \int_c^b {g\left(x \right)} dx + \left({b - c} \right)f\left(c \right).In this paper, we study the approaching of the point c towards a, when b approaches a.

scholarly journals An Integral Mean Value Theorem concerning Two Continuous Functions and Its Stability

2015 ◽  
Vol 2015 ◽  
pp. 1-4 ◽  
Author(s):  
Monea Mihai
Keyword(s):  
Continuous Functions ◽  
Mean Value ◽  
Mean Value Theorem ◽  
The Mean ◽  
The Stability

The aim of this paper is to investigate an integral mean value theorem proposed by one of the references of this paper. Unfortunately, the proof contains a gap. First, we present a counterexample which shows that this theorem fails in this form. Then, we present two improved versions of this theorem. The stability of the mean point arising from the second result concludes this paper.

scholarly journals Mean value properties of generalised eigenfunctions

1970 ◽  
Vol 17 (2) ◽  
pp. 155-158 ◽  
Author(s):  
Eberhard Gerlach
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Selfadjoint Operator ◽  
Differential Operators ◽  
Continuous Functions ◽  
Mean Value ◽  
Mean Value Property ◽  
Spaces Of Continuous Functions

Some Hilbert spaces of continuous functions satisfying a mean value property are studied in which the generalised eigenfunctions of any selfadjoint operator again satisfy the same mean value property. Applications are made to nullspaces of some differential operators. The classes of functions involved in these applications are less general than those studied by K. Maurin (6); however, the Hilbert space norms may be arbitrary, while Maurin only considered L2-norms.

LOCAL LIMIT THEOREMS FOR PARTIAL SUMS OF STATIONARY SEQUENCES GENERATED BY GIBBS–MARKOV MAPS

2001 ◽  
Vol 01 (02) ◽  
pp. 193-237 ◽  
Author(s):  
JON AARONSON ◽  
MANFRED DENKER
Keyword(s):  
Limit Theorems ◽  
Domain Of Attraction ◽  
Local Limit ◽  
Continuous Functions ◽  
Partial Sums ◽  
Lipschitz Continuous Functions ◽  
Stable Law ◽  
Lipschitz Continuous ◽  
Locally Lipschitz ◽  
Local Limit Theorems

We introduce Gibbs–Markov maps T as maps with a (possibly countable) Markov partition and a certain type of bounded distortion property, and investigate its Frobenius–Perron operator P acting on (locally) Lipschitz continuous functions ϕ. If such a function ϕ belongs to the domain of attraction of a stable law of order in (0,2), we derive the expansion of the eigenvalue function t↦λ(t) of the characteristic function operators Ptf=Pf exp [i< t,ϕ> (perturbations of P) around 0. From this representation local and distributional limit theorems for partial sums ϕ+…+ϕ◦ Tn are easily obtained, provided ϕ is aperiodic. Applications to recurrence properties of group extensions are also given.

Sign in / Sign up

Export Citation Format

Share Document