scholarly journals On the range of the derivative of a smooth mapping between Banach spaces

2005 ◽  
Vol 2005 (5) ◽  
pp. 499-507
Author(s):  
Robert Deville
Keyword(s):  
Banach Spaces ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Differentiable Mapping ◽  
Smooth Mapping ◽  
Lipschitz Continuous ◽  
Fréchet Differentiable ◽  
New Phenomena

We survey recent results on the structure of the range of the derivative of a smooth mappingfbetween two Banach spacesXandY. We recall some necessary conditions and some sufficient conditions on a subsetAofℒ(X,Y)for the existence of a Fréchet differentiable mappingffromXintoYso thatf′(X)=A. Wheneverfis only assumed Gâteaux differentiable, new phenomena appear: for instance, there exists a mappingffromℓ1(ℕ)intoℝ2, which is bounded, Lipschitz-continuous, and so that for allx,y∈ℓ1(ℕ), ifx≠y, then‖f′(x)−f′(y)‖>1.

Global bifurcation at isolated singular points of the Hadamard derivative

2021 ◽  
Vol 379 (2191) ◽  
pp. 20190379
Author(s):  
C. A. Stuart
Keyword(s):  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Global Bifurcation ◽  
Singular Boundary ◽  
Global Information ◽  
Singular Boundary Value Problems ◽  
Connected Component ◽  
Theme Issue ◽  
Fréchet Differentiable ◽  
Hadamard Derivative

Consider F ∈ C ( R × X , Y ) such that F ( λ , 0) = 0 for all λ ∈ R , where X and Y are Banach spaces. Bifurcation from the line R × { 0 } of trivial solutions is investigated in cases where F ( λ , · ) need not be Fréchet differentiable at 0. The main results provide sufficient conditions for μ to be a bifurcation point and yield global information about the connected component of { ( λ , u ) : F ( λ , u ) = 0  and  u ≠ 0 } ∪ { ( μ , 0 ) } containing ( μ , 0). Some necessary conditions for bifurcation are also formulated. The abstract results are used to treat several singular boundary value problems for which Fréchet differentiability is not available. This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.

scholarly journals Exact Filling of Figures with the Derivatives of Smooth Mappings Between Banach Spaces

2005 ◽  
Vol 48 (4) ◽  
pp. 481-499
Author(s):  
D. Azagra ◽  
M. Fabian ◽  
M. Jiménez-Sevilla
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Sufficient Conditions ◽  
Smooth Mapping ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Smooth Case ◽  
Open Set ◽  
Infinite Dimensional Banach Space ◽  
Finite Dimensional Case

AbstractWe establish sufficient conditions on the shape of a set A included in the space (X, Y ) of the n-linear symmetric mappings between Banach spaces X and Y , to ensure the existence of a Cn-smooth mapping f: X → Y, with bounded support, and such that f(n)(X) = A, provided that X admits a Cn-smooth bump with bounded n-th derivative and dens X = dens ℒn(X, Y ). For instance, when X is infinite-dimensional, every bounded connected and open set U containing the origin is the range of the n-th derivative of such amapping. The same holds true for the closure of U, provided that every point in the boundary of U is the end point of a path within U. In the finite-dimensional case, more restrictive conditions are required. We also study the Fréchet smooth case for mappings from ℝn to a separable infinite-dimensional Banach space and the Gâteaux smooth case for mappings defined on a separable infinite-dimensional Banach space and with values in a separable Banach space.

scholarly journals On projection constant problems and the existence of metric projections in normed spaces

2001 ◽  
Vol 6 (7) ◽  
pp. 401-411 ◽  
Author(s):  
Entisarat El-Shobaky ◽  
Sahar Mohammed Ali ◽  
Wataru Takahashi
Keyword(s):  
Banach Spaces ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Metric Projection ◽  
Infinite Matrix ◽  
Normed Spaces ◽  
Reflexive Banach Spaces ◽  
Linear Spaces ◽  
Sufficient And Necessary Conditions ◽  
Projection Constant

We give the sufficient conditions for the existence of a metric projection onto convex closed subsets of normed linear spaces which are reduced conditions than that in the case of reflexive Banach spaces and we find a general formula for the projections onto the maximal proper subspaces of the classical Banach spacesl p,1≤p<∞andc 0. We also give the sufficient and necessary conditions for an infinite matrix to represent a projection operator froml p,1≤p<∞orc 0onto anyone of their maximal proper subspaces.

scholarly journals New Geometric Constants in Banach Spaces Related to the Inscribed Equilateral Triangles of Unit Balls

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 951
Author(s):  
Yuankang Fu ◽  
Qi Liu ◽  
Yongjin Li
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Normal Structure ◽  
Upper And Lower Bounds ◽  
Uniformly Convex ◽  
Equilateral Triangles ◽  
Geometric Constant ◽  
Geometric Constants

Geometric constant is one of the important tools to study geometric properties of Banach spaces. In this paper, we will introduce two new geometric constants JL(X) and YJ(X) in Banach spaces, which are symmetric and related to the side lengths of inscribed equilateral triangles of unit balls. The upper and lower bounds of JL(X) and YJ(X) as well as the values of JL(X) and YJ(X) for Hilbert spaces and some common Banach spaces will be calculated. In addition, some inequalities for JL(X), YJ(X) and some significant geometric constants will be presented. Furthermore, the sufficient conditions for uniformly non-square and normal structure, and the necessary conditions for uniformly non-square and uniformly convex will be established.

Efficiency of the weak Rescaled Pure Greedy Algorithm

2021 ◽  
pp. 2150001
Author(s):  
Bing Jiang ◽  
Peixin Ye
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Error Estimate ◽  
Banach Spaces ◽  
Greedy Algorithm ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Pure Greedy Algorithm ◽  
Sufficient Conditions For Convergence ◽  
Sufficient And Necessary Conditions

We study the efficiency of the Weak Rescaled Pure Greedy Algorithm (WRPGA) with respect to a dictionary in a Hilbert space or more generally a Banach space. We obtain the sufficient and necessary conditions for the convergence of WRPGA for any element and any dictionary. This condition is weaker than the sufficient conditions for convergence of the Weak Pure Greedy Algorithm (WPGA). In addition, we establish the noisy version of the error estimate for the WRPGA, which implies the results on sparse classes obtained in [G. Petrova, Rescaled pure greedy algorithm for Hlibert and Banach spaces, Appl. Comput. Harmon. Anal. 41(3) (2016) 852–866].

Controllability for some nonlinear impulsive partial functional integrodifferential systems with infinite delay in Banach spaces

2020 ◽  
Vol 4 (2) ◽  
pp. 104-115
Author(s):  
Khalil Ezzinbi ◽  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Spaces ◽  
Integrodifferential Equation ◽  
Resolvent Operator ◽  
Measure Of Noncompactness ◽  
Infinite Delay ◽  
Sufficient Conditions ◽  
Mönch Fixed Point Theorem ◽  
The Resolvent Operator

This work concerns the study of the controllability for some impulsive partial functional integrodifferential equation with infinite delay in Banach spaces. We give sufficient conditions that ensure the controllability of the system by supposing that its undelayed part admits a resolvent operator in the sense of Grimmer, and by making use of the measure of noncompactness and the Mönch fixed-point Theorem. As a result, we obtain a generalization of the work of K. Balachandran and R. Sakthivel (Journal of Mathematical Analysis and Applications, 255, 447-457, (2001)) and a host of important results in the literature, without assuming the compactness of the resolvent operator. An example is given for illustration.

scholarly journals On a New Geometric Constant Related to the Euler-Lagrange Type Identity in Banach Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 116
Author(s):  
Qi Liu ◽  
Yongjin Li
Keyword(s):  
Banach Spaces ◽  
Product Space ◽  
Sufficient Conditions ◽  
Normal Structure ◽  
The Other ◽  
Inner Product ◽  
Equivalent Characterization ◽  
Geometric Constant ◽  
Geometric Constants

In this paper, we will introduce a new geometric constant LYJ(λ,μ,X) based on an equivalent characterization of inner product space, which was proposed by Moslehian and Rassias. We first discuss some equivalent forms of the proposed constant. Next, a characterization of uniformly non-square is given. Moreover, some sufficient conditions which imply weak normal structure are presented. Finally, we obtain some relationship between the other well-known geometric constants and LYJ(λ,μ,X). Also, this new coefficient is computed for X being concrete space.

scholarly journals No Infimum Gap and Normality in Optimal Impulsive Control Under State Constraints

2021 ◽  
Author(s):  
Giovanni Fusco ◽  
Monica Motta
Keyword(s):  
Original Problem ◽  
State Constraints ◽  
Impulsive Control ◽  
Sufficient Conditions ◽  
Continuous Data ◽  
Lipschitz Continuous ◽  
Extended Sense ◽  
Unbounded Controls ◽  
Locally Lipschitz ◽  
Optimal Impulsive Control

AbstractIn this paper we consider an impulsive extension of an optimal control problem with unbounded controls, subject to endpoint and state constraints. We show that the existence of an extended-sense minimizer that is a normal extremal for a constrained Maximum Principle ensures that there is no gap between the infima of the original problem and of its extension. Furthermore, we translate such relation into verifiable sufficient conditions for normality in the form of constraint and endpoint qualifications. Links between existence of an infimum gap and normality in impulsive control have previously been explored for problems without state constraints. This paper establishes such links in the presence of state constraints and of an additional ordinary control, for locally Lipschitz continuous data.

scholarly journals Noetherian properties in composite generalized power series rings

2020 ◽  
Vol 18 (1) ◽  
pp. 1540-1551
Author(s):  
Jung Wook Lim ◽  
Dong Yeol Oh
Keyword(s):  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Generalized Power Series ◽  
Ordered Monoid ◽  
Power Series Rings ◽  
Necessary And Sufficient ◽  
Equivalent Conditions

Abstract Let ({\mathrm{\Gamma}},\le ) be a strictly ordered monoid, and let {{\mathrm{\Gamma}}}^{\ast }\left={\mathrm{\Gamma}}\backslash \{0\} . Let D\subseteq E be an extension of commutative rings with identity, and let I be a nonzero proper ideal of D. Set \begin{array}{l}D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {E}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(0)\in D\right\}\hspace{.5em}\text{and}\\ \hspace{0.2em}D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {D}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(\alpha )\in I,\hspace{.5em}\text{for}\hspace{.25em}\text{all}\hspace{.5em}\alpha \in {{\mathrm{\Gamma}}}^{\ast }\right\}.\end{array} In this paper, we give necessary conditions for the rings D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively ordered, and sufficient conditions for the rings D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively totally ordered. Moreover, we give a necessary and sufficient condition for the ring D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively totally ordered. As corollaries, we give equivalent conditions for the rings D+({X}_{1},\ldots ,{X}_{n})E{[}{X}_{1},\ldots ,{X}_{n}] and D+({X}_{1},\ldots ,{X}_{n})I{[}{X}_{1},\ldots ,{X}_{n}] to be Noetherian.

scholarly journals On the equivalence of three-dimensional differential systems

2020 ◽  
Vol 18 (1) ◽  
pp. 1164-1172
Author(s):  
Jian Zhou ◽  
Shiyin Zhao
Keyword(s):  
Periodic Solutions ◽  
Differential System ◽  
Necessary Conditions ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Periodic Systems ◽  
Structural Form ◽  
Differential Systems

Abstract In this paper, firstly, we study the structural form of reflective integral for a given system. Then the sufficient conditions are obtained to ensure there exists the reflective integral with these structured form for such system. Secondly, we discuss the necessary conditions for the equivalence of such systems and a general three-dimensional differential system. And then, we apply the obtained results to the study of the behavior of their periodic solutions when such systems are periodic systems in t.

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